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arXiv:0802.2808v1 [cond-mat.mes-hall] 20 Feb 2008

Charge-memory polaron effect in molecular junctions

DmitryA. Ryndyk1, Pino D’Amico1, Gianaurelio Cuniberti2, and Klaus Richter1

1Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany

2Institute for Material Science and Max Bergmann Center of Biomaterials,

Dresden University of Technology, D-01062 Dresden, Germany

(Dated: February 20, 2008)

The charge-memory effect, bistability and switching between charged and neutral states of a

molecular junction, as observed in recent STM experiments, is considered within a minimal polaron

model. We show that in the case of strong electron-vibron interaction the rate of spontaneous quan-

tum switching between charged and neutral states is exponentially suppressed at zero bias voltage

but can be tuned through a wide range of finite switching timescales upon changing the bias. We

further find that, while junctions with symmetric voltage drop give rise to random switching at finite

bias, asymmetric junctions exhibit hysteretic behavior enabling controlled switching. Lifetimes and

charge-voltage curves are calculated by the master equation method for weak coupling to the leads

and at stronger coupling by the equation-of-motion method for nonequilibrium Green functions.

Memory effects and switching at the molecular scale

are in the focus of present experimental and theoretical

studies within molecular electronics [1, 2, 3, 4, 5, 6, 7].

Beside stochastic switching in single-molecule junctions

[4], recent STM experiments [2, 3] show multistability

of neutral and charged states of single metallic atoms

coupled to a metallic substrate through a thin insulating

ionic film. The switching was performed by the applica-

tion of a finite voltage to the STM tip and was explained

by the large ionic polarizability of the film [2].

The coupling of a charge to the displacement of ions

in the film can be treated as an electron-vibron inter-

action. If the energy of the unoccupied electron level

without electron-vibron interaction is ǫ0, the occupied

(charged) state of the interacting system will have the en-

ergy ǫ1= ǫ0− ǫp, where ǫpis so-called polaron shift (or

recombination energy). Neutral and charged (polaron)

states correspond to local minimums of the potential en-

ergy surface and are metastable, if the electron-vibron

interaction is strong enough. Applying an external volt-

age, one can change the state of this bistable system, an

effect that is accompanied by hysteretic charge-voltage

and current-voltage curves. In this approximation it is

not necessary to include Coulomb interaction explicitly,

though one can additionally incorporate charging effects.

It was suggested [8, 9] that bistability between charged

and neutral states can be accounted for in a single-level

model, when one electron level is coupled to one vibra-

tion (Fig.1). The same problem was also considered in

Refs. [10, 11], however with the conclusion that quan-

tum switching between bistable states results in telegraph

noise at finite voltage rather than in a memory effect. In

this Letter we show that there is no contradiction among

these two pictures, taking into account the time-scale of

the switching process. Indeed, the switching time τ be-

tween the two states of interest should be compared with

the characteristic time of the external voltage sweeping,

τs ∼ V (t)/(dV (t)/dt). For τ ≫ τs, quantum switching

can be neglected and hysteresis can be observed, while

in the opposite limit, τ ≪ τs, the averaging removes the

hysteresis. We calculate the charge-voltage curves and

describe the full crossover between two regimes.

The Hamiltonian of the single-level polaron model is

ˆH = (ǫ0+ eϕ0)d†d + ω0a†a + λ?a†+ a?d†d

+

?

ik

?

(ǫik+ eϕi)c†

ikcik+

?

Vikc†

ikd + h.c.

??

, (1)

where the first three terms describe the free electron

state, the free vibron of frequency ω0 (? = 1) and the

electron-vibron interaction. The further terms are the

Hamiltonian of the leads and the tunneling coupling

(i = L,R is the lead index, k labels electron states).

The electrical potential ϕ0 plays an important role in

transport at finite bias voltages V = ϕL− ϕR between

05 10

m

1520

0.0

0.1

0.2

0.3

0.4

|M0m|

2

right

(substrate)

ε

eV

( )

ρ ε

left

(tip)

0

n =

1

n =

1x

0x

1ε

(a)

(b)

0ε

Figure 1: (Color online) (a) The energy diagram of the single-

level electron-vibron model, coupled to left and right lead (or

tip and substrate in the case of STM). (b) Franck-Condon ma-

trix elements M0m for weak (g = 0.1, squares), intermediate

(g = 1, triangles), and strong (g = 10, circles) interaction.

Page 2

2

the left and right electrical potentials. ϕ0describes the

shift of the molecular level by the bias voltage and can

be written as ϕ0= ϕR+ η(ϕL− ϕR), η ∈ [0,1] [12].

The coupling to the leads is characterized by the level-

width function Γi(ǫ) = 2π?

coupling Vikis assumed to be energy-independent (wide-

band limit). The full level broadening is given by the

sum Γ = ΓL+ ΓR.

Consider first the case of very weak coupling to the

leads, Γ ≪ ω0,ǫp. Using the polaron (Lang-Firsov) [13,

14, 15] canonical transformation, it is easy to show that

the eigenstates of the isolated system (Γ = 0) are

k|Vik|2δ(ǫ − ǫik), where the

|ψnm? = e−λ

ω0(a†−a)d†d(d†)n(a†)m

√m!|0? (2)

with the energies

Enm= ǫ1n + ω0m, ǫ1= ǫ0−λ2

ω0, ǫp=λ2

ω0. (3)

When the system is weakly coupled to the leads, the po-

laron representation, Eqs. (2,3), is a convenient starting

point. n denotes the number of electrons, while the quan-

tum number m characterizes vibronic eigenstates, which

are superpositions of states with different number of bare

vibrons. The qualitative picture of the sequential tunnel-

ing through a polaronic state is given in Fig.1(a). Here

the potential energies of the neutral and charged states

are sketched as a function of the vibronic coordinate x.

When the external voltage is applied, the energy levels

are shifted depending on the asymmetry parameter η. It

should be noted that this type of the energy diagram is

quite general for charge-controlled bistable systems.

In the sequential tunneling regime the master equation

for the probability pnm(t) to find the system in one of the

polaron eigenstates (2) can be written as [16, 17, 18]

dpnm

dt

=

?

n′m′

Γnn′

mm′pn′m′ −

?

n′m′

Γn′n

m′mpnm+ IV[p]. (4)

Here the first term describes the tunneling transition into

the state |n,m? and the second term the transition out

of the state |n,m?. IV[p] is the vibron scattering inte-

gral describing the relaxation of vibrons to equilibrium.

The transition rates Γnn′

mm′ are found from the tunneling

Hamiltonian (the last term in Eq. (1)). Taking into ac-

count all possible single-electron tunneling processes, we

obtain the incoming and outgoing tunneling rates at zero

bias voltage as

Γ10

mm′ =

?

i=L,R

Γi(E1m− E0m′)|Mmm′|2f0

i(E1m− E0m′),

(5)

Γ01

mm′ =

?

i=L,R

Γi(E1m′ − E0m)|Mmm′|2

×?1 − f0

i(E1m′ − E0m)?. (6)

0123456

λ/ω0

0.0

0.2

0.4

0.6

(τΓ)−1

τ00

−1

τ10

−1

Figure 2: Inverse life-time (τΓ)−1of the neutral state (thin

solid line) and the charged state (thick gray solid line) as

a function of λ/ω0 at ǫ0 = λ2/2ω0; and the same at ǫ0 =

0.9λ2/ω0 (dashed lines), kT = 0.1ω0.

Here f0(ǫ) is the equilibrium Fermi function, and

?

Franck-Condonmatrix element. It is symmetric in m−m′

and can be calculated analytically. For m < m′it reads

(−g)l√m!m′!e−g/2g(m′−m)/2

l!(m − l)!(l + m′− m)!

Mmm′

=0

????

am

√m!exp

?

λ

ω0

?a†− a??

(a†)m′

√m′!

????0

?

is the

Mm<m′ =

m

?

l=0

,(7)

where g = (λ/ω0)2is the Huang-Rhys factor [20].

One characteristic feature of these matrix elements in

transport is so-called Franck-Condon blockade [18, 19]:

in the case of strong electron-vibron interaction the tun-

neling with small changes in m is suppressed exponen-

tially, as illustrated in Fig. 1(b) for the matrix element

M0m= e−g/2gm/2

energy states is possible. This is also suppressed at low

bias voltage and low temperature.

Finally, the average charge is ?n?(t) =?

the average current (from the left or right lead) reads

Ji=L,R(t) = e?

To proceed further, we calculate the characteristic life

times of the neutral and charged ground states. The life

time τnm of the state |n,m? is given by the sum of the

rates of all possible processes which change this state,

τ−1

m′m. As an example, calculating the life

time of the neutral state |0,0?, with an energy higher

than the charged ground state |1,0?, we find

√m!. Hence only tunneling through high-

mp1m, and

mm′

?Γ10

imm′p0m′ − Γ01

imm′p1m′?.

nm=?

n′m′Γn′n

τ−1

00=

?

m

?

i=L,R

Γi(E1m− E00)|Mm0|2f0

i(E1m− E00).

(8)

For energy-independent Γi(the wide-band limit) we ob-

tain the simple analytical expression

τ−1

00= Γ

?

m

e−ggm

m!f0

?

ǫ0−λ2

ω0

+ ω0m

?

. (9)

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3

-50 -25025 50

eV/ω0

0.0

0.2

0.4

0.6

(τΓ)−1

τ10

−1

τ00

−1

n

1

0

t

1

11

0

0

τ10

τ00

τ10

Figure 3: Inverse life-time (τΓ)−1as a function of normal-

ized voltage eV/ω0 for the asymmetric junction (η = 0) at

λ/ω0 = 5 and ǫ0 = λ2/2ω0 for the neutral state (thin solid

line), the charged state (thick gray solid line) and the same

for the symmetric junction (η = 0.5, dashed lines). Inset:

random switching between bistable states (dashed line) and

single switching into the stable state (full line) after a sudden

change of the voltage.

The corresponding expression for the life time of the

charged state is (assuming that the equilibrium electro-

chemical potential in the leads is zero)

τ−1

10= Γ

?

m

e−ggm

m!f0

?

−ǫ0+λ2

ω0

+ ω0m

?

. (10)

The dependence of the tunneling rates (9,10) on the

scaled electron-vibron interaction constant λ/ω0is shown

in Fig. 2. It is clearly seen that at large values of λ

the tunneling from the neutral state to the charged state

and vice versa is exponentially suppressed in comparison

with the bare tunneling rate Γ. Hence both states are

(meta)stable at low temperatures and zero voltage.

Based on the experimental parameters of Ref. [2], the

charged ground state is assumed to be below the equilib-

rium Fermi energy of the leads, while the neutral ground

state is above it. In the experiments [2] the observed re-

laxation energy ǫp≈ 2.4 eV leads to the parameter λ/ω0

of the order 5 to 10. Thus the system is in the blockade

regime at zero voltage, see Fig. 2.

Next we consider the other important question,

whether fast switching between the two states is possible.

At finite voltage the switching rates are

τ−1

00=

?

m

e−ggm

m!

?ΓLf0(ǫ1+ ω0m − (1 − η)eV )

+ ΓRf0(ǫ1+ ω0m + ηeV )?,

?ΓLf0(−ǫ1+ ω0m + (1 − η)eV )

+ ΓRf0(−ǫ1+ ω0m − ηeV )?.

(11)

τ−1

10=

?

m

e−ggm

m!

(12)

The voltage dependence of the inverse life time (τΓ)−1

is shown in Fig. 3 for a junction with the same tunnel-

ing coupling, ΓL = ΓR, but asymmetric electrical field

-100 -500 50 100

eV/ω0

0.0

0.2

0.4

0.6

0.8

1.0

p0

V

t

0

τS

Figure 4: Population of the neutral state as a function of nor-

malized voltage eV/ω0 in the asymmetrical junction (η = 0)

at λ/ω0 = 5 and ǫ0 = λ2/2ω0for fast voltage sweep (thin solid

line), slower sweep (thick gray solid line), and in the adiabatic

limit (dashed line). Inset: sketch of voltage time-dependence.

(η = 0), as well as for the completely symmetric junc-

tion (η = 0.5). The results in Fig. 3 imply that in both

cases one can tune (τΓ)−1upon sweeping the bias voltage

and thereby control the timescales for switching between

charged and neutral states. For the symmetric junction

both switching rates, τ−1

multaneously nonzero at finite voltage (eV/ω0≥ 40 for

the parameters of Fig. 3) leading to random switching

(noise) sketched as dashed line in the inset. On the con-

trary, for the asymmetric junction controlled switching

into the neutral (black solid line) and charged (grey line)

state can be achieved at large enough negative and posi-

tive voltage, respectively. This qualitatively different be-

haviour is a result of the distinct voltage asymmetry of

the two inverse lifetimes which are never both finite. The

further peculiar feature of the asymmetric case, namely

that the switching rates of the neutral and charged states

interchange their role as a function of bias, i.e., the neu-

tral (charged) state is long-lived at negative (positive)

bias, implies hysteretic behavior and a memory effect.

To this end we consider what happens, if one sweeps

the voltage with different velocity (Fig. 4) for the asym-

metic case η = 0. If the voltage is changed fast enough,

i.e. faster than the life time of charged and neutral states

(τ ≫ τs as discussed in the introduction), then both

states can be obtained at zero voltage (hysteresis). In

the opposite (adiabatic) limit the change is so slow that

the system relaxes into the equilibrium state, and the

population-voltage curve is single-valued. Note that this

controlled switching is possible only for asymmetric junc-

tions for the reason given above.

We finally compare the results with those of a fur-

ther important limiting case, namely that the level width

is finite (and possible finite dissipation of vibrons is

taken into account).Then the master equation ap-

proach can no longer be used, and we apply alternatively

the nonequilibrium Green function technique. Follow-

00and τ−1

10, (dashed lines) are si-

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4

-500 50

eV/ω0

0.0

0.2

0.4

0.6

0.8

1.0

〈n〉

Figure 5: Average number of electrons at ΓL = ΓR = 5ω0

as a function of normalized voltage eV/ω0 for the asymmet-

ric junction, η = 0 (thin solid line), and for the symmetric

junction, η = 0.5 (dashed line), for λ/ω0 = 5 and ǫ0 = λ2/ω0.

ing Refs. [21, 22], the average number of electrons is

determined by the lesser Green function G<(t1− t2) =

i?d†(t2)d(t1)?

of the Green function is a nontrivial task even in the

single-level model. It is simplified in the important limit

of low vibron frequencies, ω0 ≪ Γ < ǫp, where the

Born-Oppenheimer approximation holds true. We used

the equation-of-motion approach in this case. In Fig. 5

the charge-voltage dependence, obtained in the simplest

mean-field approximation [8, 9], is shown.

function is represented as G<(ǫ) = iA(ǫ)f(ǫ), with the

spectral and the distribution function

as ?n? = −i?G<(ǫ)dǫ

2π. The calculation

The lesser

A(ǫ) =

2Γ

(ǫ − ǫ0−2λ2

ΓLf0

ω0?n? − eϕ0)2+ Γ2,

L(ǫ − eϕL) + ΓRf0

ΓL+ ΓR

(13)

f(ǫ) =

R(ǫ − eϕR)

. (14)

The result is qualitatively the same as in the sequential

tunneling case: For electrically asymmetric junctions two

stable states exist at zero bias (memory effect), which

can be switched by the voltage. The current shows simi-

lar hysteretic behaviour as a function of voltage. For the

symmetric junction hysteresis is observed only at finite

voltage (nonequilibrium bistability [9]).

metric junctions are again preferable for a memory effect.

Finally we note that in the case ω0≪ Γ we considered

the stationary problem only, assuming that the switching

rate between the two metastable states is small (com-

pared e.g. to Γ) at large λ/ω0. The calculation of the

life times of metastable states within the Green function

appraoch and of dynamical effects arising from the com-

petition between voltage sweeping and switching times,

such as in Fig. 4, remains as a problem for the future.

To conclude, we considered a charge-memory effect and

switching phenomena in single-molecule junctions taking

into account dynamical effects such as the interplay be-

tween timescales of voltage sweeping and switching rates.

Hence, asym-

We showed that bistability arises if quantum transitions

between neutral and charged states involved are sup-

pressed, e.g. due to Franck-Condon blockade. Different

regimes, characterized by random mutual transitions and

by single switching events into a stable configuration are

identified. In the latter case controlled switching of the

molecule is achieved by applying finite voltage pulses.

We acknowledge fruitful discussions with J. Repp.

This work was funded by the Deutsche Forschungsge-

meinschaft within the Priority Program SPP 1243 and

Collaborative Research Center SFB 689 (D.A.R.).

[1] A. S. Alexandrov and A. M. Bratkovsky, Phys. Rev. B

67, 235312 (2003).

[2] J. Repp, G. Meyer, F. E. Olsson, and M. Persson, Science

305, 493 (2004).

[3] F. E. Olsson, S. Paavilainen, M. Persson, J. Repp, and

G. Meyer, Phys. Rev. Lett. 98, 176803 (2007).

[4] E. L¨ ortscher, H. B. Weber, and H. Riel, Phys. Rev. Lett.

98, 176807 (2007).

[5] P. Liljeroth, J. Repp, and G. Meyer, Science 317, 1203

(2007).

[6] M. del Valle, R. Guti´ errez, C. Tejedor, and G. Cuniberti,

Nature Nanotechnology 2, 176 (2007).

[7] G. Cuniberti, G. Fagas, and K. Richter, Introducing

Molecular Electronics (Springer-Verlag, 2005).

[8] A. C. Hewson and D. M. Newns, J. Phys. C: Solid State

Phys. 12, 1665 (1979).

[9] M. Galperin, M. A. Ratner, and A. Nitzan, Nano Lett.

5, 125 (2005).

[10] A. Mitra, I. Aleiner, and A. J. Millis, Phys. Rev. Lett.

94, 076404 (2005).

[11] D. Mozyrsky, M. B. Hastings, and I. Martin, Phys. Rev.

B 73, 035104 (2006).

[12] S. Datta, W. Tian, S. Hong, R. Reifenberger, J. I. Hen-

derson, and C. P. Kubiak, Phys. Rev. Lett. 79, 2530

(1997); T. Rakshit, G.-C. Liang, A. W. Gosh, M. C. Her-

sam, and S. Datta, Phys. Rev. B 72, 125305 (2005).

[13] I. G. Lang and Y. A. Firsov, Sov. Phys. JETP 16, 1301

(1963).

[14] A. C. Hewson and D. M. Newns, Japan. J. Appl. Phys.

Suppl. 2, Pt. 2, 121 (1974).

[15] G. Mahan, Many-Particle Physics (Plenum, N. Y., 1990).

[16] S. Braig and K. Flensberg, Phys. Rev. B 68, 205324

(2003).

[17] A. Mitra, I. Aleiner, and A. J. Millis, Phys. Rev. B 69,

245302 (2004).

[18] J. Koch and F. von Oppen, Phys. Rev. Lett. 94, 206804

(2005); J. Koch, M. Semmelhack, F. von Oppen, and

A. Nitzan, Phys. Rev. B 73, 155306 (2006).

[19] K. C. Nowack and M. R. Wegewijs, cond-mat/0506552.

[20] K. Huang and A. Rhys, Proc. R. Soc. London Ser. A 204,

406 (1950).

[21] Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512

(1992); A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys.

Rev. B 50, 5528 (1994).

[22] H. Haug and A.-P. Jauho, Quantum Kinetics and Optics

of Semiconductors, vol. 123 of Springer Series in Solid-

State Sciences (Springer, 1996).