Magnetopolaronic effects in electron transport through a single-level vibrating quantum dot

Magnetopolaronic effects are considered in electron transport through a single-level vibrating quantum dot subjected to a transverse (to the current flow) magnetic field. It is shown that the effects are most pronounced in the regime of sequential electron tunneling, where a polaronic blockade of th...

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Datum:	2011
Hauptverfasser:	Skorobagatko, G.A., Kulinich, S.I., Krive, I.V., Shekhter, R.I., Jonson, M.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2011
Schriftenreihe:	Физика низких температур
Schlagworte:	Квантовые эффекты в полупроводниках и диэлектриках
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spelling	irk-123456789-1187992017-06-01T03:04:40Z Magnetopolaronic effects in electron transport through a single-level vibrating quantum dot Skorobagatko, G.A. Kulinich, S.I. Krive, I.V. Shekhter, R.I. Jonson, M. Квантовые эффекты в полупроводниках и диэлектриках Magnetopolaronic effects are considered in electron transport through a single-level vibrating quantum dot subjected to a transverse (to the current flow) magnetic field. It is shown that the effects are most pronounced in the regime of sequential electron tunneling, where a polaronic blockade of the current at low temperatures and an anomalous temperature dependence of the magnetoconductance are predicted. In contrast, for resonant tunneling of polarons the peak conductance is not affected by the magnetic field. 2011 Article Magnetopolaronic effects in electron transport through a single-level vibrating quantum dot / G.A. Skorobagatko, S.I. Kulinich, I.V. Krive, R.I. Shekhter, M. Jonson // Физика низких температур. — 2011. — Т. 37, № 12. — С. 1295–1301. — Бібліогр.: 24 назв. — англ. 0132-6414 PACS: 73.63.–b, 71.38.–k, 85.85.+j http://dspace.nbuv.gov.ua/handle/123456789/118799 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Квантовые эффекты в полупроводниках и диэлектриках Квантовые эффекты в полупроводниках и диэлектриках
spellingShingle	Квантовые эффекты в полупроводниках и диэлектриках Квантовые эффекты в полупроводниках и диэлектриках Skorobagatko, G.A. Kulinich, S.I. Krive, I.V. Shekhter, R.I. Jonson, M. Magnetopolaronic effects in electron transport through a single-level vibrating quantum dot Физика низких температур
description	Magnetopolaronic effects are considered in electron transport through a single-level vibrating quantum dot subjected to a transverse (to the current flow) magnetic field. It is shown that the effects are most pronounced in the regime of sequential electron tunneling, where a polaronic blockade of the current at low temperatures and an anomalous temperature dependence of the magnetoconductance are predicted. In contrast, for resonant tunneling of polarons the peak conductance is not affected by the magnetic field.
format	Article
author	Skorobagatko, G.A. Kulinich, S.I. Krive, I.V. Shekhter, R.I. Jonson, M.
author_facet	Skorobagatko, G.A. Kulinich, S.I. Krive, I.V. Shekhter, R.I. Jonson, M.
author_sort	Skorobagatko, G.A.
title	Magnetopolaronic effects in electron transport through a single-level vibrating quantum dot
title_short	Magnetopolaronic effects in electron transport through a single-level vibrating quantum dot
title_full	Magnetopolaronic effects in electron transport through a single-level vibrating quantum dot
title_fullStr	Magnetopolaronic effects in electron transport through a single-level vibrating quantum dot
title_full_unstemmed	Magnetopolaronic effects in electron transport through a single-level vibrating quantum dot
title_sort	magnetopolaronic effects in electron transport through a single-level vibrating quantum dot
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2011
topic_facet	Квантовые эффекты в полупроводниках и диэлектриках
url	http://dspace.nbuv.gov.ua/handle/123456789/118799
citation_txt	Magnetopolaronic effects in electron transport through a single-level vibrating quantum dot / G.A. Skorobagatko, S.I. Kulinich, I.V. Krive, R.I. Shekhter, M. Jonson // Физика низких температур. — 2011. — Т. 37, № 12. — С. 1295–1301. — Бібліогр.: 24 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT skorobagatkoga magnetopolaroniceffectsinelectrontransportthroughasinglelevelvibratingquantumdot AT kulinichsi magnetopolaroniceffectsinelectrontransportthroughasinglelevelvibratingquantumdot AT kriveiv magnetopolaroniceffectsinelectrontransportthroughasinglelevelvibratingquantumdot AT shekhterri magnetopolaroniceffectsinelectrontransportthroughasinglelevelvibratingquantumdot AT jonsonm magnetopolaroniceffectsinelectrontransportthroughasinglelevelvibratingquantumdot
first_indexed	2025-07-08T14:40:03Z
last_indexed	2025-07-08T14:40:03Z
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fulltext	© G.A. Skorobagatko, S.I. Kulinich, I.V. Krive, R.I. Shekhter, and M. Jonson, 2011 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12, p. 1295–1301 Magnetopolaronic effects in electron transport through a single-level vibrating quantum dot G.A. Skorobagatko1, S.I. Kulinich1,2, I.V. Krive1,2,3, R.I. Shekhter2, and M. Jonson2,4,5 1B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Lenin Avenue, Kharkov 61103, Ukraine 2Department of Physics, University of Gothenburg, SE-412 96 Göteborg, Sweden 3Physics Department, V. N. Karazin National University, Kharkov 61077, Ukraine 4SUPA, Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK 5Division of Quantum Phases and Devices, School of Physics, Konkuk University, Seoul 143-701, Republic of Korea E-mail: gleb_skor@mail.ru Received 12 July, 2011 Magnetopolaronic effects are considered in electron transport through a single-level vibrating quantum dot subjected to a transverse (to the current flow) magnetic field. It is shown that the effects are most pronounced in the regime of sequential electron tunneling, where a polaronic blockade of the current at low temperatures and an anomalous temperature dependence of the magnetoconductance are predicted. In contrast, for resonant tunneling of polarons the peak conductance is not affected by the magnetic field. PACS: 73.63.–b Electronic transport in nanoscale materials and structures; 71.38.–k Polarons and electron–phonon interactions; 85.85.+j Micro- and nano-electromechanical systems (MEMS/NEMS) and devices. Keywords: magnetopolaronic effects, single-level vibrating quantum dot, sequential electron tunneling, Frank– Condon (polaronic) blockade. 1. Introduction Single-molecule transistors have been intensively stu- died in recent years (see, e.g., the review in Ref. 1). The specific feature of these nanostructures is the presence of vibrational effects [2] in electron transport. Recently, ef- fects of a strong electron–vibron interaction were observed in electron tunneling through suspended single wall carbon nanotubes (SWNT) [3,4] and in carbon nanopeapods [5]. In suspended carbon nanotubes an electron–vibron coupl- ing is induced by the electrostatic interaction of the charge on a vibrating molecule with the metal electrodes. Electri- cally excited vibrations result in such effects as phonon- assisted tunneling [6], Franck–Condon (polaronic) block- ade [7] and electron shuttling [8,9] (see also the reviews in Refs. 2, 10, 11). Much less is known about the influence of a magnetic field on electron transport in molecular transistors. One can expect that a magnetic field, interacting with the electric current flowing through the system, will shift the position of the molecule inside the gap between the leads. For point-like electrodes this could result in a change of elec- tron tunneling probabilities and as a consequence in a neg- ative magnetoconductance. For suspended carbon nano- tubes magnetic field-induced displacements (due to the Laplace force) of the center-of-mass coordinate of the wire does not influence the absolute values of tunneling matrix elements. The magnetic influence emerges from a more subtle effect — the dependence of the phase of the electron tunneling amplitude on magnetic field (Aharonov–Bohm phase). It was shown in Ref. 12 that despite the 1D nature of electron transport in SWNTs, a magnetic field applied perpendicular to the quantum wire results in negative mag- netoconductance due to quantum vibrations of the tube. At low temperatures 0( / ,BT kω where 0ω is the fre- quency of the vibrational bending mode of the tube) the conductance, ( ),G H of the tube is exponentially sup- pressed, 2exp( ),G ∝ −φ where 0= 2 /φ πΦ Φ 0( = HLlΦ is the effective magnetic flux, L is the length of the wire, 0l is the amplitude of zero-point fluctuations of the tube, 0 = /hc eΦ is the flux quantum) [12]. At high tempera- tures 0( / )BT kω the magnetic field-induced correc- tion to the tunnel conductance scales as 1/ .T The scaling properties of magnetoconductace predicted in Ref. 12 re- semble the ones known for polaronic effects in electron transport through a vibrating quantum qot (see, e.g., the review in Ref. 11) if one identifies the dimensionless flux 0φ with the electron–vibron coupling constant. So it is interesting to consider electron transport through a vibrat- G.A. Skorobagatko, S.I. Kulinich, I.V. Krive, R.I. Shekhter, and M. Jonson 1296 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12 ing nanowire in the model (single-level quantum dot (QD)) when magnetopolaronic effects can be evaluated analyti- cally. In Ref. 13 the resonant magnetoconductance of a vibrating single-level quantum dot was calculated in per- turbation theory with respect to 1.φ It was shown in particular that unlike the case of electrically induced vibra- tions, where the peak conductance is known [14] to be un- affected by vibrations, the magnetic field influences reson- ance conductance for an asymmetric junction. The purpose of the present paper is to consider how the electrical current through a vibrating molecule depends on magnetic field and temperature in the limit of strong elec- tron–vibron coupling, that is to consider magnetopolaronic effects. As in Ref. 13 we model the vibrating molecule by a single-level quantum dot in a harmonic potential. For point-like contacts both the modulus and the phase of the electron tunneling amplitudes depend on the QD position in the gap between the leads. We will assume that the “longitudinal” center-of-mass coordinate ( )cx of the QD is fixed and that the quantum dot only vibrates along the y direction, while the magnetic field H is applied along z axis (see Fig. 1). At first we calculate the tunneling current in the case when the displacement cy of the QD in a magnetic field (due to the Laplace force) is greater than the amplitude of zero-point fluctuations, 0cy l 0 0( = /l mω , m is the QD mass). Then the mechanical part of the problem can be treated classically and the dependence of electrical current on magnetic field appears due to the dependence of the electron tunneling probabilities on the equilibrium position of the current-carrying QD in the magnetic field. We show that in strong magnetic fields H (the appropriate limit for the considered classical problem) the current scales as 1/ .J H∝ In the case when the bare tunneling probabilities do not depend on the QD displacement cy ( c cy x ) the mag- netic field influences the current only through the phase factors (Aharonov–Bohm phase) in the tunneling Hamilto- nian (this is always the case for a suspended SWNT). We considered magnetopolaronic effects in the regime of se- quential electron tunneling and for resonant tunneling of polarons (polaron tunneling approximation [15]). In these cases analytical formulae for the current were derived. We predict: (i) a Franck–Condon (polaronic) blockade of mag- netoconductance in the regime of sequential electron tunneling, (ii) an anomalous temperature dependence of magnetoconductance for strong electron–vibron coupling ( 1),φ (iii) a magnetic field-induced narrowing of reso- nant conductance peaks, and (iv) an excess current at high biases, 0 .eV ω 2. Model Hamiltonian and equations of motion The Hamiltonian of a single-level vibrating quantum dot in a magnetic field takes the form ( ) ( ) = , ˆ = ( ) ,l t Dj j j L R H H H H+ +∑ (1) where ( ) † , ,,= ( )l k j j k jj k j k H a aε −μ∑ (2) is the Hamiltonian of noninteracting electrons in the left ( = )j L and right ( = )j R leads ( kjε is the energy of elec- trons with momentum ,k jμ is the chemical potential), † ,, ( )k jk ja a are the creation (destruction) operators with the standard commutation relations † ,,{ , } = ( ).q j jjk ja a k q′ ′δ δ − Furthermore, † † 0 0=DH c c b bε + ω (3) is the Hamiltonian of a single-level 0(ε is the level ener- gy) vibrating quantum dot 0(ω is the frequency of vibra- tions in the y direction), † ( )c c and †( )b b are creation (de- struction) fermionic †({ , } = 1)c c and bosonic †([ , ] = 1)b b operators. Finally, ( ) † ,ˆ ˆ= ( )exp( ) H.c.t j H k jjH t y ij y a c− λ + , (4) = ( , ) ( , ),j L R ≡ − + Fig. 1. Schematic diagram of the device geometry. A single-level quantum dot of characteristic size L is placed in the gap between two point-like metal electrodes. The QD position ,L Rx on the x axis is fixed and it vibrates (depicted as a spring) only along the y axis. An external magnetic field is directed along the z axis. L y z x V xR xL Н Each of the moving charges that contribute to the current through a conductor experiences a Lorentz force in an electromagnetic field. The Laplace force refers to the total force on the conductor. ** Here we consider spinless electrons. Notice that in strong magnetic fields studied below the Zeeman splitting is so large that at all reasonable temperatures and bias voltages one can neglect the contribution of minority spin-polarized states. The effects of Zee- man splitting on electron transport in single electron transistors with spin-polirized leads were considered in [16]. Magnetopolaronic effects in electron transport through a single-level vibrating quantum dot Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12 1297 is the tunneling Hamiltonian. Here † 0ˆ = ( ) / 2y b b l+ is the coordinate operator and = /H eHd cλ is the magnet- ic field-induced electron–vibron coupling [12] (d is a pa- rameter of dimension length; its physical meaning in our model will be clarified later). In this section, the amplitude ˆ( )jt y of the tunneling matrix element in Eq. (4) will be modelled by the expression 2 2 0ˆ ˆ( ) = exp / ,j j j tt y t x y l⎛ ⎞− +⎜ ⎟ ⎝ ⎠ (5) where tl is the tunneling length and jx is the position of the center of mass of QD along x axis, which is assumed to be fixed. We will show that our results are not sensitive to the choice of parametrization, Eq. (5). The Heisenberg equation of motion for the fermion and boson operators are ˆ ˆ= ( )exp( ) ,kj kj kj j Hi a a t y ij y c∗ε + − λ (6) 0 , ˆ ˆ= ( )exp( ) ,j H kj k j i c c t y ij y aε + λ∑ (7) 2 1 0 ˆ ˆˆ ˆ = ( ),c Ly y m F F−+ ω + (8) where the operator expressions for the cohesive ( )cF and Laplace ( )LF forces take the form ˆ † , ˆ( )ˆ = e H.c., ˆ j ij yHc kj k j t y F c a y λ∂ + ∂∑ (9) ˆ † , ˆ ˆ= ( )e H.c.ij yHL H j kj k j F i jt y c aλ− λ + =∑ ˆ ˆ= ( ).L R Hd I I c − − (10) In the last equality in Eq. (10) we introduced the standard notation for the current operator † ,, ˆ = , = .j j j k jk j k I N N a a∑ (11) At first we neglect the quantum fluctuations of the coordinate operator ŷ and derive the equation of motion for the average (classical) coordinate ˆ< > = .cy y When QD vibrations are treated as classical oscillations the eq- uations of motion for the fermion operators, Eqs. (6) and (7), become a set of first order linear differential equa- tions, which can easily be solved analytically. After straightforward calculations (see, e.g., Ref. 17, where an analogous equation was derived for the electron shuttle problem) we get the following classical equation of mo- tion (notice that we made use of the wide band approxi- mation when calculating the averages over electron oper- ators and introduced a coordinate-dependent level width 2( ) = 2 ( ) \| ( ) \| ,j c F j cy t yΓ πν ε where ( )Fν ε is the electron density of states in the leads): 2 0 ( )1 2= ( ) ( ) ,j c c c j c c cj y dy y R y J y H m y c ⎛ ⎞∂Γ ⎜ ⎟+ω + ⎜ ⎟∂⎝ ⎠ ∑ (12) where 0 2 2 00 ( ) ( )1( ) = 2 ( ) [ ( ) / 2] j j c t c f R y d y ∞ ε − ε ε ε π ε − ε + Γ∫ (13) and 0 ( ) = ( ; )[ ( ) ( )]. 2c BW c L R eJ y d T y f f ∞ ε ε ε − ε π ∫ (14) Here ( ) = ( ) ( )t c L c R cy y yΓ Γ + Γ is the total level width, and 2 2 0 ( ) ( ) ( ; ) = , ( ) [ ( ) / 2] L c R c BW c t c y y T y y Γ Γ ε ε − ε + Γ (15) 1 ( ) = exp 1j jf T − ⎡ ε −μ ⎤⎛ ⎞ ε +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ (16) are the Breit–Wigner transmission cofficient and Fermi– Dirac distribution function, respectively. Equation (14) is the standard Landauer–Büttiker formula for the resonant current through a single-level quantum dot. The first term on the r.h.s of Eq. (12) can be interpreted as the cohesive force, the second term coincides with the force on a cur- rent-carrying conductor in a magnetic field (Laplace force) if we identify 2d with the longitudinal size of the QD, 2 = .d L This is the definition of the parameter d which appears in the operator form of the Aharonov–Bohm phase in the tunneling Hamiltonian, Eq. (4). In the absence of a magnetic field, = 0,H the equi- librium position of the transverse coordinate = 0.cy One can expect that the maximal influence of the magnetic field on the electrical current through a single-level QD occurs at high voltages, , , 0( = 0) ,L R L R ceV y≥ Γ ≡ Γ that is in the regime of sequential electron tunneling. In this case Eqs. (13), (14) are strongly simplified and one finds that ( ) ( ) ( ) ( ) = , ( ) = ( ) ( ) c L c R c c c L c R c e y y y J y y y y Γ Γ Γ Γ Γ + Γ (17) and ( Fε is the Fermi energy) , 0 1= = ln , > . 2 2 L R L R F eVR R eV ⎛ ⎞ Γ⎜ ⎟π ε⎝ ⎠ (18) For simplification we consider symmetric junctions ( = = )L Rx x l for which 2 2 0 2( ) = ( ) = exp ,L c R c c t y y l y l l ⎧ ⎫⎛ ⎞Γ Γ Γ − + −⎜ ⎟⎨ ⎬⎝ ⎠⎩ ⎭ (19) G.A. Skorobagatko, S.I. Kulinich, I.V. Krive, R.I. Shekhter, and M. Jonson 1298 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12 where 2 0 0= 2 ( ) \| \| exp( 2 / )F tt l lΓ πν ε − is the level width of the symmetric junction in the absence of a magnetic field. According to Eq. (13) the equilibrium position of the QD in a constant magnetic field does not depend on time; in weak magnetic fields it scales linerly with ,H 0 0 ( ) , , 2c L H ly H l H H H L (20) where the characteristic magnetic field 0H is defined by the equation 20 0 0 0= , = ,R I H e m I c Γ ω (21) 2 2 0 0 22= ln .F R tmll eV Γ ε⎛ ⎞ω ω + ⎜ ⎟π ⎝ ⎠ We see from Eqs. (20) and (21) that in weak magnetic fields the only effect of the cohesive force is to renormal- ize the frequency 0.ω In tunnel junctions 0( 0)Γ → the renormalization is small and can be neglected. In strong magnetic fields, 0( / ) ,H l L H we can neglect the con- tribution of the cohesive force to Eq. (12) as well. The magnetic field-induced shift of the QD in this limit scales logarithmically with ,H 0 2( ) ln exp . 2 2 t c t l L H ly H l l H l ⎧ ⎫⎛ ⎞⎪ ⎪ ⎨ ⎬⎜ ⎟ ⎪ ⎪⎝ ⎠⎩ ⎭ (22) In weak magnetic fields the small shift ( )cy l of the QD position does not influence the tunnel current in the considered classical approach. We will see in the next sec- tion that in this case one has to take into account quantum effects (phase fluctuations in the tunneling Hamiltonian), which strongly modify tunnel transport. In strong magnetic fields (classical limit) the current ( ) = ( / ) ( )J H e HΓ scales as 1/ H according to Eq. (22). At the end of this section we briefly comment on the in- fluence of the magnetic field on the resonant current in the considered classical approach. In the regime of resonant electron tunneling 0( , )T eV ≤ Γ the current depends li- nearly on the bias voltage ,V 0 2 ( ) ( ) ( ) = 4 [ ( ) ( )] L R r L R H H J H G V H H Γ Γ Γ + Γ (23) 2 0( = /G e h is the conductance quantum). It is evident from Eq. (23) that the resonant current through a symme- tric junction ( = )L RΓ Γ is not affected by the magnetic field since 0( ) = (0) = .r rJ H J G V For an asymmetric junc- tion the resonant current does not depend on magnetic field in the strong-H limit when the field-induced factor in the expression for the renormalized partial widths (see Eq. (19)) is cancelled in the expression for the electrical current, Eq. (23). 3. Magnetopolaronic effects In this section we consider the influence of quantum and thermodynamical fluctuations of the coordinate opera- tor of QD, ˆ,y on electron transport in a magnetic field. Quantum effects are significant (at low temperatures) when one can neglect the dependence of the modulus of the tunneling matrix element on the QD displacement in a magnetic field. This is always the case for tunneling through a suspended SWNT [12]. When considering the quantum effects of magnetic field-induced vibrations it is convenient to introduce the electron–vibron coupling constant in the form of a di- mensionless magnetic flux, 0= 2 /φ πΦ Φ 0( = / 2,HLlΦ 0 = /hc eΦ is the flux quantum). The dimensionless elec- tron–vibron interaction constant φ determines the quan- tum phase of the tunneling matrix element † / 0ˆ( ) = exp [ ( ) / 2],L R jt y t i b bφ +∓ (24) where 0 0= /l mω is the amplitude of zero-point fluctua- tions. Resonant electron tunneling in the model Eqs. (1)–(4), (24) was studied in Ref. 13 using perturbation theory with respect to 1.φ Here we are interested in nonperturbative effects, 1.φ At first we consider the regime of sequential electron tunneling where the effects of magnetic field-induced vi- brations are most pronounced. In this regime the current can be calculated perturbatively with respect to the level width .Γ The sign of the Aharonov–Bohm phase, Eq. (24), which is opposite for left- and right-tunneling electrons, does not play any role in the considered regime of tunne- ling (which can be treated classically by using a master equation approach). So our model is equivalent to the pola- ronic model of electron tunneling through a vibrating QD (see, e.g., the reviews in Refs. 2, 11). Notice that in a gen- eral case the “magnetic” problem can not be mapped to the polaronic problem because of the above mentioned “sign” difference [13]. We show below that this difference is not essential for magnetopolaronic effects. The average electric current J in the regime of sequen- tial electron tunneling ( , )Bk T eV Γ can be calculated by using a master equation approach. It can be represented as a sum of partial currents over “vibron channels” [18], > 0 ( < 0),n n corresponding to vibron emission (absorp- tion). Hence, 0 0 0 0 0 = ( ; ) = ( ){ ( ) ( )},n L R n J V J A f n f n ∞ −∞ φ φ ε − ω − ε − ω∑ (25) where 0 = /J eΓ is the maximal current through a single- level QD ( = / ( ))L R L RΓ Γ Γ Γ + Γ and the spectral weights nA are defined by the equation ˆ ˆ( ) (0)0 = e = e e ,in t i y t i y n n A ∞ ω ± φ φ −∞ ∑ ∓ (26) Magnetopolaronic effects in electron transport through a single-level vibrating quantum dot Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12 1299 where the average < ... > is taken with respect to the Hamiltonian of noninteracting vibrons, † 0= .bH b bω The spectral weights defined by Eq. (26) coincide with the analogous quantities in the polaronic model, where they are defined through the correlation function of op- erators ˆexp ( )i pλ ˆ( p is the momentum operator). For the equilibrated vibrons with the distribution function 1 0( ) = [exp ( / ) 1]Bn T T −ω − the coefficients nA take the following well-known form (see, e.g., Ref. 19) 20 ; = exp [ (1 2 )]n BA n T ω⎛ ⎞φ −φ + ×⎜ ⎟ ⎝ ⎠ 2 0[2 (1 )]exp .n B BI n n n T ω⎛ ⎞× φ + −⎜ ⎟ ⎝ ⎠ (27) Here ( )nI z denotes a modified Bessel function (see, e.g., Ref. 20). As is evident from their definition in Eq. (26), the spectral weights nA satisfy the sum rule = = 1,n n A ∞ −∞ ∑ (28) which can be rewritten as a nontrivial mathematical identi- ty for the sum of modified Bessel functions: coth = e = e , 0. sinh nx a x n n aI a x ∞ − − −∞ ⎛ ⎞ ≥⎜ ⎟ ⎝ ⎠ ∑ (29) The unitary condition Eq. (28) plays a crucial role in the derivation of analytical formulae for the current and con- ductance at 0, .Bk T eV ω Since all the analytical for- mulas of interest for us have already been derived in the literature on the polaronic model, we here merely formu- late the results. The magnetoconductance ( )G H in the regime of se- quential electron tunneling takes the following asymptotics at low and high temperatures [21]: 2 0 2 20 0 exp ( ), ,( ) (0) 1 , . 2 TG H G T T ⎧ −φ Γ ω ⎪ ⎨ ω −φ φ ω⎪ ⎩ (30) Here 0(0) = ( / 2)( / )G G Tπ Γ is the standard formula for conductance of a single-level QD. At intermediate temper- atures, 0 ,Bk T ω∼ and for strong electron–vibron inte- ractions, 1,φ the temperature dependence of conduc- tance is nonmonotonic (anomalous) [21]. This signature of polaronic effects was observed in experiments [5] on elec- tron tunneling through a carbon nanopeapod-based single- electron transistor. The asymptotics Eq. (30) coincide (up to numerical factors) with the ones found in Ref. 12 for a different model (electron tunneling through a suspended nanotube). The high-temperature asymptotics in Eq. (30) exactly coincides with the corresponding quantity calcu- lated in Ref. 13 for resonant electron tunneling. It is inter- esting to notice that the calculations based on a full quan- tum mechanical treatment [13] of interacting electrons and the master equation approach yield exactly the same results in high-T limit. Now we consider the behavior of current, Eq. (25), at low temperatures as a function of bias voltage ( ).eV Γ At = 0T Eq. (25) takes the form 2 2 0 =0 ( ; ) = exp( ) , ! n nm n J V J n φ φ −φ ∑ (31) where 0= [ / ]mn eV ω ([ ]x denotes the integer part of ).x At low voltages 0( < )eV ω = 0mn and the current does not depend on V and is determined by a standard formula for a saturated current through a single-level QD. Howev- er, in our case the level width ( )Γ φ is renormalized by the electron–vibron interaction: 2 0 ( )( < / ; ) = exp ( ).e eJ V e Γ φ Γ ω φ −φ (32) This is the demonstration of polaronic blockade [7]. With an increase of bias voltage the current jumps by an amount determined by the Franck–Condon factors 22 1 0= = e ! n n n nJ J J J n −φ + φ Δ − (33) each time the bias voltage opens a new inelastic channel 0( ) = [ / ]n V eV ω (see Fig. 2). At high voltages ( 0eV ω ) the polaronic blockade is lifted and the current saturates at its maximum value 0J , 22 0 0( / ; ) 1 e . ( 1) nm m J V e J n −φ ⎧ ⎫φ⎪ ⎪ω φ −⎨ ⎬ Γ +⎪ ⎪⎩ ⎭ (34) Here ( )xΓ is the gamma-function and 2 0= [ / ] 1mn eV ω φ . The difference between the maximum and minimum currents 2 0= {1 exp ( )}J Jδ − −φ (35) Fig. 2. Low-temperature current ( )J V in units of 0 = /J eΓ as a function of bias voltage at 0β / 4,T= ω = for three different val- ues of the electron–vibron coupling constant :φ the solid line cor- responding to the weak ( = 0.5),φ the dash-dotted line to an inter- mediate ( = 1),φ and the dashed line to the strong ( =1.5)φ coupling regimes. J V J ( )/ 0 1 0 0.4 0.2 2 3 eV/ �0� G.A. Skorobagatko, S.I. Kulinich, I.V. Krive, R.I. Shekhter, and M. Jonson 1300 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12 is nothing but the excess current considered in Refs. 22, 23 for the model of electron tunneling through a suspended carbon nanotube. The presence of a high-temperature ( )Bk T eV →∞ excess current in electron tunneling is another demonstration of polaronic blockade effects. Although the magnetic field-induced polaronic effects are most pronounced in the regime of sequential electron tunneling (when the current through a single-level QD is maximal) we briefly comment here on polaronic effects in resonant electron tunneling. It is physically evident that an electron–vibron polaron state can be formed on a quantum dot coupled to reservoirs if the life-time of the electronic state, / tτ Γ∼ , is much longer than the characteristic time of polaron formation, 2 0/ 1/ , 1.p pτ ε ω φ φ ≥∼ ∼ The corresponding inequality 2 0tΓ φ ω allows one to consider resonant tunneling of strongly interacting elec- trons in a simple model (polaron tunneling approximation [15]). In this approximation the bare electron Green’s func- tion (GF) in the Dyson equation for the retarded (ad- vanced) GFs is replaced by the polaron GF ( ( )pG ε ) 1( ) 1 , ,= ( ) ,PTA r a p r aG G −−⎡ ⎤ε − Σ⎣ ⎦ (36) where 0 0= ( ) = n p n A G n ∞ −∞ ε ε − ε + ω∑ (37) and nA are defined in Eq. (26). In the limit of wide electron bands in the leads the imaginary part of the self-energy func- tion ( )Σ ε is not renormalized by electron–vibron interaction in the considered approach ,Im = / 2r a tΣ Γ∓ and the real part of ( )Σ ε can be neglected. Then by evaluating the spec- tral function ( ) ( ) ( )= [ ]PTA PTA PTA r aA i G G− one can find the current with the help of the Meir–Wingreen formula [24]. It takes the form [15] { } 2 2 ( ) = ( ) ( ) . 1 [ ( ) / 2] L R p L R t p GeJ d f f G Γ Γ ε ε ε − ε + Γ ε∫ (38) From Eqs. (37), (38) it is easy to show that at low tempe- ratures ( 0T ω ) the conductance 0( , )G T ε in resonant tunneling can be represented in the Breit–Wigner form with the renormalized level widths , ( ) =L RΓ φ 2 2 , , 0= exp ( ) ( ):L R L R−φ Γ Γ φ ω 0 2 2 0 ( ) ( ) (0) = . ( ) [ ( ) ( )] / 4 L R F L R G G Γ φ Γ φ ε − ε + Γ φ + Γ φ (39) According to Eq. (39) the peak conductance, 0(0, = ),r FG ε ε is not renormalized by the magnetic field even for an asymmetric junction, =rG 2 0= 4 / [ ] ,L R L RG Γ Γ Γ + Γ as is the case in the polaronic model [14]. Notice that the opposite statement, that in an asymmetric junction the peak conductance is influenced by a magnetic field [13], was obtained in perturbation theory with respect to the electron–vibron coupling constant 1φ and that it holds in another region of model parame- ters, 2 , 0.L RΓ φ ω At high temperatures 2 0( )Bk T φ ω the polaronic blockade is lifted and the formula for the conductance de- rived from Eq. (39) coincides with the corresponding for- mula (see Eq. (31)) obtained in the regime of sequential electron tunneling. 4. Conclusion In conclusion we have shown that the quantum-vibration- induced Aharonov–Bohm effect, predicted in Ref. 12 for electron tunneling through a suspended carbon nanotube in magnetic field, can be interpreted as a magnetopolaronic effect, where the dimensionless flux 0= 2 /φ πΦ Φ plays the role of a magnetic field-induced electron–vibron interaction constant. We considered a simple model in the form of a single-level vibrating quantum “dot” in a transverse (with respect to the current flow) magnetic field and evaluated the electrical current and the magnetoconductance in two cases: 1) the amplitude of electron tunneling depends on the mag- netic field-induced QD displacement (point-like contacts), and 2) the magnetic field influences only the Aharonov– Bohm phase of the tunneling matrix element. It was shown that magnetic field-induced polaronic effects are most pro- nounced: (i) in the regime of sequential electron tunneling, (ii) in high magnetic fields when the momentum /p eHL cδ ∼ of the current-carrying QD induced by the Laplace force exceeds the momentum of zero-point fluctua- tions 0 0/p l∼ , and (iii) at low temperatures, 2 /Bk T p mδ (m is the mass of the QD). Recently polaronic effects were measured in nanotube- based single–electron transistors [3–5]. In particular, a Franck–Condon blockade was observed in a suspended car- bon nanotube [4]. Electrically induced electron–vibron inte- ractions happen to be much stronger than the electron– phonon interaction in isolated carbon nanotubes. So the magnetic effects could also be enhanced in the presence of ferromagnetic leads. Although simple estimations for a mi- cron-sized nanotube-based device show that even in a very strong transverse magnetic field ( 20T)H ∼ the magneto- current is only of the order of 0.1 pA, the effect is measura- ble and its fundamental nature justifies efforts to detect it. 5. Acknowledgment The authors thank L.Y. Gorelik and F. Pistolesi for valu- able discussions. Financial support from the European Commission (FP7-ICT-FET Proj. No. 225955 STELE), the Swedish VR, the Korean WCU program funded by MEST/NFR (R31-2008-000-10057-0) and the Grant “Quan- tum phenomena in nanosystems and nanomaterials at low temperatures” (No. 4/10-H) from the National Academy of Sciences of Ukraine is gratefully acknowledged. I.V.K. and S.I.K. acknowledge the hospitality of the Department of Physics at the University of Gothenburg. Magnetopolaronic effects in electron transport through a single-level vibrating quantum dot Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12 1301 1. S.M. Lindsay and M.A. Ratner, Adv. Mater. 19, 23 (2007). 2. M. Galperin, M.A. Ratner, and A. Nitzan, J. Phys.: Condens. Matter 19, 103201 (2007). 3. A.K. Hüttel, M. Poot, B. Witkamp, and H.S.J. van der Zant, New J. Phys. 10, 095003 (2008); G.A. Steele, A.K. Hüttel, B. Witkamp, M. Poot, H.B. Meerwaldt, L.P. Kouwenhoven, and H.S.J. van der Zant, Science 325, 1103 (2009). 4. R. Leturcq, Ch. Stampfer, K. Inderbitzin. L. Durrer, Ch. 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Magnetopolaronic effects in electron transport through a single-level vibrating quantum dot

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