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II. CHARACTERISTIC ENERGY LEVELS IN THE ELECTRODE/ELECTROLYTE SOLUTION SYSTEM

II. CHARACTERISTIC ENERGY LEVELS IN THE ELECTRODE/ELECTROLYTE SOLUTION SYSTEM

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Electrochemistry of Semiconductors: New Problems and Prospects



191



lem both in purely methodological and practical respects. In particular, the concept of the electrochemical potential of electrons in

a solution which contains a redox couple, is essential for the

understanding of the laws governing electrochemical and photoelectrochemical processes. Next, the reorganization energy of a

solvent affects significantly the kinetics of reactions which involve

charge transfer. Finally, the electron energy levels, which are

directly related to the oxidized and reduced components of the

couple, determine the quantitative characteristics of the corresponding electron currents.

1. The Electrochemical Potential of Electrons in the Solution

Containing a Redox Couple



Consider an electrode immersed in a solution, which contains a

redox couple taking part in a reversible electron exchange reaction

ox + ne~ ^ r e d

(1)

Here ox and red denote the oxidized and reduced forms of the

substance (e.g., Fe3+ and Fe2+) in the solvated state, and n is the

number of electrons transferred in the electrode reaction. Under

thermodynamic equilibrium, the following relation holds:

/Zox 4- n/Ie = /Ired

(2)

where /Iox and /Zred are the electrochemical potentials of the redox

couple components in the solution (per mole), and {xe is the

electrochemical potential of electrons in the electrode. The quantities /IOx,red can be represented in the form

Mox,red = RT In C ox , red + ZOXfTed^(f) + Mox,red



(3)



Here R is the universal gas constant, T the absolute temperature,

cOx,red are the concentrations (the activities in a more general case)

of the ox and red reactants in the solution, zox red are their charge

numbers, 8F is the Faraday constant, is the electric potential, and

Mox,red a r e constants independent of cox red and .

Taking into account Eqs. (2) and (3), we can calculate the

equilibrium electrode potential
(1) occurring on an electrode:



192



Yu. V. Pleskov and Yu. Ya. Gurevich



where

of the electrode reaction and also on the choice of the reference

electrode.

It is rather important that if thermodynamic equilibrium is

established, in accordance with Eq. (1), we can introduce the

concept of the electrochemical potential of electrons in the electrolyte solution, /I*1, which is in no way related to the presence of

the electrode itself. The fact is that within the framework of thermodynamics, a specific mechanism, through which an equilibrium

is established, does not play any role, so one may assume, in

particular, that electron exchange between ox and red proceeds

without any electrode involved. In this case, /If for reaction (1) is

given by

^



= ~ (/Ired - Mox)



(5)



Ti



where /2red and /Iox are taken at the potential el'b of the bulk

electrolyte solution. Since the quantities /Ired,ox depend only on the

properties of the ox and red reactants and the solvent, /If is in no

way related to the nature of the electrode and does not depend on

its interface structure. Moreover, it is the quantity /If which determines the electrochemical potentials of electrons in a given electrode

under equilibrium conditions. In particular, this electrochemical

potential has the same value for any electrode which is in equilibrium with a given redox couple.

The above considerations show how important it is to find a

relationship between jxf and the characteristic parameters of a

redox couple. To this end, we shall consider a solution containing

one red particle. Denoting the most probable thermodynamic

energy of the electron level of the red particle by E°ed and that of

the ox particle by E°ox, we shall analyze, following papers,8'9 the

thermodynamic cycle (a-d) shown in Fig. 1 [in Eq. (1), n = 1 is

assumed for simplicity):

a. An electron is instantly transferred from a solvated red

particle (from the level E°cd) to the level Evac. Here £vac

is the potential energy of an electron in vacuum, more

precisely, in the vapor phase in a close proximity of the

electrolyte surface, yet beyond the action limits of purely

surface forces.



Electrochemistry of Semiconductors: New Problems and Prospects



193



vac



L



ox



L



red



Figure 1. Determination of the position of energy

levels of an electron in the solution containing a

redox couple. E is the internal energy and G is

the free energy.



b. The state of the solvent near the red particle, which has

now become the ox particle, changes, corresponding to

the thermodynamic equilibrium of the ox particle in the

solution.

c. The electron from the level £ vac is instantly transferred to

the ox particle formed (to the level £°ox) without any change

in the state of the solvent.

d. A certain rearrangement of the solvent occurs around the

red particle formed; as a result, the whole system returns

to the initial equilibrium state.

In this cycle, the changes in the internal energy of the system

taking place at stages a and c characterize the depths of the levels

£°ed and E°ox with respect to £ vac ; these changes are also equal to

the ionization energy of the red particle and the electron affinity

of the ox particle in the solution. The energy characteristics corresponding to stages b and d represent reorganization energies of the

solvent, £R,red and ERox, related to the red and ox particles.

For the process considered here, the quantities EKredox > 0 are

the absolute value of energies (free energies to be exact), which

are necessary to alter the state of the medium (solvent) from the

initial to the final equilibrium state when the charge of the particle

changes by unity.10"12



194



Yu. V. Pleskov and Yu. Ya. Gurevk*



Thus, by virtue of the above analysis,

£red + E R r e d ~ E°ox + E^ox = 0



(6)



The quantity Fo is defined by the relation

F0



=



E%d + ^R,red = ^ox ~ ER,OX



0)



which follows from Eq. (6) and characterizes the change in the free

energy of the system considered when an electron is transferred,

in an equilibrium manner, from the red particle in the solution into

vacuum (or when an electron is transferred, again in an equilibrium

manner, from vacuum to the ox particle). Combining Eqs. (6) and

(7), we obtain

(8)

where

AER = £ R , r e d - EKox



and



E°ox - E°Ted = E^ox + £ R , r e d



(9)



We note that relation (9) can also be derived if we consider

the equilibrium transfer of an electron directly from the red particle

to the ox particle within the solution.

In polar liquids, in particular in water, one may assume EKox =

£/?,red — ER, especially for the so-called outer-sphere reactions.13 In

this case, the main contribution to the change in state of the medium

is due to repolarization of the solvent caused by long-range electrostatic interactions.

We have then from Eqs. (8) and (9),

Fo = K^ox + £red),



E°ox ~ E°red = 2ER



(10)



It can be seen from Eq. (10) that, in the simplest case considered

here, the characteristic level Fo lies midway between E°ox and E°ed,

and the difference between E°ox and E°ed is twice the value of ER.

Thus, using Eqs. (3) and (5) and taking into account that Fo

is calculated per particle, we finally obtain for the electrochemical

potential, /I*1,8

- RTIn—

where NA is the Avogadro number.



(11)



Electrochemistry of Semiconductors: New Problems and Prospects



195



Since relation (11) is of fundamental importance, we shall give

another, independent derivation of this relation by the methods of

statistical mechanics,7'9 without using Eq. (5).

Consider a system which is a set of cells whose average number

per unit volume is N. These cells contain at most one electron per

cell, so that the average number of electrons per unit volume is

n < N. In this case, the free energy of the system with only one

electron is Fo. The free energy of a unit volume of such a system

containing n electrons, G, is given by the relation (see, e.g., Ref. 14)

G = Fon- TS



(12)



the 5 is the so-called configurational entropy, equal to

AM



S = -k\nCnN = -k\n—

-—

(13)

(N - n)\n\

where k is the Boltzmann constant (k = R/ NA) and CnN is the

number of ways according to which n particles may be placed in

N cells. The electrochemical (chemical) potential of electrons jxe

per mole of the substance is {xe = NAdG/dn. Taking into account

Eq. (13) and the formula a (In q\)/aq — In q (q » 1), we obtain

for /Ie:

fle = JfAF0-RT In ^ - ^

(14)

n

We note that the model system considered here is, in fact, a

redox couple in the solution. Indeed, electrons in this system, whose

number is, obviously, cred, are distributed over (cox + cred) positions.

Thus, substituting N = cox + cred and n = cred into Eq. (14), we

obtain an expression which coincides with Eq. (11).

Hence, the electrochemical potential of electrons in the solution can be introduced quite rigorously and consistently within the

framework of both the thermodynamic and statistical-mechanical

approach.

In analogy with the Fermi level of electrons in solids (calculated

per electron), we introduce the quantity F redox , which is denned by

the relation Fredox = {*.*!NA and represents the electrochemical

potential of electrons in the solution, per particle. The quantity

^Vedox is frequently called "the Fermi level of electrons in solution."

We note that this term (and more so the term "Fermi level of



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Yu. V. Pleskov and Yu. Ya. Gurevich



solution") should be considered as not quite adequate and often

leading to confusion. It should be stressed that the position of Fredox

in the solution is in no way related to the concentration of "free"

electrons in it, but is determined by the presence in the solution of

a redox couple, which contains only "bound" electrons. The quantity Fredox does not characterize the energy of any particle because

no electrons with the energy Fredox are present in the solution

(unlike, e.g., the case of metals where the Fermi level Fmet coincides

with the energy EF of electrons at the Fermi surface). Thus, in the

course of electrode reactions, electrons can pass neither from the

level Fredox to the electrode, nor from the electrode to this level;

transitions occur with the levels E°ed and F° x involved. At the same

time, it is the position of the level Fredox, which determines the

thermodynamic properties of a redox couple and the interface

between the electrode and solution containing this couple. For

example, proceeding from the condition of thermodynamic

equilibrium in the semiconductor/solution system,

F=Fre6ox



(15)



(F is the Fermi level of the semiconductor), the condition for an

electrochemical reaction to occur can be written in the form

F > Fredox



(cathodic reaction)



(16a)



F < Fredox



(anodic reaction)



(16b)



and

Similar expressions can be used in considering photoelectrochemical reactions (see Section IV.2).

The electrochemical potentials of redox couples in solution

are calculated in a known manner from the thermodynamic characteristics of the substances involved. For certain reactions, which

proceed with the participation of the semiconductor electrode

material, in particular anodic or cathodic decomposition reactions,

the values of Fredox are as reported in References 15-18.

2. Relationship between the Energy Scale and the Scale

of Electrode Potentials

The position of the energy levels E°ox and F° ed and the level of the

electrochemical potential F redox with respect to the level F v a c can



Electrochemistry of Semiconductors: New Problems and Prospects



197



be determined by establishing a quantitative relationship between

the "physical" energy scale, in which the energy £vac is taken as

the zero reference point, and the electrochemical scale of electrode

potentials, which is reckoned from the potential of a certain

arbitrarily chosen reference electrode. The relationship between

these scales is, in general, of the form

Fredox = - V + c o n s t



(17)



Here e > 0 is the absolute value of the electron charge.

Let us consider such an ox/red couple, for which the equilibrium potential
between the solution of this couple and an inert conductor [i.e.,

one which does not take part in reaction (1)] is, by definition, the

reference electrode. The constant in Eq. (17) coincides then with

^Vedox, thereby representing the change in the free energy in the

course of the reaction

ox + e~ac = red



(18)



In other words, this constant is the change in the free energy in

the transfer of an electron from a point in vacuum near the solution

surface to the Fermi level of the reference-electrode metal, this

change being equal to Fo for cox = cred. The quantity Fo/ e is sometimes called the "absolute potential" of an electrode.19

An attempt at determining this quantity was first made in

Reference 19 and then in Reference 20. By analyzing a rather

complicated thermodynamic cycle,t the author of Reference 20

found that the change in the free energy of the reaction

H: q +e v - ac *±!H 2



(19)



which is equal to the value of the constant in Eq. (17) for the

normal hydrogen electrode (NHE), amounted to -4.5 eV.

The results of Reference 20 have been extensively used in the

literature (in particular, in considering the photoelectrochemical

method of solar energy conversion) for determining the

electrochemical potential level of various redox couples.

t This cycle was based on the known value of the standard potential of a silver

electrode in a solution of Ag+ ions relative to the NHE and comprised the following

stages: sublimation of metallic silver, ionization of silver atoms thus formed, and

hydration of the ions.



Yu. V. Pleskov and Yu. Ya. Gurevich



198



It should be noted, however, that the method chosen in Reference 20, as well as in certain subsequent papers (see, e.g., Reference

21) for determining AG appears to be unjustifiably complicated

and gives lower accuracy. Indeed, in those calculations the values

of, for example, sublimation energy and ionization energy were

taken into account twice, and with opposite signs, so they cancel

out in the final result.

A simple and accurate method of comparing the scales of

energies and electrode potentials follows from Reference 22 and

lies in the calculation of the work function of a metal, which is in

the electrolyte solution at the reference-electrode potential. This is

illustrated in Fig. 2, which shows a metal in equilibrium with an

ox/red couple in the solution (so that their electrochemical potential

levels coincide: Fredox = Fmet). The change in the free energy of

reaction (16) is the sum of the energies of stages c and d of the

cycle shown in Fig. 1, that is the quantity F redox (which coincides,

as was already noted, with Fo for cox = cred).

It can be seen from the cycle of Fig. 2 that

(20)

=w

Here w is the work function of the uncharged metal in vacuum,

and At/us the Volta-potential difference in the metal/solution system

which arises as a result of mutual charging of "free" surfaces of

the metal and solution when they are brought into contact.

For a particular case, for example, for the normal hydrogen

electrode, it is convenient to perform calculations using data for

the Volta-potential difference in the mercury-aqueous solution system23: A

that the work function of mercury in vacuum is24 w = 4.50 ± 0.03 eV,



w

—redox



P



F



met



Figure 2. Diagram illustrating

the relationship between the

"physical" and "electrochemical"

scales.



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199



we obtain from Eq. (18)

- Fredox(NHE) = 4.50 - 0.07 = 4.43 - 4.4 eV.

Thus, the level of the electrochemical potential of the normal

hydrogen electrode, Fredox(NHE), lies 4.4 eV below the energy level

^vac-



Let us recall in this connection that the thermodynamic work

function is equal to w = x + \F - Ec\. Here x is the electron affinity

of the semiconductor, and the position of F relative to the conduction-band bottom, Ec, in the semiconductor bulk is given by the

following relations:

F-Ec



= -Eg/2 + kT\n(Nv/Nc)1/29



no = po (/-type)



(21a)



F- Ec = -kTln(Nc/n0)9



n0»p0



(n-type)



(21b)



F - Ec = -Eg + kT ln(Nv/p0),



Po » «o (p-type)



(21c)



where Eg is the band gap, Nc and Nv are the effective densities of

states in the conduction and valence bands, and n0 and p0 are

thermodynamic equilibrium concentrations of electrons and holes

in the semiconductor bulk.

3. Determination of the Reorganization Energy of a Solvent



Using the results presented above, one can suggest a method of

determining the reorganization energy from data on photoemission

of electrons from solutions into the vapor phase (see review25).

Illumination of a solution containing red particles with light

of quantum energy exceeding the electron work function leads to

photoemission of electrons into the vapor phase. Since photoionization is a fairly rapid process, an electron passes into the vapor

phase from the level E°ed, so that the "photoelectric" work function

appears to be equal to —E°ed9 if we take £ vac = 0. It follows from

•Eq. (11) that for cox = cred it differs from the "equilibrium" (thermodynamic) work function, equal to -F r e d o x = - F o , by the reorganization energy E^Ted (see Fig. 1). Thus, the latter can be found

as the difference between the photoelectric work function, wph =

-E°cd, and -F r e d o x (a similar method for determining £R,red was



200



Yu. V. Pleskov and Yu. Ya. Gurevich



employed previously26 for the case where solvated electrons served

as "emitters" in the solution).

The quantity wph can be found from the dependence of the

photoemission current on the light quantum energy by extrapolating

it to the zero current. Reference 27 reports the work function as

5.95 eV for photoemission from aqueous solutions of Fe(CN)64~.

Using the standard potential of the system Fe(CN)64"/Fe(CN)63~

as +0.36 V (relative to the normal hydrogen electrode), we obtain

for the reorganization energy of water around ferrocyanide/ferricyanide ions: 5.95 — 4.4 — 0.36 — 1.2 eV. For the system considered, the electron transfer reaction is an outer-sphere one, so

one can take E^ox = EKred = ER.



We note in this connection that reliable experimental data on

the values of ER are rather scanty.10 According to Reference 28, in

aqueous solutions at room temperatures ER amounts to about

0.5-2 eV. The problems concerning model calculations of ER are

considered in detail in the review.13 Moreover, we have to bear in

mind that the "homogeneous" values of ER (which can be determined, in particular, using electron photoemission from solutions)

are, apparently, much higher than the "heterogeneous" values

(which can be found, for example, by processing experimental data

on tunnel currents flowing from semiconductor electrodes into

solutions28). Physical reasons for this discrepancy are mainly related

to the fact that in heterogeneous charge transfer processes only the

half-space occupied by the solvent is rearranged, and, to a lesser

extent, to the contribution of image forces near the surface to the

reorganization energy, and also to the effect of specific adsorption,

structurization of the solvent near the interface, deformation of

solvation shells of reacting particles near the electrode, etc. In the

simplest model, the "homogeneous" reorganization energy exceeds

approximately twice the "heterogeneous" reorganization

energy.10'13

It is of interest that Reference 28 reports for the system

Fe(CN)64~/Fe(CN)63~ the "homogeneous" value 1.24 eV obtained

by model calculation, which agrees well with the value presented

above. As for the "heterogeneous" reorganization energy, the same

paper reports, as it might be expected, a much lower value (0.4 eV)

obtained from tunnel currents measured on heavily doped SnO2

electrodes.



Electrochemistry of Semiconductors: New Problems and Prospects



201



III. SPECIFIC FEATURES OF THE STRUCTURE OF THE

SEMICONDUCTOR/ELECTROLYTE INTERFACE

An electrical double layer arises at the semiconductor electrode/electrolyte solution interface, as in the case of the metal/ solution interface. The double layer consists of the "plates" carrying

charges of opposite sign, each "plate" being located in one of the

phases in contact. In the near-surface region of the semiconductor

the charge is formed as a result of redistribution of electrons and

holes, while in the solution it is formed as a result of ion redistribution. Under equilibrium conditions, the absolute values of these

charges are the same.

1. The Electrical Double-Layer Model

The factors which lead to the formation of an electrical double

layer are rather general. First, charges flow across the interface

when a thermodynamic equilibrium is established between the

phases in contact; second, it is charging processes, which are not

generally related to charge transfer across the interface—for

example, charging of surface states (see below), certain types of

adsorption, etc.

According to the conventional model,1'29'30 three regions can

be distinguished within the electrical double layer: the space-charge

region in the solution, the intermediate region, called the Helmholtz

layer, and the space-charge region in the semiconductor.

Helmholtz

layer

Space charge (

layer

\



Electrolyte



Semiconductor'



Figure 3. Components of potential drop at the semiconductor/electrolyte interface.



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