Shell effects in nonlinear magnetotransport through small quantum dots

Shell effects in nonlinear magnetotransport through small quantum dots

I. SandalovDepartment of Condensed Matter Physics, Royal Institute of Technology, Electrum 229, SE-164 40 Stockholm-Kista, Sweden

and Max-Planck-Institut für Physik komplexer Systeme, D-01187 Dresden, Germany

R. G. NazmitdinovDepartament de Física, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain;

Max-Planck-Institut für Physik komplexer Systeme, D-01187 Dresden, Germany;and Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia

�Received 6 December 2005; revised manuscript received 30 June 2006; published 9 February 2007�

Nonlinear transport through a quantum dot attached to two metallic contacts is studied in the limit of a weakand a strong intradot Coulomb interaction. The nonequilibrium self-consistent mean-field equations for theenergies and spectral weights of one-electron transitions are formulated in the strong-interaction regime. Theself-consistent solutions result in a decrease of bias voltage threshold for nonzero current, contrary to the weakcase. With an increase of the bias voltage window, the solutions lead to a significant deviation from the Gibbsstatistics: the populations of states involved in tunneling are equalizing even at low temperatures. For asymmetric dot-contact coupling we obtain simple analytical relations between the heights of the current stepsand degeneracies of a spectrum of a circular dot in a perpendicular magnetic field, in both regimes.

DOI: 10.1103/PhysRevB.75.075315 PACS number�s�: 73.23.�b, 73.21.La, 75.75.�a

I. INTRODUCTION

Transport measurements provide rich information on theinternal dynamics of quantum dots �QD’s�. Controlling gateand bias voltages, at different dot-contact couplings, experi-ments enable one to display various degrees of interplay be-tween electron correlations and size quantization effects atthe atomic scale.1,2 At low temperatures, the conductanceusually shows a pronounced resonant behavior as a functionof gate voltage, indicating that electrons tunnel through a dotvia discrete states.

At weak dot-contact coupling, according to the theory ofsequential tunneling,3,4 the intradot electron correlations aredominant in the tunneling. In the Coulomb-blockade regime,the charging energy EC, which is needed to add an electronto a dot with N electrons, considerably exceeds an averagelevel spacing �= ��−�� is the energy level of the ini-tially closed dot� and a thermal energy T �T��EC�. At thesame time, T and � are much larger than the width � thatdetermines the lifetime of the level.2 At lower temperature��T��EC�, it is predicted5 that a crossover from se-quential tunneling to cotunneling over high-lying, correlatedstates takes place. With a further decrease of temperature,Kondo-type effects can develop if a dot has quasidegeneratelocalized states.6 Note that numerous papers devoted toquantum effects in transport are focused on the analysis ofKondo phenomena in single-level QD models7 or are re-stricted by a linear-response regime, which is well under-stood nowadays.

Recently, in experiments with small QD’s in the nonlinearregime,8 a fine structure was observed in the conductance inthe Coulomb-blockade valley. It was suggested that this phe-nomenon is mainly due to cotunneling.5 The theory ofcotunneling5 �as well as the theory of sequential tunneling3,4�neglects, however, specific quantum effects caused by astrong Coulomb interaction �SCI�, which affects the transi-

tion energies and the tunneling rates �cf. Ref. 9�. One of ourgoals is to present a self-consistent approach to a nonlineartransport through a multilevel quantum dot in the SCI re-gime, which takes these effects into account. Treating thedot-contact coupling perturbatively at ��T��EC, simi-lar to Refs. 3 and 4, we will show that the conductance insideof the “Coulomb diamond” is governed by spectral weightsof one-electron transition energies. The transition energiesare renormalized due to the coupling as well as the weights.It turns out that a fine structure may appear even at the se-quential tunneling due to the renormalization and shell struc-ture of the dot.

To elucidate shell effects in the SCI regime, we studytransport through a dot with parabolic confinement, under aperpendicular magnetic field. An approximation of the con-finement by a parabolic potential is well justified by experi-mental observations of the magic numbers of a two-dimensional harmonic oscillator10 and a fulfillment of theKohn theorem11 in far-infrared spectroscopy experiments.12

When the confinement dominates over the Coulomb interac-tion, the electron spectrum is well approximated by Fock-Darwin levels13 �we denote such states as ��, which areeigenstates of the two-dimensional harmonic oscillator undera perpendicular magnetic field. Indeed, recent single-electronspectroscopy experiments nicely confirm this fact.14 Hereaf-ter, this regime is called the weak-Coulomb-interaction�WCI� regime. In the limit of a very strong intradot interac-tion EC→ �the SCI regime� we consider a transport win-dow, in which only transitions between empty �0� and single-electron states �� contribute to the transport. These states areseparated by a large energy gap from two-electron, three-electron, etc., correlated states. The existence of such a gapcan be easily verified within an analytical model of two-electron quantum dots.15 Note that the bare energies of thedot coincide in the two regimes. This fact presents a niceopportunity to study the effect of the Coulomb interactionstrength on the transport, within this particular charge sector.

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In the SCI regime, we employ the Hubbard operatortechnique16 which is well suited for the analysis in this case.Our approach is similar in spirit �but different in details� tothe one developed by Schoeller and Schön.17 The model con-sidered by them is restricted, however, by a two-charge stateapproximation �shell phenomenon is neglected�, and effectsproduced by a magnetic field are not analyzed.

The paper is organized as follows: in Sec. II we discussour model Hamiltonian, a nonlinear current in terms of theHubbard operators, and formulate a mean-field approxima-tion. Section III is devoted to a derivation of equations forpopulation numbers and energy shifts in the diagonal ap-proximation. The results for magnetotransport through a cir-cular quantum dot in the WCI and SCI regimes are discussedin Sec. IV. The conclusions are summarized in Sec. V. Therelations between the fermions and Hubbard operators aredescribed in the Appendix. Preliminary results of our analy-sis have been presented in Ref. 18.

II. FORMALISM

A. Model Hamiltonian and current

One of simplest quantum transport devices consists of twometallic contacts attached via insulating-vacuum barriers tothe dot. Such a system can be modeled by the followingHamiltonian:

H = Hl + Hr + HQD + Ht, �1�

where

H = �k��

�k�ck�† ck�,

HQD = ��

��n� + ��

U��n�n��, n� = d�†d�,

Ht = �k��,,�

�vk�,0� ck�

† d� + H.c.� .

The term H describes noninteracting electrons with energy�k�, wave number k, and spin � in the left/right contact�=l /r�. The Hamiltonian of the closed dot is HQD, where ��

is a single-electron energy of a state ��. Tunneling betweenthe dot and the contacts is described by the term Ht; matrixelements vk�,0�

couple the left and right contacts with thedot. A ratio of the energy-level separation � for noninteract-ing electrons in the dot to the average Coulomb energy�“Hubbard U”� determines whether the intradot electrons arein a regime of �i� the weak �� /U�1� or �ii� the strong�� /U�1� Coulomb interaction.

The dot Hamiltonian HQD is diagonalized by means of theHubbard operators19 �see the Appendix�. As a result, theHamiltonian �1� acquires the form

H = �k�

�k�ck�† ck� + �0Z00 + �

�

��Z��

+ �k�,�

�vk�,� ck�

† X0� + H.c.� , �2�

where the matrix elements vk�,� are determined by Eq.

�A11�. Following Refs. 20 and 21 and using Eq. �A7�, weobtain for the “left” steady current

Jl =ie

��

� d�

2��

l �� − ��r ��G�,��

� ��

+ ��l ��f l�� − ��

r ��fr��

��G�,��R �� − G�,��

A �� . �3�

Here f l�fr� is a standard Fermi function defined with respectof the chemical potential �l��r� of the left �right� contact.The current is expressed in terms of Green functions �GF’s�defined on the Hubbard operators:

G�,�� =� d�t − t��ei��t−t��G�,��

� �t,t�� ,

G�,�� t,t�� i�X��0�t��X0��t�� ,

G�,�� t,t�� − i�X0��t�X��0�t�� ,

G�,��R/A �t,t�� i��±t � t��X0��t�,X��0�t�� , �4�

where �= �,� ,R ,A. The GF’s depend on a time difference �orone frequency ��, since we consider the steady current only.As is well known, the Dyson equation does not exist for anarbitrary many-electron GF and, in particularly, for the GF’sdefined on the Hubbard operators. For the latter case the�anti�commutator of many-electron operators �see Eq. �A12�generates an operator again. As a result, the perturbationtheory for the many-electron GF’s generates additionalgraphs, in contrast to the conventional theory for fermions orbosons �examples are given in Ref. 16; see also Sec. II B�.Therefore, the results of Refs. 20 and 21, based on the Dysonequation for the retarded and advanced GF’s, should be re-inspected. Below we show that results, similar to those ofRefs. 20 and 21, are valid only within the well-knownHubbard-I and mean-field approximations16 �MFA� �Ref. 16will be referred below as I�. The MFA is, however, sufficientfor the analysis of transport properties at small transparenciesof the junctions.

Note that for a wide conduction band—i.e., when thebandwidth is much larger than other parameters in theproblem—the width function

�� = ��

k�v��,k� �� − �k��vk�,�

�5�

depends on � weakly. It is convenient to introduce a fullwidth � and dimensionless partial widths �:

�� = ��l + ��

r , �� = ��

/��. �6�

At � /��1, the level mixing due to nondiagonal terms doesnot influence the physics of transport �see also the discussionin Refs. 3 and 4�. Neglecting the nondiagonal terms, oneobtains a diagonal approximation ��=��,��, ��

=��,��. It yields the following expression for the current:

I. SANDALOV AND R. G. NAZMITDINOV PHYSICAL REVIEW B 75, 075315 �2007�

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J =ie

�� d�

2��

l − ��r G�,�

� �� + ��l f l�� − ��

r fr��

��G�,�R �� − G�,�

A �� . �7�

We recall that this result can be obtained only for noninter-acting conduction electrons. Our next step is to derive equa-tions for GF’s G�,�

�,R,A in the MFA.

B. Exact equations on the imaginary-time axis

Our derivation is based on an approach9,16 that employsideas suggested by Martin and Schwinger22 and developedfurther by Kadanoff and Baym.23 Since we consider only theintradot Coulomb interaction, the subsystems with conduc-tion electrons �contacts� are linear and can be integrated out.Following Ref. 23, equations for the dot will be written interms of functional derivatives with respect to external aux-iliary sources. We progress in three steps. First, we deriveexact equations for the dot GF’s on the imaginary-time axis.Second, the MFA is formulated. Third, we perform a con-tinuation of these equations on the real-time axis.

The time evolution of the Hubbard operator X0� is definedby the commutator

�X0�,H = ��0�0�X0� + �

k��,,�1

v�1,k� ��1�

Z00 + Z�1��ck�,

�8�

which contains nonlinear terms. Here, ��0�0��

�0�=��−�0 isthe transition energy between the single-electron state andthe one without electrons in the dot. We choose �0=0. Thenonlinear terms produce GF’s of the type −i�T��Z00�t�+Z��t�ck��t�ck��

† �t��. Below we omit summation signs:a summation over repeating indices is implied unless other-wise stated. To proceed further, we introduce the auxiliarysources U00�t� and U��t� into a definition of the GF’s:

GAB�t,t�� =1

i�TA�t�B�t��U =

1

i

�TA�t�B�t��t0,��T��t0,��

,

��t0,�� T exp − i�t0

t0−i�

dt1�U00�t1�Z00�t1�

+ U��t1�Z��t1�� , �9�

where A ,B=cp ,X0�, respectively. Hereafter, we use the fol-lowing notation:

�Cp,p��t,t�� Gp,��t,t��

G�,p��t,t�� G�,��t,t��

=1

i � �Tcp�t�cp�† �t��U �Tcp�t�X��0�t��U

�TX0��t�cp�† �t��U �TX0��t�X��0�t��U

� , �10�

where a composite index p= �k� is introduced.Equations of motion for the GF’s Cp,p��t , t��, Gp,��t , t��,

and G�,p��t , t�� are simple:

Cp,p��t,t�� = Cp�0��t,t��p,p� +� dt1Cp

�0��t,t1�vp,��G��,p��t1,t�� ,

�11�

Gp,��t,t�� =� dt1Cp�0��t,t1�vp,�2

G�2,��t1,t�� , �12�

G�,p��t,t�� =� dt1G�,�1�t,t1�v�1,p�Cp�

�0��t1,t�� , �13�

where a notation vp,��vk�,� is introduced and a zero GF Cp

�0�

satisfies the equation

�i�t − �pCp,p��0� �t,t�� p,p�

��t − t�� . �14�

Hereafter, the integrals are taken between t0 and t0− i�, if theother limits are not specified explicitly; t0 is an arbitrarymoment of time.

The equation for G�,��t , t�� has, however, a complexform:

��1,�i�t − ��1

�0� �t�G�1,��t,t��

= ��t − t��P��t� + v�1,p1

i�T��1�

Z00 + Z�1��

�cp�t�X��0�t��U, �15�

where

P��t� � �T�X0��t�,X��0�t��U, �16�

��1

�0� �t� = ��0��,�1

+ U��1�t� − ��,�1U00�t� . �17�

The term (U��t�−��U00�t�) is due to the differentiation of��t0 ,��.23

We use below a standard assumption that the interaction isabsent at infinitely remote time t0→− and is switched onadiabatically. This assumption does not affect our approach,since we study a stationary regime. In this case the relation

�TZ��t�X0��t�B�t��U = ��TZ��t�� + i�

�U��t��TX0��t�B�t��U

�18�

is useful, which follows from a functional derivative� /�U��t� of Eq. �9� �here �= �00 , ��. With the aid ofEqs. �12� and �18� and a definition of the effective interactionvia conduction electrons,

V�1�2�t,t1� = v�1,pCp

�0��t,t1�vp,�2, �19�

we transform Eq. �15� into the form

SHELL EFFECTS IN NONLINEAR MAGNETOTRANSPORT… PHYSICAL REVIEW B 75, 075315 �2007�

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��,�1i�t − ��1

�0� �t�G�1,��t,t��

= ��t − t��P��t� +� dt1V�1�2�t,t1�

��P��1�t� + i��,�1

�

�U00�t�+ i

�

�U�1��t��G�2,��t1,t�� ,

�20�

where we used also that �Cp�0� /�U=0. Equation �20� is exact

on the imaginary-time axis. Since all other GF’s are ex-pressed in terms of G�,��, the iteration of Eq. �20� with re-spect to the effective interaction generates a full perturbationtheory. Within this formulation, continuation from theimaginary-time axis to the real one should be performed ineach term of the expansion.

The well-known Hubbard-I �HI� approximation can beobtained from Eq. �20�, if we put �G�2,�� /�U=0. Let us de-fine a zero locator D�1,�2

�0� �t1 , t��:

� dt1��,�1i�t − ��1

�0� �t��t − t1�D�1,�2

�0� �t1,t��

� � dt1�D�0�−1�t,t1��,�1D�1,�2

�0� �t1,t�� = ��t − t��,�2.

�21�

In the HI approximation, the locator is determined by a so-lution to the equation

� dt1��D�0�−1�t,t1��,�1− S�,�1

HI �t,t1�D�1,�3

HI �t1,t��

= ��,�3��t − t�� , �22�

where

S�,�1

HI �t,t1� = P��2�t�V�2�1�t,t1� . �23�

Using the above equations, we rewrite Eq. �20� in the form

� dt1��DHI,−1�t,t1��,�1− S̃�,�1

�t,t1�G�1,��t1,t��

= ��t − t��P��t�� , �24�

with

� dt1S̃�,�1�t,t1�G�1,��t1,t��

� i� dt1V�1�2�t,t1��,�1

�G�2,��t1,t��

�U00�t�

+�G�2,��t1,t��

�U�1��t�� . �25�

This equation suggests that the full locator D can be definedby the equation

� dt1�D�t,t1�−1�,�1G�1,��t1,t�� = ��t − t��P��t�� ,

�26�

where

�D�t,t1�−1�,�1� �D�0�−1�t,t1��,�1

− S�,�1�t,t1� ,

and the full self-operator is defined as

S�,�1= S�,�1

HI + S̃�,�1. �27�

The magnitude S�,�1has been named in I as a “self-operator”

in order to distinguish it from the standard self-energy opera-tor, since it renormalizes the transition energies in the dot aswell as the end factors P��t�. Indeed, one can see from Eq.�26� that

G�1,��t,t�� = D�1,�2�t,t��P�2��t�� . �28�

As a result, the term P�2��t�� enters into equations for theGF’s under the sign of the functional derivative. Being anexpectation value of the anticommutator between many-electron operators, it is not a constant �like in case of Bose orFermi operators�. Therefore, its functional derivative is non-zero. It generates a subset of graphs that do not appear instandard techniques for fermions and bosons. These graphsdescribe kinematic interactions.

C. Energy shifts (imaginary time)

The effective interaction of the intradot states via conduc-tion electrons arises in the WCI and SCI regimes. In bothcases, the lowest order of perturbation theory contains termswith an integration over a wide energy region of a continu-ous spectrum. It results in finite widths of the intradot states.In the SCI regime, already the HI approximation providesthe width. The width depends on a population of differentelectron states, involved in the single-electron transition inquestion. This fact is a trivial consequence of non-Fermi-Bose commutation relations between the Hubbard operators.A nontrivial consequence of these relations is the kinematicinteraction that shifts the transition energies ��

�0� in the dot.The correlation-caused shift, constant for all levels, arisesnaturally within the MFA in the slave-boson theory �cf. Ref.24�. Using the Hubbard operator formalism �see also I, Ref.9� and the MFA, we will show that the shift is different fordifferent levels. Our mean-field approximation for the Hub-bard operator GF’s on the imaginary-time axis will be deter-mined by two conditions: �i� a dynamical scattering on Bose-like excitations is neglected, i.e., �P /�U=0; �ii� a full vertex��D−1 /�U is replaced by a zero one, �0��D0

−1 /�U. Inother words, we neglect fluctuations around stationary MFAsolutions.

To develop the MFA we have to extract from the self-

operator S̃ the local-in-time contribution in the lowest orderwith respect to the effective interaction V. The self-operator

S̃ �see Eq. �25� is already proportional to the interactionV��. Therefore, it is enough to calculate only the derivative


075315-4

�G /�U with respect to the interaction. We recall that theinteraction V��t , t1� does not depend on U�. The derivativeis

�G�2,��t1,t2�

�U��t�=��D�2,�5

�t1,t2�P�5��t2�

�U��t�

=�D�2,�5

�t1,t2�

�U��t�P�5��t2�

+ D�2,�5�t1,t2�

�P�5��t2��U��t�

. �29�

The second term in Eq. �29� expresses a dynamical rescatter-ing process, where the Fermi-like excitation �the locator D�scatters on the Bose-like excitation �the GF’s �P /�U�. Theterm �P�5,�� /�U� has the following structure:

�P�5,��/�U� � �TZ00Z�� − �TZ00��TZ�� + �TZ�4�5Z��

− �TZ�4�5��TZ�� .

The diagonal correlators �TZ��t�Z��t��− �TZ��t��TZ��t�� and �TZ00�t�Z00�t��− �TZ00�t��Z00�t�� describefluctuations of the population numbers �TZ��N� and�TZ00��N0, respectively. The nondiagonal GF’s

�TZ��t�Z��t�� describe the transitions between one-electron states � and ��. In the MFA one can ignore thefluctuations of the population numbers, which appear in thenext order of the perturbation theory with respect to thetransparency of junctions.

The derivative of the locator �D /�U in Eq. �29� can becalculated using the trick

�D/�U = − D��D−1/�U�D , �30�

which follows from the equation ��D−1D� /�U=0. The valid-ity of the trick is well justified, since we consider only sta-tionary processes and within the assumption that the interac-tion, which is absent at remote time t0→−, is switched onadiabatically. Taking into account only the first term in Eq.�29� �the condition �i� �P /�U=0, by means of Eqs. �28� and�30�, we obtain from Eq. �25�

S̃�,��t,t�� − i� dt1� dt3V�1�2�t,t1�D�2,�3

�t1,t3�

� ��,�1

�D�3,��−1 �t3,t��

�U00�t�+�D�3,��

−1 �t3,t��

�U�1��t�� .

�31�

Replacing the full vertexes by zero ones �condition �ii�and using Eq. �21�,

��3,��t3,t�;t�

��D�3,��

−1 �t3,t��

�U��t�

��D�0�−1�t3,t��3,��

�U��t�

= − ��t3 − t��3��

�0� �t��

�U��t�, �32�

we find with the aid of Eq. �17�

�00�3,��t3,t�;t�

� − ��t3 − t��3��

�0� �t��

�U00�t�

= ��3,��t3 − t��t − t�� , �33�

��1��3,��t3,t�;t�

� − ��t3 − t��3��

�0� �t��

�U�1��t�

= − ��3,�1��,��t3 − t��t − t�� . �34�

As a result, we have

S̃�,��1� �t,t�� = − i� dt1V��2

�t,t1�D�2,��t1,t+��t − t��

+ i��,�� dt1V�1�2�t,t1�D�2,�1

�t1,t+��t − t�� .

�35�

This contribution is, indeed, local in time and, therefore, rep-resents an effective field that shifts the transition energies inthe dot.

III. DIAGONAL APPROXIMATION

The analysis of magnetotransport through a multileveldot, even within the MFA, is still a bulky numerical task.Fortunately, we can use a diagonal approximation �see thearguments between Eqs. �6� and �7�, which enables furtheranalytical treatment and simplifies the numerical analysis. Inthe diagonal approximation the self-operator takes the form

S̃�,��1� �t,t�� = ��t − t��0�

shift�t� ,

�0�shift�t� = i� dt1V�1�1

�t,t1�D�1,�1�t1,t+��1 − ��,�1

� . �36�

Evidently, the self-interaction is canceled. We recall that inthe equation of motion for the Hubbard operator, in the firstorder with respect to the tunneling matrix elements v,the anticommutator �X0� ,v†X�0cp generates the operator

P̂0�=Z00+Z��. In order to obtain the first order in the pertur-bation theory with respect to the effective interactionV=v†C�t , t��v we have to calculate the second orderwith respect to v. This gives rise to the commutator

�X0� , P̂0�= �X0� ,Z00+ �X0� ,Z��=−X0�+X0�=0. Thus, thecancellation of the self-interaction is an inner property of thetheory, indeed.

Since there is only one type of coupling between transi-tions, �0,�⇔ �� ,0, we can use a simplified notation in thediagonal approximation:


075315-5

S�,�HI � S�

HI, P�� P�, V�� V�, S̃�,��1� � S̃�

�1�

��0� � ��, �0�

shift � ��shift, D�,�

HI � D�, . . . . �37�

Equation �24� for the GF G� ��G�,�� can be written as fol-lows:

� dt1�d�−1�t,t1� − S�

HI�t,t1�G��t1,t�� = ��t − t��P��t�� ,

�38�

where

d�−1�t,t1� = �i�t − ��t� − ��

shift�t��t − t1� . �39�

With the aid of Eq. �28�, Eq. �26� for a full locator yields

� dt1�d�−1�t,t1� − S�

HI�t,t1�D��t1,t�� = ��t − t�� . �40�

Since the equations for the GF’s and locators are derived,the auxiliary fields can be omitted—i.e., U��t�=0. As a re-sult, the Hamiltonian is time independent, while the locators

and GF’s depend only on the time difference. P��t�� does notdepend on time as well: P��t��→P�=N0+N�. The latter sim-plifies the derivation of the expressions for S�

HI�t− t1� for realtimes:

�S�HI�t − t1�� = P��V��t − t1��, � � �R,A, � , � .

�41�

For the energy shift ��shift �which also does not depend on

time at U��t�=0 the “lesser” value must be used. As a result,Eq. �40�, written in the form of the integral equation on theimaginary time axis,

D��t − t�� = d��t − t�� +� dt1� dt2d��t − t2�

�P�V��t2 − t1�D��t1 − t�� , �42�

can be continued on the real-time axis:

D�R/A�t − t�� = d�

R/A�t − t�� + d�R/A�t − t2�P�V�

R/A�t2 − t1�

�D�R/A�t1 − t�� , �43�

D��t − t�� = d�

��t − t�� + �−

dt1�−

dt2d��t − t2�P�V�

A�t2 − t1�D�A�t1 − t�� + �

−

dt1�−

dt2d�R�t − t2�P�V�

��t2 − t1�D�A�t1 − t��

+ �−

dt1�−

dt2d�R�t − t2�P�V�

R�t2 − t1�D��t1 − t�� . �44�

The matrix form enables us to simplify the latter equation:

�1 − dRPVRD� = dRPV�DA + d�PVADA. �45�

As follows from Eq. �43�, the square brackets on the left-hand side of Eq. �45� is dR�DR−1. Multiplying Eq. �45� fromthe left by �dR−1 and taking into account that �dR−1dR=1,�dR−1d�=0, we have

�DR−1D� = PV�DA. �46�

Thus, we obtain the same relation which is known in thetheories for fermions �see Ref. 23�:

D��t − t�� = �

−

dt1�−

dt2D�R�t − t2�P�

�V��t2 − t1�D�

A�t1 − t�� . �47�

The advantage of this expression consists in the fact that itcontains renormalized magnitudes. However, its validity isrestricted by �i� the MFA and �ii� stationary states �the stepfrom the differential equation �46� to the integral one, Eq.�47�, uses the boundary condition; the same statement isvalid for Eq. �43� as well�. The expression for D�

�, obtained

in a similar fashion, leads to Eq. �47�, where V�� should be

replaced by V��.

The Fourier transformation of Eqs. �43� and �47� yields

D�R/A�� =

1

�d�R/A��−1 − P�V�

R/A��

=1

� − �� − P�V�R/A�� ± i�

, �48�

D��,�� = D�

R��P�V��,��D�

A�� . �49�

The effective interactions V��,�, expressed in terms of the

width function ��, are

V�� = − 2i�

��1 − f�� ,

V�� = 2i�

��f�� ,

V�R/A�� = �

V�,R/A�� = �� i�� , �50�

where = l ,r, ��l ��+��

r ��,


075315-6

�� = �p

v�,p

P� − �p

vp,�, �51�

and P denotes the principal part of the integral. The real part�� can be neglected in the retarded and advanced inter-action V�

R/A�� in the wideband case (��v�,p�2� /Wln��W+�� / �W−��, where W is of order of abandwidth half and ��W). As a result, V�

R/A�� i��.Multiplying Eq. �48� by P�, we obtain �see Eq. �28�

G�R/A�� =

P�

� − �� ± iP��

. �52�

By virtue of this equation, the GF difference yields

G�R�� − G�

A�� = − 2i�P�L�� , �53�

where

L�� P��/�

�� − ��2 + �P��2 �54�

is the Lorentz distribution. Note that in the SCI case, P�

=N0+N�; i.e., the distribution depends on the nonequilibriumpopulation numbers. Multiplying Eq. �49� by P�, by meansof Eqs. �50� and �52�, we obtain for the GF G�

G�� = G�

R��V��G�

A�� = 2i�P�L�� f̄�� , �55�

where ��l/r��

l/r /�� and f̄�� is the weighted Fermi function:

f̄�� l f l�� + ��

r fr�� . �56�

A. Equations for population numbers

All the locators and GF’s depend on the single-time cor-relator P��X0�X�0+X�0X0��=N0+N�. The expectation val-ues �X�0X0�� and �X0�X�0� can be found by means of theGF’s G�

�,��t , t�. We recall that our approach is developed forthe stationary case only. Thus, we have

�X�0X0�� = − i�i�X�0�t�X0��t��

= − iG��t,t+� = − i� d�

2�G�

�� . �57�

Since the GF’s depend on P� themselves, Eq. �57� for thepopulation numbers turns into the self-consistency equations:

N� = − i� d�

2�G�

�� ,

N0 = i� d�

2�G�

�� . �58�

The normalization condition, which follows from the aver-aging with respect to Eq. �A5�, provides the additional equa-tion

1 = N0 + ��

N�. �59�

Substituting Eq. �55� into Eq. �58�, we obtain

N� = s�P�,s� �� d�L�� f̄�� . �60�

From Eqs. �55� and �60� one concludes that the magnitude s�is the integrated “lesser” part of the locator:

D�� = 2i�L�� f̄�� . �61�

For ��1, the Lorentzian can be approximately treated as a� function. It results in a simple expression

N� = s̃�P�, s̃� � �f l��l + fr��

r . �62�

In numerous experiments1,2,8,14 with QD’s the temperature isusually lower than the level spacing. In this condition, if thetransition energy �� is above both the left and right electro-chemical potentials, s̃� is exponentially small. This leads toexponentially small N� as well. If �� is below both �l,r, s̃��1, we obtain N0�1. Evidently, N0 and Np are finite onlywhen �� is located within the “conducting window” �CW� inthe interval of energies between �r and �l. This conclusion isvalid, of course, for a finite width �� of the transition ��

when s� is defined by Eq. �60�.Let us introduce the function

�� s�−1 − 1�−1, �63�

which couples the population numbers of empty and single-electron states:

N� = N0��. �64�

The normalization condition yields

N0 =1

1 + ��

, N� =��

1 + ��

. �65�

In order to display a nonlinear dependence of the popula-tion numbers on the applied voltage, let us consider the limits�→ s̃�. In this case, we obtain

�� → �̃� = e−��−��

�1 + e��−��cosh��eV/2� − �� sinh��eV/2�

e��−�� + cosh��eV/2� + �� sinh��eV/2�.

�66�

Here, we introduced the following notation: �= ��l+�r� /2,eV=�l−�r, �=1/T, and the degree of asymmetry of

the contacts, ��=��l −��

r . At eV=0 we have �̃�

=exp�−��−�� and, therefore, the Gibbs-ensemble limitholds:

N0 =1

1 + ��e−��−��

, N� =e−��−��

1 + ��e−��−��

. �67�

In the limit of a small bias voltage, �eV /2�1, we obtain

�̃� � e−��−��1 + ��eV

2� . �68�

The most prominent phenomenon occurs for a symmetriccoupling to the contacts, ��=0. Namely, in this case we have


075315-7

�̃� = e−��−��1 + e��−�� cosh��eV/2�e��−�� + cosh��eV/2�

. �69�

When a “level” �� gets into the CW ��eV=�r� ��−��l one obtains exp��−�+eV /2��1. At large bias

voltages, �̃�→1 and all population numbers become equalto each other:

N0 = N� =1

1 + ��eV1

. �70�

In contrast, the population numbers of states with the ener-gies outside of the interval �eV are zeros. Thus, we found aremarkable feature of the correlated transport for asymmetric-coupling design of the device: a large bias volt-age equalizes the population of conducting states. We willreturn to this result later.

B. Real-time equation for energy shifts

We have to find the renormalized transition energy ��,which is still unknown. The transition-energy shift �36� con-tains the integral

l̃� = i�t0

t0−i�

dt1V��t,t1�D��t1,t+� . �71�

We assume adiabatic switching of the interaction and,therefore, ignore the contribution from the part of thecontour �t0 , t0− i�� at t0→−. Note that considerationof the integral �71� on the Keldysh contour orapplication of Langreth’s rules to this expression�in this case limt�→t+0�V��t , t1�D��t1 , t��

=limt�→t+0�V��t , t1�D�

A�t1 , t��+V�R�t , t1�D�

��t1 , t�� leads toequivalent forms.

A real-time expression of the integral l̃� has the form

l� = i�−

t

d �V��t, �D�

�� ,t� − V��t, �D�

�� ,t�

= −1

2��

−

dE

E + i�!��E� , �72�

where we introduced the notation

!��E� = �−

d�

2��V�

��D�� + E� − V�

��D�� + E� .

�73�

The right-hand side of Eq. �72� contains dressed locatorsD�

�,�. Therefore, the system of equations

�� = ��0� + ��

shift,

��shift = �

�1

l�1− l� �74�

and Eq. �72� have to be solved self-consistently.In fact, the integration limits in Eqs. �72� and �73� are

determined by the bottom and top of the conduction bands

�“left” and “right”�. It follows from the definition of thewidth, Eq. �5�: ��

��0 only within the interval of �,where the density of states of the corresponding conductionband is nonzero. Within our approximation, by means ofEqs. �50� and of the expressions for the “lesser” �Eq. �61�and “greater” �D�=−2i�L��1− f̄� locators, we obtain fromEqs. �72� and �73� a simple expression for the partial shift ofthe transition energy:

l� = −1

��

−W

W

d��−W

W

dEL��E�� f̄�E� − f̄��

E − � + i�. �75�

Here, W is a half of bandwidth, which is the same in bothcontacts and is much larger than any other parameter in thetheory. A few remarks are in order. First, the Lorentzianwidth, Eq. �54�, is determined by the product P��= �N0

+N��. Therefore, if the transition energy �� is outside ofthe CW, the population numbers are zeros approximately. Inthis case, the dot-contact coupling does not affect the discretetransitions �their widths are zeros�. Second, the imaginarypart of Eq. �75� is exactly zero. Thus, this expression pro-vides a simple shift of the transition energy �as it should�.Third, estimating the integral �75� at low temperatures, weobtain

l� � −1

��=l,r

�� ln

W�� − ��2 + ��P��2

. �76�

Evidently, the smallness of the coupling constant � may becompensated for by a large logarithm when the transitionenergy is in the proximity of one of the electrochemical po-tentials. The shift may be large and, moreover, it is sensitiveto the bias voltage, since �l/r=�±eV /2. On the other hand,the effect is logarithmically weak and, besides, the infraredcutoff in the integral is max��P ,T; i.e., with an increase ofthe temperature the renormalization becomes less effective.Nevertheless, the effect is quite appreciable numerically �seebelow�. Thus, we conclude that the attachment of contacts tothe dot shifts the transition energies logarithmically.

C. Current in the diagonal approximation

The final expression for the current is obtained by thesubstitution of Eqs. �53�, �55�, and �56� into Eq. �7�:

JSCI =2e

��

P�� d��l ��

r �f l�� − fr��L�� .

�77�

This expression is almost identical to the one for the nonin-teracting electron problem. However, in the SCI regime theexpression contains the effective width �P, where P�=N0+N� is a combination of population numbers. All P��1 dueto the normalization condition, Eq. �59�.

At ��1 the Lorentzian can be replaced by a � function.In this case the integral yields


075315-8

JSCI =2e

��

P��l ��

r sinh�eV/2Tcosh�� − ��/T + cosh�eV/2T

.

�78�

Here, the coefficients ��l/r determine the relative transparency

of the left and right junction. The degree of the channelopening is defined by the product ��

l ��r P�. The population

of each “level” �� depends on the populations of other levelsdue to the sum rule, Eq. �59�. Assuming that only one levelcontributes to the sum and ��−��→�, at the limit of zerobias voltage, one obtains from Eq. �78� a well-known resultfor the differential conductance3:

G = limV→0

dJSCI

d�eV��

1

Tcosh−2� �

2T� . �79�

Below we apply our theory to magnetotransport through asmall vertical quantum dot under a perpendicular magneticfield.

IV. PARABOLIC QUANTUM DOT IN MAGNETIC FIELD

Shell effects are among the most remarkable phenomenaobserved in vertical quantum dots.10 By virtue of the poten-tial symmetries, the orbital motion of electrons in a perpen-dicular magnetic field could lead to the degeneracies even atthe strong intradot Coulomb interaction �see, for example, adiscussion of hidden symmetries for a parabolic potential inRef. 25�. Simple estimations indicate that orbital effects aremuch stronger than spin effects and as yet are not discussedin the literature related to quantum transport. For example, inthe two-dimensional model for QD’s upon a perpendicularmagnetic field26 the effective spin magnetic moment is �*

=gL�B with �B= �e� /2mec and the effective Landé factorgL=0.44. The effective mass m*=0.067me �for GaAs� deter-mines the orbital magnetic moment for electrons, �B

eff

�15�B, which is 30 times stronger than the spin one.In this section we discuss the effects of orbital motion on

the nonlinear �with respect to the bias voltage� magnetotrans-port in the WCI and SCI regimes. While our approach in-cludes the case ��1 ��=1/T�, the range of parameters��T��, exploited usually in experiments, will be in thescope of our interest. We consider a circular dot ��x=�y

=�0� in a perpendicular magnetic field B.26 The dot eigen-modes are �±= ��±�c /2� with �=��0

2+�c2 /4 and �c

= �e�B / �m*c�. We choose the position of the dot potential wellsuch that the first level in the dot is above �. In this case thedot is empty at zero bias voltage.

As was shown above, the dot-contact coupling produces ashift of the transition energies and non-Gibbs behavior of thepopulation numbers of the dot states. In addition, the dotspectrum in a magnetic field displays degeneracies �see Figs.2 and 3 below�. To illuminate these features, we consider asimple example of two degenerate transitions �0�→ �p1� , �p2�. The solution of self-consistent equations for thiscase is displayed in Fig. 1. The QD with bare energies isempty �N0=1� until eV reaches the bare level "p /�0=1.5,which is above the Fermi energy without the shift. However,the renormalized single-electron states are filled at much

smaller values of bias voltage �see the solid lines�, since thecorresponding transition energies ��p /�0=0.5� are below thechemical potential. One observes also that the populationnumbers are equalized �=1/3� after the voltage reaches thecorresponding transition energy. We stress that both mecha-nisms �the shift due to the attachment to the contacts and theequalization� are missing in the master-equation approach.3,4

Below we will demonstrate how these mechanisms show upin the transport properties.

A. Shell effects

The response of the dot to the applied bias voltage isdetermined by the number of electronic levels within theconducting window ��l ,�r�. This number depends on �i� therelative position of the bottom of the confinement potentialin the dot and on the equilibrium position of the electro-chemical potential �eV=0� and �ii� the magnetic fieldstrength. Although transport properties in the WCI regimeare known �cf. Ref. 4�, we shortly summarize the essentialeffects in order to compare with those in the SCI regime. Wefocus on shell degeneracies caused by the magnetic field,which are missing from the previous studies.

The expression for the WCI current is the same as Eq.�77�, but contains the normal width � instead of the effectiveone �P �P=1�. At ��1 the Lorentzian can be replaced bya � function. In the wideband case the width does not dependon energy and can be replaced by a constant—i.e., ��

l/r��→��

l/r. Then, at a weak dot-contact coupling, setting P=1 inEq. �77�, one finds

FIG. 1. �Color online� Population numbers N0 ,Np for threestates—empty state �0� and two degenerate ones �p1� , �p2�—as afunction of the bias voltage. The energies are given in units ofconfinement energy �0. The electrochemical potential is chosen at� /�0=1.0 and T /�0=0.033. The solid �dashed� line displays thebehavior of the population number with the renormalized �bare�energy. The solid and dashed vertical lines crossing the bias voltageaxis indicate the position of the bare �p /�0=1.5 and the renormal-ized �p /�0=0.5 energies, respectively.


075315-9

JWCI �2e

��

��l ��

r

�ma�f l�� − fr�� . �80�

Evidently, the dot spectrum exhibits a shell structure inthe WCI regime �see Fig. 2�. The effect of degeneracy of thespectrum becomes transparent at special values of the mag-netic field:

t0 � �c/�0 = �l − 1�/�l , �81�

which are determined by the ratio �+ /�−= l=1,2 , . . . �r=0,2 . . . � of the eigenmodes of the dot.26 We use a fixedvalue for the confinement energy �0, which may be changed,however, with an increase of the number of electrons in thedot or with an increase of the gate voltage. We believe thatfor a small number of electrons �i.e., N�20� all resultswould persist, since the magnetic field contributes addition-ally to the external confining potential.

At given �= ��L+�R� /2 �which simulates in our case thegate voltage Vgate� the number of conducting channels is de-termined by the number of the Fock-Darwin levels in theCW ��R��E��L�. A new level, entering the CW, deter-mined by �−eV /2��+eV /2, produces a step in thecurrent with height J0,�=4�e��

l ��r / �h��. In a narrow energy

interval, we may neglect the difference between the widths�� and, therefore, J0,��J0=�e� /h. The “reduced” currentJ /J0 displays integer steps and, if some level �� is n� timesdegenerate, the step increases by n� times.

The above features are blown up in Fig. 3. With an in-crease of the magnetic field the levels with higher values oforbital �and spin� momenta m move down faster than thosewith lower momenta �see also the left panel of Fig. 2�. Con-sequently, high-lying levels with large values of Lz=mshould unavoidably show up in the CW at large enoughmagnetic field. The levels with negative orbital momentaLz=−m� move up and leave the CW, decreasing the numberof conducting channels. These two processes result in oscil-lations in the current. These oscillations are greatly enhanced

at the specific values of the magnetic field, Eq. �81�. Forexample, at t0=0.7�l=2� �compare Figs. 3�a� and 3�b�the WCI current drastically increases. At values slightlylarger than t0, the negative differential conductanceG=dJWCI /d�eV� arises �see Fig. 3�c�.

The renormalized spectrum, obtained from the self-consistent solution of the system of Eqs. �58�, �59�, �72�, and�74�, is displayed in Fig. 4. As expected from the estimation�76�, the renormalization for an individual level is large anddepends on the initial conditions �� ,T ,eV�. In fact, the firstrenormalized transition energy may be shifted even to theregion of negative values with respect to the one of theclosed dot. We recall that these transition energies are thedifferences between shifted single- and zero-electron levels.

FIG. 2. �Color online� Magnetic field ��c /�0�dependence of the Fock-Darwin spectrum �inunits �0� �left� and the tunneling current JWCI �inunits J0� through the quantum dot �right�. Arrows�right panel� indicate the cuts at �c /�0

=0.0,0.34,0.7, respectively. The degeneracy at�c /�0=0.0,0.7 is clearly seen in jumps of thecurrent at T=0 Arrows 2,3,4,5 �left panel� indi-cate the level structures which correspond to dif-ferent cuts �right panel�.

FIG. 3. �Color online� Magnetic field dependence �t=�c /�0� atT=0 of �a� the Fock-Darwin spectrum �the position of the chemicalpotential � is displayed by a dashed line�, �b� the tunneling currentJWCI �� /�0=0.045� through the dot, and �c� the conductance G. Seetext for the definition of JWCI and G. The energies are given in unitsof the confinement energy �0.


075315-10

Due to a large number of the transitions contributing into theenergy shift �see Eqs. �72� and �74�, all transition energiesare shifted more or less homogeneously. The phenomenonacts in the spirit of the MFA: the more transitions contributeto the renormalization, the closer is the shift of an individuallevel to an average value.

The most prominent feature, caused by the energy shifts,is a decrease of the bias voltage threshold for a nonzerocurrent. At small eV and fixed � �or gate voltage� the firsttransition �10��r and the current, Eq. �77�, is zero, in spiteof P0��0 �N0=1� �see Fig. 5�. At higher voltages the CWcontains nW-electron states and, according to Eq. �65�, P0�=2/ �nW+1�. As a result, the SCI current is JSCI

=2J0nW / �nW+1�, where J0=�e�� / �hnW��e�̃ /h. TheWCI current, however, is JWCI=J0nW. Thus, even for a large

bias voltage eV, the SCI current is weaker than the WCI oneby a factor of �=JSCI /JWCI=2/ �nW+1� �until the othercharge sector is not switched on at eV�U�.

Let us compare the manifestation of shell degeneraciesdefined by the condition �81� in the WCI and SCI regimes.We consider first r=0—i.e., a zero magnetic field �see alsoFig. 2�. In this case, each shell k has the degeneracy gk=k+1. If all single-electron states up to the last shell n arepresent in the transport window, the total number of states,involved in the transport, is nW+1=2�k=0

n �k+1�+1=n2+3n+3 �the factor of 2 is due to the spin degeneracy�. Conse-quently, the nth step height in the SCI current is JSCI /J0=2�n+1��n+2� / �n2+3n+3� �see the right top panel of Fig.5�, which is smaller than the WCI current by a factor of �=2/ �n2+3n+3�.

We interpret this result for the SCI regime as follows.Once an electron is in the dot, it is distributed among manyavailable single-electron levels, with different populationnumbers N� �Eq. �70�. The strong intradot Coulomb inter-action prevents the second electron from entering the dot.Altering the bias voltage, we allow the in-dot electron toleave the dot from the occupied states �with different popu-lation numbers�. The step height in the current depends onthe number of the states producing shells due to the symme-tries of the confining potential. As a result, the differentialconductance exhibits sharp peaks inside the first Coulombdiamond, which decay with an increase of the bias voltage.We return to this feature shown in Fig. 6.

Since ��e−��−��1+��eV�2 at small eV �see Eq.�66�, these effects cannot be seen in the linear conductance.Another effect �which is not seen in the master-equationapproach3,4,27� follows from Eq. �74� �see also Fig. 1�: thecoupling pushes the transition energies �� down compared tothe bare energies ��. It decreases the bias voltage threshold,as was mentioned above, for the nonzero current. At r=2��c /�0�0.7�, a new shell structure �see Fig. 2� arises as ifthe confining potential were a deformed harmonic oscillatorwithout the magnetic field. The number of levels is just thenumber of levels obtained from the two-dimensional oscilla-

FIG. 4. �Color online� The bare �dashed line� and renormalized�solid line� energies as a function of the magnetic field. The param-eters of calculations �in units �0� are the same for the WCI and SCIregimes. Five lowest levels of the Fock-Darwin spectrum weretaken as the bare energies.

FIG. 5. �Color online� Population numbers Np

�left panel�, the current JS /J0, and the conduc-tance GS=dJS /d�eV� as a function of the biasvoltage �right panel� in the SCI regime at zeromagnetic field. In each shell k we have 2�k+1�degenerate orbitals characterized by the same Np

�p=k+1�. In particular, N0= �X00�, N1=Nk=0,�0=1

=Nk=0,�0=2, etc., where �k is the orbital index inthe shell k. The exact values of the bare �1 andthe renormalized �10 energies from the shell k=0 and the parameters of the calculations are dis-played in the left panel. In the right panel, therational numbers characterize the height of thenth step in the SCI current, JSCI /J0=2�n+1��n+2� / �n2+3n+3�, for the last filled shell n.


075315-11

tor with ��=2�� and �� denote the larger and smallervalues of the two frequencies�. In this case nW= �n+2�2 /2 ifthe last available shell is even and nW= �n+1��n+3� /2 if it isodd, and these numbers define the heights of steps in theWCI and SCI regimes.

The evolution of the spectrum under the magnetic fieldcan be traced by means of the conductance measurements�see Fig. 6�. In particular, the results for the differential con-ductance dJ /d�eV� in the WCI regime �Fig. 6�a� resemblesvery much the experimental conductance discussed in Ref.14. The increase of the bias voltage allows one to detect thedegeneracy of quantum transitions involved in the transportby means of the oscillations in the conductance �see Fig.6�b�. At zero magnetic field, the large conductance magni-tude corresponds to the large number of levels involved inthe transport. With an increase of the magnetic field at a fixedbias voltage, the conductance oscillates due to the appear-ance and disappearance of the shells. As was mentionedabove, an increase of the magnetic field brings into the CWstates with a large magnetic quantum number m=n−−n+ andpushes up the ones with −m. This gives oscillations in thecurrent �see Fig. 2�b�, which are suppressed, however, to-gether with the current itself in the SCI regime �see Fig.6�c�.

In the SCI regime all processes are developing under afew constraints with an increase of the bias voltage. The firstrestriction is that the dot cannot contain more than one elec-tron due to the large energy gap between single- and two-electron correlated states �our model consideration is per-formed in the limit U→�. The second constraint is due tothe normalization condition: each channel participates in thetransport with spectral weight P�=N0+N��1, whereas thepopulation numbers fulfill the sum rule N0+��N�=1. As aresult, the more levels are involved with the increase of thebias voltage, the lesser is the contribution of each channel tothe current and, correspondingly, to the conductance �seeFig. 5�. There is a saturation effect of the currentlimeV→ JSCI /J0=2 that diminishes the differential conduc-

tance. One should keep in mind, however, that this formallimit should be performed under the condition that U→and eV�U. At the neighborhood of the degeneracy pointsthe number of involved levels grows very fast. This explainsthe strong suppression of the transport by correlations in anarrow interval of the bias voltage �see Fig. 6�d�. One canobserve, however, that a fine structure, produced by the shellstructure, manifests itself at certain values of the bias voltageand magnetic field. At these values the transport exhibits arelatively strong conductance, indeed.

V. SUMMARY

We have analyzed the nonlinear transport through the dot,weakly attached to two metallic contacts via insulating-vacuum barriers, in the regime of the strong intradot Cou-lomb interaction. To carry out this analysis we have reformu-lated the results of Refs. 20 and 21 in terms of GF’s definedon the Hubbard operators. It turns out that Wick’s theorem isnot valid for these operators. We resolved this problem bymeans of an extension of the diagram technique for theHubbard-operator GF’s �developed earlier for an equilibriumin Ref. 16� for the nonequilibrium states. The extension con-sists of two steps. In the first step, the exact functional equa-tions for the many-electron GF’s are derived and used for theiterations in the spirit of the Schwinger-Kadanoff-Baym ap-proach. This gives rise to the diagrammatic expansion. In thesecond step, the analytical continuation of these equations tothe real-time axis transforms the system to the nonequilib-rium form. The solution of equations for the GF’s yieldsself-consistent equations for the renormalized transition en-ergies of the dot and their population numbers. Our equationscontain a Lorentzian broadening of the transitions due to thecontact of the dot with the continuum of conduction elec-trons. While we have considered the limit ��T, a generalscheme discussed in Sec. III allows us to analyze the oppo-site one, T��, as well. Note that the latter regime cannot betreated in the master-equation approach �see the discussion in

FIG. 6. �Color online� The magnetoconduc-tance G=dJ /d�eV� at the WCI �top� and SCI�bottom� regimes. �a� The contour plot for theWCI conductance displays the Fock-Darwinspectrum of the circular dot in the magnetic field.�b� The conductance in the WCI regime, at fixedvalues of the bias voltage. The results foreV /�0=1, 5, and 9 are presented by dot-dashed,dashed, and solid lines, respectively. �c� The con-tour plot for the conductance at the SCI regime.�d� The conductance at the SCI regime, at fixedvalues of the bias voltage. The dot-dashed,dashed, and solid lines present the results foreV /�0=1,1.25, and 1.45 respectively. The pa-rameters of the calculations are displayed inpanel �c�.


075315-12

Ref. 3�. The latter misses also the renormalization of thetransition energies �cf. Refs. 3, 4, and 27�.

One of the questions we have addressed in this paper is towhat extent the levels �the transition energies� of the dot aresensitive to the applied bias voltage eV in the SCI regime.We found that, although this dependence is logarithmicallyweak �see Eq. �76�, the energy shift for the individual levelcan be quite noticeable �see Figs. 1 and 4�. In the multileveldot, the difference between different energy shifts diminisheswith an increase of the bias voltage, and all levels becomeshifted more or less homogeneously. This mechanism affects,however, the threshold for the nonzero current, which canappear at much smaller bias voltages, in contrast to the WCIregime.

The self-consistent solutions of the equations for thepopulation numbers �Eqs. �58� and �59� and for the transi-tion energy shifts �Eqs. �72�–�74� result in a strong andhighly nontrivial dependence of the population numbers onthe bias voltage. At zero bias voltage, they display the stan-dard Gibbs’ statistics. However, at voltages that are higherthan a certain critical value �determined by the position ofthe renormalized transition energy ��, the population num-bers are equalizing for the energies within the conductingwindow ��r��l�, although the temperature is small.This result, obtained analytically for a simple case �see thediscussion related to Eq. �70�, is confirmed by numericalsolution of the complete system of nonlinear integral equa-tions.

The degree of nonlinearity of the current in the SCI re-gime �Eq. �77� is essentially determined by the spectralweights P� of the dot states �see Eqs. �60�, �63�, and �65�.Therefore, the equalization of the population numbers ismanifested in the current steps of certain heights, as well asin the saturation of the current with a growing number oflevels involved in the transport. A simple expression, whichrelates the height of the nth step in the current to the numberof states participating in the transport, is obtained for thecase of a circular dot. With the aid of this expression onemay extract the degeneracy of the dot spectrum even at thestrong Coulomb interaction.

In the WCI regime, at specific values of the magneticfield, Eq. �81�, we expect a drastic increase of the currentthrough the dot due to shell effects. In the SCI regime, thetransport is strongly suppressed due to spectral weights,regulating the contribution of each conducting channel. Wepredict, however, that the shell structure of the dot producesa fine structure in the conductance in the SCI regime. In thefine structure, the number of peaks and their heights dependsensitively on values of the bias, the gate voltage, and themagnetic field. As was underlined above, the renormalizedtransition energies are shifted with respect to the bare onesonly logarithmically, whereas the dependence of the spectralweights of these transitions is exponentially strong. Note thatthe shifts arise already in the second order of perturbationtheory with respect to the dot-contact coupling. We recallthat the cotunneling picture5 is based, however, on a higherorder of perturbation theory. It appears that the mechanism,discussed in the present paper, provides the dominant contri-bution into the fine structure at the strong intradot Coulombinteraction and the weak dot-contact coupling.

ACKNOWLEDGMENTS

We are grateful to David Sanchez for fruitful discussions.This work was partly supported by Grant No. FIS2005-02796�MEC�. I.S. thanks UIB for the support and hospitality,and R.G.N. is grateful to the Ramón y Cajal programme�Spain�.

APPENDIX: RELATIONS BETWEEN FERMI ANDHUBBARD OPERATORS

It is convenient to use different notations for the Hubbardoperators that describe Bose-like transitions �without changeof particle numbers� and Fermi-like ones �between the states,differing by one electron�. The former and latter ones aredenoted as �Z� and �X�, respectively:

Z00 = ��1

�1 − d�1

† d�1� = �0��0� , �A1�

Z�� = ��1��

�1 − d�1

† d�1�d�

†d� = �� , �A2�

X0� = ��1��

�1 − d�1

† d�1�d� = �0�� , �A3�

Z00�0� = �0�, Z�� = �� , �A4�

where Eq. �A1� defines the vacuum state and the product istaken over all single-particle states �1 of the dot confiningpotential. In this case, the expansion of unity acquires theform

1̂ = Z00 + ��

Z��. �A5�

The expectation values N0= �Z00� and N�= �Z�� with respectto the state and ensemble of interest are population numbersof corresponding many-electron states. The Hamiltonian of aclosed dot in these terms takes the form

HQDWCI = �

�

��d�r d�, HQD

SCI = �0Z00 + ��

��Z��. �A6�

The WCI regime corresponds to U→0, while the SCI re-gime takes place at U→. The energy of an empty QD, �0,is determined by the potential depth and can be chosen arbi-trarily. While the eigenvalues �� are defined by the shape ofthe potential, the transition energies are ��

�0��−�0. Werecall that the position of the single-electron transitions withrespect to the electrochemical potentials is essential for thetransport properties of the system.

To express Ht in these variables, we have to define theannihilation �creation� operator d��d�

†� in terms of the Hub-bard operators:

d� = 1̂ · d� · 1̂ = �0�d��X0�. �A7�

Evidently, the decomposition �A7� should contain the transi-tions from one- to two-electron configuration, from two- tothree-electron ones, etc.,


075315-13

d� = ��,n

��n�,n�d��n+1�,n + 1�X��n��n+1�, �A8�

which defines the anticommutator relation

1̂ = �d�,d�† = �

��,nZ��n��n�

. �A9�

However, in the limit U→ the above equation is reduced toEq. �A5�. In other words, we consider the transport withinthe first conducting “diamond” only, in coordinates of gateand bias voltages. The term Ht remains unchanged in theWCI regime, whereas in the SCI regime it can be written inthe form

HtSCI = �

k,��,,��vk�,�

ck�† X0� + H.c.� , �A10�

where

vk�;� = vk�;0�

�0�d�� ,

v�;k� = v�0;k�

��d�†�0� . �A11�

The matrix element vk�;0� contains information on specific

features of attachment between a quantum dot and contacts.Here, = l ,r denotes the left and right contacts, respectively.One may notice that the Hubbard operators take into accountcorresponding kinematic restrictions placed by the strong

Coulomb interaction. The price for this is, however, the non-trivial commutation relations

�X0�,X��0 = ��Z00 + Z��, �Z00,X0� = X0�,

�Z��,X0� = − ��X0��. �A12�

Thus, our model Hamiltonian takes the form

H = �k��,

�k�ck�† ck� + �0Z00 + �

�

��Z��

+ �k��,,�

�vk�,� ck�

† X0� + H.c.� . �A13�

We assume also that �ck� ,c�k��† =�k,k��,��,� , �ck� ,d�

†=0.

As was discussed in the Introduction, the particular sectorof the transitions �0⇔1� is remarkable due to the fact thatthe bare energies �� and the tunneling matrix elements coin-cide in the WCI and SCI regimes. This fact allows us tostudy the role played by the strong Coulomb interaction inthe formation of transport properties in a most refined form.Indeed, in the SCI case the electrons in QD can “see” theconduction electrons in the attached contacts only throughthe prism of the in-dot kinematic constraints.

1 L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen, S. Tarucha, R.M. Westervelt, and N. S. Wingren, in Mesoscopic ElectronTransport, Vol. 345 of NATO Advanced Study Institute, SeriesE, edited by L. P. Kouwenhoven, G. Schön, and L. L. Sohn�Kluwer, Dordrecht, 1997�, pp. 105–214.

2 L. P. Kouwenhoven, D. G. Austing, and S. Tarucha, Rep. Prog.Phys. 64, 701 �2001�.

3 C. W. J. Beenakker, Phys. Rev. B 44, 1646 �1991�.4 D. V. Averin, A. N. Korotkov, and K. K. Likharev, Phys. Rev. B

44, 6199 �1991�.5 D. V. Averin and Yu. V. Nazarov, Phys. Rev. Lett. 65, 2446

�1990�.6 See for a review, e.g., M. Pustilnik and L. I. Glazman, J. Phys.:

Condens. Matter 16, R513 �2004�; L. I. Glazman and M.Pustilnik, cond-mat/0501007 �unpublished�.

7 S. Hershfield, J. H. Davies, and J. W. Wilkins, Phys. Rev. Lett.67, 3720 �1991�; B. R. Bulka and S. Lipinski, Phys. Rev. B 67,024404 �2003�; R. Lopez and D. Sanchez, Phys. Rev. Lett. 90,116602 �2003�; J. Paaske, A. Rosch, and P. Wolfle, Phys. Rev. B69, 155330 �2004�.

8 R. Schleser, T. Ihn, E. Ruh, K. Ensslin, M. Tews, D. Pfannkuche,D. C. Driscoll, and A. C. Gossard, Phys. Rev. Lett. 94, 206805�2005�.

9 J. Fransson, O. Eriksson, and I. Sandalov, Phys. Rev. Lett. 88,226601 �2002�; Phys. Rev. B 66, 195319 �1996�.

10 S. Tarucha, D. G. Austing, T. Honda, R. J. van der Hage, and L.P. Kouwenhoven, Phys. Rev. Lett. 77, 3613 �1996�.

11 W. Kohn, Phys. Rev. 123, 1242 �1961�.12 Ch. Sikorski and U. Merkt, Phys. Rev. Lett. 62, 2164 �1989�.13 V. Fock, Z. Phys. 47, 446 �1928�; C. G. Darwin, Proc. Cambridge

Philos. Soc. 27, 86 �1930�.14 K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Physica B

314, 450 �2002�; T. Ota, K. Ono, M. Stopa, T. Hatano, S.Tarucha, H. Z. Song, Y. Nakata, T. Miyazawa, T. Ohshima, andN. Yokoyama, Phys. Rev. Lett. 93, 066801 �2004�.

15 M. Dineykhan and R. G. Nazmitdinov, Phys. Rev. B 55, 13707�1997�; J. Phys.: Condens. Matter 11, L83 �1999�.

16 I. Sandalov, B. Johansson, and O. Eriksson, Int. J. QuantumChem. 94, 113 �2003�; see also cond-mat/0011260 �unpub-lished�; cond-mat/0011261 �unpublished�.

17 H. Schoeller and G. Schön, Phys. Rev. B 50, 18436 �1994�.18 I. Sandalov and R. G. Nazmitdinov, J. Phys.: Condens. Matter

18, L55 �2006�.19 J. Hubbard, Proc. R. Soc. London, Ser. A 276, 238 �1963�; 277,

237 �1963�.20 Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512 �1992�.21 A. P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B 50, 5528

�1994�.22 P. C. Martin and J. Schwinger, Phys. Rev. 115, 1342 �1959�.23 L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics

�Addison-Wesley, New York, 1989�.24 D. C. Langreth, in Linear and Nonlinear Electron Transport in

Solids, Vol. 17 of Nato Advanced Study Institute, Series B:Physics, edited by J. T. Devreese and V. E. van Doren �Plenum,New York, 1976�, pp. 3–32.

25 N. S. Simonovic and R. G. Nazmitdinov, Phys. Rev. B 67,041305�R� �2003�.

26 W. D. Heiss and R. G. Nazmitdinov, Pis’ma Zh. Eksp. Teor. Fiz.68, 870 �1998� �JETP Lett. 68, 915 �1998�.

27 V. N. Golovach and D. Loss, Phys. Rev. B 69, 245327 �2004�.


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