Semiconductor Electrodes
VIII. Digital Simulation of Open-Circuit Photopotentials
Daniel Laserand Allen J. Bard*
D e p a r t m e n t of Chemistry, The University of Texas at Austin, Austin, Texas 78712
ABSTRACT
A simulation technique is described for the calculation of semiconductor
electrode properties at steady state, e.g., at e q u i l i b r i u m i n the d a r k or u n d e r
constant illumination. I n t e g r a t i o n of the c o n t i n u i t y equation with respect to
distance at the steady state yields a relation b e t w e e n the light flux a n d f r e e
carriers, which can be used i n a recursion relation to d e t e r m i n e the free
carrier concentrations and the electric field w i t h i n the space charge region
of the semiconductor electrode. The technique is used to calculate the Boltzm a n n d i s t r i b u t i o n w i t h i n the semiconductor electrode and to d e t e r m i n e the
photopotential i n the absence of faradaic c u r r e n t a n d surface states.
I n a previous p a p e r i n this series (1) we described
a digital s i m u l a t i o n method for semiconductor electrodes and the formation of the space charge region
i n a semiconductor electrode u p o n charge injection. The
m a i n m o t i v a t i o n for these simulations has been the
recent interest i n semiconductor electrodes and their
application to photoelectrochemical cells and devices
(2). One characteristic of interest is the change i n
surface potential of a semiconductor/electrolyte i n t e r face u n d e r illumination, the photopotential. Although
theoretical treatments of the photopotential have been
given (3,4), and its m a g n i t u d e related to the semiconductor properties, surface potential, and light i n tensity, these t r e a t m e n t s u s u a l l y i n v o l v e restrictive
conditions, (e.g., total light absorption i n the space
charge region, m i n o r i t y carrier diffusion length m u c h
larger t h a n the size of the space Charge region). We describe here the digital s i m u l a t i o n of the steady-state
photopotential which arises u p o n i l l u m i n a t i o n of a
semiconductor electrode previously b r o u g h t to a given
potential and now held at open circuit. Charge t r a n s f e r
reactions, e.g., open-circuit corrosion, are assumed not
to occur d u r i n g this illumination.
Physical Model
The general equation for the processes w i t h i n a
semiconductor electrode u n d e r i l l u m i n a t i o n can be obtained by considering the creation of electron/hole
pairs by light absorption, their mass t r a n s f e r b y diffusion a n d migration, and their recombination. Thus the
change i n concentration of holes, p, at a given location
i n the semiconductor is given (in o n e - d i m e n s i o n a l
form) b y
Op/Ot = Ojp(x)/Ox + Ioae -ax
-
-
R(x)
s
R ( x ) d x -- Io(1 -- e -ax)
s
R(x)dx
[4]
[5]
i.e., at s t e a d y state a c u r r e n t (which m a y be zero)
flows through the semiconductor phase. Equations [3]
and [4] are employed i n a digital or finite difference
form in the simulation. The procedure follows the
usual digital simulation approach (5). The semiconductor region of interest is divided into space elements
of width ~x which are assigned an index K, from
K ---- 1 (surface element) to K ---- K M A X (Fig. 1). The
carrier concentration w i t h i n each element is assumed
constant and represents the average value of PE and nK
for holes and electrons, respectively, at that location.
The electric field at the left b o u n d a r y (the solution
side) of element K, i.e., b e t w e e n it and e l e m e n t K -- 1,
is denoted EK and the carrier concentration at this
k-'!
k
k+l
I
Pk n k
Jm._~l
I
:%
[2]
f
:loe'~X
w h e r e jp(0) is the flux of holes crossing the interface,
or the hole c o n t r i b u t i o n to the faradaic c u r r e n t density,
ip. Thus
jp(X) : ip/e -- Io(1 -- e -ax) +
R(x)dx
3p(X) + 3n(X) = (ip + i n ) / e = i
where Io is the intensity of light incident on the semiconductor/electrolyte i n t e r f a c e (taken as x ---- 0), a is
the coefficient of light absorption of the semiconductor,
jp(X) is the total flux of holes at x, and R ( x ) the rate
of electron/hole recombination. We are concerned here
with the steady-state photopotential. Thus with Op/Ot
---- 0, i n t e g r a t i o n of Eq. [1] yields
j p ( x ) -- jp(O) =
z
where 3n (x) and in are the electron flux and the e l e c tron contribution to the faradaic c u r r e n t density, respectively. The light flux and electron flux are t a k e n
as positive going into the semiconductor from the
solution and the hole flux is positive going out of the
semiconductor to the solution. (A representation of the
fluxes is shown i n Fig. 1.) Equations [3] and [4] hold
at all x, both w i t h i n and outside of the space charge
region. The net c u r r e n t density at a n y x, 6, is given
b y E q. [5]
k=l
[1]
~
3n(X) ----in/e -Jr Io(1 -- e - a x ) --
I
I
I
[3]
I
Ax
X:O
A similar expression can be w r i t t e n for electrons
* Electrochemical Society Active Member.
K e y w o r d s : s e m i c o n d u c t o r s , d i g i t a l s i m u l a t i o n , photopotentials,
photoeffects.
Fig. 1. Digital representation of the semiconductor phase and
flux notation.
1833
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1834
J. E l e c t r o c h e m . Soc.: E L E C T R O C H E M I C A L S C I E N C E A N D T E C H N O L O G Y
December
1976
Table I. Comparison of differential and finite difference notation
Quantity
Distance, x
Migration flux of holes, jp,m
Diffusional flux of holes, 5p.d
Electric field,
Differential notation
Finite difference notation
uvEp
V2UvEE (pK + px-1)
(K-
Dp ( 0 p / a x )
E
(1/eoer)
Recombination rate, R(x) (e~g., for
excess h o l e s in an n-type semiconductor) *
Krn(X)
~)hx
Dp ( p g - 1 -
Q(x)dx
(p(X) -- p ' q )
(eAX/eoe,)
(p~ -
pK)/AX
n ~ + ND --
Na)
K--O
( n ~ / r p n b ) (pK -- pKeq)
* Op(x)/Ot
= krn(x)p(x)
-- k~, w h e r e k , i s t h e recombination rate constant g i v e n b y 1 / r p n ~ a n d k~ is t h e f o r m a t i o n constant, g i v e n
by kr~eqp eq (the s u p e r s c r i p t eq d e n o t e s the equilibrium concentrations in t h e n-type s e r e . c o n d u c t o r ) . T h e equation in t h e table results
w h e n n eq ~ ~ t ( x ) ,
b o u n d a r y is t a k e n as the average of that of the two
adjacent elements [e.g., V2 (PK -5 PK--1)]- A comparison
of the differential and finite difference notation used
i n the simulation for expressing fluxes and other q u a n tities of interest is given in Table I.
Initial conditions. T h e B o l t z m a n n d i s t r i b u t i o n . - - T h e
d i s t r i b u t i o n of the carriers and the electric field in the
semiconductor biased to a k n o w n potential (thus cont a i n i n g a k n o w n excess charge) and at e q u i l i b r i u m in
the dark serves as the initial state preceding calculation of the photopotential. This e q u i l i b r i u m distribution
is of f u n d a m e n t a l importance i n u n d e r s t a n d i n g and
predicting the electrochemical behavior of semiconductor electrodes and is assumed to be essentially
obeyed even i n n o n e q u i l i b r i u m situations (4-6). This
distribution is u s u a l l y obtained by using F e r m i statistics for the occupancy of allowed energy states for
which e x p [ ( E - E F ) / k T ] < ~ I and which physically
means that at low occupancy, spin r e q u i r e m e n t s m a y
be relaxed. In addition, the Poisson equation must be
solved using charge density terms which are based on
the e q u i l i b r i u m distribution of carriers as functions of
a coordinate which is not yet explicitly k n o w n (7). I n
general, a closed-form explicit relation b e t w e e n the
potential and its gradient cannot be obtained for a
doped semiconductor (8). At e q u i l i b r i u m in the dark,
there is no faradaic c u r r e n t and no excess free carriers to give n o n e q u i l i b r i u m r e c o m b i n a t i o n effects.
Thus Eq. [3] and [4] yield
jp(X) = i n ( x ) ~- 0
(for a l l x )
[6]
and at the b o u n d a r y of each element, the migrational
flux is compensated by the diffusional one for both
electrons and holes. E q u a t i n g these fluxes using the
digital-form equations i n Table I and rearranging, we
obtain
( Dp/ ~ x ) -- O.5UpE K
PK =
PK--1
[7]
( D p / h X ) -5 O.5UpEK
nK ----
(Dn/AX) -5 0.5UnEK
(Dn/AX) -- 0.SUnEK
nK--1
[8]
Equations [7] and [8] are used as recursion relations i n
a n iterative computation b e g i n n i n g with the second
element (K ---7- 2). For the first element (the semiconductor surface) the b o u n d a r y conditions are (9)
Pl ~- po exp ( e V s / k T )
[9a]
nl ----n ~ exp ( - - e V s / k T )
[9b]
where V~ is the applied surface potential governing the
distribution. The U and D values i n Eq. [7] and [8] are
related to each other via the Einstein relation
U ( c m 2 s e c - 1 V -1) _-- D / k T ----39D (cm 2 sec -x) [10]
at room temperature. A recursion relation for the
electric field, E~:, which takes account of the semiconductor properties (dielectric constant, doping level)
is obtained from Gauss' l a w
EK --~ EK+I -5 (eAX/eoer) (PK ~ nK -J- ND -- NA) [11]
The b o u n d a r y condition used with this relation is that
in the b u l k semiconductor (K -----K M A X ) , EK ---- O. The
simulation proceeds b y using the applied potential i n
the b o u n d a r y condition, Eq. [9], a n d t h e n calculating
nK and PK (K ~ 1 to KMAX) assuming a n y a r b i t r a r y
initial distribution (usually t a k e n as a u n i f o r m one,
i.e., a flatband condition). The EK values are t h e n calculated, using Eq. [11]. A l t e r n a t e calculations of nK,
PK, and EK are continued u n t i l the three arrays are
constant with respect to f u r t h e r iterations. The resulting values, besides satisfying Eq. [7]-[11], also show
the following features: (i) a n u m e r i c a l integration
over the electric field from the b u l k to the surface of
the semiconductor yields the surface potential governing the distribution (which enters into the s i m u l a tion only i n assigning Pl and n l ) , thus d e m o n s t r a t i n g
self-consistency in Vs; (it) the product pKnK is constant and equal to ni 2 (e.g., for G e n i 2 w_ 6.25 >< 1016
cm -6 at room t e m p e r a t u r e ) for all K.
A problem arises in the selection of KMAX. If this
value is too large, the simulation does not converge to
a constant solution. Since the v a l u e of KMAX, r e p r e senting the thickness of the space charge region, is not
k n o w n in advance, an a r b i t r a r y value which will yield
a solution is chosen and w h e n a convergent solution is
obtained, K M A X is increased. The calculation t e r m i nates with the highest value of K M A X which still
gives a constant solution. Inspection of the result shows
that only the very diffuse part of the space charge region, which contributes insignificantly ( < 1 % ) to the
electrical state (fields, potentials) of the semiconductor
cannot be displayed.
Note that i n contrast to our previous s i m u l a t i o n of
space charge region formation in a semiconductor following charge injection which portrayed the time dependence of the fields and concentration profiles (1),
the method employed here derives only the e q u i l i b r i u m
properties at a given potential. The concept of time is
omitted and the i n t e r m e d i a t e results have no physical
meaning. Typical e q u i l i b r i u m concentration profiles of
the mobile carriers in intrinsic and n - t y p e Ge at several surface potentials obtained by the simulation are
shown in Fig. 2-4. Figures 2 and 3 illustrate the final
e q u i l i b r i u m situation u n d e r the same conditions as the
relaxation results given previously (1). A comparison
of the surface potential and space charge layer thickness (or Debye length) for a highly doped semiconductor obtained by the approximate "depletion layer"
treatment, which underlies the Schottky-Mott plot
(10), and our calculation is given i n T a b l e II. Note that
the approximation becomes less applicable at low potentials w h e n the concentration of the existing carriers
i n the space charge region cannot be neglected. Other
related properties of the space charge at equilibrium,
e.g., surface conductivity and capacitance, can be calculated as well using the simulation results. The space
charge region capacitance is obtained as the additive
contribution of the i n d i v i d u a l space charge elements
connected in series, each having a capacitance of CK =
q K / E K a x where qK is the charge in element K. The
surface conductivity may be deduced b y considering
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Vol. 123, No. 12
SEMICONDUCTOR ELECTRODES
1835
1.37E1B
4.93 14
1.26E17
2.08 14
1.74E16
8.60 13
t~ 2.50 13
u
7
1.12 13
E
u~
U
3.03 ~2
U
1.1o 12
4.58 11
i
>
3l
i
31
Distance
i
/
i
i
1J2
115
Fig. 2. Carrier concentration at equilibrium for intrinsic Ge,
1 0 0 m V ; p ~ : n ~ : 2.5 X 1013cm-3; er : 16 esu.
:
~.34E10
xlo 3 / c m
Vs
I
4,95E g
4,58E 8
2.50e 16
I
L
I
e
2
I
I
t
10
Distance
x~o e
/em
Fig. 4. Carrier concentration at equilibrium far n-type Ge (sample
as in Fig. 2). Vs ---- --100 mV.
2,03E 15
5,50E14
~
ductor b u l k boundary. The dimensions of the space
charge region will change to accommodate these shifts
i n carrier concentration. The total charge, and hence
the surface field, will r e m a i n constant, so that the
6.02e13
semiconductor space charge region can be compared
to a capacitor with a given charge which undergoes a
s.
d e c r e a s e in its w i d t h r e s u l t i n g in a n i n c r e a s e in c a p a c i t a n c e and thus a decrease i n the voltage drop across it
u n d e r i l l u m i n a t i o n . T h e s t e a d y s t a t e is c h a r a c t e r i z e d
3.8 0E12
Rt
f'~
1.14e12
b y the value of Es, the surface field, k n o w n from the
initial condition (Boltzmann distribution), and by the
l 3,08Ell
~P
2.50E10
)
5
Distance
10
15
x ~oe/cm
Fig. 3. Carrier concentration at equilibrium for n-type Ge. Vs
250 mV; n ~ = 2.5 • 1016; p~ = 2.5 • 101~ er = 16 esu.
the elements as r e p r e s e n t i n g resistors connected in
parallel, each with a conductivity of ehx(UnnK
UppK) .
The photopotential ef]ect.--Assume the initially
biased semiconductor (now at open circuit) is i l l u m i nated with a constant light intensity. We now calculate
the open-circuit, steady-state photopotential using an
iterative procedure similar to that just described. A
t r e a t m e n t of the t i m e - d e p e n d e n t relaxation of carrier
concentration, field, and surface potential u n d e r closedcircuit conditions is treated in the next paper in this
series (11). The process at open circuit is a coulostatic
one; charge in the semiconductor is conserved and the
effect of light is simply to r e a r r a n g e the concentration
and field profiles w i t h i n the semiconductor. (The same
would hold true even w h e n a faradaic c u r r e n t flows, if
the n u m b e r of holes crossing the interface is balanced
by electrons extracted at the semiconductor ohmic contact or vice versa.) Consider an n - t y p e semiconductor
biased at a positive potential with respect to the soluion. U n d e r i l l u m i n a t i o n the photogenerated holes will
accumulate at the semiconductor surface and the electrons will move to the space charge r e g i o n / s e m i c o n -
"minority injection level," here, the hole concentration i m m e d i a t e l y beyond the space charge region,
which is directly proportional to i l l u m i n a t i o n intensity.
Analytical procedures exist for the rigorous d e t e r m i n a tion of the injection level of m i n o r i t y carriers (here,
holes) (12). For Ge, one can assume the existence of
a diffusion layer, Lp, t h r o u g h which excess holes diffuse
into the b u l k (where Lp _-- ~/DpTp). Lp is much wider
t h a n the space charge region thickness (L1) and if light
is a b s o r b e d m a i n l y i n t h e s p a c e c h a r g e r e g i o n , t h e e x p r e s s i o n o b t a i n e d f o r t h e f l u x of h o l e s a t t h e s p a c e
c h a r g e r e g i o n / b u l k b o u n d a r y ( x ~ Lz) is (3)
jp(Ll) :Dp(0(Ap)/0x)z=Lz-~Dp(pLI--p~
[12]
Table II. Comparison of the simulated electrical properties of a
large bandgap semiconductor electrode (n-type TiO2) and the
electrostatic approximation of the depletion layer*
Depletion layer
Simulated
approximation
LI~* *
Q,
Es,
Lz,
C/cm ~
V/cm
cm
V.,
1.56
0.100
0.200
(•
107)
1.485
2.243
3.249
4.001
4.675
5.166
5.752
6.271
(X i0 -i)
1.677
2.533
3.670
4.519
5.774
5.635
6.496
7.083
( x 105)
1.92
2.36
2.76
3.16
3.36
3.76
4.00
V
0.393
0.585
0.777
0.963
1.168
1.408
cm
V,,***
0.928
1.40
0.078
0.178
2.50
2.89
3.23
3.595
3.91
0.565
0.755
0.943
1.168
1.388
( x i0 s)
2.03
V
0.373
* Assumed to consist of only immobile donors; ND = l0 l~
cm-S; er = 100 esu.
** L1 = Q / 1 . N D (depletion layer w i d t h ) .
*** V , = E s / 2 L z ( E , , surface field; V , , s u r f a c e p o t e n t i a l ) .
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1836
SEMICONDUCTOR ELECTRODES
S?
R(x)dx
-
because in this region e l e c t r o n e u t r a l i t y essentially
holds (13). I n more complicated cases, w h e n Lp and L1
are comparable i n magnitude, light is absorbed outside
the space charge region as well and linearization of
the diffusion l a y e r is unjustified. I n this case a s i m u l a tion involving a futler t r e a t m e n t of the diffusion layer
can be u n d e r t a k e n to find the injection level.
PL1 and riLl serve as b o u n d a r y conditions for the
recursion formula b e t w e e n the elements, this time
taken from the space charge region/semiconductor b u l k
b o u n d a r y to the surface which at the steady state, according to Eq. [3] and [4] and Table I, is given by
L ( K ) -- ~ R ( K )
PK ~---
+ [0.5 UpEK+I + (Dp/AX) ]" PK+I
( Dp/AX),-- 0.5 U p E K + 1
[15a]
L ( K ) -- XR (K) -- [0.5 UnEK+I - - (Dn/&X) ]" T~K+I
~K
--
i
[13]
Equations [12] a n d [13] allow the assignment of a n
initial value to PL~ (assuming at first no recombination
i n the space charge region) which is modified d u r i n g
the simulation due to the recombination term, using
the given values of Io, a, and the semiconductor p r o p erties (DpTp). The e q u i v a l e n t expression for the electrons is
r i l l : n ~ "1- ( P L 1
pO)
[14:]
-
1976
4,12E16
F r o m Eq. [3] this same flux is given b y
jp(L1) = - - I o ( 1 - e - a ~ ) +
December
( D n / A x ) - - 0.5 UnEK + 1
J////~
e
-
3.96E15 3.80E14
•OE3,66E13
?,
3,51E12
gO
//
\
/
\
\
"7.
O 3.37Ell
\
x
T2.eoE10
I
[
>
I
I
I
a.e
1
Distance
x 105 / c m
I
I
1,5
Fig. 5. Carrier concentration at steady state with constant illumination for n-type Ge (sample as in Fig. 3). Io ----- 1016 photons/
cm2-sec, Vs (dark) = 250 mV. The dashed lines show the equilibrium concentration in the dark.
1el
612124~
[15b]
where
L ( K ) = Io(1 -- e -a(K-v=)~)
[cm-2sec-1]
[16]
E
-K=I
-gp
~x
[cm-Ssec -1]
%0
[17]
The electric fields are calculated with Eq. [11], this
time proceeding from the surface to the b u l k because
the surface field is known.
Thus, starting with the semiconductor in the dark
at a surface potential Vs, the condition u n d e r i l l u m i n a tion is simulated b y repetitively using Eq. [12]-[17]
u n t i l three n e w constant arrays (holes, electrons, and
electric field) are obtained. A n u m e r i c a l integration
over the fields yields a new surface potential, Vs',
where the photopotential, AV, is Vs -- Ys'.
Results
The simulated distribution of carriers with and w i t h out i l l u m i n a t i o n is shown in Fig. 5; Fig. 6 compares the
electric fields u n d e r these conditions. The dependence
of the photopotential on the e q u i l i b r i u m surface p o t e n tial which exists before i l l u m i n a t i o n obtained by the
simulation compared to the calculation method of
Johnson (4) is given in Fig. 7 and the relation b e t w e e n
the photopotential and the i l l u m i n a t i o n i n t e n s i t y obtained by these two methods is shown in Fig. 8. J o h n son's method of obtaining the photopotential u s u a l l y
uses the assumption that the Bo]tzmann distribution
and the same analytical expression relating the potential and the field for the semiconductor holds both in
the dark and as well as in the light (3, 4). This assumption was checked b y the digital simulation and indeed
we find that Eq. [18] and [19] hold
Ps' ~'~ PLI' exp ( e V s ' / k T )
ns' ~ n ~ exp ( e V s ' / k T )
(riLl' ~ n O)
[18]
[19]
where the primed quantities denote values u n d e r illumination. This assumption applies, as has been
pointed out previously (12), because there is fast t r a n s -
I
I
0.5-
)
I
l
I
I
,.s
DISTANCEX105/Cm.
I
~
I
2:s
I
Fig. 6. Electric fields for n-type Ge (sample as in Fig. 3): (a) in
the dark and (b) under constant illumination, Io ---- 1016 photons/
cm2-sec. Vs (dark) = 400 mV.
port w i t h i n the semiconductor phase. Thus only a very
slight imbalance b e t w e e n the diffusional and the m i grational fluxes (compared to their absolute m a g n i tude) has to exist to provide the n o n e q u i l i b r i u m flux
which corresponds to moderate illumination. Hence, in
practice, even u n d e r illumination, the carriers and
electrical field will be distributed in such a w a y that
uiniEi ~ D i ( a n i / a x ) , which leads to the same f u n c tional relation as in the dark.
Acknowledgment
The support of this research by the National Science
F o u n d a t i o n (MPS74-23210) is gratefully acknowledged.
Manuscript submitted Feb. 3, 1976; revised m a n u script received Feb. 7, 1976.
A n y discussion of this paper will appear in a Discussion Section to be published i n the J u n e 1977 JOURNAL.
All discussions for the J u n e 1977 Discussion Section
should be submitted b y Feb. 1, 1977.
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VoL I23, No. I2
SEMICONDUCTOR
1837
ELECTRODES
160
//
150
~~
/
120
.~E
so
z
60
/
/
/
./
80
0
/
40
I
1o15
W
I,-
> ~
o
o
o
"l-
I "
100
>
I
200
POTENTIAL
I
I
300
400
/mV
[
500
Fig. 7. The dependence of the photopotential for n-type Ge
(sample as in Fig. 3) on dark surface potential Vs;
Io
=
i
lO TM
/Phot.I cmysecl
5
1017
5
10 I s
Fig. 8. Dependence of photopotential for n-type Ge (sample as
in Fig. 2) on illumination intensity. Vs (dark) = 250 mV. Points
are simulations and the dashed line was calculated by Johnson's
method (4) assuming that the injection level was proportional to
the light intensity and was 1.3 • 1013 cm - ~ at an illumination
intensity of 10te photons/cm2-sec.
30
I,,,
a,
I
5
1016
photons/cm2-sec. Dashed line, calculated according to Johnson's
method (4) assuming PL1 = 1.3 X 1013 cm -'~ (see text).
6.
7.
Publication costs of this article were assisted by The
University of Texas at Austin.
8.
REFERENCES
1. D. L a s e r and A. J. Bard, This Journal, 1'23, 1828
(1976).
2. M. D. Archer, J. Appl. EIectrochem., 5, 17 (1975).
3. C. G. B. G a r r e t t and W. H. Brattain, Phys. Rev.,
99, 376 (1955).
4. E. O. Johnson, ibid., 111, 153 (1958).
5. S. W. F e l d b e r g , in " E l e c t r o a n a l y t i c a l Chemistry,"
9.
10.
11.
12.
13.
Vol. 3, A. J. Bard, Editor, chap. 4, Marcel Dekker,
Inc., New York (1969).
H. Gerischer, This Journal, 113, 1174 (1966).
V. A. M y a m l i n and Yu. V. Pleskov, "Electroc h e m i s t r y of Semiconductors," pp. 30-50, P l e n u m
Press, New Y o r k (1967).
A. M a n y , Y. Goldstein, and N. B. Graver, " S e m i conductor Surfaces," chap. 4, J o h n W i l e y & Sons,
Inc., New Y o r k (1965).
V. A. M y a m l i n and Yu. V. Pleskov, op. cir., p. 18.
Ibid., p. 55.
D. Laser and A. J. Bard, This Journal, 123, 1837
(1976).
A. R o t h w a r p and K. W. BSer, "Progress in Solid
State Chemistry," Vol. 10, p. 71, P e r g a m o n Press,
.Oxford (1975).
V. A. M y a m l i n and Yu. V. Pleskov, op. cir., p. 182.
Semiconductor Electrodes
IX. Digital Simulation of the Relaxation of Photogenerated Free Carriers
and Photocurrents
Daniel Laserand Allen J. Bard*
Department of Chemistry, The University of Texas at Austin, Austin, Texas 78712
ABSTRACT
A digital s i m u l a t i o n of the photoprocess at a semiconductor electrode is
described. The simulation m o d e l accounts for photogeneration, recombination,
and t r a n s p o r t of excess free carriers w i t h i n the semiconductor phase. The
origin of the p h o t o p o t e n t i a l in the absence of f a r a d a i c c u r r e n t is elucidated.
Q u a n t i t a t i v e c u r r e n t efficiency-potential curves for the p h o t o c u r r e n t s u n d e r a
v a r i e t y of conditions a r e calculated for n - t y p e TiO2 a n d these a r e c o m p a r e d to
e x p e r i m e n t a l results.
I n p r e v i o u s p a p e r s in this series we have i n t r o d u c e d
the use of digital s i m u l a t i o n methods for the t r e a t m e n t
of semiconductor electrodes. I n Ref. (1) t h e r e l a x a t i o n
of free carriers following charge injection, w i t h and
w i t h o u t surface states, was described. In Ref. (2) a
m e t h o d of d e r i v i n g the semiconductor electrode c h a r acteristics, e q u i l i b r i u m or s t e a d y state, at open circuit
in the d a r k or u n d e r constant illumination, was p r e sented. W h e n a semiconductor electrode at e q u i l i b r i u m
and in Contact w i t h solution is illuminated, a certain
* Electrochemical Society Active Member.
Key words: semiconductors, digital simulation, photoelectrochemistry, photogalvanic cells.
time elapses before the photoeffects a r e observed. D u r ing this time a r e d i s t r i b u t i o n of free carriers and
charges in t h e electric field in t h e space charge region
occurs. [When the semiconductor e l e c t r o d e / s o l u t i o n
interface is blocked to charge transfer, the n e w distribution of free carriers in the space charge region u n d e r
i l l u m i n a t i o n will cause a change in the potential of the
electrode (the p h o t o p o t e n t i a l effect).] F r e q u e n t l y , i l l u m i n a t i o n of the electrode is a c c o m p a n i e d b y charge
t r a n s f e r to solution species and this gives rise to a
photocurrent. F o r example, i r r a d i a t i o n of n - t y p e TiO2
w i t h light of e n e r g y l a r g e r t h a n the b a n d g a p e n e r g y
will result in the o x i d a t i o n of w a t e r (3, 4), while
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