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Electronic correlations in coherent transport through a two quantum dot system
Abstract
Coherent electronic transport through a double quantum dot (2qd) system connected in a series with electrodes is studied by means of nonequilibrium Green functions and using the equation of motion method, in which all electron correlations in the 2qd are treated exactly. For moderate Coulomb interactions we predict features in a conductance characteristics resulting from transmission through triplet states, which can be strongly activated for larger source-drain voltages. The analysis of the spin-spin correlation functions shows strong antiferromagnetic correlations arising from transport through the singlet state and a reduction of the total magnetic moment. However, when the transmission channel corresponding to the triplet state becomes activated the antiferromagnetic correlations are much weaker, the spins behave as free electrons spins and strongly fluctuate. We speculate that this effect can be seen in wide range of a gate voltage for a double electron occupancy of the 2qd.
... Since then nanoscale thermoelectricity has been addressed by an increasing number of theoretical and experimental works; a perspective of the field can be found in the focus point collection in [2] and in the articles appeared in [3]. In particular, interference Ahronov-Bohm [4][5][6][7], Fano [8][9][10][11], Dicke [12,13] and Mach-Zehnder [14,15] effects, inter-and intra-dot correlation effects [16][17][18], coherent transport modification by external magnetic fields and gate voltages. [19][20][21], have been exploited to control the performance of thermoelectric heat devices. ...
... (almost an order of magnitude) assures that in the transmission function the Lorentz lineshape and the Fano lineshape are in general well resolved, with linewidths 2 cos 4 2 g q ( ) and 2 sin 4 2 g q ( ), respectively, as it is seen from equation (16). The values of θ explored to better highlight periodicity as function of θ, are in the whole range [0, 4π], and in particular θ=0, π/2, π, 3π/2 and 2π. ...
... the bonding state, and effective width Γ eff =2γ. In the presence of one flux quantum (or any odd integer number of flux quanta) equation(16) gives the anti-bonding state, and effective width Γ eff =2γ. At semi-integer flux quanta θ=π (or any odd integer number of π) the transmission function versus E takes the symmetric structure with respect to the dot energy E d , with expression ...
Coupled double quantum dots (c-2QD) connected to leads have been widely adopted as prototype model systems to verify interference effects on quantum transport at the nanoscale. We provide here an analytic study of the thermoelectric properties of c-2QD systems pierced by a uniform magnetic field. Fully analytic and easy-to-use expressions are derived for all the kinetic functionals of interest. Within the Green 0 s function formalism, our results allow a simple inexpensive procedure for the theoretical description of the thermoelectric phenomena for different chemical potentials and temperatures of the reservoirs, different threading magnetic fluxes, dot energies and interdot interactions; moreover they provide an intuitive guide to parametrize the system Hamiltonian for the design of best performing realistic devices. We have found that the thermopower S can be enhanced by more than ten times and the figure of merit ZT by more than hundred times by the presence of a threading magnetic field. Most important, we show that the magnetic flux increases also the performance of the device under maximum power output conditions.
... Multi-level systems were started to be considered only recently [210,211]. Besides, there are some difficulties in building the lesser GF in the nonequilibrium case (at finite bias voltages) by means of the EOM method [212,213,214]. ...
... [205] Later the same method was applied to some two-site models. [206,207,208,214] Multi-level systems were started to be considered only recently. [210,211] For out-of-equilibrium situations (finite applied bias), there are some methodological unclarified issues for calculating correlation functions using EOM techniques. ...
... [210,211] For out-of-equilibrium situations (finite applied bias), there are some methodological unclarified issues for calculating correlation functions using EOM techniques. [212,213,214] We have developed an EOM-based method which allows to deal with the finite-bias case in a self-consistent way. [209] 4. 1 ...
The theoretical investigation of charge (and spin) transport at nanometer length scales requires the use of advanced and powerful techniques able to deal with the dynamical properties of the relevant physical systems, to explicitly include out-of-equilibrium situations typical for electrical/heat transport as well as to take into account interaction effects in a systematic way. Equilibrium Green function techniques and their extension to non-equilibrium situations via the Keldysh formalism build one of the pillars of current state-of-the-art approaches to quantum transport which have been implemented in both model Hamiltonian formulations and first-principle methodologies. We offer a tutorial overview of the applications of Green functions to deal with some fundamental aspects of charge transport at the nanoscale, mainly focusing on applications to model Hamiltonian formulations.
... Several theoretical methods have been proposed to study the charge transport through molecular junctions in the presence of electron-electron interaction and beyond the single-site model, such as Landauer's formula with meanfield approximation, [18][19][20] nonequilibrium Green's function (NEGF) with GW approximation, 21-23 many-body perturbation theory, 24, 25 equation of motion method, [26][27][28][29][30] master equation, [31][32][33] quantum master equation, 34,35 hierarchical equations of motion approach, 36,37 renormalized perturbation theory, 38 source-sink potential formalism, 39,40 and dynamical mean-field theory. 41 However, most methods are useful either for the Coulomb blockade regime, where the electronelectron interaction dominates, or the coherent tunneling regime, where the molecule-lead interaction dominates. ...
... we can find that the zeroth-order lesser and greater Green's functions are given by Eqs. (28) and (29), and the occupation number is solved from Eq. (A25). While the parameter m nσ is arbitrary, we can choose that ...
We study charge transport through molecular junctions in the presence of electron-electron interaction using the nonequilibrium Green's function techniques and the renormalized perturbation theory. In the perturbation treatment, the zeroth-order Hamiltonian of the molecular junction is composed of independent single-impurity Anderson's models, which act as the channels where charges come through or occupy, and the interactions between different channels are treated as the perturbation. Using this scheme, the effects of molecule-lead, electron-electron, and hopping interactions are included nonperturbatively, and the charge transport processes can thus be studied in the intermediate parameter range from the Coulomb blockade to the coherent tunneling regimes. The concept of quasi-particles is introduced to describe the kinetic process of charge transport, and then the electric current can be studied and calculated. As a test study, the Hubbard model is used as the molecular Hamiltonian to simulate dimeric and trimeric molecular junctions. Various nonlinear current-voltage characteristics, including Coulomb blockade, negative differential resistance, rectification, and current hysteresis, are shown in the calculations, and the mechanisms are elucidated.
... Electron tunneling through double quantum dot confined in the gap between two metal electrodes is the subject of a large number of studies. The contact in vacuum is usually considered [1][2][3][4][5][6][7][8][9][10][11]. However, it was shown [12][13][14][15][16][17][18][19][20] that strong interaction of electrons with vibrational modes of the solvent for the in situ systems plays a crucial role in determining the physical mechanism of transitions. ...
... Let us consider the small k limit and compare the exact expression for the tunnel current and that obtained in the "quasi-chemical approximation" in the case when U = 0. Using Eqs. (6) and (13) ...
A consistent method for the calculation of the electric current through a redox-mediated electrochemical tunnel contact with two redox groups in the bridge molecule is presented for the sequential mechanism of electron transfer in the limit of infinitely large Coulomb repulsion between the electrons located in the same redox group. It is shown that the kinetic inter-group correlation exists which results in general in the deviation of exact solution from the approximate “quasi-chemical approach” even in the absence of inter-group Coulomb repulsion. The increase of the latter leads to the appearance of the electron correlation effects which result, in particular, in the second maximum (Coulomb blockade peak) in the current/overvoltage dependence.
... For a specific form of L(ǫ), such as a resonant transmission function (RTF) (a Lorentzian), an exact solution of eq (1) exists [6,7]. Thus, if the L(ǫ) can be represented as a polynomial functional of RTFs, such as a linear superposition of multiple (Lorentzian) resonances (at the lowest order) ...
... Substituting eq (2) with A r = γ 2 r in eq (1), and transforming it to a contour integration [6], we get the following expression [7] for the current I(V, T L , T R ). ...
In this work, we have analyzed the exact closed-form solutions for transport
quantities through a mesoscopic region which may be characterized by a
polynomial functional of resonant transmission functions. These are then
utilized to develop considerably improved protocols for parameters relevant for
quantum transport through molecular junctions and quantum dots. The protocols
are shown to be experimentally feasible and should yield the parameters at much
higher resolution than the previously proposed ones.
... From the theoretical side, the electrical transport properties in DQDs have been widely studied by using various methods going from scattering matrix theory [41][42][43] for the noninteracting case to Master or Bloch equation approaches [44][45][46][47][48][49][50] and nonequilibrium Green function methods [51][52][53][54][55][56] for more general cases. A classical theory has also been developed along which the DQD is modeled as a network of resistors and capacitors which mimic the tunnel and electrostatic couplings between dots and leads [4][5][6]. ...
We study electrical and thermoelectrical properties for a double quantum dot system. We consider the cases of both single-level and multilevel quantum dots whatever the way they are coupled, either in a series or in a parallel arrangement. The calculations are performed by using the nonequilibrium Green function theory. In the case of a single-level double quantum dot, the problem is exactly solvable whereas for a multilevel double quantum dot, an analytical solution is obtained in the limit of energy-independent hopping integrals. We present a detailed discussion about the dependences of electrical conductance, zero-frequency charge susceptibility, and Seebeck coefficient on the gate voltages applied to the dots, allowing us to derive the charge stability diagram. The findings are in agreement with the experimental observations notably with the occurrence of successive sign changes of the Seebeck coefficient when varying the gate voltages. We interpret the results in terms of the bonding and antibonding states produced by the level anticrossing effect which occurs in the presence of a finite interdot coupling. We show that at equilibrium the boundary lines between the domains with different dot occupancies in the charge stability diagram take place when the bonding and antibonding state levels are aligned with the chemical potentials in the leads. Finally the total dot occupancy is found to be considerably reduced in the case in parallel compared with the case in series, whenever the level energies in each dot are equal. We interpret this dip as a direct manifestation of the interference effects occurring in the presence of the two electronic transmission paths provided by each dot.
... From the theoretical side, the electrical transport properties in DQD have been widely studied and the approaches to model them are mainly based on the use of Master equations for the density matrix [38][39][40][41][42][43] or on the use of Keldysh Green function methods [44][45][46][47][48][49] which becomes es-sential when one wants to treat out-of-equilibrium situations. The overall evolution of the linear conductance as a function of gate voltages is well understood within a classical theory along which the DQD is modeled as a network of resistors and capacitors which mimic the tunnel and electrostatic couplings between dots and leads 4-6 : (i) at weak interdot coupling, conductance peaks are observed at the nodes of a square lattice, (ii) at intermediate interdot coupling, pairs of triple-points arise at the boundaries between the regions with different electron occupancy in the dots to form a honeycomb lattice in the plane (ε 1 , ε 2 ), where ε 1 and ε 2 are the energy levels of the two dots, and (iii) at strong interdot coupling, the triple-point separation reaches its maximum and the DQD behaves as a single dot with an occupancy N 1 + N 2 , where N 1 and N 2 are the average number of electrons in the dots 1 and 2 respectively. ...
A theory is developed to model a double quantum dot in a unified way whatever the geometry of the two dots is, either in series or in parallel. In the case of a single-level double quantum dot, the problem is exactly solvable whereas for a multi-level double quantum dot, an analytical solution is obtained in the limit of energy-independent hopping integrals. After deriving the expressions for the non-equilibrium Green functions, we study the dependences of the conductance, zero-frequency charge susceptibility and Seebeck coefficient on the gate voltages applied to the dots, allowing us to derive the charge stability diagram revealing a wide variety of behaviors for the system. The findings are in agreement with the experimental observations notably with the occurrence of successive sign changes of the Seebeck coefficient when varying the gate voltages. We interpret the results in terms of the bonding and antibonding states produced by the level anticrossing effect which occurs in the presence of a finite interdot coupling. We show that at equilibrium the boundary lines between the domains with different dot occupancies in the charge stability diagram, take place when the bonding and antibonding state levels are aligned with the chemical potentials in the leads. Finally the total dot occupancy is found to be considerably reduced in the case in parallel compared with the case in series, when the level energies in each dot are equal. We interpret this dip as a direct manifestation of the interference effects occurring in the presence of the two electronic transmission paths provided by each dot.
... 到目前为止, 对纳米结构中电子自旋的电 学、 磁学和光学的调制技术已经日益成熟. 近 期 基 于 四 端 铁 磁 材 料 系 统 的 实 验 中 发 现 了 一 种 新 的 热 电 现 象, 被 称 为 自 旋 能 斯 特 效 应 (spin Nernst effect) [12] 或 横 向 自 旋 塞 贝 克 效 应 [17,18] ...
With the increase of integration scale, heat dissipation becomes one of the major problems in high density electronic devices and circuits. Controlling and reusing the heat energy in such miniaturized structures are essential topics for current and future technologies. With the development of microfabrication technology and low-temperature measurement technology in the last two decades, the thermoelectric measurement in low-dimensional sample has been feasible, and the thermal transport has received more and more attention. For the multi-terminal device, there is a novel thermoelectric phenomenon, called the spin Nernst effect, in which spin currents (or spin voltages) are generated perpendicularly to the temperature gradient. The spin Nernst effect has been confirmed experimentally, and has been theoretically studied in a variety of materials. In this paper, the spin and charge Nernst effect in a pair of vertically aligned quantum dots attached to four leads are studied in the Coulomb blockade regime based on the nonequilibrium Green's function technique. We focus on the influences of magnetic configuration and intra-dot (inter-dot) Coulomb interaction on the spin and charge Nernst effect. It is found that the signs and the magnitudes of spin and charge Nernst effect can be modulated by adjusting the magnetization directions of ferromagnetic electrodes. When the magnetic moments in the 1 and 3 electrodes are turned to antiparallel alignment, the pure spin Nernst (without charge Nernst) effect can occur by applying a transverse temperature gradient. Conversely, the spin and charge Nernst effect disappear if the magnetic moments of lead 1 and lead 3 are in the case of parallel configuration. Except for left and right thermal leads, we investigate the effect of the middle lead (lead 4) on the property of the Nernst effect. We find that when the normal metal lead 4 is transferred to ferromagnetic metal, the spin and charge Nernst effect both can be obtained simultaneously. In the end of the paper, we study the influences of intra-dot and inter-dot Coulomb interaction on the spin dependent Nernst coefficient. Through numerical calculations, we demonstrate that the magnitude of the Nernst effect is less dependent on the polarization strength of ferromagnetic electrodes, but can be remarkably enhanced by the Coulomb blockade. The spin Nernst coefficient is predicted to be more than two orders of magnitude larger than that of the case of zero Coulomb interaction. All the results indicate that the proposed four-terminal double quantum dot nano system is a promising candidate for spin caloritronic device.
... The value of t changes only the wide peak position and hardly change the height of these peaks. 58,59 The thermal conductance K as a function of ε is shown in Fig. 4(b). In thermal conductance spectrum a minimum (a valley) appear in the centre peak at ε = 0 due to the bipolar effect (i.e., the presence of electrons and holes allows heat conduction to take place even if the net electrical current is zero). ...
We theoretically analyze the properties of thermoelectric transport through a T-shaped DQD connected to ferromagnetic and superconducting electrodes by means of nonequilibrium Green function formalism. The influences of the superconducting gap, interdot tunneling coupling and asymmetry parameter on the thermoelectric properties are discussed. The large thermoelectric efficiency can be obtained by choosing small polarization of ferromagnetic electrode, small asymmetry parameter ( < 1), appropriately large gap and appropriately interdot coupling, which can be used as the optimal schemes for obtaining high thermoelectric efficiency in the device.
... In quantum dots the conductance shows universal dependences [3] in agreement with theoretical studies based on a single impurity Anderson model [4]. The problem is obviously richer for multi-dot systems where one can expect interplay of the Kondo ground state with internal magnetic orderings [5][6][7][8][9] as well as a quantum phase transition [10][11][12][13][14][15][16]. There is a competition between the Kondo effect with various intra-and inter-dot electron correlations. ...
A system of three coupled quantum dots in a triangular geometry (TQD) with electron–electron interaction and symmetrically coupled to two leads is analyzed with respect to the electron transport by means of the numerical renormalization group. Varying gate potentials this system exhibits extremely rich range of regimes with different many-electron states with various local spin orderings. It is demonstrated how the Luttinger phase changes in a controlled manner which then via the Friedel sum rule formula exactly reproduces the conductance through the TQD system. The analysis of the uncoupled TQD molecule from the leads gives a reliable qualitative understanding of various relevant regimes and an insight into the phase diagram with the regular Fermi liquid and singular-Fermi liquid phases.
... In quantum dots the conductance shows universal dependences [3] in agreement with theoretical studies based on a single impurity Anderson model [4]. The problem is obviously richer for multi-dot systems where one can expect interplay of the Kondo ground state with internal magnetic orderings [5][6][7][8][9] as well as a quantum phase transition [10][11][12][13][14][15][16]. There is a competition between the Kondo effect with various intra-and inter-dot electron correlations. ...
... 25) The lesser Green's functions hd l; d y l; i < and hn l;À d l; d y l; i < can be obtained by the equation of motion method. 26,27) In the Coulomb blockade regime we have hd l; d y l; i < ¼ f < ðÞ2 Im G r l; ðÞ and hd l; n l;À d y l; i < ¼ f < ðÞ2 Im G r l;l ðÞ, where f < ðÞ ¼ ÀiðÀ l;L ðÞ f L ðÞ þ À l;R f R ðÞÞ=ðÀ l;L þ À l;R Þ. ...
The tunneling current through multiple quantum dots (QDs) sandwiched between metallic contacts is theoretically studied in the framework of Green's function technique. The Anderson model with multiple energy levels is employed to simulate the considered system. We found that the negative differential resistance (NDR) arises from the interdot Coulomb interactions and the QDs under shell-filling conditions. Such NDRs are suppressed by the temperature effect. It is expected that multipeak NDR devices could be realized using a sophisticated layout of quantum dots.
... Here we focus on the role the inter-dot (capacitive) interaction plays in the electron transport through serially coupled DQD. The influence of the capacitive coupling was already studied by means of the equation-of-motion method 23,24,25,26 which, however, fails to capture the Kondo correlations accurately. On the other hand, the Kondo correlations were considered in the limit of vanishing tunnel coupling and strong interdot interaction where the ground state of the isolated DQD exhibits four-fold degeneracy: in addition to spin degeneracy also singly occupied states labeled with (1, 0) and (0,1) are degenerate leading to orbital Kondo behavior 27,28,29,30,31 . ...
We investigate a symmetrical double quantum dot system serially attached to the leads. The emphasis is on the numerical analysis of finite interdot tunneling in the presence of interdot repulsive capacitive coupling. The results reveal the competition between extended Kondo phases and local singlet phases in spin and charge degrees of freedom. The corresponding phase diagram is determined quantitatively.
... Assumption (c) is equivalent to treating the coupling to the leads up to the second order with respect to t σ km . It neglects processes necessary to qualitatively capture the Kondo effect, [49][50][51] yet results are predicted to be reliable for temperatures above the Kondo temperature (T K ). 24,52 The resulting equations are given by (for brevity, we omit the implicit dependence on ω): ...
We study steady state transport through a double quantum dot array using the equation-of-motion approach to the nonequilibrium Green functions formalism. This popular technique relies on uncontrolled approximations to obtain a closure for a hierarchy of equations; however, its accuracy is questioned. We focus on 4 different closures, 2 of which were previously proposed in the context of the single quantum dot system (Anderson impurity model) and were extended to the double quantum dot array, and develop 2 new closures. Results for the differential conductance are compared to those attained by a master equation approach known to be accurate for weak system-leads couplings and high temperatures. While all 4 closures provide an accurate description of the Coulomb blockade and other transport properties in the single quantum dot case, they differ in the case of the double quantum dot array, where only one of the developed closures provides satisfactory results. This is rationalized by comparing the poles of the Green functions to the exact many-particle energy differences for the isolate system. Our analysis provides means to extend the equation-of-motion technique to more elaborate models of large bridge systems with strong electronic interactions.
... Moreover, the considered situations were limited to transport through two quantum dots coupled either in series or in parallel to nonmagnetic electron reservoirs. The electron correlations in two QDs coupled in series to nonmagnetic leads have been taken into account in a recent paper [5]. As concerns transport through double quantum dots (DQDs) attached to magnetic leads, only a few papers have addressed this issue up to now [6][7][8]. ...
Spin-dependent transport through two coupled single-level quantum dots attached to ferromagnetic leads with collinear (parallel and antiparallel) magnetizations is analyzed theoretically by the nonequilibrium Green function technique. Transport characteristics, in particular, linear and nonlinear differential conductance and tunnel magnetoresistance associated with the magnetization rotation from antiparallel to parallel alignment, are calculated numerically with intradot Coulomb interaction taken into account. The relevant Green functions are derived by the equation of motion method within the Hartree-Fock decoupling scheme. The dot occupations and Green functions are calculated self-consistently, and the numerical analysis is focused on the interference (Fano antiresonance) and Coulomb interaction effects. It is shown that the presence of Fano antiresonance depends on the sign of the nondiagonal coupling elements.
We model a molecular device as a molecule attached to a set of leads treated at the tight-binding level, with the central molecule described to any desired level of electronic structure theory. Within this model, in the absence of electron-phonon interactions, the Landauer-B\"{u}ttiker part of the Meir-Wingreen formula is shown to be sufficient to describe the transmission factor of the correlated device.The key to this demonstration is to ensure that the correlation self-energy has the same functional form as the exact correlation self-energy. This form implies that non-symmetric contributions to the Meir-Wingreen formula vanish, and hence conservation of current is achieved without the need for Green's Function self-consistency. An extension of the Source-Sink-Potential (SSP) approach gives a computational route to the calculation and interpretation of electron transmission in correlated systems. In this picture, current passes through internal molecular channels via resonance states with complex-valued energies. Each resonant state arises from one of the states in the Lehmann expansion of the one-electron Green's Function, hole conduction deriving from ionised states, and particle conduction from attached states. In the correlated device, the dependence of transmission on electron energy is determined by four structural polynomials, as it was in the tight-binding (H\"{u}ckel) version of the SSP method. Hence, there are active and inert conduction channels (in the correlated case, linked to Dyson orbitals) governed by a set of selection rules that map smoothly onto the simpler picture.
We investigate two prototypical dissipative bosonic systems under slow driving and arbitrary system-bath coupling strength, recovering their dynamic evolution as well as the heat and work rates, and we verify that thermodynamic laws are respected. Specifically, we look at the damped harmonic oscillator and the damped two-level system. For the former, we study independently the slow time- dependent perturbation in the oscillator frequency and in the coupling strength. For the latter, we concentrate on the slow modulation of the energy gap between the two levels. Importantly, we are able to find the entropy production rates for each case without explicitly defining nonequilibrium extensions for the entropy functional. This analysis also permits the definition of phenomenological friction coefficients in terms of structural properties of the system-bath composite.
We extend the quasi-particle renormalized perturbation theory developed in our previous work [Y.-W. Chang and B.-Y. Jin, J. Chem. Phys. 141, 064111 (2014)] based on nonequilibrium Green’s function techniques to study the effects of electron correlation on the charge transport process in molecular junctions. In this formalism, the single-impurity Anderson’s model is used as the zeroth-order Hamiltonian of each channel orbital, and the inter-channel interactions are treated by perturbation corrections. Within this scheme, the on-channel Coulomb repulsion and the single-particle spectral line-broadening can be incorporated in the zeroth-order approximation, and thus the Coulomb blockade and coherent tunneling through individual channels can be described properly. Beyond the zeroth-order description, electron correlation can be included through the self-energy corrections in the forms of the second-Born approximation and the GW approximation. The effects of electron correlation on molecular junctions are manifested as the orbital energy correction, correlated transport process, and collisional line-broadening. As an application, we have applied the present formalism to phenyl-based molecular junctions described by the Pariser-Parr-Pople Hamiltonian. The signatures of electron correlation in the simulated current-voltage curves are identified and discussed.
We theoretically investigate the thermoelectric properties of a serial double quantum dot junction system. A two-level Anderson model including electron hoppings and intradot Coulomb interactions as well as interdot Coulomb interactions is employed to simulate the system. The charge and heat currents in the Coulomb blockade regime are calculated by Keldysh Green's function technique. The electrical conductance, Seebeck coefficient, electronic thermal conductance, and figure of merit (ZT) of the system are calculated in the linear response regime. We find that the figure of merit ZT is markedly reduced by the size fluctuation and Coulomb interactions.
Some examples of the NGF method for electron-electron and electron-vibron interactions are given. The Dyson-Keldysh equations are applied to electron-vibron interaction. The spectroscopic problems with equilibrium vibrons and the vibronic instability and strongly nonequilibrium vibrons are discussed. The equations of motion for Heisenberg operators and Green functions obtained from the Hubbard-Anderson Hamiltonian are presented.
The exchange field for molecular states of double quantum dot, induced by two ferromagnets coupled to the device in T-shaped configuration, is defined and calculated. It is found, that in the regime of strong coupling between quantum dots, the dependence of the exchange field on this coupling becomes nontrivial. In particular, it changes the sign a few times to eventually vanish in the limit of infinite inter-dot coupling. The excitation energies of double quantum dot are calculated and the results used to predict the conditions for suppression of the two-stage Kondo effect in the considered nanostructure.
We study the charge transport properties of triangular quantum dot molecule (TQDM) connected to metallic electrodes, taking into account all correlation functions and relevant charging states. The quantum interference (QI) effect of TQDM resulting from electron coherent tunneling between quantum dots is revealed and well interpreted by the long distance coherent tunneling mechanism. The spectra of electrical conductance of TQDM with charge filling from one to six electrons clearly depict the many-body and topological effects. The calculated charge stability diagram for conductance and total occupation numbers match well with the recent experimental measurements. We also demonstrate that the destructive QI effect on the tunneling current of TQDM is robust with respect to temperature variation, making the single electron QI transistor feasible at higher temperatures.
Electron transport at zero temperature through T-shaped double quantum dot attached to the non-interacting leads is studied using Keldysh non-equilibrium Green's function technique. Linear conductance profile and dot occupancies are calculated for various parameters corresponding
to non-interacting as well as interacting electrons on the dots. In case of non-interacting electrons, we observe Fano-antiresonance wherein the linear conductance vanishes (despite occupancies on the dots being finite) whenever the energy level of the quantum dot not directly attached to
the leads, aligns with the Fermi energy of the electrons in the leads at zero-bias. This is understood in terms of destructive interference between several possible Feynman paths between the source and the drain. Electron–electron correlation on the dots incorporated via intradot and
interdot interaction is investigated in Hartree-Fock as well as beyond Hartree-Fock approximation. Results obtained using present decoupling scheme for Green's functions shows that the intradot interaction on the quantum dot not directly connected to the leads removes the anti-resonance point
and leads to splitting into two dips in the linear conductance profile. Results are compared with the one obtained using Hartee-Fock approximation.
We consider a projection operator approach to the non-equilbrium Green
function equation-of-motion (PO-NEGF EOM) method. The technique resolves
problems of arbitrariness in truncation of an infinite chain of EOMs, and
prevents violation of symmetry relations resulting from the truncation. The
approach, originally developed by Tserkovnikov [Theor. Math. Phys. 118, 85
(1999)] for equilibrium systems, is reformulated to be applicable to
time-dependent non-equilibrium situations. We derive a canonical form of EOMs,
thus explicitly demonstrating a proper result for the non-equilibrium atomic
limit in junction problems. A simple practical scheme applicable to quantum
transport simulations is formulated. We perform numerical simulations within
simple models, and compare results of the approach to other techniques, and
(where available) also to exact results.
A theoretical approach to a problem of electron transport through double
quantum dot systems based on non-equilibrium Green function formalism
using equation of motion method is presented. I-V characteristics and
differential conductance are calculated and discussed in detail in the
intermediate regime with tunneling rate between the quantum dots
comparable to coupling constants with external electrodes. Effects of
inter-dot Coulomb correlations are studied for various values of
interaction parameter U. It is shown that the interaction influences
transport properties in a pronounced way and apart from the simple
Coulomb blockade additional effects can be obtained. When energy levels
of two quantum dots are not aligned, the asymmetry in conductance
characteristics is closely related to a voltage dependence of population
numbers in both quantum dots. For a one bias polarization electrons are
well localized in quantum dots in a low voltage region, whereas for the
opposite one they are partly delocalized.
We present studies of the role of charge fluctuations in transport through a quantum point contact (QPC). Our model of QPC is based on assumption of a specific electronic energy structure with a resonant level below the electronic sub-band. We show that charge fluctuations lead to a dynamical Coulomb blockade effect, which is responsible for reduction of the conductance to the value 2/3×(2e2/h). The evolution of conductance with a magnetic field and as a function of source-drain voltage is presented as well. The conductance plateau at 2/3×(2e2/h) evolves continuously with an increase of the voltage to a 0.8 or 0.4 plateau, depending on the relative position of the resonant state with respect to the Fermi energy at zero bias. Our simple model shows many similarities with experimental characteristics.
The effect of long distance coherent tunneling (LDCT) on the charge and heat
currents in serially coupled triple quantum dots (TQDs) connected to electrodes
is illustrated by using a combination of the extended Hurbbard model and
Anderson model. The charge and heat currents are calculated with a closed-form
Landauer expression for the transmission coefficient suitable for the Coulomb
blockade regime. The physical parameters including bias-dependent quantum dot
energy levels, electron Coulomb interactions, and electron hopping strengths
are calculated in the framework of effective mass theory for semiconductor
TQDs. We demonstrate that the effect of LDCT on the charge and heat currents
can be robust. In addition, it is shown that prominent heat rectification
behavior can exist in the TQD system with asymmetrical energy levels.
We theoretically investigate the thermoelectric properties of a serial double quantum dot junction system. A two-level Anderson model including electron hoppings and intradot Coulomb interactions as well as interdot Coulomb interactions is employed to simulate the system. The charge and heat currents in the Coulomb blockade regime are calculated by Keldysh Green's function technique. The electrical conductance, Seebeck coefficient, electronic thermal conductance, and figure of merit (\mathit{ZT}) of the system are calculated in the linear response regime. We find that the figure of merit \mathit{ZT} is markedly reduced by the size fluctuation and Coulomb interactions.
Electron transport properties of an isolated quantum dot sandwiched between a metallic contact and a scanning tunneling microscopy tip are theoretically investigated. Keldysh-Green’s function technique is used to calculate the tunneling current of an Anderson model with multiple energy levels. The spectral function of the quantum dot system (with arbitrary number of energy levels) embedded in a tunnel junction is derived and used to calculate the tunneling current spectra. Finally, the authors calculate the emission spectra due to the electron-hole recombination that occurs in the case of bipolar tunneling, where both electrons and holes are allowed to simultaneously tunnel into the quantum dot. The authors find dramatic changes in the emission spectra as the applied bias is varied.
The dynamics of correlated electrons in quantum impurity models is typically described within the nonequilibrium Green function formalism combined with a suitable approximation. One common approach is based on the equation-of-motion technique often used to describe different regimes of the dynamic response. Here, we show that this approach may violate certain symmetry relations that must be fulfilled by the definition of the Green functions. These broken symmetries can lead to unphysical behaviour. To circumvent this pathological shortcoming of the equation-of-motion approach we provide a scheme to restore basic symmetry relations. Illustrations are given for the Anderson and double Anderson impurity models.
Numerical analysis of the simplest odd-numbered system of coupled quantum dots reveals an interplay between magnetic ordering, charge fluctuations, and the tendency of itinerant electrons in the leads to screen magnetic moments. The transition from local-moment to molecular-orbital behavior is visible in the evolution of correlation functions as the interdot coupling is increased. Resulting Kondo phases are presented in a phase diagram which can be sampled by measuring the zero-bias conductance. We discuss the origin of the even-odd effects by comparing with the double quantum dot.
Theoretical studies of the coherent electronic transport in a system of coupled quantum wires show that switching on the conducting channel in one wire can be manifested in the other coupled wires. Interference processes and electronic correlations are taken into account in our studies on the same footing. The conductance changes depend on the interference conditions of a transmitted wave with that one reflected from the wires that indirectly influence the transport. We show that electronic correlations lead to a dynamical Coulomb blockade effect, which changes the conductance response quantitatively but its shape is still kept the same. Our results are discussed in correspondence with an experiment recently performed by Morimoto et al. Appl. Phys. Lett. 82 3952 (2003) on a system of coupled quantum wires.
We present a procedure for calculation of transport characteristics of molecular junctions. It is based on the nonequilibrium Green’s functions and exploits the Hubbard operators, which allows us to treat formally exactly all electron correlations within the molecule. The procedure reproduces exact results in the limiting cases: for a weak molecule-lead coupling and for high temperatures (i.e., Coulomb blockade limit), and for the limit of vanishing electron interactions. Between these limits the method can be applied as an interpolating scheme. As an example of an application we present the results obtained for a two-atom molecule.
We investigate linear and nonlinear transport in a double quantum dot system weakly coupled to spin-polarized leads. In the linear regime, the conductance as well as the nonequilibrium spin accumulation are evaluated in analytic form. The conductance as a function of the gate voltage exhibits four peaks of different heights with mirror symmetry with respect to the charge neutrality point. As the polarization angle is varied, due to exchange effects, the position and shape of the peaks change in a characteristic way, which preserves the electron-hole symmetry of the problem. In the nonlinear regime, various spin-blockade effects are observed. Moreover, negative differential conductance features occur for noncollinear magnetizations of the leads. In the considered sequential tunneling limit, the tunneling magnetoresistance (TMR) is always positive with a characteristic gate voltage dependence for noncollinear magnetization. If a magnetic field is added to the system, the TMR can become negative.
Pronounced effects of the interdot Coulomb repulsion on the tunnel current/gate voltage dependence at the ambient conditions are predicted for the double quantum dot system in the serial configuration immersed in the electrolyte solution in the case of the weak tunneling of electrons both between the dots and between the dots and leads. Electrons at the dots are coupled strongly to the classical phonon modes and Debye screening of the electric field is taken into account. The infinite intradot Coulomb repulsion limit is used. The effects consist of (i) a very large width of the maximum of the tunnel current/gate voltage dependence [of the order of −kBT ln(k0/k), where k0 and k are the characteristic rates of the electron tunneling between the dots and between the dots and leads, respectively] in the limit k0/k→0, (ii) the dependence of the positions of the maxima of the current/gate voltage curve and their widths on the sign of the difference of the electron energy levels δ of the quantum dots and the energy of the polaron shift, and (iii) narrow-width Coulomb blockade peaks in the tunnel current/gate voltage curve for k0≥k. The dependence of the differential conductance on the gate voltage, the energy of the interdot Coulomb repulsion, the Debye screening length, and values of k0/k and δ are studied. It is shown that the curves of the differential conductance/bias voltage dependence can be very different for different values of these parameters. These parameters also determine the position of the regions of the negative differential conductance which exist in the general case.
Transport properties of nanoscale quantum dots embedded in a matrix connected with metallic electrodes are investigated theoretically. The Green’s function method is used to calculate the tunneling current of an Anderson model with multiple energy levels, which is employed to model the nanoscale tunnel junction of concern. A closed form spectral function of a quantum dot or coupled dots (with arbitrary number of energy levels) embedded in a tunnel junction is derived and rigorously proved via the principle of induction. Such an expression can give an efficient and reliable way for analyzing the complicated current spectra of a quantum dot tunnel junction. Besides, it can also be applied to the coupled dots case, where the negative differential conductance due to the proximity effect is found. Finally, we investigate the case of bipolar tunneling, in which both electrons and holes are allowed to tunnel into the quantum dot, while optical emission occurs. We find dramatic changes in the emission spectra as the applied bias is varied.
We have studied the equilibrium and nonequilibrium electronic transports through a double quantum dot coupled to leads in a symmetrical parallel configuration in the presence of both the inter- and the intradot Coulomb interactions. The influences of the interdot interaction and the difference between dot levels on the local density of states (LDOS) and the differential conductance are paid special attention. We find an interesting zero-bias maximum of the differential conductance induced by the interdot interaction, which can be interpreted in terms of the LDOS of the two dots. Due to the presence of the interdot interaction, the LDOS peaks around the dot levels εi are split, and as a result, the most active energy level which supports the transport is shifted near to the Fermi level of the leads in the equilibrium situation.
A problem of electronic correlations is considered for two specific mesoscopic systems: the quantum point contact (QPC) and the double quantum dot (2QD) system. The systems are described using a generalized Anderson Hamiltonian. We show that charge fluctuations are relevant for electronic transport. In the QPC a local accumulation of charge and the dynamical Coulomb blockade effect lead to the 0.7 structure in the conductance characteristics. The evolution of the conductance with a magnetic field and in non-equilibrium situations is presented as well. The double quantum dot is studied in the approach, in which correlations within the 2QD are treated exactly, whereas the coupling of the 2QD to the leads is considered in the approximation valid at temperatures above the Kondo temperature. We analyse the evolution of the gate voltage dependence of the spin correlation functions and the conductance with the change of the interdot hopping. For the hopping parameter greater than a threshold value of the on-dot repulsion the physics of the device is dominated by the ground state eigenstates of the 2QD and antiferromagnetic correlations in the case of the doubly occupied 2QD. With a decrease of the interdot hopping repulsion below the threshold we observe a significant reduction of the antiferromagnetic coupling between the dots together with an enhanced occupation of the triplet states.
We present a microscopic theory of transport through quantum dot set-ups
coupled to superconducting leads. We derive a master equation for the reduced
density matrix to lowest order in the tunneling Hamiltonian and focus on
quasiparticle tunneling. For high enough temperatures transport occurs in the
subgap region due to thermally excited quasiparticles, which can be used to
observe excited states of the system for low bias voltages. On the example of a
double quantum dot we show how subgap transport spectroscopy can be done.
Moreover, we use the single level quantum dot coupled to a normal and a
superconducting lead to give a possible explanation for the subgap features
observed in the experiments published in Appl. Phys. Lett. 95, 192103 (2009).
Spin-dependent transport through two coupled single-level quantum dots attached to ferromagnetic leads with collinear (parallel and antiparallel) magnetizations is analyzed theoretically. Generally, the intra- and inter-dot Coulomb correlations are taken into account. Transport characteristics, including conductance and tunnel magnetoresistance associated with the magnetization rotation from the parallel to antiparallel configurations, are calculated by the noneqiulibrium Green function technique. The relevant Green functions are derived by the equation of motion method. To close the set of equations we employ the Hartree–Fock decoupling scheme. The dot occupation numbers and the Green functions are calculated self-consistently. We focus on the interplay of interference (Fano resonance) and Coulomb interaction effects. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
We study the thermoelectric effect in a serial-coupled two quantum dots (QDs) device in the Coulomb blockade regime. The electrical conductance, the thermal conductance, the thermopower, and the thermoelectrical figure of merit are calculated by using the Green's function method. It is found that the energy levels of the two dots are split into a series of molecular states, where the electrical and the thermal conductances show resonance peaks. These peaks in the electrical conductance are eliminated by the increase of the temperature, while those in the thermal conductance are enhanced because of the bipolar effect. In quite high temperature regime, the figure of merit has two huge peaks with maximums exceeding 20 in the vicinity of the electron-hole symmetry point. The magnitude of the figure of merit will be suppressed for unequal dots' levels, but is enhanced by the asymmetry of the dot-lead coupling strengths.
Conductance and other physical quantities are calculated in double quantum dots (DQD) connected in series in the limit of coherent tunnelling using a Green's function technique. The inter-dot Coulomb repulsion and the exchange interaction are studied by means of the Kotliar and Ruckenstein slave-boson mean-field approach. The crossover from the atomic to the molecular limit is analyzed in order to show how the conductance in the model depends on the competition between the level broadening (dot-lead coupling) and the dot-dot transmission. The double Kondo effect
was found in the gate voltage characteristics of the conductance in the atomic limit. In the case, when each dot accommodates one electron, the Kondo resonant states are formed between dots and their adjacent leads and transport is dominated by hopping between these two resonances. In the molecular limit the conductance vanishes for sufficiently low gate voltages, which means the Kondo effect disappeared. For small dot-lead coupling the transport characteristics are very sensitive on the influence of the inter-dot Coulomb repulsion and the position of the local energy level. The
resonance region is widened with increase of the inter-dot Coulomb interactions while the exchange interaction has opposite influence.
We study the splitting of the Fano resonance in a Aharonov–Bohm interferometer with a quantum dot in each of its arms. Both intra- and inter-dot Coulomb repulsions are taken into account by employing the Keldysh nonequilibrium Green’s function technique. The single narrow Fano resonance in the noninteracting case is split into two in the presence of either intra- or inter-dot Coulomb interaction. We find that four Fano peaks emerge in the conductance or local density of states spectra when the two kinds of interactions exist simultaneously. Such behavior holds true for the accompanying broad Breit–Wigner type resonance. We also show that the positions of the Fano peaks can be tuned with the aid of the magnetic flux penetrating through the ring, which might have practical applications in device design or quantum computation.
Based on the Keldysh Green's function technique and the equation-of-motion method, we investigate theoretically the electronic transport properties of an Aharonov-Bohm ring with embedded coupled double quantum dots connected to two electrodes in a symmetrical parallel configuration in the presence of strong interdot Coulomb interaction. Special attention is paid to the effects of the interdot Coulomb interaction on the transport properties. It has been shown numerically that the interdot Coulomb interaction gives rise to four electronic states in the ring. The quantum interferences between two strongly coupled electronic states and two weakly coupled ones lead to two Breit-Wigner and two Fano resonances in the linear conductance spectrum with the magnetic flux switched on or the imbalance between the energy levels of two quantum dots. The positions and shapes of the four resonances can be controlled by adjusting the magnetic flux through the device or energy levels of the two quantum dots. When the Fermi energy levels in the leads sweep across the weakly coupled electronic states, the negative differential conductance (NDC) is developed in the current-voltage characteristics for the non-equilibrium case.
The charge transport of a serially coupled quantum dots (SCQD) connected to
the metallic electrodes is theoretically investigated in the Coulomb blockade
regime. A closed-form expression for the tunneling current of SCQD in the
{\color{red} weak interdot hopping} limit is obtained by solving an extended
two-site Hubbard model via the Green's function method. We use this expression
to investigate spin current rectification, negative differential conductance,
and coherent tunneling in the nonlinear response regime. The current
rectification arising from the space symmetry breaking of SCQD is suppressed by
increasing temperature. The calculation of SCQD is extended to the case of
multiple parallel SCQDs for studying the charge ratchet effect and SCQD with
multiple levels. In the linear response regime, the functionalities of spin
filter and low-temperature current filter are demonstrated to coexist in this
system. It is further demonstrated that two-electron spin singlet and triplet
states can be readily resolved from the measurement of Seebeck coefficient
rather than that of electrical conductance.
We investigate the current and noise characteristics of a double quantum dot system. The strong correlations induced by the Coulomb interaction and the Pauli principle create entangled two-electron states and lead to signatures in the transport properties. We show that the interaction parameter Ø, which measures the admixture of the double-occupancy contribution to the singlet state and thus the degree of entanglement, can be directly accessed through the Fano factor of super-Poissonian shot noise.
Various causes for negative differential conductance in transport through an
interacting double quantum dot are investigated. Particular focus is given to
the interplay between the renormalization of the energy levels due to the
coupling to the leads and the decoherence of the states. The calculations are
performed within a basis of many-particle eigenstates and we consider the
dynamics given by the von Neumann-equation taking into account also processes
beyond sequential tunneling. A systematic comparison between the levels of
approximation and also with different formalisms is performed. It is found that
the current is qualitatively well described by sequential processes as long as
the temperature is larger than the level broadening induced by the contacts.
Motivated by activities of several experimental groups we investigate electron transport through two coherent, strongly coupled quantum dots (“double quantum dots”), taking into account both intra- and inter-dot Coulomb interactions. The shot noise in this system is very sensitive to the internal electronic level structure of the coupled dot system and its specific coupling to the electrodes. Accordingly a comparison between experiments and our predictions should allow for a characterization of the relevant parameters. We discuss in detail the effect of asymmetries, either asymmetries in the couplings to the electrodes or a detuning of the quantum dot levels out of resonance with each other. In the Coulomb blockade region super-Poissonian noise appears even for symmetric systems. For bias voltages above the sequential tunneling threshold super-Poissonian noise and regions of negative differential conductance develop if the symmetry is broken sufficiently strongly.
A double-quantum-dot coupled to electrodes with spin-dependent splitting of chemical potentials (spin bias) is investigated theoretically by means of the Green's functions formalism. By applying a large spin bias, the quantum spin in a quantum dot (the dot 1) can be manipulated in a fully electrical manner. To noninvasively monitor the manipulation of the quantum spin in the dot 1, it is proposed that the second quantum dot (the dot 2) is weakly coupled to the dot 1. In the presence of the exchange interaction between the two dots, the polarized spin in the dot 1 behaves like an effective magnetic field and weakly polarizes the spin in the nearby quantum dot 2. By applying a very small spin bias to the dot 2, the spin-dependent transport through the dot 2 can be probed, allowing the spin polarization in the dot 1 to be identified nondestructively. These two steps form a complete scheme to manipulate a trapped spin while permitting this manipulation to be monitored in the double-dot system using pure electric approaches.
Kondo conduction has been observed in a quantum dot with an even number of electrons at the triplet-singlet degeneracy point produced by applying a small magnetic field B orthogonal to the dot plane. At a much larger field B~B*, orbital effects induce the reversed transition from the singlet to the triplet state. We study the newly proposed Kondo behavior at this point. Here the Zeeman spin splitting cannot be neglected, which changes the nature of the Kondo coupling. On the grounds of exact diagonalization results in a dot with cylindrical symmetry, we show that, at odds with what happens at the other crossing point, close to B*, orbital and spin degrees of freedom are ``locked together,'' so that the Kondo coupling involves a fictitious spin 12 only, which is fully compensated for by conduction electrons under suitable conditions. In this sense, spin at the dot is fractionalized. We derive the scaling equation of the system by means of a nonperturbative variational approach. The approach is extended to the B!=B* case and the residual magnetization on the dot is discussed.
The out-of-equilibrium transport properties of a double quantum dot system in the Kondo regime are studied theoretically by means of a two-impurity Anderson Hamiltonian with interimpurity hopping. The Hamiltonian, formulated in slave-boson language, is solved by means of a generalization of the noncrossing approximation (NCA) to the present problem. We provide benchmark calculations of the predictions of the NCA for the linear and nonlinear transport properties of coupled quantum dots in the Kondo regime. We give a series of predictions that can be observed experimentally in linear and nonlinear transport measurements through coupled quantum dots. Importantly, it is demonstrated that measurements of the differential conductance G=dI/dV, for the appropriate values of voltages and interdot tunneling couplings, can give a direct observation of the coherent superposition between the many-body Kondo states of each dot. This coherence can be also detected in the linear transport through the system: the curve linear conductance vs temperature is nonmonotonic, with a maximum at a temperature T* characterizing quantum coherence between both the Kondo states.
We have studied the current vs voltage curves (I–V characteristics) of a mesoscopic device consisting of two electrodes and a molecular wire. The wire Hamiltonian includes both electronic tunneling and Coulomb repulsion within a Hubbard model that is treated at the Hartree–Fock level. The inclusion of electron repulsion is an extension of our previous work that only considered the case of noninteracting electrons. We have found several important features in the calculated characteristics of the wire. These include (1) a staircaselike structure that strongly resembles that associated with Coulomb blockade in heterostructures and quantum dots, but that in the case of the wire is associated with the discrete nature of the molecular resonances; (2) regions of negative differential resistance associated with increased localization of the molecular resonances. Our theoretical model includes a consistent treatment of the conduction in the linear and nonlinear regimes which remains valid even when the device is operated close to resonance. These results can be particularly relevant for a comparison with recent experiments on molecular wires. © 1996 American Institute of Physics.
We have combined direct nanofabrication by local anodic oxidation with conventional electron-beam lithography to produce a parallel double quantum dot based on a GaAs/AlGaAs heterostructure. The combination of both nanolithography methods allows fabrication of robust in-plane gates and Cr/Au top-gate electrodes on the same device for optimal controllability. This is illustrated by the tunability of the interdot coupling in our device. We describe our fabrication and alignment scheme in detail and demonstrate the tunability in low-temperature transport measurements. © 2003 American Institute of Physics.
We report on electron transport through an artificial molecule formed by two tunnel coupled quantum dots, which are laterally confined in a two-dimensional electron system of an AlxGa1- xAs/GaAs heterostructure. Coherent molecular states in the coupled dots are probed by photon-assisted tunneling (PAT). Above 10 GHz, we observe clear PAT as a result of the resonance between the microwave photons and the molecular states. Below 8 GHz, a pronounced superposition of phonon- and photon-assisted tunneling is observed. Coherent superposition of molecular states persists under excitation of acoustic phonons.
We apply the Hubbard Hamiltonian to describe quantum-dot arrays weakly coupled to two contacts. Exact diagonalization is used to calculate the eigenstates of the arrays containing up to six dots and the linear-response conductance is then calculated as a function of the Fermi energy. In the atomic limit the conductance peaks form two distinct groups separated by the intradot Coulomb repulsion, while in the band limit the peaks occur in pairs. The crossover is studied. A finite interdot repulsion is found to cause interesting rearrangements in the conductance spectrum.
We investigate the linear-response conductance through a pair of coupled quantum dots. The conductance spectrum under ideal conditions is shown to consist of two sets of twin peaks whose locations and amplitudes are determined by the interdot coupling and the intradot charging. We will show that the qualitative features of the spectrum survive against experimental nonidealities such as (1) detuning of the individual dots, (2) interdot charging, (3) inelastic scattering, and (4) multiple lateral states. The effect of higher lateral states depends strongly on the nature of the interaction potential, screening lengths, and exchange terms, but the lowest set of twin peaks is largely unaffected by these details.
The out-of-equilibrium transport properties of a double quantum dot system in the Kondo regime are studied theoretically by means of a two-impurity Anderson Hamiltonian with interimpurity hopping. The Hamiltonian is solved by means of a nonequilibrium generalization of the slave-boson mean-field theory. It is demonstrated that measurements of the differential conductance dI/dV, for appropriate values of voltages and tunneling couplings, can give a direct observation of the coherent superposition between the many-body Kondo states of each dot. For large voltages and arbitrarily large interdot tunneling, there is a critical voltage above which the physical behavior of the system again resembles that of two decoupled quantum dots.
The connection of electrical leads to wire-like molecules is a logical step in the development of molecular electronics, but also allows studies of fundamental physics. For example, metallic carbon nanotubes are quantum wires that have been found to act as one-dimensional quantum dots, Luttinger liquids, proximity-induced superconductors and ballistic and diffusive one-dimensional metals. Here we report that electrically contacted single-walled carbon nanotubes can serve as powerful probes of Kondo physics, demonstrating the universality of the Kondo effect. Arising in the prototypical case from the interaction between a localized impurity magnetic moment and delocalized electrons in a metallic host, the Kondo effect has been used to explain enhanced low-temperature scattering from magnetic impurities in metals, and also occurs in transport through semiconductor quantum dots. The far greater tunability of dots (in our case, nanotubes) compared with atomic impurities renders new classes of Kondo-like effects accessible. Our nanotube devices differ from previous systems in which Kondo effects have been observed, in that they are one-dimensional quantum dots with three-dimensional metal (gold) reservoirs. This allows us to observe Kondo resonances for very large electron numbers (N) in the dot, and approaching the unitary limit (where the transmission reaches its maximum possible value). Moreover, we detect a previously unobserved Kondo effect, occurring for even values of N in a magnetic field.
The behaviour of traditional electronic devices can be understood in terms of the classical diffusive motion of electrons. As the size of a device becomes comparable to the electron coherence length, however, quantum interference between electron waves becomes increasingly important, leading to dramatic changes in device properties. This classical-to-quantum transition in device behaviour suggests the possibility for nanometer-sized electronic elements that make use of quantum coherence. Molecular electronic devices are promising candidates for realizing such device elements because the electronic motion in molecules is inherently quantum mechanical and it can be modified by well defined chemistry. Here we describe an example of a coherent molecular electronic device whose behaviour is explicitly dependent on quantum interference between propagating electron waves-a Fabry-Perot electron resonator based on individual single-walled carbon nanotubes with near-perfect ohmic contacts to electrodes. In these devices, the nanotubes act as coherent electron waveguides, with the resonant cavity formed between the two nanotube-electrode interfaces. We use a theoretical model based on the multichannel Landauer-Büttiker formalism to analyse the device characteristics and find that coupling between the two propagating modes of the nanotubes caused by electron scattering at the nanotube-electrode interfaces is important.
We define two laterally gated small quantum dots with less than 15 electrons in an Aharonov-Bohm geometry in which the coupling between the two dots can be changed. We measure Aharonov-Bohm oscillations for weakly coupled quantum dots. In an intermediate coupling regime we study molecular states of the double dot and extract the magnetic field dependence of the coherently coupled states.
We report the characterization of electronic shell filling in metallic single-walled carbon nanotubes by low-temperature transport measurements. Nanotube quantum dots with average conductance approximately (1-2)e(2)/h exhibit a distinct four-electron periodicity for electron addition as well as signatures of Kondo and inelastic cotunneling. The Hartree-Fock parameters that govern the electronic structure of metallic nanotubes are determined from the analysis of transport data using a shell-filling model that incorporates the nanotube band structure and Coulomb and exchange interactions.
When an individual molecule, nanocrystal, nanotube or lithographically defined quantum dot is attached to metallic electrodes via tunnel barriers, electron transport is dominated by single-electron charging and energy-level quantization. As the coupling to the electrodes increases, higher-order tunnelling and correlated electron motion give rise to new phenomena, including the Kondo resonance. To date, all of the studies of Kondo phenomena in quantum dots have been performed on systems where precise control over the spin degrees of freedom is difficult. Molecules incorporating transition-metal atoms provide powerful new systems in this regard, because the spin and orbital degrees of freedom can be controlled through well-defined chemistry. Here we report the observation of the Kondo effect in single-molecule transistors, where an individual divanadium molecule serves as a spin impurity. We find that the Kondo resonance can be tuned reversibly using the gate voltage to alter the charge and spin state of the molecule. The resonance persists at temperatures up to 30 K and when the energy separation between the molecular state and the Fermi level of the metal exceeds 100 meV.
We demonstrate how molecular quantum states of coupled semiconductor quantum dots are coherently probed and manipulated in transport experiments. The applied method probes quantum states by the virtual cotunneling of two electrons and hence resolves the sequences of molecular states simultaneously. This result is achieved by weakly probing the quantum system through parallel contacts to its constituting quantum dots. The overlap of the dots' wave functions and, in turn, the splitting of molecular states are adjusted by the direct influence of coupling electrodes.
We theoretically study the nonequilibrium transport properties of double quantum dots, in both series and parallel configurations. Our results lead to novel experimental predictions that unambiguously signal the transition from a Kondo state to an antiferromagnetic spin-singlet state, directly reflecting the physics of the two-impurity Kondo problem. We prove that the nonlinear conductance through parallel dots directly measures the exchange constant J between the spins of the dots. In serial dots, the nonlinear conductance provides an upper bound on J.
We investigate coherent time evolution of charge states (pseudospin qubit) in a semiconductor double quantum dot. This fully tunable qubit is manipulated with a high-speed voltage pulse that controls the energy and decoherence of the system. Coherent oscillations of the qubit are observed for several combinations of many-body ground and excited states of the quantum dots. Possible decoherence mechanisms in the present device are also discussed.
We have observed asymmetric Fano resonances in the conductance of a single electron transistor resulting from interference between a resonant and a nonresonant path through the system. The resonant component shows all the features typical of quantum dots, but the origin of the non-resonant path is unclear. A unique feature of this experimental system, compared to others that show Fano line shapes, is that changing the voltages on various gates allows one to alter the interference between the two paths. Comment: 8 pages, 6 figures. Submitted to PRB
We study a small spin-degenerate quantum dot with even number of electrons, weakly connected by point contacts to the metallic electrodes, and subject to an external magnetic field. If the Zeeman energy B is equal to the single-particle level spacing $\Delta $ in the dot, the ground state of the dot becomes doubly degenerate, and the system exhibits Kondo effect, despite the fact that B exceeds by far the Kondo temperature $T_{K}$. A possible realization of this in tunneling experiments is discussed.
The conductance through two quantum dots in series is studied using general qualitative arguments and quantitative slave-boson mean-field theory. It is demonstrated that measurements of the conductance can explore the phase diagram of the two-impurity Anderson model. Competition between the Kondo effect and the inter-dot magnetic exchange leads to a two-plateau structure in the conductance as a function of gate voltage and a two or three peak structure in the conductance vs. inter-dot tunneling. Comment: 4 pages + 3 figures
Hubbard's model for studying correlation effects in systems with narrow energy bands is analyzed by means of a technique which allows the calculation of moments of the individual peaks in the spectral weight function for single-particle excitations. The analysis of the zeroth moments of the peaks shows that the total weight in the bands depends on the strength of the kinetic-energy term in the Hamiltonian even though the bands may be narrow and widely separated. This conclusion is illustrated and verified by an exact calculation for the case when there are only two lattice sites. Analysis of first and higher moments yields results for nonmagnetic or paramagnetic phases which are in qualitative agreement with Hubbard's improved solution. However, we find that (a) there occurs a spin-dependent shift in the band energies which has not been obtained by other treatments of the model and which energetically favors ferromagnetism, and (b) single-particle excitations are more heavily damped in antiferromagnetic than in isomorphic paramagnetic phases.
Transport spectra of a quantum dot dimer are studied with the assumption that energy levels are quantized in each quantum dot. The splitting of energy levels between bonding and antibonding states is modulated by the coupling with electronic states in an emitter and a collector. When the coupling with external electrodes is strong enough, the spectra of a quantum dot dimer are not necessarily observed by I-V characteristics.
The conductance through two quantum dots connected in seriesis studied below the Kondo temperature, based onthe slave boson formalism of the Anderson model.The transport properties are characterized bythe ratio of the magnitude of the tunneling coupling betweentwo dots to the width of the level broadening.When the ratio is less than unity,the Kondo resonant states are formed between each dot and an external lead, and the conductance is determined by the hopping betweenthe two resonant states.When the ratio is larger than unity, these Kondo resonances are split intothe bonding and antibonding peaks,which are located below and above the Fermi levelin the leads, respectively, for low gate voltages.As a result, the conductance is suppressed.The conductance has a maximum of 2 e2/h when the bondingpeak is matched with the Fermi level by controlling the gate voltage.
The authors review the Keldysh method of obtaining kinetic equations for normal and superconducting metals. The use of the method is illustrated by examples involving electron-impurity, electron-phonon, and electron-electron scattering, both within and beyond the quasiclassical approximation.
The Anderson model is studied in the limit U to infinity using a Green function decoupling procedure. It is shown that the solution gives the correct results in the intermediate valence case, and in the Kondo limit, at low and high temperatures: for the intermediate valence case, the position of the virtual d level is obtained as a function of temperature; for the Kondo case it is shown that the density of states has a peak of width TK at the Fermi level, which disappears above the Kondo temperature.
A double quantum dot is investigated in the few electron limit. The dots are coupled by a tunneling barrier allowing the exchange of a ``valence'' electron, leading to the formation of a molecular state. The existence of this molecular state is verified by the determination of the tunnel splitting.
We study the effects of electron correlation on transport through a small interacting system connected to reservoirs using an effective Hamiltonian which describes the free quasiparticles of a Fermi liquid. The effective Hamiltonian is defined microscopically with the value of the self-energy at ω=0. Specifically, we apply the method to a Hubbard chain of finite size N (=1,2,3,…), and calculate the self-energy within the second order in U in the electron-hole-symmetric case. When couplings between the chain and the reservoirs on the left and right are small, the conductance for even N decreases with increasing N, showing a tendency toward a Mott-Hubbard insulator. This is caused by the off-diagonal element of the self-energy, and this behavior is qualitatively different from that in the special case examined in previous work. We also study the effects of the asymmetry in the two couplings. While a perfect transmission due to the Kondo resonance occurs for any odd N in the symmetric coupling, the conductance for odd N decreases with increasing N in the case of asymmetric coupling.
The coupled double quantum dot system is modeled by a two impurity Anderson-type Hamiltonian and interdot Coulomb interaction is included. The conductance of this system is calculated using a nonequilibrium Green’s-function formalism. We have evaluated conductance, peak splitting, and amplitude of conductance peaks as a function of Fermi energy for various values of parameters of the model Hamiltonian. It is demonstrated that an interdot interaction whose strength is assumed to be 10% of the ondot Coulomb interaction produces significant changes in the conductance in a coupled double quantum dot system.
Transport properties of short molecular chains connecting metallic electrodes are studied within the one-band Hubbard model taking into account electron Coulomb interactions in the self-consistent (Hartree-Fock) approximation. The current-voltage characteristics are represented as a sequence of the voltage ranges differing in the number of channels dominating in the electron transport. These channels originate from the molecular self-consistent energy (SCE) levels trapped in the source-drain voltage window. The new qualitative effect of the electron repulsion is the pinning of the SCE levels by the Fermi energy of the source or the drain lead throughout the finite voltage ranges. The computed current-voltage curves are in a good qualitative agreement with the experimental data. For the case of the two-atom device and the weak coupling between the leads and the molecule, the analytical solution for the charge and the current is obtained.
We study transport through two quantum dots in series. Electron-electron interactions are taken into account in the capacitive model with an additional interdot capacitance. The tunneling rates between the dots and the outside reservoirs are assumed to be weak, therefore we treat it perturbatively and derive a master equation. The interdot tunneling is treated in two limits: weak interdot tunneling is included perturbatively, whereas in the opposite limit we assume that there is only one level in the dot in the relevant energy range such that the Hamiltonian of the dots can be diagonalized exactly. We calculate the current through the structure as a function of the two gate voltages. The well-known Coulomb oscillations of a single dot are changed into a characteristic structure of boomerang-like shape. The transport and gate voltages can be time-dependent, and in this case we find that the dependence of the Coulomb oscillations on the two gate voltages allows us to identify which level dominates the transport.
The effects of the electron correlation on the electronic structure and transport in a pair of vertically coupled quantum dots are studied as functions of the number of electrons and the distance between the two dots by using a numerical diagonalization method. The electron correlation drastically affects the spin structure of the ground states when the number of electrons and the distance between the dots are sufficiently large. The electronic states with the large (small) spin momentum are stable when the number of electrons is an odd (even) integer. A physical picture of this characteristic behavior can be understood by an analogy to that of the electronic states in the Hubbard model near half filling. The Coulomb oscillation can be seen in the conductance of the electric current through the double-quantum dots. When the distance between the dots is large, the amplitudes of several conductance peaks are suppressed by the electron correlation.
Electron transport through a double-quantum-dot structure with intradot and interdot Coulomb interactions is studied by a Green’s function (GF) approach. The conductance is calculated by a Landauer-Büttiker formula for the interacting systems derived using the nonequilibrium Keldysh formalism and the GF’s are solved by the equation-of-motion method. It is shown that the interdot-coupling dependence of the conductance peak splitting matches the recent experimental observations. Also, the breaking of the electron-hole symmetry is numerically demonstrated by the presence of the interdot repulsion.
The conductance through two quantum dotsconnected in a series is
examined below the Kondo temperatureas a function of the gate voltage
attached to the dots.The ratio of the tunneling coupling between two
dotsto the level broadening characterizes the transport properties.When
the ratio is less thanunity, each dot accommodates one electron and
forms the Kondo resonantstate with an external lead at a sufficiently
low gate voltage.In the valence fluctuating regime,the number of
electrons in the dots decreases from two to zero whereasthe conductance
is suppressed. The corresponding range of the gate voltageis nearly the
level broadening.When the ratio is larger than unity, the Kondo
resonances are split intothe bonding and antibonding peaks. The valence
fluctuating regime is extended over the tunneling coupling between the
two dots.
Bose-Einstein condensation (BEG) is a purely quantum phenomenon whereby a macroscopic number of identical atoms occupy the same single-particle state. Interest in this phenomenon has grown considerably following the direct demonstration of BEG in low-density gases of alkali metal atoms. It is therefore worth reconsidering the case of liquid 4He, which is generally accepted to have such a condensate, but for which similarly direct evidence is lacking. Nevertheless, theoretical models that depend on the existence of a condensate have proved successful at explaining many of the properties of this system, and BEG is considered to underlie the striking phenomena of superfluidity and quantized vorticity observed in liquid 4He. So the current issue is not whether there is a condensate in this system, but how to demonstrate its existence in a clear and simple way. Here I argue that an earlier measurement of evaporation from liquid 4He caused by a collimated beam of phonons provides such a demonstration. The calculated angular distribution of evaporated atoms agrees well with that measured if it is assumed that the atoms initially had zero momentum parallel to the surface of the liquid-this is to be expected if the atoms originate from a condensate. This process of quantum evaporation also opens the possibility for creating beams of phase-coherent atoms of short wavelength.
The equation of motion for nonequilibrium Green functions is derived within the framework of the Schwinger and Keldysh formalism of perturbation expansion. For nonequilibrium distribution Green functions, the equation of motion derived from quantum mechanics contains undefined singularities, whose explicit form depends on the specific initial or boundary condition. In the present work, the exact expression of singular terms is found in the equation of motion from the time-looped perturbation theory in which the adiabatic initial condition is implied. Unlike the usual Dyson perturbation formalism or the well known Kadanoff-Baym equation of motion, our resulting equation can be adopted directly for calculations without the graphical analysis, which depends on the specific form of the Hamiltonian. On the basis of this equation of motion, the procedure of a nonperturbative solution is outlined and potential applications are briefly discussed.
A quantum Boltzmann equation is derived which is valid for electron transport in electric and magnetic fields including all many-body effects. A solution in both d.c. and a.c. electric fields is given for electrons in simple metals. The solution for transport in large magnetic fields is also given including a theory of the Shubnikov-deHaas oscillations which includes inelastic phonon scattering rigorously.
A new formulation of the mixed-valence problem is presented in which the singlet valence state of a rare-earth ion is represented by a zero-energy boson and the spinning state by a spin-$j$ fermion. This representation avoids the need to use Hubbard operators with awkward algebras and avails itself of standard techniques for dealing with interacting quantum systems. In particular, a Feynman-diagram expansion for the thermodynamic variables and spectral functions can be developed. The advantages of the approach are illustrated for the mixed-valence impurity problem. Vertex corrections are found to be $O(\frac{1}{{N}^{2}})$, where $N$ is the degeneracy of the rare-earth ion, allowing a self-consistent calculation of the $f$-electron spectral function to order $O(\frac{1}{{N}^{2}})$ that is valid in both the mixed-valence and Kondo regimes. The extension to the lattice is outlined and some preliminary results reported.
Quantum dots by now offer a well-defined environment for studying quantum physics. Hence. various proposals have been introduced how to integrate these artificial molecules for building quantum computing devices. Crucial for operating such circuits is the realization of wave function coherence established in coupled quantum dots. Consequently, the foremost goal is to devise basic circuits for testing phase coherence and dissipation mechanisms, (C) 2002 Elsevier Science B.V. All rights reserved.
By using a generalized version of the infinite-U Anderson model, strong-coupling properties of mixed-valence systems are modeled by means of an expansion about a broken-symmetry mean-field theory. A renormalized Fermi liquid, with heavy-fermion bands in the lattice is an intrinsic feature of this mean-field theory. Strong-coupling divergence of the Kondo coupling constant arises as a direct consequence of the zero-mode fluctuations about the broken-symmetry state. In the large-degeneracy limit these fluctuations vanish and the broken-symmetry state is an exact solution, explicitly confirmed for the single-impurity case by a new Bethe-ansatz solution. The crossover to strong coupling is a vestige of the phase transition into the broken-symmetry state. Landau parameters, charge and spin correlations of the heavy Fermi liquid are directly related to the fluctuations about the broken-symmetry state. The general approach presented is applicable to an arbitrary number of impurities or a lattice. Analytic results are presented for the Landau parameters, the dynamical charge and spin correlations in the one- and the two-impurity models, and the one-impurity f spectral function.
Exact many-body eigenstates in a quantum dot formed in double-barrier heterostructures are calculated in the limit of strong confinement, and the nonlinear coherent transport through the states is studied for temperatures larger than their level broadenings. Energy splittings between many-body states due to exchange and correlation effects manifest themselves as small steps which decorate the Coulomb staircase in the current-voltage characteristics, which strongly depends on the number of electrons in the dot. Clear many-body effects are also found in peak heights and peak separations in the Coulomb oscillation of the linear conductance.
The conductance through a quantum dot is calculated via an Anderson model of a site weakly coupled to ideal leads with an on-site Coulomb interaction. As the chemical potential is varied, peaks occur periodically in the conductance whenever an electron is added to the site. The participation of multiple electronic levels in each conductance peak explains the anomalous temperature dependence of peak heights observed in recent narrow-channel experiments.
A multiterminal conductance formula describing resonant tunneling through an interacting mesoscopic system is derived and used to investigate the nonlinear conductance of a quantum dot. An explicit gauge-invariant expression for the I- V characteristic which depends sensitively on the full capacitance matrix is obtained. A voltage probe is found to have a dramatic effect on the nonlinear conductance.
Image projection relies on classical wave mechanics and the use of natural or engineered structures such as lenses or resonant cavities. Well-known examples include the bending of light to create mirages in the atmosphere, and the focusing of sound by whispering galleries. However, the observation of analogous phenomena in condensed matter systems is a more recent development, facilitated by advances in nanofabrication. Here we report the projection of the electronic structure surrounding a magnetic Co atom to a remote location on the surface of a Cu crystal; electron partial waves scattered from the real Co atom are coherently refocused to form a spectral image or 'quantum mirage'. The focusing device is an elliptical quantum corral, assembled on the Cu surface. The corral acts as a quantum mechanical resonator, while the two-dimensional Cu surface-state electrons form the projection medium. When placed on the surface, Co atoms display a distinctive spectroscopic signature, known as the many-particle Kondo resonance, which arises from their magnetic moment. By positioning a Co atom at one focus of the ellipse, we detect a strong Kondo signature not only at the atom, but also at the empty focus. This behaviour contrasts with the usual spatially-decreasing response of an electron gas to a localized perturbation.
We find that Kondo resonant conductance can occur in a quantum dot in the Coulomb blockade regime with an even number of electrons N. The contacts are attached to the dot in a pillar configuration, and a magnetic field B( perpendicular) along the axis is applied. B( perpendicular) lifts the spin degeneracy of the dot energies. Usually, this prevents the system from developing the Kondo effect. Tuning B( perpendicular) to the value B(*) where levels with different total spin cross restores both the degeneracy and the Kondo effect. We analyze a dot charged with N = 2 electrons. Coupling to the contacts is antiferromagnetic due to a spin selection rule and, in the Kondo state, the charge is unchanged while the total spin on the dot is S = 1/2.
Kondo effect in the vicinity of a singlet-triplet transition in a vertical quantum dot is considered. This system is shown to map onto a special version of the two-impurity Kondo model. At any value of the control parameter, the system has a Fermi-liquid ground state. Explicit expressions for the linear conductance as a function of the control parameter and temperature T are obtained. At T = 0, the conductance reaches the unitary limit approximately 4e(2)/h at the triplet side of the transition, and decreases with the increasing distance to the transition at the singlet side. At finite temperature, the conductance exhibits a peak near the transition point.
Double quantum dots provide an ideal model system for studying interactions between localized impurity spins. We report on
the transport properties of a series-coupled double quantum dot as electrons are added one by one onto the dots. When the
many-body molecular states are formed, we observe a splitting of the Kondo resonance peak in the differential conductance.
This splitting reflects the energy difference between the bonding and antibonding states formed by the coherent superposition
of the Kondo states of each dot. The occurrence of the Kondo resonance and its magnetic field dependence agree with a simple
interpretation of the spin status of a double quantum dot.
Tunneling conductance through a quantum dot is calculated around the local spin singlet-triplet crossover region including the Kondo effect. The calculation is carried out using the numerical renormalization group method. When the potential on the dot deepens, two electrons filling a lower energy orbital redistribute to gain Hund's coupling energy. This redistribution induces a bump in the conductance between the Coulomb peaks. The Kondo temperature on the bump is high due to the fluctuation on the singlet-triplet crossover region. The behaviors agree well with recent experiment.
The Kondo effect was analyzed in real quantum dots. The dependence of the zero-temperature conductance on the magnetic field applied in the plane of the dot was discussed. The problem of transmission through a dot was found to be similar to that of transition between channels in a multichannel scattering problem. Extensive numerical renormalization group calculations based on such a model were performed. It was predicted that if the spin of the dot S>1/2, then the dependence of conductance on temperature and in-plane magnetic field was nonmonotonic.
We discuss electronic transport through a lateral quantum dot close to the singlet-triplet degeneracy in the case of a single conduction channel per lead. By applying the numerical renormalization group, we obtain rigorous results for the linear conductance and the density of states. A new quantum phase transition of the Kosterlitz-Thouless-type is found, with an exponentially small energy scale T(*) close to the degeneracy point. Below T(*), the conductance is strongly suppressed, corresponding to a universal dip in the density of states. This explains recent transport measurements.
Using molecules as electronic components is a powerful new direction in the science and technology of nanometre-scale systems. Experiments to date have examined a multitude of molecules conducting in parallel, or, in some cases, transport through single molecules. The latter includes molecules probed in a two-terminal geometry using mechanically controlled break junctions or scanning probes as well as three-terminal single-molecule transistors made from carbon nanotubes, C(60) molecules, and conjugated molecules diluted in a less-conducting molecular layer. The ultimate limit would be a device where electrons hop on to, and off from, a single atom between two contacts. Here we describe transistors incorporating a transition-metal complex designed so that electron transport occurs through well-defined charge states of a single atom. We examine two related molecules containing a Co ion bonded to polypyridyl ligands, attached to insulating tethers of different lengths. Changing the length of the insulating tether alters the coupling of the ion to the electrodes, enabling the fabrication of devices that exhibit either single-electron phenomena, such as Coulomb blockade, or the Kondo effect.
Coulomb- and spin-blockade spectroscopy investigations have been performed on an electrostatically defined "artificial molecule" connected to spin polarized leads. The molecule is first effectively reduced to a two-level system by placing both constituent atoms at a specific location of the level spectrum. The spin sensitivity of the conductance enables us to identify the electronic spin states of the two-level molecule. We find in addition that the magnetic field induces variations in the tunnel coupling between the two atoms. The lateral nature of the device is evoked to explain this behavior.