On the Accuracy of the Noninteracting Electron Approximation for Vibrationally Coupled Electron Transport
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un On the Accuracy of the Noninteracting Electron Approximationfor Vibrationally Coupled Electron Transport
Haobin Wang
Department of Chemistry, University of Colorado Denver, Denver, CO 80217-3364, USA
Michael Thoss
Institut f¨ur Theoretische Physik und Interdisziplin¨ares Zentrum f¨ur Molekulare Materialien,Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg,Staudtstr. 7/B2, D-91058, Germany
Abstract
The accuracy of the noninteracting electron approximation is examined for a model of vibra-tionally coupled electron transport in single molecule junction. In the absence of electronic-vibrational coupling, steady state transport in this model is described exactly by Landauer theory.Including coupling, both electronic-vibrational and vibrationally induced electron-electron correla-tion effects may contribute to the real time quantum dynamics. Using the multilayer multiconfig-uration time-dependent Hartree (ML-MCTDH) theory to describe nuclear dynamics exactly whilemaintaining the noninteracting electron approximation for the electronic dynamics, the correlationeffects are analyzed in different physical regimes. It is shown that although the noninteractingelectron approximation may be reasonable for describing short time dynamics, it does not give thecorrect long time limit for certain initial conditions. . INTRODUCTION There is considerable interest in modeling charge transport in single-moleculejunctions.
From a practical perspective, this may provide insight for the developmentof molecular electronic devices. A variety of experimental techniques, such electromigration,mechanically controllable break junctions, and scanning tunneling microscopy have beenemployed to study molecular junctions.
In contrast to macroscopic conductors, molec-ular junctions typically have nonlinear current-voltage characteristics, which often show finestructures that reveal molecular details such as positions of molecular orbitals and vibrationalsignatures. From a more fundamental point of view, the experiments have also revealed manyinteresting transport phenomena such as Coulomb blockade, Kondo effect, negative dif-ferential resistance, switching and hysteresis, and quantum interference. Thesefindings have stimulated the development of physical theories and simulation techniques thatcan be used to rationalize experimental results and make predictions for improved designsof molecular junctions.A useful approach for a qualitative modeling of the conductance in molecular junctions isLandauer theory.
For noninteracting systems, such as, e.g., tight-binding based modelsof molecular junctions, it provides an exact description of steady state transport. How-ever, it does not include correlation effects due to electron-electron or electronic-vibrationalcoupling. To describe electron transport with electronic-vibrational interaction, more elabo-rate approximate theories have been used, such as the scattering theory, nonequilibriumGreen’s function (NEGF) approaches, and master equation methods.
Further-more, numerically exact simulation methods have been developed such as path integral, real-time quantum Monte Carlo, and numerical renormalization group approaches, the multilayer multiconfiguration time-dependent Hartree theory in second quantizationrepresentation (ML-MCTDH-SQR), as well as combinations of the latter method withreduced density matrix theory. In contrast to mesoscopic systems, molecular junctions of-ten exhibit strong electronic-vibrational coupling and, therefore, the vibrations have to beincluded in the theoretical treatment. This coupling may give rise to substantial current-induced vibrational excitation and thus may cause heating and possible breakage of themolecular junction. The all-importance of vibrational effects in molecular junctions havealso been confirmed by a variety of experiments. Time-dependent density functionaltheory (TDDFT) in combination with a classical treatment of the nuclear motion also be-longs to this class of approximations. In the absence of vibrational coupling this approach isexact for a tight-binding electronic Hamiltonian. When the vibrational coupling is included,both electronic-vibrational and vibrationally induced correlation effects may participate inthe real time quantum dynamics. To assess the errors introduced in this approximation,we use the multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) theory todescribe the dynamics of the vibrational degrees of freedom exactly while maintaining anoninteracting electron approximation for the electronic dynamics. The correlation effectsare analyzed in different physical regimes by comparing with the the fully correlated simula-tion employing the ML-MCTDH-SQR theory. It is hoped that this study will provide someinsight into the commonly adopted noninteracting electron approximation.The remainder of the paper is organized as follows. Section II outlines the physical model3nd the observables of interest, and briefly discusses the simulation methods. Section IIIpresents numerical results for vibrationally coupled electron transport in different parameterregimes as well as comparisons with numerically exact simulations. Section IV concludes.
II. MODEL AND SIMULATION METHODSA. Model
In this work we use a simple model for a molecular junction or a quantum dot to studycorrelation effects for vibrationally coupled electron transport. The electronic part of theHamiltonian is based on a tight-binding model, where one electronic state of the molecularbridge is coupled to two electronic continua describing the left and the right electrodes. Adistribution of harmonic oscillators is used to model the vibrational modes of the molecularbridge. The total Hamiltonian is given byˆ H = ˆ H el + ˆ H nuc + ˆ H el − nuc , (2.1a)where ˆ H el , ˆ H nuc , and ˆ H el − nuc describe the electronic, vibrational, and coupling terms, re-spectively ˆ H el = E d d + d + X k L E k L c + k L c k L + X k R E k R c + k R c k R (2.1b)+ X k L V dk L ( d + c k L + c + k L d ) + X k R V dk R ( d + c k R + c + k R d ) , (2.1c)ˆ H nuc = 12 X j ( P j + ω j Q j ) , (2.1d)ˆ H el − nuc = d + d X j γ j Q j . (2.1e)In the expression above d + /d , c + k L /c k L , c + k R /c k R are the fermionic creation/annihilation op-erators for the electronic states on the molecular bridge, the left and the right leads, re-spectively. The corresponding electronic energies E k L , E k R and the molecule-lead couplingstrengths V dk L , V dk R , are defined through the energy-dependent level width functionsΓ L ( E ) = 2 π X k l | V dk L | δ ( E − E k L ) , Γ R ( E ) = 2 π X k r | V dk R | δ ( E − E k R ) . (2.2)4mploying a tight-binding model, the function Γ( E ) is given asΓ( E ) = α e β e p β e − E | E | ≤ | β e | | E | > | β e | , (2.3a)Γ L ( E ) = Γ( E − µ L ) , Γ R ( E ) = Γ( E − µ R ) , (2.3b)where β e and α e are nearest-neighbor couplings between two lead sites and between the leadand the bridge state, respectively. I.e., the width functions for the left and the right leadsare obtained by shifting Γ( E ) relative to the chemical potentials of the corresponding leads.We consider a case with two identical leads, in which the chemical potentials are given by µ L/R = E f ± V / , (2.4)where V is the bias voltage and E f the Fermi energy of the leads.Moreover, P j and Q j in Eq. (2.1) denote the momentum and coordinate of the j th vi-brational mode with frequency ω j . The frequencies ω j and electronic-vibrational couplingconstants γ j of the vibrational modes of the molecular junctions are modeled by a spectraldensity function J ( ω ) = π X j γ j ω j δ ( ω − ω j ) . (2.5)In this paper, the spectral density is chosen in Ohmic form with an exponential cutoff J O ( ω ) = πλω c ωe − ω/ω c , (2.6)where λ is the reorganization energy. Both the electronic and the vibrational continua canbe discretized using an appropriate scheme. In this paper, we employ 200-400 states foreach electronic lead, and a bath with 900 modes. In addition, we also consider the case ofa single vibrational mode.The observable of interest in transport through molecular junctions is the current for agiven bias voltage, given by (in this paper we use atomic units where ¯ h = e = 1) I L ( t ) = − dN L ( t ) dt = − ρ ] tr n ˆ ρe i ˆ Ht i [ ˆ H, ˆ N L ] e − i ˆ Ht o , (2.7a) I R ( t ) = dN R ( t ) dt = 1tr[ ˆ ρ ] tr n ˆ ρe i ˆ Ht i [ ˆ H, ˆ N R ] e − i ˆ Ht o . (2.7b)5ere ˆ N ζ = P k ζ c + k ζ c k ζ is the occupation number operator for the electrons in each lead( ζ = L, R ) and ˆ ρ is the initial density matrix representing a grand-canonical ensemble foreach lead and a certain occupation (occupied or unoccupied) for the bridge stateˆ ρ = ˆ ρ d exp h − β ( ˆ H − µ L ˆ N L − µ R ˆ N R ) i , (2.8a)ˆ H = X k l E k l c + k l c k l + X k r E k r c + k r c k r + ˆ H . (2.8b)That is, ˆ ρ d is the initial reduced density matrix for the bridge state, which is chosen as a purestate representing an occupied or an empty bridge state, and ˆ H defines the initial bathequilibrium distribution, e.g., ˆ H nuc given above in equilibrium with an empty bridge state ora shifted bath in equilibrium with an occupied bridge state. The dependence of the steady-state current on the initial density matrix has been discussed before. In the context ofthe current work, it only affects the accuracy of the noninteracting electron approximation.To minimize the transient effects, the average current I ( t ) = 12 [ I R ( t ) + I L ( t )] , (2.9)will be used in the results presented below. B. Multilayer Multiconfiguration Time-Dependent Hartree Theory
The physical observables are calculated by solving the time-dependent Schr¨odinger equa-tion employing the multilayer multiconfiguration time-dependent Hartree (ML-MCTDH)theory.
Within the ML-MCTDH method the wave function | Ψ( t ) i is expressed in aflexible, hierarchical form | Ψ( t ) i = X j X j ... X j p A j j ...j p ( t ) p Y κ =1 | ϕ ( κ ) j κ ( t ) i , (2.10a) | ϕ ( κ ) j κ ( t ) i = X i X i ... X i Q ( κ ) B κ,j κ i i ...i Q ( κ ) ( t ) Q ( κ ) Y q =1 | v ( κ,q ) i q ( t ) i , (2.10b) | v ( κ,q ) i q ( t ) i = X α X α ... X α M ( κ,q ) C κ,q,i q α α ...α M ( κ,q ) ( t ) M ( κ,q ) Y s =1 | ξ ( κ,q,s ) α s ( t ) i , (2.10c) ... A j j ...j p ( t ), B κ,j κ i i ...i Q ( κ ) ( t ), C κ,q,i q α α ...α M ( κ,q ) ( t ), ..., are expansion coefficients of the first(top) layer, second layer, third layer, and so on; and | ϕ ( κ ) j κ ( t ) i , | v ( κ,q ) i q ( t ) i , | ξ ( κ,q,s ) α s ( t ) i , ..., aresingle particle functions (SPFs) of the respective layers. The multilayer expansion is termi-nated at a particular level by requiring the SPFs of the deepest layer to be time-independent,i.e., they are expanded in static, primitive basis functions or contracted configurations withina few degrees of freedom. SPFs of the second to last layer are then constructed using theexpansion coefficients and the (static) SPFs of the last layer. SPFs of all other layers arethen built bottom-up according to Eq. (2.10).As in the underlying MCTDH method, the ML-MCTDH equations of motion are obtained by applying the Dirac-Frenkel variational principle. The implementation ofthe ML-MCTDH method follows a systematic streamlined procedure as described in detailpreviously.
On one hand, different parts of the Hamiltonian are built “bottom-up”. Onthe other hand, reduced density matrices needed in each layer are built “top-down”. Thematrices of mean-field operators is a combination of the two procedures.The introduction of the recursive, dynamically optimized layering scheme in the ML-MCTDH wave function provides a great deal of flexibility in the trial wave function, whichresults in a tremendous gain in the ability to study large many-body quantum systems.This is demonstrated by many applications on simulating quantum dynamics of ultrafastelectron transfer reactions in condensed phases.
The ML-MCTDH work of Manthehas introduced an even more adaptive formulation based on a layered correlation discretevariable representation (CDVR).
This important development potentially extends theapplicability of ML-MCTDH theory to rather general systems described by a general formof the potential energy surface.The original ML-MCTDH method was not directly applicable to systems of identicalparticles. This is because a Hartree product in the first quantized picture is only suitableto describe a configuration for a system of distinguishable particles. To handle systems ofidentical particles explicitly, additional constraints need to be imposed since the exchangesymmetry is not accounted for in the Schr¨odinger equation or the Dirac-Frenkel variationalprinciple. To retain the multilayer form of the wave function, ML-MCTDH in the secondquantized form, the ML-MCTDH-SQR theory, was proposed, where the variation is carriedout entirely in the abstract Fock space represented by the occupation number states. TheML-MCTDH-SQR theory has seen several promising applications. . Noninteracting Electron Approximation ML-MCTDH-SQR simulations taking full account of electron-electron and electronic-vibrational correlations can be computationally demanding. Thus, it is of interest to seekless demanding approximate solutions. One approximation is to adopt a noninteractingelectron picture, that is, neglecting electron-electron correlation effects. To formulate anoninteracting electron theory of vibrationally coupled electron transport, we consider thesingle-electron Hamiltonian underlying the many-electron Hamiltonian given in Eq. (2.1),ˆ h = E d | d ih d | + X k L E k L | k L ih k L | + X k R E k R | k R ih k R | (2.11)+ X k L V dk L ( | d ih k L | + | k L ih d | ) + X k R V dk R ( | d ih k R | + | k R ih d | ) , + 12 X j ( P j + ω j Q j ) + | d ih d | X j γ j Q j , (2.12)where | k L/R i , | d i denote the electronic single particle states in the left/right leads and atthe molecule bridge, respectively. The solution of the time-dependent Schr¨odinger equationfor the single-electron Hamiltonian (2.12) represents still a many-body problem, due toelectronic-vibrational coupling. To solve it, we use also the ML-MCTDH method, similaras in our previous work on electron transfer at dye-semiconductor interfaces. To calculate transport properties using the noninteracting electron approximation, thetime-dependent Schr¨odinger equation is solved for a set of initial states. The approximationto the single electron density matrix is then obtained by summing over these wave functionsweighted according to their initial occupationsˆ ρ se ( t ) = X j p ( j ) | ψ j ( t ) ih ψ j ( t ) | , (2.13)where p ( j ) denotes the initial occupation determined by the distribution in Eq. (2.8). Theinitial wave function is given by | ψ j (0) i = | k j i| v i (2.14)if lead state | k j i is initially occupied or | ψ j (0) i = | d i| v i (2.15)8f the electronic bridge state is initially occupied. Furthermore, | v i denotes the initialvibrational state, which in all results presented below is the ground vibrational state of theoccupied or unoccupied molecular bridge. Based on the single electron density matrix (2.13),the current within the noninteracting electron approximation is given by I L ( t ) = − dN L ( t ) dt = − ddt X k L tr {| k L ih k L | ˆ ρ se ( t ) } , (2.16)and similar for I R ( t ).It is noted that the noninteracting electron approximation introduced above is exact forvanishing electronic-vibrational coupling, i.e. for the noninteracting transport problem. Inthis work, we examine the accuracy of such an approximation in the presence of electronic-vibrational coupling. It is also noted that the noninteracting electron approximation tovibrationally coupled electron transport is similar to the inelastic scattering theory approachto that problem. Both approaches treat the transport of independent electronscoupled to the vibrational degrees of freedom. In contrast to the scattering theory approach,the ML-MCTDH treatment of the noninteracting electron approximation is not limited toa few vibrational modes.
III. RESULTS AND DISCUSSION
We will assess the accuracy of the noninteracting electron transport approximation bycomparing the results from this approach with those obtained from the fully converged,numerically exact ML-MCTDH-SQR theory. In both simulations the vibrational degrees offreedom are treated via the converged ML-MCTDH approach. The difference is that in theML-MCTDH-SQR calculations the (vibrationally induced) electron-electron correlations arefully accounted for whereas the noninteracting electron approximation lacks such a treat-ment. To distinguish the two approaches, we call ML-MCTDH-SQR calculations the “full”simulation. In all results presented below, the temperature is T = 0 and the tight-bindingparameters for the function Γ( E ) are α e = 0 . β e = 1 eV, corresponding to a moderatemolecule-lead coupling and a bandwidth of 4 eV.We first consider a model, where the discrete state E d is located 0.5 eV above the Fermienergy of the leads E f . Figure 1a shows the time-dependent current for the case with asingle vibrational mode. Initially, the bridge state is occupied and the vibrational mode9s in equilibrium with the occupied bridge state. Significant oscillations of the current onthe time scale of the vibrational mode are observed in the transient current for this initialcondition, which will be quenched for long times. Compared with the full ML-MCTDH-SQR simulation, the noninteracting electron approximation reproduces I ( t ) only for veryshort time. Since it does not include vibrational nonequilibrium effect induced by electrontransport, it incorrectly predicts an increase in the amplitude of initial oscillation whereas thefull simulation predicts a damped oscillation. The noninteracting calculations also exhibitsspurious fast oscillations of the current.Figure 1b shows the time-dependent current for the same set of parameters but with adifferent initial state: an unoccupied bridge state and an unshifted vibrational mode. Withinthe same time scale as in Figure 1a, the noninteracting electron approximation providesa much better agreement with the full ML-MCTDH-SQR simulation result. Although itexaggerates the decoherence of vibrational oscillations the current, it reproduces the firstshort time transient oscillation, which is of electronic origin, and predicts a steady-statecurrent that agrees approximately with the average of the full simulation result.This observation suggests that the accuracy of the noninteracting electron approximationdepends on the initial condition. If the initial density matrix is closer to the steady state,then the effect of vibrationally induced electron correlation is smaller, which renders thenoninteracting electron approximation more accurate. In the example above, an initiallyunoccupied bridge state with an unshifted vibrational mode is closer to the steady state dis-tribution. Thus, the noninteracting electron approximation for this initial condition agreesbetter with the full ML-MCTDH-SQR simulation. One would expect that if the single vi-brational mode is replaced by a vibrational bath, the electronic coherence will be quenchedmore efficiently such that the agreement between the noninteracting electron approximationand the full ML-MCTDH-SQR simulation would improve. This is indeed the case, as shownin Figure 2.If different initial conditions give the same steady state current within a reasonably shorttime, one may argue that although the noninteracting electron approximation does not givethe correct transient dynamics, it may still predict the correct stationary current. This is tosome extent correct, as shown in Figure 3, where the parameters are the same as in Figure 2.Results for two initial conditions are plotted corresponding to an occupied or an unoccupiedbridge state. In each case the vibrational bath is in equilibrium with the bridge state. It is10een that the two initial conditions give the same stationary current within the simulationtimescale. The stationary current from the noninteracting electron approximation, as shownin Figure 2, agrees with that from the full ML-MCTDH-SQR simulation.As discussed previously, , for certain parameter regime, in particular small bias voltage,low bath characteristic frequency of the vibrational bath and strong electronic-vibrationalcoupling, the bridge state population and the time-dependent current may exhibit long-time bistability behavior. Figure 4 is an example of this phenomenon. It is seen thatthe noninteracting electron approximation incorrectly predicts that the two different initialconditions lead to the same stationary current within a very short time. Comparing withthe full ML-MCTDH-SQR simulation it can be concluded that the bistability behavior isdue to vibrationally induced correlation, which cannot be captured by the noninteractingelectron approximation. Interestingly, if one picks the “correct” initial condition based onphysical intuition (in this case an unoccupied bridge state and an unshifted vibrationalbath), then the stationary current from the noninteracting electron approximation does notdeviate much from the full ML-MCTDH-SQR value. As shown in Figure 5, the error of thenoninteracting electron approximation is only 20% for this set of parameters.The most severe failure of the noninteracting electron approximation occurs when thevibrationally induced correlation effect becomes dominant. One such example is the regimeof phonon blockade, as shown in Figure 6. The electronic parameters are the same asabove except that the energy of the discrete state E d coincides with the Fermi energy of theleads E f . The usual qualitative interpretation of the observed suppression of the currentdue to phonon blockade is that the polaron shift brings the bridge state out of the biaswindow. Figure 6 shows that this is due to vibrationally induced correlation, because thenoninteracting electron approximation predicts an incorrect value of the stationary current. IV. CONCLUDING REMARKS
In this paper, we have assessed the validity of a the noninteracting electron approxi-mation to describe transient and steady state transport in models of molecular junctionswith electronic-vibrational interaction. Within the noninteracting electron approximation,a single electron description is adopted but the interaction with the vibrational degrees offreedom is still described completely using the ML-MCTDH method. The assessment is11ased on a comparison with numerically exact results for the interacting transport problemobtained with the ML-MCTDH-SQR method.The results show that the noninteracting electron approximation provides a good repre-sentation of the short time dynamics, but may fail to describe the longer time dynamics andthe steady state current. This is particularly the case for parameter regimes that involvesignificant vibrationally induced correlation effects, such as, e.g., in the phonon blockaderegime. The validity of the noninteracting electron approximation can be improved by usingan initial state that is close to the steady state.
Acknowledgments
This work has been supported by the National Science Foundation CHE-1500285 (HW)and the Deutsche Forschungsgemeinschaft (DFG) (MT), and used resources of the NationalEnergy Research Scientific Computing Center, a DOE Office of Science User Facility sup-ported by the Office of Science of the U.S. Department of Energy under Contract No.DE-AC02-05CH11231. 12
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Time (fs) -0.10.00.10.20.30.40.50.6 I ( µ A ) FullNoninteracting
FIG. 1: Comparison of the time-dependent current I ( t ) between the noninteracting electron ap-proximation and the full ML-MCTDH-SQR simulation where a single vibrational mode is coupledto the bridge state. The frequency is ω = 500 cm − and the reorganization energy is λ = 2000cm − . The bias voltage is V = 0 .
1V and the initial condition is: (a) an occupied bridge statewith the mode’s coordinate shifted to be in equilibrium with it; (b) an empty bridge state with anunshifted mode.
20 40 60 80 100
Time (fs) I ( µ A ) FullNoninteracting
FIG. 2: Same as Fig. 1b but for a vibrational bath modeled by an Ohmic spectral density. Thecharacteristic frequency is ω c = 500 cm − and the reorganization energy is λ = 2000 cm − . Theinitial condition is specified by an empty bridge state with an unshifted vibrational bath. a) Time (fs) I ( µ A ) Initially unoccupied bridgeInitially occupied bridge (b)
Time (fs) I ( µ A ) Initially unoccupied bridgeInitially occupied bridge
FIG. 3: Time-dependent current at different initial conditions: (a) noninteracting electron approx-imation, (b) full ML-MCTDH-SQR simulation. The characteristic frequency for the vibrationalbath is ω c = 500 cm − and the reorganization energy is λ = 2000 cm − . The bias voltage is V = 0 . a) Time (fs) I ( µ A ) Initially unoccupied bridgeInitially occupied bridge (b)
Time (fs) I ( µ A ) Initially unoccupied bridgeInitially occupied bridge
FIG. 4: Time-dependent current at different initial conditions: (a) noninteracting electron approx-imation, (b) full ML-MCTDH-SQR simulation. The characteristic frequency for the vibrationalbath is ω c = 100 cm − and the reorganization energy is λ = 3000 cm − . The bias voltage is V = 0 .
25 50 75 100
Time (fs) -0.20.00.20.40.6 I ( µ A ) FullNoninteracting
FIG. 5: Comparison of the stationary current between the noninteracting electron approximationand the full ML-MCTDH-SQR simulation for the initially unoccupied bridge state and an unshiftedvibrational bath. The parameters are the same as in Fig. 4. a) Time (fs) I ( µ A ) Initially unoccupied bridgeInitially occupied bridge (b)
Time (fs) I ( µ A ) Initially unoccupied bridgeInitially occupied bridge
FIG. 6: Time-dependent current at different initial conditions: (a) noninteracting electron approx-imation, (b) full ML-MCTDH-SQR simulation. The bridge state has the same energy as the Fermilevel, E d − E f = 0. The characteristic frequency for the vibrational bath is ω c = 500 cm − and thereorganization energy is λ = 2000 cm − . The bias voltage is V = 0 .1 V.