ElectronâPhonon Coupling Effect on Charge Transfer in

Article pubs.acs.org/JPCC

Electron−Phonon Coupling Effect on Charge Transfer in Nanostructures Guangqi Li, Bijan Movaghar, and Mark A. Ratner* Department of Chemistry, Northwestern University, Evanston, Illinois 60208, United States ABSTRACT: Two simple quantum electron networks are considered: one has an interference structure, and one is a simple chain. The network is coupled at one edge site to a metal reservoir that works as a sink for arriving charges. When the electron reaches the edge site and has an energy at or above the Fermi level of the metal sink, we assume that it will be absorbed. The adiabatic electron phonon coupling will lower the energy level of the last site (before the sink) when the electron enters it. When this polarization-corrected energy is lower than the Fermi level, the absorption into the sink has to be activated above the metal Fermi sea and the absorbing rate will be slowed down. elegantly in the paper by Kopidakis et al.,31 where the mathematical derivation and explanation is given. The adiabatic limit is essentially equivalent to neglecting the kinetic energy of the lattice. This limit is in contrast to the fully time-dependent treatment with allowed dissipation.33−36 Our aim here is to examine the influence of the electron−phonon coupling on the injection process into a sink. We believe that ignoring the time delay between carrier arriving in a site and the energy shift may be a useful starting approximation for certain values of the Hamiltanian parameters. Assuming that the motion of lattice atoms is slow compared to the motion of the electrons is the basis of all semiclassical and adiabatic polaron theories and is not always justified. It depends on the systems under investigation.31,33,34 We have allowed ourselves in this paper to stretch the region of its applicability, aware of its limitations, and considering this as a useful starting point. In reality, the effect of electron phonon interactions is more complex than that and apart from the possibility of self-trapping the carrier, the coupling also gives rise to a new spectrum of eigenstates with single particle self-energies that can also have imaginary parts and give rise to broadening of the levels and energy exchange. We have neglected these subtleties here and have focused on the interplay between interference and nonlinearity, which could also find analogues in high laser power optic wire networks. Thus constructive interference can switch on the nonlinear term which slows down the flow whereas destructive interference keeps the flow . In section II we introduce the theoretical model, and section III discusses the numerical scheme used here. Section IV discusses the numerical results for the ring structure. For the

I. INTRODUCTION Bath assisted electron transfer (ET) is a process by which an electron moves from one molecule to another in a given network and environment. This process occurs in many areas of chemistry physics and biology.1−5 ET is under intense investigation, both for its intrinsic challenge and in view of application to new molecular-based devices.6−10 Many experiments have investigated the properties of different types of molecules in junctions. The research field of molecular electronics and in particular transport through molecular wires and junctions, is getting attention both experimentally and theoretically7,11−19 as we learn how to measure, prepare, and compute materials on the nanoscale. Because electron trapping and scattering at interfaces drastically affects the charge transport properties, theorists are attempting to reach higher degrees of accuracy in modeling these phenomena, now taking into account more details and effects of vibronic coupling on ET. Under the effect of the electron−phonon coupling, population can be localized to form a small polaron,20 undergoing a local site energy shift. This kind of self-trapping in molecular junction theory has been used to explore vibronically induced phenomena in molecular systems.21−30 A polaron is a quasiparticle composed of a charged particle and its accompanying polarization field. A lattice polaron is formed when a charge within a molecular chain influences the local nuclear geometry, causing an attenuation of nearby bond alternation amplitudes. Understanding charge localization and self-trapping is important in determining the electronic and optical properties and in the development of new materials. In this paper we apply the adiabatic polaron theory to both a ring and a chain, allowing 6 sites in total. At the exit edge site, the system is coupled to a metal reservoir acting as a sink, which absorbs the population. In the adiabatic limit the polaron energy is a self-consistent energy shift.31,32 The meaning of “adiabatic” and “self-consistent” limit is explained concisely and © 2012 American Chemical Society

Received: October 24, 2012 Revised: November 28, 2012 Published: December 9, 2012 850

dx.doi.org/10.1021/jp310557h | J. Phys. Chem. C 2013, 117, 850−857

The Journal of Physical Chemistry C

Article

⎧ V1π −i model I ⎪ ⎪ 4 χ=⎨ ⎪ V2π model II ⎪− i ⎩ 4

linear chain, the results are shown in section V. A conclusion follows.

II. THEORETICAL MODEL The model Hamiltonian H of the molecule (shown in Figure 1) is

Here nF(E) = 1/(1 + e(E−EF)/KT) is the Fermi function, EF is the Fermi level, E5(t) = ε5 − λn5 is the energy of site 5 at time t, which includes the transient electron−phonon induced shift. This part of the Hamiltonian is time dependent because the site energy E5(t) depends on its state of occupation by a moving charge. The exit rate is also temperature dependent but we set kT = 0.01 eV throughout. The equation signifies in particular that if the energy of the final site before the sink is above the Fermi level of the metal escape into the sink is easy. However, if the relaxed occupied polaron at site 5 is below the Fermi level, then escape into the sink becomes activated. This limit, however, has its own complications because one can now also consider the metal as emitting electrons into the occupied energy level of site 5, which has arrived below its Fermi energy. We assume that this process is prevented by Coulomb repulsion when the energy level of site 5 is occupied and negligible when site 5 is empty. The condition of interest here is that the empty level is always above the Fermi level of the metal. The fact that the sink rate depends on the energy of site 5 at any time t does not introduce new complications. The present nonlinear Schroedinger equation approach does not explicitly allow dissipation of energy to the environment (adiabatic limit). Thus there do exist apparently solitonic solutions even in the time dependent case,31 there are many possible solutions to the equation, and it is not guaranteed that a given initial condition will reach a steady state. This is because the dynamics of energy relaxation to the secondary bath is not taken into account here. But because we have included a sink, total decay of populations in the long time limit always occurs. A. Time Evolution of the Density Matrix ρ. The Liouville equation for the density matrix ρ of this 6 site system can be written as

Figure 1. Sketch of a system including 6 sites coupled by a metal reservoir.

⎧ 5 ⎪∑ (εl − λnl)cl†cl + V1(c0†c1 + c0†c 2 + c1†c3 ⎪ l=0 ⎪ ⎪ H = ⎨ + c 2†c4 + c3†c5 + c4†c5 + H.c.) model I ⎪ 5 4 ⎪ ⎪∑ (εl − ηnl)cl†cl + ∑ V2(cl†cl + 1 + H.c.) model II ⎪ ⎩ l=0 l=0

(1)

where εl is the site energy on site l, c+l (cl) creates (annihilates) an electron, λnl (model I) and ηnl (model II) are the phononinduced energy shifts with the parameter λ and η, nl = ⟨c†l cl⟩ is the on-site population, and V1 and V2 are the nearest neighbor site coupling parameters. H.c. denotes Hermitian conjugate. Beyond site 5, there is a metal reservoir electrode that is coupled to the molecule and into which the charge can transfer. The behavior due to this is16,37−40 represented as Hsink(t ) = χ {1 − nF[E5(t ) − E F]}c5†c5

(3)

(2)

with the imaginary number χ16,41−43

Figure 2. On-site population distribution P0, P1, P2, P3, P4, P5 and yield Y shown as a function of time. ε0 = ε5 = 0, ε1 = ε3 = 0.1 eV, ε2 = ε4 = 0.1 eV, V1 = 0.1 eV, EF = 0, λ = 0 (model I). 851



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Figure 3. Same as Figure 2 but with ε2 = ε4 = 0.1 eV (model I).

Figure 4. Same as Figure 2 but with λ = 0.2 eV (model I).

iℏ

dρ + = [H + Hsink , ρ] = Hρ − ρH + Hsinkρ − ρHsink dt

A (t ) = (4)

⎪

which will be solved using the Runge−Kutta nethod with the initial condition that there is one electron on site 0. Because of the complex term in eq 2 (Hsink is non-Hermitian), some diagonal elements are complex numbers. The initial condition is that at the beginning, the electron is at site 0 throughout. B. On-Site Population and the Absorbing Rate. The time dependent on-site population distribution is Pl(t ) = Tr{cl†clρ(t )}

l = 0, 1, 2, 3, 4, 5

⎪

∞

Y=

l=0

∫t =0 A(t ) dt

(8)

(5)

the relationship between Y and PT(∞) (the long time solution of PT) is Y = 1 − PT(∞).

(6)

III. NUMERICAL CALCULATION For the numerical calculation, we will set the nearest neighbor site coupling V1 = V2 = 0.1 eV. For model I, we set the site energies as ε5 = 0, ε1 = ε3 = 0.1 eV. We will vary ε2 = ε4 = 0.1 eV or ε2 = ε4 = −0.1 eV. Besides this we also vary ε0. The unit of rate A(t) is set as (e/ℏ)·fs with e one electron charge and fs 1

5

∑ Tr{cl†clρ(t )}

(7)

It is dependent on the population on site 5. The yield of the absorbed population can be obtained by defining

and the total population PT(t) on the whole system is PT(t ) =

eπ Tr{c5†c5ρ(t )}(1 − nF(E5(t ) − E F)) 2ℏ ⎧ V1 model I ⎨ ⎩ V2 model II

The time dependent absorbing rate A(t) can be obtained from the trace of eq 4 as 852



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Figure 5. Same as Figure 3 but with λ = 0.2 eV (model I).

Figure 6. Same as Figure 4 but with ε0 = 0.2 eV (model I).

calculations we will vary the energy shift parameter λ and the site energies to examine how this effects the population distribution. A. Influence from the Varying the Site Energies and λ. The electron−phonon coupling parameter λ has a direct influence on the transfer process. When the electron enters a site, its site energy will shift down. With λ = 0 (no bath coupling), part of the population can be transferred quickly from the start site to the edge site and it will be partly absorbed by the sink if the energy level of site 5 is at or above the Fermi level. From Figure 2 we can observe that with λ = 0, the on-site population disappears very quickly. This is also partially due to the symmetry of the bridge energies (1, 2, 3, 4) which allows constructive interfering pathways. In Figure 3 the site energies are chosen to form an unsymmetric bridge with ε2 and ε4. We note the change in the decay rates (Figures 2 and 3). The unsymmetric bridge has introduced dephasing, and the decay becomes much slower, as can be seen by comparing the yield Y. This example shows how lack of phase coherence can work and have useful applications.

fs. The phonon induced energy shift parameters λ (model I) and η (model II) will also be varied. For model II the site energies are set as εl = 0 for l = 0, 1, 2, 3, 4, 5. The results shown in Figures 2−8 are for model I. The results shown in Figures 9−11 are for model II.

IV. MODEL I. THE RING The interesting point about the ring model is the interferometric structure of the charge transfer pathways. Constructive interference of equivalent paths will enhance the transfer rate. Each site has added to its bare energy (an electron−phonon coupling-induced energy shift), which is selfconsistently dependent on the on-site population. When the population enters into the site, the neighboring ions move in response and the site energy will be shifted down. In principle, the electron−phonon coupling and dissipation allow the population to be “localized” before it gets to the sink. But 100% localization is not possible in these models because the sink term is a complex number and (if there is any population on this site) allows some leakage out. In the following 853



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Figure 7. Same as Figure 4 but with ε0 = 0.2 eV (model I).

Figure 8. 3-Dimensional figure showing the absorbed population yield Y as a function of both the time and the phonon coupling parameter λ. ε0 = ε5 = 0, ε1 = ε3 = 0.1 eV, ε2 = ε4 = −0.1 eV, V1 = 0.1 eV, EF = 0 (model I).

The point is that in molecular networks, control over the quantum coherence and dephasing is partially achievable. After switching on the electron phonon coupling λ, the site energies will shift down with occupation, and the transfer process will last longer as the charge partially localizes. The transfer process will require a longer time, and we can see this from the yield Y by comparing Figures 4 and 5 to Figures 2 and 3. One interesting point is that (Figure 4) the symmetry of the bridge energies (1, 2, 3, 4) is still here even after the phonon coupling energy shifts have been introduced, we can see P1 = P2 and P3 = P4. However, in Figure 5 this energy symmetry is broken and the populations are not the same any more. Another key observation is the nature of the ordered coherent oscillations, which one can see when the pathways are symmetric on the bridge. In Figure 4 the energy of site 0 is assumed lower than the bridge sites 1, 2, 3, and 4. The electron−phonon coupling will induce an energy shift on site 0 in the first few femtoseconds, resulting in a larger energy gap between sites (0,1) and (0,2). In

Figure 6 we reverse this situation, and assume the energy of site 0 to be 0.1 eV higher than sites 1, 2, 3, and 4. In the latter scenario, in the first few femtoseconds, the energy shift on site 0 will make the gap between sites (0,1) and (0,2) smaller. The closing up of initial site energy with the bridge energies enhances the quantum transfer. Thus it makes the transfer slightly faster as we can see from the yield Y by comparing Figure 6 to Figure 4. In Figure 7 we take ε0 0.3 eV lower than the bridge sites. Then most of the population will now be localized on site 0, and the transfer process to the sink will need a much longer time. This is an example of how transfer times can be manipulated within the quantum limit, a feature which has potential applications using nanogates, plasmon nanoparticle heat sources or exciton sources. B. Yield of the Absorbed Population. Under the influence of the metal reservoir working as a sink, electrons transferring from the left site 0 and arriving at site 5 will be absorbed, but during the process, there will also be some reflection, and the population will slosh about for some time 854



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Figure 9. Yield Y of the absorbed population shown as a function of time with the different phonon coupling parameter η. εl = 0 (l = 0, 1, 2, 3, 4, 5), V2 = 0.1 eV, EF = 0 (model II).

Figure 10. 3-Dimensional figure showing the absorbing rate A (unit (e/ℏ)·fs) as a function of both the time and the Fermi energy EF. εl = 0 (l = 0, 1, 2, 3, 4, 5), V2 = 0.1 eV, η = 0 (model II).

Figure 11. 3-Dimensional figure showing the absorbing rate A (unit (e/ℏ)·fs) as a function of both the time and the phonon coupling parameter η. εl = 0 (l = 0, 1, 2, 3, 4, 5), V2 = 0.1 eV, EF = 0 (model II).

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The Journal of Physical Chemistry C until everything is finally absorbed by the sink. The yields of the absorbed population are shown in Figure 8 with varying λ as a function of time. We can see that the yield decreases with increasing vibronic trapping, because escape is being blocked by self-localization.

ACKNOWLEDGMENTS

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REFERENCES

We thanks Abraham Nitzan for his insightful remarks. This work was supported by the Non-Equilibrium Energy Research Center (NERC) which is an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DESC0000989. M.A.R. thanks the chemistry division of the NSF (CHE-1058896) for support.

V. MODEL II. THE STRAND Now we turn to a simpler, more common experimental model situation in model II, a chain with 6 sites. The results are shown in Figures 9−11. In Figure 9 we show the yield of the absorbed population as a function of time while varying η in the chain model II. With η = 0 the carrier reaches the sink with minimal delay. The yield reaches its maximum value 1 very quickly. Nonzero η make this process slower, and the system needs a longer time for the population to reach the exit completely. The time scales are of course dependent on the parameters selected. The objective here is to demonstrate the order of magnitude change that one can expect. To demonstrate another interesting feature, the role played by the escape energy with respect to the metal Fermi level, we plot the absorption rate A at time t, without the electron− phonon coupling, as a function of different values of the Fermi energy EF in Figure 10. The crucial role played by the metal Fermi level can be seen by looking at the sink rate shown in eq 7. Figure 11 on the other hand demonstrates what happens to this rate when we vary the electron phonon coupling. The amplitude of A decreases with increasing λ but mainly when the coupling pulls down the energy of the last site below the metal Fermi level, blocking the exit rate.

(1) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265. (2) Barbara, P. F.; Meyer, T. J.; Ratner, M. A. J. Phys. Chem. 1996, 100, 13148. (3) Bixon, M.; Jortner, J. Adv. Chem. Phys. 1999, 106, 35−208. (4) Weiss, E. A.; Katz, G.; Goldsmith, R. H.; Wasielewski, M. R.; Ratner, M. A.; Kosloff, R.; Nitzan, A. J. Chem. Phys. 2006, 124, 074501. (5) Berlin, Y. A.; Kurnikov, I. V.; Beratan, D.; Ratner, M. A.; Burin, A. L. Top. Curr. Chem. 2004, 237, 1−36. (6) Aviram, A.; Ratner, M. A. Chem. Phys. Lett. 1974, 29, 277−283. (7) Joachim, C.; Ratner, M. A. Nanotechnology 2004, 15, 1065. (8) Cuevas, J. C.; Scheer, E. Molecular electronics: an introduction to theory and experiment; World Scientific Publishing Company Inc.: Hackensack, NJ, 2010. (9) Petty, M. C., Bryce, M. R., Bloor, D., Eds. Introduction to Molecular Electronics; Oxford University Press: New York, 1995. (10) Chen, J.; Reed, M. A.; Dirk, S. M.; Price, D. W.; Rawlett, A. M.; Tour, J. M.; Grubisha, D. S.; Bennett, D. W. Advanced Semiconductor and Organic Nano-Techniques; Academic Press: New York, 2003; Vol. 3. (11) Hänggi, P.; Ratner, M.; Yaliraki, S. Chem. Phys. 2002, 281, 111. (12) Nitzan, A.; Ratner, M. A. Science 2003, 300, 1384−1389. (13) Gerber, R.; Ratner, M. J. Phys. Chem. 1988, 92, 3252−3260. (14) Schatz, G. C.; Ratner, M. A. Quantum Mechanics in Chemistry; Dover Publications: New York, 2002. (15) Nitzan, A. Annu. Rev. Phys. Chem. 2001, 52, 681−750. (16) Li, G.-Q.; Nitzan, A.; Ratner, M. A. Phys. Chem. Chem. Phys. 2012, 14, 14270−14276. (17) Li, G.-Q.; Fainberg, B. D.; Nitzan, A.; Kohler, S.; Hnggi, P. Phys. Rev. B 2010, 81, 165310. (18) Zhao, Y.; Li, G.-Q.; Sun, J.; Wang, W.-H. J. Chem. Phys. 2008, 129, 124114. (19) Zheng, X.; Jin, J.-S.; Welack, S.; Luo, M.; Yan, Y.-J. J. Chem. Phys. 2009, 130, 164708. (20) Holstein, T. Ann. Phys. 1959, 8, 325−342. (21) Emin, D.; Kriman, A. M. Phys. Rev. B 1986, 34, 72787289. (22) Emin, D. Monatsh. Chem. 2012, DOI: 10.1007/s00706-0120836-z. (23) Mahan, G. D. Many-Particle Physics, 3nd ed.; Plenum Press: New York, 2000. (24) Galperin, M.; Ratner, M. A.; Nitzan, A. Nano Lett. 2005, 5, 125− 130. (25) Galperin, M.; Ratner, M. A.; Nitzan, A.; Troisi, A. Science 2008, 319, 1056−1060. (26) Yeganeh, S.; Galperin, M.; Ratner, M. J. Am. Chem. Soc. 2007, 129, 13313−13320. (27) Nitzan, A. Chemical Dynamics in condensed Phases; Oxford: Oxford, U.K., 2006. (28) Zhao, Y.; Luo, B.; Zhang, Y.-Y.; Ye, J. J. Chem. Phys. 2012, 137, 084113. (29) Höfer, U. Science 1998, 279, 190−191. (30) An, Z.; Wu, C. Q.; Sun, X. Phys. Rev. Lett. 2004, 93, 216407−10. (31) Kopidakis, G.; Soukoulis, C. M.; Economou, E. N. Phys. Rev. B 1995, 51, 15038−52. (32) Galperin, M.; Ratner, M. A.; Nitzan, A. J. Phys: Condens. Matter 2007, 19, 103201. (33) Lakhno, V. D. Phys. Chem. Chem. Phys. 2002, 4, 2246−50. (34) Lakhno, V. D. J. Bio. Phys. 2005, 31, 145−159.

VI. CONCLUSION In this paper, we examine two electron transfer models: a ring and a chain, both constituted by 6 sites. Each site is coupled to phonons adiabatically and thus has an electron−phonon coupling-induced energy shift. The edge site is coupled to a metal reservoir, which works as a sink to absorb the population. The absorbing rate depends on the on-site population P5 and also on the site energy at site 5, which changes because of the electron−phonon coupling induced energy variation. This time-dependent energy changes the Fermi function at that site. Escape into the sink is easy only when the Fermi level of the metal is below the effective energy of the last site. Thus if the population arriving on site 5 is not immediately absorbed, part of it will be reflected back and forth until complete decay. Two processes determine the decay process and time dependent yield: (i) the time-dependent population and energy of the last site and (ii) the nature of the quantum pathways to the last site. Both can in principle be externally controlled in designed molecular nanostructures. We only considered two situations of energy variation and dephasing, but the consequences of varying energies is clear enough from these examples. The role of the adiabatic electron phonon coupling was demonstrated clearly: the on-site populations decrease very quickly without the electron−phonon coupling. After switching on this coupling, the transfer process is delayed because of the quasi “localization” of the population.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 856



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(35) Li, G.-Q.; Movaghar, B.; Nitzan, A.; Ratner, M. A. Polaron formation: Ehrenfest dynamics vs. exact results. 2012, arXiv:1210.7234,. (36) Li, G.-Q.; Movaghar, B.; Ratner, M. A. Dynamic electron localisation initiated by particle-bath coupling. To be submitted. (37) Solomon, G. C.; Andrews, D. Q.; Duyne, R. P. V.; Ratner, M. A. Phys. Chem. Chem. Phys. 2009, 10, 257. (38) Reuter, M. G.; Hansen, T.; Seideman, T.; Ratner, M. A. J. Phys. Chem. A 2009, 113, 4665−76. (39) Rai, D.; Hod, O.; Nitzan, A. J. Phys. Chem. C 2010, 114, 20583. (40) Ben-Moshe, V.; Rai, D.; Skourtis, S. S.; Nitzan, A. J. Chem. Phys. 2010, 133, 054105. (41) Baer, R.; Seideman, T.; Ilani, S.; Neuhauser, D. J. Chem. Phys. 2004, 120, 3387. (42) Neuhauser, D.; Baer, M. J. Chem. Phys. 1989, 90, 4351. (43) Seideman, T.; William, H. M. J. Chem. Phys. 1992, 96, 4412.

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ElectronâPhonon Coupling Effect on Charge Transfer in

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