Lattice thermal conductivity of p-type III-V semiconductors and p-Si at low temperatures

HAL Id: jpa-00208183

https://hal.archives-ouvertes.fr/jpa-00208183

Submitted on 1 Jan 1974

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Lattice thermal conductivity of p-type III-V semiconductors and p-Si at low temperatures

M. Singh, G.S. Verma

To cite this version:

M. Singh, G.S. Verma. Lattice thermal conductivity of p-type III-V semiconductors and p-Si at low temperatures. Journal de Physique, 1974, 35 (7-8), pp.571-575. �10.1051/jphys:01974003507-8057100�.

�jpa-00208183�

LATTICE THERMAL CONDUCTIVITY OF p-TYPE ^III-V SEMICONDUCTORS AND p-Si ^{AT LOW} TEMPERATURES

M. SINGH and G. S. VERMA

Solid State

Section, Physics Department,

^{Banaras Hindu}

University,

^Varanasi

^221005,

^India

(Reçu

^le

10 janvier 1974)

Résumé. ²⁰¹⁴ Le rôle des interactions entre

phonons

^{et trou lié dans} la conductivité des semiconduc- teurs de type p

légèrement dopés

^tels que

p-GaSb, p-InSb, p-AlSb, p-GaP

^et

p-Si

^a été étudié à basse

température,

domaine dans

lequel les

^autres processus

importants

^de diffusion des

phonons

^{sont dus}

aux effets de

bord,

^{aux défauts}

ponctuels

^{et aux}

phonons

eux-mêmes. En

général,

les valeurs

expé-

rimentales peuvent être très bien

expliquées

par la diffusion

élastique

^des

phonons

^{par des trous} occupant l’état de base quatre ^fois

dégénéré,

^{ainsi que} l’ont discuté Suzuki et Mikoshiba. Les valeurs des constantes définissant le

potentiel

de déformation _que l’on

ajuste

pour obtenir le meilleur accord entre les valeurs

expérimentales

^et

théoriques

de la conduction des

phonons

^{sont en}

général plus

élevées _que celles _que l’on obtient pour des déformations

statiques.

Abstract. ²⁰¹⁴ The role of the bound

hole-phonon

interaction in the

phonon conductivity

^of

lightly doped

p-type ^III-V semiconductors such as

p-GaSb, p-InSb, p-AlSb, p-GaP

^{as well as}

p-Si

^{has been}

investigated

^{in the low} temperature range where other relevant

phonon-scattering

processes ^are caused

by

^the

boundary, point-defects

^and

phonons

themselves. The

experimental values,

ⁱⁿ

general

can be

explained

very well

by

the elastic

scattering

^{of the}

phonons by

the holes in the four-fold

degenerate ground

^{state as discussed}

by

Suzuki and Mikoshiba. The

adjusted

values of the deforma-

tion-potential

^constants for the best agreement between the

experimental

and theoretical values of the

phonon conductivity

are, in

general, higher

than those obtained with static strains.

Classification

Physics ^Abstracts

7.392 ^- 7.630

1. Introduction. ^- Neutral shallow

impurities

ⁱⁿ

semiconductors are very effective scatterers of thermal

phonons

^{at low}

temperatures.

^This has been observed

as a

strong

^{decrease in the thermal}

conductivity

^of

Ge and Si

by light doping

^with

n-type [1-11]

^and

p-type [11-16] impurities.

Similar effects have also been observed in the

p-type

^III-V semiconductor

compounds

^{such as GaSb}

[17,18],

^GaAs

[21],

^InSb

[19],

AlSb

[20]

^{and GaP}

[20].

^The theories of

Keyes [1],

^of

Griffin and Carruthers

[2]

^{and of Kwok}

[10],

^which

were later on extended and

simplified by Kumar,

Srivastava and Verma

[11],

^{have been used to}

explaih quantitatively

^the drastic reduction in the

phonon conductivity

^of

n-type

^{Ge and Si.}

Attempts

^{were also}

made to

explain

the similar reduction in the

phonon conductivity

^{of the}

p-type

^{Ge and Si}

by Pyle [16], by

Griffin and Carruthers

[2]

^and

by

^Shimizu

[22]

^but

they

^{failed to}

give

^a

quantitative explanation

^{of the}

effect.

Recently

^thermal

conductivity

results of the

p-type

^Ge and Si have been

explained by

^{Suzuki and}

Mikoshiba

[23]. According

^to them the decrease of the thermal

conductivity

^{is due to} the elastic

scattering

of

phonons arising

from virtual transitions of bound holes between the

split ground

^{states of}

acceptors.

The theoretical results are in

good agreement

^with the

experimental

results. Since the

acceptor ground

state in III-V semiconductor

compounds

^is

quite

similar to that of Ge and Si

[24],

^we have used their

theory

^to

explain

the thermal

conductivity

^{results of}

III-V semiconductors at low

temperatures.

^{It has} been established in the

present

work that the

theory

of Suzuki and Mikoshiba is

quite

successful in

explain- ing

the results of the

p-type

^III-V semiconductors. It is also shown that the values of the deformation

potential

^constants

Du

^and

Dû

ⁱⁿ the interaction of

acceptor

holes with thermal

phonons

^{in III-V semi-}

conductors are

larger

than those in the interaction with static strâins.

2.

Hole-phonon scattering

and thermal

conductivity.

- The

coupling

of lattice waves to the

impurities

^can

be related to the

crystal symmetry

^{at the}

impurity

^site.

For the

p-type

^{Ge and}

Si,

^the

ground

^{state of the}

acceptor

holes has the

symmetry

^{of the} valence-band

edge (at

^{the centre of Brillouin}

zone)

^and is thus four- fold

degenerate [25].

^The

ground

^{state in} semiconduc-

tors with zinc-blende structure

(III-V semiconductors)

is

quite

^{similar to that} in Ge and Si because the contri-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01974003507-8057100

572

bution of wave vector linear terms to the

ground

state is small

[24].

^One can assume the

splitting

^{of the}

ground

^state

quartet by

internal static strains due to

dislocations, impurity

^atoms

themselves,

^{lattice defects}

and lattice vibrations. These levels contribute

strongly

to the thermal

conductivity

^of semiconductors because of elastic

scattering

^of

phonons by

holes in these levels. If one assumes that _any

splitting

^of quartet

ground ^state

^caused

by

^various

perturbations

^men-

tioned above is much smaller than

4 kB T,

^{then the} relaxation rate of elastic

scattering

^{of the}

phonons by

^{holes in the four-fold}

degenerate ground

^state,

can be written

[23]

^as

where

Here

Dû

^and

Dau’

^{are the} deformation

potential constants, p

^is

density of crystal

^and

Nex

^{is the}

acceptor

hole concentration. The form factor

F(q)

^is

given by

where q

^{is the}

phonon

^{wave vector} and a* is the

acceptor

hole-radius.

wqt,

^{is the}

phonon angular

^fre-

quency for the

phonon (q, t) and vt

is the sound velo-

city.

To make the calculations

simple

^{we have not}

considered the distinctions between

longitudinal

^and

transverse

phonons.

^{This enables us to make use}

of _{v 1} ⁼ _v2 ⁼ v3 = v in

writing

the eq. ^as

or

where

Here we have defined

Wav as [29]

The

phonon conductivity

^{of the}

doped

^material

can be

expressed

ⁱⁿ

Debye approximation

^as

[26]

where

k,,

^is Boltzman’s constant, v ^{is the} average

phonon velocity

^{x =}

hwlk,

^T

and OD

^is

Debye

^tem-

perature. Le

is the combined relaxation time and in the _presence of several

phonon-scattering

processes it can be

expressed

^as

where i refers to ith

scattering

process. ^In the

present

case of the

p-type

semiconductors

t;

^{1 is}

given by

where

Here

iB 1, ipt’

^and

Tpp-1

^{1 are} relaxation rates for the

boundary scattering

^{of the}

phonons, point

^defect

(isotopic) [27] scattering

^of

phonons

^and

phonon- phonon scattering, respectively.

^{L is Casimir}

length [28]. Th!ph is

the relaxation rate due to the

hole-phonon

interaction and is

given by

^eq.

(3).

FIG. 1. ^- The phonon conductivity ^of p-type InSb. Dotted line is theoretical curve and solid circles represent ^the experimental

points.

TABLE 1

Values

of

^the parameters ^{used in the}

phonon conductivity

calculations

(a) HANSEL,

^{J. C. and}

FEHER, G., Phys.

^{Rev. 129}

(1963)

^1041.

(b) PETER,

^Y.

Yu., CORDONA,

^{M. and}

FOLLAK,

^F.

H., Phys.

^{Rev. 3}

(1971)

^340.

(C) PETER,

^Y.

Yu., CORDONA,

^{M. and}

FOLLAK,

^F.

H.,

Solid State Commun. 7

(1969)

^1113.

(d) BOLEF,

^{D. L. and}

MENES, M.,

^J.

Appl. Phys.

³¹

(1960)

^1426.

(e) WEIL,

^{R. and}

GRovEs,

^W.

0.,

^J.

Appl. Phys.

³⁹

(1968)

^4049.

3. Results and discussion. ^-

Eq. (3)-(8)

^{are used in}

the calculation of thermal

conductivity

^{of the}

p-type,

InSb

[19],

^GaSb

[17],

^AISb

[20]

^{and GaP}

[20]

^semi-

conductors. The

experimental

^{data of}

p-Si [15]

^are

also

explained

^{with the}

help

^of eq.

(3)-(8).

^{Our theo-}

retical results are similar to that of Suzuki and Mikos- hiba. The values of the various

parameters

^{used in} the calculation of the thermal

conductivity

^are

given

in table I. The

present

calculations for the

p-type

^III-V semiconductors show

good agreement

^with

experi-

mental

results,

^{as shown in}

figures

^1-3. This shows that the

hole-phonon scattering mechanism,

^which involves transitions of the

acceptor

holes between the four-fold

degenerate

^states is the relevant scat-

tering

^to

explain

the observed

conductivity.

Since

H,

^the

adjustable parameter

^{of our calcu-}

lation,

involves the deformation

potential

^constants

Dû

^and

Dau,

^one

^can

obtain the information about the deformation

potential

^{constants from the}

adjusted

values of H. The theoretical values of

Dû

^and

Dû,

^are

given

^{in table}

^I,

^where

they

^are

compared

^{with the}

experimental

values. It may be ^seen from this table that the theoretical values of

Da

^and

Dû,

^are

greater

than the

experimental

^values.

Therefore,

^{one can}

conclude that the values of the deformation

potential

constants

Dû

^and

Dû, occurring

^{in the} interaction of

acceptor

holes with the thermal

phonons

^{in III-V}

semiconductor

compounds

^{and Si are}

larger

^than

those obtained from the consideration of the interac- tion with static strains.

A similar result for Ge

[30]

^{was obtained}

by

^Suzuki

and Mikoshiba.

They

also showed that the values of the deformation

potential

^constants

D"

^and

D a1

ⁱⁿ

the interaction of

acceptor

holes with thermal

phonons

are

larger

^than those in the interaction with static strains.

However,

^{in the}

present

calculations we

have used _V1 ⁼ _V2 ⁼ _V3 ⁼ v. This leads to an error

of about 6

%

ⁱⁿ the final result. Hence the above calculation is

subject

^{to the above} limitations.

It _may be seen from

figures

^{1-3 that at very low}

temperatures

^the

theory

underestimates the

scattering

for

p-Si, p-GaP, p-InSb

and overestimates it for

p-AlSb

^and

p-GaSb.

^{It means that}

hole-phonon

interaction relaxation rate becomes ineffective at very low

temperatures

^for

p-Si, p-GaP,

^and

p-InSb.

The

discrepancy

^between

theory

^and

experiment

can be removed if one considers that the

phonons

are

elastically

^scattered

by

^the holes in the

split quartet-ground

^{state in}

place

^of

degenerate ground

state. At

higher temperatures,

^the

theory

also under- estimates the

scattering

^{as is seen from}

figures

^1-3.

It is noted that

Lh!ph ^depends

^on the effective Bohr radius a* of the

acceptor

holes via form

factor, F(q). Therefore, discrepancies

^between

theory

^and

experiment

^at

higher temperatures

^{can be removed}

by choosing

^somewhat smaller values of a* than those

given

^{in table I.}

In

conclusion,

^{one can} say that the modified eq.

(3)

^{is able to}

explain

^the

éxperimental

results of thermal

conductivity

^{of the}

p-type

III-V semiconductor

compounds along

with the results of

p-Si.

4.

Acknowledgments.

^- The authors express their thanks to Professor K. S.

Singwi

and Professor B.

Dayal

^{for their} interest in this work. One of us, M.

S.,

is indebted to Council of Scientific and Indus- trial Research for Senior

Fellowship.

LE JOURNAL DE PHYSIQUE. ^- T. 35, ^N° 7-8, JUILLET-AOUT 1974

574

FIG. 2. ^- The phonon conductivity ^of p-type GaP and AISb.

Dotted lines are theoretical curves and solid circles represent the

experimental points.

FIG. 3. ^- The phonon conductivity of p-Si ^and p-GaSb. ^{The dotted}

lines represent ^the theoretical curves and. solid circles represent ^the experimental points ^of p-Si. Solid line represents ^the experimental

curve of p-GaSb.

References

[1] KEYES, ^R. W., Phys. ^{Rev. 122} (1961) ^1176.

[2] GRIFFIN, A. and CARRUTHERS, P., Phys. ^{Rev. 131} (1963) ^1976.

[3] GOFF, J. F. and PEARLMAN, N., Phys. ^{Rev. 140} (1965) ^{A 2151.}

[4] MATHUR, ^{M. P. and} PEARLMAN, N., Phys. ^{Rev. 180} (1969) ^833.

[5] ALBANY, ^{H. J.} and LAURANCE, G., Solid State Commun. 7

(1969) 63.

[6] SINGH, ^M. and VERMA, G. S., Phys. ^{Rev. B 7} (1973) ^2626.

[7] BIRD, B. L. and PEARLMAN, N., Phys. ^{Rev. B} 4 (1971) ^4406.

[8] SINGH, M., VERMA, G. S., Phys. ^Rev. (1973).

[9] SUZUKI, ^{K. and} MIKOSHIBA, N., ^J. Phys. ^Soc. Japan ³¹ (1971) 186.

[10] KWOK, ^P. C., Phys. ^{Rev. 149} (1966) ^666.

[11] KUMAR, A., SRIVASTAVA, ^{A. K. and} VERMA, G. S., Phys. ^Rev.

B 2 (1971) ^4903.

[12] CARRUTHERS, ^J. A., GEBALLE, ^T. H., RESENBERG, ^{H. M. and} ZIMAN, ^J. M., ^{Proc. R. Soc.} A 233 (1957) ^502.

[13] CARRUTHERS, ^J. A., COCHRAN, J. F. and MENDELSOHN, K., Crysonics ² (1962) ^160.

[14] HOLLAND, ^{M. G. and} NEURINGER, ^L. S., ⁱⁿ Proceedings of

the International Conference ^on semiconductors Physics

Exter 1962 (The Physical Society, ^London 1962), p. 474.

[15] THOMSON, J. C. and YOUNGLOVE, B. A., ^J. Phys. & ^Chem.

Solids 20 (1961) ^146.

[16] PYLE, L. C., ^Phil. Mag. ⁶ (1961) ^609.

[17] POUJADE, ^{A. M. and} ALBANY, H., Phys. ^{Rev. 182} (1969) ^802.

[18] HOLLAND, ^M. G., Phys. ^{Rev. 134} (1964) ^{A 471.}

[19] KOSEEREV, ^V. V., TAMRIN, P. V. and SHALYT, S. S., Phys. ^Stat.

Sol. (b) ⁴⁴ (1971) ^525.

[20] MUZHDALA, ^V. M., YA NASHEL’SHU, A., TAMARIN, ^{P. V. and} SHALYT, ^S. S., ^Sov. Phys. ^{Solid State 10} (1969) ^2265.

[21] HOLLAND, M. G., ^Proc. Int. Conf. Physics Semiconductors, (1964) p. 173.

[22] SHIMZU, T., Phys. ^{Lett. 12} (1964) ^179.

[23] SUZUKI, ^{K. and} MIKOSHIBA, N., Phys. ^{Rev. B 3} (1971) ^2550.

[24] THAND, D., Phys. ^{Stat. Sol. 42} (1970) K 61 and K 65.

[25] KOHN, W., in Solid State Phys. ^Edited by ^{F. Seitz and D. Turn-} bull (Academic, ^New York) 1957, ^Vol. 5, p. 257.

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

[27] KLEMENS, ^P. G., in Advance in Solid State Physics ^edited by

F. Seitz and D. Turnbull (Academic Press) ^{1958, Vol. 7.}

[28] CASIMIR, ^{H. B.} G., Physica ⁵ (1958) ^595.

[29] CROSBY, C. R. and GRENIER, C. G., Phys. ^{Rev. B 4} (1971) ^1258.

[30] SUZUKI, ^{K. and} MIKOSHIBA, N., Phys. ^{Rev. B 4} (1971) ^2822.