Luttinger liquid and polaronic effects in electron transport through a molecular transistor

Electron transport through a single-level quantum weakly dot coupled to Luttinger liquid leads is considered in the master equation approach. It is shown that for a weak or moderately strong interaction the differential conductance demonstrates resonant-like behavior as a function of bias and gate...

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Date:	2008
Main Authors:	Skorobagat’ko, G.A., Krive, I.V.
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Language:	English
Published:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2008
Series:	Физика низких температур
Subjects:	Carbon nanotubes, quantum wires and Luttinger liquid
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Cite this:	Luttinger liquid and polaronic effects in electron transport through a molecular transistor / G.A. Skorobagat’ko, I.V. Krive // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1086–1093. — Бібліогр.: 23 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1175672025-02-09T13:55:08Z Luttinger liquid and polaronic effects in electron transport through a molecular transistor Skorobagat’ko, G.A. Krive, I.V. Carbon nanotubes, quantum wires and Luttinger liquid Electron transport through a single-level quantum weakly dot coupled to Luttinger liquid leads is considered in the master equation approach. It is shown that for a weak or moderately strong interaction the differential conductance demonstrates resonant-like behavior as a function of bias and gate voltages. The inelastic channels associated with vibron-assisted electron tunnelling can even dominate electron transport for a certain region of interaction strength. In the limit of strong interaction resonant behavior disappears and the differential conductance scales as a power low on temperature (linear regime) or on bias voltage (nonlinear regime). The authors would like to thank S.I. Kulinich for valuable discussions. This work was partly supported by the joint grant of the Ministries of Education and Science in Ukraine and Israel and by the grant «Effects of electronic, magnetic and elastic properties in strongly inhomogeneous nanostructures» of the National Academy of Sciences of Ukraine. 2008 Article Luttinger liquid and polaronic effects in electron transport through a molecular transistor / G.A. Skorobagat’ko, I.V. Krive // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1086–1093. — Бібліогр.: 23 назв. — англ. 0132-6414 PACS: 73.63.–b;73.63.Kv https://nasplib.isofts.kiev.ua/handle/123456789/117567 en Физика низких температур application/pdf 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	Carbon nanotubes, quantum wires and Luttinger liquid Carbon nanotubes, quantum wires and Luttinger liquid
spellingShingle	Carbon nanotubes, quantum wires and Luttinger liquid Carbon nanotubes, quantum wires and Luttinger liquid Skorobagat’ko, G.A. Krive, I.V. Luttinger liquid and polaronic effects in electron transport through a molecular transistor Физика низких температур
description	Electron transport through a single-level quantum weakly dot coupled to Luttinger liquid leads is considered in the master equation approach. It is shown that for a weak or moderately strong interaction the differential conductance demonstrates resonant-like behavior as a function of bias and gate voltages. The inelastic channels associated with vibron-assisted electron tunnelling can even dominate electron transport for a certain region of interaction strength. In the limit of strong interaction resonant behavior disappears and the differential conductance scales as a power low on temperature (linear regime) or on bias voltage (nonlinear regime).
format	Article
author	Skorobagat’ko, G.A. Krive, I.V.
author_facet	Skorobagat’ko, G.A. Krive, I.V.
author_sort	Skorobagat’ko, G.A.
title	Luttinger liquid and polaronic effects in electron transport through a molecular transistor
title_short	Luttinger liquid and polaronic effects in electron transport through a molecular transistor
title_full	Luttinger liquid and polaronic effects in electron transport through a molecular transistor
title_fullStr	Luttinger liquid and polaronic effects in electron transport through a molecular transistor
title_full_unstemmed	Luttinger liquid and polaronic effects in electron transport through a molecular transistor
title_sort	luttinger liquid and polaronic effects in electron transport through a molecular transistor
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2008
topic_facet	Carbon nanotubes, quantum wires and Luttinger liquid
url	https://nasplib.isofts.kiev.ua/handle/123456789/117567
citation_txt	Luttinger liquid and polaronic effects in electron transport through a molecular transistor / G.A. Skorobagat’ko, I.V. Krive // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1086–1093. — Бібліогр.: 23 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT skorobagatkoga luttingerliquidandpolaroniceffectsinelectrontransportthroughamoleculartransistor AT kriveiv luttingerliquidandpolaroniceffectsinelectrontransportthroughamoleculartransistor
first_indexed	2025-11-26T13:09:44Z
last_indexed	2025-11-26T13:09:44Z
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fulltext	Fizika Nizkikh Temperatur, 2008, v. 34, No. 10, p. 1086–1093 Luttinger liquid and polaronic effects in electron transport through a molecular transistor G.A. Skorobagat’ko and I.V. Krive B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Lenin Ave., Kharkov 61103, Ukraine E-mail: gleb_skor@mail.ru Received June 4, 2008 Electron transport through a single-level quantum weakly dot coupled to Luttinger liquid leads is consid- ered in the master equation approach. It is shown that for a weak or moderately strong interaction the differ- ential conductance demonstrates resonant-like behavior as a function of bias and gate voltages. The inelastic channels associated with vibron-assisted electron tunnelling can even dominate electron transport for a cer- tain region of interaction strength. In the limit of strong interaction resonant behavior disappears and the dif- ferential conductance scales as a power low on temperature (linear regime) or on bias voltage (nonlinear re- gime). PACS: 73.63.–b Electronic transport in nanoscale materials and structures; 73.63.Kv Quantum dots. Keywords: Luttinger liquid, electron transport, polaronic effects. 1. Introduction Last years electron transport in molecular transistors became a hot topic of experimental and theoretical inves- tigations in nanoelectronics (see e.g. [1,2]). From experi- mental point of view it is a real challenge to place a single molecule in a gap between electric leads and to repeatedly measure electric current as a function of bias and gate voltages. Being in a gap the molecule may form chemical bonds with one of metallic electrodes and then a consider- able charge transfer from the electrode to the molecule takes place. In this case one can consider the trapped mol- ecule as a part of metallic electrode and the corresponding device does not function as a single electron transistor (SET). Much more interesting situation is the case when the trapped molecule is more or less isolated from the leads and preserves its electronic structure. In a stable state at zero gate voltage the molecule is electrically neu- tral and the chemical potential of the leads lies inside the gap between HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) states. This structure demonstrates Coulomb blockade phenome- non [3,4] and Coulomb blockade oscillations of conduc- tance as a function of gate voltage (see review papers in [5] and references therein). In other words a molecule trapped in a potential well between the leads behaves as a quantum dot and the corresponding device exhibits the properties of SET. The new features in a charge transport through molecular transistors as compared to the well- studied semiconducting SET appear due to «movable» character of the molecule trapped in potential well (the middle electrode of the molecular transistor). Two qua- litatively new effects were predicted for molecular tran- sistors: (i) vibron-assisted electron tunnelling (see e.g. [6,7]) and, (ii) electron shuttling [8] (see also Rev. 9). Vibron (phonon)-assisted electron tunnelling is induc- ed by the interaction of charge density on the with dot lo- cal phonon modes (vibrons) which describe low-energy excitations of the molecule in a potential well. This inter- action leads to satellite peaks (side bands) and unusual temperature dependence of peak conductance in resonant electron tunnelling [10]. For strong electron–vibron in- teraction the exponential narrowing of level width and as a result strong suppression of electron transport (pola- ronic blockade) was predicted [10,11]. The effect of elec- tron shuttling appears at finite bias voltages when addi- tionally to electron–vibron interaction one takes into account coordinate dependence of electron tunnelling amplitude [8,9]. Recent years carbon nanotubes are considered as the most promising candidates for basic element of future © G.A. Skorobagat’ko and I.V. Krive, 2008 nanoelectronics. Both C60-based and carbon nano- tube-based molecular transistors were already realized in experiment [12,13]. The low-energy features of I–V char- acteristics measured in experiment with C60-based mo- lecular transistor [12] can be theoretically explained by the effects of vibron-assisted tunnelling [7]. It is well known that in single-wall carbon nanotubes (SWNT) electron–electron interaction is strong and the electron transport in SWNT quantum wires is described by Luttinger liquid theory. Resonant electron tunnelling through a quantum dot weakly coupled to Luttinger liquid leads for the first time was studied in Ref. 14 were a new temperature scaling of maximum conductance was pre- dicted: G T T g( ) /� �1 2 with interaction dependent expo- nent (g is the Luttinger liquid correlation parameter). In this paper we generalize the results of Refs. 10 and 14 to the case when a quantum dot with vibrational de- grees of freedom is coupled to Luttinger liquid quantum wires. The experimental realization of our model system could be, for instance, C60-based molecular transistors with SWNT quantum wires. In our model electron–electron and electron–phonon interactions can be of arbitrary strength while electron tunnelling amplitudes are assumed to be small (that is the vibrating quantum dot is weakly coupled to quantum wires). We will use master equation approach to evaluate the average current and noise power. For noninteracting electrons this approximation is valid for temperatures T �� 0, where �0 is the bare level width. For interacting electrons the validity of this approach (perturbation the- ory on �0) for high-T regime of electron transport was proved for g � 1 2/ (strong interaction) [15] and when 1 1� ��g (weak interaction) [16]. We found that at low temperatures: �0 0�� T �� (��0 is the characteristic energy of vibrons) the peak con- ductance scales with temperature accordingly to Furusaki prediction [14]: G T T T g( ) ( / )( / ) /� �� 1 1 (�� F is the Luttinger liquid cutoff energy). The influence of elec- tron–phonon interaction in low-T region results in renormalization of bare level width: � �� 0 2exp ( ), where � is the dimensionless constant of electron–phonon interaction. In the intermediate temperature region: � �� 0 2 0� �T , (� �� 1), Furusaki scaling is changed to G T T g( ) ( ) / /� �1 3 2 and at high temperatures when all in- elastic channels for electron tunnelling are open we again recovered Furusaki scaling with nonrenormalized level width (�0). For nonlinear regime of electron tunnelling we showed that zero-bias peak in differential conductance, presenting elastic tunnelling, is suppressed by Coulomb correlations in the leads. This is manifestation of the Kane–Fisher effect [14,15]. When interaction is moder- ately strong (1 2 1/ � �g ) the dependence of differential conductance on bias voltage is non-monotonous due to the presence of satellite peaks. For g � 1 2/ the zero-bias peak can be even more suppressed than the satellite peaks, which dominate in this case. This is the manifesta- tion of the interplay between the Luttinger liquid effects in the leads and the electron–phonon coupling in the dot. For strong interaction g � 1 2/ satellites are also sup- pressed and the differential conductance at low tempera- tures (T �� 0) scales as dI dV V g/ /� �1 2. This scaling coincides with the Furusaki prediction, where tempera- ture is replaced by the driving voltage (eV ) which be- comes the relevant energy scale for eV T�� , ,��0 �. It means that the influence of vibrons on the resonant elec- tron tunnelling through a vibrating quantum dot can be observed only for weak or medium strong interaction ( /1 2 1� �g ) in the leads. 2. The Model The Hamiltonian of our system (vibrating quantum dot weakly coupled to Luttinger liquid leads, see Fig. 1) con- sists of three parts � � � � � �LL QD T . (1) Here � �LL l j j L R ( ) , describes quantum wires adiabati- cally connected to electron reservoirs. Quantum wires (left-L and right-R) are supposed equal and modelled by Luttinger liquid Hamiltonians with equal Luttinger liquid parameters 1/ ( )gL R : 1 1 1/ / /g g gL R (see e.g. [14]) � � l L R l c k ka a kdk ( ) � � ��v 0 . (2) Here ak � (ak ) are the creation (annihilation) operators of bosons which describe the charge density fluctuations propagating in the leads with velocity v vc F� . These op- erators satisfy canonical bosonic commutation relations [ , ] ( )a a k kk k� � � �� . In what follows we consider for sim- plicity the case of spinless electrons. The Hamiltonian of vibrating single level quantum takes dot the form (see e.g. [10]) �QD if f b b f f b b � � �� 0 0( ) � , (3) where 0 is the energy of electron level on the dot, ��0 is the energy of vibrons, i is the electron–vibron interac- tion energy, f � ( f ) and b� (b) are fermionic ( f ) and bosonic (b) creation (annihilation) operators with canoni- cal commutation relations { , }f f � 1, [ , ]b b� 1. The tunnelling Hamiltonian is given by standard ex- pression �T j j L R t f j �� { ( ) , � h.c.}, (4) Luttinger liquid and polaronic effects in electron transport through a molecular transistor Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1087 where t j is the electron tunnelling amplitude and �( )j , j L R , is the annihilation operator of electron at the end point of L R( )-electrode. This operator could be written in a «bosonised» form (according to [14]) �( ( )) exp ( ) / L R dk K k a a k k k � � � � � � � � � � � � �2 2 2 0 �� e , (5) here � is a short-distance cutoff of the order of the recip- rocal of the Fermi wave number k F and K g� � �( / )2 1 1 is the interaction parameter in the «fermionic» form of the Luttinger liquid Hamiltonian (2), it defines the Luttinger liquid parameter g which is varied between 0 and 1: the case g 1 describes the «noninteracting» (Fermi-liquid) leads, than in the case g � 0 the interaction in the leads goes to infinity. Hamiltonian (3) is «diagonalized» to �d P f f �� 0b b by the unitary transformation (see e.g. [17]) U i pn f exp ( )� , where p i b b ��( ) / 2, n f ff � and the dimensionless parameter � � � 2 0i / � character- izes electron–vibron coupling. The unitary transforma- tion results in: (i) the shift of fermionic level (polaronic shift) �P i �0 2 0/ � and (ii) the replacement of tun- nelling amplitude in (3) t t i pj j� � �exp ( )� . The model Eqs. (1)–(5) can not be solved exactly and one needs to exploit certain approximations to go further. We will use «master equation» approximation (see e.g. [5]) to evaluate the average current and noise power in our model. It is in this approximation that average current separately for the model with interacting leads [14] and for vibrating quantum dot with noninteracting leads [18] was calculated earlier. Master (rate) equation approach exploits such quantities as the probability for electron to occupy level dot and the transition rates. It neglects quan- tum interference in electron tunnelling and therefore de- scribes only the regime of sequential electron tunnelling which is valid when the width of electron level �0 �� min ,( )T eV . In other words, in our case «master equation» approach is equivalent to the lowest order of perturbation theory in �0. For interacting electrons the validity of master equa- tion approach for high-T regime of resonant electron tun- nelling can be justified for strong repulsive interaction g � 1 2/ [14]. It is correct also for weak interaction 1 1� ��g as one can check by comparing the results of Refs. 14 and 16, where resonant tunnelling through a dou- ble-barrier Luttinger liquid was considered for weak elec- tron–electron interaction. Notice, that the results [18] of exact solution known for g 1 2/ , where a mapping to free-fermion theory can be used [5], do not agree with the high-T scaling of G T( ) [14] extrapolated to this special point g 1 2/ . The free-fermion scaling G T T( ) � �1 found for g 1 2/ (master equation approach predicts T -inde- pendent value [14]) could be a special feature of this ex- actly solvable case. We will assume that beyond the close vicinity to g 1 2/ the master equation approach for high-T behavior of conductance is a reasonable approxi- mation. 3. Transition rates and the average current In master equation approach the average current through a single level quantum expressed dot in terms of transition rates takes the form I e R L L R �� 01 10 01 10 , (6) where � 01 R L( ) is the rate of electron tunnelling from the to dot right (left) electrode, � 10 R L( ) describes the reverse pro- cess and � � �� 01 10, � � �if if L if R � ( , , )i f 0 1 . To 1088 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 G.A. Skorobagat’ko and I.V. Krive �t �t �QD �l�l �l�l –eV/2 eV/2 ��0 –��0 p Vg Fig. 1. The schematic picture of the two-terminal electron transport through a vibrating quantum weakly dot coupled (via narrow dielectric regions Ht ) to the Luttinger liquid leads ( ( ) � � l L R l ) with the chemical potentials F eV / 2 (V is the driving voltage). All the energies are counted from the Fermi energy, which chosen to be zero. Electrons tunnel from one lead to another by hopping on and off the level dot with the en- ergy P (elastic channel) and due to electron–vibron coupling they can emit or absorb vibrons (vibron–assisted tunnelling). Inelastic channels are represented as side-levels with energies �P P � 0. The position of the levels dot with respect to the Fermi energy can be uniformly shifted by applying voltage Vg to the «gate» electrode. evaluate these rates in our approach we will use Fermi «Golden Rule» (quantum mechanical perturbation the- ory) for tunnelling Hamiltonian obtained from Eq. (4) af- ter the unitary transformation: � �T t! �t j j L R t j f i p � � �� { ( ) exp ( ) , � � h.c.} . (7) The standard calculation procedure results in the follow- ing expressions for tunnelling rates � � � 10 2 0 0 ( ) ( ) ( ) ( ) ( ) e j j b j j f t dt V t V t " " " " " " # $ # $ �� xp ( ( ) / )i eV tP F j � � � , (8) � � � 01 2 0 0 ( ) ( ) ( ) ( ) ( ) e j j b j j f t dt V t V t " " " " " " # $ # $ �� xp ( ( ) / )� � �i eV tP F j � , (9) where V V VL R� is the bias voltage and j L R , . Notice that in the perturbation calculation on the bare level width �0 2� \| \|,t L R , we neglect the level width in the Green func- tion of the level dot. Besides, in this approximation aver- ages over bosonic and fermionic operators in formulas for tunnelling rates are factorized and, thus, the averages # $� b over bosonic variables V i p � �exp ( )� , p i b b �� 2 [ ] (10) can be calculated with the quadratic Hamiltonian �b b b � ��0 . In what follows we will assume that vib- rons are characterized by equilibrium distribution func- tion: n T Tb ( ) [ exp ( / ) ] � � ��0 11 . Averages # $� f over fermionic operators in Eqs. (8), (9) are calculated with the Luttinger liquid Hamiltonian (2). The corresponding cor- relation functions in Eqs. (8), (9) are well known in the literature (see e.g. [10,14]) # $ � � %�V t V nb B( ) ( ) exp ( ( ))0 1 22� % � � � & ' ( )( * + ( , �� I n n il t i Tl l B B[ ( )] exp [ ( / )]2 1 22 0� � � ( , (11) # $ � �� & ' ) * + , � � � � � � j j f F g t i T Tt ( ) ( ) / 0 1 � � �v � � sinh . (12) Here I zl ( ) is a modified Bessel function, � � F is a ultra- violet cutoff energy, g is the Luttinger liquid correlation parameter. By putting correlation functions (11),(12) in Eqs. (8), (9) and evaluating time integrals we get the following equations for tunnelling rates � �� ( ) ( )j j 10 and �� ( )j �- . 01 j , ( j L R , ) � � � / � 0 1 2 3 4 5 � ( ) / exp [ ( / ) /j j g jT T 2 2 2 21 1 2 0 � � � � � �coth T g ] ( / )� 1 % % � � � � � � � � � � � � I T g i l T l l j — ( / ) ( )� � � � 2 0 0 2 1 2 2sinh � � � /0 1 2 2 3 4 5 5 " " " " " " & ' ( )( * + ( ,( 2 , (13) where �L R( ) is the partial level width (see, for example, [ 1 4 ] ) � �j j jc t f f # $ �( / )2 2� � const ( j L R , ) , �L � � � �R 0, / j F P jeV � � ; here �( )z is Gamma function. At first we consider different limiting cases when it is possible to obtain simple analytical expressions for the average current (6). Notice, that electric current depends on the gate voltageVg through the corresponding depend- ence of level energy P gV( ). It is convenient for the fur- ther analysis to choose the value of gate voltage at which the current at low bias is maximum as: P g FV( ) . In what follows we also put V V VL R � / 2. For noninteracting leads ( )g 1 and noninteracting quantum dot ( )� 1 it is easy to derive from Eqs. (6), (13) the well-known formula for the maximum (resonant) cur- rent at temperatures T L R�� ( ) I V e eV T ( ) � � � tanh 4 0 1 2 3 4 5 , (14) where � � � � � �L R L R/ ( ) is the effective level width. It is evident that at high voltages: eV T�� the current through a single level dot does not depend on the bias voltage and its value is totally determined by the effective level width �. For a vibrating quantum dot ( )� 6 0 weakly coupled to noninteracting leads ( )g 1 our approach reproduces the results of Ref. 10. In the temperature region we are inter- esting in ( )( )T L R�� the general formulae derived in [10] can be rewritten in a more clear and compact form. In particular, by using for g 1 in Eq. (11) the well-known representation for Gamma function (see e.g. [19]) �( ) cosh ( ) 1 2 2 �" " " " " " iz z � � , (15) Luttinger liquid and polaronic effects in electron transport through a molecular transistor Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1089 it is easy to obtain the following expression for the maxi- mum (peak) conductance G T G T F T � � � ( ) ( ) 0 1 2 3 4 5 � 0 , (16) where G T G T ( ) � � 2 0 � , G e h 0 2 (17) is the standard resonance conductance of a single-level quantum at dot T L R�� ( ). The dimensionless function F x�( ) takes the form F x n xB� �( ) exp [ ( ( ))] � � %2 1 % � & ' ( )( * + ( ,( � I z x I z x lx l l 0 2 1 2 2 ( ( )) ( ( )) cosh ( / ) , (18) h e r e z x n x n xB B( ) ( )[ ( )]7 �2 12� a n d n xB ( ) � �( exp ( ) )x 1 1. At low temperature region � �� T �� 0, when there are no thermally activated vibrons in the dot ( )nB �� 1 only the first term in the brackets con- tribute to the sum and: F T� � �( ) exp ( )�� 0 2 � . We see, that zero-point fluctuations of the position dot result in renormalization of the level width � �� exp ( )2 . For strong electron–vibron coupling this phenomenon (polaronic narrowing of level width) leads to polaronic (Franck–Condon) blockade of electron transport through vibrating quantum dot [11]. The temperature behavior of peak conductance (16) was considered in Ref. 20. Now we will study the general case when interacting quantum dot ( )� 6 0 is connected to interacting leads ( )g � 1 . Analytical expressions for conductance in this case can be obtained in the limits of low ( ( )�L R �� T ��0 ) and high ( )T �� 0 temperatures. At low temperatures the main contribution to the sum over «l» in Eq. (13) comes from elastic transition l 0. All inelastic channels ( )l 6 0 are exponentially suppressed for eV , T �� 0. At T g�� 0 the peak conductance takes the form G T G T g g T g ( ) ( / ) ( / / ) / � � �� 2 1 2 1 2 1 2 0 1 1� � � �� 0 1 2 3 4 5 � . (19) We see that at low temperatures conductance scales with temperature according to Furusaki’s prediction [14]: G T T g( ) /� �1 2. The influence of electron–vibron coupling results in multiplicative renormalization of bare level width � �� exp ( )2 . At high temperatures: T �� 0 one can use the well known asymptotic expansion for Bessel function I z z zl ( ) exp ( ) /� 2� , which can be used in summation Eqs. (18), (13) until l z2 � . Besides, in this temperature region the summation in Eq. (13), can be replaced by inte- gration and the corresponding integral can be taken exactly \| ( )\| ( ) �� a iz dz aa2 1 22 2� . (20) This allows us to derive the following expression for the temperature dependence of peak conductance in the inter- mediate temperature region � �� 0 2 0�� T , ( )� 8 1 G T G T T T g ( ) exp ( / ) / � � � � � � � 2 4 0 2 0 0 1� � �� 0 1 2 3 4 5 � � � 1 . (21) Notice that in the considered temperature region the polaronic blockade is already partially lifted ��( )T � �� exp ( / )� �2 0 4� T at T � � �2 0� and conductance scales with temperature as G T T g( ) / /� �1 3 2. At last, at temperatures T �� 2 0� when all inelastic channels for electron transport are open, the polaronic blockade is to- tally lifted [20] and we reproduce again Furusaki scaling. It is clear from our asymptotic formulae (19),(21) that both in low- and in high- temperature regions the contri- butions of electron–electron and electron–vibron interac- tions to the conductance are factorized. In general case these contributions are not factorized, as one can see from Eqs. (8), (9) and from Eq. (13) for tunnelling rates, and we can expect interplay of Kane–Fisher effect and the ef- fect of phonon(vibron)-assisted tunnelling. To see this interplay we consider nonlinear (differen- tial) conductance G V dI dV( ) / . It is well known that Kane–Fisher effect is pronounced for the energies close to the Fermi energy. For differential conductance it means that the zero-bias (elastic) resonance peak is suppressed with the increase of electron–electron interaction, while satellite peaks are less affected by the interaction. When electron–electron interaction is weak or moderately strong ( / )1 2 1� �g the dependence of differential con- ductance on the bias voltage (for � � 1) is not a monoto- nous function due to the presence of satellite peaks (see Fig. 2,a,b). The resonant behavior disappears for strong interaction g � 1 2/ (Fig. 2,b), when at low temperatures T �� 0 differential conductance scales with bias volt- age as G V V g( ) /� �1 2 in accordance with the Luttinger liquid prediction for nonlinear electron transport through a single-level quantum dot. For instance, if we put 1 2 3/ , ( , , )g n n � and tune the level energy to the reso- nance point � �0 2 0 � — («resonant» location of the level in the presence of «polaronic» shift), we obtain for the differential conductance G V( ) the following expres- sion for eV g/ ( / )��0 1 2�� G V G n eV n ( ) ( )! � 4 1 1 2 0 2 � �� 0 1 2 3 4 5 � , (22) 1090 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 G.A. Skorobagat’ko and I.V. Krive where � �� exp ( )2 . One can readily see that expres- sion (22) reproduces Furusaki temperature scaling Eq. (19) when eV is replaced by T . Analogous interplay of Kane–Fisher and polaronic ef- fects one can see in Fig. 3,a,b, where differential conduc- tance is plotted as a function of level energy 0 (or, equiv- alently, as a function of gate voltage). For noninteracting leads ( )g 1 the resonance conductance peaks correspond to the level positions at P F eV / 2 (in our plot we put: eV 5 0�� ). This elastic resonance peak is sup- pressed by electron–electron interaction in the leads ( )g � 1 . The dependence G Vg( ) for weak and moderately strong interaction still reveals resonance structure with 4 satellites in our case, Fig. 3,a. The inelastic resonance peaks disappear at g � 1 2/ and maximum of differential conductance corresponds at g �� 1 to the level position at P g FV( ) , that is exactly in the middle between chemi- cal potentials of left and right electrodes (Fig. 3,b). It is important to stress here once more that for moder- ately strong electron–electron interaction in the leads the inelastic tunnelling can dominate in electron transport. One can see from Figs. 2,3 that there is region of coupling constants when the first satellite peak is higher than the «main» (zero-bias) resonant peak, which corresponds to elastic (l 0) tunnelling channel. It is the most significant prediction, we have made in this paper. 4. The noise power The knowledge of tunnelling rates Eqs. (8), (9) allows us to evaluate not only the average current Eq. (6) but the noise power as well. We will follow the method devel- oped in Refs. 21 and 22 where quantum noise was calcu- lated for resonant electron transport through a quantum weakly dot coupled to noninteracting electrodes. The noise power is defined (see e.g. [23]) as S dt i t I t I( ) exp ( ) ( ) ( )� � # $ �� 2 0/ / , (23) Luttinger liquid and polaronic effects in electron transport through a molecular transistor Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1091 1 d I G 0 d V 1 d I G 0 d V 0.06 0.05 0.04 0.03 0.02 0.01 0 0 0 0.5 0.5 1.0 1.0 1.5 1.5 2.0 2.0 –0.01 (eV/2)/��0 (eV/2)/��0 a b 0.04 0.03 0.02 0.01 0.05 0.06 Fig. 2. Differential conductance (in the units of G0) as a func- tion of driving voltage (in the units of ��0). Here we put � / kT 0.01; ��0 10/ kT ; �2 1 ; and tune the level energy to the resonant position � �0 2 0 � ( P 0). Solid lines corre- spond to the case of noninteracting leads g 1 (a) and (b); dot line (g 0.8), dash-line dot (g 0.6) (a), dot line (g 0.45), dash-dot (g 0.25) (b). Zero-bias (elastic) resonance peak is gradually suppressed with the increase of electron–electron correlations (decrease of Luttinger liquid parameter g) while the satellite peaks survive until g �1 2/ (a). For g � 1 2/ the res- onance-like behavior of differential conductance disappears and conductance scales as a power-law of the bias voltage (b). 0.05 0.04 0.03 0.02 0.01 0 0 0 0/�� 0 0/�� a b 0.06 0.05 0.04 0.03 0.02 0.01 0 0 1 1 2 2 3 3 4 4–2 –1 –2 –1 1 d I G 0 d V 1 d I G 0 d V Fig. 3. Differential conductance (in the units of G0) as a func- tion of level energy 0, counted from the Fermi energy. The bias voltage eV / ��0 5 is sufficiently high to excite vibrons and to support electron transport through inelastic channels. All parameters are the same as for Fig. 2,a and b, correspondingly. where /I t I t I( ) ( ) � (I is the average current). The noise defined in Eq. (23), in the case of sequential tunnelling through a quantum dot, can be expressed in terms of tun- nelling rates. For a single level quantum this dot formula for low frequency noise S S ( )� 0 takes the form S eI I e L R � �2 4 4 2 2 01 10 � � � �� , (24) here the average current I is determined by Eq. (6). The noise power Eq. (24) depends on temperature and bias vol tage S T V( , ) and conta ins both thermal (Jon- son–Nyquist) noise S T S T V TG TJN ( ) ( , ) ( )7 0 4 (G is the conductance) and the nonequilibrium (shot) noise S T Vsh ( , ). Since the thermal noise is totally determined by temperature dependence of conductance, we will study in what follows only shot noise and Fano factor F S eIsh / 2 . In particular, Fano factor in our case can be represented as follows F I e e I TG e L R � � � 0 1 2 2 3 4 5 5 & ' ( )( * + ( ,( 1 2 2 01 10 2� � � �� . (25) For noninteracting leads ( )g 1 and noninteracting quan- tum dot ( )� 0 one readily gets from Eqs. (13), (24) a sim- ple expression for the «full» noise ( )S of a single electron transistor (SET). On resonance P g FV( ) and at tem- peratures T �� one finds S e eV T eV T 0 1 2 3 4 5 � 0 1 2 3 4 5 � � � � � � � & ' 2 4 1 2 4 2� � �� tanh tanh ( )( � �� * + ( ,( e eV T eV T 2 2 4 4 � � exp ( / ) cosh ( / ) . (26) From Eq. (26) in the limit V � 0 we obtain S S TJN ( ), where S T e TGJN ( ) / 2 4� � is the thermal noise. In the opposite case eV T�� we rederive the well-known for- mulae for the shot-noise and the Fano factor of a single level quantum dot [21–23] S e sh � 0 1 22 3 4 55 2 1 22� � �� , F �1 2� �� . (27) These formulae (26), (27) can be also re-derived from the general expression for the full noise of noninteracting electrons (see e.g., Eq. (61) in Ref. 23) S V T e d T f f f f T Tt L L R R t t( , ) { ( )[ ( ) ( )] ( )[ ( � � � � �� 2 1 1 1 � )]( ) }f fL R� 2 , (28) where Tt ( ) i s the t ransmiss ion coeff ic ien t and f Tj j � � �{exp [( ) / ] } 9 1 1 is the equilibrium distribu- tion function of electrons in the leads (9 j is the chemical potential; j L R , ). In the case of single level quantum dot Tt ( ) takes the form Breit–Wigner tunnelling probability Tt L R P L R ( ) ( ) ( ) / � � � � � � �2 2 4 . (29) For a weak tunnelling when �L R( ) are the smallest energy scales in the problem the Lorentzian shape of the Breit–Wigner resonance shrinks to �-function Tt PL R ( )\| ( — ) , � � � ��0 2� . (30) With the help of Eqs. (28),(30) for the resonance condi- tion P g FV( ) we easily re-derive Eq. (26). (Notice, that in sequential tunnelling approach the tunnelling tran- sitions through the left and right barriers are assumed to be weak and uncorrelated. Therefore we can safely ne- glect Tt 2-term in Eq. (28).) It is evident from Eqs. (25), (27) that for noninteracting electrons the Fano factor is sub-Poissonian ( )F � 1 and F approaches 1 for strongly asymmetric junction � �L R R L( ) ( )�� and for eV T�� . The master equation approach we have used in our analysis holds when electron tunnelling amplitudes are small. For noninteracting electrons this assumption is sat- isfied when electron energies are far from the resonant en- ergy level, i.e. Tt ( ) �� 1. The differential shot noise in this case as a function of bias voltage or gate voltage be- haves similarly to the differential conductance. Notice however that due to different dependence on temperature the shot noise unlike the thermal one even in sequential tunnelling regime (T �� ) can not be expressed in terms of conductance. By comparing Fig. 4,a and Fig. 3,b, one can see that the above similarity is preserved for interacting electrons (g 6 1, � 6 0) as well. The corresponding Fano factor which is the «shot noise/current» ratio and thus is less sensitive to the details of tunnelling process, for strong electron–electron interaction exhibits a simple behavior (see Fig. 4,b). It dips (F � 1 2/ ) at symmetric (with respect to chemical potentials of the leads) position of the level dot. Outside this region F � 1 (Poissonian noise). The width of the dip decreases with the increase of interac- tion, Fig. 4,b. 1092 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 G.A. Skorobagat’ko and I.V. Krive 5. Summary We considered the influence of interaction on trans- port properties of molecular transistor which was modell- ed as a vibrating single-level quantum dot weakly coupl- ed to the Luttinger liquid leads. We found interesting interplay between polaronic and Luttinger liquid effects in our system. In particular it was shown that for weak or moderately strong interaction (1 2 1/ � �g ) the differential conductance demonstrates resonance-like behavior and for moderately strong interaction inelastic channels can even dominate in electron transport through a vibrating quantum dot. For strong interaction (g �� 1) the resonant character of vibron-assisted tunnelling disappears and the differential conductance scales as a power law on temper- ature (linear regime T eV�� ) or on bias voltage (nonlin- ear regime eV T�� ). The authors would like to thank S.I. Kulinich for valu- able discussions. This work was partly supported by the joint grant of the Ministries of Education and Science in Ukraine and Israel and by the grant «Effects of electronic, magnetic and elastic properties in strongly inhomoge- neous nanostructures» of the National Academy of Sci- ences of Ukraine. 1. A. Nitzan and M.A. Ratner, Science 300, 1384 (2003). 2. M. Galperin, M.A. Ratner, and A. Nitzan, J. 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Luttinger liquid and polaronic effects in electron transport through a molecular transistor Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1093 0.05 0.06 0.04 0.03 0.02 0.01 0 0 0/�� 0 0/�� a b 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0 0 1 1 2 2 3 3 4 4–2 –1 –2 –1 F 1 d S sh eG 0 d V Fig. 4. Differential shot noise power (in the units of eG0) (a) and Fano factor as functions of the level energy 0 in the non- linear transport regime eV / ��0 5 (b). Other parameters are the same as on Fig. 2,b.

Luttinger liquid and polaronic effects in electron transport through a molecular transistor

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