J. Phys. Chem. 1983, 87, 5405-5411
5405
Transport and Kinetics in Photoelectrolysis by Semiconductor Particles in Suspension J. S. Curran’ and D. Lamouche Laboratolre de Physlcochlmle des Interfaces, Ecole Centrale de Lyon, B.P. 163, 69131 Ecul/y Cedex, France (Received: February 22, 1983)
Assuming that small semiconducting particles retain the same optical and electronic properties as macroscopic samples, we obtain an expression for the quantum yield of photoelectrolysis. It is shown that small unmodified particles are in principle capable of efficient photoelectrolysis if surface recombination and the back-reaction of surface-adsorbed intermediates is suppressed. The case of metal-loaded particles is discussed in this light. Introduction Recent interest in the behavior of illuminated semiconductor particles in suspension in an electrolyte is partly due to the possible use of this sample system as a converter of solar energy to fuel (e.g., hydrogen). Some progress has been made in this domain, as recent results on Ti02,1-4 SrTi03,4+ and CdS7-9 show. In addition, some general features of such systems are beginning to emerge. Untreated semiconductor particles are generally ineffective or totally inactive, whereas when modified by metallic deposits they are more efficient a t photoelectrolysis.lJwlz The quantity of metal deposited has an important influence and it has recently been shown that a maximum in the efficiency-metal quantity curve exists.13J4 It has also been shown that small-sized particles are more efficient,13 and that the electrolyte composition has an important influence on the photoelectrolysis yield?J3 However, there has been little attempt to interpret these results quantitatively, except by Pichat et al., whose model concentrates on surface catalysis mechanisms.14 This article attempts to show that semiconductor powder behavior can be interpreted by using the conventional concepts of photoelectrochemistry provided some important differences between macroscopic “contacted” and microscopic “uncontacted” electrodes are borne in mind.
sample. The discussion is restricted to “small” particles in the first instance, since they may be considered to be uniformly illuminated. The illumination is diffuse under most experimental conditions, so uniform illumination can be assumed for ro < a-l, where ro is the particle radius and a the optical absorption coefficient. For submicron particles this condition is met unless very short wavelength light is used. The grains are not, however, considered to be so small that band theory does not apply. To take an example from the work of Pichat,14 ro N 0.01 pm, the illumination condition is met down to X = 300 nm and the particles contain N lo5 atoms, quite enough to establish a band structure. Between the initial creation of a hole-electron pair and the final appearance of the products of the photoelectrolysis reaction in solution there exist several obstacles. In chronological order these are (a) bulk recombination in the semiconductor particle, (b) surface recombination a t the electrolyte-semiconductor interface, (c) back-reaction of surface intermediates, and (d) back-reaction of products in solution. For each of these distinct processes a yield can be defined; thus the overall problem can be decomposed into four independent parts. The overall quantum yield of the system qtot is the product
T h e Model The behavior of untreated powders is considered first. The grains are assumed to be microcrystals of spherical shape which retain all the equilibrium, transport, and optical properties attributed to a macroscopic crystalline
where (a) qi is the fraction of photoexcited carriers which escape bulk recombination and reach the interface, (b) qSR is the fraction of carriers arriving a t the interface which escape surface recombination, (c) qabs is the fraction of carriers escaping the processes in (a) and (b) which react to form product molecules in solution rather than performing back-reactions with adsorbed intermediates (if any), and (d) qpr is the fraction of product molecules formed in solution which escape back-reactions a t the interface and diffuse away into the bulk of the solution. Each of these factors is dealt with in turn below, beginning with qi.
(1)S.Sat0 and J. M. White, J.Phys. Chem., 85,592 (1981). (2)T.Kawai and T. Sakata, Chem. Phys. Lett., 72, 87 (1980). (3)E. Borgarello, J. Kiwi, E. Pellizetti, M. Visca, and M. Gratzel, Nature (London),281, 158 (1981). (4)D.H. M. W. Thewissen, M. Euwhorst-Reinten, K. Timmer, A. H. A. Tinnemans, and A. Mackor, “Proceedings of the International Conference on the Photochemical Conversion and Storage of Solar Energy”, Jerusalem, 1982,p 261. (5)J. M. Lehn, J. P. Sauvage, R. Ziessel, and L. Hilaire, Isr. J.Chem., 22, 168 (1982). (6)J. M. Lehn in “Photochemical Conversion and Storage of Solar Energy”, Academic Press, New York, 1981,p 161. (7)J. F. Reber, K. Meier, and N. Buhler, “Proceedings of the International Conference on the Photochemical Conversion and Storage of Solar Energy”, Jerusalem, 1982,p 252. (8)D.Duonghong, J. Ramsden, and M. Gratzel, “Proceedings of the International Conference on the Photochemical Conversion and Storage of Solar Energy”, Jerusalem, 1982,p 251. (9)T.Kawai, “Proceedings of the International Conference on the Photochemical Conversion and Storage of Solar Energy”, Jerusalem, 1982, p 284. (10)A. R. Ellis, J. Solid State Chem., 22, 17 (1977). (11)B. Krauetler and A. Bard, J. Am. Chem. SOC.,100,4317 (1978). (12)Y. Kakato, K. Abe, and H. Tsubomura, Ber. Bunsenges. Phys. Chem., 80,1002 (1976). (13)T. Sakata, T.Kawai, and K. Hashimoto, Chem. Phys. Lett., 88, 50 (1982). (14)P. Pichat, M. N. Mozzanega, J. Disdier, and J. M. Hermann, Nouu. J. Chim., 6,559 (1982). 0022-365418312087-5405~01.5010
Vtot
= ~iqSRqabs?lpr
T r a n s p o r t Equation for Electrons a n d Holes under Illumination Four simplifications are made, discussed in parts a-d, which enable us to obtain an analytically soluble differential equation for hole and electron transport, and which lead to a simple expression for the quantum yield. ( a ) Negligible Electrical Field within Small Particles. In general, when a semiconductor is in contact with an electrolyte in the dark, its Fermi level must equalize with that of the solution. For macroscopic electrodes this leads to the formation of a space charge layer a t the interface, and a considerable potential drop in the semiconductor. The situation is quite different for a small semiconductor particle in suspension. The space charge layer must always form, but in the absence of an external contact this same 0 1983 American Chemical Society
Curran and Lamouche V
I
0.4
0.2
0.6
0.8 -8
10
:r
\E
Flgure 1. A plot which shows the criterion for the flat-bands assumption, where the potential drop between the center and the surface of a particle radius r o is 2 k B T l q . Any given powder can be placed on this diagram, and will either be in the “field-free’’ zone under the curve, where the electric field is too small to affect carrier transport, or above it, where the flat-band assumption must be otherwise justified or abandoned.
depletion layer exists under the entire surface. More importantly if the radius of the particle is smaller than the thickness of the space charge layer formed in a macroscopic electrode of the same material the potential drop in the semiconductor becomes limited by the size of the particle. Under these conditions all the donors (for the n-type case) are ionized and there are no electrons left in the conduction band. Under total depletion the potential drop A V between the center and the surface of a spherical particle can be easily calculated by applying Gauss’ theorem.15 If we assume a dielectric constant t and a donor density N
Hence
A V = V(r=O) - V(r=ro) =
qN s, E ( r ) GL r0
dr =
ro
r dr
N AV = -ro2q
(NHE)
L
0.01
0.02
0.03
0.04
-
x
Flgure 2. The energy diagram for a small TiO, particle in suspension in H 2 0 of pH 7.
zone of Figure 1 the total absence of electrons in the conduction band and the small value of potential difference that is possible inside the semiconductor has the important consequences that under most practical circumstances the Fermi level is no longer associated with the conduction band (n-type case) but is placed between the bands in line with the solution redox (Fermi) level. Figure 2 shows such a diagram for T i 0 2 in H 2 0 a t pH 7. Exactly the same is true of a p-type particle. The initial doping type for small particles on the lower left of Figure 1is immaterial to the form of the energy diagram of Figure 2. ( b ) Lack of Effect of Illumination on the Electrical Field. It is assumed that illumination of the particle does not affect the “flat-band’’condition discussed above. Since there is no reason for either carrier type to accumulate at the center, only the Dember potential, normally neglected, can cause a change in the internal field, even if the total charge on the particle is changed by illumination, as observed by Bard et ( c ) Pseudo-First-Order Bulk Recombination. The Shockley-Read-Hall treatment of bulk recombination via trapping levels gives a general expression for the recombination current per unit volume RE:
6tto
The potential difference A V is the largest that may exist inside the particle, but its value is small for small ro. In particular if AV < 2 k B T / q , its influence on carrier transport may be neglected. This inequality is satisfied for ro(
iv)1/2
~
(12kBT~o)1/2
< 407
q2
where c, and ch are the electron and hole concentrations, ci is the intrinsic carrier concentration, and 7, and are the electron and hole lifetimes. For wide bandgap n-type semiconductors under low injection conditions the expression simplifies to RE = C h / T h = kCh
Figure 1illustrates this criterion graphically, in such a way that a given particle system may be located either in the field-free zone under the curve, or in the field-present zone above it. In many cases in the literature the size of the particle is not reported, but Pichat et al.14 state that ro varies from 0.01 to 0.015 pm. The donor density for TiOz ranges from about 1015 for undoped material to 10l8 for material heat treated in hydrogen.16 Metal-loaded particles are likely to have the latter doping density, unless no heat treatment has been used. With respect to the energy diagram for a particle which lies in the field-free
where k is a first-order recombination rate constant, since c, >> c h and c, >> ci. This situation also prevails in the case of the small particle under illumination, but for an entirely different reason. In the dark as described above the particle is totally depleted, but under illumination the holes and electrons created react at the semiconductor-electrolyte interface a t different rates in the normal case. Thus, for example, ce >> ch when k h >> k,, and first-order recombination kinetics prevails. We adopt this plausible assumption for which some experimental evidence in the case of CdS particles has recently been reported18 by Rosetti and Brus, who state that luminescence from radiative re-
(15) W. J. Albery, Keynote Lecture of the 33rd Meeting of the International Society of Electrochemistry, Lyon, France, Sept, 1982. (16) R. H. Wilson, L. A. Harris, and M. E. Gerstner, J.Electrochem. SOC.,126, 844 (1979).
(17) M. D.Ward, J. R. White, and A. J. Bard, J.Am. Chem. Soc., 105, 27 (1983). (18) R. Rosetti and L. Brus, J.Phys. Chem., 88, 4470 (1982).
Photoelectrolysis by Semiconductor Particles in Suspension
The Journal of Physical Chemistry, Vol. 87, No. 26, 1983 5407
combination has a linear and not a quadratic dependence on light intensity. ( d ) First-Order Interfacial Kinetics. The interfacial reaction kinetics of both electrons and holes is assumed to be first order in electron and in hole concentrations. The respective rate constants kh and k , represent the sum of surface recombination plus forward and backward Faradaic charge transfer processes. We show below that the back-reaction of products in solution is negligible. However, in principle surface recombination and backreactions of surface intermediates may introduce a second-order term into the interfacial kinetics, with the consequence that kh (and k,) increase with illumination intensity. This would be observable as negative deviation from linearity in an experiment of the type reported by Rosetti and Brus for CdSs as we show in Appendix 11, and the same negative deviation in a plot of photoelectrolysis yield vs. illumination intensity, neither of which have been reported. Linearity up to 1015 photons cm-2 s-l has recently been confirmed for hydrogen evolution from the system Pt/ Ti02/H 2 0/ propanol. l9 ( e ) Solution of the Transport Equation. Using the simplifications explained in parts a-d, we can write an analytically soluble pair of differential equations for the steady state on for each carrier type, using the normal formula for spherical diffusion:
tlli
where I is the illumination intensity. Equation 2 can be solved directly for ch. The first boundary condition is a t the center dch/dr = 0 a t r = 0 (spherical symmetry) and the second a t the interface where the current density is attributed a value, J , eventually experimentally accessible, which must be a t steady state be the same for both electron and hole currents: Jh = J , = khch a t r = ro The mathematical details are given in Appendix I. The essential result is that 3(R coth R - 1 ) (3) 'li = kr0 R2 -(R coth R - 1) kh where R = ro/LD,L D being the diffusion length defined by L D = (D/k)lI2
+
Values of vi as a function of LD for fixed values of ro and kh are shown in Figure 3. In Figure 4 we have calculated vi as a function of ro for a fixed LDand various values of kh. Using a series expansion and retaining only the firstorder terms gives a simplified expression of eq 3 for vi 1 Vi = (4) L rc 1 + -ro 3kh This expression was used to calculate the dotted lines in (19) P. Pichat et al., unpublished results.
1
2
3
4
5
6
1
8
,
IO~L,
9
1 0 .
Flgure 3. The internal quantum yield 7, of a small semiconductor particle of radius r o = 0.03 p m as a function of the diffusion length L D (cm). The rate-limiting interfacial rate constant k, is given a value of 600 cm s-l and two values of the hole diffusion constant are used, 2.6 X lo-* and 0.39 cm2 s-', corresponding approximateb to TiOz and CdS.
.04
.12
.08
Flgure 4. The internal quantum yield 7, of a semiconductor particle as a function of it's radius r o for various values of the interfacial rate constant k,. The dotted lines show the results of applying an approximation (discussedin the text) which neglects diffusion. L , is taken as 1 0 - ~cm.
Figure 4. From a physical point of view, it is worth noticing that applying the simplified eq 4 for vi is equivalent to neglecting the diffusion (e.g., neglecting the diffusion terms in the general eq 2) and thus to considering c, and c h as constant within the particle. From Figure 4,it is clear that this approximation appears to be very useful as long as kh < lo3and for small particles say r < 0.1 km for the specific case considered here, or in general for R < 1.0. ( f ) Surface Recombination Factor vSR. By defining a first-order rate constant for surface recombination kSR,for the same reasons as discussed in parts c and d and for the forward Faradaic process kh,f,we can write the fraction of photogenerated carriers escaping surface recombination as kh,f
VSR
=
+ kh,f (g) Intermediate Back-Reaction Factor 'labs. It is impossible to write an expression for this term without adopting a specific mechanism and making some rather arbitrary assumptions about the rates of various processes. We make no attempt to estimate this parameter but simply point out that it may play a crucial role. (h)Product Back-Reaction Factor vpr., We are able to show that for small particles this factor is close to unity, which eliminates one complication. The physical reason is that enhanced diffusion at small particles is able, at the kSR
5408
Curran and Lamouche
The Journal of Physical Chemistry, Vol. 87, No. 26, 1983
low current densities involved, to remove products rapidly so that their concentration a t the interface is very low. The factor vpr can be written by inspection, taking for example the back-reaction of H2 k',,f[H,OI vPr - k H b[ 21 + k6,f[H201 where kb and k i , f are second-order interfacial rate constants (since it is necessary to take account of the concentration of the respective reactants). [H2] is estimated by equating the ideal current density J (vi, ~ S R vlab , = 1.0) to the diffusion flux in solution; it is thus an overestimate. If we assume that diffusion takes place over a stagnant layer of thickness S, beyond which [H,] is virtually zero, and with a diffusion coefficient D for H2 DW21S J(idea1) = (S - r o h It is commonly assumed that S is quite large compared to the ro values we consider in this article. Thus a simplification takes place.
Iar2 [H21 =
30
Inserting appropriate values for Iar, (lo* M cm-2 s-l for normal illumination intensities) and ro/D (0.2 cm-l), and using [H20] = 0.055 M ~ m - one ~ , finds that vpraturns out to be unity unless k , / k , , > los, a condition unlikely to be met in practice. Interim Conclusion. So far we have been able to show that for sufficiently small particles vi and qpr are near to unity in theory at least. If such small particles turn out to be poor at photoelectrolysis it is logical to conclude that either surface recombination of intermediate back-reactions are responsible, as long as it can be shown that either the k h or k , reactions are sufficiently downhill to be reasonably rapid. Theory20and experiments on meta121-23organic crystal24 and semiconductor electrodesz5show that such rate constants reach values of around lo4 cm s-l for one-electron transfers for large negative free energy changes (30.5 eV) such as could be the case for oxygen evolution steps on TiOz and similar semiconductors. Oxygen evolution on oxides may be a four-electron process26but given the very negative values for AGO = -2.0 eV individual steps can also be expected to be rapid. For the typical case of Ti02/H20 we expect from Figure 2 that kh is in the range 102-104 cm s-l ( k , is much smaller) and that curves such as in Figures 3 and 4 can be used.
Particles Modified by Metallic Deposits It is well-known that semiconductor dispersions are frequently more effective if they have been submitted to treatments which leave small quantities of metals or metallike oxides on the surface of the semiconductor particles. Small metal spots can be observed by electron (20) H. Gerischer, Top. Appl. Phys., 31, 115 (1979). (21) P. Bindra, H. Gerischer, and L. M. Peter, J.Electroanal. Chem., 57,453 (1974). (22) C. Bernstein. A. Heindrichs. and W. Vielstich, J. Electroanal. c h K , 87,81 (197sj. (23) J. Kuta and E. Yeager, J. Electroanal. Chem., 59, 110 (1975). (24) F. Willig, Adu. Electrochem. Eng., 12, 1 (1981). (25) R. Memming and F. Mollers, Ber. Bunsenges. Phys. Chem., 76, 475 .. . (1972). -, \--
(26) S. Trasatti and G. Lodi, in "Electrodes of Conductive Metal Oxides", Part B, Elsevier Scientific, Amsterdam, 1981.
1
E INHE)
4
0.0
+ 1.0
+ 2.0
+
3.0
CATHODE A
ANODE
NEITHER
T
4
1
4
C
0.02 prn
Flgure 5. The energy diagram for metallized TIO, (TiO,/Pt etc.) (a) with the metal acting as a cathode, (b) with the metal as an anode, and (c) with the metal acting neither as cathode nor as anode, but as a recombination zone. Both (a) and (b) are impossible. V , is the intrinsic barrier height between n-TiO, and Pt, estlmated as 0.5 V.,'
m i c r o s ~ o p $ ~and ~ it is generally assumed that these spots act as cathodes, the structure behaving like a Schottky barrier solar cell with efficient hole-electron pair separation by the internal field. However, it is easy to show that this can only be the case under rather restricted thermodynamic conditions. We show that a t least for Ti02/ P t / H 2 0 the metal spots can neither act as cathodes or as anodes. Figure 5a shows the energy diagram for this structure, supposing that the metal spot is a cathode. The quantity Vbi is the barrier height between Ti02 and Pt, calculated to be 0.8 V13 and measured on a single crystal to be 0.5 V.27 Clearly, the cathode hypothesis is false, since the carriers migrate in the wrong direction. Figure 5b shows the same diagram supposing the metal acts as an anode. Again it is immediately evident that this alternative is false. The only possible conclusion is that no current passes through the metal, that the bands are flat, and that the metal spot acts as an inefficient recombination zone. The logical consequence is that the beneficial effect of the metal must be explained by another mechanism, which we propose to be an increase in k,,f and/or kh,f,and/or a decrease in kSR due to submonolayer coverage of the semiconductor surface by metal atoms. It is known that submonolayer metal deposits can lead to dramatic changes of the type propo~ed.~~,~~ The same arguments applied to the SrTi03/H20case lead to precisely the same conclusion-the bands are flat for a small particle with a metal deposit. If, as argued above, the bands are flat and the metal spot causes recombination, a very simple explanation can be given for the maximum in the metal quantity-efficiency curve. On increasing the number of spots, recombination must eventually overtake the beneficial electrocatalytic effect, and the yield begins to drop. The thermodynamic condition which must be satisfied if the metal is to act as a cathode for a reduction reaction of redox potential E , is easily written: 2kBT ER > Vdn-type) + v b i + 4
(27) N. Yamamoto, S. Tonomura, and H. Tsubomura,J.Electrochem. SOC.,129, 444 (1982). (28) K. W. Frese, M. J. Madou, and S. R. Morrison, J.Electrochem. SOC., 128 1939 (1981). (29) A. Heller, "Photoeffectsat SemiconductorElectrolyte Interfaces", American Chemical Society, Washington, DC, ACS Symp. Ser. No. 146, 1980, p 57.
Photoelectrolysis by Semiconductor Particles in Suspension
The Journal of Physical Chemistty, Vol. 87, No. 26, 7983 5409
and for the anode case:
E,, < V,(n-type)
2kBT + E, - vhi - 9
where V, is the flat-band potential and E, the bandgap. The term 2kBT/q arises from the minimum value of the potential drop across the particle capable of assisting charge separation. At nonequilibrium the overvoltage on the metal must be taken into account, and effective charge separation would generally require more than a 2kT/q drop. Thus only very positive couples seem likely to satisfy the cathode condition. The reason why water-alcohol mixtures can be relatively efficiently photoelectrolyzed by T i 0 2 / P t is less likely to be due to the fulfillment of the cathode condition or the anode condition than to an increase in the surface back-reaction factor vah, for relatively subtle kinetic reasons. It should be mentioned that the extension of the above argument to particles suspended in water totally free of dissolved ions, or in very pure organic solvents, is not necessarily straightforward, due to the presence of an extended double layer in the solvent, and to difficulties in defining flat-band and redox potentials. It would seem logical to make experiments in the presence of an inert electrolyte as is the general practice in electrochemistry. Conclusions We have been able to calculate the internal quantum yield of a small uniformly illuminated semiconductor particle in suspension, as a function of it's radius, the Faradaic rate constants and the diffusion length in the semiconductor. The assumption is made that there is a considerable difference between the Faradaic rate constants for holes and electrons. Under these conditions bulk recombination becomes first order and the qusntum yield is controlled by the fastest interfacial rate constant (probably holes for Ti02) in competition with bulk recombination. We show that for particles with a radius near to the diffusion length, and for interfacial rate constants in the range expected for the rather downhill hole reactions, the internal quantum yield approaches unity. Since we have also been able to show that the back-reaction of products is negligible for small enough particles, the often low photoelectrolysis efficiency of unmodified particles is either due to surface recombination or to back-reactions of surface intermediates. It has also been suggested that unless rather restrictive thermodynamic conditions are met, semiconductor particles with metallic deposits (spots) cannot be considered as Schottky diodes. The metal spots act as recombination zones, and it is suggested that the beneficial electrocatalybic effect of metallic submonolayers on the semiconductor is responsible for increasing the quantum yield. This may be through either suppression of the intermediate backreactions or surface recombination. An extension of these arguments to larger particles requires a calculation which accounts for nonuniform illumination, and especially an examination of the conditions under which a flat-band condition can be met dynamically by the collection of majority carriers in the center of the particle. We are undertaking such a study. Acknowledgment. We thank Professor W. J. Albery for his inspiration, and Professor P. Clechet, Dr. Pichat, and Dr. J. M. Herrmann for their constructive comments. Appendix I The differential equations (see text) to be solved are
By subtracting (1-2) from (1-1)and rearranging one finds
where c(r) = DhCh - Dece. Using u(r) = rc(r) and substituting in (3), one obtains
(1-4)
(1-5) Hence
u=ar+b (1-6) where a and b are constants. When r = 0 a may be :Found from the definitions of c and u: a = DhCh - DeC, (1-7) and thus ce =
1
- a) (1-8) De This shows that the electron concentration is linearly related to the hole concentration. We now proceed ito the solution of eq 1-1using the substitution -((DhCh
w(r) = rch
(1-9)
Rearrangement gives (1-10) which is a linear second-order equation with respect to w. The general solution can be written as w = up + wh where up is a particular solution of (1-10) and wh is the general solution of the homogeneous differential equation associated with (1-10): (1-11) wh
can be found easily
a h
= A eXP[-(k/Dh)1/2r]
+ B e~p[(k/DJ'/~r]
where A and B are constants. If we take Iar wp = k
(1-12)
(1-13)
A e~p[-(k/Dh)'/~r]+ B e ~ p [ ( k / D h ) ~ / ~ r ] tr k ch(r) must remain finite when r approaches zero, hence A = -B and 2A'sinh (k/Dh)1/2r la +(1-14) ChW = r 12 The constant A'is determined from the surface boundary condition a t r = ro: Ch(d
=
5410
Curran and Lamouche
The Journal of Physical Chemistry, Vol. 87, No. 26, 1983
ch(r=ro) = ch,O
(ro/R2)(R coth R - 1)
(1-15)
where vi is the internal quantum yield. Equations 1-14 and 1-15 lead to
(kro/R2)(R coth R - 1) sinh (r/LD) J-0 (11-4’) sinh (ro/LD) r If kh can be considered constant, we obtain for the recombination current and Iem= k’RE’ =
since the diffusion length LD is defined by LD = (D/k)’I2 It is now easy to calculate the total recombination current in the particle, RE’: RE’ = k
s“ 0
4rr2ch(r)dr
(1-18)
From eq 11-4’ and 11-6 we can see that the integral term is independent of I (excitation) and thus
(2)
Using eq 1-17 we obtain kD ’r - ch,$) x sinh (ro/LD) k
( r o cosh
2
G - RE’ (1-20) =G where G is the total hole-electron pair generation rate in the particle, and using eq 1-16, 1-19, and 1-20, we obtain
By rearranging, and using R = ro/LD
(1-22)
we obtain 3(R coth R - 1) kr0
R2 + -(R
-
i)H
+
du = constant (11-7)
Ti
=
constant
- LD sinh
By defining now
Vi
= k’ta x
11lol(&
(Icu
kh
with k ” = kk’, le,being the detected fraction of the total radiative recombination current. Now, if we consider kh as dependent of C0,h and c0,, we can reasonably assume that dkh/dc0,, > 0, dkh/dCo,h > 0, and so dkh/dI > 0. If we use eq 6, differentiation with respect to I leads to
The last term on the right-hand side can be simplified in such a way that we can invert the order of differentiation and integration because the differentiation and integration parameters (Iand r) are independent. Thus
coth R - 1)
kh
Appendix I1 We can write
(11-9)
As dkh/dI > 0, F
> 0, and
SOP04rr2H dr
> 0, hence
(11-1)
(11-2) and using (R coth R - 1)
Iaro khCO,h
=
V
i
T
= ICYro
R2 +
kr0 -(R
coth R - 1)
kh
gives
where F , G, and H represent respectively
(11-3)
Equations 11-9 and 11-10reveal the nonlinearity of I,, with respect to I and the negative deviation of the emission signal in the case where k h is dependent on C?J, with respect to the emission signal when kh can be considered as constant (linearity between I,, and I). Glossary radius of particle (average), cm optical absorption coefficient, cm-l c, (ch) electron (hole) concentration, ~ r n - ~ De (Dh) diffusion coefficients, cm2 s-l k , (kh) pseudo-first-orderinterfacial rate constants, cm s-l k pseudo-first-order bulk recombination rate constant, s-l r0
a
J. Phys. Chem. 1983, 87, 5411-5417
I N 4 t
AV €0
11
light intensity, photons cm-2 s-l donor density, cm-3 electron charge, C dielectric constant, dimensionless potential drop between center and radius, V permittivity of free space, F cm-' quantum yield, dimensionless
kB
RE vfb
E,
5411
Boltzmann constant grain recombination current flat-band potential, V bandgap, eV
Registry No. Ti02, 13463-67-7; SrTiO,, 12060-59-2; Pt, 7440-06-4; hydrogen, 1333-74-0.
Electron Spectroscopy for Chemical Analysis Studies on Oxyligated Holmium Compounds W. 0. Mllllgan, D. F. Mulllca," H. 0. Perklns, C. K. C. Lok, Departments of Chemistry and Physics. Baylor Un;vers;ty, Waco, Texas 76798
and V. Young Department of Chem;stry, Texas A d M Unlverslty, College Station, Texas 77843 (Received: March 14, 1983)
ESCA studies on Hoz03,HoOOH, Ho(OH)~, and amorphous holmium hydrous oxide are reported. Experimental results imply that small amounts of 4f covalent bonding exist in these ionic compounds and that electron transfer from the conduction band to the valence band is possible. The key to understanding the catalytic power of these compounds may be related to the 4f energy level. The origin of "shake down" at 4d and 3d levels is a two-electron process which involves the creation of an n-d hole followed by the electron jump from the 5d conduction band to the 4f energy level. The 4d-4f couplings are ligand sensitive. The photoelectron signals are identical with the nonequivalent oxygens in these compounds. Using the Doniach-SunjiE line shapes folded with a Gaussian approximating the resolution of the instrument, one can completely separate the 4f band and the OZpband. The photoionization energies of the 4f band approximate the theoretical values of the major spin excitation A-(t) at 8.8 eV and the Hund's rule ground state A(J)at 5.4 eV. The peaks of the OZpband at 4.5 and 7.6 eV indicate different oxygen sites. An electron flow from OZpto 4f orbitals and the quasi-atomic-model calculation indicate a covalent effect in these ionic compounds.
el,
Introduction Investigations related to electronic structure and bonding in oxyligated lanthanide compounds have become necessary, since their catalytic power has been reported in the literat~re.l-~Attempts have been made to use the 01,binding energy to correlate EPA (electron-pair acceptor) and EPD (electron-pair donor) with the catalytic properties of heterogeneous catalyst^.^ The absence of the 5d conduction band is apparent in these insulating compounds. However, the 4f and the OZpbands play an important part in the valence level, and the shift of 01, binding energy has a more subtle meaning as this shift is governed by the external screening effects and the Madelurig potentials. The method of using Doniach-SunjiE lines folded with experimental resolution function proposed by Wertheim4 and the theoretical calculation of the coefficients of fractional parentage on the open shell by Cox5 have gained success in explaining the behavior of lanthanide alloys in solid-state physics.6-8 In light of this technique, the aim (1) S. Bema1 and J. M. Trillo, J. Catal., 66, 184 (1980). (2) B. H. Davis, S. N. Russell, P. J. Reucroft, and R. B. Shalvoy, J. Chern. SOC.,Faraday Trans. 1 , 7 6 , 1917 (1980). (3) H. Vinek, H. Noller, M. Ebel, and K. Schwarz, J. Chern. Soc., Faraday Trans. I , 73,734 (1977). (4) G. K. Wertheim and S. Hufner, Phys. Rev. Lett., 35, 53 (1975). ( 5 ) P. A. Cox, Struct. Bonding (Berlin),24, 59 (1975). (6) M. Campagna, G. K. Wertheim, and E. Bucher, Struct. Bonding (Berlin),30, 99 (1976).
of the present work was to conduct systematic studies on the energy levels of holmium trihydroxide, oxyhydroxide, sesquioxide, and amorphous hydrous oxide by employing an HP 5950A ESCA spectrometer. Experimental Section
Holmium sesquioxide used in the crystal growth of the holmium trihydroxide, oxyhydroxide, and holmium hydrous oxide was obtained commercially (Reagent grade, 99.9-99.99%) and was used without any further purification. Crystals of holmium trihydroxide were produced by placing the corresponding oxide into a Teflon bomb containing a 20 N NaOH solution. Hydrothermal aging at 180 "C for 3-5 days was required and then the Teflon bomb was transferred to a step-down oven at 80 " C for several hours before harvesting at room temperature. The crystals of oxyhydroxide were grown in a like manner except for the hydrothermal aging temperature, which was 300 "C. These crystal structures were verified by comparing obtained X-ray powder diffraction data with literature valU~S.~JO The amorphous material was prepared by dis(7) J. K. Lang, Y. Baer, and P. A. Cox, Phys. Reu. Lett., 42,74 (1979). (8) G. Kaindl, C. Laubschat, B. Reihl, R. A. Pollak, N. Mortensson, F. Holtzberg, and D. E. Eastman, Phys. Reu. B , 26, 1713 (1982). (9) G. W. Beall, W. 0. Milligan, and H. A. Wolcott, J . Inorg. Nuc2. Chern., 39,65 (1977). (10) W. 0. Milligan, G. W. Beall, and D. F. Mullica, "Crystallography
in North America", American Crystallographic Association, New York, 1983, Section F, Chapter 9, pp 362-9.
0022-3654/83/2087-5411$01.50/00 1983 American Chemical Society
