We present a quantum theory of the shuttle instability in electronic
transport through a nanostructure with a mechanical degree of freedom. A phase
space formulation in terms of the Wigner function allows us to identify a
cross-over from the tunnelling to the shuttling regime, thus extending the
previously found classical results to the quantum domain. Further, a new
dynamical regime is discovered, where the shuttling is driven exclusively by
the quantum noise.Comment: 4 pages, 2 figures; minor changes; final version published in Phys.
  Rev. Let

Donarini, Andrea

Jauho, Antti-Pekka

Novotny, Tomas

English

arXiv

Abstract: We present a quantum theory of the shuttle instability in electronic transport through a nanostructure with a mechanical degree of freedom. A phase space formulation in terms of the Wigner function allows us to identify a crossover from the tunneling to the shuttling regime, thus extending the previously found classical results to the quantum domain. Further, a new dynamical regime is discovered, where the shuttling is driven exclusively by the quantum noise

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Quantum Shuttle in Phase SpaceNovotny, Tomas; Donarini, Andrea; Jauho, Antti-PekkaPublished in:Physical Review LettersLink to article, DOI:10.1103/PhysRevLett.90.256801Publication date:2003Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Novotny, T., Donarini, A., & Jauho, A-P. (2003). Quantum Shuttle in Phase Space. Physical Review Letters,90(25), 256801. DOI: 10.1103/PhysRevLett.90.256801Quantum Shuttle in Phase SpaceToma´sˇ Novotny´,1,2,* Andrea Donarini,1,† and Antti-Pekka Jauho1,‡1Mikroelektronik Centret, Technical University of Denmark, Ørsteds Plads, Building 345 east, DK-2800 Kgs. Lyngby, Denmark2Department of Electronic Structures, Faculty of Mathematics and Physics, Charles University,Ke Karlovu 5, 121 16 Prague, Czech Republic(Received 23 January 2003; published 25 June 2003)We present a quantum theory of the shuttle instability in electronic transport through a nanostructurewith a mechanical degree of freedom. A phase space formulation in terms of theWigner function allowsus to identify a crossover from the tunneling to the shuttling regime, thus extending the previouslyfound classical results to the quantum domain. Further, a new dynamical regime is discovered, wherethe shuttling is driven exclusively by the quantum noise.DOI: 10.1103/PhysRevLett.90.256801 PACS numbers: 73.23.Hk, 05.60.Gg, 85.85.+jAdvances in microfabrication technology are pushingtoday’s microelectromechanical systems towards thenanometer regime, and the emerging new technology ofnanoelectromechanical systems (NEMS) is expected toplay an important role in the future. The ubiquitousquantum mechanical effects affecting the performanceof these devices present many theoretical challenges, onlya few of which have so far been addressed in the litera-ture. The purpose of this Letter is to present a fullyquantum theory for an electromechanical instability ina generic NEMS device, the single-electron shuttle firststudied by Gorelik et al. [1].This device consists of a movable single-electron tran-sistor (SET) and exhibits an electromechanical instabil-ity from the standard tunneling regime to a new regime inwhich the SEToscillates and carries an integer number ofelectrons per a cycle (shuttle regime). Since the originalsuggestion [1], there has been increasing interest in theshuttle phenomenon [2–10]: e.g., by incorporating theshot noise due to the electron transfer [2], gate effects[3], the coherence effects in the electronic subsystem [4],and strong dissipation of the oscillator energy [5]. Onlyvery recently the quantum mechanical treatment of theoscillations in various modifications of the shuttle setuphas been considered [6,8].The classical theory of shuttle transport has been used[3,4] to describe the experiments on C60 single-electrontransistor [11], where the oscillations of the center of massof the molecule were found to be important. However,also explanations based on incoherent phonon assistedtunneling theory, which do not take into account thecorrelation between the coherent oscillator motion andthe electron transfer, seem to yield reasonable predictionsfor the I-V curves [12,13].Therefore, a fully coherent quantum mechanical treat-ment of the oscillator, which does take into accountpossible correlations and which would, in principle, allowone to join the two approaches into a unified framework,is a most desirable theoretical task. In this study wepresent an attempt for such a treatment for the simplestmodel supporting the shuttling transition in the classicalregime. A complementary study to ours already exists[10] which uses a similar model, albeit without anymechanical damping. However, the quasiclassical expan-sion of the tunneling term used in that work does not giveaccess to the purely quantum phenomena discussed below.We demonstrate that the characteristic strong correla-tion between the oscillator motion and the electron trans-fer persists even in the quantum regime. The noisegenerated by various sources (shot and thermal, bothhaving quantum components) is found to be very impor-tant for the phenomenon. Not only does it smear theclassical transition found in [1] into a crossover with aconsiderably shifted position in parameter space com-pared to the classical ‘‘mean field’’ study [1,4] but thequantum component of the shot noise also generates theshuttle instability even in the classically stable regionwith zero electric field.Using the generalized master equation approach sug-gested in [6] we study a simple model motivated byseveral previous studies [1,4,10,13]. Namely, we consideran oscillating nanoscopic grain with only one electroniclevel (strict Coulomb blockade regime) coupled to twoleads. The oscillator degree of freedom is treated fullyquantum mechanically in our approach. We also accountfor the oscillator damping, the thermal noise due to theoscillator-bath coupling, and the shot noise due to thequantized electron transfer.The Hamiltonian of the model readsH  Hosc  0cy0c0 Xk;L;Rk cykck HB eExcy0c0 Xk;L;RTk0xcykc0  h:c: Hint;(1)where the first three terms describe the free evolution of alinear harmonic oscillator Hosc  p22m m!2x22 , the singleelectronic level on the grain, and the two noninteractingleads symmetrically biased by voltage V (i.e., L  eV2 ,R   eV2 with e; V > 0; all electronic energies are mea-sured with respect to the equilibrium chemical potentialP H Y S I C A L R E V I E W L E T T E R S week ending27 JUNE 2003VOLUME 90, NUMBER 25256801-1 0031-9007=03=90(25)=256801(4)$20.00  2003 The American Physical Society 256801-1of the leads), respectively. The term HB describes a ge-neric Ohmic heat bath [14]. The first interaction termaccounts for the electrostatic coupling between the posi-tion of the grain and the occupation of the grain elec-tronic level; E is the electric field caused by the biasapplied between the leads and/or gate voltage [3,13].This electric field causes a shift of the charged oscillatorequilibrium by d  eEm!2. The second term is the couplingof the leads to the single electronic state on the grain withoscillator-dependent hopping amplitudes TkL0x tL exp x, TkR0x  tR expx where  is the electrontunneling length. The last term Hint describes the cou-pling of the heat bath to the oscillator which is linear inthe oscillator coordinate [14]. The net effect of this cou-pling is the oscillator mechanical damping characterizedby a constant  and the Langevin random force that thebath exerts on the oscillator and which depends on thetemperature of the bath T.By projecting out the leads and the thermal bathwe arrive at a Markovian generalized master equation(GME) for the density matrix t of the system com-posed of the grain and the harmonic oscillator. Since theelectronic transfer rate in the shuttling regime is compa-rable with the frequency of the oscillations, we employedthe singular coupling limit [15,16] appropriate for rapidlydecaying systems. This approximation is valid for biasmuch larger than any energy scale of the system, i.e.,eV 	 h!; 0. Moreover, we also assume eV 	 kBTwhich is reasonable in the present physical context.The mechanical damping due to the heat bath is, on theother hand, treated within the standard weak couplingtheory [17].Our GME reads_t  Lt  Lcoh Ldriv Ldampt (2)withL coh  1i h Hosc  0cy0c0  eExcy0c0; ; (3)L driv  L2 c0cy0e2x= 2cy0ex=ex=c0  e2x=c0cy0  R2cy0c0e2x= 2c0ex=ex=cy0  e2x=cy0c0; (4)L damp   i2 h x; fp; g m!h N  1=2x; x; :(5)Here ;  and f; g denote the commutator and the anticom-mutator, respectively, and N  exp h!kBT  11 is themean oscillator occupation at the bath temperature.The first term in Eq. (2) is the free coherent evolutionof the oscillator and the grain level while the second onedescribes the transfer of electrons via the oscillator-position-dependent tunnel junctions from the left reser-voir to the grain level (the term with L) and from thelevel to the right reservoir (the term with R). The oppo-site processes can be neglected due to the assumptioneV 	 h!; 0; kBT. The transfer rates equal L;R 2h jtL;Rj2DL;R with the constant densities of states of theleads DL;R. The third term accounts for the interaction ofthe oscillator with the heat bath. Translational invarianceof the damping, positivity of the density matrix, andrelaxation towards canonical equilibrium cannot beachieved simultaneously with any Markovian dampingkernel, and one has to sacrifice at least one of theseproperties. The most physical choice is to relax the pos-itivity [17]. We checked the magnitude of breaking of thepositivity in our calculations and found it irrelevant.Moreover, it occurs only for large values of  out of theshuttling regime.It can be shown that the electronic off-diagonal ele-ments of the density matrix are decoupled from thediagonal ones and, moreover, decay to zero in the sta-tionary state. Therefore, it is sufficient to consider onlythe electronic diagonal elements: 00t  h0 jtj0i and11t  h1 jtj1i, where j1i  cy0 j0i. These objects arestill full density matrices in the oscillator space andsatisfy_00t  1i h Hosc; 00t L2e2x=00t  00te2x=  Rex=11tex= Ldamp 00t;_11t  1i h Hosc  eEx;11t  Lex=00tex=  R2 e2x=11t  11te2x= Ldamp 11t:(6)From the continuity equation for the electronic charge wemay deduce the following formula for the stationarycurrent through the grain (flowing from the left to theright lead):Istat  eLTrosce2x=stat00   eRTrosce2x=stat11 : (7)The trace is carried out over the oscillator basis andstatnn  limt!1nnt.We solved numerically the stationary version of theabove Eqs. (6): 0  P2N2l1 Lklstatl where the column vectorstatl consists of the matrix elements of stat00 ; stat11 andthe (super)matrix Lkl of the dimension 2N2  2N2 con-tains the appropriate coefficients of the linear system (6).The density matrix was represented in the harmonicoscillator basis which was truncated by taking up to N 100 lowest states which yields satisfactory numericalP H Y S I C A L R E V I E W L E T T E R S week ending27 JUNE 2003VOLUME 90, NUMBER 25256801-2 256801-2convergence in our parameters range. We determined theunique (for R  L    0) null vector of the super-matrix L by using the Arnoldi iteration [18].Our approach differs from the one used in [6] where theexplicit time integration scheme of the time-dependentequation analogous to (6) was used to determine thestationary density matrix. We applied our method to themodel of [6] and recovered the I-V curve results reportedthere. Using the phase space analysis detailed below wefully confirmed the shuttling interpretation based on theindirect evidence from changes of the I-V curves withchanging parameters.The I-V curve (or other dependencies of the current onsome parameter) alone yields only an indirect evidence ofshuttling and may actually not be decisive whether thesystem is shuttling or not (see, e.g., [12] versus [4]).Therefore, it is preferable to consider quantities whichdepend also on the state of the oscillator. An excellentvisualization tool [17] for the description of the jointelectronic and oscillator properties are the Wigner func-tions (n  0; 1)WnnX;P Z 11dy2 hX y2statnnX y2expiPyh(8)yielding the charge-resolved quasiprobability distribu-tions of the oscillator in the phase space. These functionsprovide us with a clear evidence of the transition from theincoherent tunneling to the coherent quasiclassical shut-tling behavior with decreasing damping coefficient.Our focus is in the quantum effects on the shuttlingtransition and, in particular, whether there is any tran-sition in the quantum regime at all. Therefore, we work inthe strictly quantum regime where the tunneling length iscomparable to the zero uncertainty of the oscillator, x0 													h=m!p, and where only a relatively small numberof oscillator states is excited.In Fig. 1 we depict the Wigner functions W00, W11, andWtot  W00 W11 showing the crossover from the tun-neling to the shuttling regime with decreasing damping.In the tunneling regime (large ) the oscillator is locatedaround the origin (or shifted origin when charged) withno particular correlation between its charge state andmomentum (Wigner functions are centered around theorigin with some ‘‘quantum fuzziness’’). This is consis-tent with the quantum incoherent tunneling picture. Onthe other hand, in the shuttling regime (small ) theoscillator orbits almost classically (ringlike shape ofWtot with a hole around the origin), shuttles the chargeon its way from the left to the right lead, and returnsempty back (half-moon shapes of W00, W11). The corre-lation between the charge state and the mechanicalmotion is very strong. In the crossover region (medium) we can see that both regimes of transport are contri-buting additively (ringlike shape plus an incoherent peakaround the origin of Wtot). The classical mean field sharptransition of [1] between the tunneling and the shuttlingregime is smeared into a crossover due to the noise. AlsoFIG. 1 (color online). Phase space picture of the tunneling-to-shuttling crossover. The respective rows show the Wignerdistribution functions for the discharged (W00), charged (W11), and both (Wtot) states of the oscillator in the phase space (horizontalaxis–coordinate in units of x0 													h=m!p, vertical axis–momentum in h=x0). The values of the parameters are   x0, T  0,d  0:5x0,   0:05 h!. The values of  are in units of h!. The Wigner functions are normalized within each column.P H Y S I C A L R E V I E W L E T T E R S week ending27 JUNE 2003VOLUME 90, NUMBER 25256801-3 256801-3the position of the crossover is substantially shifted withrespect to the classical values (cross is 5 times largerthan the classical value) which is attributed to the deeplyquantum (and therefore noisy) regime. The classical pic-ture is expected to emerge in the quasiclassical limitd; 	 x0.In Fig. 2 we plot the  dependence of the stationarycurrent through the grain for different transfer rates and electric fields (measured in d). For the two curveswith d  0 we can see the rise of the current in thecrossover region from the tunneling-limited values pro-portional to  to shuttling-mediated quantized value ofone shuttled electron per each cycle (1=2  0:16 in ourunits; independent of ) in agreement with the classicalresults. More surprisingly, also the results for the case ofzero electric field d  0 show clear signs of shuttlingcrossover in the I- curve for small enough mechanicaldamping. Classically there is no shuttling transition ford  0 regardless of the other parameters’ values [4], andthe I- curve is constant. Therefore, this shuttling must bedriven purely by the quantum component of the shot noise(proportional to h). Also the phase space pictures of thisregime reveal the onset of shuttling transport.Finally, we comment on the effect of the temperature.For a nonzero temperature the shuttling transition withinour model is facilitated by the increase of the mechanicalnoise driving the transition. The deteriorating effect ofthe temperature on the transition is not included in themodel due to the high bias assumption. The developmentof the theory for a finite bias is under way.To summarize, we have presented a quantum theoryof the shuttling transition in fully developed Coulombblockade regime. Using the Wigner functions as a phasespace visualization method we have exhibited a clearcrossover from the tunneling to the shuttling regime ofthe transport as a function of the mechanical dampingparameter. The effect of noise on the transition in thedeeply quantum regime ( x0 													h=m!p) is pro-nounced and can even trigger the transition in the classi-cally stable regime with the zero electric field.The authors thank the members of the Chalmers group,A. Wacker, and B. Velicky´ for helpful discussions. Advicefrom T. Eirola concerning numerical methods was indis-pensable. Support of Grant No. 202/01/D099 of the Czechgrant agency for one of us (T. N.) is also gratefullyacknowledged.*Electronic address: tno@mic.dtu.dk†Electronic address: ad@mic.dtu.dk‡Electronic address: antti@mic.dtu.dk[1] L.Y. Gorelik, A. Isacsson, M.V. Voinova, B. Kasemo, R. I.Shekhter, and M. Jonson, Phys. Rev. Lett. 80, 4526(1998).[2] A. Isacsson, L.Y. Gorelik, M.V. Voinova, B. Kasemo, R. I.Shekhter, and M. Jonson, Physica (Amsterdam) 255B,150 (1998).[3] N. Nishiguchi, Phys. Rev. B 65, 035403 (2002).[4] D. Fedorets, L.Y. Gorelik, R. I. Shekhter, and M. Jonson,Europhys. Lett. 58, 99 (2002).[5] T. Nord, L.Y. Gorelik, R. I. Shekhter, and M. Jonson,Phys. Rev. B 65, 165312 (2002).[6] A. D. Armour and A. MacKinnon, Phys. Rev. B 66,035333 (2002).[7] N. Nishiguchi, Phys. Rev. Lett. 89, 066802 (2002).[8] A.Y. Smirnov, L. G. Mourokh, and N. J. M. Horing, Phys.Rev. B 67, 115312 (2003).[9] L.Y. Gorelik, A. Isacsson, Y. M. Galperin, R. I. Shekhter,and M. Jonson, Nature (London) 411, 454 (2001).[10] D. Fedorets, L.Y. Gorelik, R. I. Shekhter, and M. Jonson,cond-mat/0212561.[11] H. Park, J. Park, A. K. L. Lim, E. H. Anderson, A. P.Alivisatos, and P. L. McEuen, Nature (London) 407, 57(2000).[12] D. Boese and H. Schoeller, Europhys. Lett. 54, 668(2001).[13] K. D. McCarthy, N. Prokof ’ev, and M.T. Tuominen,cond-mat/0205419.[14] U. Weiss, Quantum Dissipative Systems, Series inModern Condensed Matter Physics Vol. 10 (WorldScientific, Singapore, 1999), 2nd ed.[15] H. Spohn, Rev. Mod. Phys. 52, 569 (1980).[16] G. Kimura, K. Yuasa, and K. Imafuku, Phys. Rev. A 63,22103 (2001).[17] D. Kohen, C. C. Marston, and D. J. Tannor, J. Chem. Phys.107, 5236 (1997).[18] G. H. Golub and C. F.V. Loan, Matrix Computations (TheJohn Hopkins University Press, Baltimore, MD, 1996),3rd ed.0 0.05 0.1 0.15 0.2 0.2500.020.040.060.080.10.120.140.160.18γCurrentE=0.5, Γ=0.05E=0.5, Γ=0.01E=0.0, Γ=0.05E=0.0, Γ=0.01FIG. 2 (color online). I- curve. The  dependence of thestationary current through the grain for different transfer ratesand electric fields. Their values are d  0:5x0,   0:05 h!(pluses; corresponds to Fig. 1); d  0:5x0,   0:01 h!(circles); d  0:0,   0:05 h! (asterisks); d  0:0,  0:01 h (crosses). Other parameters are   x0, T  0. Thecurrent is in units of e! while  is in h!.P H Y S I C A L R E V I E W L E T T E R S week ending27 JUNE 2003VOLUME 90, NUMBER 25256801-4 256801-4

Quantum Shuttle in Phase Space

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