3040
J . Phys. Chem. 1990, 94, 3040-3045
New Aspects of Electron Transfer on Semiconductor Surface: Dye-Sensitization System T. Sakata,*.+ K. Hashimoto,*J and M. Hiramotoff Institute f o r Molecular Science, Myodaiji, Okazaki 444, Japan (Received: November 14, 1988; In Final Form: October 17, 1989)
A simple model for the electron transfer (ET) from excited dyes to the conduction band of semiconductors is proposed. It
is shown that the equation for the ET rate does not contain the FranckCondon term, but the state density of the semiconductor plays an important role. The energy gap dependence of ET rates in the dye-sensitizedsemiconductors, which is quite different from those in molecular systems, is explained well by this model. The electron exchange energy is determined for Ru complexes and Rhodamine B on oxide semiconductors. Temperature dependence of the electron exchange energy term, caused by thermal motion of adsorbed dyes, is suggested as a new origin of activation energy of ET on the surface. The temperature dependence of the ET rates is explained well by this model.
Introduction Much progress has been made in recent years on the subject of electron transfer (ET) between molecules. For instance, the inverted region, which was predicted theoretically by Marcus,' was proved experimentally in various molecular systems by Miller -~ and Mataga and Closs2 and by other several g r ~ u p s . ~Kakitani derived a general formula for the energy gap dependence of the ET rate, taking into consideration the different polarization effect of solvent on the initial and final states of ETS6 They obtained three different energy gap laws for the photoinduced charge separation, charge shift, and charge recombination reactions. According to this model, the inverted region does not appear in the case of charge separation because the potential curvature along the solvent coordinate in the initial neutral state is much smaller than that in the final ionic state. This theory explains the anomalous energy gap dependence of the ET rate observed for photochemical charge separation by Rehm and Weller.7 On the other hand, ET in heterogeneous systems is not well understood, compared with ET between molecules. However, since heterogeneous ET plays important roles in (photo)electrochemistry, photocatalysis, and photography, a lot of work has been undertaken to understand the process. The dynamics of the process has been studied by various techniques, such as picosecond time-resolved Raman,8 transient absorption9J0 and luminescence,' transient photocurrent and p h ~ t o p o t e n t i a l , ' and ~ ~ ~ ~transient grating met hods. I In the investigation of the time-resolved luminescence spectroscopy of the dye sensitization of semiconductors, we have found a new type of energy gap dependence of ET rates. The ET rates increase monotonically with increasing the energy gap.l6,l7 This energy gap dependence is quite different from that observed in molecular systems, including ET on the surface of molecular crystals,l* where an inverted region is observed. Recently we have reported temperature-independent ET in semiconductor dye sensitization systems.Ig Here we will propose a theoretical model of ET from the excited dye to the continuous levels (conduction band) of the semiconductor. The anomalous energy gap dependence and temperature-independent ET in the dye sensitization is explained well by this model. However, recently we have observed a new kind of temperature dependence in semiconductor-dye sensitization systems. It is suggested that the temperature dependence of the exchange energy, caused by the thermal motion of adsorbed dyes, is the origin of the activation energy on the solid surface. 'Present address: Department of Electronic Chemistry, Tokyo Institute of Technology, 4259 Nagatsuda, Midori-ku, Yokohama 227 Japan.
*Present address: Department of Synthetic Chemistry, Faculty of Engineering, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan. 4 Present address: Department of Chemical Process Engineering, Faculty of Engineering, Osaka University, Suita, Osaka 565, Japan.
0022-3654/90/2094-3040$02.50/0
Theory of Energy Gap Dependence on Semiconductor Surface In the dye sensitization of semiconductors, adsorbed dyes are photoexcited and they inject electrons or holes into the semiconductors.zo*zl In the case of electron transfer to the conduction band of n-type semiconductors, an anodic current flows, whereas in the case of hole transfer, a cathodic current f l o ~ s . ~ However, ',~~ the detailed mechanism of dye sensitization is not well established. There are three possible mechanisms, Le., (1) the direct ET to the conduction or valence band of the semiconductor, ( 2 ) energy transfer from the excited dye to the surface states followed by the electron (or hole) injection from the excited surface states, and (3) the ET from the excited dye to the surface states and from the surface states to the continuum of the semiconductor.
( I ) (a) Marcus, R. A. J. Chem. Phys. 1956,24,966. (b) Marcus, R. A. J. Chem. Phys. 1956, 24, 979. (2) Miller, J. R.; Calcaterra, L. T.; Closs, G.L. J . Am. Chem. Soc. 1984, 106, 3047. (3) Wasielsky, W. R.; Niemczyk, M. P.; Svec, W. A,; Pewitt, E. B. J. Am. Chem. SOC.1985, 107, 1080. (4) Irvine, I. P.; Harrison, R. J.; Beddard, G.S.; Leighton, P.; Sanders, J. K. M. Chem. Phys. 1986, 104, 315. ( 5 ) (a) Kakitani, T.; Mataga, N. J. Phys. Chem. 1986, 90, 993. (b)
Mataga, N.; Asahi, T.; Kanda, Y.; Okada, T.; Kakitani, T. Chem. Phys., in press. (6) (a) Kakitani, T.;Mataga, N. Chem. Phys. 1985,93,381. (b) Kakitani, T.; Mataga. N. J. Phys. Chem. 1986, 90, 993. (7) Rehm, D.; Weller, A. Isr. J. Chem. 1970, 8, 259. (8) Rossetti, R.; Brus, L. E. J. Phys. Chem. 1986, 90, 558. (9) Moser, J.; Gratzel, M.; Sharma, D. K.; Serpone, N. Helo. Chim. Acta 1985, 68, 1686.
(IO) Nosaka, Y.; Fox, M. A. Langmuir 1987, 3, 1147. ( 1 1 ) Kamat, P. V.;Fox, M. A. Chem. Phys. Lett. 1983, 102, 379. (12) Hashimoto, K.; Hiramoto, M.; Sakata, T.; Muraki, H.; Takemura, H.; Fujihira, M. J. Phys. Chem. 1987, 91, 6198. (13) Itoh, K.; Baba, R.; Fujishima, A. Chem. Phys. Left. 1987, 135, 521. (14) Sakata, T.; Janata, E.; Jaegermann. W.; Tributsch, H. J. Electrochem. SOC.1986, 133, 339. (15) Nakabayashi, S.; Komuro, S.; Aoyagi, Y.; Kira, A. J. Phys. Chem. 1987, 91, 1696. (16) Hashimoto, K.; Hiramoto, M.; Lever, A . B. P.; Sakata, T. J. Phys. Chem. 1988, 92, 1016. (17) Hashimoto, K.; Hiramoto, M.; Sakata, T. Chem. Phys. Lett. 1988, 148, 215. (18) Kemnitz, K.; Nakashima, N.; Yoshihara, K. J. Phys. Chem. 1988, 92, 3915. (19) Hashimoto, K.; Hiramoto, M.; Sakata, T. J. Phys. Chem. 1988, 92, 4272. (20) Gerischer,,H. Phorochem. Photobiol. 1972, 16, 243. (21) Tributsch, H.; Gerischer, H. Ber. Bunsen-Ges. Phys. Chem. 1%9, 73, 251. (22) Memming, R.; Tributsch, H. J. Phys. Chem. 1971, 75, 562.
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990 3041
Electron Transfer on Semiconductor Surface
I
Stat. Donrity of Somiconductor
.------e-
C.B. -
excited state
state
I
-e-+-
ground state
molecule
semiconductor
Figure 1. Schematic illustration of dye-sensitized electron transfer on a semiconductor surface. Under photoexcitation of adsorbed dyes, electrons transfer from the excited dyes to the semiconductor.
The recent experimental results from our laboratory suggest strongly the direct electron transfer from the excited dye to the conduction band.16J7J9.23The situation is illustrated in Figure I . Now let us consider the ET rate to the continuous states of semiconductors. The ET rate between two molecular electronic levels can be expressed as k,, = (2?r/h)uZ(FC) (1) where FC is the Franck-Condon weighted density of states. Under a high-temperature approximation it is given by the following equation.24-28 FC =
1
(4rXk T)'I2
1
(AGO + 4XkT
(2)
Here AGO is the free energy change in the electron-transfer reaction. This equation is equivalent to the one from Marcus theory.lqZi The ET rate, k(E), between a single electronic level at the energy E in the conduction band and the excited dye can be written by the same equatiomZ5
"1' -
+
2r 1 ( E - AE A)' k(E) = -uZ h (4~XkT)~l~ 4XkT
1
=
2* -u2D-(E) (3) h Here AE is the energy difference between the energy level of the excited dye and the bottom of the conduction band (see Figure I). E is the energy of an electron-accepting level in the conduction band and u is the electron exchange energy between Bloch function, qE,of the conduction band electron with an energy E and that of the dye, qD. = (@E(Hint(*D) (4) and DJE) =
1
(4rXk T ) I J 2
+
( E - AE 4XkT
1
I Figure 2. The state density, p ( E ) , of the conduction band of a semiconductor and the distribution function, DJE) of the excited dye whose peak is located at AE - A above the bottom of the conduction band. Because of a small reorganization energy, A, on the surface under vacuum, the half-width of distribution function of the dye is very small compared with the width of the conduction band.
in the electron transfer is the same between the initial and final state and a single quantum mode is involved in the transferZ5 DJE) =
1
coth (hw/2kT) X (4rhkT) 'I2 [AE - E - X - ( m - n)hwI2 CC exp 4XkT m n
I \
1
A2 exp(-hw/2kT) exp( mhw/ k T) 4 sinh (hw/2kT)
rL
m!n! ( 6 )
Here A represents a reduced displacement parameter. At lower temperatures this equation should be used instead of eq 5. However, under a high-temperature approximation, eq 6 is reduced to eq 5.25 The electron-transfer rate from the excited dye to the continuous electronic levels can be obtained by integrating k(E) over the whole energy band.26,27 k,, = S k ( E ) P ( E ) ( 1 -f(E))d E
(7)
Here, p ( E ) andf(E) represent the state density of the conduction band and the Fermi distribution function. In eq 7 p(E)( 1 -f(E)) represents the vacant-state density of the conduction band. Except for the highly doped n-semiconductors,f(E) in the conduction band is negligibly small. Under such a situationf(E) can be neglected and the following equation is obtained from eqs 3 and 7 . 2r k,, = -i;u2SD-(E) p ( E ) d E
(5)
D-(E) is the distribution function of electrons in the excited dye.2728 D-(E) denotes the electron removal ~ p e c t r u m . ~ , ~ ' The quantum mechanical theory gives the following analytical form for DJE) when the force constant of the vibration involved (23) Sakata, T.; Hiramoto, M.; Hashimoto, K. Proceedings of a Symposium on Photoelectrochemistry and Electrosynthesis on Semiconducting Materials; Electrochemical Society: Pennington, NJ, 1988; p 428. (24) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265. (25) Kestner, N. R.; Logan, J.; Jortner, J. J . Phys. Chem. 1974, 78,2148. (26) Levich, V. G. Adu. Electrochem. Electrochem. Eng. 1966, 4 , 249. (27) Hopfield, J . J . Proc. Narl. Acad. Sei. U S A 1974, 71, 3640. (28) Gerischer, H . Adu. Electrochem. Electrochem. Eng. 1961, I , 139.
The reorganization energy X is very small in a vacuum. It is estimated in the order of 0.1-0.2 eV.I8 Therefore, the distribution function D-(E) has large values only around AE - A. On the other hand, the state density p ( E ) is a monotonic function. Figure 2 illustrates the situation. Under such a situation, the right-hand side of eq 8 is approximated as k,, = S2 ;r U ~ ~ ( A E - X ) S D - ( dEE) (9) Since the distribution function is normalized
3042
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990
Sakata et al.
the following simple expression for the electron-transfer rate on the solid surface is obtained. 27r k,, = r u 2 p ( A E - X ) (11) This equation can be transformed into a more useful form. In eq 4, the Bloch function of the conduction band, \kE, is expressed by a linear combination of the Wannier function (approximately, atomic orbital) at the nth lattice point, $n.29
Here N is the number of atoms (or molecules) of the semiconductor, is the coefficient of $ p and k is the wave vector. Then, u2 =
=
(*E(ffin,l*D)2
~(~~un$nIffintl*D)2 n
(I3)
1
-($~IHi"tl\kD)~
N
1 = -u2
N
(14)
Here u is the exchange energy expressed by the following equation. =
($slHintl*D)
The exchange energy u thus defined is not dependent on N a n d has a similar meaning to the electron exchange energy in the case of ET between molecules. From eqs 1 1 and 13, the ET rate is written as
27r = -u~~~(AE-x) h
(15)
Po(E) = P ( E ) / N
(16)
excitedstate
I 1
:
ground state
~~~~
_.
1
When dSis the atomic (or molecular) wave function of the surface atom (or molecule) interacting with the adsorbed dye molecule, it is reasonable to assume that the overlaps of with the &, apart from that of &, are negligible. Therefore, eq 13 can be approximated as follows u2
1 -+-
- -
molecule I
I
ramiconductor
Figure 3. Schematic illustration of hole transfer in the dye sensitization on a semiconductor. Under excitation of the dye, holes transfer from the excited dye to the valence band of the semiconductor.
applied. The situation for hole transfer in the dye sensitization is illustrated in the Figure 3. In this case, the hole-transfer rate is written as 2* (21) ket = h P ( E )AE) d~
--u2S~+(~)
analogous to eq 8. Here, D + ( E ) is the distribution function of holes generated in the highest occupied molecular orbital of the excited state of the dye and denotes the electron insertion spectrum (see Figure 3 ) . p ( E ) represents the state density of the valence band. Under a high-temperature approximation, the D + ( E ) is written as2'
Here po is the normalized state density. Since the state density,
By use of similar procedures, the following equation, analogous to eq 15, is derived. 2* kef = -u2p0(AE+X) h
p(E), is proportional to N, the normalized state density po(E) is independent of N. Moreover, the exchange energy, u, is inde-
Near the edge of the valence band, a relation similar to eq 19 is obtained.
and
pendent of N . Consequently, eq 15 is a more useful equation than eq 11. In order to apply this equation to the experimental data, a further approximation is necessary. In the case of an isotropic crystal, the state density near the band edge has Ell2 dependenceB
k,, = C ( A E
+
X)II2
(24)
Here m h is the effective mass of the hole in the valence band. By using the relation V = Nuo (uo, unit cell volume), we get
Here m, is the effective mass of electrons in the conduction band and V is the volume. The equation ( 1 7 ) is commonly used to express the state density near the band edge. From eqs 15 and 18 we can obtain
k,, = C ( A E - A)'/*
(19)
In the case of hole transfer from the excited dye to the valence band of the semiconductor, an analogous formulation can be ( 2 9 ) For instance, Ziman, J. M. Principles of the Theory o f s o l i d s ; Cambridge University Press: London, 1965.
Analysis of the Experimental Results by the Present Theory Equation 15 has a very simple form. Let us use this equation to analyze the energy gap dependence of the electron-transfer rate obtained for Ru complexes and Rh B on various oxide semiconductors in which an electron is considered to transfer from the excited dye to the semiconductor's conduction band. Figure 4 shows the results of the analysis of the energy gap dependence of the fastest electron-transfer component for Ru(bpy)32+and Ru(bpz),2+ on various oxides at room temperature.16 The solid line shows the theoretical curve with h = 0.0 eV. In this analysis, it is assumed that me,u, and uo take common values among different oxides. In this figure, the energy gap values in the vacuum should be used, because the experiments were performed under v a ~ u u m . ' ~ *However, ~' the experimental data in vacuum are not available at present, so electrochemical data are used to evaluate the energy gap, AE, by using the following relation, A E = -e(&,* - E&) (26) In eq 26 E,,* and Em are oxidation potential for the excited state of the adsorbed dye and the flat-band potential of the semicon-
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990 3043
Electron Transfer on Semiconductor Surface
- 7 0 1
0.0
0.5
1 .o
1.5
Energy Gap, AE/eV
Figure 4. Dependence of electron-transfer rates at room temperature from photoexcited Ru( 11) complexes to semiconductors on the energy gap, AE, which is defined in eq 26. The electron-transfer rates in this figure represent those of the fastest component.I6 The solid line shows the theoretical curve calculated by using eq 19 with A = 0.0 eV and C = 6.35 X IOi4 (g cm2)-i/2.Assuming me = mo and uo = 30 A', the electron exchange energy, D, is evaluated to be 5.2 cm-' from eq 20.
ductors at pH 0, re~pectively.'~J~ The theoretical curve based on eq 19 with X = 0.0 eV is shown by a solid line. As seen in this figure, the agreement between theory and experiment is good. This fact strongly supports the mechanism of the direct ET to the conduction band of the semiconductor. From this analysis, the value of C of eq 19 is determined as 6.35 X lOI4 (g cm2)-1/2,Consequently the exchange energy u can be evaluated by using appropriate values of me and uo in eq 20. The following values for me of oxide semiconductors have been reported, 0.3mo for Sn02,3010-20mo for Ti02,3i1.8mofor wo3,32 1.2mo for NiO?, and 0 . 2 4 ~for 1 ~Zn0.34 Here mo is the electronic rest mass. Now, we use me = mo tentatively in the present calculation. As for uo,the crystallographic data give rigorous values; for example, it is 31.1 A3 for Ti02 (rutile), 32.5 A3 for TiOz (anatase), 35.8 ASfor SnO,, 32.9 A3 for WO,, 29.8 A3 for SrTi03, and 32.2 A3 for BaTiO,. Consequently 30 A3 is used here as an approximate value for uo. When these values are used, po(E)/uo becomes 0.20/eV at E = 1 .O eV. This value looks reasonable as a normalized state density, because it is expected to be in the order of 1 /eV. By use of these values, 5.2 cm-I is obtained as the value of the electron exchange energy, u, of the fastest ET component for the Ru complexes on the oxide semiconductors. Figure 5 shows a similar analysis for the fastest electron-transfer component in the case of dye sensitization of RhB on various oxide semiconductors at room t e m p e r a t ~ r e . 'The ~ solid line shows a theoretical curve with X = 0.0 eV and C = 2.92 X IOi5 (g ")-I/*. With uo = 30 A3 and m, = 1 .Omo,we find from eq 20 that the electron exchange energy u is 1 I . 1 cm-' for RhB/oxide semiconductors. This value is about 2 times larger than that of the Ru complexes on oxide semiconductors. Experimentally it is observed that RhB is adsorbed more strongly than the Ru complexes on the oxide surfaces. Since RhB is a flat molecule, its adsorption may cause a larger exchange interaction than that of the Ru complexes. As seen in Figures 4 and 5, the present theory explains well the absence of an inverted region and a monotonical increase of the ET rates with the energy gap. Marcus suggested that there would be no inverted region when there is a significant band of electronic states acceptable rather than a single electronic state.35 The present theory corresponds to this situation. In the case of RhB, the ET rates on SnO2and W 0 3 deviate from the theoretical (30) (31) (32) (33) (34) (35)
Jarzebski, Z. M.; Martan, J. P. J . Electrochem. Soc. 1976,123, 199C. Frederikse, H . P. R. J . Appl. Phys. 1961, 32, 2211. Berak, J. M.; Sienko, M. J. J . Solid State Chem. 1970, 2, 109. Spear, W. E. S.; Tannhausen, D. S . Phys. Reu. 1973, 137, 8 3 1 . Baer, W. S. Phys. Reo. 1967, 154, 785. Marcus, R . A . J . Chem. Phys. 1965, 43, 2654.
0
0.5
1 .o
1.5
Energy Gap, AE/(eV) Figure 5. Dependence of electron-transfer rates at room temperature from photoexcited Rhodamine B to various oxide semiconductors on the energy gap, AE, which is defined in eq 26. The electron-transfer rates in the figure represent those of the fastest component." The solid line shows the theoretical curve calculated by using eq 19 with X = 0.0 eV and C = 2.92 X IOi5 (g cm2)-'/*. Assuming me = m0 and uo = 30 A', the electron exchange energy, v , is evaluated to be 1 1.1 cm-I from eq 20.
curve. There may be several possibilities for this disagreement: (1) As will be discussed later, the ET rate on S n 0 2and W 0 3 shows a relatively large temperature dependence; at high temperature the exchange energy u increases because of low vibrational frequency for the motion of the adsorbed molecule relative to the surface. If this temperature dependence is taken into consideration, the deviation for S n 0 2and WO, becomes smaller. (2) The Ell2 dependence is not a good approximation for the state density of these oxides. (3) There is the possibility of the superposition of another band in this large AE region.3638 (4)Assumptions used for the theoretical analysis such as constant me and u may break down. Except for this discrepancy, the present theory explains well the experimental results. Although our experiments were carried out under vacuum (Torr), the energy gap AE estimated from the electrochemical data looks reasonable. This interesting result seems to suggest that the energetical situation, especially the relative location of energy levels, is not so different from that in aqueous medium probably because the surface of the oxides is covered with O H g r o ~ p s . ~The ~ , preliminary ~~ evaluation of AE in vacuum by ultraviolet photoelectron spectroscopy measurements supports this idea.41 We should mention the limitation of the present theory. Since the bandwidth of typical molecular crystals such as anthracene is in the order of 0.03-0.1 eV,42the present theory cannot be applied to the dye sensitization of molecular crystal^!^-^ Recently the existence of an inverted region has been reported by Kemnitz et al. and it is explained by the conventional ET theory between molecules.l* Temperature-Independent or Insensitive Electron Transfer
As seen in eqs 1, 2, and 6, the Franck-Condon term is temperature-dependent. Thus, in the case of electron transfer between molecules, the ET rate depends on temperature and the origin of the activation energy of the electron transfer is considered to ~~
~
(36) Tsukada, M.; Adachi, H.; Satoko, C. Prog. Surf. Sci. 1983, 14, 113. (37) Robertson, R. J . Phys. C 1979, 12, 4767. (38) Mattheiss, L. F. Phys. Reu. B 1972, 6, 4718. (39) Boehm, H . P. Discuss. Faraday Soc. 1971, 52, 264. (40) Morrison, S . R. Electrochemistry at Semiconductor and Oxide Electrodes; Plenum Press: London, 1980. (41) Hashimoto, K.; Sakata, T. Unpublished data. (42) Silbey, R.; Jortner, J.; Rice, S . A,; Vala, M. T. J . Chem. Phys. 1965, 42, 733. (43) Gerischer, H.; Willig, F. Topics of Current Chemistry; Boschke, F. L., Ed.; Springer: Berlin, 1976; Vol. 61, p 31. (44) Nakashima, N . ; Yoshihara, K.; Willig, F. J . Chem. Phys. 1980, 73, 3553.
3044
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990 Semiconductor
Sakata et al. the classical case, the average of the exchange energy term, D~ is expressed as
Here, p(x) is the probability of a molecule existing at a distance x from the surface. In the case of a harmonic potential, this is written as2' Molecule
~
I
\
v2=v;
exp(-ax)
P(X) =
exp{-F]a(x-xo)2
where a is the force constant and xo the equilibrium position of the adsorbed molecule. The exchange energy term is attenuated exponentially with x,24.47
\
u2(x) = cO2exp(-ax) By using eqs 29 and 30 we get = uo2 exp(-axo) exp( 7 a2kT
02
X /Distance Figure 6 . Illustration of temperature dependence of the electron exchange energy. Since the electron exchange energy is attenuated exponentially with increasing distance between the molecule and the surface, x, thermal population of the adsorbatesubstrate vibration influence the
be the Franck-Condon term. I n the case of Marcus theory, the activation energy, E,, is (AGO + 4X
Experimentally the electron-transfer rate depends strongly on t e m p e r a t ~ r e . ~On ~ the other hand, in the case of ET between molecule and solid, the state density of the conduction or valence band appears instead of the Franck-Condon term, as shown in eqs I 1 and 15. Since the state density is essentially temperature-independent, temperature-independent ET would be expected. Recently, we reported temperature-independent ET in the case of RhB/Ti02 and RhB/Zr0.6.'9 Interesting data which suggest the existence of temperature-independent ET in the photographic systems were reported by Tani et al.46 They are explained well by the present model. Temperature-independent ET rate seems one of the characteristic features specific to ET on the solid surface.
New Origin of Activation Energy on Solid Surface Interestingly, an activation energy has been observed for ET in several dye sensitization systems, even though the activation energy is generally very small, on the order of IO-2 eV. The existence of an activation process is surprising, because the excited energy level of the dyes is located significantly higher in energy than the bottom of the conduction band of the oxide semiconductors. Even in the "temperature independent" ET cases described above careful examination reveals a very small increase of ET rate above 150 K.I9 These results suggest that there is another origin of the activation energy on solid surfaces. Since the FC term disappears on the surface, the only possibility is the electron exchange energy term, u2(x). Since the exchange energy term is attenuated exponentially with increasing distance between the molecule and the surface, x,24*47 a thermal population of the adsorbate-substrate vibration influences the magnitude of the exchange energy. The situation is illustrated in Figure 6. The temperature dependence of the exchange energy term is obtained by calculating its thermal average. This is done rather easily in the case of a harmonic potential for the adsorbed molecule. In (45) DeVault, D.; Chance, B. Biophys. J . 1966, 6 , 825. (46) Tani. T. Phorogr. Sci. Eng. 1984, 28, 150. (47) Miller. J. R.; Beits, J. V.; Huddleston, R. K. J . Am. Chem. SOC.1984, 106, 5057.
(30)
)
As shown in eq 31, the exchange energy is temperature-dependent. Quantum mechanically the average of u(x) is given by $(x) =
exchange energy.
E, =
(29)
Tr u2(x) exp(-P%) Tr exp(-P%)
p=- 1
kT where 7f is the Hamiltonian for the thermal vibration of the adsorbate. In the case of a harmonic potential this is calculated as [see Appendix] (33) where a = mw2, 0 = hw/2k, and u2(xo) = vo2 exp(-ax,). And v(xo) indicates the electron exchange energy at x,,. The quantum mechanical equation (33) is reduced to the classical limit, i.e., eq 32, when T a. From eqs 15 and 33, we obtain the temperature-dependent ET rate from an excited dye to the conduction band as follows.
-
2*
k,, = -u2(xo) po(AE-X) exp h As seen in eqs 33 and 34, the temperature dependence of the ET rate is determined by three parameters, Le., (1) the vibrational quantum, hw; (2) the attenuation parameter, a;and (3) the mass of adsorbate, m. Figure 7 shows the result of the temperature dependence of the ET rate in the cases of RhB on Ti02, ZrO2,I9and Sn02. As seen in this figure, a slight increase in ET rate is observed at temperatures higher than 150 K in the case of T i 0 2 and Zr02, although the ET rates are almost temperature-independent. The solid lines in the same figure show the theoretical curve calculated by using eq 33 with a = 2.0 A-' 48 and h w = 20 cm-'. These theoretical curves agree well with the temperature dependence of RhB on T i 0 2 and Z r 0 2 . The same figure shows the temperature dependence of the ET rate in the case of RhB on Sn02. In this case a fairly large temperature dependence is observed. Such a large temperature dependence is also observed in other cases such as for RhB/W03 and R ~ ( b p y ) ? + / T i O ~ Since . ~ ~ the energy levels of these excited dyes are located above the bottom of the (48) I n a glassy solid, I .23 .&-Iis used for a by Miller et In the present case, it is in vacuum between the dye and solid surface. In the case of vacuum, a slightly larger value, 2-3 .&-I,is expected from the distance dependence of a tunneling current in scanning electron microsc~py.~~ Consequently, we use 2.0 A-' tentatively. (49) Hashimoto, K.; Hiramoto, M.; Sakata, T. Unpublished data. (50) Private communication from Dr. T. Kakitani.
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990 3045
Electron Transfer on Semiconductor Surface TEMPERATURE/K 300
100
50
10
Z = Tr exp(-0%) = S(xlexp(-PH)lx) dx 4
(A-2)
From the definition of x, the matrix element is written as m
(xlexp(-P7f)lx") =
n=O
exp(-PE,)J/,(x'? J/n(X'? (A-3)
For harmonic oscillators, the wave function is expressed as
(A-4)
0,
1
on ZrO,
1 o8
Here, H,([) is a Hermite polynomial and xo is the equilibrium position of the oscillators. From eqs A-3, A-4, and A-5, the diagonal matrix element for harmonic oscillators is calculated.
1
'
0
5
10
15
id'
20 1 0 0 2 5 0
m
(xlexp(-pWlx)
1000/T Figure 7. Temperature dependence of electron-transfer rate in the cases of Rhodamine B on b o 2 , T i 0 2 , and Zr02. The solid lines show the theoretical curve calculated by eq 33: Parameters used are a = 2.0 A-' and h w = 20 cm-I both for T i 0 2 and Z r0 2 and a = 2.0 .&-I and h a = 9 cm-' for SnO,.
conduction band of each semiconductor, the present theory may also be applied. The solid line in the case of SnO, represents the theoretical curve calculated with eq 33 with a = 2.0 and hu = IO cm-I. The theoretical curve explains fairly well the experimental results. As shown in the theoretical curve in Figure 7, the ET rate depends more strongly as the magnitude of the vibration quantum hu decreases. It is also dependent on the attenuation parameter, a, and the potential curve. A slight discrepancy between theory and experiment in the case of SnO, might be explained by taking into consideration a deviation of the potential from a harmonic one. The correct value of a on the surface exposed to vacuum is also another problem in future.
Acknowledgment. We thank Prof. K. Nasu for his helpful and stimulating discussions. Thanks are also given to Dr. S. Meech for careful reading of the manuscript. This work has been supported partly by a Grant-in-Aid for Scientific Research on Priority Areas from the Ministry of Education, Science and Culture, 1988.
Appendix Equation 32 can be derived as follows.
E n=O
exp(-PEn)J/,2(x) =
2i5u
1
}",exp(,.
h
= 2 n h sinh (flhu)
I
- u2 f 2 i t v du du
- x0), tanh
2
In the above derivation we used the formula
1:
exp(-bx2) dx = (n/b)'I2,
>0
b
(A-7)
By using eq A-6,
2
- x0), tanh P h w / dx
S A u o 2exp(-ax) exp
I
\
02(x) =
mw --(xh
= uo2 exp(-axo) e x p [ $ Y coth where A =
[
@hw
- x0), tanh 2 dx
(:)I
('4-8)
2 a h sinh ( p h w )
Equation A-8 is equal to eq 33. Here,
Registry No. Ru(bpy)32+, I5 158-62-0; Ru(bpz)32+, 75523-96-5; Rhodamine B. 81-88-9.
