- Email: [email protected]
electrolyte junctions—II. A comparison of the interfacial admittance
Ekcfrochimica
Pergamon PII: !30013-4686(!I6)oo266-6
Acfa, Vol. 42, No. 7, pp. 1135-I 141, 1997
Copyright 0 1997 Ekevier Science Ltd. Printed in Great Britain. All rights reserved 0013-4686/97 $17.00 + 0.00
Direct and surface state mediated electron transfer at semiconductor/electrolyte junctions-11. A comparison of the interfacial admittance D. Vanmaekelbergh Debye Institute,
Utrecht University,
P. 0. Box 80 000, 3508 TA Utrecht, The Netherlands
(Received 14 March 1996; accepted 24 June 1996) Abstract-The interfacial electrical admittance of a semiconductor electrode exchanging majority carriers with a simple redox system is calculated from basic assumptions presented in a previous paper. Direct exchange and surface state mediated exchange are considered. The coupling between the occupancy of the surface states and the distribution of the interfacial potential drop over the depletion and Helmholtz-layers is taken into account quantitatively using the fluctuating energy level model. The results show that direct and surface state mediated transfer can be distinguished on the basis of the electrical admittance. Some simple, but important, cases of surface state mediated exchange are reviewed in the framework of the model. The consequences of the coupling between the occupancy of the surface states and the distribution of the interfacial potential drop over depletion and Helmholtz-layer are discussed in detail. Copyright 0 1997 Elsevier Science Ltd
INTRODUCTION In this paper, the interfacial electrical admittance of a semiconductor electrode exchanging majority carriers with a simple redox system will be derived. The assumptions of the model were presented in part I, in which the steady-state results were also considered. It is assumed that majority carriers tunnel through the Helmholtz-layer isoenergetically, and that the electronic states of the redox system can be described by the fluctuating energy level model. Both direct exchange and exchange via surface states is considered. The effect of the relaxation of the occupancy of the surface states on the interfacial potential distribution and hence on the exchange rates is taken into account quantitatively. Measurement of the electrical impedance (or admittance) of junctions consisting of a semiconductor in contact with a metal or electrolyte allows one to investigate the interaction of free carriers in the semiconductor with discrete electronic levels, localized at the interface between semiconductor and metal or electrolyte, and with an energy in the forbidden gap. A harmonic modulation of the potential (with frequency o) of the semiconductor with respect to the other phase gives a harmonic current which is measured in the external circuit. The ratio of the harmonic current density to potential is called the
electrical admittance Y(w). Characteristically, the semiconductor is depleted of majority carriers at the interface. Modulation of the potential leads to modulation of the width of the depletion layer and of the concentration of majority carriers at the interface. If interface states do not play a role, Y(w) corresponds to the differential capacitance of the interface (predominantly the capacitance of the depletion layer), parallel to a differential resistance, which is determined by the rate of exchange of carriers between the semiconductor and the other phase. If interface states interact with the majority carriers, Y(w) contains an additional admittance with a time constant which is characteristic for the exchange of electrons between the majority carrier band and the interface states [ 1, 21.The density of the localized state and the rate constant of electron trapping can, in principle, be obtained from the additional admittance. On the other hand, the kinetics of electron exchange between a metal electrode and a redox system have been successfully investigated by means of the electrical admittance method [3-51. In the simple case of a single electron tunneling process between metal and redox system, the electrical admittance corresponds to the double layer capacitance, parallel to a differential resistance, from which the rate constant of electron exchange at equilibrium and the transfer coefficient can be obtained [4-51.
1135
1136
D. Vanmaekelbergh
Investigations of the semiconductor/electrolyte interface by means of electrical impedance measurements probably owe more to similar work on solid state devices than to electrochemical work with metal electrodes. In many studies the model of Nicollian and Goetzberger, [1, 2,6, 7, 81, which describes the interaction between surface states and the majority carrier band, was used as a basis for the interpretation of experimental results. In some papers (for instance [9-131) the exchange of electrons between surface states and the redox system was also taken into account. It has been recognized that localization of electrons in surface states may change the distribution of the interfacial potential drop A$ over the depletion- and Helmholtz-layers [12, 131. The effect of this was taken into account for the exchange rate between majority carrier band and surface states, but was neglected for the exchange rate between surface states and the redox system. An exception can be found in Chazalviel’s paper [12], where the exchange between surface states and the redox system is described by a Butler-Volmer type relationship. The coupling between the interfacial kinetics and the potential distribution over depletion and Helmholtz-layers has been analyzed in part I for steady-state conditions. The time resolved effect of a small modulation SA+ of the interfacial potential drop can be discussed qualitatively. In the case of direct transfer, the distribution of SA+ over the depletion and Helmholtz-layers depends on the relative values of C, and CH only, and is therefore time independent. Hence, the application of SA4 leads to an immediate change in the rate of electron transfer between the majority carrier band and the redox system. This results in a differential resistance in the interfacial admittance taking into account the immediate change of the Faradaic current due to 8A4. In the case of surface state mediated electron transfer, the distribution of the modulation 6A$ is, initially, determined by the values of C, and Cu. However, since the modulation of the electron exchange rate between surface states and majority carrier band is different from that between surface states and the redox system, the occupation of the surface states changes with time. This may affect, in turn, the distribution of SA4 over depletion and Helmholtz-layers. It can be concluded that the distribution of SA4 over depletion and Helmholtzlayer is time dependent and coupled to the interfacial kinetics. It will become clear that this results in a complex component in the interfacial admittance related to the relaxation of the occupancy of the surface states. Calculation of the interfacial admittance As in part I, we consider a three electrode electrochemical cell with an n-type semiconductor electrode as working electrode. For measurement of the electrical admittance, a sufficiently small harmonically modulated potential U(w) = U,,, exp(iwr) is
superimposed on the electrode potential U, and the harmonically modulated current density in the external circuit, j(m), is measured. The electrical admittance of the cell is j(w)/U(w). Modulation of the electrode potential leads to the modulation of the interfacial potential drop over the semiconductor/ electrolyte interface. The interfacial admittance of the semiconductor electrode, denoted in this paper as Y(w), is given by j(o)/A&o). The electrochemical cell is designed such that the impedance of the counter electrode is much smaller than that of the working electrode, which means that the potential of the counter electrode is not affected by the modulation U(o). From equation (21) of part I, it follows that the relationship between the measured admittance j(w)/U(w) and the interfacial admittance Y(w) of the working electrode is given by: @)/U(w)
= Y(o)/]1 + A&I
Y(w)l.
(1)
The measured admittance agrees with the interfacial admittance of the semiconductor working electrode if [Y(w)]-‘<
iWyHc%.
(2)
The parameter yu is defined by equation (26) of part I. The admittance EC(w) + ja(o)]/A4(w) due to electron exchange can be derived from:
+j4w)l/Ww)
L(o)
+
[WH(~)/W(~)IKKL
+jd/a&'HlA,,.
(3)
The partial derivates have been calculated in part I (equations (40-43)) using Fermi-statistics and the fluctuating energy level model. The distribution of the harmonically varying potential drop over the depletion and Helmholtz-layers is, for direct transfer, independent of the measuring frequency o, and determined by YH(see equation (26) of part I): A&(w)/A~(w)
=
YH = CH/(CH
WH(~)/&J(QJ)=
1-7~.
+ G)
(4)
It follows from equations (24) that the interfacial admittance can be represented by an electrical equivalent circuit consisting of the capacitance of the double layer given by YuCac, parallel to a Faradaic
Mediated electron transfer at semiconductor differential resistance RF (Fig. la). Substitution of equation (4) and equations (4Wl) of part I into equation (3) gives: R;’ =
(e/kBT)
YH) [(I
0.5
64
-
a&
The transfer coefficients GIand equations (42-43) of part I. equations (40,41) into equation be verified that R;’ is equal to
+ cd-jc)
1 }.
1137
/ ’
3’
o.4-
,p
2 z
’ 3’
/’ #’
0.3.
44 T 0.2~
{YH(-jr) + (1 -
electrolyte junctions-II ,
II
‘,
~~. 10’4,&
(5)
1 - t( are defined by By substitution of (39) of part I, it can dj/dA4.
IO‘*
10~’
100
10’
102
103
Figures 2 and 3(b) of part I account for surface state mediated exchange of majority carriers between a semiconductor and a redox system. The current density measured in the external circuit is given by equation (27) of part I. If the electrode potential is harmonically modulated, then dA&/dt = iWA&(O). Hence, it follows from equation (27) and from A&(w) + A&(w) = Am that the interfacial admittance can be written as:
t----II-l-
104
105
w [$~‘I
0.5
(b)
--.
,_-’
Surface state mediated transfer
Cb) ---j--i
lo’*cm’* 10” cn?
-
/ ‘,
c
o.4
2 : 44
0.3
T
,’
~.
,’
,’ /’
0.2 0.1
I j
-_
,’
h
I
7,
~---t
O.9 )
0.2
0.4
0.6
0.8
j
A
1.0
Fig. 2. (a) Complex plane representation of the harmonically varying potential drop over the Helmholtz-layer A&(W), divided by that over the semiconductor/electrolyte interface Ad(w), with the frequency w as a parameter (see equation (15)). The same conditions have been chosen as for Fig. 5 (part I). Surface state mediated electron transfer is considered. At equilibrium, the energy of the conduction band edge at the surface is 0.75 eV above the Fermi-level of the redox system. The surface states are located 0.75 eV below the conduction band edge. The plots pertain to different values of the effective surface state density (lo”, 1013, 1O“‘cm-‘). The results here pertain to the equilibrium potential. (b) Plot of the imaginary part of [A&(o)/A&o)] as a function of the frequency. It is seen that the characteristic frequency increases with increasing surface state density, due to the coupling between the surface state occupancy and the interfacial kinetics (see text).
Y(w) =
iWCH [WH(W)IA~(W)I
+
[ati:
+j,“)/~bbdw,
[WH((~YW(W)I. (6)
RL
L
Fig. 1. (a) Electrical equivalent representation of the interfacial admittance derived for a semiconductor electrode exchanging majority carriers with a simple redox system via direct tunneling of electrons through the Helmholtz-layer. C,: capacitance of the depletion layer in the semiconductor. CH: capacitance of the Helmholtz-layer. RF: differential Faradaic resistance accounting for direct majority carrier transfer. (b) Electrical equivalent representation of the interfacial admittance derived for a semiconductor electrode exchanging majority carriers with a simple redox system via surface states (see equation (16)). The (Rc, C) and (RL, f.) series connections account for the relaxation of the occupancy of the surface states. The Faradaic resistance is given by (R-l+ RF')-'.
Due to the relaxation of the occupancy of the surface states which is coupled to the interfacial kinetics, A&(w)/A~(w) is a complex quantity, in contrast to the case of direct transfer. From the combination of equations (20) and (27) of part I, and taking into account the fact that A&(W) + A&(w) = Am, it follows that: WH(W)/M(W) (1 -
= YH) + ?dkBT/e)
I--Nw)lW(w)l.
(7)
In equation (7) Q(w) stands for the harmonically modulated occupancy of the surface states. The parameter yS, defined by equation (55) of part I, describes the influence of the occupancy of the surface states on the distribution of the interfacial potential drop over depletion and Helmholtz-layers.
1138
D. Vanmaekelbergh
Equation (7) is the harmonically varying equivalent of equation (54) of part I, which pertains to the steady-state. The interfacial admittance differs essentially from that pertaining to direct transfer as a consequence of the fact that A&(w)/A&w) is not equal to (1 - mu), as it is for direct transfer, but contains a second (complex) component {y,(k~T/ e) [ -O(w)/A&o)]}, due to the relaxation of the occupancy of the surface states. This is shown by the substitution of equation (7) into equation (6):
+
Y~(~B T/e)
+
[-
xi:
[-
20
40
60
(8)
The partial derivatives in equation (8) were calculated in part I (equations (65,66) with jf + jf being equal to ~7 + ~7 and to the steady-state current density j). In order to calculate &T/e) [-O(w)/AO(w)], the function d@/dt described by equation (20) of part I must be developed as a total differential of 8, A&, and ASH. To shorten the notation the function dO/dt will be denoted as 0:
Equation (12) describes the relationship between the harmonically modulated occupancy of the surface states 6(w) and the modulation of the interfacial potential drop A#@), for a semiconductor electrode, described by the parameters Yu and yS, exchanging majority carriers with a simple redox system, the interfacial kinetics being described by r-l, [ -a0/ OA&]A.+H.~~and [O0/OA&]A&+ The latter two quantities are directly calculated from equation (20) of part I: (ke Tie) [ - a@/aA’#JsiA4H,e = -k/es
+ [O0/OA&]A+,,edA&.
(9)
+
[a@/ao]A4,,A4&+)
[aO/aL\~HiA~,.dA~H(O).
(10)
The partial derivative [O0/a&,,&+, is equal to -f-I, given by equation (50) of part I:
+
(h&)CRcd
+
61 +
(k/s)cOx.
Substitution of equations (7, 11) into equation gives after some rearrangement:
(11)
(10)
(13)
(ksTie) [a@/a&JH]A4,,e = (1 - a,) ci,“/es) + a,( -jf/es).
For the harmonically modulated occupancy B(w) of the surface states, we have: df?(CB)/df= i&(W) =
80
RdZ)(W
Fig. 3. Complex plane representation of the interfacial impedance [l/Y(o), (equation (15)] calculated with the assumptions valid for Fig. 5 (part I) and Fig. 2. The surface state density is 10’4cm-2, and the overpotential U-P is 0.5 V. From the characteristic frequency of the small semicircle, the value of the double layer capacitance can be obtained. The large semicircle at low frequencies is related to the kinetics of surface state mediated electron transfer (see text).
tTo)/A4(w)l {iwcH
+ jr” )/a~~&4H}.
0
(14)
In previous work, the relaxation of the occupancy of surface states has been calculated without taking into account the coupling between the occupancy and the potential drop over the Helmholtz-layer. The time constant for the relaxation of the occupancy of the surface states is then found to be T instead of {r-i + (hT/e)y, ([a@/a&bH]A&,o f [ - a@/
ah#&]A~H.o}}-’ derived here. It can be shown that the approximation made in previous work is valid as long as the density of the surface states, described by the parameter y, (equation (55) of part I), is not too large. The interfacial admittance can be calculated directly by substitution of eq. (12) into equation (8). First, the relationship between the harmonically varying potential drop A&(O) and the modulation Am of the interfacial potential drop over the semiconductor/ electrolyte interface will be considered in more detail.
Mediated electron transfer at semiconductor Substitution
electrolyte junctions-II
1139
of equation (12) into equation (7) gives: ~s(kBw
A’H(w)‘A’@) =(’ - YH) + iW +
{ ~H[-a@/aAbi~~~.e + (1 - YH)[-a@/awdA&+e} 7-I i- (kBT/e)y, { [a@/aA(bH]A+,.e + [ -a@/ad&]A#,.e) ’
Equation (15) is the complex equivalent of equation (56) of part I, pertaining to the steady-state. It shows that the distribution of the harmonically modulated interfacial potential drop, A&w) over depletion and Helmholtz-layers depends on the frequency, due to relaxation of the occupancy of the surface states. If the measuring frequency is much higher than T-’ + (kB7?)y,{ [a@/aA’#JH]A+,.s + [-a@/aA&Ids,.~}, .QH(w)/W(~) is equal to the high frequency limit, given by (1 - yH). Since in many cases C,<
(15)
The interfacial admittance consists of three components (Fig. lb). The first component corresponds to the double layer capacitance (1 - YH)CH= YH&. The second component ?H[aCi,"+j:/ %d&, + (1 - YH) [aCi,” +~m~~HlA#+, is Purely resistive, it will further be denoted as R. The third component is complex and accounts for the relaxation of the occupancy of the surface states. The third component corresponds to an electrical equivalent circuit of a series connection of a resistance and capacitance (Rc, C), parallel to a series connection of a resistance and inductance (RL, L) (see Fig. lb). The time constant of the (Rc, C) series connection is equal to that of the (RL, L) connection, and given by (7-l + (kBT/e)y,{ [a@/ aA&r]&,s + [-a@/aA&]AtiH,B} }-‘. The third component gives rise to an inductive loop in the impedance spectrum if:
(17)
frequency, it is not possible to split the electrical impedance into depletion layer and the Helmholtzlayer components. In Fig. 2(a), A&(w)/A~(w) is plotted in the complex plane for the case considered in Fig. 5 of part I. The plot pertains to the equilibrium potential. In Fig. 2(b), the imaginary component of A&(W)/ A&w) is plotted as a function of the frequency. The transfer function Ac#JH(o)/Ac$(w)has been calculated for different densities of monoenergetic surface states. It can be seen, by comparison with Fig. 5 (part I), that the low frequency limit equals the steady state value dA&/dA4, and that the high frequency limit is the real value (1 - Yn). From Fig. 2(b), it is clear that the characteristic frequency increases with increasing density of surface states. This is due to the increasing importance of the coupling between the surface state occupancy and the interfacial kinetics expressed in the additional term (kn T/e)ys { [a@/a&$n]&,s + [ -~Y@/&#I,]~~,.~} with respect to T-I. Substitution of equation (12) into equation (8) gives the total electrical admittance of a semiconductor electrode exchanging majority carriers with a simple redox system via surface states: y(O)
= iW(l +
-
YH)CH
%(kBT/e)
+
It can be shown that, with the assumptions of the model presented here, this condition cannot be fulfilled. It is, therefore, concluded that the third component gives rise to a capacitive loop in the impedance spectrum. When the measuring frequency w is much smaller than {?-I + (kBip)yJ{[a@/ aL!&&,,s + [ - a@/&#&,#&}), the interfacial admittance is equal to the Faraday admittance RF.: given by R-l + R,‘. In Fig. 3, the interfacial impedance [l/Y(w)] is plotted in the complex plane. The same conditions have been assumed as for Fig. 2; and the surface state density is assumed to be 10’4cm-2. The overpotential U-P is 0.5 V. From the characteristic frequency of the semicircle at high frequencies, the value for the double layer capacitance can be obtained. At such high frequencies, A&(w)/A~(co) tends to (1 - YH),(see Fig. 2). The large semicircle at lower frequencies corresponds to the third component which is related to the relaxation of the surface state occupancy, and hence to the interfacial kinetics.
yH[a@ +j,“)/aA&]A+n
(Yd-a@/a&Jsc]A.#&+
(1 -
+ (I -
YH)[aCj,H +maL%]A~,,
~H)[-a@/aA~H]~~,.B)
(16)
1140
D. Vanmaekelbergh DISCUSSION
Although the semiconductor/electrolyte double layer consists of a depletion layer and a Helmholtzlayer, from the above it is, in principle, impossible to separate the electrical impedance or admittance into frequency independent elements which pertain exclusively to each layer. Some special cases exist, however, for which the interfacial admittance is related predominantly to the depletion layer. The first is that of direct transfer if Yu -+ 1 (see equations (2,5)). The second is that of surface state mediated transfer for which “& + 1, and the density of surface states is sufficiently small, so that ys < 1. Most of the previous models imply these conditions [cl 1, 131. For surface state mediated transfer, A&(o)/A+(w) depends on the frequency. The high frequency limit is equal to (1 - YH), the low frequency limit, described by equation (56) of part I can be considerably larger. If ys is sufficiently large, the low frequency limit might be exclusively determined by the interfacial kinetics. In principle, the transfer function A&(w)/A~(w) can be used for the study of interfacial kinetics. Measurement of the electrical impedance, however, is an established technique in electrochemistry, in contrast to the measurement of A&(w)/A~(w). The latter transfer function can be measured by probing the time resolved concentration of majority carriers at the surface by adsorption of microwave radiation [ 141.In part I, it was shown that the presence of a metal layer, in which the electrons are in equilibrium with the electrons in surface states, opens the possibility of a direct and simple measurement of the potential drop over the Helmholtz-layer. Such measurements, performed under steady-state conditions, were presented in [15] for an n-type GaAs electrode and Fe(CN):-‘4- as redox system. Measurements during which the electrode potential is harmonically modulated and A&(W) is measured have not yet been performed. In part I, it has been shown to what extent the steady-state current-potential characteristics differ for surface state mediated and direct electron transfer. In this paper, it is shown that the interfacial admittance also differs markedly. The electrical admittance for direct transfer corresponds essentially to a Randles type electrical equivalent circuit, consisting of a parallel combination of the double layer capacitance and the Faradaic resistance (see Fig. la). Surface state mediated majority carrier transfer is characterized by a (complex) component in the electrical admittance (in addition to the double layer capacitance and a resistance R) which is related to the relaxation of the occupancy of the surface states. In previous work [9-l 1, 131, the coupling between the potential drop over the Helmholtz-layer and the relaxation kinetics was not taken into account. The time constant for the relaxation is then given by r (see equation (1 1)), instead of {T-I + (~BT/ deeh( wa~~H]A~,,e + ~-~@/a~~~iA~,+s~ j-’
rived here. From equations (13-14) and (14, 16, 18) of part I, it follows that the second term {(kBT/e) Ys{ [a@/aA&i]A&,~ + [ - a@/aLwkiA+,.,] } - ’ beCOmeS important with respect to T-I if the density of surface states is sufficiently large (ys > 1) and the occupancy of these states is different from zero and unity. Some typical cases, taken from the literature, will now be reconsidered within the framework of the more general theory proposed in this paper. A model, originally proposed by Nicollian and Goetzberger [ 11, has very often been used to explain experimental results (see for instance [2,68, 16, 171.In this model, it is assumed that surface states are in equilibrium with the majority carriers, and do not exchange electrons with a redox system in solution. This means that the steady-state current due to surface state mediated exchange is zero. Nicollian and Goetzberger showed that the exchange between surface states and majority carrier band leads to a complex admittance, parallel to that pertaining to the double layer capacitance (nearly equal to the capacitance of the depletion layer). This additional admittance corresponds to a series connection of a resistance and capacitance, with values determined by the density of the surface states and the occupancy. The assumption that there is no exchange between the surface states and the redox system, and hence, no steady-state current due to surface state mediated electron exchange, corresponds, within the framework of our model, to the assumption that j! and jr are zero (see eqs (1619) of part I). As then, [ati: + j,“)/aA&]A+, and [ati: + j? )/aA&]A& are both zero, and (ke T/e) [ - a@/a6&]A4,.s = j?nn,(1 0) and [a@/aA+H]A&,O= 0 (see equations (13, 14)) it follows from equation (16) that the interfacial admittance is reduced to the admittance pertaining to the double layer capacitance and an admittance corresponding to a series connection of Rc and C, with: Rc = [(e*/ke T)s/?,n,( 1 - @y,!,- ’
(18)
and: C = [(e2/keT)sP,n,(l
-
041
x [/3nn,+ En+ ys&ns(l - @I-‘. (19) These equations are essentially the same as those derived by Nicollian and Goetzberger, except that the time constant R&for exchange between the majority carrier band and surface states now contains an additional term given by ysPnns(l - 0). This additional term originates from the coupling between the occupancy of the states and the potential distribution over the Helmholtz and depletion layers, and is important if rl( 1 - 0) 2 1, hence for a sufficiently high concentration of surface states with a steady state occupancy essentially different from unity. The coupling between the occupancy of surface states and the distribution of the interfacial potential drop was not taken into account in the original work
Mediated electron transfer at semiconductor of Nicollian and Goetzberger. However, this coupling may strongly affect the value of the surface state capacitance. It follows from equation (19) that C becomes equal to YHCH Z CH if ys is sufficiently large. Such cases have been reported in literature [13, 16, 171. The case in which majority carriers flow in one direction from the semiconductor to the redox system is also interesting. With an n-type electrode, this situation is met at sufficiently negative overpotentials, such that the cathodic partial current density exceeds the anodic partial current density. For surface state mediated transfer, the relationship between cathodic current and electrode potential is essentially different from that for direct transfer, as is shown in part I. The question arises as to whether the mechanisms can also be distinguished on the basis of the electrical admittance. For direct cathodic current flow, the admittance corresponds to a simple Randles circuit of the interfacial capacitance (1 - 7~) CH, parallel to a Faraday resistance given by equation (5). For surface state mediated cathodic current flow, the admittance contains an additional complex component corresponding to a (Rc, C) series connection in parallel to a (RL, L) series connection, giving rise to an additional (capacitive) loop in the complex plane representation of the impedance. The characteristic frequency follows from equation (50) of part I and equations (13, 14) and is given by finnr + (k/ s)coX+ ?j&nr( 1 - 0) (1 + a,). The charge transfer coefficient as is described by equation (60) of part I. Similar results were obtained in [9, 131, except that the time constant for the relaxation of the occupancy of the surface states does not contain the term y&zs(l - Q) (1 + us), which is due to the coupling between the relaxation of the occupancy of the surface states and the relaxation of the potential drop over the Helmholtz-layer. It is also clear that important kinetic parameters, such as 8, s, and /& and (le/s) can, in principle, be derived from results of impedance measurements. Finally, it should be remarked that there is a strong analogy between the results obtained here, for cathodic current flow via surface states, and the electrical admittance due to the injection of holes (minority carriers) into the valence band of an n-type semiconductor by reduction of an oxidizing agent, and subsequent recombination of these holes with electrons from the conduction band via surface states [18-201. The mechanism of surface state mediated electron transfer is, electrically, equivalent to that in which holes are injected and subsequently recombine with electrons. Hence, distinction between the two mechanisms of cathodic current flow, by means of the electrical admittance solely is not appropriate. CONCLUSIONS It has been shown that the interfacial electrical admittance due to the exchange of majority carriers
electrolyte junctions-11
1141
between a semiconductor electrode and a simple redox system can be calculated exactly from basic assumptions, which have been presented in part I. It was found that direct and surface state mediated exchange can be distinguished by means of the electrical admittance. For surface state mediated transfer, an exact calculation accounts for the coupling between the relaxation of the occupancy of the surface states and the potential drop over the Helmholtz-layer. This coupling is particularly important if the density of surface states is high. Due to this coupling, the electrical impedance cannot be separated in discrete components, pertaining to the depletion and the Helmholtz-layer exclusively. Some simple but important cases, such as that in which the exchange between surface states and redox system can be neglected, and that of predominant majority carrier flow from the semiconductor to the redox system, have been analyzed in the framework of the model presented here. The results can be reduced to those of previous work if the density of surface states is sufficiently low. The analysis presented here shows how the coupling between relaxation of the occupancy of the states and distribution of the interfacial potential drop over depletion and Helmholtz-layers, affects the interfacial admittance.
REFERENCES 1. E. H. Nicollian and J. 46, 1055 (1967).
A. Goetzberger, Bell. Syst. Tech.
2. W. P. Gomes and F. Cardon, Progress in Surface Science 12, 155 (1982). 3. B. Ershler, Disc. Faraday Sot. 1, 269 (1947). 4. J. H. Sluvters. Recueil Trav. Chim. 79. 1092 (1960). 5. J. H. Sluytersand J. J. C. Oomen, Recheil Tra;. km. 79, 1101 (1960). 6. W. P. Gomes and F. Cardon, Ber. Bunsenges, Phys. Chem. 74,431 (1970). 7. K. Kobayashi, Y. Aikaw and M. Sukigara, J. Appl. Phys. 54, 2526 (1983).
8. G. Nagasubramanian, B. L. Wheeler and A. J. Bard, J. Electrochem. Sot. 130, 1680 (1983). 9. V. A. Tyagai and G. Ya. Kolbasov, Surf. Sri. 28, 423 (1971). 10. F. Cardon, Physica 57, 390 (1972). 11. K. Kobayashj, M. Takata and S. Okamoto, J. Electroanal. Chem. 185. 47 (1985). Chazalviel, J. ‘Eleckach~m. Sot. 129, 963 (1982). 13. G. Oskam, D. Vanmaekelhergh and J. J. Kelly, Eleclrochim. Acta 38, 301 (1993). 14. G. Schlichthiirl and L. M. Peter, J. Electrochem. Sot. 141, 1171 (1994). 15. G. Oskam, D. Vanmaekelbergh and J. J. Kelly, Electrochim. Acta 38, 1115 (1993). 16. H. H. Goossens and W. P. Gomes. J. Electroanal.
12. J.-N.
Chem. 278, 335 (1990). 17. G. Oskam, P. M. Hoffmann and P. C. Searson, Phys. Rev. L&r. 76, 1521 (1996). 18. D. Vanmaekelbergh and F. Cardon, J. Phys. D.: Appl. Phys. 19, 643 (1986). 19. B. P. Minks and J. J. Kelly, Ber. Bunsenges. Phys. Chem. 96, 894 (1992). 20. D. Vanmaekelbergh, W. P. Gomes and F. Cardon, Ber. Bunsenges. Phys. Chem. 89, 994 (1985).
Pergamon PII: !30013-4686(!I6)oo266-6
Acfa, Vol. 42, No. 7, pp. 1135-I 141, 1997
Copyright 0 1997 Ekevier Science Ltd. Printed in Great Britain. All rights reserved 0013-4686/97 $17.00 + 0.00
Direct and surface state mediated electron transfer at semiconductor/electrolyte junctions-11. A comparison of the interfacial admittance D. Vanmaekelbergh Debye Institute,
Utrecht University,
P. 0. Box 80 000, 3508 TA Utrecht, The Netherlands
(Received 14 March 1996; accepted 24 June 1996) Abstract-The interfacial electrical admittance of a semiconductor electrode exchanging majority carriers with a simple redox system is calculated from basic assumptions presented in a previous paper. Direct exchange and surface state mediated exchange are considered. The coupling between the occupancy of the surface states and the distribution of the interfacial potential drop over the depletion and Helmholtz-layers is taken into account quantitatively using the fluctuating energy level model. The results show that direct and surface state mediated transfer can be distinguished on the basis of the electrical admittance. Some simple, but important, cases of surface state mediated exchange are reviewed in the framework of the model. The consequences of the coupling between the occupancy of the surface states and the distribution of the interfacial potential drop over depletion and Helmholtz-layer are discussed in detail. Copyright 0 1997 Elsevier Science Ltd
INTRODUCTION In this paper, the interfacial electrical admittance of a semiconductor electrode exchanging majority carriers with a simple redox system will be derived. The assumptions of the model were presented in part I, in which the steady-state results were also considered. It is assumed that majority carriers tunnel through the Helmholtz-layer isoenergetically, and that the electronic states of the redox system can be described by the fluctuating energy level model. Both direct exchange and exchange via surface states is considered. The effect of the relaxation of the occupancy of the surface states on the interfacial potential distribution and hence on the exchange rates is taken into account quantitatively. Measurement of the electrical impedance (or admittance) of junctions consisting of a semiconductor in contact with a metal or electrolyte allows one to investigate the interaction of free carriers in the semiconductor with discrete electronic levels, localized at the interface between semiconductor and metal or electrolyte, and with an energy in the forbidden gap. A harmonic modulation of the potential (with frequency o) of the semiconductor with respect to the other phase gives a harmonic current which is measured in the external circuit. The ratio of the harmonic current density to potential is called the
electrical admittance Y(w). Characteristically, the semiconductor is depleted of majority carriers at the interface. Modulation of the potential leads to modulation of the width of the depletion layer and of the concentration of majority carriers at the interface. If interface states do not play a role, Y(w) corresponds to the differential capacitance of the interface (predominantly the capacitance of the depletion layer), parallel to a differential resistance, which is determined by the rate of exchange of carriers between the semiconductor and the other phase. If interface states interact with the majority carriers, Y(w) contains an additional admittance with a time constant which is characteristic for the exchange of electrons between the majority carrier band and the interface states [ 1, 21.The density of the localized state and the rate constant of electron trapping can, in principle, be obtained from the additional admittance. On the other hand, the kinetics of electron exchange between a metal electrode and a redox system have been successfully investigated by means of the electrical admittance method [3-51. In the simple case of a single electron tunneling process between metal and redox system, the electrical admittance corresponds to the double layer capacitance, parallel to a differential resistance, from which the rate constant of electron exchange at equilibrium and the transfer coefficient can be obtained [4-51.
1135
1136
D. Vanmaekelbergh
Investigations of the semiconductor/electrolyte interface by means of electrical impedance measurements probably owe more to similar work on solid state devices than to electrochemical work with metal electrodes. In many studies the model of Nicollian and Goetzberger, [1, 2,6, 7, 81, which describes the interaction between surface states and the majority carrier band, was used as a basis for the interpretation of experimental results. In some papers (for instance [9-131) the exchange of electrons between surface states and the redox system was also taken into account. It has been recognized that localization of electrons in surface states may change the distribution of the interfacial potential drop A$ over the depletion- and Helmholtz-layers [12, 131. The effect of this was taken into account for the exchange rate between majority carrier band and surface states, but was neglected for the exchange rate between surface states and the redox system. An exception can be found in Chazalviel’s paper [12], where the exchange between surface states and the redox system is described by a Butler-Volmer type relationship. The coupling between the interfacial kinetics and the potential distribution over depletion and Helmholtz-layers has been analyzed in part I for steady-state conditions. The time resolved effect of a small modulation SA+ of the interfacial potential drop can be discussed qualitatively. In the case of direct transfer, the distribution of SA+ over the depletion and Helmholtz-layers depends on the relative values of C, and CH only, and is therefore time independent. Hence, the application of SA4 leads to an immediate change in the rate of electron transfer between the majority carrier band and the redox system. This results in a differential resistance in the interfacial admittance taking into account the immediate change of the Faradaic current due to 8A4. In the case of surface state mediated electron transfer, the distribution of the modulation 6A$ is, initially, determined by the values of C, and Cu. However, since the modulation of the electron exchange rate between surface states and majority carrier band is different from that between surface states and the redox system, the occupation of the surface states changes with time. This may affect, in turn, the distribution of SA4 over depletion and Helmholtz-layers. It can be concluded that the distribution of SA4 over depletion and Helmholtzlayer is time dependent and coupled to the interfacial kinetics. It will become clear that this results in a complex component in the interfacial admittance related to the relaxation of the occupancy of the surface states. Calculation of the interfacial admittance As in part I, we consider a three electrode electrochemical cell with an n-type semiconductor electrode as working electrode. For measurement of the electrical admittance, a sufficiently small harmonically modulated potential U(w) = U,,, exp(iwr) is
superimposed on the electrode potential U, and the harmonically modulated current density in the external circuit, j(m), is measured. The electrical admittance of the cell is j(w)/U(w). Modulation of the electrode potential leads to the modulation of the interfacial potential drop over the semiconductor/ electrolyte interface. The interfacial admittance of the semiconductor electrode, denoted in this paper as Y(w), is given by j(o)/A&o). The electrochemical cell is designed such that the impedance of the counter electrode is much smaller than that of the working electrode, which means that the potential of the counter electrode is not affected by the modulation U(o). From equation (21) of part I, it follows that the relationship between the measured admittance j(w)/U(w) and the interfacial admittance Y(w) of the working electrode is given by: @)/U(w)
= Y(o)/]1 + A&I
Y(w)l.
(1)
The measured admittance agrees with the interfacial admittance of the semiconductor working electrode if [Y(w)]-‘<
iWyHc%.
(2)
The parameter yu is defined by equation (26) of part I. The admittance EC(w) + ja(o)]/A4(w) due to electron exchange can be derived from:
+j4w)l/Ww)
L(o)
+
[WH(~)/W(~)IKKL
+jd/a&'HlA,,.
(3)
The partial derivates have been calculated in part I (equations (40-43)) using Fermi-statistics and the fluctuating energy level model. The distribution of the harmonically varying potential drop over the depletion and Helmholtz-layers is, for direct transfer, independent of the measuring frequency o, and determined by YH(see equation (26) of part I): A&(w)/A~(w)
=
YH = CH/(CH
WH(~)/&J(QJ)=
1-7~.
+ G)
(4)
It follows from equations (24) that the interfacial admittance can be represented by an electrical equivalent circuit consisting of the capacitance of the double layer given by YuCac, parallel to a Faradaic
Mediated electron transfer at semiconductor differential resistance RF (Fig. la). Substitution of equation (4) and equations (4Wl) of part I into equation (3) gives: R;’ =
(e/kBT)
YH) [(I
0.5
64
-
a&
The transfer coefficients GIand equations (42-43) of part I. equations (40,41) into equation be verified that R;’ is equal to
+ cd-jc)
1 }.
1137
/ ’
3’
o.4-
,p
2 z
’ 3’
/’ #’
0.3.
44 T 0.2~
{YH(-jr) + (1 -
electrolyte junctions-II ,
II
‘,
~~. 10’4,&
(5)
1 - t( are defined by By substitution of (39) of part I, it can dj/dA4.
IO‘*
10~’
100
10’
102
103
Figures 2 and 3(b) of part I account for surface state mediated exchange of majority carriers between a semiconductor and a redox system. The current density measured in the external circuit is given by equation (27) of part I. If the electrode potential is harmonically modulated, then dA&/dt = iWA&(O). Hence, it follows from equation (27) and from A&(w) + A&(w) = Am that the interfacial admittance can be written as:
t----II-l-
104
105
w [$~‘I
0.5
(b)
--.
,_-’
Surface state mediated transfer
Cb) ---j--i
lo’*cm’* 10” cn?
-
/ ‘,
c
o.4
2 : 44
0.3
T
,’
~.
,’
,’ /’
0.2 0.1
I j
-_
,’
h
I
7,
~---t
O.9 )
0.2
0.4
0.6
0.8
j
A
1.0
Fig. 2. (a) Complex plane representation of the harmonically varying potential drop over the Helmholtz-layer A&(W), divided by that over the semiconductor/electrolyte interface Ad(w), with the frequency w as a parameter (see equation (15)). The same conditions have been chosen as for Fig. 5 (part I). Surface state mediated electron transfer is considered. At equilibrium, the energy of the conduction band edge at the surface is 0.75 eV above the Fermi-level of the redox system. The surface states are located 0.75 eV below the conduction band edge. The plots pertain to different values of the effective surface state density (lo”, 1013, 1O“‘cm-‘). The results here pertain to the equilibrium potential. (b) Plot of the imaginary part of [A&(o)/A&o)] as a function of the frequency. It is seen that the characteristic frequency increases with increasing surface state density, due to the coupling between the surface state occupancy and the interfacial kinetics (see text).
Y(w) =
iWCH [WH(W)IA~(W)I
+
[ati:
+j,“)/~bbdw,
[WH((~YW(W)I. (6)
RL
L
Fig. 1. (a) Electrical equivalent representation of the interfacial admittance derived for a semiconductor electrode exchanging majority carriers with a simple redox system via direct tunneling of electrons through the Helmholtz-layer. C,: capacitance of the depletion layer in the semiconductor. CH: capacitance of the Helmholtz-layer. RF: differential Faradaic resistance accounting for direct majority carrier transfer. (b) Electrical equivalent representation of the interfacial admittance derived for a semiconductor electrode exchanging majority carriers with a simple redox system via surface states (see equation (16)). The (Rc, C) and (RL, f.) series connections account for the relaxation of the occupancy of the surface states. The Faradaic resistance is given by (R-l+ RF')-'.
Due to the relaxation of the occupancy of the surface states which is coupled to the interfacial kinetics, A&(w)/A~(w) is a complex quantity, in contrast to the case of direct transfer. From the combination of equations (20) and (27) of part I, and taking into account the fact that A&(W) + A&(w) = Am, it follows that: WH(W)/M(W) (1 -
= YH) + ?dkBT/e)
I--Nw)lW(w)l.
(7)
In equation (7) Q(w) stands for the harmonically modulated occupancy of the surface states. The parameter yS, defined by equation (55) of part I, describes the influence of the occupancy of the surface states on the distribution of the interfacial potential drop over depletion and Helmholtz-layers.
1138
D. Vanmaekelbergh
Equation (7) is the harmonically varying equivalent of equation (54) of part I, which pertains to the steady-state. The interfacial admittance differs essentially from that pertaining to direct transfer as a consequence of the fact that A&(w)/A&w) is not equal to (1 - mu), as it is for direct transfer, but contains a second (complex) component {y,(k~T/ e) [ -O(w)/A&o)]}, due to the relaxation of the occupancy of the surface states. This is shown by the substitution of equation (7) into equation (6):
+
Y~(~B T/e)
+
[-
xi:
[-
20
40
60
(8)
The partial derivatives in equation (8) were calculated in part I (equations (65,66) with jf + jf being equal to ~7 + ~7 and to the steady-state current density j). In order to calculate &T/e) [-O(w)/AO(w)], the function d@/dt described by equation (20) of part I must be developed as a total differential of 8, A&, and ASH. To shorten the notation the function dO/dt will be denoted as 0:
Equation (12) describes the relationship between the harmonically modulated occupancy of the surface states 6(w) and the modulation of the interfacial potential drop A#@), for a semiconductor electrode, described by the parameters Yu and yS, exchanging majority carriers with a simple redox system, the interfacial kinetics being described by r-l, [ -a0/ OA&]A.+H.~~and [O0/OA&]A&+ The latter two quantities are directly calculated from equation (20) of part I: (ke Tie) [ - a@/aA’#JsiA4H,e = -k/es
+ [O0/OA&]A+,,edA&.
(9)
+
[a@/ao]A4,,A4&+)
[aO/aL\~HiA~,.dA~H(O).
(10)
The partial derivative [O0/a&,,&+, is equal to -f-I, given by equation (50) of part I:
+
(h&)CRcd
+
61 +
(k/s)cOx.
Substitution of equations (7, 11) into equation gives after some rearrangement:
(11)
(10)
(13)
(ksTie) [a@/a&JH]A4,,e = (1 - a,) ci,“/es) + a,( -jf/es).
For the harmonically modulated occupancy B(w) of the surface states, we have: df?(CB)/df= i&(W) =
80
RdZ)(W
Fig. 3. Complex plane representation of the interfacial impedance [l/Y(o), (equation (15)] calculated with the assumptions valid for Fig. 5 (part I) and Fig. 2. The surface state density is 10’4cm-2, and the overpotential U-P is 0.5 V. From the characteristic frequency of the small semicircle, the value of the double layer capacitance can be obtained. The large semicircle at low frequencies is related to the kinetics of surface state mediated electron transfer (see text).
tTo)/A4(w)l {iwcH
+ jr” )/a~~&4H}.
0
(14)
In previous work, the relaxation of the occupancy of surface states has been calculated without taking into account the coupling between the occupancy and the potential drop over the Helmholtz-layer. The time constant for the relaxation of the occupancy of the surface states is then found to be T instead of {r-i + (hT/e)y, ([a@/a&bH]A&,o f [ - a@/
ah#&]A~H.o}}-’ derived here. It can be shown that the approximation made in previous work is valid as long as the density of the surface states, described by the parameter y, (equation (55) of part I), is not too large. The interfacial admittance can be calculated directly by substitution of eq. (12) into equation (8). First, the relationship between the harmonically varying potential drop A&(O) and the modulation Am of the interfacial potential drop over the semiconductor/ electrolyte interface will be considered in more detail.
Mediated electron transfer at semiconductor Substitution
electrolyte junctions-II
1139
of equation (12) into equation (7) gives: ~s(kBw
A’H(w)‘A’@) =(’ - YH) + iW +
{ ~H[-a@/aAbi~~~.e + (1 - YH)[-a@/awdA&+e} 7-I i- (kBT/e)y, { [a@/aA(bH]A+,.e + [ -a@/ad&]A#,.e) ’
Equation (15) is the complex equivalent of equation (56) of part I, pertaining to the steady-state. It shows that the distribution of the harmonically modulated interfacial potential drop, A&w) over depletion and Helmholtz-layers depends on the frequency, due to relaxation of the occupancy of the surface states. If the measuring frequency is much higher than T-’ + (kB7?)y,{ [a@/aA’#JH]A+,.s + [-a@/aA&Ids,.~}, .QH(w)/W(~) is equal to the high frequency limit, given by (1 - yH). Since in many cases C,<
(15)
The interfacial admittance consists of three components (Fig. lb). The first component corresponds to the double layer capacitance (1 - YH)CH= YH&. The second component ?H[aCi,"+j:/ %d&, + (1 - YH) [aCi,” +~m~~HlA#+, is Purely resistive, it will further be denoted as R. The third component is complex and accounts for the relaxation of the occupancy of the surface states. The third component corresponds to an electrical equivalent circuit of a series connection of a resistance and capacitance (Rc, C), parallel to a series connection of a resistance and inductance (RL, L) (see Fig. lb). The time constant of the (Rc, C) series connection is equal to that of the (RL, L) connection, and given by (7-l + (kBT/e)y,{ [a@/ aA&r]&,s + [-a@/aA&]AtiH,B} }-‘. The third component gives rise to an inductive loop in the impedance spectrum if:
(17)
frequency, it is not possible to split the electrical impedance into depletion layer and the Helmholtzlayer components. In Fig. 2(a), A&(w)/A~(w) is plotted in the complex plane for the case considered in Fig. 5 of part I. The plot pertains to the equilibrium potential. In Fig. 2(b), the imaginary component of A&(W)/ A&w) is plotted as a function of the frequency. The transfer function Ac#JH(o)/Ac$(w)has been calculated for different densities of monoenergetic surface states. It can be seen, by comparison with Fig. 5 (part I), that the low frequency limit equals the steady state value dA&/dA4, and that the high frequency limit is the real value (1 - Yn). From Fig. 2(b), it is clear that the characteristic frequency increases with increasing density of surface states. This is due to the increasing importance of the coupling between the surface state occupancy and the interfacial kinetics expressed in the additional term (kn T/e)ys { [a@/a&$n]&,s + [ -~Y@/&#I,]~~,.~} with respect to T-I. Substitution of equation (12) into equation (8) gives the total electrical admittance of a semiconductor electrode exchanging majority carriers with a simple redox system via surface states: y(O)
= iW(l +
-
YH)CH
%(kBT/e)
+
It can be shown that, with the assumptions of the model presented here, this condition cannot be fulfilled. It is, therefore, concluded that the third component gives rise to a capacitive loop in the impedance spectrum. When the measuring frequency w is much smaller than {?-I + (kBip)yJ{[a@/ aL!&&,,s + [ - a@/&#&,#&}), the interfacial admittance is equal to the Faraday admittance RF.: given by R-l + R,‘. In Fig. 3, the interfacial impedance [l/Y(w)] is plotted in the complex plane. The same conditions have been assumed as for Fig. 2; and the surface state density is assumed to be 10’4cm-2. The overpotential U-P is 0.5 V. From the characteristic frequency of the semicircle at high frequencies, the value for the double layer capacitance can be obtained. At such high frequencies, A&(w)/A~(co) tends to (1 - YH),(see Fig. 2). The large semicircle at lower frequencies corresponds to the third component which is related to the relaxation of the surface state occupancy, and hence to the interfacial kinetics.
yH[a@ +j,“)/aA&]A+n
(Yd-a@/a&Jsc]A.#&+
(1 -
+ (I -
YH)[aCj,H +maL%]A~,,
~H)[-a@/aA~H]~~,.B)
(16)
1140
D. Vanmaekelbergh DISCUSSION
Although the semiconductor/electrolyte double layer consists of a depletion layer and a Helmholtzlayer, from the above it is, in principle, impossible to separate the electrical impedance or admittance into frequency independent elements which pertain exclusively to each layer. Some special cases exist, however, for which the interfacial admittance is related predominantly to the depletion layer. The first is that of direct transfer if Yu -+ 1 (see equations (2,5)). The second is that of surface state mediated transfer for which “& + 1, and the density of surface states is sufficiently small, so that ys < 1. Most of the previous models imply these conditions [cl 1, 131. For surface state mediated transfer, A&(o)/A+(w) depends on the frequency. The high frequency limit is equal to (1 - YH), the low frequency limit, described by equation (56) of part I can be considerably larger. If ys is sufficiently large, the low frequency limit might be exclusively determined by the interfacial kinetics. In principle, the transfer function A&(w)/A~(w) can be used for the study of interfacial kinetics. Measurement of the electrical impedance, however, is an established technique in electrochemistry, in contrast to the measurement of A&(w)/A~(w). The latter transfer function can be measured by probing the time resolved concentration of majority carriers at the surface by adsorption of microwave radiation [ 141.In part I, it was shown that the presence of a metal layer, in which the electrons are in equilibrium with the electrons in surface states, opens the possibility of a direct and simple measurement of the potential drop over the Helmholtz-layer. Such measurements, performed under steady-state conditions, were presented in [15] for an n-type GaAs electrode and Fe(CN):-‘4- as redox system. Measurements during which the electrode potential is harmonically modulated and A&(W) is measured have not yet been performed. In part I, it has been shown to what extent the steady-state current-potential characteristics differ for surface state mediated and direct electron transfer. In this paper, it is shown that the interfacial admittance also differs markedly. The electrical admittance for direct transfer corresponds essentially to a Randles type electrical equivalent circuit, consisting of a parallel combination of the double layer capacitance and the Faradaic resistance (see Fig. la). Surface state mediated majority carrier transfer is characterized by a (complex) component in the electrical admittance (in addition to the double layer capacitance and a resistance R) which is related to the relaxation of the occupancy of the surface states. In previous work [9-l 1, 131, the coupling between the potential drop over the Helmholtz-layer and the relaxation kinetics was not taken into account. The time constant for the relaxation is then given by r (see equation (1 1)), instead of {T-I + (~BT/ deeh( wa~~H]A~,,e + ~-~@/a~~~iA~,+s~ j-’
rived here. From equations (13-14) and (14, 16, 18) of part I, it follows that the second term {(kBT/e) Ys{ [a@/aA&i]A&,~ + [ - a@/aLwkiA+,.,] } - ’ beCOmeS important with respect to T-I if the density of surface states is sufficiently large (ys > 1) and the occupancy of these states is different from zero and unity. Some typical cases, taken from the literature, will now be reconsidered within the framework of the more general theory proposed in this paper. A model, originally proposed by Nicollian and Goetzberger [ 11, has very often been used to explain experimental results (see for instance [2,68, 16, 171.In this model, it is assumed that surface states are in equilibrium with the majority carriers, and do not exchange electrons with a redox system in solution. This means that the steady-state current due to surface state mediated exchange is zero. Nicollian and Goetzberger showed that the exchange between surface states and majority carrier band leads to a complex admittance, parallel to that pertaining to the double layer capacitance (nearly equal to the capacitance of the depletion layer). This additional admittance corresponds to a series connection of a resistance and capacitance, with values determined by the density of the surface states and the occupancy. The assumption that there is no exchange between the surface states and the redox system, and hence, no steady-state current due to surface state mediated electron exchange, corresponds, within the framework of our model, to the assumption that j! and jr are zero (see eqs (1619) of part I). As then, [ati: + j,“)/aA&]A+, and [ati: + j? )/aA&]A& are both zero, and (ke T/e) [ - a@/a6&]A4,.s = j?nn,(1 0) and [a@/aA+H]A&,O= 0 (see equations (13, 14)) it follows from equation (16) that the interfacial admittance is reduced to the admittance pertaining to the double layer capacitance and an admittance corresponding to a series connection of Rc and C, with: Rc = [(e*/ke T)s/?,n,( 1 - @y,!,- ’
(18)
and: C = [(e2/keT)sP,n,(l
-
041
x [/3nn,+ En+ ys&ns(l - @I-‘. (19) These equations are essentially the same as those derived by Nicollian and Goetzberger, except that the time constant R&for exchange between the majority carrier band and surface states now contains an additional term given by ysPnns(l - 0). This additional term originates from the coupling between the occupancy of the states and the potential distribution over the Helmholtz and depletion layers, and is important if rl( 1 - 0) 2 1, hence for a sufficiently high concentration of surface states with a steady state occupancy essentially different from unity. The coupling between the occupancy of surface states and the distribution of the interfacial potential drop was not taken into account in the original work
Mediated electron transfer at semiconductor of Nicollian and Goetzberger. However, this coupling may strongly affect the value of the surface state capacitance. It follows from equation (19) that C becomes equal to YHCH Z CH if ys is sufficiently large. Such cases have been reported in literature [13, 16, 171. The case in which majority carriers flow in one direction from the semiconductor to the redox system is also interesting. With an n-type electrode, this situation is met at sufficiently negative overpotentials, such that the cathodic partial current density exceeds the anodic partial current density. For surface state mediated transfer, the relationship between cathodic current and electrode potential is essentially different from that for direct transfer, as is shown in part I. The question arises as to whether the mechanisms can also be distinguished on the basis of the electrical admittance. For direct cathodic current flow, the admittance corresponds to a simple Randles circuit of the interfacial capacitance (1 - 7~) CH, parallel to a Faraday resistance given by equation (5). For surface state mediated cathodic current flow, the admittance contains an additional complex component corresponding to a (Rc, C) series connection in parallel to a (RL, L) series connection, giving rise to an additional (capacitive) loop in the complex plane representation of the impedance. The characteristic frequency follows from equation (50) of part I and equations (13, 14) and is given by finnr + (k/ s)coX+ ?j&nr( 1 - 0) (1 + a,). The charge transfer coefficient as is described by equation (60) of part I. Similar results were obtained in [9, 131, except that the time constant for the relaxation of the occupancy of the surface states does not contain the term y&zs(l - Q) (1 + us), which is due to the coupling between the relaxation of the occupancy of the surface states and the relaxation of the potential drop over the Helmholtz-layer. It is also clear that important kinetic parameters, such as 8, s, and /& and (le/s) can, in principle, be derived from results of impedance measurements. Finally, it should be remarked that there is a strong analogy between the results obtained here, for cathodic current flow via surface states, and the electrical admittance due to the injection of holes (minority carriers) into the valence band of an n-type semiconductor by reduction of an oxidizing agent, and subsequent recombination of these holes with electrons from the conduction band via surface states [18-201. The mechanism of surface state mediated electron transfer is, electrically, equivalent to that in which holes are injected and subsequently recombine with electrons. Hence, distinction between the two mechanisms of cathodic current flow, by means of the electrical admittance solely is not appropriate. CONCLUSIONS It has been shown that the interfacial electrical admittance due to the exchange of majority carriers
electrolyte junctions-11
1141
between a semiconductor electrode and a simple redox system can be calculated exactly from basic assumptions, which have been presented in part I. It was found that direct and surface state mediated exchange can be distinguished by means of the electrical admittance. For surface state mediated transfer, an exact calculation accounts for the coupling between the relaxation of the occupancy of the surface states and the potential drop over the Helmholtz-layer. This coupling is particularly important if the density of surface states is high. Due to this coupling, the electrical impedance cannot be separated in discrete components, pertaining to the depletion and the Helmholtz-layer exclusively. Some simple but important cases, such as that in which the exchange between surface states and redox system can be neglected, and that of predominant majority carrier flow from the semiconductor to the redox system, have been analyzed in the framework of the model presented here. The results can be reduced to those of previous work if the density of surface states is sufficiently low. The analysis presented here shows how the coupling between relaxation of the occupancy of the states and distribution of the interfacial potential drop over depletion and Helmholtz-layers, affects the interfacial admittance.
REFERENCES 1. E. H. Nicollian and J. 46, 1055 (1967).
A. Goetzberger, Bell. Syst. Tech.
2. W. P. Gomes and F. Cardon, Progress in Surface Science 12, 155 (1982). 3. B. Ershler, Disc. Faraday Sot. 1, 269 (1947). 4. J. H. Sluvters. Recueil Trav. Chim. 79. 1092 (1960). 5. J. H. Sluytersand J. J. C. Oomen, Recheil Tra;. km. 79, 1101 (1960). 6. W. P. Gomes and F. Cardon, Ber. Bunsenges, Phys. Chem. 74,431 (1970). 7. K. Kobayashi, Y. Aikaw and M. Sukigara, J. Appl. Phys. 54, 2526 (1983).
8. G. Nagasubramanian, B. L. Wheeler and A. J. Bard, J. Electrochem. Sot. 130, 1680 (1983). 9. V. A. Tyagai and G. Ya. Kolbasov, Surf. Sri. 28, 423 (1971). 10. F. Cardon, Physica 57, 390 (1972). 11. K. Kobayashj, M. Takata and S. Okamoto, J. Electroanal. Chem. 185. 47 (1985). Chazalviel, J. ‘Eleckach~m. Sot. 129, 963 (1982). 13. G. Oskam, D. Vanmaekelhergh and J. J. Kelly, Eleclrochim. Acta 38, 301 (1993). 14. G. Schlichthiirl and L. M. Peter, J. Electrochem. Sot. 141, 1171 (1994). 15. G. Oskam, D. Vanmaekelbergh and J. J. Kelly, Electrochim. Acta 38, 1115 (1993). 16. H. H. Goossens and W. P. Gomes. J. Electroanal.
12. J.-N.
Chem. 278, 335 (1990). 17. G. Oskam, P. M. Hoffmann and P. C. Searson, Phys. Rev. L&r. 76, 1521 (1996). 18. D. Vanmaekelbergh and F. Cardon, J. Phys. D.: Appl. Phys. 19, 643 (1986). 19. B. P. Minks and J. J. Kelly, Ber. Bunsenges. Phys. Chem. 96, 894 (1992). 20. D. Vanmaekelbergh, W. P. Gomes and F. Cardon, Ber. Bunsenges. Phys. Chem. 89, 994 (1985).