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III. POTENTIAL DISTRIBUTION AT SEMICONDUCTOR / ELECTROLYTE INTERFACE

III. POTENTIAL DISTRIBUTION AT SEMICONDUCTOR / ELECTROLYTE INTERFACE

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14



Kohei Uosaki and Hideaki Kita



1. Schottky Barrier

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. Open-circuit photovoltage for the a-Si: 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'-dimethyl-4,4'-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'-dimethyl-4,4'-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



(E0-ebVsc-EF)



(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 half-occupancy. 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. Open-circuit photovoltage for the n-GaAs, 64 p-GaAs, 64 and p-Si 67 in

CH 3 CN/[n-Bu 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 Mott-Schottky 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 Mott-Schottky plot is often considered to

be the evidence of the band-edge 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 Mott-Schottky 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 Mott-Schottky plot does not necessarily mean the existence

of band-edge 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 vibration-rotation 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,



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