5543
J . Phys. Chem. 1989, 93, 5543-5548
spherical micelles. Figure 5 shows V2as a function of CTAOTOS concentration. V, (456 mL/mol) remains constant up to a critical concentration and then decreases dramatically. That concentration is well above the first cmc and in the range of the sphere-to-rod transition (Figure 4 and Table 111). The drastic decrease in V, can be explained by the sphere-to-rod transition. Assuming spherical micelles (Dm) are in equilibrium with cylindrical micelles (Dmm),lo Le. nDm
Figure 6. Absorbance of CTAOTOS as a function of its concentration.
increase in relative viscosities has also been interpreted in terms of the overlap of rodlike but it is hard by now to decide which is the actual mechanism of micellar growth.2s Partial Molar Volumes. Besides viscosity increases, changes in partial molar volumes are also expected because monomers in cylindrical micelles should be more closely packed than those in (24) Hoffman, H. Ber. Bunsen-Ges. Phys. Chem. 1984,88, 1078. (25) Sakaiguchi, Y.; Shikata, T.; Urakami, H.; Tamura, A.; Hirata, H. Colloid Polym. Sci. 1987, 265, 750.
* (Dmm)
(7)
it can be expected that the transition from spherical to rod micelles might be accompanied by an expulsion of water molecules associated to the hydrocarbon chains which have a higher molecular volume (icelike) than bulk As CTAOTOS concentration increases, the equilibrium (7) is displaced to the right, producing more cylindrical micelles, with the corresponding viscosity increment and V2 decrease. Additional experimental evidence for the sphereto-rod transition of CTAOTOS is provided by the break in the absorbanceCTAOTOS concentration profiles (Figure 6), in the region where the transition is expected to occur. These changes, which indicate different molar absorptivities in the spherical and cylindrical micelles, could be interpreted as a change in the configuration of the benzene ring of tosylate when sorbed by either spherical or cylindrical micelles. Registry NO. CTAOBS, 17322-67-7; CTAOTOS, 138-32-9; CTAOEBS, 120525-93-1; CTAOIBS, 120525-94-2; IBS-, 6214-18-2; EBS-, 18777-64-5; TOS-, 16722-51-3; BS-, 3198-32-1. (26) Tanford, C. The Hydrophobic Effect; Formation of Micelles and Biological Membranes; Wiley: New York, 1973. (27) Nemethy, G.;Scheraga, H. A. J. Chem. Phys. 1%2,36, 3401,3382. (28) Paredes, S.; Tribout, M.; Sepiilveda, L. J. Phys. Chem. 1984, 88, 1871.
Dynamic Photocapacitance Measurement of ZnO Photoelectrode in Aqueous Solution Seiichiro Nakabayashi, Akira Kira,* The Institute of Physical and Chemical Research (RIKEN), Wakoshi, Saitama 351 -01, Japan
and Masamichi Ipponmatsu Osaka Gas Co. Ltd., Konohana-ku, Osaka 554, Japan (Received: November 21, 1988; In Final Form: February 15. 1989)
A dynamic photocapacitance method using the pulsed electrode potential was developed for measurement of the charge concentration of photoinduced deep traps and their recombination rates along the depth of the electrode. The validity of this method to bulk states was demonstrated by its application to a deep intervalence state which was found at 2.5 eV below the conduction band. Its absorption constant was estimated at 7.7 X cm-l on the assumption of a quantum yield of unity. The trap produced from the deep state by photoexcitation is distributed inside the crystal with a concentration of I O l 4 cm-'; thus, the observed are bulk states. A cross section of trap recombination with electrons was 3.6 X cm2, which suggests that the photoinduced trap is neutral and accordingly the original intervalence state is anionic. Disturbance by surface states was observed in the measurements of the deep state.
Introduction Intervalence states including surface states are regarded as playing an important role in electron transfer at semiconductor/electrolyte interfaces, but characterization of these states in real electrodes still remains as a problem. Bard'., and Tomkiewicz)" have studied the intervalence state of semiconductor (1) Nagasubramanian, G.;Wheeler, B. L.; Hope, G. A.; Bard, A. J. J . Electrochem. Soc. 1983, 130, 385. (2) Nagasubramanian, G.; Wheeler, B. L.; Bard, A. J. J . Electrochem. SOC.1983, 130, 1680.
0022-3654/89/2093-5543$01.50/0
electrodes by means of the frequency dispersion of electrode impedance and/or admittance. Wilson and have tried to investigate the surface state from the scan-rate dispersion of the electrochemical current and voltage curves. Photoelectrochemical _ _ _ _ _ _ _ _ _ _ _ _ ~ _ _ _ _ _ _ _ _ _ _
~~
~
(3) Tomkiewicz, M. J. Electrochem. SOC.1979, 126, 2220. (4) Tomkiewicz, M. J . Electrochem. Soc. 1979, 126, 1505. (5) Tomkiewicz, M. J . Electrochem. Soc. 1980, 127, 1518. (6) Shen, W. M.; Tomkiewicz, M.; Cahen, D. J. Electrochem. SOC.1986, 133, 112. (7) Wilson, R. H. J. Electrochem. Soc. 1980, 127, 288. (8) Sagara, T.; Sukigara, M.J. Electrochem. SOC.1988, 135, 363.
0 1989 American Chemical Society
5544
The Journal of Physical Chemistry, Vol. 93, No. 14. 1989 A 0 Modulator
\
Ar i o n b i e r
Nakabayashi et al. I
/-b---
I
I
I
I
a /
5 14.5nm
P
b
Figure 1. An experimental setup.
capacitance spectroscopy"* is also used for measurement of energy levels of intervalence states. The dynamic measurement to reveal trap distributions and recombination rates in solid/solid junctions has been developed by Kukimoto et al.13J4and also by Okushi et a1.I5 W e have recently applied the isothermal capacitance transient capacitance spectroscopy developed by Okushi to a photoelectrode system.16 The present paper describes a dynamic photocapacitance study of a deep intervalence state at a ZnO electrode in aqueous solution. All the description henceforth will refer implicitly to n-type semiconductors to which ZnO is classified. The change in the charge due to the trap formation in the space charge layer was measured in the form of the capacitance change. Traps are generated and accumulated by photoillumination under anodically biased conditions and then their recombination with electrons is enhanced under the control of the pulsed electrode potential. The basic idea of this technique has been demonstrated in previous studies of solid junction^.'^-'^ The fundamental relations concerning the capacitance of the space charge layer will be summarized. At a sufficiently high frequency at which intervalence traps cannot follow the perturbation, the capacitance can be described by an equation for a parallel plate c o n d e n ~ e r ~ ~ J l J ~ C = totA/w
(1)
where is the dielectric constant and w is the space charge layer thickness. Equation 1 immediately leads to A C / C = -Aw/w
(2)
If the charge density, N , in the space charge layer increases by AN under the same potential, both the thickness and the capacitance will change according to N w / C = ( N + AN)(w
+ A w ) / ( C + AC)
(3)
From eq 2 and 3 A C / C = AN/2N
(4)
(9) Haak, R.; Ogden, C.; Tench, D. J . Electrochem. Soc. 1982,129,891. (10) Haak, R.; Tench, D. J . Electrochem. Soc. 1984, 131, 275. (11) Haak, R.; Tench, D. J . Elecfrochem.Soc. 1984, 131, 1442. (12) Allongue, P.; Cachet, H. Ber. Bunsen-Ges.Phys. Chem. 1987, 91,386. (13) Kukimoto, H.; Henry, C. H.; Merritt, F. R. Phys. Rev. 1973,7, 2486. (14) Henry, C. H.; Kukimoto, H.; Miller, G. L.; Merritt, F. R. Phys. Rev. B 1973. 7. 2499. (15)'0kushi, H.; Tokumaru, Y.; Yamasaki, S.; Oheda, H.; Tanaka, K. Phys. Rev. B 1982, 25, 43 13. (16) Nakabayashi, S.; Kira, A. J . Phys. Chem. 1987, 91, 4660. (17) Lang, D. V. J . Appl. Phys. 1974, 45, 3014. Lang, D. V. J. Appl. Phys. 1974, 45, 3023.
-1.0
0
1.0
2.0
Electrode potontlal VS. Ct I volt
Figure 2. Mott-Schottky plots for a ZnO electrode. Frequencies of measurement: (a) 1 MHz, (b) 200 kHz, and (c) 10 kHz.
where N >> N V is assumed. Since the concentration of the donor vacancy is regarded as constant, N can be practically replaced by the donor concentration in most cases and the change in the capacitance can be ascribed to the change in the concentration of charged deep states that are not ionized by the high-frequency perturbation. Oxidation of a deep state makes ANpositive, and reduction makes it negative. It should be noted that the above relations are derived on the implicit assumption that no Fermi level pinning occurs. Experimental Section The experimental setup is schematically depicted in Figure 1. It consists of illumination sources and a capacitance meter (see below) with a pulsed potential control. For illumination a 150-W xenon arc lamp and a Jovan-Yvon H-20 monochrometer were used with appropriate filters for stationary spectroscopic measurements, and a Lexel Model-95 argon ion laser (4 W) was used for dynamic capacitance measurements. The photoelectrochemical cell was specially designed for the fast capacitance measurement, as described elsewhere.16 The capacitance change was measured as a function of time with a Boonton 72 B capacitance meter, which analyzes the impedance for 1 MHz with a time resolution of 50 ~ s A. Hewlett-Packard 3314A function generator was employed for the pulse control of the electrode potential. The laser light was modulated by a Hoya A-140 acousto-optic modulator and the timing between the electrode potential pulse and the laser irradiation was controlled by a Stanford Research System DG 535 digital delay and gate generator. The photocapacitance signals were recorded on a Leader LBO-5825 digital storage oscilloscope. Other conventional electrochemical measurements were conducted by a Toho-Giken 2020 potentiostat, 2230 function generator, and a Yokogawa Hewlett-Packard 41 92A low-frequency impedance analyzer. A ZnO single crystal was donated by Professor G. Heiland, Rheinische-Westfaelische Technische Hochschule, Germany. The working electrode was constructed so that light reaches the c plane of the crystal through the electrolyte solution. An In/Ga ohmic contact was used for connection. The electrode area was limited to ca. cm2 for measurement a t 1 MHz; the area differed slightly in each preparation. A platinum disk of 1 cm in diameter was used for the counter and reference electrode. The aqueous electrolyte solution contained sodium perchlorate at 2 M and sulfuric acid to adjust pH to 2.
The Journal of Physical Chemistry, Vol. 93, No. 14, 1989 5545
Dynamic Photocapacitance of ZnO I
I
I
I
7.5 -
. p
5.0
I- 1
I
1
1
c P
5.0
i
-
\. . 8
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e
2
a
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3.0 -
0
a
1 1
2.5-
1.
2.0-
U
0
c a
L
1.0 0
30 Time I
60
90
Figure 3. Typical rise and decay profile of the photocapacitance. The right side of the curve shows the removal of a residual capacitance induced by an cathodic electrode potential pulse.
(1 8) Gerischer, H. Physical Chemistry: An Advanced Treatise; Eyring, H., Henderson, D., Jost, W., Eds.; Academic Press: New York, 1970; Vol. 9A.
(19) Bard, A. J.; Faulkner, L. R. Electrochemical Methods; Wiley: New York, 1980. (20) Mollwo, E. ‘Zinc oxide”. In Physics of 2-6 and 2-7 Compounds, Semimagnetic Semiconductors; Hellwege, K. H., Editor-in-Chief; Madelung, 0.Ed.Landort-Boernstein,New Series, Springer-Verlag: New York, 1982; Vol. 17, p 35.
c
.
f. ’r ‘f
s,
I
see
Results Photocapacitance Spectrum. The Mott-Schottky p10ts’~J~ shown in Figure 2 were taken for a ZnO electrode in contact with the electrolute solution. The plots give a value of -0.4 V vs Pt for the flat-band potential, Elb. The straight line at 1 MHz indicates that the total experimental system works normally at this frequency. The slope of the 1 MHz line shows that the donor concentration of the electrode, Nd, is 1.77 X 10l6~ m - ~The . slope depends on the frequency, as seen in Figure 2, but the resultant deviation in the concentration is less than 20%. On illumination with 514.5-nm steady light from an argon ion laser at an electrode potential of 3.4 V vs E%, the rapid increase in the photocapacitance is followed by a slow one converging to a photostationary value, as shown in Figure 3. The band gap energy of ZnO is 3.2 eV,20 which is higher than the excitation energy; therefore, this photocapacitance relates to an intervalence state. The positive photocapacitance (AC > 0) indicates that the intervalence state is photooxidized: in other words, photoinduced electron transition from an intervalence state to the conduction band is responsible for this photocapacitance. The photocapacitance would be negative if the electron transition from the valence band to an intervalence state were concerned. As also shown in Figure 3, the photocapacitance decays spontaneously after the light is cut off and is removed by application of the flat-band potential for a certain duration. The removal of the photocapacitance is due to the recombination of the oxidized form with electrons coming to the region of the space charge layer for a potential of 3.4 V vs E,+,. Figure 4 shows an action spectrum of the photocapacitance taken a t the photostationary state. The spectrum below 3.2 eV is assigned to the intervalence state, whose energy level was estimated a t 2.5 eV below the bottom of the conduction band by the method of line-shape fitting proposed by Kukimoto et al.I4 The leading edge of the spectrum in the 1.7-2.5-eV region increases in accordance with Gaussian function and the trailing edge
;* :*
3.0
2.5
2.0
1.5
Photon Energy / ov
Figure 4. An action spectrum of the photocapacitance of a ZnO electrode.
u
*0°
t . a % .
150
1
8
100
E
a U u a
I-
a
E
50
0
U
c 0
1.0
2.0 3.0 Phototapacltance I pf
4.0
Figure 5. Time differential of the photocapacitanceplotted as a function of the photocapacitance. Light intensities (at 514.5 nm) in mW: (a) 160. (b) 80, (c) 40, and (d) 16.
from 2.5 to 3.2 eV falls very slowly. The broad line shape in the leading edge is probably due to the large phonon broadening induced by the ionic character of the ZnO lattice as in the case of the F center in alkali-metal halides.2’ On the assumption of a uniform spatial distribution of both the charged species and the light-absorbing state in the space charge layer, the initial growth in the photocapacitance, dAC/dt, can be expressed by eq 5 whose derivation is given in the Appendix dAC/dt = ($Cdfl>’Y/2Nd)
- ($Zoy)Ac
(5)
where $ denotes the quantum yield of the photoionization, Cothe electrode capacitance in the dark; ZO the incident photon flux, N: the initial concentration of the intervalence state, Nd the donor concentration, and y the optical cross section. (21) Markham, J. J. In Solid Stale Physics; Seitz, F., Trunbull, D.. Eds.; Academic Press: New York, 1966; Suppl. 8.
5546 The Journal of Physical Chemistry, Vol. 93, No. 14, 1989 0
I
Nakabayashi et al.
I
( A )
Laser
150 U
s
Electrode Potental
'*
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. J
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- 100
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~
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0.5 1.0 Laser Intensity / 10'" photon c,$ See' Capacitance
Figure 6. Slops (A) and Y intercepts (B) of the plots in Figure 5 plotted as a function of the laser intensity.
ACa
On the basis of eq 5 , the time differentials of the photocapacitance are plotted as a function of the photocapacitance in Figure 5. Both the slope and the intercept on the ordinate of these plots vary with the incident photon flux as shown in Figure 6 . Line A gives immediately a value of 4.5 X lo-'' cm-2 for 47, which corresponds to an absorption coefficient of 7.7 X 10-3/4 cm-I. cmz pF, which, combined with Line B has a slope of 1.5 X d = 700 pF,ZZindicates that N,0/2Nd the above value for 47 and c should be 4.8 X viz., A:' should be 1.7 X 1014 ~ m - ~ . Capacitance Change Caused by Electrode Potential Pulse. The photocapacitance diminishes either by a slow thermal process ( k C 1 s-') or by the fast recombination induced by the potential pulse (k > lo4 s-I), Henceforth, we will concentrate on the latter to which the dynamic photocapacitance method was applied. The pulse-induced decrement depends on both the pulse width (duration) and the pulse-top potential. The mechanism of the decrease in the photocapacitance is illustrated in Figure 7. It should be noted that the oxidized form of the intervalence state, which will be referred to as a trap, need not be positively charged, although it is represented by a positive charge in Figure 7: AN is determined by the change in the electronic charge irrespective of the initial charge of the trap. The initial stage (A) is the photostationary state under a deep anodic potential, E,, e.g., 3.4 V vs E,,,, with a space charge layer thichess wa which is determined by the Mott-Schottky w = (2cto(E- E f b ) / e N ) ' / 2
010 0 0 Q 0 I
( A ) ( C ) Figure 7. Left: the timing sequence of the laser irradiation and the electrode potential pulses. Right: schematic models of the recombination occurring at each stage A, B, and C .
over the recombination region is converted to an average over w,: (mt)a =
(ANt)ab(wa - w b ) / w a
(7)
From eq 4 and 7, and by using a relation that N , = Nd Aca/ca = ( m t ) a b x / 2 N d
(8)
where
x = (wa
- wb)/wa
(9)
Practically, only wb was changed with w, fixed through a series of measurement. By definition (mt)ab
= IT' xxmt(x) dx
(10)
From eq 8 and 10
(6)
A negative potential pulse is applied to the electrode in the coincidence with the sudden cutoff of illumination (stage B). The shift of the potential to the negative or cathodic direction decreases the space charge layer thickness to wb which corresponds to a pulse-top potential, Eb. Recombination takes place in the region between w, and wb into which conduction electrons pour, and this region will be referred to as the recombination region. The extent of the recombination depends on the pulse-top potential which determines the position and length of the recombination region and also on the pulse width if it is short. At step C, the electrode potential is returned to the initial value E,, but the space charge layer thickness, w,,is larger than the initial one (w,)because of the decrease in the number of the traps in the space charge layer. Thus, the decrease in the positive charge (or the increase in the negative charge) caused by the potential pulse is observed as the capacitance changes at a fixed potential E,. The change in the capacitance, AC,, measured at the initial (and also final) potential, E,, is related to Aw/wa caused by a change in the charge, ( m t ) , b ( w , - wb), where ( m t ) , b is an average trap concentration over the recombination region; the average is introduced because the trap distribution along the depth is unknown. On the basis of the charge conservation, the average (22) Electrode areas differ between this measurement and the measurement for the data of Figure 2.
Aca/ca = J x ( A N t ( a / 2 N d ) dx
(11)
Equation 11 indicates that the differential of the AC,/C, with respect to X gives the trap concentration. Spatial Distribution of Traps. If the pulse is long enough to remove all the traps in the recombination region, the decrease in the photocapacitance relates to the concentration of the whole traps in the recombination region. The measurement of the decrease as a function of the electrode potential affords the trap distribution along the depth. Experiments were carried out for 1-ms pulses stepping from 4.4 V vs E h to various potentials. A duration of 1 ms is about 2 orders of magnitude longer than the apparent life time of the trap decay due to recombination: the reaction must be completed within this duration. In Figure 8, the changes in the capacitance are plotted as a function of both the depth from the surface, wb, and the X calculated in terms of eq 6 and 9, respectively. The trap densities estimated in terms of eq 11 are plotted as a function of the depth in Figure 9. This result indicates that the trap is distributed in the bulk region and its concentration is of the order of lot4 Consequently, the original intervalence state is also distributed in the bulk and its concentration is practically equal to the above trap concentration, since most of the intervalence states are probably photoionized under the experimental conditions. The anomaly near the surface is mainly due to the disturbance by capacitance signals appearing a t pulse-top potentials close to
The Journal of Physical Chemistry, Vol, 93, No. 14, 1989
Dynamic Photocapacitance of ZnO
5547
X
1
0
1
0.1
I
I
I
0.2
0.3
0.4
' -,
Depth: Wb 1 pm
Figure 8. Capacitance decrements caused by 1-ms potential pulses
plotted as a function of the thickness of the space charge layer, Wbr and X. X 1.00
0.75
0.25
0.50
I t i l i
0
I
I
0
'I
Donor
Figure 10. Capacitance signals appearing at pulse-top potentials near the flat-band potential.
I
1016
2.0
1.0
Time I msec
Surface' Localizkd State
1
U
0
0.1
0.2
0.3
0.4
0.5
Depth /pm
Figure 9. A spatial distribution of traps produced from the intervalence state along the depth as calculated from the data in Figure 8 in terms of eq 11.
the flat-band potential which correspond to small space charge layer thicknesses. These signals decay faster but their tails are large enough to disturb measurement of the desired signal, as illustrated in Figure 10. This disturbing signal is identical with the one ascribed to a shallow trap in a previous study.I6 Decay Kinetics. If the pulse is short compared with the apparent lifetime of the trap due to recombination, some traps will survive recombination. From eq 4, the concentration of the trap surviving recombination during a pulse width t is proportional to y = AC, - AC,(t=m) (12) where AC,(t==) denotes the response for a sufficiently long pulse. Thus, the plot of y against the pulse width gives a decay curve of the trap. This method enables us to measure the kinetics in the time domain shorter than the resolution of the capacitance measurement which is 50 ps in the present experiment. The experiments were done for pulses with a base potential of 3.4 V vs Efband various top potentials. At each pulsetop potential,
0
I
I
I
I
5
10
15
20
Electrode Potential Pulse Width / psec
Figure 11. Capacitance decrement, y , as a function of the pulse width
(duration) of the electrode potential pulse. Pulse-top potentials (E!) are (a) 2.4, (b) 1.9, (c) 1.4, (d) 0.9, (e) 0.4, and (0 0 V vs Elb,and E, is 3.4 V vs Em.
pulse width was changed in the time region shorter than 20 ~ s . Decay profiles thus measured are depicted in Figure 11, which also demonstrates that the data fit simulations of the first-order kinetics: - d y / d t = ky (13) A rate constant for the flat-band potential pulse, kh, was 2.1 X lo5 s-l, and the rate constants relative to this value are plotted as a function of the pulse-top potential in Figure 12. The measurement near the flat-band potential may be less reliable for the same reason as described for the distribution measurement, although the disturbing signal was much weaker for these short pulses than for the 1-ms pulse used for the distribution measurement. The rate constant is associated with the capture of the conduction electron by the trap, as explained above. Since the concentration of the trap is times as large as the donor concentration of the electrode, the experimental rate constant is
5548
The Journal of Physical Chemistry, Vol. 93, No. 14, 1989 Electrode Potential vs. Efb. 0
1.0
0.5
1.5
2.0
2.5
3.0
I .o
-
\
0.8
n
L
ii:
-.. Y
W
0.6
Shallow State
1
-Y
0.4
0
0.1
0.2
0.3
0.4
Depth / pm
Figure 12. Dependence of the recombination rate relative to k , on the electrode potential and the corresponding thickness of the space charge layer.
regarded as pseudo-first-order expression for electron capture rate and expressed as
k = NdUthd
(14)
where 0th and u denote the thermal velocity of the electron and the capture cross section of the trap, respectively. In terms of cmz eq 14, a capture cross section was estimated at 3.6 X for a rate constant of 1.4 X lo5 s-l which is an average for pulse potentials above 0.4 V vs E,, where other values used were Nd = 1.77 x 1OI6 cm-3 and 0 t h = 2.13 x io7 cm/sZ3 Discussion Throughout the experiments, bias potentials vs E , exceeded the band gap energy. This condition leads to the level inversion where holes could be the charge carrier as the result of tunneling to the low-lying conduction band.24 The tunneling, however, is negligible for the donor concentration at the present electrode (1.8 X 10l6 ~ m - which ~ ) leads to a length to tunnel of 0.25 pm, which seems to be took thick to tunnel. The effects of the electric charge on the capture cross section has been discussed theoretically. Theoretical calculations predict that the electron capture cross sections should be 10-'z-10-'6 cmz for positively charged species and 10-15-10-20cm2 for neutral specie^.^^-^^ The experimental cross section in the bulk region (>0.4 pm) was 3.6 X cmz; therefore, the observed deep trap is most likely neutral, and accordingly the intervalence state which gives this trap on photoionization must be negatively charged. An (23) The thermal velocity of the conduction electron of ZnO is calculated by the relation uth = (3kT/m*)'/*, where m* is effective mass: m* = 0.3m from ref 20. (24) Zener, C. Proc. R . SOC.1934, A145, 523. (25) Jaros. M. Deep Levels in Semiconductors; Adam Hilger: Bristol, 1982. (26) Henry, C. H.; Lang, D. V. Phys. Rev. B 1977, 15, 989. (27) Lang, D. V.; Henry, C. H. Phys. Reu. Lett. 1975, 25, 1525.
Nakabayashi et al. intervalence state of crystalline ZnO has been found at 2.45 eV in a photoluminescence studyz8and tentatively assigned to a state involving Vz,-. This assignment is consistent with the above conclusion. The peculiarities that were observed near the surface in the measurement of both the trap distribution and kinetics are due to the disturbance of the capacitance signal identical with that observed previously.16 The disturbing signal is most likely associated with a shallow surface state, as has been described in a previous paper,16 which must be located within a depth corresponding to 0.2 V, viz., within 0.1 pm according to eq 6. The main signal in the presence measurements is a decrement in the capacitance caused by the cathodic pulse, whereas the disturbing signal is an increment under the same conditions. The increase, according to the model developed in this paper, leads to an improbable conclusion that the cathodic pulse would accumulate the positive charge. However, the model may be invalid if there are surface states at high density, since in that case the applied potential may also be shared by the solution layer facing the electrode and the space charge layer inside the electrode. Thus, a quite new model seems to be required for explanation of the electrode capacitance in the presence of surface states. Detailed discussion will be made elsewhere. The present study shows that the spatial distribution of the deep trap concentration and its recombination rate constant can be measured by the dynamic photocapacitance method, although measurement near the surface is not yet successful. Exclusive observation of the deep state without disturbance by the shallow surface state will probably be possible by removal of the shallow trap by illumination at appropriate wavelength; such experiments are now planned. Appendix The rate of the photoionization is given by
-dN,/dt = 4Zabs
('4.1)
where N, denotes the concentration of the intervalence state, 4 the quantum efficiency of the photoionization, and Zabs the photon density absorbed within the space charge layer. The initial concentration N: is given by N: = N,
+ Nt
(A.2)
where N, is the concentration of photoionized form. Equations A.l and A.2 combined with Lambert-Beer's law leads to -dN,/dt = (410/w)[l - exp(yNtw - yN:w)]
(A.3)
where Io,w,and y are the incident photon flux, the thickness of the space charge layer, and the optical cross section, respectively. Equation A.3 is recast by use of eq 4 in the text and an approximation that N = Nd as dAC/dt = (Co4Zo/2Ndw)[l - exp(-yN:w
+ 2yNdwAC/Co)]
('4.4) MacLaurin expansion of the right-hand side of eq A.4 with respect to AC gives eq 5 in the text. Registry No. ZnO, 1314-13-2. (28) Gopel, W.; Lampe, U. Phys. Reu. B 1977, 22, 6447.
