Thermoelectric power in semiconducting alloys of the InPxAs 1-x system

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Thermoelectric power in semiconducting alloys of the InPxAs 1-x system

N.P. Kekelidze, Z.V. Kvinikadze

To cite this version:

N.P. Kekelidze, Z.V. Kvinikadze. Thermoelectric power in semiconducting alloys of the InPxAs 1-x

system. Journal de Physique, 1975, 36 (9), pp.883-889. �10.1051/jphys:01975003609088300�. �jpa-

00208326�

THERMOELECTRIC POWER IN SEMICONDUCTING ALLOYS OF THE InPxAs1-x ^SYSTEM

N. P.

KEKELIDZE,

Z. V. KVINIKADZE Tbilissi State

University, Tbilissi,

^USSR

(Reçu

le 26 avril

1974,

^révisé le 14 mars

1975, accepté

^{le 16 avril}

1975)

Résumé. ²⁰¹⁴ On

présente

^une analyse détaillée du

pouvoir thermoélectrique

^et de l’effet Hall

au

voisinage

^{de la}

température

ambiante dans des solutions solides de

InPxAs1-x

pour les

composi-

tions x ⁼ 0 ; 0,1; 0,2 ; 0,3 ; 0,4 ; 0,5 ; 0,6 ; 0,7 ; 0,8 ;1,0

correspondant

à diverses concentrations de porteurs. Ces résultats sont discutés en fonction de la

non-parabolicité

de la bande de conduction.

Les mécanismes de diffusion des électrons dans

InPxAs1-x

^{ont été établis sur} la base de

l’hypothèse

d’Ehrenreich et à l’aide de mesures de

pouvoir thermoélectrique

^{et d’effet} Hall. Les valeurs des

masses effectives des électrons ont été calculées en fonction de la

composition

^des

alliages.

^{Les résultats}

sont en bon accord avec les valeurs calculées

d’après

^la théorie de Kane. Le taux de

non-parabolicité

a été déterminé. On a trouvé que la bande de conduction de InAs ^et des

alliages

très voisins est forte- ment non

parabolique,

^et que

la non-parabolicite

^décroît

graduellement

^{au fur et à mesure} que la

composition

^de

l’alliage s’approche

^{de InP.}

Abstract. ²⁰¹⁴ A detailed analysis ^{of the} thermoelectric power and the Hall effect near room tem-

perature ⁱⁿ

InPxAs1-x

solid solutions for

practically

all necessary

compositions

^x = 0 ; 0.1 ; 0.2 ; 0.3; 0.4; 0.5; 0.6; 0.7; 0.8; 1.0, containing ^{various current carrier} concentrations is given ^and

discussed in terms of the

nonparabolicity

^of the conduction band.

The electron scattering mechanisms in the

compound

^of

InPxAs1-x

system have been found on

the basis of Ehrenreich

assumption

and with the

help

^of measurements of the thermoelectric power and the Hall effect. The values of the electron effective masses were calculated as a function of the

alloy

composition.

The results are in good agreement with the values calculated according ^{to the}

Kane theory. ^The degree ^{of the} alloy

nonparabolicity

^was determined. It has been shown that the conduction band of InAs

compound

^{and the} alloys ^{close to it is} strongly

nonparabolic

^{and the non-}

parabolicity

gradually decreases with the

approach

^{of the}

alloy composition

^{to InP.}

Classification Physics ^Abstracts

8.228

1. Introduction. ^- A series of _papers has been devoted to the

investigation

^{of the}

physical properties

of InP-InAs solid

solutions, but,

many

important

characteristics of these

compounds

^{such as band}

structures, ^{effective mass of the current}

carriers,

etc., ^{are not}

fully determined ;

^{even in the} initial InP and InAs

components

the variation of the main

scattering

mechanisms as a function of the

impurity

concentration and the

temperature

^{remain to be} studied. Various authors have studied either

imperfect inhomogeneous compounds

^or the limited amount of

alloys

^{of the}

given

system ^and

they

obtained results far from the final ^ones. At the same time the non-

parabolicity

of the conduction band was not taken into consideration

during

^the

experimental

^data

analysis.

^For

example,

the values of the electron effective masses measured

optically

ⁱⁿ

[1, 2]

^are

explicitly higher, leading

^{to a}

large discrepancy

with the values calculated

according

^{to the Kane}

theory [3].

The main fact is that the

discrepancy

ⁱⁿ

values of the electron effective masses with the

corresponding

theoretical ones is

stronger

^{near InP}

compound.

The authors of

[4]

associate such a

discrepancy

^for

the

crystals

^{with the}

high phosphorous

^content

with the strong

degeneration

of the electron _gas.

It is

noteworthy, however,

^{that the narrower the}

energy gap of the material

(8g),

^the lower the occupa- tion of the conduction band at which the _nonpara-

bolicity

appears.

Hence,

^{the same}

picture

^would

also be

expected

^{in the case of}

alloys

^{which are close}

to InAs. At the ^same time the authors of

[4]

^use

rather inaccurate values of

8g(X)

in their calculations.

The author of

[5]

^notes that he had

investigated

^the

specimens

^{of very poor}

quality.

The authors of

[2]

also indicate that the

crystals they

studied had an

inhomogeneous

electron concentration over the area

that had to be controlled

by méasuring

^the

optical

transmission of small sections

using

^{an IKS-14}

Speçtrometer.

^’

The authors of

[6-8]

conclude that the lattice acoustic vibrations

play

^a dominant role in the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01975003609088300

884

carrier

scattering

^{at room}

temperatures

ⁱⁿ

InAs, InP,.,AsO.2, InPo.2Aso.8,

^InP

compounds

^contain-

ing

electron densities n -

1017 cm - 3

on the basis of the relevant

analysis.

^{The same} conclusion was

made in the earlier _papers of Weiss

[9-11],

^{the value}

of m ⁼ 0.064 _mo

being

^taken for the electron effective

mass at any concentration in InAs.

All these conclusions need strict verification.

2.

Expérimental

^{method. - The} measurements were

carried out in a metallic cryostat constructed on the basis of the

description

ⁱⁿ

[12]

^{with some modifica-}

tions,

^the vacuum more than

10-3

mm of

Hg.

^In

order to

provide

reliable heat and electric contact, the

specimen

^under

investigation

^{was fixed between}

brass blocks

by

^{means of}

atmospheric

pressure.

The overall

precision

^was

improved by measuring

the

temperature (Ti, T2)

and thermoelectric power

readings simultaneously

^{on three}

potentiometers.

All these measurements were carried out

using

^direct

current and

compensation

^methods.

The

specimens

^{cut in a}

plane perpendicular

^to

the direction of the

ingot growth

^{had the}

shape

^of

a

right parallelepiped

^with average dimensions of 3 x 4 x 15

mm3.

The selected

specimens

^of

n-type conductivity

^were

large-block, homogeneous

^and

perfect.

The

required alloy homogeneity

^{was achieved}

by

means of

cyclic

^zone passages and

repeated

^thermo-

annealing. Typical plots

^of

micro-roentgen analyses

performed

^{with a} MAR-1 Installation ^are shown in

figure

¹ which illustrates the

high homogeneity

^of

solid solutions studied. The

degree

^of

homogeneity

was also controlled

by electrophysical

^and

opto- physical

measurements. The main

experimental

^data

for the

specimens

and the relevant calculated values

are

given

^{in table 1.}

FIG. 1. ^- Typical graphs ^of microroentgen spectral investigations

of the alloys (InPo.,As,,.4).

3. Results and discussions. ^- 3.1 INDIUM PHOS- PHIDE. - The theoretical values of the Seebeck coefficient have been calculated

by

^{the formula}

TABLE 1

Theoretical value Thermoel. power of effective ^masses Y,

yv/K

m ^..

for the

arbitrary degree

of the electron _gas

degeneracy

but for a fixed

scattering

^mechanism

where

is the Fermi kinetic

integral

with index r.

r is the _energy

dependent

^index

designating

^the

length

of the electron free

path (1

^~

Er),

K is the Boltzman constant,

e is the electron

charge,

il ⁼

(/KT

^{is the} reduced Fermi level which was

determined

by

^{means of the}

expression

^{for the current}

carrier concentration

where m is the electron effective mass at the conduc- tion band

bottom, mo is

the free electron mass.

r

In the case of the

specimen

with the electron concen-

tration of n ⁼ 2.2 x

1017 cm-3

a more reasonable value of m ⁼ 0.073 _mo had been used for the effective

mass of the

density

^{of state. a was} calculated for different values of r

by

^{means of the}

graphical

^solution

of the _eq.

(1) (Fig. 2).

d- PYV

FIG. 2. ^- Graphical ^solution of eq. (1).

Figure

^{3 shows the}

comparison

^{of the}

experimental

values of Seebeck coefficient with the theoretical calculations ^for different

scattering

mechanisms : the

scattering

^on the acoustic lattice vibrations

(r

⁼

0),

^the

scattering

^{on the}

optical

^{branch of lattice}

vibrations

(r

⁼

2)

^{and the}

scattering

^{on the}

impurity

ions

(r

⁼

2).

^{As can be seen from the}

plot,

^the

experi-

mental

points

^{are in}

good agreement

^{with the theore-} tical curve

plotted

^{at r =}

y, indicating.the dominating

role of the electron

scattering

^{on the}

optical

^branch

of the lattice vibrations for

crystals

with electron concentrations of n -

1017 cm- 3

near room tempe-

rature.

FiG. 3. ^- Thermoelectric _power coefficient near room temperature in InP crystal : e (n ^{= 2.2 x} 101’ cm-3), + (n ^{= 2.1 x 1019} cm - 1), expérimental values. The solid lines correspond ^to the theoretical

calculation for various scattering mechanisms.

The results obtained are

important

for the final clarification of the

scattering

mechanism in InP

[6-8], [13-18]

^{near room}

temperature.

For the

specimen

with the concentration of

n ⁼ 2.1 x

1019 cm - 3, according

^to

[14]

^it may be concluded that the main

scattering

mechanism is the

scattering

^{on ionized}

impurities.

^In view of this

assumption

it is reasonable to use our

experimental

results in order to estimate the electron effective mass

since for the case of

large

carrier concentrations there are considerable

discrepancies

between the numerical values of the électron effective masses in the literature

(see

^Table

II).

TABLE II

The value of m’ ⁼ 0.11 _mo was obtained for the electron effective mass

by

^means of formulas

(1), (2)

from the

experimental

^{values of a and n at 300 K.}

The use of the theoretical results

of Kolodziejczak [23] ]

obtained on the basis of the Kane

theory [3]

^which

886

takes into consideration the

nonparabolicity

^{of the}

conduction

band, yields

where

m(ç)

is the value of the effective mass at Fermi level which is in

good agreement

with the above value m’. The Seebeck coefficient was also calculated with the

help

^of

Kolodziejczak theory.

The results agree

satisfactorily

^{with the}

experimental

^values.

The above results lead us to conclude that the

nonparabolicity

of the conduction band in InP is not very essential.

3.2 INDIUM MISENIDE. ^- In the case of

InAs,

the studies were carried out over the wide _range of electron concentrations

(1.7

^x

^1016-2

^x

1019) ^{cm - 3.}

The

analysis

^{of the}

experimental

data for the

speci-

mens with concentrations of n -

(1016-101’) ^cm-3

was carried out in the same manner as those for

InP,

^{i.e. on} the basis of a square law for the

disper-

sion.

However, starting

^{from the} concentrations

n >

1017 cm - 3

it is _necessary to take into account the

nonparabolicity

of the conduction band. In this

case the Seebeck coefficient and the concentration

were taken as calculated

according

^{to formulae}

[24]

where

Il 2,0(171 ^fl)

^{is the two}

parameters

^Fermi

integral

of the

type

where fl

⁼

KT/Eg

^{is the}

^parameter characterizing

^the

nonstandard of the

zone ; j8

^{= 0}

corresponds

^{to the}

standard zone. For a

highly degenerate

^electron gas,

e.g. il > 10,

^the

expression (4)

takes the form

where

is the

parameter

introduced

by Kolodziejczak

^and

Sosnovsky [23] characterizing

^the

degree

^{of the zone}

nonparabolicity. Assuming

^a square law for the

dispersion,

y ⁼

0 ;

the calculated concentration

depen-

dence of the

parameter

y is

given

ⁱⁿ

figure

4. The latter shows that for

specimens

with electron concentrations

FIG. 4. ^- Nonparabolicity parameter y ^vs. electron concentration n for the InAs compound.

higher

^than

1017 cm - 3

it is _necessary to take into account the deviation of the conduction band from

a

parabolic shape.

Figure

5 compares the

experimental

values of the thermoelectric _power coefficient at 300 K with the calculated values obtained from _eq.

(1), (4)

^{for diffe-}

rent

scattering

mechanisms.

- 1

FIG. 5. ^- Concentration dependence of the thermoelectric power coefficient in InAs : solid _curves, theoretical calculation for dif-

ferent values of r ; points, experiment.

In the calculations for the electron effective mass at the bottom of the conduction

band,

^{values of}

m ⁼ 0.23 _mo were

used;

but

m(ç)

^{was calculated}

according

^to eq.

(3) (see

^Table

III).

^{As the}

figure

shows for

specimens

^with concentrations n

1017 cm-3

polar scattering

dominates and this is consistent with the result obtained in

[14]

^on the basis of the

analysis

^{of the}

mobility.

^A contribution to the ionic

scattering

^{seems to be}

important

for concentrations

n >

1017 cm- 3 ;

^{while at n -}

¹⁰¹⁸ cm- 3,

^{the elec-}

trons are scattered on the ionized

impurities although

some fraction of

polar scattering

is observed. At

n >

1018 cm-3

the

only significant scattering

^mecha-

nism is that on

impurity

ions. The results obtained

TABLEAU III

are of ^a definite interest since the

scattering

^mecha-

nisms in InAs

crystals

^{near room}

temperature

^{for a} wide _range of the

impurity

concentrations has not been

completely

^defined

despite

^the

availability

^{of a}

series

of papers [6-9], [18], [25-27].

It is

noteworthy

^that

neglect

of the conduction band

nonparabolicity

^{in InAs} may lead to essential

errors in

estimating

^the

scattering

mechanisms for the current carriers as our calculations have shown.

3 . 3 SOLID SOLUTIONS. ^- As is known the chemical bond in InP and InAs

compounds

^{is of a mixed cha-}

racter but as a result of the

dominating

^{role of the}

covalent bond the system ^is

electrically

^neutral.

The presence of ^a definite shift of the electrons from the- atoms ofIII _group to those of V group adds ^some ionic character to the bond. As shown in the

analysis

of the

scattering

^mechanism

given above,

^{the elec-} trical interaction between the

charge

carriers and the intemal

polarization

^{field is}

considerably stronger

than that associated with the deformation

potential despite

^{the low}

degree

^of

ionicity

^{of these}

compounds.

From this

point

of view the

investigation

^{of the}

physical

characteristics of solid solutions of

InP xAs1-x

obtained on the basis of those materials as a function of the

composition,

^{is of a}

special

^interest.

The

analysis

^{of the}

experimental

data for the

alloys

^{whose indium}

phosphide composition

^is

higher

than 50

%

has been carried out as for

InP,

^for

speci-

mens with indium

phosphide compositions

^lower

than 50

% -

^{as for}

^InAs,

i.e. with account for the conduction band

nonparabolicity.

Following

Ehrenreich

[28]

^{we have assumed from}

the

beginning

^{that the} electron effective mass in

InP xAs1

^-x solid solutions varies

linearly

^{from InAs}

to InP.

Using

^this

assumption

^{we have}

analysed

^the

scattering

^{mechanism as for the}

previous

^{cases. The}

analysis

shows that for electron concentrations n

1017 CM-3@

^the

polar scattering

^is dominant in all

InP xÀs1-x

solid solutions. In the concentration range

(101’-1018) ^CM-3

^the

experimental

^{values of a}

are between the theoretical values calculated at

r

= §

^{and r = 2}

(see

^Table

I).

When the current carrier

density

^exceeds

1018 cm - 3

the

polar

interaction with the lattice is screened and in the process of

scattering

^{the main role is}

played ^by

the interaction of the current carriers with the ionized

impurity

^{centers ; but at n -}

1019 cm - 3

the dominant

phenomenon

^{is the}

scattering

^{on the}

impurity

^ions.

The interaction with the acoustic vibrations does not

produce

^{an essential} contribution to the

scattering

process in the whole concentration _range. Note that for InAs as well as for

compounds

^{near to the InAs}

composition, neglect

^of the conduction band non-

parabolicity

^{leads to the essential errors in the}

analysis.

After the

scattering

mechanisms have been esta- blished on the basis of Ehrenreich’s resonable assump-

tion,

the determination of the correct values of the electron effective masses in solid solutions becomes

possible

^at the second stage. ^For

this,

^{on the basis}

of the

experimental

values of a the reduced Fermi

level fi

^was determined

by

^{means of the}

graphical

solution of _eq.

(1)

^and

(4)

^{and the} electron effective

masses were calculated

by

^the substitution of this level in the

expressions

for the concentrations

(2)

and

(5).

Figure

6 shows the

experimental

values of the effective masses vs.

InP xAs1-x alloy composition

which are

compared

^{with the} theoretical values calculated

according

^{to the Kane}

theory [3].

^From

this

theory

^{a formula}

Fie. 6. ^- Values of the électron effective masses mlmo ^vs. InP,,As 1 -., alloy composition: 8 experimental values of the given paper ; A data of 1 ; ^{x data of} 2 ; solid line, the calculation in accordance

with [3].

is obtained for the effective mass value where

Ep

is the energy

parameter

^{and LI} is the value of

spin-

orbital

splitting.

^{For the}

alloy

energy gap 8g ^the data of

[5]

^and

[1]

^{were used}

by

the authors of

[2]

and

[4] respectively.

^We have used the exact values of 89 determined in references

[29, 30]

^{in which the}

nonlinearity dependence

^of cg

= f(x)

is discussed.

Note that as has been shown in

[31, 32],

the coeffi- cient of the

optical absorption

^near the threshold in

InP xAs1-x

solid solutions at values less than 1 000 cm-1 has

exponential

^{tails. As a} result of this

fact,

the reduced values of eg ^{were obtained in}

[1, 5].

888

For the

parameter Ep

^{and the} value of the

spin-orbital splitting

^we have taken the values obtained from the linear

interpolation

between the relevant values of InP and InAs

(Table IV).

TABLEAU IV

Figure

⁶ also shows the values of the effective masses

given

in references

[1, 2].

^Our

experimental

^data

provide

^{the best} agreement with the theoretical values of the effective masses.

0.

Berolo,

^J.

Woolley

and J. Vechten

[33]

^{note that}

for some solid solutions essential deviations from the Kane

theory

^are observed in the determination of m vs.

composition.

The authors of

[33]

^{feel that}

these deviations result from the disorder action of the

alloy

^and

develop

the relevant

theory.

As is

known,

^{for all}

alloys

^{of the}

^A’II

Bv system it is

experimentally

established

[29]

^{that the}

depen-

dence of the energy gap Ego ^{upon the}

composition

^x

has a form

For most

AIII Bv type

solid solutions a considerable difference between the

experimental

_eg ^and _8gv ^calculat-

ed over the dielectric model in the

virtual-crystal approximation

^is

observed, namely [33, 34]

where

CFg

^{is the}

electronegativity

différence between the mixed

elements, A

is the band

parameter

^{of the} order of 1 eV.

The authors of

[33]

substantiate that the deviation of the

dependence m(x)

from the Kane model is caused

by

some

mixing

of the conduction bands and the valence band in r

point,

^{due to} the distortion of the

alloy crystal symmetry. Hence,

^the

resulting

band effective

mass will also be determined

by

the valence band parameters.

Finally,

the authors deduce the follow-

ing

^{formula for the mass} calculation

where _{mco’ Mhh} Mlh, m.so. ^are the values of different effective masses in the absence of band

mixing.

Some

assumptions

^{are made :} mih is assumed to be

equal

^to mco calculated from the formula

(7).

^The

values _mhh and _mso are taken over the linear

extrapola-

tion between the

corresponding

values for the end

compounds (Mhh

⁼

^{0.8 ;}

mso ⁼ 0.35 for InP and Mhh ⁼

0.4;

mso = 0.154 for

InAs). CF’,IA

^{= 0.08 eV.}

The results of the calculation are

given

^{in table I.}

The corrections for the electron effective mass values obtained on

taking

^{into account the}

alloy

^disorder-

ing

^{in our}

alloys

^{are small}

compared

with the values calculated

according

^{to the Kane}

theory.

In conclusion it _may be said that unlike other

compounds,

solid solutions of the

InP xAst-x

system

are

fully

^described

by

^{the Kane model.}

FIG. 7. ^- Nonparabolicity parameter ^vs. InP xAs1-x alloy compo- sition.

We have also

analysed

^the

degree

^{of the}

alloy

^con-

duction band

nonparabolicity.

^As

figure

⁷

shows,

at x - 1 the

nonparabolicity parameter

^{y decreases}

gradually

^{and has a} minimum value for the InP

composition.

This proves that exists ^a strong ^devia- tion of the InAs conduction band

(and alloys

^close

to

it)

^from

parabolicity

while there is

only relatively

weak

nonparabolicity

^{in the}

dispersion

^{law for InP.}

At x - 1 the

polar

interaction increases and that

can be

explained by

the increase of the

degree

^of

ionicity

in the chemical bond.

4. Conclusions. ^- It is therefore established that

near room

temperatures,

ⁱⁿ

InP xAs1-x alloys

^and

their

compositions

with the electron concentration n

1017 cm - 3

the

scattering

^on

optical

^lattice

vibrations is dominant. In the concentration range

(101’-1018) ^cm-3

^the

scattering

is of a mixed character

(on

^the

optical

vibrations and

impurity ions)

^{but at}

the concentrations ^{of n -}

1019 cm-’

the electrons

are scattered

completely by

^{the ionized}

impurity

centers.

It is confirmed that the

ionicity

of the chemical bond is increased as the

composition

varies from InAs to InP.

It is

proved

that the conduction band of InAs

compound

^and

alloys

^{close to it is}

strongly

^non-

parabolic

and that the

nonparabolicity ^decreases

gradually

^{as the InP}

alloy composition

^is approached.

It is shown that

neglect

^{of the}

nonparabolicity

may lead to

major

^errors

during

^the estimation of the

scattering

mechanisms.

The calculated values of the effective masses are

in

good

agreement with those of the Kane

theory

with the consideration of the nonlinear

dependence

of the _energy gap upon the

composition ag(x).

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