J. Phys. Chem. 1991, 95, 10566-10575
10566
FEATURE ARTICLE Charge-Transfer and Chemical Bond Formation Dynamics in Electrochemical Systems: Processes on Metals, Semiconductors, and Blomembranes W. Lorenz Theoretische physikalische Chemie, Universitiit Leipzig, LinnZstrasse 2, 0-7010 Leipzig, Germany (Received: April 2, 1991; In Final Form: September 3, 1991)
Charge transfer is the fundamental characteristic of electrochemical and photoelectrochemical processes. A deeper understanding of these topics requires the introduction of partial charge transfer into electrochemical dynamics, on both a quantum and macrampic level, taking into account the quantum-chemical phenomenon of electronic partial charge transfer in local chemical bond formation events as well as its connection with macroscopic interfacial double-layer formation. Developments in this field of ‘chemical” charge transfer have been started in the 1960s and further pursued in the 1970s and 198Os, turning to the general quantum dynamics of chemical bond formation processes in condensed systems. In this paper, we address firstly the wide area of electrochemicalmultistep processes and chemisorbed intermediateson metals. On biomembranes, the attained state 0Sunderstandin.g the instability-generating mechanisms of ion channel gating is outlined. We turn then to a synopsis of developments on “chemical” Franck-Condon transitions, connected with partial charge injection, and to the introduction of charge-transfer kinetics under electronic nonequilibrium, which forms the basis of electrochemical and photoelectrochemical processes, particularly on semiconductors.
1. Introduction The traditional access to electrochemical processes, initiated 100 years ago, had started from electrochemical thermodynamics. A deeper understanding of the intrinsic features of electrochemical processes requires a general dynamic and quantum-chemical foundation of elementary chargetransfer events. Following chiefly the latter line, several extensions of the fundamentals of electrochemical processes have been introduced since the 1960s, resulting in generalized concepts like “partial charge transfer” and “electronic nonequilibrium charge-transfer kinetics”. In order to understand the theoretical roots of these developments, we remember shortly the thermodynamic constraints of electrochemical process dynamics and the impact given by quantum theory to chemical dynamics. 1.1. Electroebemical Processes: Tbermodpllmic coastmints. The kinetics of electrochemical processes had started with Tafel’s relation’ between the nonequilibrium deviation of the electrode potential E and current density j on metal electrodes: E - E”s = b log (j/ja) ( ( j / f ’ ) >> 1, b and f’parameters) (1)
The next important step was the derivation of electrochemical equilibrium (Nernst equation) from one-step redox kinetics by Butler,2 who has established the potential dependence of electrochemical forward and backward rate coefficients kf and kb:
R
a
0 + ze-; - In ( k f / k b )= z F / R T
(2)
aE R and 0 are the reduced and oxidized redox components. A first interpretation of the coefficient b in Tafel’s equation in accord with experimental data was given by V ~ l m e ralong , ~ the lines of eq 2, together with the concept of “slow discharge”. Over the years, the empirical importance of Tafel’s relation (1) has not diminished, but its mechanistic information content has proven ambiguous. The reason is that actually multistep processes (1) Tafel, J. Z . Phys. Chem. 1905, 50, 641. (2) Butler, J. A. V. Trans. Furaday Soc. 1924, 19, 729. (3) Erdey-Gruz, T.; Volmer, M. Z . Phys. Chem. 1930, A150, 203.
0022-3654/91/2095-10566%02.50/0
with incompletely known intermediates take place. The thermodynamic constraints to kinetics then become less definitive; e.g., for an interfacial multistep sequence one obtains
where the Ti are chemisorbed intermediates. Traditional electrochemistry assumes one, or possibly z, integral electron-transfer steps in an interfacial sequence, and otherwise precursor steps without charge t r a n ~ f e r . ~However, eq 3 admits more general situations: a foundation of partial charge transfer based on quantum-theoretical charge conservation in electrochemical elementary processes5-’ has been under development since the early 1960ssand will be considered more closely in sections 2 and 3. 1.2. Nonequilibrim ” d y n a m i c Cosstrai~~ts. Two rather unfamiliar results, derived from nonequilibrium entropy balance and a kinetic consideration of the Onsager symmetry of chemical elementary processes, receive attention in the present c o n t e ~ t . ~ J ~ Firstly, the kinetic independence of exchange rates of elementary processes (following from general potential surface consideration) is equivalent to the diagonality of the flux coefficient matrix
Lik = (UP/RT)sik
(4)
where up is the exchange rate of process i at equilibrium. Seoondly, the postulate of equality of produced entropy (multiplied by temperature) and dissipated energy, applied to nonstationary linear (4) Vetter, K. J. Z . Electrochem. 1951, 55, 121. (5) Lorenz, W.; Salib, G. Z . Phys. Chem. (kipzig) 1961, 218, 259. (6) Lorenz, W. Z . Phys. Chem. 1961, 218, 272. (7) Lorenz, W.; Krllger, G. Z . Phys. Chem. 1962, 221, 231. (8) Adsorbed intermediates in electrochemical multistep processes had otherwise been confmad to integral or zero charge transfer steps. Cf.: Holub, K.; Tessari, G.; Delahay, P. J . Phys. Chem. 1967, 71, 2612. (9) Lorenz, W. Z . Phys. Chem. 1973, 253, 243. (10) Lorenz, W.; Behrend, S. Z . Phys. Chem. 1974, 255, 1061.
0 1991 American Chemical Society
The Journal of Physical Chemistry, Vol. 95, No. 26, 1991 10567
Feature Article processes in frequency representation, yields
where on the left side are local entropy production contributions (absolute values of products of affinities Ai and rates vi of interfacial processes; the second term allows for entropy production connected with reactant diffusion in solution) and on the right side is the dissipated energy; Vis the applied voltage. In section 3 we shall rely on eq 5, which yields a general relation between charge balance and potential dependence of partial charge transfer processes. 1.3. Impact from Quantum Theory. Two essentially new notions have been brought forth with quantum mechanics: firstly the energy eigenvalues and secondly probability density distributions in real space, defining electronic partial charges. Quantum theory has moreover rendered possible the proper definition of basic kinetic notions: it may suffice here to remember the Born-Oppenheimer principle (1927) defining adiabatic potential surfaces and the introduction of quantum-statistical transition probabilities appearing in the Pauli master equation (1928). Inclusion of electrochemical processes in a quantum treatment has proved more difficult, however. In an approach tracing back to Gurney,]] current densities have been introduced as integrals over one-electron energy, and the integrand has been postulated to be proportional to occupied density of electronic states in the solid for a cathodic (backward) process or proportional to unoccupied density for an anodic (forward) process. Such expressions suppose a priori a transfer of one integral electron, thus disregarding chemical bond formation across the electrode interface. Even for one-electron transfer, the addressed form of the integrand has proved incompatible with electronic nonequilibrium charge-transfer kinetics considered below in section 7. The approach of electrochemical processes in terms of potential surfaces has been initiated by Horiuti and PolanyiIz in the scheme of integral-electron transfer: the representation of electrochemical processes requires a distortion of potential surfaces, due to the action of the electrode potential which is considered as an external perturbation. A more recent review which takes account of both items addressed here has been given in ref 13.
2. Kinetics and Charge Conservation: Basic Notions of Partial Charge Transfer Processes Charge conservation holds generally, in macroscopic terms, as well as in quantum-theoretical terms as conservation of probability density of charged particles. The strongest quantal pattern is exhibited by electrons. This consideration forms the background for introducing a nonintegral charge stoichiom*:try of electrochemical multistep processes R
* r,
... rrl
F;:
0 + re-
(6)
addressed already in eq 3. The r i are intermediate, physisorbed, or chemisorbed states of the reactant, which in the case of chemisorption carry an electronic partial charge. Starting from quantum-theoretical charge balance and allowing for chemical bond formation over a microscopic electrode interface, the concept of partial charge transfer processes has been introduced for metal electrodes in refs 5-7; first comprehensive treatments have been given in refs 14 and 15. Accordingly, the basic relation of current densityj to rates u of elementary, interfacial charge transfer steps is j=
CAFv + dq/dt
(7)
where the sum is the current which transgresses the electrode (11) Gurney, R. W. Proc. R . SOC.London 1931, AI34, 137. (12) Horiuti, J.; Polanyi, M.Acta Physicochim. URSS 1935, 2, 505. (13) Levich, V . G. Adu. Electrochem. 1965, 4, 249. (14) Lorenz, W. Z . Phys. Chem. 1970, 244.65. (15) Lorenz, W.; Salit, G. . I Electroanal. . Chem. Interfacial Electrochem. 1977, 80, 1.
interface. The X are “quantum-chemical” or “microscopic” charge-transfer coefficients which give the partial charge transgressing this interface in a local chemical bond formation process. (-4) is the excess charge stored on the electrolyte side of the interfacial double layer. (This notation also applies for semiconductor electrodes.) Charge transfer of A-type is accessible, in principle, by quantum-chemical calculations, but not immediately (i.e., without invoking further arguments) by measurement. The reason is that expansion of dqfdt in eq 7 contains terms d r f d t which in turn are connected with rates v by mass balance. It is however possible to transform eq 7 into a completely measurable representation in terms of partial charge transfer of 1-type: j = ElFv
+ C, d+q.,/dt
(8) The 1 are “macroscopic” charge-transfer coefficients which can be measured (most simply under the familiar kinetic condition of excess supporting electrolyte) on a macroscopic ensemble of identical intermediates. C, = a q / d e is the interfacial Helmholtz capacity at fixed adsorption densities of the reactant (i.e., in the limit of high frequency; on metals the derivative after the interfacial dipolar Helmholtz potential (PH can be replaced by a derivative after electrode potential E). The coefficients li are related to Xi, e.g., for the primary step of the sequence ( 6 ) , according to i1 = x1 + aq/Far, (9) where the latter quantity on the right side is a double-layer reorganization term. Under the condition of chemisorptive bonding of intermediates, there are indications that li is dominated by Xi. It is important to realize that both types of partial charge transfer are characteristic of kinetic elementary processes. There are simple criteria for homogeneity of macroscopic ensembles of intermediate states, based on dependences of 1on electrode potential and adsorption densities.I4 We shall return to the prospects offered by comparison of measured 1coefficients and calculated A coefficients in section 8.
3. Electrochemical Processes with Chemisorbed Intermediates on Metals The possibility of partial charge transfer forces one to replace the traditional a priori assumption of integral charge-transfer s t e p in electrochemical reactions by experimental determination of nonintegral charge-transfer coefficients of 1-type, following eq 8. For mechanistic evaluations, quantum-chemical estimates of partial charge transfer of A-type are needed as well. In the following we rely once moreI4J5on the most important first-order coefficients of electrochemical kinetics and on three main experimental topics and related techniques for determination of partial charge transfer of 1-type. 3.1. First-Order Coefficients of Electrochemical Kinetics. Rate coefficients kif and klb of interfacial multistep processes depend, in the general case of nonlinear adsorption isotherms of the intermediates, on both the electrode potential E and adsorption densities l’,. In addition to the universal charge balance expressions (7) and (8), one has, obeying the thermodynamic constraint eq 3
wheref;, are coefficients of potential dependence. In eqs 7,8, and 10 the Xi, li, andf;, are normalized to the reaction charge number Z.
n
EA, = ?li
n
=
w;.= z
(11) valid for a sequence of steps from reduced to oxidized bulk state (eq 6 ) . Thef;, can be further divided into forward and backward terms
a
a
- In kl%= -(F/RT)(rib (12) aE In kif = (F/RT)aif, aE (Yif
+ a i b =f;
10568 The Journal of Physical Chemistry, Vol. 95, No. 26, 1991
where the at and ab are so-called Tafel coefficients, recalling their first (but mechanistically not yet understood) occurrence in eq 1. For further first-order coefficients expressing dependences of kfand k b on adsorption densities, we refer to refs 14-16. The set of coefficients li andA are separately measurable. From theory, both turn out to be equal when one supposes “elementary” behavior (cf. refs 9 and 14): for the general case of multistep processes, this follows from nonequilibrium entropy balance, eq 5:17
f;. = li
(13)
The principle of determining li consists of an independent measurement of charge and mass to be supplied in an elementary step to the electrode. In general, the data required for that purpose allow also the additional evaluation off;, which however is generally less accurate because of the numerical differentiation required in eq 10. 3.2. Impedance Spectrometry. The fust example which allowed a complete evaluation of partial charge transfer was the Tl(Hg) * Tl(ad) * Tl+(aq) reaction. All required data could be obtained5J8from real and imaginary impedance components in the frequency range 1-100 kHz, independent of bulk concentrations c(T1) and c(Tl+), with the approximate result 1, =fi = 0.6 and l2 = fi = 0.4 (supported by refined eval~ation’~). In principle, impedance spectrometry in a broad frequency and reactant concentration range is a powerful technique for the examination of multistep processes, because any multistep mechanism can exactly be treated in terms of first-order coeffic i e n t ~ . ’“Anomalous” ~ 1 coefficients, i.e., 1 C 0 and/or I > z, due to “induced” adsorption, have been studied in ref 19. Recent progress in computer-assisted evaluation of impedance spectra has been discussed in ref 20. 3.3. Anion Chemisorption on Mercury Electrodes. A straightforward evaluation of partial charge trar sfer is possible on adsorption reactions, which can be considered as an electrochemical two-step or multistep sequence whose last (or first) step is suppressed for kinetic or thermodynamic reasons.2’ The charge supplied to the electrode can be determined by capacity measurements, the mass with the aid of Gibbs’ adsorption thermodynamics, likewise using capacity data. Many systems had been studied up to the late 1 9 6 0 ~ , ~including ’ , ~ ~ as “best” examples I-, Br-, or SZO?-on mercury in different ground electrolytes. I and fhave proved only weakly varying over potential ranges of about 0.5-V breadth, which supports a dominating contribution of X to 1 (eq 9). The state of comparison with quantum-chemical data will be addressed in section 8. 3.4. Metal Cation Chemisorption on Solid Metals. Progress has been attained recently in metal ion chemisorption on solid metals, known for a long time as “underpotential deposition”.23 Considerable partial charge transfer in primary and consecutive chemisorption steps has been found by SaliE23,24 in several metal cation chemisorption processes on gold, using a potentiostatic pulse technique combined with rotating ring-disk hydrodynamics. An impedance spectrometric gave comparable data. These results point to an accessibility of wider areas of chemisorption
Lorenz processes of solute species on solids. The examples addressed in sections 3.2-3.4 may suffice for illustrating the experimental access to partial charge transfer of I-type, together with further kinetic information; a comprehensive survey is not intended here.
4. Charge Transfer through Biomembranes We turn in this section to some specific problems of charge transfer through biomembranes. A widespread property of biomembranes is their excitability into unstable regimes of oscillation or saddlepoint type which in several cases exerts a specific biological function. Oscillatory dynamics in chemical and biological systems has a long history,26and we can look back to a great tradition by physicochemists at this university who contributed to this topic: Fechner (1828), Ostwald (1899), Bonhoeffer (1943). 4.1. Search for Instability-Generating Mechanisms of Ion Channel Gating. Our interest in biomembrane charge transfer was raised by the possible interference of adsorption phenomena in the so-called gating (opening and closing) of ion channels built up by membrane proteins embedded in a lipid b i l a ~ e r : ~ ’the -~~ possibility of “anomalous” charge-transfer coefficients in induced adsorption processes (section 3.2) gave occasion to ask for the instability patterns far from equilibrium of an electrochemical process subjected to an inverted potential dependence ( I C O).30 The next step was to include this possible instability-generating mechanism in the whole membrane dynamics, described in terms of a field representation of ion transfer through a membrane composed of a bulk phase and two interfaces.31 This means that the instability-generating element was assumed as part of the intrinsic transport sequence of a membrane permeant: diffusion I interfacial in 2-step process solution I with anomalous charge transfer
I
I membrane I
channel transport
1
interfacial process
I diffusion I
in
solution I
I
(14)
Further dynamical analysis of this approach32showed that, besides the necessity of certain ad hoc assumptions, one hardly escapes the conclusion of too large oscillation onset potential and too large potential oscillation amplitude, when the gating process is driven into nonlinear regimes by an interfacial dipolar Helmholtz potential variation: obviously, a larger part of the total membrane potential is needed as a driving force. 4.2. Instabirity Patterns of Cyclic Channel Transformation Mechanisms. The latter requirement is fulfilled when the whole channel is subjected to potential-dependent transformations between “open” and “closed” states. Processes of this kind are under d i ~ c u s s i o n , 3above ~ ~ ~ ~all for the gating of sodium channels. The point is that such channel transformations, which decide over operating of the intrinsic ion permeation sequence through the membrane, can become the instability-generatingelement when the channel transitions follow a cyclic mechanism. On this basis, a simulation study of cyclic channel transformations together with intrinsic permeation dynamics has been carried through by S c h u l ~ e . Cyclic ~ ~ transformations of channel states Ci following
co * c1 + ... C,I (16) Lorenz, W. Z . Phys. Chem. 1971, 248, 161. (17) Reference 10. For the special case of one-step adsorption, eq 13 can be derived from Gibbs’ adsorption thermodynamics: Damaskin, B. Elektrokhimiya 1969.5, 771. Cf.: Lorenz, W. Z . Phys. Chem. 1976, 257, 63, 74. (18) Salic, G.; Lorenz, W. Z . Phys. Chem. (Munich) 1961, 29, 408. (19) Salic, G. Z . Phys. Chem. (Leipzig) 1972, 250, 1 . (20) Salic, G. J . Electroanal. Chem. Interfacial Electrochem. 1989, 273, 1, 31. (21) Lorenz, W. Z . Phys. Chem. (Leipzig) 1962,219,421; 1%3,224, 145. (22) Lorenz, W. 2.Phys. Chem. 1964, 227, 419; 1966, 232, 176; 1969, 242, 138; 1970, 243, 356. (23) Salic, G. J . Electroanal. Chem. Interfacial Electrochem. 1988, 245, 1. 21.
I
co
(15)
are subjected t o
~
~
(26) For a review, cf.: Nicolis, G.; Portnow, J. Chem. Rev. 1973, 73, 365. (27) Hille, B. Biophys. J . 1978, 22, 283. (28) Liuger, P. Angew. Chem. 1985, 97,939. (29) Fisher, K. A.; Stoeckenius, W. In: Hoppe, W., et al., Us.Biophysics, 2nd ed.;Springer: Berlin, 1982; p 426. (30) Salic, G. Bioelectrochem. Bioenerget. 1980, 7, 625. (31) Lorenz, W.; Schulze, K. D. Z . Phys. Chem. 1981, 262, 1032. (32) Friedrich, T.; Schulze, K. D.; Lorenz, W. Z . Phys. Chem. 1990,271, 369. (33) Armstrong, C. M. Physiol. Rev. 1982, 61, 644. (34) Aldrich, R. W. Trends Neurosci. 1986, 9, 82. (35) Schulze, K. D. Chem. Phys. 1991, 156, 43.
Feature Article
The Journal of Physical Chemistry, Vol. 95, No. 26, 1991 10569
INTERFACE
Mechanisms of this kind guarantee channel transitions with different signs of potential dependence, which is a necessary condition of dpamical instability. One can introduce coefficients of potential dependence of the channel transformation steps, following
cpm is the inner membrane potential assumed as a driving force for channel transformations. From eq 17 different signs of individualf;. are obvious. The occurrence of all instability types with this mechanism has been verified in ref 35, and a preliminary figure of a lower bound of 1;. necessary for instability onset has been obtained. 4.3. M i t of Channel TransformationData. The potential dependence of channel transformations is of course related to a change of channel charge distributions. When one assumes (in advance of better knowledge) that such a change can be approximated as dipole moment difference Afii = heL, where L is the membrane thickness, then &1; is the change of charge (in units of e) on the channel orifices; with shorter dipole length, a larger change of charge would be necessary for the same effect. In previous related ~ o r k , ~ charge ’ . ~ ~ displacements in the channel interior have been proposed as a cause of channel transformations. Relatively largeh values (>3-4) apparently required for instability onset give occasion to ask for a possible interference of chemisorption processes on a channel orifice, above all of bivalent cations, eventually in a concerted process. A correspondingh value would then depend, in the simplest case, on the local charge on the attached ions, including partial charge transfer from channel protein and the location of the opposed charge. The latter point can give rise to dynamical complications. Finally in this section, we notice that experimentally observable potential oscillations on biomembranes show mostly a deterministic pattern, in accord with the macroscopic description of the elementary processes.
5. Interfacial Charge-Transfer Processes on Semiconductors A most interesting feature of interfacial electrode processes is the fact that the quantum-chemical partial charge transfer, which is caused by chemical bond formation across the interface, appears in the macroscopic current density expression, eq 7. From this view, the electrode processes on semiconductors were particularly challenging, for several reasons: (I) Local chemical bond formation on semiconductor interfaces is connected with a partial charge injection into a medium-the semiconductor space charge layer. (11) This charge injection occurs in general under electronic nonequilibrium of this medium, caused by slow carrier (electron and hole) transport and/or carrier photogeneration. (111) Developments of quantum dynamics of electrochemical systems should take advantage of the fact that the Born-Oppenheimer separation of electronic and nuclear motion is simpler on semiconductors than on metals, thus favoring the former as theoretical prototype. in order to include features 1 and 11, several extensions of charge-transfer theory become necessary and have been taken into investigation, stepwise, over the past 15 years. The main topics connected with these developments will be considered in this section and in sections 6 and 7. 5.1. Partial Charge Injection in a Medium (Space Charge Layer) at Electronic Equilibrium or Nonequilibrium. On semiconductor electrode interfaces it is useful to distinguish an excess charge -4 at the electrolyte side of the interface, a surface charge qssat the semiconductor side, and the charge Q,, of the semiconductor space charge layer. These charges are subjected to electroneutrality The general current density expression (7) now takes the form j = CXFu + d(q, + Q,)/dt (19) It is again advisable to transform the u-dependent parts of dq,/dt
I I
’
I
tqss
Qsc
I
1
-q
. I
Figure 1.
into a new kind of charge-transfercoefficient, the so-called charge injection coefficients m, leaving an intrinsic (u-independent) surface-charging term dq&/dt. This transformation of eq 19 yields j = CmFv
+ dq’Jdt
+ C,, dpS,/dt
(20)
where C,, is the space charge capacity and cpsc = cp(bu1k) cp(surface), the semiconductor space charge potential governed by Poisson’s equation. These patterns are sketched in Figure 1, where the macroscopic ,, qss,and -q are shown. Further, a microscopic sucharges Q permolecular reaction complex S consisting of a semiconductor part S1 and an adsorbate part S2 is shown. The charge injection coefficient m has been calculated in terms of local densities of states, under the condition of electronic equilibrium, in ref 36. One obtains, e.g., for an anodic primary process X + Q(S1) + lC,/C, m= (21) 1+
c,/c,
where Q(S1) is the local charge in the subsystem S1 shown in Figure 1. On systems with a space charge layer, a surface capacity C, is reasonably defined by dq’,/dt
= C,,dcp,,/dt
(22)
At electronic equilibrium, C,,takes the form
(2= 2e2Ns(EF,,)
(23)
where N,(EF,) is the total surface density of states at the surface Fermi energy EF,s: Ns, = MSS + Y
W l
(24)
No, (eV-’ cm-2) is an intrinsic surface density of states, y (cm-2) the density of reaction complexes, and ANsl (eV-l) the local difference density of states in the subsystem S1 (see eq 26). Charge injection under electronic nonequilibrium has been considered in ref 37. There is an important constraint upon the surface capacity C,,caused by a restricted movement of nonequilibrium surface Fermi energies EFs relative to the band edges (e.g., the conduction band minimum E,-) under cpsc variation (band bending). Instead of eq 23, one gets at electronic nonequilibrium
c, = a q ~ / a 9 =~ ,c w C - E ~ , s ) /acpsc e
(25)
where the factor a(Ec - E , s ) / e a9,, can be subjected to transport-controlled pinning of the surface Fermi energy. We shall consider this interesting pattern in greater detail in section 5.4. 5.2. Local Density of States Representation of QuantumChemical Charge Transfer (A- and m -Type):Energy-Resolved (36) Lorenz, W. Z . Phys. Chem. 1982, 263, I. ( 3 7 ) Lorenz, W.Phys. S t a m Solidi 1987, 8144, 585.
10570 The Journal of Physical Chemistry, Vol. 95, No. 26, 1991
Partial Charge Transfer. Quantum-chemical charge transfer of A- and m-type can properly be expressed in terms of electronic local densities of states n(E,i). E labels a one-electron orbital energy scale, and i the orbital under discussion. calculation^^*-^^ have shown that a chemisorption process on a semiconductor surface exerts influence on local densities only in tLe neighborhood of the adsorbate. This allows a straightforward choice of an interfacial reaction complex S following Figure 1. A useful pattern of a local chemisorption process is the difference density of states in the reaction complex ANs =
C [n(E,i)- no(E,i)]
iES
(26)
where n and no are local densities of states in the final and the initial, or reference, state of the process. Analogous definitions hold for subsystems S1 and S2. The relative charge of a local reaction complex (referred to as a reference state) is Q ( S ) = Q(core) - 2 S E -m F d d E ANs
(27)
written in units of e. The analogously formulated charge Q(S2) of the adsorbate determines the partial charge transfer A: one has, e.g., for an anodic primary process A = Q(S2) - ZR (28) where zR is the (integral) charge of the anodic reactant in solution. Under transport-controlled pinning of surface Fermi energy (to be considered in section 5.4),the charge injection coefficient of an anodic primary process is m = Q(S1) + X = Q(S) - ZR (29)
In the case of a displacement process with coupled charge transfer, eqs 28 and 29 must accordingly be extended. Difference densities of states ANs or ANs2following eq 26 give immediately a representation of orbital energy resolved partial charge transfer of m- or A-type, Le., an energy distribution of partial charge transfer. As a result of calculations, particularly for chemisorption of H, OH, H 2 0 , and halides on GaAs, Gap, and Si, (1 lo), (1 1l), or (100) s ~ r f a c e s , ~ *the - ~ 'chemisorptioninduced charge transfer occurs over the whole valence band region, while charge transfer in the bandgap region is negligible in the cases of the mentioned electronegative adsorbates on 111-V materials, due to negligible density and difference density of states in the bandgap region of 111-V materials. In contrast, on silicon a considerable intrinsic gap density of states is obtained which is diminished by electronegative adsorbates (negative difference density of states, to a minor extent also observed in the upper bandgap region of GaAs( 100A)). Energy-resolved partia.1 charge transfer can exert a static and dynamic action. Static action is connected with static chemical bonds; dynamic action is connected with chemical elementary processes accompanied by local density of states transfer expressed by ANs2 or Finally, a brief comment may be given on calculational conditions. (a) For comprehending the most important energy range of the bandgap of a semiconductor surface, a total energy representation of tight-binding (TB) type E t o t a ~=
CWi
(30)
is indispensable.39*@Ei are TB orbital energies (to be distinguished from Hartree-Fock-type orbital energies), and ni are occupation numbers. Equation 30 is a sufficient condition for applying (38) Engler, C.; Lorenz, W. Z . Phys. Chem. 1980, 261, 92. Lorenz, W.; Engler, C. Surf. Sci. 1980, 95, 431. Engler, C.; Lorenz, W. Surf. Sci. 1981, 104, 549; 1982, 114,607; Phys. Status Solidi 1982, B113, 227; 1983, B117, 203. Katterle, T.; Lorenz, W. Phys. Status Solidi 1985, 8132, 225. (39) Lorenz, W.; Katterle, T. Phys. Status Solidi 1987, B142, 149. (40) Lorenz, W.; Rommel, K. J . Electroanal. Chem. Interfacial Electrochem. 1988, 250, 37. (41) Rommel, K.; Engler, C.; Lorenz, W. Abstracts of the 9th Kinetics Conference, Leipzig, Feb 1989; pp 73, 74.
Lorenz one-electron Fermi statistics.42 For the present problems it suffices to assume spin degeneracy. (b) Local densities of states n(E,i) were directly calculated from Green matrix diagonal elements Gii of a TB Hamiltonian
n(E,i) = ( - l / r ) Im Gii(E)
(31)
using a local expansion (recursion) of Gii into a continued fraction; E labels the TB orbital energy scale. Calculational errors result from TB approximations and on approximations in the termination of continued fractions (the latter affecting mainly the conduction band density pattern). Reliability of local charges and of upper bounds of gap densities of states have been proved in that respect!' (c) Comparisons with other TB-like calculations on reconstructed surfaces43and on 111-V semiconductor heterojunctions" have been made. In chemisorption on 111-V materials, adsorbate charges are due to bonding orbital tails which jump beyond the bandgap into the conduction band region.@ On heterojunctions with ideal structural accommodation, these tails vanish and in turn only spurious upper bounds of interfacial charge are obtained, together with negligible gap density of states.41 5.3. Nooequilibrium Fermi Energy Profiles of Electrons and Holes. We turn now to the phenomenon of electronic nonequilibrium, addressed already in section 5.1. It is caused by slow electron or hole transport and/or electron-hole pair photogeneration in a semiconductor space charge layer (depletion layer). Our first step into this topic was an approximate but reliable calculation of nonequilibrium electron and hole density profiles n(x) and p(x) over the space charge layer, from electronic balance and transport equati0ns.4~ Such calculations can straightforwardly be carried through in the nondegenerate (depletion or weak inversion) regime of the semiconductor-in these regimes the electron and hole densities are completely decoupled from the electrical potential profile (o(x) which is governed by the ionized donor density. (We restrict ourselves here to n-semiconductors.) It is expedient to express the carrier densities relative to an arbitrary electronic equilibrium reference state, possibly at different space charge potential cc, (see the legend to eq 20), which yields the quotients
n(x)/n%)
and P(X)/P%)
The electronic nonequilibrium carrier density profiles depend on the kinetic boundary conditions of electron and hole flows at the interface; on 111-V materials, the main carrier-consuming or -producing process on the interface is charge transfer. In the nondegenerate regime, the nonequilibrium Fermi energy profiles of electrons and holes are46 EF,"(x) = EF
+ kT In [ n ( ~ ) / n ' ( ~ )-] eA&)
EF,Jx) = EF- kT In [P(x)/~%)l - e A d x )
(32)
where A&) = I&) - @ ( x ) . Equations 32 hold relative to an energy scale fixed to bulk band edges; the familiar equilibrium Fermi energy EFis constant over x . The state labeled "eq* in eqs 32 is subjected to a weaker electronic equilibrium condition, requiring only a Boltzmann relation between nq(x)/n(bulk) or pq(x)/p(bulk) and cdq(x), while otherwise the reference band bending qz can arbitrarily be chosen. For interfacial processes, the nonequilibrium surface Fermi energies EFOrelative to the band edges at the surface are relevant: these are obtained from (32) by addition of ea&) and obeying &' = V(bu1k) - @(x=O) (42) Slater, J. C. Quantum Theory of Molecules and Solids; McGrawHill: New York, 1974; Vol. 4. (43) Chadi, D. J. Phys. Rev. 1978, 818, 1800; Phys. Rev. Lett. 1984,52, 1911. (44) Pollmann, J. Adv. Solid State Phys. 1980, 20, 117. (45) Lorenz, W.; Handschuh, M. J . Electroanal. Chem. Interfacial Electrochem. 1984, 178, 197. (46) Lorenz, W.; Aegerter, C.; Handschuh, M. J . Electroanal. Chem. Interfacial Electrochem. 1986, ZOO, 83. Lorenz, W.; Sourisseau, R. Ibid. 1988, 239, 9.
The Journal of Physical Chemistry, Vol. 95, No. 26, 1991 10571
Feature Article e
EF,n,s
s
= EF(bulk) - e@ (0)
= E2s + kT
(ns/@)
ms - kT
(Ps/p?)
EF,p,s
ADIABATIC GROUND-STATE
DOUBLE-MI” POTENTIAL SURFACE
(33)
(referring to surface band edges; nondegenerate regime). 5.4. Transport-Controlled Pinning of Surface Fermi Energies. An interesting feature of electronic nonequilibrium is transport-controlled pinning of surface Fermi energies of electrons and h01esp7which follows from camer balance and transport treatment of electronic n o n e q u i l i b r i ~ m . 4 ~This ~ ~ ~ubiquitous type of Fermi energy pinning is independent of presence or absence of surface density of states in the bandgap. A relevant quantity is a(Ec EFs)/e&oSc which has been addressed already in section 5.1, eq
(b)
A - P R I O R I D I A B A T I C PROCESSES
ELTCTRONIC E X C I T A T I O N
25.
ELECTRON TRANSFER
Calculations (cf. ref 46) point out a different pattern for kinetically irreversible charge transfer (Le., backward reaction negligible) or for fast near-equilibrium charge transfer: the former is common in semiconductor electrode processes, and the latter is expected on solid-state junctions with electronic conduction throughout. (I) In the case of irreversible charge transfer, one has for both EF.n,s and EF,p,s at weak depletion d(Ec - E F , s ) / edcpsc = 1
