A reappraisal of the frequency dependence of the impedance of semiconductor electrodes

65

J. Electroanal. Chem., 315 (1991) 65-85 Elsevier Sequoia

S.A., Lausanne

JEC 01648

A reappraisal of the frequency dependence of the impedance of semiconductor electrodes G. Oskam, D. Vanmaekelbergh and J.J. Kelly Debye Research Institute, University of Utrecht, P. 0. Box 8O.oo0, 3508 TA Utrecht (Netherlank) (Received

25 February

1991; in revised form 7 May 1991)

Abstract

The frequency dependence of the impedance of polarizable semiconductor/metal and semiconductor/electrolyte solution interfaces is reconsidered. No frequency dispersion of the polarization capacitance is found with n-GaAs/Au contacts whereas the capacitance of n-GaAs/electrolyte solution interfaces show considerable dispersion. The frequency dispersion depends on the microroughness of the electrode surface and on the specific conductivity of the electrolyte solution. As for polarizable metal electrodes, the relationship between the capacitance and the frequency is closely related to the relationship between the capacitance and the specific conductivity of the electrolyte solution. It is concluded that the main origin of frequency dispersion should not be sought in the solid but is related to the development of the electric double layer at the electrolyte side of the interface,. A model is presented to account for these results.

(I) INTRODUCTION

Semiconductor electrodes are usually characterized by current-potential (I- V) and capacitance-potential (C-V) measurements. From C-I/ measurements the flat-band potential (V,) and the free carrier concentration of the semiconductor can, in principle, be obtained. Under depletion conditions, a semiconductor in an indifferent electrolyte solution behaves as a nearly ideally polarizable electrode. The impedance of the interface can be represented by a simple equivalent scheme as shown in Fig. 1. In this scheme, C is equal to (CL-’ + C,’ + C;‘)-l, and as the capacitance of the semiconductor (C,,) is usually much smaller than that of the Helmholtz layer (C,) and of the diffuse double layer (C,), C is approximately equal to C,. The resistance R, contains contributions from the bulk semiconductor, the bulk electrolyte and the contacts. The Faraday resistance for charge transfer R, is very high for nearly ideally polarizable contacts. Under depletion conditions, the 0022-0728/91/$03.50

0 1991 - Elsevier Sequoia

S.A. All rights reserved

66

Fig. 1. Equivalent circuit for a nearly ideally polarizable

semiconductor/electrolyte

interface.

space charge layer capacitance C, varies with the applied potential in accordance with the Mott-Schottky relation, which for an n-type electrode has the form 1 -= C,‘, where e is the electron charge, e is the dielectric constant, z. is the permittivity of free space, k is the Boltzmann constant, T is the temperature and ND is the effective donor density. According to the Mott-Schottky relation, C-’ varies linearly with the potential V when C = C,; the flat-band potential V, can be obtained by extrapolation to C-* = 0 on the potential axis. In theory, the capacitance is not a function of the angular frequency of the ac-perturbation signal w. In practice, however, the measured capacitance C is usually found to depend on w [l-8]. The Mott-Schottky plots show a frequency dependence which may hamper the determination of the flat-band potential and the free carrier concentration. The plots are often linear and in certain cases the lines converge at a point on the potential axis which is generally taken to be the flat-band potential. In other cases, however, the curves are not linear or the lines do not converge, in which case it is impossible to determine the flat-band potential accurately. Explanations for these phenomena were generally sought in intrinsic properties of the semiconductor such as deep donor centres [9] and dielectric relaxation phenomena in a surface layer [l]. The frequency dependence of polarizable solid metal/electrolyte interfaces has also been studied extensively [lo-121. The capacitance is found to depend considerably on the frequency. It is also observed that the capacitance depends on the specific resistivity of the electrolyte solution [lo]. The dependence on the frequency and specific resistivity is generally believed to be related to the microscopic roughness of the metal electrode [lo-151. We have re-examined the frequency dependence of the impedance of nearly ideally polar&able n-GaAs electrodes under depletion conditions. In order to see if there is an analogy between semiconductor and metal electrodes, we have investigated the influence of the electrolyte resistivity on the impedance. Furthermore, we have examined the frequency dependence of n-GaAs/Au Schottky diodes and compared the results with those obtained with n-GaAs/electrolyte junctions in

67

order to determine whether frequency dispersion at semiconductor electrodes originates from solid-state phenomena. In another paper [23] we propose a model for the frequency dispersion of the polarization impedance at metal/electrolyte interfaces. It is argued that this model can also be applied to semiconductor/electrolyte interfaces, and numerical calculations based on the model are compared with the experimental results. (II) EXPERIMENTAL

The impedance measurements were performed with a Frequency Response Analyzer (Solartron, FRA 1250) and an Electrochemical Interface (Solartron, EC1 1286). The measuring shunt resistor was adapted to the measured impedance in order to minimize errors in the data [16]. All potentials were measured versus a saturated calomel electrode (SCE). A platinum wire close to the tip of the capillary was connected in parallel to the reference electrode via a 20 PF capacitor to minimize possible errors at high frequencies, especially in solutions of low conductivity [ll]. The platinum counter-electrode had a surface area of 20 cm2 and the geometric surface area of the semiconductor electrode was 0.125 cm2. The n-GaAs crystals were obtained from MCP Electronic Materials Ltd. (UK). The samples had the (100) orientation and were doped with Si. The specifications were as follows: free carrier concentration n = (2.0-2.2) X 1Ol7 cme3; electron mobility p = 3400 cm2 V-’ s-l. At this doping level the specific bulk resistivity of the n-GaAs electrodes was always negligibly small compared to the specific resistivity of the electrolyte solution. Wafers were mechano-chemically polished by the supplier. In order to obtain reproducible results, it was necessary to clean the surface prior to the measurements. Surface preparation of the samples was carried out as follows. The surface was rinsed successively in acetone, ethanol and water. The electrode was then dipped in 8.0 mol/l HCl for 3 min to remove oxides before being etched in a 3 : 1: 1 mixture of H,SO, (98%), H,O and H,O, (30%) for 20 s. Afterwards, the electrode was dipped in 8.0 mol/l HCl for another 30 s and blown dry with N,. The roughness of the surface was studied by Nomarski microscopy (Zeiss) and profilometry (Alphastep 200, Tencor Instruments) and, as Figs. 2A and 2B show, the surface is almost flat after this etching procedure. For measurements on rough surfaces, the electrode was polarized at +0.7 V (vs. SCE) in 0.18 M NaClO, (pH 2.40) solution for 1 h. The Nomarski microscope image and the surface profile in Fig. 3A and 3B clearly show that the surface is indeed roughened by this etching method. In what follows, surfaces like those shown in Figs 2 and 3 will be indicated as “flat” and “roughened”, respectively. The following electrolyte solutions were used for the impedance measurements: 0.01 M HClO, + x M NaClO, (pH 2.40); x M NaClO, (pH = 5.0); and 0.01 M NaOH + x M NaClO, (pH 11.50) (x was varied). All chemicals were of p.a. quality. Before each measurement, oxygen was purged from the solution by bubbling through high-purity N,. All measurements were carried out at room temperature in

68

z/pm 0.04 0.02 0.00 -0.02

~.

-0.04j... 0

,_.

.._...... .i. 100

,.

,..,,. :.... 200

I.

,,

I. ,.. 300 x/pm

Fig. 2. (A) Nomarski microscope image of a “flat” electrode surface. (B) Surface profile of a “flat” surface.

the dark. The measurements were performed on stationary electrodes; rotation was found to have no influence on the results. The gold layers were prepared by electrochemical deposition from a 0.01 M KAu(CN), + 0.65 M KCN + 0.75 M KOH solution under potentiostatic conditions. The thickness of the gold layer was approximately 100 nm. Unless otherwise stated, the measured impedance was always corrected for the bulk resistance R, and the charge-transfer resistance R,,which were obtained with a non-linear least-squares fit program [17]. As a result, the series capacitance obtained from the corrected impedance pertains only to the electric double layer. The measurements were performed on six n-GaAs electrodes and four nGaAs/Au diodes.

69 (III) RESULTS

(III-I)

n-Gads/Au

contacts

In Fig. 4, the current-voltage curve of an n-GaAs/Au contact is shown. Before gold deposition the n-GaAs surface was roughened. From the forward current-voltage curve the ideality factor was found to be 1.07. The diode is nearly ideally polarizable in the potential range from 0 to 2.0 V. Impedance measurements were performed in this range. In Fig. 5, an impedance plot measured at 1.6 V is given. The equivalent circuit for this plot is the simple scheme mentioned before: a small bulk resistance R, in series with a parallel combination of a Faraday resistance R, and the capacitance of the se~conductor C,. None of the three elements is a function of the frequency w. In Fig. 6, the (C2, V) plots for different frequencies

+m

..

..I,.... ,.....: ,...

‘.

)I,.....

:..

.’

* (8)

0.04 ... .. . .,............. i

.........

;

j :

0.02 0.00

0

100

200

300 x /Pm

Fig. 3. (A) Nomarski microscope image of a “roughened” electrode surface. (B) Surface profile of a “ roughened” surface.

70

I /mA -0.4

-

-a8

-

-1.2

-

-1.6

-

-7. -

1

I

I 1

I

0

-1

I 2

v/v Fig. 4. Current-voltage ically.

characteristics

of an n-GaAs/Au

diode. The Au layer was deposited

electrochem-

300 Im(Z)/kR

0 0

100

200

300

400

500

Re (Z)/kR Fig. 5. Impedance plot of an n-GaAs/Au diode at a band bending of 2.6 V (not corrected for R, and Rb). The line is a semi-circle, according to the equivalent circuit of Fig. 1, with R, = 20 LI and R, = 450 kQ.

71

1.50.

16 %‘/f2

0.90

0.60. / / /

0.30, / / / 0.00

1_-

-1.00

-0.50

0.00

0.50

1.50

1.00

2.00

v/v Fig. 6. Mott-Schottky plot of an n-GaAs/Au kHz ( x), 2.05 kHz (+) and 0.205 kHz (0).

diode measured

at different

frequencies:

20.5 kHz (O), 6.50

are shown. A straight line is found, essentially independent of the frequency, and from the slope a free carrier concentration of n = 2.1 X 10” cmm3 is obtained (if the geometric surface area is used). This corresponds very well to the specifications of the supplier. The barrier height +a of the diode, obtained by extrapolation to Cd2 = 0, is - 1.0 eV, which agrees with literature data for electrochemically prepared n-GaAs/Au diodes [18]. From these results it can be concluded that the impedance of an n-GaAs/Au contact can be represented by a simple equivalent circuit consisting of three frequency-independent elements, and that the measured capacitance pertains to the depletion layer of the semiconductor and agrees with Mott-Schottky theory. (III.2)

n-GaAs/aqueous

electrolyte interfaces

(ZZZ.2.1) I-V curves in indifferent electrolyte solutions In Fig. 7, the reverse current-potential curve of a flat n-GaAs electrode (see Experimental section) in 1.0 M NaClO, (pH 2.40) is shown (curve A). In the potential range between 1.2 and -0.3 V (vs. SCE) the n-GaAs/electrolyte solution interface behaves as a nearly ideally polarizable junction. In this range, a small positive current is observed which increases with increasing potential. In the absence of a redox couple, this anodic current is caused by dissolution of n-GaAs by holes arising from minority carrier generation. It is generally believed that under these

72

0.50 I/PA 0.40

0.30

0.20

-0.10 -0.60

-0.30

-0.00

0.30

0.60

0.90

1 !O

V/V(SCE)

Fig. 7. Current-potential characteristics of an n-GaAs electrode in a 1.0 M NaClO, (A) Flat surface; (B) roughened surface.

(pH 2.40) solution.

conditions dissolution is not uniform but localized. This implies that the surface becomes rougher, which is confirmed by Nomarski microscopy and profilometry (see Fig. 3). Faktor et al. showed that tunnel formation in n-GaAs occurs when anodic dissolution is carried out in the dark in a 10% KOH solution [19]. Other authors also concluded that the n-GaAs surface roughens during anodic dissolution in the dark [19-211. When the electrode roughens, the surface area becomes larger and the anodic dissolution current increases. This is illustrated in Fig. 7, curve B. The electrode was polarized at V = 0.7 V (vs. SCE) for 60 min and it is clear that the current is higher than that for the flat electrode (A). In the more positive potential range, this effect is observed in all the electrolyte solutions used in these experiments. (111.2.2) The measured capacitance C as a function of the potential In Fig. 8, C-* measured at a flat electrode immersed in a solution of 4.5 M NaClO, (pH 2.40) is plotted versus V for different frequencies. The frequency dispersion is slight, although more pronounced than for an n-GaAs/Au contact. From this Mott-Schottky plot an accurate free carrier concentration can be obtained: n = (2.2-2.5) X 10” cme3. The lines converge at a point on the potential axis and this flat-band potential is in agreement with the literature value reported for this pH [22].

73

V/VCSCE, Fig. 8. Mott-Schottky plots of a flat n-GaAs electrode in 4.5 M NaClO, (pH 2.40) solution for different frequencies: 21.5 kHz (0),4.64 kHz (+), 1.00 kHz (X) and 0.215 kHz (0).

After 60 min anodic dissolution in the dark, the impedance results change dramatically. In Fig. 9, C2 is plotted versus V for different frequencies for a solution of 0.03 M NaClO, (pH 2.40). The (Cw2, V) plots are strongly dependent on the frequency. The curves do not converge at a point on the potential axis and it is impossible to determine the flat-band potential and the free carrier concentration. In Fig. 10, the (C2, V) plots are given for the same electrode now immersed in a solution of 4.5 M NaClO, (pH 2.40). The (C2, V) plots are markedly less dependent on the frequency and the curves converge at the flat-band potential. The frequency dispersion is still much larger than that for a flat electrode in the same solution (see Fig. 8). It is not possible to determine the free carrier concentration as the slope depends on the frequency. When the free carrier concentration is calculated from the high frequency curve (65 kHz), using the geometric area, the value is found to be too high compared with the suppliers specification: n = 4.5 x lOi cm -3. The behaviour presented in Figs. 9 and 10 is very often observed at semiconductor electrodes. Using the same terminology as Dutoit et al. [l], we denote the (C2, V) dependence in which the lines for different frequencies do not converge as A-type behaviour. If the (CP2, V) plots are frequency-dependent but converge at a point on the potential axis, we speak of B-type behaviour. From the foregoing results it follows that A-type and B-type behaviour can be found with the same electrode, depending on the conductivity of the electrolyte.

1.40 ,o

6c-2/F-:

1.00

0.80

0.60

J -1.50

-1.00

-0.50

0.00

0.50

1.00

vfvcscx, Fig. 9. Mott-Schottky plots of a roughened n-GaAs electrode in 0.03 M NaClG4 (pN 2.40) solution for different frequencies: 40.6 kHz (0), 10.0 kHz (+), 2.15 kHz (X) and 0.215 kHz (0).

It can be concluded that flat electrodes show nearly ideal Mott-Schottky behaviour while roughened electrodes give distinctive frequency-dependent ( C2, I’) plots. For roughened electrodes it is always impossible to determine the free carrier concentration and at low electrolyte conductivity it is also difficult to determine the flat-band potential. These effects are described in more detail in the next section. (111.2.3.) The frequemy dependence of C As shown in the previous section, the measured capacitance C of a roughened electrode depends on the frequency of the ac perturbation. To study this in more detail, the impedance was measured in the depletion region at a fixed potential at which the current density was very small (ca. 50 nA cm-‘) for frequencies between 10 Hz and 65 kHz. The measured impedance was corrected for the bulk resistance R, and the charge-transfer resistance R t. Hence, the remaining impedance pertains to the electric double layer only. In Fig. 11, log C is plotted versus log w for three different electrolyte concentrations measured at a flat electrode which showed a small frequency dispersion in the (C-*, I’) plot. From Fig, 11 it can be seen that log C decreases slightly with increasing log w, with a slope of ca. -0.011. In Fig. 12, results are given for a roughened electrode which showed a larger frequency dispersion in the (C2, V) plot; log C is again plotted versus log o for four different electrolyte concentra-

75

1.40

16 6C-2/r2

1.00

0.80 0.60

0.20. 0.40. 0.00.

pi-

>*

____ -_-__ y/ -2

-1.00

-1.50

-0.50

0.00

0.50

1.00

V/VtSCE,

Fig. 10. Mott-~hottky plots of a roughened n-GaAs electrode in 4.5 M NaCIO, (pH 2.40) sotution for different Frequencies: 40.6 kHz (0), 10.0 kHz (+), 2.15 kHz ( X) and 0.215 kHz (II).

-7.6

log

ww)

-7.7

-

-7.0

-

-7.9

I 1

I

1

I 5

3

log (w/rad.s-‘) Fig. 11. Log C vs. log w plot for a fiat n-GaAs electrode for three different electrolyte solutions (pH 2.40): 4.5 M NaClO, (A), 0.78 M NaC10, (+) and 0.03 M NaClO, (0).

76 -7.6

i

J

log (C/F)

-7.7

-

-7.0

-

0

_7.~~ 1

3

l

6

log (w/rad.s-I) Fig. 12. Log C versus log w for a roughened n-GaAs electrode for four different dectrolyte sohtions (pH 2.40): 4.5 M NaClO, (0). 0.78 M NaCIO, (+), 0.181 M NaClO, (0) and 0.03 M NaClO, (A).

tions. As is clear from Fig. 12, the slope is much more negative for this roughened electrode: ca. - 0.036. The slope of the log C vs. log o plot generally becomes more negative with decreasing electrolyte concentration or with increasing frequency. Generally, slopes in the range from -0.01 to -0.10 are found, depending on the surface roughness, the electrolyte concentration and the frequency. From Figs. 11 and 12 it can also be concluded that the capacitance at a fixed frequency increases with increasing electrolyte concentration. In Fig. 13, log C (at fixed frequency) is plotted versus log K, where K is the specific conductivity of the solution. As can be seen from Fig. 13, log C shows an approximately linear dependence on log K and the slope of the line depends strongly on the condition of the electrode surface; the slope is much larger for a roughened electrode. It is generally found that for a given electrode the (C, w) relationship is closely related to the (C, K) relationship. The magnitude of the slope is strongly dependent on the roughness of the surface of the electrode. (IV) DISCUSSION

(I?! 1) Existing models Two models have been proposed to explain the frequency dispersion of the capacitance of se~conductor electrodes. Nogami ascribed the frequency dispersion

-7.71 log (C/F) -7.72

-7.79

-7.70

1

-

I

0

014

0:fl

1;

log (KImhO

t

1:Fl

I

;

t

-

2:4

Cf+l

Fig. 13. Log C versus log K at a fixed frequency of 10 kHz for a flat (A) and a roughened (B) electrode.

to deep donor levels in the space charge layer [9]. Dutoit et al. proposed a model in which the frequency dispersion is ascribed to dielectric relaxation phenomena in a disturbed layer of the semiconductor at the semiconductor/electrolyte or semiconductor/metal interface [l]. If the width of the disturbed layer is much smaller than that of the space charge layer of the semiconductor, the model leads to A-type behaviour, i.e. the frequency dispersion of the capacitance is large and the (C2, V) plots do not converge at a point on the potential axis. If the width of the disturbed

78

layer is larger than that of the space charge layer, the model leads to B-type behaviour: while the capacitance is still frequency-dependent, the (C2, I’) plots now converge at a point on the potential axis. The authors showed that this model is able to explain the frequency dispersion found on CdS, CdSe and TiO, [l], as well as on GaAs [2,3] and GaP [4] electrodes. We have found that at a roughened n-type GaAs electrode either A- or B-type behaviour can be found depending on the specific conductivity of the electrolyte. When the electrolyte conductivity is low, A-type behaviour is observed (see Fig. 9) and when the conductivity is high, B-type behaviour is found (see Fig. 10). The frequency dispersion generally decreases with increasing electrolyte conductivity. Moreover, if, for a given n-GaAs electrode, the electrolyte is replaced by a Au layer, the frequency dispersion of the interfacial capacitance disappears (see Fig. 6). That the frequency dispersion at semiconductor electrodes depends markedly on the conductivity of the electrolyte was also reported for n-TiO, electrodes [6,8]; in 0.125 M NaNO,, A-type behaviour was found; in 0.250 M NaNO,, B-type behaviour was found; and at 1 M NaNO,, the frequency dispersion essentially disappeared [6]. Lyden et al. [7] showed that the capacitance measured at polycrystalline n-CdSe electrodes is considerably frequency-dependent, while at n-CdSe/Au interfaces no frequency dispersion could be observed. These observations show that the frequency dispersion of the capacitance does not originate essentially from phenomena in the semiconductor. The role of the electrolyte indicates that the main effect must be sought at the electrolyte side of the interface. We have therefore studied the role of the specific conductivity of the electrolyte in more detail. It was found at n-GaAs that the (C, w) relationship is closely related to the (C, K) relationship; if, at a given value of the specific conductivity, C decreases considerably with increasing frequency, it is also observed that C decreases considerably with decreasing specific conductivity at a given frequency. Similar observations have been reported for polarizable solid metal/electrolyte interfaces [lo]. This suggests that the frequency dispersion of the interfacial capacitance at semiconductor and metal electrodes has a common origin. In contrast to semiconductor electrodes, the frequency dispersion at metal/electrolyte interfaces is generally attributed to relaxation effects at the electrolyte side of microrough electrodes. In the next section, a model for the frequency dependence of the polarization capacitance of microrough solid electrodes which we propose in another paper [23] is discussed. We believe that this model can also be applied to semiconductor electrodes because of the close resemblance of the experimental results on the frequency dispersion on both types of electrode. Furthermore, numerical calculations based on this model are compared with the experimental results on n-type GaAs electrodes. (IV.2) A model for the frequency dispersion at microrough solid electrodes In ref. 23 we propose a model for the time-dependent build-up of the electric double layer at microrough electrodes. When a potential difference AV is applied to

79

A’/

; Av

I:

I I

Av

a

;Av

3A’/

461

v.0

I I

I I

I I

f Av

t Av

I I

X-+

v=o

I I

Fig. 14. The potential distribution for (a) a flat and (b) a microrough electrode at a very short time after applying a potential difference AV between the working electrode (x = 0) and reference electrode (x = 1). The Hehnholtz plane is situated at x = H. The coordinates x and y refer to directions perpendicular and parallel to the electrode surface, respectively. The arrows give an indication of the magnitude and the direction of the electric field.

a flat electrode, the initial electric field, and hence the polarization current, does not depend on the position at the electrodes surface (Fig. 14a). Consequently, the electric double layer builds up at the same rate at every point of the surface. On microrough electrodes, the initial electric field varies from place to place at the surface, which results in an inhomogeneous migration polarization current (Fig. 14b). Owing to this inhomogeneous primary current distribution [24], the double layer builds up at a different rate at different points of the electrode surface; at protrusions the migration of ions is fast, while it is slow at depressions in the surface. This results in ion concentration gradients not only perpendicular, but also parallel to the surface. This means that at shielded positions the electric double layer is established partly by tangential diffusion. As a consequence of the resulting non-uniform current distribution (called the secondary current distribution [24]), the time needed to reach equilibrium differs from place to place at the electrode surface.

80

For a metal electrode the equilibration time depends on the capacitance of the Hehnholtz layer C, and on the specific conductivity of the electrolyte K. At a flat metal electrode, the relaxation time is very short and frequency dispersion is expected only at very high frequencies [lo]. In the classical equivalent circuit for a metal/electrolyte interface, the impedance of the diffuse double layer 2, is totally capacitive: Z, = (iwCd)-’

(2)

(in which C, is the capacitance of the diffuse double layer). In reality, some energy is also dissipated in the region of the diffuse double layer before the equilibrium situation is settled. A more general expression for the impedance of the diffuse double layer 2, is [25-271: Z,=

[(iw)“A(a)]-l

(O
(3)

The exponent (Y is directly related to the phase angle $: (Y= 2+/7r. For a flat electrode, 1yis equal to 1 and A( a) is the capacitance of the diffuse double layer C,. During equilibration at a microrough electrode, dissipation may differ from point to point depending on the local geometry of the surface. Therefore, the electrode surface is divided into surface elements (index j). The elements have a Helmholtz double-layer capacitance C, identical for all elements. The elements are taken to be sufficiently small so that for a given element j one value of (Ycan be used. Hence, the interfacial impedance of one surface element consists of an impedance Z, in series with the Hehnholtz double-layer impedance (iwC,)-‘: Z,=

[(iw)a’A,(a,)]-l

(OGajG1)

In ref. 23 an attempt is made to define Ai in terms of the electrolyte parameters K, e and c (specific conductivity, dielectric constant and the electrolyte concentration). It is found that Aj(aj) can be written as

A;(a,) = (zJ,(

pi’-=

C,“/&p

In eqn. (5) L, stands for the Debye length of the diffuse double layer. At a microrough surface, the average length over which ions move to form the electric double layer is not necessarily the equilibrium length L,, but is dictated by the geometry around the surface element. Therefore a length L,, dictated by the roughness, is used in the resistive part of eqn. (5). The exponent ‘Yein eqns. (4) and (5) is characteristic for a given surface element and describes the energy dissipation during equilibration. If, for a given surface element, almost no dissipation occurs (as in the case of a flat electrode), aj 2: 1 holds. If the build-up occurs exclusively by tangential diffusion (along equipotential lines), a, = $ is expected. It is clear that the value of aj is determined by the geometry of the direct environment of the surface element. Therefore, a given surface can be described by a probability density function f( olj). This function gives

81

the probability that a given surface element has a value of aj between aj and a, + daj. As a theoretical description of the micro-geometry of an electrode surface in terms of a probability density function is not yet available, we used a j3 function in our numerical calculations. The B function is chosen because f(aj) is zero at a, = 0 and aj = 1 and peaks at an intermediate value of aj [28]:

j-(

aji>= gx-‘(l

where that: j

a and

)(

- ‘yly-’

b are any real positive

numbers

and

N is a normalization

factor

so

(7)

aj) dCY/= 1

The position of the maximum is a measure of the microroughness of the surface and the width of the function is a measure of the uniformity of the roughness. For an almost flat surface, f(cu,) peaks at a value of a, which is almost 1; for a microrough surface, ~(LY,) has a maximum at a value of ‘ye essentially smaller than 1. In our model, the total admittance of a polarizable electrode is calculated by taking the sum of the admittances of the surface elements: r=CI;=x[(iwC)-‘+ j

From eqn. (8) the total polarization impedance may be written as Z=

(8)

[(ii.d)‘llj(cl,)]-l]-’

J

impedance

can be calculated.

(9)

[iwC,(o)]-i+R,(w)

In another paper [23], this model was applied section numerical calculations on semiconductor experimental results on n-type GaAs. (IV.3) Comparison

between the experimental

The resulting

to metal electrodes

electrodes. In the next are compared with our

results and the model

In Section (III), results of impedance measurements on polarizable n-type GaAs electrodes were presented. It was shown that the frequency dispersion of the measured capacitance C is related to the roughness of the surface of the semiconductor electrode: after a pretreatment which provides an almost flat surface, the measured capacitance decreases only slightly with w; after anodic roughening, the decrease of C with o is much more pronounced. It was also shown that the specific conductivity of the electrolyte solution is a key parameter in determining the frequency dispersion: the frequency dispersion of the polarization capacitance decreases with increasing conductivity of the electrolyte. When the electrolyte is replaced by a gold contact (K + co), the frequency dispersion vanishes. Qualitatively, the behaviour of polarizable n-GaAs electrodes is similar to that of polarizable metal electrodes.

82

It must be remarked that under depletion conditions the structure of the double layer at the semiconductor/electrolyte interface is different from that of the metal/electrolyte double layer. At equilibrium, the semiconductor/electrolyte interface consists of a depletion layer in the solid (capacitance C,) in series with the Hehnholtz layer. The double-layer capacitance C, (see Section VI) is not equal to C,, as for metal electrodes, but to C,, C,/(C,, + C,). As under conditions of depletion, C,, -=zC,, the total double-layer capacitance C, is almost equal to C,. Consequently, the purely capacitive impedance (ioCj)-l of semiconductor electrodes is much larger than that of metal electrodes. The influence of the dissipative impedance 2, (see eqn. 5) in the total polarization impedance is therefore expected to be less important. It was indeed observed that the slope of log C vs. log w at roughened n-GaAs electrodes is less negative than that usually observed for roughened metal electrodes. However, caution is recommended when microrough electrodes are compared as “roughness” is difficult to quantify. Of special interest for semiconductor electrodes is the dependence of the measured capacitance on the electrode potential V. It was shown in Section (IV) that plots of C2 vs. V at different frequencies lead to a fan-shaped collection of linear curves. At low values of the electrolyte conductivity, the (Ce2, V) plots do not converge. When the electrolyte conductivity is increased, the frequency dispersion of the (CP2, V) plots is less pronounced and the plots converge at a common point. We have calculated the (C2, V) plots for different frequencies and values of K on the basis of our model. A typical example is presented in Figs. 15 and 16. It is found that the (C2, V) plots are linear at all the frequencies considered (1 rad s-l < w < 10’ rad s-‘). At low electrolyte conductivities (Fig. 15), the (C2, V) plots do not

0

0.6

1.6 V/VCSCE,

Fig. 15. Mott-Schottky plots obtained from numerical calculations for a microrough electrode. Function ~(cI,) peaks at 0.75 and a geometrical length L, =loO nm is chosen. The concentration is 10m4 mol dmU3. Frequency rad s- ‘. lo* (x). , 10’ (0). , lo6 (+); lo4 (0); 10’ (0). At lower frequencies the curves coincide with the lo2 rad SC’ line.

83

as

0

1.6 V/VCSCE,

Fig. 16. Mott-Schottky plots obtained from numerical calculations for a microrough electrode. Function f(a,) peaks at 0.75 and a geometrical length Lo = 100 nm is chosen, The concentration is now increased to 10m3 mol dme3. Frequency rad s- r: 10’ (x); 10’ (+); lo6 (0); lo4 (0). At lower frequencies the curves coincide with the lo4 rad s-r line.

converge at a common point (A-type behaviour); when the electrolyte conductivity is increased, the frequency dispersion becomes less pronounced (Fig. 16) and the (C-*, V) plots converge at a common point on the potential axis (B-type behaviour). When the conductivity is further increased, all (CT*, V) plots almost coincide (see Fig. 17). Quantitatively, the dispersion of the calculated (CT*, V) plots is less pronounced than that observed experimentally. In order to show the qualitative

1.5

lo

=C-‘/F-’

rn4

0 0

0.8

1.6

2.L V/VCSCE,

Fig. 17. Mott-Schottky plots obtained from numerical calculations for a microrough electrode. Function f( a,) peaks at 0.75 and a geometrical length Lo =lOO nm is chosen. The concentration is further increased to 10-r mol dme3. Frequency rad s-r: 10s (X); 1 (0).

84

agreement between the model and the experimental results, the frequency is increased to lo* rad s-’ and the electrolyte concentration is decreased to lop4 mol dmp3. The reason for the quantitative discrepancy between the experimental results and the results from numerical calculations is not yet clear. One might consider factors other than microroughness to account for the experimentally observed frequency dispersion, such as dielectric effects in the solid state [l]. However, the fact that the measured frequency dispersion is so strongly related to the conductivity of the electrolyte and essentially vanishes at polarizable semiconductor/metal interfaces does not favour such an explanation. (V) CONCLUSIONS

The frequency dispersion of the polarization capacitance of n-GaAs/electrolyte solution interfaces depends on the microroughness of the electrode surface and on the specific conductivity of the electrolyte solution. At flat electrodes, almost no dispersion is observed. At microrough electrodes, the dispersion increases with increasing frequency and decreases with increasing electrolyte conductivity; at very high conductivity the frequency dispersion essentially disappears. Similarly, almost no frequency dispersion was found with n-GaAs/Au diodes with a microrough interface. These experimental results resemble closely results found with metal electrodes. It is therefore argued that the dispersion is due to relaxation phenomena at the electrolyte side of the interface. A model which has been used to explain the frequency dispersion of the polarization capacitance of metal electrodes was applied to account for the results. From numerical calculations it can be concluded that the model provides a qualitative explanation for the behaviour observed at rough semiconductor/ electrolyte interfaces and for the absence of frequency dispersion at semiconductor/metal interfaces. ACKNOWLEDGEMENTS

The authors wish to thank Dr. B.A. Boukamp (University of Twente) for making available the fit program EQUIVCRT. They are also grateful to J.E.A.M. van den Meerakker (Philips Research Laboratories, Eindhoven) for help with the Nomarski microscopy and the profilometry experiments. This work was supported by the Netherlands Foundation for Chemical Research (SON), with financial aid from the Netherlands Organization for Scientific Research (NWO). REFERENCES 1 E.C. Dutoit, R.L. Van Meirhaeghe, F. Cardon and W.P. Gomes, Ber. Bunsenges. Phys. Chem., 79 (1975) 1206. 2 W.H. Laflkre, R.L. Van Meirhaeghe, F. Cardon and W.P. Gomes, Surf. Sci., 59 (1976) 401.

85 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

W.H. Laflere, R.L. Van Meirhaeghe, F. Cardon and W.P. Gomes, Surf. Sci., 74 (1978) 125. M.J. Madou, F. Cardon and W.P. Gomes, J. Electrochem. Sot., 124 (1977) 1623. M.J. Madou, K. Kinoshita and M.C.H. McKubre, Electrochim. Acta, 29 (1984) 419. G. Cooper, J.A. Turner and A.J. Nozik, J. Electrochem. Sot., 129 (1982) 1973. J.K. Lyden, M.H. Cohen and M. Tomkiewicz, Phys. Rev. Lett., 47 (1981) 961. N.J. Kiwiet and M.A. Fox, J. Electrochem. Sot., 137 (1990) 561. G. Nogami, J. Electrochem. Sot., 129 (1982) 2219. W. Scheider, J. Phys. Chem., 79 (1975) 127. E. Chassaing, B. Sapoval, G. Daccord and R. Lenormand, J. Electroanal. Chem., 279 (1990) 67. J.B. Bates, Y.T. Chu and W.T. Stribling, Phys. Rev. Lett., 60 (1988) 627. L. Nyikos and P. Pajkossy, Electrochim. Acta, 30 (1985) 1533. R. de Levie, J. Electroanal. Chem., 281 (1990) 1. U. Rammelt and G. Reinhard, Electrochim. Acta, 35 (1990) 1045. F. Mansfeld, S. Lin, Y.C. Chen and H. Shih, J. Electrochem. Sot., 135 (1988) 906. B.A. Boukamp, Solid State Ionics, 20 (1986) 31. R. Reineke and R. Memming, Surf. Sci., 192 (1987) 66. M.M. Faktor, D.G. Fiddyment and M.R. Taylor, J. Electrochem. Sot., 122 (1975) 1566. J.C. Tranchart, L. Hollan and R. Memming, J. Electrochem. Sot., 125 (1978) 1185. A. Yamamoto and S. Yano, J. Electrochem. Sot., 122 (1975) 260. S.R. Morrison, Electrochemistry at Semiconductor and Oxidized Metal Electrodes, Plenum Press, New York, 1980, p. 184. D. Vanmaekelbergh and G. Oskam, J. Phys. Chem., submitted. R. de Levie, Adv. Electrochem. Electrochem. Eng., 6 (1969) 329. K.B. Oldham and J. Spanier, J. Electroanal. Chem., 26 (1970) 331. J.R. MacDonald and W.B. Johnson in J.R. MacDonald (Ed.), Impedance Spectroscopy, Wiley, New York, 1987, p. 17. H. Fricke, Philos. Mag., 14 (1932) 310. M. Zelen and N. Severa in M. Abramowitz and LA. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1970, p. 944.