Mechanical Stabilization of Nanoscale Conductors by Plasmon Oscillations.

External driving of the Fermion reservoirs interacting with a nanoscale charge-conductor is shown to enhance its mechanical stability during resonant tunneling. This counterintuitive cooling effect is predicted despite the net energy flow into the device. Field-induced plasmon oscillations stir the energy distribution of charge carriers near the reservoir's chemical potentials into a nonequilibrium state with favored transport of low-energy electrons. Consequently, excess heating of mechanical degrees of freedom in the conductor is suppressed. We demonstrate and analyze this effect for a generic model of mechanical instability in nanoelectronic devices, covering a broad range of parameters. Plasmon-induced stabilization is suggested as a feasible strategy to confront a major problem of current-induced heating and breakdown of nanoscale systems operating far from equilibrium.

The rapid buildup of high energy density in nanoscale devices operating far from equilibrium holds promise for the discovery of new phenomena and for new applications. It is a major source of concern, however, when overheating hinders the mechanical stability and thus the deterministic input–output relation of such devices. For example, in single molecule junctions the Fermion current is coupled to mechanical degrees of freedom (molecular vibrations). In the off-resonant electronic tunneling regime, vibrational excitations are minor. Their effect is apparent in the current–voltage curves but the mechanical stability of the device is hardly affected. In contrast, in the resonant tunneling regime the molecule undergoes successive charging and discharging events which are often associated with vibrational heating, conformational changes, or chemical bond rupture, leading to a mechanical breakdown.

Mechanical instability of this sort was extensively studied and analyzed theoretically[1−9] and experimentally.[10−14] When the charged molecule is associated with a repulsive potential energy surface for the nuclei,[15−17] charge transport-induced dissociation is inevitable. In other cases where both the uncharged and charged potential energy surfaces are bonding, the successive charging and discharging events are often associated with inelastic scattering and, consequently, vibrational heating. This phenomenon is especially acute in the limits of high bias and weak vibronic coupling, where vibrational cooling (e.g., via electron–hole pair creation) becomes ineffective. Remedies for vibrational heating were proposed, for example, by manipulating the electrode density of states in the off resonant transport regime[18,19] as well as in the resonant transport regime,[7] or alternatively by increasing the ambient temperature.[8] These strategies, however, are challenging from experimental perspective, as they require careful control of the electrode’s material and/or the ambient temperature.

In this work, we propose a new strategy to control the energy content of the nanodevice by time-dependent driving of the external Fermion reservoirs, namely, inducing plasmon oscillations. In particular, we show that driving of a Fermion reservoir in a molecular junction helps to keep a low level of vibrational excitation on the molecule in the resonant transport regime. This implies that a mechanically unstable junction can be made stable under the field operation. Moreover, we show that while the mechanical stability increases significantly, the current through the device is compromised only mildly.

Our results are based on a numerical investigation and theoretical analysis. Focusing on weak molecule-lead coupling (where vibrational instability is acute[4,8]), the theoretical treatment is based on time-averaged Born-Markov master equations (see SI). This enables one to take full account for the electron interaction with molecular vibrations and with the driving field, as long as the driving period is short on the time scale of the field-free junction dynamics (for a complementary discussion of the adiabatic regime, where the junction undergoes significant changes within a single driving period, see ref (20)).

As a prototype, we consider a model system which demonstrates vibrational instability in the resonant transport regime.[1−5,7−10] A single molecular orbital coupled to a vibrational mode is represented by the Holstein Hamiltonian,[21]H = ε1a1†a1 + ℏΩc†c + λa1†a1 (c† + c) ≡ ĥ1a1†a1 + ĥ0a1a1†, where a1† creates an electron in the molecular orbital. The vibrational Hamiltonians, ĥ0 = ℏΩc†c and ĥ1 = ε1 + ℏΩc†c + λ(c† + c), correspond to the empty and charged states of the orbital, respectively, where c† creates a vibration quantum on the molecule at a frequency Ω, ε1 represents the charging energy of the molecular orbital, and λ is the vibronic coupling parameter. The corresponding eigenstates of H are products, |n⟩|ν⟩, where |n⟩ ∈ |0⟩, |1⟩, is the electronic occupation state, and the vibrational states are defined by . The molecular system is coupled to two reservoirs (“right” and “left” leads) of noninteracting Fermions, where the reservoir Hamiltonians read , and the molecule-lead couplings read, . These couplings are captured in the spectral density, , which is constant in the wide band limit, J(ε) ≈ Γ.

In order to account for the plasmon, we consider here an idealized model of adiabatic reservoir excitation,[22−30] ignoring plasmon broadening effects, for example, via direct electron–hole pair creation. This model was originally proposed in studies of photoassisted tunneling in superconducting junctions,[22−24] and was later adopted in studies of plasmon-driven atomic point contacts[25−27] and molecular junctions.[28−33] According to the model, the field induces time-dependent changes in the chemical potentials, that is, , where, g(t) = g(t + τ). In the common case of harmonic (AC) driving, g(t) = α cos(ωt), where ω ≡ 2π/τ is the field frequency. Notice that a uniform driving of the molecule itself by a time-dependent gate potential is formally equivalent to a time-dependence of the reservoir chemical potentials.[33] Therefore, such driving (when feasible, experimentally) should also suppress the vibrational instability during resonant transport. Nevertheless, in the present work we focus on stabilization induced by the lead plasmons. In particular, the source lead is driven, where the molecular charging energy is kept fixed (assuming the potential drop is restricted to the contacts). A unitary transformation shifts the time-dependence to the molecule-lead coupling operators,[33,34], where the full (transformed) Hamiltonian reads .

Our main observable of interest is the level of vibrational excitation on the molecule at steady state. In terms of the system eigenstate populations, ρ(, this reads

When dynamical changes in the populations of the molecular eigenstates are much slower than the periodic driving (see ref (34) and SI), the populations, {ρ(}, can be obtained by a quantum master equationwhere, , and is the transition rate from the (m,ν) to the (n,ν) molecular eigenstate, due to electron exchange with the Kth lead, averaged over the driving period. The respective steady state current (from the Kth lead) reads[34]

Notice that this treatment is restricted to a typical scenario in tunneling junctions, where the electronic coupling between the (molecular) system and the electrodes is weak, namely, small in comparison to the systems’ level spacing. Consequently, the system dynamics can be well approximated by a Markovian quantum master equation[35,36] in the wide band limit,[34] where the appealing result of effective time-independent rates is trivially justified in the absence of the field. However, it remains valid also with increasing intensity of the driving field, as long as two conditions are met (see SI for a detailed discussion): (i) The transformed time-dependent molecule-lead coupling functions ({H(t)}) remain band-limited in the frequency domain, complying with the wide band approximation. (ii) The driving field frequency must be larger than the rate of change in the eigenstate populations (which implies, , where ν0 is the highest occupied vibration quantum number). Notice that this limitation on the driving-field frequency does not relate to the frequency of coherent vibrational dynamics in the system (Ω), which can be ignored as long as Γ ≪ ℏΩ.

For a periodic driving field, is expressed as a sum over transition rates to virtual electrodes, which are different replicas of that lead (see Scheme , and SI). In each replica (a Floquet channel[37,38]) the Fermi distribution and the spectral density are displaced with respect to the field-free functions by an integer (l) multiplication of ℏω. The molecule-lead coupling is distributed among the replicas where the coupling matrix element is proportional to the lth Fourier component of the (transformed) periodic field. Formally, one obtainswhereis a generalized Fermi function,[23] corresponding to the nonequilibrium state of the driven lead. The displaced functions, f((E) = f((E – ℏωl) are expressed in terms of the nondriven lead Fermi function, , where μ is the lead chemical potential and T is the temperature. g̅( is the lth Fourier component of the (transformed) periodic field, , and ⟨ν|ν⟩ is the vibrational Franck–Condon (FC) overlap.

Scheme 1

Illustration of a Biased Molecular Junction Under Periodic Driving

Biased molecular junction before (left) and after (right) applying a driving field (α cos(ωt)) on the left electrode. The letters L, M, and R stand for the left electrode, the molecule, and the right electrode, respectively. The molecular energy level is represented by the solid line. μL (μR) denotes the chemical potential of the left (right) electrode in the absence of the driving field. Under the field, the left electrode is replaced by replicas associated with displaced chemical potentials, μL + ℏωl, l = 0, ±1, ±2 and so forth.

In Figure , the steady state observables are plotted for the above model in the zero-temperature limit, where vibrational instability is acute[8] and the field effect should be most pronounced. (Notice that although level broadening effects are missing in eq , the corresponding errors in the presented steady state observables are negligible even for KBT < Γ, since Γ is smaller than the energy gaps between μ and any molecular charging energy). As one can see, the onset of a plasmon driving field at the left electrode, gL(t) = αcos(ωt), leads to a significant reduction of the vibrational excitation energy. As the field intensity increases farther, the level of vibration excitation tends to increase (see Figure a) but remains lower than its field-free value throughout the sampled region.

Figure 1

Vibrational energy ⟨c†c⟩ (a) and the steady state current (b) as functions of the driving field intensity for different field’s frequencies. ℏΩ = 0.1 eV is the vibrational frequency, ε1 = 3ℏΩ is the elastic charging energy, ΓL = ΓR = ℏΩ/1000 is the coupling to the left and right leads, μL = ε1 + ℏΩ + 0.01 eV and μR = −μL are the leads’ chemical potentials, and λ = ℏΩ/10 is the vibronic coupling parameter. The lead temperature was set to 0 K.

The field-induced stabilization can be qualitatively understood by analysis of eq in the weak molecule-lead and vibronic coupling limits (see SI). Indeed, the vibrational excitation is shown to be associated with a steady state inverse temperaturewhere are rates of electron (hole) transfer into the molecule. Each transfer event is associated with a gain (loss) of the electronic charging energy, ε̅ = ε1 – λ2/(ℏΩ)2, plus p vibration quanta, where, f̅( (ε) = 1–f̅( (ε), and f̅( (ε) is the generalizd Fermi function (eq ).

The numerator and denominator refer, respectively, to cooling and heating pathways. Scheme a corresponds to one of the heating pathways included in W1W0, where an electron enters the molecule from the left electrode, and exits to the right electrode while emitting a vibration quantum. In the absence of the field, the molecular charging energy is within the Fermi conductance window set by μL. Under illumination, however, the original electrode is replaced by replicas with shifted chemical potentials, μL + ℏωl (l = 0, ±1, ±2, and so forth), such that the charging energy is outside the conductance window for all replicas with l < 0. Since the molecule-lead coupling is divided among the replicas (including those associated with l < 0), the electron hopping rate into the molecule decreases, and consequently the denominator in eq becomes smaller (heating is suppressed). Scheme b refers to the complementary cooling pathway (included in W1W0 in the numerator). In this process, the molecular charging step is associated with a lower energy, and therefore it exits the conductance window only for replicas with l < −1. Consequently, the reduction of the numerator in eq is smaller than the reduction of the denominator (cooling is less suppressed than heating). Scheme c corresponds to a cooling pathway via electron–hole pair creation at the driven electrode (also included in W1W0). Here the inelastic cooling process is Pauli blocked in the absence of the field. Under illumination, however, W1 becomes finite for replicas with l ≤ −1 where W0 is allowed for l ≥ −1 (see Scheme c), leading to an increase of the numerator (enhanced cooling) in eq . Notice that the complementary process of heating by electron–hole pair elimination (included in W1W0) is also enabled only under illumination, leading to increasing denominator (enhanced heating) but to a lesser extent, since W1 and W0 obtain finite values only for l ≥ 0 and l ≤ −2, respectively, where l = −1 is excluded (see Scheme d). Similar comparisons for the entire set of heating and cooling processes show that the net effect of the field is to increase the numerator/denominator ratio, resulting in lowering of the effective vibrational temperature.

Scheme 2

Field-induced Heating and Cooling Pathways

Representation of different molecular heating (a,d) and cooling (b,c) processes in a field-driven junction, induced by sequential electron transfer events. Blue (red) corresponds to a vibrational quantum (ℏΩ) transfer from (to) the molecule during the inelastic process. Under the field, the left electrode is replaced by replicas associated with displaced chemical potentials, μL + ℏωl, l = 0, ±1, ±2 and so forth. The energy difference between the lead chemical potential and the elastic charging energy is marked as Δ + ℏΩ.

We now return to a detailed analysis of the numerical results in Figure . While the simulations account for multiphonon transitions (FC factors in eq are calculated to all orders in the vibronic coupling), given the relatively small vibronic coupling parameter, and since the left chemical potential is just above ε̅ + ℏΩ, the results in Figure are dominated by the single-quantum vibrational heating (cooling) pathways discussed above. As the driving field intensity increases, the moleule-lead coupling is partially shifted from lead replicas which facilitate (block) heating (cooling) pathways (μL + ℏωl > ε̅ + ℏΩ, namely, , see Scheme ) toward replicas for which these pathways become closed (opened). Quantitatively, the global vibrational cooling effect, demonstrated in Figure a, is shown to depend on the specific parameters of the driving field. The relative contributions of the different electrode replicas to the charge transfer rates are proportional to the Fourier components of the (transformed) periodic field, that is, , in the present case, where the distribution over the different channels (different l’s) broadens with increasing field intensity (α) at any field frequency (ω). When the field frequency is larger than Δ, only replicas associated with non-negative l facilitate dominant heating pathways, for example, W1W0, and suppress cooling pathways, for example, W0W1. As the field intensity increases, the distribution over l is broadened such that negative l values are also included, resulting in a net relative decrease of heating versus cooling. Since the dominant replica at low intensities is associated with l = 0, the observed drop in the vibrational excitation level follows the corresponding Bessel function, , toward its first zero (at α ≈ 2.4ℏω). As the field intensity increases farther, the vibrational excitation level increases again, owing to the increase in , as well as to the suppression of vibrational cooling pathways (associated with W1W0, W0W–1) when the distribution over l is shifted toward l < −1 (see Scheme ). For field frequencies smaller than Δ, the range of replicas facilitating (blocking) the dominant heating (cooling) pathways expands to also include negative l values. Therefore, the vibrational excitation level is not suppressed as the field intensity increases until the distribution over l is broadened beyond this range. Given the Bessel weights, this occurs at α ∼ Δ, where for α ≳ Δ the vibrational excitation level drops sharply, reflecting the simultaneous oscillatory decay of several Bessel functions (associated with ). As discussed above, when the field intensity increases farther (and the distribution over l broadens) to the extent that additional cooling pathways are suppressed (in the weak vibronic coupling limit, the stability region extends over ∼2ℏΩ, see Figure a), the global cooling effect is diminished and the vibrational excitation level starts to increase.

Interestingly, the current, as seen in Figure b, is only mildly affected by the increasing field intensity. Indeed, as the field intensity increases, transport channels which are within the resonant Fermi window in the absence of the field, may exit that window for an increasing number of lead replicas, resulting in a current-drop. Notice, however, that the effect is minor, since the relative current-drop is limited to 50% (as shown by Floquet analysis, see ref (33)), whereas the relative drop in the vibrational heating is much larger (see Figure a). Moreover, unlike vibrational heating, the current itself is dominated by elastic transport (see SI, when, ). Therefore, the current drops only for replicas with μL + ℏωl < ε̅, whereas the vibrational stabilization becomes prominent already when, μL + ℏωl < ε̅ + ℏΩ. Indeed, the observed current-dependence on the field parameters (Figure b) can be explained by following the arguments used in the analysis of the vibrational stabilization (Figure a), where Δ is replaced by Δ + ℏΩ. In particular, when ℏω < ℏΩ, an “intensity window” appears (Δ < α < Δ + ℏΩ), in which the mechanical stabilization becomes effective, while the current is nearly unaffected.

The phenomenon of vibrational cooling by the driving field is also robust with respect to the static bias voltage applied on the junction as shown in Figure a. In the absence of the driving field, the excitation level increases in steps as a function of the bias voltage, which reflects increases of the left chemical potential by multiplications of ℏΩ.[39] In the presence of the field, the excitation level is shown to drop down regardless of the static bias applied (except for a minor increase at intermediate field intensities, attributed to the opening (closing) of two-phonon heating (cooling) pathways by high lying lead replicas).

Figure 2

Vibrational energy ⟨c†c⟩ as a function of the driving field intensity for different static bias voltages (a), vibronic coupling strengths (b), and asymmetric molecule-lead couplings (c). The fixed model parameters are ℏΩ = 0.1 eV, ε1 = 3ℏΩ, μR = −μL, ℏω = 0.02 eV, ΓR = ℏΩ/1000, the lead temperature was set to 0 K, and unless marked otherwise: ΓL = ΓR, λ = ℏΩ/10, μL = ε1 + ℏΩ + 0.01 eV.

In Figure b, we demonstrate the effect of the vibronic coupling strength, , on the level of vibrational excitation. For large vibronic coupling, the excitation level is kept small even in the absence of the driving field, owing to effective cooling by (multiphonon) electron–hole pair creation at the left lead. However, as decreases, the multiphonon cooling is suppressed, resulting in high levels of vibrational excitation in the absence of the driving field and a substantial field-induced cooling effect.

A similar effect is apparent also in Figure c, where the level of vibrational excitation is plotted as a function of the field’s intensity for different asymmetric couplings to the left and right electrodes. As the coupling to the drain (right) electrode becomes large in comparison to the coupling to the source (left) electrode (ΓL ≪ ΓR), inelastic processes involving the left electrode become much slower in comparison to processes involving transport to the right electrode. In particular, cooling by electron–hole pair creation is suppressed with growing asymmetry.[40,41] In the absence of the field, increasing the ratio, ΓR/ΓL therefore results in higher vibrational excitation levels (which are shown to saturate when cooling by electron–hole creation already becomes ineffective). In the presence of the field, the excitation level is suppressed regardless of the ratio, ΓR/ΓL, even when electron–hole cooling is ineffective. This demonstrates that plasmon oscillations induce cooling via transport processes involving both leads on top of enhancing cooling processes by electron–hole pair creation at the leads.

We now consider the stabilization due to plasmon oscillations at finite temperatures. In the absence of a driving field, as the temperature rises, the level of excitation diminished, as was studied before.[8] However, even high temperatures may not ensure mechanical stability, especially with increasing bias. A driving field can compensate for this instability, as demonstrated in Figure . Setting the bias to the highest value studied in Figure a, the vibrational excitation is shown to decrease with increasing field’s intensity, for a wide experimentally relevant temperature range. Notice that the zero-temperature result is reproduced also for a range of finite temperatures, as long as the Fermi function width is smaller than the other relevant energy scales.

Figure 3

Vibrational energy ⟨c†c⟩ as a function of the driving field intensity for different temperatures. The model parameters are ℏΩ = 0.1 eV, ε1 = 3ℏΩ, ΓL = ΓR = ℏΩ/1000, Φ = 10.5ℏΩ, λ = ℏΩ/10, ℏω = 0.02 eV.

In conclusion, plasmon oscillations in the leads are shown to suppress vibrational heating of a nanoscale conductor during resonant transport. Our results demonstrate that the excess energy associated with the induced plasmon oscillations does not find its way into the mechanical excitation of the conductor. On the contrary, pumping energy into the leads is shown to suppress energy flow into the conductor. The emerging physical picture is that the field stirs the distribution of charge carriers in the lead into a new nonequilibrium state, in which high-energy electrons are driven out of the effective inelastic transport “window”. In detail, the rate of inelastic resonant transport depends on both the Fermi distributions and Franck–Condon factors (eq ). Vibrational heating (cooling) in this regime is attributed to high (low) energy electrons entering the molecule and leaving it at a lower (higher) energy. The field suppresses heating by exciting high-energy electrons to yet higher-energies, making them irrelevant to the resonant transport processes through damped Franck–Condon factors. In contrast, low energy electrons (far below the chemical potential), which promote vibrational cooling, are less prone to absorb energy from the field, since transitions to (occupied) higher energy states are Pauli blocked. This redistribution of charge carriers in the leads favors resonant transport of low energy electrons over high energy electrons, resulting in a net vibrational cooling effect within the conductor. It was recently proposed that increasing the ambient temperature can lead to mechanical stabilization during resonant transport.[8] Indeed, increasing the lead temperature should promote vibrational cooling in a similar way. Nevertheless, here we show that an efficient cooling can be achieved in a controlled manner while keeping the lead temperature as low as possible. Moreover, plasmon oscillations can be designed in order to tailor specific nonequilibrium distributions to a desired cooling target. These ideas are currently under study with the aim of proposing efficient schemes for mechanical stabilization for nanoscale conductors operating under nonequilibrium conditions.