Geometric rectification for nanoscale vibrational energy harvesting
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Geometric rectification for nanoscale vibrational energy harvesting.
Ra´ul A. Bustos-Mar´un ∗ Instituto de F´ısica Enrique Gaviola (CONICET-UNC), Facultad de Matem´atica Astronom´ıa,F´ısica y Computaci´on and Facultad de Ciencias Qu´ımicas,Universidad Nacional de C´ordoba, Ciudad Universitaria, C´ordoba, 5000, Argentina.
In this work, we present a mechanism that, based on quantum-mechanical principles, allows oneto recover kinetic energy at the nanoscale. Our premise is that very small mechanical excitations,such as those arising from sound waves propagating through a nanoscale system or similar phenom-ena, can be quite generally converted into useful electrical work by applying the same principlesbehind conventional adiabatic quantum pumping. The proposal is potentially useful for nanoscalevibrational energy harvesting where it can have several advantages. The most important one isthat it avoids the use of classical rectification mechanisms as it is based on what we call geometricrectification. We show that this geometric rectification results from applying appropriate but quitegeneral initial conditions to damped harmonic systems coupled to electronic reservoirs. We analyzean analytically solvable example consisting of a wire suspended over permanent charges where wefind the condition for maximizing the pumped charge. We also studied the effects of coupling thesystem to a capacitor including the effect of current-induced forces and analyzing the steady-statevoltage of operation. Finally, we show how quantum effects can be used to boost the performanceof the proposed device.
I. INTRODUCTION
The current efforts to reduce devices’ dimensions to-wards the nanoscale cannot be fully reached withoutinnovative solutions to their power supply. For manyapplications such as biomedical, deployable sensor net-works, or autonomous nanomachines, replacement of ex-hausted batteries is not an option and wireless devicesare desirable or even required.
In this context, vi-brational energy harvesting is attracting considerable at-tention as vibrations are pervasively available in differ-ent environments.
A severe limitation of most of theproposed vibrational energy harvesters is their narrowbandwidth of operation at acceptable performance. In-deed, this has driven an active area of research in recentyears.
The problem is rendered even more compli-cated for true nanoscale energy harvesters, i.e. whenthe dimensions of the whole device lay in the nanoscale.There, quantum mechanical effects may become impor-tant. Moreover, very low output voltages are expected,which would prevent the use of conventional electric rec-tifiers.Nanogenerators made of piezoelectric nanorods havebeen proposed for nanoscale energy harvesting some timeago.
When nanorods are subjected to an externalforce a deformation occurs and this causes an electricalfield inside the structure. On the other hand, under theappropriate conditions, a Schottky contact can be formedbetween the counter-electrode and the tip of the nanorod.Both effects can be used, through a proper design of thedevice, to generate direct currents. It has been proventhat these devices can successfully produce electric powerfrom different sources of vibrations. However, even inthis case, there is a minimum amplitude of the motion ofthe nanorods needed to produce an efficient rectification.In this work, we study a mechanism that can con-vert kinetic energy into electrical work at the nanoscale, which is potentially useful for vibrational energy har-vesting. The proposed mechanism precludes the use ofelectric rectifiers of any kind. Moreover, it does not re-quire a tuning of the resonances of the system to themain contributions of the vibrational spectrum of theenvironment, as is the case for most vibrational energyharvesters. Our proposal is based on the long-time be-havior of quantum pumping induced by damped vi-brational modes. The idea is that mechanical excitations,such as sound waves traveling through the system or sim-ilar phenomena, triggers the movement of a device thathits a conductor. The kinetic energy of the impact isthen transformed directly into an electric current throughvibrational-induced quantum pumping. The whole pro-cess has a nonvanishing direct current component at longtimes which depends on the geometry of the trajectoriesin the phase space of the system’s normal modes.This work is organized as follows. In Sec. II we firstdiscuss in more detail the type of processes treated hereand then derive the general theory used to describe them.In Secs. III and IV we derive for particular (but quitegeneral) cases explicit expressions for the factors neededto evaluate the total charge pumped per hitting event. InSec. V we discuss the effect of coupling the proposed de-vices to a capacitor and derive some limit expressions forthe efficiency and the steady-state voltage of operation.In Sec. VI we analyze a simple example that shows howquantum effects can be used to improve the harvestercharacteristics. Finally, in Sec. VII we summarize themain conclusions.
II. GENERAL THEORY
Before starting with the theory, let us first clarify thetype of processes we are dealing with. Our goal is thesame as that of macroscopic vibrational energy harvest-ing but taken to the nanoscale. One wants to extract use-ful electrical power from ambient residual energies arisingfrom different mechanical excitations. Those mechanicalexcitations can emerge in principle from several sponta-neous sources such as those produced by biological activi-ties (e.g. walking) or industrial activities (e.g. vibrationsstemming from some machinery), but also from sourcespurposely generated by an external agent as a way offeeding a nanomachine wirelessly.The type of systems considered consists of a hitting de-vice that only when, for example, a mechanical wave goesthrough the device or the whole harvester is shaken, hitsin a certain way a conductor connected to two leads. Themotion of the conductor and its coupling to the electronicdegrees of freedom is what then pumps current betweenthe reservoirs. This process is depicted in Fig. 1. We willdescribe the pumping process quantum mechanically sowe are implicitly assuming that the coherence length ofthe electrons in the conductor is at least of the same or-der as its characteristic size, which is in the nanoscale. In contrast, the motion of the conductor is assumed tobe classically treatable.The starting point of our theoretical description is thewell-known formula due to Brouwer, B¨uttiker, Thomas,and Prˆetre of the adiabatic charge pumping, whichadapted to our problem reads Q r = e Z ∞ dt X i dn r dq i ˙ q i ! (1)Here, e is the charge of the electron, and Q r is whatwe call the asymptotic pumped charge (APC) from thereservoir r , where “asymptotic” refers to the long-timelimit of the pumped charge Q ( t ), i.e. lim t →∞ Q ( t ). Thisdiffers from the usual definition for Q r , referring to thecharge pumped per cycle . In our case there is not acycle but a hitting event which is unique in principle. Themodes of the mechanical part of the system are labeled q i ,and dn r dq i is the emissivity, defined in the low-temperaturelimit as dn r dq i = X β,α ∈ r π Im (cid:20) ∂S αβ ∂q i S ∗ αβ (cid:21) , (2)where S αβ is the element of the scattering matrix S thatconnects a conduction channel β belonging to some reser-voir, to a conduction channel α belonging to the reservoir r ( S αβ is a transmission amplitude for α and β belong-ing to different reservoirs or a reflection amplitude oth-erwise). To obtain a simple expression, we expand theemissivity up to linear order in q i , Q r ≈ e X i dn r dq i (cid:12)(cid:12)(cid:12)(cid:12) q Z ∞ ˙ q i dt + X i,j ∂∂q j dn r dq i (cid:12)(cid:12)(cid:12)(cid:12) q Z ∞ q j ˙ q i dt. (3)We assume the system is initially at rest and all exci-tations decay at long times to the initial condition, i.e. FIG. 1. A and B -
Schemes of the type of systems proposed.An external mechanical force stemming from the environmenttriggers the movement of a bistable tip ( A ) or shakes thewhole nanodevice ( B ). As a result of that, a tip hits the sys-tem, in this case, a conductive wire suspended over permanentcharges. This starts the oscillation of the wire, which in turnpumps electrons between the reservoirs. C -
Typical tra-jectory in the phase space of the normal modes of the wire,represented by q i and q j . The initial time and the long-timebehavior are marked by t and t ∞ respectively. The geome-try of the trajectories determine the total pumped charge at t → ∞ . D -
A typical plot of the pumped charge Q ( t ) as afunction of time t , in arbitrary units. q i (0) = q i ( ∞ ). Then, we can use integration by parts,which gives Z ∞ q i ˙ q j dt = − Z ∞ q j ˙ q i dt, (4)to obtain Q r ≈ X i III. GEOMETRIC FACTOR FOR DAMPEDHARMONIC SYSTEMSA. Impulsive initial conditions Let us assume the classical q i modes correspond to thenormal modes of a system initially at rest that suffered animpulsive initial condition. For the moment, let us alsoassume that temperature is zero. Then, we can write q i ( t ) = a i sin( ω i ω t ) e − γ i ω t (6)where t is the time, ω i is the resonant frequency of thenormal mode i in units of a reference frequency ω , and γ i is the damping factor, also in units of ω . Note thatthis equation makes explicit the meaning of the long-time limit of Q r , t ≫ max [1 / ( γ i ω )]. The value of the a i coefficients depend on the initial velocities of each normalmode, a i = ˙ q i (0) / ( ω i ω ), which, in turn, depends on thedetails of how the tip hits the system. Integrating thegeometric factor f g with q i given by Eq. 6 yields f g ( i, j ) = ( a i a j ) ω i ω j [( ω i − ω j ) + ( γ i − γ j )][( γ i + γ j ) + ( ω i + ω j )] − ω i ω j (7)Note that f g is independent of ω , which gives the timescale of the whole process. Thus, f g only depends on thegeometry of the trajectories, given by the pairs ( a i , a j ),( γ i , γ j ), and ( ω i , ω j ). From Eq. 7, it is clear that com-pletely random initial conditions, which would corre-spond to random values of a i and a j , would make the -2-1012 1 2 3 4 f g ω j /ω i γ = 0 . γ = 0 . γ → . FIG. 2. Geometric factor f g , in units of ( a i a j ), as a functionof the ratio between the frequencies of two normal modes, ω j /ω i and for different damping factors, γ i = γ j = γ . average value of f g zero. This highlight the obvious factthat it is not possible to extract energy from thermalfluctuations (if the whole system is described by a uniquetemperature). However, if the tip is moved by an exter-nal source, see the discussion at the beginning of sec. II,its shape is kept constant between hitting events, and ithits the device at the same position, all the ratios a i /a j will be the same and only the absolute values of the a i coefficients will change. If this is the case, then, the APCcan only change its magnitude but not its sign betweenhitting events. Fig. 1 shows schemes of two possible se-tups of the system. There, a tip hits a conducting wirerandomly in time but always at the same place and fromthe same direction. Different shapes of the tip or multi-ple tips can also be used to control which normal modesof the wire will be excited.In Eq. 7 one can check that decreasing the damp-ing factors increases the total pumped charge. However,there is an upper limit to the APC, given bylim γ → Z ∞ q i ˙ q j dt = ± ( a i a j )2 vuuuut (cid:16) ω j ω i (cid:17) − (cid:16) ω j ω i (cid:17) − ± corresponds to ω i ≷ ω j . Fig. 2 shows the de-pendence of f g on the damping factors and the frequencyratios between modes. We can see that the closer thefrequencies of two modes, the larger their contributionto the pumped charge. Then, considering two consecu-tive modes, which will give the largest contribution, thehigher their frequency the better.Up to this point we have only considered the zerotemperature case for the geometric factor, which implies q i ( t = 0) = 0 and the absence of stochastic forces in thetrajectories. To address the effect of the temperature wewill consider a more realistic situation where the dynam-ics of q i ( t ) is determined by a Langevin-like equation¨ q i = − (cid:0) ω i + γ i (cid:1) q i − γ i ˙ q i + ξ i . (9) -0.6-0.4-0.20 0.001 0.01 0.1 1 f g KT /E kin Numeri
Analyti FIG. 3. Effect of the temperature on f g . KT is the Boltzmannconstant times the temperature. E kin is the kinetic energyadded by the impulsive initial condition. The line marked as“analytic” corresponds to Eq. 7 ( KT = 0). The error barsshown as “numeric” are centered at the average value of f g obtained numerically for finite temperatures. The width ofthe error bars corresponds to 2 σ/ √ N where σ is the standarddeviation of the set of trajectories at the same temperatureand N is the number of trajectories run ( N = 100). f g is inunits of ( a i a j ). See text for details. Here, ξ i accounts for the stochastic forces . These forceshave zero mean h ξ i i = 0 and are assumed local intime with a correlation function given by h ξ i ( t ) ξ i ( t ′ ) i = D i δ ( t − t ′ ), where D i is chosen such as to fulfill thefluctuation-dissipation theorem, D i = 2 KT γ i . At zerotemperature and for impulsive initial conditions one re-covers Eq. 6. We are assuming that the hitting deviceis an object large enough so that its dynamics is notaffected by thermal noise. Therefore, only when, for ex-ample, some mechanical wave goes through the system orthe whole harvester is shaken, the hitting device is trig-gered. We also assume that the impact is fast comparedwith the time scales of the vibrational modes coupled tothe electronic degrees of freedom. Then, the only roleof the hitting device is to provide the impulsive initialcondition. For that reason, its dynamics will not be con-sidered explicitly.We numerically solved Eq. 9 for two modes with ω =2 ω and γ = γ = 0 . ω . The geometric factor, Eq.5, was numerically evaluated using a final time equal to10 /γ . The initial position and velocity of the modeswere chosen from a thermal ensemble and then at t =0 a quantity equal to √ E kin was added to the initialvelocities, where E kin = ˙ q i / . q i is in units of a and ˙ q i is in units of a ω ). Fig.3 shows the average value of the geometric factor (andits error) obtained from the simulations as a functionof the temperature. As can be seen, the only role oftemperature is to broaden the distribution functions of f g around the values predicted by Eq. 7. 15 26 37 481 3 5 8 z min z max (cid:1) t ini z ( t ) t FIG. 4. Scheme of the type of processes that can give riseto displacive initial conditions. Due to some external excita-tion a tip first pushes a conductive wire and then it retreatsbut pulling the wire with it in the process as consequence ofvan der Waals forces. The movement of the conductive wireduring the process depicted and its subsequent free move-ment, after the interaction finished, pump current betweentwo reservoirs. In the plot z ( t ) represents the position of thetip with respect to the wire. B. Displacive initial conditions In this subsection we analyze a complementary case tothat studied in the previous subsection. In the displaciveinitial conditions, the velocities of all normal modes arezero at the beginning of the free movement but not thepositions. A physical situation corresponding to this casemay be, for example, a tip that first pushes a conductorand then, when moving back, pulls the conductor withit, due to the van der Waals forces. At some point, therestoring forces overcome the van der Waals forces andthe conductor is released, marking the beginning of itsfree motion. This situation is depicted in Fig. 4. Notethat here the “collisional” time can be large comparedwith the system’s dynamics. The equation of motion forthe normal mode q i can be written in this case as q i ( t ) = (cid:26) a i z ( t ) for − t ini < t < q i e − iγ i ω t cos ( ω i ω t ) for 0 ≤ t < ∞ , (10)where z ( t ) is a coordinate that describes the tip’s move-ment, a i is the weight of the z coordinate on the normalmode q i , and − t ini marks the beginning of the interactionbetween the tip and the conductor. We will describe thetip’s movement by the minimal expression z ( t ) = ( z max − z min )2 cos ( ω z t ) + ( z max + z min )2 , (11)where z max and z min are the maximum and minimumvalues of z ( t ) respectively (while the tip is still in contactwith the conductor). From Eq. 11 it is clear that z max = z (0) and q i = a i z max The value of ω z is calculated sothat z ( − t ini ) = 0, i.e. ω z = (cid:18) π − arccos (cid:20) ( z max + z min )( z min − z max ) (cid:21)(cid:19) /t ini (12)The integration of the geometric factor is now split intotwo parts f g = Z ∞− t ini q i ˙ q j dt = Z − t ini q i ˙ q j dt + Z ∞ q i ˙ q j dt (13) The first integral is easy to evaluate, it gives q i q j / f g = q i q j − γ j ( γ i + γ j ) h ( γ i + γ j ) + ω i i + (cid:2) ( γ i + γ j ) ( γ i + 2 γ j ) − ω i (cid:3) ω j + ω j h ( γ i + γ j ) + ω i i + 2 ( γ i + γ j − ω i ) ( γ i + γ j + ω i ) ω j + ω j (14) -202 1 2 3 4 f g ω j /ω i γ = 0 . γ = 0 . γ → . FIG. 5. Same as Fig. 2 but for displacive initial conditions. f g is in units of ( q i q j ). The dependence of f g with γ and the ratio ω i /ω j is sim-ilar to that of impulsive initial conditions. See Fig. 5.The limit of small γ is nowlim γ →∞ f g = q i q j (cid:20) − (cid:16) ω j ω i (cid:17) (cid:21)(cid:20) − (cid:16) ω j ω i (cid:17) (cid:21) (15)As in Subsec. III B we implicitly assumed KT = 0 inEqs. 10 and 14. However, the effect of the temperatureis the same as before. It only broadens the distributionfunction of f g around the value predicted by the zero-temperature formulas. This can be seen in Fig. 6 wherewe performed the same type of calculation as that de-scribed in the previous subsection, see the text aroundEq. 9. -0.6-0.4-0.20 0.001 0.01 0.1 1 f g KT /E ini Numeri
Analyti FIG. 6. Same as Fig. 3 but for displacive initial conditions. KT is in units of the initial potential energy E ini = ( ω i + γ ) q i / f g is in units of ( q i q j ). IV. SCATTERING FACTOR OF ANOSCILLATING WIRE To analyze the effect of the scattering factor f s weneed to resort to particular examples. Let us examinethe case of a conductive wire suspended over an electretmaterial as shown in panels A and B of Fig. 1. Forsimplicity, we assume a small capacitive coupling betweenthe electrons of the wire and the permanent charges. Thepotential U sensed by the electrons traversing the wirecan now be taken as U ( x ) = U z ( x ), where z , the separa-tion between the wire and the electret material, dependson the position x along the wire. Then, the electronicHamiltonian, written in term of the transverse normalmodes of the wire, readsˆ H = ˆ p m e + U X i q i ( t ) sin (cid:18) πω i xL (cid:19) Θ( x )Θ( L − x ) , (16)where ˆ p and m e are the momentum and mass of the elec-tron, Θ is the Heaviside step function, L is the length ofthe wire, q i is the amplitude of the i -th normal mode ofthe wire (considered in this approximation as a classicalvariables), and ω i is in this case an integer between 1and ∞ . Note that, for simplicity, we excluded the elec-tron’s spin of the analysis. To solve our problem, westart by first noticing that our Hamiltonian is of the formˆ H = ˆ p / (2 m e ) + P i U q i . If we define S and S q i as thescattering matrices associated with the Hamiltonians ˆ H and ˆ H q i respectively, where ˆ H q i = ˆ p / (2 m e ) + U q i , then,by using the Fisher and Lee formula, one finds ∂ S ∂q i (cid:12)(cid:12)(cid:12)(cid:12) q = ∂ S q i ∂q i (cid:12)(cid:12)(cid:12)(cid:12) q (17)where q = q j ( t ) = 0. See App. A. ˆ H q i is the sameHamiltonian than that presented in Refs. 29 and 30for the Thouless motor. As shown there, one can ob-tain analytically the scattering matrix of the problemby linearizing the Hamiltonian for momenta close to ~ k i = ± ~ πω i /L . See App. B. Using this result, we ob-tain the derivative of the scattering matrix of the originalproblem, Eq. 16, ∂ S ∂q i (cid:12)(cid:12)(cid:12)(cid:12) q = − U L ~ v F sinc (∆ E i ) e i ∆ E i σ z (18)where ∆ E i = (cid:16) L ~ v F (cid:17) (cid:16) ε − ~ k i m e (cid:17) , σ z is the “ z ” Pauli ma-trix, v F is the Fermi velocity, and ε is the Fermi energy.The scattering factor f s is obtained by assuming the mo-mentum of the electron is close to both ~ k i = ± ~ πω i /L and ~ k j = ± ~ πω j /L . Then, one can apply Eq. 18 to thederivatives with respect to q i and q j . This results in f s ( i, j, ε ) = eπ (cid:18) U L ~ v F (cid:19) sinc (∆ E i ) sinc (∆ E j ) × sin (∆ E i − ∆ E j ) . (19)As can be noticed, the scattering factor f s ( i, j, ε ) dependson the Fermi energy and the pair of modes i and j underconsideration. Taking its maximum value for each pairof ( i, j ) modes, one can check that pairs of modes withthe closest frequencies, ω j = ω i + 1 for j > i , give themaximum contribution to the APC, Eq. 5. One can alsocheck that, among those pairs with ω j = ω i + 1, the oneswith the lowest frequencies, smallest ω i , give the largestcontribution to APC. This is the opposite of the behaviorof f g discussed in the previous section.The above result was confirmed by numerical calcu-lations based on a tight-binding model. Important de-viation were observed only for the smallest ( ω i )s, whereEq. 19 overestimate the maximum value of f s ( ε ), seeFig. 7. The tight-binding model used in the figureconsisted of a linear chain of 400 sites with site energy E n = U P i q i ( t ) sin (cid:0) πω i nL (cid:1) , where L = 400, U = 0 . q i is given by Eq. 6. Only first neighbors couplingswere considered with a coupling constant t c = 1, thus f s ( a . u . ) ε ( t c ) ( ω i , ω j ) = (1 , ω i , ω j ) = (3 , ω i , ω j ) = (5 , Numeri
Analyti FIG. 7. Comparison of the scattering factor f s , in arbi-trary units, evaluated using Eq. 19 and numerically. ε isthe Fermi energy in units of t c , the coupling constant of thetight-binding chain. See text for details. setting the energy scale. Leads were attached to sites n = 1 and n = 400 with a coupling constant equal to t c .The self-energies of the leads were taken asΣ( ε ) = lim η → + ε + i η − sgn( ε ) s(cid:18) ε + i η (cid:19) − t c , (20)where ε is the Fermi energy. The numerical value of f s was obtained from the numerical derivative of the scat-tering matrix around q . The scattering matrices werecalculated from the retarded Green’s functions as shownin App. A and Refs. 28, 30–32.To compare the maximum contribution that each pairof modes may have to the APC we rewrite Eq. 5 as Q ( ε ) = (cid:18) eπ U L ~ (cid:19) X i We have shown that it is possible to harvest mechani-cal energy from the environment by using geometric rec-tification. However, this energy has to be stored into avoltage bias, and now the problem is to understand theback action of it on the pumping process. Let us as-sume our system is connected in series with a capacitorwith capacitance C and let us simplify the analysis byconsidering only small voltages. Then, the total chargeaccumulated in the capacitor Q total R produces a voltagebias V according to V = Q total R /C , where V = V L − V R with L and R labeling the left and right leads respec-tively. The voltage bias induces, in turn, an additionalforce F i , given by F i = (cid:18) dn L dq i − dn R dq i (cid:19) eV , (22)and this force will affect the dynamics of the entire sys-tem. In principle, this force could change the equilibriumpositions and the normal modes of the system betweenhitting events or, even worse, while the system is relax-ing. This is because V varies with time. The variationof V with time is a consequence of the charge accumu-lation driven by charge pumping, Eq. 5, and the chargeleakage due to the bias current I bias . The bias currentcan be described by I bias L = e h T LR V where T LR is thetransmittance, and the factor 2 takes into account thespin multiplicity. A full treatment of the problem then requires the solution of an additional coupled equation, V ( t ) = − Z t e T LR VhC dt ′ − Z t X i C (cid:18) dn L dq i − dn R dq i (cid:19) ˙ q i dt ′ , (23)where the time t can be large enough as to include several“hitting” events of the type described by Eq. 5. To gainsome understanding of the role of current-induced forceswithout resorting to numerical simulations, we will makeadditional assumptions. First, the hitting events are ran-dom but sufficiently far apart such that they do not in-terfere with each other. Second, after waiting enoughtime such that a large number of hitting events have oc-curred, a steady state is reached where the variation of V ( t ) is small compared with its mean value h V i . The lat-ter is a good approximation when the average pumpedcharge during a hitting event and the total charge leakedbetween events are both negligible compared with thetotal charge accumulated in the capacitor. This condi-tion can be written as (cid:0) e T LR (cid:1) / ( hνC ) ≪ ν isthe frequency of events that lead to APC. Consideringthe above, we can clear the mean voltage from Eq. 23giving, h V i ≈ h h Q R i ν e T LR (24)where h Q R i is the mean value of APC. Using Eq. 24and expanding the emissivity up to linear order in q i (similarly to what we did in Eq. 3), we obtain a simpleexpression for the current-induced forces, F i = dn L dq i (cid:12)(cid:12)(cid:12)(cid:12) q − dn R dq i (cid:12)(cid:12)(cid:12)(cid:12) q ! e h V i X j ∂∂q j dn L dq i (cid:12)(cid:12)(cid:12)(cid:12) q − ∂∂q j dn R dq i (cid:12)(cid:12)(cid:12)(cid:12) q ! e h V i q j (25)The first term just redefines the equilibrium position ofthe “ q i ” modes, while the second one couples linearlythe modes among each other and changes their naturalfrequency of resonance. However, the whole system isstill harmonic. Therefore, the expression for f g , Eq. 7,remains valid even for finite voltages.If we consider an steady-state situation such as thatdescribed in the context of Eq. 24, we can readily obtainthe total work done by the current-induced forces after ahitting event. The result is simply the energy added tothe capacitor W = Z F · dq = Z X i F i ˙ q i dt = − Q R h V i . (26) VI. PERFORMANCE AND QUANTUMEFFECTS Considering that the energy of the whole process comesfrom the initial kinetic energy of the hitting device and − | Q p | / T ( a . u . ) ε ( t c ) − . 997 2 Q p / T E imp = 0 . t c E imp = 0 . t c E imp = 0 . t c FIG. 9. Effect of impurities with different energies ( E imp ) onthe ratios | Q R | /T LR and Q p /T LR in arbitrary units. ε is theFermi energy. the fact that we are interested in accumulating energy ina capacitor, it is natural to define the efficiency of theglobal process as η = h Q R V ih E kin i , (27)where h E kin i is the average initial kinetic energy. Then,assuming the validity of Eq. 24, we can write, for impul-sive initial conditions, η ≈ hν h Q R i e T LR ω P i h ω i a i i . (28)Note that the efficiency does depend on the absolute tem-poral scale of the charge pumping process (proportionalto 1 /ω ), while neither the APC, Eq. 5, nor the steady-state voltage, Eq. 24, does.Several parameters can be tuned to increase η , butparticularly interesting is the ratio h Q R i /T LR . In prin-ciple, different quantum effects can be used to reduce T LR . The question is: Will quantum effects also reducethe pumped charge? One of the simplest examples tostudy this is the use of Anderson’s localization inducedby impurities in molecular wires.To study the effect of an impurity on the ratio Q R /T LR ,we performed a tight-binding calculation of the APC sim-ilar to that described in the context of Fig. 7. The defectwas placed at site n = 200 for a chain of 400 sites. Thesite’s energy of the impurity was E = E n + E imp and onlymodes with ω i equal to 1 and 2 were excited assuming animpulsive initial condition with a = a . The geometricfactor was evaluated directly from Eq. 7 and the rest ofthe parameters of the tight-binding calculation were thesame than those of Fig. 7. Fig. 9 shows the effect ofimpurities with different energies on the ratios (cid:12)(cid:12) Q p (cid:12)(cid:12) /T LR and Q p /T LR , the latter shown in the inset. ConsideringEqs. 24 and 28, the figure shows that quantum-inducedlocalization of the electron’s wave function can increase up to three orders of magnitude the energy accumulatedin the capacitor and the efficiency of the whole process.This emphasizes the key role that quantum mechanicsmay have on nanoscale vibrational energy harvesting. VII. CONCLUSIONS We have studied a previously unreported mechanismthat can turn residual kinetic energy directly into usefulelectrical work in the nanoscale by using quantum pump-ing. As an application example, we have analyzed a solv-able system consisting of a wire suspended over perma-nent charges where we find the conditions for maximizingthe asymptotic pumped charge. We have discussed theeffects of coupling general systems to a capacitor wherewe include in the analysis the effect of current-inducedforces. We have given explicit expressions for the steady-state voltage of operation and the efficiency of the har-vesting process in the limit of small but stationary volt-ages. Finally, we have shown how quantum effects canbe used to enhance the performance of energy harvestersseveral orders of magnitude.We believe this work opens up many possibilities forthe study of asymptotic quantum pumping and its poten-tial applications. Although further work is required, theproposal seems amenable to harvesting very low kineticenergy as it avoids the use of electrical rectifiers and thenseems promising for powering nanoscale devices. In thiscontext, it would be important to test the ideas proposedin more concrete examples, such as carbon nanotubes orgraphene sheets under realistic conditions. One key as-pect that requires a deeper study is the sensitivity ofthe sign of the pumped current to potential defects inthe fabrication of the device. This can cause problemsfor parallel energy harvesting as the sign of the pumpedcurrent is not controlled externally but depends on thedesign of the device.Our proposal only requires appropriate initial condi-tions triggered mechanically and, because of that, dis-placement currents should be absent. This makes asymp-totic quantum pumping attractive as an alternative wayof experimentally studying quantum pumping. Althoughit was not the original idea, it would also be interest-ing to study asymptotic quantum pumping as a thermalmachine. For example, one can assume that the tip isexcited by thermal noise and there is a temperature dif-ference between the tip and the rest of the system. Ap-propriate working conditions should be found in this casebut the idea seems appealing. VIII. ACKNOWLEDGMENTS The author acknowledges useful comments and dis-cussions with L. H. Ingaramo, L. J. Fern´andez-Al´azar,L. E. F. Foa Torres, and H. M. Pastawski. This workwas supported by Consejo Nacional de InvestigacionesCient´ıficas y T´ecnicas (CONICET), Argentina; Secre-tar´ıa de Ciencia y Tecnolog´ıa, Universidad Nacionalde C´ordoba (SECYT-UNC), Argentina; and Ministeriode Ciencia y Tecnolog´ıa de la Provincia de C´ordoba(MinCyT-Cor), Argentina. Appendix A: Derivation of Eq. 17 The elements of the scattering matrix of a problem canbe evaluated from the Green’s function of the system byusing the Fisher and Lee formula which can bewritten as S = I − i W † G R W . (A1)Here, G R is the retarded Green’s function G R = lim η → + [( ε + iη ) I − H − Σ ] − . (A2)where H is the Hamiltonian of the system without theleads, Σ is the self-energy due to the leads, and ε is theenergy of the electrons. The matrix W comes from Γ α = W † Π α W . (A3)where Π α is the projection operator onto the channel α of some reservoir r and Γ α is the contributions, due tothe channel α , to the imaginary part of the self-energy Σ , i.e. Γ = Im( Σ ) and Γ = P α Γ α . The derivative of S with respect to a coordinate q i ,that does not affect the couplings to the leads, can bewritten as ∂ S ∂q i = − i W † G R ∂ H ∂q i G R W . (A4)Now, due to the particular choice of H , H q i , and q ( q =0) it is clear that ∂ H ∂q i = ∂ H qi ∂q i and H ( q ) = H q i ( q ),which immediately implies Eq. 17. Appendix B: Derivation of Eq. 18 To find the scattering matrix of the Hamiltonian ˆ H q i we start by linearizing it for momenta close to ~ k i = ± ~ πω i /L where ~ is the Planck constant divided by 2 π .The resulting Hamiltonian, given in terms of the coun-terpropagating linear channels and measuring momenta and energies from ~ k i and ~ k i / (2 m e ) respectively, canbe written asˆ H q i = v F ˆ p σ z + U q i ( t )2 σ y Θ( x )Θ( L − x ) , (B1)where σ i denotes the Pauli matrices in the space of thecounterpropagating channels, v F is the Fermi velocity,and we do not include the electron spin for simplicity.The transfer matrix M of a one dimensional problemcan be defined by its effect on the in- and outgoing waves( i and o respectively) as ( i R , o R ) T = M ( o L , i L ) T , where L and R stand for left and right leads here (not to beconfused with the length of the system in Eq. B1). Then,neglecting the reflections at the boundary of the system(small U ) and assuming the wave function inside of itis of the form e ikx , one can write the transfer matrix as M ≈ e iL ˆ k , which combined with Eq. B1 yields M q i = exp (cid:18) iL ~ v F (cid:20) δε i − U q i ( t )2 σ y (cid:21) σ z (cid:19) (B2)where δε i = (cid:16) ε − ~ k i m e (cid:17) . This equation can be rewrittenas M q i = e iλ L ~ σ eff = I cos λ L + i~ σ eff sin λ L (B3)where λ L = ( L/ ~ v f ) p ( δε i ) − ( U q i / ~ σ eff = [ − i ( U q i / σ x + δε i σ z ] p ( δε i ) − ( U q i / . (B4)The relation between M and S can be obtainedfrom their definitions [ ( i R , o R ) T = M ( o L , i L ) T and( o L , o R ) T = S ( i L , i R ) T ]. The result is S q i = − sin λ L ( U q i / M √ ( δε i ) − ( U q i / M M sin λ L ( U q i / M √ ( δε i ) − ( U q i / (B5)where M = cos λ L − i (cid:16) δε i / p ( δε i ) − ( U q i / (cid:17) sin λ L , (B6)Taking the derivative of Eq. B5 for q i = 0 and consider-ing Eq. 17, gives Eq. 18. ∗ [email protected] Y. Qi and M. C. McAlpine, Energy Environ. Sci. , 1275(2010). C. ´O. Math´una, T. ODonnell, R. V. Martinez-Catala,J. Rohan, and B. OFlynn, Talanta , 613 (2008). F. Balestra, Beyond CMOS Nanodevices 1, Chap. 6 Vi-brational Energy Harvesting , Beyond CMOS Nanodevices(Wiley, New York, 2014). R. L. Harne and K. W. Wang, Smart Mater. Struct. ,023001 (2013). S. R. Anton and H. A. Sodano, Smart Mater. Struct. ,R1 (2007). F. Cottone, H. Vocca, and L. Gammaitoni, Phys. Rev.Lett. , 080601 (2009). X. Wen, Q. Yang, W. Jing, and Z. L. Wang, ACS Nano , 7405 (2014). J. Yang, J. Chen, Y. Yang, H. Zhang, W. Yang, Y. Bai,P. Su, and Z. L. Wang, Adv. Energy Mater. , 1301322(2014). C. Kim, M. Prada, G. Platero, and R. H. Blick, Phys.Rev. Lett. , 197202 (2013). F. Hartmann, P. Pfeffer, S. H¨ofling, M. Kamp, andL. Worschech, Phys. Rev. Lett. , 146805 (2015). Z. L. Wang, Advanced Functional Materials , 3553(2008). S. Xu, B. J. H., and Z. L. Wang, Nature Communications , 93 (2010). M. B¨uttiker, H. Thomas, and A. Prˆetre, Z. Phys. B ,133 (1994). P. W. Brouwer, Phys. Rev. B , R10 135 (1998). J. E. Avron, A. Elgart, G. M. Graf, and L. Sadun, Phys.Rev. B , R10618 (2000). S. K. Watson, R. M. Potok, C. M. Marcus, and V. Uman-sky, Phys. Rev. Lett. , 258301 (2003). L. E. F. Foa Torres, Phys. Rev. B , 245339 (2005). M. Strass, P. H¨anggi, and S. Kohler, Phys. Rev. Lett. ,130601 (2005). J. Splettstoesser, M. Governale, J. K¨onig, and R. Fazio,Phys. Rev. Lett. , 246803 (2005). L. Arrachea and M. Moskalets, Phys. Rev. B , 245322(2006). S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa,L. Wang, M. Troyer, and Y. Takahashi, Nat. Phys. ,296 (2016). C. Schweizer, M. Lohse, R. Citro, and I. Bloch, Phys. Rev.Lett. , 170405 (2016). Interesting physical systems where this condition can befound are carbon nanotubes and graphene sheets for ex-ample . The low-temperature limit of the emissivity is used just forsimplicity. For finite temperatures an extra integral shouldbe added to the formulas of the scattering factor as now dn r dq i = − Z dfdε X β,α ∈ r π Im (cid:20) ∂S αβ ∂q i S ∗ αβ (cid:21) dε, (B7)where f is the Fermi function. To see that, take the integral R ∞ q i ˙ q j dt and divide it intotime intervals that correspond to the different closed tra-jectories, R ∞ dt = R t dt + ... + R t i +1 t i dt + ... . Then sim-ply change the variables of the integrals as R t i +1 t i q i ˙ q j dt = H q i ( q j ) dq j . The last integral is the area enclosed by theparticular segment ( i, i + 1) of the total trajectory. Displacement currents arise from the capacitive cou-pling of time-dependent gate voltages with the reservoirs.These currents typically hinder the detection of pumpingcurrents. . In our case, gate voltages are not necessary ingeneral but even in the case of using them, as may be thecase for proposals similar to those shown in Fig. 1, theyare time independent. The time dependence is in the de-formation of the system itself, which is independent of anyexternal agent. An electret is a dielectric material that has a quasi-permanent electric charge or dipolar polarization. An ex-ample of its application for energy harvesting can be foundin Ref. 38. D. S. Fisher and P. A. Lee, Phys. Rev. B , 6851 (1981). R. A. Bustos-Mar´un, G. Refael, and F. von Oppen, Phys.Rev. Lett. , 060802 (2013). L. J. Fern´andez-Alc´azar, H. M. Pastawski, and R. A.Bustos-Mar´un, Phys. Rev. B , 155410 (2017). H. M. Pastawski and E. Medina, Rev. Mex. Fis. , 1(2001). C. J. Cattena, L. J. Fern´andez-Alc´azar, R. A. Bustos-Mar´un, D. Nozaki, and H. M. Pastawski, Journal ofPhysics: Condensed Matter , 345304 (2014). To recover the expression for Q L shown in Eq. 5 we as-sumed that there is not an accumulation of charges in thesystem Q L = − Q R and that the equilibrium positions donot change ( q i ( t ) ≈ q i ( t ∞ )). The latter is reasonable for Q L much smaller than the total charge accumulated in thecapacitor, which implies V ( t ) ≈ V ( t ∞ ). N. Bode, S. Viola Kusminskiy, R. Egger, and F. von Op-pen, Beilstein Journal of Nanotechnology , 144 (2012). A reservoir r can always be described by a series of inde-pendent conduction channels α . This type of descriptionimplies that the ( Γ α )s are diagonals in a tight binding rep-resentation of the system. Then, the elements of S ,see Eq. A1, result in the more familiar expression S αβ = δ αβ − i √ Γ αi G Rij p Γ jβ . L. Foa Torres, S. Roche, and J. Charlier, Introductionto Graphene-Based Nanomaterials: From Electronic Struc-ture to Quantum Transport (Cambridge University Press,Cambridge, UK, 2014). P. W. Brouwer, Phys. Rev. B , 121303 (2001). K. Tao, S. Liu, S. W. Lye, J. Miao, and X. Hu, J. Mi-cromech. Microeng.24