Investigation of Thermal Transport in Layered Sytems and Micro-Structured Semiconductor Devices by Photothermal Techniques and Finite Element Simulations

Investigation of Thermal Transport in Layered Sytems and Micro-structured Semiconductor Devices by Photothermal Techniques and Finite Element Simulations

Dissertation

Zur Erlangung des Grades eines Doktors der Naturwissenschaften in der Fakultät für Physik und Astronomie

der Ruhr-Universität Bochum

vorgelegt von

Jean Lazare Nzodoum Fotsing

aus Jaunde Kamerun

Bochum

2004

Mit der Genehmigung des Dekanats vom 03.05.2004 wurde die Dissertation in englischer Sprache verfasst

Eingereicht am: 21. 05. 2004 Disputation am: 14. 07. 2004 1. Gutachter: Prof. Dr. J. Pelzl 2. Gutachter: Prof. Dr. A. Wieck

To my uncles

Takam André Pombo Gaston

who passed away respectively in 2003 and in 2004. May their souls rest in peace

iii

Contents

Contents...... iv

Abreviations and Nomenclature ...... viii

1 Introduction ...... 1 1.1 Motivation ...... 1 1.2 Objectives of the Work...... 2 1.3 Thesis Overview...... 2

2 Fundamentals and Goals of the Photothermal Techniques ...... 5 2.1 Introduction ...... 5 2.2 The concept of thermal waves...... 5 2.3 Relevant physical parameters...... 7 2.4 Review of the main detection methods ...... 7 2.5 Conclusions ...... 11

3 Surface and Subsurface Effects of Thermal Transport in Layered Systems...... 12 3.1 Introduction ...... 12 3.2 Photothermal IR radiometric signal...... 13 3.3 Description of the measuring system ...... 16 3.4 Description of the investigated samples...... 17 3.5 Theoretical models of thermal waves for layered systems...... 17 3.6 Interpretation of the modulated IR radiometric signals...... 21 3.7 Discussion of physical effects in the investigated samples...... 24 3.7.1 Surface effects...... 24 3.7.1.1 Optical effects: coating semi-transparency...... 24 3.7.1.2 Topological effects...... 26 3.7.2 Subsurface effects: lateral thermal transport ...... 28 3.7.2.1 Theoretical developments...... 28 3.7.2.2 Comparison of 1-D and 3-D thermal wave propagation...... 33 3.7.2.3 Effects of heating spot radius and subsurface thermal properties ...... 34 3.8 Interpretation of signal phases based on 3D thermal transport ...... 37 3.8.1 Determination of the properties of lateral heat transport...... 38 3.8.2 Discussion of results...... 40 3.9 Conclusions ...... 41

4 Determination of Thermal Transport Properties of Two-layer Structures using the Concept of the Phase Extremum...... 43 4.1 Motivation ...... 43 4.2 The concept of the Phase Extremum...... 44

4.2.1 Physical significance of the observable phase extrema...... 44 4.2.2 Theoretical background...... 46 4.3 Application to experimental measurements ...... 49 4.3.1 Methodology and discussions...... 49 4.3.2 Determination of the thermophysical properties ...... 51 4.3.3 Efficient localization of the phase extremum ...... 52 4.3.4 Application to measurements at high temperatures...... 54 4.4 Discussions on the reliability of the Extremum Method...... 57 4.4.1 Solutions from any measured point of the calibrated phase ...... 57 4.4.2 Comparison of results ...... 60 4.5 General results for on-line interpretation in industrial applications...... 61 4.5.1 Graphic of thermal reflection coefficients...... 61 4.5.2 Graphic of ratio of the effusivities...... 63 4.5.3 Graphic of thermal diffusion times...... 63 4.5.4 Example of on-line interpretation...... 66 4.6 Interpretation of modulated IR signals obscured by the background fluctuation ...... 67 4.6.1 Theory of the transformation of the inverse normalized phase ...... 68 4.6.2 Application of the functional transform to measurements: The problem of convergence ...... 71 4.6.3 Phase interpretation obscured by the background fluctuation...... 77 4.7 Conclusions ...... 79

5 Detection of Local Inhomogeneities of Thermal Transport and Localization of Heat sources in Micro-scaled Systems based on Spot Displacement...... 81 5.1 Motivation ...... 81 5.2 Review of the main results of 3-D thermal wave propagation...... 81 5.3 Displacement between heating and detection spots ...... 82 5.3.1 Theoretical background...... 83 5.3.2 Simulation of controlled displacements between the two spots ...... 85 5.3.3 Localization of heat sources ...... 87 5.3.4 Comparison of experimental measurements and theoretical results...... 88 5.4 Scaling of thermal localization of hot spots ...... 91 5.5 Comparisons with measurements based on Thermoreflectance...... 94 5.6 Conclusions ...... 95

6 Efficient Simulation of Thermal Wave Problems with the ANSYS-aided Finite Element Technique...... 97 6.1 Introduction ...... 97 6.2 Finite element concepts...... 97 6.2.1 Theoretical foundation...... 97 6.2.2 Fundamental steps...... 99 6.3 Finite element modeling...... 99 6.3.1 Pre-processing...... 100 6.3.2 Computation...... 100 6.3.3 Post-processing ...... 100

6.4 Efficient simulation of thermal wave problems ...... 101 6.4.1 Convergence of the numerical solution ...... 101 6.4.1.1 Non-uniform meshing and mesh refinement ...... 101 6.4.1.2 Control of convergence with the reference phase shift...... 101 6.4.2 Methodology of simulation...... 104 6.5 Finite element control of theoretical results ...... 106 6.6 Simulation of thermo-elastic signals ...... 110 6.6.1 Theoretical basis...... 110 6.6.2 Methodology...... 112 6.6.3 Example of application...... 113 6.7 Conclusions ...... 114

7 Finite Element Investigation of Heating Processes in Micro-structured Semiconductor Devices ...... 116 7.1 Motivation ...... 116 7.2 Features of two experimental techniques in micro-thermal analysis ...... 116 7.2.1 Scanning Thermal Microscopy (SThM)...... 116 7.2.2 Scanning Thermal Expansion Microscopy (SThEM)...... 117 7.3 FE simulation of thermal and thermo-elastic signals ...... 118 7.3.1 AC heating and DC heating...... 118 7.3.2 The ac-electrically heated conducting line ...... 119 7.3.2.1 Physical system and model...... 120 7.3.2.2 Thermal oscillations and thermal expansions...... 120 7.3.2.3 Frequency-dependent thermal expansions...... 122 7.4 Investigation of thermal and thermo-elastic signals in HEMT-devices ...... 124 7.4.1 Power dissipation in transistors ...... 124 7.4.1.1 Basic structures of transistors ...... 124 7.4.1.2 Origin and location of the power dissipation...... 124 7.4.2 Influence of the system Drain−Gate–Source on the signals...... 125 7.4.2.1 Structure exempted from the system D-G-S...... 126 7.4.2.2 Structure with D-S ...... 128 7.4.2.3 Structure with D-G-S...... 130 7.4.3 Comparison of experimental measurements and FE results...... 130 7.4.4 Influence of the modulation frequency...... 134 7.5 Contribution of the tip to the signals ...... 136 7.5.1 Position of the problem...... 136 7.5.2 Modeling of the tip structure ...... 137 7.5.3 Si-tip in mechanical contact with a GaAs-structure ...... 137 7.5.3.1 Temperature oscillations along the tip axis ...... 137 7.5.3.2 Derivation of the thermal expansion of the tip ...... 138 7.5.3.2 Thermal oscillations along the sample surface...... 139 7.5.4 Influence of the material of the tip...... 139 7.5.5 Influence of the heated substrate ...... 139 7.5.6 Influence of a fluid film between tip and sample ...... 143 7.6 Conclusions ...... 147

8 Conclusions and Future Work...... 149 8.1 Conclusions ...... 149 8.1.1 Review of the experimental work...... 149 8.1.2 Review of the numerical-analytical work...... 150 8.1.3 Review of the numerical work using finite elements ...... 154 8.2 Future work...... 157 8.2.1 Extensions of the Extremum Method ...... 157 8.2.2 Calibration procedure for determining the temperature of hot spot ...... 159 8.2.3 3-D finite element simulations...... 159

Appendix A: Characteristics of the investigated samples...... 160

Appendix B: Material properties used for FE simulations...... 161

Bibliography...... 162

Publications and Conference contributions...... 175

Curriculum vitae ...... 178

Acknowledgements ...... 180

vii

Abreviations and Nomenclature

Abreviations

1-D One dimensional 2-D Two dimensional 3-D Three dimensional D Drain CLTE Coefficient of Linear Thermal Expansion FEM Finite Element Method FET Field Effect Transistor G Gate GaAs-sample Sample of Galium arsenide material HEMT High Electron Mobility Transistor HMA High Effusivity Metallic Alloy HSS High Speed Steel InSb-sample Sample of Indium antimonide material MCT-detector Mercury Cadnium Telluride (HgCdTe) detector S Source Si-sample Sample of silicon material Si-tip Tip of silicon material SJEM Scanning Joule Expansion Microscopy SthEM Scanning Thermal Expansion Microscopy SThM Scanning Thermal Microscopy Pt-tip Tip of platinium material W-tip Tip of tungsten material

viii

Nomenclature

Symbol Unity Name

T s Time

τs s Thermal diffusion time of the coating ε Emissivity 1 β Coefficient of optical absorption m 2 -4 σSB W ⋅m ⋅K Stefan-Boltzmann constant 1 β Coefficient of optical absorption m 1 β Coefficient of linear thermal expansion e K

ηs Photothermal conversion efficiency W k Thermal conductivity m ⋅ K J c Specific heat capacity kg ⋅ K kg ρ Mass densitiy m3 k m2 α = Thermal diffusivity ρ⋅c s Ws1/2 e = k ρc Thermal effusivity m2 K

Rij Thermal reflection coefficient between layers i and j

1/ 2 1 P = α β Combined thermo-optical parameter αβ s1/2 f Hz Modulation frequency of heating

ω = 2⋅ π⋅ f Hz Angular frequency α µ = m Thermal diffusion length th π ⋅ f W

I o Incident beam power m2 ix

Radius of the heating spot rH m Radius of the detection spot rD m Displacement distance between the heating and the dHD m detection spots

T ()r,t K Absolute temperature δ T ()r, f , t K Modulated temperature ν Poisson-number r u()r,t m Displacement vector

δd z m Vertical thermal expansion r W Q()r Heat generation rate m3

Introduction

1.1 Motivation

The generic term layered systems refers to samples consisting of a basic material or substrate on which can be identified one or more layers of differing optical and thermophysical properties. Generally, these layered materials systems are built up to fulfill some specific purposes, e.g. to serve as thermal barriers or alternatively as thermal conductors, and so a true knowledge of the individual characteristics of the participating components is beneficial to find out the best structuring. Thus, the performance of the entire structure depends upon the integrity of the individual components as well as the interface between layers. This is why thermal characterization of such systems is of great importance since it allows to point out the governing thermal transport properties and to detect the eventual structuring defects. However, one of the major problems often resides in the contamination and even in the deterioration of the samples under investigation by inappropriate experimental methods. These problems are overcome or avoided by using the photothermal techniques which are more suited for performing thermal investigations since they are non-destructive and do not require any sample preparation. On the other hand, these layered systems are exploited for the design and manufacturing of micro-structured devices, e.g. micromechanical and microelectronic devices. As for the microelectronic devices, they operate very fast and thus produce high power densities, implying necessarily new challenges in the field of thermal management. The situation is even worse as the dimensions of these devices are getting drastically smaller and smaller since in this case the phenomena of transport and dissipation become more complex. For example, it has been established, that the close proximity of interfaces and the extremely small volume of heat dissipation strongly modifies thermal transport, thus aggravating problems of thermal management [Cahill et al., 2003]. Main obsessions in investigating –both in the theoretical and in the experimental point of view− these so tiny structures reside either in the determination of the local thermal conductivity/diffusivity [Milcent et al., 1995; Langer et al., 1996; Hartmman et al., 1997; Ruiz et al., 1998; Milcent et al., 1998; Hui et al, 1999; Gervaise et al., 2000] or in the localization of hot spots [Bolte et al., 1998; Bolte, 1999] or even in the calibration of the absolute temperature [Schaub, 2001], by using different photothermal methods. The main experimental approaches in thermal microscopy are reported in [Price et al., 2000; Pelzl et al., 2001].

1 1.2 Objectives of the work

However, the veritable difficulty in dealing with these high power devices is the identification and detection of the location of high temperatures (hot spots) as well as the management of the dissipated heat. Another major difficulty comes from the fact that materials which are brought into contact with each other to make up a unique device have different deposition temperatures and thermal expansion coefficients [Prorok and Espinosa, 2002]. Such a materials composition with so different characteristics can unavoidably lead to damages including for example cracking and de-lamination. There is therefore an absolute necessity to examine and study the mechanisms which can help to enhance the thermal performances of these extremely thin devices. In principle, two particular experimental techniques, namely the Scanning Thermal Microscopy (SThM) and the Scanning Thermal Expansion Microscopy (SThEM), are aimed at investigating and controlling the thermal movements in these kinds of devices but due to the limitations imposed to experimental measurements by the hostile dimensions of the structures, numerical simulations mostly based on the Finite Element Method (FEM) constitute a very important tool to predict the thermal behaviour of devices and then discuss about their performances via faithful models. This option has at least the advantage of limiting the high costs related to the production of layouts and of avoiding hazardous industrial tests.

1.2 Objectives of the work

The present research work had three main objectives: The first task was to investigate several layered structures by means of photothermal techniques and to find out concrete solutions for a more rapid and efficient quantitative interpretation of the measured signals. Another important task was to propose an alternative photothermal method which can help to get more reliable information on the lateral transport properties and allow the identification of heat sources in micro-scaled systems. The third objective of this contribution was to investigate by means of Scanning Thermal Expansion Microscopy and finite element simulations, the hot spots in some selected micro-structured semiconductor devices following their excitation by a modulated heat source.

1.3 Thesis overview

This work, which can be globally subdivided into two parts including the photothermal characterization of layered systems −from macroscopic to microscopic scale− and the finite element investigation of thermal and thermo-elastic signals in micro-structured devices, is organized in the following way: Chapter 2 recalls the basic concepts of thermal waves and reviews the different experimental configurations which are designed for the generation and the detection of photothermal signals in solids.

2 1.3 Thesis overview

Chapter 3 discusses the physical effects of thermal transport in various layered structures with the thickness of the thin film or coating ranging from about 0.8 to 3 µm. These physical effects are revealed by the phases and amplitudes of the modulated IR radiometric signals measured at the surface of the samples. The surface effects occurring in the range of high modulation frequencies, namely at low penetration depths, are found to be due either to the optical characteristics of the coating or to the topology of the sample surface. The subsurface effects, identified in the limit of very low modulation frequencies, are found to be related to lateral thermal transport in the substrate material of high thermal diffusivity and effusivity. In this framework, a 3-D theory of thermal wave propagation in two-layer systems is developed and correlated to the measured signals, which then helps to discriminate the main thermophysical property favouring the lateral heat propagation in the investigated samples. Chapter 4 introduces a new evaluation method based on the relative extremum of the calibrated measured signal phases and applies it to the determination of thermal and physical properties of two-layer structures via two combined thermophysical quantities. The proposed method, which is efficient, rapid and more accurate, allows to generate general results for on- line interpretation in industrial applications and also provides the possibility to interpret the measured signals that may be obscured by the background fluctuation. Chapter 5 describes a new photothermal method, based on controlled displacement distances between the heating and the detection spots, which helps to get more reliable information on the lateral thermal transport properties and on the localization of heat sources in micro-scaled structures. Although the principle of displacement distance between excitation and detection spots is not new, the originality of this method resides in the fact that it allows to study the measuring conditions on the signal phases and the relevant range of modulation frequencies. It is shown, that the relative extrema observable on the calibrated phases and which increase with increasing displacement distance between the two spots can be exploited to localize the heat sources with good precision. In order to find out at what distances such a localization of heat sources by means of measurements in the neighbourhood is still possible, the scaling of thermal localization of hot spots from macroscopic to microscopic is performed by adapting the sample to more realistic conditions of thermal microscopy and by decreasing gradually the values of the experimental parameters, namely the radii of the heating and the detection spots as well as the displacement distance between the centers of the two spots. As the simulation of thermal wave problems is somewhat difficult, chapter 6 is enrolled to show by means of tips and examples how such types of problems should be efficiently handled with the Finite Element Technique. In this framework, a new approach based on the phase shift of the thermal wave at the surface of a homogenous and opaque reference material (Φref = -45°), labelled as reference phase shift, is proposed to help judging the convergence of the numerical solution. In order to verify the reliability of the developed finite element schemes, some significant theoretical results presented in chapter 3 and in chapter 5 are compared with the results of finite element simulations.

3 1.3 Thesis overview

Chapter 7 deals with the investigation of heating processes in micro-structured devices by using ANSYS-aided finite element simulations. Here, the thermal oscillations in thin devices and the thermo-elastic displacements of their surface are simulated and compared with experimental results obtained with Scanning Thermal Expansion Microscopy. The contribution of the tip on hot spot on the device to the signals is also investigated. In this framework, the influence of the material of the tip on one hand, and the influence of the material of the heated substrate on the other hand, as well as the incidence of a fluid bridge linking the tip and the sample are methodically analysed and discussed. Chapter 8 concludes this work and provides insight on future challenges.

Fundamentals and Goals of the Photothermal Techniques

2.1 Introduction

The photothermal techniques use the concept of thermal waves and for semiconductor materials additionally the concept of plasma waves, generated in the samples by an intensity- modulated laser beam. Thermal waves, which are produced by intensity-modulated localized heating of a solid and which can be described by the heat diffusion equation, are temperature oscillations which vary as a function of space and time. Due to the diffusive propagation, the thermal wave’s amplitude is damped and the phase lag between the thermal wave and the modulated excitation increases with the propagation distance. In this chapter, the basic concepts of thermal waves are presented, followed by a review of the different experimental configurations that are designed for generating and detecting these thermal waves.

2.2 The concept of thermal waves

Three fundamental mechanisms for transferring heat from one region to another are worth to be mentioned: Radiation is the mechanism in which heat is transferred directly by electromagnetic waves. The radiation does not require a heat transfer medium, and can occur in vacuum. Convection refers to the transfer of heat by the flow of a hot or cold fluid. Conduction is characterized by the transfer of heat from one place to another in a material by a random movement and collision of carriers (atoms and molecules in a gas, electrons and phonons in a solid), but without the collective flow of the material itself. This third mechanism is the underlying heat transfer process in thermal waves. Thus, from the statement of energy conservation applied to a non-deformable solid, in the absence of internal heat generation, the classical form of the heat diffusion equation is given by:

∂T ρc + div(q) = 0 (2.1) ∂t where, q = −k grad(T) , according to the Fourier law of heat conduction which supposes an isotropic solid [André De Vriendt, 1982]. The negative sign in the Fourier law indicates that the heat flows from hot to cold areas. By assuming a homogenous material, equ.(2.1) becomes

5 2.2 The concept of thermal waves

1 ∂T (r,t) ∆T (r,t) = (2.2) α ∂t with α = k / ρ c . k is the thermal conductivity of the material, c and ρ represent the specific heat capacity and the mass density, respectively. α is the thermal diffusivity. This latest quantity measures the ability of the material to absorb heat on a transient basis. Thermal waves are special class of solutions to equ.(2.2) and are usually classified into two groups: periodic thermal waves and pulsed thermal waves. (i)− For the pulsed thermal waves, a very short pulse is applied uniformly over the sample surface. The propagation of such a pulse is described by a different one dimensional solution to equ.(2.2). It has the form:

x 2 I 1 − T (x,t) = o e 4αt (2.3) ρc ()4παt 1/ 2

In equ.(2.3), Io is the incident laser intensity. At the sample surface, the temperature decays monotonically like the reciprocal of the square root of time t. Beneath the surface, the situation is rather different. There, the temperature pulse starts at zero and increases more or less exponentially in time, reaches its maximum, and then decays approximately as the reciprocal of the square root of the time [Favro and Han, 1998]. (ii)− Periodic thermal waves are generated by applying a periodic heat source to the solid surface. Such periodic sources have a non-zero average value, and hence contain a dc component as well as an ac component. High sensitive detection techniques are capable to reject the dc component, in such a way that the periodic thermal waves are described as if they had no average value. If a periodic source has been uniformly applied over the front surface of a semi-infinite solid, then under consideration of the appropriate boundary conditions [Bein and Pelzl, 1989], the resulting thermal wave can be described by a one-dimensional solution to equ.(2.2),

⎡ ⎤ ηs Io 1 πf πf π δTs (x,t) = exp⎢− x⎥ cos(2πft − x − ) (2. 4) 2 (ρck)s 2πf ⎣ α s ⎦ α s 4

In equ.(2.4), f is the modulation frequency of heating. Thus, a thermal wave can be defined as the response of a medium to a periodic or pulsed heat source. Only the periodic thermal waves are matter of concern in this work. As can be seen from the solution of the thermal wave (2.4), the amplitude is considerably damped as the penetration depth increases. This penetration depth, x , is limited to a distance approximately equal to the thermal diffusion length x ≈ µth = α s /πf . The thermal diffusion length, which depends on the modulation frequency of heating, shows that thermal depth profiling studies can be achieved by varying the modulation frequency. This 6 2.3 Relevant physical parameters quantity indicates the depth at which a thermal wave technique can be effective. It is therefore regarded as the characteristic length scale of measurement [Almond and Patel, 1996]. On the other hand, one can see in equ.(2.4), that the phase lag relative to the heating modulation which has the value –45° at the solid surface varies with the propagation distance.

2.3 Relevant physical parameters

It can be observed in equ.(2.4), that both the amplitude and the phase of the thermal wave depend on thermophysical parameters. This means, frequency-dependent measurements (thermal depth profiling) of the amplitude and the phase give information on the effective properties of the investigated solid and can provide additional inquiries on possible defects affecting the thermal transport in the sample. The accessible parameters are the thermal diffusivity α s = ks / ()ρ c s and effusivity es = (ρ ck)s , where k, c, and ρ, are the thermal conductivity, the specific heat capacity and the mass density, respectively. One can see in equ.(2.4), that small values of the thermal effusivity lead to high surface temperature oscillations and that large values of the thermal diffusivity lead to a rapid attenuation of the amplitude below the surface. This certifies that the thermal diffusivity is the relevant thermophysical parameter which governs the heat propagation inside homogeneous solids whereas the thermal effusivity is the relevant parameter for transient surface heating processes [Bein et al., 1992].

2.4 Review of the main detection methods

The photothermal techniques developed for thermal characterization of materials are based on the same principle: generation of a thermal wave through periodic heating of the sample surface by a modulated laser beam, followed by detection of the signal induced by the local temperature increase. A review of the main experimental methods is presented below:

2.4.1 Photothermal Deflection Spectroscopy

The photothermal deflection technique or optical beam deflection or Mirage Effect was introduced in 1980s by Bocara and co-workers [Bocara et al., 1980]. The method relies on periodic heating of the sample surface by a modulated light beam or pump beam, e.g. an Ar ion laser of wavelength λ = 514 nm. The heat diffusion in both the sample and the surrounding medium (e.g., air) produces a temporarily varying gradient in the refractive index which can be detected by the deflection of the probe beam, e.g. a He-Ne laser of wavelength

7 2.4 Review of the main detection methods

λ = 638 nm. Analysis of the deflected beam provides information on the thermal and optical properties of the sample. The pump and the probe beams can be positioned either in a parallel or in a perpendicular configuration [Salazar et al., 1993; Salazar et al., 1996]. This technique has been widely used for the measurement of the thermal conductivity/diffusivity of composites [Inglehart et al., 1985; Macedo and Ferreira, 1999] or of CVD coatings [Fournier, 2001] but is also proposed for the measurement of the optical and electronic properties of semiconductor materials [Forget, 1993]. An important study presenting the effects of the non- linear variation of the refractive index with higher air temperature in front of the sample has been performed [Gruss, Bein and Pelzl, 1999].

2.4.2 Thermoreflectance

The Thermoreflectance technique exploits the local change in the sample’s optical reflectivity, which change is induced by the modulated temperature generated by the heating laser; and additionally for semiconductors, by the plasma waves. This experimental method consists of measuring the intensity variation of a probe beam reflected at the sample surface. The relative change of the sample reflectivity due to the modulated temperature is calculated by

δR 1 dR = δT = C δT (2.5) R R dT T

The temperature coefficient CT, depends on the probe beam wavelength and on the sample material. For semiconductor materials, a term indicating the contribution of plasma waves is included:

δR 1 ∂R 1 ∂R = δT + δn = C δT + C δn (2.6) R R ∂T R ∂n T n

The Thermoreflectance is suited for the measurement of thermal and electronic parameters of semiconductor materials [Fournier, 1992; Kiepert et al.; 1999, Dietzel, 2001] but some other extensions of the method have been designed for monitoring the sample temperature [Gruss et al., 1997; Schaub, 2001]. The thermoreflectance microscopy can deliver thermal images with high spatial resolutions by measuring the variations of the reflection coefficient with temperature [Tessier et al., 2003].

8 2.4 Review of the main detection methods

∆I ∆ϕ

Thermally modulated Reflexion Thermoelastic Effect

h • ν

∆n ∆Φ

h • ν Mirage-Effect IR-Radiometry

Microphon ∆p ∆Τ

PZT

Photoacoustic Effect Photopyroelectric Effect

Figure 2.1: Schemes of the different configurations for the detection of thermal waves through intensity-modulated laser radiation [Pelzl and Bein, 1990]. 9 2.4 Review of the main detection methods

2.4.3 Photothermal Radiometry

In this experimental technique, an IR-radiation detector monitors the variations of the infrared radiation emitted from the heated surface of a sample. In the absence of a thermal perturbation (thermal wave) in the sample, there is however a stationary radiometric signal which is formally proportional to the fourth power of the local static temperature, according to the Stefan-Boltzmann law [Smith et al., 1968; Hudson, 1969]. The modulated radiometric signal due to the variation of the infrared radiation is therefore proportional to the emissivity, to the cube of mean surface temperature, and to the thermal wave which is very small in comparison with the time-averaged surface temperature [Bein et al., 1989]. This technique is very sensitive at high temperature, but less at ambient temperature [Forget, 1993]. During experimental measurement, the MCT detector which has a wavelength sensitivity from 2 to 12 µm is cooled to liquid nitrogen temperature to avoid the generation of a photocurrent due to the absorption of thermal radiation emitted by the detector itself at ambient temperature [Almond and Patel, 1996]. The IR radiometry technique has been used for the investigation of fibre-reinforced composites [Bolte, 1995; Dietzel, 1997; Haj-Daoud, 2000] and has also been explored for the temperature measurement of wires [Borca-tasciuc and Chen, 1997].

2.4.4 Thermoelastic detection

Here, a region of the sample surface which is affected by thermal waves experiences a thermo-elastic deformation, which can be materialized by the reflection of a probe beam directed along the sample surface. The signal is measured by using a position sensitive diode. However, the measured signal only gives indirect information on the temperature distribution and therefore one must refer to theoretical models based on thermal waves to interpret the measured data [Varesi, 1998; Bolte, 1999].

2.4.5 Photoacoustic spectroscopy

The theoretical fundament of the photoacoustic signal has been developed by Rosencwaig and co-worker [Rosencwaig and Gersho, 1976]. Since then, several extensions of the theory have been proposed [Fernelius, 1980; Bennett and Patty, 1982; Pelzl, Klein, Nordhaus, 1982; Bein and Pelzl, 1983] and a wide range of experimental works using this technique have been or are still performed [Krüger et al., 1987; Gibkes, 1992; Malinski, 2002; Gibkes et al., 2004]. The Photoacoustic spectroscopy works according to the following scheme: A sample is irradiated by a modulated light beam, which is then absorbed by the material and converted into heat. The heat that diffuses to the sample surface and into the surrounding gas of the photoacoustic cell leads to a thermal expansion of the gas. The

10 2.5 Conclusions resulting pressure oscillations are detected as sound by a microphone and the photoacoustic signal is measured using a lock-in amplifier.

2.4.6 Photopyroelectric method

This detection method consists of measuring the temperature increase of a sample (excited by a modulated laser beam), by placing a pyroelectric transducer (sensor) in thermal contact with the sample [Chirtoc and Mihailescu, 1989]. If the sensor is placed at the rear face of the sample where excitation takes place, then the configuration is said inverse (front). Otherwise, the configuration is said standard (back). Theoretical developments in the one-dimensional approximation for the thermal wave propagation have earlier shown that the pyroelectric signal depends on the optical, thermal and geometrical parameters of the solid/pyroelectric system [Mandelis and Zver, 1985].

2.4.7 Near field techniques

The detection methods reviewed above are labelled as far-field techniques since no contact with the surface of the investigated sample is required. Near-field techniques have been developed to enable an efficient evaluation of the surface of materials. In these techniques, a thermal probe is scanned over the sample surface to extract the required information from the sample. More details concerning two of the near-field techniques, which are particulary suited to investigate structures in the micro- and submicroscale range, are given in chapter 7.

2.5 Conclusions

In general non-destructive techniques are modern tools suited for the investigation of samples since they don’t necessarily need solid contact and can be applied to samples without any special preparation. The amplitudes and phases of the signals measured using these techniques provide information on the solid characteristics and can additionally give inquiries on possible defects in the investigated materials. In the following chapter, we exploit this contactless property of the photothermal IR radiometric technique to characterize various layered solid structures.

Surface and Subsurface Effects of Thermal Transport in Layered Systems

3.1 Introduction

In industrial applications, several devices are built up from the assembly of many layers of different thermal and physical characteristics. Since these devices are designed to fulfill some specific requirements, e.g. to serve as thermal barriers for the prevention of damages in electronic circuits or alternatively as good conductors for the enhancement of the conducting capabilities of an entire structure, the choice of the constituting layers might not be hazardous. This is why one of the reasons of caring about these layered systems resides in the knowledge of the thermal transport and physical properties of the different components and eventually in the assessment of the possible defects which can obscure the performance of the whole system. Thus, the thermal characterization of such systems provides a large quantity of information which can help the investigator to indicate or recommend the more appropriate structuring to come out with the best device. Among the available techniques designed for thermal investigation of materials, the photothermal techniques are more suited since they are non-destructive and do not require any particular sample preparation. In this chapter, various layered systems consisting of thin films or coatings deposited on substrates are investigated with the help of the IR Radiometric technique. The average total thickness of each of these materials is about 3 mm while the thickness of the thin films ranges from about 0.8 to 5 µm. The phases and amplitudes of the modulated IR signals measured at the surface of these samples are correlated with theoretical predictions in order to determine the unknown thermal and physical properties. Some physical effects revealed by the measured signals are analysed and discussed, which effects can be classified into two groups: Surface effects at higher modulation frequencies or at smaller penetration depths, and subsurface effects at lower modulation frequencies or at larger penetration depths. In particular, the comprehension of the physical effects in the limit of very low modulation frequencies is made possible with the aid of theoretical develepments based on 3-D theory of thermal wave propagation in layered solids.

12 3.2 Photothermal radiometric signal

3.2 Photothermal radiometric signal

Since the IR radiometric technique has been mainly used in the frame of this work for several experimental investigations, we present the theoretical concepts and considerations which help to analyse and interpret the measured IR signals.

3.2.1 Stationary and modulated radiometric signals

According to Bein et co-workers [Bein et al., 1995], the measured stationary radiometric signal relative to the IR radiation emitted by a solid of stationary temperature T, considered as a gray body, can be described by

∞ M (T ) = Cε (T )∫ F(λ)R(λ)W o (λ,T )dλ (3.1) 0

In equ.(3.1), ε(T ) is the spectral emissivity of the sample within the collected solid angle, W o (λ,T ) the spectral Planck’s blackbody radiation, F(λ ) the transmittance of the IR optical system and R(λ ) the spectral responsivity of the detector. C is a constant which depends on the characteristics of the IR detector (collected solid angle of the radiant flux, maximum responsivity) and the electronic system. By introducing the quantity

∞ ∫ F(λ)R(λ)W o (λ,T)dλ 0 (3.2) γ (T ) = ∞ ∫W o (λ,T)dλ 0 equ.(3.1) can be rewritten in a simplified form

∞ M (T ) = Cε (T )γ (T )∫W o (λ,T )dλ (3.3) 0

The integral in the above equation can be analytically retrieved and equ.(3.3) is reduced as follows:

4 M (T) = Cε (T )γ (T )σ SBT (3.4)

4 In equ.(3.4), σ SB is the Stefan-Boltzmann constant and σ SBT the Planck’s blackbody radiation. The quantity γ (T) depends on the detectable wavelength interval in the infrared,

13 3.2 Photothermal radiometric signal

λ1 < λ < λ2 , which is limited by the transmittance F(λ) of the IR optical system. The quantity γ (T) can be considered as a measure of the efficiency of the used IR detection system, to convert the radiation emitted by a blackbody at a constant surface temperature into a voltage signal [Bolte et al., 1995]. Local thermal perturbations due to heating of the solid surface by a modulated laser source (Ar+ laser) induce a small variation of the stationary signal, formally given by (3.4). These perturbations or thermal waves produce a modulated radiometric signal which can be formally interpreted by using the first order Taylor expansion with respect to the temperature,

∂M ( f ,T ) M (T + δT, f ,t) − M (T ) = δM (T, f ,t) = δT ( f ,t) (3.5) ∂T

By combining equ.(3.4) and equ.(3.5), a correlation between the thermal wave generated at the sample surface δTs ( f ,t ) and the resulting modulated IR radiometric signal is obtained:

* 3 δM s (T, f ,t) = Cε s (T)γ (T)T 4σ SBδTs (xs = 0, f ,t) (3.6)

In fact, starting from the expression of the stationary radiometric signal given by equ.(3.4), equ.(3.6) results as a Taylor expansion limited to the first order of M (T +δT, f ,t ) with respect to the temperature, under the assumptions that the temperature oscillation at the sample surface δTs ( f ,t ) is very small in comparison with the time-averaged surface temperature, T, and that the temperature variation of the emissivity is negligible in comparison with the temperature dependence of the Planck black-body radiation. It can be see in equ.(3.6) that the modulated IR radiometric signal is proportional to the thermal wave at the sample surface. The quantity

T ∂γ (T ) γ * (T ) = γ (T ) + (3.7) 4 ∂T can be considered as a measure of the efficiency of the used IR detection system to detect the thermal waves [Bolte et al., 1995]. The complex modulated surface temperature,δTs ( f ,t ) , mentioned in equ.(3.6) can be written in terms of amplitude and phase,

i( 2πft+Φs ) δTs( xs = 0, f ,t ) = As( f )e (3.8)

In thermal wave application, the amplitude, δTs = As ( f ) , and the phase lag relative to the modulation heating,Φ s ( f ), are the results of physical interest since they contain the complete information on the thermal transport and physical properties of the investigated sample. However, as one can see in equ.(3.6), the modulated IR radiometric signal is also affected by the effects of electronic components due to the frequency dependence of the measured thermal

14 3.2 Photothermal radiometric signal wave by the measuring system. At this stage, any quantitative interpretation of the measured signals can only be possible through a signal calibration.

3.2.2 Signal calibration

In order to eliminate the influence of the electronic components on the measured signal and for quantitative interpretation, the measured signals are calibrated with the signals measured at the surface of a smooth and homogenous material, considered as reference and whose thermal, optical and geometrical characteristics are well known. Two typical examples of materials that are often used as reference are the Glassy Carbon (Sigradur), which is highly opaque, and the Neutral Glass (NG), which is rather transparent. The opaque reference material is adopted in this work for signal calibration. Thus, the modulated IR signals measured at the surface of the reference material can be written as follows:

* 3 δM r (T , f ,t ) = C( f )ε r (T )γ (T )T 4σ SBδTr ( xr = 0, f ,t ) (3.9)

If measurements are performed on the sample and after on the reference material under the same experimental conditions, then the combined factor C( f )γ * (T ) related to the detection system can be eliminated and the inverse normalized signal resulting from the calibration of the signals measured on the sample with the signals measured on the reference material is expressed as

−1 δ M r (T, f ) ε r Ar ( f ) Sn ( f ) = = exp{}i[]Φ r ( f ) −Φ s ( f ) (3.10) δ M s (T, f ) ε s As ( f )

In equ.(3.10), Ar( f ) and As( f ) represent the amplitudes of the thermal wave generated at the top surface of the reference material and of the sample, respectively. The inverse normalized phase is defined as the difference between the measured phase of the reference material and the measured phase of the unknown sample.

Φ n ( f ) =Φ r ( f ) −Φ s ( f ) (3.11)

Thus, through signal calibration the unknown thermal, geometrical and optical characteristics of the sample are retrieved from the available properties of the considered reference material. Before interpreting and discussing the signals measured on various solid samples, we first introduce the main components of the experimental set-up.

15 3.2 Photothermal radiometric signal

3.3 Description of the measuring system

The measuring system comprises three main components: (i) An argon ion laser beam (λ/nm = 514) which is modulated by means of an acousto- optical modulator. The laser intensity is about 1W and the modulation frequencies can be varied between 0.03 Hz and 100 kHz, allowing measurement of thin films of less than 1 µm up to layers of 3-4 mm. (ii) For the radiometric detection of the thermal response, a photoconductive mercury cadmium telluride (HgCdTe) detector of surface area of about 2 mm2 and an IR optical system consisting of two Barium difluorid (BaF2) lenses and a filter are used, which allows a detectable wavelength interval of 1 ≤ λ / µm ≤ 12. (iii) The electronic system consists of a pre-amplifier and a two-phase lock-in amplifier. By optimal electronic adaptation of detector and pre-amplifier, the thermal wave signal can be measured free of noise over a frequency interval of 0.1 Hz-100 kHz. The lock-in amplifier serves to filter the small thermal response from the relatively high IR radiation corresponding to the average surface temperature, T, and analyses the IR signal with respect to its amplitude and phase lag relative to the heating modulation. The computer is aimed at coordinating the set-up and processing all useful data.

Laser Modulator

Detector

High-Temp. Cell IR-Optics

Sample

Lock-In Preamplifier Amplitude Phase

Computer

Figure 3.1 Scheme of the experimental set-up for the measurement of IR radiometric signals

16 3.4 Description of the investigated samples

3.4 Description of the investigated samples

The samples investigated in the frame of this work, identifiable by their individual codes (see Appendix A), were globally subdivided into two groups in regard with with their substrate material. The two substrate materials consisted of high speed steel (HSS) and high effusivity metallic alloy (HMA). The coating deposited on top of these substrates was generally a diamond-like carbon (DLC) and the difference between the samples in regard with their surface layer was pointed out by the different variants of the DLC. For example, the samples codified C6 and F5 consisted of the same substrate material on which has been deposited two variants of the same material (DLC). Inversely, the sample C4 and C6 were different only by their substrate materials. Geometrically, the investigated samples had either a parallelepipedic form with a lateral size of about 10 mm or a cylindrical form with a diameter of about 10 mm. The thickness of the coatings ranged from 1 to 5 µm for a mean thickness of the substrate of about 3 mm. More details about all investigated samples are provided in Appendix A.

3.5 Theoretical models of thermal waves for layered systems

The generic term layered (or layer) systems refers to samples consisting of a basic material or substrate on which can be identified one or more layers of differing optical and thermophysical properties. Figure 3.2 a + b shows the schematic representations of a two-layer system and of a three-layer system, respectively. In each case, the top of the sample is surrounded by a thick layer of gas (air) whereas the bottom material is considered as semi- infinite, so that the temperature fluctuation at the end side is negligible. Several theoretical formulations of the thermal waves in layered structures have been proposed [Egee et al., 1986; Tilgner et al., 1986].

y y

Substrate Substrate

z z

(a) (b)

Figure 3.2 a + b:(from left to right) Scheme of a two-layer system (a) and of a three-layer system (b). The top of the sample is excited by a modulated laser beam to generate thermal waves. 17 3.5 Theoretical models of thermal waves for layered systems

But these formulations contain quantities which are useless because normally negligible and thus one cannot have a rapid view of the main factors which govern the propagation of thermal waves in the solid. The theoretical models for thermal waves we propose are free of such useless quantities. Here, the thermal waves are simply expressed as a function of combined parameters. Without further details, the expressions of thermal waves at the top surface of the samples are presented below for a two- and a three-layer model. As one can observe in equ.(3.6) or in equ.(3.9), the signal amplitude is proportional to the amplitude of the thermal wave while the signal phase is the phase of the thermal wave. For purpose of quantitative interpretation, the results of theoretical approximations (phase and amplitude) are correlated with the normalized measured data to extract the unknown properties of the sample investigated. Thus, starting from the hypothesis that the laser spot is very large at the sample surface, large enough to neglect any lateral heat diffusion in the material, the expression of the complex surface temperature for the two-layer and for the three-layer model has been analytically derived as a function of the modulation frequency and the relevant combined thermophysical quantities. For a highly opaque and semi-infinite three- layer model of sample (Fig. 3.2b), whose upper surface is subjected to plane harmonic radiation intensity in the form

I (x = 0,t) = (I o / 2)[]1 + cos(2πft) (3.12) the modulated complex surface temperature is given by

π i(2πft− ) 4 ηs Ioe δTs ( f ,t) = × 2es 2πf []−2(1+i) πfτ [−2(1+i) πfτ ] [−2(1+i)( πfτ + πfτ )] 1+ R e s + R R e i + R e s i × si si ib ib []−2(1+i) πfτ s []−2(1+i) πfτi []−2(1+i)()πfτ s + πfτi (3.13) 1− Rsie + Rsi Ribe − Ribe

In equ.(3.13), the indexes s, i, and b stay for the surface layer, the intermediate layer and the substrate, respectively. The two combined quantities Rsi and Rib are the thermal reflection coefficient between the coating and the intermediate layer, and between the intermediate layer and the substrate, respectively. These coefficients are given by

gsi −1 gib −1 Rsi = Rib = (3.14) gsi +1 gib +1

The two combined parameters defined in (3.14)

18 3.5 Theoretical models of thermal waves for layered systems

gsi = es / ei and gib = ei / eb (3.15) are respectively the ratio of effusivities surface layer−to−intermediate layer and the ratio of effusivities intermediate layer−to−substrate. The other combined parameters used in (3.13)

2 2 τ s = d s / α s and τ i = di /αi (3.16) are the thermal diffusion time of the thermal wave in the surface layer and the thermal diffusion time of the thermal wave in the intermediate layer, respectively. These characteristic times are defined by the thermal diffusivity α and the thickness d of the corresponding layer.

In the absence of an intermediate layer between the surface layer and the substrate, di = 0, andτ i = 0 , and the complex surface temperature valid for a two-layer system can be deduced from expression (3.13) by replacing index i with index s or b:

π i(2πft− ) 4 []−2(1+i) πfτ s ηs Ioe 1+ Rsbe δTs ( f ,t) = (3.17) []−2(1+i) πfτ s 2es 2πf 1− Rsbe

The thermal reflection coefficient in (3.17) is defined by

gsb −1 Rsb = (3.18) gsb +1

In (3.18), gsb represents the ratio of effusivities coating−to−substrate. The complex surface temperatures obtained for the three-layer and for the two-layer model of sample, and given respectively by (3.13) and (3.17) suppose a highly opaque sample for which the optical penetration depth µopt =1/ β s is very small and therefore negligible. However, if the thin film or coating deposited on the absorbing substrate is semi-transparent, e.g. in the visible spectrum, the term exp(−β s ds ) which did not appear for example in the expression of the thermal wave given by (3.17), due to complete absorption of the incident radiation at the surface (β s d s >> 1), cannot longer be omitted since in this case the optical thickness or optical absorption length, β s ds , becomes finite. Under consideration of coating semi- transparency, the modulated complex surface temperature can be expressed by equation

(3.19). In this equation, another combined quantity, α s β s , appears as a function of the thermal diffusivity and the optical absorption coefficient. Expressions (3.13), (3.17) and (3.19) show, that thermal waves generated at the surface or a region limited by the optical absorption length, propagate into the sample and interact with features of different thermal properties to produce interference effects which are detectable at the surface. That means, the measured signals are considerably influenced by the thermophysical properties of the sample.

19 3.5 Theoretical models of thermal waves for layered systems

π i(2πft− ) η I e 4 δT ( f ,t) = s o × s ⎧ 2 ⎫ ⎪ ⎡(1+ i) πf ⎤ ⎪ 2es 2πf ⎨1− ⎢ ⎥ ⎬ α β ⎩⎪ ⎣⎢ s s ⎦⎥ ⎭⎪

⎧ []−2(1+i) πfτ s ⎫ 1+ Rsbe (1+ i) πf ⎪ − ⎪ (3.19) []−2(1+i) πfτ s ⎪1− Rsbe α s βs ⎪ ⎪ ⎪ × ⎨ (1+ i) πf ⎬ + g ⎪ sb −(1+i) πfτ ⎪ α β e s e−βsds ⎪+ 2 s s ⎪ ⎪ g +1 −2(1+i) πfτ s ⎪ ⎩ sb 1− Rsbe ⎭

According to (3.10), the inverse normalized signal resulting from the calibration of the signal measured at the surface of an opaque two-layer system with the signal measured at the surface of the opaque reference material (Rsb = 0) is given by

[−2(1+i) πfτ s ] −1 ε r ηr es 1+ Rsbe Sn ( f ) = (3.20) ε η e []−2(1+i) πfτ s s s r 1− Rsbe

In the limit case of higher modulation frequencies, corresponding to a small diffusion length of the thermal wave and therefore to a small penetration depth, according to x∝ µth = α / πf ,

−1 ε r ηr es Sn ( f → ∞) = (3.21) ε s ηs er and in the limit case of lower modulation frequencies, corresponding to larger penetration depths of the thermal wave,

−1 ε r ηr eb Sn ( f → 0) = (3.22) ε s ηs er

−1 −1/ 2 That means, at smaller penetration depths the inverse normalized signal Sn ( f ) is proportional to the thermal effusivity, es, of the coating and at larger penetration depth it is proportional to the thermal effusivity, eb, of the substrate. In equ.(3.21) and (3.22), er stays for the thermal effusivity of the homogeneous reference material. By comparing the inverse signals at the two limit cases, the optical properties are eliminated and the ratio of thermal effusivities coating to substrate can be directly determined:

20 3.6 Interpretation of the measured modulated IR signals

−1 Sn ( f → ∞) es −1 = (3.23) Sn ( f → 0) eb

In the following section, the amplitudes and phases of the modulated IR signals measured at the surface of several structured samples are analysed, interpreted and discussed.

3.6 Interpretation of the measured modulated IR signals

Generally, two types of thermal wave measurements, giving different information on the materials’ properties can be done: In the transmission configuration of thermal waves, modulated heating and detection of the thermal wave response take place on the opposite surfaces while in the reflection configuration modulated excitation and detection take place at

20 10 10

0 -1

-10 n /deg S n

Φ -20

-30 -35 0 50 100 150 200 250 300 350

1/2 1 ( f / Hz )

0.01 0.1 1

( f / Hz )-1/2

Figure 3.3 a + b: (from left to right) Inverse normalized amplitudes (a) and normalized phases (b), calculated as a function of the modulation frequency for 1-D thermal wave propagation in a two- layer system. Variations of the ratio of the effusivities of surface layer – to – subsurface material are compared with the data measured for a coated sample (C6). the same surface. The transmission configuration is more appropriate for samples of finite thickness while the reflection configuration is adopted for the investigation of samples regarded as semi-infinite. It should be boted that the terminologies finite sample and semi- infinite sample are in connection with the thermal diffusion length that means the characteristic length at which a thermal wave measurement is effective.

21 3.6 Interpretation of the measured modulated IR signals

To be measured, the samples investigated in this work were placed in the reflection configuration since their average thickness was sufficient large.

1/2 -2 -1 1/2 -2 -1 2 -1 Rsb gsb eb / Ws m K es / Ws m K τs /s ds / m αs / m s -0.639 0.22 7530 1657 6.6 2.8 1.19×10-6

Table 3.1: Thermal transport properties of the sample C6 obtained through correlation between experimental measurements and theoretical predictions.

Figure 3.3a and Figure 3.3b show respectively the frequency-dependent amplitude and the frequency-dependent phase of the modulated IR signals measured at the surface of a sample (C6, a DLC on HSS), in comparison with theoretical approximations (full lines) based on 1-D thermal wave propagation in solids. The thermal transport and physical properties obtained via the combined thermophysical parameters τs and gsb (Rsb) according to (3.14), (3.15) and (3.16) are reported in Table 3.1. Additionally, the ratio of the effusivities coating–to−substrate is varied systematically, with the values es/eb = {9.5, 1.00, 0.41, 0.22, 0.13} of the theoretical curves from the top to the bottom. As one can observe in Figure 3.3, the further theoretical curves indicate the amplitudes and phases of the signals that can be measured at the surface of a sample where, either the thin film or coating of better thermal properties (gsb > 1) has been deposited on the substrate, or alternatively the coating has been deposited on a substrate of better thermal properties (gsb < 1). The ratio gsb = 1 corresponds to the case of a homogenous sample. However, as one can clearly observe in Figure 3.3a, there is a small deviation in the range of very low modulation frequencies between the measured amplitude and the theoretical approximation. Figure 3.4 shows the amplitude and phase of the IR signal measured at the surface of 2 another sample (F3, a DLC on HMA). One can see, that the thermal diffusion time τs = ds /αs, which gives information on the thermal thickness, or indirectly on the geometrical thickness ds of the surface layer, can be determined by approximating the normalized measured phases.

Once the thermal diffusivity αs of the coating is known, the layer thickness can be retrieved by non-destructive and non-contact thermal wave measurements; or inversely if the layer thickness is known, then the expected quantity is the thermal diffusivity. Good approximation between the theoretical curves and the measured data is observed in Figure 3.3 and Figure 3.4 at the intermediate modulation frequencies, e.g. in Figure 3.4b for the interval 15 < (f /Hz)1/2 < 150. In the low and high frequency limit, however, the theoretical phase and amplitude values based on 1-D theory of heat transport in the two-layer system, deviate considerably from the experimental results, as one can see also in Figure 3.4b.

22 3.6 Interpretation of the measured modulated IR signals

τ =4.2µs 5 s 0 τ =37.9µs s 18.05 18.05 -1

n -10

S / deg

n 4.2

10 Φ 37.9 -20

-25 0 50 100 150 200 250 300 350

5 1/2 0.01 0.1 1 ( f / Hz ) ( f / Hz )-1/2

Figure 3.4 a + b: (from left to right) Inverse normalized amplitudes and normalized phases, calculated for 1-D thermal wave propagation in a two-layer system. Variations of the thermal diffusion time of the surface layer are compared with the data measured for a coating on a metal alloy of higher thermal effusivity (F3).

20 10 15 5 10 0 5 -5

0 -10

-5 / deg / deg n

n -15

-10 Φ Φ -20 -15 -20 -25 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 1/2 (f / Hz)1/2 (f / Hz)

Figure 3.5 a + b: (from left to right) Inverse normalized phases of the modulated radiometric signal for two measured coated samples (a-A3 and b-F3) in comparison with a theoretical approximation assuming an opaque system and based on 1-D theory of thermal wave propagation.

In order to pursue the discussions about the deviations between theory and experiment at low and at high modulation frequencies, the measured phases of Figure 3.4b are once more represented in Figure 3.5b. Additionally, Figure 3.5a represents the calibrated phase of the sample A3 (another variant of DLC on HSS). While in Figure 3.5a, there is a small deviation of the measured phase, in the limit of very low modulation frequencies, with respect to the

23 3.7 Discussion of physical effects in the investigated samples

theoretical approximations based on 1- D thermal wave propagation, a rather considerable deviation at that frequency range can be observed in Figure 3.5b between the measured phases and the theoretical results. At high modulation frequencies, two types of deviation of the measured phases with respect to the theoretical approximation are visible. Whereas in Figure 3.5a the deviation of the measured phases (symbols) are directed above the theoretical curve (solid line) in the frequency interval, 50 < (f /Hz)1/2 < 350, the deviation of the calibrated measured phases of Figure 3.5b is directed below the theoretical approximation, however in a reduced frequency interval, 100 < (f /Hz)1/2 < 350. Another important remark is, that in the range of high modulation frequencies the measured phases of Figure 3.5a deviate from the limit Φn(f) = 0 and increase continuously to positive values while the measured phases of

Figure 3.5b show rather a relative maximum below the limit Φn(f) = 0 and then decrease with increasing modulation frequencies. All these deviations of the experimental phases and amplitudes with respect the theoretical approximations suggest the existence of physical effects revealed by the modulated IR signals of the corresponding investigated samples. In regard with the above observations, these physical effects can be classified in two groups: Surface effects which are localized at high modulation frequencies or at smaller penetration depths and subsurface effects which are localized at low modulation frequencies or at larger penetration depth.

3.7 Discussion of physical effects in the investigated samples

3.7.1 Surface effects

The deviations observed between the calibrated measured phases and the theoretical approximations based on 1-D thermal wave propagation in a highly opaque two-layer system can be at first sight attributed to the semi-transparency of the surface layer. However, as the existence of a relative phase minimum in the range of intermediate frequencies indicates the transition coating –to− substrate in the frame of a two-layer model of sample, the appearance of a relative phase maximum in the range of high modulation frequencies may also indicate the transition between a thin layer at the very surface of the sample and the coating. For this, the surface effects can be subdivided into two groups: optical effects and topological effects.

3.7.1.1 Optical effects: semi-transparency of the coating

The optical effects are essentially in connection with the semi-transparency of the coating. In fact, the theoretical approximations used above for the interpretation of the amplitudes and phases of the measured IR signals assume a highly opaque sample, where the heat source

24 3.7 Discussion of physical effects in the investigated samples coincides with the surface of the coating due to a very small optical absorption length or optical penetration depth. In this case, there is no contribution to heat generation caused by the absorption of light in the substrate. If the thin film deposited on the highly opaque substrate is semi-transparent e.g. in the visible spectrum, then a thermal wave generation at the substrate surface may be possible due to a considerable optical absorption length induced by the finite value of the optical absorption coefficient. Reinterpretation of the measured phase of Figure 3.5a is shown in Figure 3.6. As can be seen, the measured phases agree well with the theoretical approximations taking into account the semi-transparency in the visible spectrum of the thin film deposited on the highly opaque substrate. As said before, the level of this semi-transparency is controlled by the magnitude of the optical absorption coefficient. A complete characterization of the sample A3 is done by correlating the theoretical approximations with the calibrated measured phases and the extracted properties are reported in Table 3.2.

/ deg -5 n Φ -10

-15

-20 0 50 100 150 200 250 300 350 (f / Hz)1/2

Figure 3.6: Inverse normalized phases of the modulated IR signal for a DLC-coated steel sample (A3): Semi-transparency of the thin film in the visible spectrum. The dashed curve below the measured phases at the frequencies 50 < (f /Hz)½ < 350 represents the theoretical approximation based on 1-D thermal wave propagation assuming a highly opaque sample.

25 3.7 Discussion of physical effects in the investigated samples

2 -1 1/2 -1 eb gsb es ds / m τs / s αs / m s αs βs βs / m 7530 0.378 2850 3.1×10-6 20.67×10-6 0.465×10-6 2.74×10+3 4×10+6

Table 3.2: Characterics of the sample A3, taking into account the semi-transparency of the thin film layer deposited on the highly opaque substrate.

In Figure 3.6, the calibrated measured phases take the value Zero at (f /Hz)1/2 ≈ 125 and then increase continuously with increasing modulation frequencies. The upper limit of these measured phases depends on the magnitude of the combined thermo-optical quantity, 1/2 Pαβ = αs βs, therefore on the optical absorption coefficient of the coating. As example, for +3 Pαβ = 1.37×10 , the upper limit of the calibrated measured phases would be about Φn = 18° (upper theoretical curve in Figure 3.6). It is useful to mention, that the coating is assumed to be highly opaque in the IR spectrum (β IR d s >> 1). Otherwise a finite value of the optical absorption in that spectrum has to be considered to account for the radiative contributions both from the surface and the subsurface of the material. To indicate such a double semi- transparency of the layer in both the visible and the IR spectrum, the modulated IR signals are expressed as follows:

∞ δ M ( f ,T)=4Cε (T )γ σ T 3 ⋅ ß exp(−β x')δ T (x', f )dx' (3.24) D SB ∫ IR IR s 0

3.7.1.2 Topological effects

While in Figure 3.5a or in Figure 3.6, the calibrated measured phases affected by the coating semi-transparency (symbols) take the value Zero and increase continuously with the modulation frequency toward positive values, the calibrated measured phases represented in Figure 3.5b present rather a relative maximum in the range of high modulation frequencies, namely at about (f /Hz)1/2 = 225, and then decrease with increasing frequency. Referring to the relative phase minimum that can also be observed in Figure 3.5b at the modulation frequency of (f /Hz)1/2 ≈ 50 and which indicates the transition between the subsurface material and the coating, the appearance of a relative maximum in the range of very high modulation frequencies or at very low penetration depths, can be explained by the existence of an additional layer of reduced thermal transport properties at the very surface of the coating. Reassessment of the measured phases of Figure 3.5b, taking into account the existence of such a layer at the very surface of the coating is presented in Figure 3.7a. In addition, other calibrated measured phases for the sample F5 (DLC on HSS) are inserted for comparison. One can observe, that at high modulation frequencies the calibrated measured phases of the two samples possess the same surface characteristics (same thin layer at the very surface of 26 3.7 Discussion of physical effects in the investigated samples

5 0 0 (3)

-5 -5 (2)

-10 -10

(1) / deg n / deg

-15 n Φ Φ -15 -20

-25 -20 0 50 100 150 200 250 300 350 (f / Hz)1/2 -25 0 200 400 600 800 ( f / Hz )1/2

Figure 3.7a: Inverse normalized phases of two Figure 3.7b: Evolution of the relative phase samples (F5-×, F3-*) showing the behavior of a maximum in the range of high modulation three-layer system. The measured data are frequencies with the ratio of effusivities thin compared with theoretical approximations based on film-to-coating, gsi = 0.250 (1), 0.375 (2), 1-D thermal wave propagation. 0.625 (3)

2 -1 2 -1 es ei ebHSS ebHMA αs / m s αi / m s ds / m di / m 1150 4200 7530 13600 0.80×10-6 0.20×10-6 0.045×10-6 1.9×10-6

Table 3.3: Sample characteristics showing the existence of a thin layer at the very surface of the coating.

the coating and same coating) but differ with each other by their subsurface material. The results of the quantitative interpretation are consigned in Table 3.3. As one can see in this table, the thickness of the additional layer which has been deposited on the coating is -6 extremely small, ds /m = 0.045×10 . The difference between the substrates of the two samples is pointed out by the values of thermal effusivities, that is 7530 Ws1/2m-2K-1 for the HSS- subsurface material, and 13600 Ws1/2m-2K-1 for the HMA-subsurface material. According to the calibrated theoretical phases represented in Figure 3.7b, the higher the ratio gsi of effusivities between the thin layer at the very surface and the coating, the higher the value of the relative phase maximum. However, all relative extrema that can occur on the measured phases do not automatically signify the transition between two consecutive

27 3.7 Discussion of physical effects in the investigated samples

layers. In the next section we show, that the relative extremum observed in the limit of very low modulation frequencies is rather induced by thermal transport effects.

3.7.2 Subsurface effects: Lateral thermal transport

Since in the limit of very low modulation frequencies, corresponding to very large penetration depths, the 1-D theory of thermal wave propagation has been found insufficient to explain the reasons of deviations between layer models and the data measured for the amplitudes and phases in reality, a 3-D theory of thermal wave propagation in two-layer systems has been developed, which takes into account the finite size of the heating spot. First, the main features of this theory are presented:

3.7.2.1 Theoretical developments

Considering temperature-independent thermal properties and the superposition principle, T( r,θ ,z,t ) = T( r,θ ,z )+δT( r,θ ,z,t ) , for the solutions of the heat diffusion equation, the diffusion equation for thermal waves propagating in a solid medium can be written in cylindrical coordinates as

∂δT (r,θ, z ,t) ⎡1 ∂ ⎛ ∂δT (r,θ, z ,t) ⎞ ∂ 2δT (r,θ, z ,t) ∂ 2δT (r,θ, z ,t)⎤ δQ (r,θ, z ,t) j j = α ⎜r j j ⎟ + j j + j j + j j j ⎢ ⎜ ⎟ 2 2 2 ⎥ ∂t ⎣⎢r ∂r ⎝ ∂r ⎠ r ∂θ ∂z j ⎦⎥ (ρc) j (3.25)

In equ.(3.25), the index j refers to the gas layer (g), the surface layer (s) and the substrate (b).

The upper surface of the sample, defined by zs = 0, is heated by a Gaussian shaped laser beam of radius rH, which can be modulated at the frequency f. rH is the radius at which the Gaussian distribution has dropped to 1/e² of its peak value. Applying the superposition principle, the ac- part of the volumetric heat source in the surface layer is given by

2I η r 2 o s (3.26) δQs (r, zs ,t) = 2 exp(−2 2 ) βs exp(−βs zs ) exp(i2πft) πrH rH

In equ. (3.26), I0 is the incident beam power, ηs the photothermal conversion efficiency, and βs the optical absorption constant of the coating. Due to the cylindrical symmetry introduced by the laser geometry, the ac temperature is invariant with respect to the angular coordinate. Neglecting the effects of heat losses to the surrounding gases, which is justified for sample surfaces with effusivity values much larger than the effusivity of the gas atmosphere [Bein and Pelzl, 1989], the boundary conditions accompanying the above equation (3.25) are defined by:

28 3.7 Discussion of physical effects in the investigated samples a) the negligible temperature at outer boundaries,

δTg (r, zg = 0,t) = 0 (3.27a)

δTb (r, zb = lb ,t) = 0 (3.27b) b) the temperature continuity at the interfaces gas/coating and coating/substrate,

δTg (r, zg = lg ,t) = δTs (r, zs = 0,t) (3.28a)

δTs (r, zs = ls ,t) = δTb (r, zb = 0,t) (3.28b) c) and the heat flow continuity at the interfaces gas/coating and coating/substrate.

δqg ( r,z g = lg ,t ) = δqs ( r,zs = 0,t ) (3.29a)

δqs ( r,zs = ls ,t ) = δqb ( r,zb = 0,t ) (3.29b)

By using the Hankel transformation and subsequently its inverse [Abramowitz and Stegung, 1965; Özisik, 1997], defined respectively by:

∞ F(λ) = F(r) J (λr)rdr (3.30a) ∫ o 0 ∞ F(r) = F(λ) J (λr)λdλ (3.30b) ∫ o 0 the Laplacian operator in cylindrical coordinates, taking into account the cylindrical symmetry, is transformed according to (3.30a) as follows:

⎡1 ∂ ⎛ ∂ ⎞ ∂ 2 ⎤ ⎛ ∂ 2 ⎞ ∇2 = ⎜r ⎟ + → ⎜ − λ 2 ⎟ (3.31) ⎢ 2 ⎥ ⎜ 2 ⎟ ⎣⎢r ∂r ⎝ ∂r ⎠ ∂z ⎦⎥ ⎝ ∂z ⎠

Then, applying the Hankel transformation to equ.(3.25) yields for each medium (j = g, s, b) the following ordinary differential equation

d 2δT (λ, z ,t) 2η β I Q (λ) j j 2 j j o j −β j z j 2 − λ jδT j (λ, z j ,t) = − 2 e j = g, s,b (3.32) dz j πrH k j for which the general solution is given by

HT λ j z j −λ j z j − β j z j i 2πft δT j (λ, z j ,t) = (A j e + B j e + C j e ) e (3.33)

29 3.7 Discussion of physical effects in the investigated samples

Since the heat source is present only in the surface layer, Qg ( λ ) and Qb (λ ) are identically equals to zero and therefore such a term Cg or Cb does not exist. In equations (3.32) and (3.33),

2 2 2 λ j ( λ, f ) = λ + σ j and σ j = (1+ i) πf /α j (j = g,s, b) (3.34)

The quantities kj and σ j are the thermal conductivity and the thermal wave vector of the indexed layer, respectively. For quantitative interpretation of the measured signals, only the theoretical expression of the thermal wave at the sample surface is of interest and thus the useful integration constants are As, Bs and Cs. By inserting equ.(3.33) into equ.(3.32) for the index j = s, one obtains:

λ 2r 2 exp(− H ) 2ηs Io Qs (λ) ηs Io 8 Cs (λ, f ) = − 2 2 2 = − 2 2 (3.35) πrH ks β s 1− λs (λ, f )/ β s 2πks β s 1− λs (λ, f )/ β s

∞ r 2 r 2 λ 2r 2 with Q (λ) = exp(−2 ) J ((λr)rdr = H exp(− H ) , according to (3.30a). s ∫ 2 o 0 rH 4 8

By applying the Hankel transformation to the boundary conditions given by (3.27), (3.28) and (3.29), and then dividing the transformed equ. (3.29a) by the transformed equ.(3.29b), the three integration constants are interrelated in the following equation:

⎧ β s ⎫ ⎪ G(λ, f ) + ⎪ ⎪G(λ, f ) −1 λs (λ, f ) ⎪ (3.36) Bs (λ, f ) = −⎨ As (λ, f ) + Cs (λ, f )⎬ ⎪G(λ, f ) + 1 G(λ, f ) + 1 ⎪ ⎩⎪ ⎭⎪

k λ2 + σ 2 e α λ2 + σ 2 with G(λ, f ) = g g coth(λ l ) = g g g coth(λ l ) k 2 2 g g 2 2 g g s λ1 + σ s es α s λ + σ s

In equ.(3.36), G can be further simplified by assuming an expanded gas layer (air) in front of the sample, that means coth( λair lair ) ≈ 1, and by remarking that for λ → 0, G(λ,f)→ eg /es, λ

→ ∞, G(λ,f) → kg /ks. λ is an integration variable in the Hankel space. In general the thermal effusivity and conductivity of the gas layer (air) are very small in comparison with the same properties for most materials [Bein and Pelzl, 1989]. Following this, parameter G can be neglected and therefore equ. (3.36) reduces to

30 3.7 Discussion of physical effects in the investigated samples

λ 2r 2 exp(− H ) η I s o 8 (3.37) Bs (λ, f ) ≈ As (λ, f ) + 2 2 2πks λs (λ, f ) 1− λs (λ, f ) / β s

Taking into account the above approximations on the parameter G, the two following combinations of the Hankel transformed boundary conditions, namely [(3.28a) + (3.29b)] and

[(3.28a) − (3.29b)], give respectively the expressions of the integral constants Ab ( λ, f ) and

Bb ( λ, f ) which are then reintroduced in the Hankel transformed boundary condition (3.27b) to yield

λ 2r 2 exp(− H ) ηs I o 8 Bs (λ, f ) ≈ 2 2 × 2πks λs (λ, f )[]1− Rsb (λ, f )exp(−2λs d s ) 1− λs (λ, f ) / β s (3.38)

⎧ λs (λ, f ) / β s − g sb (λ, f ) ⎫ × ⎨1+ exp[]− (β s + λs )d s ⎬ ⎩ g sb (λ, f ) +1 ⎭ which according to (3.37) leads to the determination of another integration coefficient

λ 2r 2 exp(− H ) ηs I o 8 As (λ, f ) ≈ 2 2 × 2πksλs (λ, f )[]1− Rsb (λ, f )exp(−2λs d s ) 1− λs (λ, f )/ β s (3.39)

⎧ λs (λ, f )/ β s − g sb (λ, f ) ⎫ ×⎨R sb (λ, f )exp(−2λs d s ) + exp[]− (β s + λs )d s ⎬ ⎩ g sb (λ, f ) +1 ⎭

Since the model is considered to be highly opaque, the optical absorption coefficient is infinite (that means, βs → ∞) and therefore the heat source in the coating coincides with the sample surface. In this case, the optical penetration depth (1/β) is very small and the optical thickness βsds is very large, so that the magnitude of the integration coefficient Cs(λ, f) given by (3.35) becomes extremely small and can be neglected. As consequence, the integration constants given by equ.(3.38) and equ.(3.39) take respectively the reduced forms

λ 2 r 2 η I exp(− H ) s o 8 Bs (λ, f ) ≈ (3.40a) 2πk s λs (λ, f )[]1− Rsb (λ, f ) exp(−2λs d s )

λ 2r 2 η I R (λ, f )exp(− H ) s o sb 8 As (λ, f ) ≈ (3.40b) 2πksλs (λ, f )[]1− Rsb (λ, f )exp(−2λs d s )

31 3.7 Discussion of physical effects in the investigated samples

According to equ.(3.30a), and with respoect to equations (3.40a) and (3.40b) the Hankel transformed surface temperature is given by:

2 2 λ rH η s I o exp(− ) HT 8 ⎡1+ Rsb (λ, f )exp(−2λs d s ) ⎤ i2πft δTs (λ, zs =0, t) = ⎢ ⎥ e (3.41) 2π ksλs (λ, f ) ⎣1− Rsb (λ, f )exp(−2λs d s ) ⎦ and according to equ.(3.30b), the ac complex surface temperature can be calculated as

2 2 λ rH ∞ exp(− ) η I ⎡1+ R ( f ,λ)exp(−2λ d )⎤ δT (r, z = 0, t) = s o 8 sb s s J (λr)λdλ ⋅ei2πft s s ∫ ⎢ ⎥ o 2π ks 0 λs (λ, f ) ⎣1− Rsb ( f ,λ)exp(−2λs d s )⎦ (3.42)

In equations (3.38), (3.39), (3.40), (3.41) and (3.42), ds indicates the coating thickness. The quantity

gsb (λ, f ) −1 Rsb (λ, f ) = (3.43) gsb (λ, f ) +1 formally represents the thermal reflection coefficient at the interface surface layer – substrate, with the combined parameters gsb (λ, f ) given by

2 2 ks λ +σ s g sb ( λ, f ) = (3.44) k 2 2 b λ +σ s

Based on the expression of the thermal wave at the sample surface (3.42), the modulated IR radiometric signal measured over the detection spot area of radius rD can be calculated as:

* 3 δM s ( f ,T,t) = 4C( f )γ (T)σ SBT ε s (T)× 2 2 λ rH rD ∞ exp(− ) η I ⎡1+ R ( f , λ) exp(−2λ d )⎤ × s o 2πrdr 8 sb s s J (λr)λdλ ⋅ ei2πft ∫∫ ⎢ ⎥ o ks 00λs (λ, f ) ⎣1− Rsb ( f ,λ) exp(−2λs d s )⎦ (3.45)

For remind, the factor C( f ) describes the frequency characteristics of the detection device, and the factor γ *(T ) describes the efficiency of thermal wave detection [Bolte, 1995]. T is the time-averaged surface temperature; σ SB and ε s (T ) are respectively the Stefan-Boltzmann * 3 constant and the emissivity of the coating. The factor 4C( f )γ (T)σ SBT ε s (T) is a real

32 3.7 Discussion of physical effects in the investigated samples quantity, which can be eliminated by an appropriate calibration procedure. As a real quantity it cannot affect the phase of the measured thermal wave.

rD βs r H es

αs eb Surface layer αb Substrate

Figure 3.8: Scheme of the 2-layer sample and of the experimental conditions based on concentric heating and detection spot.

3.7.2.2 Comparison of one- and three dimensional thermal wave propagation

Here approximations based on 1-D and 3-D thermal wave propagation are compared with each other and with the data, measured for a hard coating of diamond-like carbon of 2.8 µm thickness on two substrate materials of different thermal properties – high speed steel (HSS) and a high effusivity metallic alloy (HMA). In Figure 3.9a one can see, that differences between 3-D and 1-D thermal wave propagation become remarkable only in the limit of the low heating modulation frequencies, corresponding to large penetration depths, that means in the subsurface material. While according to 1-D theory (broken line) the normalized phases clearly tend to the value Zero at the very low frequencies, the 3-D heat propagation (continuous line) induces considerable deviations of the normalized phase in that frequency range. The similarity of the two theoretical curves at the intermediate and high modulation frequencies suggests that the lateral heat transport only plays a role in the low frequency range, (f /Hz)1/2 < 20, corresponding to large penetration depths in the subsurface material. A similar behaviour can be deduced from the measured phases (Fig. 3.9b):

33 3.7 Discussion of physical effects in the investigated samples

(i) As already seen above, in the limit of high modulation frequencies, (f /Hz)1/2 > 150, corresponding to very small penetration depths, the normalized phases of the two samples indicate the presence of an additional thin layer of reduced effusivity at the very surface of the coatings, which is common to the two samples (x, ∗).

5 5

0 0

-5 -5

-10 -10

/ deg / deg n -15 n -15 Φ Φ

-20 -20

-25 -25 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 1/2 ( f / Hz ) ( f / Hz )1/2

Figure 3.9a: Normalized phases according to Figure 3.9b: Normalized phases measured for a 1-D (broken line) and 3-D thermal wave hard coating (DLC) deposited under equal propagation (full line) as functions of the conditions on two substrates of different thermal square root of the modulation frequency. – The properties (F5-HSS ×, F3-HMA ∗). – The phases radii of heating (2 mm) and detection spot (0.8 measured for the coated samples have been mm) used for 3-D thermal wave calculation calibrated using the signals of a smooth agree with the measuring conditions of Fig. homogeneous sample of glassy carbon. 3.8

(ii) Also, the differences at intermediate frequencies, 25 < (f /Hz)1/2 < 100, are due to different ratios of the effusivities coating-to-substrate, gsb= es/eb, and due to an eventually 2 different thermal diffusion time τs = ds /α s of the coating of the two samples [Bennet and Patty, 1982; Bein et al., 1989]. (iii) In the differences at the very low frequencies, (f /Hz)1/2 < 15, the 3-D character of thermal wave propagation becomes obvious, with a stronger lateral heat propagation in the substrate (∗) of comparatively larger thermal diffusivity and effusivity.

3.7.2.3 Effects of heating spot radius and subsurface thermal properties

In general, 1-D thermal wave propagation can be applied for the interpretation of modulated photothermal measurements, when the heating spot is large in comparison to the 1/2 thermal diffusion length, 2rH >> µth= [α /(π f )] . This can be verified in Fig. 3.10a and in

34 3.7 Discussion of physical effects in the investigated samples

Fig. 3.10b for theoretical approximations calculated for increasing heating spot radii. As can be seen, the results based on 3-D thermal wave propagation and calculated for a heating spot radius of rH = 32 mm perfectly agree with the approximations based on 1-D heat transport. The effect of lateral heat propagation increases with increasing values of the thermal diffusivity αb of the substrate material, as can be seen in Fig. 3.11a+b. For a low thermal -6 2 diffusivity value of αb= 7⋅10 m /s the theoretical approximation is close to the case of 1-D

10 10 r =0.8mm r =0.8mm D D 0 0 r =32mm r =32mm H H 8 8 -10 -10

2 2 / deg / / deg n -20 n

Φ -20 Φ

-30 -30 -35 -35 0 25 50 75 100 125 150 175 0 20406080 1/2 ( f / Hz ) ( f / Hz )1/2

Figure 3.10 a + b: (from left to right) Range of higher (a) and lower modulation frequencies (b) of the normalized phases measured for a DLC coating on a metallic alloy (C4-∇), in comparison with theoretical solutions based on 3-D thermal wave propagation with increasing heating spot radii (full lines). The detection spot radius rD of the three theoretical solutions is constant and agrees with the measuring conditions.

10 10

0 0 -6 2 α =7.0 10 m /s α =7.0 10-6m2/s b b -10 -10 22.0 10-6 22.0 10-6

/ deg -6 / deg -6 n -20 66.0 10 n -20 66.0 10 Φ Φ

-30 -30 -35 -35 0 255075100125150175 0 20406080 ( f / Hz )1/2 ( f / Hz )1/2

Figure 3.11a+b: (from left to right) Range of intermediate (a) and lower modulation frequencies (b) of the normalized phases measured for a DLC coating on a metallic alloy (C4-∇), in comparison with theoretical solutions based on 3-D thermal wave propagation at increasing values of the thermal diffusivity αb of the substrate material. The theoretical approximations rely on constant heating and detection spot radii (rH = 2 mm, rD = 0.8 mm), in agreement with the measuring conditions. 35 3.7 Discussion of physical effects in the investigated samples

-6 2 thermal wave propagation, for an intermediate value of αb= 22⋅10 m/s the theoretical approximation is in good agreement with the measured data, and for a larger value of the -6 2 thermal diffusivity, e.g. αb = 66⋅10 m /s, the the effect of 3-D thermal wave propagation becomes even more significant, and the relative phase maximum in the low frequency range, which is characteristic for 3-D thermal wave propagation, shifts to higher modulation frequencies, e.g. to (f /Hz)1/2 ≈ 8 (Fig. 3.11b). As shown in Figure 3.10a+b and Figure 3.11a+b, the deviations from 1-D thermal wave propagation increase with decreasing heating spot radii and with increasing thermal diffusivities αb of the subsurface material and, as shown in Figure 3.11b, the normalized phases are rather sensitive to variations of the thermal diffusivity αb of the subsurface material.

-10

6500

/ deg -20 n

Φ 13100 -30 e =26200 Ws1/2m-2K-1 b -40 0 50 100 150 200 250 300 350 ( f / Hz )1/2

Figure 3.12: Normalized phases calculated for 3-D thermal wave propagation (full lines) at different effusivity values eb of the subsurface material, in comparison with data measured for a DLC coating on a metallic alloy of high effusivity (C4-∇).

In Figure 3.12 the normalized phases, measured for another type of hard coating on a high effusivity metallic alloy (∇), are compared with theoretical solutions calculated for 3-D thermal wave propagation (full lines) at different effusivities eb of the subsurface material.

The variations of the effusivity eb lead to variations of the relative phase minimum at intermediate modulation frequencies (Fig. 3.12), where – according to Fig. 3.9a – the differences between 1-D and 3-D thermal wave propagation are negligible. From the point of view of thermal properties, the coating of this sample is homogeneous and can perfectly be approximated by a 2-layer solution, even in the high frequency limit (Fig. 3.12). The results of Figure 3.12 confirm that the effusivity is the relevant thermal parameter both for transient and harmonic surface heating processes and for the heat transition between the different layers of 36 3.8 Interpretation of signal phases based on 3D thermal transport a solid, whereas the results of Figure 3.11 confirm that the thermal diffusivity is the relevant parameter governing time-dependent heat propagation inside a solids and inside a layer of constant thermal diffusivity. By combining measurements of the normalized phases at intermediate frequencies, where the relative phase minimum gives information on the ratio of the effusivities coating-to- substrate, with phase measurements at the very low frequencies, where information on the thermal diffusivity of the subsurface material is obtained from the relative phase maximum, local inhomogeneities of the thermal properties of the coating and of the substrate can be identified:

(i) In the case of measured constant thermal diffusivity αb of the substrate, one can assume that measured local variations of the ratio of the effusivities coating-to-substrate refer to local variations of the coating. (ii) If the locally measured thermal diffusion time of the coating is constant, one can assume on the other hand, that measured local variations of the ratio of the effusivities coating-to-substrate refer to local variations of the substrate. Assuming locally invariant thermal properties of the coating, a complete thermal characterization of the subsurface material can then be obtained, with separate information on the thermal conductivity,

1/2 kb = eb⋅αb (3.46) and the volume heat capacity

1/2 Cb = ρ b⋅cb = eb/α b (3.47)

3.8 Interpretation of signal phases based on 3D thermal transport

Figure 3.13a and Figure 3.13b show the inverse normalized phases measured for two couples of samples, (C4, C6) and (X4, X7), respectively. The measured phases indicate that for each couple of samples, the same coating has been deposited on substrates of different thermal properties. Such a remark has already been done somewhere above. That the two samples possess the same surface characteristics provides the possibility to obtain information about the thermal properties of the substrate material and in the meantime information about the layer growth and substrate preparation. In order to show how such an information can be extracted, we refer to the theoretical expressions of the modulated IR signals based on 1-D thermal wave propagation and establish the ratio of signals Sample A−to−Sample B (equ. 3.48).

37 3.8 Interpretation of signal phases based on 3D thermal transport

1 exp[ 2(1 i) / ] * 3 η sA + Rsb A − + d s A µ s A ε s A (T ){γ (T )T }A ∆M (T , f ) e 1− Rsb A exp[−2(1+ i)d s A / µ ] −1/ 2 A s A s A S A / B ( f ) = = ∆M B (T , f ) * 3 η sB 1+ Rsb B exp[−2(1+ i) d s B / µ s B ] ε s B (T ){γ (T )T }A es B 1− Rsb B exp[−2(1+ i)d sB / µ s B ] (3.48)

In the limit case of small penetration depth, that means at higher modulation frequencies, x ∝ µs = αs /(π f ) → 0 , the ratio of signals takes the form f →∞

* 3 η sA ε s A (T ){γ (T )T }A ∆M (T , f ) e S ( f −1 / 2 ) = A = s A (3.49) A / B f →∞,x→0 M (T , f ) * 3 η ∆ B f →∞,x→0 sB ε sB (T ){γ (T )T }B es B

Since the surface characteristics cancel each other; that means, εs A = εs B, and ηs A = ηs B for the optical properties, and es A = es B for the thermal effusivities, the only difference of the -1/2 ratio SA/B(f )| f→ ∞, x→ 0 from the value 1 can be attributed to the possible temperature dependence of the detection process.

∆M (T , f ) {γ * (T )T 3 } S ( f −1 / 2 ) = A = A (3.50) A / B f →∞,x→0 * 3 ∆M B (T , f ) f →∞,x→0 {γ (T )T }B

Under this consideration, the sample B with lower effusivity of the substrate muss then have a relative high mean temperature, so that according to (3.50) a value close to 1 can be expected.

5 5

0 C6 X7 C4 0 X4 3D theory -5 3D theory -5 -10 -10 -15 / deg / deg -15 n -20 n Φ Φ -25 -20

-30 -25 -35 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 1/2 (f / Hz) (f / Hz)1/2

Figure 3.13 a + b :(from left to right) Inverse normalized phases measured for two sets of samples, C4 & C6 (a) , and X4 & X7 (b), in comparison with theoretical approximations based on 3-D heat propagation.

38 3.8 Interpretation of signal phases based on 3D thermal transport

3.8.1 Determination of the properties of lateral thermal transport

By comparing the theoretical approximations with the experimental results in the entire measured domain, Figure 3.14b and Figure 3.15b show a slope in the frequency range 0.2 < (f /Hz) -1/2 < 1.0. The observed slope is attributed to 3-D propagation effects which are very important in the sample with higher effusivity of the substrate. From these normalized amplitudes (Fig. 3.14b and Fig. 3.15b), the ratio of effusivities of the two substrates (High Speed Steel-HSS and High Effusivity Metal alloy-HMA) is retrieved:

C4/C6 eb HSS / eb HM = 0.575 (3.51a)

X4/X7 eb HSS / eb HM = 0.571 (3.51b)

1.2 1.2

1.0 1.0

0.8 0.8 C4/C6 0.6 0.6 C4/C6

A/B

A/B S S 0.4 0.4

0.2 0.2

0.0 0.0 0.0 0.1 0.2 0.3 0.0 0.2 0.4 0.6 0.8 1.0 -1/2 ( f / Hz )-1/2 ( f / Hz )

Figure 3. 14 a + b (from left to right) Normalized amplitude C4/C6 at smaller penetration depths (a), and in the total measurable domain (b), in comparison with a theoretical approximation according to a two-layer model taking into account the 3D heat propagation.

1.2 1.2

1.0 1.0

0.8 0.8 X4/X7 X4/X7 0.6 0.6

A/B

A/B S S 0.4 0.4

0.2 0.2

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 ( f / Hz)-1/2 ( f / Hz)-1/2

Figure 3. 15 a + b (from left to right) Normalized amplitude X4/X7 at smaller penetration depths (a), and in the total measurable domain (b), in comparison with a theoretical approximation according to a two-layer model taking into account the 3D heat propagation.

39 3.8 Interpretation of signal phases based on 3D thermal transport

From theoretical approximations of the measured phases in the mean frequency domain; we obtain according to Figure 3.13a and to Figure 3.13b, the following values for the ratio of effusivities of the two substrates:

C4/C6 eb HSS / eb HM = 0.575 (3.52a)

X4/X7 eb HSS / eb HM = 0.60 (3.52b)

From approximation of the normalized phases at very low modulation frequencies, (Fig.3.13a and Fig. 3.13b), the ratio of thermal diffusivities of the two substrates is obtained:

C4/C6 αb HSS / αb HM = 0.318 (3.53a)

X4/X7 αb HSS / αb HM = 0.350 (3.53b)

According to the results (3.51a), (3.52a) and (3.53a), the ratio of thermal conductivities of the substrates for the samples C4 and C6 gives:

kb HSS eb HSS αb HSS = =0.32 (3.54a) kb HM eb HM αb HM

According to the results (3.51b), (3.52b) and (3.53b), the ratio of thermal conductivities of the substrates for the samples X4 and X7 gives:

kb HSS eb HSS αb HSS = =0.346 (3.54b) kb HM eb HM αb HM

On the other hand, the theoretical ratio is given by:

kb HSS 23 W/(K m) = =0.2875 (3.55) kb HM 80 W/(K m)

3.8.2 Discussion of results

The ratio of thermal conductivities of the substrates for the samples C4 & C6 (3.54a), and for the samples X4 & X7 (3.54b) is relatively high with respect to the theoretical ratio given by equ. (3.55). By comparing the ratio (3.54a) with the theoretical value on one hand, and the ratio (3.54b) with the theoretical ratio on the other hand, the relative incertitude is given respectively by 11.3% and 20%. These relative discrepancies between derived and 40 3.9 Conclusions theoretical ratios might be attributed to the nature of the common coating deposited on the substrates of samples C4 and C6, and on the substrates of samples X4 and X7, which can play a major role in the modification of the thermal properties of the substrates. In the meantime, these differences between experimental and theoretical results can also be explained by possible changes in the properties of the substrates −thermal and mechanical− during their tailoring or during coating deposition.

3.9 Conclusions

In this chapter, the modulated IR radiometric signals measured at the surface of various layered systems have revealed three types of effects of thermal transport that have been classified into two major categories: The surface effects observed in the range of high modulation frequencies have been found to be in connection with either the optical characteristics (semi-transparency in the visible spectrum) of the surface layer or the topology (presence of a thin layer of reduced thermal transport properties at the very surface of the coating). In the case of coating semi-transparency, it has been observed, that the inverse normalized phases deviate from the limit Zero and increase continuously toward positive values whose maximum depends on the magnitude of the optical absorption coefficient, β. As for the topological effects, the existence of a very thin layer at the very surface of the coating is manifested on the calibrated measured phases by the appearance of a relative maximum in the range of high modulation frequencies. The subsurface effects, which are also materialized by the existence of a relative maximum on the calibrated measured phases in the limit of very low modulation frequencies, have been found to be induced by lateral thermal transport in the subsurface material. Observations have been made, that these lateral heat losses become more important with increasing values of the thermal diffusivity of the substrate material and that the variations of the effusivity of the subsurface material produce changes mainly at the intermediate frequencies, where the transition between the coating and the substrate is identified on the measured phases by a relative minimum. These results have confirmed that the thermal diffusivity is the relevant thermophysical parameter which governs time- dependent heat propagation inside solids and that the thermal effusivity is the relevant thermophysical parameter for both transient and harmonic surface heating processes. Thus, the appearance of relative extrema on the normalized measured phases can provide a large quantity of information about the sample characteristics and so the locations where these extrema occur on the frequency-dependent phases may help to extract the unknown thermal transport and physical properties of the investigated sample. Starting from these important observations, a new evaluation method has been developed, which is based on the relative extremum of the calibrated measured phase lag between the periodical modulated excitation of the thermal wave and the detected thermal response. In the next chapter, we

41 3.9 Conclusions

introduce the new concept of the Phase Extremum of thermal waves and apply the method, labelled as Extremum Method, to the analysis, interpretation and discussion of the phases of the modulated IR signals measured at the surface of opaque two-layer structures.

Determination of Thermal Transport Properties of Two-layer Structures using the Concept of the Phase Extremum

4.1 Motivation

We have seen in the previous chapter that the presence of several physical effects – thermal, topological and optical− on the modulated IR signals measured at the surface of various layered systems, makes difficult the quantitative interpretation of the measured data. It has been particularly noted, that in addition to the relative extremum which occurs on the calibrated measured phases in the range of intermediate frequencies and which indicates the transition coating-to-substrate, the appearance for certain types of samples of further relative phase extrema respectively in the range of low and of high modulation frequencies, reveal respectively the existence of thermal effects –3D thermal wave propagation− and topological effects–thin film on top of the coating. Thus, these relative phase extrema are the possible locations from which complete or at least essential information about the investigated sample can be obtained. In this chapter, we introduce a new concept based on the relative minimum or maximum of the modulated IR signal phases measured at the surface of two-layer systems. We demonstrate, that the measurable phase extremum and the modulation frequency at which the relative extremum occurs, which are the key parameters governing the method, lead to determination of the relevant thermal transport and physical properties of the investigated sample via two combined thermophysical quantities, namely the thermal reflection coefficient and the thermal diffusion time. In section 4.2, the main features of the Extremum Method are presented. In section 4.3, the method is applied to examples of real two-layer systems: hard coatings the thermal effusivity of which is smaller than the effusivity of the substrate consisting of tool steel and a metal alloy sample (NiTi shape memory alloy), which due to surface polishing and local demixing exhibits a two-layer structure with the effusivity of the surface layer above the effusivity of the bulk material. In section 4.4, discussions about other alternative approaches are performed, which confirm the reliability of the Extremum Method. Then general results generated for on-line interpretation in industrial applications are presented in section 4.5. In section 4.6, a functional transform is developed, which allows to obtain information on the thermal transport properties from any other value of the phase lag and the corresponding heating modulation frequency in the neighbourhood of the relative minimum or maximum of the calibrated phase. For this, the functional transform method is

43 4.2 The concept of the Phase Extremum

applied to examples of real coatings which are perfectly described by the two-layer model or which considerably deviate from the model of an opaque two-layer system due to the coating’s transparency or due to the presence of an additional layer of reduced thermal transport properties at the very surface of the coating. Furthermore the functional transform is applied to examples of measurements, which – at higher frequencies at the position of the relative minimum of the calibrated phase – may be obscured by the background fluctuation limit of modulated IR radiometry or by the cell resonance effect in photoacoustic detection. The last section discusses the application potential of both the Extremum Method and the functional transform method.

4.2 The concept of the Phase Extremum

4.2.1 Physical significance of the observable phase extrema

Figure 4.1 shows the phases of the measured modulated IR signals for three coated samples, calibrated with the measured phases of a homogenous and opaque sample used as reference, here the glassy carbon. One can immediately remark, that each of these inverse normalized phase lags possesses at least one relative extremum, which according to Figure 4.1a can be approximately identified for the samples A3() and C6(∇) at the points [(45)2 Hz, -15.4°] and [(75)2 Hz, -24.3°], respectively and for F3() at about {[(7)2 Hz, -10.21°]; [(50)2 Hz, -20.3°]; [(228)2 Hz, -1.7°]}. The relative extremum −a maximum− exhibited by the inverse normalized phase of the polished NiTi sample (x) is identifiable in the neighbourhood of [(4)2 Hz, 3.44°]. The tangents of the measured phases described in Figure 4.1a are reported in Figure 4.1b. In Fig. 4.1 the measured phases of the samples A3() and C6(∇) and the polished NiTi sample (x) are characteristic of a typical two-layer system, showing in addition that the sample A3 is semi-transparent by its coating while C6 can be assumed opaque. Each of these measured phases presents a relative minimum in the range of intermediate frequencies, which indicates the transition coating-to-substrate. In comparison with the measured phases of the samples A3 and C6, the three relative extrema observed on the measured phase of sample F3() have different significations: The first relative maximum recorded in the range of very low modulation frequencies is attributed to 3-D heat propagation in the subsurface material of high thermal diffusivity. The relative minimum occuring in the range of intermediate frequencies attests of the transition between the subsurface material and the coating while the second relative maximum which occurs in the range of high modulation frequencies proves the existence of a thin layer at the very surface of the coating. The phase extremum which is

44 4.2 The concept of the Phase Extremum of concern in this study relies on the transition between the coating and the substrate in the framework of a highly opaque two-layer structure.

-10 / deg n Φ -20

-30 0 50 100 150 200 250 300 350 (f / Hz)1/2

0.2

0.0 n Φ

-0.2 tan tan

-0.4

-0.6 0 50 100 150 200 250 300 350 ( f / Hz )1/2

Figure 4.1 a + b: (from top to bottom) Inverse normalized phases (a) and the corresponding tangents (b) of the measured modulated IR signals for different layered systems: Identification and characterization of the phase extrema.

45 4.2 The concept of the Phase Extremum

4.2.2 Theoretical background

As we have seen in chapter 3 (section 3.5, equ. 3.17), the expression of the thermal -1 wave at the surface of a two-layer system consisting of a highly opaque (βs ≈ 0) coating of -1 thickness d s on an absorbing semi-infinite substrate (βb ≈ 0), can be written as:

ηs I o 1+ Rsb exp[−2(1+ i) πfτ s )] δTs (xs = 0, f ,t) = exp[i(2πft − π / 4)] (4.1) 2es πf 1− Rsb exp[−2(1+ i) πfτ s )]

Here, ηs is the photothermal conversion efficiency which determines the fraction of the total incident light intensity Io transformed into heat, f the heating modulation frequency, t the time, and es the effusivity of the coating. The two combined thermophysical parameters used in equ.(4.1) and which have already been described in Chapter 3 are given by

1− es / eb Rsb = − (4.2) 1+ es / eb for the thermal reflection coefficient of the thermal wave and

2 τ s = d s / α s (4.3)

for the thermal diffusion time of the coating. αs is the thermal diffusivity of the coating. The amplitude of the thermal wave, which can be calculated from equ.(4.1), is given by

1− 2R2 exp(−4 πfτ )cos(4 πfτ ) + R4 exp(−8 πfτ ) ηs Io sb s s sb s δTs ( f ) = (4.4) 2e 2πf 2 s 1− 2Rsb exp(−2 πfτ s )cos(2 πfτ s ) + Rsb exp(−4 πfτ s ) and the phase shift relative to the heating modulation is given by

2 1+ 2Rsb exp(−2 πfτ s )sin(2 πfτ s ) − R exp(−4 πfτ s ) tanΦ ( f ) = − sb (4.5) s 2 1− 2Rsb exp(−2 πfτ s )sin(2 πfτ s ) − Rsb exp(−4 πfτ s )

As can be seen in equ.(4.5) the phase shift of the thermal wave relative to the heating modulation only depends on the two combined thermophysical parameters, Rsb and τs. Since the measured signals, both the amplitudes and the phases are affected by the frequency characteristics of the measurement device, a calibration with the help of a homogeneous reference sample, described by the two-layer system with equal effusivity values for coating and substrate is necessary (compare Chap.3). According to equ.(4.2), the

46 4.2 The concept of the Phase Extremum

thermal reflection coefficient is then Rsb = 0 and according to equ.(4.5) the phase shift of the reference sample is described by

tanΦr ( f ) = −1 (4.6)

In inverse normalization, the expression for the calibrated phase is given by

tanΦ − tanΦ 2R exp(−2 πfτ )sin(2 πfτ ) tanΦ ( f ) = tan[Φ ( f ) −Φ ( f )] = r s = sb s s n r s 1+ tanΦ tanΦ 2 r s 1− [Rsb exp(−2 πfτ s )] (4.7)

The existence of a relative extremum, a relative minimum or maximum, in the representation 1/2 of the inverse normalized phase lags tanΦn versus (f /Hz) as shown in Figure. 4.1b implies, that the minimum or maximum should fulfill the condition,

⎧[1 − R 2 exp(−4 π fτ )]cos(2 πfτ ) − ⎫ ∂ tan Φ n ( f ) 4Rsb πτ s exp(−2 πfτ s ) ⎪ sb s s ⎪ = ⎨ ⎬ = 0 ∂( f ) [1 − R 2 exp(−4 πfτ )]2 2 sb s ⎩⎪ − [1 + Rsb exp(−4 πfτ s )] sin(2 πfτ s )⎭⎪ (4.8) which can be written as

1− R2 exp(−4 πτ f ) tan(2 πτ f ) = sb s extr (4.9) s extr 2 1+ Rsb exp(−4 πτ s fextr )

at the frequency fextr of the relative minimum or maximum. According to equ. (4.7) the value of the relative minimum or maximum Φn(fextr) = Φn extr is then described by

2R exp(−2 π f τ )sin(2 π f τ ) sb extr s extr s (4.10) tanΦn ( fextr ) = tan (Φn extr) = 2 1−[Rsb exp(−2 π fextr τ s )]

The two equations (4.9) and (4.10) depend on two measurable quantities, namely the value tan(Φn extr ) and the frequency fextr of the extremum. On the other hand, the two equations depend on two combined thermal parameters: the thermal reflection coefficient Rsb (4.2) and the thermal diffusion time τs (4.3) of the surface layer of different thermal properties. Inverse solutions of equ.(4.9) and (4.10), which describe the two combined thermophysical parameters Rsb and τs as functions of the two measurable quantities tan(Φn extr ) and fextr can easily be obtained in analytical form. To this aim, equ.(4.9) is resolved for

2 1− tan(2 πτ s fextr ) [Rsb exp(−2 πτ s fextr )] = (4.11) 1+ tan(2 πτ s fextr )

47 4.2 The concept of the Phase Extremum

By inserting equ.(4.11) in the square of equ.(4.10)

4[R exp(−2 π f τ )]2[sin(2 π f τ )]2 2 sb extr s extr s (4.12) [tan (Φn extr)] = 2 2 {1−[Rsb exp(−2 π fextr τ s )] } the thermal reflection coefficient can be eliminated, and the resulting equation

2 cos(4 πτ s f extr ) = [tan (Φ n extr )] (4.12b) can be resolved for the thermal diffusion time, given as a function of the two measurable quantities tan(Φn extr ) and fextr :

1 2 2 τ s = {arccos[(tan Φ n extr ) ]} (4.13) 16π fextr

Alternative forms of this solution are, e.g.

2 1 ⎡ 1− (tanΦ )2 ⎤ τ = ⎢arctan n extr ⎥ (4.13a) S 4π f 1+ (tanΦ )2 extr ⎣⎢ n extr ⎦⎥

2 1 ⎧ 2 ⎫ τ s = ⎨arccos [1+ (tanΦn extr ) ]/ 2⎬ (4.13b) 4π fextr ⎩ ⎭

2 ⎡ ⎤ 1 ⎢ 1 ⎥ τ s = arctan − 1 (4.13c) 16π f ⎢ (tanΦ )4 ⎥ extr ⎣ n extr ⎦

Once the thermal diffusion time has been calculated according to equ.(4.13), the square of the thermal reflection coefficient can be determined from equ.(4.12)

2 1− tan(2 πτ s fextr ) Rsb = exp(4 πτ s fextr ) (4.14) 1+ tan(2 πτ s fextr )

As can be seen from equ.(4.13), the combined quantity (τs⋅fextr) is only a function of the value of the relative phase extremum Φn extr, and thus the thermal reflection coeffient Rsb only depends on one of the measurable parameters, namely the value of the phase extremum Φn extr:

1− tan{0.5⋅ arccos[(tan Φ )2 ]} R 2 = n extr exp{arccos[(tanΦ )2 ]} (4.15) sb 2 n extr 1+ tan{0.5⋅ arccos[(tan Φn extr) ]}

48 4.3 Application to experimental measurements

If the effusivity of the surface layer is smaller than the effusivity of the substrate, es < eb, e.g. in the case of a hard coating on tool steel or on a metal alloy of high effusivity, and a relative minimum is found for the calibrated measured phases (Cf. Figure 4.1), the solution of equ.(4.15) is given by

1− tan{0.5⋅ arccos[(tan Φ )2 ]} R = − n extr ⋅ exp{0.5⋅ arccos[(tanΦ )2 ]} sb 2 n extr 1+ tan{0.5⋅ arccos[(tan Φn extr) ]} (4.15a)

In the case of the polished surface layer on the shape memory alloy NiTi, where the effusivity of the surface layer is larger than the effusivity of the bulk of the material, es > eb, so that a maximum is observed for the calibrated measured phases, the solution for the thermal reflection coefficient is given by

1− tan{0.5⋅ arccos[(tan Φ )2 ]} R = + n extr ⋅ exp{0.5⋅ arccos[(tanΦ )2 ]} sb 2 n extr 1+ tan{0.5⋅ arccos[(tan Φn extr) ]} (4.15b)

Thus, the knowledge of the two measurable quantities tan(Φn extr ) and fextr allows to get access to the two combined thermophysical quantities [equ.(4.13) and equ.(4.15)], which according to equ.(4.2) and equ.(4.3) lead to the extraction of the thermal transport properties of the investigated sample. In section 4.3, the developed theory is applied to experimental measurements presented in Figure 4.1.

4.3 Application to experimental measurements

4.3.1 Methodology and discussions

In Figure 4.2a theoretical solutions according to equ.(4.7) are compared with the phases measured as a function of the heating modulation frequency for several hard coatings on tool steel (,∇) and on a metallic alloy of high effusivity (). Additionally the phases measured for the polished surface of a NiTi shape memory alloy sample (x) are shown. In fact, the frequency and the value of the phase extremum are obtained from the calibrated measured phases represented in Figure 4.1b. Then the combined thermophysical parameters are determined as described above and introduced in equ.(4.7) to generate the theoretical solutions. The results are reported in Table 4.1. The reasons of deviations between theory and experiments have already been studied and presented in chapter 3. It can be observed in Figure 4.2b, that the relative maximum of the inverse normalized phases for the polished NiTi 49 4.3 Application to experimental measurements

-10 / deg n Φ -20

-30 0 50 100 150 200 250 300 350 (f / Hz)1/2

0.2

0.0 n Φ

-0.2 tan

-0.4

-0.6 0 50 100 150 200 250 300 350 ( f / Hz )1/2

Figure 4.2 a+ b: (from top to bottom) Theoretical solutions for one-dimensional thermal wave propagation in a two-layer model, in comparison with the inverse normalized data measured for real two layer systems: hard coatings (∇,, ) on two different substrate materials and a polished NiTi shape memory alloy sample (x).

50 4.3 Application to experimental measurements

Sample Symbol tan Φn extr fextr / Hz τs / µs Rsb gsb NiTi x 0.060 224.882 217.3 0.093 1.206 A3 -0.276 2034.551 21.85 -0.412 0.417 F3 -0.371 2455.30 16.63 -0.538 0.300 C6 ∇ -0.452 5817.418 6.38 -0.636 0.223

Table 4.1: Frequency and value of the relative phase extremum allowing to get access to the thermal diffusion time and the thermal reflection coefficient for different samples.

samples (x), which is characteristic for a first layer with an effusivity above that of the bulk material, is found at rather low frequency, fmax ≈ 225 Hz, while the relative minima of the coated samples are found at considerably higher frequencies, e.g. at fmin ≈ 2.5 kHz. This is due to the fact, that the coatings considered here are relatively thin with low thermal diffusion 2 times τs = ds /αs while the first layer of the NiTi sample affected by polishing and de-mixing of the alloy is comparatively large.

4.3.2 Determination of the thermo-physical properties

Once the combined parameters τ s and gsb (Rsb) have been determined, the thermophysical properties of the investigated sample can be easily retrieved according to equ.(4.2) and equ.(4.3). For example, from the results mentioned in Table 4.1, and knowing 1/2 -2 -1 the value of substrate effusivity for the sample C6(∇), ebS2 = 7530 Ws m K and its coating thickness, dsC6 = 2.8 µm; the thermal diffusivity and the thermal effusivity of the coating are 2 determined according to αsC6 = (dsC6) /τs and esC6 = gsb•ebC6, respectively. In Figure 4.3, the measured phases of the samples C4(∆) and C6(∇) are represented and compared with the corresponding theoretical approximations. As already seen in chapter 3, the measured phases indicate that the two samples have similar surface characteristics. Correlations between experimental measurements and theoretical approximations based on the Extremum Method allow to remark, that the common coating of the two samples is slightly 1/2 transparent (in the range of modulation frequencies (fmin/Hz ) > 125, the calibrated measured phases are slightly above the theoretical curve which assumes a highly opaque system). The data calculated for the sample C6 are available in Table 4.1. For the sample C4, the relative minimum given by tanΦn extr = -0.587 is localized at the frequency fmin ≈ 4.7 kHz and the ratio of effusivities coating-to-substrate is given by gsb = 0.127. Since the two samples differ with each other only by their subsurface materials, that means esC6 = esC4, the unknown effusivity of the subsurface material for the sample C4 can be directly extracted according to ebC4 = (gsbC6/gsbC4).ebC6. In the meantime, the knowledge of the thermal diffusion time of the coating for this sample (C4) leads to the determination of the related thermal diffusivity.

51 4.3 Application to experimental measurements

Table 4.2 summarizes the main characteristics –given and extracted– of the two investigated samples. In this table, dT represents the total thickness of the sample.

0.0

-0.2 n

Φ -0.4 tan

-0.6 0 50 100 150 200 250 300 350 ( f / Hz )1/2

Figure 4.3: Inverse normalized phases of the IR signals for two samples (hard coating on steel-C6, and hard coating on high metal alloy-C4) presenting the same surface characteristics. The experimental measurements are correlated with theoretical approximations based on the Extremum Method.

2 1/2 -2 1 1/2 -2 1 sample τ s / µs αs / m /s ds / µm gsb es/Ws m K eb/Ws m K dT / mm C4 6.3 1.24×10-6 2.8 0.127 1679 13222 3.185 C6 6.4 1.23×10-6 2.8 0.223 1679 7530 3.505

Table 4.2: Thermal and physical properties of two samples presenting the same surface characteristics

4.3.3 Efficient localization of the phase extremum

The value and the frequency of the relative extremum, respectively tan Φn extr and ƒextr, are the key parameters governing the Extremum Method. This is why a poor estimation of these important parameters can lead to considerable errors in the interpretation of the measured phases. For illustration we reconsider the measured phases of Figure 4.3: It appears at first sight, that the relative minimum of the two calibrated measured phases can be found at 52 4.3 Application to experimental measurements

the same modulation frequency fextr = 5.160 kHz, corresponding to thermal diffusion times of values τsC4 = 5.7µs and τsC6 = 7.2µs, respectively. Theoretical curves (dashed lines) based on these values of the thermal diffusion time are also represented in Figure 4.4. As one can see, e.g. for the sample with larger ratio of effusivities (∇), the theoretical curve (dashed line) is shifted to the left hand side with respect to the measured phase while for the sample with smaller ratio of effusivities (∆) the theoretical curve (dashed line) is considerably shifted to the right hand side. Such a situation can lead to a confusion concerning the actual level of the coating semi-transparency and also to a poor appreciation of the sample characteristics, namely the coating thickness or its thermal diffusivity.

0.0

-0.2 n Φ

-0.4 C4 C6 tan poor estimation of the minimum -0.6 Theory 0 50 100 150 200 250 300 350 ( f / Hz )1/2

Figure 4.4: Inverse normalized phase of the modulated IR signals. Theoretical approximations based on the Extremum Method. Comparison of results: Efficient localization (full lines) and poor estimation of the phase minimum (dashed lines).

In order to avoid such errors, a four-point based interpolation involving the significant measured data of the inverse normalized phase at the location of the relative extremum, has to be performed to find out the appropriate and exact values of the frequency and the phase extremum. Thus, by comparing the thermal diffusion times of Table 4.2 for the samples C4 and C6, with the data obtained through a poor assessment of the coordinates of the relative extremum, namely τsC4 = 5.7µs and τsC6 = 7.2µs, the relative error made on the thermal diffusion time is estimated as ∆τ /τ = 9.5% and ∆τ /τ = 12.5% . s s C 4 s s C6

53 4.3 Application to experimental measurements

4.3.4 Application to measurements at high temperatures

In Figure 4.5, the calibrated measured phases of a layered structure are represented as a function of the modulation frequency for different measuring temperatures, and compared with theoretical approximations based on the Extremum Method. From the observations, the following remarks can be made: (i) At room temperature and at 118 °C, the measured phases (a & b) exhibit the behaviour of a two-layer structure with a semi-transparent coating. According to Fig. 5 a + b, the calibrated measured phases take the value Zero at about (f /Hz)1/2 = 100 and then increase continuously with increasing modulation frequencies (comp. Chap. 3). If the behaviour of the measured phase lags is similar from one temperature to another, one can however see that the obtained values for the thermal reflection coefficient and the thermal diffusion time vary from room temperature to 118°C. That means, the level of coating semi-transparency is modified as the temperature changes. This assertion is proven by the position of the minimum (a & b). (ii) At 248°C, the calibrated measured phases also take the value Zero at the frequency of about (f /Hz)1/2 = 100. However, instead of increasing continuously, these phases reach a maximum at about (f /Hz)1/2 = 175 before decreasing with increasing modulation frequencies. Such a behaviour of the calibrated measured phases in the limit high modulation frequencies has already been seen and described in chapter 3. It indicates the presence of a very thin layer at the very surface of the coating. With increasing temperature, namely T = 418 °C and T = 518 °C, the measured phase maximum decreases while the modulation frequency at which the extremum occurs shifts from (f /Hz)1/2 = 175 to (f /Hz)1/2 = 150. At T = 518 °C particularly, the relative maximum is recorded below the value Zero. This variation of the relative phase maximum also means a variation of the ratio of effusivities thin layer (on top of the coating)

−to− coating. In the meantime, the phase minimum oscillates around the value Φn = -15 °C as temperature changes, thus proving that the ratio of effusivities coating-to-substrate also varies with the temperature.

54 4.3 Application to experimental measurements

10 10

5 5

0 0

/ deg / deg

n -5 n -5 ϕ ϕ

-10 -10

-15 R = -0.421 τ = 31µs -15 R = -0.408 τ = 32.2µs sb s sb s

0 100 200 300 0 100 200 300 (f / Hz)1/2 (f / Hz)1/2

(a) (b)

10 10

5 5

0 0

/ deg / deg n -5 n -5 ϕ ϕ

-10 -10

-15 R = -0.406 τ = 33µs -15 R = - 0.41 τ = 32.6µs sb s sb s

0 100 200 300 0 100 200 300 1/2 (f / Hz) (f / Hz)1/2

5 Figure 4.5: Calibrated phases of a layered sample (A5) measured at different temperatures.

0 (a) T = Room temperature (RT) (b) T = 118 °C

/ deg (c) T = 248 °C n -5 ϕ (d) T = 418 °C

-10 (e) T = 518 °C

-15 R = -0.409 τ = 31.2µs sb s

0 100 200 300 1/2 (f / Hz)

(e)

55 4.3 Application to experimental measurements

4.3.4.1 Correlation thermal reflection coefficient − temperature

In Figure 4.5, the ratio of effusivities coating-to-substrate is represented as a function of temperature. As one can remark, there is a random dependence of this ratio on temperature but two sub-domains can be distinguished in which a regular distribution of the ratio with respect to the temperature is effective, namely the interval RT (room tempe.) to T = 248 °C and the interval T = 248 °C to T = 518 °C. This can be explained by the nature of the two physical effects that are revealed by the measured phases, namely the semi-transparency of the coating and the emergence of a very thin layer at the very surface of the coating as the temperature increases.

0.500

0.475

0.450 sb

g 0.425

0.400 0 100 200 300 400 500 600 Temperature / oC

Figure 4.5: Variation of the ratio effusivities coating-to-substrate as a function of temperature

4.3.4.2 Correlation thermal diffusion − temperature

Figure 4.6 shows the variation of the thermal diffusion time of the coating as a function of temperature. Here also, there is a random dependence of τs on the temperature. Since the thermal diffusion time is a function of the thermal diffusivity and the layer thickness, one can conclude that the variation of this combined quantity is in connection with the change in the topology −with increasing temperature, the sample transits from a two-layer to a three-layer model− and with the change of the thermal properties of the sample as the temperature increases.

56 4.4 Discussions on the reliability of the Extremum Method

34 / µs s τ 32

30 0 100 200 300 400 500 600 Temperature / oC

Figure 4.6: Variation of the thermal diffusion time of the coating as a function of temperature

4.4 Discussions on the reliability of the Extremum Method

The aim of this section is to check out whether there is another alternative approach better than the Extremum Method, which can lead to more reliable information about the sample characteristics from other points of the measured phase. This implies the possibility to get access to the thermal wave reflection coefficient and the thermal diffusion time from any measured point of the calibrated phase.

4.4.1 Solutions from any measured point of the calibrated phase

Starting from equ.(4.7) derived above, two solutions giving the thermal reflection coefficient as a function of the thermal diffusion time, the calibrated measured phase and the modulation frequency can be obtained:

2 ⎧ ⎛ ⎞ ⎫ ⎪ sin()2 πfτ s sin()2 πfτ s ⎪ R = − − ⎜ ⎟ +1 exp 2 πfτ (4.16a) sb ⎨ ⎜ ⎟ ⎬ ()s ⎪ tan[]Φ n ( f ) tan[]Φ n ( f ) ⎪ ⎩ ⎝ ⎠ ⎭

2 ⎧ ⎛ ⎞ ⎫ ⎪ sin()2 πfτ s sin()2 πfτ s ⎪ R = − + ⎜ ⎟ +1 exp 2 πfτ (4.16b) sb ⎨ ⎜ ⎟ ⎬ ()s ⎪ tan[]Φ n ( f ) tan[]Φ n ( f ) ⎪ ⎩ ⎝ ⎠ ⎭

57 4.4 Discussions on the reliability of the Extremum Method

The function (4.16a) describes the negative reflection coefficients which indicate the deposition of a coating on substrate of better thermal properties whereas the function (4.16b) describes the positive reflection coefficients which certifies the deposition of a coating of better thermal transport properties on a substrate. To simplify our investigations, we only consider the negative thermal reflection coefficients. The objective of these investigations is to find out whether from two random points of the measured phase, it is possible to obtain values of the thermal reflection coefficient and of the thermal diffusion time which are more reliable than the results provided by the Extremum Method, this later method being essentially based on one fundamental point of the measured phase.

4.4.1.1 Study of the function Rsb = Rsb(τs)

In Figure 4.7b, two measured points are considered at the left hand side of the phase extremum and the representative curves Rsb = Rsb(τs) generated by each of the measured points is shown in Figure 4.7a. As one can clearly see, there is a very small intersection angle between the two curves.

-0.50 0 -0.55 -5 -0.60 -10 -0.65 sb

/ deg

R -15 n

-0.70 Φ -20 -0.75 -25 -0.80 2.0x10-6 4.0x10-6 6.0x10-6 8.0x10-6 1.0x10-5 0 50 100 150 200 250 300 350 τ ( f / Hz )1/2 s

(a) (b)

Figure 4.7 a + b: Determination of the combined parameters from two measured points at the left hand side of the phase extremum (no observable intersection point).

As second test, two other measured points are considered at the left hand side of the phase extremum (Figure 4.8b) and the corresponding representative curves provide an intersection point I (Figure 4.8a). The coordinates of this point are given by I(τs = 6.6 µs and

Rsb= -0.638). For comparison, the results obtained from the Extremum Method are (τs = 6.4 µs and Rsb = -0.636. At first sight, the two results seem to be close but one can however remark in Figure 4.8b, that the theoretical curve generated by the two measured points is somewhat shifted to the left hand side with respect the measured phase.

58 4.4 Discussions on the reliability of the Extremum Method

-0.55 0

-0.60 -5

I -10 -0.65 sb

/ deg

R -15 n Φ -0.70 -20

-25 -0.75 2.0x10-6 4.0x10-6 6.0x10-6 8.0x10-6 1.0x10-5 0 50 100 150 200 250 300 350 τ ( f / Hz )1/2 s

(a) (b)

Figure 4.8 a + b: Determination of the combined parameters from two measured points at the right hand side of the phase extremum. Intersection point I(τs=6.6 µs, Rsb= -0.638). Results from the Extremum Method: τs=6.4 µs, Rsb= -0.636.

Now, by considering two other measured points located respectively at the left hand side and at the right hand side of the phase extremum (Figure 4.9b), the intersection point I of the representative curves (Figure 4.9a) is identified at τs = 6.4 µs, Rsb = -0.626. Despite the fact that the value of the thermal diffusion time is quite in agreement with that obtained from the Extremum Method (τs = 6.4 µs, Rsb= -0.636), there is a disagreement about the value of the thermal reflection coefficient. The disagreement between the theoretical solution generated by taking into τs = 6.4 µs, Rsb = -0.626 and the experimental measurement can be clearly seen in Figure 4.9b, namely in the frequency range 25 < (f / Hz)1/2 < 150.

-0.50 0

-0.55 -5

-0.60 -10 I

-0.65 / deg sb -15

n R Φ -0.70 -20

-0.75 -25 -0.80 2.0x10-6 4.0x10-6 6.0x10-6 8.0x10-6 1.0x10-5 0 50 100 150 200 250 300 350 τ 1/2 s ( f / Hz )

(a) (b)

Figure 4.9 a + b: Determination of the combined parameters from two measured points at the left and right hand side of the phase extremum. Intersection point I(τs = 6.4 µs, Rsb = -0.626). Results from the Extremum Method: τs = 6.4 µs, Rsb = -0.636.

59 4.4 Discussions on the reliability of the Extremum Method

0 -0.2

-0.3 -5

-0.4 -10 -0.5

I / deg -15 sb n -0.6 R Φ

-0.7 -20

-0.8 -25

-0.9 -6 -6 -6 -6 -5 0 50 100 150 200 250 300 350 2.0x10 4.0x10 6.0x10 8.0x10 1.0x10 1/2 τ ( f / Hz ) s

(a) (b)

Figure 4.10 a + b: Determination of the combined parameters from two other measured points at the

left and right hand side of the phase extremum. Intersection point I(τs = 6.3 µs, Rsb = -0.575). Results from the Extremum Method: τs = 6.4 µs, Rsb = -0.636.

For the latest investigation two other measured points are considered; one point at the left hand side, very far from the location of the relative minimum, in the range of low modulation frequencies and the other point in the range of high modulation frequencies

(Figure 4.10b). The intersection of the two representative curves is given by I(τs = 6.3 µs and

Rsb = -0.575). Here also, the thermal diffusion time can be considered to be in agreement with the value obtained from the Extremum Method. However, the obtained value for the thermal reflection coefficient is rather far away from the expected value. This disagreement can be explained by the fact that the two considered measured points are located in two frequency domains where the modulated IR signals are subjected to physical effects (thermal and optical), namely 3-D thermal wave propagation in the limit of very low modulation frequencies and semi-transparency effects at high modulation frequencies (Compare Chap.3).

4.4.2 Comparison of results

A small intersection angle as shown in Figure 4.7a already indicates that an alternative method for the determination of the combined thermophysical quantities, based on any point of the measured phases cannot be reliable. Occasionally, acceptable solutions using such an approach can be obtained (Figure 4.8b). However, according to Figure 4.9b and Figure 4.10b, the large disagreement between the measurements and the generated theoretical curves in the range of intermediate frequencies does not favour this alternative method which considers any point of the measured phases. It is therefore justified to conclude, that determining the two combined quantities Rsb and τs from any point of the measured phases different from the relative phase extremum is a random procedure which rather leads to contradictory results.

60 4.5 General results for on-line interpretation in industrial applications

Only the Extremum Method provides the more reliable information about the thermo-physical properties of the investigated sample. Following this comparison, we demonstrate in the next section how experimental measurements can be directly handled by using the proposed evaluation method.

4.5 General results for on-line interpretation in industrial applications

There is an increasing need for on-line interpretation. The industrial investigator or the experimentalist would like to make a rapid report on the unknown investigated samples with respect to their thermal and physical parameters. For this, a variety of questions are often of concern, e.g.: Does the coating fulfil the requirement of thermal barriers or alternatively of good conductors? Is the coating opaque or semi-transparent? What are then the thermal and physical properties of the unknown structure? What about the coating thickness?, etc…The Extremum Method gives answers to such questions by proposing general results which help to get access to the thermal transport properties of two-layer structures.

4.5.1 Graphic of thermal reflection coefficients

In Figure 4.11a, the thermal reflection coefficient is represented as a function of the measured phase extremum. As expected the thermal reflection coefficient is limited by the values, -1 and +1. The lower limit corresponds to the situation for which a coating of extremely poor thermal properties has been deposited on the substrate while the upper limit corresponds to the situation for which the thermal properties of the substrate are extremely poor in comparison with the properties of the deposited coating.

One can observe, that the phase extremum (tanΦn extr) is also limited by the same values, −1 and + 1. Mathematically, the origin of this limitation can be clearly seen in equ.(4.13). Depending on the value of the phase extremum, the graph of Figure 4.11a can be separated into two parts. For negative phase extrema (phase minima), the thermal reflection coefficient is negative, -1 < Rsb < 0, and for positive phase extremum (phase maxima), the thermal reflection coefficient is positive, 0 < Rsb < 1. As the measurable quantity tanΦn extr approaches the value Zero, the curve Rsb = Rsb(tanΦn extr) experiences a discontinuity, which is quite comprehensible since the Extremum Method strictly applies to two-layer structures. As can be seen in equ.(4.15), if the phase extremum takes the value Zero, then the thermal reflection coefficient will be also reduced to Zero, corresponding to the case of a homogenous sample, which is out the context in this study. The two solutions represented in Fig. 4.11a are characteristic for the sample C6 and for the NiTi sample (see Table 4.1).

61 4.5 General results for on-line interpretation in industrial applications

1.00

0.75

0.50

0.25

0.00

sb R -0.25

-0.50

-0.636 -0.75

-1.00 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 -0.452 0.06 tan Φ n extr

Fig. 4 11a: Thermal reflection coefficient Rsb as a function of the relative phase extremum tanΦn extr of the inverse normalized phase lag.

1.205 1 b e / s e 0.223

0,1

0,01 -1,00 -0,75 -0,50 -0,25 0,00 0,25 0,50 0,75 1,00 -0.452 0.06 tanΦ n extr

Fig. 4 11b: Ratio of effusivities coating-to-substrate (es/eb) as a function of the relative phase extremum tanΦn extr of the inverse normalized phase lag.

62 4.5 General results for on-line interpretation in industrial applications

4.5.2 Graphic of the ratio of effusivities

The graphic giving the ratio of effusivities coationg−to−substrate as a function of the relative phase extremum tanΦn extr of the inverse normalized phase lag is presented in Figure 4.11b. In analogy to the graphic of thermal reflection coefficients, two main domains can be distinguished: For negative relative phase extrema, the ratio of effusivities is valid between 0 and 1, which means that coatings are deposited on substrates of better thermal properties. For positive relative extrema, the ratio of effusivities is larger than the value 1, which represents the situation for which a coating of better thermal effusivity is deposited on a substrate. Here, the ratio has been limited to 10 to account for the usual coating materials.

0.05

0.04 0.037

0.03

K 0.02

0.01

0.00 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 -0.452 tan Φ n extr

Figure 4.12: Key function Kf as function of the measured phase extremum.

4.5.3 Graphic of thermal diffusion times

In comparison with the thermal reflection coefficient or the ratio of effusivities which are essentially generated by different values of the relative phase extremum, the thermal

63 4.5 General results for on-line interpretation in industrial applications

diffusion time of the coating is rather determined by the value and the frequency of the phase extremum (4.13). To make easier the comprehension and the readability of the graphic of thermal diffusion times, the thermal diffusion time can be calculated from equ. (4.17)

K (tan Φ n extr ) τ s = (4.17) f extr

with the key-value K(tanΦn extr)

{arccos[(tanΦ )2 ]}2 K(tanΦ )= n extr (4.18) n extr 16π

The key-function defined by equ.(4.18) is represented in Figure 4.12. In this figure, one can observe, that the key-function varies between 0 and 0.05 within the entire interval range of the possible phase extrema. The upper key-value is reached as |tanΦn extr| approaches the value

Zero. On the other hand, as the measurable quantity tanΦn extr approaches its lower or upper limit, |tanΦn extr| = 1, the key-function seems to adopt a value far below 0.005, which is already ten times smaller than the maximal value, 0.05. That means, for a given frequency of the relative extremum, the thermal diffusion time decreases with increasing absolute values of the measurable quantity tanΦn extr. An illustration of this assertion is shown later on in Figure 4.14a. Thus, once a particular key-value is derived from a given value of the measured phase maximum or minimum, the thermal diffusion time of the coating can be easily determined, as equ.(4.17) suggests. Figure 4.13 presents a graphic of thermal diffusion times represented as a function of the frequency of the relative phase extremum for different values of the relative phase extremum. In this Figure namely, the upper curve is generated by |tanΦn extr| = 0.150 for different values of the modulation frequency comprised between 0.1 kHz and 10 kHz. In the same way, the lower curve is generated by |tanΦn extr| = 0.900 for different values of the modulation frequency. Between the upper and the lower curves, and from top to bottom, the other curves are induced by |tanΦn extr| = 0.300, 0.452, 0.600, 0.750, respectively. For all curves and from top to bottom, the key-values are given in Table 4.3 as a function of the measured phase extremum. Figure 4.13 also shows, that for a given value of the measured phase extremum, a set of thermal diffusion times can be determined, depending on the modulation frequency of heating. In this case, the difference between the calculated thermal diffusion times relies on the variation of the coating thickness.

64 4.5 General results for on-line interpretation in industrial applications

103

tan Φ = 0.150 n extr 0.300 0.452 0.1 50 0.600 0.750 2 10 0.900

0 .9 00 / µs s τ 101 6.38

1 102 103 5817 104 f / Hz extr

Figure 4.13: Thermal diffusion time as a function of the frequency measured for the relative minima or maxima of the inverse normalized phase lag, considering different values of the relative extrema.

|tan Φ n extr| 0.15 0.30 0.452 0.60 0.75 0.90 K 0.048 0.044 0.037 0.029 0.019 0.0078

Table 4.3: Values K of the key function as a function of the relative extremum

The theoretical results presented in Figure 4.14a indicate that the thermal diffusion time of the thermal wave decreases with increasing absolute value of the relative phase extremum and are in good agreement with the representative curves of Figure 4.12 and of 1/2 Figure 4.13. In Figure 4.14b, values of (fextr /Hz) = 50, 75 and 100 for the frequency of the relative extremum induce respectively values of τs /µs = 14.83, 6.59, and 6.4 for the thermal diffusion time of the coating. For remind, the set of upper curves (blue color) refers to a coating of better thermal properties with respect to the substrate and the set of lower curves (red color) refers to substrate of better thermal properties with respect to the coating.

65 4.5 General results for on-line interpretation in industrial applications

1.0 0.6

0.4 0.5 0.2

0.0 0.0 n extr

n extr ϕ

ϕ -0.2 tan tan tan -0.5 -0.4

-0.6 -1.0 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 1/2 1/2 ( f / Hz ) ( f / Hz ) Figure 4.14a: Evolution of the thermal diffusion Figure 4.14b: Evolution of the thermal diffusion time of the coating with the phase extremum for a time of the coating with the frequency of the constant value of the frequency of the relative relative extremum, for a constant value of the 1/2 extremum. | tanΦn extr| = 0.25, 0.65, 0.95 gives phase extremum. (fextr /Hz) = 50, 75 and 100 respectively τs / µs = 8.05, 4.55, and 0.70 for gives respectively τs / µs = 14.83, 6.59, and 6.4 1/2 (fextr / Hz) = 75 for | tanΦn extr| = 0.452

4.5.4 Example of on-line interpretation

The aim of this section is to show briefly how experimental measurements performed on two-layer structures can be rapidly and methodically interpreted by using the established graphics and the above defined fundamental equations:

tanΦn extr Rsb K fextr / kHz τs / µs Example of comments Substrate of better thermal -0.55 -0.74 0.032 4.900 6.5 transport properties. Coating: Micro- or nano-layer? Coating of better thermal 0.38 0.55 0.040 4.225 9.6 transport properties. Thick surface layer?

Table 4.4: Example of on-line report about the experimental measurements on two-layer structures.

(i)− If we suppose, that the value of the relative minimum on the calibrated measured phases is given by tanΦn extr = -0.55, then the graphic of thermal reflection coefficients (Figure

4.11a) will indicate Rsb = -0.74 (substrate of better thermal properties) as corresponding thermal reflection coefficient. According to equ.(4.2), from the obtained combined quantity

66 4.6 Interpretation of phase measurements obscured by the background fluctuation one can immediately get access to the coating effusivity if the substrate effusivity is known or alternatively to the substrate effusivity if there is any available information about the coating effusivity. On the other hand, the considered relative minimum yields a key-value K = 0.032, which is then combined with the frequency of the relative minimum, fmin /kHz = 4.9, to determine the effective thermal diffusion time, τs /µs ≈ 6.5. The thickness or alternatively the thermal diffusivity of the coating can be extracted according to equ. (4.3).

(ii)− In the case a relative maximum of tanΦn extr = 0.38 is rather observed on the calibrated measured phases (exclusively for a two-layer system!), then a value of Rsb = 0.55 (coating of better thermal properties) will be identified on the graphic of thermal reflection coefficients. In the same way, the effusivity of either the coating or the substrate is extracted from the available information about one of the two layers. In the meantime, the related key- value (K = 0.04) helps to derive a thermal diffusivity of the coating, τs /µs ≈ 9.6, if the modulation frequency at which the relative maximum occurs is given for example by the value fmin/kHz = 4.225. For the two configurations, an example of short on-line report about the experimental measurements is given in the table 4.4.

4.6 Interpretation of phase measurements obscured by the background fluctuation

Since measurements based on modulated IR radiometry may be obscured at low frequencies due to 1/f –noise and at high frequencies due to the background fluctuation limit [Bolte et al., 1997], an alternative interpretation method has to be developed, which may be based on any other value of the measured inverse normalized phase lag and the corresponding value of the modulation frequency. The method presented in Sect. 4.2.2, to determine the two relevant thermal transport parameters τs and Rsb, the thermal diffusion time and the thermal reflection coefficient, from the measured relative minimum or maximum of the inverse normalized phase lag, is equivalent to a method based on the values of the measured inverse normalized phase lag and of its slope, which at the position of the relative minimum or maximum is Zero. Thus, an alternative method should rely on the value of the inverse normalized phase lag measured in the neighbourhood of the relative minimum or maximum and on the slope of the inverse normalized phase lag. In general the derivation of the slope from a curve of measured values generates numerical errors, leading to large errors of interpretation. Alternative methods which avoid such numerical errors of differentiation may rely on the solution of a problem-related integral equation, such as proposed by Bein [Bein, 1986] for the derivation of heat fluxes from time- resolved thermographical surface temperature measurements. Another alternative may be the calculation of problem-related functional moments such as the temporal moment method

67 4.6 Interpretation of phase measurements obscured by the background fluctuation

proposed by Balageas and co-workers [Balageas et al., 1987] for the interpretation of pulsed photothermal radiometry. Here a solution method is proposed which relies on a functional transformation of the measured inverse normalized phase lag by applying a multiplication based on the frequency 1/2 dependence of the thermal diffusion length µth ∝ 1/ f and which substitutes the minimum or maximum of the inverse normalized phase lag by the minimum or maximum of the transformed phase lag function at a modulation frequency value in the neighborhood.

4.6.1 Theory of transformation of the inverse normalized phase

The method used in Sect. 4.2.2 to determine the two combined thermal parameters, the thermal diffusion time τs and the thermal reflection coefficient Rsb, from the value and the frequency of the relative phase extremum is extended here to other frequency values, especially to lower frequencies and the corresponding phase values. To this finality a multiplicative transformation of the inverse normalized phase lag is introduced

tanΦ ( f ) 2R exp(−2 πfτ )sin(2 πfτ ) tanΨ ( f ) = n = sb s s (4.19) n q q q 2 ( f ) ( f ) {1−[Rsb exp(−2 πfτ s )] } with the exponent q a small real number in the neighbourhood of zero and the exponent q = 0 corresponding to the not transformed inverse normalized phase

tanΨ n q=0 ( f ) = tanΦ n ( f ) (4.19a)

The derivative of the transformed phase lag with respect to the square root of the modulation frequency is calculated as

∂ tanΨ ( f ) ∂ ⎡tanΦ ( f )⎤ 1 ∂ tanΦ ( f ) ∂ ( f )−q nq = n = n + tanΦ ( f ) = ⎢ q ⎥ q n ∂( f ) ∂( f ) ⎣⎢ ( f ) ⎦⎥ ( f ) ∂( f ) ∂( f )

⎧ 2 ⎫ 1 4Rsb πτ s exp(−2 πfτ s ) ⎪[1− Rsb exp(−4 π fτ s )]cos(2 πfτ s )− ⎪ = ⎨ ⎬ + ( f )q [1− R 2 exp(−4 πfτ )]2 2 sb s ⎩⎪ −[1+ Rsb exp(−4 πfτ s )] sin(2 πfτ s )⎭⎪

(−q) 2R exp(−2 πfτ )sin(2 πfτ ) + sb s s = q+1 2 ( f ) [1− Rsb exp(−4 πfτ s )]

68 4.6 Interpretation of phase measurements obscured by the background fluctuation

1 4R πτ exp(−2 πfτ ) = ⋅ sb s s cos(2 πfτ ) − q 2 [ s ( f ) [1− Rsb exp(−4 πfτ s )] 1+ R 2 exp(−4 πfτ ) q − sb s sin(2 πfτ ) − ⋅sin(2 πfτ ) (4.20) 2 s s ] 1− Rsb exp(−4 π fτ s ) 2 πfτ s

At the relative extremum ∂ tanΨ n q ( f ) / ∂( f ) = 0 of the transformed phase lag function the condition

⎡ 2 ⎤ 2 q 1− Rsb exp(−4 πfextr qτ s ) 1− Rsb exp(−4 πfextr qτ s ) tan(2 πfextr qτ s )⎢1+ ⎥ = 2 πf τ 1+ R 2 exp(−4 πf τ ) 1+ R 2 exp(−4 πf τ ) ⎣⎢ extr q s sb extr q s ⎦⎥ sb extr q s (4.21) has to be fulfilled, or alternatively

2 1− Rsb exp(−4 π fextr qτ s ) tan(2 πfextr qτ s ) = q q 2 1+ +[1− ]Rsb exp(−4 πfextr qτ s ) 2 π fextr qτ s 2 π fextr qτ s (4.21a) which substitutes the condition (4.9) of the relative extremum of the original, not transformed phase lag function. According to equ.(4.19) the transformed phase lag function at the new position of the extremum is given by the equation

2Rsb exp(−2 πfextr qτ s )sin(2 πfextr qτ s ) tanΨ ( f ) = tanΨ = (4.22) nq extr q nq extr q 2 ( fextr q ) [1− Rsb exp(−4 πfextr qτ s )] which substitutes equation (4.10) of the not transformed original phase lag. Similar to the equations (4.9) and (4.10) for the not transformed phase lag, the two equations (4.21a) and (4.22) depend on the thermal reflection coefficient and the thermal diffusion time and have to be resolved for these two quantities, to be given as functions of the relative extremum and the corresponding frequency of the transformed phase lag function. To this finality equ.(4.21a) is resolved for

q 1−[1+ ]tan(2 πf extr qτ s ) 2 π f extr qτ s R 2 exp(−4 π f τ ) = (4.23) sb extr q s q 1+ [1− ]tan(2 πf extr qτ s ) 2 π f extr qτ s

By inserting equ.(4.23) in the square of equ.(4.22)

69 4.6 Interpretation of phase measurements obscured by the background fluctuation

2 2 4Rsb exp(−4 πf extr qτ s )[sin(2 πf extr qτ s )] (tanΨ ) 2 = (4.24) n q extr 2q 2 2 ( f extr q ) [1− Rsb exp(−4 πf extr qτ s )]

the thermal reflection coefficient Rsb is eliminated and an equation is obtained, which can be interpreted as an extension of equ.(4.12b)

2q 2 q ( f extr q ) (tanΨ n q extr ) = cos(4 πf extr qτ s ) − [ ]sin(4 πf extr qτ s ) + 2 π f extr qτ s

q 2 2 + [ ] [sin(2 πf extr qτ s )] (4.25) 2 π f extr qτ s

and which – by a numerical procedure – is resolved for the thermal diffusion time τs = τs (tan

ψn q extr, fextr q), given as a function of the relative extremum tan ψn q(fextr q) = tan ψn q extr of the transformed phase lag and of the corresponding value fextr q of the modulation frequency. Once, the thermal diffusion time is known, equ. (4.23) can be resolved for the thermal reflection coefficient

q 1 −[1 + ]tan(2 πf extr qτ s ) 2 π f extr qτ s R = ± ⋅ exp(2 π f τ ) (4.26) sb q extr q s 1 + [1 − ]tan(2 πf extr qτ s ) 2 π f extr qτ s

An alternative solution can be derived from equ.(4.22) and (4.24) which allows an iterative solution and which in the case of the exponent q=0 coincides with solution (4.13a) of the not transformed phase lag function.

tan(2 πf extr qτ s ) =

q 2 2q 2 2 2q q 2 − ( ) + 1−[(tanΨ n q extr ) ( fextr q ) ] +[(tanΨ n q extr ) ( fextr q ) ]( ) 2 π fextr qτ s 2 π fextr qτ s = q 2 2 2q 1− ( ) + (tanΨ n q extr ) ( fextr q ) 2 π fextr qτ s (4.27)

Although a resolution for τs in analytical form is not possible, equation (4.27) offers several advantages: It can be resolved by a numerical procedure or by an iterative process with the (1) first step τs calculated from the minimum of the transformed phase lag function according to

70 4.6 Interpretation of phase measurements obscured by the background fluctuation

1−[(tanΨ )2 ( f )2q ]2 1−[(tanΨ )2 ( f )2q ] (1) n q extr extr q n q extr extr q tan(2 πfextr qτ s ) = = 1+ (tanΨ )2 ( f )2q 2 2q n q extr extr q 1+ (tanΨ n q extr ) ( fextr q ) (4.28a) ⎧ 1−[(tanΨ )2 ( f )2q ] ⎫ (1) ⎪ n q extr extr q ⎪ 2 πfextr qτ s = arctan⎨ ⎬ ⎪ 1+ (tanΨ )2 ( f )2q ⎪ ⎩ n q extr extr q ⎭ (4.28b) and the further steps with q≠0 calculated according to

(i) tan(2 πfextr qτ s ) =

q 2 2q 2 2 2q q 2 − ( ) + 1−[(tanΨ n q extr ) ( f extr q ) ] + [(tanΨ n q extr ) ( f extr q ) ]( ) (i−1) (i−1) 2 π f extr qτ s 2 π f extr qτ s = q 2 2 2q 1− ( ) + (tanΨ n q extr ) ( f extr q ) (i−1) 2 π f extr qτ s (4.29)

Additionally solution (4.27) can be interpreted as an extension of solution (4.13) for the relative minimum of the not transformed measured phase lag function (q=0).

4.6.2 Application of the functional transformation to measurements: The problem of convergence

In order to test the transformation method and to get information on the exponents q leading to converging solutions for the thermal diffusion time and thermal reflection coefficient, the method is applied to the inverse normalized phase lag measured for a sample consisting of a diamond-like carbon coating (∇ C6) on tool steel which according to Figure 4.2b perfectly agrees with an opaque two-layer system in the measured frequency interval. Figure 4.15 shows that with increasing positive q-values the minimum of the transformed calibrated phases is shifted towards lower modulation frequencies with respect to the frequency of the minimum related to the not transformed phase (q = 0). One can see in Table 4.5, which reports on parameters and results of this transformation, that only certain q- values lead to results very close to the result of the not transformed case. Thus, considering q- values up to about 0.140 yields a frequency range, in which the thermal diffusion time is determined with a relative error of less than 1%, 2 % or 3 %. For q-values ≥ 0.175, there is a discrepancy between the solutions of the transformed and the not transformed phases (q = 0). Figure 4.16 shows, that with decreasing negative q-values the minimum of the transformed calibrated measured phases is rather shifted towards higher modulation frequencies with respect to the frequency of the minimum corresponding to the not transformed calibrated phases. The results of the functional transformation, reported in Table 71 4.6 Interpretation of phase measurements obscured by the background fluctuation

4.6 indicate also, that only certain negative q-values are appropriate to keep constant the thermal diffusion time and the ratio of effusivities. In this example, q-values ≤ -0.3 induce considerable deviations while q-values ≥ -0.250 reproduce the expected result. Figure 4.17 and Figure 4.18 show the calibrated measured phases and the subsequent transformations obtained by operating on positive (Fig. 4.17) and negative (Fig. 4.18) q- values. Parameters and results of this functional transformation are reported in Tables 4.7 and 4.8. One can see in Table 4.7, that only a very small and positive q-value (q = 0.01) keeps approximately constant the value of the thermal diffusion time. The results consigned in Table 4.8 indicate also, that a very reduced number of negative q-values (q ≥ -0.025) contributes to reproduce a thermal diffusion time close to that obtained for q = 0.

1/2 1/2 q tan ψn q extr (fextr q) /Hz τs / µs Rsb gsb 0.000 -0.452 76.3 6.38 -0.636 0.223 0.035 -0.389 72.9 6.43 -0.636 0.223 0.070 -0.335 70.4 6.39 -0.636 0.223 0.105 -0.289 67.7 6.34 -0.636 0.223 0.140 -0.250 64.7 6.33 -0.636 0.223 0.175 -0.216 62.5 6.16 -0.638 0.221 0.200 -0.195 60.9 6.04 -0.638 0.221 0.250 -0.159 57.4 5.82 -0.641 0.219 0.300 -0.130 54.0 5.18 -0.643 0.217 0.500 -0.060 41.8 3.41 -0.682 0.189

Table 4.5: Parameters and results of the inverse normalized phase lag measured for a diamond- like coating on tool steel (C6) and after transformation (q > 0) of the phase lag.

1/2 q tan ψn q extr (fextr q/Hz) τs / µs Rsb gsb 0.000 -0.452 76.3 6.38 -0.636 0.223 - 0.035 -0.526 79.4 6.34 -0.636 0.223 - 0.070 -0.614 82.3 6.35 -0.636 0.223 - 0.140 -0.837 87.5 6.37 -0.636 0.223 - 0.175 -0.980 89.9 6.41 -0.636 0.223 - 0.250 -1.373 95.5 6.42 -0.636 0.223 - 0.300 -1.725 98.8 6.47 -0.636 0.223 - 0.500 -4.376 111.1 6.65 -0.638 0.221

Table 4.6: Parameters and results of the inverse normalized phase lag measured for a diamond-like coating on tool steel (C6) and after transformation (q < 0) of the phase lag.

72 4.6 Interpretation of phase measurements obscured by the background fluctuation

0.0

-0.1 0.250

n 0.200 Φ -0.2 0.175

tan tan

q 0.140 ) -0.3

-1/2 0.105 (f 0.070

-0.4 0.035

q= 0.000 -0.5 0 50 100 150 200 250 300 (f / Hz)1/2

Figure 4.15: Transformation of the inverse normalized phase: Shifting of the phase extremum towards smaller modulation frequencies.

0.0

-0.4 0.000 -0.070 n Φ -0.8 -0.140

tan q

) -0.175

-1/2 -1.2 (f

-0.250 -1.6 q = -0.300

0 50 100 150 200 250 300 (f / Hz)1/2

Figure 4.16: Transformation (q < 0) of the inverse normalized phase lag, shifting the relative extremum of the transformed function towards higher modulation frequencies, applied to the inverse normalized phase lag measured for a diamond-like coating on tool steel (C6). 73 4.6 Interpretation of phase measurements obscured by the background fluctuation

1/2 q tan ψn q extr (fextr q/Hz) τs / µs Rsb gsb 0.0 -0.194 59.84 13.06 -0.295 0.544 0.010 -0.186 59.53 12.97 -0.295 0.544 0.020 -0.179 59.22 12.88 -0.295 0.544 0.035 -0.168 58.76 12.75 -0.295 0.544 0.070 -0.146 58.00 12.28 -0.294 0.545 0.105 -0.127 56.91 12. -0.293 0.546 0.140 -0.110 55.76 11.60 -0.292 0.548 0.200 -0.087 53.73 11. -0.298 0.541 0.360 -0.046 45.89 10.03 -0.302 0.537 0.500 -0.027 39.56 08.45 -0.311 0.526

Table 4.7: Parameters and results of the inverse normalized phase lag measured for sample F5 and after transformation (q > 0) of the phase lag.

1/2 1/2 q tan ψn q extr (fextr q) /Hz τs / µs Rsb gsb 0.000 -0.194 59.84 13.06 -0.295 0.544 - 0.025 -0.215 60.58 13.29 -0.295 0.544 - 0.050 -0.239 61.31 13.5 -0.295 0.544 - 0.070 -0.259 61.88 13.69 -0.296 0.543 -0.105 -0.299 62.85 14.01 -0.294 0.545 - 0.140 -0.347 63.79 14.31 -0.297 0.542 - 0.200 -0.445 66.58 14.3 -0.296 0.543 - 0.250 -0.546 68.38 14.49 -0.296 0.543 - 0.360 -0.867 70.38 15.65 -0.300 0.538 - 0.500 -1.584 80.54 13.96 -0.290 0.550

Table 4.8: Parameters and results of the inverse normalized phase lag measured for sample F5 and after transformation (q < 0)of the phase lag.

The diversity observed in the results of the functional transformation of the inverse normalized phase lags for the two different samples (C6 and F5), implies the analysis of the convergence problem: – (i) Limits of convergence in the case of a coating-on-substrate sample, fulfilling the conditions of an opaque two-layer system with one-dimensional thermal wave propagation, and – (ii) Limits of convergence in the case of a coating-on-substrate sample, presenting additionally reduced thermal properties at the very surface of the coating and three-dimensional thermal wave propagation in the substrate. According to Figure 4.19, which represents the thermal diffusion time and the ratio of effusivities coating-to-substrate as a function of the q-value, the horizontal panel observed in the range (-0.250 < q ≤ 0.140) for the sample C6 means that the four-point based interpolation and the q-transformation reproduce the result, since the thermal diffusion time and the ratio of effusivities are kept constant. This result has a double significance: (i) – It provides the proof that the transformation applied to the phases correctly works over a certain range of q values.

74 4.6 Interpretation of phase measurements obscured by the background fluctuation

0.00 0.200

-0.05 n Φ

-0.10

0.140

tan tan 0.105 q ) 0.070 -0.15 -1/2 (f

-0.20 q = 0.000

0 50 100 150 200 250 300 ( f / Hz)1/2

Figure 4.17: Transformation (q > 0) of the inverse normalized phase: Shifting of the phase extremum towards lower modulation frequencies (sample F5)

0.0

-0.1 q = 0.000 n Φ -0.2

tan -0.050

q -0.3 )

-1/2 -0.140 (f -0.4

-0.200 -0.5 0 50 100 150 200 250 300 ( f / Hz)1/2

Figure 4.18: Transformation (q < 0) of the inverse normalized phase: Shifting of the phase extremum towards higher modulation frequencies (F5) 75 4.6 Interpretation of phase measurements obscured by the background fluctuation

(ii) – Although not necessary for quantitative information with respect to τs and gsb, which in the case of sample C6 are sufficiently well determined at the original measured position of the minimum, the (1/f1/2)q -transformation can give additional information about what range of frequencies the measured sample can be sufficiently well approximated by an opaque two- layer model facing 1D heat transport, and where deviations related to an effective three-layer structure or due to semi-transparency in the range of higher frequencies, and where deviations due to three-dimensional heat transport in the range of lower frequencies affect the measured signals and therefore lead to a more complex scheme of thermal wave propagation in layered systems. This is the case for the sample F5 for which it can be observed that the transformed inverse normalized phase lag produces non convergent solutions in terms of the thermal diffusion time, although a constant ratio of effusivities is recorded for q-values within the interval (-0.250 < q ≤ 0.035). In fact, as far as convergence is concerned, it may imply both the thermal diffusion time and the thermal reflection coefficient and not exclusively one of the two quantities.

10.0 0.25 20.0 0.60

0.24 17.5 0.55 C6 0.23 7.5 15.0 0.50 0.22 12.5 0.45 0.21 F5 b 5.0 0.20 b 10.0 0.40

/e /e s s /µs /µs s e e s τ τ 0.19 7.5 0.35

0.18 2.5 5.0 0.30 0.17 2.5 0.25 0.16

0.0 0.15 0.0 0.20 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 q q

Figure 4.19: Thermal diffusion time and ratio of effusivities coating-to-substrate as a function of the

transformation exponent q – Problem of convergence of the relevant thermal transport parameters τs, Rsb, or gsb.

That the functional transform applied to the sample F5 does not produce convergent solutions can be explained by the two physical effects occurring at high modulation frequencies, namely the existence of a very thin layer on top of the coating −three-layer model− and at low modulation frequencies, namely the 3-D thermal wave propagation in the subsurface material, all aspects which can contribute to reduce the frequency range of the convergence which is more extended in the case of an opaque two-layer structure with 1-D thermal wave propagation.

76 4.6 Interpretation of phase measurements obscured by the background fluctuation

4.6.3 Phase interpretation obscured by the background fluctuation

Figure 4.20 shows an example of measurement on a coated cutting tool after exposure to friction wear. Due to a highly reflecting surface, the measurement of the phase lag is already affected by the background noise before the minimum occurs. As one can observe in this figure, the measurement is already disturbed by the background fluctuation above heating modulation frequencies of about (f /Hz)1/2 > 55, so that the minimum can not easily be localized. The four measured values between 60 <(f /Hz)1/2 < 100 and all measurements above (f /Hz)1/2 > 100 can not be interpreted.

1/2 q tan ψn q extr (fextr q/Hz) τs/µs Rsb gsb Remarks 0 In this range 0.16 of q-values, 0.20 no possible 0.24 solutions are 0.28 found. 0.32 -0.034 52.72 8.97 -0.192 0.678 Minimum 0.33 -0.032 52.32 8.87 -0.191 0.679 Minimum

Table 4.9: Shifting of the transformed normalized phase minimum as a function of the exponent q (22P1)

1/2 q tan ψn q extr (fextr q/Hz) τs /µs Rsb gsb Remarks 0 In this range 0.150 of q-values, 0.225 no possible 0.300 solutions are 0.375 found 0.450 -0.024 99.54 1.58 -0.327 0.507 Minimum 0.500 -0.019 90.29 1.59 -0.33 0.504 Minimum

Table 4.10: Shifting of the normalized phase minimum as a function of the exponent q (119P2)

In order to detect the position of the minimum, several values of the exponent q have been tested. As indicated in Table 4.9, in the range of q-values < 0.32, no possible solutions can be found. A relative minimum for the transformed phase lag function has been obtained for rather large q-exponents (0.32; 0.33). The thermal diffusion time (τs ≈ 8 µs) and the thermal reflection coefficient (Rsb = -0.192) obtained from the functional transformation based on these two values −which lead to the same results− of the exponent q are then used to generate a theoretical solution according to equ. (4.10), which solution is correlated with the not transfor-

77 4.6 Interpretation of phase measurements obscured by the background fluctuation

0.00

-0.04 n Φ

q = 0.33 tan -0.08 q ) -1/2

(f 0.24 -0.12

0.16 -0.16 0 50 100 150 200 250 300 350 ( f / Hz)1/2

Figure 4.20: Inverse normalized phase lag measured for a friction wear affected coated cutting tool surface (22P1), in comparison with transformed phase lag functions and theoretical approximations.

0.00 0.50

-0.04 0.30

-0.08 n

Φ 0.15

-0.12 tan tan q )

-1/2 -0.16 0.50 (f

-0.20 q = 0.00 -0.24 0 50 100 150 200 250 300 (f /Hz)1/2

Figure 4.21: Inverse normalized phase lag measured for a friction wear affected coated cutting tool surface (119P2), in comparison with transformed phase lag functions and theoretical approximations. 78 4.7 Conclusions

-med experimental inverse normalized phase lags (Figure 4.20). Through this correlation, the frequency and value of the phase extremum that could be derived if the modulated IR signal measurements were not affected by the background fluctuation limit are estimated as follows: 1/2 (fextr /Hz) = 71.23 and tanΦnextr = -0.126, respectively. Table 4.10 reports on parameters and results obtained for the functional transformation in connection with another type of friction wear (Fig. 4.21). As in the previous case, there exists a range of q-values for which no possible solutions can be retrieved. Most interesting is however, that the interval range of q-values for which there is no recordable minimum, is enlarged. As indicated in Table 4.10, a relative minimum of the transformed phase lag function has been achieved only for q-values = 0.45 and 0.50. These two q-exponents lead to constant values of the thermal diffusion time and the thermal reflection coefficient.

Proceeding in the same way as above, the combined parameters, τs ≈ 1.6 µs and Rsb = -0.33, obtained from the functional transformation based on q = 0.45 and q = 0.50 are used to generate a theoretical solution according to equ. (4.10), which solution is correlated with the not transformed experimental inverse normalized phase lags (Fig. 4.21). Thus, the probable frequency and value of the relative phase extremum that should have been detected if the modulated IR signal measurements were not affected by the background fluctuation are 1/2 respectively estimated as follows: (fextr /Hz) = 176.85 and tanΦnextr = -0.218.

4.7 Conclusions

Thermal waves applied to control online-thicknesses of coatings or paint layers [Petry, 1998] require short measurement times and fast numerical routines. Measurements based on thermal wave methods are known to give more reliable results with the measurement at only three modulation frequencies, one below the expected minimum frequency, one above the expected minimum frequency and another in the neighbourhood. In this chapter, we have proposed and applied a new and fast procedure for the determination of thermal transport properties of two-layer structures via two combined thermophysical quantities, namely the thermal diffusion time and the thermal reflection coefficient, which are extracted from one fundamental point of the calibrated measured phase, indexed as relative phase extremum. It has been demonstrated, that the frequency and value of the relative phase extremum of the calibrated measured phases, if correctly identifed, lead to a more reliable information about the sample properties. The Extremum Method has also provided general results for on-line interpretation in industrial applications by allowing the establishment of two main graphics, namely the graphic of thermal reflection coefficients as a unique function of the relative extremum and the graphic of thermal diffusion times as a function of the frequency at which the phase extremum occurs, for different values of the relative extremum. Then a functional transform

79 4.7 Conclusions

method, originating from the Extremum Method, has also been developed to enable the interpretation of measurements based on modulated IR radiometry, which may be obscured at low frequencies due to 1/f −noise and at high frequencies due to the background fluctuation limit. In order to validate this transformation method and subsequently to study the convergence problem, several tests have been performed on phase measurements of two different samples, which measurements are free of noise or of background fluctuation. For one sample (C6), which fulfills the conditions of an opaque two-layer system facing 1D thermal wave propagation and whose minimum is easily identifiable, the functional transform method based on both negative and positive q-values has reproduced results converging to that obtained in the case q = 0, by keeping constants the thermal diffusion time and the thermal reflection coefficient within a certain frequency range. For larger absolute q-values however, deviations between solutions of transformed and not transformed phase measurements were noted, providing the proof that the q-transformation applied to the inverse calibrated phases correctly works over a certain range of q-values. Although a relative minimum was clearly identified on the calibrated measured phases of another sample (F5), successive q- transformations of these phase measurements have not provided solutions of τs and Rsb both convergent to the original solutions (q = 0). Such a situation has been explained by the fact, that the calibrated phase measurements of the considered sample (F5) are affected by physical effects such as the 3-D thermal wave propagation at low frequencies and surface structure – additional thin layer on top of the coating– or semi-transparency at high modulation frequencies, thus giving a proof that the functional transformation works very well for measurements involving typical two-layer systems facing 1-D thermal wave propagation. Finally, the functional transform method has been applied to some examples of measurements which may be obscured by the background fluctuations so that no phase extremum can be observed. Here, the q-values which led to transformed inverse normalized phase lags exhibiting a minimum, produced values of the thermal diffusion time and the thermal reflection coefficient, which were then used to estimate the probable frequency and value of the relative phase extremum that should have been detected if the modulated IR signal measurements were not affected by these background fluctuations. In the next chapter, we also demonstrate that the relative extrema appearing on the calibrated measured phases in the low modulation frequency range can contribute in thermal microscopy to get more reliable information about the lateral heat transport and the identification and detection of hot spots in the structures.

Detection of local Inhomogeneities of Thermal Transport and Localization of Heat Sources in Micro-scaled Systems based on Spot Displacement

5.1 Motivation

We have shown in chapter 3, that the relative phase maximum observed on the normalized signal phases of various layered systems, in the limit of very low modulation frequencies, is due to the effects of lateral heat propagation in the subsurface material, and that these thermal effects increase with decreasing heating spot radii and with increasing thermal diffusivities of the subsurface material. That means, by considering smaller heating spot radii, the relative phase maximum can help to detect local variations of the thermal diffusivity of the substrate. In addition to the thermal transport properties, which play a key role in such layer systems used e.g. as devices in microelectronics or micromechanics, the localization of hot spots is particularly important in microelectronics where the reduction of size is accompanied by a growing power dissipation. In this scope, numerical simulations of heating processes and of photothermal experiments become more and more important, both for the interpretation of measurements and the development of measurement methods, applied for the design and quality control of micro-structured materials. In the present chapter, starting from concrete examples of photothermal measurements of two-layer systems with film thicknesses in the range of 1 to 3 µm, we show with further simulations of photothermal experiments of two-layer systems from macroscopic to microscopic scale, that by using 3-D thermal wave propagation and considering controlled displacement distances between the excitation and the detection spot [Ikari et al., 2003], the relative extrema generated on the calibrated signal phases in the low frequency range can contribute in thermal microscopy to get more reliable information about the lateral heat transport and the identification of hot spots in the structures.

5.2 Review of main results of 3-D thermal wave propagation

We have demonstrated in chapter 3, that the modulated IR signals measured over the detection spot area of radius rD (comp. Figure 3.8, Chap. 3), resulting from the generation of a thermal wave at the surface of a two-layer system, can be calculated as

81 5.3 Displacement between heating and detection spots

* 3 δM s ( f ,T,t) = 4C( f )γ (T )σ SBT ε s (T ) ⋅

η I rD ∞ exp(−λ2 r 2 / 8) ⎡1 + R ( f ,λ) exp(−2λ d )⎤ ⋅ s o 2πrdr H sb s s J (λr)λdλ ⋅ exp(i2πft) (5.1) ∫∫ ⎢ ⎥ o 2πk s 00λs (λ, f ) ⎣1 − Rsb ( f ,λ) exp(−2λs d s )⎦

* 3 The factor 4C( f )γ (T )σ SBT ε s (T ) in equ. (5.1) is a real quantity, which cannot affect the phase of the measured thermal wave response. For recall, both the measured and the theoretical data are presented in the form of calibrated signal phases,

Φ n ( f ) =Φ ref ( f )−Φ s ( f ) (5.2)

where the signal phases Φref ( f ) refer to a homogeneous reference body of known thermal and optical properties, e.g. glassy carbon, and the signal phases Φs ( f ) refer to the sample, e.g. a two-layer system. This means, the results based on the interpretation of the signal phases, are independent of the specific detection process and thus also apply to other detection techniques, e.g. thermoreflectance. It has been observed (comp. Section 3.6.2.3, chap. 3), that the deviations from 1-D thermal wave propagation increase with decreasing heating spot radii and with increasing thermal diffusivities αb of the subsurface material and that the normalized phases are rather sensitive to variations of the thermal diffusivity αb of the subsurface material. This means, by working with smaller heating spot radii in thermal microscopy, the normalized phases with their relative phase maximum in the low frequency range can be used to detect local variations of the thermal diffusivity in the subsurface material.

5.3 Displacement between heating and detection spots

In the simulations of 3-D thermal wave propagation, concentric heating and detection spots, as schematically shown in Figure 3.8 (Chap. 3) have been considered. If heating spot and detection spot are not well focused in the experiment, errors of interpretation may occur due to deviations between the model of concentric excitation and detection and the experimental reality. On the other hand, a controlled pump-probe beam offset, intentionally introduced between heating and detection spots, as shown in Figure 5.1, can contribute in thermal microscopy to get more reliable information on the lateral heat transport properties and on the localization of heat sources (hot spots), by comparing the data measured at two or three neighbored positions. In such experimental configurations, which can be used in connection with both modulated thermoreflectance and modulated IR radiometry as detection techniques [Milcent et al., 1995; Ikari et al., 1999] and which have already been applied to coatings on metal 82 5.3 Displacement between heating and detection spots substrates and semiconductor materials [Milcent et al., 1998; Ikari et al., 2003], the centres of the heating and detection spots are shifted against each other by a distance of dHD. The heating and the detection spots may have the different radii rH and rD.

Figure 5.1: Schematic of the 2-layer sample and of the experimental configuration, based on a finite displacement dHD between the heating and the detection spot.

5.3.1 Theoretical background

According to Figure 5.2, the maximal angle under which the upper or the lower border of the detector area of radius rD can capture the modulated IR radiation induced by a thermal wave at the sample surface, localizable by its variable position r from the center of the heating spot, is described by

sin[]θmax(r) = BD(r)/r (5.3)

The term at the right hand side of equ.(5.3) can be determined by using the Pythagore’s theorem of geometric analysis (see Fig. 5.2):

83 5.3 Displacement between heating and detection spots

AD2 ( r ) = AB2 ( r ) + BD2 ( r ) (5.4a) DC 2 ( r ) = DB2 ( r ) + BC 2 ( r ) (5.4b) From equations (5.4), the following relations are extracted:

2 2 2 AB( r ) = (r + d HD − rD )/ 2d HD (5.5a)

2 2 2 2 2 2 DB( r ) = r − (r + dHD − rD ) / 4dHD (5.5b)

By introducing equ.(5.5b) into equ.(5.3) one obtains,

2 2 2 2 2 2 sin []θmax (r) = 1− ()d HD + r − rD / 4r d HD (5.6) and by considering a surface element (see Fig. 5.2) on the upper hemisphere of the detection spot area, ds = r dθ dr , and taking into account the contribution of the lower hemisphere to the detection (hence the numerical factor 2 to account for symmetry), the signal measured over the detection spot area, e.g. by modulated IR radiometry, is then described by:

* 3 δM s ( f ,T,t) = 4C( f )γ (T )σ SBT ε s (T ) ⋅ dHD +rD ⎛θmax ⎞ ⋅ 2 ⎜ dθ ⎟δT ( r,z = 0, f ,t )rdr ∫∫⎜ ⎟ s s (5.7a) dHD −rD ⎝ 0 ⎠

Evaluating the angular integral in equ. (5.7a) by taking into account the expression of the position-dependent angular upper bound given by (5.6), the modulated IR signal takes the following form:

* 3 δM s ( f ,T,t) = 4C( f )γ (T )σ SBT ε s (T ) ⋅

dHD +rD ⋅ 2 rdr arc sin[ 1− (d 2 + r 2 − r 2 )2 /(4r 2d 2 )] δT (r, z = 0,t) ∫ HD D HD s s (5.7b) dHD −rD with the thermal wave [comp. equ.(3.42) to equ.(3.44), and see also equ.(3.34), Chap. 3] given by ∞ ηs Io λ 2 2 δTs (r,0, t) = ∫ dλ J0 (λ)exp(−λ rH /8) ⋅ 2πks 0 λs

⎡1+ Rsb (λ, f )exp(−2λs d s )⎤ ⋅⎢ ⎥ ⋅ exp(i2πft) (5.8) ⎣1− Rsb (λ, f )exp(−2λs d s )⎦ at the variable position r inside the detection spot area. One can remark, that the thermal wave carries the characteristics of the excitation source (radius of the heating spot, rH) while the

84 5.3 Displacement between heating and detection spots resulting signal carries in addition the characteristics of the detection system (radius of the detection spot, rD, and displacement distance between the heating and the detection spot, dHD).

Figure 5.2: Top view of the sample showing displacement arrangements between excitation and detection spots.

5.3.2 Simulation of controlled displacements between the two spots

In Figure 5.3 the effects of defocusing, respectively controlled displacement are simulated for a heating spot radius of rH = 2000 µm and a detection spot radius of rD = 800 µm, using increasing displacement distances between the centres of the two spots. The case of well focused detection (dHD = 0), which is in good agreement with the normalized phases 1/2 -2 -1 measured for a hard coating on a metallic alloy of high effusivity (eb = 13100 Ws m K ) -6 2 -1 and thermal diffusivity (αb = 22⋅10 m s ) is compared with theoretical approximations based on dHD = {2400 µm, 3000 µm, 3400 µm}. With increasing displacement distance dHD larger deviations of the phases are found at the very low frequencies, (f /Hz)1/2 ≈ 3, with the relative phase maximum reaching positive values.

85 5.3 Displacement between heating and detection spots

20 d HD 0

/ deg n -20 Φ

-40

-60 0.8 1 10 80 (f / Hz )1/2

Figure 5.3: Effects of controlled displacement on the normalized phases, simulated for constant heating and detection spot radii (rH = 2000 µm, rD = 800 µm). The displacement distance varies from bottom-to-top by dHD = {0 µm, 2400 µm 3000 µm, 3400 µm}. The experimental data have been measured for a hard coating on a metallic alloy of high effusivity (∇) at dHD = 0 µm.

40 22×10-6 d 20 HD

0 / deg n -20 Φ

-40 -6 2 α = 66×10 m /s b -60 0.8 1 10 80 ( f / Hz )1/2

Figure 5.4a: Relative maxima of the normalized phases calculated in the range of low modulation frequencies for different displacement distances dHD = {2400 µm, 2800 µm, 3000 µm, 3200 µm, 3400 µm} and two thermal diffusivity values αb. rH = 2000 µm, rD = 800 µm.

86 5.3 Displacement between heating and detection spots

5.3.3 Localization of heat sources

Using heating and detection spots of fixed sizes, simulations are shown in Figure 5.4a and Figure 5.4b for a two-layer system with different thermal diffusivity values of the subsurface material, αb. Considering constant thermal diffusivity αb, the increasing displacement distances generate a set of relative phase maxima, which increase with the displacement distance and are found at nearly the same modulation frequency. For a value of -6 2 1/2 αb = 22⋅10 m /s the relative phase maxima are found in Fig. 5.4a at about (f /Hz) ≈ 3 and -6 2 for a higher value of the thermal diffusivity of the substrate, αb = 66⋅10 m /s, the relative phase maxima shift to higher modulation frequencies, (f /Hz)1/2 ≈ 5. Using constant thermal -6 2 diffusivity values of the substrate material (in this case, αb = 22⋅10 m /s), but smaller heating spot radii, e.g. rH = 1000 µm in Fig. 5.4b, the relative phase maxima also shift to higher modulation frequencies, (f /Hz)1/2 ≈ 6. By collecting and representing the data of the different relative phase maxima as a function of the position of the detection spot, it becomes possible to localize the heat sources. As one can see in Figure 5.5, which shows the relative phase maxima as a function of the corresponding displacement distances, extrapolation onto the heating spot is nearly independent of the thermal diffusivity of the subsurface material. Practically, the results of these simulations indicate that – when three or four measurements are performed in the neighborhood of a heat source – the hot spot can be localized with good precision.

20 d HD 0

/ deg n -20 Φ

-40 -6 2 α = 22×10 m /s b -60 0.8 1 10 80 ( f / Hz )1/2

Figure 5.4b: Relative maxima of the normalized phases for different displacement distances from bottom-to-top dHD = {1400 µm, 1500 µm, 1600 µm 1700 µm} with the heating and detection spot radii given by rH = 1000 µm and rD = 400 µm, respectively, and constant thermal diffusivity of the -6 2 subsurface material, αb = 22⋅10 m /s.

87 5.3 Displacement between heating and detection spots

40 / deg

30 n max

Φ 20

0 0 500 1000 1500 2000 2500 3000 3500 4000 Displacement distance / µm

Figure 5.5: Phase maxima as a function of the displacement distance between detection spot and heating spot. – The data points (o,∆) refer to the relative phase maxima found in Fig. 5.4a for different values of the thermal diffusivity αb of the substrate. The points (◊) refer to the relative phase maxima of Fig. 5.4b, found for smaller radii of the heating and detection spot.

5.3.4 Comparison of experimental and theoretical results

Figure 5.6 and Figure 5.7 show respectively the phases and amplitudes of the modulated IR signals measured for a diamond-like carbon on high effusivity metallic alloy and calibrated with the signals measured for a homogenous material (glassy carbon). The experimental measurements are compared with the theoretical approximations. Globally, the experimental phases and amplitudes reproduce the behavior of theoretical predictions. To perform these measurements, a gaussian shaped modulated laser beam (Argon ion laser) of radius rH ≈ 2000 µm (heating spot) was used for the excitation of thermal waves and detection of the thermal response was performed with the help of a MCT detector having a detection spot area of s = 2 mm2. As for the measuring method, the detection spot was maintained at the same position while the heating spot was scanned over the sample surface. Starting from a concentric configuration between the heating spot and detection spot, displacement distances of dHD = 1000, 2000, 2500 and 3000 µm were then considered. However, correlation between experimental and theoretical approximation leads to the following data: dHD = 1000, 1900, 2300 and 2800 µm. The discrepancies between theoretical and experimental values are not large, and it may be possible that some unavoidable reading errors occurred during experiment and so the experimental distances could have been magnified.

88 5.3 Displacement between heating and detection spots

As can be seen in Figure 5.6 and in agreement with the results presented in Figure 5.3, both the experimental and theoretical relative phase maximum recorded in the limit of very low modulation frequencies, increase with increasing displacement distance. In the same time, one can clearly see that with increasing displacement distance between the two spots, the relative phase maximum shifts to lower modulation frequencies. This experimental result is in conformity with the theoretical predictions consigned in Figure 5.4a which attest, that the relative phase maximum shifts to lower modulation frequencies with decreasing thermal diffusivity of the subsurface material and shifts to higher modulation frequencies with increasing thermal diffusivity of the subsurface material. The results of Figure 5.6 indicate therefore a local variation of the substrate thermal diffusivity. Correlation between experiment and theory yields different values of the substrate thermal diffusivity as a function of the displacement distance between the heating spot and the detection spot, e.g., for the distance -6 2 dHD = 0 (concentric heating and detection spots) αb(0) = 20⋅10 m /s and for dHD = 2800 µm, -6 2 αb(2800) = 12⋅10 m/s. The calculated values of αb as a function of the displacement distance dHD are reported in Table 5.1. The modulation frequency at which the relative phase maximum occurs is also mentioned as a function of the displacement distance. These numerical values provide the proof, that the lateral inhomogeneities of the subsurface material can be pointed out when either the detection spot or the heating spot is canned over the sample surface.

-10

/deg -20 n Φ

-30

-40 110100 ( f/Hz)1/2

Figure 5.6: Normalized phases (symbols) of the modulated IR signals, measured for a diamond-like carbon on high effusivity metallic alloy, in comparison with theoretical approximations. The displacement distance between the heating and the detection spots (respectively of radii rH = 2000 µm and rD = 800 µm) varies from bottom-to-top by dHD = {0 µm, 1000 µm, 1900 µm, 2300 µm, 2800 µm}. 89 5.3 Displacement between heating and detection spots

dHD /µm 0 1000 1900 2300 2800 1/2 (fmax /Hz) 5 4.47 3.16 2.83 2.45 -6 2 -1 αb ×10 /m s 20 19.4 18.6 16 12

Table 5.1: Lateral variations of the substrate thermal diffusivity with the displacement between the heating spot and the detection spot

A careful observation of Figure 5.6 in the range of high modulation frequencies 60 < (f /Hz)1/2 allows to establish, that there are also some sensitive variations of the calibrated measured phases as the displacement distance between the two spots varies. Since the position of the measured relative phase minimum, localizable by its modulation frequency in the range 1/2 60 < (fmin /Hz) < 80 is relatively stable so that no change in the thickness of the thin film deposited on the substrate is of concern, these variations of the measured phases at small penetration depths can be attributed to local variations of the coating thermal diffusivity as the heating spot is scanned over the sample surface. Following this, it is justified to conclude, that by performing modulated excitation of the sample surface at one point and then detecting the thermal response in the neighborhood, more reliable information about the thermal properties of the investigated sample can be easily obtained. In Figure 5.6, the disagreements observed between theory and experiment at modulation frequencies in the ranges 1 < (f /Hz)1/2 < 2 and 5 < (f /Hz)1/2 < 30 can be explained by the internal structuring of the subsurface material. In Figure 5.7 the calibrated measured amplitudes are presented, which show, that although the deviation of the signal is more pronounced in the limit of very low modulation frequencies, there exists a variation of the inverse signal in the entire frequency range with increasing displacement distances. Such a behavior can be explained by the fact, that on the contrary to the signal phase the signal amplitude is proportional to the optical absorptivity or photothermal conversion efficiency η which is defined as the fraction of the laser intensity transformed into heat and thus any variation of η induces necessarily variations of the signal amplitude. A correlation between experimental and theoretical inverse signal amplitudes gives in the case of concentric heating and detection spots (dHD = 0), an optical absorptivity of about

ηs (0) = 0.37 while at dHD = 2300 µm, ηs (2300) = 0.42. Table 5.2 presents the values recorded for the photothermal conversion efficiency ηs as a function of the displacement distance. As can be seen in this table, the optical absorptivity experiences a random variation as the heating spot is scanned over the sample surface, e.g., at dHD = 1900 µm, ηs (1900) = 0.43 and at dHD = 2800 µm, ηs (2800) = 0.418. These results are quite comprehensible since the measured signals were obtained by scanning the heating spot, instead of the detection spot, over the sample surface. Such a configuration also provides the possibility to obtain more reliable information about the lateral inhomogeneities of the optical absorptivity. Finally, by exciting the sample at one location of the sample surface and then detecting in the neighborhood, it is possible to obtain complete information about the local variations of the thermal and optical properties of the sample.

90 5.4 Scaling of thermal localization of hot spots

-1 n S

0 0.01 0.1 1 ( f/Hz)-1/2

Figure 5.7: Inverse normalized amplitude (symbols) of the modulated IR signals, measured for a diamond-like carbon on high metallic alloy, in comparison with theoretical approximations. The displacement distance between the heating and the detection spots (respectively of radii rH = 2000 µm and rD = 800 µm) varies from top-to-bottom dHD = {0 µm, 1000 µm, 1900 µm, 2300 µm}.

dHD /µm 0 1000 1900 2300 2800

ηs 0.370 0.400 0.430 0.420 0.418

Table 5.2: Lateral variations of the photothermal conversion efficiency with the displacement between the heating and the detection spot.

5.4 Scaling of thermal localization of hot spots

Starting from the results obtained for displacement distances in the millimeter range, further simulations are performed while decreasing gradually the values of the heating and detection spot radii as well as the displacement distance between the centers of the two spots.

91 5.4 Scaling of thermal localization of hot spots

Furthermore, the size of the sample is adapted to more realistic conditions of thermal microscopy, with a thin film of a thickness of ds = 0.8 µm at the surface and a substrate thickness of db = 500 µm. In this case the thermal wave solution [comp. equ.(5.8)] is described by

∞ ηs Io λ 2 2 δTs (r,0, t) = ∫ dλ J0 (λ)exp(−λ rH /8) ⋅ 2πks 0 λs

⎡1+ Rsb (λ, f )[exp(−2λsds ) + exp(−2λbdb )]+ exp(−2λsds − 2λbdb )⎤ ⋅⎢ ⎥ ⋅exp(i2πft) ⎣1− Rsb (λ, f )[exp(−2λsds ) − exp(−2λbdb )]− exp(−2λsds − 2λbdb ) ⎦ (5.9)

and the quantity λb of equ.(5.9) is given by

2 2 2 λb = λ + σ b (5.10)

120

r =800 µm 80 D 200 50

0 / deg n Φ -40 5 -80

0.1 1 10 100 1000 (f / Hz )1/2

Figure 5.8: Normalized phases for detection spots of different size versus the square root of the modulation frequency. From left to right, the detection spot decreases, rD = {800 µm, 200 µm, 50 µm, 5µm}. The ratio of displacement distance to detection spot radius, dHD/rD = 2, and that of heating to detection spot radius, rH/rD = 0.8, are constant.

92 5.4 Scaling of thermal localization of hot spots

100 80 d / r = 2 60 HD D 40 20 1.5

0 1 / deg n -20 Φ -40 -60 -80 0.1 1 10 100 (f / Hz)1/2

Fig. 5.9: Effect of the displacement distance on the normalized phases. The displacement distance varies according to dHD/rD = {1.0, 1.5, 2.0}, while the radius of the detection spot (rD=800 µm) and the ratio heating to detection spot radius, rH/rD = 0.8, are constant.

100

80 0.7 0.8 60 r / r = 0.9 H D 40

/ deg 0 n ϕ -20

-40

-60

0.6 1 10 60 (f /Hz)1/2

Fig. 5.10: Effect of the ratio heating to detection spot radius rH/rD = {0.9, 0.8, 0.7} on the relative phase minima and maxima. The radius of the detection spot (rD = 800 µm) and the displacement distance (dHD/rD = 2) are maintained constant.

93 5.5 Comparisons with measurements based on Thermoreflectance

The aim is to find out, at what distances such a localization of heat sources by means of measurements in the neighborhood is possible.

According to the ratio of heating to detection spot radius rH/rD = 0.8 and according to the ratio of displacement distance to detection spot radius dHD/rD = 2, the theoretical solutions in Figure 5.8 describe experimental configurations, where the displacement distance between the two spots is larger than the two spot radii. Each phase solution is characterized by a large relative minimum followed by a large relative maximum, and with decreasing spot sizes and displacement distance, the relative minima and maxima shift to higher modulation frequencies, e.g. for a heating spot of rH = 4µm, a detection spot of rD = 5µm, and a displacement distance of dHD = 10µm, the relative minimum would be found at rather high frequencies, e.g. at about (f / Hz)1/2 ≈ 400. Figure 5.9 presents the results of simulation at given constant radii of the detection and heating spot, while the displacement distance between heating and detection spots varies systematically to investigate its influence on the phase signals. As can be seen, the differences between the relative phase minima and maxima increase considerably with the displacement distance. Such a result has already been found in Figure 5.3 and confirmed by experimental measurements in the millimetre range presented in Figure 5.6. In Figure 5.10, the results of simulations are presented for a given radius of the detection spot and a constant displacement distance. The radius of the heating spot is varied systematically. One can observe, that the differences between the relative phase minima and phase maxima are rather large and that they sensitively vary with the size of the heating spot. The results of Figure 5.9 and of Figure 5.10 allow to conclude that the more appropriate configuration in displacement experiments is reached for larger displacement distances between heating and detection spots, and for radii of the heating spot smaller than the detection spot. With decreasing spot radii (Fig.5.8 and Fig.5.10), the relative phase minima and the phase maxima shift to higher modulation frequencies.

5.5 Comparisons with measurements based on Thermoreflectance

Measurements based on modulated thermo-reflectance [Dietzel et al., 2003] with an increasing displacement distance between the constant excitation and detection spots are shown in Figure 5.11. According to Dietzel et al., the deviations between measured amplitudes and theoretical approximations (Fig. 5.11a) at shorter distances, ∆x < 20 µm, can be explained by lateral inhomogeneities of the optical reflectance and do not affect the quality of the phase measurement. Such a result has also been found in section 5.3.4, Figure 5.7. As can be seen in Fig. 5.11b the signal phases measured at the modulation frequency of 20 kHz are in very good agreement with the theoretical approximation up to a displacement distance of about 60 µm, and cover a detection range as calculated in Figure 5.6. According to the ratio

94 5.6 Conclusions

dHD/rD = 2 (Fig. 5.8), a displacement distance of about 60 µm at the modulation frequency of 20 kHz corresponds to a detection spot radius in the range between 5 µm and 50 µm.

Fig. 5.11: Amplitudes (a) and phases (b) of the modulated optical reflectance measured for a Cu-C interface system (thin film of Cu of 0.8 µm on glassy carbon with an additional thermal contact resistance between Cu film and C substrate) at 20 kHz as a function of displacement distance between heating and detection spot. In these measurements, the heating spot has been scanned over the sample surface, while detection spot and sample remained at a constant position. This configuration is especially sensitive to local inhomogeneities of the optical absorptivity.

5.6 Conclusions

Thermal wave simulations considering controlled displacement distances between heating and detection spots have shown the potential of the method in detecting and identifying heat sources by measurements in the neighborhood. Owing to the fact that only the thermal waves’ phase retardation has mainly been considered, the results presented here should apply to thermal microscopy, independent of the applied detection technique. The first experimental measurements performed in the millimeter range using the modulated IR radiometric technique (Fig.5.6 and Fig.5.7) have confirmed the reliability of the proposed photothermal method which provides better information on the lateral inhomogeneities of the 95 5.6 Conclusions

thermal transport properties. On the other hand, correlations between measured and theoretical signal amplitudes have also revealed the lateral variations of the photothermal conversion efficiency. In particular we have demonstrated, that the relative maximum occuring in the limit of very low modulation frequencies on the calibrated phases and which has been found to be a very useful tool for the determination of the effective thermal diffusivity of the sample (comp. Chap 3), can also help to identify the heat sources in the structures. Detecting and identifying the heat sources (hot spots) in micro-structured (electronic) devices remains one of the great challenges in thermal management since the experimental techniques find their measuring capabilities limited with the drastic reduction of size of devices. This is why numerical methods are more and more required to help predicting and understanding for example the mechanisms of thermal transport which can occur in so tiny devices. In chapter 7, we will use the finite element method to calculate the temperature oscillations in several micro-structured devices as well as the thermo-elastic displacements of their surface and then compare the obtained results with some available experimental measurements based on scanning thermal microscopy (STM) and scanning thermo-elastic microscopy (SThEM). However, going from the fact that the simulation of thermal wave problems is somewhat complex, chapter 6 is enrolled to show by means of tips and examples how these types of scientific problems should be efficiently handled with the ANSYS-aided finite element technique.

Efficient Simulation of Thermal Wave Problems with the ANSYS-aided Finite Element Technique

6.1 Introduction

It is well known, that a numerical method might not reproduce the true solution if the scheme is not properly designed. In general, numerical methods are governed by many criteria including consistency, stability and convergence [Isaacson and Keller, 1966]. For example, in a numerical scheme based on the Finite Difference Method (FDM), the consistency is obtained when the global truncation error due to the approximation of derivatives by differences varnishes, and the stability is observed when the numerical errors generated during the simulation are not magnified but rather minimized. A consistent and stable numerical scheme is convergent. All these criteria also apply to the Finite Element Method (FEM) [Bathe, 1974]. Since the simulation of thermal wave problems is somewhat complex, the present chapter is enrolled to show by means of tips and examples how such types of problems should be efficiently handled with the Finite Element technique. Another purpose is to check the reliability of the numerical schemes that will be used in the next chapter, by comparing the results of the ANSYS-aided finite element simulations with some significant theoretical results, already presented in chapter 3 and in chapter 5 and applied to experimental measurements.

6.2 Finite Element concepts

6.2.1 Theoretical foundation

The Finite Element Method (FEM) is a numerical technique for solving problems which are described by partial differential equations or can be formulated as functional minimization. If the physical formulation of the problem is described as differential equation, then the solution method is the Weighted Residual Method (WRM) [Ames, 1977]. If the physical problem can be formulated as functional minimization, then the Variational Formulation (VF) is required [Norries and de Vries, 1973]. For example, the total potential

97 6.2 Finite Element concepts

energy of a system is minimal at equilibrium. Here, the potential energy constitutes the functional. Considering the variational formulation, the extremum condition imposed on the functional, e.g., the potential energy Ep,

∂ E {}T p = 0 (6.1) ∂{}T yields a system of equations from which the unknown primary variables at nodes are derived. The element equations can be assembled as follows:

[]B ⋅{}T =− {}q (6.2)

In equ.(6.2), []B is the matrix (comprising the material properties, the geometrical data, etc), {}T is for example the vector of nodal temperatures, and {q } is the load vector. The nodal temperatures are retrieved by numerical inversion of equ.(6.2):

{}T =−[]B −1 ⋅{q } (6.3)

As for the weighted residual method, one considers e.g. the following partial differential equation

∂T − LT = 0 x ∈ D, t > 0 (6.4) ∂t where the unknown and exact solution, T(x,t ) , is approximated by a trial solution

N uT (x,t) = ∑ N j (x)T j (t) (6.5) j=1

In equ.(6.4), L denotes a differential operator in the space derivatives of T, x is a vector of space variables, and D is a space domain. In equ.(6.5), the N j (x) are known basic functions or shape functions, and the Tj(t) are the (time-dependent!) constants. By inserting the trial function (6.5) into equ.(6.4), a residual subsists,

∂u R(u )= T − Lu (6.6) T ∂t T which can be reduced to Zero if the trial solution coincides with the exact solution. By choosing the optimal constants,Tj ()t , the different residuals are Zero in some average sense.

98 6.3 Finite Element Modeling

This objective is reached by selecting N weighting functions Wj , j = 1,2…N, then introducing the spatial average or inner product

()A,B = ∫ ABdV (6.7) V and setting the weighted integrals of the equation residual to Zero.

(W j , R()uT ) =0, j =1,2⋅⋅⋅ N (6.8)

Equation (6.8) represents a set of equations for the Tj(t). If the constants Tj are effectively time-dependents, then the equations are ordinary differential equations. Otherwise, the N equations are algebraic. In the Galerkin’s approach, the weighted functions are the same shape functions which have been used to define the trial solution (6.5) [Amès, 1977].

6.2.2 Fundamental steps

In the Finite Element Method, six fundamental steps can be mentioned: Step 1: The domain of interest is discretized into simple geometric shapes or elements. Step 2: The individual element equations are developed by using the weighted residual method or the variational principles. Step 3: The generated element equations are assembled in a matrix form. Step 4: The boundary conditions are imposed on the entire system, and thus modify the global equations. Step 5: The modified global equations are then solved for the primary unknowns (e.g. nodal temperatures). Step 6: Other desired secondary variables are calculated by using nodal values of the primary variables (e.g., thermal gradients and thermal fluxes from nodal temperatures).

6.3 Finite Element Modeling

In order to improve the comprehension and the use of the Finite Element technique, a large variety of commercial software packages are available (e.g. ADINA, ANSYS, MARK, LARSTRAN, etc). The ANSYS software has been used in the frame of this work. Three main component products derived from ANSYS/MULTIPHYSICS (A/M) are worth to be mentioned: A/M/MECHANICAL (structural and thermal capabilities), A/M/Emag (electromagnetic capabilities), A/M/FLOTRAN (Computational Fluid Dynamic capabilities).

99 6.3 Finite Element Modeling

The used versions of ANSYS/MULTIPHYSICS, namely ANSYS 5.7 and ANSYS 6.1, differ with each other by the organization of their interface but are subjected to the same limitations concerning the maximum number of nodes (128000) that can be generated through the meshing. In general, all FEM software packages are built up around three standard phases.

6.3.1 Pre-processing

It is the most important step in the elaboration of the finite element calculation. This phase is aimed at preparing the considered model by defining the geometry, the thermal and physical properties, the element type, and finally by performing the meshing. Meshing the model consists of subdividing the considered geometry into elements connected at nodes: The physical system is said to be transformed into a finite element model. Then, the boundary conditions are applied to the model or directed imposed on the nodes and elements. A poor choice of the element types as well as a poor meshing of the model leads unavoidably to considerable inadequacies between the model and the actual physical system. The pre- processing phase involves the steps (1) and (4) defined above in section 6.2.2.

6.3.2 Computation

In this phase, the set of equations generated by the meshed model is solved. The results consist of a large set of numerical values representing the primary variable (e.g., nodal temperatures). To check the reliability of the numerical solution, a quantitative and qualitative assessment is highly recommended. The computation phase regroups the steps (2), (3) and (5) defined above.

6.3.3 Post-processing

In this phase, the primary variables calculated in the computation phase are collected and used for further processing. For this, ANSYS offers two types of post-processors: The general post-processor or POST1 allows to list the results of the simulation at a given time or frequency over the model while the time-history post-processor or POST26 allows to review the results over time in a transient analysis or over frequency in a harmonic analysis, at a particular location of the model. In thermal analysis, the post-processing consists of determining the thermal gradients, the thermal fluxes and of evaluating the heat losses or gains in the simulated model, from the calculated nodal temperatures. In particular for thermal wave problems, the attention is focused on the simulated amplitude and phase of the temperature oscillations, which in their

100 6.4 Efficient simulations of thermal wave problems turn are exploited to assess the amplitude and phase of the photothermal signals (e.g., IR radiometry, Thermoreflectance, Thermoelastic deformation). The post-processing phase refers to step (6) of section 6.2.2. However, one should keep in mind that using a software package to perform these numerical simulations does not automatically prevent the solver from obtaining suspicious results. This is why it is always of great interest to check the convergence of the numerical solution. For thermal wave problems especially, we make some propositions which can help to obtain more accurate numerical results.

6.4 Efficient simulations of thermal wave problems

6.4.1 Convergence of the numerical solution

6.4.1.1 Non-uniform meshing and mesh refinement

Most often, all results of the numerical simulation are not necessary. There are particular regions of the simulated model where information is needed and therefore a mesh refinement is required at these locations to generate more accurate results. For example, in a simple thermal wave problem where the frequency-dependent amplitude and the frequency- dependent phase of the surface temperature are of interest, the model must be non-uniformly meshed from the bottom to the top of the sample with more mesh refinement beneath the sample surface. Non-uniform meshing is useful for two reasons: (i)− It allows to avoid the computation of too much data which are useless. (ii)− It contributes to speed the simulation by minimizing the numbers of nodes involved in the differential equations. Concretely, starting from the fact that the diffusion length of the thermal wave, µ = α / πf , is proportional to the spatial coordinate, e.g. z, in the main direction of thermal wave propagation, and that it would be absurd to have a mesh size larger than the diffusion length, the mesh density must be relatively high at large penetration depths corresponding to low modulation frequencies, and very small at low penetration depths corresponding to high modulation frequencies. The mesh refinement is said optimal when there are no or little changes in the solution. In this case the stability of the numerical solution is established and the convergence criteria are fulfilled.

6.4.1.2 Control of convergence with the reference phase shift

In addition to the principle of mesh refinement, we propose another powerful tool to control the convergence of the numerical solution. This method is specific for thermal wave problems and based on the phase shift of the thermal wave at the surface of a homogeneous

101 6.4 Efficient simulations of thermal wave problems

material, according to the theory of 1-D thermal wave propagation in solids. The value of this phase shift (reference phase shift) is given by Φref = -45° or tanΦref = -1. Thus, to simulate the propagation of thermal waves in a layered structure, it is advised to affect the poorest thermal properties (all belonging to one of the layers!) to all other layers of the system in order to fulfill the requirement of a homogenous sample. In the considered interval of modulation frequencies, the numerical data recorded for the phase shift of the thermal wave at the sample surface must be equal or at least very close to the reference value Φref = -45°. If it is not the case, then the meshing is poor and a further mesh refinement is necessary until this value is reached. As can be seen in Figure 6.1a, which shows the frequency-dependent phase of a Si- based homogenous sample, simulated by the FEM, the reference value Φref = -45° is reached in the interval of modulation frequencies, 1.00 Hz < f < 100 kHz. This is the proof, that the meshing has been very well performed and that the convergence criteria are fulfilled since the numerical and the analytical solutions are in very good agreement. To obtain this result, a value of d = 6000 µm has been affected to the sample thickness. The meshed model is presented in Figure 6.1b, where a mesh refinement is observable near the top of the sample.

-15

-30

-45

[deg]

ref -60 Φ

-75

-90 0 50 100 150 200 250 300 350 ( f / Hz )1/2

Figure 6.1a: Phase shift thermal wave at the Figure 6.1b: Portion of the model showing a surface of a smooth and homogenous solid, non-uniform meshing from the bottom to the simulated by finite elements: top, with mesh refinement near the surface

According to Fig. 6.1b, the lateral mesh size is not of great importance since the thermal waves propagate exclusively in the vertical direction due application of an uniform thermal load on top of the sample (e.g. excitation of the entire surface by an intensity- 102 6.4 Efficient simulations of thermal wave problems modulated laser beam). On the contrary, the vertical mesh density is non-uniform, going from large values at large penetration depths (low modulation frequencies) to very small values at small penetration depths (high modulation frequencies). To obtain the above result, a ratio of Rt = 500 between the largest and the smallest mesh size has been considered. However such a result is not hazardous. The modeller should be aware of the fact that the value Φref = -45° is typical for a model regarded as semi-infinite. In Figure 6.2a, the effect of sample thickness on the phase shift is investigated. With decreasing value of the sample thickness, e.g. d = 500 µm, the phase shift deviates from Φref = -45° to Φref = –90° in the limit of very low modulation frequencies or at very large penetration depth while no change is recorded at the intermediate and high modulation frequencies.

0 0

-6 2 -15 d = 500 µm -15 α = 90 10 m /s 2000 44 0.6 -30 6000 -30

-45

[deg] [deg] ref ref -60 -60 Φ Φ

-75 -75

-90 -90 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 1/2 ( f / Hz )1/2 ( f / Hz )

Figure 6.2a: Phase shift of thermal wave at the Figure 6.2b: Phase shift of thermal wave at the surface of a smooth and homogenous solid, surface of a homogenous solid, simulated by the simulated by the FEM as a function of the FEM as a function of the modulation frequency modulation frequency for different values of the for different values of the thermal diffusivity. thickness. Thermal diffusivity α = 90×10-6 m2/s Sample thickness: d = 500 µm

In Figure 6.2b, the sample thickness is maintained at d = 500 µm and the thermal -6 2 -6 2 diffusivity (α = 44×10 m /s for GaAs material, and α = 0.6×10 m /s for SiO2 material) is varied systematically. One can remark, that with smaller values of the thermal diffusivity (e.g., α = 0.6×10-6 m2/s), the phase shift recovers from the deviation observed in Figure 6.2a for the thickness d = 500 µm and the thermal diffusivity α = 90×10-6 m2/s. These deviations of the phase shift from the value Φref = -45° as the sample thickness decreases or as the thermal diffusivity of the material varies can be better understood if one recalls the analytical expression of the complex modulated surface temperature for a homogenous sample,

103 6.4 Efficient simulations of thermal wave problems

η I exp(−iπ / 4) 1+ exp[−2(1+ i)d πf /α ] δT( f ,t) = o (6.9) 2e 2πf 1− exp[−2(1+ i)d πf /α ]

In equ.(6.9), ao = d πf /α represents the thermal thickness, which is a measure of the ability of the thermal wave to propagate throughout the entire sample. Table 6.1 indicates the values calculated for the thermal thickness, at the modulation frequency of f = 1.9 Hz and for the thermal diffusivity of α = 90 10-6 m2/s (Si-material). As can be seen in this table, the sample is thermally thick for a thickness of d = 6000 µm. That means, the thermal diffusion length, calculated according to µ = α /πf and which has the value 3900 µm, is smaller than the sample thickness. In this case, the thermal wave can be completely damped before reaching the rear face of the sample and therefore the assumption of semi-infinite sample is justified [Bein et al., 1989]. For a smaller sample thickness, e.g. d = 500 µm, the obtained value for the thermal thickness indicates a very thermally thin sample since the thermal diffusion length is rather larger than the sample thickness. In this case, the thermal wave can easily reach the rear face of the sample before been completely damped. Once the convergence criteria are fulfilled, the actual thermal and physical properties of each layer of the modelled structure are reinserted into the program to perform the desired numerical simulations.

Sample thickness: d /µm Thermal thickness: ao Remarks 6000 1.545 Thermally thick 2000 0.515 Thermally thin 500 0.129 Very thermally thin

Table 6.1: Evaluation of the thermal thickness at a given frequency for different values of the sample thickness.

6.4.2 Methodology of simulation

6.4.2.1 Thermal filtering and physical units

The first thing to do before starting to build up a program is to filter the thermal entries from other entries (e.g., structural, electromagnetic and fluid menu topics) proposed by the software. This task is achieved by turning on the specified dialog box of the Graphical User Interface (GUI). Such a filtering enables for example the discrimination of thermal problems from structural problems. On the other hand, it should be noted, that ANSYS does not specify the units for any variable and so it is at the charge of the modeller to ensure that the units of the variable are consistent. A good tip consists of first defining the units of the primary variables so that the units of other variables are automatically identified in the considered system of units. For

104 6.4 Efficient simulations of thermal wave problems example, by defining the units of the thermal conductivity k, the mass density ρ, and the specific heat capacity c, the units of the thermal diffusivity, α (α = k/ρ.c), and the units of the thermal effusivity e ( e= ρ ck ), are automatically defined. If for example the International System of Units (SI) is preferred, then one should usually write the following command at the beginning of the program: /Units, SI.

6.4.2.2 Parameters, material properties and element types

A good organization of the numerical work consists of defining a group of parameters in order to avoid writing the same physical quantities several times along the program. Although parameters can be defined anywhere, the readability of the program is more facilitated by entering all necessary parameters just before the prep-processing phase

(PREP7). Parameters are entered in the following way: *SET, Io, 5. This means, the value 5 is affected to the parameter Io, which will be later on recalled at one or many locations in the program as intensity of the heat source. The material properties are also entered in the program by using appropriate codes. For example, mp, kxx, 1, k1, means that the material property (mp) is the thermal conductivity (kxx) of the material numbered 1, whose value is given by k1 supposed to have been defined earlier as parameter. Selecting the appropriate element type secures the solver from obtaining poor results. Each element type is identified by a category name, e.g. 2D solid elements have the category name PLANE. PLANE 42 is a four-node quadrilateral element with each node having two degrees of freedom [Moaveni, 1999].

6.4.2.3 Geometry definition, meshing

The geometry of the model and that of the real physical domain can be similar. In such a case, there are no observable centres or axes of symmetry in the physical domain and so all geometrical parameters of the domain are transferred to the model. If the physical domain presents some points or axes of symmetry, then the model will be the capture of a part of the domain, generally the half part. Then the meshed model will be simulated and the obtained results will be interpreted while taking into consideration the entire physical domain. Examples are shown in Figure 6.5 and in Figure 6.8. The existence or not of symmetries depends also on the way the load is applied to the physical domain. In order to mesh the model, the ANSYS software offers two possibilities to the solver: (i) Free meshing: Yields a random distribution of elements in the model. In this case, elements with different morphologies can cohabit, e.g., when meshing the model with the element type PLANE 75 both triangular (three-node) and tetrahedral (four-node) elements are generated.

105 6.5 Finite Element control of theoretical results

(ii) Mapped meshing: Yields a regular distribution of elements in the model. In this case, the morphology of the elements is uniform, e.g., when meshing the model with the element type PLANE 75, either triangular or exclusively tetrahedral elements are generated. We have essentially used the mapped meshing since it allowed us to easily control the mesh size and to correctly define the inputs of the heat generation rate as well as the vertical thermal expansion.

6.4.2.4 Solution and interpretation

The solution phase begins with the description of the analysis method. Modulated thermal wave problems must be solved by specifying the harmonic analysis. This is done by issuing the following command: ANTYPE, HARMIC (analysis type, harmonic). This instruction is directly followed by the command, HROPT, method,…, which specifies the option. The full method is generally chosen as option. Then, another command, HROUT, key,…, is written which indicates the output printing option. Here, two possibilities are offered: If the complex modulated temperatures are to be printed as real and imaginary components, then the key ON is chosen. If the output results must be printed as amplitude and phase angle, then the key OFF is suited. The desired harmonic frequency range is established by issuing the command, harfrq, fmin, fmax and the number of sub-steps between the minimal and the maximal frequencies is defined by writing, nsubst, nmax. In the case of an unique harmonic frequency, e.g. for the simulation of the amplitude and phase of the thermo-elastic signals, fmin = fmax and nmax = 1. After having calculated the primary variables in the solution phase, namely the real and the imaginary components or the amplitudes and phases of the nodal temperatures (temperatures oscillations), the close task is the calculation and collection of the secondary variables (in this work: amplitudes and phases of the thermo-elastic displacements).

6.5 Finite Element control of theoretical results

In order to check the reliability of the finite element scheme which consists of using the reference phase shift to control the convergence of the numerical solution, we compare the results of FE simulations with some theoretical results, which have been found to be in good agreement with the experimental measurements (cf. Chap. 3 and Chap. 5), and which are based on the 3-D thermal wave propagation in opaque two-layer systems where concentric heating and detection spot are first considered followed by a displacement distance between the two spots.

106 6.5 Finite Element control of theoretical results

6.5.1 3-D thermal wave propagation

Figure 6.3 shows the frequency-dependent normalized amplitude (a) and phase (b) of the thermal wave at the top surface of a two two-layer system. One can observe in Figure 6.3a, that the inverse normalized signal amplitude increases (the signal decreases!) in the range of low modulation frequencies with increasing values of the thermal diffusivity of the subsurface material. In Figure 6.3b, the inverse normalized phase experiences a relative maximum in the limit of very low modulation frequencies, which varnishes for relative small values of the thermal diffusivity and adopts the general behaviour observed in the case of 1-D thermal propagation (compare, Fig. 3.13a, Chap. 3). Such a result has already been found and applied to experimental measurements (compare, Fig. 3.15, Chap. 3). One can observe, that the analytical and the numerical results are in good agreement. To obtain such an agreement between analytical and numerical solution, the model was meshed with the element type PLANE 75 (4-node quadrilateral element, axisymmetry, harmonic analysis). In the direction of high thermal gradients, along the depth, a very small mesh density was used in the top layer

25 5 solid line: Theory symbols: FE 0 solid line: Theory symbols: FE 20 -5 -6 2 -1 -6 2 -1 α = 6 10 m s α = 66 10 m s b b -10 22 22 15 6 66 -15 -1 n

/ deg -20 10 n Φ -25

5 -30

-35

0 0.01 0.1 1 -40 0 25 50 75 100 125 150 175 ( f / Hz)-1/2 ( f / Hz )1/2

Figure 6.3 a + b: (from left to right) Normalized amplitude (a) and phase (b) of the thermal wave at the top surface of a two two-layer system, according to 3-D thermal wave propagation, for different values of the thermal diffusivity of the subsurface material. There is a very good agreement between the results of FE simulations (symbols) and the analytical predictions (solid lines).

(coating) and the mesh size was progressively enlarged from the top to the bottom of the substrate. First, by considering a laser spot very large at the top surface of of the sample to

107 6.5 Finite Element control of theoretical results

ensure 1D heat propagation and by affecting the same thermal properties to the two layers of the model, values of the phase shift very close to Φref = –45° were obtained in the entire frequency range, attesting therefore that the mesh refinement is optimal and that the convergence of the numerical solution is established. Thus, changes in the numerical solution were no more observed when the mesh size (along the depth) in the surface layer reached the value 2×10-7 and the ratio of the largest to the smallest mesh size in the substrate was adjusted to 500. Then considering the realistic thermal load (heating of the top surface of the sample by a Gaussian-shaped laser beam of radius rH, time modulated at the frequency f ), the real and imaginary components of the nodal temperatures along the sample surface (see the meshing in Figure 6.5b) were collected as function of the modulation frequency, and transferred into the graphical commercial program ORIGIN 6.0 for further processing. The real and imaginary components of the modulated IR signals were then obtained with the help of a numerical integration [Davis and Rabinowitz, 1967]. For this, the expression [comp. equ. (3.45), chap.3],

rD rD δM ( f ) = Const ⋅ rdr δTs (r, f ) =Const ⋅ dr δU s (r, f ) (6.10) ∫0 ∫0

was decomposed by using e.g. the trapezoid rule as follow:

r =r Num N D δM ( f ) =Const ⋅ drδU s (r, f ) ∫r =0 1

⎡δU s (r1, f ) δU s (rN , f )⎤ = Const ⋅ hr ⋅ ⎢ + δU s (r2 , f ) + ⋅⋅⋅⋅⋅ + δU s (rn−1, f ) + ⎥ (6.11) ⎣ 2 2 ⎦

In equ.(6.11), hr = (rN –r1)/(N-1) is the mesh size along the radial (lateral) axis at the sample surface, this means the separation between two consecutive nodal temperatures. N represents the number of nodes and δU s (rk , f )= rk δTs (rk , f ) , k =1⋅⋅⋅ N . The rk are the variable radial positions inside the detection area. Then the signal amplitude and the signal phase were obtained by operating on the real and the imaginary components as follows:

Num Num 2 Num 2 S ( f ) = [][]δM Real ( f ) + δM Im ( f ) (6.12a)

Num Num ⎧δM Im ( f )⎫ 180 Φ ( f )=atan⎨ Num ⎬⋅ (in degrees) (6.12b) ⎩δM Re al ( f )⎭ π

In equ.(6.12), the lower subscripts Real and Im refer to the real and the imaginary components, respectively while the upper subscript means: numerical values. To eliminate the constant coefficient carried by the modulated signal as shown e.g. in equ.(6.10), the same

108 6.5 Finite Element control of theoretical results operations were performed to obtain the amplitude and phase of the reference material and the obtained numerical results were normalized with the results of the two-layer sample.

5 5

Theory 0 Theory 0 Symbols: FE Symbols: FE 1/2 -2 -1 e =7000 Ws1/2m-2K-1 e =7000 Ws m K -5 b -5 b 13100 13100 26000 26000 -10 -10

-15 -15

n / deg -20 n / deg n / -20 ϕ ϕ

-25 -25

-30 -30

-35 -35 -40 -40 0 50 100 150 0 50 100 150 200 250 1/2 1/2 (f / Hz) (f / Hz)

Figure 6.4 a + b: (from left to right) Frequency-dependent normalized phase of the thermal wave at the surface of a two two-layer system, according to 3-D thermal wave propagation, for different values of the thermal effusivity of the subsurface material. Comparison between the results of FE simulations (symbols) and the theoretical results (solid lines) indicates a good agreement.

Fig. 6.5 a: 3D view of the physical two-layer Fig. 6.5 b: Finite Element model showing the system showing a thin film deposited on a semi- repartition of the mesh density in the surface infinite substrate. layer and in the substrate

109 6.5 Finite Element control of theoretical results

6.5.2 Displacement between the heating and the detection spots

To obtain the numerical solutions for the modulated IR signals taking into account controlled displacement distance between the excitation and the detection spot, the following expression [compare equ.(5.7), Chap.5]

dHD +rD δM ( f ) =⋅Const ⋅ rdr F(r,r ,d )δT (r,r , z = 0) ∫ D HD s H s (6.13) dHD −rD was decomposed according to the trapezoid rule as follows:

⎡δVs (r1 , f ) δVs (rN , f )⎤ δM ( f ) =⋅Const ⋅hr ⋅⎢ + δVs (r2 , f ) + ⋅⋅⋅⋅⋅ + δVs (rn−1 , f ) + ⎥ (6.14) ⎣ 2 2 ⎦

Here, δV(rk , f) = rk F(rk ,rD ,d HD )δTs (rk ,rH ) . The rk are the local positions of the nodal temperatures in the detection area of radius rD, with respect to the centre of the heating spot of radius rH, and are limited by r1 = dHD−rD and rN = dHD+rD. F(rk , rD , d HD ) is a function depending on the detection spot radius and on the distance dHD between excitation and detection spot [comp. equ. (5.7b), chap.5].

Symbols: FE 40 Solid line: Theory

20 d HD

0 / deg n

Φ -20

-40

1 10 100 400 (f /Hz)1/2

Figure 6.6: Evolution of the normalized phases with controlled displacement distances (from bottom-to-top by dHD = 0 µm, 3000 µm 3200 µm, and 3400 µm) between the heating and the detection spot of radii rH = 2000 µm and rD = 800 µm, respectively. Comparison between the results of Finite Element simulations (symbols) and the theoretical results (solid lines).

110 6.6 Simulation of thermo-elastic signals

6.6 Simulation of thermo-elastic signals

6.6.1 Theoretical basis

In general, the excitation and propagation of thermal waves in solids can be accompanied by the generation of thermo-elastic stresses and strains [Bein and Pelzl, 1989]. The vector of elastic displacement, ur(r,t), and the temperature distribution, T ()r,t , are interrelated by the Navier-Stokes equation of Continuum Mechanics [Nowacki, 1986],

∂ 2 ur(r,t) ()()λ + µ ∇()∇ ⋅ur r,t + µ ∇ 2 ur ()()()r,t − β 3λ + 2µ ∇T r,t = ρ (6.15) L L L e L L ∂t 2

In equ.(6.15), βe is the coefficient of linear thermal expansion (CLTE) and ρ is the mass density. λL and µL are the so-called Lamé’ constants given respectively by

ν E E λ = µ = (6.16) L ()()1+ν 1− 2ν L 2()1+ν

The Lamé constants are described by the Poisson number,ν, and the Young’s modulus, E. The Poisson number is defined as the absolute value of the ratio of lateral strain to longitudinal strain. In quasi-stationary approximation, the inertia term at the right hand side of equ.(6.15) can be neglected [Rousset et al, 1983] and by taking into account the above expressions of the Lamé constants, one obtains

2 r r r r r (1−2ν )∇ u ()r,t + ∇()()∇⋅u ()r,t = 2 1+ν βe ∇T(r,t) (6.17)

Equ.(6.17) shows that the thermo-elastic response of a heated sample can be calculated by solving simultaneously the heat diffusion equation and the equation of elastic displacement. However, due to the complexity of this equation, further simplifications are made (e.g., by neglecting the Poisson ratio, that means by neglecting the lateral strain with respect the longitudinal strain) and a simple relation between the thermal oscillations (thermal waves) and the thermo-elastic displacement of the sample surface is established [Bolte, 1999]. The thermal expansion is therefore reduced to dilation of the sample in the vertical direction:

∞ δu x, y,t = β δT x, y, z,t d z (6.18) z ()e ∫ ( ) 0

In the case of a layered structure, the total thermal expansion results as the sum of thermal i expansions of the individual layers; each layer i of thickness di = li+1-li having the CLTE βe .

111 6.6 Simulation of thermo-elastic signals

n l i i+1 δuz ()x, y,t = βe δTi (x, y, zi ,t )d zi (6.19) ∑ ∫l i=1 i

For complex structures, the simulation of thermo-elastic displacement is more difficult. Consequently, the following methodology is presented to indicate how the vertical thermal expansion can be calculated from the numerical thermal oscillations in the sample.

6.6.2 Methodology

In order to perform the simulation of thermo-elastic displacements, the real and imaginary components of the nodal temperatures are first calculated. According to equ.(6.19), the thermal expansions of the individual layers of the structure are derived through a numerical integration and then added to obtain the resulting thermal expansion. We have found, that the ANSYS software offers a special command which allows to calculate such an integral, providing that this integral is simple and that the modeller clearly identifies the parameters involved in the integration, otherwise the obtained results would be hazardous. This command, available in the general post-processor POST1, is defined as follows: PCALC, Oper, LabR, Lab1, Lab2, FACT1, FACT2, CONST. In details, Oper indicates the type of operation. Examples of operation include: addition, multiplication, division, integration, derivation, etc. LabR is a label assigned to the resulting path item. Lab1 and Lab2 are respectively the first and the second labelled path items in the operation. FACT1 and FACT2 are the factors applied to Lab1 and Lab2, respectively. CONST stays for the constant value. Concretely the numerical integration works in the following way: Set,,,,0 pdef, R, temp pcalc, intg, intR, R, s, beta_the set,,,,1 pdef, I, temp pcalc, intg, intI, I, s, beta_the The path of the real (resp. imaginary) component of the nodal temperatures is created with the command Pdef and labeled R (resp. I). The operation, pcalc, intg,…, finds the integral (operator intg) and the resulting path is labeled IntR (resp. IntI). IntR and IntI are therefore the paths of the real and imaginary components of the integrals, respectively. The integration is performed along the path s (Lab2) whose distance is automatically calculated by the program during creation of the path item, pdef, R (resp. I), temp (the degree of freedom temp refers to temperature). Beta_the (FACT1) stays for the coefficient of linear thermal expansion (CLTE) and is the factor associated to the path R (resp. I) of the real (resp. imaginary) nodal temperatures. The collected data for the real and the imaginary components of the integrals

112 6.6 Simulation of thermo-elastic signals are then transferred into ORIGIN to assess the amplitude and phase of the vertical thermo- elastic displacements as a function of the lateral position along the sample surface.

6.6.3 Example of application

Figure 6.7 shows the amplitude (a) and the phase (b) profiles of the vertical thermal expansion, calculated by FE simulations. The considered structure consisted of three superposed materials (from top to bottom Au-GaP-Si) of thicknesses, d1 = 8 µm, d2 = 6 µm, and d3 = 400 µm, respectively. To simplify the calculations, a common value of the coefficient of linear thermal expansion was used for the three materials constituting the model. A very small area (about 1µm², see Figure 6.8) on the front side of the top layer (Au) was considered to be subjected to an ac-Joule heating, thus yielding a modulated line heat source through that portion of the layer.

1.1 1.0 0

-20 0.8 -40 0.6

-60 / deg 0.4 n S [a.u] Φ -80

0.2 -100

-120 0.0 0 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160 Position [µm] Position [µm]

(a) (b) Figure 6.7 a + b: Amplitude (a) and phase (b) profiles of the vertical thermal expansion in a micro- sample, simulated by FE simulations. The amplitude and the phase are normalized to the value obtained at the center of the hot spot

Heat source x

(a) (b) Fig. 6.8 a + b: Front view of the model with a point heat source induced by Joule heating (a). Due to symmetry, only the half part (b) of the model has been considered to perform the FE simulations 113 6.7 Conclusions

The intensity of the heat source was assumed constant along each cross-section of the hot line, so that the heat propagation inside the sample was described as a 2-D problem with a point heat source as thermal load. Due to symmetry, only the half part of the 2-D model was meshed with the element type PLANE 55 (four-node element). From the meshing, 6380 elements for 6549 nodes were generated, which ensured the convergence of the numerical solution. To get access to the amplitudes and phases of the thermo-elastic displacements, a numerical integration involving the real and the imaginary components of the nodal temperatures (thermal oscillations) in the different layers was performed according to the methodology described in section 6.6.2. Of course, the nodal temperatures involved in this integration were captured in various longitudinal paths defined by a position on top of the model (direction x) and the same position at different depths, d1, d1+d2, and d1+d2+d3 (direction z). For a better understanding, the scheme of this numerical integration can be described as follows:

Integral [x, 0 → d1 + d2 + d3] =

= βe1 × Integral [x, 0 → d1] +

+ βe2 × Integral [x, d1 → d1+ d2] +

+ βe3 × Integral [x, d1 + d2 → d1 + d2 + d3]

This scheme means: the integral calculated over the depth of the entire model at each position x on top of the sample, results as sum of the integrals calculated at the same position, over the depth of the first, the second and the third components, respectively. Each component of the model having a coefficient of linear thermal expansion, βei . As the example shown in Figure

6.7 assumes the same value of βe for the three layers, the integration was directly calculated according to the left hand side of this scheme.

6.7 Conclusions

It has been seen in this chapter, that the convergence of the numerical solution can be generally ensured by keeping the mesh refined until there are no or little changes in the solution. Especially for thermal wave problems, we have shown, that the value Φref = -45° which represents the phase shift of the thermal wave at the top surface of a homogenous sample, is a powerful tool for controlling the convergence of the numerical solution. On the other hand, the very good agreement between the results of finite element simulations and some theoretical results, namely based on 3-D thermal wave propagation, has confirmed the reliability of our approach based on the control of convergence by using the reference phase shift. Using this approach we have also shown, how the amplitudes and phases of the vertical thermo-elastic displacements of the surface of structured devices can be efficiently calculated

114 6.7 Conclusions with the help of the ANSYS-aided finite element simulations. This methodology is adopted in chapter 7 for the simulation of heating processes in various models of microstructured devices.

115

Finite Element Investigation of Heating Processes in Micro-structured Semiconductor Devices

7.1 Motivation

The continuous miniaturization of semiconductor devices brings with it several problems due to size reduction. Such low dimensional devices operate very fast but dissipate more heat which can lead to overheating effects. While operating, these thin electronic components are subjected to temperature fluctuations which can unfortunately lead to high thermal stresses and strains in the device and produce failure mechanisms such as cracking or de-lamination due to the differences in the coefficient of thermal expansion between the constituting materials. Therefore, several fundamental questions are matter of concern: e.g., which locations of the high power device experience a high temperature (hot spot)? How much the device will then expand? How to evacuate the dissipated heat from the device to its environment while keeping a safe operating temperature? This is why design and thermal management strategies are required to ensure the reliability of these electronic components which are limited by thermal death. However, due to time and cost constraints related to the production of layouts and to industrial tests, and due to the limited capabilities of the experimental methods for measuring thermal transport as the dimensions of structures are getting drastically smaller, numerical simulations are more required to predict the thermal behaviour of such tiny electronic packages. In this chapter, numerical simulations based on the finite element method (FEM) are performed to investigate heating processes in several models of micro-structured semiconductor devices. The ANSYS- aided FE calculations consist essentially of determining the thermal oscillations in the devices and the related thermo-elastic expansions of their surface and of evaluating the influence of tip thermal expansion due to the heat transferred from the hot area to the tip. In some cases, the results of simulations are compared with experimental measurements.

7.2 Features of two experimental methods in micro-thermal analysis

7.2.1 Scanning Thermal Microscopy (SThM)

The field of micro-thermal analysis has benefited among others from design of the scanning thermal microscope [Price et al., 1999 and 2000; Pelzl et al., 2001]. In this

116 7.2 Features of two experimental methods in micro-thermal analysis experimental approach, a temperature sensitive tip is brought in the close vicinity to a solid sample and then scanned over the sample surface. Thermal mapping can be done in two ways: In the passive mode, the sample is heated and the measured heat flow from sample to tip provides information about the temperature oscillations across the sample. In the active mode, the tip is heated and the measured heat flow from tip to sample provides information about the local thermal properties of the sample. Here, the tip is used simultaneously as excitation source and as detector. The tip-sample heat transfer mechanism includes solid-solid conduction through the contact, liquid conduction through a liquid film bridging the tip and the sample, and air conduction [Shi et al., 2000]. Some studies have been performed to determine the predominant process of thermal transfer between the tip and the sample by considering a DC approach where no modulated temperature is used [Gomès et al., 1999]. But as mentioned by Pollock and co-worker [Pollock and Hammiche, 2001], modulation techniques have clear advantages over DC methods for quantitative interpretations. However, the main experimental difficulty is the reproducibility of the tip since the topography of the sample surface can suffer from some irregularities [Bolte et al., 1998]. In this case, the tip-sample heat transfer mechanism cannot be monitored with accuracy and precision. Two propositions of solution are recorded: a) Modeling of the effect of topography through a knowledge of tip-sample heat transfer. b) Null-point measurement that helps to minimize the temperature difference between the tip and the sample by eliminating any heat flow during thermal imaging. But more sophisticated sensors are required to achieve such an objective [Cahill et al., 2003].

7.2.2 Scanning Thermal Expansion Microscopy (SThEM)

To overcome these disadvantages in connection with the surface topography, Varesi and co-worker [Varesi and Majumdar, 1998] have proposed an alternative experimental approach named Scanning Joule Expansion Microscopy (SJEM), for which the application of a Joule heating is done by means of a modulated electric current passing through a conducting sample. The resulting thermal expansion is detected by using an atomic force microscope probe. This technique has been applied for the detection of hot spots in an IPG-transistor [Bolte et al., 1998]. A similar approach, the Scanning Thermal Expansion Microscopy (SThEM) also monitors the vertical thermal expansion of the sample and thus the measured signal is proportional to temperature oscillations over the sample depth. In comparison with the SThM signal, the SThEM signal is independent of the thermal contact resistance at the interface tip-sample. However, in addition to the vertical thermal expansion of the sample, there can exist a supplementary signal contribution due to the electrostatic interaction tip- sample, to the piezo-electric deformation of the sample and to the expansion of the tip [Bolte, 1999]. In a relatively recent report [Pelzl et al., 2001], it has been shown, that the electrostatic interaction can contribute to the signal only in the case of non-conducting tips. 117 7.3 FE Simulation of thermal and thermo-elastic signals

In a general way, the drastic size reduction of devices imposes some limitations to experimental measurements and this is why one refers more and more to numerical methods such as for example the Finite Element Method (FEM) in order to predict the thermal behaviour of these structures and verify the reliability of the eventual experimental results. In the following, we use the ANSYS-aided finite element technique to investigate the thermal oscillations in some models of devices as well as the thermo-elastic displacements of the sample surface following their excitation by a modulated excitation source.

7.3 FE Simulation of thermal and thermo-elastic signals

7.3.1 AC heating and DC heating

The generic terms AC and DC refer respectively to Alternating Current and Direct Current. By definition, current does not change direction in the DC mode while it regularly changes direction in the AC mode. Thus, by applying a constant voltage to a conducting sample of resistance R, the power dissipation which results as consequence of the Joule heating is also constant and given by:

V 2 P = 0 (7.1) R

If a sinusoidal voltage V(t) is rather applied to the sample, then the power dissipation takes the following form:

2 2 2 V (t) []V0 ⋅sin(2⋅π ⋅ f ⋅t) V0 1 ⎡ ⎛ π ⎞⎤ P()t = = = ⋅ ⋅ ⎢1− sin⎜2⋅2⋅π ⋅ f ⋅t − ⎟⎥ (7.2) R R R 2 ⎣ ⎝ 2 ⎠⎦

Equ.(7.2) shows that the thermo-elastic signals oscillate with the double modulation frequency of the applied sinusoidal voltage. That means, the power dissipation which is time modulated with the frequency f induces ac temperatures and thermal expansion in the heated sample at the frequency 2f. The choice of the optimal modulation frequency for generating the thermo-elastic signals is not hazardous. As mentioned by [Bolte et al, 1998] and [Pelzl et al., 2001], the modulation frequency has to be chosen in the range 5 kHz to 15 kHz well above the resonance frequency of the piezo-elements.

118 7.3 FE Simulation of thermal and thermo-elastic signals

7.3.2 The ac-electrically heated conducting line

y x

z1 a

Figure 7.1a: Simplified schematic cross-section of the composed structure consisting of a Si-based substrate, a SiO2-based surface layer and of a Si-based thin conductor.

Si (line source)

y x d

z1 a SiO2

Figure 7.1b: Half part of the structure due to symmetry with respect to the y-direction.

In order to illustrate the phenomenon of local heat dissipation in a device which leads to thermal oscillations and thus to thermo-elastic displacements of the surface, we consider a composed structure consisting of a very thin conducting Si-layer, separated from the Si-

119 7.3 FE Simulation of thermal and thermo-elastic signals

substrate material by an insulating SiO2-layer. The conducting line is subjected to a modulated Joule heating through application of an ac-voltage at its ends (with respect to the y direction as mentioned in Fig. 7.1a). The resulting power dissipation induces production of localized heat in the device and the immediate consequence is the appearance of thermal disturbances.

7.3.2.1 Physical system and model

As can be seen in Fig. 7.1a, the Si-hot line (of thickness dz1 = 150 nm and of width 2a

= 20 µm) crosses the surface of the poor conducting SiO2-layer (of thickness dz2 = 300 nm) at its middle. This latest layer is deposited on a conducting Si-substrate of thickness db = 500 µm which can be regarded as semi-infinite since the used excitation frequency (from 5 to 15 kHz) is high enough to induce a thermal diffusion length very small in comparison with the sample thickness. The fabrication process of such a structure is presented in [Bolte, 1999]. For reasons of symmetry, only a half part of the device is useful for calculations (Fig. 7.1b). Furthermore, the amplitude of the power density is constant at any cross-section of the hot line (along the y direction), thus allowing the heat propagation to be described in the frame of a 2-D problem as can be seen in Figure 7.1c. Since the thin Si-conducting layer at the very surface behaves like a time−dependent surface heat source, it cannot be considered as separate layer (see Fig. 7.1b).

10µm 150nm x

a 300nm z

500µm

Figure 7.1c: Model considered for FE simulations

7.3.2.2 Thermal oscillations and thermal expansions

Figure 7.2a and Figure 7.2b show respectively the amplitudes and the phases of the surface temperature and of the thermal expansion, represented as a function of the lateral dis-

120 7.3 FE Simulation of thermal and thermo-elastic signals

1.0 Th. expansion f = 5 kHz surf. temperature 0.8

0.6

0.4

Normalized AmplitudeNormalized /a.u 0.2

0.0 -160 -120 -80 -40 0 40 80 120 160 Position / µm

Figure 7.2a: Normalized amplitude of the thermal expansion and of the surface temperature.

-160 -120 -80 -40 0 40 80 120 160

0 f = 5 kHz -25

-50

-75

-100 Thermal expansion

Normalized Phase [°] Surface temperature -125

-150 -160 -120 -80 -40 0 40 80 120 160 Position / µm

Figure 7.2b: Normalized phase of the thermal expansion and of the surface temperature

121 7.3 FE Simulation of thermal and thermo-elastic signals

tance from the center of the hot line. Both the amplitudes and the phases are normalized to the values at the center of the hot spot. As one can see, the amplitude and the phase are maximal at the center of the hot line but the surface temperature falls off very rapidly as one moves away from the heat source and the shape of the thermal expansion is much broader than that of the thermal signal. Two explanations are valid: (i) −The thermal signal (reduced to the surface temperature) is considerably affected by the low conductivity of the SiO2-layer which behaves as thermal barrier and thus at the level of the hot spot, the thermal wave propagates more vertically than laterally to reach the Si-substrate which is more conductive. (ii) −As for the thermal expansion, the situation is rather different. In addition to the surface layer where information about the surface temperature is obtained, the thermal expansion integrates the contribution of the subsurface layer which is reached by the thermal wave. Here, the coefficient of thermal expansion of each participating layer plays a determinant role. In the above example, the used coefficients of linear thermal expansion -6 -1 -6 -1 (CLTE) are βe(SiO2) = 0.59 × 10 K and βe(Si) = 2.60 × 10 K . In regard with its thermal expansion coefficient, the contribution of the SiO2 −layer to the thermo-elastic expansion can be considered as negligible.

7.3.2.3 Frequency−dependent thermal expansion

In Figure 7.3a and in Figure 7.3b, the amplitude and phase of the vertical thermal expansion are represented as a function of the lateral position with respect to the hot spot, for different values of the heating modulation frequency. One can remark, that the thermal expansion of the hot area is nearly insensitive to the variation of the modulation frequency and that the discrimination between the three calculated thermal expansions becomes more visible as one moves away from the hot spot. This can be explained by the fact that the temperature decays rapidly due to strong damping of the thermal wave as the distance from the heat source increases and as the modulation frequency increases.

122

1.0 f = 5 kHz 10 0.8 15

0.6

0.4

Normalized Amplitude /a.u 0.2

0.0 -160 -120 -80 -40 0 40 80 120 160 Position / µm

Figure 7.3a: Frequency-dependent normalized signal amplitude of the thermal expansion

-25

-50

-75

-100

-125

-150 f = 5 kHz

-175 10 15 Normalized Phase [°] Phase Normalized -200

-225

-250 -160 -120 -80 -40 0 40 80 120 160 Position / µm

Figure 7.3b:Frequency-dependent normalized signal phase of the thermal expansion

123 7.4 Investigation of thermal oscillations and thermoelastic displacements in HEMT-structures

7.4 Investigation of thermal oscillations and thermoelastic displacements in HEMT-structures

7.4.1 Power dissipation in transistors

7.4.1.1 Basic structures of transistors

Basically, the transistor is a three terminal semiconductor amplifying device that was invented in 1947 by Drs. John Bardeen, Walter Brattain, and William Shockley at Bell laboratories in Murray Hill, New Jersey, USA. The discovery occurred as the three physicists were looking for a better amplifier to replace the vacuum tube. Two types of transistors are worth to be mentioned: The Bipolar Transistor (BT) and the Field Effect Transistor (FET). The FET presents a region of donor material with two terminals called the source (S) and the drain (D). The region of acceptor material between these two regions is called the gate (G). The source, the drain and the gate represent respectively the emitter (E), the collector (C), and the base (B) of a bipolar transistor. The FETs are subdivided into two groups comprising the Junction Field Effect Transistors (JFETs) and the Metal Oxide-Semiconductor FETs (MOSFETs). FETs are suited to make up circuits with very low power consumption since no current flows through the gate. High Electron Mobility transistors (HEMTs) are FETs devoted to build up low noise amplifiers at a high frequency, e.g. 12GHz: This is why they are widely used for example in terrestrial and space telecommunications systems, and radio telescopes in the area of astronomy.

7.4.1.2 Origin and location of the power dissipation

In a FET, if the gate is connected to the source or if the gate voltage is negative with respect to the source, then the resistance between drain and source is extremely high so that the channel disappears. By applying a positive voltage to the gate (see red arrow on Figure 7.4), electrons are attracted to the interface between the gate insulator (e.g., GaAs) and the substrate (e.g. Si) and thus form a conductive channel which connects the source and the drain (see green arrow on Figure 7.4). The transistor turns on and current flows between source and drain (generally, source and drain are provided with electrical contacts connected to an AC power supply). The immediate consequence of this current flow between source and drain is the production of a power dissipation which is localized near the gate, between gate and drain. This power dissipation results in turn to the production of heat which induces a temperature rise in the device. The location of the power dissipation has been confirmed by measurements of the surface temperature which indicate a maximal value near the gate, at the drain side [Majumdar et al., 1995].

124 7.4 Investigation of thermal oscillations and thermoelastic displacements in HEMT-structures

SOURCE (−) GATE (+) DRAIN (+)

Location of the power dissipation

Gate insulator

Substrate

−

Figure 7.4: Simplified scheme of the front view of a FET showing the mechanism of current flow between source and drain. The current flow induces production of a heat source localized near the gate, at the drain side.

Thus, the application of a sinusoidal voltage (amplitude Vo, modulation frequency f) to the gate yields a modulated power dissipation (Joule heating), which in turn leads to the production of thermal oscillations (modulated temperatures) in the device and of thermo- elastic displacements of the surface of the device. Since the hostile size of the micro- transistors affects the capability of experimental measurements, the results of FE simulations help to get insight on the thermal phenomena taking place in such very thin devices. In section 7.4.2, we adopt a step by step approach to calculate the thermal expansion of a simplified HEMT-structure with main emphasis on the impact of the drain, the source and the gate. For this, a modulated power dissipation will be generally imposed on a small point near the gate.

7.4.2 Influence of the system Drain-Gate-Source on the signals

Figure 7.5 shows the simplified topology of a double gate transistor where the source, the gate and the drain are constituted of a gold material separated from the substrate material by a semi-insulating layer (e.g., GaAs, GaN). On a very small point near the gate, between gate and drain, a modulated Joule heating is applied which produces a hot line along the gate. By considering a constant amplitude of the heat source, the heat propagation can be described by a 2D problem (Fig. 7.6a). Following the modulated excitation, the system will be then subjected to temperature fluctuations and the sample surface will experience thermo-elastic

125 7.4 Investigation of thermal oscillations and thermoelastic displacements in HEMT-structures

displacements. In order to calculate the thermal expansion of the surface as a function of position for a heating modulation frequency of f = 5 kHz, we use a step by step approach to investigate the three following situations: (i) −Only the hot line (point source in 2D, comp. Fig. 7.6a) is first considered (Fig. 7.6b). (ii) −Then the drain (D) and the source (S) are added (Fig. 7.7). (iii) −Drain, gate (G) and source are all considered (Fig. 7.8).

S G D G S

heat source

Figure 7.5: Simplified scheme of a double gate transistor .

7.4.2.1 Structure exempted from the system D-G-S

heat source

Figure 7.6 a + b (from left to right): Front (a) and top (b) view of the device exempted from the drain, the source and the gate. An ac-electrical heating applied on a point of the top layer (a) produces a hot line (b) in that layer.

126 7.4 Investigation of thermal oscillations and thermoelastic displacements in HEMT-structures

1.2

thermal expansion 1.0 surface temperature

0.8 f = 5 kHz 0.6

0.4 Normalized Amplitude 0.2

0.0 0 20 40 60 80 100 120 140 160 Position / µm

Figure 7.6 c: Normalized amplitude of vertical thermal expansion and of the surface temperature.

thermal expansion 0 surface temperature

-20 f = 5 kHz

-40

Normalized Phase [°] Phase Normalized -60

-80 0 20 40 60 80 100 Position / µm

Figure 7.6 d: Normalized phase of vertical thermal expansion and of the surface temperature

Figure 7.6c and Figure 7.6d show respectively the amplitudes and the phases of the thermal expansion (•) and of the surface temperature (ο), calculated at the modulation frequency f = 5 kHz as a function of lateral position from the center of the hot spot, when the structure is exempted from the terminals, gate, source and drain. As expected, both the

127 7.4 Investigation of thermal oscillations and thermoelastic displacements in HEMT-structures

amplitude and the phase are maximal at the center of the hot spot. In particular, the thermal signal (surface temperature) drastically decreases as one moves away from the hot spot while the thermal expansion is much broader than the thermal signal. The reasons of such a behaviour have already been explained in section 7.3.

7.4.2.2 Structure with D-S

As can be seen in Figure 7.7, the gold based source and drain are now added to the structure. The calculated amplitude and phase of the thermal expansion are respectively represented in Figure 7.9a and in Figure 7.9b (line-dot curve in blue), in comparison with the thermal expansion calculated when only the hot line is taken into account. Globally, one can observe that the addition of gold film representing the drain and the source increases the thermal expansion of the sample surface. It is observed that the thermal expansion of the gold film dominates the expansion of other constituting layers of the structure.

Drain Source Drain Gate Source

heat source

Figure 7.7 a + b (from top to bottom) Front (a) Figure 7.8 a + b (from top to bottom): Front (a) and top (b) view of the structure showing the hot and top (b) view of the structure showing the hot line and the gold-based drain and source. line and the gold-based drain, gate and source.

128 7.4 Investigation of thermal oscillations and thermoelastic displacements in HEMT-structures

2.0

without D-G-S 1.6 with D-G-S with D-S

1.2 f = 5 kHz

0.8

0.4 Normalized expansion Amplitude

0.0 0 20406080100 Position / µm

Figure 7.9a Normalized amplitude of the thermo-elastic signal. The influence of the gold film is investigated by considering the following three cases: (a) Dot points: without the terminals (D, G, S). (b) Line–dot curve (blue): with Drain (D) and Source (S). (c) Red curve: with D, G, and S

20 without D-G-S with D-G-S with D-S 0

-20 f = 5 kHz

-40

Normalized Phase [°] Phase Normalized -60

-80 0 20 40 60 80 100 Position / µm

Figure 7.9b Normalized phase of the thermal expansion. The influence of the gold film is investigated by considering the following three cases: (a) Dot points: without the terminals (D, G, S). (b) Line–dot curve (blue): with Drain (D) and Source (S). (c) Red curve: with D, G, and S.

129 7.4 Investigation of thermal oscillations and thermoelastic displacements in HEMT-structures

7.4.2.3 Structure with D-G-S

The addition of a very thin gold film representing the gate (Fig. 7.8) produces an effect on the integral thermal expansion only at the location near the hot line (red curve in Figure 7.9 a + b). The results of the corresponding simulations show that the thermal expansion of the gold film is very pronounced and that this expansion is also proportional to the film thickness.

7.4.3 Comparison of experimental measurements and FE results

Figure 7.10a presents the topological image of a HEMT−structure limited to the active area where one can identify the source, the gate and the drain. One can see in Figure 7.10b, that the power dissipation generates a hot line near the gate, at the drain side. By considering the realistic parameters of the active area of the HEMT-device (*), namely the gate width, the gate length, the gate height, the gates separation, the drain-gate distance, and the source-gate distance, FE simulations −essentially limited to the active area− (see Fig. 7.11) have been performed and the results of calculations compared with experimental measurements. The model as shown in Figure 7.12a was meshed using the element type PLANE 55, which is a four-node 2D thermal solid with a single degree of freedom, .i.e. temperature, at its nodes. After the meshing, first simulations considering the same thermal properties for all layers of the structure −to fulfill the requirement of a homogenous sample− and the applica−

Figure 7. 10a: Topographic image of the HEMT- Figure 7.10b: Schematic representation of the structure limited to the active area active area showing location of the hot line.

(*) Any disclosure on device parameters is subjected to authorization from THALES Research & Technology.

130 7.4 Investigation of thermal oscillations and thermoelastic displacements in HEMT-structures

Drain Gate Source

Au B A GaN

Power dissipation

Figure 7.11: Schematic representation of device cross-section, limited to the active area. Scan direction: From A to B

tion of a thermal load (e.g., heat flux) along the entire top surface of the model −to fulfill the requirement of 1D thermal wave propagation− were performed in view of checking the convergence criteria based on the reference phase shift (Φref = -45°). At the considered frequency f = 5 kHz, phase solutions very close to the reference phase shift were obtained for the thermal wave at the sample surface when the number of elements reached the value 9660 for 9954 nodes. Following this verification, the thermal properties of the different layers composing the structure were then reintroduced in the ANSYS program and the actual thermal load (heat generation rate) was applied near the gate to produce a hot point (see Figure 7.11, scan direction from A toward B). In the calculations, priority was given to the assessment of thermal oscillations in the model as well as the thermoelastic displacement of the top of the model −representing the sample surface− at the used modulation frequency. One can observe in Figure 7.12b that the thermal wave, which originates from the hot spot near the gate, propagates into the gate and that the heat spreads more into the drain than into the source since this last gold film is obscured by the presence of the gate. As can be seen in Figure 7.13, the numerical results are in good agreement with the experimental measurements. While the experimental thermal expansion (black curve) takes into account the contribution of all layers involved in the structured device, the numerical results (red curve) indicate only the thermal expansion of the gold films.

131 7.4 Investigation of thermal oscillations and thermoelastic displacements in HEMT-structures

Figure 7.12a: Meshed finite element model. Figure 7.12b: View of active area showing the location of hot spot near the gate, at the drain side.

By comparing the measured with the calculated relative amplitude of the thermal expansion and with the calculated amplitude of the surface temperature (Figure 7.13), one can make the following observations: (i) −The contribution of the gold films to the thermal expansion dominates that of the subsurface layers. (ii) −The FE simulations suppose a very flat surface for the gold films, but the experimental measurements point out the fact that these films on top of the structure present roughness effects. (iii) −The maximum temperature is recorded at the center of the hot spot near the gate while the maximum thermal expansion is detected at about 5 µm from the hot spot, precisely on the drain. This is justified by the fact that the thermo-elastic signal strongly depends on the layer thickness. In this case for example, the gate thickness is about two times smaller than that of the drain. Thus, comparison between experimental and numerical results leads to the remark that the deposition of the gold film on top of the structure is not homogeneous and has considerable roughness and that the thickness or the roughness of this gold film may be used for a calibration of the temperature of the hot spot. Figure 7.14a and Figure 7.14b show respectively the normalized amplitude of the modulated thermal signal and of the total thermo-elastic signal −taking into account the contribution of the subsurface layers, and the corresponding normalized phases. According to Figure 7.14a, the thermal expansion amplitude (red curve) is maximal on the drain (see location in Fig. 7.13) but in Figure 7.14b, it is shown that the thermal expansion phase (red curve) is maximal near the hot spot. Figure 7.15a and Figure 7.15b compare the amplitude and phase of the thermal signal and the amplitude and phase of the thermo-elastic signal, respectively. This comparison confirms the uniqueness of the hot spot but in the meantime it

132 7.4 Investigation of thermal oscillations and thermoelastic displacements in HEMT-structures confirms that the thermo-elastic signal depends on both the thermal signal and the device thickness.

1.2

Thermal expansion (Exp) 1.0 Thermal expansion (FE) Surface temperature (FE)

0.8

0.6

S /a.u S 0.4

0.2

0.0 Source Drain

0 1020304050607080 x /µm

Figure 7.13: Lateral variation of the thermo-elastic signal. Comparison between experimental measurements and results of FE simulations.

10 1.4 1.4 0 1.2 1.2 -10

1.0 1.0 -20

0.8 0.8 -30

/ deg 0.6 0.6 n

Φ -40

Thermal expansion 0.4 0.4 -50 Surface temperature Surface

0.2 0.2 -60

0.0 0.0 -70 0 1020304050607080 0 1020304050607080 x /µm x /µm

Figure 7.14 a + b (from left to right): Normalized amplitude (a) and normalized phase (b) of the calculated thermal expansion (red curve) in comparison with the surface temperature (blue curve).

133 7.4 Investigation of thermal oscillations and thermoelastic displacements in HEMT-structures

1.2 0 0 1.4 1.0 -10

1.2 -20 0.8 -20 -30

1.0 0.6

-40

-40 / deg

n / deg S /a.u n Φ S /a.u

Φ -50 0.4 0.8 -60 -60

0.2 0.6 -70

-80 -80 0.0 0 1020304050607080 0 1020304050607080 x /µm x /µm

Figure 7.15a: Normalized amplitude (red curve) Figure 7.15b: Normalized amplitude (red curve) and normalized phase (blue curve) of the surface and normalized phase (blue curve) of the thermal temperature, as a function of position, calculated at expansion, as a function of position, calculated at the modulation frequency f = 5 kHz the modulation frequency f = 5 kHz.

7.4.4 Influence of the modulation frequency

1.0 0

0.8 -20

0.6 -40

/deg n S /a.u S 0.4 Φ -60

0.2 -80

0.0 -100 0 1020304050607080 0 1020304050607080 x /µm Position / µm

Figure 7.16 a + b (from left to right): Normalized amplitude (a) and normalized phase (b) of the surface temperature as a function of position, calculated for different values of the modulation frequency of heating, f = 5 kHz (red curve), f = 10 kHz (blue curve) and f = 15 kHz (black curve).

Another important aspect of this study is the investigation of the influence of the heating modulation frequency on signals. For this, in addition to the frequency f = 5 kHz which has been used to obtain the previous results, two other modulation frequency values, namely f = 10 kHz and f = 15 kHz are considered. Figure 7.16a and Figure 7.16b present respectively the relative amplitude and the normalized phase of the thermal signal at different distances, for the three values of the modulation frequency. One can see in Figure 7.16 a + b, that the thermal signal at the hot spot is practically insensitive to the variation of the 134 7.4 Investigation of thermal oscillations and thermoelastic displacements in HEMT-structures modulation frequency. However, as one moves from the hot spot, towards drain or source, the surface temperature decreases with increasing modulation frequency since the thermal wave originating from the hot spot is affected by damping. The same observation is valid for the frequency-dependent amplitude and phase of the thermo-elastic signal presented in Figure 7.17 a + b.

1.4 0

1.2 -20

1.0 -40 0.8

/ deg

n -60 S /a.u S 0.6 Φ

0.4 -80

0.2 -100

0.0 0 1020304050607080 0 1020304050607080 x /µm x /µm

Figure 7.17 a + b (from left to right): Normalized amplitude (a) and normalized phase (b) of the thermal expansion as a function of position, calculated for different values of the modulation frequency of heating, f = 5 kHz (red curve), f = 10 kHz (blue line) and f = 15 kHz (black curve).

135 7.5 Contribution of the tip to the signals

7.5 Contribution of the tip to the signals

7.5.1 Definition of the problem

Figure 7.18 presents a Wollaston probe which was developed by Dinwiddie et al. As described by Hammiche and co-workers [Hammiche et al., 1996] or by Gomès and co- workers [Gomès et al., 1999], for such a probe the arms of the cantilever are made of Wollaston process wire consisting of silver wire 75 µm in diameter containing a platinium/10% rhodium core 5 µm in diameter. The temperature-dependent resistive element at its end comprises a 200 µm of platinium core that has been freed from silver and bent to enable a V-like shape. The mirror fixed to the cantilever arms is devoted to reflect a laser beam onto a photodiode, so that the cantilever deflection can be sensed. While moving across the sample surface, the tip helps to draw a thermal map of the sample.

. temperature dependent resistance

Widerstand

temperaturabh Tip

Wollastondraht Wollaston-wire MirrorSpiegel

PlatiniumPlatinseele core Sample

Spitzetip

Figure 7.18: Example of probe used for thermo-elastic measurements.

But, a major difficulty that can occur during signal measurement resides in the evaluation of the tip contribution. In the contact mode for example, where the tip is in direct contact with the sample surface, a heating of the tip particularly in the centre of the hot spot could be problematic [Pelzl et al., 2001]. This is why it is useful to evaluate the thermal expansion of the tip due to the heat transferred from the hot area to the tip. For this, some aspects are taken into consideration: The tip is assumed to be in direct contact with the 136 7.5 Contribution of the tip to the signals sample. Still in the contact mode, the influence of the material of the tip and of the heated substrate is also investigated. Then the influence of a fluid film (water or air), interfacing the tip and the sample, is studied.

7.5.2 Modeling of the tip structure

Tip

20°

Sample 40 nm

Figure 7.19a: Schematic representation of the tip Figure 7.19b: Schematic representation of the in mechanical contact with a heated substrate. model of the tip used for FE simulations

Figure 7.19a represents a Si-tip in mechanical contact with a GaAs-sample. The tip is approximated by a truncated cone with a diameter of dst = 40 nm at its smallest end. The diameter (dlt) of the largest end and that of the smallest end are interrelated by the following equation: dlt = dst +2×h×tan(20°), with h is the height of the tip. A schematic representation of the model of the tip used to perform the FE calculations is shown in Figure 7.19b. It derives from the cylindrical symmetry imposed by the geometry of the tip. To run the numerical simulations, the entire model (tip and sample) was meshed with the element type PLANE 78, which generates eight-node elements and helps to solve axisymmetric-harmonic problems.

7.5.3 Si−tip in mechanical contact with a GaAs−structure

7.5.3.1 Temperature oscillations along the tip axis

Figure 7.20a shows the distribution of the thermal wave amplitude in the structure, when the Si-tip is in mechanical contact with the GaAs-sample. As thermal load, a circular

137 7.5 Contribution of the tip to the signals

heat source of radius 500 nm and of intensity 3.15 ×104 W/cm2, time-modulated with a frequency f = 5 kHz, was applied at the center of the top surface, just before the contact tip- sample is established. The results of simulations indicate that the maximum temperature is not recorded at the center of the hot spot as one could have expected. The temperature amplitude recorded for the Si-tip at the center of the hot spot is given by δTmax = 2.0 K. This means, the presence of the tip on hot spot contributes to cool down the sample at the heated area. But, as one can clearly see in Figure 7.20b, the temperature amplitude along the middle axis of the tip falls off very rapidly as a function of distance from the hot spot surface.

δTmax = 2.85 K

2.0

1.5

1.0

Amplitude /K Amplitude 0.5

0.0

01234 x /µm

Figure 7.20a: FE simulation of the distribution of Figure 7.20b: Amplitude profile of the thermal the thermal wave amplitude, when the Si-tip at the wave along the middle axis of the Si-tip. center of the hot spot is in mechanical contact with the GaAs-sample.

7.5.3.2 Derivation of the thermal expansion of the tip

That the temperature amplitude along the middle axis of the Si-tip falls off very rapidly is a proof that only the external part of the tip has really experienced heating. Integration of the thermal expansion along the tip yields a vertical thermal expansion of about δd = 1.3 pm. This value of the thermal expansion is however negligible in comparison with the expansion of the sample which is of the order of some Å.

138 7.5 Contribution of the tip to the signals

7.5.3.3 Thermal oscillations along the sample surface

Figure 7.21 shows the amplitude profile of the thermal wave at the sample surface. One can already recognize the cooling effect of the tip at the sample surface, which leads to a decrease of the surface temperature. This is why the maximal temperature (δTmax = 2.85 K) at the sample surface is recorded at a lateral position of about 140nm from the hot spot (see also Fig. 7.20a). Since the alteration of the surface temperature due to the presence of the tip is limited to a very small region in the hot spot, a notable change of the thermal expansion of the sample cannot be expected.

3.0

2.5

2.0

1.5

Amplitude /K 1.0

0.5

0.0 012345 x /µm

Figure 7.21: Amplitude profile of the thermal wave at the surface of the GaAs-sample in mechanical contact with the Si-tip. The temperature decrease at the contact area does not affect significantly the thermal expansion.

7.5.4 Influence of the material of the tip

In order to investigate the influence of the material of the tip on the heat transfer sample-tip, the silicon material (Si-tip) is substituted successively by platinum (Pt-tip) and tungsten (W- tip). Figure 7.22a and Figure 7.22b show the distribution of temperature amplitude in both the tip and the GaAs-sample for a Pt-tip and a W-tip, respectively. One can remark, that considering a Pt-tip in contact with the sample results in a relative increase of the temperature

139 7.5 Contribution of the tip to the signals

amplitude in the hot spot (Fig. 7.22a) whereas the use of a W-tip rather contributes to lower this amplitude (Fig. 7.22b). Most important is that the change in the surface temperature of the sample with the material of the tip is strictly limited to the hot spot as one can clearly see in Figure 7.23b. Such a result confirms that the presence of the tip at the sample surface contributes to modify the temperature only in a very limited region in the hot spot. According to the results of Figure 7.23a, and in comparison with the results obtained when considering a Si-tip, the surface temperature of the tip (temperature at the contact point of the middle axis) in the center of hot spot increases for a Pt-material and decreases for a W-material.

δTmax = 2.91 K δTmax = 2.81 K

Figure 7.22a: FE simulation of the distribution of Figure 7.22b: FE simulation of the distribution the thermal wave amplitude, when the Pt-tip in the of the thermal wave amplitude, when the W-tip center of the hot spot is in mechanical contact with in the center of the hot spot is in mechanical the GaAs-sample. contact with the GaAs-sample.

3.0 3.0

2.5 2.5

2.0 2.0

1.5 1.5

Amplitude /KAmplitude 1.0 1.0 Amplitude /K

0.5 0.5

0.0 0.0 012345 01234 x /µm x /µm

Figure 7.23a: Amplitude profile of the thermal Figure 7.23b: Amplitude profile of the thermal wave along the middle axis of the tip. Influence of wave at the surface of the GaAs-sample in the material of the tip in mechanical contact with mechanical contact with the tip. Influence of the GaAs-sample: a) Si-tip (black curve), b) Pt-tip the tip material: a) Si-tip (black curve), b) Pt- (blue curve), W-tip (red curve). tip (blue curve), W-tip (red curve). 140 7.5 Contribution of the tip to the signals

These different thermal behaviours have the following explanations: (i)−Since the tungsten material is very conductive with respect to silicon which on its turn is more conductive than the platinium material, the heat flow from the GaAs-sample to the W-tip is more intense than the heat flow from the sample to the Si-tip or to the Pt-tip. (ii)− In regard with their effusivities, where e(Pt) < e(Si) < e(W), the surface temperature of the Pt-tip is relatively higher than that of the Si-tip which on its turn is higher than the surface temperature of the W-tip (δTmax = 2.47 K, 2.00 K, and 1.69 K, respectively) since the thermal wave amplitude is inversely proportional to the sample effusivity. (iii)− In the frame of a heat flow from sample to tip (passive mode), the tip behaves as heat sink which contributes to cool down the sample. This is why the tip of higher thermal conductivity induces a lower temperature amplitude at the hot spot surface. The calculated thermal expansion of the tip is given by δd = 4.7 pm in the case of platinium (Pt) and δd = 1.11 pm in the case of tungsten (W). It can be therfore observed that the thermal expansion of the Pt-tip is relatively large due to the high surface temperature combined with its high coefficient of linear thermal expansion, given by CLTE = 8 × 10-6 K-1.

7.5.5 Influence of the substrate material

In connection with Figure 7.20a, which presents the distribution of the thermal wave amplitude in the GaAs-sample and in the Si-tip, Figure 7.24a and Figure 7.24b indicate the distribution of the thermal wave amplitude when the Si-tip is on top of hot spot on Si-sample and InSb-sample, respectively. One can already see, that the maximal temperature in the hot spot varies with the substrate material but these maxima are however recorded at different lateral positions in the hot spot. Thus, for the Si-sample the maximal temperature in the hot spot is found at 100 nm (δTmax = 1.33 K) while for the InSb-sample it is localized at 160 nm

(δTmax = 8 K). For comparison, the maximal temperature in the hot spot for the case of the

GaAs-material is found at about 140nm (δTmax = 2.85 K). In the meantime, for the three types of material considered for the substrate, the temperature of the Si-tip at the contact point behaves as follows: δTmax = 1.13 K (Si-tip on Si-sample), δTmax = 2.00 K (Si-tip on GaAs- sample), and δTmax = 3.54 K (Si-tip on InSb-sample). These observations have the following explanations: (i)−With decreasing effusivity, the amplitude of the thermal wave at the surface increases. In the present case, e(Si) > e(GaAs) > e(InSb). (ii)−With increasing surface temperature of heated substrate, in the hot spot, the tip on top of the sample also experiences a high surface temperature. This is why the temperature of the Si-tip is increased when it is brought into contact with the InSb-sample and lowered when the contact is rather established with a Si-sample. As consequence of this situation, the thermal expansion of the tip also increases. The thermal expansion calculated for the Si-tip is given by: δd = 0.73 pm (Si-tip on Si-sample), and δd = 2.36 pm (Si-tip on InSb-

141 7.5 Contribution of the tip to the signals

sample). It appears that probing substrate materials possessing low effusivity results in increasing the thermal expansion of the tip.

δTmax = 1.33 K δTmax = 8 K

Figure 7.24a: FE simulation of the distribution of Figure 7.24b: FE simulation of the distribution of the thermal wave amplitude, when the Si-tip at the the thermal wave amplitude, when the Si-tip at the center of the hot spot is in mechanical contact with center of the hot spot is in mechanical contact the Si-sample.. with the InSb-sample

3 6

4 Amplitude /K Amplitude /K 1 2

0 0 0246810 01234 x /µm x /µm

Figure 7.25a: Amplitude profile of thermal wave Figure 7.25b: Amplitude profile of thermal wave along the tip axis. The lower the thermal effusivity at the surface of the sample in mechanical of the heated substrate material, the higher the contact with the Si-tip. Influence of the substrate surface temperature of the tip and the larger the material: InSb (red curve), GaAs (black curve), thermal expansion of the tip. and Si (blue curve).

142 7.5 Contribution of the tip to the signals

On the contrary to the results of Figure 7.20a and Figure 7.24b, which show a real discrimination between the surface temperature of the tip (contact area) and the surface temperature of the sample in the hot spot, the results of Figure 7.24a shows that the tip and the sample share practically the same surface temperature in the hot spot. Such a thermal behaviour can be explained by the fact that both the tip and the sample are constituted of the same material (Si) and thus possess the same heat capacity which is regarded as the material’s ability to absorb heat from its surroundings.

7.5.6 Influence of a fluid film between the tip and the sample

In this section, we consider that the tip is no more in direct mechanical contact with the sample surface but rather that the heat flows from sample to tip through a thin film of fluid (water or air, see Fig. 7.26b) linking the two entities. Such a study has been performed by Depasse and co-workers [Depasse et al., 1997]. However, in their approach they used a homogenous tip of cylindrical geometry and assumed that the heat source is not located at the lower surface of the tip but rather placed in the tip in the plane z = s with respect to that surface. Their investigation was therefore based on the active mode (heat flow from tip to sample). In the present study we still consider the same geometry for the tip (truncated cone, see Fig. 7.26a) and the passive mode (heat flows from sample to tip).

Tip Tip

Heat source

Air or water dtip-sple

Sample

Figure 7.26a: Schematic representation of the tip Figure 7.26b: Cross-section view of Figure maintained at a distance dtip-sple above of the 7.26a showing a fluid bridge (air or water) sample surface by a fluid (air or water). linking the tip and the sample.

143 7.5 Contribution of the tip to the signals

4 2 A circular plane heat source, of radius a = 500nm and of intensity Io = 3.15 ×10 W/cm , time modulated with the frequency f = 5 kHz is applied on top of the sample surface (see Fig. 7.26a). In the following, the influence of the distance tip-to-sample on the thermal exchange between the sample and the tip is investigated, by taking into account the nature of the coupling medium (air and water). Our interest is mainly focused on the behaviour of the mean surface temperature of the tip and of the maximal temperature amplitude in the hot spot.

δTmax = 0.32 K δTmax = 5.97 K δTmax = 0.18 K

δTmax = 6.00 K

water

sample

Figure 7.27 a + b : Distribution of the thermal wave amplitude in the sample (GaAs), the coupling medium (water) and the tip (Si), captured at two different distances tip-to-sample: (a) dtip-sple = 10 nm and (b) dtip-sple = 60 nm.

2.0 6

5 1.5 4

1.0 3 Amplitude /K

Amplitude /KAmplitude 2 0.5 1

0.0 0 012345 01234 x /µm x /µm

Figure 7.28a: Amplitude of the thermal wave along Figure 7.28a: Amplitude of the thermal wave the middle axis of the tip, when the tip is in at the sample surface, when the tip is in mechanical contact with the sample (black curve) mechanical contact with the sample (black and when the heat flows from sample to tip through curve) and when the heat flows from sample a water film of different thicknesses: dtip-sple = 5 nm to tip through a water film of different (blue curve), dtip-sple =10nm (green curve), and dtip-sple thicknesses: dtip-sple = 5 nm (blue curve), dtip-sple = 20 nm (red curve). =10nm (green curve), and dtip-sple = 20 nm (red curve).

144 7.5 Contribution of the tip to the signals

Figure 7.27 a +b shows the distribution of the thermal wave amplitude in the sample (GaAs), the coupling medium (water) and the tip (Si), captured at two different distances tip- to-sample, namely at dtip-sple = 10 nm and 60 nm respectively. As one can immediately remark, there exists a thermal gradient in the water film, which means that the water film behaves as -1 heat carrier despite its low thermal conductivity (kwater = 0.6 WmK ).

1.0 2.0 Tip thermal expansion Coupling medium: water Tip surface temperature 6 0.8 1.5 5 0.6 Coupling medium: water 1.0 4 0.4

0.5 3 0.2 Normalized amplitude /a.uNormalized amplitude Tipsurface temperature /K Max. temperatureMax. in the hot spot /K

0.0 2 0.0 0 102030405060 0 102030405060 d /nm d /nm tip-sple tip-sple

Figure 7.29a: Tip surface temperature (red curve) Figure 7.29b: Normalized surface and maximal temperature amplitude in the hot spot temperature (blue curve) and normalized (blue curve), calculated as a function of the distance thermal expansion (red curve) of the tip as a tip-to-sample, when a water film is considered as function of the distance tip-to-sample, when a coupling medium. water film is considered as coupling medium.

2.0 coupling medium 6

water air 1.5 coupling medium 5 water air

1.0

0.5

Tip surface temperature /K 3 Max. temperatureMax. in the hot spot /K 0.0

0 102030405060 0 102030405060 d /nm d /nm tip-sple tip-sple

Figure 7.30a: Evolution of the surface temperature Figure 7.30b: Evolution of the maximal of the Si- tip as a function of the distance tip-to- temperature in the hot spot as a function of the sample (GaAs), for different coupling media: distance tip-to-sample (GaAs) for different air(blue curve) and water (red curve). coupling media: air (blue curve) and water (red curve).

145 7.5 Contribution of the tip to the signals

According to Figure 7.28a, the presence of a water film at the interface tip-sample contributes to attenuate the thermal flow between the two entities. This is why the temperature along the middle axis of the tip decreases with increasing thickness of the water film. Thus, the mean surface temperature of the tip falls from δT = 2 K for dtip-sple ≈ 0 to δT = 0.54 K, δT

= 0.32 K, and δT = 0.22 K, for dtip-sple ≈ 5, 10, 20 nm respectively. On the contrary to the mean surface temperature of the tip which drastically decreases due to the presence of the water film at the sample surface, the surface temperature of the sample (Fig. 7.28b) rather increases, meaning that the thermal flow from sample to tip, which was optimal in the contact mode, is actually affected by the presence of the fluid film. As one can remark in Figure 7.28b, with increasing thickness of the water film, namely dtip-sple = 5, 10, 20 nm, no further notable changes are noted on the sample surface temperature, excepted some small modifications near the center of the hot spot, where the cooling effect of the tip is manisfested. Figure 7.29a presents the evolution of the tip surface temperature and of the maximal temperature amplitude in the hot spot as a function of the thickness of the water film linking the tip and the sample. The distance dtip-sple ≈ 0 (less than 1Å) corresponds to the configuration where the lower end of the tip is in mechanical contact with the sample surface at the hot spot. While the tip temperature drastically decreases with increasing thickness of the water film, the maximum temperature in the hot spot experiences first a jump before remaining practically constant with increasing film thickness. In Figure 7.29b, the thermal expansion of the tip is compared with the mean surface temperature of the tip, for several thicknesses of the water film. Thermal expansion and temperature are normalized to the values recorded when the tip is in direct contact with the sample. As expected, the normalized thermal expansion is larger than the normalized temperature. The reasons for that discrepancy have already been explained in section 7.3.2.2. Figure 7.30a and Figure 7.30b show respectively the mean surface temperature of the tip and the maximal temperature in the hot spot, calculated as a function of the film thickness for two different fluids, namely air and water. It can be already observed in Figure 7.30a, that using air as coupling film between the tip and the sample contributes to lower considerably the temperature of the tip. For a film thikcness of 5 nm, the mean surface temperature of the tip falls from 2 K to 0.023 K. This latest value of the temperature is negligible in comparison with that obtained for the same thickness of water film (δT = 0.54 K). For an air film of thickness larger than 10 nm, no thermal information coming from the sample reaches the -1 -1 tip.This can be comprehensible since air, in regard with its conductivity (kair = 0.026 Wm K ) is not a very good heat carrier. In the case of a water film, the tip continues to receive information from the sample whatever the distance tip-to-sample. Thus, for a water film of thickness larger than 40 nm, the thermal flow between the tip and the sample remains constant. The results of Figure 7.30a indicate that in comparison with air, water, in regard -1 -1 with its thermal conductivity (kwater = 0.6 Wm K ), enables a better thermal coupling between the tip and the sample. Of course this affirmation does not take into account the contact mode

146 7.6 Conclusions

(tip in mechanical contact with the sample surface in the hot spot) for which the thermal flow from the tip to the sample is optimal.

7.6 Conclusions

In this chapter, temperature oscillations in some models of devices and thermoelastic displacements of their heated surface have been assessed with the help of finite element simulations. First calculations involving an ac heated conducting Si-line separated from the Si-substrate by an insulating layer have compared the modulated surface temperature with the thermal expansion. It has been shown that both the surface temperature and the thermal expansion are maximal at the center of the hot spot and that the thermo-elastic signal is broader than the thermal signal, which falls off rapidly as one moves away from the hot spot while the thermal expansion accumulates the information of the surface temperature and takes into account the expansion of all subsurface layers reached by the thermal wave. By varying the modulation frequency of heating, changes of the thermal expansion have been recorded mainly in the region far away from the hot spot where the surface temperature has drastically decayed. Comparing the results of finite element simulations with the results of experimental measurements performed on a HEMT-structure and based on SThEM, we have found that the deposition of the gold films on top of the structure to make up the terminals source, gate and drain is not homogeneous and has considerable roughness and that the thermal expansion of the gold film is dominant with respect to the expansion of other sub-layers of the structure. This expansion of the gold film may be used for a calibration of the temperature of the hot spot. Subsequently, comparisons between the calculated surface temperature and the calculated thermal expansion taking into account the contribution of all layers of the structure allowed to find out that the maximal temperature is recorded at the center of the hot spot, near the gate and at the drain side, while the maximal thermal expansion is rather recorded elsewhere on the drain. Then further numerical simulations aimed at investigating the influence of the tip on the heat transfer sample-to-tip were performed, which allowed to remark, that the presence of the tip on hot spot on a sample contributes to cool down the sample surface. Simulations considering a Si-tip on top of a GaAs-sample led to a thermal expansion of the tip of the order of a pm, a value negligible in comparison with the expansion of the sample surface which is of the order of some Å. Most interesting is that, the temperature along the middle axis of the tip which was about 2K at the contact point tip-sample falls off very rapidly. By substituting the material of the tip, silicon with a Pt-material and a W-material successively, change in the surface temperature of the sample was strictly limited to a small region in the hot spot while the surface temperature of the tip increased with decreasing effusivity. Considering a Si-tip and changing the material of the sample, namely Galium

147 7.6 Conclusions

arsenide (GaAs) by Silicon (Si) and Indium antimonide (InSb), the maximal temperature in the hot spot and the maximal amplitude of the thermal wave along the middle axis of the tip increased for the InSb-sample and decreased for the Si-sample. Such a result has been explained by the fact that the amplitude of the surface temperature is inversely proportional to the sample effusivity. An interesting configuration was that of Si-tip on Si-sample for which both the sample and the tip shared practically the same surface temperature at the contact area, in the hot spot. This thermal behaviour has been attributed to the fact, that both the sample and the tip consist of the sample material and thus possess the same heat capacity, which is regarded as material’s ability to absorb heat from its surroundings. Considering a fluid film (water or air) between the tip and the sample contributed to lower the temperature of the tip and inversely to increase the surface temperature of the sample, thus indicating a poor thermal exchange between the tip and the sample. Nevertheless, we noted that in comparison with air, water by its thermal conductivity enables a better thermal coupling between the tip and the sample. Finally we came out with the conclusion, that sensing materials of lower effusivity results in increasing the thermal expansion of the tip and that sensing substrate materials with tips of higher effusivity material contributes rather to lower the expansion of the tip.

148

Conclusions and Future Work

8.1 Conclusions

8.1.1 Review of the experimental work

The experimental measurements have been performed with the help of IR photothermal radiometry, based on surface heating by means of a modulated laser beam (argon ion) and detection of the thermal wave response by a MCT (HgCdTe) detector. The investigated samples consisted of diamond-like carbon (DLC) on high speed steel (HSS) or on high effusivity metallic alloy (HMA). To be measured, the samples were placed in the reflection configuration for which the photothermal signals are excited and detected at the same surface. Then considering displacement distances between the heating and the detection spots, experimental measurements in the millimeter range were also performed. For this, the detection spot was maintained at a stable position while the heating spot was scanned over the sample surface. The measured phases and amplitudes of the modulated IR signals have revealled, that certain couples of samples (e.g., C4 & C6, F3 & F5, X4 & X7) possess the same coating deposited on different substrates. This means, the surface characteristics for the samples of each group are similar, thus giving the possibility to get access to the thermal properties of the subsurface material and in the meantime to obtain information about the different conditions of layer growth and the substrate preparation. By comparing the signal amplitudes of each couple of samples, the normalized signal was equal or close to unity in the limit of high modulation frequencies, giving a proof that the surface characteristics of the two samples cancel with each other. For each of these couples of samples, the similarity of the surface characteristics was also pointed out by the two calibrated measured phases. In a general way, two types of substrate material were identified and the obtained ratio of thermal conductivities corresponding to these two substrates, namely the high speed steel (HSS) and the high effusivity metallic alloy (HMA), was found relatively large in comparison with the expected ratio. This discrepancy between calculated and expected ratio has been explained by the possible changes in the properties of the substrates during their tailoring which can lead to some thermal strains and stresses or during the deposition process of the coating on these substrates. While applying the Extremum Method on certain signal phases measured at high temperature, we remarked that these phases reveal other characteristics which could not be

149 8.1 Conclusions

easily detected at room temperature. In the range from intermediate to high modulation frequencies or from relatively low to very low penetration depths, the calibrated measured phases which primarily exhibited the characteristics of a semi-transparent coating between 25° C and 118° C (comp. Fig. 4.5 a + b, Chap. 4), changed suddenly their behaviour as the measuring temperature increased. We observed namely, that at the measuring temperatures 248° C, 418° C and 518° C the calibrated measured phases were modified by the appearance of a relative maximum –which decreased with increasing temperature− in the range of high modulation frequencies, indicating the gradual emergence of an extremely thin layer at the very surface of the coating (comp. Fig. 4.5 c + d + e, Chap. 4). Concretely, with increasing measuring temperature, the surface characteristics of the sample moved successively from an optical to a topological aspect. These different behaviours shown by the sample as the temperature increases confirm that the photothermal radiometry technique may be more sensitive at high temperature [Forget, 1993]. IR photothermal measurements performed on samples in the millimeter range by considering displacement distances between heating and detection spots have allowed to confirm the reliability of the theory developed for the purpose. Thus, with increasing displacement distance between the two spots the relative extremum of the calibrated measured phases in the very low frequency range, which provides information about the thermal diffusivity of the subsurface material, shifted to smaller frequency values, indicating local variations of the substrate thermal diffusivity as shown by the theoretical predictions (comp. Fig. 5.6 and Table 5.1, Chap. 5). These experimental phases have proven that measurements based on displacements between heating and detection spots can also help to detect local variations of the thermal diffusivity of the coating. This is visible on the calibrated measured phases (comp. Fig. 5.6, Chap. 5) where the relative minimum relatively remains at a stable position (which means no variation of coating thickness) while some sensitive variations in the range of modulation frequencies higher than that of the minimum are pointed out as the displacement distance changes, indicating also possible local variations of the coating diffusivity. As for the experimental amplitudes, we observed from correlations with theoretical predictions that the optical absorptivity or alternatively the optical reflectivity of the coating also varied with the displacement distance (compare Table 5.2 and Fig. 5.7, Chap. 5). Finally, we have found out that one of the advantages of measuring in the neighbourhood resides in the detection of local inhomogeneities of both the thermal properties and the optical properties of the investigated sample.

8.1.2 Review of the analytical work

Several theories have been derived to enable the qualitative and the quantitative interpretation of experimental measurements. The main significant developments are reported below: 150 8.1 Conclusions

8.1.2.1 3-D thermal wave propagation in layered systems

A 3-D theory of thermal wave propagation in opaque layered systems has been developed in response to the inability of the 1-D theory to explain the reasons of discordances between the measured modulated IR signals and the theoretical predictions in the limit of very low modulation frequencies. By correlating the experimental measurements with the theoretical predictions according to the 3-D theory of thermal wave propagation in layered systems, we have found in the case of a two-layer system, that the lateral thermal transport in the samples are rather pronounced in the subsurface material and become more important with increasing thermal diffusivity of the substrate. To run the theoretical calculations, considered radii of the heating and the detection spots were respectively rH = 2000 µm and rD = 800 µm, in accordance with the measuring conditions. We have found, that for large radii of the heating spot, e.g. rH = 32 mm, the results based on 3-D thermal wave propagation perfectly agree with the approximations based on 1-D heat transport. This result has confirmed, that the 1-D thermal wave propagation can be applied for the interpretation of photothermal measurements when the diameter of the excitation spot is very large in comparison with the diffusion length of the thermal wave, that 1/2 means 2rH >> µth = [α/(πf)] .

8.1.2.2 Extraction of the thermal transport properties from the relative extremum of the calibrated measured IR signal phases

The Extremum Method has been introduced in chapter 4 and applied to experimental measurements. The main motivation behind the developed method came from the observation, that the calibrated amplitudes and phases of the measured modulated IR signals were generally characterized by the appearance of physical effects, namely the 3-D heat propagation in the limit of low modulation frequencies and the optical and topological effects at high modulation frequencies (comp. Chap. 3), which can contribute to some inaccuracies of the quantitative interpretation. We have shown, that the relative extremum and the frequency at which this extremum occurs on the calibrated measured phase can be exploited to get access to the thermal (diffusivity, effusivity, and implicitly to the thermal conductivity and the heat capacity of either the coating or the substrate) and physical properties (thickness of the coating) of an opaque two-layer structure via the thermal reflection coefficient Rsb and the thermal diffusion time τs, two combined thermophysical quantities. Thus, by simple identification of the relative extremum and the modulation frequency at which this extremum occurs on the measured phase, the above combined parameters (Rsb and τs) are determined, which then help to extract the thermal and physical properties of the unknown investigated sample. This new method is rapid, efficient and very accurate in regard with the processing time and the very good agreement between theoretical and experimental

151 8.1 Conclusions

data (representative curves). In order to confirm the reliability of this very promising method, comparisons have also been done with other alternative approaches based on the determination of the combined parameters from any point of the calibrated measured phase. By considering several measured points on the calibrated phase, different from the phase extremum, contradictory values of the thermal reflection coefficient and of the thermal diffusion time were obtained. With such a result, we came out with the conclusion that only the Extremum Method yields more reliable thermophysical quantities, therefore more reliable thermal and physical properties of the unknown sample. Then the Extremum Method has been generalized to enable the on-line interpretation of measurements performed on two-layer systems. Starting from the fact, that the thermal reflection coefficient Rsb only depends on the relative extremum of the calibrated measured phase, a graphic of thermal reflection coefficients as a function of the relative phase extremum was constructed. Such a graphic (Fig. 4.9, Chap. 4) can provide the investigator with the opportunity to have a direct and rapid information about the thermal effusivity of either the coating or the substrate of the sample by simply picking up the value of the relative extremum (either a minimum or a maximum) observed on the calibrated measured phase and then retrieving the corresponding thermal reflection coefficient. To build up the graphic of thermal diffusion times, an intermediate graphic representing a key-function (K) as a function of the relative extremum, that is K = K(tanΦn extr), was first constructed. We have observed, that for

|tanΦn extr| → 0 (but different from Zero, otherwise it corresponds to the case of a homogenous sample where Rsb = 0, which is out of context in this study), the limit value for the key- function is given by K = 0.05 (Fig. 4.10, Chap. 4). This means the key-function decreases with increasing absolute values of the measurable quantity tanΦn extr. This also means, that at a given modulation frequency the thermal diffusion time decreases with increasing absolute values of tanΦn extr. Following this demonstration, a graphic of thermal diffusion times was established as a function of the modulation frequency at which the extremum can occur, for a given value of the key-function which is induced by the relative phase extremum (Fig. 4.11, Chap. 4). As proof of its rich potential, the Extremum Method has been extended to the interpretation of the measured signal phases that may be obscured at low frequencies due to 1/f −noise and at high frequencies by the background fluctuation limit. For this, a functional transform was used by multiplying the calibrated measured phase shift with the variable, ( f −1/ 2 )q , where f is the excitation modulation frequency and q a real number close to Zero, to extend the inverse solution to other values of the calibrated phase lag measured in the neighbourhood of the relative extremum. First, the functional transform method was applied to examples of real coatings which are perfectly described by the two-layer model or which considerably deviate from the model of an opaque two-layer system due to the coating’s transparency or due to the presence of an additional layer of reduced thermal transport properties at the very surface of the coating. Then the functional transform was applied to

152 8.1 Conclusions examples of measurements, which −at higher frequencies at the position of the relative extremum of the calibrated phase− may be obscured by the background fluctuations.

8.1.2.3 Scaling of thermal localization of hot spots from macroscopic to microscopic range

While studying the 3-D propagation of thermal waves in two-layer systems (chap. 3), we have observed, that the deviations from 1-D thermal wave propagation increase with decreasing heating spot radii and with increasing thermal diffusivity of the subsurface material. This behaviour has suggested that by working with smaller heating spot radii in thermal microscopy, the calibrated phases with their relative phase maximum in the low frequency range can be used to detect local variations of the thermal diffusivity in the subsurface material. We have also remarked that errors of interpretation may occur due to the deviations between the model of concentric excitation and detection and the experimental reality, if the heating and the detection spot are not well focused in the experiment. It’s on the basis of these observations and remarks, that a 3D theory of thermal wave propagation in two-layer systems assuming a distance offset between the heating and the detection spots has been developed. In this frame, the modulated IR signals measured over the detection spot area, have been expressed as a function of radii of the heating and the detection spots (resp. rH, rD), and as a function of the displacement distance between the two spots, dHD.

By considering a constant thermal diffusivity αb of the subsurface material we noted that, the increasing displacement distances generate a set of relative phase maxima, which increase with the displacement distance and are found at nearly the same modulation frequency and that with a higher value of the thermal diffusivity, the relative phase maxima shift to higher modulation frequencies. We found out, that by collecting and representing the data of the different relative phase maxima −obtained for a constant value of the thermal diffusivity of the subsurface material− as a function of the position of the detection spot with respect to the heating spot, it becomes possible to localize the heat sources. Practically, the obtained results have shown, that an extrapolation onto the heating spot is nearly independent of the thermal diffusivity of the subsurface material and that the hot spot can be localized with good precision when three or four measurements are performed in the neighbourhood of a heat source. In order to find out at what distances such a localization in the neighbourhood is still possible, further simulations were performed while decreasing gradually the values of experimental parameters, namely the radii of heating and detection spots, and the displacement distance between the two spots. For this, the size of the sample was adapted to more realistic conditions of thermal microscopy −thin film of thickness ds = 0.8 µm and substrate of thickness db = 500 µm. By decreasing the spot sizes and the displacement distance between the spots while keeping a constant ratio radius of heating spot-to-radius of

153 8.1 Conclusions

detection spot rH/rD = 0.8 and a constant ratio of displacement distance-to-radius of detection spot dHD/rD = 2, the relative phase extrema shift to higher modulation frequencies, e.g. for heating spot of rH = 4 µm and a detection spot of rD = 5 µm, and displacement distance of 1/2 about dHD = 10 µm, the relative minimum is found at (f /Hz) = 400. Then considering different ratios of heating-to-detection spot radii rH/rD = {0.7, 0.8, 0.9} for a constant ratio of displacement distance-to-detection spot radius dHD/rD = 2 on one hand, and considering different ratios of displacement distance-to-detection spot radius dHD/rD = {1.0, 1.5, 2} for a constant ratio of heating-to-detection spot radii rH/rD = 0.8 on the other hand, the difference between the generated relative phase minima and maxima increase with increasing displacement distance and are rather large and vary with the size of the heating spot. With these results we came out with the conclusion that the more appropriate configuration in displacement experiments is reached for larger displacement distances between heating and detection spot, and for radii of the heating spot smaller than the radii of the detection spot.

8.1.3 Review of the numerical work using finite elements

8.1.3.1 Control of convergence of the numerical solution with the Reference Phase

In addition to the procedure of mesh refinement, which is recommended to enhance the convergence of a numerical solution based e. g. on the finite element method (FEM), we have proposed and applied an alternative approach, specific for thermal wave problems, which is based on the known value (Φref = −45° or tanΦref = –1) of the phase shift of the thermal wave at the top surface (where modulated excitation takes place) of a homogenous sample. We have demonstrated that this value of the phase shift, labelled as reference phase, is a powerful tool to judge the convergence of the numerical solution. The proposed method works according to the principle of mesh refinement but allows to obtain a more accurate solution. Thus, the convergence of the numerical solution can be established when the calculated phase shift of the thermal wave at the top surface of a homogeneous model is equal or very close to the reference phase shift in the considered frequency range. For a model consisting of a superposition of layers with different thermal properties, the poorest thermal properties (all belonging to one of the layers!) are first considered for all layers in order to fulfill the requirement of a homogenous sample. Then, first simulations are performed after the meshing of the model. If the phase of the thermal wave at the top surface (where the modulated thermal load has been applied) of the model reaches the reference phase lag within the entire frequency range, then the meshing is optimal, otherwise another meshing is required until the calculated phase is equal or very close to the reference phase. Once the agreement between the calculated phase and the reference phase is established, the thermal parameters of each participating layer are reintroduced in the program in order to run new simulations at the

154 8.1 Conclusions end of which the actual phases and amplitudes of the photothermal signal are recorded and collected for discussion and interpretation. Of course, the reference phase approach is also valid even if only one modulation frequency is concerned. If the heating modulation frequency is very high, e.g. f = 100 kHz, which means a small value of the thermal diffusion length, the mesh size at small penetration depth has to be very small while it can be considered large at large penetration depths. If the modulation frequency is rather very low, e.g. f = 1.00 Hz, then the mesh size at both the small and large penetration depths have to be considered large. However, the value Φref = -45° for the reference phase shift is typical for a homogeneous solid regarded as semi-infinite. This is why for a small thickness of the sample (model), e.g. d = 500 µm, the phase shift deviates from Φref = -45° towards Φref = –90° in the limit of low modulation frequencies or at large penetration depths. The reasons of these deviations have been explained with the help of the thermal thickness, a = d πf /α , which is a measure of the ability of the thermal wave to propagate throughout the entire sample. This way of judging the convergence of the numerical solution has been adopted to perform and control the simulations and the obtained results were in conformity with the expectations.

8.1.3.2 Simulation of thermal and thermo-elastic signals in micro-structured semiconductor devices

The finite element (FE) simulations performed in this work were aimed at determining the thermal oscillations in micro-structured devices and the thermo-elastic displacements of their surface as consequence of modulated heating. To simplify the calculations, the lateral thermal strains in the sample were assumed so small in comparison with the longitudinal thermal strain (Poisson number ν negligible) that the thermo-elastic expansions were only confined in the vertical direction, along the depth of the modelled structures. By definition, the vertical thermo-elastic expansion is proportional to the integral of the temperature oscillations over the sample depth and the proportionality factor is the coefficient of linear thermal expansion (CLTE), meaning that a thermal analysis is coupled with a structural analysis. In this case, two resolution methods are possible: In the direct method, degrees of freedom (thermal oscillation and thermal expansion) of the coupled field analysis (thermal and structural) are simulated simultaneously. In the indirect method, the results of one analysis are used as entries of the following analysis. This latest method was preferred in this work. Using the software ANSYS (MULTIPHYSICS/Thermal), versions 5.7 and 6.1, the real and the imaginary components of the modulated temperatures in the models of devices were first simulated for a given modulation frequency as a function of lateral distance from the hot spot. Then the thermo-elastic displacements of the sample surface were deduced by integrating the calculated modulated temperatures along the depth of the sample. During our 155 8.1 Conclusions

investigations, we found out that ANSYS offers a special command (PCALC), valid in the general post-processor (POST1), which enables such a numerical integration of the nodal temperatures over a defined path, providing that the parameters involved in this integration are clearly identified and defined by the solver, otherwise the obtained results would be hazardous. Going from the consideration that a particular location of the microstructure was subjected to an external excitation source, e.g. ac-electrical heating, modulated laser beam, the main tasks were to detect the location of the hot spot and to assess the thermal expansion of the hot area. Experimental thermal expansions performed on a HEMT-structure were compared with the results of FE simulations, and allowed to find out that the deposition of a gold film on top of the structure to coat the transistor terminals, namely the source, the gate and the drain, was not homogenous and had considerable roughness. We particularly observed that the thermal expansion of the gold film dominates the contribution of the subsurface layers of the microstructure. From this observation, we point out the fact that the thermal expansion of the gold film could be exploited to get access to the temperature of the hot spot but the way is not yet clear and the approach is under study. Another important point was the investigation of the tip contribution to the thermo- elastic signal. For this, the tip structure was approximated by a truncated cone with a diameter of 40 nm at its smaller end. The following aspects were checked: (i)−Nature of the contact tip-sample. Here, the tip was first considered to be in mechanical contact with the sample surface. Due to the contact, the thermal flow from sample to tip was optimal but the calculated thermal expansion of the tip was found very small (value δd = 1.3 pm) in comparison with that of the sample surface which was in the order of some Å. (ii) −Impact of the material of the tip on the heat transfer sample-to-tip. In this frame, the Si-tip was successively substituted by a Pt-tip and a W-tip. The material of poorer thermal properties (with respect to silicon), namely Pt (Platinium), contributed to increase relatively the temperature amplitude of the sample surface in the hot spot while the material of better thermal properties, namely W (Tungsten), induced a decrease of that temperature amplitude.

Most important is that, with respect to the surface temperature of the Si-tip (δTmax = 2.00 K) at the contact point, the temperature of the Pt-tip increased (δTmax = 2.47 K) while that of the W- tip rather decreased (δTmax = 1.69 K). These observations were explained by the fact that lower thermal effusivities lead to higher surface temperature. In fact, the change in the material of the tip brought little changes in the surface temperature of the sample only in a limited region of the hot spot, thus giving the proof that the cooling effect of the tip on the substrate material is not so severe. (iii) −Influence of the substrate material in mechanical contact with the tip. For this case, the following observations were made: a)− Using a Si-tip and a GaAs-sample, the maximal amplitude of the thermal wave at the sample surface was given by δTmax = 2.85 K while a temperature amplitude of δTmax = 2.0 K was recorded at the center of the contact area tip-sample. b)− By replacing the GaAs-sample with the InSb-sample, the maximum amplitude

156 8.2 Future work

of the thermal wave at the sample surface moved to a value of δTmax = 8.0 K while the maximum amplitude of the thermal wave along the middle axis of the Si-tip moved to the value of δTmax = 3.54 K. c)− For the configuration Si-tip on Si-sample, the maximal amplitude of the temperature at the sample surface took a value of δTmax = 1.33 K while the maximal temperature of the tip at the contact point was δTmax = 1.13 K. d)− The calculated thermal expansion of the tip varied as follows: δd = 0.73 pm (Si-tip on Si-sample), δd = 1.30 pm (Si- tip on GaAs-sample) and about δd = 2.36 pm (Si-tip on InSb-sample). These results have indicated that, since the surface temperature of the sample increases with decreasing thermal effusivity then the temperature of the Si-tip at the contact point of the hot spot and therefore its thermal expansion also increase. (iV)−Presence of a fluid film between the tip and the sample: In this investigation the interface tip−sample was considered to be occupied by either a liquid-film (water) or a gas (air). In comparison with the contact tip-sample configuration, the presence of a fluid bridge linking the tip and the sample contributed to alter the thermal exchange between the tip and the sample. Nevertheles, with respect to air the water film was found to ensure a better thermal coupling between the tip and the sample. In fact, in regard with their thermal conductivity, water constitutes a better heat carrier than air. Globally, the presence of the tip in the center of the hot spot contributes to cool down the heated substrate material. This is why the surface temperature of the sample experiences a decrease at the contact area tip-sample in the hot spot. The most interesting configuration was that of Si-tip on Si-sample for which we remarked that the tip and the sample practically shared the same surface temperature (δT = 1.13 K for the tip and δTmax = 1.33 K for the sample) in the hot spot. Such a thermal behaviour has been explained by the fact that the tip and the sample consist of the same material and thus possess the same heat capacity.

8.2 Future work

8.2.1 Extensions of the Extremum Method

The Extremum Method developed and applied to experimental measurements for the determination of thermal transport properties was restricted to opaque two-layer systems. However, this very promising evaluation method has to be extended to other sample configurations. In this frame, we are looking forward to investigating the applicability of the method to the following cases:

157 8.2 Future work

8.2.1.1 Two-layer systems with semi-transparent coating

In the case of two-layer systems presenting a coating, which is semi-transparent e.g. in 1/2 the visible spectrum, an additional combined (thermo-optical) parameter Pβ = α β, has to be associated to other combined parameters, namely the thermal reflection coefficient Rsb and the thermal diffusion time τs. This means, on the contrary to the case of opaque two-layer systems which necessitates only two governing equations involving the coordinates of the extremum, namely the frequency and the value of the measurable phase extremum (comp. equations 4.10 and 4.11, Chap. 4) as known parameters and the combined quantities, Rsb and τs as unknown parameters which are then determined to extract the thermophysical properties of the sample, a further equation has to be added to account for the coating semi-transparency. This would certainly be feasible by considering that the calibrated measured phases take the value Zero at a given high frequency fβ (comp. for example Fig. 3.6, chap. 3). The objective is to determine the following combined thermophysical quantities Pβ, Rsbβ and τsβ. (read: combined thermophysical parameters induced by the finite value of the optical absorption coefficient).

8.2.1.2 Three-layer systems

Another case of figure which would certainly benefit from the Extremum Method is the opaque three-layer system. As seen in chapter 3 and in chapter 4, such a sample structure is characterized by the appearance of two relative extrema on the calibrated measured phases of the modulated radiometric IR signals, namely in the range of intermediate and high modulation frequencies. Each of the two relative extrema is described by the frequency and the value of the measurable phase extremum. Generally, the relative extremum occurring in the range of intermediate frequencies is a minimum, and that occurring in the limit of high modulation frequencies is a maximum. In order to get access to the thermal transport and physical properties of the sample (thin layer on top of the coating, coating and subsurface material), it will be necessary to solve simultaneously four governing equations, each couple of equations being in connection with a measurable phase extremum.

8.2.1.3 Generalized Extremum Method

If the extension of the Extremum Method to the case of a two-layer system with a semi-transparent coating and to the case of an opaque three-layer structure is successful, then the next step will be focused on the generalization of the technique by taking into account the relative extremum which often occurs in the limit of very low modulation frequencies on the calibrated measured phases and which has been found to be induced by lateral heat transport in the subsurface material of high thermal diffusivity (comp. Chap. 3). In this way, the surface (optical and topological) and the subsurface (3-D heat propagation) effects of thermal

158 8.2 Future work transport in layered systems as discussed in chapter 3 could be globally analysed in the frame of a generalized Extremum Method. Such an extended method could then contribute to a more accuracy of the quantitative interpretation of the measured modulated IR signals and certainly provide investigators with new possibilities in the field of thermal microscopy.

8.2.2 Calibration procedure for determining the temperature of hot spot

While comparing experimental results performed on a HEMT-structure with the results of finite element simulations, we have seen that the deposition of a gold layer on top of the structure to make up the terminals source, gate and drain is not homogenous and has considerable roughness. In particular, we have noted that the effect of the gold film is very pronounced and thus can be exploited for further investigations. In this frame, we are looking forward to finding a calibration procedure based on the thermal expansion of the gold film in order to determine the temperature of the hot spot.

8.2.3 3-D finite element simulations

FE simulations performed in this work were essentially reduced to 1-D or to 2-D models in accordance with the nature of the thermal loads applied on the structures. For example, in the case of HEMT-structures, where the main interest was limited to the active area, the line source in the very neighborhood of the gate, induced by a modulated Joule heating applied on a point near the gate, at the drain side, had a constant intensity across each cross-section of the line, thus giving the possibility to treat the heat transport as 2-D problem. However, if the hot line adjacent to the gate does not cover the entire gate width, a formulation of the 3-D thermal transport in the microstructure is more appropriate. We are also looking forward to building up numerical programs aimed at investigating such kinds of thermal wave problems.

159 Appendix

Appendix A: Characteristics of the investigated samples

Samples Substrate Coating Codes Form Material Thickness Material Thickness /mm /µm C6 cylindrical HSS (1) 3.18 DLC (1)-var 1 (2) ≈1-3 C4 parallelepipedic HMA 3.50 DLC-var 1 ≈1-3 F5 cylindrical HSS 3.51 DLC-var 2 ≈4 F3 parallelepipedic HMA 3.19 DLC-var 2 ≈4 X4 parallelepipedic HMA 3.19 DLC-var 3 ≈4 X7 cylindrical HSS 3.51 DCL-var 3 ≈4 A3 cylindrical HSS ≈1-2 DLC-var 4 ≈2-3 A5 cylindrical HSS ≈1-2 DCL-var 5 ≈2-3 119P2 cylindrical HSS ≈1-2 DLC-var 6 ≈2-3 22P1 cylindrical HSS ≈1-2 DLC-var 7 ≈2-3

(1) see the list of abbreviations

(2) Variants of the DLC

160 8.2 Future work

Appendix B: Materials and their properties (RT) used for FE simulations

Sample Mass density Specific heat Thermal Coefficient of linear Chemical symbol and capacity conductivity thermal expansion denomination kg m-3 J kg-1 K-1 W m-1 K-1 /K-1 Si Silicon 2328 700 115.0 2.60×10-6 SiO2 Silicon oxide 2200 745 1.400 0.50×10-6 GaAs Galium arsenide 5320 385.80 52.30 5.9×10-6 Au Gold 19300 130 320.0 14.2×10-6 GaN Galium nitride 6100 34 200.0 5.59×10-6 InSb Indium antimonide 5770 200 18.00 5.00×10-6 Pt Platinium 19970 144.10 52.08 8.00×10-6 W Tungsten 19345 132 176.0 4.50×10-6

H2O Water 1000 4186 0.600

Air (N2, O2, others) 1.16 1002 0.026

161

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Publications and conference contributions

Conference contributions

Eurotherm Seminar No 75, Microscale Heat Transfer 2, July 8-10, 2003, Reims, France Potentialities of photothermal displacement experiments in micro-scaled systems J. L. Nzodoum Fotsing, B. K. Bein, J. Pelzl.

6th International Conference on Quantitative Infrared Thermography (QIRT), September 25-27, 2002, Dubrovnik, Croatia Identification and Quantification of Heat Sources and Sinks, based on IR Thermography and Inverse Solutions of the heat diffusion equation J. Gibkes, J. L. Nzodoum Fotsing, J. Pelzl, B.K. Bein.

11th International Conference on Photothermal and Photoacoustic Phenomena (ICPPP), June 24-27, 2002, Toronto, Canada Optical Properties in the Visible and Infrared Spectrum of Fibre-reinforced Composites Determined by Combined Photoacoustic and IR Detection J. Gibkes, J. L. Nzodoum Fotsing, D. Dietzel, B.K. Bein, J. Pelzl.

Surface modifications and surface-protective coatings analysed by means of thermal waves B. K. Bein, J. L. Nzodoum Fotsing, J. Gibkes, I. Delgadillo-Holfort, D. Dietzel, J. Pelzl.

German Acoustical Society Meeting (DAGA), March 4-8, 2002, Bochum, Germany Calibration of Infrared Signals by the Photoacoustic Effect J. Gibkes, J. L. Nzodoum Fotsing, K. Simon, B.K. Bein, J.Pelzl.

Deutsche Physikalische Gesellschaft (DPG) Frühjahrstagung, March 26-30, 2001, Hamburg, Germany Transmission Thermal Wave Measurements of Superinsulation Foils J. L. Nzodoum Fotsing, A. Haj-Daoud, I. Delgadillo-Holtfort, D. Dietzel, J. Gibkes, N. Marquardt, J. Pelzl, B.K. Bein.

175 Publications and conference contributions

Publications

(1) Emissivity measurements by means of combined photoacoustics and modulated IR radiometry J. Gibkes, J.L. Nzodoum Fotsing, B.K. Bein, J. Pelzl, accepted for publication in J. Phys. IV, 2004.

(2) Evaluation of active semiconductor structures by combined scanning thermo-elastic microscopy and finite element simulations D. Dietzel, S. Chotikaprakhan, R. Meckenstock, J. L. Nzodoum Fotsing, J. Pelzl, R. Aubry, J.C. Jacquet, and S. Cassette, accepted for publication in J. Phys. IV, 2004.

(3) Analysis of semiconductor structures and devices by combined scanning thermal microscopy and scanning thermo-elastic microscopy D. Dietzel, S. Chotikaprakhan, J. L. Nzodoum Fotsing, B.K. Bein, X. Filip, J. Pelzl, accepted for publication in J. Phys. IV, 2004.

(4) Study of optical properties of Zn1-XBeXTe mixed crystals by means of combined modulated IR radiometry and photoacoustics M. Pawlak, J. Gibkes, J.L. Nzodoum Fotsing, J. Zakrzewski, M. Malinski, B. K. Bein, J. Pelzl, F. Firszt, A. Marasek, accepted for publication in J. Phys. IV, 2004.

(5) The Model of a thin semiconductor layer on a thermally thick semiconductor backing M. Maliński, L. Bychto, J. L. Nzodoum Fotsing, K. Junge, A. Patryn, accepted for publication in J. Phys. IV, 2004.

(6) Inverse solution of the thermal wave problem for two-layer systems J. L. Nzodoum Fotsing, J. Gibkes, B. K. Bein, J. Pelzl, J. Appl. Phys, (2004), submitted

(7) Potentialities of photothermal displacement experiments in micro-scaled systems J. L. Nzodoum Fotsing, B. K. Bein, J. Pelzl, Superlattices and Microstructures, Elsevier Ltd, (2003), in print.

(8) Laser modulated optical reflectance of thin semiconductor films on glass J L. Nzodoum Fotsing, M. Hoffmeyer, S. Schotikaprakhan, D. Dietzel, J. Pelzl, B. K. Bein, F. Cerqueira, F. Macedo, J.A. Ferreira, Rev. of Sci. Instruments, Vol. 74 (1), pp. 873-876, Jan. 2003.

176 Publications and conference contributions

(9) Calibration of Infrared Signals by the Photoacoustic Effect J. Gibkes, J. L. Nzodoum Fotsing, K. Simon, B.K. Bein, J.Pelzl, in Proc. DAGA 2002

(10) Characterization of Si-based Films by Laser Modulated Optical Reflectance and IR Radiometry D. Dietzel, F. Cerqueira, M. Hoffmeyer, I. Delgadillo-Holtfort, J. L. Nzodoum Fotsing, J.A. Ferreira, J. Pelzl, B.K. Bein, In: ISPC 15, Symposium Proceedings 15th Intern. Symp. on Plasma Chemistry (Eds. A. Bouchoule, J.M. Pouvesle, A.L. Thomann, J.M. Bauchire and E. Robert, GREMI-ESPEO, Orleans, 2001), Vol. VIII (Post deadl. papers), p.3175-3180.

(11) Charakterisierung von Halbleiter-Schichten mittels Laserstrahl-modulierter IR- Radiometrie und Thermoreflexion F. Cerqueira, M. Hoffmeyer, D. Dietzel, I. Delgadillo-Holtfort, J. L. Nzodoum- Fotsing, J.A. Ferreira, J. Pelzl, B.K. Bein, In: Verbundwerkstoffe und Werkstoffverbunde (eds. B. Wielage und G. Leonhardt), Wiley-VCH Weinheim, 2001, 593-599.

(12) Untersuchung von beschichteten Festkörperoberflächen mittels thermischer Wellen J. Gibkes, J. L. Nzodoum Fotsing, I. Delgadillo-Holtfort, J. Pelzl, B.K. Bein, in: Verbundwerkstoffe und Werkstoffverbunde (eds. B. Wielage und G. Leonhardt), Wiley-VCH Weinheim, 2001, 600-607.

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CURRICULUM VITAE

Personal data

Name: NZODOUM FOTSING First Name: Jean Lazare Date, place of birth: May 08, 1968 at Bamendjou, Cameroon Nationality: Cameroonian Civil Status: Married, three children Adress: Auf der Prinz 89, D-44791 Bochum, Germany Home adress c/o Dr. J. L. Tamesse, ENS, Po.Box, 47, Yaoundé, Cameroon Electronic mail: [email protected], [email protected] Telephone: 0049-(0)234-32-23610, 0049-(0)234-50-71460

Academic education, extra-academic, pedagogical and scientific activities

From Aug. 2002 to present Research Assistant

Aug. 2001 − July 2002 Member of the Graduate Research Center (GRK 384) (Nanoelectronics, Micromechanics, and Microoptics)

Sept. 08th, 2000 Enrolment at the Institute for Experimental Physics III, Solid State Spectroscopy, Ruhr – University Bochum, Germany

1998 – 1999 Teaching Assistant, Department of Physics, University of Yaounde I, Yaoundé − Cameroon

Nov. 1997 – Oct. 1999 Doctorate student, Department of Physics, University of Yaounde I, Yaoundé, Cameroon Preliminar subject: Assessment of the thermophysical properties of materials under micro-gravity conditions

1997- 1998 Part time work at the universities printings University of Yaoundé I and Catholic University, Yde, Cam

Sept. 1995 – June 1997 Master of Physics student (Master Degree, 1997), Laboratory of Thermics and Energetics – Department of Physics, Faculty of Science, University of Yaoundé I (Yaoundé, Cameroon).

178 Curriculum vitae

Title of thesis and date of disputation: "Influence of the relaxation time on the energy stored by a conducting spherical particle irradiated by a short laser pulse"; April 30th, 1997.

1994 – 1995 Teacher of Physics, Chemistry and Mathematics at Private High School ISEIG, Yaoundé – Cameroon Member of the Cameroon’s Association of Volunteers, AVC

1990 – 1994 Physics and Chemistry student (Bachelor Degree in Physics, 1994)

October 1990 Admission to the Faculty of Science, University of Yaoundé (Cameroon)

June 1990 General Certificate of Advanced Level (Scientific Baccalaureat) Yaoundé – Cameroon

1986 - 1988 Vice – President (86/87) and President (87/88) of the Bamendjou association of students - N’kongsamba branch

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Acknowledgements

A scientific work results as fruit of permanent guidance and collaboration. The present thesis would have not been possible without the personal implication of my supervisor, Mr. Prof. Dr. J. Pelzl, who accepted me in his laboratory and provided me with necessary means to achieve this work. To you Professor, I would like to express my sincere gratitude for your guidance and encouragements. My second thanks go to Mr. Dr. B. K. Bein, the leader of the Photoacoustic and Photothermal Research Department, a very experienced scientist with whom several tasks in connection with my reseach work were conducted. From our frequent discussions and by its constant critics I usually came out with new ideas and a larger view of my work. I would like to thank Mr. Prof. Dr. A. Wieck, the speaker of the Graduate Research Center (GRK 384), for giving me the opportunity to achieve a part of the present work in the frame of a scholarship of the German Research Foundation (DFG). Certainly without the contribution of Mrs. Dr. Isabel Delgadilio-Holfort, a Research Scientist who shared the room with me during my stay at the Bochum University, I would have not succeed to dominate and overcome some difficulties encountered during the achievement of this thesis. I particulary enjoyed the good working atmosphere in our office. Special thanks to the whole group, especially Mrs. V. Kubiak and Mr. D. Krüger, respectively the secretary and the technician of the group, Mr. Dr. D. Spoddig and Mr. Dipl.-Phys. M. Möller for helping me to solve unavoidable computer problems, Mr. Dipl.- Phys. J. Gibkes for sharing with me its long experience in the field of Photoacoustics and Photothermics, and Mr. Dr. R. Meckenstock, for having several useful discussions with me. I am particulary pleased to address my sincere thanks to: Mr. Dr. F. Macedo and Mr. Prof. J.A. Ferreira, Department of Physics, University of Braga, Portugal, who introduced me to the Mirage Effect technique and helped me during my four stays in Portugal. Mr. Eng. Jean-Claude Jacquet, the responsible in charge of Modeling at Thalès Research & Technology, Paris, France, with whom I had first discussions related to the ANSYS software during a stay in Paris. Mrs. Dr. Xenia Filip, Institute of Isotopic and Molecular Technology, Cluj, Rumania, who arrived twice in our laboratory and worked with the ANSYS software. We had the opportunity to discuss about the difficulties in connection with that software. Mr. Dr. M. Malinski, Department of Electronics, Technical University of Koszalin, Poland, who welcomed me and provided me with all necessary documents during my stays in Poland. Finally, this work can formally be revendicated by my lovely wife, Stéphanie, who, by its patience, comprehension and several advices helped me to go ahead despite many difficulties. My dear love, you have sacrified yourself for me and our children, Erika Charlene, Hendrik Nelson and Ulrick Rayan. May the present thesis serve as a mark of courage, tenacity and perseverance to our progeniture.

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