Temperature independent current deficit due to induced quantum nanowire vibrations

We consider electronic transport through a suspended voltage-biased nanowire. By coupling the tunneling current to a transverse magnetic field, vibrational modes of the wire are excited which influences the current-voltage characteristics of the system in novel ways through the induced electron-vibron interaction. From this analysis, we find that at high voltages the current through the suspended nanowire is offset from its non-vibrating ohmic value by an amount that is independent of both bias voltage and temperature. We also show that the corrections to the current from the back-action of the vibrating wire decay exponentially in the limit of high voltage, a result that holds even if the nanowire vibrational modes have been driven out of thermal equilibrium.

We consider electronic transport through a suspended voltage-biased nanowire. By coupling the tunneling current to a transverse magnetic field, vibrational modes of the wire are excited which influences the current-voltage characteristics of the system in novel ways through the induced electron-vibron interaction. From this analysis, we find that at high voltages the current through the suspended nanowire is offset from its non-vibrating ohmic value by an amount that is independent of both bias voltage and temperature. We also show that the corrections to the current from the back-action of the vibrating wire decay exponentially in the limit of high voltage, a result that holds even if the nanowire vibration modes have been driven out of thermal equilibrium. Nanoelectromechanical systems (NEMS) are mesoscopic devices whose functionality depends on the possibility to induce mechanical vibrations or displacements of one or several of their components 1 . Examples of such setups are numerous and include shuttling of single electrons and Cooper pairs 2,3,4,5 , tuning of mechanical bending vibrations of suspended nanowires 6,7,8 and mechanically mediated superconducting and magnetic proximity effects 9,10,11 to name but a few. Crucial advantages of the downscaling implied by the acronym NEMS are the high (RF) vibration frequencies and the unprecedented sensitivity to external stimuli that can be achieved. This is due in turn to the low masses of these systems and to the strong coupling between mechanical and electrical degrees of freedom at the nanometer length scale, see, e.g., Refs. [12,13,14] and references therein. Also, nanoelectromechanical systems border the world of quantum mechanics, which opens up the possibility to experimentally study quantum effects on the interaction between electrical and mechanical degrees of freedom in mesoscopic systems 15,16,17,18,19 .
In this paper we will consider the nanoelectromechanical system studied by Shekhter et al. 20 , who analyzed the conductance through a suspended voltage biased singlewalled carbon nanotube (SWNT). In Ref. [20] the authors showed that by applying a transverse static magnetic field an effective entanglement between the electronic and quantum nanomechanical degrees of freedom in the system can be achieved. This in turn results in a finite magnetoconductance at low temperatures due to the induced quantum mechanical suppression of the elastic electronic tunneling channel together with Pauli restrictions on the inelastic channels due to the confined geometry of the system. At high temperatures however, no such restrictions on the inelastic transport channels are found, and the conductance exactly coincides with the transmission through the non-vibrating wire.
Here we will consider the same system as in Ref. [20] and in particular we will show that at large enough voltages there also exists a current deficit, as compared to the non-vibrating wire. Furthermore, this current deficit is shown to be independent of both the temperature and The effective multiconnectivity of the system leads to both a temperature-and bias voltage-independent current deficit (see text) as well as to a previously reported negative magnetoconductance 20 . Amplitude shown is greatly exaggerated.
the bias voltage, making this system a good candidate device for detection of quantum vibrations in nanoscale systems. Also, we find that this reduction to the current is in general not dependent on strong coupling of the nanowire vibrational modes to the thermal bath as previously reported in Ref. [21] where the current and conductance through a carbon nanotube containing an encapsulated fullerene was analyzed in the ballistic transport regime. Hence, the results presented are shown to be quite general also for oscillator distributions out of equilibrium.
The system considered is shown Fig. 1, and comprises a doubly clamped carbon nanotube suspended over a trench of width L, subject to a transverse magnetic field, H. In Ref. [20] it was shown that when the SWNT is biased by a voltage V the induced mechanical oscillations of the tube lead to intermediate "swinging states" through which electrons can tunnel between the leads. By restricting the analysis to the fundamental mode (which gives the most important contribution) the authors showed that the system can be described by an effective Hamiltonian (Eq. (7) in Ref. [20]), which describes nonresonant charge transfer through the suspended SWNT. In (1), the first term, describes the electrons in the leads;â † l/r,k [â l/r,k ] are creation [annihilation] operators for electrons in state k in the left/right lead with energy ε l/r,k respectively. The second term in the Hamiltonian, describes the oscillating wire whereb † [b] is a boson operator that creates [annihilates] one vibrational quantum and ω = (k/m) 1/2 is the frequency of the fundamental mode of oscillation with k the rigidity and m the effective mass of the wire (typically ω is of the order of 10 8 s −1 if L ∼1 µm). The third term in (1) describes the interaction between the electrons and the oscillating wire, Here, φ = 4gπx 0 LH/Φ 0 is the dimensionless electronvibron coupling strength, Φ 0 = h/e is the flux quantum, g is a geometric factor of order unity and x 0 is the amplitude of the zero-point fluctuation of the fundamental mode. T ef f (k, k ′ ) is the effective overlap integral between electrons in state k and k ′ in the left and right lead respectively.
To calculate the charge transport through the system for the case when the density matrix is not in thermal equilibrium, we first consider the time rate of change of the total density matrix for the system,σ(t), which is given by the Liouville-von Neumann equation (see Refs. [22,23,24] for a similar analysis). To evaluate this we switch to the interaction picture with respect to the noninteracting Hamiltonian,Ĥ 0 =Ĥ leads +Ĥ osc , for which the evolution of the density matrix is given by, whereˆ A(t) = e iĤ0t/ Â e −iĤot/ is any operatorÂ in the interaction picture. Since we are interested in the energy exchange between the electrons and the oscillating wire we need only to know the evolution of the reduced density matrix, which is found by tracing out the degrees of freedom of the leads,ρ(t) = Tr leads (σ(t)). By treating the electrons in the two leads as fermionic baths whose equilibrium distributions are virtually unaffected by the charge transfer to the wire we approximatê σ(t) ∼ˆ ρ(t)⊗σ leads whereσ leads is the equilibrium density matrix for the leads. Evaluating Eq. (5) to the lowest order in the tunneling probability enables us find find the equation of motion for the reduced density matrix in the Heisenberg picture, From this we derive the stationary equation for the reduced density of the system under the assumption made in Ref. [20] that the overlap integral T ef f (k, k ′ ) is independent of the momenta k and k ′ , In (7), χ = √ 2φ/x 0 ,x is the deflection operator of the oscillating wire and the operatorsĴ i take on the form below, Here, ν is the density of states in the leads and f l,r (ǫ l,r ) are the Fermi distribution for electrons in the left/right lead kept at chemical potential µ l,r = ±eV /2 respectively. Multiplying Eq. (7) by the position and momentum operator and tracing out the oscillator degree of freedom we find the following expression for the deflection, x , and momentum, p , expectation values, Eq. (9a) gives the force balance between the elastic force on the wire (left hand side) and the force induced by the charge transfer (right hand side). Since χ ∝ HLg we interpret the latter as the Lorentz force, F = HILg, hence evaluating this equation will allow us to find the current, I, over the leads. Also, we note thatĴ r,l =Ĵ 1,2 +Ĵ † 1,2 models electronic tunneling from the right/left lead respectively, and that the combination of these two terms effectively defines the current operator of the system. To calculate the current we evaluate Eq. (9a) by dividing the operatorsĴ r,l into their diagonal and non-diagonal parts (subscripts d and n respectively), Tr(Ĵ rρ ) = Tr(Ĵ r,dρd ) + Tr(Ĵ r,nρn ), with respect to the eigenbasis of the oscillating wire. From this analysis we find that to the zeroth order in the operatorsx and p the operatorsĴ r,l are diagonal and the expression for the force is proportional to the current, I 0 , through the system (Eq. (8) in Ref. [20]). The higher order terms inx andp, corresponding to the non-diagonal parts of J r,l , are collected in the current, I 1 , which, in the high bias limit, gives exponentially small corrections to total current I (see below).
In Eq. (11), G 0 = 2e 2 W/ is the zero field conductance with W = |T ef f | 2 ν 2 the probability of electron tunneling and P (n) is the probability that the fundamental mode is in quantum state |n with energy n ω.
Consider now the expression for the total current, Eqs. (11) and (12). The current I 0 can be understood by considering that the combinations of the Fermi functions puts restrictions on the allowed transmission channels (through the Pauli principle) for electrons as they exchange energy with the vibrating wire. Furthermore, an analytic expression to (11) can be found in the limit of high bias voltage, eV ≫ eV 0 (eV 0 ∝ max{k B T, ℓ ω}). Integrating over the electronic energy this equation can be expressed as, which, in the limit of high bias voltage, reduces to an expression for the current that is independent of temperature and bias voltage, To evaluate the non-diagonal contribution to the current we expand the exponentials in Eq. (8) in power of H 0 τ and integrate over the electronic energies, with q indicating the number of commutators to be evaluated and β = (k B T ) −1 . Evaluating (12) with this expansion we find that all contributions to the current I 1 decay exponentially in the high bias limit (βeV ≫ 1) as all correction terms will be of the form,   Thus we find that in the limit of high bias voltage, the current goes as (14), which differs from the ohmic behavior for the non-vibrating wire, I = G 0 V , by an amount that is independent of both the bias voltage and temperature as shown in Fig. 2. This can be understood from the fact that the increase in the current due to a further increment in the bias voltage under the conditions when V ≫ V 0 is fulfilled is not affected by the Pauli restrictions on the electron-vibron energy exchange. Nevertheless, the current deficit at large voltage biases is a true quantum-mechanical effect on transport that originates from the Pauli restrictions, however, these restrictions only affect the tunneling probability of low energy electrons close to the Fermi level.
In contrast to the analysis of Ref. [21] where the distribution function P (n) was considered to be only slightly out of equilibrium, our results for the current deficit survives independently of the form of P (n), even for highly excited distributions. As an example, we have analyzed Eqs. (7) and (8) separately to orderp, an analysis which shows that in the high bias limit the distribution function is indeed far from equilibrium as, e.g., the magnitude of the two lowest non-zero moments are 25 , Finally, we show the current deficit as a function of the bias voltage for realistic experimental parameters, Fig. 3. Here, the influence of the multiconnectivity of the elec- tron tunneling paths in the elastic channel and the Pauliprinciple restrictions on the available inelastic channels are clearly visible at low bias voltages as an increasing current deficit. Eventually, when the new inelastic tunneling channels that are added by a further increment of the bias voltage are not affected by the Pauli restrictions -which occurs at large enough bias voltages -the cur-rent deficit saturates to a constant value. This constant value of the current deficit depends on the magnetic field H and the mechanical properties of the nanowire through the parameter φ as illustrated in Fig. 3.
To conclude we have shown that for the system originally considered by Shekhter et al. 20 not only will the conductance be altered due to the multiconnectivity of the electronic transport through the system, but also the current. In particular we find that even for vibrational mode distributions out of thermal equilibrium the system is predicted to show a current offset which is independent of bias voltage and temperature. This is a clear manifestation of quantum mechanical effects on transport not previously considered which should be experimentally observable for magnetic fields of the order of H ∼20 T (see also discussion in Ref. [20]). Also, we have shown that the influence of internal damping in this nonresonant charge transfer process decays exponentially to all orders in the moments of the position and momentum, thus making the system considered a very good candidate for direct observation of quantum mechanical effects in mesoscopic systems.
Discussions with Leonid Gorelik, Robert Shekhter and Mats Jonson are gratefully acknowledged. This work was supported in part by the Swedish VR and SSF.