Fast electron transfer across semiconductor-molecule interfaces: GaAs/Co(Cp)2+/0

2122

J. Phys. Chem. B 1999, 103, 2122-2141

+/0

Fast Electron Transfer Across Semiconductor-Molecule Interfaces: GaAs/Co(Cp)₂

Andreas Meier,* Donald C. Selmarten, Kerstin Siemoneit,^†Barton B. Smith,* and

Arthur J. Nozik*

Center for Basic Sciences, National Renewable Energy Laboratory, 1617 Cole BouleVard,

Golden, Colorado 80401

ReceiVed: July 30, 1998; In Final Form: January 20, 1999

+

The kinetics of majority electron transfer in the dark from n-GaAs electrodes to cobaltocenium (Co(Cp)₂)

acceptors in acetonitrile has been studied in detail, both experimentally and theoretically. The experimental

results were obtained from electrochemical impedance spectroscopy, quartz crystal microbalance (QCM and

EQCM) studies, and current-potential characteristics. The theoretical work involved calculating the adsorption

energy and molecular configuration of the cobaltocenium acceptors at the GaAs surface using high level

density functional theory (B3LYP and variations thereof) as well as semiempirical methods. The QCM

+

0

experiments showed that both Co(Cp)₂and Co(Cp)₂are physisorbed at GaAs surfaces, with adsorption

energies of about 0.2 and 0.4 eV, respectively. The theoretical results are consistent with these experimental

results. They indicate that adsorption of the Co(Cp)₂^+/0redox system occurs on GaAs, with Co(Cp)₂⁰somewhat

+

+/0

more strongly adsorbed than Co(Cp)₂; the Co(Cp)₂molecules were found to adsorb with the cyclopen-

tadienyl rings parallel to the GaAs surface. A model for the overall electron-transfer process was developed

that incorporates Co(Cp)₂⁺adsorption. Analysis of the detailed impedance spectra over the range of 1 Hz to

600 kHz showed that the sequential electron-transfer steps in the model (i.e., electron transfer from the GaAs

+

0

conduction band to adsorbed Co(Cp)₂, followed by electron transfer from the adsorbed Co(Cp)₂to free

Co(Cp)₂⁺in solution) are very fast and that the observed overall rate of electron transfer is limited by the rate

of thermionic emission from the GaAs bulk region to the surface. The implications of these results for the

theory of electron transfer at semiconductor-liquid interfaces, and the associated controversies surrounding

theory and various experimental results for GaAs-metallocenium systems, are discussed.

I. Introduction

redox electrolyte, and its associated beneficial effect on energy

conversion efficiency,²³depends on the rate of ET being faster

The kinetics of charge transfer at semiconductor-molecule

interfaces (SMI) (also commonly referred to as semiconductor-

electrolyte interfaces) is a very important issue that is currently

receiving much attention.^1-13The charge transfer can involve

either electrons residing in the conduction band or positive holes

residing in the valence band of the semiconductor; in this work

we will only discuss electron transfer from the conduction band.

Furthermore, the electron transfer (ET) can be from the

semiconductor to redox acceptor molecules at the semiconductor

surface, or it can involve electron injection from photoexcited

adsorbed molecules into the semiconductor conduction band.^14-20

Finally, the former process includes electron transfer in the dark

from n-type semiconductors (majority electron transfer) as well

as photoinduced electron transfer from p-type semiconductors

(minority electron transfer). Here, we will only discuss our work

on the dark electron transfer of majority electrons from n-type

semiconductors to redox acceptors. The acceptor species is the

cobaltocenium cation in acetonitrile (ACN); the semiconductor

is epitaxially grown GaAs.

than the rate of hot electron relaxation (i.e., cooling) via

electron-phonon interactions.²²

The dynamics of electron transfer across semiconductor-

electrolyte interfaces is controversial, both with respect to theory

and experiment. One approach to the theory has been to adapt

a model that was developed for ET between two immiscible

liquids^24,25(which treats the redox species as hard spheres with

fixed radii) to the semiconductor-liquid interface by modeling

electrons in the semiconductor as hard spheres with a radius

related to the Bohr radius of an electron bound to an impurity

donor site in the semiconductor.¹⁰This model is constrained to

the regime of nonadiabatic ET, i.e., to very weak electronic

coupling between the semiconductor and the acceptor molecule

and thus to nonadsorbing and ideal outersphere acceptors. The

model has been challenged²⁶and defended.^4-9Experimental

studies of ET kinetics have also been reported that both

support^4-10and apparently contradict^1,2,13the theoretical predic-

tions of this model.

The range of experimental results on the kinetics of ET at

various semiconductor-liquid interfaces is very large, with the

reported heterogeneous second-order rate constants (k_ET) varying

by up to 6-7 orders of magnitude. The model proposed in refs

4-9 places an upper limit of about 1 × 10^-17cm⁴/s for the

rate constant for ET to nonadsorbed outersphere redox acceptors

at semiconductor surfaces; this is presented as a general limit

that is valid for all semiconductors and outersphere redox

molecules. Experimental results reported in refs 4-10 are

apparently consistent with this proposed upper limit. However,

ET kinetics are important both from the point of view of

photoelectrochemical devices and for the fundamental science

of these interfaces. For example, the efficiency of photoelec-

trochemical energy conversion systems depends on the rate of

ET compared to the rates of electron-hole recombination at

the surface and in the semiconductor bulk. Also, the possibility

of hot electron transfer^1,3,21,22from semiconductors to liquid

^†Institut fu¨r Solarenergieforschung, Hannover, Germany.

Published on Web 03/11/1999

Semiconductor-Molecule Interfaces

J. Phys. Chem. B, Vol. 103, No. 12, 1999 2123

values of k_ETreported in refs 2 and 13 range from 1 × 10^-10to

1 × 10^-18cm⁴/s; these values are based on the assumption that

the acceptor concentration is the value for the bulk solution.

In this study we examine the reasons for the apparent

discrepancy in reported k_ETvalues. We use quartz crystal

microbalance (QCM) techniques, detailed impedance spectro-

scopic studies at low and high current flow, and theoretical

calculations to examine the ET dynamics. We find that

adsorption of the cobaltocenium redox acceptor plays a critical

role in the dynamics, notwithstanding the general previous belief

that metallocenes and their cations are outersphere, nonadsorbing

molecules.^6,8,10,27,28These results have important implications

for the various models of ET at the SLI.

using a Solartron 1260-FRA in conjunction with a Solartron

1287-EI potentiostat. To avoid artifacts in impedance measure-

ments in the high-frequency range (>50 kHz), the platinum wire

of the reference electrode was short circuited via a capacitance

(10 nF) and a platinum wire in series.²⁹Only the tip of the

platinum wire was exposed to the cell solution and placed close

to the frit of the reference electrode. The majority of experiments

were conducted at rotating disk electrodes (RDE) at rotation

velocities between 50 and 2000 rpm (Rotator AFASR, Pine

Instrument).

C. Quartz Crystal Microbalance (QCM). The QCM allows

in situ detection of mass changes at an electrode surface with

submonolayer precision; it is actually a quartz crystal nanobal-

ance. The basis for the QCM’s mass sensitivity is the piezo-

electric effect for materials that crystallize with noncentrosym-

metric space groups; an electric potential causes a physical

distortion of the crystal lattice. A crystal of given mass will

resonate at a specific frequency. The resonant frequency is

determined by the acoustic properties of the particular crystal.

Changes in crystal mass or thickness (e.g., from adsorption of

material from solution or vapor deposition) result in a change

in the resonant frequency. This frequency change is described

by the Sauerbrey equation:

II. Experimental Section

A. Chemicals and Electrodes. The cobaltocene-cobalto-

cenium redox couple used in the electrochemical investigations

was purchased from Strem Chemicals. Prior to use, the

0

cobaltocene (Co(Cp)₂) was purified through sublimation in a

vacuum at 70 °C and the cobaltocenium (Co(Cp)₂⁺) hexafluo-

rophosphate was recrystallized from ethanol. The acetonitrile

(ACN, Burdick & Jackson) was refluxed for 1 day and distilled

under nitrogen atmosphere over calcium hydride. Tetrabutyl-

ammonium hexafluorophosphate (TBAPF₆, Aldrich) was used

as supporting electrolyte after it was recrystallized from ethanol

three times and dried under vacuum at 180 °C. In all

electrochemical studies the TBAPF₆concentration of the ACN

solutions was 0.3 M. All chemicals were stored in a glovebox

under an atmosphere of helium gas.

2f ²₀∆m

A(F_qµ_q)^1/2

∆f ) -

(1)

where ∆f is the measured frequency change, f⁰is the funda-

mental frequency of the oscillator, ∆m is the change of mass,

A is the piezoelectrically active area, F_qis the density of quartz

(2.648 g cm^-3), and µ_qis the shear modulus for quartz (2.947

× 10¹¹dyn cm^-2).³⁰

The GaAs electrodes used for the electrochemical investiga-

tions presented here consist of a 2-µm-thick epilayer of GaAs

grown on heavily doped GaAs substrates. The GaAs epilayers

were grown in an atmospheric pressure metallorganic chemical

vapor deposition (MOCVD) reactor. The doping density of the

GaAs samples ranged between 1 × 10¹⁷and 2 × 10¹⁷cm^-3for

the studies presented in this paper. For the electrochemical

studies the GaAs epilayers were cut into small square pieces

(≈0.6 × 0.6 cm), contacted with Ga-In alloy, and mounted

with conductive silver epoxy (Epoxy-Technology, 415G or H31)

as rotating disk electrodes. To provide an approximately circular

geometry of the electrode surface and to seal the electrodes,

the edges of the GaAs epilayer were covered with Epoxy

(HYSOL Epoxi-Patch). The electrode areas were usually

between 0.23 and 0.15 cm². Prior to the experiments the

electrodes were either treated with a half-concentrated HCl or

half-concentrated NH₄OH solution for 60 s to remove gallium

or arsenate oxides from the GaAs surface that developed during

the preparation procedure. These etching procedures produced

the highest rates of ET. GaAs electrode samples that had been

sitting in air for several days and were unetched, or had been

exposed to an oxidizing etch like H₂O/H₂SO₄/H₂O₂, showed

slower ET rates by up to 6 orders of magnitude. The slow ET

rates are attributed to oxide barrier layers on the surface.

The reference potential (measured with a platinum wire) was

either the solution redox potential or the redox potential of a

ferrocene-ferrocenium solution (Fc^+/0, molar ratio 1:1) sepa-

rated by two frits (glass and “Vycor”) in a different cell

compartment. The counter electrode was a circular bent platinum

wire placed on the flat bottom of the cone shaped cell.

The quartz crystal nanobalance instrument used in the

measurements was an Elchema model EQCN-700. When

electrochemical experiments were performed, the above instru-

ment was used in conjunction with an Elchema Model PS-605

high-precision potentiostat/galvanostat. The working electrode

was the GaAs epilayer (with potential necessarily held at

ground), and Pt wires were used for the counter and reference

electrodes. The 10 MHz quartz crystals were obtained from

International Crystal Manufactures. The crystals were first

cleaned with dichloromethane to remove any organic contami-

nants prior to use.

In order for the mass sensitive surface to consist of GaAs, it

was necessary to position a piece of single crystal GaAs onto

the active surface of the nanobalance crystal. Since GaAs could

not be epitaxially grown directly onto a substrate of gold, the

following procedure was performed. A multiepilayer structure

was prepared with the intention of removing all material but

the epilayer of interest from the structure and introducing this

epilayer onto the nanobalance crystal. The initial multilayered

structure consisted of a substrate of bulk GaAs, a stop-etch layer

of 1 µm of GaInP₂, and a 1 µm epilayer of GaAs. The stop

layer and the GaAs epilayer of interest were grown epitaxially

in an MOCVD reactor. This device was then placed (epilayer

down) in black wax in order to protect it from etching. The

substrate was then etched with a solution of phosphoric acid/

peroxide/water (3:4:1), which effectively removed the bulk

GaAs, but was unreactive to GaInP₂(the stop-etch layer). The

GaInP₂stop-etch layer was then removed by a hydrochloric acid/

acetic acid/peroxide (2/20/1) solution. This solution was chosen

for its selective reactivity toward GaInP₂over GaAs. With only

the epitaxial GaAs remaining, a thin layer of Au (400 Å) was

added by vapor deposition. This was accomplished in a Key

B. Electrochemical Apparatus. The electrochemical mea-

surements were performed in a glovebox (Mbraun, Labmaster

130) under an atmosphere of helium gas and using a three-

electrode setup. Current-voltage (I-V) and impedance spec-

troscopic measurements (1 Hz to 600 kHz) were performed

2124 J. Phys. Chem. B, Vol. 103, No. 12, 1999

Meier et al.

High Vacuum Products model KV-301 gold evaporator (base

line pressure 1 × 10^-6Torr). This gold layer was added to

improve the nanobalance/GaAs contact and to provide ohmic

contact to the circuit. The black wax was removed from the

GaAs epilayer with dichloromethane, and the epilayer was then

“floated” onto the quartz crystal. Edges of the GaAs and any

exposed gold were then coated with protective nylon (solvated

in phenol). Because the 10 MHz crystals are physically too

fragile to be used in reusable cells in conjunction with O-rings,

the nanobalance crystal with the GaAs epilayer was permanently

glued with silicone to a glass cell, with the epilayer facing the

solution side of the cell. The QCM measurements were made

both at open circuit (no current flow), and with current flow to

determine if there are mass changes associated with ET and

faradaic current. With current flow QCM is referred to as an

electrochemical quartz crystal Microbalance (EQCM) experi-

ment.

If the electrode shows a weight increase during current flow,

then ET has caused either (a) adsorption of the reduced acceptor

species or (b) multilayer formation of the reduced acceptor (if

the acceptor was already adsorbed prior to current flow). If there

is a weight loss during current flow, then ET has resulted in a

desorption of the reduced acceptor species. Finally, if there is

no change in electrode mass, then (a) the ET is mediated either

through a self-exchange reaction between adsorbed and free

acceptors in solution or (b) the rate of desorption and adsorption

are comparable.

D. Blank Preparation. SiO₂was deposited onto a nanobal-

ance crystal by e^-beam evaporation. The resistive quality of

the overlayer was tested with an I-V experiment. The gold

blank was prepared by vapor depositing approximately 500

angstroms of gold onto the quartz crystal in a gold evaporator

(Key High Vacuum Products model KV-301). For the nylon

blank, nylon suspended in a volatile organic matrix was simply

applied to the surface of the quartz crystal; it was then cured at

elevated temperatures. In this case the integrity (insulating value)

of the nylon coverage was tested with an I-V experiment. In

all cases the treated nanobalance crystals were glued to a glass

cell with silicone.

Figure 1. (a) QCM response of GaAs upon the addition of aliquots

+

of 5 µL of 100 mM Co(Cp)₂in ACN to a cell containing 1 mL of

electrolyte (100 mM TBAPF₆in ACN). (b) QCM response of GaAs

0

upon the addition of aliquots of 5 µL of 44 mM Co(Cp)₂in ACN to

Viscosity measurements were performed with a Cannon-

Ubbelolde viscometer, model 25 B114. The average of at least

five room-temperature measurements was taken to be the

kinematic viscosity for a given solution concentration of analyte.

3 mL of electrolyte.

change at the GaAs electrode.³²Note that in this figure the scale

is an order of magnitude larger than the cobaltocenium case,

and the concentration of the added species is several times more

dilute. This indicates qualitatively that while both oxidized and

reduced forms of the redox species display an affinity for the

GaAs surface, the neutral reduced species is bound more

strongly. In fact, in our preliminary experiments, cobaltocene

additions after the initial aliquot render only minor mass

inflections. This behavior suggests that a monolayer may have

formed with the initial addition of cobaltocene. This monolayer

formation may have an inhibiting effect on further binding

events.

III. QCM Results

A. Cobaltocenium Response. As can be seen in Figure 1a,

GaAs shows a mass response to changes in the solution

concentration of cobaltocenium. An electrode mass change

(when it has been experimentally verified that there is no

viscosity change) indicates that cobaltocenium is binding to the

surface of the GaAs electrode. The supporting electrolyte in

this experiment is 100 mM TBAPF₆; this was chosen to most

closely match the experimental conditions in the electrochemical

experiments. In other experiments, the addition of two aliquots

of blank electrolyte solution was investigated to ascertain the

systems susceptibility to the mechanical stresses of aliquot

addition. Although the system did show a slight perturbation

to the mechanical addition of electrolyte, it was seen that this

is a transient response and that the equilibrium resonant

frequency was quickly reestablished. This method of adding a

blank solution to the QCM cell has been similarly employed in

the literature³¹to establish unequivocally that the analyte was

binding to the electrode.

C. EQCM Results. With current flow the quartz crystal

electrode showed no change in mass even as the diffusion-

limited current was approached (potential more negative than

-320 mV) (see Figure 2). When the current was driven at the

highest values (in a galvanostatic experiment), a mass change

could be observed after tens of seconds. Concurrent with this

mass change was a dramatic shift of potential (more than 800

mV). On the basis of literature reports of cobaltocene reduction

of platinum microelectrodes,³³we believe this mass change at

high potential and current to be associated with the further

reduction of neutral cobaltocene and subsequent adsorption.

B. Cobaltocene Response. As seen in Figure 1b, addition

of the neutral cobaltocene species to the cell also evokes a mass

Semiconductor-Molecule Interfaces

J. Phys. Chem. B, Vol. 103, No. 12, 1999 2125

mental behavior indicates that, for cobaltocenium on GaAs, a

monolayer of cobaltocenium is bound more strongly than any

subsequently bound material.

It has been shown that changes in solution viscosity can have

an effect on the resonant frequency of a QCM.³⁴The contribu-

tions of solution viscosity (η_L) and density changes (F_L) are

shown in eq 2. Here, f_sis the fundamental frequency of the

crystal. Careful viscosity measurements exclude any significant

viscosity changes within the concentration range of cobaltoce-

nium used in these studies.

∆f_s) -[2f_s²/(µ_qF_q)^1/2][(∆m/A) + [F_Lη_L/4πf_s]^1/2] (2)

It is reasonable to assume that in our experiments, the mass

change response is from the GaAs surface. In an AT cut piece

of quartz there is a differential sensitivity constant, c_f:

Figure 2. EQCM response (mass and current) of n-GaAs as a function

of electrode potential. Electrolyte is 100 mM TBAPF₆with 0.33 mM

Co(Cp)₂and 0.03 mM Co(Cp)₂.

c_f) df/dm ) S

(3)

+

0

This represents the differential frequency change for a given

mass change at a particular region of the crystal surface. That

is, the change in frequency per unit mass is a function of position

on the crystal. Essentially this sensitivity function looks like a

Gaussian centered at the gold flag overlap regions and decreases

monotonically to the point of being negligible beyond the

electrode boundary. The instrument reads the integral sensitivity

constant (the differential sensitivity constant integrated over the

surface of the crystal) when it is measuring a frequency change.

Energy trapping, the effect of the crystal oscillations being

confined to a region of higher density (defined by the electrode

region), tends to confine the oscillations of the quartz crystal

in the region that has the gold electrodes deposited. Hence,

considering that nylon is a nonresponsive material and coats

everything but the exposed GaAs, the mass response observed

in these experiments arises from the GaAs electrode-electrolyte

interface.

Figure 3. Comparison of adsorption isotherms for Co(Cp)₂⁺on GaAs,

SiO₂, and Au surfaces.

E. Absorption Isotherm. Langmuir Isotherm. Thermody-

namic information about the adsorption process may be obtained

from the adsorption isotherms. The free energy of adsorption

is defined as the difference in the standard electrochemical

potentials between the adsorbed and free species, µj_i^0,Aand µj_i^0,b

respectively, as shown in eq 4.

Whereas this reaction is not directly involved with the scope

of this paper, it is interesting that we are able to observe it with

EQCM.

D. Blanks. To verify that the results in our experiments were

exclusively from activity at the GaAs-electrolyte interface, we

performed several control experiments. In each case we

configured the cell in such a way that only one material was

exposed to solution. Each material was chosen to represent any

possible surface that may be exposed to the electrolyte solution

in the actual GaAs experiment. Figure 3 shows the response

from the different blanks (and GaAs) to changing solution

concentration of Co(Cp)₂⁺. The first thing to note is that all

surfaces (except nylon, not shown) show some affinity for the

cobaltocenium species. Whereas each surface (except for nylon)

does show a response, each isotherm is different in both shape

and slope. Au shows a linear mass response with changing

solution concentration. A linear isotherm indicates physical

adsorption with no monolayer distinction; i.e., the surface shows

the same affinity for the solution species after an entire

monolayer has been formed. This activity excludes the pos-

sibility that a chemical bond forms between the surface and

the species. The response of SiO₂to changing cobaltocenium

concentrations has a much more shallow slope and displays a

slight change in slope after the solution concentration reaches

10 mM. The different slopes indicate that cobaltocenium has a

differentially higher affinity for gold than quartz. In contrast,

the GaAs surface becomes relatively unresponsive to changes

in solution concentrations greater than 10 mM. This experi-

∆Gh⁰_i) µj⁰_i^,A- µj⁰_i^,b

(4)

A number of different isotherms have been proposed

(logarithmic Temkin, Frumkin, etc.), and there have been a

number of different experimentally determined isotherm shapes.³⁵

Here we use the simple Langmuir isotherm to interpret our

experimental data. When employing this model, certain as-

sumptions are involved; specifically, there is no interaction

between adsorbed species on the electrode surface, the surface

is completely uniform, and at high bulk activities a monolayer

is formed. Although the Langmuir isotherm involves several

assumptions, some of which are not precisely valid, and at best

may be considered to be less than rigorous, it has been proven

in the past to be a useful semiquantitative tool in the analysis

of adsorption events:

-∆Gh⁰_i

RT

â ) exp

(5)

(6)

(

)

θ

) â_ia^b

1

1 - θ

2126 J. Phys. Chem. B, Vol. 103, No. 12, 1999

Meier et al.

In the above expressions, the free energy of adsorption is

shown in terms of the fractional coverage of the surface, Γ_i/Γ_s

) θ. Where Γ_iis the surface excess of species i at equilibrium,

and Γ_sis the surface excess of species i at saturation. The above

expression can also be written in terms of solution concentration

of species if the activity coefficients are included in the â term.

In this simplistic treatment of the isotherm, we are not

accounting for any competitively adsorbed/desorbed species in

the QCM studies

Γ_sâ_iC_i

Γ_i)

(7)

1 + â_iC_i

In our analysis of the experimental isotherms for GaAs/Co-

+

(Cp)₂(see Figure 4), we assume that monolayer coverage

coincides with the first plateau of the electrode mass/solution

concentration plot. At this inflection point, the change in the

GaAs electrode mass equals 0.065 µg. This corresponds to 7.5

× 10¹³molecules/cm²(or equivalently 1.15 nm separation from

center to center). In an ideal close-packed surface (assuming a

molecule diameter of 4.6 Å) the molecule density would be

4.7 × 10¹⁴particles per cm². Since Co(Cp)₂⁰forms a monolayer

on GaAs at much lower bulk solution concentrations (i.e., 0.1

+

Figure 4. Adsorption isotherm for Co(Cp)₂on GaAs.

species and the electron density at the semiconductor surface.

However, this model leads to an unusually large ET rate constant

if the bulk concentration of Co(Cp)₂is used in the analysis,²

+

and it also does not satisfactorily describe all the experimental

results. Further investigations reported here reveal that electron

transfer between the n-GaAs electrode and the Co(Cp)₂^+/0redox

0

+

mM), it is assumed that with both Co(Cp)₂and Co(Cp)₂

present in the electrochemical experiments at 0.9 and 1.3 mM,

+/0

system in solution occurs via Co(Cp)₂molecules that are

0/+

respectively, a full monolayer of Co(Cp)₂is present. This

adsorbed on the semiconductor surface.

total coverage is taken to be approximately 4 × 10¹⁴molecules/

cm². As discussed in the next section, at equilibrium the ratio

of the oxidized and reduced species will be the same as in the

bulk solution.

Depending on the adsorption strength, it is reasonable to

distinguish two parallel reaction mechanisms that occur after

the first electron-transfer process from the semiconductor to the

adsorbed redox molecules. These two mechanisms, illustrated

in Figure 5, are (1) adsorption and desorption reactions of the

redox molecules on the semiconductor surface and (2) a self-

exchange (i.e., mediated ET) reaction between the adsorbed

redox molecules and the redox species in solution.

Depending on the binding strength of the adsorbed redox

molecules, either one of the two parallel mechanisms can

become a rate-determining reaction. This will be the adsorption

and desorption of the redox molecules if these reactions are

much faster than the self-exchange. However, if the self-

exchange rate is much faster than the adsorption and desorption

processes, then the rate-determining reaction will be the self-

exchange reaction between adsorbed cobaltocene-cobaltoce-

nium molecules (Co(Cp)₂^+/0(ad)) and the cobaltocene-cobal-

tocenium molecules in solution (Co(Cp)₂^0/+(sol)).

Heiland, Gileadi, and Bockris have studied the adsorption of

benzene on platinum;³⁶they used a value of 1.5 × 10¹⁴particles/

cm²as one monolayer. Also, Shimazu et al. have reported the

adsorption of a monolayer of ferrocenyldodecanethiol on gold

that consisted of a density of 2.6 × 10¹⁴molecules cm^{-2 31}

. Thus,

the number we have used here for monolayer coverage seems

reasonable. On the basis of the above Langmuir treatment of

+

our adsorption data for Co(Cp)₂on GaAs (see Figure 4), the

free energy of adsorption is approximately 20 kJ/mol (0.2 eV);

0

for Co(Cp)₂the adsorption free energy is estimated to be 0.4

eV.

IV. Impedance Measurements

A. Theory and Models. The kinetics of majority charge

+/0

carrier (electron) transfer for the Co(Cp)₂redox reaction at

To develop a model that describes the electrochemical

current-voltage (I-V) and impedance behavior of the system,

the charge-transfer steps have to be quantified. In the following,

the chemical reactions and the corresponding expressions for

the reactions rates are described

n-GaAs in ACN was previously studied and described based

on a simple heterogeneous charge-transfer reaction and applying

a second-order rate law.²For a second-order rate law the electron

transfer rate is proportional to the concentration of the redox

chemical reactions

reaction rates

majority charge carrier transfer

-

k_c

R ^-_c) N^maxk^-_cn_sθ_ox

R ⁺_c) N^maxk⁺_cN_cθ_red

+

0

(8)

(8a)

(8b)

Co(Cp)_{2 (ad)}+ e- y\z Co(Cp)₂

(ad)

k_c⁺

ad

self-exchange reaction

-

k_se

R ^-_se) N^maxk^-_seθ_redc_o^s_x

R ⁺_se) N^maxk⁺_seθ_oxc_r^s_ed

0

+

0

(9)

(9a)

(9b)

Co(Cp)_{2 (ad)}+ Co(Cp)_{2 (sol)}y

\z Co(Cp)_{2 (ad)}+ Co(Cp)₂

k⁺_se

ad

(sol)

ad

adsorption and desorption reactions

(10)

+

k_ox

R ^-_ox) N^maxk_o^-_x(1 - θ)c^s_ox

R ⁺_ox) N^maxk⁺_oxθ_ox

R ^-_red) N^maxk^-_redθ_red

R ⁺_red) N^maxk_r⁺_ed(1 - θ)c^s_red

+

(10a)

(10b)

(11a)

(11b)

Co(Cp)_{2 (ad)}y

\

k^-_ox

z Co(Cp)₂

ad

(sol)

ad

-

k_red

0

(sol)

(11)

Co(Cp)_{2 (ad)}y z Co(Cp)₂

ad

k⁺_red

ad

Semiconductor-Molecule Interfaces

J. Phys. Chem. B, Vol. 103, No. 12, 1999 2127

The kinetic model for the adsorption and desorption reactions

yields a Langmuir isotherm for the adsorption behavior of two

competitively adsorbed species³⁷(see Appendix A).

Assuming that the self-exchange reaction is the fastest of all

ET reaction steps and that the rate constants for the self-

-

exchange reaction are similar (k_se≈ k_se⁺), the ratio θ_ox/θ_redis

defined by the ratio of the redox concentrations at the surface,

s

θ_ox/θ_red≈ c_ox^s/c_red

(15)

Since the potential of the layer of redox molecules adsorbed at

the semiconductor surface is defined by the ratio θ_ox/θ_red

+

(identical with the ratio of the number of Co(Cp)₂and Co-

(Cp)₂molecules), the potential of the layer is determined by

the redox potential of the solution if the self-exchange reaction

is fast, i.e., significantly faster than the adsorption or desorption

reactions (see Appendix A).

0

Figure 5. Energy diagram for an n-type semiconductor (left side) and

+/0

kinetic reaction scheme for charge transfer to the Co(Cp)₂redox

+/0

couple in solution via adsorbed Co(Cp)₂molecules (right side).

max

ad

where R is the reaction rate; N is the maximum number of

For calculating the impedance of this system, it is important

to define the quantities that depend on the electrode potential

and therefore become time dependent when a small sinusoidal

voltage perturbation is superimposed on the electrode potential.

The potential-dependent parameters in general are the electron

concentration at the semiconductor surface (n_s), the degrees of

coverage (θ, θ_ox, θ_red), the concentrations of the redox couple

(c_ox^s, c_red^s) at the surface if mass transport of the redox species

has a limiting influence on the overall charge-transfer process,

and the rate constants k_c^-and k_c⁺if a movement of the energy

bands at the semiconductor surface occurs. As shown later, the

influence of the diffusion of redox ions as well as a movement

of the energy bands gain importance if the reduction current

exceeds 10% of the diffusion-limited current. However, in this

paper we will restrict the impedance analysis to a potential range

where band movement has no major influence on the parameters

adsorption sites, N_cis the effective density of states in the

0

conduction band; θ_redis the degree of coverage of Co(Cp)₂;

θ

_oxis the degree of coverage of Co(Cp)₂⁺; θ is the total degree

0

+

of coverage of Co(Cp)₂and Co(Cp)₂(θ ) θ_ox+ θ_red); n_sis

the electron density at the semiconductor surface; c_oxand c_red

are the concentrations of Co(Cp)₂⁺and Co(Cp)₂, respectively;

0

and k are rate constants (k_c^-, k_c⁺, k_se^-, k_se⁺, k_ox^-, k_redin cm³

+

s^-1and k_ox⁺, k_redin s^-1).

-

In the next section, the kinetic expressions that follow from

eqs 8-11 are derived. Since the results of the QCM measure-

0

ments show that Co(Cp)₂is significantly more strongly

adsorbed to GaAs than Co(Cp)₂⁺, it is reasonable to assume

that the self-exchange reaction is much faster than the adsorption

+

and desorption processes and that the Co(Cp)₂reduction

therefore proceeds via the self-exchange reaction rather than

via adsorption and desorption.

-

+

of interest. For this reason we will treat k_cand k_cas time-

independent constants in the following calculations. Considering

now the time-dependent quantities, the complex faradaic ac

current ∆I_Ffollows from eq 12 and is given by eq 16.

1. Charge Transfer Via a Self-Exchange Reaction. The

faradaic current for the majority charge carrier transfer across

the semiconductor interface is obtained from the reaction rates

(eq 8a and b). Considering that the reduction current is negative

and the oxidation current positive, the expression for the faradaic

current (I_F) is

∆I_F) eAN^max(k_c⁺N_c∆θ_red- k^-_cn_s∆θ_ox- k_c^-θ_ox∆n_s) (16)

ad

I_F) e A(R _c⁺- R ^-_c) ) e A N^max(k⁺_cN_cθ_red- k_c^-n_sθ_ox) (12)

The ∆ terms in eq 16 are the complex values of the time-

dependent parameters, and the notation as well as the procedure

for their evaluation is explained in Appendix B.

ad

where e is the elementary charge and A is the area of the

electrode surface.

The time-dependent occupation of the adsorption sites with

oxidized and reduced redox molecules is defined by the mass

and charge balance for the adsorption and desorption processes

and the charge-transfer events (eq 13a-c).

The time-dependent variations of the degrees of coverage can

be calculated from eq 13. EQCM measurements have shown

that the total degree of coverage θ does not change in the

potential range of interest, in the depletion region and under

current flow. This result suggests that either the rates for the

adsorption and desorption reactions are slow compared to the

potential changes during the potential scan or the total degree

of coverage θ is close to unity. However, with the assumption

that ∆θ ) 0, the appropriate rearrangement leads to the relation

17a and b for ∆θ_oxand ∆θ_red, respectively (it also has to be

_maxdθ

N_ad

) R ^-_ox- R ⁺_ox- R ^-_red+ R _r⁺_ed

(13a)

dt

_maxdθ_ox

N_ad

) R _s^-_e- R _s⁺_e- R ^-_c+ R _c⁺+ R ^-_ox- R _o⁺_x(13b)

) R ⁺_se- R ^-_se- R ⁺_c+ R ^-_c+ R _r⁺_ed- R _r^-_ed(13c)

s

considered that that ∆c_ox) -∆c_red^s).

dt

[k^-_scθ_red+ k⁺_seθ_ox+ k^-_ox(1 - θ)]∆c_o^s_x- k^-_cθ_ox∆n_s

_maxdθ_red

N_ad

∆θ_ox

)

iω + k^-_sec^s_ox+ k_s⁺_ec_r^s_ed+ k_c⁺N_c+ k_c^-n_s+ k_o⁺_x

dt

(17a)

Equations 13a-c include the assumption that the Co(Cp)₂⁺and

Co(Cp)₂⁰molecules adsorb at the same kind of adsorption sites

on the semiconductor surface, i.e., for stationary conditions

-[k^-_seθ_red+ k⁺_seθ_ox+ k⁺_red(1 - θ)]∆c_o^s_x+ k^-_cθ_ox∆n_s

∆θ_red

)

iω + k_s^-_ec^s_ox+ k_s⁺_ec^s_red+ k_c⁺N_c+ k_c^-n_s+ k_r^-_ed

θ ) θ_ox+ θ_red

(14)

(17b)

2128 J. Phys. Chem. B, Vol. 103, No. 12, 1999

Meier et al.

(21a)

According to the assumption that the Co(Cp)₂⁰and Co(Cp)₂

molecules adsorb competitively at the same kind of adsorption

+

m₄) k⁺_cN_c+ k^-_cn_s

(m_1,ω, m₂, and m₃are defined in eq 19a-c)

sites, and that ∆θ is zero, the time-dependent variations of θ_ox

must to be equal to the variations of θ_red, i.e., ∆θ_ox) - ∆θ_red

.

-

+

The frequency characteristics of the faradaic impedance Z_F

are identical to the frequency characteristics of the impedance

for the equivalent circuit in Figure 6 that is drawn in solid lines.

Because of the same frequency dependence, each element of

the equivalent circuit in Figure 6 (R_a, R_b, C_b, and Z_w) can be

correlated with a kinetic expression that follows from eq 21

through a comparison of the coefficients. The results from the

comparison of the coefficients are listed in the following four

eqs (22a-d).

For this reason the terms k_ox(1 - θ) and k_red(1 - θ) that

describe the adsorption processes have to be negligibly small

compared to the other terms in the sum in square brackets in

eq 17a and b. This will be the case if either the rate constants

-

+

k_oxand k_redfor the adsorption rates are small or if the total

degree of coverage θ is close to unity. If it is further assumed

+

-

that the rates k_oxand k_redfor the desorption processes are

negligibly small, the complex time-dependent variations of the

degrees of coverage is described by eq 18.

(k_s^-_eθ_red+ k⁺_seθ_ox)∆c^s_ox- k_c^-θ_ox∆n_s

iω + k^-_sec^s_ox+ k_s⁺_ec^s_red+ k_c⁺N_c+ k_c^-n_s

kT

1

R_a)

) R_ct

(22a)

e²AN^maxk_cn_sθ_ox

-

(

)

∆θ_ox) - ∆θ_red

)

ad

k⁺_cN_c+ k^-_cn_s

(18)

kT

R_b)

(22b)

e²AN^maxk_cn_sθ_ox(k_sec_ox+ k_sec_red)

-

s

+ s

[

]

Equation 18 includes the variation of the surface concentration

of the redox couple and the electron density as time-dependent

quantities. Consideration of the solution for the variation of the

surface redox concentration with the time-dependent faradaic

current as given in Appendix C yields eq 19.

ad

e²AN^maxk^-_cn_sθ_ox

ad

C_b)

(22c)

(22d)

+

-

(

)

kT

k_cN_c+ k_cn_s

Z_W) N^maxN_ω(k^-_seθ_red+ k_s⁺_eθ_ox)R_b

∆I_F

∆n_s

ad

m_1,ω

eAN_ad

_max- m₂

n_s

To derive the total electrochemical impedance at the interface,

the nonfaradaic currents ∆I_Cmust also be taken into account.

In general, the capacity of the double-layer capacitance on the

electrolyte side of the interface can be assumed to be much

larger than the space charge capacitance on the semiconductor

side. For this reason, the nonfaradaic current is only due to the

charging of the space charge capacitance, i.e., ∆I_C) iωC_sc∆U_e.

In an approximation, we assume that the faradaic current is

decoupled from the nonfaradaic current.³⁸The total current is

then the sum ∆I_C) ∆I_F+ ∆I_Cand the total impedance Z at

the semiconductor-electrolyte interface is

∆θ_ox) -∆θ_red

)

(19)

iω(1 + m_1,ω) + m₃

with

m

_1,ω) N^maxN_ω(k^-_seθ_red+ k_s⁺_eθ_ox)

(19a)

(19b)

(19c)

ad

m₂) k^-_cn_sθ_ox

m₃) k_s^-_ec_o^s_x+ k_s⁺_ec^s_red+ k_c⁺N_c+ k_c^-n_s

1

The time dependence of the faradaic current ∆I_F(eq 16) on

a small perturbation voltage ∆U_ethat is superimposed on the

electrode potential U_eis buried in the time-dependent electron

concentration ∆n_sat the semiconductor surface. If the electron

concentration n_sat the surface is in equilibrium with the electron

concentration n₀in the semiconductor bulk, then according to

the Boltzmann relation

Z )

(23)

1

Z_F

+ iωC_sc

The impedance of the equivalent circuit that shows the same

frequency behavior as the total impedance Z is shown in Figure

6 (solid and dashed lines).³⁹Accordingly, this model that

invokes a self-exchange reaction would be expected to exhibit

an impedance spectrum for a plot in the complex plane that

exhibits three features (2 semicircles and 1 asymmetric arc);

these are: (1) one semicircle for the charge-transfer reaction

between the semiconductor and the adsorbed redox molecules

with the time constant τ_ct) R_aC_sc) R_ctC_sc, (2) one semicircle

for the self-exchange reaction between the adsorbed molecules

and the redox molecules in solution with the time constant τ_se

e

kT

n_s) n₀exp - φ_sc

)

[

]

e

∆n_s) -n ∆U_e(20)

^skT

n₀exp (U_fb- U_e)

and

[

]

kT

where φ_scis the potential drop across the space charge layer

and U_fbis the flatband potential.

The combination of eqs 16, 19, and 20 leads to the faradaic

impedance Z_Fof the charge-transfer process as defined in eq

21.

) R_bC_b) (k_sec_ox+ k_sec_red)

^{s -1}, and (3) one asymmetric arc

(arc starting with a line of a 45° angle and turning into a

semicircle at lower frequencies (usually <1 Hz)) for the

diffusion or Warburg impedance Z_w.

As an instructive example, three impedance spectra calculated

with different values for the rate constant k_sefor the self-

exchange reaction are shown in Appendix D; a second semicircle

-

s

+

m₄m_1,ω

1 +

∆U_e

∆I_F

iω(1 + m_1,ω) + m₃

m₄m₂

kT

Z_F)

)

(21)

e²AN^max

+

-

ad

in the complex impedance plot develops when k_se) k_se

m -

[

]

2

iω(1 + m_1,ω) + m₃

becomes slow.

2. Charge Transfer Taking Thermionic Emission into Account.

It has to be considered that the overall charge-transfer event

with

Semiconductor-Molecule Interfaces

J. Phys. Chem. B, Vol. 103, No. 12, 1999 2129

The back reaction, that is the electron injection from the

adsorbed molecules into the conduction band of the n-type

semiconductor, is already defined in the kinetic expression 12

and has to be the same for the thermionic emission current.

Since both currents (I_kinand I_th) contain the same kinetic

expression for the back reaction, a clear separation of both

reaction paths will be difficult. For the sake of simplicity, we

assume a perfect rectifying behavior of the semiconductor

electrode in the following considerations and neglect the anodic

current. With this, the thermionic current can be written as

Figure 6. Equivalent circuit that shows the same frequency charac-

teristic for the impedance as the impedance calculated from the kinetic

model for charge transfer via a self-exchange reaction (Z, eq 23), solid

and dashed lines. The faradaic impedance (Z_F, eq 21 and eq 33) for

the charge transfer is presented by the circuit in solid lines.

n_s

m*

m_e

1

nkT

I_th) - AA*

T²

exp -

(E_F- E_c)

(27)

( ) ( )

[

]

n₀

can be limited by the transport of majority charge carriers

through the space-charge region if the heterogeneous charge-

transfer (eq 8) and the self-exchange reaction (eq 9) are very

fast. The overall measurable faradic current I_Fcannot be higher

than the thermionic emission current. The theory according to

Bethe assumes that every electron that reaches the semiconduc-

tor surface and has the appropriate energy to overcome the

potential barrier will cross the interface with a tunnel probability

of unity.⁴⁰However, if the kinetics for the heterogeneous charge-

transfer reaction is not infinite, some of the charge carriers will

be reflected at the interface. As shown in eq 24, the measurable

faradaic current I_Fwill be smaller than the thermionic emission

current I_thby the fraction r.⁴¹

and

∆n_s

∆I_th) I_th

(28)

n_s

(The introduction of the carrier concentrations n₀and n_sinto

the thermionic emission current assumes the applicability of eq

20. Restrictions are discussed later in this section.)

In eq 27, A* is the Richardson constant (120 A K^-1cm^-2),

m*/m_eis the relative effective electron mass in the conduction

band, T is the temperature, and n is the quality factor. For the

sake of simplicity, the quality factor is considered to be 1 in

the following theoretical considerations.

I_F) I_th(1 - r)

(24)

The fraction r can vary between 0 and 1. For r f 0 none of the

electrons reaching the interface is reflected; every electron at

the semiconductor surface is transferred to an acceptor molecule

adsorbed at the semiconductor surface according to the chemical

reaction given in eq 8. Since the charge-transfer step is not

limiting the current in this case, the kinetic current I_kin(defined

as the current determined by eq 12) has to be much larger than

the measured faradaic current I_F, i.e., I_kin. I_F. In contrast, for

r f 1 nearly every electron reaching the semiconductor surface

is reflected. According to the model the electrons are reflected

if the charge-transfer reaction (see eq 8) is very slow (e.g., low

acceptor concentration or low charge-transfer probability). In

this case the kinetic current I_kinwill become equal to the

measured faradaic current I_F, i.e., I_kinf I_Fas r f 1.

Accordingly, the simplest, reasonable expression for the fraction

r that describes the discussed boundary conditions the proper

way is r ≡ I_F/I_kin. Including the thermionic emission, the

measurable, stationary faradaic current is now

The time-dependent kinetic current is

∆I_kin) -eAN^maxk_c^-(θ_ox∆n_s+ n_s∆θ_ox) )

ad

∆n

∆θ_ox

I_kin^s+ I_kin

(29)

n_s

θ_ox

Inserting eqs 28 and 29 into eq 26 yields eq 30 for the time-

dependent measurable faradaic current

∆n_s

I²_F∆θ_ox

I_kinθ_ox

∆I_F) I_F

+

(30)

n_s

For the calculation of the time-dependent degree of coverage

from eq 13b, it has to be taken into account that reaction rate

-

R _cfor the heterogeneous electron transfer is now defined by

eq 31.

1

)

+

(25)

I_FI_thI_kin

I_F

I_thI_kin

1

R ^-_c) -

) -

(31)

(

)

eA

eA I_th+ I_kin

and the time-dependent measurable faradaic current is

The combination of eq 13b (with R _c≈ 0, R _ox^-) 0 and R _ox

) 0), eq 31, and eq C4 (Appendix C) yields for the time-

dependent degree of coverage eq 32.

+

∆I_th∆I_kin

∆I_F) I²_F

+

(26)

2

(

)

I_th

I_kin

It should be noted that for all expressions in this section (eqs

25-34), the various relationships between the dependent and

independent variables and parameters differ from those in the

previous section (eqs 12-23) because eqs 25-34 include the

effects of thermionic emission in the model. Equations 12-23

are only valid for the case where the rate of thermionic emission

is always faster than the rate of electron transfer.

∆I_F

∆n

m_1,ω

_max- m′

²n_s

eAN_ad

∆θ_ox) -∆θ_red

)

(32)

iω(1 + m_1,ω) + m′

3

with

2130 J. Phys. Chem. B, Vol. 103, No. 12, 1999

Meier et al.

I_F

m′ ) -

(32a)

(32b)

2

max

ad

eAN

2

I_F

m′ ) k^-_sec^s_ox+ k⁺_sec^s_red

+

k^-_cn_s

3

( )

I_kin

(m_1,ω: see eq 19a)

For the calculation of the faradaic impedance it has to be

assumed that the Fermi level is constant from the interior of

the semiconductor to the surface. This will be approximately

the case if the diffusion of the electrons is very fast.⁴⁰Then,

the electron concentration n_sat the surface can be calculated

from the Boltzmann relation (eq 20). As it will be shown later

in the discussion of the experimental results, a net current flow

across the interface does not significantly alter the concentration

profile that is found under equilibrium conditions. This conclu-

sion follows from the fact that the I-V (and R-V) character-

istics of the n-GaAs electrodes still show a linear behavior in a

semilogarithmic plot with a slope of approximately 60 mV per

decade. This ideal behavior is not expected if the diffusion

perturbs the Fermi level at the surface. For this reason, eq 20 is

also assumed to be valid even for currents in the order of the

thermionic emission current. The calculation of the faradaic

impedance from the combination of eqs 30, 32, and 20 yields

eq 33.

Figure 7. Stationary current-potential curve for the Co(Cp)₂⁺reduc-

tion at an n-GaAs RDE (200 rpm) in ACN-TBAPF₆(0.3 M). Other

0

parameters are [Co(Cp)₂⁺] ) 1.3 mM and [Co(Cp)₂] ) 0.9 mM, and

the electrode area is 0.152 cm². For the four potentials marked with

filled circles the EIS are shown in Figure 8.

The time constant τ_ct[τ_ct) R_aC_sc) (R_ct+ R_th)C_sc] is here

defined by the space charge capacitance and the sum of the

resistance R_ctfor the heterogeneous charge transfer and the

resistance R_thfor the thermionic emission. A single impedance

measurement does not allow one to distinguish whether the

charge transfer is limited by thermionic emission or the charge-

transfer step. But according to eq 34a, the resistance R_awill

only show a dependence on the degree of coverage θ_oxor θ_red

and subsequently a concentration dependence if the charge

transfer is limited by the heterogeneous charge-transfer reaction

and not by thermionic emission. The time constant τ_se[τ_se

)

R_bC_b) (k_sec_ox+ k_sec_red)

^{s -1}] for the self-exchange reaction

-

s

+

m′m_1,ω

4

is not affected by the current limitation through thermionic

emission, as expected. The expression 34d for the Warburg

impedance Z_Wis identical with eq 22d although the correct term

for the resistance R_bhas to be considered for the calculations.

1 +

∆U_e

∆I_F

iω(1 + m_1,ω) + m′

3

kT

Z_F)

)

(33)

e²AN^max

4

2

m′m′

ad

m′ -

[

]

2

iω(1 + m_1,ω) + m′

3

B. Results And Discussion of Impedance Data. Figure 7

shows an I-V curve for a stationary current-voltage measure-

with

0

ment for Co(Cp)₂⁺reduction ([Co(Cp)₂⁺] ) 1.3 mM, [Co(Cp)₂]

) 0.9 mM) at an n-GaAs RDE in ACN-TBAPF₆(0.3 M) at

200 rpm. The measurement covers the potential range of the

depletion region (+0.5 V to -0.1 V), the current onset region

(from -0.1 V to -0.2 V), the region of sharp current rise (from

-0.2 V to -0.4 V), and the diffusion-limited region (from -0.4

V to -0.7 V). The diffusion-limited current here amounts to

71 µA. For each I-V-data point, the frequency dependence of

the impedance was recorded between 1 Hz and 600 kHz. For

the four black marked I-V data points in Figure 7, the EIS are

shown as a plot in the complex impedance plane in Figure 8a-

d. In the current onset region at -0.1 V and -0.2 V (Figure 8a

and b), where the faradaic current is small compared to the

diffusion-limited current, the EIS show one single semicircle.

The EIS recorded at -0.3 V and -0.4 V (Figure 8c and d) in

the current rise region, where the faradaic currents exceed 10%

of the diffusion-limited current, show as an additional feature

in the spectra, the Warburg impedance (Z_W), that is due to the

diffusion of the redox species.

2

I_F

m′ ) -

)

k^-_cn_s

(33a)

4

( )

^maxI θ_ox

I_kin

eAN_adkin

(m_1,ω: see eq 19a. m′₂: see eq 32a. m′: see eq 32b.)

3

The faradaic impedance Z_Fof eq 33 shows the same

frequency dependence as the faradaic impedance Z_Fof eq 21

or the equivalent circuit displayed in Figure 6 (solid and dashed

lines). However, the comparison of the coefficients yields for

the elements R_a, R_b, C_b, and Z_Wa different set of kinetic

expressions, which are listed in eq 34a-d.

kT 1

e -I_F

kT

1

R_a)

)

+

) R_th+ R_ct(34a)

(

)

e -I_th-I_kin

2

I_F

k^-_cn_s

( )

I_kin

kT

e²AN^max

R_b)

(34b)

The dashed lines in Figure 8 are the best fits according to

the equivalent circuit in Figure 6 assuming that the self-exchange

rate is infinitely fast, i.e., the resistance R_bapproaches such small

values that the time constant τ_secannot be resolved. None of

the EIS that were recorded for the Co(Cp)₂⁺reduction at GaAs

revealed a third time constant that would indicate that the

adsorption of redox molecules is an important reaction step in

the charge-transfer mechanism. This lack of a third relaxation

time in the EIS led us to the previous assumption that no other

more complicated mechanism than a simple second-order

reaction was involved in the total charge-transfer reaction.²

-I_F

(k^-_sec^s_ox+ k⁺c^s)

ad

[

]

se red

max

ad

eAN

-I_F

e²AN^maxeAN^max

ad

C_b)

(34c)

(34d)

2

kT

I_F

-

[

]

k_cn_s

( )

I_kin

Z_W) N^maxN_ω(k^-_seθ_red+ k_s⁺_eθ_ox)R_b

ad

Semiconductor-Molecule Interfaces

J. Phys. Chem. B, Vol. 103, No. 12, 1999 2131

Figure 9. Mott-Schottky plot of the capacitance versus the electrode

potential (upper part, left scale) as evaluated from the semicircles of

the EIS. Comparison with the current-potential data (lower part, right

scale) reveals that the energy bands shift to negative potentials when

the faradaic current exceeds 10% of the diffusion-limited current.

k_c^-n_sthe term k_c⁺N_cbecomes negligibly small in the current

onset region, and the resistor R_bbecomes independent of the

quantities k_c^-, k_c⁺, and n_s. Using the experimental values for

the quantities of interest (A ) 0.152 cm², θ_ox) 0.59 (for θ )

1), and N_ad^max) 4 × 10¹⁴cm^-2) the resistor R_bis calculated to

be smaller than 0.05 Ω. A semicircle with a diameter of 0.05

Ω is far too small to be resolved in the EIS shown in Figure 8.

The tendency of the shape of the impedance spectra for

increasing values of the rate constant k_seis shown in Figure 15

(Appendix D). According to the simulations, k_semust be smaller

than 10^-15cm³s^-1to resolve the relaxation time for the self-

exchange reaction in the EIS. Therefore, the semicircle in the

EIS of Figures 8a to 8d has to be attributed to the relaxation

time τ_ct) R_ctC_scfor the heterogeneous charge transfer between

electrons in the GaAs conduction band and the adsorbed

+/0

Co(Cp)₂molecules.

The capacitance values as calculated from the semicircles of

the EIS are plotted in the Mott-Schottky manner (1/C²vs U_e)

in the upper part of Figure 9 (left scale). The data points fit

very well on a straight line in the depletion region. According

to the Mott-Schottky theory the linear part of the plot yields

information about the doping density N_dand the flatband

potential U_fb(see eq 35).

Figure 8. Four EIS recorded at (a) -0.1 V, (b) -0.2 V, (c) -0.3 V,

and (d) -0.4 V (see Figure 7 for corresponding I-V data). The EIS

are shown in the complex plane in the frequency range between 1 Hz

to 600 kHz. The dashed lines show the best fit according to the model

1

2

kT

e

)

φ_sc-

with

φ_sc) -(U_fb- U_e) (35)

(

)

eꢀꢀ₀N_dA²

C_sc

assuming a very fast self-exchange reaction with k_se> 10^-14cm³s^-1

.

For other parameters see caption of Figure 7.

The rate constant for the homogeneous self-exchange reaction

where ꢀ is the relative dielectric constant, ꢀ₀is the permittivity

of free space, and N_dis the doping density.

+

0

between Co(Cp)₂and Co(Cp)₂redox molecules in ACN

amounts to k_se) 7 × 10^-14cm³s^-1as reported in the

literature.^42-44Assuming that the adsorption of the redox

molecules does not have a drastic impact on the rate constant

for the homogeneous self-exchange reaction, the value of k_se)

7 × 10^-14cm³s^-1can be used to calculate the relaxation time

using eq 22b and c. This calculation yields τ_se≈ 10^-5s ([Co-

The value for the doping density calculated from the slope

of the linear region matches well with the value determined

from Hall measurements. This confirms that the measured

capacitance is the space charge capacitance of the n-GaAs

electrode.

A comparison of the Mott-Schottky data with the current-

voltage data, which are plotted in the lower part of Figure 9

(right scale), reveals that the Mott-Schottky data fit well on

the straight line only if the faradaic currents do not exceed 10%

of the diffusion-limited current. This indicates that no band

movement, and hence no surface state charging, occurs at the

GaAs-electrolyte interface in the current onset region. However,

for faradaic currents higher than 10% of the diffusion-limited

current (see vertical dashed line in Figure 9), the flatband

0

-

(Cp)₂⁺] ) 1.3 mM, [Co(Cp)₂] ) 0.9 mM; k_se) k_se⁺); thus,

the frequency marking the semicircle top is expected to be 16

kHz. As seen from the impedance spectra in the current step

region (Figure 8c and d), τ_seis very similar to the time constant

τ_ctwhich makes a resolution of both charge-transfer steps

impossible. The equation for the self-exchange resistor R_b(eq

22b) allows for an estimation of whether the relaxation time

τ

_seor τ_ctdominates in the region of current flow. Compared to

2132 J. Phys. Chem. B, Vol. 103, No. 12, 1999

Meier et al.

Figure 10. Comparison of the flatband potential (calculated from the

Mott-Schottky plot) with the surface redox potential (calculated from

the diffusion-limited current using eq 36) as a function of electrode

potential. The flatband potential and hence the energy bands at the

surface shift negative at the same potential where the surface redox

potential starts to move to negative potentials, i.e., when the faradaic

current exceeds 10% of the diffusion-limited current (see vertical dashed

line).

Figure 11. Faradaic current and resistance R_a, as obtained from the

semicircles of the EIS, plotted versus the electrode potential (cross

marked data points, upper x-scale) and the band bending in the GaAs

electrode (circles, lower x-scale). The solid lines are calculated

according to the thermionic emission theory.

potential (and hence the position of the energy bands) starts to

move toward negative potentials. The flatband potential shift

increases with negative bias into the diffusion-limited current

region. This “unpinning” of the energy bands in the region of

the current rise provides evidence that the charge-transfer

reaction in this region does not follow a simple model with

fixed band edges and a second-order rate law.²

A more detailed evaluation of the flatband potential shift in

the current step is presented in Figure 10. Here, the flatband

potential shift and the surface redox potential are plotted versus

the electrode potential. The surface redox potential was calcu-

lated using eq 36 which is derived from the Nernst relation³⁷

and eq 37 (considering that c_ox^sis zero in the diffusion-limited

region).

the surface redox potential and the flatband potential indicates

that a strong coupling exists between the GaAs electrode and

the redox couple molecules. Such an “unpinning” of the energy

bands occurs if the redox species interacts more strongly with

the GaAs surface than with the solvent molecules.⁴⁶Because

the concentration of the redox molecules ([Co(Cp)₂^+/0] ≈ 10^-3

M) is 4 orders of magnitude smaller than the concentration of

the solvent molecules ([ACN] ≈ 19 M), it is likely that the

redox molecules have to be adsorbed at the semiconductor

surface to cause an unpinning of the energy bands. The

unpinning of the energy bands (Fermi-level pinning) at the

n-GaAs-ACN interface through Co(Cp)₂^+/0has been reported

in the literature at various times,^47-50but to our knowledge it

has never been reported to be caused by adsorption of the

+/0

U^s_redox

)

Co(Cp)₂molecules at the GaAs surface.

In our model, the potential barrier at the semiconductor-

electrolyte interface is defined by the adsorbed layer of redox

molecules. Therefore, an equal shift of the flatband and the

surface redox potential is expected if the potential barrier does

not change in the considered potential range.

I_F

1 -

I_F,dl

kT

e

U⁰_redox

+

ln

(36)

I_F

e

kT

exp (U⁰_redox- U_r^b_edox) +

{

}

[

]

I_F,dl

Knowledge of the flatband potential U_fballows one to

calculate the bending of the energy bands φ_scat each electrode

potential U_e(according to eq 35). In Figure 11 the resistance

as obtained from the EIS (open circles, upper part, right scale)

and the faradaic current (full circles, lower part, left scale) are

plotted versus the band bending φ_sc(bottom x-axis scale). In

this figure, the R_aand I_Fdata points marked as crosses are also

plotted versus the electrode potential U_e(x-axis scale on top),

i.e., assuming that the energy bands do not move and that the

flatband potential remains constant at U_fb) -0.842 V as

determined from the Mott-Schottky plot in the depletion region.

Without considering the band movement, only the data points

at high band bendings and low faradaic currents show a slope

of approximately 60 mV per I_For R_adecade, as theoretically

expected. However, if the band movement is taken into account

(according to φ_sc) -(δU_fb+ U_fb- U_e)), then nearly all I_F

and R_adata points (filled and open circles) exhibit a potential

dependence of 60 mV per I_For R_adecade, as theoretically

expected. A potential dependence of 60 mV per decade is other

unequivocal evidence that the semicircle in the EIS of Figure 8

is due to the relaxation time τ_ct) R_ctC_scfor the heterogeneous

charge-transfer reaction.

where U_redox^sis the surface redox potential, U_redox⁰is the standard

redox potential, U_redoxis the redox potential in solution bulk,

and I_F,dlis the diffusion-limited faradaic current, and

b

D_ox

D_red

I_F) eA

(c^s_ox- c^b_ox) ) eA

(c^b_red- c_r^s_ed) (37)

(

)

(

)

δ_N,ox

δ_N,red

where A is electrode area, D_oxor D_redare the diffusion

coefficients of the redox species, c_oxor c_redare the concentrations

of the redox species, and δ_N,oxor δ_N,redare the thicknesses of

the Nernst diffusion layers.

In Figure 10 the flatband potential shift in the current rise

and the diffusion-limited current region was calculated for each

Mott-Schottky data point with the reasonable assumption that

the slope of the Mott-Schottky line does not change. The graph

shows that the energy bands at the GaAs-electrolyte interface

start to move when the redox potential at the electrode surface

shifts to negative potentials due to the influence of diffusion

limitation of the redox species at the electrode surface (see

vertical dashed line in Figure 10).⁴⁵This simultaneous shift of

Semiconductor-Molecule Interfaces

J. Phys. Chem. B, Vol. 103, No. 12, 1999 2133

δ_N,ox) 1.61 ν^1/6D_ox^1/3ω^-1/2

We further assume that the potential barrier for the majority

charge carriers at the n-GaAs-Co(Cp)₂^+/0(ad) interface does

not change within the current step. According to the model, a

constant potential barrier means that the shift of the flatband

potential and the potential of the adsorbed layer of redox

molecules is identical and equal to the shift of the surface redox

potential. The coincidental shift of the flatband potential and

the surface redox potential has been (qualitatively) shown in

Figure 10.

(38)

where I_F,dlis the diffusion-limited current, ν is the kinematic

viscosity, and ω is the angular velocity.

Figure 12 shows the dependence of the limited current for

the Co(Cp)₂⁺reduction in ACN on the rotation velocity of the

n-GaAs-RDE between 50 and 1200 rpm. The Levich plot in

Figure 12a fits very well on a straight line that crosses the origin.

With c_ox) 1.5 mM, A ) 0.152 cm², and ν ) 5.8 × 10^-3cm²

The calculation of the rate constant k_c^-from the I_Fand R_avs

s^{-1 51}the diffusion coefficient for the Co(Cp)₂⁺ion, as calculated

,

from the slope, amounts to D_ox) 1.3 × 10^-5cm²s^-1. This

value is in good agreement with the value for the diffusion

coefficient obtained from similar studies at metal-RDE (plati-

num, gold).

-

φ_scplot according to eqs 12 and 22a, respectively, yields k_c

)

1.2 × 10^-7cm³s^-1. This value can only be exactly assigned to

k_c^-if the charge transfer between the n-GaAs and the adsorbed

+/0

Co(Cp)₂

molecules is the rate-limiting step in the overall

Figure 12b shows a plot of the reciprocal of the limited current

versus the reciprocal of the square-root of the angular frequency.

For this type of plot a finite intercept is expected if any processes

other than the diffusion of the active redox species becomes

the limiting mechanism for the overall current (e.g., diffusion

at partially blocked electrodes,^52-54or preceding heterogeneous

(adsorption) or homogeneous chemical (dissociation) reac-

tions³⁷). However, the linear fit in Figure 12b does not show

an intercept within the range of error, and therefore the limited

charge-transfer reaction; otherwise, it is a lower limit. However,

it will be shown in the following paragraph that the thermionic

emission of majority charge carriers is limiting the current rather

than the electrochemical charge-transfer reaction. Our previous

published results²on the concentration dependence of the

current-voltage characteristics of the n-GaAs/Co(Cp)₂^+/0system

also indicates that the rate of thermionic emission limits the

observed faradaic current when the concentration of the Co-

+

(Cp)₂acceptor is sufficiently high. In that study, the current

+

current for the Co(Cp)₂reduction at n-GaAs is considered to

be the diffusion-limited current.

was found to depend approximately linearly on the Co(Cp)₂

concentration only when it was between 0.27 and 1.9 mM; at

higher concentrations the current started to become independent

of Co(Cp)₂⁺concentration and saturated at 17 mM. Thus, while

in the previous study thermionic emission began to limit the

current above 1.9 mM, we show below that in the present study

thermionic limitation of the current is present at 1.3 mM Co-

The observation of the theoretically expected rotation de-

pendence of the diffusion-limited current allows an estimation

of a lower value for the rate constant k_se, i.e., a lowest value,

which provides a minimum current flow, such that the self-

exchange reaction does not show a significant limiting influence

on the overall charge transfer. The rotation-speed-dependent

quantities are the gradients of the redox concentrations at the

electrode according to eq 37 and subsequently the degrees of

coverage θ_oxand θ_red. For this estimation it is sufficient just to

consider θ_ox. θ_oxis rewritten in the form of eq 39 that follows

+

(Cp)₂⁺. We attribute the somewhat lower value of Co(Cp)₂

concentration at which thermionic emission is rate limiting to

0

the higher [Co(Cp)₂/Co(Cp)₂⁺] ratio in the present experiments;

because of the Nernst relationship, the higher ratio produces a

more negative redox potential for the solution, a smaller band

bending, a greater surface electron concentration, and hence

faster ET rates for given electrode potentials.

+

-

from eq 13b with I_F) eA(×c2_c+ - ×c2_c^-), ×c2_ox) ×c2_ox

-

+

) 0 and k_se) k_se) k_se

.

The solid lines in Figure 11 have been calculated according

to the thermionic emission theory (using an ideality factor of n

) 1.04); the justification of n is discussed in ref 2, assuming

that the transport of majority charge carriers in the space charge

layer of the semiconductor is limiting the overall charge-transfer

process. Both the I_Fand R_adata points fit very well on the solid

line, which supports the assumption that thermionic emission

is the current-limiting transport mechanism. As explained in

more detail in the theory part of this section (IV.A.2.), the overall

current cannot be higher than the thermionic current. But since

the thermionic emission current is limiting the charge transfer,

the rate constant k_c^-for the kinetic current has to be larger than

1.2 × 10^-7cm³s^-1according to eqs 25 and 34a for the faradaic

current I_Fand the resistance R_a, respectively.

I_F

k_sec^s_oxθ +

max

ad

eAN

θ_ox

)

(39)

k_se(c^s_ox+ c^s_red)

The surface concentrations (c_ox^s, c_red^s) in eq 39 are replaced

by the appropriate term of eq 37. With the assumption that D

) D_ox) D_redand δ_N) δ_N,ox) δ_N,red, the rearrangement of eq

39 leads to

I_Fδ_N

I_F

max

k_se

θ +

)

(

eA D

eAN

k_se(c^b_ox+ c^b_red)

ad

θ_ox

)

(40)

(41)

To confirm that the faradaic current at negative potentials

(from -0.4 V to -0.7 V in Figure 7) is indeed limited by the

diffusion of the redox active species, the limited current was

recorded at different rotation velocities. According to the Levich

theory for rotating disk electrodes, the plot of the diffusion-

limited current versus the square root of the angular velocity

yields a straight line that crosses the origin (eq 38).³⁷

and

δ

^maxk θ ^N+ 1

c^b_oxθ

I_F

N

(

)

ad

se

D

θ_ox

)

+

k_se(c^b_ox+ c^b_red) eAN^max

b

(

)

c_ox+ c_red

ad

The rotation speed dependence of θ_oxis buried in δ_N(see eq

38) and according to eq 41 is only expected if

D_ox

I_F,dl) eA

c_ox

(

)

δ_N,ox

δ

N^m_ad^axk θ ^N> 1

(42)

se

with

D

2134 J. Phys. Chem. B, Vol. 103, No. 12, 1999

Meier et al.

calculations were applied to the GaAs/Co(Cp)₂^+/0/ACN system

to determine whether the model proposed here, which involves

+/0

adsorption of the Co(Cp)₂

redox system onto the GaAs

surface, and the corresponding experimental results are consis-

tent with theoretical expectations. This is necessary because

metallocenes, such as Co(Cp)₂^+/0, have previously been gener-

ally presumed to be nonadsorbing, outersphere types of redox

molecules at SMIs.^6,8,10,27,28

A. Theoretical Methods. Our goal is to calculate the

+

adsorption energetics and molecular configuration of Co(Cp)₂

0

and Co(Cp)₂on GaAs surfaces with the redox molecules

dissolved in ACN. Calculations of adsorption energies and

molecular configurations for semiconductor-vacuum interfaces

are very challenging primarily because electron correlation and

exchange must be accurately addressed; the SMI with liquid

present is even more challenging primarily because near-surface

solvent molecule orientation, adsorbed solvent, and adsorbed

oxygen affect the adsorption of the redox species. Notwithstand-

ing, we show below that adsorption energies and geometries

can be calculated for the SMI system of interest in this paper.

For adsorption at vacuum interfaces, density functional theory

(DFT) methods have been used predominantly. The latest DFT

methods use nonlocal pseudopotentials^57-61(including general-

ized gradient (GGA) functionals) and some methods also include

HF exchange. The error in the calculated adsorption energies,

relative to experiment for systems explored thus far, is typically

in the range of 0.1-0.5 eV. We have adopted a cluster

calculation approach to the adsorption problem at liquid

+/0

interfaces, which is described below. We found the Co(Cp)₂

/

GaAs system extremely computationally demanding at the DFT

BLYP (Becke-Lee-Yang-Par) and B3LYP level (B3LYP is

one of the most accurate DFT methods) when used with the

high-level quantum-chemistry basis sets implemented in the PS-

GVB program.⁶²Thus, our combined requirements of accuracy

Figure 12. (a) Levich plot of the diffusion-limited currents (I_dl)

recorded for the Co(Cp)₂reduction at n-GaAs at different rotation

+

speeds between 50 and 1200 rpm (ω ) 2πf). The electrode area was

+

0.152 cm², and the Co(Cp)₂concentration was 1.5 mM. (b) Plot of

the reciprocal of the diffusion-limited current (1/I_dl) versus the reciprocal

and computational tractability necessitated that we employ an

x

of the square root of the angular velocity (1/ ω).

+/0

assortment of methods to address adsorption for the Co(Cp)₂

These methods are

.

regardless of the current-limiting transport mechanism. That is,

for N_ad^max) 4 × 10¹⁴cm^-2, θ ) 1, δ_N) 1.43 × 10^-3cm (for

1200 rpm), and D ) 1.3 × 10^-5cm²s^-1, the rate constant k_se

for the self-exchange reaction has to be at least larger than 3 ×

10^-17cm³s^-1. This result is qualitatively in agreement with

the results from the EIS measurements where we concluded

that k_seis larger than 10^-14cm³s^-1. However, it has to be noted

that the EIS analysis is in general more sensitive than the

electrochemical analysis of the diffusion-limited current. There-

fore it is reasonable to assume that the rate constant k_sefor the

Full B3LYP. Full B3LYP calculations were performed on

small clusters.

B3LYP-PSCF. In this method we first performed with PS-

GVB, full ab initio Hartree-Fock (with the high-level basis

set denoted as LACVP**) calculations on larger clusters.⁶²We

then evaluated the LACVP** HF post SCF wave functions via

the high-level DFT method (B3LYP) provided in the PS-GVB

program suite. This post self-consistent field (PSCF) method

represents an approximation to the full B3LYP method; other

PSCF methods are possible within PS-GVB.

self-exchange reaction is in the order of 10^-14cm³s^-1

.

In conclusion, the electrochemical studies of the reduction

of Co(Cp)₂at n-GaAs electrodes in ACN show that both

CI. CI calculations were also performed on small systems.

R-PM3. Small cluster B3LYP and larger scale B3LYP-

PSCF results were fit to the PM3 Hamiltonian of MOPAC by

a method described below. The reparametrized PM3 (R-PM3)

method was subsequently used on very large systems.

First Principles Molecular Dynamics. We also performed

preliminary first principles MD simulations, of the same kind

we have done for another system,⁵⁶using the R-PM3 electronic

structure method to calculate the interatomic forces for the fully

solvated Co(Cp)₂^+/0/GaAs/ACN system. We also addressed

solvation with higher level methods as discussed below.

Regarding the R-PM3 method, we chose standard PM3

parameters for all elements in the GaAs/CO(Cp)₂^+/0/ACN

system except Co. Since we have not yet completed our own

s-, p-, and d-orbital-based PM3 program, we chose a PM3 s-

and p-orbital model for cobalt. We denote our model R-PM3

metallocene as M(Cp)₂^+/0. We then varied PM3 parameters for

+

charge-transfer steps, namely, the heterogeneous charge transfer

from the n-GaAs electrode to the adsorbed Co(Cp)₂^+/0molecules

and the self-exchange charge transfer from the adsorbed

+/0

Co(Cp)₂molecules to the Co(Cp)₂molecules in solution

are considered to be fast. The rate constant k_sefor the self-

exchange reaction has to be larger than 10^-14cm³s^-1(concluded

from the fact that the EIS only exhibit two relaxation times)

and since the overall current is limited by thermionic emission,

the rate constant k_c- for the heterogeneous charge transfer must

be larger than 1.2 × 10^-7cm³s^-1. Consequences of these results

for the dynamics of ET will be discussed below in section VI.

+/0

V. Theory of the GaAs/Co(Cp)₂Interface

Since new rigorous theoretical approaches to the description

of the SMI have been recently developed,^55,56these types of

Semiconductor-Molecule Interfaces

J. Phys. Chem. B, Vol. 103, No. 12, 1999 2135

M to fit Co(Cp)₂^+/0/GaAs results obtained from full B3LYP,

LACVP**, and B3LYP-PSCF calculations for adsorbate

geometry, binding energy, and other energetics. The orbital

structure of the cobalt complex is obviously only roughly

characterized without true d-orbitals. But for present purposes

we are most interested in fitting B3LYP energetics and

geometry, and our results indicate the R-PM3 approach is able

to reproduce these features qualitatively. The PM3 parameters

ultimately chosen for M, are closest to PM3 Ga. The optimized

geometry of Ga(Cp)₂^+/0, using even essentially standard PM3

Ga, is remarkably similar to that for Co(Cp)₂^+/0, and the match

to B3LYP adsorption energy profiles as a function of distance

is similar. This R-PM3 electronic structure method was useful

for qualitative adsorption geometry and energetics and also to

explore the effects of very large semiconductor and ACN

clusters which would be computationally prohibitive with the

other methods. Further progress along these lines must await

completion of our d-orbital PM3 code.

The GaAs (100) surface in UHV conditions undergoes a

number of reconstructions, as has been analyzed in detail both

experimentally and theoretically.^61,63-65It is computationally

prohibitive for us to let the surface reconstruct self-consistently

via BLYP methods to examine the effect of the solvent on the

reconstruction. Thus, in our calculations we simply chose two

representative surface structures. One of these is the (2 × 1)

dimer UHV reconstruction, while the other is cleanly terminated

(100) GaAs. We examine both As and Ga-terminated surfaces.

Our cleanly terminated surface has the main characteristics of

the UHV (100) GaAs reconstructions (e.g., all Ga or As on the

surface), but the arrangement of the atoms on the surface is not

of the proper reconstructed geometry, and this can have profound

effects.

Other complications of our system include oxidation of the

GaAs surface and solvent adsorption. We do not include oxide

layers in the calculations, but we do address solvent (ACN)

adsorption (it is found to be weaker than Co(Cp)₂^+/0adsorption).

Figure 13. Orientation and geometry of Co(Cp)₂⁺adsorbed on (100)

GaAs: (a) side view looking along the cleanly terminated (100) GaAs

surface with Ga termination and full hydrogen passivation (H atoms

not shown) and (b) top view looking down on reconstructed (2 × 1)

dimer (100) GaAs surface. Cyclopentadienyl rings are parallel to the

GaAs surface.

B. Results. We find below that the adsorption energy for all

+

methods employed is less than 1.0 eV both for Co(Cp)₂and

0

Co(Cp)₂, and for both the reconstructed (2 × 1) dimer⁶⁵((2 ×

to the surface (Figure 13). For the (2 × 1) D surface two of the

carbons of the cyclopentadienyl (Cp) ring prefer to be close to

a Ga dimer of the Ga-terminated (2 × 1) D surface, as shown

in Figure 13b. The As-terminated (2 × 1) D surface undergoes

less electron density exchange with the Co(Cp)₂^+/0than the Ga

termination, and the adsorption is weak enough that the

minimum energy surface site is not clear, but the preferred

geometry is still parallel. In the parallel geometry the adsorption

primarily involves π-system interaction of the Cp groups with

the GaAs. For the CT surface the adsorption energy is roughly

the same as for the 2 × 1 D and with the parallel geometry

preferred. There is some artifactual binding of the complex with

the passivating hydrogen atoms in the CT case. Ideally the H

atoms should be indifferent spectator atoms only serving to

passivate dangling bonds. The dipole resulting from H-Ga or

H-As bonds is different than the Ga-As dipole and can

significantly affect the adsorption energy of the charged

cobaltocenium. While these geometries seem to minimize the

energy, a thorough test was simply computationally prohibitive

with the higher level methods. Thus our adsorption energy

estimates reported below with the high level methods are, again,

lower bounds. The adsorption could be stronger for different

redox species/GaAs substrate geometries.

1) D) and cleanly terminated (CT), hydrogen-passivated (100)

GaAs surfaces. In these cases there is so little charge exchange

with the surface, and small enough polarization, that the cobalt

complexes essentially maintain their gas-phase geometry (the

rings themselves can freely rotate). Only in the case where one

of the CT surface Ga atoms was left unpassivated by hydrogen,

+/0

did the geometry of adsorbed Co(Cp)₂significantly distort.

In light of these results, we take a conservative viewpoint and

use only the (2 × 1) D or fully hydrogen-passivated CT (100)

GaAs surface in the discussions below.

With the two types of surfaces described above, we performed

adsorption calculations both for a UHV system (no solvent, only

GaAs and Co(Cp)₂^+/0) and the solvated system (GaAs/Co-

(Cp)₂^+/0/ACN).

We calculated the minimum energy adsorption configuration

at the B3LYP-PSCF level for the UHV system using an 18-

atom GaAs(100) CT cluster. For this case, upright (end on) and

parallel face configurations were examined. A much larger UHV

geometry optimization search was performed with MOPAC’s

BFGS method using our R-PM3 with a large 50-atom CT

cluster. The lowest energy configuration for the solvated system

was determined with our R-PM3 first-principles MD simula-

tion.

The adsorption energy for Co(Cp)₂^+/0on (2 × 1) D and CT

100 GaAs is of order 1.0 eV or less for all theoretical methods

applied to both the oxidized and reduced cobalt species, and in

The preferred adsorption geometry, in both the UHV and

solvated case, appears to be with the rings of Co(Cp)₂^+/0parallel

2136 J. Phys. Chem. B, Vol. 103, No. 12, 1999

Meier et al.

solvent molecules were used. Near-surface solvent was treated

quantum mechanically (via R- PM3), while the remaining

solvent was addressed with conventional classical MD methods.

This hybrid approach was implemented in exactly the same

fashion as in ref 56. All semiconductor atoms were treated

quantum mechanically. The polarization of the surface by the

ACN and the polarization of and by the Co(Cp)₂^0/+both affect

the adsorption energy. The aVerage adsorption energy in the

solvated R-PM3 system relative to the R-PM3 UHV adsorp-

tion energy is, remarkably, not dramatically affected by the

presence of the solvent. This kind of phenomenon has been seen

in other solid-liquid-interface adsorption studies.⁶⁶In our case

this appears to be primarily a result of the absence of solvent

molecules between the semiconductor and the redox species and

the fact that dipolar and charge exchange interactions mutually

+/0

perpendicular to the surface and the Co(Cp)₂rings (rather

than lateral interactions) are most important in the adsorption.

This is also consistent with our high level UHV calculations.

In summary, the maximum adsorption energy occurs with

both Co(Cp)₂⁺and Co(Cp)₂⁰having their cyclopentadienyl rings

parallel to the GaAs surface, rather than edge on, and in all

cases the calculated adsorption energy is of order 1.0 eV or

+/0

less. The possibility that Co(Cp)₂dissociates on the (100)

GaAs surface is remote based on our calculations. The rings

are rather strongly bound, and our calculated electron density

exchange is rather small.

C. Possible Adsorption-Mediated ET Scenarios. On the

basis of our experimental and calculated adsorption energies,

we assume both cobalt complexes are adsorbed, and a steady-

state coverage is formed on the surface; thus, the primary role

played by the adsorbed species is mediating the ET to redox

species in solution. The latter species may also adsorb onto the

+/0

previously established Co(Cp)₂

layer, and with the face

+

Figure 14. Energy vs Ga-Co distance for Co(Cp)₂at cleanly

terminated (100) GaAs with Ga termination: (a) BLYP calculation and

(b) R-PM2 simulation (the energies are averages over time).

parallel geometry it is possible that stacking of Co(Cp)₂^+/0could

occur. However, our EQCM results indicate that stacking (i.e.,

multilayer formation) does not occur If an adsorbed monolayer

does not act as a favorable substrate for further cobalt species

adsorption, then we effectively have a modified electrode surface

that can mediate ET to cobalt species near the surface.

both the UHV and solvated cases. Our calculations in Figures

14a and 14b show that the adsorption energy seems somewhat

greater for the cobaltocene species than cobaltocenium both in

the UHV and solvated cases. But with such a complicated

interface, the error in the calculations could easily be judged to

be a few tenths of an eV, which is about equal to the maximum

estimated difference in the adsorption energy. The BLYP

calculations are, of course, the most accurate. These adsorption

energy values (about 0.55 and 0.35 eV, for cobaltocene and

cobaltocenium respectively, see Figure 14a), are in reasonable

agreement with the experimental results (solvation does not

significantly affect the average adsorption energy, as discussed

further below). However, all DFT methods such as B3LYP are

expected to underestimate the actual adsorption energy since

the calculated correlation energy is usually too small. The actual

surface may also be significantly different than the theoretical

ones. The most important point is that high level theory (BLYP)

does predict that metallocenes can, indeed, adsorb on the (100)

GaAs surface. Note that the adsorption energy of the lowest

level method employed (R-PM3) is about twice that of the

B3LYP calculations (Figure 13a), but the qualitative trends are

the same, and the quantitative trends are similar. While the

R-PM3 method was fit to the BLYP methods by varying the

model Co parameters, it is difficult to offset our R-PM3’s

apparent tendency to overestimate the adsorption energy.

Figure 14b shows the results of a R-PM3 first-principles

MD simulation for the average adsorption energy in the solvated

system as a function of distance from the surface. About 200

Another possibility is that an irreversibly adsorbed layer could

form a cobalt metal-semiconductor interface (Schottky junc-

tion). Our experiments do not support this possibility, and our

calculations also indicate that it is unlikely either cobalt redox

species dissociates.

When both redox species adsorb, there are a number of

possible ET scenarios. Since the Co(Cp)₂⁺is positively charged

and its charge exchange on adsorption to the semiconductor is

calculated to be relatively small, it could form a long-range

Coulombic trap for electrons. If a solvated Co(Cp)₂⁺in solution

approaches the adsorbed and reduced Co(Cp)₂⁺(i.e., Co(Cp)₂),

0

then the latter species could transfer the transiently trapped

electron to the former species. In other words, the adsorbed

cobalt species could form a spatially local density of states, i.e.,

a surface-localized density of states (surface state) that could

transfer charge to acceptors in solution. This is similar to

+/0

homogeneous ET between the Co(Cp)₂species in solution,

but in this case coupling to semiconductor eigenstates can have

profound effects on the ET, and reorganizational modes can

also be dramatically affected. The trapped electron would have

a lifetime determined primarily by the strength of the electronic

interaction of the orbital (in which it is trapped) to the

semiconductor (roughly, lifetime ) p/|V|, where |V| is the

electronic coupling matrix element).

Semiconductor-Molecule Interfaces

J. Phys. Chem. B, Vol. 103, No. 12, 1999 2137

0

Adsorbed (neutral) Co(Cp)₂could also trap electrons for a

of about 3 orders of magnitude can be attributed both to the

degree to which the GaAs/Co(Cp)₂system falls outside the

+

sufficient period of time to pass them to a neighboring Co-

(Cp)₂in solution. As is well-known from UHV studies,

+

limits of conventional outersphere ET and deficiencies of the

model of refs 9 and 10.

quasistationary states of this kind can have lifetimes over the

femtosecond-to-nanosecond range. In UHV studies on metal

electrodes it is thought that the exit of the electron from the

neutral back into the metal can sometimes induce desorption.

Here the trapped state would dissipate some of its probability

density into the redox species, and some into the semiconductor.

+

The second-order rate constants for the GaAs/Co(Cp)₂

system can be converted into an electron-transfer velocity (S_ET

,

cm/s) or into a cross section (σ_ET, cm²) using the following

simple relationships:

S_ET) k_ETc_ox) k_c^-N_ad≈ 5 × 10⁶cm/s

ox

VI. Concluding Discussion

and

Traditional models for ET from semiconductors to redox

molecules in solution^1,3,10,67,68assume nonadsorbing outersphere

redox species;^3,10the metallocenes and their cations were

generally believed to fall into this class of redox mol-

ecules.^6,8,10,27,28

σ_ET) S_ET/V_thN_ad^ox) k_c^-/V_th) 1.2 × 10^-14cm²

where V_this the thermal velocity of electrons (1 × 10⁷cm/s)

ox

However, the results reported here show that adsorption of

the Co(Cp)₂metallocene redox system occurs on GaAs and

and N_adis the areal density of acceptors.

+/0

Both of these numbers indicate ultrafast ET dynamics. The

electron-transfer velocity is comparable to the highest values

of the recombination velocity reported for electrons for nonra-

diative recombination dynamics at surface states and/or adsorbed

species on semiconductor surfaces.^71,72Within experimental

error, the cross section is in the range of the physical area of

the Co(Cp)₂⁺acceptor molecule. A simple-minded conversion

of S_ETinto a photoinduced electron-transfer time (obtained by

dividing S_ETinto the absorption depth) yields a few ps. However,

the actual physical significance of the electron-transfer velocity,

the cross section, and the electron-transfer time is more

complicated than that implied by the simple interpretation

implied above, and a deeper theoretical analysis is required for

the interpretation of these numbers.⁶⁹

those adsorptions plays a critical role in the dynamics of ET.

The adsorption energies are found to be relatively small

+

(experimentally, about 0.2 and 0.4 eV for Co(Cp)₂and Co-

0

(Cp)₂, respectively), and therefore the adsorption type is closer

to physisorption than chemisorption. The electronic coupling

+

matrix element (|V|) between Co(Cp)₂and GaAs has been

estimated to be about 0.01 eV.⁶⁹This magnitude of coupling is

much too high to safely classify the ET as nonadiabatic, and it

is too low to be categorically described as adiabatic. It is best

categorized as an intermediate regime. It should be realized that

the electronic coupling can change significantly with configu-

ration of the redox species relative to the surface (especially

with respect to surface site location), and with surface recon-

struction and contamination. Thus, the actual value for the

experimental system could deviate from the calculated value

by a significant factor. Simple models indicate full electronic

adiabaticity might be achieved with a |V| as low as 0.03 eV.

The ultralarge second-order heterogeneous rate constants (k_ET

1 × 10^-10cm⁴/s) reported previously for ET across the GaAs/

It is of interest to compare the values of the ET rate constants

+

that we report here for GaAs/Co(Cp)₂to related III-V

semiconductor-metallocene-metallocenium systems in the

literature.^1,3,13,4-10In all these previous experiments the met-

allocenium electron acceptors or metallocene hole acceptors

were assumed to be nonadsorbed to the semiconductor surface.

In ref 8, experiments based on dark I-V and Mott-Schottky

plots reported an upper limit for k_ETfor the n-GaAs/ACN/Fe-

(Cp)₂⁺system to be <1 × 10^-14cm⁴/s; for p-InP/CH₃OH/Co-

Co(Cp)₂interface²were calculated assuming bulk solution

+/0

concentrations of Co(Cp)₂⁺, i.e., adsorption was not considered.

From the present analysis of our impedance and current-

potential data based on the model in Figure 5, we estimate the

minimum heterogeneous rate constant for ET from GaAs to

adsorbed Co(Cp)₂⁺to be k_c^-) 1.2 × 10^-7cm³/s. The units of

+

(Cp)₂the upper limit for the rate constant for hole transfer

was reported to be <1 × 10^-13cm⁴/s. For n-InP in methanol

with Fe(Cp)₂and several substituted ferrocenium acceptors,⁶

+

-

k_care cm³/s because the concentration of adsorbed acceptors

k_ETwas reported to be 1 × 10^-16to 10^-17cm⁴/s.

is expressed as an areal density. If one takes the thickness of

the layer containing the adsorbed Co(Cp)₂⁺to be 8 Å (the sum

of the molecule thickness and the distance from the surface at

which the adsorption energy was calculated to be a maximum

(see Figure 14), then the effective value of the second-order

Rate constants for photoinduced ET from p-type GaAs and

InP electrodes with metallocenium acceptors in ACN have also

been reported based on PL quenching experiments.^1,13In these

experiments it was again assumed that the metallocenium

acceptors are not adsorbed. For p-GaAs sulfided with Na₂S to

reduce intrinsic surface recombination, the reported k_ETvalue

rate constant using volumetric units for the acceptor is k_ET

)

(1.2 × 10^-7)(8 × 10^-8) = 1 × 10^-14cm⁴/s.

was 4 × 10^-13cm⁴/s for Co(Cp)₂and 2 × 10^-12cm⁴/s for

+

Since the electron coupling between Co(Cp)₂and GaAs is

believed to be relatively small (estimated to be 0.01 eV) and

the acceptor is weakly adsorbed, the k_ETvalue (1 × 10^-14cm⁴

/s) can be roughly compared to the value of 1 × 10^-17cm⁴/s,

which has been proposed as the upper limit for ET to

nonadsorbed, ideal outersphere acceptors.^4-10Our k_ETvalue

estimated here is about 3 orders of magnitude larger than the

supposed upper limit and is more consistent with upper-limit

values reported by Smith et al.^26,56Thus, the discrepancy

between our reported k_ETvalues and the upper limit values

predicted in refs 9 and 10 is reduced from about 6 orders of

magnitude (reported in ref 2) to about 3 orders of magnitude as

adsorption is taken into account.⁷⁰The remaining discrepancy

Fe(Cp)₂⁺; S_ETwas 4 × 10⁴and 2 × 10⁵cm/s for 0.5 mM

solutions, respectively.¹³A recent reexamination of these data

(Poles, E.; Smith, B.; Nozik, A. J. Unpublished results.) using

a Monte-Carlo model yields S_ETand k_ETvalues of about 7 ×

10³cm/s and 1 × 10^-14cm⁴/s, respectively, for Co(Cp)₂⁺(1.0

+

mM) and 10⁴cm/s and ∼4 × 10^-14cm⁴/s for Fe(Cp)₂(0.5

mM), respectively. In a very recent study¹using pristine GaAs

surfaces with Fe(Cp)₂⁺, S_ETwas reported to be 7 × 10³cm/s

for 10 mM Fe(Cp)₂⁺, and the cross section was 2 × 10^-15cm²

(compared to our estimate of 1.2 × 10^-14cm²); these values

lead to a k_ETof 2 × 10^-15cm⁴/s. Again, no enhancement of

acceptor concentration by adsorption was believed to occur, and

it was estimated that the S_ETvalue would increase to 1 × 10⁷

2138 J. Phys. Chem. B, Vol. 103, No. 12, 1999

Meier et al.

+

cm/s if a full monolayer of adsorbed Fe(Cp)₂was formed on

dt ) 0), eqs 13b, 13c, and 14 yield

θ_oxâ_ox(1 - θ)c^s_ox

the GaAs surface.¹

A major outstanding issue for the comparison of the various

reported values of S_ET, k_ET, and cross section is whether the

acceptor is adsorbed or not. Our QCM experiments show that

in the case of our etched epilayer GaAs electrodes, physisorption

of both Co(Cp)₂⁺and Co(Cp)₂occurs; Co(Cp)₂⁺was also found

to adsorb to quartz and gold surfaces, but did not absorb onto

nylon surfaces. Our preliminary QCM experiments with Fe-

(A1)

(A2)

θ_red) â_red(1 - θ)c^s_red

θ

) â_oxc^s_ox) â_redc_r^s_ed

(A3)

(1 - θ)

+

(Cp)₂indicates that this molecule is also adsorbed to GaAs.

with â_ox) k^-_ox/k⁺_oxand â_red) k⁺_red/k^-_red. The â-term can be

identified with the definition given in eq 5. The adsorption

strength of each species in this kinetic evaluation is expressed

through the ratio of rate constants for the adsorption and

desorption reactions.

The combination of the balance eqs 13b, 13c, and 14 gives

the relations A4 and A5 for the dependence of the degrees of

coverage θ_oxand θ_redon the rate constants, the electron density

at the surface, and the concentration of the redox species.

Thus, if adsorption is present in the experiments of refs 1 and

13, then the calculated k_ETcould be reduced by several orders

of magnitude depending upon the acceptor concentration and

adsorption isotherm. We note that since in ref 1 the projected

value of S_ETfor a full adsorbed monolayer of Co(Cp)₂⁺is 1 ×

10⁷cm/s, which is consistent with what we find here with the

known adsorption of Co(Cp)₂⁺, this suggests that the Co(Cp)₂

+

is not adsorbed in that specific experimental system. The factors

controlling whether acceptors are adsorbed or not on III-V

surfaces are not understood, but this factor will make very large

differences in the rates of ET and the calculated rate constants.

Determination of adsorption in any given system will require

an independent measurement using, for example, QCM, IR

reflection spectroscopy, or SHG.

(k_s^-_ec_o^s_x+ k⁺_cN_c)θ + k^-_oxc³_ox(1 - θ)

k_s^-_ec^s_ox+ k_s⁺_ec_r^s_ed+ k_c⁺N_c+ k_c^-n_s+ k_o⁺_x

θ_ox

)

(A4)

(A5)

(k_s⁺_ec_r^s_ed+ k^-_cn_s)θ + k_r⁺_edc_r^s_ed(1 - θ)

k_s^-_ec^s_ox+ k_s⁺_ec_r^s_ed+ k_c⁺N_c+ k_c^-n_s+ k_r^-_ed

Concerning the issue of hot electron transfer across semi-

conductor surfaces, what matters here is the net rate of ET, not

the rate constant; that is, the rate of ET has to be faster than

the rate of hot electron cooling (reported hot electron cooling

times range from several hundred fs to several hundred ps). In

refs 4 and 10, the analysis of ET kinetics led to the reported

upper limit for k_ETof about 1 × 10¹⁷cm⁴/s (and also assuming

nonadsorption of acceptors); this then led to the conclusion that

hot electron transfer is not possible with nonadsorbed accep-

tors.¹⁰This is because with a maximum k_ETof 1 × 10^-17cm⁴/s

and an acceptor concentration of say 100 mM, S_ETwould only

θ_red

Relations A4 and A5 allow a discussion about the state of

occupation as defined by the ratio of the degrees of coverage

θ_ox/θ_red. The ratio θ_ox/θ_redat a given ratio of the redox

concentrations depends on the chemical reactions as defined in

eqs 8 to 11. Assuming first that no charge transfer occurs via

the adsorbed layer (k_c+ ) k_c- ) 0) and that the self-exchange

+

-

reaction is negligibly slow (k_sec_red^s, k_sec_ox^s, k_red^-, k_ox⁺), i.e.,

only adsorption and desorption reactions take place, then the

ratio θ_ox/θ_redis given by

be (1 × 10^-17cm⁴/s) × (6 × 10¹⁹cm^-3) ) 600 cm/s, and the

-1

ET time would be roughly (RS_ET

)

∼ 50 ns, not fast enough

to compete with hot electron cooling. However, as we show

here, adsorption of so-called outersphere acceptors, such as

metallocenes and their cations does occur, and this can increase

the ET rate or S_ETby 3-5 orders of magnitude compared to

the case where there is no adsorption; thus, in the case of

adsorbed acceptors, the ET rate can be 500 fs to 50 ps, which

is competitive with hot electron cooling. Furthermore, as seen

in the present experiment, the ET process can be mediated

through the adsorbed molecules (i.e., through a self- exchange

θ_oxâ_oxc^s_ox

)

(A6)

s

θ_red

â

^redc_red

According to eq A6, the ratio θ_ox/θ_reddepends strongly on

the ratio â_ox/â_redthat takes the difference of the adsorption

energies into account. If charge transfer is allowed and assuming

that the charge exchange via the adsorbed layer of redox

molecules is the fastest of all ET reaction steps, whereas

+/0

+

ET reaction between adsorbed Co(Cp)₂and free Co(Cp)₂

-

adsorption and desorption reactions are negligibly slow (k_red

,

in solution), preventing the buildup of more than one monolayer

of reactant and product and allowing for ultrafast steady-state

ET and current flow.

k_ox⁺, k_red⁺, k_ox^-f 0), then the ratio θ_ox/θ_redis defined through

θ_ox

k^-_sec^s_ox

k⁺_sec^s_red

f

(A7)

θ_red

Acknowledgment. This work was supported by the U.S.

Department of Energy, Office of Science, Office of Basic

Energy Sciences, Division of Chemical Sciences. K.S. was

supported by the Deutsche Forschungsgemeinschaft and super-

vised at ISFH by R. Memming and R. Reineke-Koch. We thank

Mark C. Hanna for providing the epilayer GaAs samples, A.

Duda for advice on sample preparation for QCM measurements,

and R.J.D. Miller for useful comments.

Appendix B

In this publication the complex notation is used to calculate

the impedance of the electrochemical system. The purpose of

Appendix B is to impart the basics of this mathematical

treatment to the reader to understand the derivations. A more

detailed treatise is given in text books.^73,74

For the measurement of the impedance the dc voltage U¨

applied to a system is superimposed by a sinusoidal ac

perturbation voltage U_t(frequency, ν ) ω/(2π); phase angle,

φ_U, amplitude Uˆ ). The mathematical treatment using the Euler

identity is shown in eq B1.

Appendix A

The Langmuir isotherm follows from eq 13 if only the mass

balance is considered (i.e., none of the charge-transfer events

is admitted). For stationary conditions (dθ/dt ) dθ_ox/dt ) dθ_red

/

Semiconductor-Molecule Interfaces

J. Phys. Chem. B, Vol. 103, No. 12, 1999 2139

molecules. For this calculation we define the number of

molecules that are exactly at the electrode surface but not

adsorbed as N_redox(the unit of N_redoxis molecules × cm^-2). The

time-dependent changes of N_redoxcan be expressed in terms of

the self-exchange rate:⁷⁵

U ) U¨ + U_t) U¨ + Uˆ exp[i(φ_U+ ωt)] )

U¨ + ∆U exp(iωt) (B1)

with

∆U ) Uˆ exp[iφ_U]

dN

^ox) -

dt

dN_red

dt

) R ⁺_se- R ^-_se(C1)

(i is the imaginary number, i ) (-1)^1/2; ω is the angular

frequency of the voltage perturbation, and t is the time)

The current response of the system is modulated with the

same frequency and can be separated into a dc current part ¨I

and into an ac current part ˆI as shown in eq B2.

ν_ox) -ν_red

≡

Assuming that the self-exchange reaction is much faster than

the adsorption and desorption reaction, the rate ν_oxequals the

difference between the flux of charge carriers through the

interface (i.e., I_F/(eA) ) ×c2_c⁺- ×c2_c^-) and the time-dependent

changes of the coverage of adsorbed acceptor molecules (i.e.,

I ) ¨I + I_t) ¨I + ˆI exp[i(φ_I+ ωt)] ) ¨I + ∆I exp(iωt) (B2)

max

-

+

N_ad

dθ_ox/dt ≈ ×c2_se- ×c2_se+ ×c2_c- ×c2_c^-). An

with

analogous relation for ν_redfollows:

∆I ) ˆI exp[iφ_I]

ν_ox

_max) -

N_ad

ν_red

I_F

dθ_ox

dt

I_F

dθ_red

dt

)

-

)

+

max

(C2)

max

According to the definition, the impedance of the system is

defined through the ratio of the complex quantities ∆U and ∆I.

If the Euler identity is used, the impedance can be separated

into a complex number as shown in eq B3.

N

eAN

ad

Keeping in mind that d(θ + ∆θ exp[iωt])/dt ) iω∆θ exp[iωt],

the variations of the rates ν_oxand ν_redare

∆U Uˆ

∆ν_ox

_max) -

N_ad

∆ν_red

∆I_F

_max- iω∆θ_ox

eAN_ad

∆I_F

_max+ iω∆θ_red

eAN_ad

Z )

)

exp[i(φ_U- φ_I)] )

Uˆ

)

∆I

ˆI

max

N

ad

Uˆ

cos(φ_U- φ_I) + i sin(φ_U- φ_I) (B3)

ˆI

(C3)

ˆI

The variation of the rate has to be coupled with the diffusion

equations; the boundary conditions of the laatter problem are

defined by the experimental setup. For a rotating disk electrode,

the solution is^74-76

In eq B3 the impedance is the sum of the real part Z′ (cos

term) and the 90° phase-shifted imaginary part Z′′ (sin term).

Z ) Z′ + iZ′′

(B4)

1

Most of the electrochemical systems are nonlinear, but if the

perturbation signal (voltage) is kept sufficiently small (∆U <

e/(kT) ≈ 25 mV), the variation of the quantity of interest can

be calculated from a Taylor series that is developed around the

stationary value and truncated after the second term. This

linearization procedure is shown in eq B5 for the faradaic current

I_F.

∆ν_redox

)

∆c^s_redox

with

N_ω,redox

iω

tanh δ_N,redox

(

)

D

x

redox

N_ω,redox

)

(C4)

iωD

x

redox

which holds either for the reduced or oxidized species if the

diffusion coefficients or thicknesses of the Nernst diffusion

layers are different (i.e., D_ox* D_redand δ_N,ox* δ_N,red). However,

assuming that D_ox) D_redand δ_N,ox) δ_N,red, the combination

of eqs C3 and C4 yields the relationship between the time-

dependent variations of the degrees of coverage, the faradaic

current, and the surface concentration of the redox system as

shown in eq C5.

dI_F

I ) ¨I +

∆U exp(iωt)

(B5)

( )

F

dU

and with consideration of eq B2

dI_F

∆I )

∆U

(B6)

( )

F

dU

The capacitive charging currents I_Care expressed as time-

dependent voltage variations across a capacitance C, i.e., I_C)

(dU/dt)C. Considering that the time-independent dc currents do

not show any contribution, the linearization procedure yields

∆I_F

∆c_o^s_x) -∆c_r^s_ed) -N^maxN_ωiω∆θ_ox

-

ad

max

ad

(

)

eAN

∆I_F

max

ad

) N N_ωiω∆θ_red

+

(C5)

d

max

ad

(

)

eAN

∆I_Cexp(iωt) ) C [∆U exp(iωt)]

dt

(B7)

Appendix D

and

Figure 15 shows the experimental impedance spectra (M,

open circles) of Figure 8c recorded at -0.3 V vs solution redox

potential and three simulated impedance spectra (S(1) to S(3),

lines with black dots) calculated for different rate constants k_se

for the self-exchange reaction. The values of k_seused for the

simulations are listed in Table 1. Table 1 also contains the

calculated values for the faradaic current I_F, the degrees of

coverage θ_oxand θ_red, the time constant τ_ct) R_aC_sc, and the

∆I_C) iωC∆U

(B8)

Appendix C

Appendix C shows a solution for the calculation of the

complex, time-dependent variations of the surface concentration

due to a time-dependent perturbation of the electrode potential

if the electrode surface is covered with a layer of adsorbed redox

2140 J. Phys. Chem. B, Vol. 103, No. 12, 1999

Meier et al.

(19) Heimer, T. A.; Heilweil, E. J. J. Phys. Chem. B 1997, 101, 10990.

(20) Lanzafame, J. M.; Miller, R. J. D.; Muenter, A. A.; Parkinson, B.

A. J. Phys. Chem. 1992, 96, 2820.

(21) Boudreaux, D. S.; Williams, F.; Nozik, A. J. J. Appl. Phys. 1980,

51, 2158.

(22) Nozik, A. J.; Memming, R. J. Phys. Chem. 1996, 100, 13061.

(23) Ross, R. T.; Nozik, A. J. J. Appl. Phys. 1982, 53, 3813.

(24) Marcus, R. A. J. Phys. Chem. 1990, 94, 4152.

(25) Marcus, R. A. J. Phys. Chem. 1990, 94, 7742.

(26) Smith, B. B.; Halley, J. W.; Nozik, A. J. Chem. Phys. 1996, 205,

245.

(27) Taube, H. Electron-Transfer Reactions of Complex Ions in Solution;

Academic Press: New York, 1970.

(28) Koval, C. A.; Austermann, R. L.; Turner, J. A.; Parkinson, B. A.

J. Electrochem. Soc. 1985, 132, 613.

(29) Herrmann, C.; Perault, G.; Pilla, A. Anal. Chem. 1968, 56, 1173.

(30) Buttry, D. A.; Ward, M. D. Chem. ReV. 1992, 92, 1355.

(31) Shimazu, K.; Yagi, I.; Sato, Y.; Uosaki, K. Langmuir 1992, 8, 1385.

(32) We note that some QCM experiments that were performed with

aged, unsublimed and sublimed Co(Cp)₂⁰crystals from Aldrich did not show

any response (i.e., mass change). NMR studies on these sublimed chemicals

reveal that the Co(Cp)₂⁰crystals from Aldrich contained a higher concentra-

tion, as well as different impurities, than the Co(Cp)₂⁰from Strem. However,

these impurities are believed to be of an organic nature because the

electrochemical analysis at a gold electrode did not reveal any differences

Figure 15. Experimental EIS (M, open circles, recorded at -0.3 V)

and three simulated impedance spectra (S(1) to S(3), lines with black

dots) calculated for different rate constants k_sefor the self-exchange

reaction (see Table 1).

TABLE 1: Faradaic Current (I_F) Degrees of Coverage

(θ_ox/θ_red), and the Relaxation Times for the Heterogeneous

Charge Transfer and the Self-Exchange Reaction (τ_ctand

τ_se) Calculated for Three Different Rate Constants for the

Self-Exchange Reaction

0

between solutions of the two Co(Cp)₂sources.

(33) Cooper, J. B.; Bond, A. M. Anal. Chem. 1993, 65, 2724.

(34) Ward, M. D.; Lee, W. W.; White, H. S. J. Am. Chem. Soc. 1993,

65, 3232.

(35) Giles, C. H.; D’silva, A. P.; Easton, I. A. J. Colloid Interface Sci.

1974, 47, 11.

+

k_se

)

curve k_se^-[cm³s^-1

]

I_F[A]

θ_ox/θ_red

τ

_ct[s]

τ_se[s]

S1 <1.0 × 10^-14-2.7 × 10^-50.59/0.41 1.9 × 10^-57.6 × 10^-5

S2

S3

1.0 × 10^-16-2.1 × 10^-50.46/0.54 2.4 × 10^-57.6 × 10^-3

5.0 × 10^-17-1.7 × 10^-50.38/0.62 3.0 × 10^-51.5 × 10^-2

(36) Heiland, W.; Gileadi, E.; Bockris, J. J. Phys. Chem. 1966, 70, 1207.

(37) Bard, A. J.; Faulkner, L. R. Electrochemical Methods; John Wiley

& Sons: New York, 1980.

(38) According to the model, ∆I_Fcan be considered as being decoupled

from ∆I_Cas long as surface concentrations of the redox system are equal

to the bulk concentrations (c_redox^s) c_redox^b, ∆c_redox^s) 0). This is in general

the case if the faradaic current does not exceed ≈10% of the diffusion-

limited current.

(39) Equation 23 for the total impedance Z is not quite complete to

describe the impedance of the whole electrochemical setup. For this a series

resistance R_shas to be added that takes into account the frequency and

potential independent resistivity of the electrolyte solution and the external

circuit, i.e., the impedance of the whole electrochemical setup is R_s+ Z.

(40) Rhoderick, E. H.; Williams, R. H. Metal-Semiconductor Contacts;

Clarendon Press: Oxford, 1988.

time constant τ_se) R_bC_b. The spectra were calculated for a

frequency range from 1 Hz to 1 MHz.

Other parameters are the following: φ_sc) 0.542 V (+U_e)

-0.3 V); N_d) 1.0 × 10¹⁷cm^-3; A ) 0.152 cm²; c_ox) 1.3

mM; c_red) 0.9 mM; D ) 1.3 × 10^-5cm²s^-1; ν ) 5.8 × 10^-5

cm²s^-1; 200 rpm; N_ad^max) 4 × 10¹⁴cm^-2; θ ) 1; k_c^-) 1.21

× 10^-7cm³s^-1; k_c) 0 cm³s^-1; and R_s) 44.7 Ω.

+

With k_c⁺) 0 cm³s^-1, we assume a perfect rectifying diode

behavior which is in agreement with the experimental result:

the saturation current density at 0.4 V (φ_sc) 1.242 V) amounts

to 30 nA cm^-2. For a rotation velocity of 200 rpm, the diffusion-

limited current amounts to I_F,dl) - 7.1 × 10^-5A.

(41) Ashcroft, N. W.; Mermin, N. D. Solid State Physics; W. B. Saunders

Company: New York, 1976.

(42) Nielson, R. M.; McManis, G. E.; Golovin, M. N.; Weaver, M. J. J.

Phys. Chem. 1988, 92, 3441.

(43) Nielson, R. M.; McManis, G. E.; Weaver, M. J. J. Phys. Chem.

References and Notes

1989, 93, 4703.

(1) Diol, S. J.; Poles, E.; Rosenwaks, Y.; Miller, R. J. D. J. Phys. Chem.

1998. In press.

(2) Meier, A.; Kocha, S. S.; Hanna, M. C.; Nozik, A. J.; Siemoneit,

K.; Reineke-Koch, R.; Memming, R. J. Phys. Chem. B. 1997, 101, 7038.

(3) Miller, R. D. J.; McLendon, G.; Nozik, A. J.; Schmickler, W.;

Willig, F. Surface Electron-Transfer Processes; VCH Publishers: New

York, 1995.

(4) Lewis, N. S. J. Phys. Chem. 1998, 102, 4843.

(5) Fajardo, A. M.; Lewis, N. S. J. Phys. Chem. B. 1997, 101, 11136.

(6) Pomykal, K. E.; Lewis, N. S. J. Phys. Chem. B 1997, 101, 2476.

(7) Fajardo, A. M.; Lewis, N. S. Science 1996, 274, 969.

(8) Pomykal, K. E.; Fajardo, A. M.; Lewis, N. S. J. Phys. Chem. 1996,

100, 3652.

(9) Royea, W. J.; Fajardo, A. M.; Lewis, N. S. J. Phys. Chem. B 1997,

101, 11152.

(10) Lewis, N. S. Annu. ReV. Phys. Chem. 1991, 42, 543.

(11) Gu, Y.; Waldeck, D. H. J. Phys. Chem. 1996, 100, 9573.

(12) Wang, D.; Buontempo, J.; Li, Z. W.; Miller, R. J. D. Chem. Phys.

Lett. 1995, 232, 7.

(13) Rosenwaks, Y.; Thacker, B. R.; Ahrenkiel, R. K.; Nozik, A. J. J.

Phys. Chem. 1992, 96, 10096.

(14) Ellingson, R. J.; Asbury, J. B.; Ferrere, S.; Ghosh, H. N.; Sprague,

J.; Lian, T.; Nozik, A. J. J. Phys. Chem. 1998, 102, 6455.

(15) Tachibana, Y.; Moser, J. E.; GrA¨ tzel, M.; Klug, D. R.; Durrant, J.

R. J. Phys. Chem. 1996, 100, 20056.

(44) McManis, G. E.; Nielson, R. M.; Gochev, A.; Weaver, M. J. J.

Am. Chem. Soc. 1989, 111, 5533.

(45) The reason that the slopes of the curves for the flatband and the

surface redox potential are not exactly identical is due to the fact that during

the long stationary measurements the solvent partially evaporates. This led

to an increase of redox concentration (between 10% and 20%) with time

and to an increase of the diffusion limited current. For the calculation of

the surface redox potential with eq 18 the value of the diffusion limited

current has to be known the more exactly the closer the ratio of I_F/I_F,dl

approaches unity. This time-dependent concentration change (or increase

of the diffusion-limited current) can be proven by starting the measurement

at negative potentials instead of starting in the depletion region.

(46) Memming, R. Topics in Current Chemistry; Springer-Verlag:

Berlin, 1994; Vol. 169, p 106.

(47) Ba, B.; Cachet, H.; Fotouhi, B.; Gorochov, O. Electrochim. Acta

1992, 37, 309.

(48) Ba, B.; Fotouhi, B.; Gabouze, N.; Gorochov, O.; Cachet, H. J.

Electroanal. Chem. 1992, 334, 263.

(49) Ba, B.; Cachet, H.; Fotouhi, B.; Gorochov, O. J. Electroanal. Chem.

1993, 351, 337.

(50) Ba, B.; Cachet, H.; Fotouhi, B.; Gorochov, O. Semicond. Sci.

Technol. 1994, 9, 1529.

(51) The value for the kinematic viscosity of ν ) 5.8 × 10^-3cm²s^-1

is an interpolated value between ν ) 5.8 × 10^-3cm²s^-1(21 °C) measured

for an ACN-0.5 M TBAPF₆solution and ν ) 4.9 × 10^-3cm²s^-1(21 °C)

for an ACN-0.1 M TBAPF₆solution. The kinematic viscosity for ACN as

calculated from the viscosity and the density amounts to ν ) η/F ) 4.3 ×

10^-3cm²s^-1(Handbook of Chemistry and Physics, 74th ed.; CRC Press:

Boca Raton, FL, 1993-1994).

(16) Hannappel, T.; Burfeindt, B.; Storck, W.; Willig, F. J. Phys. Chem.

B 1997, 101, 6799.

(17) Moser, J. E.; Noukakis, D.; Bach, U.; Tachibana, Y.; Klug, D. R.;

Durrant, J. R.; Humphry-Baker, R.; GrA¨ tzel, M. J. Phys. Chem. B 1998,

102, 3649.

(18) Fessenden, R. W.; Kamat, P. V. J. Phys. Chem. 1995, 99, 12902.

(52) Landsberg, R.; Thiele, R. Electrochim. Acta 1966, 11, 1243.

Semiconductor-Molecule Interfaces

J. Phys. Chem. B, Vol. 103, No. 12, 1999 2141

(53) Scheller, F.; MU¨ ller, S.; Landsberg, R.; Spitzer, H.-J. J. Electroanal.

Chem. 1968, 19, 187.

(54) Scheller, F.; Landsberg, R.; MU¨ ller, S. J. Electroanal. Chem. 1969,

20, 375.

(55) Smith, B. B.; Nozik, A. J. Chem. Phys. 1996, 205, 47.

(56) Smith, B. B.; Nozik, A. J. J. Phys. Chem. 1997, 101, 2459.

(57) Becke, A. D. J. Chem. Phys. 1993, 98, 1372.

(58) Becke, A. D. J. Chem. Phys. 1993, 98, 5648.

(59) Perdew, J. P. In Electronic Structure Theory of Solids; Ziesche,

P., Eshrig, H., Eds.; Akademie Verlag: Berlin, 1991.

(60) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.;

Pederson, M. R.; Singhm, D. J.; Fiolhais, C. Phys. ReV. B 1992, 46, 6671.

(61) Becke, A. D. Phys. ReV. A. 1988, 38, 3098.

(62) Schrodinger, Inc. Jaguar, version; 3.0; Portland, OR, 1997.

(63) Qian, G.; Martin, R. M.; Chadi, D. J. Phys. ReV. B. 1988, 37, 1303.

(64) Biegelsen, D. K.; Bringans, R. D.; Northrup, J. E.; Swartz, E. L.

Phys. ReV. B. 1990, 5701.

(65) Qian, G.; Martin, R. M.; Chadi, D. J. Phys. ReV. B 1988, 38, 7649.

(66) Johnson, M. A.; Stefanovich, E. V.; Truong, T. N. J. Phys. Chem.

1998. In press.

(67) Marcus, R. A. Annu. ReV. Phys. Chem. 1964, 15, 155.

(68) Gerischer, H. Z. Phys. Chem. N. F. 1961, 27, 40.

(69) Smith, B. B.; Nozik, A. J. To be published.

(70) In another previous publication (Rosenwaks, Y.; Thacker, B. R.;

Nozik, A. J.; Ellingson, R. J.; Burr, K. C.; Tang, C. L. J. Phys. Chem.

1994, 98, 2739) ultrafast rate constants were also reported for photoinduced

ET at p-InP in contact with Fe(CN)₆^4-/3-in aqueous electrolyte when large

electric fields were produced in the semiconductor space charge layer. The

S_ETvalues were reported to show an electric field dependence, and they

increased from about 10⁴cm/s to 5 × 10⁷cm/s as the band bending in the

space charge layer increased from <0.1 to >0.5 eV. However, additional

experimental studies of this system (Ellingson, R. J.; Nozik, A. J.

Unpublished data.) showed a very pronounced dependence of the PL decay

characteristics and its reversibility (with respect to the presence or absence

of applied potential) on sample history and whether the laser excitation

spot was stationary or moving during the experiment. With the laser spot

moving, the PL decay was reversible; however, if the laser spot were

stationary, then the fast PL decay with applied negative potential became

constant and independent of potential. This indicates that some irreversible

chemical changes are occurring at the p-InP surface with applied negative

potential in the presence of Fe(CN)₆^4-/3-. Thus, the conclusions in this

manuscript that a field dependence of the ET rate constant was observed

and that ultrafast interfacial ET rates occurred at high fields are not supported

by new data.

(71) Ahrenkiel, R. K. Minority-Carrier Lifetime in III-V Semiconduc-

tors. In Minority Carriers in III-V Semiconductors: Physics and Applica-

tions; Ahrenkiel, R. K., Lundstrom, M. S., Eds.; Academic Press: San

Diego, 1993; Vol. 39, p 39.

(72) Rosenwaks, Y.; Shapira, Y.; Huppert, D. Appl. Phys. Lett. 1990,

57, 2552.

(73) Macdonald, J. R. Impedance Spectroscopy; John Wiley & Sons:

New York, 1987.

(74) Gabrielli, C. Identification of electrochemical processes by fre-

quency response analysis. Schlumberger Technologies Technical Report

Number 004/83; Solartron/Schlumberger Instruments, Farnborough, Eng-

land, 1984.

(75) Uhlendorf, I. Thesis, University of Hamburg, 1994.

(76) Uhlendorf, I.; Reineke-Koch, R.; Memming, R. J. Phys. Chem.

1996, 100, 4930.

Article Doi

Fast electron transfer across semiconductor-molecule interfaces: GaAs/Co(Cp)2+/0

DOI: 10.1021/jp983230x

Source and publish data:

Authors:

Article abstract of DOI:10.1021/jp983230x

Full text of DOI:10.1021/jp983230x

Products guided by the article

R&D Labs maybe for 1277-43-6

Relevant to this article

Hot Product