THERMAL PROPERTIES OF SINGLE CRYSTAL
LANTHANUM ALUMINATE
by
Peter C. Michael ProQuest Number: 10783713
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ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 A thesis submitted to the Faculty and the Board of
Trustees of the Colorado School of Mines in partial fulfillment of the requirements for the degree of Master of
Science (Physics).
Golden, Colorado Date b - 1 % 7 1
Peter C. Michael
Approved:
Dr. Baki Yarar Thesis co-advisor
Golden, Colorado
Physics Department ABSTRACT
The thermal diffusivity, thermal conductivity, and
specific heat capacity of single crystal (100) lanthanum
aluminate (LaAl03) have been determined at various temperatures ranging from 77K to 3 53K. The thermal diffusivity was measured using a transient heat-pulse technique, the thermal conductivity was measured using a
steady state d.c. technique and the specific heat capacity was
calculated from the measured thermal diffusivity and thermal conductivity. The temperature dependence of the thermal diffusivity was found to be of the form [T exp(TD/bdT)] and that of the thermal conductivity of the form [T3 exp(TD/bcT)] which are consistent with the expected behaviour of the thermal transport, limited by phonon-phonon collision
("Umklapp") processes, in high purity dielectric single crystals. The Debye temperature (TD) was calculated to be 72 0
± 22 K and the parameters bd and bc were found to be 1.4 ±
0.1 and 0.7 ± 0.1 respectively. The average sound velocity was found to be (5.36 ± 0.61)xl05 cm/s. The phonon mean-free- path at room temperature was found to be (2.22 ± 0.20)xl0'7 cm.
iii T—4029
TABLE OF CONTENTS page
ABSTRACT ...... iii
LIST OF FIGURES ...... vi
LIST OF TABLES ...... ix
ACKNOWLEDGEMENTS ...... X
Chapter 1. INTRODUCTION...... 1
Chapter 2. XRD CHARACTERIZATION OF SAMPLE ...... 3
Chapter 3. THERMAL DIFFUSIVITY ...... 9
3 .1 Theory ...... 9
3.1.1 Diffusion Phenomena in General ...... 9 3.1.2 Thermal Diffusion Equation ...... 10 3.1.3 Solution to the Heat Conduction Equation for the Sample Geometry at Hand ...... 12
3.2 Experimental Apparatus ...... 17 3.3 Sample Preparation ...... 2 0
3.3.1 Sample Configuration ...... 20 3.3.2 Deposition Process ...... 23 3.3.3 Sample Mounting ...... 27
3.4 Heat Pulse Measurement ...... 32 3.5 Data Analysis ...... 34
Chapter 4. THERMAL CONDUCTIVITY ...... 44
4.1 Theory ...... 44 4.2 Experiment ...... 4 6 4.3 Data Analysis ...... 46 4.4 Radiation Corrections ...... 48
iv T—4029
P .ag e
Chapter 5. SPECIFIC HEAT CAPACITY ...... 53
5.1 Calculated Specific Heat ...... 53 5.2 Debye Theory ...... 5 6
Chapter 6. THERMAL TRANSPORT LIMITED BY "UMKLAPP" PROCESSES ...... 60
6.1 Introduction ...... 60 6.2 Phonon-phonon Scattering Processes .... 61
Chapter 7. SUMMARY ...... 7 6
7.1 Comparison with other Dielectrics .... 7 6 7.2 Conclusions ...... 81 7.3 Suggestions for Future Work ...... 82
REFERENCES CITED ...... 8 3
Appendix I : Solution for a One-dimensional Semi-infinite Solid ...... 85 Appendix II : Radiation Correction Calculation 88
v T-4029
LIST OF FIGURES
Figure page
2.1 X-ray diffraction data of (100) lanthanum aluminate single crystal ...... 4
2.2 Rocking curve analysis of the 200 peak of lanthanum aluminate single crystal ...... 5
2.3 Plot of am vs. cos20/sin0 ...... 8
3.1 Sample geometry (10x10x0.5) mm3 ...... 13
3.2 Schematic of the experimental apparatus used for the transient heat-pulse thermal diffusivity measurements ...... 18
3.3 Sample holder and masks used to deposit the heater, thermocouple, and the thermal pads onto the sample ...... 21
3.4 Mask and sample configuration during the different stages of deposition ...... 22
3.5 Sample at the end of the deposition cycle ...... 24
3.6 Schematic of the vapour deposition system ...... 2 6
3.7 Sample mount used in the measurement of thermal diffusivity and thermal conductivity ...... 2 8
3.8 The main holder which accomodated the sample mount and housed the electrical connectors ...... 2 9
3.9 Copper can, enclosing the apparatus, immersed in a double-walled glass dewar containing the refrigerant ...... 31
3.10 Circuit diagram of the Perry amplifier ...... 33
3.11 Flow diagram of the program sequence in a heat-pulse measurement ...... 3 5
vi T-4029
Figure page
3.12 Typical temperature vs. time profile following a heat-pulse to the sample ...... 3 6
3.13 Linearization of the temperature vs. time profile ...... 37
3.14 Extrapolation to zero time of thermal diffusivities determined by fitting the data over progressively shorter time intervals ...... 3 8
3.15 Extrapolation to zero pulse width of thermal diffusivities determined from heat pulses of varying input pulse widths ...... 4 0
3.16 Thermal diffusivity of single crystal lanthanum aluminate ...... 4 3
4.1 Geometry of the one dimensional heat conduction problem ...... 4 5
4.2 Sample configuration used in the steady state d.c. thermal conductivity measurement ...... 47
4.3 A linear plot of input power (dQ0/dt) vs. temperature difference (T^Tj) ...... 50
4.4 Thermal conductivity of single crystal lanthanum aluminate ...... 52
5.1 The calculated specific heat capacity of single crystal lanthanum aluminate ...... 5 5
5.2 Temperature variation of the specific heat in the Debye model ...... 57
5.3 An example of the variation of the Debye characteristic temperature of Nal with temperature ...... 58
6.1 (a) N-process where phonon momentum is conserved ...... 66 (b) U-process where phonon momentum is not conserved ...... 66
6.2 A linearized plot of ln(D/T) vs. 1/T for single crystal lanthanum aluminate ...... 7 0
vii T-4029
Figure page
6.3 Linearized plot of ln(K/T3) vs. l/T for single crystal lanthanum aluminate ...... 7 3
6.4 Dominant phonon scattering mechanisms as revealed in the thermal conductivity of a non-metallie crystal ...... 74
7.1 Comparison of the thermal diffusivities of various dielectrics with lanthanum aluminate ..... 77
7.2 Comparison of the thermal conductivities of various dielectrics with lanthanum aluminate ..... 78
7.3 Comparison of the thermal conductivities of strontium titanate and lanthanum aluminate .... 79
7.4 Comparison of the atomic specific heat capacities of various dielectrics with lanthanum aluminate ...... 80
II. 1 Sample geometry ...... 89
viii T-4029
LIST OF TABLES
Table Page
2.1 Calculated Peak Positions for Lanthanum Aluminate ...... 6
2.2 Observed Peak Positions and Lattice Constants ...... 6
3.1 List of Thermostat Baths ...... 32
3.2 Thermal Diffusivity of Lanthanum Aluminate ...... 42
4. Thermal Conductivity of Lanthanum Aluminate ...... 51
5.1 Specific Heat Capacity of Lanthanum Aluminate .... 54
5.2 Specific Heat Capacity per Atom-mole of Lanthanum Aluminate and Debye Temperatures...... 59
6. Phonon Mean-free-path (X) at Different Temperatures ...... 7 5
ARTHUR LAKES LIBRARY
GOLDEN, CO S0401 T-4029
ACKNOWLEDGEMENTS
It is my pleasure to have this opportunity to acknowledge the support of my superiors and peers who were involved in the
successful completion of this project.
- I thank my advisor Dr. J. U. Trefny for his expert advice
and encouragement at all times. It has been both an enjoyable
and a learning experience working under his supervision.
I thank Dr. Baki Yarar, Dr. D. Williamson and Dr. J. T. Brown
for their kind cooperation. I also thank my parents for their
continued prayerful support from afar. I thank Dr. Christine
Platt for making the lanthanum aluminate samples available for
study. I appreciate Tom Haard's help in assembling the
peripherals of the diffusivity apparatus and the useful
discussions with Abdel Outzourhit. I thank Luigi Greco and
John McCullough for their assistance in 'transportation',
Philip Ahrenkiel for his assistance in the use of the X-ray
diffractometer, and Mrs. Li Wang for making the glass dewar
available. The technical expertise of Mr. Rex Rideout on the
electronics side and Mr. Jack Kintner on the engineering side was invaluable. I am thankful for the financial scholarship
provided in part by the TRW Space and Technology Group under
subcontract No. HTS. MA. DARPA/ONR 51606.01 and the Colorado
School of Mines.
x T-4029
Above all I thank God the Father, God the Son, and God the Holy Spirit for guidance, protection, and strength. (The
Lord is my strength and my shield; my heart trusted in him, and I am helped: therefore my heart greatly rejoiceth; and with my song will I praise him [Psalms 28:7, The Holy Bible,
King James Version, published by The Zondervan Corporation,
Grand Rapids, Michigan 49506]).
xi T-4029 1
Chapter 1
INTRODUCTION
Lanthanum aluminate (LaA103) has in the last two years gained importance as a substrate material in the field of high temperature superconductivity(1). In addition to having a good lattice match to the a-axis of the yttrium-barium-copper-oxide superconductor, it has a favorably low dielectric constant.
It is the latter property that proves it to be superior to the much celebrated strontium titanate (SrTi03) . It is a suitable substrate for supporting well-oriented superconducting thin films, especially in high frequency device applications such as interconnects, superconducting quantum interference devices
(SQUIDs) and passive microwave components.
Since the recognition of lanthanum aluminate as a substrate material for high-Tc thin films in 1-2-3 compounds such as Y1Ba2Cu307.x and E r 1Ba 2Cu 307.x(1,2), well c-axis oriented thin films of the thallium- and bismuth-based superconducting compounds have also been grown successfully on this m a t e r i a l (3,4). Thus lanthanum aluminate has proven to be a compatible substrate for a variety of superconducting thin films.
In view of potential electronic device applications, it T-4 02 9 2
is imperative that the thermal properties of lanthanum
aluminate be understood. One can imagine the importance of
knowing the thermal transport properties of the substrate material used in electronic circuits. In a cryogenically
cooled operating device, inadvertent heating would lead to
deterioration or even loss of the superconducting properties.
A catastrophic dissipation of power in the circuits could then
result in the destruction of the device. Efficient heat
transfer through the substrate to a heat exchanger can be
designed only with a knowledge of the thermal transport
properties of the substrate. It is therefore the objective of
this project to determine the thermal diffusivity and thermal
conductivity of lanthanum aluminate.
This research was conducted in conjunction with the study
conducted by C. C . P o i r o t (5,6) on r. f.-sputtered lanthanum
aluminate thin films, in terms of their applications as buffer
layers in multilayered superconductor-insulator-superconductor
device applications, in our Superconductivity Research group
at the Colorado School of Mines. T-4029 3
Chapter 2
XRD CHARACTERIZATION OF SAMPLE
The (100) lanthanum aluminate single crystal sample was obtained from AT&T through the TRW Space and Technology group.
An x-ray diffraction (0-20) scan showed only the (hOO) peaks without any impurity peaks (Figure 2.1).
A rocking curve analysis of the (200) peak yielded a rocking curve with a full width at half maximum of 0.04 5 degrees indicating good crystallinity (Figure 2.2).
The crystal structure of lanthanum aluminate as determined by Geller and Bala^ is pseudocubic with lattice constant a=3.79oA. The angle (a) between the crystallographic axes is 90°5'.
With the cubic indexing of the Bragg planes, the peak positions as calculated are listed in Table 2.1. The
Molybdenum x-ray source used has two main wavelengths in the beam, namely, KalXMo - 0.70930A and Kcx2XMo = 0.71359A. These give rise to two different peaks for each Bragg diffraction plane but the resolution is greater at higher angles, as can be seen in Figure 2.1.
The observed peak positions and the lattice constants as measured are listed in Table 2.2. T-4029 Intensity (Cts/sec) lOOOO- 12000 140OO?3-07"1991 /ia* 33 * 4000 2000 6000 8000 Figure 2.1 X-ray diffraction data of (100) lanthanum lanthanum (100) of data diffraction X-ray 2.1 Figure - lmnt snl crystal. single aluminate Phi--39LOI Chi- .24 fNngr* O fNngr* O Thatn/ZThato «eon Phi--39LOIChi- .24 lanthanum aluminate single crystal single aluminate lanthanum Ei\XRDDATA\LAOSNGI\LAO1. 200 2Theta (Degrees) 20
Nn/-SD «rf/-30 *1/1 Nn/-SD 4 T-4029
Intensity (Cte/eec) 20000 16000 4000 8000 iue . Rcig uv nlss f h 2 of the of peak 00 analysis curve Rocking 2.2 Figure -.5 >03-13-1991/22. 18. ] 32 3 atau lmnt snl crystal. single aluminate lanthanum lanthanum aluminate single crystal single aluminate lanthanum h—37.ZThttto- 37Phi— Chi-.26 21.64 Owaqo E. \XRDDATA\LA0SNC1\LA02R. 200 E. \XRDDATA\LA0SNC1\LA02R. I mg (Degrees) Omega FWHM - FWHM . 045 0 . 1
.3 tin/-50 KV/-30 »A/1 tin/-50KV/-30 .5 5 T-4029 6
Table 2.1 Calculated Peak Positions for Lanthanum Aluminate h k 1 e 20 1 0 0 5.3693 10.7386 2 0 0 10.7865 21.5731 3 0 0 16.3035 32.6070
Table 2.2 Observed Peak Positions and Lattice Constants
x k 1 26 6 a (A) 1 0 0 10.84 5.420 3.7547 2 0 0(Kal) 21. 68 10. 84 3.7715 2 0 0 (K^) 21.80 10.90 3.7737 3 0 0(Kal) 32.68 16. 34 3.7818 3 0 O(K^) 32.92 16.46 3 . 7777 T-4029 7
The discrepancy in the observed peak positions and in the lattice constants is due primarily to the misalignment between the effective center and the mechanical center of the sample.
This can be accounted for by applying the following correction. The error in the lattice constant (Aa) as measured i s (8),
Aa _ A d cos26 (2-1) a R sin0
where Ad is the sample displacement and R is the distance between the sample and the detector. It is evident from the above expression that the error in determining the lattice constant is greater at small angles. Now Aa = a0-ara, where aQ is the true lattice constant and am is the measured value.
A plot of am vs. (cos20/sin0) has aG as the intercept
(Figure 2.3). The lattice constant aQ was found to be 3.791
± 0.002 A in agreement with the determination of Geller and
Bala™. 4029
<0 (A) 3.76 3.77 3 3 3.80 iue . Po f mv. cos20/sin0 amvs. of Plot 2.3 Figure . . 78 79 2.0 4.0 6 . 0 () ( n I 8.0
10.0 12 . T-4029 9
Chapter 3
THERMAL DIFFUSIVITY
3.1 Theory
3.1.1 Diffusion phenomena in general
The phenomenon of diffusion involves the transfer of
substance, through a medium, from one point to another. The
law governing the rate of diffusion is known as Fick's l a w (9):
J=-DV p (3-1) where, J is the current density (the amount of substance
passing per unit time through a unit area normal to the
direction of flow) , p is the density of the diffusing
substance, and D is the diffusion coefficient characteristic
of the properties of the medium.
Conservation of the amount of substance results in a
relationship between the density of the diffusing substance
and the current density, known as the continuity equation:
(3-2) T-4029 10
A source term is added for the case of emission or absorption in the medium and the continuity equation becomes,
(3-3) where 's' is the source density (the amount of substance created or destroyed per unit volume per unit time).
Equations (3-1) and (3-3) combined reduce to the
following non-homogeneous diffusion equation:
(3-4) | £ = ^ P+S
3.1.2 Thermal diffusion equation
The diffusion equation as applied to heat conduction in solids is derived as follows.
The heat diffusion law as established by Fourier is
q =-&7t (3-5) where, Q is the heat current density, K is the thermal conductivity, and T is the temperature. (The Fourier law, as as expressed in Equation (3-5) is valid for an isotropic solid.)
The heat influx, per unit time, into a medium of volume V, enclosed by a surface area A, is T-4029 11
jjQ .dA -jjjv . QdV (3-6) where, dA is the outward unit normal to the surface.
Additional heat energy generated per unit time by an external source is
(3-7) where, s is the source density (the amount of heat generated per unit volume per unit time). The amount of heat required to raise the temperature of the medium by AT is
jjjc p A T d V (3-8) where, C is the heat capacity per unit mass of the medium and p is its density. The law of conservation of energy requires that the total heat input into the system, in a time At, equal the heat required to raise the temperature of the system by
AT. Thus,
(3-9)
cP_ = - v . e + s (3-10)
This is the continuity equation for heat conduction.
OF MINES T-4029 12
The Fourier law and the continuity equation can be
combined to form the heat conduction equation as follows:
4 ^ = — (KtPT+s) (3-11) dt Cp
Comparison of the above heat conduction equation (3-11) with
the diffusion equation (3-4) leads to the identity,
(3-12)
3.1,3 Solution of the heat conduction equation for the sample
geometry at hand
The partial differential equation (PDE) to be solved is,
s 4 I ’=d v 2t + (3-13) dt Cp where, T=T(x,y,z,t). The sample geometry under consideration
is shown in Figure 3.1. T-4029 13
-z A
999999999999
□ sample 1 ine heater source
copper clamps
Figure 3.1 Sample geometry (10x10x0.5)mm3 T—4029 14
We first find the Green's function (G), satisfying the following PDE, which would yield the response of the system to an instantaneous point source of unit strength:
(3-14) where G = G(x|x•;y|y';z|z';t|t1).
The solution to (14) is then given by,
T(x,y, z, t) =f jj G(x|x/;y|y/; z\z;; t\ t') s ^ ^ dx'dy'dz'dt1
(3-15) Now, the Green's function satisfies the homogeneous PDE for x;*x', y^y', ' , t?*t' (G=0 for t + (3-16) dt dx2 dy2 dz2 Seeking solutions of the form G(x,y,z,t) = Gx(x,t)Gy(y,t)Gz(z,t), for t>t', equation (3-16) reduces to, dG d2Gx . _. — ^ =D (3-17) dt dx2 dGy = D d2Gv (3-18) dt dy: T—4029 15 dG'=D ^ <:B —19) at dy“ The solution of equation (3-17) for a one dimensional semi- infinite solid is (Appendix I), - fyX-X') 2 ~ {X+X') 2 Q ______1______(e 423(t-t7) + e 4Z3(t-t7) ) (!3-20) \f4nD (t-t7) which satisfies the boundary condition, dG X 3-21) dx But we also have the other boundary condition, as. 'X 3-22) dx I x=L ® that is to be satisfied. Thus , _ (x-xJ-2nL)A £.LLLu) 2 —- \(x+x/-2nL) AfA G . . 4 T - - ______--- 1 (e (e iDU-e,) + e ) (3-23) yJ&nD( t-t1) The solution of equation (3-18) is, - (y-y‘) 2 r> - X T ______1______- 4i?(t-t7) 3-24) y n=-°° ------v/4t:D( t-t7) which satisfies the boundary condition G(y=±»,t)=0. The solution of equation (3-19) is, T-4029 16 - (z-z'-inM) 2 - (z + z/+2M-4,nM) 2 G 1 (e +e wit-t') ) (3-25) y/4nD{t-tf) which is similar to that of equation (3-17) , but with the boundary conditions applied at z=-M and z=M. Now the Green*s function (G=GxGyGz) can be used to solve the problem at hand. The source term for an infinitesimally short duration pulse is, six1 ,y‘, z’, t1) =— 6 (x'-O) 6 iy‘-0) 6 (t'-0) [S(z'+M) -Siz'-M) ] w (3-26) where Q is the heat input, w is the sample width (along the heater length) , and S is a step function. From Equation (3-15), T(x,y, t) =x/,0> z/ tj‘j‘j‘j>"'tG(Ar|x/;y|y/;z|z/; tit7) S ' dx/dy,dz'dtf (3-27) Integration over the spatial and temporal coordinates using Equations (3-23), (3-24) and (3-25) results in the following solution. T-4029 17 L2 -2L2/l(fl+l) AT(x=L, y=0 , z=0, t) =---£-- e 4Z?t(l +Le Dt ) (3“28> wCp7tJDt With respect to the first term in the expansion, the error in neglecting the subsequent terms is less than 1% if the data are analysed over the time scale t < (.87L2/D). Thus, equation (3-28) reduces to, -L2 A T(x=L,y=0, z=0, t) =---—-- e'ADt '(3-29) wCpnDC 3.2 Experimental Apparatus A schematic diagram of the experimental setup is shown in Figure 3.2. The Apple lie personal computer is equipped with a parallel interface that enables it to transmit trigger pulses and also to read and store the final output data. The trigger pulse from the Apple is used both to trigger the Datapulse 100A pulse generator and to ready the Biomation 610 transient recorder. A digital time-delay generator delays the trigger pulse to the pulse generator, with a delay adjustable in the range 1 microsecond to 1 second. An initial baseline is thus obtained with reference to which the actual pulse is measured. Details of the interface and the digital time-delay T—4 02 9 18 APPLE lie EPSON RX-80 PRINTER DIGITAL DATA PULSE DELAY NO BIOMAT ION TIME-DELAY I OOA DELAY 610A GENERATOR POWER PERRY TRANSISTOR AMPLIFIER CIRCUIT HP 6286A RM 56 I A POWER OSCILLOSCOPE SUPPLY Figure .2 Schematic of the experimental apparatus used for the transient heat-pulse thermal diffusivity measurements. T-4029 19 generator can be found in Madsen's t h e s i s (10). A pulse with a user-defined width from the Datapulse generator switches a transistor circuit which is energized by a Hewlett Packard 62 8 6A power supply. A short duration current pulse is then sent to a heater strip on one face of the sample. A thermocouple sensor on the opposite face registers the resulting temperature excursion of the sample. The thermoelectric e.m.f. thus generated is first pre-amplified before being subsequently recorded by the transient recorder. The transient recorder temporarily records the output pulse in digitized form which the Apple lie registers and stores for later analysis. A Type RM 561A Oscilloscope is used to set the pulse width and also to monitor the output response of the thermocouple. An Epson RX-8 0 printer was later used to display the data in printed form. T-4 02 9 20 3.3 Sample Preparation 3.3.1 Sample confiquration .The lanthanum aluminate (100) single crystal slab (10.03mm x 10.03mm x 0.53mm) was first run through a cleaning cycle. A three minute rinse in trichloroethylene was followed by acetone, methanol, and de-ionized water. The substrate was finally dried with dry nitrogen gas. In perfecting the thermocouple deposition technique, in the early stages, dilute nitric acid (10%) was used first to clean the sample. For the deposition of the heater and thermocouple, a sample-holder was designed to support the sample and also the masks used to cover selected areas of the sample. A simple design that could accomodate samples of various sizes and also different masks was accomplished as shown in Figure 3.3. In addition to the stainless steel sheets used, mylar sheet was also used to mask one half of the thermocouple strip while the other half was being deposited. The slot in the sample holder enabled the alignment of the thermocouple-strip with the heater-strip. The different stages of the sample preparation before each deposition are shown in Figure 3.4. A copper heater strip was first deposited across one T-4029 21 CLAMP t Figure 3.3 Sample holder and masks used to deposit the heater, thermocouple, and thermal pads onto the sample. T-4 02 9 22 L -I. J (a) (b) 1 r . i _ j . _ i m 1 1 (c) I _ sample Issa exposed half-face of sample C D mask □ mylar sheet over one-half of sample Figure 3.4 Mask and sample configuration during the different stages of deposition. T-4 02 9 23 face of the sample (Figure 3.4a). Copper thermal pads were deposited next on both ends of the sample (Figure 3.4b). This was to ensure good thermal contact between the sample and the copper clamp of the sample-holder. Thermally conducting Apiezon (N) grease was also used to this effect. A second copper strip constituting one-half of a copper-constantan thermocouple was deposited directly opposite the heater strip on one-half of the other face. A constantan strip was then deposited overlapping the copper half-strip at the sample center, providing the copper-constantan thermocouple junction (Figure 3.4c). It was observed that depositing constantan first resulted in preferential re-evaporation of copper from the constantan strip during the subsequent copper deposition cycle, so that the thermocouple was no longer continuous on the constantan side. A better thermocouple is thus obtained by depositing constantan last. An example of a typical sample after the deposition process is shown in Figure 3.5. 3.3.2 Deposition process The evaporator system used for the deposition process is essentially a vacuum chamber (Figure 3.6) containing a resistive heater, in the form of a coiled boat, containing pellets of the material to be evaporated. The sample was T-4029 copper-constantan thermocouple heater thermal end-pads copper □ sample constantan Figure 3.5 Sample at the end of the deposition cycle. T-4029 25 placed face down, directly above the evaporator source in the sample holder. The pumping-down procedure was as follows. A mechanical pump was used to pump down the foreline to less than 5 0 mTorr as measured by the thermocouple gauge (TCI). The water-cooled diffusion pump, with a warm-up time of 10 minutes, was then turned on. The vacuum chamber was roughed down to less than 50 mTorr as measured by the thermocouple gauge (TC2). The liquid nitrogen trap was then filled with about 4 litres of liquid nitrogen. With the diffusion pump hot and the chamber roughed down, the high-vac valve was opened and the chamber was evacuated to less than 5xl0'6 Torr as measured by the ionization gauge. The power supply was next turned on and the current was slowly ramped up until a melt was obtained, soon after which the material evaporated. A deposition time of twenty minutes was found to be sufficient to produce good conducting strips. This was found to be true only after the sample-holder was lowered close (12cm) to the source, thus increasing the deposition rate. Once the deposition was completed the power supply was slowly ramped down to zero. The ionization gauge was then turned off, the high-vac valve was closed, the diffusion pump was turned off, and the sample was allowed to cool in the vacuum chamber. A few hours later the system was brought to atmospheric pressure through the -4029 26 vacuum chamber view vent (air) port argon ion TC! guage hi-v3c gate valve CD liquid nitrogen vent (rough) fill, port cold trap V2 diffusion (foreline) pump w a te r TC2 cooling coils Figure 3.6 Schematic diagram of the vapour deposition system. T-4029 27 vent valve. The sample was then reconfigured in the sample- holder and the deposition process repeated until the copper heater-strip, the copper thermal pads, and the copper- constantan thermocouple had all been deposited. 3,3.3 Sample mounting Once the sample had been configured with the heater and the thermocouple it was mounted onto the thermal diffusivity apparatus. There were two main considerations in designing the sample mount (Figure 3.7). One was that it should provide a good thermal sink and the other that it should be able to accomodate samples of various sizes. Easy access to the sample while in the holder was naturally desired, especially during the process of attaching the electrical leads. Ceramic feed-throughs provided adequate thermal isolation for the copper and constantan leads. A platinum resistance thermometer (Lake Shore Cryotronics, Model PT-102, Serial No.P5595) was embedded in one of the copper clamps to monitor the ambient temperature in the system. The sample mount was itself supported by the main holder which housed the electrical connectors (Figure 3.8). Apiezon (N) grease was applied to the ends of the sample before T-4029 28 ceramic copper feedthrough platinum sample clamp resistance thermometer (embedded in copper ____ clamp) aluminum base Figure 3.7 Sample mount used in the measurement of thermal diffusivity and thermal conductivity. T-4 02 9 29 SIDE VIEW sample mount (Fig.3 7) fits here FRONT VIFW set screw coaxial cable connector vent pump. valve valve Figure 3.8 The main holder which accomodated the sample mount and housed the electrical connectors. T-4 02 9 30 clamping them down in the sample mount. The next step was to silver paint the copper heater-wire (AWG No.40, 0.00315") leads and the constantan (AWG No.40, 0.002") - copper (AWG No.40, 0.00315") thermocouple leads onto the sample. This assembly was then inserted into a resistive heater oven. Trial runs at atmospheric pressure resulted in oxidation of the thermocouple and heater wires leading to their degradation. In addition it was preferrable to have an evacuated system in order to eliminate errors due to convection and thermal conduction of air. The oven was made vacuum tight so that, under experimental conditions, it could be pumped down to pressures of less than lmTorr. It was convenient to use the oven to run experiments at elevated temperatures relative to room temperature. For low temperature measurements, however, it was necessary to design and make a copper can to house the diffusivity apparatus. The can was also made vacuum tight to enable measurements to be made under similar conditions. Indium wire provided a good vacuum seal at low temperatures as opposed to a silicone rubber o-ring which would freeze and thus deteriorate as a sealant. The can was customarily immersed in a double-walled glass dewar containing a solid-liquid mixture of a fluid having the desired low melting point (Figure 3.9). A list of the thermostat baths used is given in Table 3 (11). T-4029 31 n n n e lectrical connectors plexiglass cover- copper can housing the apparatus douDie-wa! led glass dewar containing the refrigerant m 1xture Figure 3 .9 Copper can, enclosing the apparatus, immersed in a double-walled glass dewar containing the refrigerant mixture. T-4029 32 Table 3.1 List of Thermostat B a t h s (11) Thermostat bath M.P./Normal B.P. (K) liquid nitrogen 77 dry ice + acetone 195 ice + water 273 3,4 Heat Pulse Measurement The sample was placed in the oven or in the cold temperature bath and was allowed to equilibrate to the temperature of its surroundings. The Perry amplifier was biased to balance the input offset until the output signal was zero. This null-biasing was routine when the experiment was conducted at room temperature. At lower or elevated temperatures, the amplifier required modification to accomodate larger thermocouple e.m.f input offsets. This was achieved by suitably varying the series resistance (Rg) at the negative input (Figure 3.10). Test runs were conducted next to optimize the output pulse by varying the input power, pulse width, baseline time delay, Biomation sweep time, and Biomation voltage sensitivity. The sweep time was chosen to record a large part of the data at early times, the region of interest for later data analysis. The Digital Time-delay Generator was used to T-4029 33 28K IN Q 725 725PERR' + 12V 33 12V NO 001 AMP 0.02 70K- -,-0.01 200 28K-1 OOK- 47 0.047 1 7 OK' -12V d -12V Figure 3.10 Circuit diagram of the Perry amplifier. T-4029 34 record a baseline before each pulse. The software used for the experiment could accomodate repeated measurement of several pulses with the desired settings. An additional time delay (tD) was included in the software. This programmable delay (usually about nine seconds) was more than adequate to ensure that the sample was restored to its original state before each data-pulse. A flow diagram of the program sequence of the Data Acquisition software used is shown in Figure 3.11. 3.5 Data Analysis A typical response of the thermocouple entailed a plot of the temperature (T) against time (t) (Figure 3.12). Linearizing equation (3-29), a plot of ln(t AT) versus 1/t yields a straight line with slope -L2/4D from which the diffusion coefficient D can be extracted (Figure 3.13). In order to account for the finite sample size, the fit to the early time approximation derived above was repeated over progressively shorter time intervals and the value of D was found by extrapolation to the early-time limit (Figure 3.14). Equation (3-29) is valid for an input-pulse of infinitesimal width. The experiment, however was conducted with pulses of finite pulse-width. The corrections applied to the data Colorado sownm n r v iin e s buLutN,uu 80401 - T-4029 35 Apple lie Data Acquisition Program NO Yes n=n+ No Delay Trigger Pulse Time Delay (tD) Pulse Generator Sample Biomation Transient Recorder Store Data End Figure 3.11 Flow diagram of the program sequence in a heat-pulse measurement. T-4029 36 I I • • I ···•--.•··· . _...... - · .. .· -...... ·-·.·· • ...·• ·· ... . .,,...... ···•···• ...... · •::·· • .. I • 10 50 t (ms) Figure 3.12 Typical temperature vs. time profile following a heat-pulse to the sample. ARTH UR LAKES LIB-RARY COLORADO SCH GO OOL OF MIN($' LDEN, CO 80401 T-4 T-4 029 In (t b T ) Figure 3.13 Linearization of the temperature vs. time time vs. temperature the of Linearization 3.13 Figure profile. 1 t / 37 T-4029 iue .4 xrplto o eotm o thermal of time zero to Extrapolation 3.14 Figure D (cm /s) 026 2 .0 0 0 3 .0 0 034 3 .0 0 10.0 diffusivities determined by fitting the by data fitting determined diffusivities vrporsieysotrtm intervals. time shorter progressively over 20.0 Ie RaTIme (nge ms) .0 0 3 .0 0 4 0 . 0 5 60.0 38 T—4 02 9 39 derived from the early-time approximations are as follows. The source term for a finite pulse width (tp) is, six', y '", t') = — 3 (x '-O ) 3 ( y '- 0 ) [S(z'+M) -S(z'-M) ] [S( t'-O) -S( t ' - f c j ] w * (3-30) where S is a step function and P is the input power. By integrating equation (3-27) using the new source term (Equation (3-30) it can be shown that, _£2 AT(x=L, y=0 , z=0, t) =---^-- e ~^t (1+_^) (3-31) wcpnDt 2 1 to lowest order in tp. Taking the natural logarithm of Equation (3-31) and expanding the logarithmic term as a Taylor series, it can be shown that to lowest order in tp/ 1 _ 1 ^ 2 . ,^ — +— - t (3-32) D D0 L2 p where D0 is the corrected value of the diffusion coefficient. The diffusion coefficient was thus measured for a range of pulse widths and the data were extrapolated to zero pulse width (Figure 3.15). The diffusivity data corrected for the non-ideal case of finite sample size and finite pulse width yielded a value for the diffusion coefficient characteristic of the sample at the given ambient temperature (T). T-4029 Figure 3.15 Extrapolation to zero pulse width of thermal of thermal width zero pulse to Extrapolation 3.15 Figure 1 1 /D (cm /$ ) 34.0 .0 0 3 .0 2 3 0.2 varying input pulse width. pulse input of pulses heat varying from determined diffusivities 0 . 4 . 0 us Wdh (Pulse Width ms) 0 . 6 0.8 1 . 0 1 . 2 40 T-4029 41 The results obtained over the temperature range 77 K to 3 53 K are shown in Table 3.1. A plot of the same is shown in Figure 3.16. The uncertainties in the temperature measurement are due to the . varying sensitivities of the platinum resistance thermometer in the different temperature ranges. The error in the diffusion coefficient (D) is the standard deviation of the mean of the intercept as obtained from the least squares fit. T-4029 42 Table 3.2 Thermal Diffusivity of Lanthanum Aluminate Temperature (K) D (cm2/s) 77.9 ± 1.3 0.1976 ± 0.0108 192.4 ± 0.9 0.0654 ± 0.0030 275.3 ± 0.7 0.0437 ± 0.0014 303.2 ± 0.4 0.0397 ± 0.0006 353.2 ± 0.4 0.0346 ± 0.0001 T-4029 43 0 .2 5 0 THERMAL DIFFUSIVITY OF SINGLE CRYSTAL LANTHANUM ALUMINATE 0.200 i 0) 0. 1 50 \ CM E o 0.100 O 0 .0 5 0 0 0 0 0 J J 1 1 1 1 111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i j 50.0 100.0 150.0 200.0 250.0 300.0 350.0 400. 0 Temperature ( K) Figure 3.16 Thermal diffusivity of single crystal lanthanum aluminate. T-4029 44 Chapter 4 THERMAL CONDUCTIVITY 4.l Theory The heat conduction equation (3-11) is, ■f?=-r- (jflPr+s) (4-i) ot Cp For the case of a steady state (dT/dt = 0) and in the absence of a source (s=0), for a one dimensional problem (Figure 4.1), the above equation reduces to, d2r=0 (4-2) dx2 Integrating the above equation we get, ——riT =constant (4-3) dx which is in agreement with the Fourier law of heat conduction for a constant heat flux: d0° = -KA-2£ (4-4) dt dx T-4029 45 T2 heat heat in out x l x2 x Figure 4.1 Geometry of the one dimensional heat conduction problem. T-4029 46 4.2 Experiment Thermal conduction in the steady state was achieved by heating the sample at one end and having a thermal sink at the other (Figure 4.2). Heat was provided by a constantan wire- wound resistive heater and copper clamps provided the thermal sink. A differential copper-constantan thermocouple was used to monitor the temperature gradient along the sample. Two strips of copper film were deposited onto the sample to improve the thermocouple contact to the sample. The copper strips also helped to define the locations of the thermocouples. Copper thermal pads were also provided at the heat-sink end of the sample. The dimensions of the thermocouple wire were suitably chosen to avoid thermal shorting of the sample and to reduce errors due to thermal conduction away from the sample through the leads. 4.3 Data analysis In the steady state, equation (4-4) can be rewritten as, T-4029 47 platinum resistance (copper-constantan-copper thermometer differential thermocouple) embedded in copper copper copper clamp cons. constantan w ire heater copper clamps of sample mount (Fig. 3.7) Figure 4.2 Sample configuration used in the steady state d.c. thermal conductivity measurement. T-4 02 9 48 dQ =KA ^Tl Tz- (4-5) dt ix2-xx) A linear plot of dQ/dt vs. (Tj - T2) has a slope KA/(x2 - Xj) . With A and (x2 - x2) known, K is determined from the slope. An example of the typical data obtained for lanthanum aluminate is shown in Figure 4.3. 4.4 Radiation corrections Since the experiment was performed under vacuum, the effects of convection or extraneous conduction through the surrounding medium were negligible. However, it was necessary to account for heat loss by radiation. The correction to the thermal conductivity(Km) obtained from the measurement is (Appendix II), e - where a and (3 are defined in Appendix II and K0 is the value for the thermal conductivity corrected for radiative heat loss. The radiation correction to the measured thermal conductivity was of the order of a few percent for the temperature range studied. The thermal conductivity of single crystal lanthanum aluminate as determined over the temperature range 7 7K to 4 5 3K T-4029 49 is shown in Table 4.1. A plot of the same is shown in Figure 4.4. The uncertainties in the temperature measurement are due to the varying sensitivities of the platinum resistance thermometer in the different temperature ranges. The error in the thermal conductivity (K) is the standard deviation of the mean of the slope as obtained from the least squares fit. T-4029 0. ^ 0 0 iue . A ierpo o iptpwr d/t vs. (dQ/dt) power input of plot linear A 4.3 Figure dQ/dt (Watts) 0.20 30 .3 0 0. 40 0. 0 . 00 T i ” T 2C degreesabsolute) 4. 00 4. temperature difference difference temperature 8 . 00 12.00 (Tl-T2) • 6. 0 .0 16 0 2 . 00 50 T-4 02 9 51 Table 4. Thermal Conductivity of Lanthanum Aluminate Temperature (K) K (W/cm/K) 77.2 ± 1.3 0.1855 ± 0.0005 197.4 ± 0.9 0.1434 ± 0.0003 273.0 ±0.7 0.1201 ± 0.0001 296.8 ± 0.7 0.1168 ± 0.0002 351.4 ± 0.4 0.1066 ± 0.0001 473.6 ± 0.9 0.1006 ± 0.0001 T-4029 .080 0 8 .0 0 ( Ul/c m/ K) 0 0. 1 60 0.200 Figure 4.4 Thermal Conductivity of Single Crystal Crystal Single of Conductivity Thermal 4.4 Figure . 120 0 200. 0 400.0 0 0 4 .0 0 0 3 .0 0 0 2 .0 0 0 1 0 . 0 -- 1 -- 1 __ 1 1 1 1 ~ L __ 1 __ THERMAL CONDUCTIVITY OF SINGLE CRYSTAL SINGLE OF CONDUCTIVITY THERMAL 1 __ Lanthanum Aluminate. Lanthanum 1 __ I __ 1 __ 1 __ 1 __ 1 __ 1 T e m p e r a l u( r e K ) __ LANTHANUM ALUMINATE LANTHANUM 1 __ 1 __ 1 __ 1 __ I __ I I I I I IL __ 1 I I I I 1I ...... I I I I I I I I I I I I I I i-.-l » A .0 0 0 5 52 T-4 02 9 53 Chapter 5 SPECIFIC HEAT CAPACITY 5.1 Calculated Specific Heat The specific heat capacity of a solid of density p is related to the thermal diffusivity and thermal conductivity by the following relation (Equation 3-12). The specific heat capacity of lanthanum aluminate as calculated from the measured values of thermal conductivity and thermal diffusivity over the temperature range 77K to 3 53K is shown in Table 5.1. A plot of the same is shown in Figure 5.1. The uncertainty in the temperature is due to the platinum resistance thermometer sensitivity. The error in the specific heat capacity (C) was determined using the standard propagation of errors method(12). The error in a quantity f(x,y,...) is given by, A f = [ ( | ^ A x ) 2+ ( | ^ A y ) 2+. . J 1'2 (5-2) ox oy where, Ax, Ay, . . . are the uncertainties in each of the variables. T-4029 54 Table 5.1 Specific Heat Capacity of Lanthanum Aluminate Temperature (K) C (J/kg/K) 77 ± 1 141 ± 8 197 ± 1 349 ± 17 273 ± 1 409 ± 9 297 ± 1 448 ± 11 351 ± 1 467 ± 13 T-4 02 9 U iue . Te acltd pcfcha cpct of capacity heat specific calculated The 5.1 Figure ( J/k g / K) 200 500 300 400 600 100 00 0. 100 0. 200 300.0 250.0 200.0 150.0 100.0 50.0 ' i ‘ i ' » i i ' i ■ i i i ■ i i i i i i i i i i i i i i i i i i t 11 i i i i i t i i i i i i i i 1 1 i i 1 1 . m ACLTDSEII ET AAIY OF CAPACITY HEAT SPECIFIC CALCULATED SINGLE CRYSTAL LANTHANUM ALUMINATE LANTHANUM CRYSTAL SINGLE single crystal lanthanum aluminate. lanthanum crystal single Temperature ( K) 350.0 1 I 1 L.l lJ 4 00 . 0 . 00 4 55 T-4029 56 5,2 Debve Theory According to the Debye theory, the specific heat per mole of atoms of any solid should fall on the 'universal' Debye curve(13) (Figure 5.2). Experimental fits to the Debye curve are in good agreement only if the characteristic Debye temperature (TD) is allowed to vary with temperature(14) (Figure 5.3). We can estimate the Debye temperature for lanthanum aluminate by first normalizing the calculated specific heat data to one mole of atoms (molecular weight of LaA103 = 0.214 kg/mole) and comparing them with the 'universal' Debye curve (Table 5.2). The average Debye temperature in the high temperature limit is 720 ± 22 K. The error in the Debye temperature (TD) was determined using the propagation of errors method (Equation 5-2) . T-4029 Figure 5.2 Temperature variation of the specific heat heat specific the of variation Temperature 5.2 Figure CW/3R 0.2 1.0 0.6 0.2 in the Debye model<14). Debye the in Debye 0.6 T'*o 1.0 1.4 57 T-4029 58 200, . . . • .. I . ! . . --; ' • # >I•••• ... ,...... I .. . I .. ~ I .• ,. , •••• & 1 aoi ,• . aJo ., a, ... I :::, .. ~ -tU . ...a, . ~ . E . Sodium Iodide Q,) 160 .... ~I OJ >- .0 ~ I 0 . , . ] 140 ~ :J I I I 0 ,OQ 200 300 Temperature , K Figure 5.3 An example of the variation of the Debye characteristic temperature of NaI with temperature<14>. T-4029 59 Table 5.2 Specific Heat Capacity per Atom-mole of Lanthanum Aluminate and Debye Temperatures Temperature (K) C (J/atom-mole/K) TD (K) 77 ± 1 6.1 ± 0.2 483 ± 10 197 ± 1 14.9 ± 0.7 667 ± 34 273 ± 1 17.5 ± 0.4 754 ± 26 297 ± 1 19.2 ± 0.5 702 ± 35 351 ± 1 20.0 ± 0.6 757 ± 51 T-4029 60 Chapter 6 THERMAL TRANSPORT LIMITED BY "UMKLAPP" PROCESSES 6.1 Introduction Thermal conduction in a solid involving the transfer of heat from the hot end to the cold end is a diffusion process. The heat carriers suffer collisions along their path with scattering centers such as phonons, electrons, impurities, point defects, and boundaries. For single crystal lanthanum aluminate, a dielectric solid, the heat carriers are phonons and the scattering mechanism at high temperatures is governed mainly by phonon-phonon collisions ("Umklapp" processes055) . From the kinetic theory of gases, the thermal conductivity (K) of a solid of density p can be related to the average phonon velocity (v) , the mean-free-path (X) of a phonon, and the specific heat capacity (C) of the solid as follows: K=— Cpvk (6-1) 3 Assuming a constant average velocity (v) and using Equation (3-12), the temperature dependence of the thermal diffusivity can be related directly to that of ARTHUR LAKES LIBRARY COLORADO SCHOOL OF MINEa GOLDEN, CO 80401 T-4029 61 the mean-free-path (X) : D(T) =±vX(T) (6-2) 6*2 Phonon-phonon Scattering Processes The motion of the atoms about their equilibrium positions in a solid is governed by the interatomic forces that bind them together. In the harmonic approximation, the force that one atom exerts on another is directly proportional to the relative displacements from their equilibrium positions. The allowed modes of vibration, where all the atoms vibrate independently with the same frequency (co) , are the "Normal" modes. The energy (En(w)) in a vibrational mode of frequency a) is quantized as follows: ■En( For a system of identical harmonic oscillators in thermal equilibrium at temperature T, the average occupation number (N°) of phonons in a mode of frequency o) is given by the T-4029 62 Planck distribution (16), O— 1 N°- h(ji ( 6 — 4) e 2«kBT_1 where, h is the Planck constant, and kB is the Boltzmann constant. The heat current (H) due to a mode ' q' is given by, /f=2W( composed of a range of normal modes. N(q) is the phonon occupation number of the mode ' q' in the wave packet(17). At equilibrium, in the absence of a thermal gradient, N(q)=N(-q) at all points in the crystal. Since w (q) =co (-q) and vG(q)=~v g (”3) the net heat current is zero. However in the presence of a thermal gradient, the phonon number density is different at different points in the crystal ( N(q)?*N(-q)). The change in phonon density due to the temperature gradient (dN/dt)drift is limited by scattering processes (dN/dt) scatt, so that in the steady state the phonon density becomes independent of time. Thus, < ! f > dri«+ (Jf)sca«=° (6-6) which is the Boltzmann equation, where T-4029 63 and, in the "relaxation time approximation", {dN) = N ^ N . (6. g) K dt scatt - If only harmonic forces were present in the crystal, the non-interacting normal modes would sustain a heat current even in the absence of a thermal gradient once the phonon distribution is disturbed from its equilibrium value. This implies an infinite value for the thermal conductivity. In practice, single crystals of high purity still exhibit a finite conductivity. This is attributed to the fact that the interatomic forces are not strictly linear in the atomic displacements (or equivalently the crystal potential is not strictly quadratic in the atomic displacements.) It appears that for most crystals, the thermal conductivity can be modeled by retaining the next higher order term in the atomic displacement, namely, the quadratic term in the force and the cubic term in the crystal potential. “The crystal hamiltonian, in simplified form, can be written as(l8), T—4.029 64 tf=^SAf(D2 (q) a (g) a (-q) +— — ^—22 c (grx , gr2 , gr3 ) a (g1) a (q2) a (g2) (6-9) where a(q) and a(-q) are the annihilation and creation operators. The non-zero matrix elements of the annihilation and creation operators linking states N to N-l and N to N+l are, a(g)„„.= [ h , ' (N) ] ^ (6-10) 1 2nMu> (g) and i a(- In Equation (6-9) the first term is the unperturbed hamiltonian and the second term is the perturbation. The coefficient fc' in the perturbation term contains a phase factor of the form exp[ix. (q1+q2~q3 ) ] which is zero unless, (6- 12 ) where the q's are the wave vectors of three different modes, and G is a reciprocal lattice vector. The transition probability for a three-phonon process can be calculated using second order time-dependent perturbation theory as follows. Consider the transition from the state T-4029 65 (N^N^Nj) to the state (N^l,N2-l,N3+l) , for the forward process, ^ + ^ 2 ^ 3 ( 6 - 1 3 ) The transition probability (dP/dt) is given by, J f = l l i26 {co1+ o j 2 - o j 3 ) ( 6 - 1 4 ) OU M u )^ u )2 ^ 3 which is zero unless, a>i + a>2 =co3 ( 6 - 1 5 ) For a three-phonon process, two cases arise from equation (6-12) (Figure 6.1). When G=0, the process is termed a "Normal" (N) process; when G^O the process is an "Umklapp" (U) process. The thermal energy is transported in the direction of the phonon group velocity. It is evident (Figure 6.1(a)) that in an N-process, the direction of the mode ' q3' is similar to that of ' q/ and ' q2'» In the U-process however the direction of 'qj' is quite different from that of 'q/ and ^ 2 *. Thus it is the U-process that provides the thermal resistance and is responsible for restoring the local equilibrium phonon distribution. (The N-process enables energy transfer between different modes and plays an important role in the establishment of thermal equilibrium.) T-4029 66 (a) (b) Figure 6.1 (a) N-process where phonon momentum is conserved. (b) U-process whee phonon momentum is not conserved. T-4029 67 For the three-phonon process in question, the rate of change in the number of phonons in mode q: is proportional to the difference in the probabilities of the processes which tend to increase N(qt) and those which decrease N(qj) (i.e. anc* respectively.) Using equation (6-14), the total rate of change of the phonon population in mode qL is obtained by summing over all the interacting modes q2 and q3: dN(gr) _ „ h \c(qr,q2,qj |2 6 ((01 + (o2-o>3) [{N^l) (Nz + 1 ) N2~N1N2 (iV3 + l) ] at w1 <*)2 u>3 (6-16) The only temperature dependent term on the right hand side is the last term. Thus, (6-17) To calculate the rate of change of the occupation number in mode qlf we assume that all other modes have the equilibrium phonon distribution (N°) (i.e. (Nj—N^) , N2 =N2°, N3 =N3°) and using Equation (6-15), we can deduce that, (6-18) At sufficiently high temperatures (T>TD) , where U- processes are frequent due to the abundance of large ’ q' phonons, hcj/27rkBT is small for all modes. Thus, T-4029 68 ( 6 - 1 9 ) ho> and 3 iV ( g 1 ) (6-20) At low temperatures (T (i.e. q2 must be about the maximum that can occur in a crystal, which in the Debye approximation is qD (qD=27rkBTD/hc) . Thus for small ulf and co2 «o)3, (6- 21) and dNiq^ 2nkgT (6-22) Since, haj/27rkB«TD, we can express the exponential term as exp(-TD/bT) , where, ,b* is a parameter of order unity. From Equation (6 -8 ) and (6-21), the mean-free-path of a phonon (X) can be related to the relaxation time (r) and the phonon group velocity (vG) : T-4029 69 X(T) =A2l k?Te b°T (6-23) Ao>l where A is a constant independent of temperature. Klemens(16) has estimated A to be of the order MvG2 /8 Y2 kBTDG, where M is the atomic mass and 7 is the Grueneisen constant. From Equations (6-1) and (6-22), the thermal diffusivity at low temperatures is given by, 0 D(T) = ~ v A fTe bdT (6-24) 3 where A 1 =A(^Trkg/ha^) . The exponential temperature dependence is strongest at low temperatures so that the linear term is not as significant. A linearized plot of ln(DT'1) vs. 1/T has a slope TD/bd. The data obtained for lanthanum aluminate are shown in Figure 6.2. The Debye temperature (TD) was calculated, from the Debye theory on the specific heat of solids (Sec. 5.2), to be 720 ± 22 K. Thus from the slope and Td, the value of ' bd* was found to be 1.4 ± 0.1. Now, the temperature dependence of the thermal conductivity (K) can also be analysed. At high temperatures (T^>Td) , K oc 1/T. An estimate given by Leibfried and Schloemann has been reported by Klemens<18): 4029 Figure Figure In (D/T) -5.00 - -7.00 - -9.00 -7.60 6.00 - 8.00 -8.40 -9.20 8.00 000 .00 .00 0.0080 0.0060 0.0040 00l020 6 .2 A linearized plot of In(D/T) vs. 1/T for for 1/T vs. In(D/T) of plot linearized A .2 single crystal lanthanum aluminate. lanthanum crystal single 1/T 05 0•0055-.0050 9.61 GOLDEN, CO 80401 COLORADO SCHOOL OF MINES ARTHUR LAKES LIBRARY 70 T-4029 71 (6-25) At lower temperatures (T thermal conductivity to be: (6-26) A linearized plot of ln(KT'3) vs. 1/T has a slope TD/bc (Figure 6.3). The value of 'bc' was found to be 0.7 ± 0.1. From Figures 6.2 and 6.3, it is evident that the data point at 77K deviates from the observed *umklapp' behaviour over the remaining temperature range. It is well known that at very low temperatures, other scattering mechanisms such as defect scattering and boundary scattering become more dominant17 (Figure 6.4). It therefore seems plausible that the rapid exponential rise in the thermal diffusivity and the thermal conductivity due to U-processes is limited by defect scattering; the defects in this case being most likely of the order of the crystallinity in the sample. From the Debye temperature we can calculate the average sound velocity (vG) (i.e. the average group velocity of a phonon wave packet) from the following relationship0^: 2 n T jz B (6-27) h[6n2 (N/V) ] 1/3 T-4029 72 where N/V is the atomic density of LaAl03 (= 9.178xl028 atoms/m3) . The average sound velocity for single crystal lanthanum aluminate was found to be (5.36 ± 0.16)xl05 cm/s. From equation (6-1) the mean-free-path (X) of a phonon at a given temperature can be determined. The values at several different temperatures ^re given in Table 6 . The errors in the sound velocity (vQ) and the mean-free-path (X) were determined using the propagation of errors method (Equation 5-2). T-4029 73 - 14.00 - 16.00 I— \ c - 20.00 - 22 J00 .0010 0.0030 0.0050 .0070 0.0090 1 /T -17.50 -18.50 I— \ w - 19.50 c 0.0020 0.0030 0.0040 .0050 0.0060 1 /T Figure 6 .3 Linearized plot of ln(K/T3) vs. 1/T for single crystal lanthanum aluminate. T-4029 > u h*-C Boundaries Defects Umklapp processes Temperature Figure 6.4 Dominant phonon scattering mechanisms as revealed in the thermal conductivity of a non-metallie crystal. T-4 02 9 75 Table 6 . Phonon Mean-free-path (X) at Different Temperatures Temperature (K) X (xlO*7 cm) 77.9 ± 1.3 11.1 ± 1.17 192.4 ± 0.9 3.66 ± 0.37 275.3 ± 0.7 2.44 ± 0.23 303.2 ± 0.4 2 . 2 2 ± 0 . 2 0 353.2 ± 0.4 1.94 ± 0.18 T-4029 76 Chapter 7 SUMMARY 7.1 Comparison with other Dielectrics The following is a comparison of the lanthanum aluminate data with the available, published data on other substrate materials used for supporting High-Tc superconducting thin films (Figures 7.1 - 7.4). The thermal diffusivity (D) of lanthanum aluminate is lower than that of aluminium oxide and magnesium oxide. The thermal conductivity (K) of lanthanum aluminate is also lower than that of aluminium oxide and magnesium oxide but is comparable to that of strontium titanate. The atomic specific heat (Cat) of lanthanum aluminate is lower than that of lanthanum oxide. It is comparable to that of strontium titanate and is higher than that of magnesium oxide and aluminium oxide. T-4029 D (cm /s) 0 0 0.30 - 0.20 iue . Cmaio o te hra dfuiiis of diffusivities thermal the of Comparison 7.1 Figure . . 0 0 10 0.0 - —!—I—J- - »'i ■' 11 0. 200 0. 400 0. 600.0 500.0 400.0 300.0 200.0 100.0 aiu deetis ih lanthanum with dielectrics various aluminate(19)...... i Tempera lure ( K) THERMAL DIFFUSIVITY THERMAL A A A ***** ***** La A l 03 A Ai A A ...... A A A ZA * ...... AiA ..... [£b ...... 111 111 11i 1111 A Z& AAAAAMgO I I 1 1 I I A i 203 M M A 1 11111 □ 77 T-4029 0.40 ^ iue . Cmaio o teteml odciiis of conductivities thermal the of Comparison 7.2 Figure ( U/c m / K ) 0 0.20 0.60 0.80 1 1 . .00 .20 00 0 100 0. 300 0. 500.0 400.0 300.0 200.0 100.0 0 r 1 1 1 1 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 aiu deetis ih lanthanum with dielectrics various aluminate00*. 0 <> 0 eprtr (Temperature K) THERMAL CONDUCTIVITY THERMAL « A □ 0 0 A □ 0 0 □ * $ □ A A □ AAA6AMg AAA6AMg 0 I I 1 I I I A l 00000 ***** ***** Lo AI 03 A □ S A □ p 1 ' ' T 203 io 1 1 3 ' 1 11 600. 0 1 i 78 T-4029 79 THERMAL CONDUCTIVITY 0.20 - 0 0 0 0 0 0 SrTio3 0 ***** Lo AI 03 \ 0 ^ 0 . 1 6 O 0 \ 0 0 0 0 . 12 - 0 ° * o 0 , 0 0 I I I I ■ I I I ■ I I, I ,1.1_I I I I I I I I <1 I I t 1 I I I I I I I 1 I I I 1 I I_I I I I I I I 0.0 100.0 200.0 300.0 400.0 500.0 Tempera lure ( K) Figure 7.3 Comparison of the thermal conductivities of strontium titanate with lanthanum aluminate*20*. T-4029 Figure 7.4 Comparison of the atomic specific heat heat specific atomic the of Comparison 7.4 Figure C ( J/a tom-mo I /K) 00 1 20.00 25.00r 0 0 . 0 1 15.00 - 0.00 - 5.00 0 . 1 capacities of various dielectrics with with dielectrics various of capacities lanthanum lanthanum I I 1 I I I I I I I I 1 I I 1 I 1 I I 1 I I I I I I I I I I I I I I 1 I I 1 I I I I J 0 . 0 0 1 eprtr (Temperature K) SPECIFIC HEAT CAPACITY HEAT SPECIFIC (21). e t a n i m u l a ^ < 0 . 0 0 2 0 0 °. 300.0 $&&&& hgO \ W V A 0 0 0 0 0 ***** Lo ***** A l 03 m n r SrTi Lq a a i c3 3c 203 ------0 3 3 0 1 ___ I ___ L_ 400. 400. 0 80 T-4 029 81 7.2 Conclusions We have completed the characterization of the thermal properties of single crystal lanthanum aluminate over the temperature range 77 K to 353 K. The thermal properties of lanthanum aluminate are well behaved with, D (LaA103) < D(A1203) < D(MgO) K(LaA103) < K(A1203) < K(MgO) Cat(LaA103) > Cat(MgO) > Cat(Al203) The thermal conductivity of lanthanum aluminate is comparable to that of strontium titanate. And with the better dielectric properties, lanthanum aluminate is still the favoured substrate material for high frequency device applications. The Debye temperature for lanthanum aluminate was estimated to be 720±22K. The average sound velocity was calculated to be (5.36±0.16)xlO5 cm/s. The phonon mean-free-path (X) varied from ll.lxlO'7 cm at 77.9 K to 1.94xl0*7 cm at 3 53.2 K. The temperature dependence of the thermal diffusivity was found to be of the form [T exp (TD/bdT) ] with bd=1.4±0.1 and that of the thermal conductivity of the form [T3 exp(TD/bcT) ] with bc=0.7±0.1 which is the expected behaviour of thermal transport limited by phonon-phonon scattering in pure dielectric single crystals. T-4029 82 7.3 Suggestions for Future Work The thermal conductivity data were taken manually. A first step towards further improvement would be to automate the data acquisition procedure. The specific heat capacity data as calculated from the measured thermal diffusivuty and the thermal conductivity was satisfactory. A direct measurement of the specific heat capacity would serve as a good comparison. Finally, a smaller temperature interval between data points would be beneficial. T-4029 83 REFERENCES CITED 1. R.W.Simon, C.E.Platt, A.E.Lee, G.S.Lee, K.P.Daly, M.S.Wire, A.J.Luine, and M.Urbanik, Appl. Phys. Lett. 53, 2 677 (1988) . 2. A.Mogro-Campero, L.G.Turner, E.L.Hall, M.F.Garbauskas, and N.Lewis, Appl. Phys. Lett. 54 (26), 2719 (1989). 3. G .Subramanyam, F.Radpour, and V.J.Kapoor, Appl. Phys. Lett. 56 (18), 1799 (1990). 4. R .Sobolewski, P .Gierlowski, W.Kula, G.Jung, S .Zarembinski, M.Berkowski, A.Dabkowski, A.Pajalkowska, Physica C (Amsterdam), 162-164 (Pt.I), 631-2 (1989). 5. C.Poirot, J.Trefny, B.Yarar, J.Ahn, P.Michael, L.Greco, N.Wada, P.Ahrenkiel, C.Platt, A.Lee, "Characterization of RF Sputtered Lanthanum Aluminate Thin Films", Proceedings of the Conference on the Science and Technology of Thin Film Superconductors at SERI, April 1990, Denver, Colorado. 6 . C.C.Poirot, The Preparation and Characterization of Lanthanum Aluminate Powder and RF Sputtered Thin Films. Master's Thesis, Colorado School of Mines (1990). 7. S.Geller and V.B.Bala, Acta Cryst. 9, 1019 (1956). 8 . L.H.Schwartz and J.B.Cohen, Diffraction from Materials, p.198, second edition, Springer-Verlag (1987). 9. E.Butkov, Mathematical Phvsics. Addison-Wesley Publishing Company, Inc. (1968). 10. J.M.Madsen, Fast-time Heat Pulses in High- and Low Diffusivity Solids. Ph.D Thesis, Colorado School of Mines (1987) . 11. CRC Handbook of Chemistry and Physics, 69th edition, CRC Press, Inc. (1988-1989). 12. F.R.Yeatts and F.E.Cecil, Concepts of Physical Measurements. Department of Physics, Colorado School of Mines (1987). 13. AIP Handbook, third edition, Me Graw-Hill Book Co., p.4-113 (1972). T-4029 84 14. Specific heat of solids, CINDAS Data Series on Material Properties, Volume 1-2, edited by C.Y.Ho, Hemisphere Publishing Corportion (1988). 15. R.E.Peierls, Quantum Theory of Solids. Clarendon Press, Oxford (1965). 16. C.Kittel, Introduction to Solid State Physics, fifth edition, John Wiley & Sons, Inc., p.128 (1976). 17. R.Berman, Heat Conduction in Solids. Clarendon Press, Oxford (1975). 18. P.G.Klemens, Solid State Physics (eds. F.Seitz and D.Turnbull), Vol.7, p.l, Academic Press, New York (1958). 19. Thermophysical Properties of Matter, Thermal Diffusivity, Volume 10, Y .S .Touloukian (Series editor), C.Y.Ho (Technical editor), OFI/Plenum, New York-Washington (1973) . 20. Thermophysical Properties of Matter, Thermal Conductivity, Non-metallic Solids, Volume 2, Y .S .Touloukian (Series editor), C.Y.Ho (Technical editor), OFI/Plenum, New York -Washington (1970). 21. Thermophysical Properties of Matter, Specific Heat, Non-metallic Solids, Volume 2, Y.S.Touloukian (Series editor), C.Y.Ho (Technical editor), OFI/Plenum, New York -Washington (1970). T-4 029 85 Appendix I The solution of equation (3-17) for a one dimensional semi-infinite solid can be derived using the method of i m a g e s (9). The PDE to be solved is, a<3* = D i ! £ s (i-D dt dx2 The Green's function (g) for the problem of an infinite one dimensional solid, satisfying the above PDE, with the initial condition g(x|x';t= 0 |t') = 0 and the boundary conditions g (x= ± o o | x ' ;t 1 1 ') = 0 is derived as follows. The Green's function [g(x|x*;t|t')] is the response of the system to an infinitesimal input heat-pulse at x=x' and t=t'. The PDE to be solved is, ■If--0-^=6 (x-x')h (t-tO (1-2) ot dx2 Taking the Fourier transform of the spatial part and the La Place transform of the temporal part of Equation (1-2) , we get C T = f----- f (1-3) &FL (2tc)1/2 s+Dk2 where, k is the transform variable of x and s is the transform variable of t. Using the convolution theorem, T-4029 86 m(a) n(a) =fm(a/) nia-a') daf (1-4) the inverse La Place transform of gFL (Equation [1-3]) is gr,-g( t-t') (I_5) (2tc) 1/2 where, S(t-t') is a step function. Again, using the convolution theorem, the inverse Fourier transform of gF (Equation [1-5]) yields the Green's function (g), - (x-x*) 2 g(x\x'; t\ t') = - e 4DU fc/) S( t-t1) (1-6) yj 4 tzD (t-t1) where, T(x, t) =JJg{x\xf; t\t/) six', tl) dx'dt' (1-7) For a semi-infinite solid, we have the new boundary condition at x= 0 , namely, - | U .C= 0 d - 8 ) This can be satisfied by adding an image source in the region x< 0 at x=-x'. s(x',t') = s(x',t') x > 0 =s(-x',t') x < 0 T-4029 Thus, Tix, t) -jJ [six', t') +s{-x1, t') ] gix\x'; t| t') dx'dt' Letting x-*x', we get, Tix, t ) = f f sW , t>) [gix|x'; t|t<) +g(x\-x'; t\t') ] dx'dt' Returning to our original Green's function, Gx=g{x\x't\t') +gix\-x!; t\tf)- T-4029 88 Appendix II RADIATION CORRECTION CALCULATION Consider an elemental volume of area (ab) and length 'dx* ( Figure II.1). In the steady state, the heat leaving the elemental volume is equal to the heat entering the same volume element less the heat radiated from the elemental surface. (x+dx) —02 (X ) =eo2 (a+b) (dx) (T4 (x) -1 *) < U - 1 ) at at Here, e is the emissivity of the sample, o is Stefan's constant, a and b are the lateral dimensions of the sample, T(x) is the temperature of the sample at the position 'x', and T0 is the ambient temperature of the surroundings. For T(x)»T0, (Tl(x)-T^)~ATi(T(x)-T0) ( H - 2 ) Using the Fourier law of heat conduction (4-4), equation (II-l) can be rewritten as, *T' ( II —3 ) dx2 where, T* = T(x)-T0, T —4029 89 heat loss by radiation dx heat out (a) x=0 x I x2 x=L ± a T Figure II.1 Sample geometry. T —4029 90 and 2 - €a2 (a+jD) 47^ (II —4) r Kab The solution to the above differential equation for positive 7 is/ T'(x) =Aefx+Be~vx (H-5) The constants A, B can be determined by applying the following boundary conditions: At x=L, T=T0 => T'=0. At x=0, T ^ T ^ => T^ T ^ - T ^ At x=0, dQ°— K a b d T \ S t ~&c '*-• Thus equation (II-5) becomes, T>(X) =-(.rp2) -JL. ---i [eix_ey(2L-x) ] (II-6 ) dt Kab y (i+eY2L) Expanding the exponential terms up to the third power in y, it can be shown that, dQ T'(x) =-(— ^2) — (x-L) (l+ — [ (x-L) 2 +2 L2] +0 (y3)} (II-7) dt Kab 6 For the sample configuration used (Figure II.1), T —4029 91 r(jg) -r(x2) =_ ( dOoj i i (Xj-X. dc ab K, where, -~=~- (II-9) 6 eo2(a+b) 4 ah. and [ ix^-L) 3- (x 2-L) 3+2L2 ( -x2) ] (x2-x2) In Equation (II-9), is the measured thermal conductivity and K0 is the thermal conductivity corrected for radiative heat loss.