Kondo and Fano effect in side attached double quantum-dot molecule

ELECTRONIC AND MAGNETIC PROPERTIES OF NANOSCOPIC SYSTEMS

Kondo and Fano effect in side attached double quantum-dot molecule

Gustavo A. Lara^I; Pedro A. Orellana^I; Julio M. Yáñez^I; Enrique V. Anda^II

^IDepartamento de Física, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile. Departamento de Física, Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile

^IIDepartamento de Física, P. U. Católica do Rio de Janeiro, C.P. 38071-970, Rio de Janeiro, RJ, Brazil

ABSTRACT

Electron tunneling through a double quantum dot molecule side attached to a quantum wire, in the Kondo regime, is studied. The mean-field finite-U slave-boson formalism is used to obtain the solution of the problem. We investigate the many body molecular Kondo state and its interplay with the inter-dot antiferromagnetic correlation as a function of the parameters of the system.

Keywords: Quantum dots; Kondo effect; Fano effect

I. INTRODUCTION

The Kondo effect in quantum dots (QDs) has been extensively studied in the last years [1]. The QDs allow to study systematically the quantum-coherence many-body Kondo state, due to the possibility of continuous tuning the relevant parameters governing the properties of this state, in equilibrium and non-equilibrium situations. Recently Kondo effect has been studied in double quantum dot molecule in series [2, 3]. This system allows the study of the many body molecular Kondo states in equilibrium and non-equilibrium situation. The type of coupling between the QDs determines the character of the electronic states and the transport properties of the artificial molecule. In the tunneling regime, the electronics states are extended across the entire system and form a coherent state based on the bonding or anti-bonding levels of the QDs.

An alternative configuration consists of a side-coupled QDs attached to a perfect quantum wire (QW). This structure is reminiscent of T-shaped quantum wave guides known as electron stub tuners [4]. In this case, the QDs act as scattering centers in close analogy with the traditional Kondo effect [5].

Recent electron transport experiments showed that Kondo and Fano resonances occurs simultaneously [6]. Multiple scattering of traveling electronic waves on a localized magnetic state are crucial for a formation of both resonances. The condition for the Fano resonance [7] is the existence of two scattering channels: a discrete level and a broad continuum band [8, 9].

In this work we study the transport properties of a double quantum dot molecule side attached to a quantum wire in the Kondo regimen. We use the finite-U slave boson mean-field approach which was initially developed by Kotliar and Ruckenstein [10] and used later by Bing Dong and X. L. Lei to study the transport through coupled double quantum dots connected to leads [11], which enforces the correspondence between the impurity fermions and the auxiliary fermions to a mean-field level to release the U = ¥ restriction. In double quantum dot molecules this approach allows one to treat both dot-lead coupling and the interdot tunneling nonperturbatively at an arbitrary strengths of the Coulomb correlation U, and thus an antiferromagnetic exchange coupling parameter appears naturally [11]. We investigate the many body molecular Kondo ground state and the competing singlet ground state that results from the anti-ferromagnetic correlation between the dots, as a function of the parameters of the system. In the Kondo regime, each dot results to have its own Kondo temperature as a consequence of their different connections to the conduction electrons.

II. MODEL

Let us consider a double quantum dot (DQD) side coupled to a perfect quantum wire (QW) (see Fig. 1). We adopt the two-fold Anderson Hamiltonian. Each dot has a single level e_di (with i = 1, 2), intra-dot Coulomb repulsion U, is coupled to each other with inter-dot tunneling coupling t_c. The side attached double quantum-dot is coupled to the QW with coupling t₀. The QW has a site energy e_i,s and a hopping parameter t.

In analogy with the infinite-U slave boson approach [12] the Hilbert space is enlarged at each site, to contain in addition to the original fermions a set of four bosons [10] represented by the creation (annihilation) operators (e_i), (p_i,s), and (d_i) for the i-th dot, which act, respectively, as projectors onto empty, single occupied (with spin up and down) and doubly occupied electron states. The Hamiltonian is written as,

where (c_i,s) is the creation (annihilation) operator of an electron with spin s in the i-th site on quantum wire; (f_i,s) is the creation (annihilation) operator of an electron with spin s in the i-th QD, , and , and we set e_i,s = 0 in the QW. The operator _i,s is chosen to reproduce the correct U ® 0 limit in the mean field approximation,

The constraint, i.e., the completeness relation and the condition for the correspondence between fermions and bosons , have been incorporated with Lagrange multipliers and into the Hamiltonian. We use the mean-field approximation in which all the boson operators are replaced by their expectation value. As we work at zero temperature these expectation values and the Lagrange multipliers are then determined by minimization of the energy with respect to these variables.

We resolve the problem using the method developed in Ref[3]. Resolving the equations of motion, we find the transmission amplitude,

where the renormalized bonding and anti-bonding energies (_±) are defined by _± = (_d1+_d2)/2 ± and = is the renormalized coupling between the double quantum-dot and the leads of density of states r₀(w). The transmission amplitude can be written in the polar form as,t = |t| e^if, where f is the transmission phase and it is given by,

The transmission probability is T = |t|²,

Moreover we can obtain the density of states of the quantum dots system,

III. RESULTS

We take typical values for the parameters that define the system, µ = 0,t = 25 G, t₀ = 7.07G, where G = .

We consider first the situation where the two dots local state energies are varied simultaneously by a gate voltage V_g, i.e. e_d1 = e_d2 = V_g.

The transmission probability T is displayed in Figure 2 for various values of U and two values of t_c. The transmission probability always reaches zero at w = _ and ₊ and a unitary value at w = _d2. The figure allows to study the interplay between the Kondo effect and the inter-dot anti-ferromagnetic correlation. Increasing U, a sharp feature develops close to the Fermi energy (w = 0), indicating the appearance of a Kondo resonance. We can see that this process is more rapidly defined for the case where t_c = 0.5G than for t_c = G. This behavior is due to the anti-ferromagnetic interaction between the spins of the dots, proportional to t_c²/U [11], that destroys the Kondo effect when it is greater than the Kondo temperature creating a singlet ground state for the dot molecule.

For U sufficiently large the transmission can be written approximately as the superposition of two Fano-Kondo line shapes, one of them has zero q factor,

where e' = (w- _d2)/ , q' = (_d2 - _)/, e'' = (w- ₊)/G and q'' = 0, with .

The DOS of the quantum-dot molecule give us a better understanding of the transport properties of the system. The DOS is shown in Fig. 3 for various values of U, for the case t_c = 0.5, where the Kondo spin correlation is dominant over the anti-ferromagnetic one. We can observe the superposition of a narrow and a broad Kondo peak with widths and respectively. In fact, the DOS can be written as the superposition of two Lorentzian,

As it can be seen from the figure there is no Kondo peak for U = 2G, as it should because in this case the QDs are with two electrons each.

In order to confirm the destructive effect of anti-ferromagnetism on the Kondo regime, we display in the inset of the figure the DOS of the DQD for the same parameters as before but with t_c = G. The DOS shows two peaks at the bonding and anti-bonding energies with no Kondo peak, showing that for this value of t_c anti-ferromagnetism is dominant.

On the other hand, we can evaluate the transmission phase at the Fermi energy,

Figure 4 displays the behavior of the transmission phase evaluated at the Fermi energy as a function of the gate voltage, for the on-site energy U = 8G. For large negative V_g the energies _,₊ and _d2 are negative and f is negative and tends to zero. When ₊ goes to zero, f tends to -p/2. However, when ₊ becomes positive, f jumps to +p/2. For V_g = U/2, _d2 crosses the Fermi energy and the phase vanishes. At this gate voltage the transmission amplitude evaluated at the Fermi energy is a real number. Increasing V_g when _ approaches to the Fermi energy, f tends to -p/2. At this gate voltage the phase suffers an abrupt change and jumps to +p/2. For large positive values of V_g, the renormalized energies _,₊ and _d2 are all positive and f is positive and tends to zero.

IV. CONCLUSIONS

We have studied the transport through a side-coupled double quantum-dot molecule using the finite-U slave boson mean field approach. We have found that the transmission spectrum shows a structure with two Fano anti-resonances localized at the bonding and anti-bonding renormalized energies of the quantum-dot molecule, and one resonance at the renormalized site energy of the outside quantum-dot. The DOS shows that when the Kondo correlations are dominant both dots are at the Kondo regime each one with its own Kondo temperature. Increasing the inter-dot interaction, the anti-ferromagnetic correlation becomes dominant destroying the Kondo effect and the physics associated with it. These phenomena have been analyzed as a function of the relevant parameters of the system.

Acknowledgments

G.A.L. and P.A.O. would like to thank financial support Milenio ICM P02-054F, P.A.O. also thanks FONDECYT (grants 1020269 and 7020269), and J.M.Y. and G.A.L. thank U.A. (PEI-1305-04). E.V Anda acknowledges support from the brazilian agencies CNPq (CIAM project) and FAPERJ.

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Received on 8 December, 2005

Publication Dates

• Publication in this collection
  29 Nov 2006
• Date of issue
  Sept 2006

History

• Received
  08 Dec 2005