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
, 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 .
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
ikd + h.c.
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
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.
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] .
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
with the energies
Enm= ǫ1n + ω0m, ǫ1= ǫ0−λ2
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]
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
Γi(E1m′ − E0m)|Mmm′|2
×?1 − f0
i(E1m′ − E0m)?. (6)
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
l!(m − l)!(l + m′− m)!
where g = (λ/ω0)2is the Huang-Rhys factor .
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
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,
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-
imm′p0m′ − Γ01
For energy-independent Γi(the wide-band limit) we ob-
tain the simple analytical expression
-50-250 25 50
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)
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. , 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  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
?ΓLf0(ǫ1+ ω0m − (1 − η)eV )
+ ΓRf0(ǫ1+ ω0m + ηeV )?,
?ΓLf0(−ǫ1+ ω0m + (1 − η)eV )
+ ΓRf0(−ǫ1+ ω0m − ηeV )?.
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 -50050 100
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-
10, (dashed lines) are si-
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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) =
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
(ǫ − ǫ0−2λ2
ω0?n? − eϕ0)2+ Γ2,
L(ǫ − eϕL) + ΓRf0
R(ǫ − eϕR)
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 ).
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.
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
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