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THE EFFECT OF DISLOCATIONS ON THERMAL CONDUCTIVITY
R. Brown
To cite this version:
R. Brown. THE EFFECT OF DISLOCATIONS ON THERMAL CONDUCTIVITY. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-271-C6-273. �10.1051/jphyscol:1981679�. �jpa-00221615�
JOURNAL DE PHYSIQUE
CoZZoque C6, suppl6ment au n012, Tome 42, acernbre 1981 page C6-271
T H E E F F E C T OF D I S L O C A T I O N S ON THERMAL C O N D U C T I V I T Y
R.A. Brown
School o f Mathematics and Physics, Macqtlarie University, North Ryde, N.S. W.
2113, A u s t r a l i a
Abstract: Phonon scattering by the strain fields of deformed crystals is formulated in terms of the third order elastic constants. The thermal restivity due to dislocations is calculated. Theory and experiment are in accord for Cu, Al, Ge and Si but not for LiF.
1. Aims: To formulate the problem of phonon scattering by the static strain fields of crystal defects in terms of the third order elastic (TOE) constants, and to calculate the thermal resistivity due to dislocations.
2. Formulation: A classical formulation of the equations of motion of a statically deformed continuum was presented by Kogure and Hikill/. The system may be quantized to yield the hamiltonian
1 1
H = IKhwK(bpK+ I ) + yIK(hKK.bKb:. -+ hZK.bpK, )
where V is the volume of the deformed body and K : (j,k_) = (polarization, wavevector). The operators br satisfy the usual commutation rules and
, .
+
-
hKKl = (hw/4) ;iiea( K' ) + (kBk&/ pow2)ea(
K' ) NaBy6 + CaByGS,,nSn , (1) where repeated suffices are summed. The e (K) are polarization vectors and p is
a -
the density of the undeformed crystal. The n(k--k_') and N"(k_-k_') are Fourier transforms of the static strains nij(&) and of Nijkl = -Cijklnpp + Cijkpnlp +
Cijplxp + Cipklnjp + Cpjklnip, respectively. the C's are the usual elastic moduli/2/. Due to cancellation of cross-terms when (1) is squared we have, to a good approximation,
lhKK, (2 = (h2/16p:w2) kikp:( ()e:r K') %p:u4lKii
l2
(:e ~)ei( K')(2) The last term is negligibly small in practice and will be dropped. If we follow Klemens/3/ and set = 1/43 = ka/k eqn. (2) becomes
lhKK, 12 = where, for a cubic crystal P = ~(Ig~~l' + I.",212 + Ic,312) + ~ ( 1 ~ 1 2 1 ~ + )""2312 + 1""33112)
, *
+ ~ ( ; ~ ~ g ~ ; + K ~ ~ c ~ : + g33nll + complex conjugate) (3)
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981679
C6-272 JOURNAL DE PHYSIQUE
A = (c
,,
+ 3cl1I2 + 4(Cl12 + Cl2I2 + 2(C123 - cl2I2 + 2(Cl12 - Cl1I2+ 8(C166 + C44)2 + 4(Cls4 - C4,+I2
B = 8(2C166 + C l l + 2 ~+ 16(C14,, ~ ~+ c) ~+ ~ 32(C654 ~ ) + ~C4,,12
c = (Clll + 3Cl1)(Cll2 - Cll) + 4(Cl12 + C12)* + 2(c123 - C12)(Cl12 + C12) + 8(C166 + c44)2 + 4(C166 + C44)(C144 - C44)
The C's are the elastic constants in Voigt notation.
3. Dislocation Thermal Resistivity: In cases where the frequencies w = v.k are j J not degenerate and the dislocations not all parallel it is necessary to use the variational formula/4/
wdis = ( ~ / 2 k ~ ~ ~ ) l ~ ~ ~ ( $ ~ - $Kl)2 pKKl/1 lKxK$Kdno/d~12,
where k is Boltzmann's constant. The sum in (5) is over all dislocations in B
the crystal, which are assumed straight but of random orientation and
arbitrary character. When the resistivity tensor is isotropic, we expect the standard trial function 4 = k_*VT to furnish a good approximation, particularly since it removes the singularity from the forward scattering amplitude for each dislocation. It yields in ( 4 )
where Le,Ls are the total lengths of edge and screw dislocation, respectively, f = B/4,fe = (~-V)-~{A(V~ - V + 318) + ~ / 1 6 + 2c(v2 - v/2 + 1/16)}
in a crystal of (average) Poisson's ratio v, and
In = dmxnex(ex - = 26.0, 124 for n = 4,5 respectively.
We have followed previous authors/3,5,6/ in averaging (3) over directions in the plane normal to each dislocation. The corresponding relaxation rate for a density N (length/volume) of randomly oriented dislocations is ~ - 1 = r.No where
j J
r . = b2S4(fs + 2fe)/1296pz
-
, I' independent of jJ ( 7 )
For random characters we take L = 2L/3 = 2L in (6) for a total length L of dislocation line. The elastic constants of 17-101 were adopted. The sums Sn were evaluated for the [loo], [I101 and [Ill] directions and the arithmetic mean was taken. The results are shown in the table. To check the reliability of the trial function we repeated the calculation for the case of parallel dislocations and degenerate polarization branches, a case which leads to a tractable Boltzmann
e q u a t i o n . The r e s u l t s s u g g e s t t h a t t h e upper bound ( 6 ) o v e r e s t i m a t e s t h e t r u e r e s i s t i v i t y by a f a c t o r 413 when e i t h e r d i s l o c a t i o n s c a t t e r i n g o r electron-phonon s c a t t e r i n g dominate, and i s e x a c t i f boundary s c a t t e r i n g dominates. On t h e o t h e r hand t h e p r o p o r t i o n of edge d i s l o c a t i o n s i n deformed c r y s t a l s i s normally much g r e a t e r t h a n 2 / 3 of t h e t o t a l s o t h i s o v e r e s t i m a t i o n w i l l be l a r g e l y
compensated.
* We have doubled e t e h - p i t c o u n t s t o g e t l e n g t h p e r u n i t volume.
A d e t a i l e d d i s c u s s i o n of t h e d a t a w i l l be g i v e n e l s e w h e r e .
4. Conclusion: The t h e o r y i s i n agreement wish d a t a f o r Cu, Al, S i and Ge. For LiF t h e t h e o r y u n d e r e s t i m a t e s t h e r e s i s t i v i t y by a f a c t o r of 5 ( d a t a of / 1 1 / ) o r 40 ( d a t a of /12/; a s i m i l a r d i s c r e p a n c y a p p e a r s f o r o t h e r a l k a l i h a l i d e s / 1 3 / ) . R e f e r e n c e s :
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/4/ J.M. Ziman, E l e c t r o n s and Phonons (Clarendon P r e s s , Oxford, 1960).
/ 5 / P. C a r r u t h e r s , Rev. Mod. Phys. 2, 92 (1961).
/6/ K. Ohashi, J. Phys. Soc. Japan 24, 437 (1968).
171 K. Salama and G.A. A l e r s , Phys. Rev. 161, 673 (1967).
/ 8 / H.J. McSkimmin and P. Andreatch, J. Appl. Phys. 35, 3312 (1964).
/ 9 / V.K. Garg e t a l . , Phys. S t a t . S o l . ( b ) 80, 6 3 (1977).
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