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Photoinduced electron transport process in electrochemical cell
MOLSTR 10321
Journal of Molecular Structure 450 (1998) 155–161
Photoinduced electron transport process in electrochemical cell I. Phenomenological description Tadeusz Hoffmann a, Danuta Wro´bel b,* a
Institute of Applied Mechanics, Poznan˜ University of Technology, Piotrowo 3, 60-965 Poznan˜, Poland b Institute of Physics, Poznan˜ University of Technology, Piotrowo 3, 60-965 Poznan˜, Poland Received 12 October 1997; revised 16 March 1998; accepted 16 March 1998
Abstract The process of electron transport plays an essential role in the fundamental phenomena of life like photosynthesis, respiration and vision as well as in photoelectronic devices. However the molecular mechanisms of the electron way and factors governing the transport rate in such systems are still unclear. Several groups have reported theoretical approaches for searching the mechanisms by using statistical mechanics, coherent dynamics and quantum mechanics. The current density vector inside the semiconducting layer is determined. In this paper we consider the problem of transport of electron promoted in the electrochemical cell constructed of two electrodes with the dye molecules immersed in. We describe the process of electron promotion by refractive light wave on the vacuum–semiconductor boundary as well as on the semiconducting electrode and the dye molecule layer in terms of extended phenomenological electrodynamics formalism. The results of our theoretical model show that such a theoretical approach will give more information on the mechanism of electron transport and will give insight in the determination of some electric features of materials. q 1998 Elsevier Science B.V. All rights reserved Keywords: Electrochemical cell; Electron density vector; Extended electrodynamics; Phenomenological description
1. Introduction A number of different techniques can be used to convert light/solar energy into electric energy [1–3]. When a photon in the wavelengths of UV or visible light is able to free an electron from a molecule or to remove it from the stable state under a proper condition, the chain of electron transfers can create useful electric current. The photoelectric effect can take place everywhere a photon can deeply penetrate a material. Before a * Corresponding author. Tel.: +48 61 878 2344; Fax: +48 61 878 2324; e-mail: [email protected]
freed electron can return to a stable state and release its energy randomly, the process of electron transport has to be organized in a particular direction under established conditions. Such a condition can be achieved by sticking together two substances which possess different electron spacing features – one substance should easily reject the electron and the other accept it. An electron released by a photon is able to move from one substance to the other before recombination. One of the widely applied photovoltaic technologies is a construction of an electrochemical cell made of two thin transparent electrodes with a semiconductor between them creating Schottky
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T. Hoffmann, D. Wro´bel/Journal of Molecular Structure 450 (1998) 155–161
barrier-like junction. In such an electrochemical cell it is possible to excite an electron in a molecule of semiconductor itself or in a molecule of another substance doped in a semiconducting material and to allow to create an electron transport which can result in an electric current. The initiation of the electron freeing can be caused by a photon of light which penetrates the surface deeply enough to give energy to the electron. In our and other laboratories a lot of experiments have been done with various materials and the photoelectric response has been observed in a photoelectrochemical cell constructed with thin semiconducting and metal electrodes [4–12]. The observed electron transport depends on the kind of the dye molecule used in the experiment and of medium in which the molecules are immersed. However, the mechanism of the electron motion process in the medium between metal and semiconducting electrodes in such a system is still unclear and under discussion. Several groups have reported theoretical approach for seeking the mechanism of electron transfer reaction in photovoltaic cells and other models [13–15]. One of the first theoretical treatments in searching the mechanisms of oxidation–reduction reactions in solvent medium treated as a dielectric continuum was presented by Marcus [13] and calculation of the free energy of reorganization prior to the electronic jump process has been developed. The mechanism of electron transfer was later examined using potentialenergy surface and the study was extended to the unifield theory of homogeneous and electrochemical electron transfer by using statistical mechanics [14] and to the coherent dynamics of the electron transfer involving strong electronic coupling [15]. In Ref. [16] the chemical model of the geometrical orientation of donor and acceptor with a well-defined structure is presented. Theoretical analyses of the electron transfer reactions with a description from the classical Marcus equation to the quantum mechanical equation were gathered by Parson and Warshel [3]. In this paper the simple, flat 2-dimensional model of the photoelectrochemical cell with the refraction of the optic wave on the vacuum and semiconductor boundary as well as on the semiconducting electrodes and dye molecules layer will be described in terms of the extended phenomenological electrodynamics. Such a theoretical approach seems to be a good
approximation of the determination of the density vector of the ‘excited’ electrons by the incident optic wave and can give more information on the mechanism of the electron transport as well as to give input in determination of some electric features of materials.
2. Results 2.1. Theoretical model – extended phenomenological electrodynamics Let us introduce the fundamentals of the idea exposed in Ref. [17] and, in order to describe the electromagnetic effects, let us choose two vector field functions b s(x,t) [A s/m], d s(x,t) [kg m/A s]. Consider a continuous, dynamic physical system described by the action functional t2 ÿ s s (1) L bs ; d s ; b˙ ; d˙ ; bs, k ; d,sk dV dt, W= t1
V
where ]bs s , b˙ = ]t bs = ]t
bs, k = ]k bs =
]bs : ]xk
The Euler–Lagrange equations of this system have the form [18]: ! · ]L ]L ]L − − − − = 0, ]bs ]bs ]bs, k ,k ! · ]L ]L ]L − − − = 0: 2 s− ]d s ]d,sk ]d˙ ,k We define: the magnetic induction vector Bs ≡
]L ; ]b˙ s
(3a)
the electric induction vector Ds ≡
]L ; ]d˙ s
(3b)
the electric field intensity vector «slm Em ≡
]L ; ]d,sl
(3c)
(« slm denotes the antisymmetric Ricci symbol) the
T. Hoffmann, D. Wro´bel/Journal of Molecular Structure 450 (1998) 155–161
accepted form
magnetic field intensity vector; «slm Hm ≡
]L ; ]d,sl
(3d)
]L , ]bs
jM s = ]t (m0 Ms ),
m0 = 1:2566371 × 10 − 6 [kg m=A2 s2 ];
(3e)
(M s [A/m] denotes magnetization) the electric current density vector jEs ≡ −
]L , ]d s
jEs = js + ]t Ps
(3f )
( j s denotes the conductivity current density, P s [A s/m 2] denotes the electric polarization). From the above definitions the Lagrangian properties follow, ]L ]L = = 0, ]bs, s ]bs, s
]L ]L =− l , ]bs, l ]b, s
]L ]L =− l: ]d,sl ]d, s (4)
If we add supplementary conditions to Eq. (2) and definitions ]L = rM , rM = − ]k (m0 Mk ), rM = 0, ]b˙ s , s ]L = r, r = rc + ( − ]k Pk ), 5 ]d˙ s , s c
where r denotes the electric charge density, it means that the dynamic system (1), (2) models the electromagnetic field described by the equations ]t Bk + «klm ]l Em + jM k = 0,
] k B k = rM ,
]t Dk − «klm ]l Hm + jEk = 0,
]k Dk = r:
6
It will be assumed, in agreement with Ref. [17], that the electric and magnetic inductions at a given point of the field were a sum of a free field induction and contributions from the polarization P k and magnetization M k B k 9 = B k + m0 M k ,
Dk 9 = Dk + Pk ,
«klm ]t Em + ]t Bk 9 = 0, «klm ]t Hm − ]t Dk 9 = jk ,
the magnetic current density vector jM s ≡ −
157
(7)
where B k is the magnetic induction vector of a free field, and D k the electric induction vector of a free field. This allows to write Eq. (6) in the commonly
]k Bk 9 = 0, ]t Dk 9 = r:
8
The equations introduced in this section describe the generally acknowledged experimental results. In accordance with the suggestion contained in the introduction we will consider semiconductor being magnetized. It must be, therefore, a material in which neither conductivity currents nor charges occur. According to Ref. [17] in such a material there are polarization ‘charges’ and ‘currents’ as well as magnetization ‘charges’ and ‘currents’. Because of the lack of the conductivity currents, the magnetization effect is entirely of atomic origin. This leads to the assumptions: ]L = − jM k , ]bk
]L = − (jpk + jck ): ]d k
(9)
The Lagrange function depends, therefore, on both dynamic variables and their spatial and time derivatives. The fundamental requirement in construction the Lagrange function is that it must be a scalar function with the physical dimension of energy density. Following the above condition and aiming at linear field equations, we will assume the Lagrange function to be in the form L = 12Isr b˙ s b˙ r + 12Jrs d˙ s d˙ r − 12Asrmn bs, r bm, n − 12Csrmn ds, r dm, n + 12Ksr bs br + 12Nsr ds dr ,
10
which is invariant under space and time-coordinates reversals. The tensor coefficients appearing in (10) may be, in general, the functions of space coordinates and time. Let us assume, however, that they are constant and represent isotropic tensors [19], respectively of the 4th, 3rd and 2nd order, Asrmn = A0dsr dmn + A9dsm drn + Adsn drm , Isr = Idsr :
R ijk = R«ijk , 11
By virtue of the theorems on representation of isotropic tensor functions [20–23] and Lagrangian properties (4), the function L = 12I b˙ r b˙ r + 12J d˙ r d˙ r − 12Abs, r (br, s − bs, r ) − 12Cds, r (dr, s − ds, r ) + 12Kbr br + 12Ndr dr
12
158
T. Hoffmann, D. Wro´bel/Journal of Molecular Structure 450 (1998) 155–161
results from (10). This leads to the Lagrange equations in the form A(bs, r − br, s ), r − Kds = I b˙ s , C(ds, r − dr, s ), r − Nds = J d˙ s :
13
The above field equations expressed in terms of dynamic variables constitute a set of partial differential equations of the second order. For the complete formulation of the initial-boundary problem, the relevant initial and boundary conditions for the field functions and their first derivatives are needed. To reach full agreement with experiment, the boundary conditions should have the form of classical jump conditions resulting from Eq. (8).
3. Electron creation by the refractive optic wave Let us consider the problem of the refraction of the optic wave on the boundary of the vacuum and semiconductor in the photoelectrochemical cell (Fig. 1). The right-handed Cartesian coordinate system will be taken in such a way that the plane which separates the semiconducting semispace and the vacuum will be represented by the equation x 3 = 0. In the two semispaces, the electromagnetic phenomena will be described according to the equations of the extended electrodynamics. The free field exists in the semispace of the vacuum, whereas in the semispace of semiconductor, the electric polarization, magnetization and current flow will occur. The problem under consideration is taken as flat (2-dimensional), thus all polar quantities depend on
Fig. 1. Scheme of the refraction of optic wave on the semiconductor–vacuum boundary. Semiconductor 1 and 2: A9, A0, C9, C0, I9, I0, J9, J0, K9, K0, N9, N0 are the modelling constants. «9,«0 – permitivity; m9, m0 – permeability; j9, j0 – conductivity. Vacuum: A, C, I, J are the modelling constants. « 0 – permitivity; m 0 – permeability.
T. Hoffmann, D. Wro´bel/Journal of Molecular Structure 450 (1998) 155–161
(x 2,x 3,t) whereas they are independent of x 1. The optic wave coincides from the vacuum side and is described by the vector ~ b (b 1,0,0): n1 −~ rqt, b1 = b0 expi k~
(14)
with the directional vector, ~ e2 − n13~ e3 , n1 = n12~
(15)
where ~ n12 = cos Q1 ,
~ n13 = sin Q1 :
It means that optic wave incidents in the plane x 1 = 0, under the angle Q 1 with respect to the plane x 3 = 0. The wave b 1 satisfies an equation A (16) c21 (b1, 22 + b1, 33 ) − b¨ 1 = 0, c21 = I and its parameters are known. A and I as indicated in Fig. 1. For simplification we will neglect the field quantities which result from the equations of constrains. In semispace filled with semiconductor the following equations will be satisfied: c2 (b1, 22 9 + b1, 33 9) − b¨ 1 9 − q20 b1 9 = 0, c2 (d2, 33 9 + d3, 23 9) − d¨ 2 9 − q20 d2 9 = 0, c
2
(d3, 22 9 + d2, 32 9) − d¨ 3 9 − q20 d3 9 = 0,
17
where c2 =
A9 C9 = , I9 J 9
q20 =
K9 N 9 = I9 J 9
with A9, I9, C9, J9, K9 and N9 as indicated in Fig. 1. The quantities d 29, d 39 have to arrive on the basis of the equations of constrains: N 9d2 9 = 12j9A9b1, 3 9, N 9d3 9 = − 12j9A9b1, 2 9, I9d2 9 = −
j2 9 = j9E2 9, j3 9 = j9E3 9,
1 2«9A9b1, 3 9,
D2 9 = «9E2 9,
I9d3 9 = − 12«9A9b1, 2 9,
D3 9 = «9E3 9:
18
On the plane x 3 = 0 which separates semispaces the following continuity conditions have to be satisfied: I I9 H1 = H1 9 → b˙ 1 = b˙ 1 9, m m9 E2 = E2 9 → Ab1, 3 = A9b1, 3 9, D3 = D3 9 → «Ab1, 2 = «9A9b1, 2 9:
19
159
Function (14) satisfies Eq. (16) when the following relation is fulfilled: n212 + n213 = 1:
(20)
The least numbers of waves which satisfy the boundary conditions (19) are expressed by the functions: incident wave: q ~ r − qt , n ·~ b1 = b01 exp i c1 1 refractive waves: q9 ~ n2 ·~ r − q9t , b1 9 = b01 9 exp i c9 q9 ~ r − q9t , n ·~ d2 9 = d02 9 exp i c9 2 q9 ~ r − q9t : n2 ·~ d3 9 = d03 9 exp i c9
21
The functions (21) satisfy the boundary conditions (19) when following relations on the plane x 3 = 0 are true: q = q9,
q q9 n , n = c1 12 c1 9 22
(22)
which are the refraction principles. The simple form of the continuity Eq. (19) and of the equations of constrains suggests that the amplitudes of refractive waves are determined unambiguously by the parameters of incident waves. On the basis of the definition Eq. (3a), (3b), (3c), (3d), (3e) and (3f) we can determine the components (j 29,j 39) of the density vector of the electron flow: 1 1 q9 ~ n·~ r − q9t , j2 9 = j9A9b1, 3 9 = j9A9in13 b01 9 exp i 2 2 c 1 1 q9 ~ n·~ r − q9t : j3 9 = j9A9b1, 2 9 = jA9in12 b01 9 exp i 2 2 c 23 The multiplicators ‘i’ and ‘ − i’, respectively are interpreted as the phase shifts of p=2(~j2 ) and (3=2)p(~j3 9) of the components of the electron density vector with respect to the optic wave vector. Let us suppose that the molecular system which is embedded between two electrodes in the
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T. Hoffmann, D. Wro´bel/Journal of Molecular Structure 450 (1998) 155–161
electrochemical cell is the material of the semiconducting properties. If we assumed the known thickness of the semiconducting electrode, it is possible to calculate the value of the solutions (23) on the boundary of semiconductors. Going further, on the basis of the conditions of continuity of the tangential component of the electric field and the normal component of the electric induction, we can try to determine the components j 20 and j 30 in the second semiconductor. And thus: continuity conditions: E2 9 = E2 0,
j2 9=j9 = j2 0=j0,
D3 9 = D3 0,
«9E3 9 = «0E3 0:
24
From (23) and (24) we have: E3 0 = «9E3 9=«0 = «9j3 9=«0j9
(25)
and j2 0 = j2 9j0=j9,
j3 0 = j0«9j3 9=j9«0:
(26)
On the basis of (25) and (26) one can notice that the components of the current density vector get the jump on the boundary of two semiconductors. In the case of the components ( j 29, j 20), the jump is dependent on the relation between the conductivities j9, j0 while for the components ( j 39, j 30) it depends additionally on the electric permitivities «9, «0. The two jumps one can interpret as the surface currents in the phenomenological point of view, whereas in the microscopic description it can be interpreted as the motion of the charges bound in the layers of semiconductors.
4. Discussion The photocurrent in the photoelectrochemical cell involves elementary electron transfer process: photoexcitation of the dye molecule, charge separation in the excited dye molecule and charge transfer to the electrode, discharged of the carriers and the back electron motion which returns the system to its original state. Photoinduced electron transfer can be described most simply by the motion of electron, caused by the absorption of light, from an electron rich molecules to the electron-deficient system. As it has been recently shown, the charge carrier flow has
been found to vary greatly with the chemical nature of the interfacial systems [24,25] and as reported in [5–7,26] the variations of the intensity of photocurrent generated in an electrochemical cell for dyes of different molecular structure and embedded in the different media have been observed. In electron transfer, solvation of the charged particles is often critical in describing the electron motion process. Strong solvent dependence from low to high polarity, which can cause the electron transfer rate variations even of several order of magnitude is caused by large changes in the molecular Franck–Condon factors [13]. For the electrochemical cell to work efficiently, the photogenerated electrons must move through an electric circuit to the electrode. This electron motion and density of charged carriers in a photovoltaic cell is a classic topic, in the molecular aspect, of the light energy to the electric energy conversion in which the electrodes and molecules are involved and it is expected to occur when the molecules in their excited states are in close contact with the semiconductor layer. In this paper we have presented the results of the first approach for the description of ‘optically excited electrons’ using the mathematical model with the refractive wave. We have determined by the formalism of extended phenomenological electrodynamics, the current density vector of electrons which are ‘excited’ by the incident optic wave and we have shown the occurring of the phase shifts of the electron density vector components with respect to the optic wave vector on the vacuum–semiconductor boundary. Moreover it has been also shown that in the semiconductors, the jump of the current density vector components depends on either the relation of the conductivities or of the conductivities as well as of the electric permitivities of materials. These mentioned electric features are very closely connected with the molecular structure of the dye molecules and the medium in which they are dissolved.
Acknowledgements The paper was supported by Poznan˜ University of Technology – grants TB 21-811/97 DS (TH) and DS-62-133/2 (DW).
T. Hoffmann, D. Wro´bel/Journal of Molecular Structure 450 (1998) 155–161
References [1] Molecular Electronics, Biosensors and Biocomputers, Plenum Press, New York, London, 1989. [2] M.R. Wasielewski, in: Chlorophylls, CRC Press, Boca Raton, Ann Arbor, Boston, London, 1991, Ch. 1.10. [3] W.W. Parson, A. Warshel, in: Anoxogenic Photosynthetic Bacteria, Kluwer Academic Publishers, 1995, Ch. 25. [4] D. Frackowiak, J. Goc, H. Malak, A. Planner, A. Ptak, B. Zelent, J. Photochem. Photobiol. A: Chemistry 94 (1996) 43. [5] A. Ptak, E. Chrzumnicka, A. Dudkowiak, D. Frackowiak, J. Photochem. Photobiol A: Chemistry 98 (1996) 159. [6] A. Ptak, E. Chrzumnicka, A. Planner, D. Frackowiak, Biophys. Chem. 65 (1997) 165. [7] A. Ptak, A. Der, R. Toth-Boconadi, N.S. Naser, D. Frackowiak, J. Photochem. Photobiol. A: Chemistry 104 (1997) 133. [8] J. Miyake, M. Hara, J. Goc, A. Planner, D. Wro´bel, Spectrochim. Acta A 53 (1997) 1485. [9] M. Yagi, K. Nagai, A. Kira, M. Kaneko, J. Electroanal. Chem. 349 (1995) 169. [10] K. Akiyama, S. Nishikawa, S. Ueyama, S. Isoda, Jpn. J. Appl. Phys. 34 (1995) 3942. [11] A.N. Macpherson, P.A. Liddell, S. Lin, L. Noss, G.R. Seely, J.M. DeGraziano, A.L. Moore, T.A. Moore, D. Gust, J. Am. Chem. Soc. 117 (1995) 7202.
161
[12] A.G. Volkov, M.I. Gugeshashvili, B. Zelent, D. Cote, G. Munger, A. Tessier, P.-F. Blanchet, R.M. Leblanc, Bioelectrochem. Bioenergetics 38 (1995) 333. [13] R.A. Marcus, J. Chem. Phys. 24 (1956) 966. [14] R.A. Marcus, J. Chem. Phys. 43 (1965) 679. [15] R.A. Marcus, R. Almeida, J. Phys. Chem. 94 (1990) 2973. [16] A.D. Joran, B.A. Leland, P.M. Felker, A.H. Zewail, J.J. Hopfield, P.B. Derran, Nature 327 (1987) 508. [17] T. Hoffmann, J. Techn. Phys. 32 (1991) 1. [18] A.A. Bogus and L.G. Moroz, Introduction to Classical Field Theory, Nauka i Technika Publishers, Minsk, 1968 (in Russian). [19] J.I. Sirotin and M.P. Saskolskaya, Fundaments of Crystals Physics, Nauka Publishers, Moskva, 1979 (in Russian). [20] G.F. Smith, Int. J. Eng. Sci. 9 (1971) 899. [21] A.J.M. Spencer, in: Continuum Physics, Vol. 1., Academic Press, 1971. [22] C.C. Wang, Arch. Rat. Mech. Anal. 36 (1970) 166. [23] C.C. Wang, Arch. Rat. Mech. Anal. 43 (1971) 392. [24] A. Mills, S. LeHunte, J. Photochem. Photobiol. A: Chemistry 108 (1997) 1. [25] P. Avouris, B.N.J. Persson, J. Phys. Chem. 88 (1984) 837. [26] D. Wro´bel, J. Goc and R.M. Ion, J. Mol. Struct. (1998), in press.
Journal of Molecular Structure 450 (1998) 155–161
Photoinduced electron transport process in electrochemical cell I. Phenomenological description Tadeusz Hoffmann a, Danuta Wro´bel b,* a
Institute of Applied Mechanics, Poznan˜ University of Technology, Piotrowo 3, 60-965 Poznan˜, Poland b Institute of Physics, Poznan˜ University of Technology, Piotrowo 3, 60-965 Poznan˜, Poland Received 12 October 1997; revised 16 March 1998; accepted 16 March 1998
Abstract The process of electron transport plays an essential role in the fundamental phenomena of life like photosynthesis, respiration and vision as well as in photoelectronic devices. However the molecular mechanisms of the electron way and factors governing the transport rate in such systems are still unclear. Several groups have reported theoretical approaches for searching the mechanisms by using statistical mechanics, coherent dynamics and quantum mechanics. The current density vector inside the semiconducting layer is determined. In this paper we consider the problem of transport of electron promoted in the electrochemical cell constructed of two electrodes with the dye molecules immersed in. We describe the process of electron promotion by refractive light wave on the vacuum–semiconductor boundary as well as on the semiconducting electrode and the dye molecule layer in terms of extended phenomenological electrodynamics formalism. The results of our theoretical model show that such a theoretical approach will give more information on the mechanism of electron transport and will give insight in the determination of some electric features of materials. q 1998 Elsevier Science B.V. All rights reserved Keywords: Electrochemical cell; Electron density vector; Extended electrodynamics; Phenomenological description
1. Introduction A number of different techniques can be used to convert light/solar energy into electric energy [1–3]. When a photon in the wavelengths of UV or visible light is able to free an electron from a molecule or to remove it from the stable state under a proper condition, the chain of electron transfers can create useful electric current. The photoelectric effect can take place everywhere a photon can deeply penetrate a material. Before a * Corresponding author. Tel.: +48 61 878 2344; Fax: +48 61 878 2324; e-mail: [email protected]
freed electron can return to a stable state and release its energy randomly, the process of electron transport has to be organized in a particular direction under established conditions. Such a condition can be achieved by sticking together two substances which possess different electron spacing features – one substance should easily reject the electron and the other accept it. An electron released by a photon is able to move from one substance to the other before recombination. One of the widely applied photovoltaic technologies is a construction of an electrochemical cell made of two thin transparent electrodes with a semiconductor between them creating Schottky
0022-2860/98/$19.00 q 1998 Elsevier Science B.V. All rights reserved PII S 0 02 2- 2 86 0 (9 8 )0 0 42 4 -4
156
T. Hoffmann, D. Wro´bel/Journal of Molecular Structure 450 (1998) 155–161
barrier-like junction. In such an electrochemical cell it is possible to excite an electron in a molecule of semiconductor itself or in a molecule of another substance doped in a semiconducting material and to allow to create an electron transport which can result in an electric current. The initiation of the electron freeing can be caused by a photon of light which penetrates the surface deeply enough to give energy to the electron. In our and other laboratories a lot of experiments have been done with various materials and the photoelectric response has been observed in a photoelectrochemical cell constructed with thin semiconducting and metal electrodes [4–12]. The observed electron transport depends on the kind of the dye molecule used in the experiment and of medium in which the molecules are immersed. However, the mechanism of the electron motion process in the medium between metal and semiconducting electrodes in such a system is still unclear and under discussion. Several groups have reported theoretical approach for seeking the mechanism of electron transfer reaction in photovoltaic cells and other models [13–15]. One of the first theoretical treatments in searching the mechanisms of oxidation–reduction reactions in solvent medium treated as a dielectric continuum was presented by Marcus [13] and calculation of the free energy of reorganization prior to the electronic jump process has been developed. The mechanism of electron transfer was later examined using potentialenergy surface and the study was extended to the unifield theory of homogeneous and electrochemical electron transfer by using statistical mechanics [14] and to the coherent dynamics of the electron transfer involving strong electronic coupling [15]. In Ref. [16] the chemical model of the geometrical orientation of donor and acceptor with a well-defined structure is presented. Theoretical analyses of the electron transfer reactions with a description from the classical Marcus equation to the quantum mechanical equation were gathered by Parson and Warshel [3]. In this paper the simple, flat 2-dimensional model of the photoelectrochemical cell with the refraction of the optic wave on the vacuum and semiconductor boundary as well as on the semiconducting electrodes and dye molecules layer will be described in terms of the extended phenomenological electrodynamics. Such a theoretical approach seems to be a good
approximation of the determination of the density vector of the ‘excited’ electrons by the incident optic wave and can give more information on the mechanism of the electron transport as well as to give input in determination of some electric features of materials.
2. Results 2.1. Theoretical model – extended phenomenological electrodynamics Let us introduce the fundamentals of the idea exposed in Ref. [17] and, in order to describe the electromagnetic effects, let us choose two vector field functions b s(x,t) [A s/m], d s(x,t) [kg m/A s]. Consider a continuous, dynamic physical system described by the action functional t2 ÿ s s (1) L bs ; d s ; b˙ ; d˙ ; bs, k ; d,sk dV dt, W= t1
V
where ]bs s , b˙ = ]t bs = ]t
bs, k = ]k bs =
]bs : ]xk
The Euler–Lagrange equations of this system have the form [18]: ! · ]L ]L ]L − − − − = 0, ]bs ]bs ]bs, k ,k ! · ]L ]L ]L − − − = 0: 2 s− ]d s ]d,sk ]d˙ ,k We define: the magnetic induction vector Bs ≡
]L ; ]b˙ s
(3a)
the electric induction vector Ds ≡
]L ; ]d˙ s
(3b)
the electric field intensity vector «slm Em ≡
]L ; ]d,sl
(3c)
(« slm denotes the antisymmetric Ricci symbol) the
T. Hoffmann, D. Wro´bel/Journal of Molecular Structure 450 (1998) 155–161
accepted form
magnetic field intensity vector; «slm Hm ≡
]L ; ]d,sl
(3d)
]L , ]bs
jM s = ]t (m0 Ms ),
m0 = 1:2566371 × 10 − 6 [kg m=A2 s2 ];
(3e)
(M s [A/m] denotes magnetization) the electric current density vector jEs ≡ −
]L , ]d s
jEs = js + ]t Ps
(3f )
( j s denotes the conductivity current density, P s [A s/m 2] denotes the electric polarization). From the above definitions the Lagrangian properties follow, ]L ]L = = 0, ]bs, s ]bs, s
]L ]L =− l , ]bs, l ]b, s
]L ]L =− l: ]d,sl ]d, s (4)
If we add supplementary conditions to Eq. (2) and definitions ]L = rM , rM = − ]k (m0 Mk ), rM = 0, ]b˙ s , s ]L = r, r = rc + ( − ]k Pk ), 5 ]d˙ s , s c
where r denotes the electric charge density, it means that the dynamic system (1), (2) models the electromagnetic field described by the equations ]t Bk + «klm ]l Em + jM k = 0,
] k B k = rM ,
]t Dk − «klm ]l Hm + jEk = 0,
]k Dk = r:
6
It will be assumed, in agreement with Ref. [17], that the electric and magnetic inductions at a given point of the field were a sum of a free field induction and contributions from the polarization P k and magnetization M k B k 9 = B k + m0 M k ,
Dk 9 = Dk + Pk ,
«klm ]t Em + ]t Bk 9 = 0, «klm ]t Hm − ]t Dk 9 = jk ,
the magnetic current density vector jM s ≡ −
157
(7)
where B k is the magnetic induction vector of a free field, and D k the electric induction vector of a free field. This allows to write Eq. (6) in the commonly
]k Bk 9 = 0, ]t Dk 9 = r:
8
The equations introduced in this section describe the generally acknowledged experimental results. In accordance with the suggestion contained in the introduction we will consider semiconductor being magnetized. It must be, therefore, a material in which neither conductivity currents nor charges occur. According to Ref. [17] in such a material there are polarization ‘charges’ and ‘currents’ as well as magnetization ‘charges’ and ‘currents’. Because of the lack of the conductivity currents, the magnetization effect is entirely of atomic origin. This leads to the assumptions: ]L = − jM k , ]bk
]L = − (jpk + jck ): ]d k
(9)
The Lagrange function depends, therefore, on both dynamic variables and their spatial and time derivatives. The fundamental requirement in construction the Lagrange function is that it must be a scalar function with the physical dimension of energy density. Following the above condition and aiming at linear field equations, we will assume the Lagrange function to be in the form L = 12Isr b˙ s b˙ r + 12Jrs d˙ s d˙ r − 12Asrmn bs, r bm, n − 12Csrmn ds, r dm, n + 12Ksr bs br + 12Nsr ds dr ,
10
which is invariant under space and time-coordinates reversals. The tensor coefficients appearing in (10) may be, in general, the functions of space coordinates and time. Let us assume, however, that they are constant and represent isotropic tensors [19], respectively of the 4th, 3rd and 2nd order, Asrmn = A0dsr dmn + A9dsm drn + Adsn drm , Isr = Idsr :
R ijk = R«ijk , 11
By virtue of the theorems on representation of isotropic tensor functions [20–23] and Lagrangian properties (4), the function L = 12I b˙ r b˙ r + 12J d˙ r d˙ r − 12Abs, r (br, s − bs, r ) − 12Cds, r (dr, s − ds, r ) + 12Kbr br + 12Ndr dr
12
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results from (10). This leads to the Lagrange equations in the form A(bs, r − br, s ), r − Kds = I b˙ s , C(ds, r − dr, s ), r − Nds = J d˙ s :
13
The above field equations expressed in terms of dynamic variables constitute a set of partial differential equations of the second order. For the complete formulation of the initial-boundary problem, the relevant initial and boundary conditions for the field functions and their first derivatives are needed. To reach full agreement with experiment, the boundary conditions should have the form of classical jump conditions resulting from Eq. (8).
3. Electron creation by the refractive optic wave Let us consider the problem of the refraction of the optic wave on the boundary of the vacuum and semiconductor in the photoelectrochemical cell (Fig. 1). The right-handed Cartesian coordinate system will be taken in such a way that the plane which separates the semiconducting semispace and the vacuum will be represented by the equation x 3 = 0. In the two semispaces, the electromagnetic phenomena will be described according to the equations of the extended electrodynamics. The free field exists in the semispace of the vacuum, whereas in the semispace of semiconductor, the electric polarization, magnetization and current flow will occur. The problem under consideration is taken as flat (2-dimensional), thus all polar quantities depend on
Fig. 1. Scheme of the refraction of optic wave on the semiconductor–vacuum boundary. Semiconductor 1 and 2: A9, A0, C9, C0, I9, I0, J9, J0, K9, K0, N9, N0 are the modelling constants. «9,«0 – permitivity; m9, m0 – permeability; j9, j0 – conductivity. Vacuum: A, C, I, J are the modelling constants. « 0 – permitivity; m 0 – permeability.
T. Hoffmann, D. Wro´bel/Journal of Molecular Structure 450 (1998) 155–161
(x 2,x 3,t) whereas they are independent of x 1. The optic wave coincides from the vacuum side and is described by the vector ~ b (b 1,0,0): n1 −~ rqt, b1 = b0 expi k~
(14)
with the directional vector, ~ e2 − n13~ e3 , n1 = n12~
(15)
where ~ n12 = cos Q1 ,
~ n13 = sin Q1 :
It means that optic wave incidents in the plane x 1 = 0, under the angle Q 1 with respect to the plane x 3 = 0. The wave b 1 satisfies an equation A (16) c21 (b1, 22 + b1, 33 ) − b¨ 1 = 0, c21 = I and its parameters are known. A and I as indicated in Fig. 1. For simplification we will neglect the field quantities which result from the equations of constrains. In semispace filled with semiconductor the following equations will be satisfied: c2 (b1, 22 9 + b1, 33 9) − b¨ 1 9 − q20 b1 9 = 0, c2 (d2, 33 9 + d3, 23 9) − d¨ 2 9 − q20 d2 9 = 0, c
2
(d3, 22 9 + d2, 32 9) − d¨ 3 9 − q20 d3 9 = 0,
17
where c2 =
A9 C9 = , I9 J 9
q20 =
K9 N 9 = I9 J 9
with A9, I9, C9, J9, K9 and N9 as indicated in Fig. 1. The quantities d 29, d 39 have to arrive on the basis of the equations of constrains: N 9d2 9 = 12j9A9b1, 3 9, N 9d3 9 = − 12j9A9b1, 2 9, I9d2 9 = −
j2 9 = j9E2 9, j3 9 = j9E3 9,
1 2«9A9b1, 3 9,
D2 9 = «9E2 9,
I9d3 9 = − 12«9A9b1, 2 9,
D3 9 = «9E3 9:
18
On the plane x 3 = 0 which separates semispaces the following continuity conditions have to be satisfied: I I9 H1 = H1 9 → b˙ 1 = b˙ 1 9, m m9 E2 = E2 9 → Ab1, 3 = A9b1, 3 9, D3 = D3 9 → «Ab1, 2 = «9A9b1, 2 9:
19
159
Function (14) satisfies Eq. (16) when the following relation is fulfilled: n212 + n213 = 1:
(20)
The least numbers of waves which satisfy the boundary conditions (19) are expressed by the functions: incident wave: q ~ r − qt , n ·~ b1 = b01 exp i c1 1 refractive waves: q9 ~ n2 ·~ r − q9t , b1 9 = b01 9 exp i c9 q9 ~ r − q9t , n ·~ d2 9 = d02 9 exp i c9 2 q9 ~ r − q9t : n2 ·~ d3 9 = d03 9 exp i c9
21
The functions (21) satisfy the boundary conditions (19) when following relations on the plane x 3 = 0 are true: q = q9,
q q9 n , n = c1 12 c1 9 22
(22)
which are the refraction principles. The simple form of the continuity Eq. (19) and of the equations of constrains suggests that the amplitudes of refractive waves are determined unambiguously by the parameters of incident waves. On the basis of the definition Eq. (3a), (3b), (3c), (3d), (3e) and (3f) we can determine the components (j 29,j 39) of the density vector of the electron flow: 1 1 q9 ~ n·~ r − q9t , j2 9 = j9A9b1, 3 9 = j9A9in13 b01 9 exp i 2 2 c 1 1 q9 ~ n·~ r − q9t : j3 9 = j9A9b1, 2 9 = jA9in12 b01 9 exp i 2 2 c 23 The multiplicators ‘i’ and ‘ − i’, respectively are interpreted as the phase shifts of p=2(~j2 ) and (3=2)p(~j3 9) of the components of the electron density vector with respect to the optic wave vector. Let us suppose that the molecular system which is embedded between two electrodes in the
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electrochemical cell is the material of the semiconducting properties. If we assumed the known thickness of the semiconducting electrode, it is possible to calculate the value of the solutions (23) on the boundary of semiconductors. Going further, on the basis of the conditions of continuity of the tangential component of the electric field and the normal component of the electric induction, we can try to determine the components j 20 and j 30 in the second semiconductor. And thus: continuity conditions: E2 9 = E2 0,
j2 9=j9 = j2 0=j0,
D3 9 = D3 0,
«9E3 9 = «0E3 0:
24
From (23) and (24) we have: E3 0 = «9E3 9=«0 = «9j3 9=«0j9
(25)
and j2 0 = j2 9j0=j9,
j3 0 = j0«9j3 9=j9«0:
(26)
On the basis of (25) and (26) one can notice that the components of the current density vector get the jump on the boundary of two semiconductors. In the case of the components ( j 29, j 20), the jump is dependent on the relation between the conductivities j9, j0 while for the components ( j 39, j 30) it depends additionally on the electric permitivities «9, «0. The two jumps one can interpret as the surface currents in the phenomenological point of view, whereas in the microscopic description it can be interpreted as the motion of the charges bound in the layers of semiconductors.
4. Discussion The photocurrent in the photoelectrochemical cell involves elementary electron transfer process: photoexcitation of the dye molecule, charge separation in the excited dye molecule and charge transfer to the electrode, discharged of the carriers and the back electron motion which returns the system to its original state. Photoinduced electron transfer can be described most simply by the motion of electron, caused by the absorption of light, from an electron rich molecules to the electron-deficient system. As it has been recently shown, the charge carrier flow has
been found to vary greatly with the chemical nature of the interfacial systems [24,25] and as reported in [5–7,26] the variations of the intensity of photocurrent generated in an electrochemical cell for dyes of different molecular structure and embedded in the different media have been observed. In electron transfer, solvation of the charged particles is often critical in describing the electron motion process. Strong solvent dependence from low to high polarity, which can cause the electron transfer rate variations even of several order of magnitude is caused by large changes in the molecular Franck–Condon factors [13]. For the electrochemical cell to work efficiently, the photogenerated electrons must move through an electric circuit to the electrode. This electron motion and density of charged carriers in a photovoltaic cell is a classic topic, in the molecular aspect, of the light energy to the electric energy conversion in which the electrodes and molecules are involved and it is expected to occur when the molecules in their excited states are in close contact with the semiconductor layer. In this paper we have presented the results of the first approach for the description of ‘optically excited electrons’ using the mathematical model with the refractive wave. We have determined by the formalism of extended phenomenological electrodynamics, the current density vector of electrons which are ‘excited’ by the incident optic wave and we have shown the occurring of the phase shifts of the electron density vector components with respect to the optic wave vector on the vacuum–semiconductor boundary. Moreover it has been also shown that in the semiconductors, the jump of the current density vector components depends on either the relation of the conductivities or of the conductivities as well as of the electric permitivities of materials. These mentioned electric features are very closely connected with the molecular structure of the dye molecules and the medium in which they are dissolved.
Acknowledgements The paper was supported by Poznan˜ University of Technology – grants TB 21-811/97 DS (TH) and DS-62-133/2 (DW).
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References [1] Molecular Electronics, Biosensors and Biocomputers, Plenum Press, New York, London, 1989. [2] M.R. Wasielewski, in: Chlorophylls, CRC Press, Boca Raton, Ann Arbor, Boston, London, 1991, Ch. 1.10. [3] W.W. Parson, A. Warshel, in: Anoxogenic Photosynthetic Bacteria, Kluwer Academic Publishers, 1995, Ch. 25. [4] D. Frackowiak, J. Goc, H. Malak, A. Planner, A. Ptak, B. Zelent, J. Photochem. Photobiol. A: Chemistry 94 (1996) 43. [5] A. Ptak, E. Chrzumnicka, A. Dudkowiak, D. Frackowiak, J. Photochem. Photobiol A: Chemistry 98 (1996) 159. [6] A. Ptak, E. Chrzumnicka, A. Planner, D. Frackowiak, Biophys. Chem. 65 (1997) 165. [7] A. Ptak, A. Der, R. Toth-Boconadi, N.S. Naser, D. Frackowiak, J. Photochem. Photobiol. A: Chemistry 104 (1997) 133. [8] J. Miyake, M. Hara, J. Goc, A. Planner, D. Wro´bel, Spectrochim. Acta A 53 (1997) 1485. [9] M. Yagi, K. Nagai, A. Kira, M. Kaneko, J. Electroanal. Chem. 349 (1995) 169. [10] K. Akiyama, S. Nishikawa, S. Ueyama, S. Isoda, Jpn. J. Appl. Phys. 34 (1995) 3942. [11] A.N. Macpherson, P.A. Liddell, S. Lin, L. Noss, G.R. Seely, J.M. DeGraziano, A.L. Moore, T.A. Moore, D. Gust, J. Am. Chem. Soc. 117 (1995) 7202.
161
[12] A.G. Volkov, M.I. Gugeshashvili, B. Zelent, D. Cote, G. Munger, A. Tessier, P.-F. Blanchet, R.M. Leblanc, Bioelectrochem. Bioenergetics 38 (1995) 333. [13] R.A. Marcus, J. Chem. Phys. 24 (1956) 966. [14] R.A. Marcus, J. Chem. Phys. 43 (1965) 679. [15] R.A. Marcus, R. Almeida, J. Phys. Chem. 94 (1990) 2973. [16] A.D. Joran, B.A. Leland, P.M. Felker, A.H. Zewail, J.J. Hopfield, P.B. Derran, Nature 327 (1987) 508. [17] T. Hoffmann, J. Techn. Phys. 32 (1991) 1. [18] A.A. Bogus and L.G. Moroz, Introduction to Classical Field Theory, Nauka i Technika Publishers, Minsk, 1968 (in Russian). [19] J.I. Sirotin and M.P. Saskolskaya, Fundaments of Crystals Physics, Nauka Publishers, Moskva, 1979 (in Russian). [20] G.F. Smith, Int. J. Eng. Sci. 9 (1971) 899. [21] A.J.M. Spencer, in: Continuum Physics, Vol. 1., Academic Press, 1971. [22] C.C. Wang, Arch. Rat. Mech. Anal. 36 (1970) 166. [23] C.C. Wang, Arch. Rat. Mech. Anal. 43 (1971) 392. [24] A. Mills, S. LeHunte, J. Photochem. Photobiol. A: Chemistry 108 (1997) 1. [25] P. Avouris, B.N.J. Persson, J. Phys. Chem. 88 (1984) 837. [26] D. Wro´bel, J. Goc and R.M. Ion, J. Mol. Struct. (1998), in press.