Pramioa, Vol. 7, No. 4, 1975, pp. 236-244. © Printed in India
Comparison of high-temperature three-phonon resistivities from different theoretical models
G P SRIVASTAVA
Department of Theoretical Physics, The University, Newcastle-upon-Tyne NE1 7RU, England
MS received 13 February 1976; after revision 2 April 1976
Abstract. We present a comparison of high-temperature three-phonon resistivities from the relaxation time methods due to Klemens, Callawoy, and Debye, and the variational method due to Leibfried and Schl/Smarm. All the calculations are based upon the same choice of anharmonic crystal potential; and hence the results are suitable for comparison on quantitative basis. We firld that the resistivities in increasing order of magnitude come from the methods due to Klemens, Callaway Debye, and Leibfrie',l and Schl/3mann respectively.
K~yward~. Lattice the,'tnal conductivity; phonon relaxation.
1. Introduction
Because of lack of reliable resttlt for three-phozlon relaxatio~ times it has. Jtot been possible so far to give a'ty qua,atitative mmlysis for three-pholto~t resistivities due to various models and this is due to two mafit reaso~ts. Firstly, different workers ill the field have co~sJdered different crystal anharmonic potential. Sit'~ce three- phoao.'t scatteri,Jg strengths are decided by anharmonic pom,atial, the magnitude of three-phoao'~ relaxation thnes obtained by different workers differ from each other. Secondly, differet~.t workers have described different ways o f dealing with the co~'lditions imposed by energy a:~d momentum ce_~servatiom This yields diffe- reJtt f, eque:~cy and temperature depe~det~ce for three-pho,to~'t relaxatio~t times.
For example, iu the pioneer works of Klemens (1951) a:~d Leibfried aud Schl6- mamt (1954) the magltitude of the crystal anharmoaic potential is not similar.
This disagreeme_-.t has beeJ~ pointed out at many places
(see e.g.,
Roufosse a~d Klemens 1 9 7 3 - - a b b r e v i a t e d hereafter as RK). What is more i m p o r t a n t is that ~either of these is correct (see Hamiltoat a,.td P a t r o t t 1969; Parrott and Stuckes 1975). Probably the most reliable estimateis due to Hamilton at~d P a r r o t t (1969). Parrott (1972) has correlated the result of Hamilto~t and Parrott with a corrected form of Leibfried aud Sch61marm,The seco~:d fi~ctor deserves equally importa~tt, attentiom Let us, first co~'tsider the umklapp relaxatio~t times. Freque~tcy depe~tde~,.ce tbr the same has been considered as proportional to q"~ by KlemeJts (1951) aitd others
(e.g.,
RK), but q in 1958; aad rece:ttly Mikhail and Simo~ts (1975) have show11 it to be p r o p o r -236
tional to q3. So there can be a considerable amount of unreliability i~ theresults o f calculations from these different pictures. For normal processes Callaway ,(1959) took qO dependence, while Parrott (1963, 1972) and Jackson aald Walker (1971) have suggested q dependence to be more appropriate. Herr,,'ng (1954) however obtained a general picture q", n = 1, 2, 5]2, 3 or 4 depending upon the particular combination of phonon polarizations involved in three-phonon processes.
Also, different authors have considered different combinations of polarizations to give the most importartt corttributiom For example, Benin (1972) found the process t + t ~ 1 to dominate the thermal resistance, while Mikhail and Simons (1975) accept that t + 1 ~ 1 is the dominant process. Author's recent calculations
(1975), however, support Benin's finding.
Thus it is desirable to follow a systematic approach for calculating normal at~d umklapp relaxation times, and thence compare the results of resistivity calcu- lations from different models. In the present paper we consider our recent formulation (1974, 1976) of phonon anharmonic relaxation times, and evaluate Klemens, Callaway and Debye terms for the conductivity, and compare them with the Ziman limit (which is the same as the treatment Leibfr!ed and Sch[6mmm's) calculatio~ts presented by the author elsewhere (1975). Also we point out that the treatment of the variational Ziman limit result in the spirit of the single-mode relaxation time picture does not give the same result. We evaluate the latter, viz., K ' , , and then compare it with Klemens result. The ratio thus obtained is explained theoretically and the disagreement with the recent finding of RK is discussed.
Separate contributions from longitudinal and transverse phonons towards the Debye term is evaluated and some earlier and contemporary works are criticised.
2. Relaxation time results for the conductivity
A generalized form of the conductivity result due to Debye is
KD -- 3No g2 k T 2 ('r°j2 c2) (1)
where NoD is volume of the crystal, k is the Boltzmarm constant, /~ is Planck's constant divided by 2zr, T is absolute temperature, c --- c, (q) is velocity for phonons with frequency ~o---o~, (q), ~- Js combined single-mode relaxation time ~.-1=
.r~l + ~1 (we consider in this paper only three-phonon processes), and ) = ~'f~ N (N + 1) ; q = (q, s) and _N is Planck's distribution function for
q
phonons.
Klemens' conductivity result follows from eq. (1) if we simply ignore the normal (N) processes:
KK -- 3No~2kT2 (~u ~2 ca). (2)
Callaway took into account the momentum conservation property of normal processes and improved the Debye single-mode relaxation time (abbreviated as smrt) result by adding a term to eq. (1) which is really a contribution from the off- diagonal N collision operator (see Simons 1975)"
238 G P Srivastava
~'~
[ ( rqc°J/~-)~-.1
Kc = 3NoDkT" (r°J~cg} q- (q2 rN-1 (1 ~ ~" ~'N-I>J " (3) Going to high temperature approximation _N = kT/ItoJ we can express the ab eve results as follows:
1
I) o
1
and
Kc
=(k/No.Q)I~, c'~
o jdx(x2/~'-1)
(z ¢.xx',,.-')' l
+ , o ( 3 3
(ZC, -= f dx x 2 ( l - - T[TN) "rN - 1 )
# 0
with x = I q [/qo, and qD as Debye radius.
Following our previous work (1974) we write the results for r-a:
r~-x = 32 fro a a' a "~ dx' x 'z a + 2 tz
s s "
E
× 2 (Cx -t- Dx') (1 - - • -F • (Cx -F Dx') (/9 -t- 1) N ' N "
+ ( C x - - D x ' ) [ ( 1 - - • + • ( C x - - D x ' ) ] - ~ - } . (4) Here c~ = c,/cl, ed = ce/cl, a" = c,,,/cx, C = c,[c,,,, D = G,/c,,,, cl = ~/(h + 2ix)/p, h and t~ beittg Lame's second-order elastic constants, and [ A~,,a,, 12 are called throe- p h o a o n scattering strengths. The symbol • takes values 1 and - - i for N and U processes respectively. The first and second t e r m s ! n eq. (4) come from three- p h o a o a events of the types q -}- q' ~ ( - - q " ) and ( - - q) ~- q' + q" respectively.
Further, we express, following Parrott (1972)
I A ~ , ~ , , i 2 ! AL :2 ~-" ~'~ ~"~ (5)
la-~- 2~ = !a + 2~,L
for an isotropic continuum model, with
High ten~verature three-phonon resistivities
239Az = ~/10/3 2p (y -k 2 x) cf~, (6)
a~%
where p is mass density, a is the cubic lattice constant, wD is the Debye frequency and y is the Griiueisen constant for the material. With eqs (5) and (6) the resu It in the high temperature approximation for r -1 becomes
-1 kTqD ~ 1 AL ~ 1/eJ f dx'xx'
r~ -- 327rpcl ~+-2-/~
E
× {2 [1 - - , -? ,
(Cx -k Dx')] q-
(1 - - • q - ,(Cx --
Dx')l}. (7) It can be noticed, however, that the above result for the relaxation times corres- ponds to our earlier set of results (1976).For the isotropic continuum model the energy conservation condition,
viz.,
co ~ o J ' = co" and the momentum conservation condition,
riz., q q-q' + q " =
¢7, G being a reciprocal lattice vector, allow the following processes:
(--q) ~- q' + q":
t ~- t -k t (N only)
l ~ - t + t ( N a n d U b o t h )
l ~ l - q - 1 ; l ~ l - k t ;
q-k q' ~-(--q"):
1 - k l ~ - l ; 1 - k t ~ - l ; t-k 1 ~- 1;
t q- t ~ t (N only)
t -q- t ~ 1. (N and U both).
It should be noticed here that processes t + t~- t; t , ~ t - b t; and 1 -k 1 ~ 1;
1 ~ 1 q- 1 are allowed only in the isotropic continuum model where broadening effect, as suggested by Bross (1962) and Simons (1963), may favour them. However, Hamilton and Parrott suggest that scattering strength for t q- t ~ t (and hence t ~- t -b t) is zero. Therefore we simply ignore them. The limits of integrations over the variable x' for these processes are presented in author's previous papers (1974, 1975, 1976).
From eqs (1') and (2') it is clear that we can separate contributions from sepa- rate polarizatiorts towards Ko a~d KK. However, the normal-drift term in eq. (3') does not permit us to express Kc as being contributed by separate polarizations explicitly.
3. Comparison of relaxation time results with the variational Ziman limit resa It The variational result for the conductivity due to Leibfded arid Schl6mann is irt fact the Ziman limit. The high temperature expression for the same is presented in our previous paper (1975). It is
pc13 h -k 2tz 2
Ko : 3rrqo T
.ff dx dx' x x' {t2 - - r ( x + x')] + (2 - - 1 rx - - x ' ) }(8)
P--3
240 G P Srivastava
where r = ct/cl. This takes into account the full phonon collision operator an d hence is expressed in such a way that only processes of the type q + q' (--q~) (U only) contribute. Further, symmetry and indistinguishability consi- derations allow only the processes t + 1 ~ 1 (U) and t + t ~ 1 (U) to describe
the situation.
If we express the Ziman limit in the spirit of the smrt picture, then it can be written at high temperatures as (see e.g., RK)
Kz' : k/(NoO) 1 x (9)
Z 1/(c?) S d x x ~ ~ u - 1 "
$ 0
With this, we rewrite, with the help of eqs (I'-Y),
K,, = (l + A) K~ (10)
Kc = KD
q-- A/(1 + A)
Kz"---- 1/(1 + A) [KK + AKz'I
(11) (12) where A = {,~-:}~,/{zd:},v, which cast at least be used at high temperatures. Express- ing Kc in terms of A is not necessary at all, but helps in understanding the same in terms of relative strengths of N and U processes. Defining K~r as in eq. (10) not only helps in its interpretation at high temperatures, but also is useful in performing calculations as we shall see in the next section. KK comes from the smrt result, viz., Ko uuder the condition when the role of N processes can be neglected in comparison to U processes. Kc makes a balance between the two extreme situations, viz., N processes very weak (K~r)and N processes infinitely strong (Kzg.
4. Results and discussion
We evaluate KD, Kc, K <, Kz', numerically, using eqs (1'), (Y), (8) and (9) respec- tively, with eq. (7). K~r is evaluated, however, using eq. (10). The reason why we use eq. (10) instead of eq. (2') is that numerical evaluation of KK from eq. (2') gives computing interruption. This is obvious, because for x < x0 (the reduced wave vector at which an U process starts) the integrand in eq. (2') becomes .nfinite. Use of eq. (10) saves us from that difficulty and at the same time does not
~
ead to any inaccuracy for the high temperature results. The parameters used i n the calculations for Ge are taken from our previous papers. The results of calculations are presented in tables 1 and 2.We note that the calculated results are in the order W~: < We < We < W >.
This means when N processes are neglected one gets a lower estimate (W•) for the smrt resistivity (WD). In the Callaway approach the N-drift term is an improve- ment over the result Ko ; and hence We < We. The Ziman limit result from the Leibfried and Schl~mann approach, which considers N processes infinitely strong, brings the highest measure of resistivity.
Our calculations give W > = 2 " 8 5 W~r, and W > = 1"59 W ' z , so that iWz'/Wg= 1"79 at high temperatures. Let us obtain this ratio theoretically.
We cast write, at high temperatures,
Table 1. Comparison of conductivity results (only three-phonon processes a unit : Watt cm -x K-
0.142 0"226 0"211 0'318 0'405
considered)
Table 2. Contribution from separate polarization modes towards the conductivity in the start picture, and the parameter A
KD(1) Kn(t) KD A
0"059 0- 152 0.211 -91
I X
Wz' _ ( 2 1/c= = I dx x ~ -c~ ~) (Sc= ~ I dx x z re)
8 O $ 0
w~
1(21 ax x~)=
8 0
1 1
(]L)..
q'
- - = ( ~ l ! ¢ . = f d x x 2 * u - ' ) ( ~ c . = f d x x ~ . u )
tl 0 • 0
F r o m here it follows that
W z ' / W K = I ' 8 when ~'u-laco 2. (13)
Therefore, theoretically, we see that in the smrt picture the ratio of lattice thermal resistances in the limit of N processes infinitely strong to N processes negligibly small comes out to be 1" 8. Our calculated results give almost the same ratio and hence cart be regarded as very convincingly correct.
However, R K have shown that their approach gives for WK a result WR~r which is 6"8 times Wzs (a result obtained by Leibfried and Schl6mann for W>). We attribute two factors in explaining the wrong finding o f R K in comparison to our convincingly correct result, viz., W > = 2" 85 W~. As we pointed out in our previous paper (1976), these authors cortsider a result for ~.-1 which is 16 times greater than what is normally expected. Consideration of this will make RK's ratio as WLs/WnK = 2" 35. Also, let us compare RK's summation procedure with ours (GP) :
oO(, + a)= f a t , , t
-+ 8x/2 ~r=
q'
Nol2 qo G ~
f
-+ 8 ~/2 ~ dx'
SoOqo3z=c,,fdx ,
8,r3 x '=f
do, .
sm 0' ~ (& oJ)(14)
(where cos 0' = q . q')
+ goOq~= f a x '
(151242 G P Srivastava so that
( Z )~
q ' . . . G 2
( s ) ~ , 2 ,/2 q~
q
= ~ / 2 for I G I : Z q D . (16)
Therefore, assuming t h a t our summation procedure is fairly reasonable for the isotropic continuum model of the crystal, we a~ote that RK's summation proce- dure should be divided by a factor of ` / 2 in order to yield correct result.
Employing this correction their ratio should finally become
Wzs/Wmc : 2' 3 5 . (`/2) : 3" 31. (17)
This ratio, viz., 3" 31, differs from our result, viz., 2" 85, by a factor of 1" 16. We again attribute this discrepancy to three factors. First, R K consider that the summatiort over the polarization indices s' and s" introduces a factor of 4: S - + 4
$'8't
[their assumption (ii) on page 5832 of their paper]. Let us see whether or n o t it is the case. We write, symbolically,
KK -~ ~ l / ~ ' v -1
8
E 1
-->" . ~ ['7"U -1 ( S @ S t ~ - st') ~ - ~'U -1 ( S ~ - S t @ s t ' ) ]
II 8 r 8 tt
Now we can sum over s' and s" to write the denominator as (see Srivastava 1974)
S [7~ -~ (s + s' ~ s") + ~-~ (s ~ s' + s")]
8 p $~P
= ~ u l ( t + 1~- 1 ) + 2 [ r u -1(1 + t ~ 1)-k-ru - l ( t + t ~ 1)]
+ 4 [ r u - 1 ( 1 ~ - 1 q - t ) + r u - l ( l ~ - t q - t ) ] .
This suggests that a factor of 4 comes only for type II processes (s ~- s' q- s"), and n o t for type I processes. Therefore we suggest that R K have over-counted the occurrence of type I processes. Secondly, R K neglect the cross term while averaging over the angles between q and G: they write
(q -}- G) z : q~ q- G ~ (18)
G 2, (19)
i.e., they further neglect q2 relative to G 2. But since q should make an acute angle with G (see Simons 1975) for a U process to take place, we can write
(q + G)Z = q2 + G 2 +
2lq [I G
11 cos w 1= qD 2 (x 2 q- 4 q- 4x [ cos ~/ 1).
Taking [ c o s ~ [ = 1/`/3, as the RMS value for an angle between two random
directions. We can approximate the above to 5.4 q 2 for x = 1/2, and 7.3 qo z f o r x = 1. These values are less than G 2 - - 4 q c 2 by factors of 1"35 and 1.83 respectively. This illustration shows that neglect o/'the cross term by R K is also responsible for their wrong finding. Thirdly, Leibfried and Schl6mann use some averaging procedure i~1 obtaining their result WLs. In particular, we like to mention that they approximate their integral at high temperatures by Z J,,,,,,
G (G) = 1, without evaluating the same over suitable integration domains. Proper evaluation of J may modify the numerical factor in WLs.
Table 2 shows that at 900 K lo~,lgitudinal p h o n o n s contribute about 2 8 ~ of the total heat predicted by the Debye term Ko. This finding suggests that in the smrt treatment longitudinal p h o n o n s play quite important part ill heat conduc- tion at high temperatures. This is in disagreement with a qualitative picture by Sharma e t a l ( 1 9 7 1 ) a n d D u b e y (1976a) who assume that at high temperatures almost all heat is transported by transverse phonons. Also, in the high tempera- ture approximation the parameter A is obtained to be 0.91 at 900 K. This suggests that at high temperatures N processes are almost as strong as U processes.
With this value of A we obtain the normal-drift to be half in magnitude of KD.
This overrules the suggestion by Sh~rma et al (1971) and Dubey (.1976 b) that normal-drift is negligible in Ge. Table 2 gives coatributiolls from separate
polarization modes, and the parameter A.
The codclusions drawn in this paper, although exhibited for Ge alone, should come true at least for crystals showing cubic symmetry, where use of an isotropic model cart be made without introducing appreciable error. However, use o f the continuum model at high temperatures should n o longer be regarded as fair. A suitable dispersion law must be used. But a proper use of a dispersive model will not be so easy, and as far as qualitative p~cture for the present intercomparisor~
of results is cor~cerned n o t much difference should be expected. The work was completed at UWIST, Condiff.
Acknowledgements
Interest o f J E Parrott, R A H Hamilton, M Jaros and G S Verma ( ~ d i a ) and G S Rushbrook are thankfully acknowledged.
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