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On the evaluation theory of CV measurements on
narrow gap semiconductor MIS structures
K.G. Germanova, E.P. Valcheva
To cite this version:
K.G. Germanova, E.P. Valcheva. On the evaluation theory of CV measurements on narrow gap
semiconductor MIS structures. Revue de Physique Appliquée, Société française de physique / EDP,
1987, 22 (2), pp.107111. �10.1051/rphysap:01987002202010700�. �jpa00245521�
107
REVUE
DE
PHYSIQUE
APPLIQUÉE
On
the
evaluation
theory
of
CV
measurements
on
narrow
gap
semiconductor
MIS
structures
K.
G.
Germanova
and
E.
P.
Valcheva
Solid State
Physics
Department,
Sofia
University,
1126
Sofia,
Bulgaria
(Reçu
le
5
août
1986,
révisé
le
24
octobre,
accepté
le
13
novembre
1986)
Résumé.
2014
On
a
développé
et
utilisé
un
modèle
théorique
pour
évaluer
les
courbes
expérimentales
capacitévoltage
dans
des
structures
MIS
réalisées
sur
des
semiconducteurs
avec
une
bande
interdite
étroite.
On
a
discuté
et
démontré
l’influence
de
certains
facteurs
sur
le
comportement
des
courbes
CV
théoriques
et
sur
la
densité
des
états
d’interface,
comme
l’utilisation
de
la
statistique
de
FermiDirac,
l’ionisation
incomplète
et
la
recharge
des
dopants,
le
caractère
non
parabolique
de
la
zone
de
conduction.
Abstract.
2014
A
theoretical
model
for
evaluating
experimental
capacitance
2014voltage
curves
on
narrow 2014
gap
semiconductor
(NGS)
MIS
structures
is
developed.
The
features
of
NGS
are
taken
into
account.
Demonstrated
and
discussed
is
the
effect
of
utilizing
FermiDirac
statistics,
incomplete
ionization
and
recharging
of
dopants
and
conduction
band
nonparabolicity
on
the
behaviour
of
theoretical
CV
curves
and
interface
state
density
assessment.
The
analysis
so
conducted
shows
that
these
features
must
be
accounted
in
CV
analysis
of
NGS
MIS
structures.
Otherwise
incorrect
densities
of
interface
states
distributed
across
the
bandgap
of
the
semiconductor
are
obtained.
Tome
22
N°
2
FÉVRIER
198’/
Revue
Phys.
Appl.
22
(1987)
107111
FÉVRIER
1987,
Classification
Physics
Abstracts
73.40Q
1.
Introduction.
The
preparation
of
metalinsulatorsemiconductor
(MIS)
structures
based
on
narrowgap
semiconductors
(NGS)
has
considerably
succeeded
recently.
This
is
stimulated
by
the
efforts
for
producing
basic
elements
of
various
infrared
imaging
devices
[1,
2].
From
a
fundamental
point
of
view
MIS
structures
appear
as
a
tool
for
investigating
the
quasitwodimensional
systems
in
the
potential
well
at
NGSinsulator
interface
[3,
4]
and
the
interface
itself
[57]
whose
nature
is
far
from
common
satisfactory
explanation
yet
[8].
The
performance
of
MIS
devices
relies
strongly,
on
the
electrical
properties
of
the
insulatorsemiconductor
interface.
Commonly
as
an
interface
characteristics
is
utilized
the
density
distribution
of
the
localized
states
across
the
bandgap
of
the
semiconductor.
The
interface
state
spectra
are
obtained
from
the
comparison
of
a
measured
CV
curve
with
a
theoretical
one.
The
computation
of
the
theoretical
CV
curve
requires
a
model
of
an
ideal
structure
to
be
developed
assuming
all
basic
semiconductor
features
and
differing
from
the
real
one
through
the
absence
of
interface
states
only.
Thus
in
the
case
of
NGS
must
be
considered
the
nonparabolicity
of
the
conduction
band,
the
recharging
of
the
dopants
at
low
temperatures,
degenerate
statis
tics,
tunnelling
and
surface
quantization
[6,
9,
10].
Hence,
the
development
of
a
model
for
the
analysis
of
the
spacecharge
region
in
an
ideal
NGS
MIS
structure
considers
effects
that
in
the
theory
of
conventional
silicon
MOS
devices
need
not
be
taken
into
account.
In
this
paper
we
propose
a
theoretical
model
of
NGS
MIS
structure
which
is
applied
to
evaluate
CV
data
obtained
on
InSb
MIS
structures.
Both
low
frequency
(LF)
and
high
frequency
(HF)
occasions
are
considered.
A
convenient
algorithm
and
computer
program
were
elaborated
to
calculate
the
ideal
MIS
C V
curves
and
to
evaluate
the
measured
data.
Discussed
and
demonstrat
ed
is
the
effect
of
the
NGS
features
taken
into
account.
The
analysis
so
conducted
shows
that
these
features
must
be
accounted
in
CV
analysis
of
NGS
MIS
structures.
Otherwise
incorrect
densities
of
interface
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:01987002202010700
108
states
distributed
across
the
bandgap
of
the
semicon
ductor
are
obtained.
2.
Basic
considerations.
For
the
calculation
of
the
theoretical
CV
curve
it
is
necessary
to
know
the
dependence
of
the
total
charge
density
Qsc in
the
semiconductor
spacecharge
region
on
the
surface
potential
03C8S.
The
dependence
Qsc( I/Is) is
determined
from
the
numerical
solution
of
the
one
dimensional
Poisson’s
equation
[11]
where
p
(x )
is
the
charge
density
in
the
semiconductor
spacecharge
région, 03B5s
is
the
semiconductor
permit
tivity,
1/1 (x)
is
the
normalized
electrostatic
potential
in
kT
units,
(Eib
is
the
bulk
intrinsic
Fermilevel).
The
boundary
conditions
for
the
solution
of
equation
(1)
of
zero
electric
field
in
the
substrate
and
a
specified
potential
at
the
insulatorsemiconductor
interface
are
The
charge
density
in
the
semiconductor
spacecharge
region
that
enters
Poisson’s
equation
(1)
is
where p
and
n
represent
the
free
hole
and
electron
concentrations,
respectively.
NÓ
and
NA
are
the
ionized
donor
and
acceptor
impurities
concentrations,
respectively.
Due
to
the
specific
conductionband
structure
of
NGS
these
materials
become
degenerate
at
moderate
bulk
or
surface
electron
concentrations
so
that
Fermi
Dirac
statistics
should
be
used
when
calculating
n.
Furthermore,
the
concentration
of
electrons
is
calcu
lated
taking
into
account
the
nonparabolicity
of
NGS
conductionband
in
the
Kane
approximation
[12]
where
tion
function.
mn
is
the
electron
effective
mass
at
the
conductionband
edge
and
Eg
is
the
bandgap.
So,
we
obtain
from
equations
(4)
and
(5)
where
03BC(03C8) = EF kT+03C8
is
the
normalized
Fermilevel
in
the s p ace
char g e
région,
b
=
kT E
is
the
nonparabolici
ty
parameter
and
J n (IL ,
(3)
is
generalized
Fermiinte
al
of
order n
= 3
[13].
With
the
use
of
FermiDirac
statistics
the
concen
tration
of
free
holes
is
given
by
where
mp
is
the
hole
effective
mass
at
the
valenceband
edge,
F 112 (IL )
is
Fermiintegral
of
order
1/2.
As
the
temperature
decreases
impurity
freeze
out
can
occur
when
band
bending
at
the
semiconductor
surface
forces
the
donor
or
acceptor
impurity
levels
near
to
the
Fermilevel.
So,
the
incomplete
ionization
and
recharging
of
dopants
with
the
Fermilevel
move
ment
should
be
considered
through
the
use
of
Fermi
Dirac
statistics
in
the
calculation
of
the
ionized
dopant
concentrations
in
the
space
charge
region,
i.e.
where
03B5A
and
ED
are
the
normalized
acceptor
and
donor
level
energy
positions
and g
is
the
spin
degenera
cy
factor.
All
energy
positions
are
counted
from
the
bandgap
middle,
positive
towards
conductionband.
So,
including
equations
(6)(8)
in
equation
(1)
results
in
more
complicated
solution
of
Poisson’s
equation.
The
first
integration
of
Poisson’s
equation
gives
Qx (Ws)
where
109
03BCb
is
the
bulk
Fermilevel
and
03BCs
is
Fermilevel
at
the
semiconductor
surface.
The
simulation
of
the
capacitance
of
MIS
structure
is
obtained
from
the
dependence
Qsc(l/Is)
using
the
well
known
relations
between
capacitance,
surface
potential
and
gate
voltage
in
low
frequency
and
high
frequency
approaches
[11].
The
LF
CV
curves
were
computed
under
the
assumption
that
minority
carriers
contribute
fully
to
the
capacity.
Thus
the
LF
semiconductor
capacitance
is
The
most
commonly
accepted
approach
for
accurate
HF
case
calculations
is
used,
i. e.
the
depletioncharge
approximation.
In
this
approach
the
semiconductor
depletion
charge
calculated
from
a
solution
of
Poisson’s
equation
is
treated
as
a
stepfunction.
Once
strong
inversion
is
reached
the
charge
per
unit
area
due
to
the
depletion
of
minority
carriers
saturates
and
the
HF
capacitance
approaches
a
minimum
value
asymptotical
ly
[11].
Calculation
of
the
surface
potential
1/1 sand
its
depen
dence
on
the
applied
voltage
Vg
is
an
essential
step
in
the
analysis.
The
LF
Berglund’s
graphical
integration
[14]
method
is
based
on
where
C LF
is
the
LF
capacitance
and
C ox
is
the
insulator
capacitance.
The
additive
constant 4
may
be
evaluated
by
calculating
the
flatband
voltage
and
by
alignment
of
the
surface
potentials
corresponding
to
flatbands
in
the
experimental
and
theoretical
curves.
The
HF
method
for
obtaining
I/Is(V g) is
based
on
numerical
inversion
of
the
theoretical
formula
In
both
LF
and
HF
methods
every
deviation
of
the
measured
data
from
the
ideal
theoretical
values
is
attributed
to
interface
states
distributed
across
the
bandgap.
The
evaluation
of
interface
state
density
distribution
follows
the
wellknown
differentiation
and
integration
methods
in
the
AF
and
LF
cases,
respect
ively
[11].

Some
basic
semiconductor
parameters
are
needed
for
the
computations.
These
are
the
semiconductor
dielectric
permittivity
and
the
permittivity
of
the
in
sulator,
the
semiconductor
bandgap
Eg,
the
Fermi
level
in
the
bulk
of
the
semiconductor,
the
effective
masses
of
electrons
and
holes
and
acceptor
and
donor
energy
level
position.
Bulk
Fermilevel
position
in
any
particular
case
of
fixed
doping
level
ND
and
NA
and
temperature
in
the
range
4.277 K
is
calculated
from
the
solution
of
electroneutrality
equation
in
the
semiconductor
bulk.
Moreover,
FermiDirac
statistics,
conductionband
nonparabolicity
and
incomplete
ionization
and
recharg
ing
of
dopants
are
considered.
The
intrinsic
Fermilevel
position
for
each
temperature
considered
is
calculated
too.
The
computation
used
includes
iterative
techniques
for
numerical
solution.
The
température
dependences
of
Eg,
m.
and
mp
must
also
be
considered.
We
have
utilized
the
graphical
presentation
of
Eg(T)
from
reference
[15]
and
tempera
ture
dependences
of
the
effective
masses
values
from
references
[16,
17]
in
the
range
of
4.277
K.
The
intrinsic
carrier
density ni
is
utilized
in
the
calculations
and
its
temperature
dependence
is
taken
into
account.
A
numerical
problem
arizes
as
temperature
is
reduced.
The
quantities
n,
p
and
ni
are
all
upon
terms
of
the
form
exp(E/kT).
At
T
=
30
K,
kT
=
0.002585
eV
and
the
normalized
gap
is
approximately
Eg
=
93.46
kT
placing
exp
(Eg 2)
outside
the
range
of
most
computer
arithmetic.
The
use
of
logarithms
and
proper
ordering
of
multiplications
and
divisions
has
allowed
our
pro
gramme
to
operate
down
to
4
K.
3.
Results
and
discussion.
There
are
many
factors
that
may
introduce
error
into
determination
of
interface
characteristics.
One
of
these
is
an
incorrect
ideal
C sc ( 1/1 s)
curve
for
evaluating
CV
analysis.
We
have
studied
what
sort
of
error
can
introduce
an
ideal
C V
curve
when
NGS
features
here
discussed
are
not
taken
into
account.
An
illustration
of
the
result
of
applying
degenerate
statistics
on
the
CV
behaviour
is
given
in
figure
1.
The
figure
presents
the
capacitance
in
accumulation
for
n
type
InSb
MIS
structure
calculated
using
both
Fermi
Fig.
1.
z
Capacitance
in
accumulation
versus
surface
poten
tial
for
different
statistics

curve
1 . 
FermiDirac
statistics ;
curve
2

x 
MaxwellBoltzmann
statistics.
110
Dirac
and
MaxwellBoltzmann
statistics
for
com
parison.
The
particular
parameters
for
the
calculations
are
temperature
of
77 K,
acceptor
concentration
NA
=
5.3
x
1014
cm 3
and
oxide
thickness
do,,
=
1 400
A.
The
surface
charge
concentration
in
the
Max
wellBoltzmann
approximation
grows
exponentially
towards
infinity
with
I/Is
when
in
the
FermiDirac
case
the
surface
charge
concentration
saturates.
The
corre
sponding
width
of
the
spacecharge
region
in
the
MaxwellBoltzmann
approximation
is
larger,
the
capacitance
of
the
layer
is
smaller
and
the
CV
curve
in
accumulation
saturates
slowlier.
Analogous
is
the
be
haviour
of
the
CV
curve
in
inversion
in
ptype
substrate
in
the
low
frequency
regime.
The
effect
of
conductionband
nonparabolicity
on
the
CV
behaviour
is
shown
in
figure
2.
Ptype
material
is
considered
in
the
LF
regime.
The
two
curves
in
the
figure
are
calculated
using
parabolic
and
nonparabolic
dispersion
laws
E(k).
The
nonparabolicity
of
the
conductionband
makes
the
increase
of
the
capacitance
in
inversion
up
to
the
oxide
capacitance
to
begin
earlier
in
respect
to
I/Is
comparing
with
the
CV
curve
calcu
lated
for
parabolic
conduction
band.
The
results
displayed
in
figure
1
and
figure
2
show
that
the
use
of
FermiDirac
statistics
and
conduction
band
nonparabolicity
leads
to
pronounced
differences
in
the
theoretical
C sc ( 1/1 s)
dependences
as
compared
to
the
corresponding
curves
when
NGS
features
are
not
accounted.
It
follows
to
be
expected
that
this
is
going
to
have
an
effect
upon
the
interface
spectra
evaluated
with
Fig.
2.
 Theoretical
CV
curves
for
different
dispersion
laws

curve
1 :
parabolic
conduction
band ;
curve
2 :
non
parabolic
conduction
band.
Fig.
3.

Interface
states
distributions

curve
1 .

with
FermiDirac
statistics
and
nonparabolic
conduction
band ;
curve
2

x 
with
MaxwellBoltzmann
statistics
and
para
bolic
conduction
band.
the
use
of
such
theoretical
curves.
This
is
actually
seen
from
figure
3.
The
figure
presents
two
interface
spectra
calculated
following
two
different
approaches.
A
measured
curve
at
1
MHz
and
77
K
on
the
same
ntype
InSb
sample
as
in
figure
1
is
utilized.
Curve
1
is
obtained
using
MaxwellBoltzmann
statistics
and
para
bolic
conductionband
dispersion
law.
Curve
2
is
ob
tained
using
FermiDirac
statistics
and
nonparabolic
conduction
band.
From
the
comparison
of
the
two
curves
it
is
evident
that
degenerate
statistics
and
conduction
band
nonparabolicity
if
no
accounted
(curve
1)
would
lead
to
a
larger
apparent
interface
state
density.
4.
Conclusion.
A
theoretical
model
for
evaluating
experimental
capacitancevoltage
curves
on
narrowgap
semiconduc
tor
MIS
structure
is
developed.
Theoretical
capacitancevoltage
curves
of
MIS
struc
tures
on
NGS
considering
FermiDirac
statistics,
incom
plete
ionization
and
recharging
of
dopants
and
non
parabolic
conduction
band
are
calculated
in
the
tem
perature
range
of
4.2
to
77
K
and
employed
to
evaluate
interface
state
density
distribution.
The
computed
results
are
concerned
to
InSb
MIS
structures.
Discussed