fulltext

Heat transport in dielectric solids with diamond structure

M. OMINIand A. SPARAVIGNA

Dipartimento di Fisica and Istituto Nazionale di Fisica della Materia (INFM) Politecnico di Torino - C.so Duca degli Abruzzi 24, 10129 Torino, Italy

(ricevuto il 29 Luglio 1997; approvato il 2 Settembre 1997)

Summary. — Through an iteration procedure the phonon transport equation is solved for solids with diamond structure. The interaction between nearest and next-nearest neighbours is expressed in terms of a pair potential by which an accurate description of the phonon spectrum and of the anharmonic properties of the crystal is obtained. In this way the role of the various scattering mechanisms contributing to thermal resistance can be investigated without the use of the continuum approximation or of the relaxation time approximation for phonon-phonon interaction. The theory is applied to Ge and Si, for which the predicted temperature behaviour of thermal conductivity is found in excellent agreement with experiment.

PACS 66.70 – Nonelectronic thermal conduction and heat-pulse propagation in solids; thermal waves.

PACS 63.20.Kr – Phonon-electron and phonon-phonon interactions. PACS 63.20.Mt – Phonon-defect interactions.

1. – Introduction

The theory of lattice thermal conductivity in germanium and silicon is still confined to the works of Srivastava [1, 2], which will be referred to as I and II, respectively. From these fundamental works two kinds of problems arise: the first one is connected with the use of a relaxation time for all the phonon scattering processes; the second one with the simplified description of the crystal lattice, assumed by Srivastava as an isotropic continuum carrying only acoustic phonons in I, both acoustic and optical phonons in II. The relaxation time approximation, already introduced by Callaway [3], Holland [4], Parrott [5], Tiwari and Agrawal [6], as well as the various approximations inherent in the continuum hypothesis (use of the Debye model for acoustic branches and of the Einstein model for the optical branches; Hamilton and Parrott [7] scheme for the description of the Umklapp processes) make it very hard to draw definite conclusions as to the role played by phonon-phonon scattering, isotope scattering and

boundary scattering, as a function of temperature. In particular, the contribution of optical phonons to the thermal transport is not clear: in I Srivastava overlooked the weight of these phonons, while in II he deduced that their inclusion should significantly change the magnitude of the conductivity. How strongly is this conclusion affected by the use of the Einstein model, by the absence of dispersion, by the isotropic model or by an approximate description of phonon-phonon interaction?

A final problem emphasized by Srivastava concerns the high-temperature behavior of the conductivity k: the experimental data (Holland [4], Slack and Glassbrenner [8]) show that the simple law k AT21 _{is not obeyed, and should be substituted by a power}

law k AT2n _{with n slightly higher than unity. In II Srivastava did not succeed in}

explaining such an effect, and attributed the failure of his calculation both to the simplified treatment of ahnarmonicity and to the absence of phonon dispersion.

The aim of the present paper is to provide an answer to all the above problems thanks to the same technique by which the authors calculated the thermal conductivity of rare-gas solids [9, 10]: in ref. [10], hereafter referred to as III, the use of an iteration procedure was shown to be quite useful to arrive at the solution of phonon Boltzmann equation without any model approximation. By the same procedure it will be possible to describe, in the frame of the real Brillouin Zone of the lattice, and without unjustified assumptions, the role of three-phonon collisions (N- and U-processes) and of the isotope scattering, and to account for the real form of the dispersion curves, both for acoustic and optical phonons. The only approximation maintained in the description of scattering processes refers to boundary scattering, for which a rigorous treatment would be too cumbersome and of some interest only for the very low-temperature range: in agreement with the choice of previous authors, the presence of boundary is simply accounted for by a relaxation time, so that our conclusions are rigorous in the whole range above A30 K, where boundary scattering is not present, while they are affected by some uncertainty below this temperature. This means, in particular, that the problem of the role played by optical phonons is completely solved, because these phonons are really effective only for T D30 K.

2. – Phonons in the harmonic lattice

The first task in a theory of thermal transport is to express the matrix elements for three-phonon scattering in terms of measurable anharmonic parameters. Within the continuum approximation, which represents the only framework so far considered for the solids of our interest in the literature, the above parameters are the Fourier transforms A_{qq 8 q 9} of the 3rd-order elastic constants: since, however, either elastic constants are not available for various crystal structures, or different experimental results disagree widely, the coefficients A_{qq 8 q 9} are usually expressed, by a further approximation, in terms of the Grüneisen constant. Such a procedure is open to criticisms because the continuum hypothesis is not consistent with the occurrence of

U-processes, nor with the presence of phonon dispersion: also the use of the Einstein

model to account for optical modes cannot be satisfactory, these modes being intrinsically extraneous to a continuum solid.

Our approach to the problem is through a model calculation where the interaction between atoms is expressed by a volume-independent central potential. We recall that a potential of this kind gives rise to the relation [11] ¯K/¯P 42g01 2.36 between the

isotropic solid: such a relation is in fair agreement with experiment for many metals, and shows that the model represents a reliable tool for a simple description of anharmonicity even in solids where pseudopotential theory should be applied.

In the present case, the parameters of the potential will be chosen so as to provide a satisfactory description of phonon dispersion curves, and of the experimental temperature dependence g(T) of the Grüneisen constant. In this way we move within a framework where optical modes are not introduced as an extraneous presence (Einstein model), but follow as a natural consequence of the crystal Hamiltonian. It is clear that a calculation in terms of pseudopotential theory would be more fundamental. However, apart from its complexity, we point out that the aim of the present work is not to provide a first-principle theory of crystal anharmonicity, but to solve the phonon Boltzmann equation for a solid having the structure and the vibrational (harmonic and anharmonic) properties of the real lattice. The solid with our model potential has precisely the required properties: more specifically, we anticipate that by such a model we will be able to explain, in an unexpected way, even the minimum of g(T) in the low-temperature region. It is reasonable, therefore, to expect from the same model a reliable information on the weight of the various scattering mechanisms contributing to thermal resistance.

In the following we will denote by l the position vector of the origin of the generical cell and by b the vector describing, within the cell, an atom belonging to the basis. For Ge and Si b has two possible determinations, which will be labelled by 0 and B, respectively. We will refer to Cartesian coordinates xi(i 41, 2, 3) with unit vectors ui chosen in such a way that vectors l are represented by the equation

l 4 h1

k2

!

i

Niui, (1)

where N1, N2, N3 are integers satisfying the condition N11 N21 N34 even integer

(including zero) and h1 is the distance between the origins of two adjacent cells. By

such a choice of axes, the two atoms of the basis 0 and B are represented by vectors 0 4 ( 0 , 0 , 0 ) and B 4 1 2 h1 k2

!

i ui, (2)

respectively. Assuming a central potential V(r), where r is the interatomic distance, and callinghlbthe displacement of atom (l,b) from its average position in the vibrating

lattice, one can expand in terms of the displacements the function V(Nl81b81h_{l8 b82}

l 2b2hlbN): in this way one obtains the following expression for the second-order

term of the potential energy of interaction between atoms (l , b) and (l8, b8):

V_{lb , l8 b8}( 2 ) ₄ 1

2]gbb8(l82l) [hl8 b82hlb] Q [hl8 b82hlb] 1 (3)

where

.

`

/

`

´

g_{bb8(l82l) 4}

k

1 r dV dr

l

_{r 4Nl82l1b82bN}, b_{bb8(l82l) 4}

k

1 r d dr

k

1 r dV dr

ll

_{r 4Nl82l1b82bN}. (4)

In the harmonic approximation, the total potential energy of the reticular center (l , b) is Ulb( 2 )4

!

l8(cl)

!

b8 V_{lb , l8 b8}( 2 ) ₁

!

b8(cb) V_{lb , lb8}( 2 ) . (5)

The force acting on (l , b) is 2˜U( 2 )

lb , where the derivatives required by the operator ˜

must be evaluated with respect to the components of vector hlb; consequently the

equation of motion can be written in the form

Mh n n lb4

!

l8(cl)

!

b8]gbb8(l82l)( h_{l8 b82}hlb) 1 (6) 1bbb8(l82l)[ (l82l1b82b)Q (hl8 b82hlb) ](l82l1b82b)(1 1

!

b8(cb)]gbb8 (0)(h_lb82hlb) 1bbb8(0)(b82b)Q (hlb82hlb)(b82b)( ,

where M is the mass of an atom. The sum over l8 will be transformed into a sum over a cell index t labelling the point with b 40 of the generic cell; more precisely, introducing the vector h_{t4 l8 2 l and looking for a solution of eq. (6) in the form}hlb4

ebexp [iq Q l2ivt], we easily obtain from (6) (written for b40 and b4B) the following

system of equations for the amplitudese0andeB:

!

t(c 0 )]g t 00(jt2 1 )e01 bt00(jt2 1 )(htQe0) ht( 1 (7) 1

!

t ]g t 0B(jteB2e0) 1bt0B(ht1 B) Q (jteB2e0)(ht1 B)( 4 2Mv2e0,

!

t ]g t B0(jte02eB) 1btB0(ht2 B) Q (jte02eB)(ht2 B)( 1 (8) 1

!

t(c 0 )]g t BB(jt2 1 )eB1 btBB(jt2 1 )(htQeB) ht( 4 2Mv2eB, where gt

bb8and btbb8stand for gbb8(ht) and bbb8(ht) , respectively, and jt4 exp [iq Q ht]. The constraint t c 0 simply means that an atom cannot interact with itself. According to (1), vectors ht can be expressed by the linear combination

ht4

h1

k2

!

i

xtiui, (9)

where xt1, xt2, xt3 are of the form ( 1 , 1 , 0 ), ( 1 , 0 , 1 ) R etc. for the twelve cells lying closest to the cell at l (that is to the cell with h 40). We will use the same description of the 1st Brillouin Zone (BZ) of the reciprocal lattice as introduced in III. Such a lattice is reciprocal with respect to the fcc lattice determined by points with b 40. The

components qi of vector q will be written in the form qi4 ( 2 pk2/h1) hi, where the dimensionless variables hi are expressed in terms of cylindrical coordinates u , h , z through the relations h14 h cos u, h24 hsin u, h34 z. In this way one obtains

jt4 exp [ 2 pi(hxt1cos u 1hxt2sin u 1zxt3) ] . (10)

The coupling coefficients (4) take values that will be denoted by g and b, respectively, for atoms at the nearest-neighbour distance R and by g 8 and b8 for atoms at the next-nearest-neighbour distance R 8. Since coefficients gt

0B and bt0B in eqs. (7), (8) connect

atoms at a relative distance Nht1 BN, they take values g and b when t runs over sites for which Nht1 BN 4 R and cannot take values g 8 and b 8 because the condition Nht1 BN 4 R 8 is never satisfied. Similarly, coefficients gtB0 and btB0 connect atoms at

relative distance Nht2 BN, so that they take values g and b when t satisfy the condition Nht2 BN 4 R, and again, they cannot take values g 8 and b 8. On account of the rela-tions R 4 (k3/2k2) h1and R 84h1, one can also say that gt0Band bt0Btake values g and

b, respectively, when

!

i

g

x_ti1 1 2

h

2 4 3 4 (11) and gt B0and btB0 when

!

i

g

xti2 1 2

h

2 4 3 4 . (12)

The index t satisfying eq. (11) labels the three lattice points characterized by the following values of xti: (21, 21, 0); (21, 0, 21); (0, 21, 21); conversely, the lattice points satisfying eq. (12) are ( 1 , 1 , 0 ); ( 1 , 0 , 1 ); ( 0 , 1 , 1 ).

Coefficients gt00, bt00and gtBB, btBBconnect atoms at a relative distance NhtN so that, since NhtN D R, they can never take values g, b, but take values g 8, b 8 when t satisfies the condition NhtN 4 R 8 4 h1, or

!

i x2 ti4 2 . (13)

The lattice points satisfying such a condition are the twelve neighbours of an atom in a fcc lattice with nearest-neighbour distance h1.

In the following we will denote by

!

1_{a sum over the three lattice points satisfying}

eq. (11) and by

!

2_{a sum over the three lattice points satisfying eq. (12). A sum over t}

without any superscript will be referred to the twelve lattice points satisfying eq. (13). In the frame of such notations, it is useful to define the following dimensionless coefficients, depending on the phonon wave vector q

m6 ki4

!

t 6

g

_x tk6 1 2

hg

xti6 1 2

h

jt, (14) n6 ki4

!

t 6

g

_x tk6 1 2

hg

xti6 1 2

h

, (15) hki4

!

t (jt2 1 ) xtkxti, (16)

l6 4

!

t 6_j t, (17) h 4

!

t (jt2 1 ) , (18)

and the following dimensionless parameters, depending on the features of the pair potential, r 4 g bh2 1 , _{r 84} g 8 bh2 1 . (19)

It is also convenient to introduce a reduced frequency v connected to v by the relation

v 4h1

o

b

2 M v . (20)

In this way, putting e14 e01, e24 e02, e34 e03, e44 eB1, e54 eB2, e64 eB3, where e0i

and eBi are the i-th component of vectors e0 and eB, respectively, one can easily

transform eqs. (7)-(8) into a linear homogeneous system of equations of the form

!

6 k 41

aikek1 v2ei4 0 , (21)

where the coefficients aikhave the expressions given in table I.

Since matrix ]aik( is Hermitian, the six solutions vp (p 41, . . . 6) of the equation det ]aik( 4 0 are real and correspond to six-dimensional vectors with components epi TABLEI. – Matrix elements aikentering eq. (21).

a114 2n1112 1 O4 1 ( b 8 Ob) h112 8 r 1 2 r 8 h a124 2n1212 1 O4 1 ( b 8 Ob) h21 a134 2n1312 1 O4 1 ( b 8 Ob) h31 a144 m1111 1 O4 1 2 r( 1 1 l1) a154 m1211 1 O4 a164 m1311 1 O4 a314 2n1132 1 O4 1 ( b 8 Ob) h13 a324 2n1232 1 O4 1 ( b 8 Ob) h23 a334 2n1332 1 O4 1 ( b 8 Ob) h332 8 r 1 2 r 8 h a344 m1131 1 O4 a354 m1231 1 O4 a364 m1331 1 O4 1 2 r( 1 1 l1) a514 m2121 1 O4 a524 m2221 1 O4 1 2 r( 1 1 l2) a534 m2321 1 O4 a544 2n2122 1 O4 1 ( b 8 Ob) h12 a554 2n2222 1 O4 1 (b 8 Ob) h222 8 r 1 2 r 8 h a564 2n2322 1 O4 1 (b 8 Ob) h32 a214 2n1122 1 O4 1 ( b 8 Ob) h12 a224 2n1222 1 O4 1 ( b 8 Ob) h222 8 r 1 2 r 8 h a234 2n1322 1 O4 1 ( b 8 Ob) h32 a244 m1121 1 O4 a254 m1221 1 O4 1 2 r( 1 1 l1) a264 m1321 1 O4 a414 m2111 1 O4 1 2 r( 1 1 l2) a424 m2211 1 O4 a434 m2311 1 O4 a444 2n2112 1 O4 1 ( b 8 Ob) h112 8 r 1 2 r 8 h a454 2n2212 1 O4 1 ( b 8 Ob) h21 a464 2n2312 1 O4 1 ( b 8 Ob) h31 a614 m2131 1 O4 a624 m2231 1 O4 a634 m2331 1 O4 1 2 r( 1 1 l2) a644 2n2132 1 O4 1 ( b 8 Ob) h13 a654 2n2232 1 O4 1 ( b 8 Ob) h23 a664 2n2332 1 O4 1 ( b 8 Ob) h332 8 r 1 2 r 8 h

which can be normalized so as to satisfy orthonormality and completeness relations: in terms of the corresponding vectorsebp(b 40, B) these relations are

!

b e*bpQebp 84 dpp 8, (22)

!

_p e *_{b8 pj}ebpi4 dijdbb8. (23)

An elementary excitation of the crystal in the harmonic approximation will be a phonon described by wave vector q and polarization index p: it will be denoted by the compact notation Q. Consequently, its frequency will be indicated by vQ and its polarization vector associated to atom b byeQb: the full notations would be vp(u , h , z) andebp(u , h , z), the dependence on the cylindrical coordinates u , h , z being explicitly

given through eq. (10). If N is the total number of cells in the crystal, the expansion of the displacement fieldhlbin terms of phonon absorption and creation operators aQ, aQ†, can be written in the form

hlb4 i

g

ˇ 2 MN

h

1 /2

!

Q 1 kvQ

[

e*Qbe2iq Q laQ2eQbeiq Q la † Q

]

. (24)

This corresponds to the representation used by Srivastava [12], (p. 94) for the choice

ebp(2q) 4e*bp(q).

3. – Phonon-phonon interaction

The third-order contribution to the interaction energy between the two reticular centers at l , b and l8, b8 is [9, 10] V_{lb,l8 b8}( 3 ) ₄ 1 6abb8(l82l)[ (l82l1b82b)Q (hl8 b82hlb) ] 3 1 (25) 11 2bbb8(l82l)[ (l82l1b82b)Q (hl8 b82hlb) ] 3 [ (hl8 b82hlb) Q (hl8 b82hlb) ] , where a_bb8(l82l) 4

k

1 r d dr

k

1 r d dr

k

1 r dV dr

lll

_{r 4Nl82l1b82bN}. (26)

The corresponding contribution to the total energy of the crystal is

H( 3 ) 4 1 2

!

l l8(cl)

!

b , b8

!

V( 3 ) lb , l8 b81

!

l V( 3 ) l0 , lB. (27)

In terms of the vector ht4 l8 2 l, where t becomes the index labelling the cells in the neighbourhood of the cell at l, we obtain the following result for the matrix element aQ 9 NH( 3 )NQ , Q 8 b required to describe a scattering processes in which two phonons Q

and Q 8 annihilate and give rise to an emerging phonon Q 9: aQ 9 NH( 3 )NQ , Q 8 b 4 i

g

ˇ 2 MN

h

3 O2 _N 2 d_{q 1q82q9, g} (vQvQ 8vQ 9)1 O2 HQQ 8 Q 9, (28) where HQQ 8 Q 94

!

t(c 0 )b , b8

!

at bb8htbb8Q (e*Qb8j *tQ2eQb* ) htbb8Q (e*Q 8 b8j *tQ 82e*Q 8 b) 3 (29) 3htbb8Q (eQ 9 b8jtQ 92eQ 9 b) 1

!

t(c 0 )b , b8

!

bt bb8[htbb8Q (e*Qb8j *tQ2e*Qb) 3 3(e*_{Q 8 b8}j *_{tQ 82}e*_{Q 8 b}) Q (e_{Q 9 b8}j_{tQ 92}e_{Q 9 b) 1h}t bb8Q (e*Q 8 b8j *tQ 82e*Q 8 b) 3 3(e*_Qb8j *tQ2e*Qb) Q (eQ 9 b8jtQ 92eQ 9 b) 1htbb8Q (eQ 9 b8jtQ 92eQ 9 b) 3 3(e*_Qb8j *tQ2eQb) Q (* e*Q 8 b8j *tQ 82e*Q 8 b) ] 3 12 a00B[B Q (e*QB2e*Q0) ] [B Q (e*Q 8 B2e*Q 8 0) ] [B Q (eQ 9 B2eQ 9 0) ] 1 12 b00B[B Q (e*QB2e*Q0)(e*Q 8 B2e*Q 8 0) Q (eQ 9 B2eQ 9 0) 1 1B Q (e*_{Q 8 B2}e*_{Q 8 0})(eQB* 2e*Q0) Q (eQ 9 B2eQ 9 0) 1 1B Q (eQ 9 B2eQ 9 0)(e*QB2e*Q0) Q (e*Q 8 B2e*Q 8 0) ] , where we have put ht

bb84 ht1 b8 2 b and jtQ4 jt(q) and g is any reciprocal lattice vector.

Denoting by a and a 8 the values taken by (26) when the two atoms are at the nearest-neighbour distance R, and at the next-nearest-neighbour distance R 8, respectively, we will use the dimensionless anharmonic parameters

e 42 h 2 1a b , e 842 h12a 8 b . (30) Putting E2

tQi4 eQ0ijtQ2 eQBi, EtQi14 eQBijtQ2 eQ0i, FQi4 eQBi2 eQ0i, xt4

!

ixtiui and introducing the dimensionless functions

A(6)₄

!

t 6

!

ijk

g

xti6 1 2

h

(

E 6 tQi

)

*

g

xtj6 1 2

h

(

E 6 tQ 8 j

)

*

g

xtk6 1 2

h

E 6 tQ 9 k, (31) C 4

!

i F *Qi

!

j F *_{Q 8 j}

!

k F_{Q 9 k}, (32) B(6)₄

!

t 6

!

ij

k

g

x_ti6 1 2

h

(

E 6 tQi

)

*

(

EtQ 8 j6

)

* EtQ 9 j16

g

xti6 1 2

h

3 (33) 3

(

E6 tQ 8 i

)

*

(

EtQ j6

)

* EtQ 9 j16

g

xti6 1 2

h

E 6 tQ 9 i

(

EtQ j6

)

*

(

EtQ 8 j6

)

*

l

, D 4

!

i F *Qi

!

j F *_{Q 8 j}F_{Q 9 j1}

!

i F *_{Q 8 i}

!

j F *Q jFQ 9 j1

!

i F_{Q 9 i}

!

j F *Q jF *Q 8 j, (34)

M 4

!

t (j *tQ2 1 )(j *tQ 82 1 )(jtQ 92 1 ) ](xtQe*Q0)(xtQe*Q 8 0)(xtQeQ 9 0) 1 (35) 1(xtQe*QB)(xtQe*Q 8 B)(xtQeQ 9 B)( , N 4

!

t (j *tQ2 1 )(j *tQ 82 1 )(jtQ 92 1 )](xtQe*Q0)(e*Q 8 0QeQ 9 0) 1 (xtQe*Q 8 0)(e*Q0QeQ 9 0) 1 (36) 1 (xtQeQ 9 0)(e*Q0QeQ 8 0* ) 1 (xtQe*QB)(e*Q 8 BQeQ 9 B) 1 (xtQe*Q 8 B)(e*QBQeQ 9 B) 1 (xtQeQ 9 B)(e*QBQe*Q 8 B)( , one easily obtains

aQ 9 NH( 3 ) NQ , Q 8 b 4 i 4k2

g

ˇ 2 MN

h

3 O2 _Nh3 1a (vQvQ 8vQ 9)1 O2 M(Q , Q 8, Q 9) , (37) where M(Q , Q 8, Q 9) 4A(2) 1 A(1)1 1 4C 1 e 8 e M 2 2 e

k

B (2) 1 B(1)1 D 1 b 8 b N

l

(38)

and vectors q , q8, q9 are subjected to the conservation equation

q 1q84q91g .

(39)

Let us introduce the dimensionless functions

N1_{(p , u , h , z ; p 8, u8, h8, z8; p 9, u9, h9, z9) 4n}o QnQ 8( 1 1no Q 9o ) /(vQvQ 8vQ 9) , (40) N2_{(p , u , h , z ; p 8, u8, h8, z8; p 9, u9, h9, z9)4n}o Q( 11nQ 8)( 11no Q 9o ) /(vQvQ 8vQ 9) , (41)

where nQois the Bose-Einstein distribution function.

For a process described by (39) the probability rate of interest for transport properties is QQ 9 QQ 84 2 p ˇ NaQ 9 NH ( 3 ) NQ , Q 8 b N2d(ˇv_{Q 92 ˇv}Q2 ˇvQ 8) nQonQ 8( 1 1no Q 9) 4o (42) 4 pˇh 2 1 32 MN a2 b2 N 1 N M (Q , Q 8 , Q 9 ) N2d(v_{Q 92 vQ2 vQ 8}) . It is easy to see that the analogous rate related to the process Q KQ 81Q 9 is

QQ 8 Q 9 Q 4 pˇh2 1 32 MN a2 b2 N 2 N M (Q 8 , Q 9 , Q) N2d(v_{Q 81 vQ 92 vQ}) , (43)

where vectors q , q8 q9 satisfy the conservation equation

q 4q81q91g

(44)

and M(Q 8, Q 9, Q) is the function that can be obtained by performing the substitutions

4. – Transport equation for phonons

In terms of the deviation function cqp, linked to the perturbed and unperturbed

phonon distributions nqpand nqpo by the relation

nqp4 nqpo2 cqp ¯no qp ¯(ˇvqp) , (45)

the linearized Boltzmann equation for a solid subjected to a thermal gradient can be written in the form [13]

kBTvqpQ ˜T ¯no qp ¯T 4q8 p 8

!

q9 p 9

!

Qq9 p" qp , q8 p 8[cq9 p 92 cq8 p 82 cqp] 1 (46) 11 2 q8 p 8

!

q9 p 9

!

Qqpq8 p 8, q9 p 9[cq9 p 91 cq8 p 82 cqp] 1 1

!

q8 p 8 Qq8 p 8 qp [cq8 p 82 cqp] 2 1 tqp cqpnqpo( 1 1nqpo) ,

where the 1st and the 2nd term on the right-hand side describe three phonon scattering processes, the 3rd term the elastic scattering due to impurities, and the 4th term provides a phenomenological description of boundary scattering in terms of relaxation times tqp. By vqp we have denoted the phonon group velocity ¯vqp/¯q. In terms of the reduced components hi of vector q, as introduced in sect. 2, and frequencies vqp_{4 v}p(u , h , z) defined by eq. (20), the left-hand side of eq. (46) can be written lhs of eq . ( 46 ) 4 ˇh 3 1b 4 pk2 MT vp eb vp (eb vp 2 1 )2

!

i ¯vp ¯hi

g

¯T ¯xi

h

, (47) where b 4 ˇh1 kBT

o

b 2 M . (48)

To evaluate the first and the second term on the right-hand side of eq. (46), it is convenient to transform each sum over q9 into a sum over the reciprocal lattice vectors

g

(

see eqs. (39)-(44)

)

. As in III, these vectors will be denoted by two indices, n and k, the former being referred to the shell of neighbours around the origin of the reciprocal lattice, the latter to the point of the reciprocal lattice belonging to this shell. In terms of the unit vectors ui introduced in eq. (1) one has

gn k4 pk2 h1

!

i mn k , iui, (49) where mn

k , 1, mnk , 2, mnk , 3 are of the form ( 1 , 1 , 1 ), (21, 1, 1), R for n41 (1st shell), ( 2 , 0 , 0 ), ( 0 , 2 , 0 ), R for n 42 (2nd shell), ... etc. The case n40 corresponds to g40 (normal processes) and, consequently, mn

In III it was shown that the momentum and energy conservation equations can be written in the form

hi6 hi8 2 lnk , i4 hi9 , (50)

vp(u , h , z) 6vp 8(u 8, h8, z8) 4 vp 9(u 9, h9, z6z82lnk3) , (51)

where ln

k , i4 mnk , i/2 and the double sign corresponds to + for scattering processes described by eq. (42) and to 2 for scattering processes described by eq. (43). Let us fix g, namely (n , k): for a given choice of the variables u , h , z , u 8, h8, eqs. (50) provide

u 9, h9

(

see eqs. (18) and (19) in III

)

and consequently eq. (51) can be solved with respect to z 8. Let z86( j) be the j-th solution satisfying the condition 2M(u8, h8) G

z 8GM(u8, h8), where 6M(u8, h8) are the values of z for the points lying on the

boundary of the BZ having polar coordinates u 8 and h8: the value of z9 corresponding to z 86( j) will be z 96( j) 4z6z86(j) 2l

n

k , 3. Let V be the volume of the crystal. Transforming, as in III, the sum over q8 into an integral over the BZ through the substitution

!

q8K V ( 2 p)3

g

2 pk2 h1

h

3



0 2 p du 8



0 H(u 8) h 8 dh8



2M(u 8 , h 8 ) M(u 8, h8) dz 8 , (52)

where H(u 8) is the function explicitly given in III, and eliminating the integration over

z 8 thanks to the presence of the delta-functions in (42) and (43), we easily obtain the

contribution of the 1st and 2nd term to the right-hand side of eq. (46):

1st term 12nd term4 pk2 16 Vˇ NMh1 a2 b2 p 8 p 9

!

!

nk

!

j



0 2 p du 8



0 H(u 8) h 8 dh83 (53) 3

{

k

N1

N

¯_v8 ¯_{z 8} 2 ¯_v9 ¯_{z 9}

N

21 NF1N2(c 92c82c)

l

z 84z81( j), z 94z91( j) 1 11 2

k

N 2

N

¯¯v8_{z 8} 2 ¯_v9 ¯_{z 9}

N

21 NF2N2(c 91c82c)

l

z 84z82( j), z 94z92( j)

}

.

Here, for each of the three functions c, n0_{, v we have used the compact notations}

cp(u , h , z) 4c, cp 8(u 8, h8, z8) 4c8, cp 9(u 9, h9, z8) 4c9, etc.; moreover, according to eqs. (42) and (43), we have put

F1_fF1 pp 8 p 94 M (p , u , h , z ; p 8 , u 8 , h 8 , z 8 ; p 9 , u 9 , h 9 , z 9 ) , (54) F2_fF2 pp 8 p 94 M (p 8 , u 8 , h 8 , z 8 ; p 9 , u 9 , h 9 , z" ; p , u , h , z) ; (55)

finally, it has to be pointed out that V/N represents the cell volume, namely

Vc4 h13/k2.

Let us now consider the 3rd and the 4th term appearing on the right-hand side of eq. (46). The 3rd term will be discussed in connection with isotope scattering, due to a difference of mass DM which is assumed to be present in a fraction fi of atoms belonging to the crystal. Since the total number of atoms is twice the total number of

cells N, there are 2 Nfi centers producing elastic scattering, and these are statistically distributed in such a way that Nfi are of type 0 and Nfiare of type B. The probability rate ( Pq 8

q )0for a scattering Q KQ 8 due to a single center at (l, 0) can be deduced by

evaluating the matrix element aQ 8 NH 8isoNQb, where H 8iso4 DM(h

n

l0)2/2 and h

n

l0 is the

operator corresponding to the velocity of atom (l , 0) (see Srivastava [12], p. 179): one easily obtains ( PQ 8 Q )04 p 2

g

DM M

h

2 ₁ N2vQvQ 8Ne*Q0QeQ 8 0N 2_d(v Q 82 vQ) nQo( 1 1nQ 8o ) . (56)

The probability rate ( PQ 8

Q )B related to a single-scattering center at (l8, B) is

described by a similar expression, obtainable from (56) by the substitution 0 KB in the indices of polarization vectors. Since the scattering centers are randomly distributed and, on the other hand, fiis assumed to be so small that any interference effect between any two centers is excluded, the total probability rate Qq8 p

qp appearing in eq. (46) is QQ 8 Q 4 Nfi( PQQ 8)01 Nfi( PQQ 8)B4 (57) 4 p 2 Nfi

g

DM M

h

2 vQvQ 8n o Q( 1 1n o Q 8)[Ne*Q0QeQ 8 0N 2 1 Ne*QBQeQ 8 BN 2_{] d(v} Q 82 vQ) . Consequently, thanks to (52), the 3rd term on the rhs of eq. (46) can be written in the form 3 rd term 4fip

g

D M M

h

2 h1

g

b 2 M

h

1 /2

!

jp 8



0 2 p du 8



0 H(u 8) h 8 dh83 (58) 3

m

v v8 N¯v8 ¯_{z 8} N 21_no ( 1 1no 8 )(c 8 2 c)[Ne*0Qe80N21 Ne*BQe8BN2]

n

z 84z8j ,

wheree0,e80, etc., are compact notations foreQ0,eQ 8 0, etc., and z 8j is the j-th solution of the energy conservation equation vp_{(u , h , z) 4 vp 8(u 8, h8, z8).}

Let us now introduce the dimensionless parameter

l 4 8k2 h 3 1fi ˇ_e2 kbM

g

DM M

h

2 (59)

and the characteristic time

U 4 16 Mh

2 1

pˇe2 .

(60)

Let us also define the functions

Api(u , h , z) 4 ¯vp ¯hi , (61) u_{pi(u , h , z) 4 vp} e b vp

(

eb vp 2 1

)

2 Api (62)

and use the notations

K6

4 ]NFpp 8 p 96 N2NAp 83(u 8, h8, z8)2Ap 93(u 9, h9, z9)N21(z 84z86( j), z 94z86( j),

(63) Z 4

{

no p(u , h , z)[np 8o(u 8, h8, z8)11] vp(u , h , z) vp 8(u 8, h8, z8) NAp 83(u 8, h8, z8)N 3 (64) 3

k

N

!

k e *p0k (u , h , z) e_{p 8 0k(u 8, h8, z8)}

N

2 1

N

!

ke *pBk (u , h , z) e_{p 8 Bk(u 8, h8, z8)}

N

2

l

n

_{z 84z8j}. Finally, let us introduce the function fp(u , h , z) related to cp(u , h , z) by the equation

cp4 2

2k2 h1b3

p2_Ta2 fp.

(65)

In this way it is easy to see that the transport equation (46) can be rewritten in the following way: f 4 1 Q

!

i upi ¯T ¯xi 1 1 Qp 8 p"

!

!

nk

!

j



d2 h 83 (66) 3

m

[K1_N1_{( f 92f 8) ]z 84z8} 1( j); z 94z91( j)1 1 2[K 2 _N2_{( f 91f 8) ]z 84z8} 2( j ); z 94z92( j)

n

1 1 l Q

!

p 8

!

j



_{]Zf 8 (z 84z8} jd 2 h 8 ,

where again we have used the compact notation for which fp(u , h , z) 4f,

f_{p 8(u 8, h8, z8) 4f 8, fp 9(u 9, h9, z9) 4f 9. The symbol}

s

d2

h 8 stays for the double integral

s

0 2 p

du 8

s

0H(u 8)h 8 dh8. Q stands for the function

Qp(u , h , z) 4

!

n , k p 8, p"

!

!

j



d2 h 83 (67) 3

m

[K1N1]z 84z81( j); z 94z91( j)1 1 2[K 2 _N2_] z 84z82( j); z 94z92( j)

n

1 1l

!

p 8

!

j



_{Z d}2 h 81 U tp(u , h , z) no p(u , h , z)[ 1 1npo(u , h , z) ] . Equation (66) is quite similar to eq. (40) of III: its solution can be obtained through an iteration procedure, by assuming the 1st term on the r.h.s. as the zero-order approximation of the unknow function f. The result is

fp(u , h , z) 4

!

i Fpi (u , h , z) ¯T ¯xi , (68)

relation (which is now written in full notation) Fn 11 pi (u , h , z) 4 upi(u , h , z) Qp(u , h , z)1 1 Qp(u , h , z)

!

nkp 8 p 9

!

!

j



0 2 p du 8



0 H(u 8) h 8 dh83 (69) 3][K1 N1_{[ F}n p 9 i(u 9, h9, z" )2 Fp 8 in (u 8, h8, z8) ] ]z 84z81( j); z 94z91( j)1 11 2[K 2 _N2_{[ F}n p 9 i(u 9, h9, z9)1 Fp 8 i(u 8, h8, z8) ] ]z 84z8n 2( j); z 94z92( j)( 1 1 l Qp(u , h , z)

!

p 8

!

j



0 2 p du 8



0 H(u 8) ]Z Fp 8 i(u 8, h8, z8)(z 84z8n jh 8 dh8 .

The heat current density

U 4 1

V

!

qp

ˇ_vqpvqpnqp (70)

can now be easily evaluated. Owing to eqs. (45), (65) and (68), one obtains the following expression for the n-th component of U with respect to the Cartesian reference frame with unit vectors ui:

Un4 2

!

i kni ¯T ¯xi , (71) where kni4 k2 p3 ˇ_h15_b2 MkBT2e2

!

p



_d3_{h v} pApn Fpi eb vp (eb vp 2 1 )2 , (72)

the symbol

s

d3_{h staying for the triple integral appearing in eq. (52).}

5. – Connection with thermal expansion

The anharmonic parameters e and e 8 can be deduced from the experimental behavior of the Grüneisen parameter as a function of temperature. To this purpose it is necessary to write the explicit expression of the free energy of the perfect lattice. A first contribution to this expression is represented by the potential energy U(o)

corresponding to hlb4 0, that is the energy of interaction between frozen

(non-vibrating) ions: this is directly obtainable from (27) by substituting to V( 3 )

lb , l8 b8 the pair

potential V(r) evaluated for r 4Nl82l1b82bN. Introducing, as usual, the index t to label the cell in the neighbourhood of the cell at l, and writing V(Nl82l1b82bN) 4

Vt bb8, we obtain U(o) 4 1 2

!

l t(c 0 )

!

]V t 0B1 V00t 1 VBBt 1 VB0t ( 1

!

l V0 0B. (73)

next-nearest–neighbour distance R 8, respectively, one gets from (73) U(o)4 4 N(V 1 3 V 8 )

(74)

having considered that there are three values of t for which both Vt

0Band VB0t take the

value V, and twelve values of t for which both Vt

00and VBBt take the value V 8. A second

contribution to the free energy is from vibrations connected with the harmonic phonon system, namely [14] Fvib4 kBT

!

qpln

[

1 2exp [2ˇvqp/kBT]

]

1 1 2

!

qp ˇ_vqp (75) or also, owing to (52) Fvib4 2 NkBT

!

p



d3_h

m

ln [ 1 2exp [2bvp] 1 1 2b vp

n

. (76)

A third contribution is from anharmonic terms containing 3rd- and 4th-order derivatives of the pair potential, such as discussed by Wallace [14] (p. 180): this, however, can be neglected for crystals where anharmonic effects are small.

To evaluate the pressure P 42(¯F/¯V)T, where F 4 U(o)

1 Fvib, we need the

derivatives with respect to V of V , V 8 and of all the dimensionless coefficients (b 8/b, r, r8) entering the expression of the reduced phonon frequencies vp. One obtains

g

¯V ¯V

h

T 4 dV dR R 3 V 4 gR2 3 V 4 gh12 8 V , (77)

g

¯_{V 8} ¯V

h

T 4 dV 8 dR 8 R 8 3 V 4 g 8 R 82 3 V 4 g 8 h12 3 V (78)

and in a similar way

g

¯g ¯V

h

T 4 bh 2 1 8 V ,

g

¯_{g 8} ¯V

h

T 4 b 8 h 2 1 3 V ,

g

¯b ¯V

h

T 4 ah 2 1 8 V ,

g

¯_{b 8} ¯V

h

T 4 a 8 h 2 1 3 V . (79)

Owing to eqs.(22) and (30) the final result is

g

¯F ¯V

h

T 4 4 bh 4 1 Vc (r/8 1r8)1Nˇ

o

2 M

!

p



_d3 h

k

npo1 1 2

lk

¯ ¯V

(

h1kb vp

)

l

T 4 (80) 4 2 bkBT 3 Vc

m

g

1 2 3 16e

h

F ( 1 ) 1 3 8

g

1 1re2 16 3 r

h

F ( 2 ) 1 1

g

b 8 b 1 3 8 r 8 e22r8

h

F ( 3 ) 1

g

2e 8 1 3 8e b 8 b

h

F ( 4 ) 1 ub

g

r 81 r 8

h

}

,

where we have used the following definitions: no p4

[

exp [b vp] 21

]

21, u 4 12 Mh12kBT/ˇ2, Sp( 1 )4 vp, Sp( 2 )4 (¯ vp/¯r)T, Sp( 3 )4 (¯ vp/¯r 8)T, Sp( 4 )4 [¯ vp/¯(b 8 /b) ]T and F(i) 4

!

_p



d3_h

g

_no p1 1 2

h

S (i) p . (81)

From (80) we obtain the thermal expansion coefficient through the relation (Wallace [14], p. 5) a 42 1 KT

k

¯ ¯T

g

¯F ¯V

h

T

l

V , (82)

KTbeing the isothermal bulk modulus. Introducing the functions

G(i)₄

!

p



_d3

h npo(npo1 1 ) vpSp(i) (83)

and considering that the unit volume heat capacity at constant volume is simply

CV4 2 b2kBG( 1 )/Vc, we easily deduce from (80) and (82) the following relation involving

the Grüneisen parameter gG4 KTa /CV:

23 gGG( 1 )4

g

1 2 3 16e

h

G ( 1 ) 1 3 8

g

1 1re2 16 3 r

h

G ( 2 ) 1 (84) 1

g

b 8 b 1 3 8 r 8 e22r8

h

G ( 3 ) 2

g

e 82 3 8e b 8 b

h

G ( 4 )_.

At room pressure, the curly bracket of (80) can be equated to zero: if one neglects zero-point motion and the temperature-dependent terms (1st to 4th term in the bracket) which are vanishingly small for T K0, the relation r842r/8 is obtained. For a given choice of r at 0 K (say, r0), one can use such a relation to solve the secular equation for

the phonon frequencies and determine the corresponding parameters b0and b 80so as to

obtain the best agreement between theoretical and experimental dispersion curves at low temperatures. In order to determine eo and eo8 (that is, the values of e and e 8 at

P 40, T40, corresponding to a volume V0), let us point out that for a

thermo-dinamical state in which the volume differs by DV from V0, one can approximately

write r 4r01 (¯r/¯V)0DV, etc., or also, owing to (79)

r 4ro1 1 8

g

1 1eoro2 16 3 ro

h

DV Vo , (85) r 842ro 8 1 1 3

g

b 8o bo 2 3 64eoro1 ro 4

h

DV Vo , (86)

b 4bo2 1 8eobo DV Vo , (87) b 8 b 4 b 8o bo 2 1 3

g

e 8o2 3 8eo b 8o bo

h

DV Vo . (88)

Since in deducing the equation of state (80) we neglected volume derivatives of any anharmonic free energy term (in particular of terms containing the anharmonic parameters a and a 8), the values of e and e8, for consistency of the theory, can be treated as volume-independent and, therefore, coincident at any temperature with eo and e 8o. Some arbitrariness is connected with this approximation, because, on alternative ground, one could assume a and a 8 as volume-independent parameters, and deduce e and e 8 from the relations following from the volume dependence of h1 and b

(

(see eqs. (79)

)

. Actually, the discrepancy between the numerical results corresponding to the two choices turns out to be of the order of 1 % of the thermal conductivity at the highest temperatures, and consequently of no weight for low-anharmonicity solids like Ge and Si.

In principle, one could insert expressions (85)-(88) into eq. (80), which represents the equation of state of the solid. Equating the r.h.s. of this equation to zero it would be possible to obtain a relation expressing in terms of eo and eo8 , at any desired temperature, the volume change DV at room pressure (P A0). At this point the two parameters could be determined by imposing the best fit of the resulting function DV 4DV(T) to the experimental expansion data.

Instead of following this procedure, we prefer to start from eq. (84), which, having been derived from (80), represents an alternative form of the equation of state. Inserting into (85)-(88) the experimental values of DV/Vo as a function of T at room pressure, and substituting into eq. (85), we will determine eo and e 8o by imposing the best fit of this equation to the experimental behavior of the Grüneisen parameter gG(T)

above 100 K.

At this point we can evaluate the pressure derivative of the bulk modulus at 0 K through the relation (see appendix)

g

¯K ¯P

h

o 4 21 3 1 eo 8 1

k

1 192 1 1 9

g

b8o bo 1 eo8 2 3 8eo b8o bo

h

lk

1 64 1 1 3 b8o bo

l

21 (89)

and obtain a result which depends on the value of ro chosen at the beginning of the calculation. We then vary rountil we reach the experimental value of (¯K/¯P)o. In this way all the parameters ro, r8 , bo o, b8, eo o, e8 are determined and eq. (72) can be used too calculate the thermal conductivity.

6. – Numerical results for germanium and silicon

The values of ro, r8 , bo o, b8 , eo o, eo8 for Ge, as determined through the above procedure, are listed in table II. They refer to (¯K/¯P)_{o4 4.6, a value deduced from} Ziman [13] (p. 151) and to h14 5.65 /k2 31028cm at 0 K [2].

The corresponding phonon dispersion curves for acoustic and optical branches are given in fig. 1, and turn out to be in good agreement with experiment [15, 16]. The continuous curve of fig. 2 displays the temperature behavior of the Grüneisen parameter for Ge at room pressure, as obtained by inserting into eq. (84) expressions

TABLE II. – 0 K values of the coefficients describing the interaction between nearest neighbors (ro, bo, eo) and next-nearest neighbors (r8 , bo o8 , e 8o). The values of boand bo8 are given in units of 1020_{g cm}22_s22_. ro r8o bo bo8 eo e 8o Ge Si 0.052 0.060 20.0062 20.0075 1.9 2.2 4.0 31022 4.5 31022 36. 25. 0.4 0.2

(85)-(88) with the numerical values of table II. Also in this case the comparison with experimental data [17] is highly satisfactory.

It is interesting to point out that we determined the two parameters eoand e8 byo fitting the behavior of gG(T) above 100 K: no parameter was determined by using the

experimental behavior of gG(T) in the low temperature region. Nevertheless, one sees

from fig. 3 that the resulting curve (continuous line) shows not only the imposed decrease of gG(T) when temperature is lowered from 900 to 100 K, but also the

presence of the minimum at about 30 K. The capability of the theory of predicting the minimum of gG(T) is considered a serious test of the reliability of our model potential in

the description of anharmonic effects.

By the data listed in the 1st row of table II we calculated the thermal conductivity of natural Ge at 900 K, using for fi(DM/M)2 the value 3.68 31025[1]. In this way we obtained k 40.164 W cm21_K21_{, a value in full agreement with the corresponding}

! (10 13 c=s)

Germanium

, X K , L

1

2

3

4

5

6

0

3 3 3 3 3 3 33 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

Fig. 1. – Theoretical phonon dispersion curves (continuous lines) for germanium, in comparison with experimental data [15, 16].

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

20

40

60

80 100 120 140 160 180 200

G

T (K)

Germanium

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 + + + +

Fig. 2. – Temperature dependence of the Grüneisen parameter for natural Ge, with and without the volume corrections (85)-(88) (continuous and dotted curve, respectively). The experimental data are from [17].

! (10 13 c=s) Silicon , X K , L 2 4 6 8 10 12 03 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

Fig. 3. – Theoretical phonon dispersion curves (continuous lines) for silicon, in comparison with experimental data [22, 23].

experimental condutivity kexp4 0.177 W cm21 K21[18]. The result is highly

satisfactory because it follows from a first-principle calculation, all the parameters employed in the theory having been independently justified through the harmonic and anharmonic vibrational properties of the crystal: conversely, in all the previous calculations the presence of adjustable parameters was unavoidable [4, 6, 19-21].

For Si, having not at disposal the value of (¯K/¯P)o, we determined ro, r8, bo o, b8,o

eo, eo8 (table II) by imposing the agreement between the experimental and the theoretical conductivity at 200 K. We put h14 5.43 /k2 31028cm at 0 K [2] and used for

fi(DM/M)2the value 2.7 31024[13]. The resulting dispersion curves are reproduced in fig. 3 and fit well the experimental data of Broughton and Li [22] and of Li et al. [23]. The continuous curve of fig. 4 provides the corresponding optimized behavior of gG(T):

the general trend of the experimental points [17] is reproduced by our theoretical curve, although this fails to explain the deep negative minimum of gG(T) in the

low-temperature region. Clearly, the model potential works better for Ge than for Si. The same conclusion, however, would be reached in the frame of a pseudopotential approach: the calculation of Soma [24], which approximately explains the behavior of

gG(T) for Ge, is far from predicting the minimum of Si.

The dotted curves of figs. 2 and 4 were obtained by putting r 4ro, r 84r8o, b 8 /b4

b 8o/bo at any temperature. Their closeness to the continuous curves shows that the volume corrections in expression (85)-(88) are very small, and supports our low-anharmonicity approximation, for which we neglected volume derivatives of the anharmonic free-energy terms.

For the numerical evaluation of functions Fn

pi

(

see eq. (69)

)

we subdivided the

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 50 100 150 200 250 300 G T(K) Silicon 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 + + + + 22 2 2 2 2 2

Fig. 4. – Temperature dependence of the Grüneisen parameter for Si, with and without the volume corrections (85)-(88) (continuous and dotted curve, respectively). The experimental data are from [17].

Brillouin zone into 7488 cells and followed the same procedure as described in III. Particular care was taken to perform the derivatives ¯ vp/¯hi: these were obtained numerically, thanks to a program exploring the effect of a small change dhi on the solutions of the secular equation det ]aij( 4 0. We checked this program by comparing its results with the analytical expressions of the derivatives

(

see eq. (34) of III

)

in the simple case of a fcc lattice without basis. In our investigation of three phonon processes, we found, as in III, the absence of any contribution from the reciprocal lattice shells with n F2: this means that for any kind of scattering involving acoustic or optical phonons, normal collisions and umklapp collisions are described by the terms with n 40 and n41, respectively. We also found that after 10 steps the iteration process leads to a satisfactory convergence of the numerical iteration.

To present the resulting behavior of the deviation function it is convenient to write, as in III, cqpin the form

cqp4 2kBh1Fp(q) uqQ ˜T 42kBh1Fp(q) ¯T

¯x3

cos a , (90)

where u_{q4q/q and the last step has been accomplished by assuming ˜T along the x}3

-axis (a being the angle formed by this -axis with the selected direction of q). The functions Fp(q) as defined by (90) can easily be derived from eq. (65) and (68):

Fp4 2k2 bh4 1 p2_e2_k BT Fp3 cos a . (91)

In fig. 5 we reproduce for Ge the behavior of functions Fpin the plane u 40 for a40, corresponding to the [ 100 ] direction of the Brillouin zone. The dotted curve in this figure shows the zero-order approximation in the iteration procedure for the longitudinal acoustic branch

(

corresponding to Fpi4 upi/ Qp, see eq. (69)

)

: this fails to describe the singularity for q K0, which is required by general considerations on the Boltzmann equation (Omini and Sparavigna [25]). The singularity can be obtained only by increasing the iteration order: see curve 1, which refers to the same branch and to the 10th step of the iteration procedure.

In fig. 6 the continuous curves 1 and 2 show the theoretical behavior of thermal conductivity for natural and enhriched Ge as a function of temperature at room pressure. The low-temperature part was determined by expressing the relaxation time for boundary scattering in the form

tp4 LF/sp, (92)

where L is the Casimir length, F a correction factor depending on the width-to-length ratio of the sample [26, 27] and sp the velocity of sound for a phonon with polarization index p. In principle, sp depends on the particular direction of q but, owing to the roughness of the model, we take for sp the value referring to the isotropic model, in which the two transverse acoustic branches have a common value, sT, to be

distinguished from the value sLof the longitudinal acoustic phonons, and both sT and sL

are independent of q. We neglect boundary scattering of optical modes, which are not excited at low temperatures

(

npoA 0 in the last term of (67)

)

. It has to be pointed out that eq. (92), though strictly reasonable only in the long-wavelength limit, can however be used for any q because boundary scattering is confined to a temperature range where only the low q-phonons are called into play.

-100 0 100 200 300 400 500 -0.4 -0.2 0 0.2 0.4 F p  1 1 2 2 3 3 4 4

Fig. 5. – Natural Ge at 900 K: behaviour of functions Fp(see eq. (91))vs. the reduced distance z

from the origin of the BZ, along the [100] direction: curve 1, 2, 3 and 4 correspond to LA, TA, LO, TO, respectively, the results for the two transverse branches being coincident. The dotted curve shows the zero-order approximation in the iteration procedure for LA.

The values for sL( 5.208 3105cm/s) and sT (3.165 3105cm/s) for Ge are taken from

Srivastava [1]. For L and F we use the Casimir length (L 40.24 cm) and the correction factor (F 40.8) corresponding to the Ge sample employed by Holland [4]. The circumstance that Holland’s data for natural Ge agree very well with the corresponding Geballe-Hull’s (GH) data below A30 K [18], shows that the above values of L and F can be used to interpretate both the sets of data and, consequently, also to describe the low-temperature behavior of k for enriched Ge, as obtained by GH themselves for samples of unchanged geometry. The value of fi(DM/M)2corresponding to enriched Ge is 5.72 31024_{[1]. The upper continuous curve of fig. 6 (curve 3) was calculated with}

reference to an ideal one-isotope crystal (that is, without isotope scattering) having the same geometry of the previous samples.

For Si, we take from Srivastava [2] sL4 8.2 3 105cm/s, sT4 5.41 3 105cm/s, L 4

0.51 cm, F 40.1. The resulting thermal conductivity is shown in fig. 7 by the lower continuous curve (1), while the upper continuous curve (2) is calculated in the absence of isotopic effect. The experimental points referring to natural Si are deduced from Holland [28] and from Holland and Neuringer [29].

A first comment to our results is that the theory is capable of explaining not only the absolute values of the conductivity of natural Ge, but also, in a satisfactory way, the difference of conductivity between natural and enriched Ge.

A second comment concerns the role of three-phonon processes. In the last row of table III we list the values of k referring to a calculation for natural Ge in which only

0.1

1

10

100

1

10

100

1000



T (K)

Germanium

1

2

3

3 3 3 3 3 3 3 3 3 3 3 + + + + + + + + + + + +++ + + + + + + + + + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Fig. 6. – Thermal conductivities (in W cm21_K21_{) for a natural sample (curve 1), for an enriched}

sample (curve 2) and for an ideal one-isotope sample (curve 3) of Germanium as a function of tem-perature. The experimental points are from Geballe and Hull [18] (p,1) and from Holland [4] ({).

isotope and boundary scattering are taken into account. Such values are compared to the corresponding ones accounting for all phonon processes (2nd row). The comparison shows that the effect of three-phonon processes is present down to about A10 K: such a result confirms the conclusions reached by Srivastava in the frame of his approximate theory.

A third comment refers to the contribution brought to heat transport by longitudi-nal phonons. According to Sharma, Dubey and Verma [20] at high temperatures it should be negligible: such an assumption was strongly criticized by Srivastava [1], who found an effect of longitudinal phonons of about A25% in the range between 4 and 900 K. The correct answer is easily obtained from table IV, where we list the four contributions to k represented by kLA, kLO, kTA, and kTO. These refer to longitudinal

acoustic, longitudinal optical, transverse acoustic and transverse optical phonons, re-spectively, and are deduced by collecting into four corresponding groups the various terms of the sum over p appearing in expression (72). One sees that the effect of longitudinal acoustic phonons is remarkably high at 900 K, its weight being even higher than that predicted by Srivastava (about 46 % at 900 K, about 53 % at 100 K).

A fourth comment emerges from the 2nd row of table V, which provides the values (kacoust) of thermal conductivity in the absence of all the scattering processes involving

optical phonons. At high temperatures these values turn out to be remarkably higher than the corresponding ones of the last row, where the above processes are included. Thus optical phonons are effective in providing the main source of the thermal

1 10 100 10 100  T(K) Silicon 1 2 3 3 3 3 3 3 3 3 3 3 333 3 3 3 3 3 3 3 3

Fig. 7. – Thermal conductivity (in W cm21_K21_{) for a natural sample (curve 1) and for an ideal}

one-isotope sample (curve 2) of Silicon as a function of temperature. The experimental points are from-Holland [28] and Holland and Neuringer [29].

resistance in the range above A200 K. Because of their small group velocity they cannot carry but a negligible amount of heat (see the low values of kLO and kTO in

table IV): they only scatter acoustic phonons, to which heat transport is committed. A final important comment is related to the behavior of product Tk : as anticipated in the Introduction, the high-temperature experimental data show for this function a small decrease vs. T, for which no satisfactory explanation was found by previous authors. Srivastava [1, 2] suggested the possibility of explaining the above decrease through a better description of the roles played by a) optical phonons b) phonon dispersion c) crystal anharmonicity. The present theory, in which both the effects connected to points a) and b) are treated in a rigorous way, suggests on the contrary that only point c) must eventually be considered. This conclusion is reached by assigning to all the microscopic parameters involved by the theory (r , b , r 8, b8, h1)

their values at 0 K, and determining the corresponding temperature behavior of Tk. The result of this procedure in which one accounts for all the scattering processes

TABLE III. – Thermal conductivity of natural Ge (in W cm21K21) with and without

three-phonon processes, as represented by the 2nd and 3rd row, respectively.

T ( K ) 10 20 30 40 60 80 100 300 500 900 kall k_{I 1B} 9.9 10.0 10.3 12.1 8.6 14.5 6.8 16.1 4.8 17.7 3.1 18.3 2.2 19.5 0.55 23.0 0.31 23.4 0.16 24.1

TABLE IV. – Contributions of longitudinal-acoustic, longitudinal-optical, transverse-acoustic

and transverse-optical phonons to the thermal conductivity k of natural Ge (in W cm21_K21_).

T (K) kLA kLO kTA kTO k 60 80 100 200 300 500 900 2.469 1.690 1.182 0.450 0.272 0.146 0.075 0.0001 0.0005 0.0010 0.0018 0.0015 0.0015 0.0005 2.145 1.365 0.988 0.372 0.235 0.134 0.070 0.013 0.032 0.048 0.058 0.040 0.027 0.015 4.63 3.08 2.21 0.88 0.55 0.31 0.16

TABLEV. – Thermal conductivity of natural Ge (in W cm21K21) as obtained by neglecting the

presence of optical phonons (2nd row). The 3rd row gives the values of k including all the phonon contributions. T (K) 40 60 80 100 300 500 900 kacoust kall 6.78 6.78 4.90 4.63 3.52 3.08 2.49 2.21 0.85 0.55 0.52 0.31 0.29 0.16

involving not only acoustic, but also optical phonons, with the correct description of the phonon dispersion, is that no appreciable decrease is found at high temperatures. Conversely, if the above parameters are duly allowed to change with temperature according to the temperature change of DV

(

see eqs. (85)-(88)

)

, one obtains the expected decrease. Our numerical analysis leads to a theoretical law k PT2n _{with n 4}

1.14, a value slightly lower than that deducible from the experiments of Slack and Glassbrenner [8] (n 41.2). If such a discrepancy is really out of the experimental uncertainties, the conclusion is that higher-order anharmonic terms should be introduced in order to arrive at a full explaination of the effect. Considering, however, that the discrepancy is very small and, on the other hand, that Geballe-Hull’s [18] data between 500 K and 900 K (leading to n A1.1) are closer to our numerical predictions, we think that a refinement of the theory in this direction would be of little interest, and essentially extraneous to the aim of the present work.

7. – Conclusions

The improvements presented by this paper with respect to previous works are: i) The rejection of any continuum approximation, the solid being now described according to its real structure, both in the direct and in the reciprocal space.

ii) The expression of the matrix elements for phonon-phonon interaction in terms of the geometrical parameters related to the Brillouin zone, and of the microscopic parameters entering the pair potential: these follow from an independent study of lattice dynamics, and are determined so as to account for the experimental vibrational properties of the crystal, as well as for its basic anharmonic properties. As a result, all the quantities involved in the probability rates for phonon scattering are justified through arguments that are independent of the transport properties to be explained.

The situation is quite different from that of previous theories, where some parameters entering the final expression of thermal conductivity were either adjusted to fit the experimental conductivity curve (see [2], p. 359, and more extensively refs. [3, 4, 6, 19-21]) or determined through criticizable arguments linked to the continuum hypothesis: for instance, Srivastava [1] used for umklapp processes a reciprocal lattice vector of magnitude equal to the Debye diamater, while a direct numerical calculation shows that by such a large value of g (which is 1.43 times the real value of interest for these processes), the number of allowed U-processes, as well as their contribution to thermal resistivity, is negligibly small at any temperature.

iii) The description of the vibrational properties of the lattice in terms of realistic phonon dispersion curves, both for acoustic and optical modes: this is a consequence of point i), in contrast with all previous treatments, where dispersion was either ignored [3, 19] or introduced in an improper way [1, 2, 4, 6, 20, 21], being essentially inconsistent with the continuum hypothesis used to describe phonon-phonon interaction. As to optical modes, only Srivastava [1, 2] realized the importance of their contribution, but his use of the Debye-Eistein model could not account in a satisfactory way for the coupled dynamical behavior of the two atoms pertaining to the unit cell. The correct description of this behavior in the frame of the theory of lattice dynamics, leading to our matrix element for phonon-phonon interaction, shows the complexity of the role played by the two corresponding polarization vectors (eQ0 and eQB) in determining the transition probabilities (see sect. 3). The absence of these vectors in Srivastava’s formulation means that some unclear average over the polarizations is implicit in his model.

iv) The solution of the transport equation without the use of any relaxation time approximation for three phonon scattering and isotope scattering processes: these are all rigorously treated in the frame of the iteration procedure employed to solve the equation.

To summarize, we have presented a first-principle calculation of thermal conductivity for a solid having the same structure and the same harmonic and anharmonic properties of the real diamond-like crystals: owing precisely to the fact that our model potential proves to be sufficiently reliable to account for all these properties, we believe that no substantially new result could be obtained by adopting more sofisticated descriptions of pair interaction, like that deriving from pseudopotential theory.

AP P E N D I X

Neglecting zero-point motion, we express the energy of the crystal at 0 K through eq. (74) and use eqs. (77)-(79) to evaluate its volume derivatives: we obtain

¯_U(o) ¯V 4 4 N h12 V

g

g 8 1 g 8

h

(A.1)

and for the bulk modulus

K 4V¯ 2 U(o) ¯V2 4 4 N V h 2 1

k

2 1 3

g

g 8 1 g 8

h

1 bh2 1 64 1 b 8 h2 1 3

l

(A.2)