vibration induced memory effects and switching in ac-driven

8/13/2019 Vibration Induced Memory Effects and Switching in Ac-driven

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Eur. Phys. J. B(2012) 85: 316DOI:10.1140/epjb/e2012-30407-5

Regular Article

THEEUROPEANPHYSICAL JOURNALB

Vibration induced memory effects and switching in ac-drivenmolecular nanojunctions

A. Donarinia, A. Yar, and M. Grifoni

Institute of Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany

Received 21 May 2012 / Received in final form 18 July 2012Published online 17 September 2012 cEDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2012

Abstract. We investigate bistability and memory effects in a molecular junction weakly coupled to metallicleads with the latter being subject to an adiabatic periodic change of the bias voltage. The system isdescribed by a simple Anderson-Holstein model and its dynamics is calculated via a master equationapproach. The controlled electrical switching between the many-body states of the system is achieveddue to polaron shift and Franck-Condon blockade in the presence of strong electron-vibron interaction.Particular emphasis is given to the role played by the excited vibronic states in the bistability and hystereticswitching dynamics as a function of the voltage sweeping rates. In general, both the occupation probabilitiesof the vibronic states and the associated vibron energy show hysteretic behaviour for driving frequenciesin a range set by the minimum and maximum lifetimes of the system. The consequences on the transportproperties for various driving frequencies and in the limit of DC-bias are also investigated.

1 Introduction

Quantum switching, bistability and memory effects

provide potential applications for molecular electron-ics [14]. Recent scanning-tunneling microscopy (STM)experiments[59] have shown bistability and multista-bility of neutral and charged states. Random and con-trolled switching of single molecules [1012], as well asconformational memory effects [6,9,13,14] have been re-cently investigated. Other groups have observed memoryeffects in graphene [1517] and carbon nanotubes [1820].Motivated by the experimental achievements, severalgroups [2128] have attempted to theoretically explainthese striking features invoking a strong electron-vibroncoupling. In reference [21] charge-memory effects havebeen investigated in a polaron-modeled system using the

equation-of-motion method for the Greens functions inthe strong tunnel coupling regime. Similarly, in refer-ence [23] these effects are associated with a polaron sys-tem treated within a simple mean-field approach. How-ever, the hysteresis effects in reference [23] may be anartefact of the mean-field approximation as pointed out byAlexandrov and Bratkovsky[29]. In reference [24] memoryeffects have been found in a 1 polaron-modeled system tak-ing the quantum dot as a d-fold-degenerate energy levelweakly coupled to the leads and accounting for attrac-tive electron-electron interactions. However, here a multi-ple degenerate energy level (d >2) is required. In contrast,in reference [27], again the situation of weak coupling tothe leads but with repulsive electron-electron interaction

a e-mail:[email protected]

is considered. In this work, bistability, charge-memory ef-fects and switching between charged and neutral statesof a molecular junction have been explained within the

framework of a polaron model, where an electronic stateis coupled to a single vibronic mode. These features havebeen associated with the asymmetric voltage drop acrossthe junction and the interplay between time scales ofvoltage sweeping and quantum switching rates betweenmetastable states in the strong electron-vibron couplingregime. In the weak tunnel coupling limit, a perturbationtheory in the tunneling amplitude between the moleculeand leads is appropriate to describe electronic transport.In particular, such a perturbative treatment is valid if thetunneling-induced level width is small enough com-pared to the thermal energy kBT. The lowest order inthis expansion leads to sequential tunneling, which cor-responds to the incoherent transfer of a single electronfrom a lead onto the molecule or vice versa. Moreover,it is known from transport theory that sequential tunnel-ing is dominant as long as the dot electrochemical poten-tial (i.e. the differenceEN EN1between eigenvalues ofthe many-body Hamiltonian corresponding to states withparticle number differing by unity) is located between theFermi energies of the leads.

A strong electron-vibron coupling can in turn quali-tatively affect the sequential tunneling dynamics [3035].For strong coupling, the displacements of the potentialsurfaces for the molecule in a charged or neutral configu-ration are large compared to the quantum fluctuations ofthe nuclear configuration in the vibrational ground state.As a result, the overlap between low-lying vibronic statesis exponentially small. This leads to a low-bias suppression
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Page 2 of14 Eur. Phys. J. B(2012) 85: 316

of the sequential transport known as Franck-Condon (FC)blockade, which in turn is responsible for bistability effectsin[27].

In this paper we extend and improve the ideas of ref-

erence[27]. Specifically, we include the time dependenceof the bias voltage explicitly, and derive a time-dependentmaster equation for the reduced density matrix of a singlelevel molecule coupled to a vibrational mode and weaklycoupled to metallic leads. Moreover, we relax the assump-tion of fast relaxation of vibrons into their ground statesand discuss the role played by the vibronic excited statesin the switching dynamics. As in reference [27], we findthat controlled electrical switching between metastablestates is achieved due to polaron shift and Franck-Condonblockade in the presence of strong electron-vibron inter-action. Moreover, we find that the hysteresis effects canbe observed in the switching dynamics only if the timescale of variation of the external perturbation, Tex, is con-strained into a specific range set by the minimum, min,and maximum, max, charge lifetimes of the system as afunction of the applied bias. With being the dimen-sionless electron-vibron coupling, it holds min 1,max 1e2 . Hence, a strong electron-vibron cou-pling ( 1) is a necessary condition for the openingof this time scale window and thus of hysteresis. Such alarge dimensionless electron-vibron coupling is not rarein conjugated molecules with soft torsional modes (e.g.biphenyl with different substituents, azobenzene) whichhave been experimentally proven to behave as conforma-tional switches [12,13]. Very large reorganization energies(of the order of 1 eV) attributed to a polaron effect havealso been observed in STM single atom switching de-vices [5]. Also in this case the electron-phonon couplingshould be large ( 1) to justify the bistability. Outsidethis range the averaging over multiple charging events inthe slow driving case or multiple driving cycles in the fastcase removes the hysteresis.

The paper is organized as follows: in Section 2 themodel Hamiltonian of a single level molecule coupled toa vibronic mode is introduced. A polaron transformationis employed to decouple the electron-vibron interactionHamiltonian and obtain the spectrum of the system.

In Section3 we derive equations of motion for the re-duced density matrix for the case in which the leads are

subject to an adiabatic bias sweep. The time-dependentmaster equation is solved in the limit of weak coupling tothe leads and important time scale relations are derived.

In Sections4,5and6,our main results of the memoryeffects are presented and analyzed for a sinusoidal pertur-bation of period Tex = 2/.

In Section4 the lifetimes of the many-body states ofthe system are calculated. We show that, for the case ofasymmetric voltage drop across the junction, at small biasvoltages a bistable configuration is achieved which plays asignificant role in the hysteretic dynamics of the system.Bistability can involve also vibronic excited states of thesystem.

In Sections5 and 6 we give an explanation of the hys-teretic behavior of the system in terms of characteristic

time scales, in particular, the interplay between the timescale Tex of variation of the external perturbation and ofthe dynamics of the system set by switch min 1.

In Section5focus is on the regime while in Sec-tion6 is . In the latter case the features observedin reference [27] can be successfully reproduced.

In Section7,the consequences on the transport proper-ties in the DC-limit are presented as a special case. Finally,we conclude in Section8.

2 Model Hamiltonian

We consider a simple Anderson-Holstein model where theHamiltonian of the central system is described as

Hsys= Hmol+ Hv+ He-v, (1)

where Hmol represents a spinless single molecular levelmodeled by the Hamiltonian

Hmol = (0+ eVg)dd, (2)

where d(d) is the creation (annihilation) operator of anelectron on the molecule and 0is the energy of the molec-ular level, and Vg accounts for an externally applied gatevoltage. For simplicity we assume a spinless state describ-ing the molecular level with strong Coulomb interactionwhere only one excess electron is taken into account. Thespin degeneracy would not qualitatively change the resultsof the paper. The vibron Hamiltonian can be written as

Hv = 0

aa +12

, (3)

where a(a) creates (annihilates) a vibron with energy0. Finally, the electron-vibron interaction Hamiltonianis expressed as

He-v= gdd

a + a

, (4)

whereg is a coupling constant.

2.1 Polaron transformation

In order to decouple the electron-vibron interactionHamiltonian, we apply the canonical polaron unitary

transformation[36]. Explicitly, we set Hsys =e

SHsyseS,

where

S=dd

a a, (5)with = g

0as the dimensionless coupling constant. The

transformed form of the electron operator is

d= dX, (6)

where X= exp

a a

. In a similar way, the vibron

operator is transformed as

a= a dd. (7)
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Eur. Phys. J. B(2012) 85: 316 Page 3 of 14

Now the transformed form of the system Hamiltonianreads

Hsys= d

d +0aa+

1

2, (8)where = 0+eVg g

2

0is the polaron energy with polaron

shiftp = g2

0. The polaron eigenstates of the system are

|n, m1:= eS|n, m, (9)wheren denotes the number of electrons on the molecularquantum dot, while the quantum number m characterizesa vibrational excitation induced by the electron transferto or from the dot.

3 Sequential tunnelingWe analyze the transport properties of the system in thelimit of weak coupling to the leads. The Hamiltonian ofthe full system is expressed as

H(t) = Hsys+ HT+

H(t), (10)

where = s, d, denotes the source and the drain contacts,respectively. The tunneling Hamiltonian is given by

HT=

t

c

d +dc

, (11)

where c(c) creates (annihilates) an electron in lead.The coupling between molecule and leads is parametrizedby the tunneling matrix elements ts andtd. Here, we con-sider the weak coupling regime so that the energy broad-ening of molecular levels due to HT is small, i.e., 0, kBT, and a perturbative treatment for HT inthe framework of rate equations is appropriate. For sim-plicity, we assume that the tunneling amplitude ts/d oflead s/d is real and independent of the momentum ofthe lead state. In addition, we consider a symmetric devicewithts = td. Finally, the time dependent lead Hamiltonianis described by

H(t) =

+ (t)

cc. (12)

The above equation describes the lead Hamiltonian of non-interacting electrons with dispersion relation . The time-varying chemical potential (t) of lead depends onthe applied bias voltage, and yields a -independent shiftof all the single-particle levels.

3.1 Time dependent master equations for the reduceddensity matrix

In this section, we briefly derive the equation of motion forthe reduced density matrix (RDM) of the molecular junc-tion accounting for the time-dependence, equation (12),

of the lead Hamiltonian H(t). We restrict to the lowestnonvanishing order in the tunneling Hamiltonian. Never-theless, due to the explicit time dependance in the leadsHamiltonian, this work represents an extension of previous

studies on similar systems (see e.g., Refs. [34,35,3751]).The method is based on the well known Liouville equationfor the time evolution of the density matrix of the full sys-tem consisting of the leads and the generic quantum dot.To describe the electronic transport through the molecule,we solve the Liouville equation

iIred(t)

t = Trleads

HIT(t),

I(t)

(13)

for the reduced density matrix red(t) = Trleads {(t)} inthe interaction picture, where the trace over the leads de-grees of freedom is taken. In the above equation, HIT(t) isthe tunneling Hamiltonian in the interaction picture to be

calculated as below:

HIT(t) =

t

cd(t)e

i[t+(t)] + h.c.

, (14)

where (t) =tt0

(t)dt. We make the following ap-

proximations to solve the above equation: (i) the leadsare considered as reservoirs of noninteracting electrons inadiabatic thermal equilibrium. Note that this implies thatthe time scale of variation of the external perturbationhas to be large compared to the relaxation time scale ofthe reservoirs (cf. Eq. (19) below). We assume the cou-pling between system and reservoirs has been switched

on at time t = t0 and consider a factorized initial con-dition. Thus at times t t0 it holds I(t) = Isys(t)sd + (t t0)O(HT) :=Isys(t) leads + (t t0)O(HT),where the correction in the tunnelling Hamiltonian dropsin the second order master equation (see Eq. (16)). Here

s/d = 1Zs/d

e(Hs/d(t)s/d(t)Ns/d) denotes the thermal

equilibrium grandcanonical distribution of lead s/d, Zs/dis the partition function, the inverse of the thermal en-ergy, Ns/d the electron number operator, and s/d(t) =0 + s/d(t) is the time dependent chemical potentialof lead s/d which depends on the applied bias voltage.Note that the levels shift is taken into account by the

time-dependent perturbation s/d(t), while the changein chemical potential is taken into account accordingly viathe chemical potential s/d(t) so that the net positive ornegative charge accumulation in the leads is avoided. Con-ventionally, we take the molecular energy levels as a fixedreference and let the bias voltage drop across the sourceand drain contacts through the Fermi energies as [52]

s(t) =0+ (1 )eVb(t),d(t) =0 eVb(t), (15)

where 0 1 describes the symmetry of the volt-age drop across the junction. Specifically, = 0 corre-sponds to the most asymmetric situation, while = 1/2represents the symmetric case. In addition, we consider asinusoidally-varying bias voltage, i.e., Vb(t) = V0sin(t),


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where is the frequency of the driving field. (ii) Sincewe assume weak coupling of the molecule to the leads, wetreat the effects ofHT perturbatively up to second order.Accounting for the time-evolution as in equation (14) of

the leads creation/annihilation operators, we find:

Ired(t) =

|t|22

tt0

dt

f( 0)d(t)d(t)Ired(t)

e i [(tt)+(t)(t)] + [1 f( 0)] d(t)d(t)Ired(t)e

i[(tt

)+(t)(t)]

[1 f( 0)]d(t)Ired(t)d(t) e i [(tt)+(t)(t)] f( 0)d(t)

Ired(t)d(t)ei[(tt

)+(t)(t)] + h.c.

.

(16)

In the derivation of the above equation we have usedthe relation: Trleads

cc sd

= f( 0),

where f( 0) is the Fermi function, and the cyclicproperty of the trace. By summing over we obtain thegeneralized master equation (GME) for the reduced den-sity matrix in the form

Ired(t) =

|t|22

tt0

dt

F(t t, 0)d(t)d(t)

Ired(t)ei[(t)(t

)] + F(t t, 0)

d(t)d(t)Ired(t

)ei[(t)(t

)]

F(t t, 0)d(t)Ired(t)d(t) e i [(t)(t)] F(t t, 0)d(t)

Ired(t)d(t)ei[(t)(t

)] + h.c.

, (17)

where the correlation functionF(t t, 0) of lead(seeAppendix A) has, in the wide band limit, the followingform:

F

t t, 0

= Dei0 (tt)

t t i

sinh

(tt

)

, (18)which decays with the time difference t t approximatelyas exp

(tt)

on the time scale

. Here Dis the den-

sity of states of lead at the Fermi level. (iii) Since we areinterested in the long-term dynamical behavior of the sys-tem, we set t0 in equation(17). Furthermore, wereplacet bytt. We then apply the Markov approxima-tion, where the time evolution of Ired is taken only localin time, meaning we approximate red(t t) red(t)in equation (17). In general the condition of time localityrequires that[53]

,

. (19)

Here we defined from equation (17) together with equa-tion (18), =

2|t|2D as the bare transfer rates and

=

as the tunneling-induced level width. No-tice that the validity of the Markov approximation, justi-

fied in this case, is crucially depending by the order of thecurrent cumulant and the order of the perturbation expan-sion in the tunnelling coupling[54]. Finally, the conditionof adiabatic driving equation (19) allows to approximate(t) (t t) = (t)t. Taking into account thesesimplifications, the generalized master equation (GME)for the reduced density matrix acquires the form

Ired(t) =

|t|22

0

dt

F[t, (t)]d(t)d(t t)

Ired(t) + F[t, (t)]d(t)d(t t)Ired(t) F[t, (t)]d(t)Ired(t)d(t t)

F[t, (t)]d(t)Ired(t)d(t t) + h.c.

,

(20)

where F[t, (t)] = F(t, 0)ei(t)t

. Since the

eigenstates|n, m1 ofHsys are known, it is convenient tocalculate the time evolution of Ired in this basis. For ageneric quantum dot system, this projection yields a setof differential equations coupling diagonal (populations)and off-diagonal (coherences) components of the RDM.For the simple Anderson-Holstein model, equation (1), co-herences and populations are, however, decoupled. In thesequential-tunneling regime, the master equation for theoccupation probabilities Pmn = 1n, m|red|n, m1 of find-ing the system in one of the polaron eigenstates assumesthe form

Pmn =n,m

mm

nn (t)Pm

n n,m

mm

nn (t)Pmn , (21)

where the inequality 0 ensures the applicability ofthe secular approximation, i.e., the separation between thedynamics of populations and coherences. In the numeri-cal treatment of these equations we truncate the phononspace. Convergence is reached already with 40 excitations.In equation (21) the coefficient m

mnn denotes the transi-

tion rate from|n, m1 into the many body state|n, m1,while mm

nn describes the transition rate out of the state|n, m1to |n, m1. Taking into account all possible single-electron-tunneling processes, we obtain the incoming andoutgoing tunneling rates, in the wide band limit, as

mm

01 (t) =

Fmmf+

+0 (m m) (t)

mm

,01 (t), (22)

mm

10 (t) =

Fmmf

+0 (m

m) (t)

m

m,10 (t), (23)


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where the terms describing sequential tunneling from andto the lead are proportional to the Fermi functionsf+(x ) =f(x ) andf(x ) = 1 f(x ),respectively. Notice that the integrations over energy and

time introduce the explicit time dependance in the Fermifunctions. The factorFmm = |m|X|m|2 is the Franck-Condon matrix element which can be calculated, with Xdefined in Section 2.1, explicitly using Appendix C. Thesum rules

m Fmm =

mFmm = 1 are well satis-

fied because of the completeness of each vibrational basisset|0, m and|1, m1. This factor describes the wave-function overlap between the vibronic states participat-ing in the particular transition. It contains essential in-formation about the quantum mechanics of the moleculeand significantly influences the transport properties ofthe single-molecule junction. Within the rate-equation ap-proach, the (particle) current through lead is deter-

mined by

I(t) =mm

mm

,01 (t)Pm0 (t) m

m,10 (t)P

m

1 (t)

(24)

and it is in general time dependent. Moreover, differentlyfrom the stationary case, in general IL(t)=IR(t). Thecharge is though not accumulating on the dot since, forthe average quantities

I,av = limt

t+Text

dtI(t) (25)

it holds IL,av=

IR,av, as it can be easily proved consid-

ering that the average charge on the dot oscillates withthe same period Tex of the driving bias. Finally, in theDC limit 0 the relation IL(t) =IR(t) holds as thefully adiabatic driving allows to reach the quasi-stationarylimit at all times.

4 Lifetimes and bistability of states

In this section, we show that when the bias voltage drop isasymmetric across the junction, upon sweeping the bias,one can tune the lifetime of the neutral and charged statesto achieve a bistable system. The lifetime of a state isobtained by calculating the switching rate of that state.The lifetime nm of a generic quantum state|n, m1 isgiven by the sum of the rates of all possible processeswhich depopulate this state, i.e.,

1nm =n,m

mm

nn , (26)

and it defines, at least on a relative scale, the stability ofthe state|n, m1. Thus, at finite bias voltage, the inverselifetime of the 0-particlemth vibronic state is given by therelation

10m =,m

Fmmf++ 0 (m m) . (27)

In a similar way, the inverse lifetime of the 1-particle andmth vibronic state is expressed as

11m = ,m

Fmmf

+0 (m m) . (28)

A consequence of equations (27) and(28) is that, due tothe characteristic features of the Franck-Condon matrixelements, in the strong electron-vibron coupling regime,the tunneling with small changes in m m is suppressedexponentially. Hence only some selected vibronic statescontribute to the tunneling process. However, tunnelingalso depends on the bias voltage and temperature throughthe Fermi function. To proceed further, let us focus firston the lifetime of the 0- and 1-particle ground states forthe case of fully asymmetric coupling of the bias voltageto the leads, i.e., = 0:

100 =m

e22m

m!

sf

+ ( + m0 0 eVb)

+ df+ ( + m0 0)

, (29)

110 =m

e2

2m

m!

sf

( m0 0 eVb)

+ df ( m0 0)

. (30)

One can see from equation (29) that if in the con-sidered parameters range is + m0 0, i.e.,f( + m

0 0) 0, then the second term in thebracket is negligible. The first term is nonzero at largepositive bias, while at large negative bias it remains neg-ligible. In a similar way one can analyze the behavior of110 in which the first term on the r.h.s. of equation (30)will be dominating at large negative bias. In order to un-derstand the mechanism of this process the energy-levelscheme for the relevant transitions in a coordinate systemgiven by the particle number N and the grandcanonicalenergy E 0N shown in Figure 1. We choose Vg = 0and 0 = 0. Moreover, the polaron energy levels are atresonance with the 0-particle states for our chosen set ofparameters: we setp=0and hence = 0. Then the onlytransitions allowed at zero bias are ground state

ground

state transitions. At finite bias also transitions involvingexcited vibronic states become allowed. In particular, atVb= 0 it follows from equations (29), and(30) that

100 (Vb = 0) = 110 (Vb = 0) = e

2(s+ d)/2, (31)

while at|Vb| it holds

100 (Vb ) =110 (Vb )= s+

d2

e2 s 1min, (32)

whereas

100 (Vb ) =110 (Vb ) = d

2 e

2 1max.(33)


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Fig. 1. (Color online) (a) Energy-level scheme for the rele-vant transitions in a coordinate system given by the particlenumber N and grandcanonical energy E 0N at Vb = 0.The red lines represent the transitions threshold, where thethickness of each transition line gives the strength of the tran-sition. The polaron energy levels are aligned with the 0-particlestates for our chosen set of parameters (0 = 0, Vg = 0, 0 =250, = 5) yielding the polaron shift p =0. (b) Inverselifetimes (n0)

1 on logarithmic scale as a function of nor-malized bias voltage eVb/0. The red thick line representsthe inverse lifetime of the 1-particle ground state, while thethin blue line refers to the 0-particle ground state.

In practice the asymptotic behaviors are already reachedat e|Vb|/0 22 as observed in Figure 1b. Note thatmax and min set the maximum and minimum achiev-able lifetimes which, due to max/min e2 , can differby several orders of magnitude for > 1. Note also thatnear zero bias the lifetimes are so long that the systemnever likes to charge or discharge and a bistable situationis reached. A selective switching, however, can occur uponsweeping the bias voltage. Hence min also sets the timescale for switching:min switch.

Analogously, we can explain the behavior of the life-times of the excited states (see Fig. 2). It follows that inthe considered parameters range, in general, the 0-particle

Fig. 2. (Color online) Inverse lifetime (nm)1 as a func-

tion of normalized bias voltage eVb/0 for (a) vibronicground states, (b) first excited states, (c) second excited states,(d) third excited states, (e) fourth excited states, (f) fifth ex-cited states when Vg = 0. The blue thin line represents the

inverse lifetime of the 0-particle state (n

= 0), while the thickdashed red line refers to the 1-particle state (n= 1). The asym-metry parameter is = 0 and we fix the zero of the energy atthe leads chemical potential at zero bias: 0 = 0. The energy ofthe molecular level is 0 = 250. The electron-vibron couplingconstant is = 5 yielding a polaron shift p = 0. Finally, thethermal energy is kBT = 0.20, the frequency of the drivingfield is = 0.0020, and s = d = 0.0060.

vibronic states are stable at large enough negative biasvoltage, while the 1-particle vibronic states are stable atlarge positive bias. There is, however, an interval of biasvoltage, the so-called bistable region, where both states

|1, m

1 and|0, m1 are metastable for not too large mand m, as shown in Figure 2. Moreover,m steps are ob-served in the inverse lifetimes 1nm (see Figs. 2b2f) be-cause for certain values of the coupling constant someof the FC factors Fmm vanish or are exponentially smallsuch that the additional channels opened upon increasingthe bias voltage do not have pronounced contribution. Forinstance, the FC factor for the first excited vibronic statecan be described as

F1m =e2

2(m1)

m! (m 2)2, (34)

which vanishes form =2. That is why a plateau around

eVb/0 = 25 in Figure 2b is observed for our chosenparameters. Analogously, using equation (D.1), one canfind (cf. AppendixD) thatF2m has two minima at

m1= 1 + 22 +

1 + 42

2 , m2=

1 + 22 1 + 422

.

(35)

Hence two plateau can be observed (see Fig. 2c) aroundeVb/0 = 20 and eVb/0 = 31. Similar arguments canbe extended to explain the steps in the inverse lifetimes ofhigher excited states. This also implies that the bias win-dow for bistability shrinks for excited states and even dis-appears for large enoughm. It follows that the major con-tribution in bistability is coming from low excited vibronicstates. Note that the bistability of the many body states is


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Fig. 3. (Color online) Inverse lifetime (nm)1 as a function

of normalized bias voltage eVb/0 for (a) vibronic groundstates, (b) first excited states, (c) fifth excited states wheneVg/0 = 8, while (d) shows vibronic ground states, (e) firstexcited states, and (f) fifth excited states, when eVg/0 = 8.

The remaining parameters are the same as used in Figure 2.

crucial for the hysteresis and hence memory effects whichis discussed in the next section. Finally, a closer inspectionof Figure2reveals that the minimum of the inverse life-time increases with the vibronic quantum numberm. Thiseffect can be understood easily by analyzing the minimumof the inverse lifetime for each particle state. For examplethe minimum of the inverse lifetime for the 0-particle vi-bronic ground state is, cf. equation (33), whereas for the0-particle vibronic first excited state is

101 (Vb

) =

d

2

(1 + 4)e2

. (36)

From equations (33) and (36), one can conclude that100 (Vb ) < 101 (Vb ). A similar explana-tion can be extended to the higher excited states. Forgate voltages such that eVg > 0, the 1-particle vibronicexcited states are becoming unstable faster than the 0-particle states (see Figs. 3a3c), while for large negativegate (eVg < 0), the 0-particle states are getting unstablefast (see Figs. 3d3f). In order to explain this effect, weanalyze the shift of the inverse lifetime of the 0-particlevibronic first excited state, 101 , as follows:The maximum of the inverse lifetime for Vg= 0 is

1

01

(Vb

) = s+ dm

F1mf(eVg+ 0(m

1)),

(37)

whereas the minimum is given by

101 (Vb ) = dm

F1mf(eVg+ 0(m 1)). (38)

Equations (37) and (38) imply that both minimum andmaximum of101 shift by an equal amount and the condi-tion of the bistability region can be tuned by setting Vg.

5 Quantum switching and hysteresis

Neutral and charged (polaron) states correspond to dif-ferent potential energy surfaces and transitions between

Fig. 4. (Color online) (a)(b) Occupation probabilities P0 and

P1of the 0- and 1-particle electronic states as a function of nor-malized time dependent bias voltage eVb/0, (c) populationof the 0-particle configuration as a function of time; (d) nor-malized bias voltage as a function of time. The parameters arethe same as used in Figure2.

low-lying vibronic states are strongly suppressed inthe presence of strong electron-vibron interaction. Thisleads to the bistability of the system. Upon applyingan external voltage, one can change the state of thisbistable system obtaining under specific conditionshysteretic charge-voltage and current-voltage curves.

Here it is crucial to point out that only if the time scaleof variation of the external perturbation is shorter thanthe maximum lifetime but longer than the minimumlifetime of the system hysteresis can be observed, i.e.,min switch < Tex < max. Due to max > Tex, thesystem stays in the stable state during the sweepinguntil the sign of the perturbation changes, the formerstable state becomes unstable and, due to Tex < min,a switching to the new stable state can occur. In thissection we now consider the situation when , i.e.,Tex switch while in Section 6 the regime , i.e.,Tex switch is addressed.

In Figures 4 and5we present the populations of theelectronic states,Pn =

m Pmn , as well as of the vibronic

states, Pm =

n Pmn , respectively. Specifically, in Fig-

ures4a4b, we have plotted the populations of the 0- and1-particle electronic states as a function of normalized biasvoltage, where hysteresis loops can be seen. In Figure 4c,instead, we have shown the population of the 0-particleelectronic state as a function of time. The latter can beused to determine the timeswitchof switching between theneutral and charged states. In a similar way, the sweep-ing time Tex of the bias voltage can be calculated usingFigure 4d. By comparison of these two time scales, it isapparent that the switching time is of the same order asthe sweeping time and much shorter than the lifetime inthe bistable region (see Fig.1). The relation switch Texalso explains why the switching between the neutral and


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Fig. 5.(Color online) Populations Pm of the vibronic states as

a function of normalized time dependent bias voltage eVb/

0for (a) ground state, (b) first excited state, (c) second excitedstate, and (d) fifth excited state. The parameters are the sameas used in Figure2.

charged state is on average never complete (P0 oscillatesbetween 0.2 and 0.8).

In Figure5,the populations of the vibronic states as afunction of the normalized bias voltage are shown, whilein Figure6the populations of the different vibronic statesresolved for different charges have been plotted. Clearlynot only the vibronic ground states (which were consid-ered in Ref. [27]) show hysteretic behavior but the vibronic

excited states also exhibit these interesting features. Fur-thermore, inspection of these figures reveals that even af-ter relaxation on the stable limit cycle, the vibronic ex-cited states are highly populated in the non-stationarycase in contrast to the stationary case 0 (see e.g.,Figs.15and17) where the population of the excited statesis strongly suppressed. Finally, while the general trend isa reduction of the population, the higher the excitationand the populations are negligible for m 40, an inter-esting behaviour can be recognized in the form of the limitcycles. Namely, upon sweeping the bias we find that, form 8 the probability grows at large biases, it stays es-sentially constant for m 8 and it decreases at largerbiases for m < 8. The interpretation of this behaviour isstill unclear to us. All these observation confirm, though,that it is natural to take into account the vibronic excitedstates in the dynamics of the system.

5.1 I-V characteristics

The hysteretic behavior of the bistable system is also re-flected in the current as a function of normalized bias (seeFig. 7) where a hysteresis loop (single loop) is observedin the current calculated both at the left and the rightlead. Interestingly, the left and the right currents differ bymore than a sign, in contrast to the stationary case. Thisbehavior is understandable again in terms of relaxationtime scales. In fact, for voltages|Vb| outside the bistable

Fig. 6. (Color online) Plots of the population Pmn

as a func-tion of normalized time dependent voltage eVb/0 for the0-particle vibronic (a) ground state, (b) first excited state, (c)fifth excited state, and for the 1-particle vibronic (d) groundstate, (e) first excited state, (f) fifth excited state. The param-

eters are the same as used in Figure2.

Fig. 7. (Color online) Time dependent current as a functionof normalized voltage for (a) left lead, (b) right lead. The pa-

rameters are the same as used in Figure 2.

region the system relaxes to the stationary regime on atime scaleswitch. Though, since the driving time Tex hasthe same order of magnitude, the stationary regime can-not be reached. Yet, no net charge accumulation occurssinceIL,av= IR,av.

In Figure 8, we plot the left time dependent currentas a function of the normalized bias for different valuesof the electron-vibron coupling constant. An inspection of

this figure reveals that the width of the hysteresis loop de-creases and shifts from zero bias upon decreasing the cou-pling constant . This feature can be understood by ob-serving that for = 5 the polaron shift p does not longercompensate the energy of the molecular level0, and hencethe polaron energy = 0. In other words, the system isno longer behaving symmetrically upon exchange of thesign of the bias voltage. If we consider e.g. the case = 1is, forVg= 0= 0, /0= 24. In turn this implies that

100 (Vb = 0) 0 and 110 (Vb = 0) s+ d, i.e., the re-gion around zero bias is no longer bistable as for the case = 5. Hence the dot is preferably empty at zero bias.Switching however can be reached upon increasing Vb inthe region around eVb

. Overall however the bistabil-

ity region has shrunk. Similar considerations apply to theother considered values of.


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Fig. 8. (Color online) Time dependent current for the left leadas a function of the normalized voltage for coupling constants(a) = 4, (b) = 3, (c) = 2, and (d) = 1. The remainingparameters are the same as used in Figure2.

5.2 Vibron energy

In this section, we illustrate the role played by the vibronicenergy in the hysteretic behavior of the system. The vi-bron energy of the whole system can be expressed as

Ev = Trsys

red0

aa +

1

2

, (39)

where the trace is taken over the system degrees of free-dom. The normalized vibronic energy as a function of nor-malized bias voltage is depicted in Figure9a, where hys-teretic loops are also observed. The value of the vibronicenergy, together with the observation that the probabil-ity distribution is relatively flat over the excitations (seeFig. 6) ensures that, depending on the bias, between 10and 20 vibronic excited states are considerably populated.Further insight in the dynamics of the system is obtainedby considering the correlation between the vibronic energyand the charge occupation.

The vibron energy associated with the 0-particle stateis determined by the relation

Ev,0= Trsys

00

aa+

1

2

, (40)

with 0= red|0, m110, m|. In Figure9b, the normalizedvibronic energy as a function of normalized bias voltagefor the 0-particle configuration has been plotted. The hys-teresis loop resembles that of Figure4a implying a directcorrelation between the vibronic energy and the popula-tion of the neutral state i.e., the more the neutral stateis occupied the higher is the associated vibronic energy.Qualitatively the result can be explained as follows: tran-sitions from the charged to the neutral states are predom-inantly involving low energy charged states and highlyexcited neutral states. Due to energy conservation andasymmetric bias drop these transitions are confined to

Fig. 9. (Color online) (a) Total vibron energy as a functionof the time dependent bias voltage. (b) Vibron energy for the0-particle, and (c) for the 1-particle configuration only. Param-eters are the same as used in Figure2.

the large negative biases where the highly excited neutralstates show also a long life time. This situation remainsroughly unchanged during the up sweep of the bias untilthe symmetric condition is obtained at high positive biasand the charged excited states are maximally populated.

Finally, the bistability around zero bias explains the hys-teresis. The analytical expression for the vibronic energyof the 1-particle state is given by

Ev,1= Trsys

10

aa +

1

2

, (41)

with 1 = red|1, m111, m|. The normalized averagevibron energy as a function of normalized bias voltagefor the 1-particle configuration is sketched in Figure 9c,where we can observe a hysteresis loop resembling that ofFigure4b.

In conclusion, the vibron energies also show hys-teretic behavior, in analogy to the population-voltage andcurrent-voltage curves, in the non-stationary limit.

6 Testing lower driving frequencies

When lowering the driving frequency ( ) of theexternal perturbation, we choose = 21060, ourmodel displays features similar to those presented in ref-erence [27]. In more detail, we show the population ofthe electronic states as a function of normalized bias andtime in Figures10a,10b,10c, respectively, whereas in Fig-ure10d the normalized bias as a function of time is shown.

In this case the population-voltage curve is slightly differ-ent from Figure4 because the transition between 0 and 1occurs more abruptly as a function of Vb and it is com-plete. Indeed, for the parameter chosen in Figure 10 is= 0 and1max min 1switch. In other wordsthe frequency is small compared to the charge/dischargerate. The system thus follows adiabatically the changes ofthe bias voltage and only switches at those values of thebias wheren0 switch. The time-dependent left currentas a function of normalized bias is shown in Figure 11agiving two loops, one for positive bias sweeping and theother for negative sweeping. The right current is shownin Figure 11b. Due to the extremely low frequency thecurrents substantially fulfill the quasi-stationary relationIL(t) = IR(t) associated to a fully adiabatic regime.In Figure12, we present the populations of the vibronic


10/14


Fig. 10. (Color online) (a)(b) Population of the 0- and 1-particle electronic states as a function of the bias voltage, (c)population of the 0-particle electronic state as a function oftime, and (d) normalized bias voltage as a function of time.The frequency of the driving field is = 2 1060. Theother parameters are the same as used in Figure2.

Fig. 11. (Color online) (a) Plot of the time dependent currentas a function of normalized voltage, (b) current averaged overone driving period as a function of normalized voltage. Thevalue of gate voltage isVg = 0 and the frequency of the drivingfield is = 2 1060. The other parameters are the same asused in Figure2.

states and hysteretic loops are visible. Vibronic states withquantum numbers up to all display nonvanishing popu-lations, much less than in the case Tex switch.

7 The DC-case (

0)In this section, we consider the limit ( 0) of DC-bias as a special case of the master equation presented inthe previous section and compare the results. Even if thesystem still exhibits the bistable properties discussed inSection4(they are in fact not related to the sweeping timeof the bias) the hysteretic behavior cannot be observedanymore. In Figure13, we present the population of theelectronic states for gate voltageVg= 0.

Analogously, in Figure14,the population of electronicstates as a function of normalized bias is depicted for gatevoltage eVg/0 = 8. Due to a finite , the transition0

1 occurs at positive bias voltages. Moreover, the pop-

ulations of the vibronic states are sketched in Figure 15for gate voltageVg= 0, which clearly shows that, for the

Fig. 12. (Color online) Plots of populations of vibronic(a) ground state, (b) first excited state, (c) second excitedstate, and (d) fifth excited state. The frequency of the driv-ing field is = 21060. The other parameters are the sameas used in Figure2.

Fig. 13. (Color online) Population of (a) the 0-particle elec-tronic state, (b) the 1-particle electronic state. The value ofgate voltage is Vg = 0, and the frequency of the driving fieldis 1/max. The other parameters are the same as used inFigure2.

Fig. 14. (Color online) (a) Population of the 0-particle elec-tronic state, (b) population of the 1-particle electronic state.The value of the gate voltage is eVg/0 = 8, and the frequencyof the driving field is 1/max. The other parameters arethe same as used in Figure2.

considered parameters, only the vibronic ground state andfirst excited state are populated, whereas the populationsof higher excited states are very small. This is in contrastto the non-stationary case where the excited states arehighly populated (see Fig.5). In a similar way, the popu-lations of the vibronic states for gate voltageeVg/0= 8are presented in Figure16where higher excited states alsoget populated. Finally, in Figures17and18we show the


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Fig. 15. (Color online) Populations as a function of normal-ized bias voltage for (a) vibronic ground state, (b) first excitedstate, (c) second excited state, and (d) fifth excited state. Thefrequency of the driving field is 1/max. The other param-eters are the same as used in Figure2.

Fig. 16. (Color online) Population as a function of normal-ized bias voltage for (a) vibronic ground state, (b) first excitedstate, (c) second excited state, and (d) fifth excited state. Thevalue of the gate voltage is eVg/0 = 8, and the frequency ofthe driving field is 1/max. The other parameters are thesame as used in Figure2.

populations of the 0- and 1-particle vibronic states for gatevoltagesVg= 0 and eVg/0= 8, respectively, which ba-sically provide the same information as mentioned before.

7.1 I-V characteristics for the DC-case

In the DC-case the analytical expression for the currentremains the same as given by equation (24) taking intoaccount a time independent bias. Let us first discuss thesituation when the 0- and 1-particle states are in reso-nance, = 0

p= 0 andVg= 0. In this particular case,

an interesting behavior of theI-Vcharacteristics with twoopposite current peaks around zero bias can be observed

Fig. 17. (Color online) Populations Pm0 as a function of nor-malized bias voltage for the 0-particle (a) vibronic groundstate, (b) first excited state, (c) second excited state, (d) thirdexcited state, (e) fourth excited state, (f) fifth excited state,(g) sixth excited state, (h) seventh excited state, and (i) eighthexcited state. The frequency of the driving field is 1/max.

The other parameters are the same as used in Figure 2.

Fig. 18. (Color online) Population as a function of normal-ized bias voltage for the 0-particle (a) vibronic ground state,(b) first excited state, (c) fifth excited state and (d) 1-particleground state, (e) first excited state, (f) fifth excited state. Thevalue of the gate voltage is eVg/0 = 8, and the frequency ofthe driving field is 1/max. The other parameters are thesame as used in Figure2.

(see Fig. 19). In order to understand the mechanism ofthis process, we consider the source current which can beexpressed in the form

Is = smm

Fmm

f+

+0 (m m) eVbPm0 f +0(m m) eVbPm1 . (42)

At Vb = 0 only ground to ground state transitions areopen and P00 = P

01 =

12

. Hence, from equation (42) onededuces that in this region the current is zero. At largepositive bias, i.e., Vb , the current is zero becausethe system is in a 1-particle stable state and no newtransition channel is available. For finite bias, the behav-ior of the Franck-Condon factor Fmm is of importance.In particular, it suffices to investigate the classically al-lowed transitions as determined by the Franck-Condon

parabola [33,55]. The minimum of the parabola is form = m

2

2, i.e., m, m 0.

(B.3)

Using equation (B.3), we can write equation (B.2) as

I =D

1 + tanh

(t) E2

. (B.4)

After some calculations, we obtain

I = 2DfE (t)

, (B.5)

where(t) =0+ (t).

Appendix C: Evaluation of transition matrixelements of the electron operator

To determine the transition rates, we need to calculate thematrix elements

r|d|s =e 12 ||2F(,m,m) , (C.1)

where |r and|srepresent the eigenstates given by equa-tion (9). The functionF(,m,m

) determines the couplingbetween states with a different vibronic number of excita-tions with effective coupling and is expressed as[44,56]

F(,m,m) =(m m)mm + (m m)mm

mmin!

mmax!

mmini=0

||2ii!(i + mmax mmin)!

mmax!

(mmin i)! , (C.2)

where mmin/max = min/max(m, m). The coefficient

Fmm in equations (27) and (28) is defined as Fmm =

e2

F2(,m,m).

Appendix D: Expression for the FC factorF2m

Using AppendixC the expression for F2m follows to be

F2m =e2

1

26 3

24 + 22

+2(m

2)

2!m!m2 m(1 + 22) + 4

2. (D.1)


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