Solvent induced current-voltage hysteresis and negative differential resistance in molecular junctions
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Solvent induced current-voltage hysteresis and negative differential resistance inmolecular junctions
Alan A. Dzhioev ∗ and D. S. Kosov Department of Physics, Universit´e Libre de Bruxelles, Campus Plaine,CP 231, Blvd du Triomphe, B-1050 Brussels, Belgium
We consider a single molecule circuit embedded into solvent. The Born dielectric solvation model iscombined with Keldysh nonequilibrium Green’s functions to describe the electron transport proper-ties of the system. Depending on the dielectric constant, the solvent induces multiple nonequilibriumsteady states with corresponding hysteresis in molecular current-voltage characteristics as well asnegative differential resistance. We identify the physical range of solvent and molecular parame-ters where the effects are present. The position of the negative differential resistance peak can becontrolled by the dielectric constant of the solvent.
PACS numbers: 05.30.-d, 05.60.Gg, 72.10.Bg
The use of molecules – either singly or in small ensem-bles – as the elements of electronic circuits holds sub-stantial promise in the fields of informational technology,biological and environmental nanosensors, and energyharvesting.[1] For the science of molecular electronics tobe transformed into a technology it is not only importantto fabricate stable molecular junctions but also to be ableto efficiently control and manipulate their electric proper-ties. In the silicon-based microelectronic technology thegate voltage regulates the flow of electrons, but placinga third gate electrode has proven to be difficult in singlemolecular size devices. The negative differential resis-tance (NDR) also plays an important role in semiconduc-tor devices, because circuits with complicated functionscan be implemented with significantly fewer componentswith its help. On the other hand, instead of copying theexisting paradigms, such as, for example, gate voltage orresonant tunneling diode structure for NDR, the molecu-lar electronics create new and unique opportunities. The”wet” molecular electronics, where solvent controls theelectric behavior of an electronic circuit, may open a newchapter in device engineering. Indeed, some molecularelectronic devices already exploit the solvent around themolecule to modulate conductance through alteration ofthe charge state or polarizability of the molecule.[2–4]Let us consider a ”wet” molecular circuit – a moleculeattached to two macroscopic metal electrodes and em-bedded into solvent (Fig. 1). The total Hamiltonian is H = H L + H R + H M + H T + H MS . (1)The left and right electrodes contain free electrons andare described by the following Hamiltonians: H L = X lσ ε l a † lσ a lσ , H R = X rσ ε r a † rσ a rσ . (2)Here a † lσ/rσ creates an electron with spin σ in the single-particle state l/r of the left/right electrode and a lσ/rσ is the corresponding electron annihilation operator. The molecule is described by a single spin degenerate elec-tronic level with energy ε H M = ε X σ a † σ a σ . (3)The operator a † σ ( a σ ) creates (destroys) an electron withspin σ on the molecular level. The tunneling couplingbetween the molecule and electrodes is H T = X lσ t l ( a † lσ a σ + h.c ) + X rσ t r ( a † rσ a σ + h.c ) . (4)The interaction between the molecule and the surround-ing solvent, H MS , will be discussed below. We use nat-ural units in equations throughout the paper: ¯ h = k B = | − e | = 1, where −| e | is the electron charge.We describe the interaction between the molecule andthe solvent based on the following simple model. Themolecule is considered as a conducting sphere of radius R and the solvent is macroscopically uniform and char-acterized by dielectric constant ǫ . The work needed toplace charge q M on a conducting sphere in the dielec-tric environment is given by the Born expression for thedielectric solvation energy [5]: W = q M q S R (cid:18) − ǫ (cid:19) , (5)where q S is the induced charge in the solvent ( q M = − q S ). The model can be easily extended to the moleculesof complex shapes (the so-called generalized Born model,which represents the molecule as a number of overlap-ping spheres of different radii).[6] The (generalized) Bornmodel is quite simple yet is very successful in comput-ing the electrostatic contribution to the solvation freeenergy.[6, 7] The solvent dynamics is slow in comparisonwith the electron tunneling time scale. For example, thedielectric relaxation of the solvent is diffusive and occurson the picosecond or slower time scales since dipolar sol-vent molecules generally respond to the change of the L R0
FIG. 1. Schematic illustration of the model. The molecule isattached to two metal electrodes and surrounded by solvent.The solvent is described by uniform dielectric constant ǫ . molecule junction charging state by rotating.[5] There-fore, we can assume that the induced charge q S corre-sponds to the average electronic population of the molec-ular junction. Then, the dielectric solvation energy canbe directly associated with the interaction of the moleculewith the surrounding solvent: H MS = − U ( ǫ )( N − δ )( h N i − δ ) , (6)where U ( ǫ ) = R (cid:0) − ǫ (cid:1) is an effective, local and sol-vent controlled electron-electron attraction, and N = P σ a † σ a σ . The charge of the molecule due to nonequi-librium tunneling of electrons is ( N − δ ), while ( δ − h N i )is the corresponding induced charge in the solvent. Theparameter δ is the equilibrium molecular electronic pop-ulation which depends on the position of the molecu-lar level ε relative to the electrode Fermi energy ε f . If ε corresponds to the highest occupied molecular orbital(i.e., ε < ε f ), then, without the applied voltage bias, themolecular level is double occupied and δ = 2. If ε is thelowest unoccupied molecular orbital (i.e. ε > ε f ), thenthe molecular level is empty in equilibrium and δ = 0.It is known that such model Hamiltonians generally leadto bistable solutions.[8, 9] We emphasize that the modelis not only applicable to the solvated molecular junctionbut also to the often employed experimental setting whenthe junction is embedded into isolating or semiconductormolecular film. In this case the surrounding molecularfilm can be considered as a macroscopic dielectric envi-ronment.The similar mean-field-type interaction between themolecule and the solvent (Eq. 6) can be also obtainedwithin the polaron model in the limit ω/ Γ << ω is the frequency of a characteristic vibrational mode cou-pled to the electrons and Γ is the broadening of the molec-ular level due to coupling to the metal electrodes).[10, 11]In our case ω is related to the dielectric relaxation of thesolvent, which occurs on the picosecond and slower time scales, so ω ∼ .
001 eV. For molecules interacting withthe metal electrodes, Γ ∼ . − H M + H MS is exactly reduced to a spin degenerate single-level model with a local mean-field attractive interactionbetween electrons, which can be controlled by the dielec-tric constant of the environment. To describe electrontransport through the system we use Keldysh nonequi-librium Green’s-function formalism.[12, 13] The exactnonequilibrium molecular population h N i and electriccurrent J become: h N i = 2 π Z dω Γ L ( ω ) f L ( ω ) + Γ R ( ω ) f R ( ω )( ω − ε − ω )) + (Γ( ω )) (7) J = 4 π Z dω Γ L ( ω )Γ R ( ω )( f L ( ω ) − f R ( ω ))( ω − ε − ω )) + (Γ( ω )) . (8)Here f L/R ( ω ) = [1 + e ( ω − µ L/R ) /T ] − is the Fermi-Diracdistribution for electrons in the electrodes, ε = ε − U ( ǫ )( h N i − δ ) is the effective energy of the molecularlevel, and Λ = Λ L + Λ R , Γ = Γ L + Γ R are the real andimaginary parts of the electrode self-energyΣ L/R = X k ∈ l/r t k ω − ε k + iη = Λ L/R ( ω ) − i Γ L/R ( ω ) . (9)The electrodes are modeled as a semi-infinite chain ofatoms, characterized by the voltage-dependent on-site -1 0 10.00.51.01.52.0 -1 0 1 (eV) = -1.5 eV = -1.3 eV FIG. 2. Graphical solution of Eq. (7). The straight lines aregiven by equation h N i = − εU ( ǫ ) + (cid:16) ε U ( ǫ ) (cid:17) . Depending on ǫ and ε there can exist one, three, or five fixed point nonequi-librium molecular populations. Filled circles represent sta-ble steady state populations, while open ones correspond tounstable fixed point solutions of Eq. (7). Parameters: ap-plied voltage bias V = 1 . T = 300 K, Γ = 0 . R = 10 Bohr, ǫ = 50 (dotted lines), and ǫ = 3 (dashed lines). energy µ L,R = ± V / V h = 2 . µ L/R − V h , µ L/R + 2 V h ] is half filled, so the Fermienergy coincides with the one-site energy. The couplingbetween the left/right electrode edge and the moleculeis taken to be √ V h Γ , where Γ = Γ L ( µ L ) = Γ R ( µ R ) isthe maximal broadening of the molecular electronic leveldue to the coupling to the electrodes. Below we focus onthe case when ε is lower than the electrode equilibriumFermi energy. All our results also remain qualitativelyvalid when ε is above the Fermi level.To compute the current, we first should determine thenonequilibrium molecular population h N i . Since Eq. (7)is nonlinear, it generally has multiple solutions. Fig. 2shows the graphical solution of this equation. As we see,likewise for the electron transport in the polaron model[9], depending on values of U ( ǫ ) and the molecular levelenergy ε , Eq. (7) can have one, three, or even five solu-tions (the nonequilibrium fixed points). These multiplesolutions may or may not be steady states (i.e., the sta-ble fixed point). Following our method described in [15]we obtain the stability matrix and analyze the real partof its spectrum to assess the asymptotic time behavior ofthe fixed points. We find that only the two outer and themiddle solutions are stable in the five-solution case (rightpanel in Fig.2); i.e., they correspond to physically real-izable nonequilibrium steady-state populations. In thecase of three solutions (left panel in Fig.2), the middlesolution is unstable and the other two fixed points arestable. We note that our approach is immune from thecriticism that the observed multiple steady states are ar- ) / V / V FIG. 3. The white domain corresponds to the values of ε and U ( ǫ ) where the multiple steady state solutions exist(Γ /V << ε = − . V , ε = − U + 0 . V , ε = − U − . V , ε = − U + 0 . V , and U = 1 / R . P opu l a t i on C u rr en t ( A ) V (eV)
FIG. 4. Population-voltage and current-voltages characteris-tics. Parameters are T = 300 K, ε = − . R = 10 Bohr, ǫ = 50, Γ = 0 . tifacts of the mean-field and electron self-interaction.[16]The effect of self-interaction is physically present in ourcase, since an electron in the molecule interacts with itsown induced charge in the solvent.Let us now establish the range of key physical parame-ters – dielectric constant ǫ , molecular size R , and molec-ular level energy ε , which allow the existence of multiplenonequilibrium steady states. For presentation purposeswe assume that the molecular level broadening, Γ , aswell as the temperature are much smaller than appliedvoltage V . Therefore the molecular population (Eq. 7)(solid lines in Fig. 2) can be approximated by a step likefunction of energy ε . Then, we can readily determine an-alytically the conditions on ε and U ( ǫ ) when Eq. (7) hasonly one solution. In Fig. 3 we show the domain wheremultiple steady states exist for the case ε <
0. The case ε > V pea k ( e V ) = V (eV) c u rr en t ( A ) FIG. 5. The NDR peak voltage value as a function of thedielectric constant for two different temperatures. Parametersare ε = − . R = 10 Bohr, Γ = 0 .
01 eV. Inset: NDRin current-voltage characteristic for ǫ = ∞ . lation and the electron current demonstrate a hysteresisbehavior. The width of the hysteresis loop is propor-tional to U ( ǫ ) and, therefore, it can be controlled bythe dielectric constant. It should be emphasized thatthe solvent-induced hysteresis loop can be observed atmoderate applied voltages where the molecular device isstill mechanically stable. Moreover, the nonlinearity inthe molecule-solvent interaction leads to NDR featuresin the current - voltage characteristic (the drop in thecurrent represented by the dashed line at around 1 eV ofapplied voltage in Fig. 4). The NDR appears when oneof the electrode chemical potentials crosses the positionof the molecular level. Then, due to the subsequent shiftin the level energy caused by the electronic populationchange, the level moves away from the current-carryingwindow between the chemical potentials. In the case of ε < − . U ( ǫ ) shown in Fig. 4 the NDR takes placewhen we begin with the empty level. When ε lays above − . U ( ǫ ) ( ε <
0) the NDR also takes place, but in thiscase we need to start from the initially fully occupiedlevel.The NDR in the ”wet” molecular circuit turns out tobe sensitive to the dielectric constant of the environment.Figure 5 shows the dependence of the NDR peak positionon the dielectric constant of the solvent. The increase ofthe solvent polarity shifts the peak toward the highervoltages. This effect is very robust. It does not requirean artificial tuning of the model parameters and holdsat very large ranges of temperatures. The temperaturedependence of the NDR peak (inset in Fig.5) is consistentwith experimental observations,[17, 18] and in contrastto the polaron model explanation of NDR [9] does notrequire unphysical values for the parameters. We would like to comment here on the importance ofthe time scales. Depending on the relative time scalesof measurements and transitions between stable fixedpoints, the multistability can result in merely noise as-sociated with the jumps between steady states or it canlead to hysteresis and NDR [19]. To be experimentally re-solved the transition rate between multiple steady statesshould be smaller than the typical observation time. Inour case the transition between steady states is deter-mined by the very slow diffusive reorganization of thesolvent, which opens a possibility for experimental real-ization of the proposed effects.In conclusion, we have presented a theoretical modelto describe the environmental control of the electron-transport properties of ”wet” molecular junctions. Theinteraction between the molecule and solvent leads to ef-fective attraction between electrons which is governed bythe dielectric constant of the surrounding solvent. Thenatural separation of electronic and solvent time scalesmakes the mean-field consideration exact for our model.We used Keldysh nonequilibrium Green’s functions to ob-tain a nonlinear equation for molecular population andelectric current. Depending on the dielectric constant,the inherent nonlinearity of molecule-solvent interactionsinduces multiple nonequilibrium steady states with cor-responding hysteresis in molecular I-V characteristics aswell as NDR. We identify the physical range of solventand molecular parameters which allows the appearanceof multiple steady states. The temperature effects on theNDR peak are in qualitative agreement with the availableexperimental data. We demonstrated that the dielectricconstant of the solvent can be used as a control parameterwhich regulates the position of the NDR peak.This work has been supported by the Francqui Foun-dation, Belgian Federal Government, under the Inter-University Attraction Pole project NOSY and Pro-gramme d’Actions de Recherche Concert´ee de la Commu-naut´e Fran¸caise (Belgium), under project ”Theoreticaland experimental approaches to surface reactions”. ∗ On leave of absence from Bogoliubov Laboratory of Theo-retical Physics, Joint Institute for Nuclear Research, RU-141980 Dubna, Russia[1] M. Galperin, M. A. Ratner, A. Nitzan, and A. Troisi,Science , 1056 (2008).[2] X. Y. Xiao, L. A. Nagahara, A. M. Rawlett, and N. J.Tao, J. Am. Chem. Soc. , 9235 (2005).[3] F. Chen, J. He, C. Nuckolls, T. Roberts, J. Klare, andS. Lindsay, Nano Letters , 503 (2005), ISSN 1530-6984.[4] G. Morales, P. Jiang, S. Yuan, Y. Lee, A. Sanchez,W. You, and L. Yu, J. Am. Chem. Soc. , 10456(2005).[5] A. Nitzan, Chemical Dynamics in Condensed Phases (Oxford University Press, Oxford, 2006).[6] D. Bashford and D. A. Case, Annu. Rev. Phys. Chem. , 129 (2000).[7] A. R. Leach, Molecular Modelling. Principles and Appli-cations (Pearson, Harlow UK, 2001).[8] A. S. Alexandrov, A. M. Bratkovsky, and R. S. Williams,Phys. Rev. B , 075301 (2003).[9] M. Galperin, M. A. Ratner, and A. Nitzan, Nano Letters , 125 (2005).[10] A. M. Kuznetsov, J. Chem. Phys. , 084710 (2007).[11] M. Galperin, A. Nitzan, and M. A. Ratner, J. Phys.:Cond. Matt. , 374107 (2008).[12] L. V. Keldysh, [Zh. Eksp. Teor. Fiz. 47, 1515 (1965)] Sov.Phys. JETP , 1018 (1965).[13] H. Haug and A. Jauho, Quantum Kinetics inTransport and Optics of Semiconductors (Springer, Berlin/Heidelberg, 2010).[14] U. Peskin, Journal of Physics B: Atomic, Molecular andOptical Physics , 153001 (2010).[15] A. A. Dzhioev and D. S. Kosov, J. Chem. Phys. ,174111 (2011).[16] A. S. Alexandrov and A. M. Bratkovsky, arXiv:cond-mat/0603467v3 (2006).[17] J. Chen, M. A. Reed, A. M. Rawlett, and J. M. Tour,Science , 1550 (1999).[18] J. Chen and M. A. Reed, Chem. Phys. , 127 (2002),sp. Iss. SI.[19] E. L¨ortscher, J. W. Ciszek, J. Tour, and H. Riel, Small2