Interlayer Bonding and the Lattice Vibrations of b-GaSe.pdf

7/24/2019 Interlayer Bonding and the Lattice Vibrations of b-GaSe.pdf

1/7

FLUORESCENCE

OF

GRADED-

BAND-GAP. . .

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and

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Yokozawa,

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Takayanagi,

Japan.

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1019

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Strehlow, J.

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PHYSIC

AL

REVIEW

B

VOLUME

5,

NU MB ER

4 15

F

EBRUARY

1972

Interlayer

Bonding

and

the Lattice

Vibrations

of P-GaSe

T.

J.

dieting*

and

J. L.

Verble

Naval

Research

Laboratory,

Washington,

D.

C.

20390

(Received 29 September

1971)

The

lattice

vibrations of

the

layer

compound

P-GaSe have been

investigated

by

means

of

in-

frared and Raman experiments. Ref

lectivity

Ineasurements

for

Eic

over the

range

175

100

cm

have

shown one infrared-active

mode at 213.

9

cm

.

Six Raman

lines

have been

observed

at

19.

1,

60.

1,

134.

6,

213.

1,

249,

and

307.

8

cm

. A

group-theoretical

analysis

of the

lattice

vibrations, which shows the

origin

of

the

conjugate

modes

in

layer

compounds,

is

presented.

Electrostatic

as

well

as

van

der

Waals

coupling

between

the

layers

is

indicated

by

the

large

localized

effective

charge

obtained from

the

infrared measurements.

I.

INTRODUCTION

Layer

compounds

are

characterized

by

highly

anisotropic bonding

forces

and

structurally

iden-

tical

layers.

The

forces

between

the

layers

are

known to be

weak

by

comparison

with the forces

within the

layers.

One

therefore

expects

that the

interlayer

coupling

will have a

small

effect

upon

the vibrational

frequencies

of the

lattice. A

spe-

cial

case

of

some importance arises when the

primitive

unit

cell

contains

two

layers

and

there

is

an inversion center between the

layers.

Then

the

normal modes occur in

pairs

that

are either

symmetric

or

antisymmetric

with

respect

to

in-

version.

These

so-called

conjugate

modes can

be

easily studied, provided

that

they

are

optically

active,

by

means of

a

combination of infrared

and

Raman

experiments.

Such

experiments'

have

recently

been carried

out

for

the

layer

compounds

MoS2,

As2SS,

and

AsSe3,

and the

conjugate

optical

modes

have

been

observed

to

be

nearly degenerate

in

energy.

The

hexagonal

layer

compound

Mo82

has

only

one

pair

of

conjugate

modes

that

are

optically

active:

the

Ej

infrared-active

and the

8+,

Haman-active

pair.

A

frequency

difference of

=1

cm

was

previously

reported

by

the

authors

for these

two vibrations.

For

As28,

and

As2Sehowever,

several

pairs

of

conjugate

modes are

optically

active;

Zallen

et

a/.

'

have

observed

frequency

splittings

as

large

as

6 cm

'

between

conjugate

pairs.

In

this

paper

we

present infrared

and

Raman

data

on

another semiconducting

layer compound

of

wide

interest,

GaSe.

In

the

previously

mentioned

investigations it

was

assumed

that the

interlayer

forces

were

primarily

of

the

van der

%aals

type.

This

is a

common

assumption

for

layer

compounds

and one

which

clearly

needs

justification.

Indeed,

we shall

propose

that

the interlayer interaction in

GaSe

contains

an

ionic

or Coulomb contribution.

The

principal

evidence for this is

the

large

local-

ized effective

charge

which has been determined


2/7

T.

J.

WIE

TING AND

J.

I

. VEHBI

E

from

the

present

infrared measurements.

The

value

of the

charge

is

in

reasonable

agreement

with the ionic

charge required

by

the

bonding

scheme of Fischer and

Brebner.

'

In

Sec. II

we describe the

crystal

structure

and

present a group-theoretical

analysis

of

the

vibra-

tional

modes of GaSe; The correlation method

of

group

theory

is used to relate the vibrational

modes

of

an

isolated

layer

to those of the

crystal.

This

analysis allows

the

irreducible

representa-

tions of

the

modes to

be

assigned to specific

sets

of

displacement

vectors.

In

Sec. III the experimental infrared

and

Raman

results

are

given,

and the

frequencies

of

the

modes

are

assigned

to

the

displacement vectors

of Sec.

II.

The infrared ref

lectivity obtained for

E i

c

is

significantly

different from that

reported

by

I

eung

et

al.

'

In

Sec.

IV

the

GaSe

data

are

interpreted

interms

of the

nature of

the

interlayer interactions,

and

the

effects of ionic

bonding

are discussed. It is

predicted

that

long-range

electrostatic

forces

will

be

important

in

force-model

calculations

of

the

lattice

vibrations.

II.

CRYSTAL STRUCTURE

AND GROUP

THEORY

Each

layer

of

GaSe

is

structurally

identical

and

is

composed

of

single

planes

of selenium

atoms

on

either

side

of

a double

plane

of

gallium

atoms.

Figure

I

illustrates

the

coordination of

the

gallium

and selenium

atoms

in a

single

layer.

The

coordina-

tion numbers of the

gallium

and

selenium

atoms are

four and

three, respectively.

Within a

layer

the

crystal

bonding

is

thought

to

be

primarily

covalent.

The

fact that mechanical

cleavage

occurs

easily

along

the

basal

plane

indicates

that

the bonding

TABLE

I.

Long-wavelength lattice vibrations

of P-GaSe.

rep.

A2

2

2g

B2g

1

Ag

2

Ag

2

Bf,

Trans.

properties

Activity

acous

tie al

inactive

infrared

inactive

n,

+o,

~Raman

inactive

n

+~,

e~

Haman

inactive

Direction

of

vibration

C

BXlS

C

BXlS

C

BXlS

C

BXls

C

BXlS

c

axis

C

BXls

C

BXlS

V

(cm

~)

237.

0

134.6

307. 8

E~u

2

1

E

1

E(

2

2

E~e

Tg j

Ty

XX

g

&

3QP

Tg)

Ty

xx

pV

&

3gp

acoustical basal

plane

Haman

infrared

Raman

inactive

Haman

inactive

Raman

basal

plane

19.

1

basal

plane

213.

9

basal

plane

213.

1

basal

plane

basal

plane

60.

1

basal

plane

basal

plane 249

~See

Ref. 5,

between

the

layers

is

relatively

weak.

Three

crystallographically

distinct

polytypes

of

GaSe

have been

reported: The

p

and

e

structures

have

two-layer

hexagonal stacking

s'equences,

whereas

the

y

structure

has a

three-layer

rhom-

bohedral

stacking

sequence.

'

Carter

(of

this

lab-

oratory)

has shown

by

means

of

x-ray diffraction

that

the

GaSe

Bridgman-grown

crystal used in this

investigation

has

the

hexagonal

P

structure.

The

P

structure

is

similar to

that

of

GaS'

and

has the

space

group

D,

(PG~/mme).

Two layers

and

four

molecular

units

are

contained within

the

primitive

unit

cell.

Thus,

24

normal

modes are

allowed.

The

normal vibrations

of

GaSe

decompose into

irreducible

representations

at

the Brillouin-zone

center

as

follows:

FIG. 1.

The coordination of

the gallium and selenium

atoms in

a

single

layer

of

GaSe. The

gallium

atoms

are

represented

by

the black

spheres.

I'=

2A.

~+

2A.

gg+

28'+

282g+

2E)

+

2+

~

+

2Egg

+

2E

2.

Table

I

gives

the transformation

properties

of

the

representations.

Since the

representations

of

the

acoustical

modes must

transform as

T,

T,

,

or

T

these

modes are

represented

by Aaand

the

doubly

degenerate

E&

.

In addition. there

are two

infrared-active

modes of

A2and

Ej

symmetry

and six

Haman-active

modes of

A.

]

E&g

and

Epg

symmetry.

The

infrared-

and

Haman-active

modes

are

mutually exclusive

because

of the

inversion

center

between the

layers. The

remaining modes


3/7

INTERLAYER

BONDING

AND THE LATTICE

VIBRATIONS

OF.

.

.

1475

Ga

(C~)

LAYER

(Dqh)

CRYSTAL

(Deh)

Aq

III.

INFRARED AND

RAMAN

MEASUREMENTS

A. Reflectance

and Transmittance

Ai

E

Se

(C

Ai

E

Ai

E

Eiu

Eig

Ep

FIG.

2. Correlation

diagram

for the irreducible

repre-

sentations

of

the

site

groups,

the

point

group

of

an isolated

layer,

and the

crystal

factor

group

of P-GaSe.

are

optically

inactive.

In

order to

assign

the

representations

to

specific

sets of

atomic

displacements,

we

have

used the

correlation

method

of

group

theory.

'

Figure

2

shows the correlations

among

the

representations

for the

site

groups,

the

point

group

of

an

isolated

layer,

and the

crystal

factor

group.

An

important

feature

of

the

correlation

diagram

is

that

each of

the

irreducible

representations

for the

isolated

layer

divides

into

two

pairs

of

representations

for

the

crystal.

Each

pair

consists

of an

odd and even

representation,

and

we

therefore

refer to

the

modes

which

they

represent

as

conjugate

modes.

The

correlation diagram

indicates

further

that

the

gal-

lium

and

selenium

atoms

vibrate

in

all

of the

nor-

mal

modes of

the crystal.

The

assignments

of

representations

to

sets of

atomic displacements

can

now be carried

out.

These

are

shown

in

Fig.

3. Odd

modes

are

shown

in

the

top

row

of the

figure,

and

even modes are

shown in

the bottom

row. Conjugate

pairs

are

therefore

grouped

vertically.

Because

of the

in-

version

center between

the

layers,

the

conjugate

modes

differ

from each

other

by

an

interlayer

phase

shift

of

180'.

Moreover,

the

conjugates

of

the

acoustical

modes

(Amand

E,

)

are optical

modes

(Bz

and

Ez,

)

in

which entire

layers

vibrate

rigidly

out

of

phase

with

their

neighbors.

We

shall

refer

to

this

class

of

optical

modes

as

rigid-layer

modes.

The

A&and

Erepresentations

belonging

to

the

infrared

modes are assigned

to the sets

of atomic

displacements

shown

in

Fig.

3,

because

these

are

the

only

antisymmetric

modes

which

generate

a

net

electric-dipole

moment

within the

unit cell.

The

remaining

assignments

of

the

representations

are

unambiguous,

apart

from

the

superscripts

on

the

representations,

which have

no

physical

sig-

nific

ance.

where 8

and

T

are

the

measured

reflectance and

transmittance.

Between 1250 and 4100 cm

',

the

high-frequency

dielectric

constant

is

6.

71+0.

1,

and

the index

of

refraction

for

the

ordinary

ray

is 2. 59+0.02.

The

curve

shown

in

Fig.

4

represents

a

least-

squares

fit

to the

ref

lectivity

data. In generating

this

curve

the

well-known

Lorentzian-oscillator

model was

used. The

dispersion

parameters

of

the model are

as

follows: the high-frequency

di-

electric

constant

e

the

oscillator

strength

p,

the

damping

constant

y,

and

the resonance

fre-

quency

of

the oscillator

~0.

Table II

gives

the

values of

the

parameters

obtained

from the

least-

squares

fit

to

the

measured ref

lectivity.

Note

that

the

high-frequency

dielectric

constant

for

the

least-squares

fit

agrees

with

that determined

from

reflectance

and

transmittance

measurements

in

6

,

o-

6

o-

6

i

6'

o-

bI

o-

q~

,

6

o-

6

o-

6

o-

I

6

o-

'

6

Aau

EIu

Aau

EIu

BIu

61

o-

6

o-

bl

6

o-

6

I

o-

(p

f

B~g

E~g B~g

E~g

AIg

0-

Eau

0-

E

Ig

6

o-

fb

o-

Blu

Eau

6

o-

6

o-

A'

E'

FIG. 3.

Displacement

vectors

and

representations

for

the long-wavelength

vibrational

modes of

P-GaSe.

Near-normal-incidence measurements of the

reflectance

and transmittance

of

a

large

Bridgman-

grown

crystal'0

(1.

5&&1.

0-cm

surface

area)

were

made at

room

temperature using

a

Perkin-Elmer

model

No.

301

spectroghotometer.

No

measure-

ments were

made

for

E

II

c,

because

the

crystal

was

too thin

(0.

067

cm).

The experimental

tech-

niques

employed

were

identical to

those

used

in

an

earlier

study

and will

not be

discussed

here.

The

range

of

the

measurements was 175

4100

cm

'.

Figure

4

shows

the

ref lectivity

of

P-GaSe in the

region

below 450

cm,

where the

crystal

was

op-

tically

opaque.

Above

450

cm

'

the ref

lectivity

was

calculated from

the

equation

'

1+2'

g~

+

T2

2(2-R)

4

R(2

-f~.

)

(t

~ mz.

-~'+T.

)')


4/7

T. J. WIE TING AND

J.

L. VE

RB

LE

90

I

I

I

I

I

I

I

I

)

I

I

I

I

)

I I

I

I

j

I I

I

I

f

I

I I I

80

70

60

I

50

l

~0

w

50

K

20

IO

P-GOSe

o

experiment

oscillator

model

FIG. 4.

Ref

lectivity

of

P-GaSe

at room

temperature

for E~c.

The solid curve

represents a

least-

squares

fit to the

ref lectivity

data.

0

I

I

I I

I I

I

I I

l

50

200

250

500

550

400

450

WAVE

NUMBER

(cm

~)

the

1250

4100-cm

region.

The last

column

of

Table

II shows

the longitu-

dinal optical-phonon

frequency

v

(in

units

of

cm

'),

which

was

calculated

from the

Lyddane-

Sachs

Teller relation:

&Lo

&o

1

4

P

v

To

6E

where

co

is

the

low-frequency dielectric constant

for

Elc.

The infrared

ref

lectivity of

GaSe

was

previously

measured

by

Leung

et

a/.

They

reported

one

pair

of

infrared-active

modes,

which

is consistent with

the

hexagonal

P

structure.

Their

data are

given

in

Tables I and

II. The

dispersion

parameters

deter-

mined

by

the

method of interference

fringes

were

regarded

by

these authors

as

more accurate

than

those

obtained

from

their ref

lectivity

measure-

ments.

However,

the

values

which

we

have

ob-

tained

are

substantially

different,

particularly

the

value of

the transverse optical-phonon

frequency

vyo

Leung et

al

. reported

variations

in

the

re-

f

lectivity

of

10/g

from

sample

to

sample,

due

to

surface

preparation,

and

this

may

account

for the

differences

in

the

high-frequency

dielectric

con-

stant,

as well

as

the

oscillator strength and

damp-

ing

constant. Qn

the other

hand,

we

observed

no

frequency dependence

of the ref

lectivity

above

1250

cm,

where

dispersion

in the

index

of

re-

fraction

should be

negligible

and

the

condition of

the

surface

should be

important. We

did

not

at-

tempt

to

polish

or

cleave

the

natural

surface

of

the

GaSe

sample.

The

natural

surface

appeared

visually

to be

in excellent condition.

The

dispersion

parameters of

Table

II

can

be

used

to

determine the localized

effective

charge

on the

gallium

and

selenium

atoms.

Huang's

classical

theory'

of

effective

charge

applies

pri-

marily

to diatomic

crystals

which

have

tetra-

hedral

site

symmetry.

In

this

case

the

local

ef-

TABLE II. Dispersion parameters

for the

E

infrared-active

mode

of

P-GaSe.

Present work

Leung et

al.

See Ref. 5.

Lorentz

oscillator

Reflectance and

transmittance

Lorentz

oscillator

Inter

ferenc e

fr

lnges

6.73

(+

0.

1)

6.

71

(+

0.

1)

8.

4

0.

224

(+

0.

02)

0.

143

0.

187

0.

0125

(+

0.

001)

0.

0087

V

TO

(cm-')

213.

9

(+ 1)

230.7

230.7

VLo

(cm

~)

254. 7

(+

5)

254.

2

264.

6


5/7

INTERLAYER

BQNDING

AND THE LA TTIC

E

VIBRA

TIQN S

Q

F

fective

field has

a

simple form,

and

the

Szigeti

charge

can

be written as

where

e*

is

the effective

charge,

e

is

the

electronic

charge,

c

is

the

speed

of

light,

M

is the reduced

mass

of

the

ion

pair,

and

N

is the

number of ion

pairs per

unit

volume.

Since

P-GaSe is

a

complex

anisotropic crystal,

we

can obtain

only

an

ap-

proximate

measure of the

localized

charge

from

Eq.

(4).

The most important

source

of

error

comes

from the

local-field

factor

3/(e+

2),

which

may

be

appreciably

different

for

GaSe. However,

if

we

assume

that

Eq. (4)

is

approximately

cor-

rect,

we

obtain

an effective

charge

of

0.

74e.

Leung

et

al.

's

data

from

interference

fringes

(Table

II) give

an

effective

charge

of

0. 68e.

B. Raman Scattering

The

first-order

Raman spectrum

of P-GaSe was

obtained

using

an argon-ion

laser,

a

double

mono-

chromator,

photon-counting

electronics,

and

backscatter

geometry.

Six Raman

lines

were

observed

with

5145-A

laser

light

of about

500-mW

intensity.

The

unanalyzed

spectrum

is shown in

Fig.

5.

Since

the

GaSe crystal

was too

thin

to

prepare

and

polish

a

surface parallel

to

the

c

axis,

light

scattering

was measured

only

from

the basal

plane

(xy

plane).

For normal

incidence and

scat-

tering

[z(xx)z

or

z(xy)Z],

the

polarization

prop-

erties

of

the

Raman

lines

at

19.

1,

134.

6,

213.1,

and

307.

8 crn

'

matched

the transformation

prop-

erties

of

the

A&,

and

E3,

representations

of Table

I.

The lines

at

60.

1

and

249

cm

'

were

either

very

weak

or

absent.

For

grazing

incidence,

in

which

components

of

the incident electric

vector were

along

z

as

well

as

x

or

y,

the

lines

at

60.

1

and

249

cm

'

were

greatly

strengthened. We

there-

fore

assign

these two frequencies to

the

E&,

rep-

resentations.

The line

at

249

cm

'

remained weak

and relatively

broad

under all

conditions

of

ex-

citation.

Although

there

are two Raman

frequencies

for

each

irreducible

representation,

the following

con-

siderations permit

a

unique

determination

of

the

frequency

for

each set of

displacement

vectors

(see Fig. 3).

We

first

note

that the

E~a,

mode

is

the

conjugate

of the

infrared-active

Eqmode

and

that

the

frequencies

of the

E@,

modes are

very

different,

213.

1

as

compared

with

19.

1

cm

'.

We

further

note

that the

E@~.

mode

is

a rigid-layer

mode.

Now

since

the

interlayer

interaction in

GaSe

is

relatively weak,

we

shall

assign

the

19.

9-cm

frequency

to

the

rigid-layer

E@

mode,

and

the 213.

1-cm

'

frequency

to

the

E2,

mode.

The

assignment

of the

A&,

and

E&,

frequencies,

however,

requires

another

kind of argument. In

the

E~,

normal

mode

(see Fig.

3)

the

restoring

force

is

primarily

due

to

the gallium-gallium

bond,

whereas

in

the

E&,

mode

the

gallium-sele-

nium

as

well

as

the

gallium-gallium

bonds

are

involved.

Thus we shall

assign

the

higher

fre-

quency

to

the

Ej,

mode.

The

same argument

applies

to

the

frequencies

of

the

A&,

modes. Table

I

summarizes

the

frequency

assignments

of

the

Raman-active modes.

IV.

DISCUSSION

OF

INTERLAYER INTERACTION

In

P-GaSe

the

E~and

E'2, conjugate

optical

modes

are nearly

degenerate

in

energy.

Moreover,

since

the

acoustical-mode

frequency

is

zero

at the

Bril-

louin-zone

center,

the frequencies

of the

E&

(acoustical)

and

E22,

conjugate

modes

differ

by

19.1

cm

.

These

two experimental

facts are

re-

lated

to

the

nature of the

interlayer

forces

in

P-GaSe.

If

we assume

that

the

interlayer forces

are

ex-

clusively

of

the van

der

Waals

type,

the

symmetric

E2,

mode

will

have

a

highex

frequency

than

that

of

the

antisymmetric E&mode.

The

reason for

this

is

that the

layers

in the

unit

cell vibrate

out

of

phase

in

the

symmetric

mode

(see

Fig.

3),

and

the

additional

van

der Waals

interaction

between

the

selenium atoms

in

adjacent

layers

will

raise

the

frequency

of the

E2,

mode.

The

frequency

of

the antisymmetric

Eqmode,

however,

will not

be

affected

by

the van der

Waals

interaction,

be-

cause

the

layers

vibrate

in

phase.

The

result

in

this

case

is

that

v(E&)

&

v(E,

).

A

similar

argu-

ment

applies

to

the relative

frequencies

of the

Eqacoustical

and

E2,

rigid-layer

modes.

Using

a

simple model

for the

van

der Waals

inter-

action

between

the

layers,

we

can estimate

the

in-

crease in

frequency

of the

E&~

mode. Itis

well known

that

two identical

coupled

oscillators

have vibration-

al

frequencies

given

by

Po

and

(vo+

&v

)'

2,

where

vo

is

the

frequency

of

the

isolated oscillator, and

hv

is

the

coupling frequency,

which

depends only upon

the

coupling

force

constant

and

the

mass

of the

oscil-

lators. The

frequency

vo

represents

the

mode

in

which

the

oscillators

vibrate

in

phase,

so

that

the

coupling

force

constant

has

no effectonthe

frequency

of

this

mode.

In

the

other

mode,

the

oscillators

vibrate

180'

out of

phase.

The

coupling

frequency

for

P-GaSe,

using

this

simple

model,

is the

same

as

the

E3~

rigid-layer

mode

frequency,

19.

1 cm

Thus the frequency

of the

E~,

mode is

given

by

(213.

92+19.

1

)

'2=

214.

8

cm

~,

and

the calculated

difference

v(E~,

)

v

(E~)

=+0.

9

cm

.

Although

the experimental

error

is

+1 cm

'

for

the

conjugate-

mode

frequencies,

the important

fact

is that

the

symmetric

E3

mode

was

observed

to

have

a louex

frequency

than

that of the antisymmetric E&mode


6/7

1478

T. J. WIE TING

AND

J. L. VE

RB

LE

M

LLI

1.

0

9

.8

t

E2

2g

7

A,

g

p-GaSe

I

E~g

FIG. 5. The

unanalyzed

Raman

spectrum

of

P-GaSe

at room

tem-

perature.

0 50

100

150 200

WAVE NUMBER

(cm

')

250

500

[observed

difference

v

(E2,

)

(E&)

=

-0.

8

cm

'].

It

is improbable that

the

experimental

error alone

can account

for the

observed difference

in

the

con-

jugate-mode

frequencies. The

interlayer

inter-

action

cannot

therefore

be

exclusively of

the

van

der

Waals

type.

Another kind

of interlayer

force for

GaSe is

suggested

by

the

bonding

scheme

of Fischer and

Brebner.

'

Their

scheme assumes

an electron

pair

for

each

of the

bonding

atoms. If

during

bonding

one

electron

is transferred from the

selenium

to

the

gallium

atoms,

the

hybridized

orbitals

of

the Ga

'

and

Se'

ions

are

tetrahedral

sp'

and

trigonal

p',

respectively. These config-

urations

agree

with the coordination

numbers

of

the

gallium

and

selenium

atoms.

Moreover,

the

Szigeti

charge

for

GaSe,

calculated in

Sec.

III,

is

approximately

equal

to

the

one

electron

re-

quired

by

the

bonding

scheme. Thus

the

long-

range

Coulomb forces

between

the

ions constitute

an

electrostatic

interaction

between the

layers.

Additional

indirect evidence

of the

electro-

static interaction

comes from

the

size

of

the

van

der Waals

radius

for the selenium

atoms.

If the

selenium

4t)

orbitals

are used

in the

covalent

bonds

to

the

gallium

atoms,

then the van

der Waals

interaction

between

the selenium

atoms

in

GaSe

is

primarily

due

to the

4s

electrons.

Since these

are closer

to

the

nucleus

than

the

4&

electrons,

the

interlayer

distance

between the

selenium

atoms

should

be

smaller

than

twice the van

der Waals

radius. The

actual interlayer

distance

(4.

04

A)

is

slightly larger

than

twice

the

van der

Waals

radius

given

by

Pauling

(2x 2.00

A).

This

dif-

ference can be

qualitatively

explained

by

a

Coulomb

repulsion

between the

layers.

By

contrast,

in

MoS&

the

sulfur-sulfur

distance across the

gap

is

3.

37

A,

which

is

appreciably

less

than twice

the

Pauling

van

der

Waals

radius

(2x1.

85

A).

Using

the

authors'

previous

measurements

on

MoS~

and

Eq.

(4),

we

find that the

effective

charge

(for

both

E J.

c

and

E

tl

c)

is 0. lie.

Consequently,

the

elec-

trostatic

interaction

between

the

layers

is

con-

siderably

smaller than that in

GaSe.

Finally,

since

the

van

der

Waals

interaction in

MoS2

is

primarily

caused

by

the

3s,

rather than

3p,

electrons,

a

reduction

of

the

interlayer

van

der

Waals

gap

is

to

be

expected.

In

conclusion,

the

arguments presented

here

suggest

that

long-range Coulomb

forces

are

a

significant

part

of the lattice

dynamics

of

GaSe.

Moreover,

they

represent

the first

evidence of

electrostatic

interactions

in

layer compounds.

Thus

a

simple

force model for

GaSe,

based on

nearest-neighbor

interactions within

a

single

layer,

such

as

that

applied

to

MoS2,

will

not

be

adequate.

The

success of

this

model

for

MoS2

appears

to

be

directly

related

to

the

small

value

of

the

effective

charge,

which

implies

that

the

Coulomb

forces

are

negligible.

A

more

complete

understanding

of

the

lattice

dynamics of

GaSe

must await

further

experimental

and

theoretical

investigations.

ACKNOWLEDGMENT

The

authors

wish

to thank

M.

Schlueter for

pro-

viding

the

GaSe crystal

used

in

this

investigation.

*National

Res

earch

Council

Research

Associate.

~J.

L.

Verble and T.

J. Wieting,

Phys.

Rev.

Letters

25,

362

0.

970).

T.

J. Wieting and

J. L.

Verble, Phys. Rev. B

3,

4286


7/7

INTERLAYER

BONDING AND THE LATTICE

VIBRATIONS OF

1479

(1971).

R.

Zallen,

M.

L.

Slade,

and A. T.

Ward, Phys.

Rev.

B

3,

4257

(1971).

G.

Fischer and

J. L. Brebner,

J.

Phys.

Chem.

Solids

23,

1363

(1962).

P.

C.

Leung,

G. Andermann,

W.

G. Spitzer, and

C.

A.

Mead,

J.

Phys.

Chem.

Solids

27,

849

(1966).

6F.

Jellinek

and

H. Hahn, Z.

Naturforsch.

16b,

713

(1961).

R.

W.

G.

Wyckoff,

Crystal

Structures

(Interscience,

New

York, 1965),

2nd ed.

,

Vol.

1, p.

145.

D. F.

Hornig,

J.

Chem.

Phys.

16,

1063

(1948); E.

B.

Wilson, Jr.

,

J.

C.

Decius, and

P.

C. Cross,

Molecu-

lar

Vibrations

(McGraw-Hill, New York,

1955),

Appendix

X,

pp.

312

40.

In

previous

papers

we have

used the term

quasi-

acoustical

instead of

rigid-layer.

However,

thelatter

term

is

perhaps

clearer and

more

descriptive,

since

in

the

long-wavelength

limit

there

is

no

relative displacement

of the atoms

within

the

layers.

The

GaSe crystal

was

grown

by

J. P.

Voitchovsky of

the

Ecole

Polytechnique Fdddral

de

Lausanne, Switzerland.

I

H. O.

McMahon.

,

J.

Opt.

Soc.

Am.

.

40,

376

(1950).

~~M.

Born and

K.

Huang,

Dynamical

Theory

of

Crystal

I.

attices

(Oxford

U.

P.

,

Oxford,

England,

1968),

Sec.

9.

G.

B.

Wright

and

A. Mooradian

[Bull.

Am.

Phys.

Soc.

11,

812

(1966)]

have

observed

Baman lines

in

GaSe

at

59.

6,

133.

8,

209.

5,

253.

8,

and

308.6 cm

~.

However,

no

polarization

properties

were

given,

and the

low-frequency

mode at 19.

1

cm

~

was not

reported.

~4K.

R.

Symon,

Mechanics

(Addison-Wesley,

Beading,

Mass.

,

1957),

p.

165.

L. Pauling,

The

Nature

of

the Chemical Bond

(Cornell

U.

P.

,

Ithaca,

New

York, 1960),

p.

257.

~6R.

A.

Bromley,

Phil.

Mag.

23,

1417

(1971).

PHYSIC

AL

REVIEW

B

VOLUME

5,

NUMB

ER

4

15

F

EBRUARY

1972

Experimental

Observation of Wannier Levels in Semi-Insulating

Gallium

Arsenide

Robert

VV.

Koss

University

of

&ermont,

Burlington,

V'ermont

and

L. M.

Lambert*

Norwegian

Institute

of

Technology,

Trondheim,

Norway

(Received

3

August 1971)

Optical

absorption

in

an

electric field

has

been

of

increased interest

in recent

years

since

Callaway

predicted

that

the

Wannier levels

may

be observable

in

direct-transition

semiconduc-

tors

such

as GaAs.

In

this

work,

such

levels have been observed for

the first time and found

to

be

in

substantial agreement

with

the

Callaway

theory.

INTRODUCTION

2e'

I

g

~

p.

i'

7t'5'P

flC

pC

x

=

i'aK'/P' ,

P

=2vF/@',

and where

A,

(x)

is

the usual

Airy

function,

~ is

the

photon

frequency,

E

is the width

of

the

Bril-

This

paper

reports

on

an

experimental

investiga-

tion of

the

effect of

a

uniform

external

electric

field on

optical absorption

in semi-insulating

GaAs.

Experimental

data

were

obtained

from room

tem-

perature

to

24

'K

for

electric fields

up

to 1.

6x

10~

V/cm.

The

existence

of Wannier

levels

was

clearly

evident

at

24

'K

and

agreed

well

with

the

Callaway

theory.

The direct-transition

absorption

coefficient

predicted

by

Callaway'

using Kane

functions was

given

as

478K/

P

g

2

27Tg

P

f=&p

where

K

5

K

qp

=

+E~

k~

(2)

To increase

the value

of

qp

by 1,

it is necessary

to decrease the photon

energy

by

b.

h

w=

2'/K

.

Thus,

when

n

is

plotted as a function

of

increasing

photon

energy

for

a

given

electric

field,

the

re-

sult is

a

monotonically

increasing

staircase.

The

width

of

each

step

is

proportional

to

the

electric

field

as

given

by

Eq. (3).

For

small,

uniform

electric

fields,

theory

pre-

louin zone

along

a

principal

lattice direction

of the

applied

field,

I is the

electric

field

force,

p,

is the

reduced effective

mass,

p

.

is

the

interband-mo-

mentum matrix

element,

is

the

photon

pola.

riza-

tion

vector,

and

n

is

the index

of refraction.

The

lower

limit

of

the

summation

is

given

by

jp,

where

jp

is

dependent

upon

the photon

energy

A~,

and

is the

next

integer greater

than

qp,

where

qp

is

given

by

Documents

Interlayer Bonding and the Lattice Vibrations of b-GaSe.pdf