J . Phys. Chem. 1985,89, 3783-3791
3783
FEATURE ARTICLE The Fermi Level and the Redox Potential H.Reiss Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90024 (Received: March 19, 1985)
This paper presents a simple didactic discussion of the Fermi level as the electrochemical potential of the electron in both solids and nonmetallic liquid solutions. Its identity with the electrochemical potential is demonstrated whether or not (1) free electrons are present or (2) the spectrum of energy levels over which the electrons are distributed is, or is not, invariant to the distribution itself. It remains a matter of individual taste as to whether or not the term “Fermi level” should be used in solution. The equivalence of the Fermi level to the redox potential is demonstrated by simple thermodynamic arguments, uncluttered by the possibly confusing paraphernalia of specialized cycles. The meaning of the absolute redox potential is clarified. The main thrust of the paper is toward the elimination of the various controversies which seem to have arisen concerning these subjects.
I. Introduction In recent years considerable attention has been focused on the electrolyte-semiconductorjunction.’ Among other things this interest has been generated by the many investigations*-* of the photovoltaic characteristics of such junctions. A useful theory of these characteristics requires the knowledge of the relative positions of the electrochemical potentials of an electron in the semiconductor on the one hand and in the electrolyte on the other. Unfortunately, electrochemical potentials in the semiconductor are available as “Fermi levels” which are already referenced to the vacuum, since they are determined by the measurement of “work functions”, while the electrochemical potentials of the electron in redox couples (electrolyte), although available, are usually referenced to the standard hydrogen electrode (SHE). The necessary establishment of the relative positions of these quantities in the two phases clearly requires that they be located on a common scale. Ultimately, the problem of a common scale can be seen to be identical with that of determining the so-called absolute half-cell a concern which predated the development of the electrolyte-semiconductor junction. Interest in this junction generated problems involving a ‘‘social’ as well as a “physical” interface. Chemists and physicists were forced to learn each other’s language and to reconcile versions of the same concept couched, alternately, in these separate languages. Although, by this time, many workers on both sides of the disciplinary interface have (1) Dewald, J. F. BellSysr. Tech. J. 1960, 39, 615.
(2) Gerischer, H. In “Physical Chemistry, An Advanced Treatise”; Henderson, D., Jost, W., Eds.; Academic Press: New York, 1970; Vol. IX. (3) Gerischer, H. In “Advances in Electrochemistry and Electrical Engineering”; Delahay, P., Ed.; Interscience: New York, 1961; Vol. I. (4) Gerischer, H. J. Elecrrochem. Soc. 1966, 113, 1174. (5) Chang, K.C.; Heller, A.; Schwartz, 9.; Menem, S.; Miller, 9. Science 1977, 116, 1097. (6) Butler, M. A. J. Appl. Phys. 1977, 48, 1914. (7) Laser, D.; Bard, A. J. J. Electrochem. Soc. 1976, 123, 1833. (8) Reins, H. J. Elecrrochem. SOC.1978, 125, 937. (9) Lohmann, F. 2.Naturforsc., A 1967, 224A, 843. (1 0 ) Gomer, R.; Tryson, G. J . Chem. Phys. 1977, 66,441 3. (11) Gerischer, H.; Ekardt, W. Appl. Phys. Let?. 1983,43, 393. (12) Trasatti, S. J. Electroanal. Chem. 1982, 139, 1. (13) Gurevich, Yu. Ya.; Pleskov, Yu. V. Electrokhimlya 1982, 18, 1477; Sov. Electrochem. 1982, 18, 1315. (14) Reiss, H.; Heller, A. J. Phys. Chem., in press.
0022-3654/85/2089-3783%01 .SO10
managed to dispel their own confusions, papers and controvers i e ~ ’ continue ~J~ to appear which indicate that some confusion still persists. The present paper is a didactic review designed to aid in the removal of this last confusion. It deals primarily with the conceptual relation between the Fermi level and the redox potential. They are really the same thing, namely, the electrochemical potential of the electron, but the weight of tradition makes it difficult for some workers to be comfortable with this fact. Perhaps one should therefore reserve the term “Fermi level” for the electrochemical potential of the electron in solids alone. We discuss this issue further below, but we take no sides in this matter. In order to maintain a clarity of presentation suitable for all of the readers who may be interested in this subject we are forced to deal, in places, with some elementary subjects, e.g., the derivation of Fermi statistics. Although some apology to the more sophisticated reader is therefore appropriate, it is worth emphasizing that the subject can rapidly develop subtle conceptual aspects in connection with which convenient reference to such elementary ideas may be very helpful. 11. Fermi Statistics Fermi statistics is usually derived for a system of particles distributed, at equilibrium, over a spectrum of quantum states which is invariant to the distribution; i.e., energy levels do not change as the distribution changes. In other words the particles are “weakly coupled”; they do not interact. However, from one point of view the interaction is very strong; the Pauli exclusion principle must be obeyed, and each state can be occupied by no more than a single particle. It is the very “strength” of the interaction which allows us to treat the system as “weakly coupled”, because, once it is known that only one particle can occupy a state, this restriction can be built into the statistics, at the outset. Then there are no further restrictions, and the particles may be treated as noninteracting. Another (cumbersome) approach would be to have the level of a state make a transition to infinite energy when more than one particle occupies the state. Then the system would appear to be “strongly coupled”, and the spectrum of levels would be strongly dependent on the distribution. However, the resulting (15) Khan, S. U. M.; Bockris, J. O M . J. Phys. Chem. 1985, 89, 555. (16) Scherson, D.; Ekardt, W.; Gerischer, H. J. Phys. Chem. 1985, 89, 554.
0 1985 American Chemical Society
3784
The Journal of Physical Chemistry, Vol. 89, No. 18, 1985
statistics would prove to be the same as in the former case. Electrons on ions, say in aqueous solution or, for that matter, in solids, would represent an intermediate case. For example, the energy level of an electron on a ferrous ion, Fe*+, may depend on the concentration of these ions, since they interact. If ferrous ions are in equilibrium with ferric ions the number of ferrous and ferric ions in the solution may vary, at equilibrium, according to the reaction (where e- denotes an electron) Fe3+
+ e-
P
Fe2+
Reiss A = N p - k T C w j In i
chem pot. =
E =
En)”) i
T,V
chem pot. = N( $ ) T , v
(4)
J
wi
(5)
1 + &-P)
where /3 and /3p are associated with undetermined multipliers introduced in the process of maximization. In eq 5, the superscript D has been dropped in the designation of nj, the equilibrium distribution. In the usual manner, the entropy is given by
(6)
S=klnQ
where k is the Boltzmann constant and Q is the value of QD obtained by substituting eq 5 into eq 2. The quantity /3 is identified with l/kT, where T i s the temperature, in the standard manner,’* and we do not reproduce the details here. However, it is worthwhile to explore the method which ultimately identifies p as the chemical or electrochemical potential of a particle, depending on whether the particle is uncharged or charged. Introduction of eq 5 into eq 2, and the result into eq 6 gives (writing 1/ k T for 8, and employing Stirling’s approximation)
Using eq 3 and 4 in eq 7 yields -TS = -E
+ N p - kT‘&,J
WJ
In @J
- ‘J
The Helmholtz free energy is A=E-TS
(8)
(9)
and substitution of this relation into eq 8 gives (17) Mayer, J. E.; Mayer, M. G . ‘Statistical Mechanics”; Wiley: New York, 1940; p 111. (18) Reference 17, p 117.
dN
In ___ w, - n,
j
wJ
)T,V
Then
(
= E -nJ j
=
k T dN
”)
kT d N
(14)
T,V
Substitution of this relation into eq 12, using eq 3, gives
Maximizing QD subject to the constraints eq 3 and 4 leads to the standard Fermi-Dirac distribution
nj =
+ p - kT(
From eq 5 , we have
(3)
ztjn)”)
(g)
Substitution of eq 10 into eq 11 yields
Adopting the microcanonical ensemble in which the energy of the system is fixed at E and the particle number at N , we require
N =
w j - nj
Now the chemical potential of a particle is given by
(1)
which signifies a redistribution of electrons, and a possible change in the amount of interaction between ions. Thus the spectrum may depend on the distribution. Should the electrochemical potential of the electron be called the Fermi level under these circumstances? Returning to the most elementary, weakly coupled example, we consider a system (not necessarily electrons) in which the spectrum of energy levels is characterized by there being wj states with energy level tj. If there are N particles in the system, and if, in a particular distribution D, there are nfD) particles in the j t h level, the number of arrangements corresponding to this distribution (only one particle, at the most in any state) is easily shown” to be
“‘J I _
chem pot. = N ( $ ) T , v
+ p - N(
$)
=
p
(15)
T.V
Thus p is the chemical potential. Clearly, if the particles are charged, the local energy levels (which are the cj in the preceding discussion) are increased by qV where q is the charge on a particle and Vis the local electrostatic potential. Then cj in the preceding discussion should be replaced by ti qV. When this is done the preceding argument leads to the result chem pot. = p - qV
+
or p
= chem pot.
+ qV
(16)
from which it is apparent that p represents the electrochemical potential of the particle. If the charged particle is an electron it is customary to refer to 1.1as the “Fermi level”, at least in a solid. Some chemists claim that the term “Fermi level” should only be used for the electrochemical potential of the electron in solids where there are “free electrons”. However, the physicists have used it for years to deal with electrons in bound states in semiconductors. Then, the fact that it has been used for the electrochemical potential of electrons in aqueous solutions where there are no free electrons should not really occasion objection. In the case of an aqueous solution, in which a redox reaction of the type of eq 1 occurs, the simple Fermi-Dirac formula, eq 5, may no longer hold, because of the interaction between ions which could render the energy levels dependent on distribution. We consider a system in which the degeneracies of all levels save one, say the Zth level, are independent of the distribution. In the Ith level, however, there are two sublevels corresponding respectively to states (or atoms) which are either singly occupied or doubly occupied. For concreteness, assume that we deal with atoms and that the particles are electrons. Suppose that we have wI atoms. The energy level of a single electron on an atom will be denoted by e)’). If a second electron is added to an atom, the electron’s energy is e)’), but the number of states of energy e)*) then depends on the number
The Journal of Physical Chemistry, Vol, 89, No. 18, 1985 3785
Feature Article
et1).
of electrons in the state having energy Thus the spectrum of levels depends on distribution. Furthermore, denote the number of electrons in singly occupied atoms by nl(l) and the number of doubly occupied atoms by nf2) (for notational simplicity we now omit the D denoting the distribution). Then it is an easy matter to show that the number of arrangements on the atoms is
‘“T I
W/!
0,= (wI - nl(1) - n/(2) )!n/(l)!nl(2)!
A
1
(17)
The total number of arrangements over A11 states in the system is
Q = Q/rI’ J
T
Wj!
1
(w - nj)!nj!
where the prime on the product sign indicates that the Ith level is not included in the product. The conservation of both electrons and energy now requires
and
+
E = nl(l)eI(l) nl(*)(el(l) + e,(2))
+ E’njej
(20)
j
where again the prime indicates that the Ith level is omitted. In eq 20 nl(2)is multiplied by the sum el(1) el(*) because the second electron (energy level = el(2)) on a doubly occupied atom requires the presence of the first electron (energy level = #)). Substituting eq 17 into eq 18, taking logarithms, using Stirling’s approximation, and maximizing In Q,subject to the constraints eq 19 and 20 yields for the equilibrium distribution
+
w/
1 + e(#)-fi)/kT+ e(*r(’)-fi)/kT wg-(c~2’-d
nl(2)
I
DISTANCE
+
Figure 1. (A) Energy level diagram for an n-type semiconductor, (B) energy level diagram for a p-type semiconductor, (C) energy level diagram for the p,n junction formed on contact. ev, eA, cD, ec and ,’,e eva/,
Wi
nl(1) =
”T
(22)
/kT
= 1 + e(#’-W)/kT+ e(d2)-fi)/kT
(23)
where p is a parameter, again associated with an undetermined multiplier. Equation 21 is the normal Fermi-Dirac expression, but eq 22 and 23 are not. Yet, by a procedure almost identical with that employed in arriving at eq 16, and which the reader is invited to attempt, we can show that p is still the electrochemical potential. Because eq 22 and 23 are not of the usual Fermi-Dirac form, should we cease to refer to p as the “Fermi” level? The answer can only be a matter of taste. The same is true in the absence of free electrons. In any event, p is alwuys the electrochemical potential. GerischerI9 has pointed out that eq 22 and 23 can be interpreted as referring to the redox equilibria
+ e- AA- + e- s A2A
if n{l) and nl(2)are the concentrations of A- and A”, respectively, and wI is interpreted as the sum of the concentrations of A, A-, and A2-. In fact these equations are equivalent to
where wI - nl(’) - nk2) is the concentration of the species A. They are therefore the proper classical expressions for the equilibrium ratios of these species. This does not detract from the fact that they correspond to a Fermi-Dirac distribution over a spectrum (19) Gerischer, H., private communication to the author.
e,, denote, respectively, valence band edge, acceptor level, donor level, conduction band edge, and vacuum levels. Plus and minus signs denote the polarity of the field in the p,n junction. p is the Fermi level.
of energy states which changes with occupation, i.e., the degeneracies (the concentrations of the various species) change with occupation. The example is an extreme one, and the change in the spectrum can be represented by changes in the degeneracies of the various energy levels. This extreme example was chosen in the interest of expositional simplicity. However, consider a less extreme case in which the ions in solution interact by means of Coulombic and other forces. This interaction gives rise to a change in energy levels which can still be described by a change in degeneracies, using a variable, possibly continuous, density of states. These changes occur when changes in concentration, temperature, etc., cause changes in the correlation between ions. One change in concentration may be due to the shift of redox equilibria, such as the above. Whatever the cause, the Fermi-Dirac distribution of electrons can still be expressed in terms of an invariant set of energy levels (and variable degeneracies), and the distribution will not have the usual simple form, but rather a more complicated form not unlike that in eq 22 and 23. Of course, as in those equations, I.L is still the electrochemical potential and there is no reason beyond that of “taste” for not calling it the Fermi level. 111. Location of the Fermi Level, Relative to Vacuum From now on, we shall use the more conventional symbol fi for the electrochemical potential in place of p, reserving the latter symbol for chemical potential. The normal procedure for determining the Fermi level relative to the energy levels in the system involves substituting eq 5 into the conservation condition, eq 3, and solving the resulting transcendental equation for p. How do we locate the Fermi level relative to vacuum? Consider Figure la. This exhibits the energy level diagram of an n-type semiconductor not in contact with any other phase.
3786 The Journal of Physical Chemistry, Vol. 89, No. 18, 1985 The horizontal coordinate denotes position in space, as usual, and ec, eD, and eV are respectively the conduction band edge, the donor level, and the valence band edge. The Fermi level is denoted by p , and e, is the vacuum level just outside of the semiconductor. Figure 1b exhibits the energy level scheme for the same semiconductor, this time doped p-type. Here we show, in addition to the levels already defined in Figure la, the acceptor level e*. Figure I C shows the equilibrium situation which results when the semiconductors of parts a and c of Figure 1 are brought into contact so that a p,n junction is formed. The previously disparate (relative to vacuum) Fermi levels are brought into coincidence, when equilibrium is achieved, by the development of the built-in electric field at the junction which, as the and - signs in the figure indicate, results in the n-type semiconductor acquiring a positive charge, while the p-type specimen acquires a negative one. and eva/ The built-in field gives rise to two new vacuum levels just outside the n-type and p-type semiconductors respectively. Which is the “true” vacuum level? It might seem reasonable to choose evac in Figure la,b. But how is the Fermi level in Figure I C related to this original e,,? It cannot be related as it was related originally, in both parts a and b of Figure 1, because in Figure 1c it is common while in the other parts the Fermi levels are disparate. The answer to this question is not easy. However, it is unimportant! What really counts is the relation between p and the vacuum level just outside the surface of the particular phase we are interested in. For example, if it is the n-type phase then we would like to know the difference eva: - p . This difference which, as we shall see later, is the work function for the n-type phase is identical with the difference, evac - p in Figure la. If we are interested in the p-type phase, then the difference eva/ - p , which is the same as evac - p in Figure 1b, is the work function of that phase. We made our point by referring, in Figure 1, to semiconductor phases. However, the same point could have been established by using any two phases, e.g., semiconductor and metal, metal and aqueous solution, etc. The “contact potential” phenomenon, implicit in the development of a built-in field, will always give rise to the same subleties. It should be clear that the “work function” is going to be important in the discussion which follows. Consequently we proceed to discuss it in the next section.
+
IV. Revisiting the Work Function One’s first acquaintance with the work function usually occurs in a course on elementary physics where one learns from the “photoelectric effect” that the work function is a threshold energy which accounts for the difference between the total energy of an incident photon and the residual kinetic energy of the ejected electron. Later one learns that the work function is a thermodynamic quantity and not simply an energy, but can be associated with a free energy in which effects due to entropy may be. involved. This point and many others are made clear in a landmark 1949 paper (on thermionic emission) by Herring and Nichols.2o This paper bears directly on the problem under consideration, and is superior in depth and scope to any other with.which we are familiar, yet it does not seem to have noticed by electrochemists. It is rarely or never referred to in the electrochemical literature. Nevertheless it is recommended as required reading for students and investigators concerned with absolute redox potentials. In anticipation of the phenomena and measurements in which it appears, Herring and Nichols define the “true work function”, ’$, by the relation 4 = -‘$a - @ / e (24) in which e is the electronic charge, and where ‘$a is the electrostatic potential in the vacuum just outside of an electronic conductor. p is the electrochemical potential of the electron in the bulk of the conductor. Both ’$, and p are conveniently referenced to the same level. Note that this means that the “energy” term in p is (20) Herring, C.; Nichols, M. H. Rev. Mod. Phys. 1949, 21, 185.
Reiss referenced to the same level as ‘$a. The “entropy“ component cannot, of course, be so referenced. If the reference level is in the vacuum, ‘$a = 0. Now, if & is the electrostatic potential in the bulk of the conductor, we may write, using the definition of the electrochemical potential
P = 1 - e’$b (25) where 1 is the chemical potential of the electron. When substituted into eq 24, eq 25 yields ’$ = d’b -
‘$a
-r/e
(26)
where &I - ‘$a, if nonzero, is the potential across the surface dipole layer. This potential may be extremely sensitive to the peculiar condition of the surface because of adsorption, orientation, relaxation, and other effects. On the other hand M/e represents a true bulk property. Thus, ‘$ is clearly a function of the condition of the surface. Free energies are usually determined by the performance of equilibrium (reversible) measurements, yet work functions are usually thought of in the context of irreversible processes such as thermionic or photoemission. How then does the work function involve a free energy? Perhaps the most direct answer can be found in the simplified derivation of the formula for the thermionic emission current.21 In this derivation an ambient electron vapor is constrained to equilibrate with the solid through one particular crystal face with fixed surface dipole, and for which we choose 4ato be zero. The electrochemical potential for an ideal gas of electrons just outside of this crystal face is given (in the most elementary statistical thermodynamic analysis) by the expression p = 1 - e$a = p = kT In (A3p/2) where p is the number density of electrons in the gas, and A-’ is the momentum partition function A = h/(2*mkT)‘I2 (28) h being Planck’s constant and m the electronic mass. The numeral 2 appearing in eq 27 accounts for the two possible spin states of the electron. If the electrons in the gas are in equilibrium with those in the crystal, then the p appearing in eq 27 is also the electrochemical potential within the crystal bulk. Now, from simple kinetic theory, the number of gaseous electrons striking unit area of crystal surface in unit time is J = p[
E]’/’
If the reflection coefficient is zero, then, at equilibrium, J must also be the flux density of emitted electrons, Le., the emission current. Solving eq 27 for p, and substituting the result in eq 29 yields, for this emission current
If we assume, invoking detailed balancing, that this result continues to hold for the nonequilibrium case, we arrive at an expression of the current which contains the work function. This result contains many approximations beyond the assumption of detailed balance (which one hopes will be fairly accurate under conditions of saturation). For example, the assumption of ideal gas behavior on the part of the electrons neglects interaction between them. This is usually justified2*on the basis of their low number density and of the presence of positive ions which provide a screening function23without themselves introducing interaction terms. Furthermore, the effects of image forces are not apparent in eq 30. Such effects can, however, be intro(21). Fowler, R. H.; Guggenheim, E. A. “Statistical Thermodynamics”; Cambridge University Press: New York, 1939; Chapter IX. (22) Reference 19, see footnote on p 191. (23) Reference 19, section I, Id.
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The Journal of Physical Chemistry, Vol. 89, No. 18, 1985 3787
Finally, the reflection coefficient is by no means precisely zero. It should be noted that the measurement of 4, via thermionic emission, at no point involves an electron in a vacuum for which p = 0. In fact, as eq 27 indicates, for p = 0, the chemical potential p would be infinite. The overall scheme allows p to be eliminated between eq 27 and 29, leading eventually to eq 30, by means of which the “true work function” Cp can be measured. Herring and Nichols defined the “true work function” in this manner so that, once measured by whatever means, it could be used, via eq 24, for the determination of the electrochemical potential, p, with respect to vacuum. In spite of all this, eq 30 is useful because it demonstrates how a measurement of the work function (i.e., through the measurement of thermionic saturation currents) yields a quantity p involving free energy. Furthermore, eq 30 shows that $, which is ideally a reversible quantity, must nonetheless appeal to an essentially irreversible measurement for its determination. One could object and point to the facts that the work function is involved in both contact potentials and in the threshold for photoelectric emission, both of which are much more arguably reversible (the contact potential is in fact reversible) than the measurement of saturation current. However, both of these measurements, like the standard electrode potentials, are always referenced to another potential which, itself, cannot be determined in a reversible manner. To gain some insight into this problem, and to learn of a few more subtle features, we discuss contact phenomena and photoelectric emission a bit further. When two phases, denoted by subscripts 1 and 2, are placed in contact and are in equilibrium, pI must equal p 2 . Making use of this requirement, and using eq 24 for each phase, we find where 111,2is the contact, potential between phases 1 and 2. We notice that is the potential between points lying in the common vacuum phase just outside of the respective phases, and, as such, is the well-known measurable Volta potential. One method of measurement involves the Kelvin technique in which an external compensating electromotive force balances the contact potential, so that a change of capacitance between the phases causes no flow of charge. The electromotive force required for such balancing then measures II,,2. We now need to deal with a somewhat vexing and sometimes unrecognized problem. According to eq 31, the contact potential can be computed from the work functions. However, the work functions are surface and dipole dependent and may even vary over a single surface or because of the necessary several exposed surfaces of a single crystal. This “patch effect” is discussed at length by Herring and Nichols.20 Because of it we have to ask what work functions are to be used in eq 31? In their elegant analysis, Herring and Nichols demonstrate that where (33) with a similar expression for $2, wherefii is the fraction of the ith surface exposed on crystal 1 while &i is the work function going with the ith surface. Furthermore, they demonstrate that given by eq 32 is the contact potential which would be measured by the Kelvin method. All of this demonstrates that the dipole layer presents a problem in establishing the absolute location of p even within the same single crystal solid, and not only between different crystals of the same solid. Ultimately, as we show below, methods for canceling the effect of the dipole layer must be found. Even more important is the point which stimulated this discussion of the contact potential, in the first place; it is evident from eq 31 and 32 that the reversible measurement of 111,2does not (24) Apker, L.;Taft, E.; Dickey, J. Phys. Rev. 1948, 74, 1462.
by itself yield a work function! Only a difference between work functions is obtained. Unless one of the functions in this difference is known, cannot yield the value of an unknown work function. To obtain one of the two terms in it is necessary to perform an irreverisible measurement, such as that implied by eq 30. Now consider photoelectric threshold measurementsz4with the emitter electrode and a collector electrode connected without any external electromotive force in the circuit. For convenience let us measure all energy levels from a low lying reference level in the emitter. Consider an electron in the emitter in a state of energy 6. If this electron absorbs a photon of energy, hu, where u is the photon frequency, its total energy will be e hu. Now the collector is in contact with the emitter so that its electrochemical potential is equal to p, the electrochemical potential of the emitter. Thus according to eq 24, the potential just outside of the collector is
+
4ac
=
- F/e
(34)
where $c is the work function of the collector. This potential is of course referenced to the same low lying level in the emitter. In order for an emitted electron to be “collected” it must reach this potential. This requires that e hu = -e& = p e& (35)
+
+
As an example, in a metal at 0 K, the highest occupied electron state has e = p. Substituting this condition into eq 35 gives for the threshold photon of frequency u0 hvo = e&
(36)
Thus, observation of the threshold frequency allows one to determine the work function of the collector at zero degrees. However, at zero degrees there is no entropy and the work function is strictly an energy, not afree energy. At a higher temperature it becomes a free energy, and the highest t is not p. Thus the equality in eq 36 is no longer true. The conclusion, therefore, is that one cannot rigorously determine the work function by a threshold measurement alone: an irreversible measurement is always necessary. It is somewhat paradoxical that as soon as the work function becomes a thermodynamic free energy it can no longer be fully measured by means of an equilibrium thermodynamic method. Nevertheless, in the case of a metal, the perturbation of the zero temperature distribution is not large at room temperature. As a result one can almost obtain the work function from the photoelectric threshold. The work function of a semiconductor may then be obtained from eq 32 through the determination of the contact potential of the junction between the semiconductor and the metal in question.
V. Electrons and Holes as Chemical Entities in Solids Just as the “Fermi level” might be viewed as a physicist’s idea invading the chemist’s domain when it is used in connection with the redox potential, the ”law of mass action” could be viewed as a chemist’s idea invading the physicist’s domain when it is used to characterize the distribution of electrons over the energy level spectrum of a solid. Nevertheless this latter invasion not only occurred but has been very useful. It is instructive to chemists to have an example, in solids, in which the Fermi level is an important feature, but where a more traditional alternative method accomplishes the same end. The inverse situation, i.e., the ”Fermi level” as an alternative in electrochemistry might then achieve greater acceptability. Furthermore, in this example, the distribution of ”bound”, as well as ”free” electrons, is involved, yet the term “Fermi level” is used for the electrochemical potential of the electron. An exactly similar problem is treated in the following section dealing with solutions where “Fermi level” is not commonly used. These sections serve to indicate how much the debate is a matter of taste. The treatment of electrons and “holes”, as well as other defects in solids, as chemical-like entities was pioneered largely by Wagner.25 It has been well reviewed in a book by Kroger.26 An (25) Wagner, C. Z . Phys. Chem., Abt. E 1933, 21, 25; 1936, 32, 441.
3788 The Journal of Physical Chemistry9 Vol. 89, No. 18, 1985
Reiss are the conduction and valence band edges. Clearly the product of n by p is np = NCNVe-((C-W)/kT = n,2
Figure 2. Energy level diagram for lithium and boron in silicon. Arrows
indicate the various electronic transitions corresponding to the “reactions” of eq 49. Reprinted with permission from ref 27. Copyright 1956 Bell
Laboratories Incorporated. especially clear example is presented by the behavior of donor and acceptor impurities in elemental semiconductors such as germanium and ~ i l i c o n . ~For ~ ~ didactic ~* reasons we describe an example of this sort.*’ In this example lithium is dissolved in molten tin. A silicon crystal doped with boron is immersed in the tin, and the lithium which diffuses very rapidly in solid silicon distributes itself between the two phases. In effect, the boron in the silicon does not diffuse at all, and remains in place. Now lithium in silicon behaves as a donor, and ionizes to contribute a motlile electron to the conduction band of the crystal (transition 1 in Figure 2). Boron is an acceptor and ionizes to produce a positive “hole” in the valence band (transition 2 in Figure 2). Electrons and holes can also recombine by means of a transition sparining the semiconductor band gap (transition 3 in Figure 2). Figure 2 diagrams this situation. The equilibrium distribution of electrons over the energy levels of the semiconductor is determined by Fermi statistics, i.e., by eq 5. We define the number of “holes” in the j t h level by u/e(erfi)lkT
pJ = wJ - nJ = 1
+ e(cj-b)/kT
(37)
where eq 5 has been used. When the Fermi level lies far enough below cJ, Le., when c, - p >> kT (38) eq 5 reduces to nl = uJe-(cj-P)/kT (39) and the classical result is recovered. Similarly when the Fermi level lies sufficiently above the j t h level, eq 37 reduces to p = u.e-(P-cj)/kT (40) J
J
These results are used with some approximationz9to prescribe the concentrations of electrons and holes in the conduction and valence bands, respectively, as n = NCe-((c-fi)/kT
(41)
p = NVe-(b--Iv)/kT (42) where Nc and Nv are the so called effective densities of states in the conduction and valence bands, respectively, and ec and tV (26) Kr6ger, F. A. “The Chemistry of Imperfect Crystals”;North-Holland: Amsterdam, 1964. (27) Reiss, H.; Fuller, C. S.; Morin, F. J. Bell Sysr. Tech. Journ. 1956, 35, 535.
(28) Reiss, H. Proc. Robert A. Welch Found. Con$ Chem. Res. 1970,14, 145.
(29) Grove, A. S . “Physics and Technology of Semiconductor Devices”; Wiley: New York, 1967; p 101.
(43)
where q is called the intrinsic concentration of carriers, and where, significantly, the Fermi level has disappeared! This is an example of the law of mass action applied to electrons and holes, since n,2 is clearly the equilibrium constant for the “dissociation” of an electron-hole complex. Thus, whereas one can learn something about the distribution of electrons over the energy levels by knowing p and applying eq 41 and 42, the same feat can be accomplished by using eq 43 without having to know the value of ,a. In a sense, the law of mass action replaces the use of the Fermi level. This is an advantage because the Fermi level must be determined by solving the transcendental equation obtained by substituting eq 5 into eq 3. However, it will be noted that the simple equilibrium constant, eq 43, is obtained when the statistics of the conduction and valence bands become classical. Otherwise, thermodynamic activities must be used in place of concentrations, and the activity coefficients then depend on p,30 and reflect the interaction between electrons implicit in the Pauli exclusion principle. Applying eq 5 and 37 to the lithium levels in Figure 2, we find (44)
CLi+
NLie(Q-P)IkT = 1 + e(fo-fi)/kT
(45)
in which cLi and c ~ i are + the concentrations of un-ionized and ionized lithium, respectively, and CD is the energy of the lithium level, while NLi is the total concentration of lithium in the silicon and, for this case, takes the place of uj.Equations 41, 44, and 45 can be combined to yield
which is obviously an equilibrium constant for the ionization of lithium, namely Li (Si)
F?
Li+ (Si)
+ e-
(47) where e- is a conduction band electron. Again the Fermi level has disappeared. The entire process of distribution of lithium between molten tin and boron-doped silicon mentioned above can be described by the following process in which electronic transitions are described by chemical-like reactions Li (Sn)
2 Li (Si) 2 Li+ t e-
+
B(Si)
2 B- + e-
(48)
tJ e+e-
in which e+ is a valence band hole and e+e- is a “recombined” holeelectron pair. Since e’e- is “weakly dissociated”, one expects, on the basis of mass action, that the solubility of lithium will be increased by the presence of boron. The entire process is the analogue of neutralization in an aqueous solution with e- taking the place of the hydroxyl ion, e+ taking the place of the hydrogen ion, and the donor and acceptor playing the roles of base and acid. Thus eq 48 represents the dissolving of a base by an acid. The equilibria in eq 48 can be quantified by using equilibrium constants such as eq 43 and 46 together with the requirement of electroneutrality. How well this works can be judged from a comparison of experiment with theory (mass action), reproduced*’ in Figure 3. Here the points are experimental data, and the curves are (30) Rosenberg, A. J. J . Chem. Phys. 1960, 33, 665.
The Journal of Physical Chemistry, Vol. 89, No. 18, 1985 3789
Feature Article
Solving this equation for the ratio
d’
cFe2+/cFe3+
= e-(f2-p)/kT
(55)
Comparison with eq 53 indicates that the two equations are the same with
ld’
pFeZ+’
n
3
- PFe3+’ =
62
(56)
an entirely reasonable result since the cz should be the work expended in adding an electron to Fe3+. Thus, even in aqueous solution, Fermi statistics leads to the correct equilibrium constant. The “electrochemical potential” of the electron, p , is used in eq 54 in place of the Fermi level and is obviously the same thing. Arguments of the type presented in this section were first given by G e r i ~ c h e r . ~ ~
.y 101‘ &
n ton
mu cd)
CF~~+/CF yields ~~+
VII. The Fermi Level as the Redox Potential w’4
bow
IO”
A-PER
cy1
m”
d’
Figure 3. Effect of boron on the solubility of lithium in silicon. Ordinate is NLi = cLi+in particles per cubic centimeter while abscissa is NB= C,. Symbols D+ and A- are the notation in ref 26. Reprinted with permission from ref 27. Copyright 1956 Bell Laboratories Incorporated.
computed from the various equilibrium constants. Boron is assumed to be fully ionized. In the following section we examine the relation to the Fermi level of familiar equilibrium constants on the other side of the “social” interface, Le., in aqueous solution.
VI. Equilibrium in Solution and the Fermi Level We now transfer attention to a half-cell containing an aqueous phase in which a redox equilibrium involving ferrous and ferric ions, eq 1, is established. For convenience we rewrite this equation here
+
Fe3+ e- e Fe2+
(1)
The species e- is an electron in the electrode of this half-cell while the ions are in the aqueous solution. Since there is a Galvani potential difference between these two phases, the condition of equilibrium corresponding to eq 1 must involve electrochemical rather than chemical potentials. Thus we write
p=p+qv
(49)
where q is the charge on a species, and Vis the local electrostatic potential. Then, corresponding to eq 1, we have, at equilibrium FFe3+
+ F.
pFe2+
(50)
where we now use p without a subscript to denote the electrochemical potential of the electron. Now, assuming that concentrations, c, can be used in place of thermodynamic activities, we may write
+ kT hl C F ~ ~ + = p~~3+ +’ kT In
p ~ ~ 2 +p ~ ~ 2 + ’
(51)
p ~ ~ 3 +
(52)
where the quantities with superscript zero are the electrochemical potentials in the standard state. Substituting eq 51 and 52 into eq 50 gives cFe2+ /cFe3+=
~-(PF~~+O-PF.~+’-P)/~T
(53)
W e may calculate this ratio (equilibrium constant) in an alternative manner using Fermi statistics (in the solution). The Fe3+ ion may be regarded as an empty electron state of energy t2. When filled with an electron it becomes Fe2+. The total number of such states (assumed noninteracting, since concentrations are to be used + CFe3+. Then eq 5, for this state, in place of activities) is cFCz+ becomes CF$+
CFe2+
= 1
+ cFe3+
+ e(f2-B)IkT
(54)
where we use the electrochemical potential as the Fermi level.
The last few sections have been concerned with the appropriateness of using the term “Fermi level” outside of solids, and especially under circumstances where (1) there are no free electrons, and (2) there is enough interaction between occupied states so that the spectrum of levels, itself, depends on the distribution of electrons among the states. From the relevant discussion, it should be clear that the Fermi level is the electrochemical potential of the electron and that the question of its use in aqueous solutions, for example, is only a matter of taste. It has been used for many years, in solids, under the very circumstances delineated by points 1 and 2 above. On the other hand there is a more serious controversy underway. This concerns the question of whether the Fermi level is identical, except for reference level, with the redox potential. Gomer and Trysonlo and especially Gerischer*>” have presented rational arguments which confirm this equivalence, while many other authors1Zl5have debated it. The present author must go on record, at the outset, as affirming that the Fermi level is the same as the redox potential. However, this assertion by itself cannot dispel the widespread confusion which seems to exist within the electrochemical community. Therefore this section will be devoted to a, hopefully, transparent elaboration of the matter. The explanation of the equivalence is really very simple, depending only on straightforward thermodynamic reasoning. Unfortunately, most of the attempts to make the situation clear have relied on specific cycles in which the elementary steps accumulate an abundance of intermediate species together with their many (not always the same) symbols and the thermodynamic quantities associated with them. In addition, the descriptions of these intermediate steps involve quantities such as solvation energies, reorganization energies, surface dipoles, etc., which, although useful in establishing an absolute scale for the redox potential, are not necessary for demonstrating the Fermi level’s equivalence with the redox potential and seem to have added to the confusion. In the following we explain the equivalence in a simple thermodynamic manner, dispensing with this unnecessary paraphernalia. If we clear of exponentials in eq 53 we obtain
p = pFe2+0 - pFe3c0
+ kT In
C F ~ ~ + / C F ~ ~ +
(57)
Now, the conventional expression for the redox potential of the Fe2+/Fe3+couple isj2 kT Vrdox(Fe2+/Fe3+)= Vrdoxo(Fe2+/Fe3+) - In CFc2+/CFe3+ e (58) where e is the charge on the electron and Vrdol(Fe2+/Fe3+)is the redox potential in the standard state. If eq 57 is divided by e, it becomes identical with eq 58 to within a constant term so that p / e can be made the same as Vrdoxby no more than a choice of reference level. (31) Gerischer, H. J . Phys. Chem. 1960, 26, 223. (32) Koryta, J.; Dvorak, J.; Bohackova, V. “Electrochemistry”; Methuen: London, 1966; pp 173-183.
3790 The Journal of Physical Chemistry, Vol. 89, No. 18, 1985 However it does not require a specialized argument, such as a comparison of eq 57 and 58 to demonstrate that the Fermi level can serve as a redox potential. The Fermi level is the electrochemical potential of the electron, as has been amply demonstrated in the preceding sections, and it is the electrochemical potential which determines when, and in what direction, a charged species will be transferred between two points in a system. We make no attempt to prove this here; the proof can be found in almost any text on thermodynamics. When, for example, an electron can be transferred between two systems it will move from the system of higher electrochemical potential (for the electron) to that of lower electrochemical potential. Thus the system of higher electrochemical potential (Fermi level) “reduces” that of lower potential. The former phase therefore possesses the higher redox potential. When the Fermi levels are equal, no electron transfer occurs. Furthermore, it follows, from the manner of definition of the electrochemical potential (see thermodynamics texts), that the difference in electrochemical potential between the two systems measures the reversible work which must be performed on or by the system (depending on direction of transfer) in effecting the transfer. To make things concrete, consider the simple example in which the two systems are two aqueous solutions of CuS04, each in contact with a copper electrode but containing different concentrations of C U S O ~ ( C U ~ + , S O Relative ~ ~ - ) . to a common reference level, the electrochemical potential of electrons (bound or free-it does not matter-it is still the Fermi level) in these two will be different since the concentrations of CuS04 are different. However, the Fermi level within each system is uniform, the same in both the copper and the solution, since these two phases are in equilibrium with each other, in each system. Suppose that we now bring the two electrodes into contact. Since the Fermi levels are initially disparate, some electrons will be transferred from one system to the other, copper will dissolve in one system and deposit in the other, and there will be a net transfer of charge between the two systems, ultimately represented by a deficiency of sulfate ions in one system and an excess in the other. An electrostatic potential difference will develop because of this transfer of charge until the Fermi levels are brought into coincidence, at which point the system will achieve equilibrium. Using the subscripts 1 and 2 for the two solutions, we then have for the Fermi levels
PI = w1 - eVl = b2 - eV2 = p 2
(59)
from which V2 - VI =
(112
- &)/e
(60)
so that the potential difference between the solutions is the ”unmeasurable” Galvani potential and is given by the difference in chemical rather than electrochemical potentials of the electron in the solutions. There will also be a Volta potential (contact potential) difference given, according to eq 31, by the initial difference in work functions between the solutions. It is to be noted that V2- Vl in eq 60 is not the electromotive force (eMF) of the cell which would be formed by establishing a liquid junction between the two solutions. In order to understand this distinction, assume that a liquid junction is established having (as usual) no liquid junction potential (normally eliminated by theoretical consideration). Now, upon transfer of electrons through the copper conductor, no net charge can be transferred since it is relieved by transport of sulfate ions through the junction. An electrical potential necessary for bringing the systems into contact equilibrium cannot therefore develop. In order to bring the cell to equilibrium an externally supplied electrical potential must be introduced into the copper conductor of such magnitude and sign that the initially disparate Fermi levels are brought into coincidence. Thus
Reiss an electron between the two systems. This merely reaffirms the earlier statement that this reversible work corresponds to the initial difference between electrochemical potentials. By definition E2,1is the cell e M F and corresponds to the net cell reaction34 CU(1)
+ CuSOh(2)
-+
Cu(2) + CuS04(1)
(62)
where Cu refers to the electrode and CuS04to the solution while 1 and 2 denote the respective systems. Before the two systems are formed into a cell, equilibrium in system 1, corresponding to the reaction
+
Cu s Cu2+ 2e-
(63)
requires or pcu = pcU2+O
+ k T In cCu2++ 2Op
(65)
where again (for simplicity) we use the concentration of Cu2+in place of the thermodynamically precise activity, and where pCuz+O again refers to the standard state. Rearranging eq 65, and dividing by e, gives Op/e = ( F , - ~- pCu2+O)/2e- kT In ccu2+= VrdOx (66) 2e where Vrdonis meant to indicate the redox potential of the system in question. If it is a redox potential it should be useable in the conventional manner in the construction of the cell eMF. For system 1 (now half-cell 1) we may write eq 66 as
where Vl,redo~, the redox potential in the standard state, has been introduced in an obvious way. A similar expression holds for system 2 . Substituting these equations into eq 61 we obtain
(68) This is the standard expression for the cell potential. It should now be clear that the Fermi level is equivalent to the redox potential. Ultimately, the proof hinges on the fact that the cell potential measures the reversible work required for the transfer of an electron from one half-cell to another and that this reversible work is, on general thermodynamic grounds, nothing more than the difference between the initial Fermi levels (referred to a common reference) of the isolated redox couples. In the case of the Fe2+/Fe3+couple, to which eq 57 and 58 refer, the initial Fermi level can be taken in the isolated solution phase since both the reductant and oxidant reside there. In the case of Cu/Cu2+ the solution in contact with the copper electrode constitutes the initial system, and the “initial” Fermi level must refer to the solution in contact with Cu. In the case of the standard hydrogen electrode, the reductant is hydrogen vapor and the oxidant H+ ions in solution, so the initial Fermi level must involve the solution in contact with the vapor phase containing HZ. However, the overall principle remains the same; the Fermi level is equivalent to the redox potential. One conventional redox potential, for a given couple, is defined as Vcouple,the e M F of the cell formed from the half-cells of the couple and the standard hydrogen electrode (SHE). If p (couple) and p (SHE) are the Fermi levels relative to a common zero (say vacuum) in the two initially isolated half-cells, then for this couple If p(SHE) is somehow known, then from eq 69
where Op, and OpI are the initial electrochemical potentials. It is also apparent that since, under E2:lthe system is brought into equilibrium, measures the reversible work required to transfer
p(coup1e) = p(SHE)
+ eVmuple
(70)
Note that p(SHE), since it is referred to vacuum, is the absolute redox potential of the SHE. Correspondingly the absolute redox
J . Phys. Chem. 1985,89, 3791-379s potential of the couple, p(coup1e) can be determined by measuring Vcouplcand applying eq 70, once p(SHE) is known. Then p(couple)/e in eq 69 is a particular example of Vrdox,now referenced to the vacuum. As indicated in the discussion following eq 24, this means that the energy component of p is referenced to the vacuum level. We do not speak of the electrochemical potential in the vacuum. This point is elaborated in the discussion following eq 30. Then from eq 24 Unfortunately +H cannot be measured directly. It must be estimated by an indirect means. Several such estimates have been made9J1J3and a new estimate (which avoids a cycle) is described in a following paper14 in this journal. The reader is referred to these various investigations (all of which give closely agreeable results) for the elaboration of this problem. Also, in closing, it should be pointed out that Farrell and M c T i g ~ have e ~ ~ succeeded in performing accurate measurements of the Volta potential difference between a mercury surface and an electrolyte. On the basis of these data, the difference between the Fermi level in an electrolyte and a point close to the surface of the liquid can be determined, provided that the work function of mercury is known under the same ambient conditions.
VIII. Summary In the foregoing, the following points have been developed: (33) Farrell, J. R.; McTigue, P. J . Electroanol. Chem. 1982, 139, 37. (34) Although the eMF of the cell is not the “unmeasurable” Galvani potential, it is essentially the ”measurable” Volta potential. However we do not press this issue here, because of some complications associated with the liquid junction potential (an irreversible quantity), and because nothing is gained by pressing it. Such complication might only increase the confusion.
3791
(1) There is no essential reason, beyond that of individual taste, for not using the term “Fermi level” for the electrochemical potentiai of the electron in an aqueous solution. There is nothing compelling about restricting the term to phases in which the distributed electrons are “free”. Nor is there any reason for not using the term for systems in which the spectrum of energy levels depends on the distribution. Ample precedent exists in solids for its use in both of these situations. (2) The law of mass action has been applied successfully to equilibria involving electrons in solids for many years. In such cases the law is established, using the identity of the Fermi level as the electrochemical potential of the electron. Equilibrium constants between ions and electrons in redox couples in solution are derived in exactly the same way. The Fermi level in solution has the same meaning as in solids. (3) The Fermi level can be identified with the redox potential. In its absolute form it is referenced to the vacuum. However this standard cannot be established unless the work function of the reference cell is known. This can only be estimated indirectly. However, the fact that the Fermi level is the redox potential requires no complicated argument for its proof. It is almost obvious on the basis of simple thermodynamics. The use of cycles with individual intermediate steps such as solvation, solvent reorganization, etc., although useful in the estimation of the work function for the reference cell, may be confusing when used to demonstrate the identity of the Fermi level with the redox potential. Acknowledgment. This work was supported by the National Science Foundation under Grant No. CHE82-07432. I thank Dr. Adam Heller for many helpful discussions in connection with this paper. Registry No. Si, 7440-21-3; Li, 7439-93-2; B, 7440-42-8.
ARTICLES Solvation Thermodynamics of Completely Dissociable Solutes A. Ben-Naim Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem 91 904, Israel (Received: April 23, 1985)
A new definition of the solvation thermodynamics of completely dissociablesolutes is suggested. A relation is derived between the solvation thermodynamical quantities and measurable quantities. Some illustrative results obtained for simple electrolytes are presented and compared with the conventional quantities of solvation.
1. Introduction The problem of solvation thermodynamics is perhaps as 01’ as physical chemistry of ionic solutions.’-3 Yet, in spite of its importance for the understanding of almost any process that takes Place in a solvent (Particularly aqueous there is no agreement on a unified definition of the solvation process. As (1) Conway, B. E. “Ionic Hydration in Chemistry and Biophysics”; Elsevier: Amsterdam, 1981. (2) Bockris, J. OM.;Reddy, A. K. N. ‘Modem Electrochemistry”; Plenum Press: New York, 1970. (3) Franks, F. ”Water, A Comprehensive Treatise”; Plenum Press: New York, 1975; Vol. 2, 3.
a result, there exist many different, sometimes confusing, definitions of standard quantities that are presumed to convey information on free energy of solvation, entropy of solvation, volume of solvation, etc. We believe that this multitude of definitions is a result of the fact that thermodynamics alone cannot offer a molecular interpretation of various standard thermodvnamic ouantities. Therefore, the choice of different standard stales made dy different authors is based on arguments of convenience or simplicity, but not on their merit as bona fide measures of the solvation properties of a YSOluten in a usolventn. With this situation in mind we have recently suggested a unified approach to the problem of solvation processes.e6 This approach
0022-3654/85/2089-3791$01.50/00 1985 American Chemical Society
