JOURNAL O F T H E AMERICAN CHEMICAL SOCIETY Registered i n
U.S. Paten1 Ofice
@
Copyrighl, 1964, by the American Chemical Socicly
SEPTEMBER 4, 1964
VOLUME86, NUMBER17
PETER J . W. DEBYE
80 TH
ANNIVERSARY ISSUE
[CONTRIBUTION FROM WESTINGHOUSE
RESEARCH LABORATORIES,
PITTSBURGH
35, PENNSYLVANIA]
The Lattice Thermal Conductivity of an Isotopically Disordered Crystal' BY A. A. MARADUDIN RECEIVED APRIL 12, 1964 Starting from the Kubo expression for thermal conductivity a calculation of the lattice thermal conductivity of an isotopically disordered cubic Bravais crystal is carried out to lowest order in the concentration of the minority constituent ( t h e impurities). I t is shown t h a t to this order in the concentration the expression for the conductivity has the form predicted by simple kinetic theory arguments, in the case t h a t the impurity atoms are heavier than the atoms of the host crystal. T h e reciprocal of the phonon lifetime in this case has a resonance character of the type discussed recently by Brout and Visscher.
1. Introduction In 1914, in the published proceedings of the Wolfskehl Conferences of the preceding year, Debye* suggested that anharmonic terms in the expansion of the potential energy of a crystal in powers of the displacements of the atoms from their equilibrium positions would lead to the exchange of energy between the normal modes of the harmonic approximation, and therefore provided a mechanism which could explain the thermal resistance of insulators. Fifteen years later Peierls3 derived a quantum mechanical transport equation for the phonon distribution function for a crystal possessing cubic anharmonicities, from the solution of which the lattice thermal conductivity could be calculated. The next major contribution to the theory of lattice thermal conductivity is to be found in the work of Klemens4 who first emphasized the importance of the scattering of phonons by crystalline defects as a mechanism giving rise to thermal resistance. In the succeeding years the theory of thermal resistance due to point defects has been discussed by a number of a ~ t h o r s . All ~ of these authors have based their treatments on the Peierls transport equation. In 1957 a new approach to the calculation of transport coefficients was presented by Kubo and his co(1) T h i s research was supported b y t h e Advanced Research Projects Agency, Director f o r Materials Sciences, a n d technically monitored b y t h e Air Force Office of Scientific Research under C o n t r a c t A F 49(638)-1245 ( 2 ) P D e b y e in "Vortrage uber die Kinetische Theorie der Materie,"
B G T e u b n e r , Leipzig a n d Berlin, 1914. (3) R I? Peiel-Is, A n n P h y s i k , 3 , 1055 (1929). ( 4 ) P G Klemens, Proc R o y Soc ( L o n d o n ) , A108, 108 (19511, Proc P h r s S o c ( L o n d o n ) , 868, 1113 (1958). ( 5 ) K B e r m a n , P. T N e t t l e y , F W S h e a r d , A N Spencer, R W H. Stevenson, and J ?K Z i m a n , Proc R o y Soc. ( L o n d o n ) . A263, 403 ( 1 9 5 9 ) ; J C a l l a w a y , Phys R e u , 113, 1046 ( 1 9 5 9 ) , P C a r r u t h e r s , Rev Mod P h y s S3, 92 (18611, H Rross. P h y r . S f o f u s Solidi, 1,481 ( 1 9 6 2 ) ; P G Klemens, Phys R P ~119, , ,507 (1960); P G Klemens. G K W h i t e , a n d R . J T a i n s h , Phil M a g , 7 , 1323 (1862)
workers.6 The starting point of this approach is an expression for the desired transport coefficient as a Fourier transform of the two time correlation function of the current operators which appear in the macroscopic equations by which the coefficient is defined. I t was hoped that the use of such correlation function expressions for the calculation of transport coefficients would yield results which the conventional approach v i a a transport equation could not reproduce. Recent experimental results' for the thermal conductivity of ionic crystals containing point defects can be explained rather well if i t is assumed that the inverse relaxation time for the scattering of phonons by point defects, in terms of which the collision term in the Peierls transport equation is usually approximated, has a resonance character in its dependence on the frequency of the phonon impinging on the defect. The subsequent demonstration by Brout and Visschers and by othersQ t h a t a heavy mass defect in a crystal can give rise to a low frequency "resonance mode" of vibration of the perturbed crystal has led to a number of a t tempts to incorporate this phenomenon into calculations of lattice thermal conductivity. This is usually donelo by calculating the cross section for the scattering (6) R . Kubo, M Y o k o t a , a n d S N a k a j i m a , J P h y s . Soc J a p a n . 11, 1203 (1957). I n this paper a q u a n t u m mechanical derivation of formal expresA classical derivation of these sions for kinetic coefficients is presented expressions h a d been given several years earlier b y M S G r e e n , 3 C h e m . Phys , 11,398 (1954). (7) R . 0 P o h l , P h y s . Rev. Lellcrs, 8 , 481 (1962), C T Walker a n d R 0 P o h l , P h y s . R e v , 131, 1433 (1963). (8) R B r o u t a n d W Visscher, P h y s Rev L e l l c r s . 9 , 54 (1962). (9) S T a k e n o , P r o g r . Theorel P h y s . ( K y o t o ) , 19, 191 (1963); ibid , 99, 328 ( 1 9 6 3 ) ; Yu M K a g a n a n d Y a A. Iosilevskii, Zh Ekspcrim i Tcor F i z , 42, 259 (19621, Soviel P h y s J E T P . 18. 182 (1962) (10) M . V Klein, P h y s Reu., 131, 1500 ( i 9 6 3 ) . J K r u m h a n s l , P r o ceedings of t h e 1963 I n t e r n a t i o n a l Conference o n Lattice D y n a m i c s , Pergamon Press, L o n d o n , t o b e published; J C a l l a w a y , .Vuouo Cimento, 19, 883 ( 1 0 6 3 ) ; S T a k e n o , Progr Theorel P h y s . ( K y o t o ) , S O , 144 (1963).
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of phonons by a point defect not in the Born approximation but by the use of the T-matrix of generalized scattering theory.]] A% relaxation time is then calculated from the scattering cross section and used in the relaxation time approximation to the Peierls transport equation. In the present paper we calculate the lattice thermal conductivity of a disordered crystal starting from a correlation function expression for the thermal conductivity. il-e consider for simplicity an arbitrary cubic Bravais crystal a fraction ( I - p ) of whose lattice sites are occupied by atoms of mass J f , wide a fraction p ( 19 > 1, t h e conditional probability distribution function l V ( i - 3 , .Ya. 1 X g , X2,1 2 ; Xt, XL,11) is independent of t h e values of a n d A' a t time l i , i . e . , if ~ ( i i ) , ta i 2 , ~ 2 l t. ; Xi.S I . t i ) = W C X . J XS,. 1% 1 22, X Z , i d otherwise t h e variables a r e said t o be n o n - M a r k o m a n . la
la
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