J. Phys. Chem. 1988, 92, 2493-2501
2493
Kinetics of Electron Transfer from Photoexcited Superlattlce Electrodes A. J. Nozik,* J. A. Turner, and M. W. Peterson Solar Energy Research Institute, Golden, Colorado 80401 (Received: July 7, 1986; I n Final Form: October 5, 1987)
A kinetic model has been developed that quantitatively describes electron transfer from photoexcited superlattice electrodes
into liquid solutions. The model permits electron transfer from all quantum levels as well as from surface states; it also takes into account recombinationin the bulk, space charge layer, and surface states, and band-edge movement. The model calculations define the values of the rate constants for heterogeneous electron transfer and hot electron thermalization among the various energy levels in the superlattice quantum wells that are necessary to achieve hot electron transfer from excited quantum states. The question of whether hot electron transfer is manifested by a dependence of the photocurrent action spectra on acceptor redox potential is examined in detail.
Introduction The photocurrent action spectra of lattice-matched superlattice electrodes in photoelectrochemical cells' show structure with stepped waves that correspond very well with the theoretical energy level scheme of the quantum states in the quantum wells of the superlattice. Data have also been published for strained-layer superlatticesa that show structure in the photocurrent spectra that also agree with the predicted energy levels of the quantum wells, but these data were subsequently found to be clouded by unrecognized contributions to the photocurrent from buffer layers present in the electrodes.2b An important question regarding the photocurrent behavior of superlattice electrodes is whether a significant fraction of the total electron transfer from the quantum wells into solution occurs from the upper quantum states (quantum number ne > 1; a hot electron-transfer process) or only from the lowest (ne = 1) state and/or from surface states (thermalized electron-transfer processes). In previous excited-state (i.e., hot) electron transfer from the high-lying quantum states with ne > 1 was believed to be reflected by a dependence of the shape of the photocurrent action spectra on acceptor redox potential. In this paper, we examine this hypothesis further and develop a detailed quantitative kinetic model for electron transfer from photoexcited superlattice electrodes into l i p i d electrolytes; we then consider two problems: (1) What are the conditions required for excited-state electron transfer from quantum states with ne > l ? (2) Is a shape dependence of photocurrent spectra on acceptor redox potential indeed a valid indicator of hot electron transfer from high-lying, excited quantum states? Kinetic Model A kinetic model was developed for p-type superlattice electrodes that permits electron transfer into solution from any of the quantum states in the superlattice as well as from surface states in the semiconductor band gap. A schematic of the model is presented in Figure 1; it is based on a more simple kinetic model previously developed for conventional p-type electrodes3 The model assumes that electron transfer into solution is controlled by the kinetics of charge transfer from the last quantum well nearest to the solution interface. The outermost layer of the superlattice electrode can be either this last quantum well or a barrier; in the latter case electron transport through the barrier is assumed to be possible from each quantum state in the last quantum well. The electron flux into each quantum state of the outermost quantum well is calculated according to the Gaertner model! In (1) Nozik, A. J.; Thacker, B. R.; Turner, J. A.; Klem, J.; Morkoc, H. Appl. Phys. Lett. 1987, 50, 34. (2) (a) Nozik, A. J.; Thacker, B. R.; Turner, J. A.; Olson, J. M. J . A m . Chem. Soc. 1986, 107, 7805. Nozik, A. J.; Thacker, B. R.; Olson, J. M. Nature (London) 1985,316,6023. (b) Nozik, A. J.; Thacker, B. R.; Olson, J. M. Nature (London) 1987,326,450. Nozik, A. J.; Thacker, B. R.; Turner, J. A. to be submitted for publication in J. A m . Chem. SOC. (3) Kelly, J . J. Memming, R. J . Electrochem. SOC.1982, 129, 730.
0022-3654/88/2092-2493$01 S O / O
the simplest approximation, the normal Gaertner expression is used except that the exponential term is divided by the ratio 6 of the sum of the thicknesses of the well plus barrier to the thickness of the well to take into account the fact that the superlattice consists of alternating layers of quantum wells and barriers and it is excited with light that is only absorbed in the quantum well layers. Thus
where g = electron generation rate, e-/(cm2-s); R = reflectivity; Io = incident photon flux, photons/(cm*.s); CY = absorption coefficient, cm-I; w = space charge layer width, cm; and L, = minority carrier diffusion length perpendicular to the superlattice, cm. The wavelength dependence of the electron flux into each quantum state of the outermost well next to the solution will depend upon whether the quantum wells in the electrode couple to form minibands or the wells are sufficiently isolated from each other (Le., thick barriers are present) such that the set of quantum states in each of the wells are independent; the latter configuration is termed multiple quantum wells (MQW). The situation is depicted in Figure 2. If the barriers are thin (e50 A), then minibands are formed which tilt with the electric field; electrons created in a given quantum level anywhere in the superlattice will remain in that level as they traverse the electrode. Thus, photoexcitation at the wavelength required for the population of a given quantum state will produce electrons at the last quantum well in the same quantum state. This situation does not hold for the isolated quantum well case where the barriers are thick. Here, the electric field causes each set of quantum states in each well to have different absolute energies (Figure 2a). This situation leads to a condition wherein electrons created at a given quantum state in one well can be transferred to higher quantum states in adjacent wells as the electrons move toward the surface. The result is that photoexcitation of a given quantum state in a well that is removed from the surface by several periods can create electrons that populate all the quantum states in the last well at the surface. As will be shown below, this latter case greatly complicates the analysis of hot electron-transfer processes. This effect for the MQW case is accounted for in our model by assigning a distribution among the quantum states for the electrons arriving at the last quantum well that were created in the superlattice at a given photoexcitation wavelength. Also, as seen in Figure 2b, the miniband case will convert into the MQW case when the electric field per well, Ew, exceeds the miniband width, AEn. A more rigorous description of the electron flux to the surface also takes into account recombination in the space charge layer (SCL) and carries out the calculation of the space charge contribution to the current by a stepwise integration over the GaAs ~
~~
~
(4) Gaertner, W. W . Phys. Rev. 1959, 116, 84.
0 1988 American Chemical Society
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Nozik et al.
The Journal of Physical Chemistry, Vol. 92, No. 9, 1988
I‘
in the SCL, cm2;x = distance from liquid interface, cm; p ( x ) = po exp[-V‘(x)/krj; po = bulk doping density, ~ m - k~ =; Boltzmann constant; 6(b) = barrier-well 6 function (0 for barriers, 1 for wells); T = temperature, K; n’(x) = (1 - R)Zoa exp(-ax)6(b) n(x); and n ( x ) = electron density in SCL from diffusive contribution in field-free region.
+
ne = 151-
V(X) =
- x2/2) = band bending as function of distance ( x ) , V
where V, = band bending at surface, V; w = [2cVB/(qp0)]’/* = SCL width, cm; t = dielectric constant; and qe = electron charge, coulombs. The net current produced by photon absorption in the SCL CI’SCL) is
l /
;i
v, + (qgo/t)(wx
Solution
jscL =
f jwwl [ ( I - R)zoaexp(-ax)h(b) -
q=o
WQ
~ R S C L ~ ’ ( X ) ~ ( X+)~/ ( ~ 1’ (dx 1 ~1 )(2)
Figure 1. Model for electron transfer from superlattice electrodes into solution.
h a) MQW
where Q = total number of superlattice layers (barriers + wells) within the depletion width and wZqand w2q+l= distances from liquid interface of front and back surfaces of qth layer of the superlattice. Equation 2 takes into account the fact that photons are only absorbed in the wells (for hv < barrier band gap) while the space charge is developed across both wells and barriers; equal doping densities are assumed for wells and barriers. The current produced by photon absorption in the field-free region G o i f f ) is4 jDiff
= (1 - R)zO(aLp/(l + OtLp)) exp(-aw/6)
(3)
Thus, the total electron flux to the last quantum well for a given transition wsvelength and corresponding a is
g = i S C L -k iDiff
Minibands Break Up Into Localized
- e-
U
Minibands
Tilt With Electric Field
(E,
1, where ne is the quantum number of the quantum states in the well) can either thermalize down to the next lowest quantum state or be transferred into solution to do redox chemistry. We assume that electrons with ne > 2 cannot thermalize directly to the lowest ne = 1 state or to surface states but must cascade down the energy level ladder through each quantum state. This path is more probable since it minimizes the phonon interaction requirements for energy dissipation. Our model also neglects radiative electron-hole recombination from any of the quantum states in the last well since this process is slow compared to electron transfer, thermalization, or trapping. Electrons populating the lowest quantum state are either trapped into surface states or transferred into solution. Electrons trapped in surface states are then either transferred into solution from the surface state or recombine with valence-band holes. All of these competing rates have associated rate constants as indicated in Figure 1, and they can be written as
where tit= electron-transfer rate from ith quantum state, e-/ (cm2.s); k:, = rate constant for electron transfer from ith quantum gscL(x) = state, cm3/s); n, = electron density in ith quantum state, e-/cm2; (1 - W o a exp(-ax)Nb) - ~ R S C L P ( X ) ~ ’ + ( ~n’(x)l ) / ~ ( ~ ) c = acceptor concentration in solution, ~ m - t-i ~ ;= thermalization rate from ith quantum state to the (i - I ) state, e-/(cm2.s); k; (1b) = rate constant for thermalization from ith quantum state to the where gscL(x) = electron generation rate in SCL, e-/(cm2.s.cm); ( i - 1) state, s-I; = electron-transfer rate from surface states, kSCL = electron-hole recombination rate constant (cross section) e-/(cm%); k? = rate constant for electron transfer from surface states, cm3/s; f = fraction of surface states filled; Nss = surface-state density, cm-2; r,,,, = electron trapping rate into surface ( 5 ) Sah, C . T.; Noyce, R. N.; Shockley, W. Proc. IRE 1957, 45, 1228.
The Journal of Physical Chemistry, Vol. 92, No. 9, 1988 2495
Electron Transfer from Superlattice Electrodes states, e-/(cm2.s); ktmp= rate constant for electron trapping, cm2/s; rrccomb = electron-hole recombination rate from surface states, e-/(cm2,s); kp = rate constant for electron-hole recombination, cm3/s; and ps = hole concentration at surface, ~ m - ~ . The rate constants for electron transfer from the quantum states in the last quantum well or from the surface states are calculated by using the well-known Gerischer-Marcus a p p r ~ a c h .We ~ use one intrinsic maximum rate constant ktt for a given acceptor and modify its value as a function of the energy level of the electron transfer, assuming a Gaussian distribution of electron acceptor states in solution. Thus
k:, = ktt eXp[(Ej - Eo
- AVH
X ~ ) ' / ( 4 k m ~ ) ] (10)
where kf, = maximum rate constant for electron transfer; Ei = energy level of ith quantum state, V (vs SCE); E= = energy level of surface state, V (vs SCE); AVH = potential drop in Helmholtz layer due to surface states, V; and XR = solvent reorganization energy, V. As seen in eq 10, the calculation of k:, takes into account band-edge movement due to occupancy and charging of surface states. The amount of band movement, AVH, is determined by the fraction of surface states occupied:
= fNsSqe/CH
AVH
(12)
where CHis the capacitance of the Helmholtz double layer at the electrode surface. In our calculations we can either have the band movement not affect the energy level (and hence ks?) of the surface state (AVH = 0), or we may let the surface state energy level move with the band edges (eq 11). Equation 12 requires a value for f. However, AVH must first be calculated to obtain a value for f. The value off at a given potential is typically taken from the previous potential; however, this requires that we use a very small potential step, and in some cases this step must be infinitely small to avoid artificial shifts of the onset current or oscillations (due to charging and discharging of surface states) in the value of AVH. To avoid such problems, we have used a self-consistentcalculation of the AVH. This involves the use of the previously calculated AVH.old as the initial value at the considered potential. We then calculate a new value of the AVH at the considered potential; the two values are then compared and if they are not within 0.1%of each other, a weighted average of the two is used as the new and we recalculate the AV at this potential. The process is continued until convergence is achieved. The number of iterations needed to reach convergence is strongly dependent on the potential step size, the location of the surface states relative to the redox couple, and whether or not the surface states are allowed to move. In our calculations the number of iterations has ranged from 200 (worst case) to 1 (best case). If one has M quantum states, ranging from ne = 1 to ne = M , and level ne = L is excited with light, then from a balance of the electron fluxes in the quantum well
then substituting eq 19 into eq 18 leads to r
If one divides all terms in eq 17 and 20 by k$)cnl (rate of electron transfer from the lowest (ne = 1) quantum state), one can then define a new set of rate expressions that are normalized to the rate of electron transfer from the lowest quantum state: A = ktra&.snl/(k$)cnl)5 normalized rate of trapping
(21)
Bi = kL,ni/(kaf)nl) = normalized rate of electron transfer from ith quantum state (22) C = ~FNssc/(k$)nlc) normalized rate of electron transfer from surface states (23)
D = kd\'sspo/(ki!)cnl)
= normalized rate of electron-hole recombination from surface states (24)
To evaluate these normalized rate parameters, it is necessary to calculate the equilibrium electron population of each quantum state or energy level, ni. This is done for each level by equating the total input electron flux to the total exit flux (i.e., dn,/dt = 0). For the quantum state that is directly excited by the incident light (ne = L ) , the equilibrium electron density, nL, is nL
= g(A)/(kttc + 6 )
(25)
For lower quantum states (ne = 2 to L - 1)
Finally, for the lowest quantum state (ne = 1)
where Also, from a balance of electron flux around the surface state
(1 -ANSSktrapnl =
~~~W + k Sf lS~ C ~ ~
L
cz kitni + @FcNsJ i= 1
Substituting eq 15 into eq 13 and rearranging terms
+ kgs
ktrap
+ kppsNss -
k:FNssc
+
kbf)c
s(14)
The total photocurrent iphfrom all quantum states and the surface state is iph =
kzFC
P=[
(15)
The expression for y depends upon the quantum state being excited. If this is the ne = 1 state, then y = g(X); for excitation of all other states (ne > l ) , y = ki2)n2. Finally, eq 20 and 17 can be respectively rewritten simply as A
'=
D eXp(-l/,/kr)
4- A
+c
(29)
L
iph _ g
Solving for f
fC
+ i-2CB,+ 1
A(1 -f) +
(30)
L
CBi +
i=2
1
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The Journal of Physical Chemistry, Vol. 92, No. 9, 1988
Nozik et al.
TABLE I: Energy Levels and Thermalization Rate Constants for First Five Levels in 250-A GaAs Quantum Wells" quantum photon energy above level energy, eV A, nm conduction band, meV k;, s-' 4, ps 1 1.43 865 8.8 4.35 x 109 219 2 1.46 849 34.2 3 1.50 826 79.2 1.33 X 1O'O 71.5 2.0 x 10'0 47.6 4 1.55 800 140.8 5 1.63 760 220.0 3.33 x 10'0 28.6
The total quantum yield (QY)can then be easily calculated from eq 30 and 4:
QY =
(:)(E)
The quantum yields from each quantum state and from the surface state can also be calculated: R?
(33)
QYss = fC
L + JZB, + 1(Qy)
(34)
Model Calculations Using our kinetic model, theoretical photocurrents as a function of electrode potential and excitation wavelength can be calculated for superlattice electrodes. To make these calculations, it is necessary to specify many parameters and constants. We are presently not attempting to fit our model to experimental data in order to determine the values of certain unknown parameters. Rather, using the model, we want to explore the range of the values of certain critical parameters (such as, for example, kzt and kk) that are required to produce hot electron-transfer processes from superlattice electrodes; we also wish to determine whether our model predicts that hot electron-transfer processes produce a dependence of the photocurrent action spectra on acceptor redox potential. In the calculations reported here, we use the simple expression (eq la) for the electron flux into the last outermost quantum well. The more complicated and rigorous treatment, taking into account recombination in the SCL (eq 1b-4), was found to simply decrease the quantum yield in the saturated region of the photocurrentpotential (I-V) curve depending upon the value of kRSCL;the general shape and the potential of the photocurrent onset was not greatly affected by the value of kRscr. Since the calculations that include SCL recombination take 20-30 times longer than those that do not, and we are presently not interested in fitting experimental data to the model, we do not use the more rigorous treatment. This simplification does not affect our conclusions in this paper. Finally, we note that the inclusion of SCL recombination introduces oscillations in the calculated current-voltage curves because of the alternating GaAs layers. The significance of this effect is not related to the present problem and will be discussed in a future publication. We make our calculations for superlattices with GaAs wells and 2.0-eV barriers; thus, light is only absorbed in the GaAs quantum wells for X > 6200 A and we assume that the absorption coefficient for the GaAs wells is the same as that for bulk GaAs. The optical constants ( n and k ) for GaAs were obtained from ref 6. The reflectivity was calculated from the expression
R =
+ k2 ( n + 1)* + k2
re1 thermalzn rates 1.o
3.06 4.60 7.65
The calculations were made fixing the following conditions: Io = 1.0 mW/cm2, kT = 0.025 eV, c = 6 X lOI9 (0.1 M), and po = 1 X 10l6(3x11~~. The surface-state density (Nss)and Helmholtz shift (AV,) were assumed to range from 1 X IOI3 c d and 0.3 eV, respectively, to 2 X 10l3 and 0.6 eV, respectively;
these values span ranges corresponding from moderate to high levels of surface-state densities. The Helmholtz shifts are obtained from eq 12 assuming a Helmholtz layer capacitance of 5 ~ F / c m ~ . ~ The reorganization energy, XR, was assumed to range from 0.6 to 1.3 eV. These values are based on the approximation that the reorganization energy for redox species at solid electrodes is half that for redox species in homogeneous s ~ l u t i o n . ' ~Values ~ for the latter range from 2.6 eV for slow redox couples (Eu3+/Eu2+)to 1.1 eV for fast redox couples (ferrocene/ferrocenium).*-I0 One set of critical parameters in the model are the rate constants for thermalization of electrons from the various quantum states in the GaAs wells. These parameters are not known and therefore were determined independently in experiments by using a picosecond nonlinear hot luminescence correlation technique." The thermalization times for GaAs/Al,-,Ga,As lattice-matched superlattices and GaAs/GaAso,SPo,S strained-layer superlattices were measured and found to depend upon the excitation energy. Electrons created at the higher energy levels in the quantum well have faster thermalization times than those created near the bottom of the well; the data are summarized in Table I. The nonlinear hot luminescence correlation technique utilizes a focused laser beam, resulting in very high light intensity. The photoelectrochemical experiments use much lower light intensities. Since the thermalization rates were found to increase with decreased light intensity," we cannot use the absolute values of the experimentally measured thermalization rate constants in our model calculations. Instead, we parameterize the thermalization rate to see what values of thermalization rates combined with what values of electron-transfer rates are necessary to produce hot carrier injection. For calculations involving multiple quantum states in a well we use experimental values for the relative rates of thermalization at the specific energy levels in the quantum well and assume that they are independent of light intensity. We discuss below in more detail the results of the calculations defining the conditions required for hot electron transfer and for seeing a shape dependence of the photocurrent action spectra on acceptor redox potential. 1 . Electron Transfer and Thermalization Rates Required for Hot Electron Transfer. The first calculation involves determining the range of thermalization rates and electron-transfer rates that permit significant hot electron transfer out of a single quantum state into solution. The standard potential and reorganization energy of the redox couple are selected such that the peak of the acceptor distribution is isoenergetic with the quantum level under consideration. Figure 3 shows the energy diagram for the system for this calculation. In addition to the values of Io, c, and p o specified above, the following values were used for the remaining parameters: E" = -0.65 V, Ess = -0.12 V, XR = I .O eV, E , = -0.178 V, AV, = 0, Nss = 1 X 10'' L, = 0.1 fim, ktrap=
( n - 1)2
where n = refractive index and k = absorption index = 4 4 a . Also, for GaAs, E = 13.1.
(7) Marcus, R. A. J . Chem. Phys. 1965, 43, 679. Alberry, W. J. Annu. Rev. Phys. Chem. 1980, 31, 227. (8) Frese, K. W. J . Phys. Chem. 1981.85, 3911. (9) Hupp, J. T.; Weaver, M. J. J . Phys. Chem. 1984,88, 6128. Ibid. 1985, 89, 2795. Yee, E. L.; Hupp, J. T.; Weaver, M. J. Inorg. Chem. 1983,22, 3465. (10) Memming, R.; Mollers, F. Ber. Bunsen-Ges. Phys. Chem. 1972, 76, 47s.
(6) Seraphin, B. 0.; Bennett, H. E. In Semiconductors and Semimetals; Willardson, R. K., Beer, A. C., Eds.; Academic: New York, 1967.
( 1 1) Edelstein, D.; Tang, C. L.; Nozik, A. J. Appl. Phys. Lett. 1987, 51, 48. J . Opt. S o t . A m . B Opt. Phys. 1986, 3 (8, Part 2), 32.
7'he Journal of Physical Chemistry, Vol. 92, No. 9, 1988 2491
Electron Transfer from Superlattice Electrodes
-225r
\
.Eox eV
I
Ot --RED
L
Electron Transfer Rate Constant kg, (cm3/sec)
I
Figure 3. Energy diagram for calculations of electron transfer from a single excited state in the well matched to the peak of the distribution of acceptor states in solution. ~0 25
Figure 5. Three-dimensional plot of total fractional hot current quantum yield (summed over four excited-state levels) versus rate constants for electron transfer and thermalization from ne = 2. The wells have five states, and the relationship between the energies and thermalization rates for the quantum levels is given in Table I. r DI 0 7 2 -
Quantum Yield
Electron Transfer Time, ns
Figure 4. Three-dimensional plot of fractional hot current quantum yield versus electron-transfer time and thermalization time for a single excited-state energy level. lo-' cm3/s, k , = lo-' cm3/s, GaAs well width = 250 A, AlGaAs barrier width = 40 A, superlattice periods = 20, a = 1.5 X lo4 cm-l, X = 760 nm, and electrode potential ( E ) = -1.0 V. The results are summarized in a three-dimensional plot, shown in Figure 4, of the fraction of the quantum yield that occurs as hot electron transfer as a function of both the characteristic electron-transfer and thermalization times. The thermalization time is the reciprocal of the first-order rate constant for thermalization, and the electron-transfer time is the reciprocal of the product of the second-order electron-transfer rate constant multiplied by the acceptor concentration. As seen in Figure 4, greater than 10% of the photocurrent will flow via hot electrons if the thermalization time ( T ~ is) longer than about 60 ps and the ~ )faster than about 500 ps. The electron-transfer time ( T ~ is fraction of hot electron photocurrent increases rapidly with decreasing electron-transfer times and increasing thermalization time; it becomes about 25% with T~ = 130 ps and T , ~= 167 ps. The requirements for hot electron transfer become less severe when all the quantum states in a well are considered. This is because the electrons have multiple chances to be injected into the solution as hot electrons as they cascade down the quantum levels in the well. That is, an electron in a given excited quantum state can be injected into solution from that state or thermalize down to the next lowest state; from there it can again either be injected into solution or thermalize down to the next level. Thus, the probability that the electrons will be injected before they fully thermalize to the lowest ne = 1 state is much higher for a given level if multiple levels exist below it in the quantum well compared to the case of a single level of the same initial energy. Since all electrons with quantum numbers ne 2 2 are defined as hot electrons, the probability of hot electron transfer is higher for the multiple level case. The calculations for a multiple level case are summarized in Figure 5 . The quantum well is 250 A wide and contains five levels; these levels have been calculated on the basis of the analysis
1x10'2 1
16 7
40
8 0 1x10"
40
Electron Transfer Rate Constant cdlsec 1 67
80 1 ~ 1 0 ' ~ 1
0 167
Electron Transfer Time ns
Figure 6. A slice of Figure 5 at constant k$-') = 4.6 X lo9 s-' showing the distribution of hot current between ne = 2 and n, = 5 as a function of ko,,.
in ref 12 to be at the following energies above the bottom of the well: 8.8, 34.2, 79.2, 140.8, and 220 meV. The thermalization times for each of the levels are assumed to have the same relative values as measured experimentally by using the nonlinear hot luminescence corrtlation technique (Table I). The first excited ne = 2 state has the slowest thermalization rate; if it is normalized to 1.0, then the thermalization rates for the next three successively higher quantum levels are 3.06, 4.60, and 7.66 times faster." In the three-dimensional plot shown in Figure 5, the fractional hot electron photocurrent is plotted against ktt, the maximum second-order electron-transfer rate constant, and against k$'), the first-order thermalization rate constant for the ne = 2 quantum level. The thermalization rate constants for the levels above ne = 2 are proportionally higher by the ratios indicated above. As seen from Figure 5, greater than 10% of the photocurrent will be injected as hot electrons if ktt is greater than about 1 X lo-" cm3/s ( T C~ 1.7 ~ ns) and k$') is less than about 8 X lo9 s-l (T$*) > 125 ps) with corres onding thermalization rate constants for the higher levels of k$! = 2.4 X 1OIos-l ( T $ ~ )= 41 ps), ki4) = 3.7 X 1Olos-l (TY) = 27 ps), and ki5) = 6.1 X 1Olo s-l (TP) = 16 ps). If ko > 1.8 X lo-'' cm3/s ( T C~ 0.9 ~ ns and k v ) < 4.6 X 1Olos-l ( T @ > 22 ps), then the fractional hot current is greater than about 50%. An indication of the distribution of the hot electron photocurrent among the quantum levels in the well is given in Figure 6. Figure 6 is a slice through the three-dimensional plot of Figure 5 at a ( 12) Miller, R. C.; Kleinman, D. A,; Gossard, A. C. Phys. Reu. E : Condens. Matter 1984, 29, 7085.
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The Journal of Physical Chemistry, Vol. 92, No. 9, 1988
constant set of thermalization rate constants (ki’l = 4.6 X lo9 s-l). In addition to the total fraction of the photocurrent injected as hot electrons, the hot photocurrent fraction for quantum levels with ne = 2 and ne = 5 (lowest and highest excited quantum states, respectively) are also plotted against ktt in Figure 6. The highest hot electron-transfer fraction always comes from level 2, and the fraction decreases with increasing quantum number. This is because of the increasing rate of thermalization with increasin !i energy above the well bottom. As seen from Figure 6, with k,, = 4 X lo-” cm3/s, about 6.4% of the total hot photocurrent comes from level 5 compared to about 20% from level 2 . The distribution of the thermalized photocurrent between the ne = 1 quantum level and the surface states depends upon the values of ktiapand kY In our calculations, these two rate constants have been set equal to each other. At trapping and recombination rate constant values 10% of the photocurrent will flow from this level as hot electrons if the thermalization time ( r T )is >60 ps and the electron-transfer time ( T ~ Jis 10% of the photocurrent will be hot (Le., flowing from levels with ne > 1 ) if the electron transfer time is 125 ps for ne = 2 and > 16 ps for ne = 5. These numbers for the multiple energy level case are based on the assumption that thermalization proceeds stepwise through the various levels in the well and not by a direct pathway to the ne = 1 state or to surface states. The validity of this assumption will be examined in future studies. The required thermalization and electron-transfer times described above for hot e- injection to occur are attainable in real systems. Direct measurements of the thermalization time of hot carriers in superlattices have been made by using the nonlinear hot luminescence correlation technique." These data show that for lattice-matched superlattices (GaAs/Alo,3sGao,62As)with 250-A wells and either 40- or 250-A barriers, the thermalization times range from 350 ps near the well bottom (ne = 2) to 50 ps near the top of the well (ne = 5). For strained-layer superlattices (GaAs/GaAso.5Po,5)with the same dimensions, the corresponding thermalization times were about 150 and 30 ps, respectively. However, it must be noted that these times correspond to very high light intensities (photodoping levels of 5 X lo1*~ m - ~and ), they decrease with decreasing light intensity." The thermalization times for bulk GaAs at the above energies are 40 and 5 ps, respectively.'' Measurements of the electron-transfer times from semiconductor electrodes are sparse. Willig and c o - ~ o r k e r sand ' ~ Miller14 have measured charge-transfer times at semiconductor electrodes in the 1-ns to 100-ps range. Thus, the thermalization and electron-transfer times specified above from model calculations for significant hot electron transfer have indeed been experimentally found in some systems. However, it will be necessary to make such measurements on the same system being specifically investigated for hot electron-transfer effects. Our calculations show that band-edge-movement (BEM) effects can be very important in determining the degree of hot electron transfer. This is true for the specific case of high-lying redox acceptors (negative Eo values) where the initial alignment of quantum states and acceptor states in solution is poor but becomes favorable for excited-state electron transfer when the band edges move upon surface-state charging. Since BEM is strongly dependent upon ktrapand kp, we find a critical value of these parameters where hot electron transfer becomes important (Figure 7). The degree of hot electron transfer to high-lying acceptors also depends upon whether the surface-state energies move with the band edges; fixed surface states lead to much higher (2-3X) levels of hot electron transfer. This is because with fixed surface states, BEM leads to better overlap of the quantum well states with the acceptor states while the poor surface-state overlap remains constant. On the contrary, surface-state movement with BEM (13) Bitterling, K.; Willig, F. J . Elecfroanal. Chem. Interfacial Elecrrochem. 1986, 204, 21 1. Willig, F.; Bitterling, K.; Charle, K.-P.; Decker, F. Ber. Bunsen-Ges. Phys. Chem. 1984, 88, 374. (14) Miller, R. J. D., personal communication.
J. Phys. Chem. 1988, 92, 2501-2506 leads to better surface-state overlap, which can dominate the electron-transfer kinetics. Little is known about the nature of surface states and whether one has the type that moves with BEM (derived from the solid) or does not move (derived from the liquid). In our calculations, we also briefly studied the effects of XR on the hot electron-transfer processes. Preliminary calculations indicated that, within the range of the model parameters we studied and described above, the results and conclusions are not sensitive to XR over the range 0.6-1.3 eV. An important question we considered is whether hot electron transfer from superlattice electrodes can be detected through a dependence of the shape of the photocurrent action spectrum on acceptor redox potential. Our model calculations show that this is indeed the case for a very restricted region of parameter space, namely, where the degree of hot electron transfer is very high for the high-lying acceptors (Le., where >75% of the photocurrent is via hot electron injection) and the superlattice is in the miniband regime such that the bulk of the electrons photogenerated at a given energy level remain in that level as they traverse the superlattice. However, as the fraction of hot electron transfer decreases (below about 40-50%) or miniband formation is destroyed (such that electrons generated at low energies in the superlattice interior can be distributed at higher energies in wells near the surface), then a dependence of the action spectra on redox potential is not an appropriate signature of hot electron-transfer effects. In other words, one can have significant hot electron photocurrent (up to 50%) and not see a dependence of photocurrent spectral shape on redox potential. The latter situation
2501
represents a much larger region of parameter space and will be present generally with lower values of kt,, higher values of k i , and thicker barriers. In our model calculations, this situation also arises with lower values of ktrapand k,. The values of ktrapand k , used for Figure 9 correspond to extremely high rates of surface trapping and surface recombination. Lower values of k,,,, and k, could be used in combination with different values of other parameters (such as k$ AVH, k i , and E o ) to maintain good overlap of the excited quantum states with the acceptor levels in solution under illumination, such that a high degree of hot electron transfer and the corresponding dependence of photocurrent spectral shape on acceptor redox potential are also maintained. In our previously published experimental results with latticematched GaAs/Alo,38GaAso,6z superlattices and cobaltocenium and ferrocenium in acetonitrile,’ we did not observe a dependence of the photocurrent action spectrum on acceptor redox potential. We interpreted this’ to indicate that hot electron transfer was not a dominant process in that system. However, in light of the present detailed calculations, we cannot make this conclusion. Additional experiments designed to detect hot electron injection into solution with specificity are required to establish the importance of this process in superlattice electrodes.
Acknowledgment. This work was funded by the US. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences; M.W.P. was supported by the SERI Director’s Development Fund. Registry No. GeAs, 1303-00-0.
Fractal Character of Compact Layers of Tris(2,2‘-bipyridlne)cobalt( I I ) Perchlorate Formed at the Mercury/Solutlon Interface by the Electrochemical Nucleation and Growth Mechanism Lubodr Pospkil The J. Heyrovskjj Institute of Physical Chemistry and Electrochemistry, Czechoslova& Academy of Sciences, Wazskd 9, 11840 Prague, Czechoslovakia (Received: March 16, 1987; In Final Form: July 20, 1987)
The ac technique has been applied to the investigation of the growth of a compact film of tris(2,2’-bipyridine)cobalt(II) perchlorate at the mercury-aqueous solution interface. The previously reported deviations from the nucleation and growth models at long growth time are scrutinized by means of time-resolved FFT impedance measurements, which yield the effective fractal dimension D of the film. The maximum value D = 2.16 is interpreted in terms of a three-dimensional, hexagonally packed cluster-cluster aggregation model.
Introduction The mechanism of formation and growth of a new phase at the interfaces or in the bulk is a subject of current interest motivated by an effort to understand the organization and properties of small structures of matter and stochastic laws governing the chaotic processes, like the aggregation of particles in a diffusion field or random walks of reactants on lattices. The effects of surface structure play an important role in the heterogeneous kinetics of catalysts and their overall selectivity.’l The electrochemical phase formation was investigated mainly with respect to metal deposition where the formation of a new metallic phase is connected with the electron transfer from the ele~trode.~” Both the theoretical ~~
(1) Anacker, L. W.; Parson, R. P.; Kopelman, R. J . Phys. Chem. 1985, 89, 4758. (2) Meakin, P. Chem. Phys. Lett. 1986, 123, 428. (3) Bewick, A.; Fleischmann, M.;Thirsk, H. R. Trans. Faraday Soc. 1962, 58. 2200.
0022-365418812092-2501$01SO10
background and experiments are well worked outs3” During the past decade many other electrochemical systems were described which form deposits of solid phase at the electrode-solution interface without participation of a redox p r o c e s ~ . ~This - ~ ~ type (4) Rangarajan, S. K. Faraday Symp. Chem. SOC.1977, No. 12, 101. (5) Bosco, E.; Rangarajan, S. K. J . Chem. SOC.,Faraday Trans. 1 1981, 77, 1673. (6) Bosco, E.; Rangarajan, S. K. J . Electroanal. Chem. Interfacial Electrochem. 1981, 129, 25. (7) Vetterl, V. Collect. Czech. Chem. Commun. 1966, 31, 2105. (8) Sathyanarayana, S.; Baikerikar, K. G. J . Electroanal. Chem. Interfacial Electrochem. 19.10, 25, 209. (9) Kuta, J.; PospiSil, L.; Smoler, J. J . Electroanal. Chem. Interfacial Electrochem. 1977, 75, 407. (10) Retter, U.J . Electroanal. Chem. Interfacial Electrochem. 1978, 87, 181. (1 1 ) Pospkl, L. J . Electroanal. Chem. Interfacial Electrochem. 1981, 123, 323. (12) Gierst, L.; Franck, C.; Quarin, G.; Buess-Herman, CI. J . Electroanal. Chem. Interfacial Electrochem. 1981, 129, 3 5 3 . (13) Sridharan, R.; de Levie, R. J . Phys. Chem. 1982, 86, 4489.
0 1988 American Chemical Society
