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A model for bridge-assisted electron exchange between two electrodes
Chemical Physics 289 (2003) 349–357 www.elsevier.com/locate/chemphys
A model for bridge-assisted electron exchange between two electrodes W. Schmickler * Department of Chemistry, University of Florence, I-50019 Sesto Fiorentino, Italy Received 12 September 2002
Abstract We consider electron exchange between two metal electrodes mediated by two intermediate states. Our model Hamiltonian accounts for coupling to solvent and vibrational modes, and describes both elastic and inelastic processes. For the case of elastic transitions, we derive an explicit formula for the current that also includes the effect of an overlap between the solvation spheres of the two bridge states. For inelastic transitions we treat the case where one quantum mode interacts with one of the two bridge states; model calculations for the current as a function of the bias show a characteristic rise in the current whenever a new inelastic channel is opened. The system may show several features desirable for molecular electronics: rectification, negative differential resistance, and triode characteristics. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction Recent progress in nanotechnology has made it possible to study electron transfer through one or several intermediate states in well-defined molecular junctions. Much of the interest in this, currently very active, field resides in the prospect of building electronic devices on a molecular scale [1,2]. Indeed, as early as 1974 Aviram and Rattner [3] had pointed out that electron transfer through donor–acceptor systems may exhibit current rectification.
*
Permanent address: Abteilung Elektrochemie, University of Ulm, D-89069 Ulm, Germany. Fax: +49-731-502-5409. E-mail address: [email protected]
A good view over the current activities in this field can be obtained from the collection of papers in [4], and the theory is well summarized in the recent review by Nitzan [5]. Several mechanisms have been proposed for electron exchange along a series of intermediate states: electron hopping, superexchange or resonant transitions, or simply a series of chemical reactions. Indeed, depending on circumstances, on or the other mechanism may dominate, or they may act in parallel (see e.g. [6]). In this paper, we focus on electrochemical systems, on one particular case and one mechanism: resonant exchange between two metals mediated by two intermediate states. The electron transfer is coupled to a phonon bath which represents both solvent modes and local vibrations. This coupling is treated exactly, but explicit model calculations are restricted to relatively simple cases. The same
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situation has recently been discussed by Kuznetsov and Ulstrup [7], but in contrast to these authors we treat also the effect of an overlap of the solvation spheres and, more importantly, inelastic resonant transitions, which have been little discussed in the chemical community. In addition, we derive a compact and transparent expression for the electron transfer rate in the simple case, where the phonon bath is purely classical. These theoretical results are illustrated by model calculations for various cases. The calculated current–potential curves show features desirable for molecular electronics: current rectification and negative differential resistance. In fact, since two potential drops can be controlled separately – the bias and the electrode potential – the system may be operated as a triode. The rest of this paper is organized as follows: In the following section we present our model Hamiltonian, formulated in second-quantitized form. Thereupon we present explicit expressions for the rate of exchange under various circumstances, relegating the mathematical details to Appendices A–C. Results of numerical model calculations are shown and discussed in Section 4, and we finish with a short Section 5. 2. The model Hamiltonian We consider electron exchange between two electroactive species, labeled Ô1Õ and Ô2Õ; species 1 is adsorbed on a metal, whose electronic states are labeled by l, species 2 on a metal with electronic states k. Let ni ; i ¼ 1; 2; l; k denote the corresponding number operators, and i the associated energies; then the Hamiltonian for the non-interacting electrons is X X Hel ¼ 1 n1 þ 2 n2 þ l nl þ k nk : ð1Þ l
k
Electrons can be exchanged along the chain: k ! 1 ! 2 ! l and in the reverse direction. This is effected by the transfer Hamiltonian X HT ¼ Vk1 ck c1 þ Vk1 c1 ck þ V12 c1 c2 þ V12 c2 c1 k
þ
X l
Vl2 c2 c2 þ Vl2 c2 cl ;
ð2Þ
where c and c denote creation and annihilation operators, and V an appropriate coupling strength. The electronic states 1 and 2 are coupled to a phonon bath, which comprises a classical part, the solvent, and perhaps a few localized quantum modes. The phonons and their interactions are represented by the terms " # X 2 1X 2 Hph ¼ hxm pm þ qm n1 hxm gm1 qm 2 m m " # X hxm gm2 qm : n2 ð3Þ m
Here, pm and qm denote the dimensionless momenta and coordinates of the phonons, and gm1 ; gm2 are the coupling constants for the indicated states. For the two statesP we define reorganization energies P 2 2 through: k1 ¼ m hxm gm1 =2, k2 ¼ m hxm gm2 =2. The interaction of the two reactive species with the metals on which they are adsorbed can be characterized by X jVk1 j2 dðx k Þ; D1 ¼ p k
D2 ¼ p
X
ð4Þ
2
jVl2 j dðx l Þ:
l
We employ the so-called wide band approximation [8], in which D1 and D2 are taken as constant. This is a good approximation when these quantities are much smaller than the electronic bands in the two metals, which is generally the case. As is common in electron-transfer theory, we have neglected spin. This is permissible if the energy widths D1 and D2 are much smaller than the repulsive interaction energy between two electrons on the same valence orbital. Our model Hamiltonian is given by the sum of the terms presented above H ¼ Hel þ HT þ Hph ¼ H0 þ H 0 ; H 0 ¼ V12 c1 c2 þ V12 c2 c1 ;
where ð5Þ
where in the last step we have singled out the part H 0 responsible for electron exchange between the two species 1 and 2. We now proceed to calculate the rate of this exchange from scattering theory.
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3. Elastic and inelastic scattering The rate of electron exchange between the two metals can be calculated from scattering theory using the T -matrix or, equivalently, the Greens function G. The relevant matrix elements are hkjT þ ðzÞjli ¼ Vl2 Vk1 h1jGþ ðzÞj2i:
ð6Þ
When the phonon bath is purely classical, it is convenient to define qm -dependent electronic energies through X X ~1 ¼ 1 hxm gm1 qm ; ~2 ¼ 2 hxm gm2 qm : m
m
ð7Þ In the semiclassical case the Greens function can be calculated for fixed values of the phonon coordinates; this is done in Appendix A and gives h1jGþ ðzÞj2i ¼
V12 : 2 ðz ~1 þ iD1 Þðz ~2 þ iD2 Þ jV12 j ð8Þ
The rate of electron exchange between two levels k and l on the two metals, for given values of the phonon coordinates, is then Wkl ¼
2p jVk1 j2 jV12 j2 jVl2 j2 dðk l Þ ; 2 h jDj
ð9Þ
where D is the denominator in Eq. (8). Note that only elastic scattering is allowed when all the phonon modes are classical. The forward current due to electrons flowing from states k to states l is obtained from a thermal average over the initial states and summing over the final states: X i f ¼ e0 f ðk Þ½1 f ðl Þ hWkl i; ð10Þ ¼
Z
k;l
dl
Z
dk f ðk Þ½1 f ðl Þ hWkl iqðk Þqðl Þ; ð11Þ
where f ðÞ denotes the Fermi–Dirac distribution, and qðÞ the density of states. An equivalent expression holds for the reverse current. Eqs. (9) and (11) can form the basis for a completely numerical treatment of the problem. However, more insight is gained by proceeding analytically. For this purpose we assume that the coupling jV12 j is much smaller
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than one of the two energy broadenings D1 or D2 , and can hence be neglected in the denominator. This condition will be met in most cases of practical interest, since the two active species are usually connected by a bridge whose energy levels are far from the energy range in which electron transfer occurs. As shown in Appendix B the thermal average of the rate of exchange can then be written in the form 2p dðk l Þ 2 2 2 jVk1 j jV12 j jVl2 j h 4D1 D2 Z dt ds exp f D1 jtj D2 jsjg exp itðk 1 Þ isðl 2 Þ kT ½k1 t2 þ k2 s2 þ 2k12 ts ; ð12Þ P where k12 ¼ m hxm gm1 gm2 =2 measures the overlap of the two reorganization spheres. As is often the case in semi-classical treatments, the transition rate does not depend on the coupling constants of the individual modes, but only on the collective properties expressed through k1 ; k2 ; k12 . The integrals in Eq. (12) are easily programmed. In the case where the overlap k12 is negligible the integrals factorize and can be calculated explicitly. We choose the Fermi level of the first electrode (states k) as our energy zero, and define the bias Vb as the potential difference between the second electrode and the first. For simplicity, we replace the Fermi–Dirac distribution by step functions and obtain Z 0 2p 2 e0 jV12 j if ¼ d D1 ðÞD2 ðÞ; ð13Þ h e0 Vb hWkl i ¼
where the density of states, in GerischerÕs terminology [10], of species i; ði ¼ 1; 2Þ is given by
1=2
p i þ iDi pffiffiffiffiffiffiffiffiffi R W Di ðÞ ¼ : ð14Þ kT ki 2 kT ki Here, R denotes the real part, and WðzÞ ¼ expðz2 Þ erfcðizÞ is the scaled complex error function [9]. Eqs. (13) and (14) have a transparent physical interpretation: The total current is obtained as an integral over the electronic energy between the two Fermi levels. The contribution of each electronic energy is proportional to the coupling jV12 j2 between the two molecules, which is the weakest link in the chain metal–molecule–
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molecule–metal, and to the densities Di of the two molecules. These densities of states are broadened both by the electronic interaction Di and by the coupling ki to the phonon bath. A few examples for the resulting current – potential curves will be presented in the following section. When the overlap k12 between the solvation spheres is not negligible, the integral in Eq. (12) no longer factorizes into the product of the density of states. We consider the case where the electronic broadening is small comparedpffiffiffiffiffiffiffiffiffiffi to the thermal pffiffiffiffiffiffiffiffiffiffi width, i.e. D1 kT k1 ; D2 kT k2 , in greater detail. The integrals over s and t are easily performed, resulting in Z 0 2p 2 p 2 e0 jV12 j K1=2 if ¼ d exp k1 ð 2 Þ h kT e0 Vb þ k2 ð 1 Þ2 k12 ð 1 Þð 2 Þ =4kT K; ð15Þ k212
model calculations for spewhere K ¼ k1 k2 cific cases will be presented below. When the electron exchange couples to quantum modes, with hx > kT , inelastic processes may occur, in which the transferring electron excites a mode and loses energy. When the quantum modes interact only with one of the two electroactive species, i.e. either with species 1 or with 2, the situation is similar to inelastic resonant tunneling through one species [11,12], and the pertinent results can be used (see Appendix C). This should be the typical case, since quantum modes in solutions are localized and hence not likely to interact strongly with two species. We consider explicitly the case in which one quantum mode interacts with species 1; the extension to several quantum modes interacting either with 1 or 2 is straightforward but tedious to write down. Further, we assume that the quantum mode is initially in its ground state – transitions from excited states are much less probable – and that the two solvation spheres do not overlap. The forward current can then be written as a sum over the number n of quanta hx which are excited Z 0 X 2p if ¼ e0 D1 jV12 j2 dAn ðÞD2 ð nh xÞ: h x e0 Vb þnh n ð16Þ
The argument of D2 and the lower limit of the integral take account of the fact that the tunneling electron has lost an energy nhx. Explicit expressions for the amplitudes An have been given by us in a previous publication [12].
4. Numerical results We first consider the simplest case: elastic scattering in the absence of quantum modes and negligible overlap of the solvation spheres. In this case the current is governed by the densities of states Di ðÞ as given by Eq. (14). In general, the widths of these densities are determined both by the energy of reorganization k and the electronic broadening D – in fact, they are essentially the convolution of a Gaussian of width ð1=2Þ ð4kkT Þ and a Lorenzian of width D. An example is shown in Fig. 1(a); there the parameters have been chosen such that D1 ðÞ is mainly determined by the electronic broadening, and D2 ðÞ by the interaction with the solvent. Since a Lorenz distribution does not fall off as sharply as a Gaussian, the center of the product D12 ðÞ ¼ D1 ðÞD2 ðÞ is close to that of the Gaussian-like distribution D2 ðÞ. The current is proportional to the integral over that part of the product which lies between the two Fermi levels. In Fig. 1(b) both densities are practically Gaussians, but with different widths; in this case the product has two peaks. The current–potential characteristics are affected by the way in which these densities of states shift with potential. In the electrochemical situation, two potential drops can be controlled separately: the bias between the two metals, and the potential of one of the electrodes with respect to the solution – in practice, this means with respect to a reference electrode. A fair number of situations are possible; here we consider a few representative examples. When the space between the two electrodes is narrow the potential is mainly determined by the bias. In Fig. 2 we consider the limiting case in which the solution potential plays no role, while the electronic levels are shifted by the bias. Specifically we have taken
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Fig. 2. Current–potential curves under bias control. System parameters: 1 ¼ 0:2 eV, 2 ¼ 0:3 eV. Full line: k1 ¼ k2 ¼ 0:2 eV, D1 ¼ D2 ¼ 0:1 eV. Dashed line: k1 ¼ k2 ¼ 0:2 eV, D1 ¼ D2 ¼ 0:3 eV. Dotted line: k1 ¼ k2 ¼ 0:4 eV, D1 ¼ D2 ¼ 0:1 eV.
Fig. 1. Examples for the density of states. (a) System 1: 1 ¼ 0:2 eV, k1 ¼ 0:1 eV, D1 ¼ 0:3 eV. System 2: 2 ¼ 0:3 eV, k2 ¼ 0:3 eV, D2 ¼ 0:01 eV. The dashed curve is the product. (b) System 1: 1 ¼ 0:2 eV, k1 ¼ 0:3 eV, D1 ¼ 0:01 eV. System 2: 2 ¼ 0:3 eV, k2 ¼ 0:1 eV, D2 ¼ 0:01 eV. The dashed line is the product D12 and has been multiplied by 10 for better visibility.
1 ¼ 1 ðVb ¼ 0Þ þ aVb ;
2 ¼ 1 ðVb ¼ 0Þ þ bVb ð17Þ
with a ¼ 0:1 and b ¼ 0:9 so that each level is shifted by 10% of the bias with respect to the Fermi level of the metal on which it is adsorbed. this corresponds to the case where each level is closely attached to its electrode. With the parameters chosen, a positive bias shifts the two levels towards each other (see Fig. 1) and enhances the overlap of the two densities of states, while a negative bias shifts them away from each other. Consequently the current– potential curves show a strong rectification behavior, which is typical for this situation. For a very high bias, the maxima of the two densities
have passed each other, and the overlap starts to decrease when the bias is raised further. In this region the system shows a negative resistance. When the two reactants are separated from the electrodes by spacers, they will roughly experience about half the bias potential (a ¼ b ¼ 0:5). In addition, the electronic interaction will be small, so that the densities of space are governed by the reorganization energies. Under these circumstances, the two densities are not shifted with respect to each other, and an increasing bias simply means an that the integral is performed over a larger portion of the product D12 ðÞ. When the two energies k1 and k2 are equal , this product has a single peak. In this case the current first increase with the bias, until saturation is reached, because the integral extends practically over the whole range where D12 is nonvanishing (see full line in Fig. 3). If k1 and k2 are not equal, D12 has two peaks. In this case the current–potential curves show an inflection point after the first maximum has been passed (broken line in Fig. 3). In other experimental arrangements, in particular if one of the two metals is the tip of a scanning tunneling microscope, the region between the two electrodes may be controlled by the solution potential. It is then possible to keep the bias fixed, and shift the potential of both electrodes with respect to the reference electrode. The two adsorbate
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W. Schmickler / Chemical Physics 289 (2003) 349–357
Fig. 3. Current–potential curves under bias control, with a ¼ b ¼ 0:5 (see Eq. (17)). System parameters: D1 ¼ D2 ¼ 0:01 eV, 1 ¼ 0:2 eV, 2 ¼ 0:3 eV. Full line: k1 ¼ k2 ¼ 0:2 eV. Dashed line: k1 ¼ 0:3 eV, k2 ¼ 0:1 eV.
levels may then experience the whole shift, or only a fraction of it. Specifically we may set: 1 ¼ 01 a0 e0 g;
2 ¼ 02 b0 e0 g;
where g is the deviation from an arbitrary reference potential. The full line in Fig. 4 shows the case where a0 ¼ b0 ¼ 1, i.e. where the adsorbate levels experience the whole potential shift. The current is again determined by that part of D12 ðÞ that lies between the two Fermi levels. As g is varied D12 ðÞ is shifted with respect to the Fermi levels, and if the bias is small the current in effect
Fig. 4. Examples of current–potential curves. System parameters: k1 ¼ 0:3 eV, k2 ¼ 0:1 eV, D1 ¼ D2 ¼ 0:01 eV, 1 ¼ 0:2 eV, 2 ¼ 0:3 eV, c1 ¼ 1:0. Full line: c2 ¼ 1:0. Dashed line: c2 ¼ 0.
images D12 ðÞ. Fig. 4 shows the more interesting case in which the two energies of reorganization are not equal, and a double peak is obtained. We consider also another limiting case in which only one of the adsorbate levels is shifted: a0 ¼ 1; b0 ¼ 0; this is the dashed line in Fig. 4. Now the overlap D12 between the two densities of state D1 ðÞ and D1 ðÞ changes with g, and as a result only a single peak is observed. Since the two potential drops can be controlled separately, the system can be operated as a triode: The dependence of the current on the bias depends critically on the energies 1 and 2 of the two participating levels, and these, as discussed above, are affected by the electrode potential. As an example, Fig. 5 shows current–bias curves for various values of the electronic energies 1 and 2 keeping their relative values fixed. This corresponds to the situation, where both levels are equally shifted by the electrode potential. The resulting curves are easy to understand in terms of the position of the levels and their overlap. The important point is that the current drops drastically as the initial level 1 is lifted above the Fermi level. In fact, the current can be switched on or off by shifting this level below or above the Fermi level of the first electrode.
Fig. 5. Current–bias curves for various positions of the two electronic levels, which are assumed to be shifted by the electrode potential. System parameters: k1 ¼ 0:2 eV, k2 ¼ 0:2 eV, D1 ¼ D2 ¼ 0:01 eV, a ¼ 0:1, b ¼ 0:9 (see Eq. (17)). Curve (1) solid line: 1 ¼ 0:4 eV, 2 ¼ 0:1 eV. Curve (2) dotted line: 1 ¼ 0:2 eV, 2 ¼ 0:3 eV. Curve (3) dashed line: 1 ¼ 0:0 eV, 2 ¼ 0:5 eV. Curve (4) long dashes: 1 ¼ 0:2 eV, 2 ¼ 0:7 eV.
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A variety of other cases are possible, but these examples should suffice to present the main principles. An overlap of the solvation spheres is likely when the two reactants are close to each other, so that they interact with the same modes. The resultant effect on the electron exchange is complex, since the solvation that is most favorable for the reaction changes with the reaction free energy and with the electrode potential. This is illustrated in Fig. 6, which shows current–potential curves under bias control for a few different values of the overlap energy k12 . For the parameters investigated, it depends on the bias if a particular overlap enhances or inhibits the reaction. In most cases we expect solvation overlap to be unfavorable, since usually the two states require rather different solvent configurations for the transfer. Finally we consider inelastic tunneling, where one quantum mode interacting with reactant 1 couples to the transfer. For the case of one reactant sandwiched between two metals this has been extensively discussed by us before [12]. Here, in effect, the second metal is replaced by the second reactant, whose energy level is broadened by the interaction with the second electrode and by the interaction with the classical phonon bath. As an example we consider the case where the shift of the levels is controlled by the bias, and assume that at zero bias both densities D1 ðÞ and D1 ðÞ are
Fig. 6. Examples of current–potential curves under bias control, for various values of the reorganization overlap k12 . System parameters: 1 ¼ 0:1 eV, 2 ¼ 0:1 eV, k1 ¼ k2 ¼ 0:3 eV; a ¼ 0:1, b ¼ 0:1. Full line: k12 ¼ 0. Dashed line: k12 ¼ 0:2 eV. Dotted line: k12 ¼ 0:2 eV. The energy broadening was neglected.
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Fig. 7. Illustration of the inelastic tunneling current shown in Fig. 8. (a) Situation at vanishing bias; (b) inelastic transition for a bias Vb > n hx.
centered at the Fermi levels (see Fig. 7), i.e. 1 ¼ 2 ¼ 0. For simplicity we assume that both states follow the Fermi levels of their metals and take a ¼ 0; b ¼ 1 in Eq. (17). The current due to elastic tunneling first shows an increase because the bias gets larger, but then drops off rapidly as the overlap between the two densities decreases (see Fig. 8). However, when Vb > nhx, an additional number n of inelastic channels are open, which may contribute appreciably to the current. In the calculated curves the opening of each new inelastic channel is clearly visible. In an experiment the onset may be easier to see in the derivative of the current with the bias. For obvious reasons the inelastic channels become important at a large bias; the situation can therefore be compared to inner sphere excitation in ordinary electron-transfer reactions, which may contribute appreciably at high reaction energies.
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So we believe that the value of this work lies in the advances that we have made in the theory. An extension of this work to other cases, in particular to three or more intervening states, should be possible. Acknowledgements This work was performed when I was on a sabbatical leave at the University of Florence. I would like to thank Prof. R. Guidelli for his hospitality and for valuable comments. Financial support by the Volkswagenstiftung is gratefully acknowledged. Fig. 8. Current–bias curves in the presence of inelastic transitions. The upper curve gives the total current, the other curves correspond to the excitation of n quanta, n ¼ 0; . . . ; 6. System parameters: solvent kout ¼ 0:2 eV, D1 ¼ D2 ¼ 0:01 eV, quantum mode kin ¼ 0:2 eV, hx ¼ 0:1 eV.
5. Conclusions In a certain sense, we have studied a particular realization of the original Aviram–Rattner proposal [3]. In our system, both acceptor and donor are adsorbed on metal electrodes immersed in an electrolyte solution, which provides coupling to a bath. Electron transfer was assumed to be by resonant exchange. The current–bias curves can exhibit features desirable for molecular electronics, in particular current rectification and regions with a negative differential resistance. In fact, since two voltages – bias and electrode potential – can be controlled individually, the system can be considered as a molecular triode. Whether this fact has any practical consequences, remains to be seen, but this comment applies to the whole field of molecular electronics. Current rectification has been observed in a number of systems – a thoughtful review has recently been given by Metzger [13]. However, we feel it is too early for a discussion of any experimental data: The experimental systems are not as well defined as our theoretical one, and for a meaningful comparison we would need to know the important parameters a priori. A fitting of our theoretical curves to experimental data is meaningless; given the number of available parameters we can reproduce any observed behavior.
Appendix A. Quasiclassical Greens functions In the semiclassical case the electronic energies ~1 and ~2 depend parametrically on the solvent coordinates qm , and we calculate the electronic transitions for fixed values of these coordinates. For this purpose we divide the Hamiltonian into a term H0 , which is the same as the electronic Hamiltonian Hel of Eq. (1), but with 1 and 2 replaced by their renormalized values ~1 and ~2 , and the transfer Hamiltonian HT of Eq. (2). Using the operator identity þ þ Gþ ¼ Gþ 0 þ G HT G0 ;
ðA:1Þ
we derive the following relations: h1jGþ ðzÞj2i 1 ¼ z ~2 þ id (
þ
V12 h1jG ðzÞj1i þ
X
) þ
Vl2 h1jG ðzÞjli ;
l
h1jGþ ðzÞj1i 1 ¼ ~ z 1 þ id ( ) X þ þ 1þ V1k h1jG ðzÞjki þ V12 h1jG ðzÞj2i ; k
Vl2 h1jGþ ðzÞj2i; z l þ id Vk1 h1jGþ ðzÞj1i h1jGþ ðzÞjki ¼ z k þ id þ
h1jG ðzÞjli ¼
ðA:2Þ from which Eq. (8) is easily derived.
W. Schmickler / Chemical Physics 289 (2003) 349–357
Appendix B. Transition rate
H ¼ H0 þ
X
357
Vl2 cl c2 þ Vl2 c2 cl ¼ H0 þ H1
l
Here we outline the derivation of the transition rate given by Eq. (12). We start from Eq. (9), ne2 glecting jV12 j in the denominator. We first introduce the Fourier transform of the Lorenz distribution: Z 1 1 expðipx ajpjÞ dp: ðB:1Þ ¼ x2 þ a2 2a This gives Wkl ¼
and using the operator identity: T þ ¼ T0þ þ T þ Gþ 0 H1 :
ðC:2Þ
We introduce quantum numbers m1 and m2 for the quantum modes; these corresponds to eigenstates of the undisturbed mode, when the orbital 1 is empty. We obtain hk; m1 jT þ ðzÞjl; m2 i ¼
2p dðk l Þ 2 2 2 jVk1 j jV12 j jVl2 j h 4D1 D2 Z h i exp D1 jtj itðk ~1 Þ h i exp D2 jsj isðl ~2 Þ :
ðC:1Þ
Vl2 hk; m1 jT þ ðzÞj2m2 i z l m1 ðC:3Þ
ðB:2Þ
The energies ~1 and ~2 depend on the phonon coordinates. Next we perform thermal averages over these coordinates; the integrals are elementary and result in Eq. (12). When the overlap k12 can be neglected the integrals factorize, and each can be expressed through the scaled complex error function.
Appendix C. Inelastic transitions We consider the case in which the two solvation spheres do not overlap, and in which a single quantum mode is coupled to reactant 1. The process is then quite similar to the transition metal– molecule–metal, which we treated in [12], but with the second metal replaced by reactant 2. Eq. (16) can then be written down by inspection. Formally, it can be derived by dividing the Hamiltonian into two parts:
from which Eq. (16) can be derived by noting that the thermal average for state 2 can be performed independently since the solvation spheres do not overlap. References [1] F.L. Carter (Ed.), Molecular Electronic Devices, Marcel Dekker, New York, 1982. [2] A. Aviram, M. Rattner (Eds.), Molecular Electronics: Science and Technology, Ann. N. Y. Acad. Sci. 852 (1998). [3] A. Aviram, M. Ratner, Chem. Phys. Lett. 29 (1974) 277. [4] A. Aviram, M. Rattner, V. Mujica (Eds.), Molecular Electronics II, Ann. N. Y. Acad. Sci. 960 (2002). [5] A. Nitzan, Ann. Rev. Phys. Chem. 52 (2001) 681. [6] C. Lambert, G. N€ oll, J. Schelter, Nat. Mater. 1 (2002) 69. [7] A.M. Kuznetsov, J. Ulstrup, J. Chem. Phys. 116 (2002) 2149. [8] R. Brako, D.M. Newns, Rep. Prog. Phys. 2 (1989) 655. [9] M. Abramowitz, I. Stegun, in: Handbook of Mathematical Functions, Applied Mathematics Series, vol. 55, National Bureau of Standards, Washington, DC, 1966, p. 297. [10] H. Gerischer, Z. Phys. Chem. NF 6 (1960) 223. [11] N.S. Wingreen, K.J. Jacobsen, J.K. Wilkin, Phys. Rev. B 40 (1989) 11834. [12] W. Schmickler, Surf. Sci. 95 (1993) 43. [13] R.M. Metzger, Acc. Chem. Res. 32 (1999) 950.
A model for bridge-assisted electron exchange between two electrodes W. Schmickler * Department of Chemistry, University of Florence, I-50019 Sesto Fiorentino, Italy Received 12 September 2002
Abstract We consider electron exchange between two metal electrodes mediated by two intermediate states. Our model Hamiltonian accounts for coupling to solvent and vibrational modes, and describes both elastic and inelastic processes. For the case of elastic transitions, we derive an explicit formula for the current that also includes the effect of an overlap between the solvation spheres of the two bridge states. For inelastic transitions we treat the case where one quantum mode interacts with one of the two bridge states; model calculations for the current as a function of the bias show a characteristic rise in the current whenever a new inelastic channel is opened. The system may show several features desirable for molecular electronics: rectification, negative differential resistance, and triode characteristics. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction Recent progress in nanotechnology has made it possible to study electron transfer through one or several intermediate states in well-defined molecular junctions. Much of the interest in this, currently very active, field resides in the prospect of building electronic devices on a molecular scale [1,2]. Indeed, as early as 1974 Aviram and Rattner [3] had pointed out that electron transfer through donor–acceptor systems may exhibit current rectification.
*
Permanent address: Abteilung Elektrochemie, University of Ulm, D-89069 Ulm, Germany. Fax: +49-731-502-5409. E-mail address: [email protected]
A good view over the current activities in this field can be obtained from the collection of papers in [4], and the theory is well summarized in the recent review by Nitzan [5]. Several mechanisms have been proposed for electron exchange along a series of intermediate states: electron hopping, superexchange or resonant transitions, or simply a series of chemical reactions. Indeed, depending on circumstances, on or the other mechanism may dominate, or they may act in parallel (see e.g. [6]). In this paper, we focus on electrochemical systems, on one particular case and one mechanism: resonant exchange between two metals mediated by two intermediate states. The electron transfer is coupled to a phonon bath which represents both solvent modes and local vibrations. This coupling is treated exactly, but explicit model calculations are restricted to relatively simple cases. The same
0301-0104/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0301-0104(03)00063-6
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situation has recently been discussed by Kuznetsov and Ulstrup [7], but in contrast to these authors we treat also the effect of an overlap of the solvation spheres and, more importantly, inelastic resonant transitions, which have been little discussed in the chemical community. In addition, we derive a compact and transparent expression for the electron transfer rate in the simple case, where the phonon bath is purely classical. These theoretical results are illustrated by model calculations for various cases. The calculated current–potential curves show features desirable for molecular electronics: current rectification and negative differential resistance. In fact, since two potential drops can be controlled separately – the bias and the electrode potential – the system may be operated as a triode. The rest of this paper is organized as follows: In the following section we present our model Hamiltonian, formulated in second-quantitized form. Thereupon we present explicit expressions for the rate of exchange under various circumstances, relegating the mathematical details to Appendices A–C. Results of numerical model calculations are shown and discussed in Section 4, and we finish with a short Section 5. 2. The model Hamiltonian We consider electron exchange between two electroactive species, labeled Ô1Õ and Ô2Õ; species 1 is adsorbed on a metal, whose electronic states are labeled by l, species 2 on a metal with electronic states k. Let ni ; i ¼ 1; 2; l; k denote the corresponding number operators, and i the associated energies; then the Hamiltonian for the non-interacting electrons is X X Hel ¼ 1 n1 þ 2 n2 þ l nl þ k nk : ð1Þ l
k
Electrons can be exchanged along the chain: k ! 1 ! 2 ! l and in the reverse direction. This is effected by the transfer Hamiltonian X HT ¼ Vk1 ck c1 þ Vk1 c1 ck þ V12 c1 c2 þ V12 c2 c1 k
þ
X l
Vl2 c2 c2 þ Vl2 c2 cl ;
ð2Þ
where c and c denote creation and annihilation operators, and V an appropriate coupling strength. The electronic states 1 and 2 are coupled to a phonon bath, which comprises a classical part, the solvent, and perhaps a few localized quantum modes. The phonons and their interactions are represented by the terms " # X 2 1X 2 Hph ¼ hxm pm þ qm n1 hxm gm1 qm 2 m m " # X hxm gm2 qm : n2 ð3Þ m
Here, pm and qm denote the dimensionless momenta and coordinates of the phonons, and gm1 ; gm2 are the coupling constants for the indicated states. For the two statesP we define reorganization energies P 2 2 through: k1 ¼ m hxm gm1 =2, k2 ¼ m hxm gm2 =2. The interaction of the two reactive species with the metals on which they are adsorbed can be characterized by X jVk1 j2 dðx k Þ; D1 ¼ p k
D2 ¼ p
X
ð4Þ
2
jVl2 j dðx l Þ:
l
We employ the so-called wide band approximation [8], in which D1 and D2 are taken as constant. This is a good approximation when these quantities are much smaller than the electronic bands in the two metals, which is generally the case. As is common in electron-transfer theory, we have neglected spin. This is permissible if the energy widths D1 and D2 are much smaller than the repulsive interaction energy between two electrons on the same valence orbital. Our model Hamiltonian is given by the sum of the terms presented above H ¼ Hel þ HT þ Hph ¼ H0 þ H 0 ; H 0 ¼ V12 c1 c2 þ V12 c2 c1 ;
where ð5Þ
where in the last step we have singled out the part H 0 responsible for electron exchange between the two species 1 and 2. We now proceed to calculate the rate of this exchange from scattering theory.
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3. Elastic and inelastic scattering The rate of electron exchange between the two metals can be calculated from scattering theory using the T -matrix or, equivalently, the Greens function G. The relevant matrix elements are hkjT þ ðzÞjli ¼ Vl2 Vk1 h1jGþ ðzÞj2i:
ð6Þ
When the phonon bath is purely classical, it is convenient to define qm -dependent electronic energies through X X ~1 ¼ 1 hxm gm1 qm ; ~2 ¼ 2 hxm gm2 qm : m
m
ð7Þ In the semiclassical case the Greens function can be calculated for fixed values of the phonon coordinates; this is done in Appendix A and gives h1jGþ ðzÞj2i ¼
V12 : 2 ðz ~1 þ iD1 Þðz ~2 þ iD2 Þ jV12 j ð8Þ
The rate of electron exchange between two levels k and l on the two metals, for given values of the phonon coordinates, is then Wkl ¼
2p jVk1 j2 jV12 j2 jVl2 j2 dðk l Þ ; 2 h jDj
ð9Þ
where D is the denominator in Eq. (8). Note that only elastic scattering is allowed when all the phonon modes are classical. The forward current due to electrons flowing from states k to states l is obtained from a thermal average over the initial states and summing over the final states: X i f ¼ e0 f ðk Þ½1 f ðl Þ hWkl i; ð10Þ ¼
Z
k;l
dl
Z
dk f ðk Þ½1 f ðl Þ hWkl iqðk Þqðl Þ; ð11Þ
where f ðÞ denotes the Fermi–Dirac distribution, and qðÞ the density of states. An equivalent expression holds for the reverse current. Eqs. (9) and (11) can form the basis for a completely numerical treatment of the problem. However, more insight is gained by proceeding analytically. For this purpose we assume that the coupling jV12 j is much smaller
351
than one of the two energy broadenings D1 or D2 , and can hence be neglected in the denominator. This condition will be met in most cases of practical interest, since the two active species are usually connected by a bridge whose energy levels are far from the energy range in which electron transfer occurs. As shown in Appendix B the thermal average of the rate of exchange can then be written in the form 2p dðk l Þ 2 2 2 jVk1 j jV12 j jVl2 j h 4D1 D2 Z dt ds exp f D1 jtj D2 jsjg exp itðk 1 Þ isðl 2 Þ kT ½k1 t2 þ k2 s2 þ 2k12 ts ; ð12Þ P where k12 ¼ m hxm gm1 gm2 =2 measures the overlap of the two reorganization spheres. As is often the case in semi-classical treatments, the transition rate does not depend on the coupling constants of the individual modes, but only on the collective properties expressed through k1 ; k2 ; k12 . The integrals in Eq. (12) are easily programmed. In the case where the overlap k12 is negligible the integrals factorize and can be calculated explicitly. We choose the Fermi level of the first electrode (states k) as our energy zero, and define the bias Vb as the potential difference between the second electrode and the first. For simplicity, we replace the Fermi–Dirac distribution by step functions and obtain Z 0 2p 2 e0 jV12 j if ¼ d D1 ðÞD2 ðÞ; ð13Þ h e0 Vb hWkl i ¼
where the density of states, in GerischerÕs terminology [10], of species i; ði ¼ 1; 2Þ is given by
1=2
p i þ iDi pffiffiffiffiffiffiffiffiffi R W Di ðÞ ¼ : ð14Þ kT ki 2 kT ki Here, R denotes the real part, and WðzÞ ¼ expðz2 Þ erfcðizÞ is the scaled complex error function [9]. Eqs. (13) and (14) have a transparent physical interpretation: The total current is obtained as an integral over the electronic energy between the two Fermi levels. The contribution of each electronic energy is proportional to the coupling jV12 j2 between the two molecules, which is the weakest link in the chain metal–molecule–
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molecule–metal, and to the densities Di of the two molecules. These densities of states are broadened both by the electronic interaction Di and by the coupling ki to the phonon bath. A few examples for the resulting current – potential curves will be presented in the following section. When the overlap k12 between the solvation spheres is not negligible, the integral in Eq. (12) no longer factorizes into the product of the density of states. We consider the case where the electronic broadening is small comparedpffiffiffiffiffiffiffiffiffiffi to the thermal pffiffiffiffiffiffiffiffiffiffi width, i.e. D1 kT k1 ; D2 kT k2 , in greater detail. The integrals over s and t are easily performed, resulting in Z 0 2p 2 p 2 e0 jV12 j K1=2 if ¼ d exp k1 ð 2 Þ h kT e0 Vb þ k2 ð 1 Þ2 k12 ð 1 Þð 2 Þ =4kT K; ð15Þ k212
model calculations for spewhere K ¼ k1 k2 cific cases will be presented below. When the electron exchange couples to quantum modes, with hx > kT , inelastic processes may occur, in which the transferring electron excites a mode and loses energy. When the quantum modes interact only with one of the two electroactive species, i.e. either with species 1 or with 2, the situation is similar to inelastic resonant tunneling through one species [11,12], and the pertinent results can be used (see Appendix C). This should be the typical case, since quantum modes in solutions are localized and hence not likely to interact strongly with two species. We consider explicitly the case in which one quantum mode interacts with species 1; the extension to several quantum modes interacting either with 1 or 2 is straightforward but tedious to write down. Further, we assume that the quantum mode is initially in its ground state – transitions from excited states are much less probable – and that the two solvation spheres do not overlap. The forward current can then be written as a sum over the number n of quanta hx which are excited Z 0 X 2p if ¼ e0 D1 jV12 j2 dAn ðÞD2 ð nh xÞ: h x e0 Vb þnh n ð16Þ
The argument of D2 and the lower limit of the integral take account of the fact that the tunneling electron has lost an energy nhx. Explicit expressions for the amplitudes An have been given by us in a previous publication [12].
4. Numerical results We first consider the simplest case: elastic scattering in the absence of quantum modes and negligible overlap of the solvation spheres. In this case the current is governed by the densities of states Di ðÞ as given by Eq. (14). In general, the widths of these densities are determined both by the energy of reorganization k and the electronic broadening D – in fact, they are essentially the convolution of a Gaussian of width ð1=2Þ ð4kkT Þ and a Lorenzian of width D. An example is shown in Fig. 1(a); there the parameters have been chosen such that D1 ðÞ is mainly determined by the electronic broadening, and D2 ðÞ by the interaction with the solvent. Since a Lorenz distribution does not fall off as sharply as a Gaussian, the center of the product D12 ðÞ ¼ D1 ðÞD2 ðÞ is close to that of the Gaussian-like distribution D2 ðÞ. The current is proportional to the integral over that part of the product which lies between the two Fermi levels. In Fig. 1(b) both densities are practically Gaussians, but with different widths; in this case the product has two peaks. The current–potential characteristics are affected by the way in which these densities of states shift with potential. In the electrochemical situation, two potential drops can be controlled separately: the bias between the two metals, and the potential of one of the electrodes with respect to the solution – in practice, this means with respect to a reference electrode. A fair number of situations are possible; here we consider a few representative examples. When the space between the two electrodes is narrow the potential is mainly determined by the bias. In Fig. 2 we consider the limiting case in which the solution potential plays no role, while the electronic levels are shifted by the bias. Specifically we have taken
W. Schmickler / Chemical Physics 289 (2003) 349–357
353
Fig. 2. Current–potential curves under bias control. System parameters: 1 ¼ 0:2 eV, 2 ¼ 0:3 eV. Full line: k1 ¼ k2 ¼ 0:2 eV, D1 ¼ D2 ¼ 0:1 eV. Dashed line: k1 ¼ k2 ¼ 0:2 eV, D1 ¼ D2 ¼ 0:3 eV. Dotted line: k1 ¼ k2 ¼ 0:4 eV, D1 ¼ D2 ¼ 0:1 eV.
Fig. 1. Examples for the density of states. (a) System 1: 1 ¼ 0:2 eV, k1 ¼ 0:1 eV, D1 ¼ 0:3 eV. System 2: 2 ¼ 0:3 eV, k2 ¼ 0:3 eV, D2 ¼ 0:01 eV. The dashed curve is the product. (b) System 1: 1 ¼ 0:2 eV, k1 ¼ 0:3 eV, D1 ¼ 0:01 eV. System 2: 2 ¼ 0:3 eV, k2 ¼ 0:1 eV, D2 ¼ 0:01 eV. The dashed line is the product D12 and has been multiplied by 10 for better visibility.
1 ¼ 1 ðVb ¼ 0Þ þ aVb ;
2 ¼ 1 ðVb ¼ 0Þ þ bVb ð17Þ
with a ¼ 0:1 and b ¼ 0:9 so that each level is shifted by 10% of the bias with respect to the Fermi level of the metal on which it is adsorbed. this corresponds to the case where each level is closely attached to its electrode. With the parameters chosen, a positive bias shifts the two levels towards each other (see Fig. 1) and enhances the overlap of the two densities of states, while a negative bias shifts them away from each other. Consequently the current– potential curves show a strong rectification behavior, which is typical for this situation. For a very high bias, the maxima of the two densities
have passed each other, and the overlap starts to decrease when the bias is raised further. In this region the system shows a negative resistance. When the two reactants are separated from the electrodes by spacers, they will roughly experience about half the bias potential (a ¼ b ¼ 0:5). In addition, the electronic interaction will be small, so that the densities of space are governed by the reorganization energies. Under these circumstances, the two densities are not shifted with respect to each other, and an increasing bias simply means an that the integral is performed over a larger portion of the product D12 ðÞ. When the two energies k1 and k2 are equal , this product has a single peak. In this case the current first increase with the bias, until saturation is reached, because the integral extends practically over the whole range where D12 is nonvanishing (see full line in Fig. 3). If k1 and k2 are not equal, D12 has two peaks. In this case the current–potential curves show an inflection point after the first maximum has been passed (broken line in Fig. 3). In other experimental arrangements, in particular if one of the two metals is the tip of a scanning tunneling microscope, the region between the two electrodes may be controlled by the solution potential. It is then possible to keep the bias fixed, and shift the potential of both electrodes with respect to the reference electrode. The two adsorbate
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W. Schmickler / Chemical Physics 289 (2003) 349–357
Fig. 3. Current–potential curves under bias control, with a ¼ b ¼ 0:5 (see Eq. (17)). System parameters: D1 ¼ D2 ¼ 0:01 eV, 1 ¼ 0:2 eV, 2 ¼ 0:3 eV. Full line: k1 ¼ k2 ¼ 0:2 eV. Dashed line: k1 ¼ 0:3 eV, k2 ¼ 0:1 eV.
levels may then experience the whole shift, or only a fraction of it. Specifically we may set: 1 ¼ 01 a0 e0 g;
2 ¼ 02 b0 e0 g;
where g is the deviation from an arbitrary reference potential. The full line in Fig. 4 shows the case where a0 ¼ b0 ¼ 1, i.e. where the adsorbate levels experience the whole potential shift. The current is again determined by that part of D12 ðÞ that lies between the two Fermi levels. As g is varied D12 ðÞ is shifted with respect to the Fermi levels, and if the bias is small the current in effect
Fig. 4. Examples of current–potential curves. System parameters: k1 ¼ 0:3 eV, k2 ¼ 0:1 eV, D1 ¼ D2 ¼ 0:01 eV, 1 ¼ 0:2 eV, 2 ¼ 0:3 eV, c1 ¼ 1:0. Full line: c2 ¼ 1:0. Dashed line: c2 ¼ 0.
images D12 ðÞ. Fig. 4 shows the more interesting case in which the two energies of reorganization are not equal, and a double peak is obtained. We consider also another limiting case in which only one of the adsorbate levels is shifted: a0 ¼ 1; b0 ¼ 0; this is the dashed line in Fig. 4. Now the overlap D12 between the two densities of state D1 ðÞ and D1 ðÞ changes with g, and as a result only a single peak is observed. Since the two potential drops can be controlled separately, the system can be operated as a triode: The dependence of the current on the bias depends critically on the energies 1 and 2 of the two participating levels, and these, as discussed above, are affected by the electrode potential. As an example, Fig. 5 shows current–bias curves for various values of the electronic energies 1 and 2 keeping their relative values fixed. This corresponds to the situation, where both levels are equally shifted by the electrode potential. The resulting curves are easy to understand in terms of the position of the levels and their overlap. The important point is that the current drops drastically as the initial level 1 is lifted above the Fermi level. In fact, the current can be switched on or off by shifting this level below or above the Fermi level of the first electrode.
Fig. 5. Current–bias curves for various positions of the two electronic levels, which are assumed to be shifted by the electrode potential. System parameters: k1 ¼ 0:2 eV, k2 ¼ 0:2 eV, D1 ¼ D2 ¼ 0:01 eV, a ¼ 0:1, b ¼ 0:9 (see Eq. (17)). Curve (1) solid line: 1 ¼ 0:4 eV, 2 ¼ 0:1 eV. Curve (2) dotted line: 1 ¼ 0:2 eV, 2 ¼ 0:3 eV. Curve (3) dashed line: 1 ¼ 0:0 eV, 2 ¼ 0:5 eV. Curve (4) long dashes: 1 ¼ 0:2 eV, 2 ¼ 0:7 eV.
W. Schmickler / Chemical Physics 289 (2003) 349–357
A variety of other cases are possible, but these examples should suffice to present the main principles. An overlap of the solvation spheres is likely when the two reactants are close to each other, so that they interact with the same modes. The resultant effect on the electron exchange is complex, since the solvation that is most favorable for the reaction changes with the reaction free energy and with the electrode potential. This is illustrated in Fig. 6, which shows current–potential curves under bias control for a few different values of the overlap energy k12 . For the parameters investigated, it depends on the bias if a particular overlap enhances or inhibits the reaction. In most cases we expect solvation overlap to be unfavorable, since usually the two states require rather different solvent configurations for the transfer. Finally we consider inelastic tunneling, where one quantum mode interacting with reactant 1 couples to the transfer. For the case of one reactant sandwiched between two metals this has been extensively discussed by us before [12]. Here, in effect, the second metal is replaced by the second reactant, whose energy level is broadened by the interaction with the second electrode and by the interaction with the classical phonon bath. As an example we consider the case where the shift of the levels is controlled by the bias, and assume that at zero bias both densities D1 ðÞ and D1 ðÞ are
Fig. 6. Examples of current–potential curves under bias control, for various values of the reorganization overlap k12 . System parameters: 1 ¼ 0:1 eV, 2 ¼ 0:1 eV, k1 ¼ k2 ¼ 0:3 eV; a ¼ 0:1, b ¼ 0:1. Full line: k12 ¼ 0. Dashed line: k12 ¼ 0:2 eV. Dotted line: k12 ¼ 0:2 eV. The energy broadening was neglected.
355
Fig. 7. Illustration of the inelastic tunneling current shown in Fig. 8. (a) Situation at vanishing bias; (b) inelastic transition for a bias Vb > n hx.
centered at the Fermi levels (see Fig. 7), i.e. 1 ¼ 2 ¼ 0. For simplicity we assume that both states follow the Fermi levels of their metals and take a ¼ 0; b ¼ 1 in Eq. (17). The current due to elastic tunneling first shows an increase because the bias gets larger, but then drops off rapidly as the overlap between the two densities decreases (see Fig. 8). However, when Vb > nhx, an additional number n of inelastic channels are open, which may contribute appreciably to the current. In the calculated curves the opening of each new inelastic channel is clearly visible. In an experiment the onset may be easier to see in the derivative of the current with the bias. For obvious reasons the inelastic channels become important at a large bias; the situation can therefore be compared to inner sphere excitation in ordinary electron-transfer reactions, which may contribute appreciably at high reaction energies.
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So we believe that the value of this work lies in the advances that we have made in the theory. An extension of this work to other cases, in particular to three or more intervening states, should be possible. Acknowledgements This work was performed when I was on a sabbatical leave at the University of Florence. I would like to thank Prof. R. Guidelli for his hospitality and for valuable comments. Financial support by the Volkswagenstiftung is gratefully acknowledged. Fig. 8. Current–bias curves in the presence of inelastic transitions. The upper curve gives the total current, the other curves correspond to the excitation of n quanta, n ¼ 0; . . . ; 6. System parameters: solvent kout ¼ 0:2 eV, D1 ¼ D2 ¼ 0:01 eV, quantum mode kin ¼ 0:2 eV, hx ¼ 0:1 eV.
5. Conclusions In a certain sense, we have studied a particular realization of the original Aviram–Rattner proposal [3]. In our system, both acceptor and donor are adsorbed on metal electrodes immersed in an electrolyte solution, which provides coupling to a bath. Electron transfer was assumed to be by resonant exchange. The current–bias curves can exhibit features desirable for molecular electronics, in particular current rectification and regions with a negative differential resistance. In fact, since two voltages – bias and electrode potential – can be controlled individually, the system can be considered as a molecular triode. Whether this fact has any practical consequences, remains to be seen, but this comment applies to the whole field of molecular electronics. Current rectification has been observed in a number of systems – a thoughtful review has recently been given by Metzger [13]. However, we feel it is too early for a discussion of any experimental data: The experimental systems are not as well defined as our theoretical one, and for a meaningful comparison we would need to know the important parameters a priori. A fitting of our theoretical curves to experimental data is meaningless; given the number of available parameters we can reproduce any observed behavior.
Appendix A. Quasiclassical Greens functions In the semiclassical case the electronic energies ~1 and ~2 depend parametrically on the solvent coordinates qm , and we calculate the electronic transitions for fixed values of these coordinates. For this purpose we divide the Hamiltonian into a term H0 , which is the same as the electronic Hamiltonian Hel of Eq. (1), but with 1 and 2 replaced by their renormalized values ~1 and ~2 , and the transfer Hamiltonian HT of Eq. (2). Using the operator identity þ þ Gþ ¼ Gþ 0 þ G HT G0 ;
ðA:1Þ
we derive the following relations: h1jGþ ðzÞj2i 1 ¼ z ~2 þ id (
þ
V12 h1jG ðzÞj1i þ
X
) þ
Vl2 h1jG ðzÞjli ;
l
h1jGþ ðzÞj1i 1 ¼ ~ z 1 þ id ( ) X þ þ 1þ V1k h1jG ðzÞjki þ V12 h1jG ðzÞj2i ; k
Vl2 h1jGþ ðzÞj2i; z l þ id Vk1 h1jGþ ðzÞj1i h1jGþ ðzÞjki ¼ z k þ id þ
h1jG ðzÞjli ¼
ðA:2Þ from which Eq. (8) is easily derived.
W. Schmickler / Chemical Physics 289 (2003) 349–357
Appendix B. Transition rate
H ¼ H0 þ
X
357
Vl2 cl c2 þ Vl2 c2 cl ¼ H0 þ H1
l
Here we outline the derivation of the transition rate given by Eq. (12). We start from Eq. (9), ne2 glecting jV12 j in the denominator. We first introduce the Fourier transform of the Lorenz distribution: Z 1 1 expðipx ajpjÞ dp: ðB:1Þ ¼ x2 þ a2 2a This gives Wkl ¼
and using the operator identity: T þ ¼ T0þ þ T þ Gþ 0 H1 :
ðC:2Þ
We introduce quantum numbers m1 and m2 for the quantum modes; these corresponds to eigenstates of the undisturbed mode, when the orbital 1 is empty. We obtain hk; m1 jT þ ðzÞjl; m2 i ¼
2p dðk l Þ 2 2 2 jVk1 j jV12 j jVl2 j h 4D1 D2 Z h i exp D1 jtj itðk ~1 Þ h i exp D2 jsj isðl ~2 Þ :
ðC:1Þ
Vl2 hk; m1 jT þ ðzÞj2m2 i z l m1 ðC:3Þ
ðB:2Þ
The energies ~1 and ~2 depend on the phonon coordinates. Next we perform thermal averages over these coordinates; the integrals are elementary and result in Eq. (12). When the overlap k12 can be neglected the integrals factorize, and each can be expressed through the scaled complex error function.
Appendix C. Inelastic transitions We consider the case in which the two solvation spheres do not overlap, and in which a single quantum mode is coupled to reactant 1. The process is then quite similar to the transition metal– molecule–metal, which we treated in [12], but with the second metal replaced by reactant 2. Eq. (16) can then be written down by inspection. Formally, it can be derived by dividing the Hamiltonian into two parts:
from which Eq. (16) can be derived by noting that the thermal average for state 2 can be performed independently since the solvation spheres do not overlap. References [1] F.L. Carter (Ed.), Molecular Electronic Devices, Marcel Dekker, New York, 1982. [2] A. Aviram, M. Rattner (Eds.), Molecular Electronics: Science and Technology, Ann. N. Y. Acad. Sci. 852 (1998). [3] A. Aviram, M. Ratner, Chem. Phys. Lett. 29 (1974) 277. [4] A. Aviram, M. Rattner, V. Mujica (Eds.), Molecular Electronics II, Ann. N. Y. Acad. Sci. 960 (2002). [5] A. Nitzan, Ann. Rev. Phys. Chem. 52 (2001) 681. [6] C. Lambert, G. N€ oll, J. Schelter, Nat. Mater. 1 (2002) 69. [7] A.M. Kuznetsov, J. Ulstrup, J. Chem. Phys. 116 (2002) 2149. [8] R. Brako, D.M. Newns, Rep. Prog. Phys. 2 (1989) 655. [9] M. Abramowitz, I. Stegun, in: Handbook of Mathematical Functions, Applied Mathematics Series, vol. 55, National Bureau of Standards, Washington, DC, 1966, p. 297. [10] H. Gerischer, Z. Phys. Chem. NF 6 (1960) 223. [11] N.S. Wingreen, K.J. Jacobsen, J.K. Wilkin, Phys. Rev. B 40 (1989) 11834. [12] W. Schmickler, Surf. Sci. 95 (1993) 43. [13] R.M. Metzger, Acc. Chem. Res. 32 (1999) 950.