Brief summary of thermal conductivity of solids

T T4!cesn3T42 K

2. Brief summary of thermal conductivity of solids

The total thermal conductivity lCtot of insulators, metals and semiconductors consists of the following contributions:

insulators: /('tot = lgL~

m e t a l s : ~Ctot = ~cL + x e ,

semiconductors: /('tot = KL -[- ICe -~- /£b -~- K'ph -~- /£exc,

where bc L is the lattice, G is the electron, ~c u the bipolar, bCph the p h o t o n and Gxc the exciton contribution to the thermal conductivity. A contribution ~c m, the magnon

112 I.A. SMIRNOV and V.S. OSKOTSKI

thermal conductivity, is possible in magnetic insulators, metals and semiconductors at low temperatures due to heat transport by spin waves. At present the main theoretical propositions of the thermal conductivity K of solids are obtained, compre- hended and described in Oskotski and Smirnov (1972), Berman (1976), Smirnov and Tamarchenko (1977) and other monographs, and many reviews.

2.1. Thermal conductivity o f a crystal lattice

A thermal flow is carried by phonons and its value and temperature behaviour depend on the contribution of different scattering mechanisms. IcL(T ) of semiconductors and insulators is similar, while ~c L of metals is significantly smaller in the low- temperature region because of strong p h o n o n - e l e c t r o n scattering (fig. 1). At low temperatures ( T ~ O, O is the Debye temperature) phonons are scattered, mainly, by crystal boundaries and ~L oCT 3. At T > O p h o n o n - p h o n o n scattering results in a temperature dependence tCLOC T 1. At T < O exists a temperature region with

~c L oc e x p ( O / a T ) . In metals at T ~ O due to p h o n o n - e l e c t r o n scattering t% oc T 2. As it has been noted above, different temperature dependences could occur in other temperature regions of ~cL(T ) due to contributions from different scattering mechanisms.

For three-phonon interactions one distinguishes two types of collisions: normal processes (N processes), in which the total momentum is conserved and the direction of flow does not change (these processes lead to infinite thermal conductivity); and Umklapp processes (U processes), in which the sum of the wave vectors is not conserved and changes sharply, leading to a finite thermal resistivity of a crystal. In U processes the following conditions are fulfilled:

col + co2 = c03, (1)

ql + q2 = q3 ⁼ 2~zb, (2)

where co is an angular frequency, ql, q2 and q3 are p h o n o n wave vectors, and b is a reciprocal lattice vector. The theory of lattice thermal conductivity meets with difficulties with N processes, especially at low temperatures, when long-wavelength phonons prevail and N processes are the main type of p h o n o n - p h o n o n scattering.

Although N processes themselves do not produce a thermal resistance, they influence

~L indirectly by redistributing phonons of different modes. Three-phonon N and U processes are mainly anharmonic processes. However, processes of higher orders are

X L

2 .),,I..X.j- ex P O/aT

I Z ,

1,3//-._~

Fig. 1. Schematic shape of the tempcrature dependence o f l ¢ L for metals T (1) and insulators (semiconductors) (2).

T H E R M A L C O N D U C T I V I T Y O F RECs 113

possible when simultaneously four of more phonons interact (Oskotski and Smirnov 1972, Berman 1976, Smirnov and Tamarchenko 1977). To satisfy condition (2), interacting phonons are to have wave vectors near rib. At T > O there are many such phonons, and U processes occur often, their number increases proportional to T, and lCLOCT *. At T ~ O the number of phonons with wave vectors near nb decreases exponentially giving rise to a fast increase of ~CL by the law tc Loc e x p ( O / a T ) (a is a coefficient of order two). At low and medium temperatures the acoustic phonons of the spectrum participate in the heat transport and scattering processes. At high temperature optical phonons are excited. Then t% can increase due to additional heat transport by optical phonons, or decrease due to additional scattering of acoustic phonons by optical ones (Blackman 1935, Gurevich 1959, Leroux-Huqou and Yeyssie 1965, Devyatkova and Smirnov 1962, Slack 1979).

Usually theoretical analysis of tel data is carried out in the Debye model in the relaxation-time approximation, where 1eL has the form

[ o / r siXnh2 (½x)dx.

tel = GT3 1 (3)

20 ~ -

Here x = h~O/ko T, G is a constant, T- ~ is the sum of the reciprocal relaxation times of phonons in different scattering processes: r - 1 = E~i- 1, where ~i = %, *a, re, VD, %, Z~., Z[, rp, %, ZN, ZF, the relaxation times of phonon scattering, correspondingly, on crystal boundaries, point defects and impurities, conduction electrons, dislocations, colloids, molecular impurities, quasi-local vibrations, paramagnetic ions, U, N and four-phonon processes. Each z- ~ has its own frequency and temperature dependence (Oskotski and Smirnov 1972, Berman 1976, Smirnov and Tamarchenko 1977). In fig. 2 the temperature regions are marked in which the various scattering mechanisms are significant. Callaway (1959) has shown how it is possible to account in eq. (3) for the relaxation time of N processes. Holland (1963) has proposed a mechanism which, for ~ 1 and ~R 1, takes into account some details of phonon spectra and the separated heat flows transferred by transverse and longitudinal acoustic phonons. To conform X L

X rnQ x

T d , T u , T N ,7"R' , ' g e , T p

/ \

T b ~ , ~ u ,'t'N, TF

I ^~_

o T

F'ig. 2. Schematic shape of the temperature dependence of ~c L for solids. The temperature regions are marked, where various p h o n o n scattering processes play an appreciable role. O is the Debye temperature.

114 I.A. SMIRNOV and V.S. OSKOTSK1

experimental and theoretical values of ~CL, according to eq. (3) one has to introduce a great number of fitting constants (for example, A1-Edani and Dubey (1986) have introduced twenty constants for ~c L of GdS). The more mechanisms of p h o n o n scattering are introduced, the greater is the number of fitting constants. Sometimes some of the constants can be evaluated by independent theoretical calculations or measured experimentally. In this review we will try to use this method of analysis as seldom as possible and limit ourselves only to the physical nature of the behaviour of

ICL(T).

Thus, analysing experimental data on teL(T) of RECs we consider the results

"standard" if ~CL(T ) has a value and a temperature dependence predicted by the theory for ordinary solids (Oskotski and Smirnov 1972, Berman 1976, Smirnov and Tamarchenko 1977), and "nonstandard" if there are deviations from the theory.

2.2. Electron contribution to the thermal conductivity

The electron thermal conductivity of metals and semiconductors is determined by the W i e d e m a n n - F r a n z law

K e = LT/p, (4)

where L is the Lorentz number and p is the electrical resistivity. In the low- temperature region ( T < O/10) the electron thermal resistivity, We= 1/G , can be presented in the form (Smirnov and Tamarchenko 1977)

G = Wo + (5)

where Wo is the thermal resistivity caused by electron scattering by impurities and defects (Wo = po/Lo Z with Po the residual electrical resistivity, L o the Sommerfeld value of the Lorentz number and Lo = ½(kc/e)Z~r 2 = 2.45 x 10 -a W •/K 2) and Wi is the ideal thermal resistivity caused by electron scattering of phonons (Oskotski and Smirnov 1972, Berman 1976, Smirnov and Tamarchenko 1977)

( T ~ [ 0 D 0 [2~2 0 1 0 ) 1

- - - 5 17 (6)

w , = B T .

Here B and D are constants, eV is the Fermi energy, and 15 and I7 are integrals of the type

fo E z" dz

I,(x) = (e z - 1)(1 - e z)" (7)

These integrals are tabulated in Smirnov and Tamarchenko (1977). Equation (5) is obtained by analogy with the case of the electrical resistivity Ptot = Po + Pi (where Pi is the ideal electrical resistivity caused by electron scattering by thermal vibrations).

At low temperatures WE oCT 2, so eq. (5) can be presented in the form

W~ = po/Lo T + ocT 2 = fi/T + c~T 2, (8)

We T = ~T 3 + ft. (9)

THERMAL CONDUCTIVITY OF RECs 115 The temperature dependence, We = f ( T 3 ) , is a straight line with a slope of c< The intercept on the We-axis is/L The parameter c~ must be a constant for a certain class of materials and independent of sample purity (Smirnov and Tamarchenko 1977).

The constant /~, on the other hand, changes from sample to sample and depends indirectly (through Pc) on its purity. Figure 3 shows schematically the temperature dependence of the different contributions to eq. (8). Since W~ increases and Wo decreases with T, We has a minimum at a certain temperature. The Lorentz number of metals of any purity for T >> O, and in samples with impurities and defects for T ~ O, is a constant and equals Lo. At T = 0 for pure metals L = 0 (see fig. 4 and eq. (10)). In the intermediate temperature range, L is determined by the temperature and the purity of the sample. At temperatures both lower as well as higher than O the ratio

L/Lo

in metals could be presented in the form (Oskotski and Smirnov 1972)

L [ p o

( T ) 5

(O)l/[po

( T , D ~ ] , (10)

Lo = 7~0 + I5 ~

4AEo+X\o ~FJ]

where the function

= 1 + 1 5 -

with 15 and 17 integrals of the type (7), A o is a constant which depends on an interaction constant between electrons and the lattice, the lattice constant, @, atomic mass and O. D is also a constant. According to Olsen and Rosenberg (1953)

D /@ = 2-1/3 N.

2/3 (11)

kk...d'//

y . . . . j W o

r Fig. 3. The temperature dependence of different contributions to eq. (8).

1.0

o 0 . 6

-4 d 0.2

~ l I I I I l~----I'~'--'~ "--

.II ?'-'-,d" I

/ / I

_b,? ',

., "~1 I

z ~ I I I I I I

0.4 0.8, 1.2 1.6

T/0

Fig. 4. The dependence of

L/Lo

T/O

(Oskotski and Smirnov 1972) calculated using eq.(10) for N, = 1.

Curve 1:

po/4A=O,

curve 2: 0.0038, curve 3: 0.0191, curve 4: 0.0957.

116 1.A. S M I R N O V a n d V.S. O S K O T S K I

where N a is the number of free electrons on an atom. F o r T/O > 0.6 (Olsen and Rosenberg 1953)

Pi = Ptot PO -- Ao r i o . (12)

The parameter po/4Ao in eq. (10) is a measure of the purity of a material. F o r very pure metals po/4Ao = 0. Figure 4 shows, e.g., dependences L/L o = f ( T / O ) calculated using eq. (10) for a number of values of po/4Ao. A decrease of L compared to L o is connected with the inelastic character of electron scattering by phonons in this temperature region. At very low temperatures in metals with impurities and for T > O electrons are scattered elastically (Smirnov and Tamarchenko 1977). In some metals (e.g., monovalent and noble ones) for T > 0 one finds that L < Lo. One connects this effect with an additional inelastic scattering mechanism electron electron scattering (Smirnov and Tamarchenko 1977). In the thermal conductivity theory any deviation of L from Lo indicates the presence of some new elastic or inelastic scattering mechanism for the current carriers.

A more complicated behaviour of the Lorentz number is observed in semiconductors. L/L o behaves differently for elastic and inelastic scattering of electrons and in the presence of complicated energy band and interband scattering.

The Lorentz number of a semiconductor with a parabolic band, due to elastic scattering of electrons can be written as (Oskotski and Smirnov 1972, Smirnov and Tamarchenko 1977)

= IFr+ - (r + )2FL

where F represents the Fermi integral Fro(#* ) = ~ x ' ( e U * + 1) 1 dx (the integrals are tabulated in Smirnov and Tamarchenko (1977)). /x* is a reduced Fermi level, # * = ev/ko T [#* is determined from experimental data on the thermal electromotive force, c~, by the equation

c~ = \ ( r + ~-)F~ + 1/2(#*) - #* (13a)

(Oskotski and Smirnov 1972)], r is a scattering parameter (r is an exponent in the dependence of electron relaxation time on energy, e.g., r = 0.5 and - 0 . 5 , respectively, for scattering by optical and acoustic phonons). In nondegenerated samples (#* < 0, 1#1 ~> 1) the Lorentz number is equal to

L = (r + 2 . 5 ) ( k o / e ) 2

(14)

and depends only on the scattering parameter. In the case of strong degeneration (/~*>> 1) L = L o . Figure 5 (curve 1) shows the dependence L/(ko/e) 2 = f ( # * ) for r = - 0 . 5 .

In the case of a nonparabolic band described by the Kane model (Kane 1957) for the elastic scattering mechanism of electrons by acoustic vibrations, the Lorentz

T H E R M A L C O N D U C T I V I T Y O F R E C s 117

3.0

1.0

f i [ I"

_ . L . _ _ _ _ L I A _ _ I - - - L

- 4 0 4 8 12 16

/z*

Fig. 5. The dependences L/(ko/e) 2 - f(ll*) calculated by eq.(13) (curve 1) and eq.(15) (curves 2 and 3) for r = -0.5. Curve 1: /3=0, parabolic band (a); curve 2:/3 = 0.05, nonparabolic band (b); curve 3:/3 = 0.2, nonparabolic band (b).

number has a m o r e complicated form (Oskotski and Smirnov 1972)

2LI °LI-- (1L2)2 (15)

LN = OL 1 ,

- 2

where "L m (z, #*) are generalised Fermi integrals:

These integrals are tabulated in Smirnov and T a m a r c h e n k o (1977). Here

= ko Te*, z = e/ko T, f is the Fermi function and e* is the effective width of the forbidden band of interaction, which is near to the real forbidden band width, eg, for a n u m b e r of n a r r o w - b a n d semiconductors. #* is also determined from c~ by equations taking account of the band nonparabolicity. F o r r = - 0 . 5

1L1 _ # * OL1

2 --2

c~N = (ko/e) 0L 1 (15a)

--2

Figure 5 (curves 2 and 3) shows the dependences L/(k/e) 2 = f ( f f * ) calculated using eq. (15) for a n u m b e r of parameters/3. A nonparabolic band appreciably changes the L value except for the case of strong degeneration.

All inelastic scattering mechanisms of electrons (electron electron scattering, scattering by optical lattice vibrations, intervalley transitions and so on) lead to a decreasing Lorentz n u m b e r c o m p a r e d with the case of elastic scattering. F o r example, in degenerated semiconductors (and metals) we always have L < L 0 (Oskotski and Smirnov 1972, Smirnov and T a m a r c h e n k o 1977).

In materials with a complex electron band structure (subbands of "heavy" and

"light" carriers) strong interband scattering (e.g., s d ) is possible at a certain position of the chemical potential relative to the b o t t o m of the "heavy" subband. This scattering strongly influences the value of L (fig. 6) (Kolomoets 1966). The value of

118 I.A. S M I R N O V a n d V.S. O S K O T S K I

4 0 3.5 3 0 2.5

- a - 4 0 4 B 12

1 + I +

o I b I

Fig. 6. The dependence of the Lorentz number on 7 for the case of high degeneration of current carriers at S = 32 (1), 8 (2). (a), (b), and (c) depict different positions of the Fermi level, eF, relative to the bottom of the band of heavy carriers (eo) , 7 = (eF -- eo)/ko T.

L depends on the p a r a m e t e r S:

IMlhl2m~ 3/2

S = iMl12m,3/2, ⁽¹⁷⁾

where Mlh a n d M1 are matrix elements of transitions between states of c o r r e s p o n d i n g s u b b a n d s a n d m* a n d m* are the effective masses of h e a v y a n d light carriers.

Let us consider briefly other c o n t r i b u t i o n s to the thermal conductivity. Bipolar thermal conductivity, tCblp, appears in s e m i c o n d u c t o r s in the region of the intrinsic c o n d u c t i v i t y due to diffusion of electron hole pairs fi'om the h o t to the cold end of a sample.

T h e p h o t o n contribution, Kphot, to the thermal conductivity, Ktot, can be significant at m e d i u m a n d high temperatures in materials which are semi-transparent a n d t r a n s p a r e n t in the infrared region. In insulators 1%hot oc T 3, a n d in s e m i c o n d u c t o r s

tCphot(T )

shows a curve with a m a x i m u m . At low t e m p e r a t u r e s K'phot = 0. At high temperatures, due to the intrinsic conductivity, the c o n c e n t r a t i o n of carriers increases exponentially, and the a b s o r p t i o n coefficient increases in s e m i c o n d u c t o r s so that 1Cphot decreases to zero. The m a g n o n c o m p o n e n t , ~m, of the thermal c o n d u c t i v i t y will be considered in detail in section 3.4. Exciton thermal c o n d u c t i v i t y (~cexo) - diffusion of excitons f r o m the h o t to the cold end with a consecutive r e c o m b i n a t i o n - could be expected at high temperatures. This has been predicted by Pikus (1956).

T h e following i n f o r m a t i o n a b o u t defects in solids can be obtained from ~:(T) measurements:

(1) T y p e of defects: vacancies, bivacancies, complexes, clusters, impurity atoms, dislocations, isotopes and so on.

THERMAL CONDUCTIVITY OF RECs 119 (2) A value for the impurity and defect concentrations down to small values

(~1012_1013 c m - 3).

(3) Kinetics of ordering processes of defects in solids.

(4) Kinetics of the formation and decay of bivacancies, complexes and clusters.

F r o m analysis of KL(T) one can obtain information about the contributions of different p h o n o n groups (optical or acoustic) to the heat transport and about the nature of p h o n o n interactions with free carriers. The electron c o m p o n e n t of the thermal conductivity contains information on the scattering mechanism and its character (elastic or inelastic), and allows one to determine the kind of energy band structure (parabolic, nonparabolic), the presence of additional subbands with light and heavy carriers, and the nature of the electron interaction with phonons and other electrons. F r o m data on the bipolar thermal conductivity one can determine the forbidden-band width at medium and high temperatures. Useful information on material properties can be obtained from ^{tgphot ,}/£exc and ~c m.

In recent years a new class of materials rare earth c o m p o u n d s (RECs) has been investigated intensively. These materials possess n o n s t a n d a r d physical properties, including the thermal conductivity. New features were observed in ~c L, ~co and lCm. Analysing data on the thermal conductivity of RECs, in this review we will try to attract more attention to n o n s t a n d a r d effects characteristic of the solids containing ions with f-electrons.

We will consider in m o r e detail results for ~CL, Kc, Km. Table 1 shows materials, effects and parameters, which can be analysed using experimental data on ~cL, ~c e and

~m' This review is devoted, mainly, to the discussion of situations enumerated in table 1.

TABLE 1

Effects influencing the thermal conductivity of rare earth compounds.

Ktot Superconducting systems d

with heavy fermions [ I superconduetin9 systems [ I Systems with coexistencel I

of magnetism and

superconductivity I

= ~L + Ke +

Influence of magnetic ~__

phase transitions Phonon scattering on ~_

spin disorder

Phonon scattering }_

on paramagnetic rare earth ons

Influence of

L

Jahn-Te er effect | I Spin g'asses

I R . . . rth glasses }'--

Systems with homo- geneous and nonhomo- geneous intermediate va ency of rare earth ions

Systems with Kondo impurities I Concentrated Kondo

lattices; systems with heavy fermions

I Systems with

intermediate valency of rare earth ions I Electron scattering

on spin disorder

Scattering of current carriers on ground

state levels split by crystal fie d

-- I Magnon heat transport I Phonon scattering

by magnons I Scattering of current

__ carriers by magnons

K m

120 1.A. SMIRNOV and V.S. OSKOTSKI

3. Influence of the magnetic structure on the thermal conductivity of lanthanide

Dalam dokumen Handbook on the Physics and (Halaman 119-128)