Contribution to the thermal conductivity due to the correction term in the frame of the generalized callaway integral : application to Ge

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Contribution to the thermal conductivity due to the correction term in the frame of the generalized callaway

integral : application to Ge

K.S. Dubey

To cite this version:

K.S. Dubey. Contribution to the thermal conductivity due to the correction term in the frame of the generalized callaway integral : application to Ge. Journal de Physique, 1976, 37 (3), pp.267-269.

�10.1051/jphys:01976003703026700�. �jpa-00208419�

267

CONTRIBUTION TO THE THERMAL CONDUCTIVITY DUE

TO THE CORRECTION TERM IN THE FRAME OF THE GENERALIZED

CALLAWAY INTEGRAL : APPLICATION TO Ge

K. S. DUBEY

Department

^of

Physics, College

^of

Science, University

^of

Basrah, Basrah, Iraq

(Reçu.le ^{6 juin 1975,}

^{révisé le 18}

^septembre

^{et le 13 novembre}

^1975, accepte

^{le 20 novembre}

1975)

Résumé. ²⁰¹⁴ La contribution à la conductivité thermique de réseau due au terme correctif a été étudiée à hautes températures dans le cadre de l’intégrale ^de Callaway généralisée ^en établissant son

expression analytique. ^En utilisant les expressions obtenues, ^on calcule la conductivité

thermique

^de

réseau due au terme correctif dans le domaine de température 100 K-1000 K pour différentes lon- gueurs de diffusion. On ^trouve que la conductivité thermique ^{due au terme correctif} apporte ^une contribution négligeable à la conductivité thermique totale à hautes températures.

Abstract. ²⁰¹⁴ The contribution to the lattice thermal conductivity ^{due to} the correction term has been studied at high temperatures in the frame of the generalized Callaway integral by obtaining ^an analytical expression for it. These expressions, have been used to calculate the lattice thermal conduc-

tivity of Ge in the temperature range 100 to 1 000 K for the different scattering strengths. ^{It is found}

that the contribution to the thermal conductivity arising from the correction term is negligible ^at high temperatures.

LE JOURNAL DE PHYSIQUE TOME 37, ^MARS 1976, ^F

Classification Physics ^Abstracts

7.392

1. Introduction. ^- The

generalized

form of the

Callaway [1]

^{lattice thermal}

conductivity expression,

based on two mode conduction of

phonons

^has

been

recently published [2-4] by

Kosarev et al.

[2], [4]

and Parrott

[3]. However,

ⁱⁿ

generalizing

^{the Calla-}

way

integral,

these authors did not take the

disper-

sive nature of

phonons

^{into account. Later on, it}

was modified

by Dubey [5]

who did consider the

dispersion

^of

phonons [6]. Thus,

^the

Callaway integral

of lattice thermal

conductivity

^is

having

three diffe- rent forms of the correction term

(due

^{to three}

phonon

normal

processes).

The first form is

given by

^Calla-

way himself. The second form is the

generalized

^form

based on two mode conduction of

phonons ^whereas,

the third form is the modified form of the

generalized expression given by Dubey.

The lattice thermal

conductivity

^{due to the correc-}

tion term

(W

^has been studied

by

^{several wor-}

kers

[7-9] using

^the

Callaway expression

^{of the}

correction term in which no distinction is made

between transverse and

longitudinal phonons:

^The

study

^{of the}

generalized

form of the

Callaway integral

also allows one to calculate the contribution to the total lattice thermal

conductivity

of the correction term

(M)

^{due to three}

phonon

^normal processes.

In the

present

paper, ^we have used a modified form of the

generalized expression given by Dubey

^to

calculate the contribution of correction ^term

(AK)

and to obtain

analytical expressions

^{for it. The entire}

study

^{is confined to}

high

temperatures

only

^{and the}

expressions

^{obtained are valid}

only

^{in the limit of the}

high

temperatures. ^A

simple expression

^{has also}

been obtained

by introducing

^{a new}

parameter

p which is the ratio of the

scattering strengths

^{of three}

phonon

normal and

umklapp

processes.

Using

^the

expressions obtained,

^the contribution of the correc-

tion term

(AK)

^to the total lattice thermal conduc-

tivity

of Ge has been calculated in the temperature range 100-1 000 K for the different values

of p (101

to

10-3).

2.

Theory.

^{- The}

generalized

^{form of the}

Callaway

^{lattice thermal}

conductivity expression

^{can be}

given

as

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01976003703026700

268

where kB

is the Boltzmann

constant, 1i

is the Planck constant divided

by

² 03C0,

CN

^1,s is the three

phonon

^normal

process

scattering

relaxation rate,

CR 1’s

is the combined

scattering

relaxation rate of

umklapp

processes

given by

CB 1

^{is the}

boundary scattering

relaxation rate,

Cpt 1

^{is the}

^point

^defect

scattering

relaxation rate and

C3 1

is the three

phonon umklapp

process

scattering

relaxation rate. R is a constant whose value can be

calculated _[6]

from the

dispersion

^{curve, v is the}

velocity

^of

phonons,

^{0 is the}

temperature corresponding

^to

zone-boundary

of the

crystal

and the suffix s is taken to differentiate between transverse and

longitudinal phonons (s

repre- sents mode of

phonons).

^4K is the contribution towards total lattice thermal

conductivity

^{due to the correc-}

tion term. x is

given by x

⁼

(hcolkb T ), w

^{is the}

phonon-frequency.

The

expression

^{for AK can} be written as

where

Similar

types

^of

expressions

^can also be written for

AT2’

^and

AL’ 2

^{with the}

help

^of eq.

(5), (7), (8)

and

(9),

where suffix T and L represents ^transverse and

longitudinal phonons respectively.

Since our interests are confined to

high

tempera- tures, ^the

following expressions

have been used in the present

analysis.

For transverse

phonons

In the above

expressions ^the boundary scattering

relaxation rate

CB- 1

^is

neglected

^at

high temperatures

due to its _very low contribution. Here 0 is the

Debye temperature

^{and a is a constant. The term}

e- 01,xT

is absorbed in the constant

PT u.

^These

expressions

are the same as those

given by

^Klemens

[10].

^Simi-

larly,

^the

expressions

used for the

longitudinal pho-

nons are

given by

One can

therefore,

^write

and

where

At

high temperatures,

it is found that x is a very small

quantity

^{i.e. x 1}

^{and x2} ex(e" - 1) - 2 _+

^1.

Using

^{the above}

approximations

^{and the}

expressions

stated above for the

scattering

relaxation rates and

neglecting

lower order terms, ^one obtains the follow-

ing expressions :

269

where _p is the ratio of the

scattering strengths

^{of the}

normal and

umklapp

processes and it is

given by

and

If one does

not

differentiate between

longitudinal

and transverse

phonons

^i.e.

assuming

^{that the are}

non-degenerate,

eq.

(26)

^{reduces to}

which tells us that AK a

(pE) -’.

This is similar to the

previous findings

^of

Dubey [8]

^{based on the}

Callaway integral.

3. Results and discussions. ^-

Using

^the

expressions

obtained above

(10) (17)

^and eq.

(2).

AK has been calculated in the

temperature

range 100 to 1 000 K for Ge for the different values

of p (ratio

^{of the scatter-}

ing strengths)

in the framework of the

generalized

form of the

Callaway integral.

The results are

reported

in table I

along

^{with K. The} calculations are made for the different values

of p ( 103

^to

10 - 3)

^{also but}

only

the results for _p ⁼

103

and _p ⁼

10-3

are

reported

here. The constants used in the

present study

^are

taken from the earlier

report

of Sharma et al.

[11],

and are therefore not

reported

here. With the

help

TABLE I

The values

of

^AK

of

^{Ge in the} temperature range 100-1000 K obtained with the

help of expressions (10)- (17)

^and eq.

(2)

Units of K and AK are

watt/deg/cm.

of table

I,

^{it can} be concluded that the contribution of AK towards the total lattice thermal

conductivity (K)

^is very small in

comparison

^{with K. It is also}

found that for _any value

of p (103

^to

10-3),

^{AK is}

showing negligible

contribution towards K in the entire temperature range 100-1000 K. With the

help

of the obtained

expressions (10)-(17),

^{one can}

see that these are very

simple expressions

^{and that}

they

^are

easily computable.

^{At the same}

time,

^{it can} also be concluded that the effect of _p on AK can be studied with the

help

of above obtained

expressions

at

high temperatures.

^{Thus one can} conclude that the contribution of AK is

negligible

ⁱⁿ the frame of the

generalized Callaway integral

^at

high tempera-

tures and one can calculate the lattice thermal conduc-

tivity

^{of an} insulator in the frame of the

generalized Callaway equation

^without

considering

the contri- bution of the correction term due to the three

phonon

normal _processes at

high temperatures.

Acknowledgments.

^- The author wishes to express his thanks to Dr. G. S. Verma and Dr. R. H. Misho for theire constant encouragements. ^{He is also} thankful to Dr. K. L. Yadav for his useful discussions.

References

[1] CALLAWAY, J., Phys. ^{Rev. 113} (1959) ^1046.

[2] KOSAREV, ^V. V., TAMARIN, ^{P. V. and} SHALYAT, ^S. S., Phys.

Stat. Sol. (b) ⁴⁴ (1971) ^525.

[3] PARROTT, ^J. E., Phys. ^{Stat. Sol.} (b) ⁴⁸ (1971) ^139.

[4] KOSAREV, ^V. V., TAMARIN, ^{F. V. and} SHALYAT, ^S. S., Phys.

Stat. Sol. (b) ⁵⁰ (1972) ^595.

[5] DUBEY, ^K. S., Phys. Stat. Sol. (b) ⁶³ (1974) ^39.

[6] SHARMA, ^P. C., DUBEY, ^{K. S. and} VERMA, ^G. S., Phys. ^{Rev. B 3} (1971) ^1985.

[7] JOSHI, Y. P., TIWARI, ^{M. D. and} VERMA, ^G. S., Physica ⁴⁷ (1970) 213.

[8] DUBEY, ^K. S., ^{Indian J. Pure.} Appl. Phys. 11 (1973) ^265.

[9] DUBEY, ^{K. S. and} VERMA, G. S., ^Proc. Phys. ^Soc. Japan ³² (1972) ^1202.

[10] KLEMENS, ^P. G., Solid State Phys. ⁷ (1958) ^1.

[11] SHARMA, ^P. C., DUBEY, K. S. and VERMA, G. S., Phys. ^{Rev. B 4} (1971) ^1306.