Role of different polarisation branches in the phonon conductivity of NaCl, KCl, KBr and KI in the temperature range 1-100 K

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Role of different polarisation branches in the phonon conductivity of NaCl, KCl, KBr and KI in the

temperature range 1-100 K

M. P. Singh, G.S. Verma

To cite this version:

M. P. Singh, G.S. Verma. Role of different polarisation branches in the phonon conductivity of NaCl, KCl, KBr and KI in the temperature range 1-100 K. Journal de Physique, 1974, 35 (3), pp.263-269.

�10.1051/jphys:01974003503026300�. �jpa-00208148�

ROLE OF DIFFERENT POLARISATION BRANCHES

IN

THE PHONON CONDUCTIVITY OF NaCl, KCl, KBr AND KI IN THE TEMPERATURE

RANGE 1-100 K

M. P. SINGH and G. S. VERMA

Department

^of

Physics,

^{Banaras Hindu}

University, Varanasi-221005,

^India

(Reçu

^{le 26}

juillet 1973,

révisé le 19 octobre

1973)

Résumé. 2014 On utilise, pour expliquer la conductivité par phonons ^des halogénures alcalins, ^un modèle précédemment ^introduit qui distingue les diverses branches de polarisation ^et qui ^{donne un} comportement ^{correct en} fonction de la température ^{des taux} de relaxation à 3 phonons ^{dans tous}

les domaines de température. ^On introduit, pour les halogénures alcalins, la distinction entre pola-

risations différentes, ^ainsi qu’entre événements à 3 phonons de classes I et II. On utilise les courbes

de dispersion

expérimentales pour les diverses branches de polarisation pour ^trouver les diverses limites des 4 intégrales de conductivité

(KT)003C903C91, (KT)03C9103C903C92, (KL)003C903C94

^et

(KL)03C9403C903C93.

^Les

calculs ont été effectués pour NaCl, KCl, ^{KBr et} KI, ^{donnant un bon accord avec}

l’expérience,

^sauf

aux températures près du maximum de conductivité. La contribution des phonons transverses domine celle des phonons longitudinaux, ^en dehors d’une zone de températures limitée pour NaCl à 19-31 K, pour KCl à 13-22 K, pour KBr à 10-18 K et pour KI à 12-18 K.

Abstract. ²⁰¹⁴ SDV model of phonon conductivity ^{of an} insulator, which distinguishes ^between

different polarisation branches and which not only gives ^correct temperature dependence ^{of three-} phonon relaxation rates in the high ^{as well as low} temperature regions but is also valid for interme- diate

temperatures,

^{is used to} explain ^the phonon conductivity of alkali halides. Distinction between different polarisations ^{as well as between} three-phonon class-I and class-II events is introduced in the case of alkali halides. Experimental dispersion ^curves for the different polarisation ^branches

are used to find out the different 03C9 limits for the four conductivity integrals

Calculations have been performed ^for NaCl, KCl, KBr and KI. Except ^{for the} temperature ^{near the} conductivity maximum the agreement ^between theory ^and experiment ^is good. Except ^{for a limited} temperature range which is 19-31 K for NaCl, 13-22 ^{K for} KCl, ^{10-18 K} for KBr and 12-18 K for KI, the contribution of transverse phonons ^{dominates over that of} longitudinal phonons.

Classification

Physics Abstracts 7.660

1. Introduction. ^- So far there has been no

theory

for

three-phonon scattering

processes, which could

consistently explain

^the

phonon conductivity

^results

of an insulator in both the

high

^{and low}

temperature regions.

In the low temperature

region

^one

usually

takes the

help

^of well known

Herring’s [1]

^relations

regarding

^the

frequency

^and temperature

dependences

of the

three-phonon

relaxation rates. For

example,

in the low temperature

region, T- 1

^oc

^{W2 T3}

^for

longitudinal phonons

^and

i3ph

^{oc coT 4for transverse}

phonons.

^{In the}

high

temperature range, ^one

usually

uses

T-dependence

^{or Klemens’}

[2]-[3]

relation for

three-phonon

relaxation rate. Several workers

[4]-[5]

have assumed low temperature ^{relations of}

Herring

to be valid in the

high temperature region.

It may be noted that the distinction between the low and

high

temperature

regions

^for

three-phonon scattering

processes is based _{upon the} fact that the

high

tempe-

rature

region corresponds

^to the range where K oc T y 1 and the low temperature

region

^to the range where

Herring’s

^{relations are valid. Guthrie}

[6]

^{has dis-}

cussed the

validity

^{of the} temperature

dependences .

of

three-phonon

relaxation rate in the different tem-

perature

regions.

^For

example,

^{he has shown that}

Herring’s

^{relations for}

longitudinal phonons

^{are not}

valid

beyond

^{20 K} in Ge. These difficulties have been very

successfully

^{resolved in the}

Sharma-Dubey-

Verma model

[7]-[8]. (Hereafter

^{we will use the nota-}

tion SDV in

place

^of

Sharma-Dubey-Verma).

^This

model has been _very

successfully applied

^to

explain

the

phonon conductivity

^{of Ge}

[8],

Si

[9],

^{InSb and}

GaAs

[7].

^{In the}

present

paper ^an

attempt

^{has been}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01974003503026300

264

made to

apply

^{this model for the} first time to

explain

the

phonon conductivity

^{of alkali}

halides. ^Recently

Dubey

^{has also}

applied

this model

successfully

^to

NaF

[10].

^{In the SDV model} distinction has been made between

three-phonon

class-I and class-II

events as well as between

longitudinal

^{and transverse}

phonons.

^Guthrie

[6]

^while

discussing

^{the bounds}

on the temperature

dependences

^of

three-phonon

relaxation rate for Si, Ge, InSb and GaSb was the first to

point

^{out that}

three-phonon glass-I

^and

class-II events have différent temperature

depen-

dences and deserve separate considération. The dis- tinction between

longitudinal

^and transverse

phonons

was used for the first time

by

^Holland

[5].

^{Later on,}

this idea was used

by

^{several workers}

[4], [10a]-[19]

in

explaining

^the

phonon conductivity

^{of solids.}

Unlike other workers who used relaxational

approach,

Hamilton and Parrott

[14]

^{and also} Srivastava and Verma

[17]

^have

used

^the variational

approach

^to

establish the separate contributions of different

pola-

risations of

phonons.

^{Joshi and Verma}

[3]

^{and also}

Guillon and

Albany [19]

^used Guthrie’s idea

partially by using

^different

temperature dependences

^{of three-}

phonon

relaxation rate in the different temperature ranges. For

example,

^for

three-phonon

^relaxation

rate

i3ph

^{oc Tm}

^they

^{used m =}

4, 3, 2, 1

^{in the diffe-}

rent temperature ranges.

Singh

and Verma

[18]

applied

Holland’s relaxational

approach

^{to obtain}

the separate contributions of

longitudinal

^{and trans-}

verse

phonons

in alkali halides.

However, they

^have

used

Herring’s

relations for

three-phonon

^relaxation

rate in the

high

temperature

region,

^{which is not}

valid. Guthrie

[20]

^has

recently

^{reviewed the} present situation in this field and some of the issues raised

by

him have been

already

discussed in a series of letters

[21 ]-[23]

ⁱⁿ

Physical

^Reviews.

In the

original

SDV model the

dispersion

^effects

of the different

phonon

^{branches has} been also consi-

dered,

but in the present paper ^we have

neglected

the

dispersion

effects with a view to

simplify

^the

calculations. Thus the model in the present ^form refers to

purely isotropic

materials. The present

model,

like other models of

phonon conductivity

^of

^solids,

is

phenomenological

^{in the sense that}

three-phonon scattering strengths

^{are treated as}

adjustable

para- meters. The reason for this

difficulty

lies in the fact that there is no exact

theory

^for

three-phonon

^scat-

tering strengths. Only recently

^some attempt ^{has been} made to find

thè

relevant

expressions

^for

isotropic [14]

and cubic materials

[24]. ^However,

^the temperature

dependence

^of

three-phonon

relaxation rate in the SDV model conforms to

theory

^{as well as}

experimen-

tal results both in the

high

^{and low} temperature

regions.

^The

exponent m(T)

^of

varies

continuously

^from

high

^values

(3

^or

4)

^{in the}

low temperature

region

^to

unity

^{in the}

high

tempera-

ture range. As

argued by

^{us in reference}

[22], analysis

of the

high-temperature

^{data must be consistent}

with that in the

low-temperature

range. An

attempt

is made first to

explain

^{the low} temperature ^{data on} the basis of two separate modes. Once the three-

phonon scattering strengths

^for

longitudinal phonons

as well ^as transverse

phonons

^are

adjusted

^{for the}

low-temperature

range, the ^same

scattering strengths

are used to calculate the contributions of different models in the entire temperature range. The theore- tical values of

phonon conductivity

^for

longitudinal phonons

fall far below the observed values in the

high

temperature

region indicating

^that their contri- bution in the

high

temperature

region

^is

negligible

and heat transport ^is

mainly

^due to transverse

phonons.

2. Salient features of the SDV model. ^- In the SDV model the

three-phonon scattering

^relaxa-

tion rates for the

longitudinal

^and transverse

phonons

are

given by

Here the suffixes L and T refer to the

longitudinal

and transverse

phonons.

The suffixes 1 and II refer to class-I and class-II

three-phonon scattering

^events,

respectively.

^{In class-I}

scattering

^{events the carrier}

phonon

is annihilated

by

combination with other

phonons

^{and in class-II events the carrier}

phonon

^is

annihilated

by splitting.

This idea of classification is borrowed from Guthrie’s work. In terms of wave- vector restrictions this classification is

represented

as follows :

Further the exponent m ^{is a} continuous function of temperature ^{i. e. m =}

m(T).

This basic idea is

again

borrowed from Guthrie’s work

although

^{this is used}

in calculations of

phonon conductivity

for the first time. Guthrie has assumed that both normal and

umklapp

processes have the same bounds on the temperature

dependence

^of

three-phonon

^relaxation

rate. In this model there is no

explicit

distinction between normal and

umklapp

processes.

However,

one can argue that because of the factor

e -8/aT,

^{, our}

expressions actually

^{refer to}

umklapp

processes.

However,

^{in this}

model,

^{one does}

distinguish

^between

longitudinal phonons

^and transverse

phonons

^and

also between class-I and class-II events.

In view of the fact that m

continuously depends

upon temperature, ^the

expressions

for the three-

phonon

relaxation rates reduce to the well-known

T-dependence

^{for both}

longitudinal

^{and transverse}

phonons

^{in the}

high

temperature range. In the

high

temperature

region, e - (J/rxT

tends to

unity.

^The

beauty

of these

expressions

^{also lies in} the fact that at low

temperatures

they

^{reduce to} the well-known expres- sions of

Herring

^for

three-phonon

relaxation rates.

According

^{to SDV}

model,

^the contributions of

longitudinal

^and transverse

phonons

towards ther- mal

conductivity

^are

given by

The total

conductivity

^is

given by

^{the sum of}

KL

and

KT,

^{i. e.}

Here

is called the correction factor and is

simply equal

^to

(Vg/Yp)

^{for the first}

^portion

^{of the} transverse

phonon

branch in the

region

^{0 0) 0)1. The second} por- tion of transverse

phonon

branch is covered

by

^the

region

col ⁰⁾ 0)2 where ₀₎₂ is the zone

boundary frequency

^{for the} transverse

phonon

^branch.

Similarly

other correction factor involve _{y2, 73} and _74. The value of _yi is obtained from the

dispersion

^relation

and

Thus

As usual

rB ’

^{is the} relaxation rate

[25]

^{for the}

boundary scattering

^of

phonons

^and

T;t l .

^{is the}

relaxation

^rate

[2]-[3]

^{for the}

point-defect scattering

of

phonons.

^The limits of the

conductivity integrals 81, 82, 83

^and

e4

^are obtained from the

experimental dispersion

^{curves for the différent}

polarisation

branches. The values of the

three-phonon scattering strengths

^{are treated as}

adjustable

parameters ^for

the best fit between the theoretical and

experimental

values of

phonon conductivity.

^As mentioned in the introduction we

neglect dispersion

effects and

keep

The

procedure [7]-[9]

^for

calculating

^the

phonon conductivity

^on the basis of SDV model is discussed elsewhere. Here we shall

simply

report ^{the results} for few

typical

alkali halides.

3. Contribution of

longitudinal

^and transverse

pho-

nons to

phonon conductivity.

^{- 3.1} NACI. - The results of calculations of the

phonon conductivity

of NaCI on the basis of SDV model are shown in

figure

^{1 where} the theoretical values are

compared

with the

experimental

^{results in the} température

range

^{0.6 to} 100 K. The

experimental

^{results are}

taken from the paper of

Rosenbaum,

^{Chau and} Klein

[26].

^The

experimental points

^{are marked as}

solid circles. The limits of the different

conductivity integrals

^{are taken from the}

experimental dispersion

curves studied

by

^{Raunio and Stedman}

[27] by

^neu-

tron

scattering experiments.

The values of the various parameters ^{used in the} evaluations of different conduc-

tivity integrals

^are

given

in table I. The theoretical values are

represented by

^solid lines. It may be ^seen from

figure

^{1 that} except ^{near the}

conductivity

^maxi-

mum excellent agreement ^between

theory

^and expe- riment has been obtained.

Figure

^{1 also shows the}

values of

phonon conductivity

^{due to}

longitudinal phonons

^and transverse

phonons. Figure

^{2 shows the} percentage contribution of transverse and

longitudi-

nal

phonons

^{towards the total thermal}

conductivity.

It may be ^seen from this

figure

^that except ^{for the} temperature range 18-32

K,

the contribution of trans-

verse

phonons

^{dominates over that of}

longitudinal

phonons. Figure

3 shows the

plot

^{of m vs. T in the}

266

FIG. 1. ^- Comparison of theoretical values of phonon ^conducti- vity ^{of NaCI with the} experimental ^{values in the} range 0.6 ^to 100 K.

TABLE 1

Values

of

^the

different

parameters used ⁱⁿ the cal- culation

of phonon conductivity of

^NaCl

from

^0.6

to 100 K.

FIG. 2. ^- Percentage contribution of longitudinal phonons ^and

transverse phonons towards thermal conductivity ^{of NaCl.}

temperature range 20 ^to 100 K. It may be seen from

figure

^{3 that for} transverse

phonons

^{and class-I}

events

m(T) approaches

^{to the} value 1.2 at 100 K.

Similarly

^for

longitudinal phonons

^{and class-I events}

m(T) approaches

^to the value 1.7 at 100 K. For the convenience of calculations

ML,,,(T)

^{is taken to be}

unity throughout

^{the entire} temperature range.

According

^to Guthrie this is the maximum value of

ML,,,(T).

FIG. 3. ^- Temperature dependence ^{of the} exponent m ^for longi-

tudinal phonons ^and transverse phonons.

3.2 KCI. ^- The

comparison

^{between the theore-}

tical values of

phonon conductivity

of KCI calcu- lated on the basis of SDV

model,

^{and the}

experimen-

tal values is shown in

figure 4,

^where

phonon

^conduc-

tivity

^{K has been}

plotted

^{at différent} temperatures in the range 1 ^to 100 K. The

experimental

^results

are taken from the paper of Pohl and Walker

[28].

Figure

⁴ also shows the calculated values of

phonon conductivity

^{due to}

longitudinal phonons KL

^and

due to transverse

phonons KT.

^{The total}

conductivity

is

plotted

^{as the sum of}

KL

^and

KT.

^{The values of the}

various parameters used in the calculation of

phonon conductivity

^are

given

^{in table II. The} limits of the different

conductivity integrals

^{are taken from the}

experimental dispersion

^{curves as obtained}

by Copley

and Timusk

[29].

It may be seen from

figure

^{4 that}

except ^{near the}

conductivity

^{maximum excellent} agreement ^between

theory

^and

experiment

^{has been}

FIG. 4. ^- Comparison ^of theoretical values of phonon ^conduc- tivity of KCI with the experimental ^{values in} the range 1 ^to 100 K.

TABLE Il

Values

of

^the

differént

parameters used in the cal- culation

of phonon conductivity of

^KCI

ji’0l11

^{1 to}

100 K.

obtained.

Figure

^{5 shows the}

percentage

contribution of

longitudinal phonons

^and transverse

phonons

towards total thermal

conductivity.

It may be seen

from this

figure

^that except ^{for the} temperature range

13-22 K,

^the transverse

phonons

^{make the}

major

contribution towards total thermal conduc-

tivity. Figure

^{6 shows} the values of

m(T )

^{at différent}

’temperatures

^{in the} range 1 ^to 100 K.

FIG. 5. ^- Percentage contribution of longitudinal phonons ^and

transverse phonons towards thermal conductivity ^{of KCI.}

FIG. 6. ^- Temperature dependence ^{of the} exponent m ^for longi-

tudinal phonons ^and transverse phonons.

3.3 KBr. - The results of calculation of

phonon conductivity

^{of KBr on the basis of SDV model are}

shown in

figure

7 where K vs. T has been

plotted

^for the temperature range 1 ^to 100 K. The separate

FIG. 7. ^- Comparison of theoretical values of phonon ^conduc- tivity ^{of KBr with the} experimental values in the range 1 to 100 K.

contributions of

KL

^and

KT

and the resultant

are

plotted

^{in this curve. The}

experimental points

are taken from the _paper of

Rosenbaum,

^{Chau and} Klein

[26]

^{and are shown as solid}

points.

^{The values}

of the various parameters ^{used in the} calculation of

phonon conductivity

^are

given

^{in table III. The limits}

of the different

conductivity integrals

^{are taken from}

the

experimental dispersion

^{curves obtained}

by Copley

and Timusk

[29].

^It mùy ^{be seen from}

figure

^{7 that}

except ^{near the}

conductivity

^{maximum the SDV}

model

gives

^excellent àgreement ^between

theory

^and

experiment. Figure

8 shows the

plot

^of percentage contribution of

longitudinal phonons

^{and transverse}

phonons

^{towards the total}

phonon conductivity.

^It

may be seen from

figure

^{8 that} except ^{for the} tempe-

rature range 10.5 ^to 18 K the transverse

phonons

TABLE III

Values

of

^the

different

parameters ^{used in the} cal- culation

of phonon conductivity of

^KBr

from

^{1 to}

100 K.

268

Flc. 8.

FIG. 9. ^- Temperature dependence ^{of the} exponent m ^for longi- tudinal phonons ^and transverse phonons.

make a

major

contribution towards transport ^of thermal _energy.

Figure

9 shows the

plot

^{of m at diffe-}

rent

temperatures

T. It may be seen from

figure

⁹

that

m(T)

^for transverse

phonons

and class-I events

approaches

^{1.12 at 100 K.}

Similarly m(T)

^for

longi-

tudinal

phonons

and class-I events

approaches

^1.4

at 100 K.

Again mL,II(T)

^{is taken to be}

unity.

3.4 KI. ^-

Figure

^{10 shows the}

plot

^{of K vs. T in}

the temperature range 0.2 ^to 100 K. The

experimental points

^{are taken from} the paper of

Rosenbaum,

^Chau

and Klein and are shown as solid circles. The theore- tical curves of

KT, KL

^{and K =}

KL

⁺

KT

^{have been}

calculated on the basis of SDV model. The values of the various

parameters

^{used in the} present

analysis

are

given

^{in table IV.} The values of the limits of the different

conductivity integrals

^are taken from the paper of

Dolling

^and

Cowley [30].

It may be seen

from this

figure

^that except ^{near the}

conductivity

maximum excellent agreement ^between

theory

^and

experiment

^has been obtained.

Figure

11 shows the percentage contribution of

longitudinal phonons

^and

transverse

phonons

^towards

phonon conductivity.

Except

^{for a} very ^narrow temperature range from 13-20 K the contribution of transverse

phonons

^domi-

FIG. 10. ^- Comparison of theoretical values of phonon ^conduc- tivity ^{of KI with the} experimental ^{values in} the range 0.2 to 100 K.

TABLE IV

Values

of the diffèrent

parameters ^{used in the cal-} culations

of phonon conductivity of

^KI

from

^0.2

to 100 K.

FIG. 11. ^- Percentage contribution of longitudinal phonons ^and

transverse phonons towards thermal conductivity ^{of KI.}

nates over that of

longitudinal phonons. Figure

¹²

shows the

plot

^of

m(T)

^{at different}

temperatures.

It may be seen from this

figure

^that

m(T)

^{for trans-}

verse

phonons

^{and for class-I events}

approaches

^1.13

at 100 K.

Similarly m(T)

^for

longitudinal phonons

and class-I

scattering

^events

approaches

^{1.17 at}

100 K.

mL,II(T)

^is

again

^{taken to be}

unity.

FIG. 12. ^- Temperature dependence ^{of the} exponent ^{m for} longi-

,

tudinal phonons ^and transverse phonons.

4. Results and discussion. ^- The SDV model in which there is a distinction between

longitudinal phonons

^and transverse

phonons

^{and also between}

three-phonon

class-I and class-II events and in which the temperature

dependence

^of

three-phonon

^relaxa-

tion rate is consistent with

high

^{and low} temperature

theoretical

results,

^{can account for the observed}

phonon conductivity

^{results of alkali halides} very well. The

discrepancy

^between

theory

^and

experiment

lies in the

region

^of

conductivity maximum,

^where

the

point-defect scattering

^of

phonons plays

^{a domi-}

nant

role.

The values of

m(T),

^{both for}

longitudinal

^and

transverse

phonons,

^{are in}

general

^{in the close}

neigh-

bourhood of the maximum values of

m(T),

^{i. e.}

m(T)max

Guthrie for three

phonon

^{class-I events. For}

class-II

events,

^{which takes}

place only

^for

rongitudi-

nal

phonons, m(T)

has been taken to be

equal

to

m(T)max,

^{L, II, Guthrie}

In the

present

calculations a is almost

equal

^{to 2}

for the four alkali halides

which

have been studied.

This value of _oc, which is used in the factor

e - O/aT

is close to the theoretical estimate oc - 2.

Acknowledgments.

^- The authors wish to express their thanks to Professor B.

Dayal

^{and Professor}

K. S.

Singwi

for their interest in the present ^work.

One of us

(M.

^P.

S.)

^is

grateful

^{to Dr. K. S.}

Dubey

for the valuable discussions

during

^the present ^work.

References

[1] HERRING, C., Phys. ^{Rev. 95} (1954) ^954.

[2] KLEMENS, ^P. G., ^{Adv. in Solid State} Physics, ^edited by ^{F. Seitz}

and D. Turnbull (Academic, New York) 1958, vol. 7.

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