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Differential capacitance of the P-type silicon—electrolyte solution interface
DIFFERENTIAL CAPACITANCE OF THE P-TYPE SILICON-ELECTROLYTE SOLUTION INTERFACE S. MINCand K. JACKOWSKA Laboratory
of Electrochemistry, Institute of Fundamental Problems of Chemistry, Warsaw University, 02-093 Warszawa, Poland (Received 18 Ocrober 1974; in revisedform 16 May 1975)
Abstract-The results of differential capacitance measurements of P(1I I)-type silicon electrodes (10 n cm) placed in 0.1 M NazS04 solutions with various pH are given in the paper. The capacitance were determined with the pulse techniques in the frequency range from I to 100 kHz. The silicon electrode potential, U, relative to an ace was varied from -0.05 to -0.9 V. It was found that a dispersion of differential capacitance appeared for potentials up to an U, value. For the potentials U < U,, the experimental differential capacitance values are independent of frequency and of potential variation. The experimental results were interpreted hy comparing them with the space charge region capacitance values of silicon.
measurements are one of the commonly used methods of study on the semiconductor-electrolyte solution interface. Silicon in aqueous solution is usually covered with a layer of hydrated oxide the composition of which depends on solution composition and on electrode preparation. The ability of silicon to undergo oxidation makes it troublesome to obtain reproducible experimental results. Work on the differential capacitance of the silicon-electrolyte solution system is sparse. Mostly it concerns measurements carried out at a constant frequency in HF[l,2], NaOH[3,4], and H,SO,[5-71 solutions. Hurd and Wrottenbery[I] found that capacitance of the silicon electrodes was pH dependent. The relationship between capacitance and frequency was studied by Efimov and Jerusalimchik in the silicon/HF solution system[9] and by Izidmov in the silicon/H,SO, solution system[lO]. In our work the dependences of the differential mpacitancc on pH and on frequency for the hydrated oxide covered silicon electrode-electrolyte solution systems have been investigated. The measurements were made with the pulse method in 0.1 M Na2S04 solutions for the frequencies ranging from 1 to 100 kHz. The measured capacitance values were separated into the space charge region capacitance and that of the surface state. The potential distribution at a silicon-electrolyte solution interface, which was previously determined under the same conditions[ll], was used. Some conclusions concerning the surface states on the silicon electrodes were reached from these results.
The ditTerential capacitance
contacts were applied on them electrolytically and the electrode was then insulated with an epoxy resin and with paraffin leaving a free surface of 1 cm’. Before each measurement the electrode was etched in a HF t HNO, (1:3) solution and rinsed with triply distilled water. The capacitance measurements were made at 22-23°C. The capacitance measurements as well as those of the surface photovoltage[l I] were carried out at the same range of the slowly varied potential. The capacitance values were not precise enough for U > 0 (where U denotes difference between Si and the SC?). Apparatus
The system for the pulse measurements of the differential capacitance is presented diagrammatically in Fig. 1. The rectangular current pulse from the generator (G) were fed in between the silicon electrode (1) and a large surface platinum electrode (2). Variations of the studied electrode potential with respect to a platinum wire (3), which appear during the pulse were
EXPERIMENTAL Electrodes (111) oriented P-type silicon electrodes with a lo-Qcm-resistance (acceptor concentration NA u 1.7 x lOI4 cm-‘) were used in the differential capacitance measurements. The electrodes were shaped into 20 x IO x 3 mm cuboids. The ohmic
Fig.
I. Diagram of the apparatus for pulse measurements of differential capacitance.
219
S.
220
MING
and K. JACKOWSKA system (3) was applied.
E
E,,,=E-
1~E-E,lexp~-jl(:
(3)
yielding the expression for E, at t = Ti E,=IR,~l
Ex ---~
-expl-$11;
since Ei = E, + E,. R,, and R, were determined from the formula (1) and (4). The differential C?qXiCihnCe of siliconelectrolyte solution interface was-determined from the time constant (7) of the studied system:
----
r = RZCp, where
R, = ~‘~R’+R~
P Fig. 2. The shape of rectangular current pulse perturbed by a silicon electrode in the potential E us time t coordinates.
recorded by means of an oscilloscope (0). The silicon electrode was polarized relative to the see (4) by a constant voltage feeder (F); its potential was measured against the other see (5) with a tube voltmeter (V). The amplitude of the current pulses was 13 /.L4. Calculation
(4)
ofequivalent
circuit elements
It can be inferred from the analysis of a rectangular current pulse perturbed by a silicon electrode that the silicon-electrolyte solution system may be represented by a simple electric equivalent circuit. The shape of a perturbed pulse is shown in Fig. 2; such a shape can be observed if the Faradaic processes are negligible The electric equivalent circuit of the studied system is shown in Fig. 3. Here R, is a resistance connected in series with the generator. R, is assumed to be the resistance of metallic contacts and of electrolyte solution, R, is the resistance of silicon-electrolyte solution interface, and C, is the differential capacitance of silicon-electrolyte solution interface. The electric elements of this circuit were determined from potential transients during the pulse duration (Fig. 2). In the limiting cases: (a) for 7; = 0, E, = I R,; (b) for T-co,
E = I(R, + R,);
(1) (2)
I-denoting the current value during the pulse. For the intermediate range the transient equation of a RC
Fig. 3. The electric equivalent circuit applied for the determination of silicon-electrolyte solution interface capacitance.
r = C,R,
since
8
; p
R, + R, $ R,.
Thus, the capacitance is given by the formula (5) c =+-exp:-:~l I, RX
(5)
This method can be applied to the capacitance determinations in the range from OQO5to 5 pF/cm’. Accuracy of the measurements decreases with increasing capacitance values. It is controlled by the accuracy of time constant readout and was estimated to be + 10% in the capacitance range from BOO5 to 1 pF/cm’.
RESULTS The differential capacitance was measured with the P( 111) type silicon electrodes of a 10 R cm resistivity, placed in 0.1 M Na,S04 solutions of various pH values: I-2; 26; 6-O; 9.3; 11.1. The measurements were carried out at frequencies : 1, 2, 5, 10, 20, 50, 1oOkHz. The roughness coefficient of silicon equal to 2[1] has been introduced in the calculation of differential capacitance, C, The experimental C, values were plotted vs the CJ potentials of the silicon electrodes against see. The results obtained with @1M Na,SO, solutions (pH = 6.0) are illustrated in Fig. 4. As it follows from the plots shown in Fig. 4, the differential capacitance does not depend on the silicon electrode potential. above a certain Ii value (here denoted by U,). This ekt is observed within the whole frequency range. The differential capacitance C, us LJ and us fmquency plots of the similar character have also been found for the solutions with other pH values, The curves just gradually shift along the U axis as pH value changes (Fig. 5). The dependence of the U, potential on pH values of 01 M Na,SO, solution is presented in Fig. 6. The same, minimum C, value has been obtained for II = U, in all the solutions studied. It amounts OGO6~F/cn?. Monotonic increase of C, (Figs. 4 and 5) suggests that there is no inversion state in the space charge region of silicon in the applied potential range, The analogous shapes of C, vs U curves and similar C,
221
Differential capacitance -0 8
zf z
-0.4 -06 -0
J’
/ 2
.-.A’
PH
Fig. 6. The dependence of the P-type (10 R cm) silicon clcctrode potential U, on pH values, measured in 0.1 M Na,SO_, solutions with respect to a see.
plots have been reported by Hurd and Wrottenbery[X]. The C, values obtained by us are however much lower. The shape of C, vs U curves is different too.
--u [v] Fig. 4. The dependence of differential capacitance, C,, on the P-type (10 fI cm) silicon electrode potential, U, measured in a 0.1 M Na,SO, solution with pH = 6.0 with respect to an see, 0 100 kHz, 0 50 kHz, + 10 kHz, x 2 kHz, 0 1 kHz, the arrow indicates the direction in which the silicon electrode potential was varied.
values were obtained by the ac-method for H,SO, solutions[5,6]. As is seen in Fig. 6 the U,, potential
gradually
shifts
to the more negative values with increasing pH. The relationship between U,, or any other potential U/C,constant/and pH is not linear. Analogous C, us pH
DISCUSSION
The basic problems is the physical significance of the measured capacitance values. The silicon-electrolyte solution interface can be considered as an arrangement of capacitors interconnected in parallel and in series. An equivalent circuit of the silicon electrode is presented in Fig. 7, which corresponds to the generally accepted electric equivalent circuit of the semiconductor electrode[12]. This circuit is applicable where surface state occur at the semiconductor surface and where the Faradaic current may be neglected. The resulting capacitance of the circuit shown in Fig. 7 is given by equation (6): cz
lc,+csslcH
c, + c, + c, ; where
-
C,, denotes the diflerential capacitance of the space charge region, C,, is the differential capacitance of the surface states existing at the silicon electrode, CH is the differential capacitance of the Helmholtz layer.
r
gE
The C, values are considtirably larger than the C, values and at sufficiently low surface state densities are also larger than C,. ‘Thus, C, may be represented by the sum of C, and C,
LY
u” 0.10 008 006 0.04 002 0
0.2
0.4
-”
0 6
0 6
[VI
Fig. 5. The dependence of differential capacitance, C,, on the silicon electrode potential. U. measured in 0.1 M Na,SO,solutionwithvariouspHvalues: 0 1,2;02.6; + 6.0; x 9.3; q 11.1. The frequency-20 kHz.
csc Fig. 7. The equivalent circuit of silicon electrode.
222
S.
MING
and K. JACKOWSKA
The differential capacitance of space charge region was calculated from the relationship (7) given by Myamlin and Pleskov [13]: 2 c,, = V064 ~
l,Z
I I 2kT
a _
a-1
+
a-1
go - ae-yo
I,l(e-yo - 1) + Am‘(eyu- 1) + (A - R-‘)Y01”2
(7)
where q-the electron charge. e-the relative dielectric permittivity of the semiconductor, c0 the dielectric permittivity of vacuum, A-pO/ni, 1 N N,/ni
p,-hole concentration of the bulk semiconductor at equilibrium, n,--electron concentration in the intrinsic semiconductor, Y,-the electrostatic potential of the space charge region in kT/q units; Y, = (q/kT)I cps - (~~1, p,-the inner potential of the semiconductor surface, rp,the inner potential of the bulk semiconductor. All values are given in the SI unit system. The C, values P-type silicon (10 fi cm) were calculated assuming the following parameters: E - 11.5, T - 295°K ni - 0.74 x 10’6m-3, 1 N 2.32 x 104. The Y, parameter was varied in the direction corresponding to the formation of exhaustion and inversion layers in P-type silicon (YO> 0). The theoretical C,, values are the maximal ones corresponding to the pulses of zero frequency. Any dispersion of C,, has been neglected in the discussion of the experimental results. Figure 8 presents the theoretical values of the differential capacitance, C,, plotted vs the (kT/q)Y, po-
9 Yo' w-%[Vl I I I I I III 1,
024
0.04
0 16
I-@36
-056
I I,
-0.76
[VI
NJ
Fig. 8. Dependence of the theoretical differential capacitance values, C,, and C$, of the P-type (IO fi cm) silicon electrodes on (kT/q)Y,;~~-AC,,;.>C$’ ‘(dashed lines). Dependences of the experimental differential capacitance values, C,, of the P-type (10 0 cm) silicon electrodes placed in a 0.1 M Na,SO, solution with pH = 6 on kT/q Y, and on U; l 100 kHz; 0 50 kHz; x 20 kHz; + IO kHz (solid lines).
tential for P-type silicon/lo 0 cm/electrodes. In order to compare the theoretical value of C, with the experimental results, C, should be plotted as a function ofpotential(kT/q)Yo = 1rp,- c,o,,/; this in turn required the knowledge of the dependence of this potential on U. These data were obtained from pulse measurements of a large surface photopotential[ll]. Figure 8 presents also the experimental capacitance values, C, plotted US(kT/q)Y, = 1(p. = cph( potential and USthe U potential. Since, in the semiconductorelectrolyte solution system the change of potential distribution could be expected for tower frequencies[l4], only the C, us U plots for the frequencies ranging from 10 to 100 kHz has been chosen for a comparison of the theoretical and experimental capacitance values. As is seen in Fig. 8 the U values <(U, - 0.1) corresponds to the (kT/q)Y, values at which there should be an inversion state in the space charge region of silicon. The experimental capacitance determined in this potential range is lower than the C, values. On this ground it was concluded that the inversion state does not appear in the space charge region of a P type silicon electrode placed in an electrolyte. This conclusion was confirmed by a good agreement between the experimental and theoretical values which had obtained assuming the exhaustion state to be the only one existing in the space charge region of silicon. In the depletion approximation[ZJ the space charge capacitance is given by (8):
all values being in the SI system. The calculated C$p values are plotted in Fig. 8 as functions as (kT/q)YO. In all the potential range U < U, the experimental C, capacitance is identic with the differential capaci_ tance of the exhaustion layer in silicon. The current has been observed for U < -0.35 V in 01 M Na2S04 with pH values 1.2; 2.6 and for U < -0.5 V in the solution with pH from 6 to 11-2. Thus, it seems probable that the inversion state does not occur on account of an exhaustion of the minority carriers (electron) in course of electrochemical reaction. This behaviour has not been observed in the potential range of -0.05 Z U > U,. As is seen in Fig. 8 for Ii > U,Xthe experimental capacitance values are considerably larger than the theoretical C, and C$ values. The capacitance found in this potential range is determined mainly by the surface state capacitance, c SS The differential capacitance of surface states is related to the variations in occupation of surface levels by electrons. As the potential values approach U,, the surface state capacitance tends to zero. Thus, the potential the should correspond ~?/q)Y~ = 1p, - (P,,)potential value at whichothe occupation of surface levels is invariant. Although the experimental U,, values depend on pH, the corresponding (kT/q)Y, = 1rp, - tph1should be independent of this parameter. The Ii’,, value (in the frequency range from 10 to 100kHz) and the corresponding (kT/q)Yo = 1‘pA- ,pnI values for 0.1 M Na,SO, solutions of various pH are
Differential capacitance Table 1. Potentials
of silicon electrode placed in 0.1 M Na,SO, solutions
IJSSOI) lC%lO kHz
&T/g) yo (V)
6.0 9.3
-025 -025 -@30 -0.50
034 0.40 037 0.35
11.1
-0.65
0.33
PH ::;
collected in Table 1. The Y. values differs of about three kT/q units in the applied pH range. Figure 9 presents the experimental values of log C, plotted us log f for various U and pH values. The C, andfvalues are given in pF/cm’ and kHz respcctively. As is seen, the dependences are linear in the frequency range from I kHz to IO kHz. In this range the relationship between C, andfis given by the formula :
c, = uf”
06
IO log
Harten[17] and Memming[ 181 have shown using the surface recombination velocity technique, that recombination centres for fast surface states exist at the silicon -electrolyte interface. Basing on the experimentally observed dependence of C, onfone can suppose that there are also surface states with non-monoenergetic levels. Summarizing: (a) the capacitance C, of the silicon-elecctrolyte interface is the sum of space charge region capacitance, that of the surface states, for and C, -0.05 2 II > u,,; (b) C, is identical with capacitance of the exhaustion layer in silicon when U < I/,,; (c) the parameter Ui, corresponds to the potential $;;vsia;t which occupation of the surface levels (d) the erperimentally observed dispersion of C, can not be described by the formulas applied to the electrodes with monoenergetic surface levels.
REFERENCES
where K and a are the parameters dependent on U and pH. The K values ranges from 0.05 to 05. The deviation from linearity occurs for f > 10 kHz. The experimentally observed dispersion of the differential capacitance C, can not be described by the formula concerning the electrodes with monoenergetic surface levels[l3, 15, 161.
02
223
14
18
F
Fig. 9. The dependence of differential capacitance (log Cp) on frequency (log fl for various U values measured in 0.1 M Na,SO, solutions. pH values: 0 1.2; 0 6.0; + 9-3: x 11.1.
I. R. Memming and G. Schwandt, Surf Sci. 5, 97 (1966). 2. R. L. Meek, Surf: Sci. 25, 526 (1971). 3. M. Z. Seipt, Z. Naturf A 14, 926 (1959). 4. S. 0. Izidinov and A. N. Blokhina, Elektrokhimiya 8, 1138 (1973). 5. E. A. Efimov and I. G. Jerusalimchik, Dokl. &ad. Nauk SSSR 124, 609 (1959). 6. W. I. Zwiagin and A. S. Liutowicz, Elektrokhimiya Germania and Kremniu (wited by E. A. Efimov and I. G. Jerusahmchik), p, 16. Moscov (1963). 7. H. Gobrecht and 0. Meinhardt, BPT. Brmsunges. phys. Chem. 67, 142 (1963). 8. R. M. Hurd and P. T. Wrotenbery, Ann. N.Y. Acud. Sci. 101, 876 (I 963). 9. E. A. Efimov and I. G. Jerusalimchik. Zh. jz. Khim. 35, 384 (1961). 10. S. 0. Izidinov, Elektrokhimiya 4, 1027 (1968). 11. K. Jackowska, Electrochim. Acta Xl, 27 (1975). 12. H. Gerischer, Advances in Electrochemical and Electrochemical Engineering (Edited by P. Delahay), Vol. 1. Interscience, New York (1961). 13. V. A. Myamlin and Y. V. Pleskov, Electrochrmistry of Srmicowluctors, Plenum Press, New York (1967). 14. H. Gobrecht, M. Schaldach, F. Hein, R. Diaser and H. G. Wagemann, Ber. Bunsenges, phys. Chem. 70, 646 (1966). 15. H. Gobrecht and 0. Meinhardt, I+. Bunxng-nyra.phys. Chem. 67, 151 (1963). 16. H. Gerischer, Physical Chemistry. (Edited by H. Eyring, D. Henderson and W. Jest), Vol. 9A. Academic Presr, New York (1970). 17. H. U. Harten, Z. &turf: A. 16, 459 (1961). IX. R. Memming, Surf: Sci. 1, S8 (1964).
of Electrochemistry, Institute of Fundamental Problems of Chemistry, Warsaw University, 02-093 Warszawa, Poland (Received 18 Ocrober 1974; in revisedform 16 May 1975)
Abstract-The results of differential capacitance measurements of P(1I I)-type silicon electrodes (10 n cm) placed in 0.1 M NazS04 solutions with various pH are given in the paper. The capacitance were determined with the pulse techniques in the frequency range from I to 100 kHz. The silicon electrode potential, U, relative to an ace was varied from -0.05 to -0.9 V. It was found that a dispersion of differential capacitance appeared for potentials up to an U, value. For the potentials U < U,, the experimental differential capacitance values are independent of frequency and of potential variation. The experimental results were interpreted hy comparing them with the space charge region capacitance values of silicon.
measurements are one of the commonly used methods of study on the semiconductor-electrolyte solution interface. Silicon in aqueous solution is usually covered with a layer of hydrated oxide the composition of which depends on solution composition and on electrode preparation. The ability of silicon to undergo oxidation makes it troublesome to obtain reproducible experimental results. Work on the differential capacitance of the silicon-electrolyte solution system is sparse. Mostly it concerns measurements carried out at a constant frequency in HF[l,2], NaOH[3,4], and H,SO,[5-71 solutions. Hurd and Wrottenbery[I] found that capacitance of the silicon electrodes was pH dependent. The relationship between capacitance and frequency was studied by Efimov and Jerusalimchik in the silicon/HF solution system[9] and by Izidmov in the silicon/H,SO, solution system[lO]. In our work the dependences of the differential mpacitancc on pH and on frequency for the hydrated oxide covered silicon electrode-electrolyte solution systems have been investigated. The measurements were made with the pulse method in 0.1 M Na2S04 solutions for the frequencies ranging from 1 to 100 kHz. The measured capacitance values were separated into the space charge region capacitance and that of the surface state. The potential distribution at a silicon-electrolyte solution interface, which was previously determined under the same conditions[ll], was used. Some conclusions concerning the surface states on the silicon electrodes were reached from these results.
The ditTerential capacitance
contacts were applied on them electrolytically and the electrode was then insulated with an epoxy resin and with paraffin leaving a free surface of 1 cm’. Before each measurement the electrode was etched in a HF t HNO, (1:3) solution and rinsed with triply distilled water. The capacitance measurements were made at 22-23°C. The capacitance measurements as well as those of the surface photovoltage[l I] were carried out at the same range of the slowly varied potential. The capacitance values were not precise enough for U > 0 (where U denotes difference between Si and the SC?). Apparatus
The system for the pulse measurements of the differential capacitance is presented diagrammatically in Fig. 1. The rectangular current pulse from the generator (G) were fed in between the silicon electrode (1) and a large surface platinum electrode (2). Variations of the studied electrode potential with respect to a platinum wire (3), which appear during the pulse were
EXPERIMENTAL Electrodes (111) oriented P-type silicon electrodes with a lo-Qcm-resistance (acceptor concentration NA u 1.7 x lOI4 cm-‘) were used in the differential capacitance measurements. The electrodes were shaped into 20 x IO x 3 mm cuboids. The ohmic
Fig.
I. Diagram of the apparatus for pulse measurements of differential capacitance.
219
S.
220
MING
and K. JACKOWSKA system (3) was applied.
E
E,,,=E-
1~E-E,lexp~-jl(:
(3)
yielding the expression for E, at t = Ti E,=IR,~l
Ex ---~
-expl-$11;
since Ei = E, + E,. R,, and R, were determined from the formula (1) and (4). The differential C?qXiCihnCe of siliconelectrolyte solution interface was-determined from the time constant (7) of the studied system:
----
r = RZCp, where
R, = ~‘~R’+R~
P Fig. 2. The shape of rectangular current pulse perturbed by a silicon electrode in the potential E us time t coordinates.
recorded by means of an oscilloscope (0). The silicon electrode was polarized relative to the see (4) by a constant voltage feeder (F); its potential was measured against the other see (5) with a tube voltmeter (V). The amplitude of the current pulses was 13 /.L4. Calculation
(4)
ofequivalent
circuit elements
It can be inferred from the analysis of a rectangular current pulse perturbed by a silicon electrode that the silicon-electrolyte solution system may be represented by a simple electric equivalent circuit. The shape of a perturbed pulse is shown in Fig. 2; such a shape can be observed if the Faradaic processes are negligible The electric equivalent circuit of the studied system is shown in Fig. 3. Here R, is a resistance connected in series with the generator. R, is assumed to be the resistance of metallic contacts and of electrolyte solution, R, is the resistance of silicon-electrolyte solution interface, and C, is the differential capacitance of silicon-electrolyte solution interface. The electric elements of this circuit were determined from potential transients during the pulse duration (Fig. 2). In the limiting cases: (a) for 7; = 0, E, = I R,; (b) for T-co,
E = I(R, + R,);
(1) (2)
I-denoting the current value during the pulse. For the intermediate range the transient equation of a RC
Fig. 3. The electric equivalent circuit applied for the determination of silicon-electrolyte solution interface capacitance.
r = C,R,
since
8
; p
R, + R, $ R,.
Thus, the capacitance is given by the formula (5) c =+-exp:-:~l I, RX
(5)
This method can be applied to the capacitance determinations in the range from OQO5to 5 pF/cm’. Accuracy of the measurements decreases with increasing capacitance values. It is controlled by the accuracy of time constant readout and was estimated to be + 10% in the capacitance range from BOO5 to 1 pF/cm’.
RESULTS The differential capacitance was measured with the P( 111) type silicon electrodes of a 10 R cm resistivity, placed in 0.1 M Na,S04 solutions of various pH values: I-2; 26; 6-O; 9.3; 11.1. The measurements were carried out at frequencies : 1, 2, 5, 10, 20, 50, 1oOkHz. The roughness coefficient of silicon equal to 2[1] has been introduced in the calculation of differential capacitance, C, The experimental C, values were plotted vs the CJ potentials of the silicon electrodes against see. The results obtained with @1M Na,SO, solutions (pH = 6.0) are illustrated in Fig. 4. As it follows from the plots shown in Fig. 4, the differential capacitance does not depend on the silicon electrode potential. above a certain Ii value (here denoted by U,). This ekt is observed within the whole frequency range. The differential capacitance C, us LJ and us fmquency plots of the similar character have also been found for the solutions with other pH values, The curves just gradually shift along the U axis as pH value changes (Fig. 5). The dependence of the U, potential on pH values of 01 M Na,SO, solution is presented in Fig. 6. The same, minimum C, value has been obtained for II = U, in all the solutions studied. It amounts OGO6~F/cn?. Monotonic increase of C, (Figs. 4 and 5) suggests that there is no inversion state in the space charge region of silicon in the applied potential range, The analogous shapes of C, vs U curves and similar C,
221
Differential capacitance -0 8
zf z
-0.4 -06 -0
J’
/ 2
.-.A’
PH
Fig. 6. The dependence of the P-type (10 R cm) silicon clcctrode potential U, on pH values, measured in 0.1 M Na,SO_, solutions with respect to a see.
plots have been reported by Hurd and Wrottenbery[X]. The C, values obtained by us are however much lower. The shape of C, vs U curves is different too.
--u [v] Fig. 4. The dependence of differential capacitance, C,, on the P-type (10 fI cm) silicon electrode potential, U, measured in a 0.1 M Na,SO, solution with pH = 6.0 with respect to an see, 0 100 kHz, 0 50 kHz, + 10 kHz, x 2 kHz, 0 1 kHz, the arrow indicates the direction in which the silicon electrode potential was varied.
values were obtained by the ac-method for H,SO, solutions[5,6]. As is seen in Fig. 6 the U,, potential
gradually
shifts
to the more negative values with increasing pH. The relationship between U,, or any other potential U/C,constant/and pH is not linear. Analogous C, us pH
DISCUSSION
The basic problems is the physical significance of the measured capacitance values. The silicon-electrolyte solution interface can be considered as an arrangement of capacitors interconnected in parallel and in series. An equivalent circuit of the silicon electrode is presented in Fig. 7, which corresponds to the generally accepted electric equivalent circuit of the semiconductor electrode[12]. This circuit is applicable where surface state occur at the semiconductor surface and where the Faradaic current may be neglected. The resulting capacitance of the circuit shown in Fig. 7 is given by equation (6): cz
lc,+csslcH
c, + c, + c, ; where
-
C,, denotes the diflerential capacitance of the space charge region, C,, is the differential capacitance of the surface states existing at the silicon electrode, CH is the differential capacitance of the Helmholtz layer.
r
gE
The C, values are considtirably larger than the C, values and at sufficiently low surface state densities are also larger than C,. ‘Thus, C, may be represented by the sum of C, and C,
LY
u” 0.10 008 006 0.04 002 0
0.2
0.4
-”
0 6
0 6
[VI
Fig. 5. The dependence of differential capacitance, C,, on the silicon electrode potential. U. measured in 0.1 M Na,SO,solutionwithvariouspHvalues: 0 1,2;02.6; + 6.0; x 9.3; q 11.1. The frequency-20 kHz.
csc Fig. 7. The equivalent circuit of silicon electrode.
222
S.
MING
and K. JACKOWSKA
The differential capacitance of space charge region was calculated from the relationship (7) given by Myamlin and Pleskov [13]: 2 c,, = V064 ~
l,Z
I I 2kT
a _
a-1
+
a-1
go - ae-yo
I,l(e-yo - 1) + Am‘(eyu- 1) + (A - R-‘)Y01”2
(7)
where q-the electron charge. e-the relative dielectric permittivity of the semiconductor, c0 the dielectric permittivity of vacuum, A-pO/ni, 1 N N,/ni
p,-hole concentration of the bulk semiconductor at equilibrium, n,--electron concentration in the intrinsic semiconductor, Y,-the electrostatic potential of the space charge region in kT/q units; Y, = (q/kT)I cps - (~~1, p,-the inner potential of the semiconductor surface, rp,the inner potential of the bulk semiconductor. All values are given in the SI unit system. The C, values P-type silicon (10 fi cm) were calculated assuming the following parameters: E - 11.5, T - 295°K ni - 0.74 x 10’6m-3, 1 N 2.32 x 104. The Y, parameter was varied in the direction corresponding to the formation of exhaustion and inversion layers in P-type silicon (YO> 0). The theoretical C,, values are the maximal ones corresponding to the pulses of zero frequency. Any dispersion of C,, has been neglected in the discussion of the experimental results. Figure 8 presents the theoretical values of the differential capacitance, C,, plotted vs the (kT/q)Y, po-
9 Yo' w-%[Vl I I I I I III 1,
024
0.04
0 16
I-@36
-056
I I,
-0.76
[VI
NJ
Fig. 8. Dependence of the theoretical differential capacitance values, C,, and C$, of the P-type (IO fi cm) silicon electrodes on (kT/q)Y,;~~-AC,,;.>C$’ ‘(dashed lines). Dependences of the experimental differential capacitance values, C,, of the P-type (10 0 cm) silicon electrodes placed in a 0.1 M Na,SO, solution with pH = 6 on kT/q Y, and on U; l 100 kHz; 0 50 kHz; x 20 kHz; + IO kHz (solid lines).
tential for P-type silicon/lo 0 cm/electrodes. In order to compare the theoretical value of C, with the experimental results, C, should be plotted as a function ofpotential(kT/q)Yo = 1rp,- c,o,,/; this in turn required the knowledge of the dependence of this potential on U. These data were obtained from pulse measurements of a large surface photopotential[ll]. Figure 8 presents also the experimental capacitance values, C, plotted US(kT/q)Y, = 1(p. = cph( potential and USthe U potential. Since, in the semiconductorelectrolyte solution system the change of potential distribution could be expected for tower frequencies[l4], only the C, us U plots for the frequencies ranging from 10 to 100 kHz has been chosen for a comparison of the theoretical and experimental capacitance values. As is seen in Fig. 8 the U values <(U, - 0.1) corresponds to the (kT/q)Y, values at which there should be an inversion state in the space charge region of silicon. The experimental capacitance determined in this potential range is lower than the C, values. On this ground it was concluded that the inversion state does not appear in the space charge region of a P type silicon electrode placed in an electrolyte. This conclusion was confirmed by a good agreement between the experimental and theoretical values which had obtained assuming the exhaustion state to be the only one existing in the space charge region of silicon. In the depletion approximation[ZJ the space charge capacitance is given by (8):
all values being in the SI system. The calculated C$p values are plotted in Fig. 8 as functions as (kT/q)YO. In all the potential range U < U, the experimental C, capacitance is identic with the differential capaci_ tance of the exhaustion layer in silicon. The current has been observed for U < -0.35 V in 01 M Na2S04 with pH values 1.2; 2.6 and for U < -0.5 V in the solution with pH from 6 to 11-2. Thus, it seems probable that the inversion state does not occur on account of an exhaustion of the minority carriers (electron) in course of electrochemical reaction. This behaviour has not been observed in the potential range of -0.05 Z U > U,. As is seen in Fig. 8 for Ii > U,Xthe experimental capacitance values are considerably larger than the theoretical C, and C$ values. The capacitance found in this potential range is determined mainly by the surface state capacitance, c SS The differential capacitance of surface states is related to the variations in occupation of surface levels by electrons. As the potential values approach U,, the surface state capacitance tends to zero. Thus, the potential the should correspond ~?/q)Y~ = 1p, - (P,,)potential value at whichothe occupation of surface levels is invariant. Although the experimental U,, values depend on pH, the corresponding (kT/q)Y, = 1rp, - tph1should be independent of this parameter. The Ii’,, value (in the frequency range from 10 to 100kHz) and the corresponding (kT/q)Yo = 1‘pA- ,pnI values for 0.1 M Na,SO, solutions of various pH are
Differential capacitance Table 1. Potentials
of silicon electrode placed in 0.1 M Na,SO, solutions
IJSSOI) lC%lO kHz
&T/g) yo (V)
6.0 9.3
-025 -025 -@30 -0.50
034 0.40 037 0.35
11.1
-0.65
0.33
PH ::;
collected in Table 1. The Y. values differs of about three kT/q units in the applied pH range. Figure 9 presents the experimental values of log C, plotted us log f for various U and pH values. The C, andfvalues are given in pF/cm’ and kHz respcctively. As is seen, the dependences are linear in the frequency range from I kHz to IO kHz. In this range the relationship between C, andfis given by the formula :
c, = uf”
06
IO log
Harten[17] and Memming[ 181 have shown using the surface recombination velocity technique, that recombination centres for fast surface states exist at the silicon -electrolyte interface. Basing on the experimentally observed dependence of C, onfone can suppose that there are also surface states with non-monoenergetic levels. Summarizing: (a) the capacitance C, of the silicon-elecctrolyte interface is the sum of space charge region capacitance, that of the surface states, for and C, -0.05 2 II > u,,; (b) C, is identical with capacitance of the exhaustion layer in silicon when U < I/,,; (c) the parameter Ui, corresponds to the potential $;;vsia;t which occupation of the surface levels (d) the erperimentally observed dispersion of C, can not be described by the formulas applied to the electrodes with monoenergetic surface levels.
REFERENCES
where K and a are the parameters dependent on U and pH. The K values ranges from 0.05 to 05. The deviation from linearity occurs for f > 10 kHz. The experimentally observed dispersion of the differential capacitance C, can not be described by the formula concerning the electrodes with monoenergetic surface levels[l3, 15, 161.
02
223
14
18
F
Fig. 9. The dependence of differential capacitance (log Cp) on frequency (log fl for various U values measured in 0.1 M Na,SO, solutions. pH values: 0 1.2; 0 6.0; + 9-3: x 11.1.
I. R. Memming and G. Schwandt, Surf Sci. 5, 97 (1966). 2. R. L. Meek, Surf: Sci. 25, 526 (1971). 3. M. Z. Seipt, Z. Naturf A 14, 926 (1959). 4. S. 0. Izidinov and A. N. Blokhina, Elektrokhimiya 8, 1138 (1973). 5. E. A. Efimov and I. G. Jerusalimchik, Dokl. &ad. Nauk SSSR 124, 609 (1959). 6. W. I. Zwiagin and A. S. Liutowicz, Elektrokhimiya Germania and Kremniu (wited by E. A. Efimov and I. G. Jerusahmchik), p, 16. Moscov (1963). 7. H. Gobrecht and 0. Meinhardt, BPT. Brmsunges. phys. Chem. 67, 142 (1963). 8. R. M. Hurd and P. T. Wrotenbery, Ann. N.Y. Acud. Sci. 101, 876 (I 963). 9. E. A. Efimov and I. G. Jerusalimchik. Zh. jz. Khim. 35, 384 (1961). 10. S. 0. Izidinov, Elektrokhimiya 4, 1027 (1968). 11. K. Jackowska, Electrochim. Acta Xl, 27 (1975). 12. H. Gerischer, Advances in Electrochemical and Electrochemical Engineering (Edited by P. Delahay), Vol. 1. Interscience, New York (1961). 13. V. A. Myamlin and Y. V. Pleskov, Electrochrmistry of Srmicowluctors, Plenum Press, New York (1967). 14. H. Gobrecht, M. Schaldach, F. Hein, R. Diaser and H. G. Wagemann, Ber. Bunsenges, phys. Chem. 70, 646 (1966). 15. H. Gobrecht and 0. Meinhardt, I+. Bunxng-nyra.phys. Chem. 67, 151 (1963). 16. H. Gerischer, Physical Chemistry. (Edited by H. Eyring, D. Henderson and W. Jest), Vol. 9A. Academic Presr, New York (1970). 17. H. U. Harten, Z. &turf: A. 16, 459 (1961). IX. R. Memming, Surf: Sci. 1, S8 (1964).