interlayer bonding and the lattice vibrations of b-gase.pdf
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FLUORESCENCE
OF
GRADED-
BAND-GAP. . .
and
R. Stille,
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Status Solidi
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K71
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Williams, J.
Luminescence
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T.
Handelman
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W. Kaiser,
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Appl.
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351e
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H.
Woodbury
and
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M.
Yokozawa,
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Takayanagi,
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1019
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1279
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L.
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Strehlow, J.
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Phys.
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2928
(1969).
N.
I.
Vitrikhovskii
and
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B. Mizetskaya,
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397
(1959} [Sov.
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1,
358
(1959}].
~SA.
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~
F. L.
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see D. C.
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W. Litton,
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T. C. Collins,
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Phys. Chem.
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D. C.
Reynolds,
C.
W. Litton,
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C.
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156,
881
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C. E.
Hurwitz,
II-VI
Semiconducting
Compounds
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Ibuki
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Ohso,
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Berkowitz
and
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R.
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J.
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A.
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Brooks, J.
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(McGraw-Hill,
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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
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7/24/2019 Interlayer Bonding and the Lattice Vibrations of b-GaSe.pdf
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
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7/24/2019 Interlayer Bonding and the Lattice Vibrations of b-GaSe.pdf
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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.
)')
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7/24/2019 Interlayer Bonding and the Lattice Vibrations of b-GaSe.pdf
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
-
7/24/2019 Interlayer Bonding and the Lattice Vibrations of b-GaSe.pdf
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
-
7/24/2019 Interlayer Bonding and the Lattice Vibrations of b-GaSe.pdf
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
-
7/24/2019 Interlayer Bonding and the Lattice Vibrations of b-GaSe.pdf
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