C 2016 Subhadeep De THEORY and SIMULATION of SURFACE EFFECTS on INTRINSIC DISSIPATION

c 2016 Subhadeep De THEORY AND SIMULATION OF SURFACE EFFECTS ON INTRINSIC DISSIPATION

SUBHADEEP DE

THESIS

Submitted in partial fulﬁllment of the requirements for the degree of Master of Science in Mechanical Engineering in the Graduate College of the University of Illinois at Urbana-Champaign, 2016

Urbana, Illinois

Adviser:

Professor Narayana R. Aluru ABSTRACT

The goal of this research is to study the role of surface on intrinsic dissipation in silicon (Si) nanostructures. We look at two different cases namely (i) Si(100) surface with (2x1) reconstruction and (ii) Hydrogen (H) terminated Si surface. We utilize molecular dynamics (MD) simulations and show that the two surfaces play opposing role on intrinsic dissipation. While surface defects always aid in the entropy generation process, the scattering of phonons from rough surfaces can suppress Akhiezer damping. For the case of (2x1) reconstructed silicon surface, the former dominates and the inverse quality factor (Q−1) is found to increase with the decrease in size. However, different scaling trends are observed in the case of a H-terminated silicon surface with no defects and dimers. Particularly, in the case of a H-terminated silicon, if the resonator is operated with a frequency Ω such that Ωτph < 1, −1 where τph is the phonon relaxation time, Q is found to decrease with the decrease in size. The opposite scaling is observed for Ωτph > 1. A simplified model, based on two phonon groups (with positive and negative Grüneisen parameters), is considered to explain the observed trend. We show that the equilibration time between the two mode groups decreases with the decrease in size for the H-terminated structure. We also study the scaling of Q factor with frequency for these cases.

Key Words: intrinsic dissipation, Akhiezer, molecular dynamics, surface, phonon, relaxation time

ii ACKNOWLEDGMENTS

I would like to extend my sincere gratitude towards Prof. N. R. Aluru for giving me the opportunity to work on an exciting problem and supporting me during the course of my master’s degree. His keen insights and critique has been invaluable towards the successful execution of this research.

I would like to thank Kunal for the countless enlightening discussions and valuable suggestions through out the course of this research. I would also like to thank all my research group members for providing a stimulating work environment.

A special note of thanks for my parents, for their tireless support and constant encouragement, which, even from thousands of miles away, inspired me to move forward, no matter the obstacle. I am also indebted to the beautiful group of friends in Urbana Champaign, who made my life exciting and enjoyable outside the research lab.

I would like to acknowledge the support of National Science Foundation under grant number 1420882 and 1506619.

iii TABLE OF CONTENTS

LIST OF FIGURES ...... vi

LIST OF ABBREVIATIONS ...... viii

CHAPTER 1 INTRODUCTION ...... 1

1.1 Background and motivation ...... 1

1.2 Objectives ...... 6

1.3 Thesis overview ...... 7

CHAPTER 2 DYNAMIC ANALYSIS OF NANORESONATORS . . 8

2.1 Continuum description ...... 8

2.2 Atomistic description ...... 9

2.3 Phonon dynamics ...... 12

2.4 A simplistic dissipation model ...... 14

CHAPTER 3 MOLECULAR DYNAMICS SIMULATION AND COMPUTATIONAL METHODS ...... 16

3.1 Simulation setup ...... 16

3.2 Computational framework ...... 19

CHAPTER 4 RESULTS AND DISCUSSIONS ...... 20

iv CHAPTER 5 CONCLUSION ...... 27

REFERENCES ...... 28

v LIST OF FIGURES

1.1 Vapor sensor fabricated from single layer MoS2 [1]...... 2

1.2 Mass sensing using cantilever nanoresonator ...... 2

2.1 Beam dynamics ...... 8

2.2 (a) Atomistic picture of thermal motion, (b) One mode of vibration, in a nanoresonator...... 10

2.3 Illustration of intrinsic dissipation mechanisms. Sinusoidal strain field in the structure is coupled with different intrinsic dissipation mechanisms. This leads to irreversible flow of mechanical energy into internal energy of the system. The schematic on the left shows mechanism underlying dissipation due to phonon interactions. The reference configuration of all the phonons of the structure in thermal equilibrium at temperature T in the unstrained condition is shown by yellow solid line. Upon straining, these phonons absorb different amount of energy and undergo change in temperature accordingly. The mean change in temperature of the ‘hot’ phonon group under strained condition is shown with a red solid line and that of the ‘cold’ phonon group with a blue solid line. The two groups subsequently exchange energy with each other and relax. The schematic on the right shows the two-configurational minima with a separation d, barrier height v, and asymmetry ∆, which can form due to surface defects. Strain can modulate transition between these two minima with an

amplitude ∆0 and dissipate energy...... 14

3.1 (a) MD simulation setup for bulk silicon. (b) Internal energy vs periods of deformation...... 17

3.2 (a) MD simulation setup for H-terminated silicon nanostructure. (b) (2x1) reconstruction on the free silicon surface at 300 K ...... 18

vi 4.1 Scaling of energy dissipated per unit period with frequency for bulk silicon and H-terminated silicon of diﬀerent thicknesses. The data points are dissipation values obtained from NEMD simulations. Er- rors in the data are shown using the black bars. The colored dashed lines are the curve ﬁtting with a double-relaxation time (DRT) Akhiezer model...... 20

4.2 Energy correlation of the phonon group for bulk silicon and H-terminated silicon of different thicknesses. The colored dashed lines represent the data obtained from EMD simulations. The biexponential fits to the data are shown using the black solid lines. The slow (τ s) and fast (τ f ) timescales of decay of the biexponential fits are shown individually using black dashed and dotdashed lines, respectively...... 21

4.3 Diﬀerent phonon scattering processes...... 22

2 4.4 (a) τph vs thickness for H-terminated silicon. (b) Variation of ∆γ with thickness of H-terminated silicon obtained using the DRT model (c) PCA of displacements of the surface atoms for free and H-terminated silicon...... 23

4.5 Scaling of dissipation with size for diﬀerent frequencies. The data points are dissipation values obtained from nonequilibrium MD simulations. Errors in the data are shown using the black bars...... 25

vii LIST OF ABBREVIATIONS

NEMS Nanoelectromechanical system

SVR Surface to volume ratio

TED Thermoelastic damping

MFP Mean free path

SRT Single relaxation time

DRT Double relaxation time

MD Molecular dynamics

EMD Equilibrium molecular dynamics

NEMD Nonequilibrium molecular dynamics

QH Quasiharmonic

QHMK K-space quasiharmonic method

QHMR Real space quasiharmonic method

viii CHAPTER 1

INTRODUCTION

1.1 Background and motivation

1.1.1 Nanoelectromechanical systems

An electromechanical system couples a mechanical element to an electronic circuit. Miniaturization of these electromechanical devices to nanometer dimension has led to the emergence of a class of devices called Nanoelectrome- chanical systems (NEMS) [2]. A principal component of these NEMS devices is a nanomechanical resonator [3]. Typically these nanoresonators are char- acterized by very low mass, very high mechanical resonance frequencies and stiﬀness and large surface-to-volume ratio (SVR) [4]. These attributes of NEMS devices have been exploited to achieve previously unprecedented sen- sitivity in measurement of fundamental quantities like position, mass, charge etc. [5, 6, 7, 8].

1.1.2 Factors inﬂuencing NEMS performance

The performance of these nanomechanical resonator for the aforementioned applications is limited by mechanisms that dissipate the mechanical energy [9]. This can be demonstrated by considering the property of nanoresonators as precision mass sensors. A schematic of single layer MoS2 mass sensor is shown in Fig. 1.1. The resonant frequency f0, of the nanoresonator which is related to its mass m0, shifts on addition of any foreign mass ∆m. The shift

1 Figure 1.1: Vapor sensor fabricated from single layer MoS2 [1]. in resonant frequency ∆f ≈ R∆m is detected, and, using this priniciple, mass sensing to the level of individual atom has been achieved [10, 11, 12, 6]. Here R ∝ − f0 is the mass responsivity of the resonator and it is a material 2m0 constant. Thus, the minimum detectable mass ∆m critically depends on the minimum measurable frequency shift ∆f. It can be shown that ∆f increases with the increase in dissipation and thus degrades the performance of a mass sensor [13]. This is not particular to precision mass measurements. In general, dissipation limits the ability of NEMS to sense changes in its environment.

(a) Cantilever resonator (b) Amplitude response curve

Figure 1.2: Mass sensing using cantilever nanoresonator

2 1.1.3 Energy losses in NEMS

In order to improve the performance of NEMS, it is required to identify different dissipation mechanisms associated with its operation and avoid them [14, 15]. Dissipation is generally quantified by a dimensionless parameter Q- factor (Q ∝ 1/Dissipation), called quality factor. Loss of vibrational energy of the resonator can take place due to coupling with the external environment which is an extrinsic dissipation mechanism. Most common extrinsic dissipation mechanisms are air damping and clamping loss. Air or fluid damping refers to the loss of mechanical energy due to the colliding fluid molecules [16, 17]. This loss is not present when operated in vacuum. Energy can also be lost to the resonator anchor from the resonator, which is referred to as clamping or anchor loss [18, 19, 20, 21]. This energy loss can be substantially mitigated by carefully designing the geometry of the resonator.

1.1.4 Intrinsic dissipation

Intrinsic dissipation, on the other hand, is an energy loss mechanism that involves coupling between mechanical deformation and internal thermal vibration of atoms in the resonator [22]. It should be pointed that the micro- scopic thermal oscillation of the atoms is often viewed in terms of modes of vibration with a particular frequency ω, called ‘phonon modes’. Classically, the temperature of the resonator is the measure of the average energy of all the phonon modes.

Diﬀerent intrinsic dissipation mechanisms are thermoelastic damping (TED) [23, 24]; Akhiezer damping [25, 26]; nonlinear coupling between mechanical modes [27, 28, 29]; surface mediated defects [30, 31]; and, internal defects [32, 33, 34]. These loss mechanisms are characteristic of the material property and cannot be eliminated. Thus, intrinsic damping mechanisms provide a fundamental upper limit to the Q factor of the resonator.

3 Thermoelastic damping

In the case of single crystal bulk silicon nanoresonators, TED and Akhiezer damping are the dominant intrinsic loss mechanisms at room temperature. Both the dissipation mechanisms can be explained by classical theory. In the case of real isotropic solids (where perfectly elastic idealization does not hold true), any strain field (uniaxial/volumetric) developed is coupled to a temperature field by the parameter α, called thermal expansion coefficient ∂ (uniaxial/volumetric) according to the relation [35] α = ∂θ , where and θ could be space and time dependent. Thus, spatial variation in strain in a structure will be accompanied by a local temperature gradient. In order to equalize the temperature difference, internal heat flow occurs which leads to entropy generation, and consequently dissipation. The Q factor due to TED is given by [36, 37, 38]

2 −1 α ET ωτd Q = 2 (1.1) Cv 1 + (ωτd) where α is the coeﬃcient of thermal expansion, E is the Young’s modulus, T is the mean temperature, Cv is the speciﬁc -heat capacity at constant volume,

ω is the angular frequency of oscillation, and τd is the thermal diffusion time. If the temperature gradient is considered across a width, w in a material with 2 w Cv thermal conductivity, κ, then τd = π2κ . Thus the strength of TED depends on the length across which temperature/strain gradient is developed apart from the material property [23]. In a system isolated from its surrounding, TED will be operative only when (i) a spatially inhomogeneous strain field exists; (ii) And, the temporal variation of the strain field is slow enough to allow sufficient local thermalization of the phonons.

Akhiezer damping

If the timescale of deformation is comparable to phonon relaxation time, dissipation due to Akhiezer mechanism starts playing a major role [39]. Akhiezer dissipation is a phonon mediated loss mechanism which takes place as a result of heat flow between different phonon modes. Any applied strain field inter-

4 acts with the ensemble of thermal phonons and modulate their frequencies.

For a particular mode µ, the fractional change in frequency ∆ωµ/ωµ due to a locally defined uniaxial strain is given by a mode-dependent Grüneisen number γ as ∆ωµ = γ [35]. Now, energy of a phonon mode is proportional µ ωµ µ to its frequency. The applied deformation, therefore, results in a spectral energy distribution which tries to relax towards a mean energy. This in- tramodal exchange of energy due to anharmonic coupling of phonon modes is an entropy generating process. This is the basis of Akhiezer theory of dissipation. Unlike TED which depends on strain gradient, Akhiezer dissipation is more fundamental. Under the condition of uniform strain field, dissipation in a bulk nanoresonator at gigahertz frequency is governed by Akhiezer mechanism [22].

Extensive studies on Akhiezer mechanism started as early as 1950. Back then, most of the studies were on attenuation of accoustic (sound) waves in dielectric crystals [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. In case of very high frequency nanomechanical resonators, the eigen modes of deformation can be described as stationary acoustic wave. Thus, the theories of absorption of sound waves have a direct correspondence to Akhiezer damping in nanoresonators.

Dissipation due to Akhiezer mechanism involves anharmonic phonon coupling and redistribution of phonon energy. This is often referred to as scattering processes and classified as phonon-phonon scattering, phonon-boundary scattering, etc. Scattering of phonon due to three phonon interaction known as three phonon scattering is predominant in bulk structures. But, for a finite-sized nanostructure, boundary or surface scattering plays a major role due to the high surface area to volume ratio (SVR). Although it is apparent that the dissipation rate is a strongly size dependent phenomena, it is not well understood how it scales with size. Numerous studies have been conducted on the size dependence of thermal conductivity. It has been shown that surface reduces the average mean free path (MFP) of thermal phonons and thus the thermal conductivity [51, 52, 53] in silicon nanostructures. However, the role of surface and the finite size effect on dissipation is relatively less ex- plored. Some experimental observations reveal that the Q factor of flexural modes decreases, approximately, linearly with the decrease in resonator size

5 [14]. This has been interpreted as a signature of surface related dissipation mechanism including defects and (2x1) reconstruction [54]. In this research, we focus on investigating, in a detailed way, the role of surface on intrinsic dissipation using non-equilibrium molecular dynamics simulations. We note that in silicon nanostructures, with hydrogen termination, two opposite size scaling of dissipation can be observed. This scaling of Q factor with size depends on the operation frequency and is explained using Akhiezer mechanism. Therefore, we assert that, contrary to the conventional idea, surface contributions can both aid and abate dissipation in a hydrogen terminated silicon nanostructure. However, (2x1) reconstruction, dangling bonds and defects in a bare silicon surface introduce additional dissipation mechanisms. This overshadows the scaling relation expected from the Akhiezer theory.

1.2 Objectives

In the current study, we first introduce, based on classical mechanics, a simplistic model to describe Akhiezer dissipation. Two parameters of signifi- cance associated with this proposed model are the time-scale of energy exchange between the phonon modes, τph, and the mode Grüneisenparameter,

γµ. We outline briefly, the underlying physics and the strategies to extract these parameters from the interatomic potential using different quasiharmonic techniques. We, then, perform a frequency study of dissipation for bulk silicon under stretching deformation. Stretching mode gives the simplicity of a nearly uniform strain field in a nanometer scale structure at the desired frequencies [26]. Dissipation in a bulk structure is governed by the Akhiezer mechanism and is described using the reduced order model. We, next, perform the frequency and size study of dissipation for hydrogen (H) terminated silicon nanostructures. Akhiezer damping is found to be the dominant mechanism. Surface scattering manifests itself in the Akhiezer damping by reducing timescale of energy relaxation of the thermal phonons. Our observation discloses contrary roles of the surface in assisting dissipation. We also study the scaling of dissipation with size for silicon structure with free surfaces. We try to ascertain the additional dissipation in bare silicon sur-

6 faces with (2x1) reconstruction as a strain driven transition between diﬀerent conﬁgurational minima.

1.3 Thesis overview

The thesis is organized as follows:

Chapter 2 aims towards developing a simplistic model for intrinsic dissipation in nanoresonators. In the beginning of the chapter, using continuum equation of motion, the macroscopic deformation of the nanoresonator is described. Then, general overview of the thermal motion of the atoms and phonon modes is presented. It is shown that due to anharmonicity in the interatomic potential, macroscale deformation interacts with dynamics of phonons. This interaction term is introduced in classical energy equation of each phonon. Then, making some simplifying approximation, a closed form expression is derived for Akhiezer dissipation.

Chapter 3 describes the simulation setup for studying intrinsic dissipation and outlines the computational framework. First part of the chapter presents the method to study intrinsic dissipation using molecular dynamics. It also provides the details of the structure, interatomic potential and ensembles pertaining to the simulations in our current study. Second part of the chapter discusses the quasiharmonic techniques and strategies employed to extract the parameters for the proposed simplistic dissipation model.

Chapter 4 presents the results from the molecular dynamics simulation and compares it with the proposed model. In this chapter, scaling of dissipation with frequency and size for bulk and ﬁnite-sized silicon structures is discussed. Dissipation is compared for diﬀerent surface conditions and their role on intrinsic dissipation is also discussed.

Chapter 5 concludes the discussion and summarizes the ﬁndings.

7 CHAPTER 2

DYNAMIC ANALYSIS OF NANORESONATORS

2.1 Continuum description

Let us consider a perfectly elastic beam of length L0 with uniform cross- sectional area S0 as shown in Fig. 2.1. The equation of motion of the beam under application of periodic axial force f(x, t) is given by Eq. (2.1)

∂2u(x, t) ∂2u(x, t) ES + ρS = f(x, t) (2.1) 0 ∂x2 0 ∂t2

If the periodic force is applied to the end with a frequency Ω, then f(x, t) −jΩt can be written as f(x, t) = Re[F0δ(x − L0)e ], where F0 is the magnitude of the applied load and δ(x) is the Dirac delta function. The solution to the

Figure 2.1: Beam dynamics

πx −jΩt above equation of motion can be written as u(x, t) = Re[Am sin( 2L∗ )e ]. ∗ π q E Here Am is the measure of amplitude of oscillation and L = 2Ω ρ . The π q E fundamental mode of vibration of the beam has a frequency ω0 = . 2L0 ρ If the beam resonator is operated at a frequency much less than the fun- ∗ damental frequency (Ω ω0), then it would imply L0 L . Thus the

8 displacement ﬁeld u(x, t) can be approximated as

h πx i u(x, t) ≈ Re A e−jΩt (2.2) m 2L∗

The linearity in the displacement field results in spatially uniform strain field in the structure. The periodic strain field can be represented as =

−jΩt πL0 Re[0e ], with amplitude 0 = Am 2L∗ . The maximum elastic energy stored in the structure during a period of deformation is given as

1 ES0 2 U = 0 (2.3) 2 L0

2.2 Atomistic description

An ideal crystal is an inﬁnite periodic array of group of atoms in three- dimensional space. The group of atoms, which may consist of one or more atoms, form the basis. This basis can be attached to a set points in three- dimension called the lattice. The crystal structure is deﬁned by addition of the basis to every point in the lattice [55].

2.2.1 Thermal oscillation

In a crystal structure, the atoms are bound together due to presence of interatomic potential. At zero temperature, all the atoms rest in their equilibrium positions, in a classical solid. The equilibrium position of each atom, in this case, corresponds to a configuration that minimizes the total potential energy of the structure. As the temperature is raised, the atoms are set to motion. At any instant, the inertia force of the atom due to finite temperature is counteracted by the restoring force due to interatomic potential. Thus, the atoms exhibit an oscillatory motion about their equilibrium position at finite temperature, which is known as thermal oscillations. In the case of real solids, the bonded interactions between the atoms are nonlinear. Thus the equilibrium spacing between atoms varies with temperature. At

9 a given temperature and pressure, the equilibrium conﬁguration is the one which minimizes the total potential and entropic energy.

We suppose the potential energy of a system of N atoms at the equilibrium conﬁguration is given by V0. The total energy V of the system as result of atomic displacements ui,α about the equilibrium can be expressed by Taylor series expansion as

X X ∂V 1 X X ∂2V V =V0 + ui,α + ui,αuj,β ∂ui,α 0 2 ∂ui,α∂ui,β 0 i α i,j α,β 1 X X ∂3V + ui,αuj,βuk,γ + ··· (2.4) 6 ∂ui,α∂ui,β∂uk,γ 0 i,j,k α,β,γ

Here, i, j and k denotes sum over atoms in the system, and α, β and γ denotes sum over the x, y and z directions. The ﬁrst order term in the expansion which represents the negative of force acting on the atoms is zero at equilibrium.

(a) (b)

Figure 2.2: (a) Atomistic picture of thermal motion, (b) One mode of vibration, in a nanoresonator.

2.2.2 Normal modes

At room temperature, the displacements due to thermal oscillation of atoms remain small. Under quasiharmonic approximation [56], the Taylor series can be truncated at the second order term and the phonons can be treated

10 independently. The total vibrational energy of the system, then, takes the form

2 1 X X 2 1 X X ∂ V Evib = miu˙ i,α + ui,αuj,β (2.5) 2 2 ∂ui,α∂ui,β 0 i α i,j α,β

Since the vibrational energy has a quadratic dependence in phase-space variables (u ˙ i,α, ui,α), equations of atomic motions can be decoupled by suit- able transformation of coordinates. The transformation from 3N real space coordinates to a set of 3N new coordinates Aµ gives us the normal modes of vibration, where m denotes the mode index. Each of these normal modes has ~ an associated mode shape (or polarization Pµ), direction (wave vector kµ) and frequency (ωµ). Each normal mode is equivalent to a harmonic oscilla- ˙ tor in a 3N dimensional space with phase space variable (Aµ(t)Pµ, Aµ(t)Pµ). The energy of the system in terms of modal coordinates can be written as

1 X 1 X ∂2V E = m¯ A˙ 2 + A˙ 2 (2.6) vib 2 µ µ 2 ∂A2 µ µ µ µ

Herem ¯ µ is the eﬀective mass of the oscillator and m sums over 3N oscillator. These normal modes of thermal vibration are called phonon modes.

2.2.3 Anharmonicity

The higher order terms in the expansion of potential energy in Eq. (2.4) give rise to phonon-phonon interaction which results in a shift in phonon frequencies and finite lifetimes of phonons [57]. Quasiharmonic (QH) lattice dynamics incorporates the anharmonic effect of the potential ‘partially’, by allowing dependence of the phonon frequencies on the volume of material. ∂2V Thus, under QH approximation, 2 in Eq. (2.6) is a function of volume ∂Aµ of the structure, v. Thermal expansion of solids, which is a result of anharmonicity in potential can be explained using this framework [58]. The dependence of modal frequencies on volume is captured by mode Grüneisen

11 parameter deﬁned by

v ∂ω (v) γ = 0 µ (2.7) µ ω (v ) ∂v µ 0 v=v0

Here, v0 is the volume of the structure in the reference conﬁguration. In general both the frequencies and eigen vectors of the normal mode will change with an applied strain, but often the change in the eigen vectors may be quite small [59, 60].

2.2.4 Phonon lifetimes

The anharmonic terms (order ≥ 3), which are excluded with the QH approximation, result in mode mixing and finite lifetimes of phonons [58]. Because of the anharmonic forces, the phonons interacts and exchange energy with each other. Small perturbations in the energy of the phonon modes takes a finite time to decay due to this intermixing of energy, which is termed as phonon relaxation time. The relaxation time of a phonon can be defined by [61, 58]

R ∞ 0 hδEµ(t)δEµ(0)i dt τµ = 2 (2.8) h(δEµ) i

2.3 Phonon dynamics

Under uniaxial straining of magnitude , the fractional change in frequency of any phonon mode µ, can be written as (∆ω/ω)µ = −γµ, where γµ is the strain coupling factor (Grüneisenparameter) and ω is the frequency of the phonon mode. This modulation of phonon frequencies with strain is a consequence of anharmonicity in the interatomic potential of crystals. For small strains about reference configuration, γµ is a constant and independent of strain. Strain field resulting from periodic deformations with frequency Ω in a structure is likely to change all the phonon frequencies periodically.

12 For simplicity, we can assume a time varying strain ﬁeld of the form (t) = jΩt Im[0e ], where 0 is the complex strain amplitude.

For a vibrating harmonic oscillator, whose frequency is slowly changing E with time (Ω << ω), the ratio ω is a constant,[62] E being the energy of the oscillator. This invariant enables us to relate the fractional change in frequency of any phonon mode to the fractional change in the energy as ∆E ∆ω E µ = ω µ = −γµ(t). The mean vibrational energy of any phonon mode in a structure in thermal equilibrium at temperature T , is given by kBT , where kB is the Boltzmann constant. This relation can be used to assign a temperature change (∆T )µ of the phonon mode with strain, given near equilibrium condition is maintained (by keeping |0| << 1). Depending on the sign of γµ, some phonon modes may have ∆T > 0 (red circles in Fig 2.3), and others ∆T < 0 (blue circles in Fig 2.3).

The time rate of change in energy of the µth mode due to a periodic straining can be expressed as

∂Eµ = −γµEµ˙(t) ∂t strain

1 jΩt ∗ −jΩt where ˙(t) = 2 0e + 0e Ω. The frequency modulation due to the strain wave disturbs the equilibrium condition. The phonon modes are no longer in thermal equilibrium but relax towards a mean energy of the system due to anharmonic coupling between the modes. The exchange of energy between the phonon modes can be described as a relaxation process. The energy exchanged by the µth mode due to phonon collision can be mathe- matically written as ¯ ∂Eµ Eµ − E = − . ∂t exc τµ ¯ Here, E is the mean energy and τµ is the relaxation time for mode µ. The diﬀerential form of time-dependent modal energies which appropriately describes the coupling of mechanical deformation with thermal vibration is, then, given by

¯ jΩt ∗ −jΩt ∂Eµ Eµ − E 0e + 0e + = −γµEµ Ω (2.9) ∂t τµ 2

13 Figure 2.3: Illustration of intrinsic dissipation mechanisms. Sinusoidal strain field in the structure is coupled with different intrinsic dissipation mechanisms. This leads to irreversible flow of mechanical energy into internal energy of the system. The schematic on the left shows mechanism underlying dissipation due to phonon interactions. The reference configuration of all the phonons of the structure in thermal equilibrium at temperature T in the unstrained condition is shown by yellow solid line. Upon straining, these phonons absorb different amount of energy and undergo change in temperature accordingly. The mean change in temperature of the ‘hot’ phonon group under strained condition is shown with a red solid line and that of the ‘cold’ phonon group with a blue solid line. The two groups subsequently exchange energy with each other and relax. The schematic on the right shows the two-configurational minima with a separation d, barrier height v, and asymmetry ∆, which can form due to surface defects. Strain can modulate transition between these two minima with an amplitude ∆0 and dissipate energy. 2.4 A simplistic dissipation model

Instead of looking at each phonon mode, we can group them into a set of two phonon bunches. All the phonon modes that undergo positive change in temperature (γµ < 0) with tensile strain form a hot phonon bunch (denoted by subscript ‘h’). And the rest with zero or negative change in temperature

(γµ > 0) form a cold phonon bunch (denoted by subscript ‘c’). This approximation simpliﬁes the idea of energy exchange between the two phonon bunches with some relaxation time, τph (schematically shown in Fig 2.3). The time rate of change of the energy diﬀerence, ∆E, between the two mode groups is obtained as

d∆E ∆E ejΩt + ∗e−jΩt + ≈ −∆γE¯ 0 0 Ω (2.10) dt τph 2

Eh−Ec γh−γc ¯ Eh+Ec γh+γc Here, ∆E = 2 , ∆γ = 2 , E = 2 andγ ¯ = 2 . γh/c is the mean γ and Eh/c is the mean energy of the respective phonon bunch. The time

14 rate of change of mean energy of the system is given as

dE¯ ejΩt + ∗e−jΩt = − γ¯E¯ + ∆γ∆E 0 0 Ω (2.11) dt 2

The net change in mean energy of the system over a period of oscillation will give us the amount of dissipated energy. Taking E¯ to be approximately constant and equal to kBT , the energy dissipated over a period of oscillation 1 R Tp jΩt ∗ −jΩt is given by Ediss = 2 0 ∆γ∆E(0e + 0e )Ω dt. The energy dissipated attains a steady state after few periods of oscillation (nTp >> τph), where n is the number of oscillation periods. Using equation [2.10], the analytical solution for Ediss at steady state is obtained as[22]

2 2 Ωτph Ediss = πkBT |0| ∆γ 2 . (2.12) 1 + (Ωτph)

This Akhiezer dissipation model is based on a single timescale of energy relaxation of the phonon groups. If the energy relaxation exhibits multiple i timescales, τph, the model can be extended as

i X Ωτph E = πk T | |2 ∆γ2 . (2.13) diss B 0 i 1 + (Ωτ i )2 i ph

In our study, we observe two well separated timescales of energy relaxation. s f We denote the slow timescale as τph and the fast timescale as τph. We use a double relaxation time (DRT) Akhiezer model to explain the frequency scaling of dissipation. In order to study the size scaling of dissipation, we investigate the variation of size dependent material parameters, ∆γ and τph, with thickness.

15 CHAPTER 3

MOLECULAR DYNAMICS SIMULATION AND COMPUTATIONAL METHODS

3.1 Simulation setup

In our studies, classical molecular dynamics simulations are performed using large-scale atomic/molecular massively parallel simulator (LAMMPS) [63] package. Visual Molecular dynamics [64] is used for visualization. Three body Tersoﬀ potential is used to model the bonded interaction between the atoms in all the three cases namely (i) bulk silicon, (ii) silicon with free surface and (iii) silicon with hydrogen terminated surface. Equilibrium molecular dynamics (EMD) simulations are carried out for equilibration and calculation of phonon properties. Nonequilibrium molecular dynamics (NEMD) simulations are carried to calculate dissipation.

3.1.1 Bulk

In the case of bulk silicon, a cubic block is generated by arranging atoms in FCC-diamond lattice with a lattice spacing of 5.427 A.˚ Each side of the block is fixed to 54.27 A˚ which corresponds to 10 unit cells in the [100] direction as shown Fig. 3.1 a. One of the sides of the cubic block is chosen to be aligned along X-axis and other two along Y and Z axes respectively. Periodic boundary conditions (PBC) are applied in all the three directions. To describe the Si-Si interaction, we use Tersoff potential [65], which is sufficiently accurate in prediction of mechanical [66] and thermodynamic properties [67]. We start with a fully relaxed structure, which correspond to minimum potential energy configuration at 0 K. Nosé-Hoover thermostat [68] with a time

16 constant of 1 ps is used to equilibrate the structure at 300 K. The structure is relaxed at 300 K by running NPT simulation for 2ns. Diﬀerent ensembles of this equilibrated and relaxed structure at 300 K is generated for further simulations.

(a) (b)

Figure 3.1: (a) MD simulation setup for bulk silicon. (b) Internal energy vs periods of deformation.

Phonon properties are calculated based on EMD trajectories. For this, the equilibrated ensembles are time integrated with a time step of 1 fs using NVE simulation without any thermostat and the atomic displacements and velocities are dumped every 20 fs. In order to estimate dissipation, we resort to NEMD simulations. The simulation box is periodically deformed along x direction and equations of motion of each atom is integrated based on Verlet scheme using NVE simulation. The system is evolved for a simulation time equivalent to 100 periods of deformation. The total energy of the system, averaged over each period, is found to increase linearly with the periods of deformations as shown in Fig. 3.1 b. The rate of increase of internal energy of the system per unit period gives the average energy dissipated over one period, Ediss

Dissipation is measured in terms of the dimensionless Q factor deﬁned as

Esto Q = 2π . Here, Esto is the maximum elastic energy stored during a pe- Ediss riod of deformation. Esto is obtained using separate equilibrium simulations. Starting with the equilibrated ensembles, the structure is deformed and allowed to relax for 1 ns for different values of strain. Esto in the structure varies as the square of strain. Using curve fit, we obtain the constant of proportionality, which is the stiffness of the structure, k. Esto at any stain 2 amplitude, 0 can then be calculated as 1/2 k0. For frequency study of dissipation, the deformation frequency is varied from 5 GHz to 150 GHz at 2%

17 strain amplitude.

3.1.2 Finite size

For MD simulations of finite-sized silicon nanostructures, periodic boundary condition is eliminated from Z direction. The structure now resembles an infinite silicon slab with a low index surface Si(100) and finite thickness along Z axis. Free Si(100) surface of silicon is not stable at room temperature [69]. It has two dangling bonds [70] which undergoes surface reconstruction [71, 72]. The symmetric dimer formation for (2x1) reconstruction on Si(100) surface is well captured by tersoff potential [73, 74]. Fig. 3.2 b shows the formation of (2x1) reconstruction in Si(100) surface at 300 K. In the free surface case, dissipation is studied for thickness ranging from 25.6 A˚ to 86.5 A.˚ NEMD simulations are carried out at 10 GHz and 100 GHz with 2% strain amplitude. In case of silicon nanostructures with free surface, intrinsic dissipation can be influenced by both finite size and surface reconstruction. In order to isolate these to effects, we consider the case of Hydrogen passivated silicon nanostructures.

(a) (b)

Figure 3.2: (a) MD simulation setup for H-terminated silicon nanostructure. (b) (2x1) reconstruction on the free silicon surface at 300 K

Termination of the two dangling bonds in Si(100) surface with two Hydro- gen (H) atoms can prevent the surface reconstruction [75]. In the present case, hybrid Tersoff potential [76] is used to model the force field in H- terminated silicon nanostructures. The parameters describing Si-H interaction in the Tersoff potential is taken from Ref. [77]. A H-terminated silicon

18 nanostructure at 300 K is shown in Fig. 3.2 a. Extremely small time step of 0.1 fs is used during MD simulations in order to preserve the Si-H bonds. Dissipation is studied at 10 GHz and 100 GHz deformation frequencies with 2% strain amplitude. Using equilibrium simulations, atomic trajectories are dumped for calculation of phonon properties.

3.2 Computational framework

The parameters τph and γ for the proposed dissipation model described by Eq. (2.12) are extracted using EMD trajectories and finite temperature quasi- continuum method. For the bulk structure, K-space quasi-harmonic method is employed to calculate the phonon mode shapes, Φµ and the corresponding frequencies, ωµ [78, 79, 80, 81]. The phonon frequencies are function of lattice spacing, which is in turn a function of strain. By differentiating the phonon frequencies with respect to strain, gives the Grüneisenconstants, γµ. Once c calculated, we can identify the cold phonons (µ 3 µ : γµ > 0) and hot h phonons (µ 3 µ : γµ < 0) and group them. 2 ns long EMD trajectories are projected onto the phonon mode shapes to get the time history of their modal displacements, Aµ(t) and modal velocities, vµ(t) [82, 83]. The total vibrational energy, Eµ(t) of each phonon mode is then obtained as summation 1 2 1 2 2 of the modal kinetic energy, 2 mvµ(t) and modal potential energy, 2 mωµAµ(t). Finally, mean energy, Ei(t) of a phonon bunch is calculated by averaging P Eµ(t) Eµ(t) over all the modes, µ in that bunch as Ei(t) = i , where i µ∈µ Ni denotes the hot/cold phonon bunch and Ni is the count of phonon modes in the hot/cold bunch. Normalized autocorrelation of the mean energy of 2 a phonon bunch is calculated as Ri(t) = hδEi(0)δEi(t)i/hδEi i [61], where

δEi(t) = Ei(t)−hEi(t)i and the angle brackets refer to the ensemble average.

In the case of finite-sized structures, real-space quasiharmonic method (QHMR) [78] is employed. QHMR can capture the influence of surface effects on the normal modes and frequencies. Using QHMR, Grüneisenparameters are obtained and the phonons are grouped. EMD trajectories are then used to calculate the average energy of each phonon group and subsequently compute τph as a function of size.

19 CHAPTER 4

RESULTS AND DISCUSSIONS

Dissipation is studied for frequencies ranging from 5 GHz to 150 GHz for the bulk structure. In Fig. 4.1, the red filled circles show the dissipation for the bulk structure obtained using NEMD simulation at different frequencies. The dissipation response has a well defined peak at a particular frequency and it sharply decays on either side of it. This Lorentzian nature of dissipation follows directly from its functional form of the Akhiezer model (in Eq. (2.12)). i i 2 For any functional dependence of the kind Ωτph/(1 + (Ωτph) ), the metric i Ωτph becomes crucial. Because the function goes through its maximum value i for Ωτph=1. Fitting the dissipation vs frequency data for bulk silicon with

Figure 4.1: Scaling of energy dissipated per unit period with frequency for bulk silicon and H-terminated silicon of diﬀerent thicknesses. The data points are dissipation values obtained from NEMD simulations. Errors in the data are shown using the black bars. The colored dashed lines are the curve ﬁtting with a double-relaxation time (DRT) Akhiezer model.

s f a DRT Akhiezer model yields τph=14.09 ps, ∆γs=4.01, τph=2.64 ps, and 2 2 ∆γf =1.86. Since ∆γs ∆γf , a single peak is observed which corresponds

20 to the slower timescale, i.e., 14.09 ps. Dissipation in bulk silicon in Fig. 4.1 attains maximum at a deformation frequency of Ω/2π =12 GHz which is s s f related to τph as Ωτph ≈ 1. Only at high frequencies(> 60 GHz), τph starts playing a role and its contribution to net dissipation is signiﬁcantly less when s compared to τph. This bulk dissipation response was found to be independent of the system size over a critical dimension of 4 lattice units.

From Fig. 4.2, Ri(t) is seen to exhibit two separate timescales of decay. The faster timescale is of the order ∼ 1 ps and the slower timescale is of the order ∼ 10 ps. Ri(t) can be ﬁt to a biexponential decay [58] of the

−t/τ1 −t/τ2 form Ce + (1 − C)e , where τ1, τ2 and C are the ﬁtting constants.

Performing the curve ﬁt in the case of bulk silicon, τ1 and τ2 are estimated

Figure 4.2: Energy correlation of the phonon group for bulk silicon and H-terminated silicon of different thicknesses. The colored dashed lines represent the data obtained from EMD simulations. The biexponential fits to the data are shown using the black solid lines. The slow (τ s) and fast (τ f ) timescales of decay of the biexponential fits are shown individually using black dashed and dotdashed lines, respectively. to be 13.45 ps and 1.725 ps. These timescales are in very good agreement s f with τph and τph obtained using the DRT Akhiezer model. Therefore, the distinct timescales of energy relaxation in a system can be inferred from the nature of decay of the energy autocorrelation of the phonon bunch. In bulk silicon, the intermixing of modal energies is solely governed by anharmonic

21 Figure 4.3: Diﬀerent phonon scattering processes. coupling of three (or more) phonon modes. This corresponds to the three- phonon scattering mechanism due to Normal and Umklapp processes. A pictorial representation of 3-phonon scattering is shown in Fig. 4.3. Due to periodicity in all the three directions, no boundary scattering is involved.

Studies on bulk silicon elucidate the underlying mechanism and provide us with a model to quantify dissipation. We will now investigate the role of surface on intrinsic dissipation. Free surface of silicon has dangling bonds and undergoes (2×1) reconstruction (as shown in Fig. 3.2b) at room temperature [69, 74]. Often, these dangling bonds are terminated with hydrogen atoms to avoid surface reconstruction and other forms of surface defects. We first look into the case of hydrogen (H) terminated silicon surface. The simulation setup is displayed in Fig. 3.2a. We considered H-terminated silicon nanostructures with thicknesses varying from 25.6 A˚ to 86.5 A˚ along the Z axis. Free surface boundary condition is applied in the Z direction. The other dimensions along the X and the Y axis are kept the same as the bulk case. The two dangling bonds [70] corresponding to each silicon atom on the surface were terminated by two H atoms. Hybrid Tersoff potential [77] is used to model the interaction between silicon and hydrogen atom. A very small time step of 0.1 fs is used to preserve the Si-H bond during simulations. Principal components of displacement of the surface Si atoms in the H-terminated silicon show a Gaussian distribution as shown in Fig. 4.4c. This confirms the absence of any surface defects or dimer formation in the H-terminated case.

Fig. 4.1 shows dissipation as a function of frequency for H-terminated silicon nanostructures of diﬀerent thicknesses. The nature of the dissipation response closely resembles the bulk silicon case. With the decrease in thick-

22 ness, there is a shift in the dissipation peak towards higher frequencies. Also, the magnitude of maximum dissipation decreases slightly with the decrease in thickness. It is clear from the trend in Fig. 4.1, that on increasing the thickness of H-terminated silicon nanostructure, its dissipation response will gradually merge with that of the bulk silicon. For a particular thickness, ﬁtting the dissipation vs frequency data with a DRT Akhiezer model yields i corresponding τph and ∆γi. Fig. 4.4a shows that as we decreases the thick- s s f ness, τph decreases signiﬁcantly. Compared to τph, change in τph is minimal. We have also seen that the frequency corresponding to the maximum dissi- s pation is inversely proportional to τph. Because of this, for a lower thickness value, maximum dissipation takes place at a higher frequency. Again, the 2 decrease in the peak value of dissipation is captured by the reduction in ∆γs with thickness as shown in Fig. 4.4b.

(a) (b)

(c)

2 Figure 4.4: (a) τph vs thickness for H-terminated silicon. (b) Variation of ∆γ with thickness of H-terminated silicon obtained using the DRT model (c) PCA of displacements of the surface atoms for free and H-terminated silicon.

i We, also, computed τph for diﬀerent thicknesses of H-terminated silicon nanostructures using QHMR technique and EMD simulations. R(t) for dif-

23 ferent thicknesses is shown in Fig. 4.2. With the decrease in thickness, the energy autocorrelation decay is faster. The timescales are obtained from biexponential ﬁt and are compared with the ones obtained from the DRT i model. Fig. 4.4a shows that the estimated τph are in good agreement for diﬀerent thickness values.

It is well known that surface acts as an additional means of phonon scattering [84, 85]. Surface of a finite-sized nanostructure causes boundary reflection of the phonon modes [86]. The nature of the reflected wave depends on the extent of surface roughness (schematically shown in Fig. 4.3). Because of the amorphous nature of the H-terminated surface at 300 K, densely popu- lated by high-frequency light-weight hydrogen atoms, diffused reflection takes place. The timescale of energy relaxation of the phonon group is, therefore, reduced drastically. This is also evident from the noticeable shift in the dissipation peak towards the high frequency end.

For a H-terminated silicon nanostructure of given thickness, the phonon group is comprised of bulk ‘like’ modes and surface modes [26]. The ratio of surface to bulk modes, η is inversely proportional to thickness, t i.e. η ∝ 1/t. The energy relaxation of the surface modes is a combined eﬀect of phonon-phonon scattering and phonon-boundary scattering because of which the surface modes relax faster. This is manifested in the decay of energy autocorrelation of the phonon group. As we decrease t, η increases which implies that the contribution of surface modes compared to bulk ‘like’ modes in the phonon group increases. This leads to a reduction of τph with the decrease in thickness [51].

To this end, it is quite intriguing to pose the question of whether a smaller size enhances or inhibits dissipation. We studied dissipation as a function of size for the hydrogen terminated silicon structure. A range of frequency values between 5 GHz and 170 GHz were considered. Fig. 4.5a shows the scaling of dissipation and Q−1 with size in two separate plots for the range of deformation frequencies. We observe opposite scaling of dissipation with size on the diﬀerent sides of 20 GHz frequency. This observed behavior can be understood using the Akhiezer theory. In our DRT Akhiezer model, the s parameter τph is seen to be a strongly varying function of thickness. There-

24 (a) H-terminated surface

(b) Free surface (with (2×1) reconstruction)

Figure 4.5: Scaling of dissipation with size for diﬀerent frequencies. The data points are dissipation values obtained from nonequilibrium MD simulations. Errors in the data are shown using the black bars. fore, operating the nanostructure at a particular frequency, the scaling of energy dissipated with size is a manifestation of change in the phonon relaxation time. The scaling trend of dissipation is elucidated by the functional Ωτph dependence of dissipation: Ediss ∝ 2 . On the left side of the Debye 1+(Ωτph) s −1 peak (Ωτph < 1) the Akhiezer expression predicts an increase in Q with s the increase in τph. So, operating in this range with constant frequency (Ω), dissipation will increase with the increase in thickness (τph ∝ t). This indeed is observed and is shown in Fig. 4.5a. Similarly, we can explain the opposite trend i.e. the decrease in dissipation with the increase in size on the other s side of the Debye peak (Ωτph > 1). Thus, depending on the frequency of

25 operation surfaces can either be a bane or boon for the Q−1 factor in the case of Akhiezer dominated dissipation.

We now look into the scaling of dissipation with size for a silicon structure with free surface. We considered the same dimensions and the range of frequencies as in the case of H-terminated silicon. Fig. 4.5b shows dissipation and Q−1 as a function of size in two separate plots for the case of free surface. For all the deformation frequencies, dissipation decreases with the increase in size. The results indicate the presence of a surface damping mechanism [87]. The ideal Si (100) surface is not stable when the temperature is higher than 200 K and the symmetric (2x1) reconstructed structure is automatically formed [69, 74]. This phenomenon is captured by the Tersoff interatomic potential. The silicon dimer atoms on the surface {100} align themselves in the diagonal direction as shown in Fig. 3.2a and breaks the symmetry of the bulk. Thus, atoms cluster into different potential energy ‘bands’ [54] and an irreversible energy flow is associated with co-ordinated hopping motion of the atoms between these energy bands. Experimental observations also indicate the presence of several defects, dangling or broken bonds on the surface due to the obvious termination of the crystal structure on the surface. The degrees of freedom corresponding to these intrinsic defects or clusters of atoms, exhibiting two energy minima separated by an energy barrier in their configurational space, can couple with external strain field and exchange energy [32, 14, 88]. The jumping between potential wells is evident from non-Gaussian probability distribution function of the principal components of displacement of the surface atoms as shown in Fig. 4.4c. The various dissipation processes add incoherently and take over Akhiezer component, altering the scaling behavior.

26 CHAPTER 5

CONCLUSION

In summary, we use a theoretical model and MD simulations to investigate the role of surface on intrinsic losses. MD simulations demonstrate an opposite scaling of dissipation with size for two diﬀerent deformation frequencies in the case of a silicon nano-structure with a perfect surface. This is found to be consistent with our dissipation model based on the Akhiezer theory. Ad- ditional dissipation mechanisms play a signiﬁcant role in free silicon surface with (2x1) dimers and defects. As surface to volume ratio increases, these become dominant. We, therefore, observe dissipation to increase with the decrease in size independent of the operation frequency.

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