PHYSICAL
REVIEW B
VOLUME
30,
NUMBER 6
15 SEPTEMBER
1984
Band-gap widening
in
heavily
Sn-daped
In203
I.
Hamberg
and
C. G.
Granqvist
Physics Department,
Chalmers University
of
Technology,
S-412
96
Goteborg,
Sweden
K.
-F.
Berggren,
B.
E.
Sernelius,
and
L.
Engstrom
Theoretical
Physics
Group,
Department
of
Physics
and
Measurement
Technology,
Linkoping
University,
S-581
83
Linkoping,
Sweden
(Received
17
April
1984)
Films
of
pure
and Sn-doped semiconducting
Inz03
were
prepared
by
reactive
e-beam
evaporation.
The
spectral
absorption
coefficient was
evaluated
by
spectrophotometry
in the
(2
—
6)-eV
range.
The
extracted band
gap
increases with electron
density
(n,
)
approximately
as
n,
for
n,
(10
'
cm
This result is
interpreted
within an
effective-mass
model
for
n-doped
semiconductors well above the
Mott
critical
density.
Because
of
the
high
degree
of
doping,
the impurities
are ionized and
the
asso-
ciated electrons
occupy
the
bottom
of
the conduction
band in the
form
of
an
electron
gas.
The
model accounts
for a
Burstein-Moss shift
as
well
as
electron-electron
and electron-impurity
scatter-
ing
treated
in
the
random-phase
approximation.
Experiments
and
theory
were
reconciled
by
assum-
ing
a parabolic
valence
band
with an effective
mass
-0.
6m.
Earlier work on
doped
oxide
semicon-
ductors
are assessed
in the
light
of the
present
results.
I. INTRODUCTION
AND
SUMMARY
In
this
paper
we
investigate
the
optical
properties
of
evaporated
films
of
doped
semiconducting
In203
in the
(2
—
6)-eV
range,
i.
e.
,
around the
fundamental band
gap.
The
study
serves
two
main
purposes:
to elucidate basic
properties
of
a
heavily
n-doped semiconductor, and
to
im-
prove
our
understanding
of
a
technologically
important
material which
is
widely
used
when
transmittance
of
visi-
ble
or
solar
radiation
needs to be combined with
good
electrical
conduction or low thermal
emittance.
We are interested
in
semiconductors
which are n
doped
so
that the Mott critical
density
is exceeded and
for
which
electrons
occupy
the host
conduction
band
in
the form of
an electron
gas.
The
energy
gap
is shifted
as
a result of
the
doping.
The
magnitude
of
the shift is
determined
by
two
competing
mechanisms.
There is
a band-gap
nar~om-
ing
which
is
a
consequence'
of
many-body
effects on
the
conduction and
valence
bands.
This
shrinkage
is
coun-
teracted
by
the
Burstein-Moss
effect3
which
gives
a
band-
gap
widening
as
a result
of the
blocking
of
the
lowest
states
in the conduction
band.
For
doped
In203,
the net
effect
is to
increase the
gap.
We
report
below in Sec.
II on the
production
and
analysis
of
evaporated
In203
films
with
up
to 9
mol%%uo
Sn02.
The
spectral
absorption
coefficient
is evaluated
for
films with electron
densities
up
to
—
10
'
cm;
the data
in-
dicate
a
band-gap
widening
by
as
much as
-0.
8
eV. To
understand
this
widening
from basic
principles,
we
outline
in
Sec.
III
a
theory
which includes the
Burstein-Moss
shift and
self-energies due to
electron-electron
and
electron-impurity scattering.
The calculations
are per-
formed
within the framework of the
random-phase ap-
proximation
(RPA),
along
the lines
given
in
an
earlier
pa-
per.
A
proper
comparison
of
theory
and
experiments
must
account for the
shape
of
the
spectral absorption
coeffi-
cient
around the
energy
gap.
The
analysis
procedure
is
presented
in
Sec.
IV.
We are
able to reconcile
theory
and
experiments
by
having
effective
electron masses of
-0.
3m for
the
conduction
band and
-0.
6m
for
the
valence
band (where
m
is
the
free-electron
mass).
Our
techniques
for
evaluating
and
interpreting
the
band
gaps
go
beyond
what is
normally
done
for
oxide
semiconduct-
ors. Hence it is of interest to consider the
earlier work in
the
light
of the
present
results. This is
done
for
In203,
CdO,
and
Sn02
in Sec.
V. As far as
we
know,
all
earlier
work
on these
materials have
neglected
the
self-energies.
It
is
demonstrated
that this
may
lead to a
qualitatively
in-
correct
interpretation
of
the shifted
band
gaps.
The
present
analysis
provides
a consistent
model
for
the
optical
properties
of
doped
In203
around its
fundamental
band
gap.
It
embraces a
doping
of
a host semiconductor
to
achieve a
high
density
of
electrons
described within
the
RPA
—
and
accounts
for
the
ensuing
ionized impuri-
ties. It is
gratifying
that the
same model can
be used
'
for
the
optical
performance in the
infrared, where the
properties
are
governed
by
a
degenerate
electron
gas
with
a
plasma
energy
—
1
eV
for
the
highest
doping
levels. We
are thus
establishing
a
model
capable
of
explaining
the
key
properties
of
transparent
conductors and
transparent
heat
mirrors:
a
high
transmittance
between a
properly
shifted
and broadened
energy
gap
and
the
plasma
wave-
length,
and a
high
reflectance
and concomitant
high
con-
ductivity
beyond
the
plasma
wavelength.
Coatings
with
these
properties,
applied
to substrates of
glass
and
plastic,
are
extensively
used
for
a
large
number
of
applications;
some of the
most
important
are
front-surface
electrodes
on
solar
cells and
display
devices,
and
low-emittance
coat-
ings
for
windows.
30 3240
1984
The
American
Physical Society
30
BAND-GAP
WIDENING
IN HEAVILY
Sn-DOPED
In203
3241
II.
FILM
FABRICATION
AND
OPTICAL
DATA
The
thin
films studied
in this
work were
produced
by
reactive
e-beam
evaporation
of
pure
In203
and
of
In203
with
up
to
9
mo1%
Sn02
onto substrates
of
CaFz
in a sys-
tem with
accurate
process
controls.
The
deposition
ma-
terials
were hot-pressed pellets
with
99.
99%
purity
sup-
plied
by
Kyodo
International,
Japan.
The evaporation
rate was
kept
at
a
constant value
in
the
0.
2
—
0.
3 nm/s
in-
terval
by
using
the
output
from a
vibrating
quartz
micro-
balance to feedback
control
the
electrical
power
of the
e-
beam
source. A constant
oxygen
pressure
in
the
(5
—
8)X10
-Torr
range
was
maintained
by
continuous
gas
inlet
through
a precision
valve. The
substrate,
posi-
tioned
3S cm
above
the
source,
was
kept
at
-300'C
dur-
ing
the
evaporation.
These
parameters
are known
to
yield particularly
good
optical
performance.
Film
thicknesses were monitored
on
the
vibrating
quartz
micro-
balance
during
the
evaporation
and
were
subsequently
determined
to
an
accuracy
of,
typically,
+2
nm
by
a
com-
100
80—
V
c
C$
V
O
~
60-
I
'D
C
lg
O
v
40—
c
cj
~
~
E
m
20—
C
I-
bination of
optical
interference
methods.
The
analyzed
films were
between
75 and
1200 nm
thick.
Optical properties
of
the films were
determined
by
spectrophotometry.
Normal transmittance
(
T)
and
near-
normal
reflectance
(R)
were recorded at
room tempera-
ture
as a
function
of
photon
energy
(fico)
on a
Beckman
ACTA
MVII
double-beam instrument
interfaced to
a
computer.
Figure
1 shows
typical
data; (a)
refers to
a
75-
nm-thick
film
made
by
evaporation
of
pure
In203,
and
(b)
refers
to
a
110-nm-thick
film made
by
evaporation
of
In203+9
mo1%
Sn02.
A
large
decrease
in
T
as
the
pho-
ton
energy
is
increased
signifies
the band
gap.
Oscilla-
tions
in
R indicate
optical
interference.
Spectral
transmittance and
reflectance data
were used
to
coinpute
the
complex
refractive
index,
n+ik,
of
the
films
by
use
of
Fresnel's
equations'
and
known
properties
of
the
CaF2
substrates.
"
Spectrophotometric
data were
selected
in such a
way
that the
ensuing
optical
constants
were
accurate to
within
prescribed
limits,
and so
that
spurious
solutions
were avoided.
It was also
verified
that
the
Kramers-Kronig
relation was
obeyed.
These
aspects,
which
fall outside
the
scope
of this
paper,
are
discussed
in
some detail
elsewhere.
'
The
optical
constants n
and
k
were
derived at
—
100
equally
spaced
energies
in the
(2
—
6)-eV interval.
Figure
2 shows
smooth
curves
drawn
through
these data
points
for the two
films
reported
in
Fig.
1.
For
films
of
pure
In203
(solid
curves),
we
find
that k
goes
up
rapidly
at
duo-3.
8
eV and
that n
is rough-
ly
2
with a
small
peak
at
Acu-4
eV.
For Sn-doped
In203
(dashed
curves),
the
increase
in
k
is
displaced
towards
higher
energy
and
now
occurs at
%co-4.
3
eV;
the
corre-
sponding
curve
of
n
again
shows
a
small
peak
at
fico-4
eV.
The
absorption
coefficient a will be analyzed
in
detail
below.
This
quantity
is
defined
by
a=4irk/A,
,
0
2.
0
100
I
3.
0
I
4.0
Energy (eV)
I
5.0
6.
0
where
A.
is
the
wavelength.
We
computed
a at
—
100
equally
spaced
energies
for
each
sample,
and
joined
the
in-
80—
I
V
C
tg
V
I
60—
I
'a
c
lO
I
40—
C
lg
O~
E
th
20—
f-
110nm
3.
0
c
2.
5—
I
'a
C
~
~
o
2.
0—
V
o
1.
5—
0
lg
~
1.
0—
I
K
0.
5—
—
0.
8
~
I
C
~
~
—
0.6
~
I
V
lg
Le
I
V
—
0.4
o
tO
CL
i
lg
—
0.
2-
Ql
0
2.
0
3.
0
6.
0
5.
0
4.0
Energy
(eV)
FIG.
1.
Spectral
normal
transmittance
and
near-normal
re-
flectance for films
of
(a) pure
In&03
and
(b)
Sn-doped
In2O3
on
substrates of
CaF2.
Estimated
experimental
errors
and the
reproducibility
among
measurements on
the same
sample
are
considerably
less than one
percent
unit.
0
2.
0
5.
0 6.
0
3.0
4.0
Energy (eV)
FIG. 2.
Optical
constants,
n
and
k,
versus
photon
energy
for
films of
pure In203
(solid
curves)
and
Sn-doped
Inq03
(dashed
curves).
Data
for
n
are less
accurate
than
those
for
k.
3242 I. HAMBERG et al.
30
4,
$ I
I I
1
i i
i i
)
i
I
Sample
Material
n~(10
crn
3)
(b)
E3-
Q
C
I
.
2
2—
O
0
O
C
0
~ ~
CL
0
tD
E~(k)=
*'E
o
rnc
E'„(k)=
2FAy
Ego
jl
~)I
kF
E,
(k)=
E',
(k)+t
Z,
(k)
E„(k)=
E'„(k)+RE„(k)
k
0
——
I
3 4 5
Energy
(eV)
FIG.
3.
Absorption
coefficient
versus
photon
energy
for
films of
pure In203
and
Sn-doped
In203.
Solid
curves
were
drawn
between
individual
data
points
whose
scatter
was
less
than
the
width
of
the
lines.
The
curve denoted
D
was
obtained
for
a
thick
film
which became
virtually
opaque
for
Ago
)
4.7
eV.
Inset
table
shows
the
electron
densities
in the
various
films.
dividual
points
by
smooth
curves.
Figure
3
shows such
data for four
different films. Curves
3
and
C
pertain
to
the
films
of
pure
In203
and
Sn-doped
In203,
respectively,
which were
reported
earlier
in
Figs.
1
and 2.
It is
ap-
parent
that
the
band
gaps
lie at
different
energies,
and
that the
onset
of
strong
absorption
is
rather
gradual.
For
low
energies
it
was
found
that
1na
versus fico
displayed
an
approximately
linear relation.
'
The
data
in
Fig.
3
can be
systematized
by
considering
the electron
density,
n„
in
the various
films. Its magni-
tude
was obtained from
determinations
of
the
plasma
en-
ergy
(i.
e.
,
the
energy
for
which
the real
part
of the
dielec-
tric function
equals
zero),
as
described
in
earlier
papers.
'
For films of
pure
Inz03,
we
have
n,
=(0.
4+0.
05)&&1
0
cm
as
a result
of
doubly
charged
oxygen
vacancies.
'
For
Sn-doped
In203,
we
have
electron densities between
(1.
7+0.
2)
and (8.
0+0.
5)
&&
10
cm
primarily
as a
consequence of
the Sn
atoms which enter
substitutionally
as Sn
+
on In
+
sites,
and hence act as
singly
charged
donors.
'
Actual
values of
n,
are
given
in
the
inset of
Fig.
3.
It is concluded
that
increasing
electron
densities
lead to
a
progressive
enhancement of the
energy
gap.
III. THEORY OF
SHIFTED BAND
GAPS
As
a starting
point
for
discussing
the
relation between
optical
band
gap
and
electron
density
we
consider the
band structure
of
undoped
In203.
The band
structure
is
unknown
in most
respects.
The
only
available
informa-
tion concerns
the direct and
indirect band gaps'
and the
region
around
the bottom of the
conduction
band,
which
is
thought
to be
parabolic
with an
effective
mass
(m,
')
of'
-0.
3m,
where
m
is the
free-electron mass.
We
can
only
make an
assumption
regarding
the
shape
of the
valence
band;
we take
it to
be parabolic
and characterized
by
an
effective mass
(
m,
*)
of
unknown
magnitude. Figure
4(a)
illustrates this
band structure.
With
the
top
of
the
FIG. 4.
(a)
shows
the
assumed
band
structure
of
undoped
In203
in the vicinity
of
the
top
of
the
valence
band and
the
bot-
tom
of the
conduction
band.
(b)
describes
the
effects
of Sn dop-
ing:
The
valence
band
is
shifted
upward
by
many-body
effects
while the
conduction
band
is
shifted
downward.
Shaded areas
denote
occupied
states.
Band
gaps,
Fermi wave number,
and
dispersion
relations are
indicated.
valence
band as
reference
energy,
the dispersions
for the
unperturbed
valence
and
conduction bands
are
E„(k)=
—
A'
k
/2m,
'
(2)
E,
(k)
=Ego+fi
k
/2m,
*,
(2')
respectively.
Ego
is the band
gap
of the
undoped
semicon-
ductor,
k
is
the
wave number,
and
superscript
0 denotes
unperturbed
bands.
Single
crystalline plates
of
In203
have'
E~o
3.
75
eV.
Po—
—
lycrystalline
thin
films
may
have
a somewhat
different
Eso,
its
actual magnitude
is depen-
dent
on the
detailed
preparation
conditions.
A detailed
discussion of
these shifts
is
not
possible,
but
we
note that
the
band
gap
can be
altered
by
local
strain induced
by
im-
purities,
point
defects,
and
poor
crystallinity.
In the
doped
material
we
have
to
consider three
dif-
ferent
effects:
First,
the
shapes
of
the valence
and
con-
duction bands
may
not
be accounted
for
by
precisely
the
same
effective
masses as
in
the
undoped
material.
Indeed,
it
has been
found'
'
that
m,
is
weakly
dependent
on
the
electron
concentration
and
goes
up
to & 0.
4m
at
n,
&3&10
cm
.
A
corresponding
variation
for
m,
*
cannot be
ruled
out,
but
nothing
is known. Second,
above
the Mott
critical
density'
the
partial
filling
of
the
con-
duction band
leads
to
a
blocking
of the
lowest states and
hence a widening
of the
optically
observed
band
gap.
This is
the
well-known
Burstein-Moss
(BM)
shift.
Third,
again
above the
Mott
critical
density
the
valence
and
con-
duction
bands
are shifted
in
energy
as a result of
electron-electron
and
electron-impurity
scattering.
In
In&03,
these tend
to
partially
compensate
the BM
shift.
Figure 4(b)
shows
schematically
the
roles
of the
second
and
third
effects.
We
first
neglect
the
role
of
electron-electron
and
electron-impurity
scattering.
The
energy
gap
for direct
transitions
in the
doped
material
is then
given
in terms
of
the unperturbed
bands
as
30
BAND-GAP
WIDENING
IN
HEAVILY
Sn-DOPED
In203
3243
Eg
E—
—
,
(kF )
—
E„(kF
),
(3)
where
kF
—
—
(3m.
n,
)'
is
the
Fermi
wave
number.
Alter-
natively,
we
may
write
0
BM
Eg
—
—
Egp+
AEg
where the
BM
shift is
given
by
g2
gEBM
(3+
)2/3
2muc
with the reduced effective mass
1 1
1
~+
Pl~ Plv
mc
(3')
E,
(k,
co)
=E,(k)+iriX,
(k,
co),
(6')
where iiiX„and
iiiX, are self-energies
due to
electron-
electron and electron-impurity
scattering.
We now
obtain,
instead
of
Eqs.
(3)
and
(3'),
the shifted
optical
gap
Es
E,
(kF,
co)
E,
(—
—
kF,
co),
—
or,
alternatively,
(7)
Equation
(4)
predicts
an
energy-gap
shift
proportional
to
~
2/3
With the
purpose
of
giving
a
more correct theoretical
model
for the shifted
band
gaps,
we
now
include electron
interactions and
impurity
scattering.
The free
electrons
in
the
doped
material cause
a
downward
shift
of the
conduc-
tion band as
a
result
of
their mutual
exchange
and
Coulomb
interactions.
This
shift is further
accentuated
by
the attractive
impurity
scattering.
The
valence band
is
infiuenced
in
the
opposite way.
The
effect
of
the various
interactions can be described
simply
by
replacing
the
bare-band
dispersions
in
Eqs.
(2)
and
(2')
by
the
corre-
sponding
quasiparticle
dispersions
E„(k,
co)
=E„(k)+iriX„(k,co)
and
where
Rg
runs
over a
random
distribution of Sn
+
on
In
+
sites.
We
estimate
V;
„by
the
Heine-Abarenkov
pseudopotentials'
appropriate
to the
two
ionic
species.
Figure
5 shows
the
Fourier transform
of
the unscreened
difference
Vs„(
r
)
—
V,
„(
r
)
divided
by
the bare Coulomb
potential
—
4me
/q
for
a unit
point
charge.
This ratio is
close to
unity
for k &
2kF,
which
is
the
pertinent
range
in
the
computations,
thus
proving
that
the
scattering
centers
behave as screened
point
charges.
We stress that this
simplifying
feature would
break
down for
electron
con-
centrations much
higher
than those of
the
present
films as
well as
for
doping
elements
whose
pseudopotentials
are
drastically
different
from
that of the substituted ion.
Ex-
plicit
results for
fiX"
,
and
A'X"
,
were obtained
by
using
m,
from Ref. 17
(together
with a reasonable
extrapolation
to-
ward
high
electron
densities).
The
computations
of
A'X„"
and
A'X,
"
require
in
principle
that
the
full
frequency
dependence
of the
background
is
included. For
simplicity
we
suppress
this
dependence.
In
all calculations we set
the dielectric constant
for the
In203
host
equal
to
the
stat-
ic value.
'
The
computations
now
proceed
as discussed at
length
in
Ref. 2.
Thus
the
screening
properties
were
included
by
using
the random-phase
approximation.
As
commonly
done,
irido
in
the
expressions
for the
self-energies was
put
equal
to
fPkF
/Zm,
~,
i
The in.
teractions
do
not
only
shift
the
positions
of the
valence
and conduction bands but also
distort
them
to
some
degree,
and
hence
the
k
dependence
of the self-energies should
be
retained
in
principle.
As
an
application
of the
above
formulas, we
now
report
on
computed band-gap
shifts
versus
n,
The eff.ective
valence-band
mass remains
as an unknown
parameter.
Figure
6 contains results for three different values
of
m„.
It
is
found that an
increasing
electron
density
leads to an
enhanced
gap
shift,
and that
at constant electron
density
the
gap
shift
is
largest
for
the smallest
m„*.
The
shifted
band
gap
is further
elucidated
in
Fig.
7. It
shows
b,
Ez
for
m„'=0.
6m
as well as
its three
contribu-
tions:
Eg
—
—
Eg
0+
AEg,
with
(7')
b,
Es
bEg
+AX,
(k—
—
F,
co)
fiX,
(kF,
co)
.
—
Electron-electron
(ee)
and electron-impurity
(ei)
scatter-
ing
are taken
as
additive
processes
within our
perturbation
treatment,
i.
e.
,
iriX,
(k,
co)
=iiiX„''(k,
co)+A'X",
(k,
co)
V;m~(r)
=+[Vs„(r
—
Rg)
—
V&„(r
—
Rg)],
(10)
AX,
(k,
co)
=AX,
"(k,
co)+A'X,
"(k,
co)
.
We refer
to
Ref. 2 for
details.
We first
consider
fiX'„'
and
fiX,
".
The Sn
+
ions
—
which
replace substitutionally some
of the In
+
ions
—
act
effec-
tively
as
singly
charged
scattering
centers.
Their
scatter-
ing
potential
can
be
written
0.
5
I
) I I
1.
0
1.
5
a(au)
FIG. 5.
F(q)
as a
function of
q
in atomic
units
(a.
u.
).
F(q)
is the ratio between
the Fourier
transform of the
bare scattering
potential
Vs„(r)
—
Vi„(r)
and the Fourier transform of the
Coulomb
potential
for
a
point
charge.
F(q)
was
evaluated for
n,
=
10
'
cm,
but
is not sensitive to
the electron
concentration.
Arrow
points
at
k~
for
n,
=7)&10 cm
3244 I. HAMBERG
et
al.
30
IV. COMPARISON OF
THEORY AND EXPERIMENTS
~
10
E
V
O
Ol
C
CO
Q
'U
C
0
4
Q
LLI
and
gEBM
g
A'X"(kF)=I)i&,
"(kF)
—
IIi&",
(kF),
2—
I I I I I
I I I l I
I
I
I
I
I I I I
0
0.
2
0.4
0.6 0.8
Energy
gap
shift
hEg
(eV}
FIG. 6.
AE~
versus
n,
computed
for
three
magnitudes
of
the
effective
valence-band
mass.
,
X
I
(I
I
I'If&
I'
(N
—
CO,
f
)
+T
where
i.
accounts
for
the
broadening of
the initial
and
fi-
nal
states,
P is
the
probability
that the state
is
occupied,
and
cof;
=(Ef
E;
)/fi.
—
In
the
limit
i
~
ac
Eq.
(11)
goes
over
to the usual golden-rule
expression.
We
now
identify
the
initial
states with
the
filled
valence
band and the
final
states with the
partially
filled
conduc-
tion band.
It
is
straightforward
to
prove
that
R
cc
I
dx(x+fico
—
W)'i
(1
P, ),
—
Xp
x
+I
(12)
where
we
have
introduced the notation
The
experimental
data
on
0,
versus
%co show a
gradual
onset
of
strong absorption,
and hence
it is
not
obvious
how
to
locate
a
unique
optical
band
gap
which
can
be
compared
with
Eg
as
derived
in
the
preceding
section. In
order
to
understand
the
experimental
broadening,
we
con-
sider
the
quantum-mechanical
transition rate
R
for
transi-
tions
between an
initial
(i)
and
a
final
(f)
state.
Accord-
ing
to time-dependent perturbation
theory,
we have
fiX"(kF):
—
fiX,
"(kF)
—
I)i'",
(kF)
.
Ak
x:—
+
O'
—
Ac@,
2&i
(13)
The BM
shift is dominating
except
for the lowest
electron
concentrations.
AEg
versus
n,
does
not
give
a
BM
2/3
straight
line,
which
is
a
consequence
of
the
empirical
rela-
tion'
between
m,
and
n,
[and
(Ref.
21)
between
the
stat-
ic
dielectric
constant
and
n,
]
The
s.
elf-energies
are
seen
to
vary
approximately
as
n,
;
this
dependence
is
rather
fortuitous
and
is
the net
result
of several
competing
ef-
fects.
xo
=
4Eg +
8'
—
Ace,
I
=&/w.
An
analogous expression for R has been
given
by
Finken-
rath who
used classical
arguments.
The minimum
dis-
tance
between
the valence
and conduction
band
in
the
doped
material
—
denoted
W
in
Fig.
4(b)
—
is
given
by
the
approximate
relation
W=EsP+fiX, (kF)
—
I)iX„(kF)
.
10—
CV
I
E
O
~
8—
I
I
I
I
I
I
I
I
I
I
I
I
I
In
writing
R
as
in
Eq.
(12)
we
have
also
ignored
the
k
dependence
of
I
.
Thermal
excitations above the
Fermi
energy
are
represented
by
a
Fermi function
according
to
A'
k
P,
=
exp,
—
p
+1.
2m,
'
C
6—
'Q
C
0
Le
V
O
LLl
where
k&T
is
Boltzmann's
constant times the tempera-
ture,
and
p
is
the chemical
potential.
At
kg
T
«A'
kF/2m,
':
—
eF
we have
2'
I I I I I I I
I
I
I I I I I I
0
0.
5
1.0
Energy (eV)
FIG. 7.
AE~
versus
n,
computed
for
m„*=0.
6m,
and
analogous
plots
for the
contributions
to
EE~
from the
Burstein-
Moss
shift
(EEL
),
electron-impurity
scattering
[fiX"(kF
)],
aud
electron-electron
scattering
[A'X"(kF
)].
2
kmT
P
~QF
3
2CF
In
the limit
I ~0
and
T~O,
Eq.
(12)
goes
to
(fico
W)'~
for
fico)
W+b,
E—
0
for
~&
g
+kg
(19)
(19')
![FIG. 7. AE~ versus n, computed for m„*=0.6m, and analogous plots for the contributions to EE~ from the BursteinMoss shift (EEL ), electron-impurity scattering [fiX"(kF)], aud electron-electron scattering [A'X"(kF )].](/figures/fig-7-ae-versus-n-computed-for-m-0-6m-and-analogous-plots-1stpcpu8.webp)