Pergamon
PII:
Solid-Stare Ekcrronics Vol. 41, No. 4, pp. 575-583, 1997
B 1997 Elsevier Science Ltd
Printed in Grea;Britain.
All rights reserved
s0038-1101(%)00148-7
0038-I 101/97 $17.00 + 0.00
A NEW GENERATION-RECOMBINATION
MODEL FOR
DEVICE SIMULATION INCLUDING THE
POOLE-FRENKEL
EFFECT AND
PHONON-ASSISTED
TUNNELLING
0. K. B. LUI and P. MIGLIORATO
University of Cambridge, Department of Engineering, Trumpington Street, Cambridge CB2 IPZ, U.K.
(Received 23 May 1996; in revised form 8 August 1996)
Abstract-Polycrystalline
silicon transistors (poly-Si TFTs) are very attractive devices for large scale
integration (LSI) on glass with regards to reliability, compactness and low cost. The correct modelling
of leakage currents in poly-Si TFTs is important for designers and technologists. Amongst other problems,
a prevailing problem is the anomalous leakage current, which can degrade the pixel voltage in an active
matrix display. This paper presents a new generation-recombination (G-R) model for the leakage current
which is suitable for implementation in a device simulator. 0 1997 Elsevier Science Ltd. All rights reserved
NOTATION
RT
due to the presence of traps
eV-‘)
Trap density of states (cm-” eV-I)
G-R
rate
(cm-’ s-’
NT(ET)
n,
Fieid-enhanced density of electrons (cm-‘)
Field-enhanced density of holes (cm-‘)
Pt
Capture coefficient (cm’ s-l)
Cn.p
Emission coefficient (s-l)
en+
Cno.pO Capture coefficient at equilibrium conditions
(CrnJ
s-‘)
Emission coefficient at equilibrium conditions
(s-l)
Intrinsic carrier concentration (cm - ‘)
n,
Trap energy level (eV)
ET
Intrinsic energy level (eV)
E,
Poole-Frenkel barrier lowering (J)
AErp
F
Electric field (Vm-‘)
Electron charge (C)
9
t
Dielectric constant (Fm-‘)
T
Temperature (K)
k
Boltzmann constant (JK-‘)
J-CO”,
Tunnelling integral for Coulombic wells
“4
Poole-Frenkel enhancement factor
:E”
Energy range for tunnelling (eV)
Tunnelling effective mass for electrons (kg)
m.*
Reduced Planck constant (h/272) (Js)
A
Jmrvc
Tunnellina integral for Dirac wells
n;;,
Convent&al d&sity of electrons in the conduction band (cm-‘)
Quasi-Fermi level (eV)
FN(x)
Tunnelling probability
T(E)
G-R rate due to acceptor states (cm-’ s-’ eV-‘)
RA
G-R rate due to donor states (cm-’ s-’ eV-‘)
RD
Total G-R rate (cm-’ s-‘)
UG R
Density of acceptor states (cm-” eV-I)
N,(ET)
ND(ET) Density of donor states (cm-) eV-I)
Conduction band maximum (eV)
EC
Valence band maximum (eV)
EV
EA%
ET””
r\n
.._
RElIW
EG
Deep states trap
(cm-l eV-I)
Tail states trap
(cm-’ eV-I)
Characteristic energy
Characteristic energy
G-R
rate using-(cm-‘s-l eV-I)
Band gap (eV)
concentration
parameter
concentration
parameter
width for deep states (eV)
width for tail states (eV)
model of reference il]
I. INTRODUCTION
A G-R model for single-crystal silicon which takes
into account
band-to-band
and trap-to-band
phonon-assisted
tunnelling in both forward and
reverse bias was successfully developed by Hurkx
et al.[l]. This model is presently widely used in
commercial device simulators. However, whilst the
band-to-band
tunnelling mode1[2-4] is of general
validity, the trap-to-band phonon-assisted tunnelling
model takes only into account Dirac wells and,
therefore, neglects the Poole-Frenkel (PF) effect.
Trap-to-band
phonon-assisted
tunnelling
has
been studied in both single crystal[l] and polysilicon
devices[5-%]. The need to include PF barrier lowering
in trap-to-band
phonon-assisted
tunnelling
was
demonstrated for poly-Si pn junctions in Refs[9,10]
and more recently, in poly-Si TFTs in Ref.[l 11.
Without the PF effect the emission rate is at least one
order of magnitude lower than what is needed to fit
the experimental data[l 11. By using a commercial
device simulator, based on the model of Ref.[l], we
575
576
0. K. B. Lui and P. Migliorato
were unable to simulate the leakage currents in
poly-Si TFTs accurately.
Apart from further enhancing the emission rate
for trap-to-band
phonon-assisted
tunnelling, the
PF effect also plays a significant role in enhancing
pure thermal emissions at low fields. This was
demonstrated
in Ref.[l2], but their model is
limited to low fields since it does not include
trap-to-band
phonon-assisted
tunnelhng which is
significant at moderate to high fields. So presently, a
complete model for trap-to-band phonon-assisted
tunnelling
inclusive
of the PF effect and
suitable for implementation in a device simulator is
not available.
We have developed a new G-R model which
incorporates the PF effect in combination
with
trap-to-band phonon-assisted tunnelling. The model
is not restricted to simulating leakage currents in
poly-Si TFTs but has general validity. It takes into
account both forward and reverse-biased tunnelling,
allowing the simulation of both forward and reverse
currents, and includes both Dirac and Coulombic
traps. Finally, the expression we deduce for the G-R
rate can be easily incorporated into the continuity
equations and implemented in a device simulation
package. Since the new model is particularly
important for poly-Si, we shall refer to this case in the
treatment which follows.
Application of this model to the case of poly-Si
TFTs gives good agreement with the experiments
and explain why the inclusion of PF barrier
lowering is much more important in the case of a
material with a continuous density of states in the
band gap.
2.1. G-R rate expression
To obtain an expression for the G-R rate due to
the presence of traps, RT, the general expression for
the net recombination rate resulting from a dynamic
balance between emission and capture of electrons
and holes is used. This expression is:
cncpnlpl- e,ep
c& + c,pt + e, + ep’
(1)
where NT(ET) is the trap density of states, while n, and
densities of electrons and
holes which have the capture coefficients c. and cp
respectively. The quantities e, and ep are the emission
coefficients of an electron and a hole respectively. The
derivation of eqn (1) can be found in Refs[l3,14].
p, are the field-enhanced
2.2. Principle of detailed balance
The principle of detailed balance gives:
en0= c,0 x c,nl,
ep0= CNPI= c,pl,
Note that in eqn (2) the assumption is that all
capture coefficients remain equal to their equilibrium
values under non-equilibrium conditions. The subscript ‘0’ means equilibrium conditions. In eqn (3) n,
is the intrinsic carrier concentration, ET is the trap
level and E, is the intrinsic level. Derivation of
eqns (2) and (3) can be found in Ref.[l3].
2.3. The Poole-Frenkel
effect
The PF effect[l5] consists of the lowering of a
Coulombic potential barrier due to the electric field
applied to a semiconductor. For a trap to experience
the effect, it must be neutral when filled (charged
when empty). A trap that is neutral when empty will
not experience the effect because of the absence of the
Coulomb potential[l4].
For a Coulombic well, the barrier lowering, AEr,
(Fig. l), is given by the expression:
where F is the electric field, q is the electron charge,
and t is the dielectric constant of the material. The
derivation of eqn (4) can be found in Refs [14,16].
2.4. Emission ratio for a Coulombic well
2. MODEL
RT = IWET)
where
(2)
The expression deduced by Vincent et a1.[17] for
the ratio of the field-enhanced thermal emission rate
over the zero-field emission rate, for a Coulombic
well, can be written as:
$ = exp($$)+G:exp{z
x
;T)[l
- z3j2
-(z>x’]}dz.
(5)
The first term on the right hand side of eqn (5) is
the contribution of the PF effect responsible for
enhancing pure thermal emission. The second term is
the contribution of the phonon-assisted tunnelling,
with the PF barrier lowering taken into account. The
latter is responsible for enhancing the emission rates
via trap-to-band tunnelling.
After some algebraic manipulation,
by letting
z = (AE,/kT)u, and substituting into eqn (5), we get:
$q+y
(6)
511
A new generation-recombination model
2.5. Carrier density ratio for a Coulombic well
where
pul
n
_
-
’
E,
kT sAEr$AE.
lm u - K,,u f,
x exp k~
xF =
_ (_!%.>‘:‘I
exp 3
(
du,
(7)
(8)
>
and
_4J%s
K
n
n,(.x)=n(l)+[(-y)Y_,,T(x-o)da
(12)
where n(x) is the conventional
the condution band:
density of electrons in
(9)
qtiF
3
Our approach follows the one originally developed
by Hurkx et al.[ 181for the case of a Dirac well, based
on the formalism first developed by Slater[l9]. In
order to extend the treatment to the case of a
Coulombic well, with PF barrier lowering, we start
from the expression:
A& is simply the energy range for which tunnelling
can occur and m.* is the tunnelling effective mass for
electrons. The expression for the emission ratio for
holes is similar.
Equation (5) is general. The case of a Dirac well is
simply obtained by assuming AE~P= 0. Thus, we get
for a Dirac well:
5&=1+rp
”
FN(x)- Edx)
kT
(13)
and FN(x) is the quasi-Fermi level at x. T(x - a) is
the probability that an electron at x tunnels to ‘a’
(Fig. 1). Equation (12) is a generalisation of eqn (5)
of Ref.[l8]. An expression for the tunnelling
probability for the case of a Coulombic well,
including PF barrier lowering, was deduced by
Vincent et a/.[l7]:
where
$$
K,u”*
u -
du.
(11)
Equations (10) and (11) are the same as in Ref.[l].
p
region
...a&a
......
FN
_ ($T]}.
Substituting eqn (14) into eqn (12) and replacing E
n region
8.
. .
(14)
K
dcpktion ngion
.
-7 “+)[1
T(E) = exp
.
Fig. 1. Schematic energy-band diagram of a reverse-biased pn junction. Both phonon-assisted tunnelling
and pure thermal excitation (conventional SRH) are illustrated. The barrier lowering (dotted lines) due
to the PF effect is also shown. A Dirac well situation (solid lines) is also represented. A linearly varying
potential is assumed.
0. K. B. Lui and P. Migliorato
578
with qF(x - a) (linearly
obtains:
x exp
x
I(
--
varying
potential)
4 J2mt[qF(x
3
@I
Table
one
E,
EV
LAp-0
L:“”
Lp=P
.P
p
P’
G=p
ToA”
- a)])
2.8
3.2
2.8
3.2
4
;
T
C”
cP
m.’
mh’
K
n,
for calculations
1.60218 x IOm’yC
11.9 x 8.854 x IO-‘2 Fm-’
1.38066 x 1O-2” J K-’
300K
1 x IO’cm/s x 3 x 10-‘scm’
I x lO’cm/s x 3 x IO-“cm’
0.25 x 0.91095 x IOm’O kg
0.25 x 0.91095 x IOm”kg
I .05458 x IO-” Js
1.18 x 10tOcm-
and for the case of Dirac wells, eqn (18) simply
reduces to:
n, e.
-=-=
The limits t, and t, define the range of ‘a’ values for
which tunnelling to or from a trap at x is possible. It
is assumed that for x > n > ter T(x - a)= I and that:
.
$
(
Values employed
>
[I - (qFr:a,>‘:‘]}da.
(15)
n(&)=n(x)exp
I.
0.56 eV
-0.56 eV
x IO”cm-’
x 10zOcm-’
x IO”cm-’
x 10xccm0.23 eV
0.038 eV
0.23 eV
0.038 eV
(16)
>
Note that in eqn (15) the integral represents the
contribution of phonon-assisted tunnelling, modified
to include the PF lowering of the tunnelling barrier.
This is the first important generalisation of the
treatment of Ref.[l8]. Also note that in eqn (15) the
first term on the right hand side of the equation is the
conventional density of electrons modified by the PF
enhancement factor, as shown in eqn (16). This is the
second important generalisation which reflects the
enhanced thermal injection of carriers due to the PF
barrier lowering. The relative weight of the two terms
Rr = RA + R.
n
y&~*+rpc
1+rgi”,
e,0
(19)
where Tzr is given by eqn (11).
Equation (17) is very important. It forms the core
of this new G-R model. Basically it shows that the
general expression Vincent et af.[17] derived for the
emission rates is essentially the same as for the carrier
densities.
2.6. Modification of conventional SRH expression
By substituting eqns (18) and (19) into eqn (2) and
the result into eqn (l), taking into account that:
WE,)
= N*(ET)+ND(ET)
where the subscript ‘A’ denotes acceptor-like trap
states and subscript ‘D’ denotes donor-like trap
states, we get the final modified SRH expression:
where:
np - nf
RA =
cp(xF: j-y,
(n + “‘)+c”(,
NA(ET)
+&ac)
@ + PI)
np - n,’
Ro =
cp(l :yc,
ND(&).
(20)
(n + nl)+c”(XF : rf”“‘) (P + PI)
varies with the applied electric field as will be
discussed later.
Assuming that the quasi-Fermi levels are approximately constant in the depletion region[l,20] we get,
after some algebraic manipulation (see appendix),
The total G-R rate is obtained
throughout the band gap:
s
s
by integrating
EC
UGR =
RT(ET)
dEr
EV
=
‘;‘[ 1 - (-$>11’]}du.
(17)
The right hand sides of eqns (17) and (5) are
identical. This implies that:
n,
__=-=en
n
e,.
XF +
rp’,
$
=
2
=
XF +
Ec(MET)+
RA(ET)) dET.
(21)
Ev
r?
(18)
Equation (20) is a general expression. The various
mechanisms proposed to explain poly-Si TFT leakage
currents, like thermal emission, PF effect[21-251 and
phonon-assisted tunnelling from traps[5-8,1 l] are
included as particular cases.
A new generation-recombination
579
model
5 .OE+Z1
4 .OE+Z1
G
,t
3.OE+21
u
3
2.OE+2 1
1.OE+Z1
O.OE+OO
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
lkap level (ev)
Fig. 2. G-R rate Rnnc vs trap level ET. The major contribution of generated electron-hole pairs comes
from the mid-gap.
3. RESULTS
No
3.1. G-R rate us trap energy
= Nimp exp
We now evaluate the new recombination
model.
Following previous analysis[26,27], we approximate
the density of states (DOS) as:
NT(&) =
N@T)+k(ET)
N&ET) =
NF exp
+
W,""
(22)
where the subscripts ‘A’ and ‘D’ refer to acceptor and
donor states. The values employed in the following
calculations are shown in Table 1 and are obtained
from the literature[ 1,7,28].
In order to show the importance of the PF effect
in the case of polysilicon we have analysed two cases,
one where all Dirac centres are assumed and the
exp
1.8E+22
1.6E+22
1.4E+22
a
_%
_ _ - RA
1.2E+22
l.OE+22
‘.‘_.-
B
3 8.OE+21
&
6.OE+2 1
-
Rn
RI
4.OE+2 1
2.OE+21
O.OE+OO
-0.5
-0.4
-0.3
-0.2
-0.1
lhp
0.0
0.1
0.2
0.3
0.4
0.5
Level (ev)
Fig. 3. G-R rate RT vs trap level ET. Note the presence of the twin peaks. The major contribution to
the generation comes from these two peaks that are shifted slightly away from the mid-gap.
580
0. K. B. Lui and P. Migliorato
1.OE+4
1.OE+2
1.OE+O
l.OE-2 4
O.OE+O 2.OE+7 4.OE+7 6.OE+7 8.OE+7 l.OE+8
Fig. 4. Comparison amongst the field effect integrals r$‘, r:y and the Poole-Frenkel
level ET = 0 eV (mid-gap).
other, where the true situation
is considered,
that is
acceptors (donors) are Dirac centres for electrons
(holes) and Coulombic for holes (electrons).
The G-R rate RT has been evaluated from eqn (20)
as a function of ET for an electric field F = 5 x 10’
V/m and n = p = 0. This can be viewed as the
situation occurring in the depletion region of a
poly-Si pn junction. RDlrac
is the G-R rate using the
model developed by Hurkx et al.[l]. The results are
shown in Figs 2 and 3.
As one can see the generation rate peaks for
ET x EG/2 only in the case of Dirac centres (Fig. 2).
term XFat trap
For the combined Dirac-Coulombic
case (present
model) twin peaks are observed equidistant from
mid-gap (Fig. 3). The presence of the twin peaks can
be explained in the following way. The G-R rate is
maximum when electron and hole emission probabilities are roughly equal. Since for monovalent levels,
in the absence of the PF effect, the tunnelling barrier
of an electron and a hole bound to a trap are ET and
EG - ET respectively, the maximum occurs for
ET z Eo/2. If the PF effect is taken into account, the
tunnelling barrier for an electron from a donor state
is reduced by AErr, while the hole barrier remains
4.OE+20
3 SE+20
3.OE+20
P
$
____._.._...
RA
2SE+20
B
a 2.OE+20
‘...-‘--Rn
r
pl 1.5E+20
3
1.OE+20
RT
5.OE+19
O.OE+OO
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
Fig. 5. G-R rate RT vs trap level ET. Note the smaller separation of the twin peaks, due to the lower
fields (F = 1 x 10’ Vm-’ in this case).
A new generation-recombination
581
model
1.OE+23
-Nmvamdeldhpoty-si
DOS
.
-Modcloffefwnce
I
-
-
.I* l.OE+21
‘Q
--o-
a
[l]wul
powi Do6
bnd-to-baad tumeNhg
mu&l
-FbWllXld~lWiUldiicrctc
~-VP m
. . . . . -.-.Modelofmsfaran~[1]wirh
dimetellid-gqDm
3
3 l.OE+l’
l.OE+l
l.OE+lS
O.OE+O2.OE+7 4.OE+76.OE+78.OE+7 l.OE+8
Ekctric field (V/m)
Fig. 6. G-R rates for the new recombination model, the model of Ref.[l] and the pure band-to-band
tunnelling mode1[2-4] for continuous poly-Si DOS and discrete mid-gap DOS.
unchanged. Hence the maximum generation rate
occurs below mid-gap and the peak value is
increased. The situation is reversed for acceptor
states, hence the twin peaks. Quantitatively,
the
above arguement is borne out, in eqn (20), by the fact
that Fy’>>Fqinc for a wide range of applied fields, as
shown” Pn Fii.‘4.
We wish now to discuss the role of the term IF.
This term is also plotted in Fig. 4 and it is observed
that xF is more significant than the tunnelling integral
Fr at low fields. This indicates that at low fields,
thermal emission dominates and therefore the main
effect of the electric field is to enhance the thermal
emission through the pure PF effect. At higher fields
TF>>xF and the effect of the applied field is both to
induce tunnelling and to reduce the tunnelling
barrier.
These effects have been separately recognised in
various works. The effect of the PF barrier lowering
on tunnelling was considered in Ref.[ll] to explain
the leakage current behaviour in polySi TFTs. The
importance of the pure PF effect was pointed out in
Ref.[l2] for the case of Sic diodes at low fields. The
present model however includes both effects in a
self-consistent way, suitable for implementation in a
device simulator.
Finally it is interesting to note that the separation
of the two peaks decreases for decreasing fields, until
they merge into a single peak as observed for a
conventional SRH generation mechanism (Fig. 5).
3.2. Comparison with the case of a single trap level
In Fig. 6 the results of the present model are
compared with those obtained by using the model of
Ref.[ 1] for the case of poly-Si (continuous DOS) and
for the case of single-crystal silicon, where a single
trap
level
at
mid-gap
is assumed
with
NT = 1 x 10” cm-3. The results for sinlge-crystal
silicon confirm the original conclusion in Ref.[ 1] that
the inclusion of the PF effect increases the generation
only by a maximum factor of 2. The situation is
markedly different in the case of a material with a
continuous DOS such as poly-Si, where the difference
is nearly one order of magnitude. Therefore the PF
effect must be taken into account to model this type
of device.
Also shown in Fig. 6 is the band-to-band tunnelling
contribution. This eventually dominates in singlecrystal silicon, where the defect density is less than
1 x 10’) crne3. This is not so in the case of poly-Si due
to the much higher density of traps.
8.0
7.0
6.0
5.0
a 4.0
3.0
2.0
1.0
0.0
O.OE+Ol.OE+7 2.OE+7 3.OE+7 4.OE+7 5.OE+7
metric field (V/m)
Fig. 7. GIvs electric field F for poly-Si DOS.
582
0. K. B. Lui and P. Migliorato
3.m
2‘
3
v/m
1.m v/o
OAR7v/m
- - - - -smohIionwmg add
l.OE-11
l.OE-12
-6.0 -5.0 -4.0 -3.0 -2.0 -1.0
0.0
Fig. 8. Leakage currents for Vc = -I, - 3, and - 5 V.. Vdr= 2.6 V. The corresponding electric fields
obtained via simulation are included. The simulation results obtained using the model of Ref.[l] are
compared with those obtained using the new model.
3.3. Comparison
with experiment
emission/capture steps which constitute an individual
generation/recombination
process, such an effect
The implementation
of the present model in a
must
always
be
taken
into
account
to ensure a correct
device simulator requires analytic approximations.
device
simulation.
Our
results
indicate
that, while the
These will be dealt with in a forthcoming paper. In
influence of the PF effect is relatively small (a factor
order to compare the present theory with the
of 2) in the case of single-crystal silicon, as pointed
experiment, we proceed in the following way. We
out
previously[l], it is much greater in the case of
calculate the ratio a = RT/Roirac for a range of values
polycrystalhne
silicon. This is the direct consequence
for F for the case of poly-Si. The results are shown
of the presence of a continuous density of states, as
in Fig. 7.
shown in this work. A very good agreement between
In Fig. 8 we plot a typical leakage current
theory
and experiment is obtained with our approach
characteristic, Id,exp
for poly-Si TFTs and that of Id,usac
in
the
case
of the anomalous off-current of poly-Si
simulated by a commercial device simulator based on
TFTs. The model has general validity and, like its
the model of Ref. [l] (Dirac centres only). For each
predecessor[l], is expected to be very successful in
V, the peak electric field near the drain is calculated
commercial
device simulators, expecially in those
in the simulation.
The leakage current Id,cOul,
including
thin-film
transistor models.
calculated by the present model (DiraccCoulombic
centres), is obtained by multiplying I.,uiracby the a
value corresponding to the appropriate electric field. Acknowledgements-The
authors wish to thank Dr T.
This calculation assumes that the dominant contriShimoda, Mr H. Ohshima and Mr M. Miyasaka of the
bution to RT comes from the region where F is Seiko Epson Base Technology Research Center, Suwa,
Japan, for providing the samples used for the off-current
maximum. Considering this approximation,
the
measurements. The authors also wish to thank Professor G.
agreement between theory and experiment can be Vincent, France Telecom CNET, Grenoble, for making
considered excellent.
available a copy of his doctoral dissertation. This work has
been supported by the Seiko Epson Corporation.
Note that in Fig. 8, the leakage current simulated
using the model of Ref.[ l] is nearly constant with V,,
in spite of the increase of RT with applied field, as
REFERENCES
shown in Fig. 6 (dashed line). This is due to the fact
that in the absence of a PF barrier lowering, the
1. Hurkx, G. A. M., Klaassen, D. B. M. and Knuvers, M.
P. G., IEEE Trans. Electron. Devices, 1992, ED-39,33 1.
field-induced generation is swamped by the thermal
2. Keldysh, L. V., Sov. Phys.-JETP, 1958, 33, 763; 1958,
generation, which occurs over a much larger volume.
4. CONCLUSION
We have presented a new G-R model for
trap-to-band phonon-assisted tunnelling inclusive of
the PF effect applicable in both forward and reverse
bias conditions. The model is based on the formalism
developed by Hurkx et a1.[18]. Since the PF reduction
of the tunnelling barrier is expected for one of the two
34, 665.
3. Kane, E. O., J. Appl. Phys., 1961, 32, 83.
4. Hurkx, G. A. M., Solid-St. Electron., 1989, 32, 665.
5. Bhattacharya, S. S., Banerjee, S. K., Nguyen, B. and
Tobin, P. J., IEEE Trans. Electron. Devices, 1994,
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lowering.
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n(x) = n, exp(v),
since FN is assumed to be a constant, we have:
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_-kT
_-kT
n, A&(x) exp(y).
x--t,
n, AE.(x)
x-11,
x exp
F, - E,(a)+ E,(x)kT
E,(x)
= n, AE.(x)
exp( v)
I=L1 kT x- I,
x exp( E’(x)&‘(a))
and
APPENDIX
Carrier density expression
We now obtain a compact expression for the carrier
density under the assumption that the quasi-Fermi levels are
constant.
The carrier density is given by:
n&x) = n(x)exp($)
x exp
where a new variable u = (x - a)/(x - t,) has been
introduced. Since -(x - t,) du = da, qF = AE.(x)/(x - t,)
and t, = x - (AEr,/AE.)/(x - r,), substituting into eqn (Al)
we have:
+ [r (+$)...
4 J2rn.l [qF(x - a)]’
-3
qFfi
)
AEn
x exp mu-
K.u ‘a[
1 - (%>“‘I}
du.
(A2)