J . Phys. Chem. 1991,95, 1356-1359
1356
spectra in solution were measured on the HP 8450A diode-array spectrophotometer. Acknowledgment. We thank T. M. Bockman for helpful suggestions and the National Science Foundation, R. A. Welch
Foundation, and the Texas Advanced Research Program for financial support. Registry NO. MV2+, 4685-14-7; DQ2+,2764-72-9; C P', 13441-45-7; TR+,2681 1-28-9; AC+, 13367-81-2; I-, 20461-54-5; Nal, 7681-82-5.
Electron-Transfer Kinetics of Redox Reactions at the Semiconductor/Electrolyte Contact. A New Approach Heinz Gerischer Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D- 1000 Berlin 33, F.R.G. (Received: May 16, 1990)
The model used for electron-transfer kinetics between the electronic charge carriers of a semiconductor and the species of a redox couple in an electrolyte has been refined by taking into account the statistics of forming a reaction pair at the interface. Electron transfer within such a reaction pair is described by the semiclassical theory. A comparison is made between the electron transfer in the forward direction over a semiconductor-metal and a semiconductor-redox electrolyte Schottky barrier of equal height. The result indicates that electronic equilibrium between bulk and surface is maintained in the electrochemical system over the entire current voltage range. The rate of electron transfer via surface states is compared with the transfer via the band edge within the same model. Conditions that can favor the reaction path via surface states are specified.
Introduction
Electron-transfer reactions at semiconductor electrodes in contact with redox electrolytes can follow two different pathways. They can be performed by electrons of the conduction band or by electrons of the valence band, the latter case corresponding to a transfer of holes.'-3 Due to the relatively small concentration of electrons or holes in semiconductors, they can be treated in interfacial reactions as individual reactantsS4v5 This situation is very different from a metal electrode where the density of states at the Fermi energy is high and correspondingly the concentration of both electrons and holes in this energy range is very large. Consequently, it was not necessary to use individual electrons or holes in the theoretical description of charge-transfer kinetics for redox reactions, as worked out by Marcus.6 Electron transfer has been treated in this case as an adiabatic process in which an "electron cloud" interacts with the electron acceptor in the electrolyte, if the latter has achieved a favorable configuration for such an interaction. Or alternatively, an electron donor interacts with the multiplicity of vacant electronic states in the metal if a suitable configuration has been reached. This is not different in the quantum mechanical approach of Levich and Dogonadze,'** although the density of electronic states appears explicitly in their formulas, but in their final results, only the energy position of the electrons or the holes around the Fermi level remains as a decisive parameter. This difference between semiconductors and metals becomes most obvious if one wants to discuss a redox reaction with the majority carriers of a semiconductor, the concentration of which can be varied over a wide range by the bias applied. In this case the kinetics in the forward direction of the majority carrier flux ( 1 ) Beck, F.; Gerischer, H. 2.Elektrochem. 1959, 63, 500. (2) Dewald, J. F. In Semiconducotrs; Hanney, N. B., Ed.; ACS Monograph No. 140; Reinhold: New York, 1959; p 727. (3) Gerischer, H. Z . Phys. Chem. (Munich) 1961, 27, 48. (4) Gerischer, H. In Physical Chemistry: An Aduanced Treatise; Eyring, H., Henderson, D., Jost, W.,Eds.; Academic Press: New York, 1970; Vol. I X , A, p 463. (5) Myamlin, V. A.; Pleskov, Yu. V. Electrochemistry of Semiconductors; Plenum Press: New York, 1967. (6) Marcus. R. A. J. Chem. Phys. 1956,24,966; Can. J . Chem. 1959,37, 138. (7) Dogonadze, R. R.; Chizmadzev, Yu. A. Dokl. Akad. Nauk SSSR 1962, 145, 848. (8) Levich, V. G . In Aduances in Electrochemical Engineering, Delahay, P., Ed.; Wiley: New York, 1966; Vol. 4, p 249.
can be described as a bimolecular reaction which occurs at the interface between these electronic charge carriers and the respective redox species in the electrolyte, provided the concentration of these change carriers is still small in comparison to the effective density of states in the respective band. This situation is found if, at equilibrium with the redox system, a depletion layer is formed at the semiconductor contact to the electrolyte. In this case, the electronic charge carriers have to overcome the energy barrier of the depletion layer in order to pass current in the forward direction. This is fully analogous to the current through a semiconductor-metal Schottky barrier:,'0 the only difference being that in an electrolyte the concentration of acceptor states for the charge carriers of the semiconductor is orders of magnitude smaller than in a metal and that these states are localized and their energy levels fluctuate over a wide energy range." In this paper, a comparison will be made of the current in the forward direction through these two types of Schottky barriers, one to the electrolyte and the other to a metal contact. A bimolecular reaction model will be consequently used for the redox reaction in analogy to electron-transfer reactions in homogeneous solutions. In order to be on safe ground, the discussion will start with the role of electron exchange at equilibrium where electronic equilibrium is guaranteed between the bulk and the surface of the semiconductor. Under bias in the forward direction, the assumption will be made that the potential drop in the Helmholtz double layer remains constant. Concentration polarization in the electrolyte shall also be excluded. Finally, the contribution of surface states to charge transfer in comparison to the reaction via the band edges will be analyzed within the same model. Only the situation for n-type semiconductors will be explicitly outlined. There is no principal difference for p-type semiconductors. However, due to their susceptibility to corrosion, systems where the forward reaction between holes and redox systems can be experimentally studied, without chemical changes of the interface, are very rare. Electron Transfer via the Conduction Band Edge to a Redox System at Equilibrium
The redox reaction Ox+.solv + e- Red-solv may occur between redox species Ox+ at a distance d from the interface and ~
~~
~
~
~
(9) Rideout, V R Thin Solid Films 1978, 48, 261 (IO) Rhoderick, E H , Williams, R H Mefal-Semiconductor Contacts, 2nd ed ; Clarendon Press Oxford, 1988 ( I I ) Gerischer, H 2 Phys Chem (Munich) 1960, 26, 223
0022-3654/9l/2095-1356$02.50/00 1991 American Chemical Society
Electron-Transfer Kinetics of Redox Reactions
The Journal of Physical Chemistry, Vol. 95, No. 3, 1991 1357 with ueI the electron-transfer frequency
redox
electrolyte
t
lHid is the matrix element of electronic interaction. In this model, the electron-transfer rate at equilibrium is given by
I
semiconductor
semiconductor redox electrolyte
10,redox
--
(b)
(a)
Figure 1. (a) Model for the interaction of an electron at the surface of a semiconductor with an electron acceptor in the electrolyte. (b) Energy correlations for an n-type semiconductor in equilibrium with a redox
system.
electrons at a distance p from a point at the interface opposite to these species. This model is shown in Figure la. The parameter p is the maximum distance from the interface to which the wave function of an electron extends into the electrolyte while still being able to interact sufficiently with the redox species. This is presumably only a few atomic distances of the crystal lattice. The spherical distribution assumed in the model shown in Figure l a is a very crude approximation. The rate is proportional to the number of reaction pairs P and to a rate constant k- in the forward direction. We only consider this path. The reverse reaction must proceed at equilibrium with the same rate, and we shall assume that the concentrations of Ox+ and Red are equal. io = eok-P
(1)
P = 6No,(2n/3)p3n,
(2)
with
No, is the concentration of Ox+ species per cubic centimeter, and n, is the concentration of electrons at the surface, which is in this situation in equilibrium with the bulk concentration: n, = no exp(
Comparison with the Electron-Exchange Rate in a Semiconductor/Metql Schottky Barrier The current through a solid-state Schottky barrier can be described by the thermionic or the diffusion theory.9*10,14-16At equilibrium, both give the same result since equilibrium is maintained up to the interface and eq 3 is valid. The current in forward direction, assuming a Maxwell distribution of the electron energies, is for a barrier height equal to that of the redox reaction at equilibrium:
(9)
meffis the effective electron mass and ii, the average velocity in the x direction, Le., normal to the interface. The comparison of both exchange currents gives
lometal
(
&)'I2
-$)= Nc exp( - 7 ) In order to get a numerical comparison, we shall use some estiSEC
(3)
EoB is the barrier height at equilibrium, EF the Fermi level (in the case of equilibrium being controlled by the Fermi level of the redox system, EF = Eoredon),N c is the density of states in the conduction band, and sEc is the energy position of the band edge at the surface. For the rate constant k- we use a semiclassical formulation given by SutinI2 for homogeneous redox reactions k- =
V,K,K~~
(4)
where v, is an averaged nuclear frequency for the ligand vibrations and the solvent orientation around the redox ion. K, takes account of the activation energy for the reorganization of the ligandsolvent structure:
mated numbers for the parameters of the redox reaction. The maximal number of u, will be an average frequency of the ligand vibrations in the order of K , = IO" s-l. 6 is a distance over which electron tunneling is still efficient. It is the sum of the distance over which tunneling occurs with high efficiency, which is between 0.8 and 2.4 A, depending on the reactants and their orientation," and the extension of the redox orbitals of the electron acceptor in solution. We shall use 6 = 3 X IO8 cm. The parameter p is the most uncertain one. We can only make a guess and will use p = lo-' cm as a rather large distance. The numerator of eq I O has the value
where m is the normal electron mass. The result with these parameters is X is the reorganization energy, and Eooxis the electronic energy level of Ox+ in its most probable configuration: Eoo, = Eorcdox + A. K , ~is an electronic factor which depends on the electronic interaction between the reaction partners. It can be expressed in the Landau-Zener approximation1' by
(12) Sutin, N. Prog. Inorg. Chem. 1983. 30, 441. (13) Sutin, N.; Brunschwig, 8. S. ACS Symp. Ser. 1982, 198, 105.
The factor IO2' could, however, be For K~~ = 1 and 1 M solution with No, = 6 X 1020,the factor in front of the exponential term is even with mer = m larger than 2. Since mer is for electrons of most semiconductors smaller than m (up to a factor of IO-' (14) Schottky, W.;Spenke, E. Wiss. Veroff Siemens-Werken 1939, 18, 225. (15) Bethe, H. A. MIT Radiation Lab. Report, 43-12, 1942. (16) Sze, S. M. Physics of Semiconductor Deoices, 2nd ed.; Wiley: New York, 198 1. (17) Newton, M. D.; Sutin. N. Annu. Reo. Phys. Chem. 1984, 35. 437.
1358 The Journal of Physical Chemistry, Vol. 95, No. 3, 1991
to and considering the uncertainty in v, and p, the factor in front of the exponential term could easily be 1 order of magnitude larger. The exponential factor of eq 11 is one for Eoox= sEc and larger than one for all redox systems with EO,, > sEc. The case of E O , , < sEc is in the inverted region,I9 where the expression for K , (eq 5) is not valid. The barrier height for the electrons at equilibrium with the redox system would in this case be sEc - Eoredox> X. It depends then on the Franck-Condon factors how much the rate constant for electron transfer to the Ox+ species decreases. But even if the rate constant would remain as high as for sEc = E O , , , the exchange currents for both types of Schottky barriers would decrease in parallel with the barrier height (cf. eqs 8 and 9). Therefore, the case sEc = Eoo, in eq 1 1 gives the lowest ratio between the two exchange currents at equal barrier height. Equation 1 1 is independent of the reorganization energy which should be larger in front of a semiconductor than at a metal electrode, as MarcusZorecently pointed out. This is due to the repulsive image forces at the semiconductor contact where the static dielectric constant in the depletion layer is much smaller than in the solvents of good electrolytes. The repulsive interaction between ions of the solution and a semiconductor with a Schottky barrier leads to an increase of the work term for ions approaching the interface. If the electronic factor should be K , ~ < 1, the exchange current at the semiconductor redox electrode decreases further. All these effects reduce the exchange current at the semiconductor-redox electrolyte contact. The conclusion is that the exchange current of a redox reaction at a semiconductor electrode can even at the highest possible concentrations of the redox systems hardly reach the exchange current of a semiconductor-metal Schottky barrier with the same barrier height.
Gerischer redox electrolyte
0 x+
O - Y+
---:q,---- -,q,,,-* .
I .
.
Q
L
b
A
semiconductor
Figure 2. Model for t h t forward current of an n-type semiconductor in contact with a redox system.
the barrier. Their concentration remains in equilibrium with the bulk. Assuming that the bias changes only the barrier height, we obtain for the redox reaction the same type of current voltage curve as for the semiconductor-metal contact.
Consequently, as long as 4kT. The larger that sEc - E,, is, the more likely it is that the reaction proceeds via surface states, provided the electron-transfer rate between the band edge and the surface states is fast enough. This will normally be the case, since the coupling between band states and surface states should generally be stronger than between surface states and redox species in the electrolyte. There is another factor that can favor the reaction path via surface states. There is some experimental evidence2s-26that KcI,cb in redox reactions at several semiconductors is much smaller than I . If the electronic coupling between surface states and the redox species is much stronger, i.e., K , ~ , , >> K,l,$,, then the reaction via surface states can predominate even with relatively small concentrations N,. Catalysis of electron-transfer reactions at semiconductors by deposition of small amounts of metal atoms could be explained by such an improvement of the electronic coupling. The case where E O , , IE,, is not represented by eq 18 or 19. The electron transfer occurs here in the inverted region for both reaction paths. Assuming that the rate constants decrease very little in this region, the nuclear activation factors of eqs 8 and 17 can be neglected, and the relation between the exchange currents becomes
This case is more favorable for the path via surface states. In this case, however, the transfer of electrons from the band edge to the surface states or from the bulk to the surface can become rate determining, if the electron exchange between surface states and the redox species is fast. Whether electronic equilibrium between bulk and surface remains established at cathodic bias or not depends on the distance sEc - E, and the transfer rate from the band edge to the surface states. If the latter reaction is fast, the transport of electrons from the bulk to the surface can be rate determining as at the semiconductor-metal Schottky barrier. This case, however, requires a barrier height at equilibrium E o B> X and a sufficiently large N,, as well as sEc - Ess. In 01 der to check this theoretical analysis, careful experiments are needed where the rate of simple redox reactions in the forward direction is studied at various concentrations of the redox species and the band bending can be well enough determined. Very few data exist in the literature, and this will be discussed together with some new experimental results in another paper. Acknowledgment. The model used here for the electron-transfer reaction was presented in September 1989 at a conference entitled "Frontier in Electrochemistry" at Sendai, Japan. I appreciate the helpful comments of Professor N. S. Lewis, Pasadena, and 1 am very grateful for the hospitality of Professor Adam Heller and his colleagues in the Department of Chemical Engineering. thc University of Texas at Austin, where I completed this paper. Thc support of DOE, which made by stay at Austin possible. is gratefully acknowledged.
e) kT
(18)
Inserting the numbers p,, = 5 X IO-* cm, pel = IO-' cm, 6 = 3 cm, and Nc = l O I 9 ~ m - one ~ , obtains X iO.cb
( 2 5 ) Unpublished results of redox reactions ill Ti02 iind %nO clcctrtdcs.
(26) Lewis, N. S. Personal communication.
