From Eq. (3), the Galvani potential difference between the semiconductor and the metal, A M ^ , is given by
A s ^ = (
(25)
where O sc is the work function of the semiconductor. This potential
difference can be also written as
(26)
where A Vsc is the potential difference developed within the semiconductor which is equivalent to the Schottky barrier height, B,56 and
AV/ is the potential difference developed at the interface due to
the surface states. The effect of the metal work function on the
barrier height or A Vsc is easily understood from Eqs. (25) and (26)
as
<*(eAVsc)
djeHV,)
+
d
M
W
_
9
)
1
When the surface state concentration is low, the variation of work
function of the contact metal, 4>M, does not affect the interfacial
potential drop, AVj, i.e., d(eAV7)/dM = 0. In this case,
d(ehVsc)/d<&M = 1. This means that the change of the work function affects only the potential drop within the semiconductor, and
the energy of the band edge at the surface before and after the
contact is constant with respect to the Fermi level of the contact
metal (band edge pinning). On the other hand, if the amount of
the surface state is large and d(eA V/)/d3>M # 0, this simple model
does not apply. At the extreme situation, M does not affect A Vsc
at all, i.e., d\ekVsc)/d$M
= Q. This is the situation where thechange of the work function affects only the potential drop at the
interface and the potential drop within the semiconductor is
independent of the contact metal. Thus, the presence of a high
density of surface states on the semiconductor usually "pins" the
Fermi level at the surface,t resulting in the formation of a Schottky
barrier even before the contact of the semiconductor with a metal.
t This situation is sometimes called a "Fermi level pinning."
Theoretical Aspects of Semiconductor Electrochemistry
15
When the semiconductor/metal contact is formed, any charge
needed to achieve the equilibrium of the Fermi levels can be
supplied mainly from the surface states without shifting the pinning
of the Fermi level at the interface with respect to the band edges.57
Figure 6 shows the barrier height, <£B, i.e., eAVsc, at semiconductor/metal junctions as a function of electronegativity (work function) of the contact metal.58 In the case of ZnS, the barrier height
correlates linearly with the electronegativity of the metal but almost
constant barrier height is shown for GaAs/metal interfaces. This
is known to be due to the high concentration of surface states at a
GaAs surface,58'59 which is mainly associated with an As deficit.60
2. Effect of the Redox Potential on the Potential Drop
in the Semiconductor at the Semiconductor/Electrolyte Interfaces
The Galvani potential difference between the semiconductor bulk
and the electrolyte bulk, As c ^, is given by Eq. (28) in analogy to
Eq. (25) and to Eq. (26)50:
= Vredox(vac. scale)  O s c /e
(28)
Figure 6. Barrier height of
(electronegativity controlled)
GaAs (surface state controlled)
function of electronegativity of
tact metals.57
ZnS
and
as a
con
1.0
2.0
ELECTRONEGATIVITY
16
Kohei Uosaki and Hideaki Kita
where A VH and A VG are the potential drop in the Helmholtz layer
and that in the Gouy layer, respectively [cf. Eq. (26)]. A VG can be
minimized by using sufficiently concentrated solutions where most
of the solution charge is squeezed into the Helmholtz plane and
little charge is scattered diffusely into the solution.21'61 In most
experiments, a very high concentration of electrolyte is added, and,
therefore, AVG is neglected in the following discussions.50 Then
Eqs. (27) and (28) become
Assc* = Vredox(vac. scale)  O s c /e
= AVSC + AV7 + AVH
(29)
As one can easily recognize by comparing Eqs. (26) and (29), the
one big difference between the semiconductor/electrolyte and the
semiconductor/metal interfaces is the existence of the potential
drop in the Helmholtz layer, A VH, even if A VG can be neglected.
When one considers the effect of the redox potential on A Vsc, it is
better to use Vredox(ref. scale) rather than Vredox(vac. scale) since
only the former values can be obtained directly by experiments.
From Eqs. (7) and (9)
Vrredox(ref. scale) = Vredox(vac. scale) + const.
(30)
From Eqs. (29) and (30),
. scale)
dVndox(ref. scale)
d
f
4>sc
T
V' reaoxv
***" scale)
"""**v/  —
redox (vac.
.scale)L
e
! \
(31)
(i) Ideal Situation
As is the case at a metal/semiconductor interface, when the
surface state density is very low and the carrier density of the
semiconductor is not too high,
^ r e d o x ( r e f . scale)
Theoretical Aspects of Semiconductor Electrochemistry
17
and
dVredox(ref. scale)
(Ref. 62).
In this case, the semiconductor/electrolyte interface behaves
as the Schottky barrier. Thus, A Vsc gives the barrier height in the
semiconductor and should be equal to the photovoltage, Vph.
Therefore, the photovoltage should vary with the redox potential
following
dVtPh
•= 1
dVredox(ref. scale)
One experimental result proving this relation is shown in Fig. 7.63
(ii) Fermi Level Pinning
With few exceptions, most of the papers published before 1980
treated the semiconductor/electrolyte interface as an ideal Schottky
barrier and relations similar to Fig. 7 were often presented and
used to predict the photovoltage of photoelectrochemical cells.
Recently, however, many experimental results which do not follow
the above prediction were reported 6470 and it became clear that
the other terms in Eq. (31), i.e.,
—
and
^
f. scale)
dVredox(ref. scale)
should be taken into account, although the importance of these
terms was stressed before.3 Bard et al.71 considered the importance
of the surface states on the potential distribution at the semiconductor/electrolyte interface and applied the concept of Fermi level
pinning developed for a semiconductor/metal interface to this
system. Qualitatively Fermi level pinning is important when the
charge in the surface states, qss, is appreciably larger than that in
the space charge region, qsc.71 A qsc value can be calculated as a
function of carrier level, N, and A Vsc from the equation7'72
qsc = (2kTniee0)l/2F(\,y)
y
F(\,y) = [X(e~  1) + A" V
\ = nJN,
(32)
l
 1) + (A  \~ )y]
l/2
y = eAVJkT
(33)
(34)
18
Kohei Uosaki and Hideaki Kita
0.4
0.0
Vredox/V VS. SCE
Figure 7. Opencircuit photovoltage for the aSi: H/CH 3 OH interface as a function
of solution redox potential. 63 The redox couples employed are N,N'dimethyl4,4'bipyridinium +/0 (0.95 V); N,7V'dimethyl4,4'bipyridinium 2+/+ (0.45 V);
[77 5 C 5 (CH 3 ) 5 ] 2 Fe +/0
(0.21 V);
N,JV,N',JV'tetramethylphenylenediamine +/0
(—0.11 V); l,l'dimethylferrocene +/0 (+0.10 V); ferrocene +/0 (+0.20 V); acetylferrocene +/0 (+0.39 V). All determinations were in 1.0 M LiClO4 except for the
N,N'dimethyl4,4'bipyridinium system, which required 1.0 M KC1 for solubility.
Measurements are the result of several independent determinations for a number of
anodes.
where k is the Boltzmann constant, nt is the intrinsic carrier density,
e is the dielectric constant of the semiconductor, and e0 is the
permittivity of free space. Also, qss can be correlated with A Vsc but
the form of the equation depends on the nature of the distribution
of the surface state energies, i.e., whether the surface states are
uniformly distributed in energy or are localized at a single energy
level. In the case of uniformly distributed acceptor surface states
with centering energy of Eo, qss is given by
(E0ebVscEF)
(35)
On the other hand, in the case of acceptor surface states at a single
energy level Ess, qss is given by
exp[(£ ss 
 EF)/kT]
(36)
where gss is the degeneracy of the energy level. The total charge in
the electrolyte, qel, is equal in magnitude to gss + gsc. By assuming
19
Theoretical Aspects of Semiconductor Electrochemistry
qcX is arranged at a distance d from the electrode surface, A VH is
given by
(37)
where eH is the dielectric constant in the Helmholtz layer. When
the concentration of surface states is very high so that qss becomes
larger than qsc at a given A Vsc, Fermi level pinning by surface states
occurs. In this case, even before the contact with an electrolyte
solution, band bending within the semiconductor occurs (qei = 0,
qsc = qss)71 and the charge required to attain an equilibrium with
a redox couple upon contact is provided by the surface states. This
situation is quite similar to the metal/semiconductor junctions with
a high density of surface states discussed before. Bard et al.71
calculated qsc and qss to demonstrate the conditions required for
this situation with values of n, = 1.3 x 106cnT3, e = 12, kT =
0.0257 eV and with the assumption of half occupancy of Nss (Fig.
8). One must note that when Nss 71 becomes larger than ca. 1012 cm"2,
which represents only ~ 1 % surface coverage, qss is greater than
qsc for any moderately doped semiconductors, leading to Fermi
level pinning. Experimental results at several semiconductor elec
10"
10"
1
10* o
O
Figure 8. Surface charge in an ntype semiconductor: (a) space
charge, qsc, at various doping levels
(N) at A Vsc of 0.3 and 1.0 V; (b)
surface state charge (qss) as a function of surface state density, 7VSS,
assuming halfoccupancy. Potential drop across Helmholtz layer
due to semiconductor charge can
be calculated by Eq. (37).
10"
1
S
1O'
1
10'
4
1
10'
s
1
10' 
10° <%\
cr 10"' 
o
/
o
io«
cr
3

10"
i
1
10"
10" 10"
N/crrf
10"
10
i
17
20
Kohei Uosaki and Hideaki Kita
10
0
Redox Potentlal/V vs. SCE
Figure 9. Opencircuit photovoltage for the nGaAs, 64 pGaAs, 64 and pSi 67 in
CH 3 CN/[nBu 4 N]CIO 4 solutions as a function of solution redox potential.
trodes have shown that this is a rather common situation.64"70 Some
typical results of Fermi level pinning are shown in Fig. 9.
Gerischer73 and Nozik74 have stated that the formation of an
inversion layer can also cause a shift of band edges. For small
bandgap semiconductors, this effect may be more important. The
formation of an inversion layer has been shown experimentally at
several semiconductor/electrolyte interfaces.75"78
3. Distribution of Externally Applied Potential at the
Semiconductor/Electrolyte Interfaces
(i) Surface States of Free Semiconductors
So far only the effect of the surface states is considered, but
if the carrier density is high, even when the surface state density is
low, the potential drop in the Helmholtz layer may be more significant than that in the semiconductor.
Uosaki and Kita calculated A V H /(A Vsc + AVH) as a function
of total potential change, A Vsc + A VH, by using Eqs. (32)(34) and
(37) and demonstrated that the contribution of A VH is surprisingly
high79 (Fig. 10).t
t Genscher demonstrated the effect of the dielectric constant of the semiconductor
on[AV H /(AV s c
by using the MottSchottky approximation. The comparison is
made between AVH/(AVSC + AVH) calculated by using Eqs. (37)
and (38) and that obtained by using Eqs. (32)(34) and (37) in Fig.
11. When the potential change, (AVSC + AV H ), is relatively large,
the two calculations give similar results, but the difference is significant when (AV SC +AV H ) < 0.15 V.79
The linearity of the MottSchottky plot is often considered to
be the evidence of the bandedge pinning. 82 The measured capacitance, C, can be written as
C
1
= C" 1 + C~Hl
(39)
Figure 11. Dependence of the ratio of the potential change in the Helmholtz layer
to the total potential change and of C~2 as a function of total potential change. 79
e = 173. Carrier density is 1019 cnT 3 . (1) A VH/(A VH + A V J vs. AVH + AVSC calculated by using Eqs. (32)(34) and (37). (2) As curve 1 but calculated by using Eqs.
(37) and (38). (3) C" 2 vs. A VH + A Vsc calculated by using Eqs. (32)(34), (37), (39),
and (40) and C H = 10 juFcm" 2 . (4) As curve 3 but C H = oo.
where CH and C sc t are the differential capacitance of the Helmholtz
layer and the semiconductor, respectively, and the electrode potential of the semiconductor electrode with respect to the flat band
potential can be written as
V
VFB = AVSC + AVH
(40)
The effect of CH on the MottSchottky plots obtained by using
Eqs. (32)(34), (37), (39), and (40) is shown in Fig. 11. The plots
are linear when potential change is large but curved when (A Vsc +
A VH) < ca. 0.3 V. The slope of the linear portion of the relation is
almost identical whether A VH is neglected or not. Thus, the linearity
of the MottSchottky plot does not necessarily mean the existence
of bandedge pinning.
Theoretical Aspects of Semiconductor Electrochemistry
23
(H) Effect of Surface States
The argument developed in Section III.2 can be applied to the
potential distribution when an external potential is applied to the
semiconductor/electrolyte interface.71 Thus, at a semiconductor
with a high density of surface states, the variation of the potential
between the semiconductor and the solution leads to a relatively
constant A Vsc and a significant change in A VH. Green 31 ' 83 has shown
that if the surface state density becomes very high, the behavior of
the semiconductor becomes that of a metal.
IV. DISTRIBUTION OF ENERGY STATES FOR IONS OF
REDOX SYSTEM IN SOLUTION
1. Importance of the Energy Levels of Redox Couples
Usually it is thought that electron transfer can take place only
between electronic energy states of equal energies, one being
occupied and the other vacant.5'61'84'85 Gurney 86 applied this concept
for the electrode kinetics at a metal electrode and obtained the
following rate expression:
I
N(E)W(E)D(E)dE
(41)
where N(E), W(E), and D(E) are the number of electrons in the
metal, tunneling probability, and the number of acceptor states in
the solution, respectively, between E and E + dE. Gerischer extended this equation for electrode reactions at semiconductor electrodes in the dark. Thus, the total current, i, is devided into two
fractions, namely the conduction band current, /CB, and the valence
band current, JVB These currents can be written as follows8:
I
ICB = ice  «"CB = W l 
N'(E) W(E)DO(E) dE
J Ec
N(E)W(E)DR(E)
dE (42)
24
Kohei Uosaki and Hideaki Kita
*VB = WB " iwB =
fcve4
JJo
N'£( W) W(E)DO(E) dE
 kWBc% I
Jo
Jo
N(E) W(E)DR(E) dE (43)
where fCB and fCB are the anodic and cathodic component of the
conduction band current, the rate constants of which are icCB and
fcCB, respectively. Similar notation is used for the valence band
current. N(E), N'(E), W(E), DO(E)9 and DR(E) represent the
number of occupied and unoccupied states in the semiconductor,
tunneling probability, and the distribution of occupied (reduced)
and of unoccupied (oxidized) state of redox couple, respectively,
between energy E and E + dE. E is taken as zero at the bottom
of the valence band, cR and c% are the surface concentrations of
reduced and oxidized species, respectively.
In Sections II and III we correlated the electronic energy level
of the solid phase, EF, and that of electrolyte, Vredox. This comparison is useful with regards to understanding the potential
difference between the two phases and the potential distribution
of this interface. However, to describe the kinetics of the electron
transfer reactions at the surface, it is necessary to know the energy
distributions of occupied and unoccupied levels of the semiconductor and those of the redox couple, as one can see from Eqs. (42)
and (43). It is well known87 that the energy distribution in a
semiconductor in the dark is described by Fermi statistics with a
function for density of states. The energy distributions of redox
couples are, however, still a very controversial issue.88 There are
essentially two major approaches to describe this problem. One
approach was originated by Gurney.86 In this model, the energy
distribution is considered to be due to the vibrationrotation interaction between the particular ion and solvent molecules. According
to the second model,61 the energy level of the ion fluctuates around
its most probable value due to the thermal fluctuation of solvent
dipoles.
2. Gurney's Model
Gurney86 developed a model for the neutralization of H+ and Cl~.
Let us first consider the H+ case. In vacuum the ionization potential,