Single charge-current in a normal mesoscopic region attached to superconductor leads via a coupled Poisson Non-equilibrium Green Function formalism
SSingle charge-current in a normal mesoscopic region attached tosuperconductor leads via a coupled Poisson Non-equilibrium GreenFunction formalism
David Verrilli , F. P. Marin , Rafael Rangel Laboratorio de F´ısica Te´orica de S´olidos (LFTS).Centro de F´ısica Te´orica y Computacional (CEFITEC).Facultad de Ciencias. Universidad Central de Venezuela.A.P. 47586. Caracas 1041-A. Venezuela, Departamento de F´ısica.Universidad Sim´øn Bol´ıvar.A.P. 89000. Caracas 1080-A. VenezuelaSeptember 14, 2018
We study the I − V characteristic of a mesoscopic systems or quantum dot (QD) attached to a pair ofsuperconducting leads. Interaction effects in the QD are considered through the charging energy of the QD, i.e.,the treatment of current transport under a voltage bias is performed within a coupled Poisson Non-equilibriumGreen Function (PNEGF) formalism. We derive the expression for the current in full generality, but consideronly the regime where transport occurs only via a single particle current. We show for this case and for variouscharging energies values U and associated capacitances of the QD , the effect on the I − V characteristic. Alsothe influence of the coupling constants on the I − V characteristic is investigated. Our approach puts forwarda novel interpretation of experiments in the strong Coulomb regime. The overall shape of the I − V characteristic of a variety of systems (metals, semiconductors, molecular conduc-tors) in the nanometer scale sandwiched between metallic or superconductors leads has been recently a matterof study (see [1, 2] and references there in). In these systems, the energy level discreteness is quite importantsince level spacing is comparable with other energy scales [3, 4]. Indeed, the coupling with the bath modifiesdrastically the properties of an otherwise uncoupled nanometer system in a sharp contrast with similar non-1 a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec igure 1: Set-up: single level quantum dot connected with two superconducting leads via coupling constants Γ s equilibrium macroscopic systems [5, 6, 7, 8, 9, 10, 11, 12, 13]. They constitutes hybrid systems. Theoreticalstudies [14, 15, 16, 17, 18, 19, 22] as well as experimental measurements have been done by many researchgroups [1, 2, 3] on such systems mostly at low enough temperature with negligible thermal and non-equilibriumfluctuations.All the systems mentioned above underlay universal common features with the hybrid superconductorquantum dot devices we want to address in this work [2, 4, 23, 24] (i) Broadened energy levels of the quantum-dot due to hybridization with the leads. (ii) Spatial potential profile. (iii) A charging energy U due to thepotential profile. An insight behind these issues have been highlighted recently [25, 26] for molecular dots. Thedevice we study in this work is shown in Figure 1. It constitutes a spin degenerated quantum dot level, which iscoupled to a pair of biased superconductors contacts or leads (source and drain). When a source-drain voltage V d is applied, an electric current flows between the leads and across the quantum dot. The biasing defines a non-equilibrium steady state situation. Such situation is coming from the frustration to establish simultaneously anequilibrium configuration with both leads under a given bias. In addition, a gate voltage V g sets the quantum dotspectrum. However, the charge energy can modify it whenever the density of states is significant. In response tothe applied voltages, an actual potential develops inside the dot, i.e., an effective electrostatic profile potentialinside the mesoscopic region exists in such a way, that it couples to both the electronic non-equilibrium statepopulation and the non-equilibrium electric current. That approach, as introduced by S. Datta [4], links theelectrostatic profile to the electronic population of the quantum dot [4, 27] via the non-equilibrium Keldyshformalism (NEGF) [28, 29]. The whole system is modeled by coupling capacitances which represents the drain,source and gate contributions to the self-consistent electrostatic problem. Incoming electrons have to overcomean energy barrier (Coulomb blockade). On the other hand, gate or source-drain voltage can lower or increasethis energy barrier. These source, drain and gate electrodes capacitances (see Figure 2) constitute a simplecapacitive model (in experiments [2, 30], these capacitances are measured) from which U L , the Laplacian partof the potential, can be obtained. In addition, the charge in dot can be expressed as the sum of the chargesin the coupling capacitances. It yields the Poisson contribution U P , to the total potential U , as a function2igure 2: Equivalent capacitive circuit with coupling capacitances C s , C g and C d , corresponding to the capaci-tances in the source, gate and drain respectively.of the dot population. In other words, we solves the self-consistency (SC) of the total electrostatic potential U = U L + U P together with the dot population. After that, the electric current is evaluated.Previous to the self-consistent program, the non-equilibrium current through the dot and electronicoccupation in the dot are worked out. We emphasize that the calculation is carry out in a general framework.However, we confine our attention to the single particle current contribution. We adapt the SC to two differentapproximation regimes. In section (4), the equivalent capacitive circuit (Figure 2) is introduced, the spatialpotential profile U is calculated within the capacitive model. The SC scheme is applied for two cases [31, 32].First, the so called restricted case, where the gap is the bigger energy scale and the coupling QD-Leads is of theorder of the charging energy (∆ (cid:29) Γ L,R (cid:39) U ). In this case, quantitative results are expected to be accurate.We also make calculations for the so called unrestricted case, where the charging energy is the dominant energyscale ∆ (cid:39) U (cid:29) Γ L,R . In this case the results are quantitatively less accurate. The experiments of Ralphet al [30] were done in this regime. Their I − V characteristic shows that the spacing of the energy levelsare subjected to strong fluctuations. According to our model, the fluctuations are due to complex multilevelcharging effects. Our hybrid S/QD/S system has been studied in previous theoretical works [16, 17, 18, 19].However, to our knowledge, the coupled SC scheme which describe charging effects has not been considered sofar. This is an important step, then, gauge invariant independence of the results as well independence of thezero reference voltage is fulfilled [33, 34]. Our model use experimental values of the equivalent capacitances [4].To this respect, pioneering work done by Meir, Wingreen and Lee [20, 21] for a N/QD/N systems, consider theinteratomic Coulomb term U n ↑ n ↓ as a measure of the charging energy e /C . His purpose was to find the mainobject of the non-equilibrium formalism, namely, the QD Green-Keldysh function, in which the influence of theleads on the QD is taken into account. Due to the presence of the Coulomb term, its equation of motion generatesa two particle Green-Keldysh function. By ignoring correlations with the leads, the equation of motion for the3D Green function closes after truncation of higher order equations of motions. This solution (their Equation(8)), has two resonances, one at the energy level weighted by the probability that the other spin degeneratelevel (raised by U) is vacant and another one at the energy level raised by U weighted by the probability thatthe level is occupied. It is correct for temperatures higher than the Kondo temperature and is exact in thenon-interacting limit ( U = 0) and the isolated limit. Analogously, for a S/QD/S hybrid systems Kang [16] hasobtained an expression for the current through the QD (his Equation (8)), which is evaluated in the U → ∞ limit(his Equation (13)). The QD Green function from the very beginning does not contain off-diagonal terms thatinvolves superconducting pairing, which excludes the possibility of Andreev reflection processes. The presencein the equation for the current (his Equation (14)) of terms proportional to (1 − (cid:104) n − σ (cid:105) ) affects the contributionto the current of the considered level. In order to complete the outlined program one has to calculate (cid:104) n − σ (cid:105) self consistently which is not carried out. Instead, Kang calculate the current (his Equation (8)) where thespectral function is calculated in the limit of zero coupling with the leads via a model taken from literature (hisreference [18]) and without taken into account the dependence of the contribution of one level to the currenton the occupancy of the other. The point of view which neglects the unavoidable influence of the bath (theleads) on the small system (the QD) is accomplished by factorizing the density matrix ( ρ ( t ) = ρ QD (cid:78) ρ Baths )and integrating out the leads degrees of freedom which simplifies the Lioville-von Neumann equation (Equation3.140 in [35]). This program is carried out by Kosov et al. for a S/QD/S system [36]. In this way, a Markovianmaster equation is obtained and an expression for the current is calculated. In their Figure 2, they show theI − V characteristic of a non-degenerated QD for a given set of parameters. In this case, the Cooper pair densityin the QD is zero [37]. For the sake of comparison, we restrict our calculations to this case. A similar butnot identical approach was done by Pfaller et al. [38]. Also, the approach of both Kosov et al. and Pfaller etal. misses the energy levels broadening as discussed in the introduction. This lack of broadening is a generaldeficit of Quantum Markov approach [39]. In particular, Pfaller et al. [38], introduce a phenomenologicalbroadening while our approach derives it from first principles. In fact, within the Keldysh formalism, thisbroadening appears naturally (see our Equation (69) below). Yeyati et al. [17] writes an expression for thecurrent (his Equation (2) and Figure 2). They use that expression to explain the experimental results of Ralphet al. [30]. Their calculation where done in the U → ∞ limit. In addition, they include charging effects,although they do not say explicitly in which way these effects are included. In this respect, one has to realizethat U has important contributions from the QD mesoscopic charging effect. In the t → −∞ the leads and theQD maintain independent thermal equilibrium, i.e., are uncoupled systems. When they become coupled theKeldysh formalism yields the general behavior of the system. After a long enough time, this particular systemreaches a steady state.Our point of view is taken form the fact, that the charging of the QD is the origin of the Coulombrepulsion between two electron occupying a two fold degenerate level. Therefore, we study the behavior of anoninteracting QD at t → −∞ where exact expressions are found. In this way, we obtain a formally similarexpression (Equation (72) below) for the current as Equation (12) in the work of Meir et al. [20]. Later on,4oulomb repulsion is introduced via a self-consistent field (SCF) that depends dynamically on the applied bias( H QD + U SCF ) and, in consequence, on the actual number of electrons in the QD. This approach constitutesthe coupled Poisson NEGF formalism that has been discussed in the context of molecular conductors by Datta[4, 26]. We use a capacitive model in section 4 to calculate U SCF and as discussed above, a numerical procedureis used to evaluate the current. Our approach has the known disadvantage, of ignoring correlations in theQD (as pointed out in [39]). In that sense, there is a proposal by Datta (Equation 3.4.9 in [4]) that improvesthe SCF method and permits more accurate quantitative results. In section 4 we apply this improvement forthe case when the Coulomb charging is greater than the value of the coupling constants. We discuss possibleimprovements of our approach in section 6.
In macroscopic systems the task of deriving transport equations or generalized Ginzburg-Landau equations relieson quasi-classical Green functions [7]. In addition, recently non-equilibrium transport in dirty Aluminium quasione dimensional nanowires coupled with normal reservoirs [11] was studied experimentally and theoreticallywith quasi-classical Green functions [13]. As we want to include the possibility of particle interference effects,we do no resort to such objects. This point of view has been discussed in [14]. Instead, we use the equationof motion method (EOM) technique of Keldysh formalism for generating non-equilibrium states (see references[8, 9, 10, 28]). We consider a spin degenerated single orbital as a quantum dot connected to superconductorsleads. The hamiltonian which describes this system is a generalized Anderson model [40]. It reads H = H S + H QD + H T , (1)where H S , H QD and H T stand for the superconducting leads, the dot and the tunneling term, respectively. H S = (cid:88) η H η = H L + H R where H L and H R are the left and right lead Hamiltonians, respectively. They aregiven, within the BCS model [41], by H S = (cid:88) η(cid:126)kσ Ψ † η(cid:126)kσ H η(cid:126)k Ψ η(cid:126)kσ , (2)with H η(cid:126)k = ε η(cid:126)k ∆ η(cid:126)k ∆ ∗ η(cid:126)k − ε η(cid:126)k , (3)where ε η(cid:126)k is the conduction electron energy, ∆ η(cid:126)k is the superconductor gap, of the lead η = L, R . Ψ † η(cid:126)kσ andΨ η(cid:126)kσ are the Nambu spinors.Ψ † η(cid:126)kσ = (cid:16) a † η(cid:126)kσ a η, − (cid:126)k, − σ (cid:17) , Ψ η(cid:126)kσ = a η(cid:126)kσ a † η, − (cid:126)k, − σ . (4)Here a † η(cid:126)kσ (cid:16) a η(cid:126)kσ (cid:17) denotes the creation (annihilation) operator for a conduction electron with wave vector (cid:126)k andspin σ in the η = L, R superconductor lead. 5 QD is the hamiltonian for the single-level quantum dot of energy E : H QD = (cid:88) σ φ † σ H QD φ σ , (5)with H QD = E − E . (6)The model QD does not contain the Hubbard Coulomb repulsion interaction term. As explained inthe introduction, Coulomb repulsion is modeled by means of the inclusion of capacitances, which are takenindependent of the charge in the QD. The model also ignores possible superconducting correlations in the QD.For sufficiently small QDs, the discreteness of the single energy levels suppress these correlations [37]. Theposition of the energy level will be treated first as fixed by the gate potential with respect to the left lead,while the effect of the applied voltage is taking into account by the coupled Poisson scheme. The tunnelinghamiltonian H T is given by: H T = (cid:88) η(cid:126)kσ Ψ † η(cid:126)kσ H Iη(cid:126)k φ σ , (7)with H Iη(cid:126)k = V η(cid:126)k − V η(cid:126)k . (8) H T connects the dot to the biased superconducting leads and it allows the electric charge flow. V η(cid:126)k is thehybridization matrix element between a conduction electron in the η = L, R superconductor lead and a localizedelectron on the dot with energy E . φ † σ and φ σ are the dot spinors φ † σ = (cid:0) d † σ d − σ (cid:1) , φ σ = d σ d †− σ , (9)here, d † σ ( d σ ) is the creation (annihilation) operator for an electron on the dot.The flow of electric charge from the terminal η is given byI η ( t ) = ( − e ) (cid:20) − d (cid:104) N η ( t ) (cid:105) dt (cid:21) = i e ¯ h (cid:104) [ H T ( t ) , N η ( t )] (cid:105) , (10)where − e is the electron charge. (cid:104)· · ·(cid:105) is the thermodynamical average over the biased L and R leads at thetemperature T , taken at time t → −∞ , as indicated in the Keldysh contour in appendix A. (cid:104)· · ·(cid:105) ≡ T r ( ρ ( t ) ... ) , ρ ( t ) ≡ e − β ( H − µN ) T r ( e − β ( H − µN ) ) , (11)and N η = a † η(cid:126)kσ a η(cid:126)kσ is the “number of particle” operator. Book keeping calculations using Equation (10), leadsto I η ( t ) = 2 e ¯ h V η (cid:60) (cid:88) (cid:126)kσ F <η(cid:126)kσ ( t, t ) . (12)6 <η(cid:126)kσ ( t, t (cid:48) ) = i (cid:104) d † σ ( t (cid:48) ) a η(cid:126)kσ ( t ) (cid:105) is the lesser Keldysh Green function,F η(cid:126)kσ ( t, t (cid:48) ) ≡ − i (cid:104) T K a η(cid:126)kσ ( t ) d † σ ( t (cid:48) ) (cid:105)≡ − iΘ ( t, t (cid:48) ) (cid:104) a η(cid:126)kσ ( t ) d † σ ( t (cid:48) ) (cid:105) + iΘ ( t (cid:48) , t ) (cid:104) d † σ ( t (cid:48) ) a η(cid:126)kσ ( t ) (cid:105)≡ Θ ( t, t (cid:48) ) F >η(cid:126)kσ ( t, t (cid:48) ) + Θ ( t (cid:48) , t ) F <η(cid:126)kσ ( t, t (cid:48) ) , (13)and T K is the time-ordering operator, the action of which is to rearrange product of operators, such that operatorwith later times, on the Keldysh contour are placed to the left of the product. Hereafter, for simplicity, wereplace V η(cid:126)k by an average V η at the Fermi surfaces( V η(cid:126)k ≡ (cid:113) (cid:104)| V η(cid:126)k | (cid:105) F S ) of the leads L and R . Using the schemegiven in appendix A for the rate of change of Equation (13), we proceed to obtain the equation of motion:i ∂ F η(cid:126)kσ ( t, t (cid:48) ) ∂t = δ ( t, t (cid:48) ) (cid:68)(cid:110) a η(cid:126)kσ ( t ) , d † σ ( t ) (cid:111)(cid:69) − i (cid:68) T K (cid:104) a η(cid:126)kσ ( t ) , H (cid:105) d † σ ( t (cid:48) ) (cid:69) . (14)Which leads to (cid:18) i ∂∂t − (cid:15) η(cid:126)k (cid:19) F η(cid:126)kσ ( t, t (cid:48) ) = − σ ∆ η F η(cid:126)kσ ( t, t (cid:48) ) + V η G σ ( t, t (cid:48) ) , (15)where F η(cid:126)kσ ( t, t (cid:48) ) = − i (cid:68) T K a † η(cid:126)k, − σ ( t ) d † σ ( t (cid:48) ) (cid:69) , (16)G σ ( t, t (cid:48) ) = − i (cid:10) T K d σ ( t ) d † σ ( t (cid:48) ) (cid:11) . (17)Note that G σ ( t, t (cid:48) ) is the QD single particle Green’s function. Similarly, F η(cid:126)kσ ( t, t (cid:48) ) satisfies the equation ofmotion: (cid:18) i ∂∂t + (cid:15) η(cid:126)k (cid:19) F η(cid:126)kσ ( t, t (cid:48) ) = − σ ∆ η F η(cid:126)kσ ( t, t (cid:48) ) − V η G σ ( t, t (cid:48) ) , (18)where G σ ( t, t (cid:48) ) = − i (cid:68) T K d †− σ ( t ) d † σ ( t (cid:48) ) (cid:69) . (19)Here G σ ( t, t (cid:48) ) is the QD of two-particle Green’s function.Equations (15) and (18) can be written in a compact form as follows (see Appendix B): i ∂∂t − (cid:15) η(cid:126)k σ ∆ η σ ∆ η i ∂∂t + (cid:15) η(cid:126)k F η(cid:126)kσ ( t, t (cid:48) ) (cid:101) F η(cid:126)kσ ( t, t (cid:48) ) F η(cid:126)kσ ( t, t (cid:48) ) (cid:101) F η(cid:126)kσ ( t, t (cid:48) ) = V η σ z G σ ( t, t (cid:48) ) (cid:101) G η(cid:126)kσ ( t, t (cid:48) ) G σ ( t, t (cid:48) ) (cid:101) G η(cid:126)kσ ( t, t (cid:48) ) . (20)We introduce the tilde Keldysh-Green functions: (cid:101) F η(cid:126)kσ ( t, t (cid:48) ) = − i (cid:68) T K a η(cid:126)kσ ( t ) d − σ ( t (cid:48) ) (cid:69) , (cid:101) F η(cid:126)kσ ( t, t (cid:48) ) = − i (cid:68) T K a † η − (cid:126)k − σ ( t ) d − σ ( t (cid:48) ) (cid:69) , (cid:101) G σ ( t, t (cid:48) ) = − i (cid:104) T K d σ ( t ) d − σ ( t (cid:48) ) (cid:105) , (cid:101) G σ ( t, t (cid:48) ) = − i (cid:68) T K d †− σ ( t ) d − σ ( t (cid:48) ) (cid:69) . (21)7onsider the following 2 × V η = 0, g η(cid:126)kσ ( t, t (cid:48) ) f η(cid:126)kσ ( t, t (cid:48) ) (cid:101) f η(cid:126)kσ ( t, t (cid:48) ) (cid:101) g η(cid:126)kσ ( t, t (cid:48) ) where g η(cid:126)kσ ( t, t (cid:48) ) ≡ − i (cid:68) T K a η(cid:126)kσ ( t ) a † η(cid:126)kσ ( t (cid:48) ) (cid:69) f η(cid:126)kσ ( t, t (cid:48) ) ≡ − i (cid:68) T K a η(cid:126)kσ ( t ) a η − (cid:126)k − σ ( t (cid:48) ) (cid:69) (cid:101) f η(cid:126)kσ ( t, t (cid:48) ) ≡ − i (cid:68) T K a † η − (cid:126)k − σ ( t ) a † η(cid:126)kσ ( t (cid:48) ) (cid:69) (cid:101) g η(cid:126)kσ ( t, t (cid:48) ) ≡ − i (cid:68) T K a † η − (cid:126)k − σ ( t ) a η − (cid:126)k − σ ( t (cid:48) ) (cid:69) (22)According to appendix A, their equations of motions are given by: (cid:18) i ∂∂t − (cid:15) η(cid:126)k (cid:19) g η(cid:126)kσ ( t, t (cid:48) ) + σ ∆ η (cid:101) f η(cid:126)kσ ( t, t (cid:48) ) = δ ( t, t (cid:48) ) , (23) (cid:18) i ∂∂t − (cid:15) η(cid:126)k (cid:19) f η(cid:126)kσ ( t, t (cid:48) ) + σ ∆ η (cid:101) g η(cid:126)kσ ( t, t (cid:48) ) = 0 , (24) (cid:18) i ∂∂t + (cid:15) η(cid:126)k (cid:19)(cid:101) f η(cid:126)kσ ( t, t (cid:48) ) + σ ∆ η g η(cid:126)kσ ( t, t (cid:48) ) = 0 , (25) (cid:18) i ∂∂t + (cid:15) η(cid:126)k (cid:19) (cid:101) g η(cid:126)kσ ( t, t (cid:48) ) + σ ∆ η f η(cid:126)kσ ( t, t (cid:48) ) = δ ( t, t (cid:48) ) . (26)These equations can be written in matrix form as follows: i ∂∂t − (cid:15) η(cid:126)k σ ∆ η σ ∆ η i ∂∂t + (cid:15) η(cid:126)k g η(cid:126)kσ ( t, t (cid:48) ) f η(cid:126)kσ ( t, t (cid:48) ) (cid:101) f η(cid:126)kσ ( t, t (cid:48) ) (cid:101) g η(cid:126)kσ ( t, t (cid:48) ) = δ ( t, t (cid:48) ) 00 δ ( t, t (cid:48) ) . (27)The Equation (20) can be written as an integral along the Keldysh contour C K , (for an explanationsee Appendix B). F η(cid:126)kσ ( t, t (cid:48) ) (cid:101) F η(cid:126)kσ ( t, t (cid:48) ) F η(cid:126)kσ ( t, t (cid:48) ) (cid:101) F η(cid:126)kσ ( t, t (cid:48) ) = (cid:90) C K d t (cid:48)(cid:48) g η(cid:126)kσ ( t, t (cid:48)(cid:48) ) (cid:101) f η(cid:126)kσ ( t, t (cid:48)(cid:48) )f η(cid:126)kσ ( t, t (cid:48)(cid:48) ) (cid:101) g η(cid:126)kσ ( t, t (cid:48)(cid:48) ) × V η σ z G σ ( t (cid:48)(cid:48) , t (cid:48) ) (cid:101) G σ ( t (cid:48)(cid:48) , t (cid:48) ) G σ ( t (cid:48)(cid:48) , t (cid:48) ) (cid:101) G σ ( t (cid:48)(cid:48) , t (cid:48) ) . (28)From the last expression one can read for F η(cid:126)kσ ( t, t (cid:48) ) the equation:F η(cid:126)kσ ( t, t (cid:48) ) = V η (cid:90) C K d t (cid:48)(cid:48) (cid:104) g η(cid:126)kσ ( t, t (cid:48)(cid:48) )G σ ( t (cid:48)(cid:48) , t (cid:48) ) − (cid:101) f η(cid:126)kσ ( t, t (cid:48)(cid:48) ) G σ ( t (cid:48)(cid:48) , t (cid:48) ) (cid:105) . (29)8e now apply the procedure explained in appendix C, in order to obtain the F η(cid:126)kσ ( t, t (cid:48) ) lesser com-ponent, we obtain:F <η(cid:126)kσ ( t, t (cid:48) ) = V η (cid:26)(cid:90) ∞−∞ d t (cid:48)(cid:48) (cid:104) g (r) η(cid:126)kσ ( t, t (cid:48)(cid:48) )G <σ ( t (cid:48)(cid:48) , t (cid:48) ) − (cid:101) f (r) η(cid:126)kσ ( t, t (cid:48)(cid:48) ) G <σ ( t (cid:48)(cid:48) , t (cid:48) ) (cid:105) + (cid:90) ∞−∞ d t (cid:48)(cid:48) (cid:104) g <η(cid:126)kσ ( t, t (cid:48)(cid:48) )G (a) σ ( t (cid:48)(cid:48) , t (cid:48) ) − (cid:101) f <η(cid:126)kσ ( t, t (cid:48)(cid:48) ) G (a) σ ( t (cid:48)(cid:48) , t (cid:48) ) (cid:105)(cid:27) . (30)Furthermore, the superscripts ( < ) , ( > )(r) , (a) correspond to lesser, greater, retarded, advanced Green’s functionsrespectively.Therefore, from Equation (12), I η ( t ), can be written as:I η ( t ) = I (1) η ( t ) + I (2) η ( t ) , (31)with I (1) η ( t ) = 2 e ¯ h (cid:60) (cid:88) σ (cid:90) ∞−∞ d t (cid:48) V η (cid:88) (cid:126)k g (r) η(cid:126)kσ ( t, t (cid:48) ) G <σ ( t (cid:48) , t ) + V η (cid:88) (cid:126)k g <η(cid:126)kσ ( t, t (cid:48) ) G (a) σ ( t (cid:48) , t ) , (32)I (2) η ( t ) = − e ¯ h (cid:60) (cid:88) σ (cid:90) ∞−∞ d t (cid:48) V η (cid:88) (cid:126)k (cid:101) f (r) η(cid:126)kσ ( t, t (cid:48) ) G <σ ( t (cid:48) , t ) + V η (cid:88) (cid:126)k (cid:101) f <η(cid:126)kσ ( t, t (cid:48) ) G (a) σ ( t (cid:48) , t ) . (33)When applying the Fourier transformations, the Equations (32) and (33) can be expressed asI (1) η ( t ) = 2 eh (cid:60) (cid:88) σ (cid:90) ∞−∞ d ω (cid:90) ∞−∞ d ω (cid:48) π e − i( ω − ω (cid:48) ) t × (cid:104) Σ (r) η ( ω )G <σ ( ω, ω (cid:48) ) + Σ <η ( ω )G (a) σ ( ω, ω (cid:48) ) (cid:105) , (34)and I (2) η ( t ) = − eh (cid:60) (cid:88) σ (cid:26) e − µ η t (cid:90) ∞−∞ d ω (cid:90) ∞−∞ d ω (cid:48) π e − i( ω − ω (cid:48) ) t × (cid:104)(cid:101) Ξ (r) η ( ω ) σ G <σ ( ω, ω (cid:48) ) + (cid:101) Ξ <η ( ω ) σ G (a) σ ( ω, ω (cid:48) ) (cid:105)(cid:111) , (35)with V η (cid:88) (cid:126)k g (r) η(cid:126)kσ ( t, t (cid:48) ) ≡ (cid:90) ∞−∞ d ω π e − i ω ( t − t (cid:48) )Σ (r) η ( ω ) , (36) V η (cid:88) (cid:126)k g <η(cid:126)kσ ( t, t (cid:48) ) ≡ (cid:90) ∞−∞ d ω π e − i ω ( t − t (cid:48) )Σ <η ( ω ) , (37) V η (cid:88) (cid:126)k (cid:101) f (r) η(cid:126)kσ ( t, t (cid:48) ) ≡ (cid:90) ∞−∞ e − µ η t σ d ω π e − i ω ( t − t (cid:48) ) (cid:101) Ξ (r) η ( ω ) , (38) V η (cid:88) (cid:126)k (cid:101) f <η(cid:126)kσ ( t, t (cid:48) ) ≡ (cid:90) ∞−∞ e − µ η t σ d ω π e − i ω ( t − t (cid:48) ) (cid:101) Ξ <η ( ω ) . (39)9n appendixes D to G we evaluate the unperturbed Green’s functions g (r) η(cid:126)kσ ( t, t (cid:48) ),g <η(cid:126)kσ ( t, t (cid:48) ), (cid:101) f (r) η(cid:126)kσ ( t, t (cid:48) ) and (cid:101) f <η(cid:126)kσ ( t, t (cid:48) ) in the wide band limit.We summarize these results:Σ (r) η ( ω ) = − Γ η (cid:20) ω − µ η ∆ η ζ (∆ η , ω − µ η ) + i ζ ( ω − µ η , ∆ η ) (cid:21) , Σ <η ( ω ) = 2iΓ η ζ ( ω − µ η , ∆ η )f( ω − µ η ) , (cid:101) Ξ (r) η ( ω ) = Γ η (cid:20) ζ (∆ η , ω + µ η ) + i ∆ η ω + µ η ζ ( ω + µ η , ∆ η ) (cid:21) , (40) (cid:101) Ξ <η ( ω ) = − η ∆ η ω + µ η ζ ( ω + µ η , ∆ η )f( ω + µ η ) ,ζ ( ω, ω (cid:48) ) ≡ Θ( | ω | − | ω (cid:48) | ) | ω |√ ω − ω (cid:48) . All these expressions will used below.
We need to evaluate the most important objet for calculations, namely the QD Green’s functions given byEquation (17) and Equation (19), as well their respective tilde functions: (cid:101) G σ ( t, t (cid:48) ) = − i (cid:68) T K d †− σ ( t ) d − σ ( t (cid:48) ) (cid:69) , (cid:101) G σ ( t, t (cid:48) ) = − i (cid:104) T K d σ ( t ) d − σ ( t (cid:48) ) (cid:105) . (41)Again using the scheme given in Appendix A, their equation of motion are:i ∂∂t G σ ( t, t (cid:48) ) (cid:101) G σ ( t, t (cid:48) ) G σ ( t, t (cid:48) ) (cid:101) G σ ( t, t (cid:48) ) = δ ( t, t (cid:48) ) − i (cid:10) T K [ d σ ( t ) , H ] d † σ ( t (cid:48) ) (cid:11) − i (cid:104) T K [ d σ ( t ) , H ] d − σ ( t (cid:48) ) (cid:105)− i (cid:68) T K [ d †− σ ( t ) , H ] d † σ ( t (cid:48) ) (cid:69) δ ( t, t (cid:48) ) − i (cid:68) T K [ d †− σ ( t ) , H ] d − σ ( t (cid:48) ) (cid:69) . Which develops to:i ∂∂t G σ ( t, t (cid:48) ) = δ ( t, t (cid:48) ) − iE (cid:10) T K d σ ( t ) d † σ ( t (cid:48) ) (cid:11) − i (cid:88) η(cid:126)k V η (cid:68) T K a η(cid:126)kσ ( t ) d † σ ( t (cid:48) ) (cid:69) = δ ( t, t (cid:48) ) + E G σ ( t, t (cid:48) ) + (cid:88) η(cid:126)k V η F η(cid:126)kσ ( t, t (cid:48) ) , (42)10 ∂∂t G σ ( t, t (cid:48) ) = iE (cid:68) T K d †− σ ( t ) d † σ ( t (cid:48) ) (cid:69) + i (cid:88) η(cid:126)k V η (cid:68) T K a † η − (cid:126)k − σ ( t ) d † σ ( t (cid:48) ) (cid:69) = − E G σ ( t, t (cid:48) ) − (cid:88) η(cid:126)k V η F η(cid:126)kσ ( t, t (cid:48) ) , (43)i ∂∂t (cid:101) G σ ( t, t (cid:48) ) = − iE (cid:104) T K d σ ( t ) d − σ ( t (cid:48) ) (cid:105) − i (cid:88) η(cid:126)k V η (cid:68) T K a η(cid:126)kσ ( t ) d − σ ( t (cid:48) ) (cid:69) = E (cid:101) G σ ( t, t (cid:48) ) + (cid:88) η(cid:126)k V η (cid:101) F η(cid:126)kσ ( t, t (cid:48) ) , (44)i ∂∂t (cid:101) G σ ( t, t (cid:48) ) = δ ( t, t (cid:48) ) + iE (cid:68) T K d †− σ ( t ) d − σ ( t (cid:48) ) (cid:69) + i (cid:88) η(cid:126)k V η (cid:68) T K a † η(cid:126)kσ ( t ) d − σ ( t (cid:48) ) (cid:69) = δ ( t, t (cid:48) ) − E (cid:101) G σ ( t, t (cid:48) ) − (cid:88) η(cid:126)k V η (cid:101) F η(cid:126)kσ ( t, t (cid:48) ) . (45)This can be written as: i ∂∂t − E
00 i ∂∂t + E G σ ( t, t (cid:48) ) (cid:101) G σ ( t, t (cid:48) ) G σ ( t, t (cid:48) ) (cid:101) G σ ( t, t (cid:48) ) = δ ( t, t (cid:48) ) 00 δ ( t, t (cid:48) ) + (cid:88) η(cid:126)k V η σ z F η(cid:126)kσ ( t, t (cid:48) ) (cid:101) F η(cid:126)kσ ( t, t (cid:48) ) F η(cid:126)kσ ( t, t (cid:48) ) (cid:101) F η(cid:126)kσ ( t, t (cid:48) ) . (46)When V η = 0 one has: i ∂∂t − E
00 i ∂∂t + E G ( t, t (cid:48) ) 00 (cid:101) G ( t, t (cid:48) ) = δ ( t, t (cid:48) ) 00 δ ( t, t (cid:48) ) , (47)with: G ( t, t (cid:48) ) = − i (cid:10) T K d σ ( t ) d † σ ( t (cid:48) ) (cid:11) , (cid:101) G ( t, t (cid:48) ) = − i (cid:68) T K d †− σ ( t ) d − σ ( t (cid:48) ) (cid:69) . G ( t, t (cid:48) ) ≡ G σ ( t, t (cid:48) ) | V η =0 , (cid:101) G ( t, t (cid:48) ) ≡ (cid:101) G σ ( t, t (cid:48) ) | V η =0 . (48)11he last two equations can be written as: i ∂∂t − E
00 i ∂∂t + E G σ ( t, t (cid:48) ) − G ( t, t (cid:48) ) (cid:101) G σ ( t, t (cid:48) ) G σ ( t, t (cid:48) ) (cid:101) G σ ( t, t (cid:48) ) − (cid:101) G ( t, t (cid:48) ) = (cid:88) η(cid:126)k V η σ z F η(cid:126)kσ ( t, t (cid:48) ) (cid:101) F η(cid:126)kσ ( t, t (cid:48) ) F η(cid:126)kσ ( t, t (cid:48) ) (cid:101) F η(cid:126)kσ ( t, t (cid:48) ) . (49)We write the last equation in its equivalent convolution integral along the Keldysh contour (seeAppendix B): G σ ( t, t (cid:48) ) − G ( t, t (cid:48) ) (cid:101) G σ ( t, t (cid:48) ) G σ ( t, t (cid:48) ) (cid:101) G σ ( t, t (cid:48) ) − (cid:101) G ( t, t (cid:48) ) = (cid:90) C K d t (cid:48)(cid:48) G ( t, t (cid:48)(cid:48) ) 00 (cid:101) G ( t, t (cid:48)(cid:48) ) × (cid:88) η(cid:126)k V η σ z F η(cid:126)kσ ( t (cid:48)(cid:48) , t (cid:48) ) (cid:101) F η(cid:126)kσ ( t (cid:48)(cid:48) , t (cid:48) ) F η(cid:126)kσ ( t (cid:48)(cid:48) , t (cid:48) ) (cid:101) F η(cid:126)kσ ( t (cid:48)(cid:48) , t (cid:48) ) . (50)An equivalent way to write the last equation (using equation (27)) as a convolution of Σ σ ( t, t (cid:48) ) and G σ ( t, t (cid:48) ) is: G σ ( t, t (cid:48) ) = G ( t, t (cid:48) ) + (cid:90) C K d t (cid:48)(cid:48) G ( t, t (cid:48) ) Σ σ ( t (cid:48) , t (cid:48)(cid:48) ) G σ ( t (cid:48)(cid:48) , t (cid:48) ) , (51)with: G σ ( t, t (cid:48) ) ≡ G σ ( t, t (cid:48) ) (cid:101) G σ ( t, t (cid:48) ) G σ ( t, t (cid:48) ) (cid:101) G σ ( t, t (cid:48) ) , G ( t, t (cid:48) ) ≡ G ( t, t (cid:48) ) 00 (cid:101) G ( t, t (cid:48) ) . (52)and Σ σ ( t, t (cid:48) ) ≡ (cid:90) C K d t (cid:48)(cid:48) V η (cid:88) η(cid:126)k g η(cid:126)kσ ( t (cid:48)(cid:48) , t (cid:48) ) − V η (cid:88) η(cid:126)k (cid:101) f η(cid:126)kσ ( t (cid:48)(cid:48) , t (cid:48) ) − V η (cid:88) η(cid:126)k f η(cid:126)kσ ( t (cid:48)(cid:48) , t (cid:48) ) V η (cid:88) η(cid:126)k (cid:101) g η(cid:126)kσ ( t (cid:48)(cid:48) , t (cid:48) ) (53)12e are interested in two regimes: A first regime in which U ∼ Γ < ∆ and the Coulomb blockadeeffects is neglected because in this case the couplings to the leads are not extremely small and the dot capacitanceis large enough. A second regime for U ∼ ∆ > Γ where Coulomb blockade effects must be taken into account.For both regimes and from now on, we are interested in the case eV > ∆, where multiple Andreev reflection[42] processes is strongly suppressed. Therefore only the single particle current ( SP ) have to be considered I SP .From the above considerations we have that the Keldysh Green function G σ ( ω ), which carries information ofthe quantum dot two-particle Green’s function can be neglected and all relevant information is contained inG σ ( ω ). The Keldysh Green function becomes spin independent, G σ ( ω ) ≡ G ( ω ). The element 11 of Equation(51) is given by: G ( t, t (cid:48) ) = G ( t, t (cid:48) ) + (cid:90) C K d t (cid:48)(cid:48) (cid:90) C K d t (cid:48)(cid:48)(cid:48) G ( t, t (cid:48)(cid:48) ) Σ ( t (cid:48)(cid:48) , t (cid:48)(cid:48)(cid:48) ) G ( t (cid:48)(cid:48)(cid:48) , t (cid:48) ) . (54)Again, using the recipe given in appendix C, we obtain for G < ( t, t (cid:48) ) and G (a) ( t, t (cid:48) ):G < ( t, t (cid:48) ) = G < ( t, t (cid:48) ) + (cid:20)(cid:90) ∞−∞ d t (cid:48)(cid:48) (cid:90) ∞−∞ d t (cid:48)(cid:48)(cid:48) G (r)0 ( t, t (cid:48)(cid:48) ) Σ (r) ( t (cid:48)(cid:48) , t (cid:48)(cid:48)(cid:48) ) G < ( t (cid:48)(cid:48)(cid:48) , t (cid:48) ) +G (r)0 ( t, t (cid:48)(cid:48) ) Σ < ( t (cid:48)(cid:48) , t (cid:48)(cid:48)(cid:48) ) G (a) ( t (cid:48)(cid:48)(cid:48) , t (cid:48) ) +G < ( t, t (cid:48)(cid:48) ) Σ (a) ( t (cid:48)(cid:48) , t (cid:48)(cid:48)(cid:48) ) G (a) ( t (cid:48)(cid:48)(cid:48) , t (cid:48) ) (cid:105) . (55)G (a) ( t, t (cid:48) ) = G (a)0 ( t, t (cid:48) ) + (cid:90) ∞−∞ d t (cid:48)(cid:48) (cid:90) ∞−∞ d t (cid:48)(cid:48)(cid:48) G (a)0 ( t, t (cid:48)(cid:48) ) Σ (a) ( t (cid:48)(cid:48) , t (cid:48)(cid:48)(cid:48) ) G (a) ( t (cid:48)(cid:48)(cid:48) , t (cid:48) ) . (56)Taking the Fourier transform of Equations (55) and (56), results in a set of algebraic equations:G < ( ω, ω (cid:48) ) = 2 πδ ( ω − ω (cid:48) ) G < ( ω ) + G (r)0 ( ω ) Σ (r) ( ω ) G < ( ω, ω (cid:48) ) +G (r)0 ( ω ) Σ < ( ω ) G (a) ( ω, ω (cid:48) ) + G < ( ω ) Σ (a) ( ω ) G (a) ( ω, ω (cid:48) ) . (57)G (a) ( ω, ω (cid:48) ) = 2 πδ ( ω − ω (cid:48) ) G (a)0 ( ω ) + G (a)0 ( ω ) Σ (a) ( ω ) G (a) ( ω, ω (cid:48) ) . (58)Dot Keldysh Green’s functions G <σ ( ω, ω (cid:48) ) and G (a) σ ( ω, ω (cid:48) ) are below straightforward evaluated. In this regimequantities such as currents are independent of time. Therefore, we have:G < ( ω, ω (cid:48) ) = 2 πδ ( ω − ω (cid:48) ) G < ( ω ) , G (a) ( ω, ω (cid:48) ) = 2 πδ ( ω − ω (cid:48) ) G (a) ( ω ) . Therefore the Equations (57) and (58) resultG < ( ω ) = G < ( ω ) + G (r)0 ( ω ) Σ (r) ( ω ) G < ( ω ) +G (r)0 ( ω ) Σ < ( ω ) G (a) ( ω ) + G < ( ω ) Σ (a) ( ω ) G (a) ( ω ) . (59)G (a) ( ω ) = G (a)0 ( ω ) + G (a)0 ( ω ) Σ (a) ( ω ) G (a) ( ω ) . (60)13olving (60), G (a) ( ω ) = 1G (a)0 ( ω ) − − Σ (a) ( ω ) = 1 ω − E − Σ (a) ( ω ) = G (r) ( ω ) ∗ . (61)Moreover, we know that G < ( ω ) ∝ δ ( ω − E ) and (62)G (a)0 ( ω ) = (cid:0) ω − E − i0 + (cid:1) − , (63)resulting G < ( ω ) Σ (a) ( ω ) G (a) ( ω ) = − G < ( ω ) . (64)The Equation (59) is reduced toG < ( ω ) = G (r)0 ( ω ) Σ (r) ( ω ) G < ( ω ) + G (r)0 ( ω ) Σ < ( ω ) G (a) ( ω ) , (65)G < ( ω ) = Σ < ( ω ) G (a) ( ω )G (r)0 ( ω ) − − Σ (r) ( ω ) = Σ < ( ω ) | G (r) ( ω ) | = π Σ < ( ω ) −(cid:61) G (r) ( ω ) /π (cid:61) (cid:0) G (r) ( ω ) (cid:1) − = π Σ < ( ω ) −(cid:61) Σ (r) ( ω ) ρ ( ω ) . (66)Here ρ ( ω ) is the so-called quantum dot spectral function which is given in terms of the imaginary part ( (cid:61) ) ofthe retarded Keldysh Green function G (r) ( ω ), ρ ( ω ) = − π (cid:61) G (r) ( ω ) = − π (cid:61) Σ (r) ( ω ) (cid:2) ω − E − (cid:60) Σ (r) ( ω ) (cid:3) + (cid:2) (cid:61) Σ (r) ( ω ) (cid:3) . (67)From Equation (32) the single particle current (I SP ) results in:I ηSP ( V, E ) = 4 eh (cid:60) (cid:90) ∞−∞ d ω (cid:104) Σ (r) η ( ω ) G < ( ω ) + Σ <η ( ω ) G (a) ( ω ) (cid:105) . (68)Substituting (61) and (66) in (68)I ηSP ( V, E ) = 4 eh (cid:90) ∞−∞ d ω (cid:20) π (cid:61) Σ (r) η ( ω ) (cid:61) Σ < ( ω ) (cid:61) Σ (r) ( ω ) ρ ( ω ) + (cid:61) Σ <η ( ω ) (cid:61) G (r) ( ω ) (cid:21) = 4 πeh (cid:90) ∞−∞ d ωρ ( ω ) (cid:20) (cid:61) Σ (r) η ( ω ) (cid:61) Σ < ( ω ) (cid:61) Σ (r) ( ω ) ρ ( ω ) − (cid:61) Σ <η ( ω ) (cid:21) = 4 πeh (cid:90) ∞−∞ d ω ρ ( ω )Γ ( ω ) (cid:2) Γ η ( ω ) (cid:61) Σ < ( ω ) − Γ ( ω ) (cid:61) Σ <η ( ω ) (cid:3) . (69)with Γ η ( ω ) = −(cid:61) Σ (r) η ( ω ) = Γ η ζ ( ω, ∆ η ) and Γ ( ω ) = (cid:80) η Γ η ( ω ). In our regime, eV > ∆, therefore, (cid:60) Σ (r) ( ω ) inthe above equations is zero. We use the expression for Σ (r) ( ω ) from appendix D, and obtain the single particlecurrent I SP ≡ (I R,SP − I L,SP ) / SP ( V, E ) = 8 πeh (cid:90) ∞−∞ d ω Γ L ( ω ) Γ R ( ω + eV )Γ L ( ω ) + Γ R ( ω + eV ) ρ ( ω ) [f ( ω ) − f ( ω + eV )] . (70) − eV = µ L − µ R correspond to the applied voltage between the superconductors electrodes with chemicalpotential µ η . In the following, we fix the chemical potential µ L = 0 and use eV as a measure of µ R . In addition,14he QD energy E is measure with respect to µ L . On the other hand, the limits of integration are given by thefunctions Γ L ( ω ) and Γ R ( ω + eV ). The extra 2 π factor arises from the dot Keldysh Green functions. ρ ( ω ) andΓ ( ω ) are given by ρ ( ω ) = Γ ( ω ) /π ( ω − E ) + Γ ( ω ) , (71)Γ ( ω ) = Γ L ( ω ) + Γ R ( ω + eV ) . (72)At steady state there is no net flow into or out of the mesoscopic channel or quantum dot which yieldsa stationary particle number in it. The population number N , at the dot, is given by N = 2 (cid:2) − iG < ( t, t ) (cid:3) = 2 (cid:90) ∞−∞ d ω π i G < ( ω ) , (73)which becomes a weighted average over the L and R contacts N = 2 (cid:90) ∞−∞ d ω ρ ( ω ) (cid:20) Γ L ( ω )Γ ( ω ) f ( ω ) + Γ R ( ω + eV )Γ ( ω ) f ( ω + eV ) (cid:21) . (74)For the N/QD/N case, Γ R,L are just constants. This case was study in the context of the generalized quantummaster approach (section IV in [39]). That approach permits the inclusion of broadening in a natural way. Theyobtained Equations similar to our Equations (70)-(74).
So far, we are not included the side effects of a potential profile inside the mesoscopic channel. On the onehand, its inclusion takes in order zero or Hartree approximation the electron-electron interaction in the QD. Itsinclusion also guarantees current independence from the choice of zero potential [34]. Such potential is inducedby the action of source, drain and gate applied voltages. In principle, we have to couple the number populationequation Equation (73), with electric field U . However, since the number of quantum levels in the channel issmall the particle number variation is negligible, the potential profile variation inside the channel is negligible.Then it is appropriate visualize the channel as an equivalent circuit framework (Figure 2). In this framework weassociate capacitances C d , C s and C g to the drain, source and gate, respectively. Whenever drain, source andgate bias potentials V d , V s and V g , respectively, are present, there is an electrostatic potential V QD inside theQD, it induces an energy shift of the QD energy level U = − e ( V QD − V ), V are channel electrostatic potentialbefore we apply the source and drain biases, respectively.The electronic population before and after we apply the biases mentioned above are given by − eN = C d V + C s V + C g V , (75) − eN = C d ( V QD − V d ) + C s ( V QD − V s ) + C g ( V QD − V g ) , (76)respectively. It leads us to − e ∆ N ≡ − e ( N − N ) = C E ( V QD − V ) − C d V d − C s V s − C g V g , (77)15here C E = C d + C s + C g . Therefore, the energy shift U is given by: U = U L + e C E ∆ N, (78)and where U L ≡ C d C E ( − eV d ) + C s C E ( − eV s ) + C g C E ( − eV g ) . (79)In the the expression for U , U L represents a uniform shift for all levels, whereas the second term (the Poissoncontribution denoted U P in the introduction) represents a level repulsion which is proportional to the averagedoccupation of the QD level refereed to N , and proportional to the charging energy U = e /C E .On the other hand, one has ∆ N , from Equations (73) and (77) given by:∆ N = 2 (cid:90) ∞−∞ d ω π i (cid:2) G < ( ω, U ) − G < ( ω, − eV ) (cid:3) . (80)In the expression for G < ( ω, U ) (Equation (66)), the energy level shifts only, ( E ⇒ ( E + U )) in the expressionfor the QD spectral function ρ ( ω − U ). Equations (78) and (80) are coupled non-linear equations with unknowns U and ∆ N . We solve the coupled equations via an iteration procedure. First we guest a value for ∆ N , plugthis value in U , then we calculate ∆ N with the equation:∆ N = 2 (cid:90) ∞−∞ d ω ρ ( ω − U ) Γ L ( ω ) f ( ω ) + Γ R ( ω + eV ) f ( ω + eV )Γ L ( ω ) + Γ R ( ω + eV ) , (81)and so on until convergence is achieved. With the final value of U obtained for a given bias voltage V , I SP iscalculated via the equation:I SP ( V, U ) = 8 πeh (cid:90) ∞−∞ d ω Γ L ( ω ) Γ R ( ω + eV )Γ L ( ω ) + Γ R ( ω + eV ) ρ ( ω − U ) [f ( ω ) − f ( ω + eV )] . (82)In summary, the procedure for computing I consists of the following steps. i) Determine the spectraldensity. ii) Specify V g , V d and V s and coupling constants. iii) Iteratively solve (81) and (78). iv) Evaluate thecurrent from (82) for the V g , V d and V s . Once a converged U has been found, the current is finally evaluated.The way we consider electron-electron interactions, imposes restrictions on the possible values of thecharging energy U . For the self-consistent scheme to be valid, we have to assume, that ∆ (cid:29) Γ L,R (cid:39) U .However, less precisely quantitative results, although qualitative correct results can be obtained if ∆ (cid:39) U (cid:29) Γ L,R , when the called Coulomb Blockade energy dominates over the coupling constants. For this case, we usethe improvement of the SCF method discussed in the introduction [4, 31, 32]. The self consistent generalizesto: U ↑ = U L + e C E ( N ↓ − N ) , (83) U ↓ = U L − e C E ( N ↑ − N ) . (84)where the up-spin level feels a potential due to the down-spin electrons and viceversa. Notice the different signs,which reflects the Coulomb repulsion between otherwise degenerate levels. N ↑ = (cid:90) ∞−∞ d ω ρ ( ω − U ↑ ) Γ L ( ω ) f ( ω ) + Γ R ( ω + eV ) f ( ω + eV )Γ L ( ω ) + Γ R ( ω + eV ) , (85)16 ↓ = (cid:90) ∞−∞ d ω ρ ( ω − U ↓ ) Γ L ( ω ) f ( ω ) + Γ R ( ω + eV ) f ( ω + eV )Γ L ( ω ) + Γ R ( ω + eV ) . (86) N = N ↑ + N ↓ . (87)Here, N ↑ and N ↓ are the population of the spin-up and spin-down levels. Once the values of U ↑ and U ↓ arecalculated, I SP is calculated from:I SP ( V, U ↑ , U ↓ ) = 4 πeh (cid:90) ∞−∞ d ω Γ L ( ω ) Γ R ( ω + eV )Γ L ( ω ) + Γ R ( ω + eV ) (cid:110) ρ ( ω − U ↑ ) + ρ ( ω − U ↓ ) (cid:111) × (cid:104) f ( ω ) − f ( ω + eV ) (cid:105) . (88)As Datta has pointed out [4], the approach described above (called unrestricted SCF) can lead to a betterquantitative agreement in comparison with a conceptually correct multi-level Master equation calculation. ∆ (cid:29) Γ L,R (cid:39) U In this regime there are the Multiple Andreev reflections [42] for voltages such that, eV < ∆ (MAR). Also,there is the possibility for quasi-particle co-tunneling current for energy levels far from µ L . These cases will beconsidered in a future work and involves the whole expressions we have derived for the currents (Equations (32)and (33)) and eventually more accurate Green-Keldysh functions and the use of master equations [17]. This casewas studied experimentally in [43]. For given values of capacitances and source voltage, we iterate Equations (78)and (80), in order to find the potential U . Then the single particle current, I SP ( V, U ) is evaluated (Equation(82)). We put the charge before biasing N = 0, such that Coulomb repulsion with the QD-energy level isabsent. Anyhow, in this regime, the effect of the second term in Equation (86) is negligible. Consequently,the Laplace term U L , essentially position the QD degenerate energy level (with respect µ L = 0). In Figure3 we show I − V characteristics for gate voltage values V g = 0 and K B T (cid:28) ∆, whereas Figure 4 shows theoccupation number ∆ N . These curves are symmetric, due to the assumed equality of the coupling capacitances( C d /C E = 0 . − V shifts to right or to the left for C d /C E = 0 . > . < . ρ ( ω ) for eV d = − − . E + U L . Qualitatively, these results are similar of Yeyatiet al. [17]. Characteristic is the broadening of the BSC singularity. The effect of bigger values of Γ R,L is a morepronounced round off the BCS-type singularity. We discuss this issue below. For large enough bias the currentapproaches the normal saturation value I
Sat . 17igure 3: Zero temperature I − V characteristics for superconductor-quantum-dot-superconductor system, cal-culated using the self consistent field (SCF) method, with E = 1 . eV g = 0 . U = 0 .
005 ∆, C d /C E = 0 . L = Γ R = 0 .
005 ∆. 18igure 4: Zero temperature Number of electrons- eV d / ∆ graph for superconductor-quantum-dot-superconductorsystem, calculated using the self consistent field (SCF) method, with E = 1 . eV g = 0 . U = 0 .
005 ∆ , C d /C E = 0 .
5, Γ L = Γ R = 0 .
005 ∆. 19 .2 Second case: ∆ (cid:39) U (cid:29) Γ L,R
In this regime the charging energy acts effectively in lifting the degeneracy of the otherwise single degenerateQD-energy level. For this regime, we use the couple system defined by Equations (83) - (87) and calculate thecurrent according to Equation (88). This is the unrestricted SCF method mentioned in the introduction. Thetransport begins through one level as long as there is in average less than one electron in it. For the givenparameters the onset of current is similar to the first case (No interaction with residual charge in the QD isconsidered). However, when the average occupation exceeds one, the other degenerate levels floats according tothe resulting values values U ↑ and U ↓ . This values push down the position of this second level and push up thealready occupied energy level. In Figures 5 - 6, we show I − V and the number of electrons. In Figure 8 it isshown the spectral density for eV d = − E + U L = − . − . − . ∼ V ∼ . V ∼ . SP (cid:39) U (cid:39) . ∼ . E < E c , therefore another current signal mayoccurs before and a quantitative proper description would be a multilevel QD- model. One notice, however, thestrong fluctuations in the spacing in Figure 2 [30], indicating complex charging many levels phenomena. Noticethat the theoretical explanation of Yeyati et al. [17, 18], does not contain our prediction. The influence of the coupling constant is to broaden the otherwise sharp energy QD level. However, thebroadening is not equally strong, and depends on the relative values of Γ R / Γ L . Ralph et al. [30] determined forthe sample in his Figure 3 Γ R / Γ L (cid:29)
1. Figure 10 shows the I − V characteristics for the restricted case whenboth coupling constants are equal. For larger values of the coupling constants the broadening is stronger. Ifthey are dissimilar in value, the broadening is stronger when Γ R > Γ L . This effect is shown in Figures 9 - 14.This effect is due to the stronger involving of the BSC-DOS singularity of the left lead in the integral expressionfor the current (Equation (70)) and the particular choice of the zero bias voltage (see Figure 2). We have studied the single particle current through a quantum dot coupled with two superconductor leads via acoupled Poisson Non- equilibrium Green Function (PNEGF) formalism. In a systematic and self contained way,we derived the expressions for the current in full generality. In this work we focused only in the weak couplingregime where single particle current is dominant one. The QD is a single degenerate energy level systems modeledvia a capacitive circuit. The influence of the Potential on the QD, on the I − V characteristic is calculated for20igure 5: Zero temperature I − V characteristics showing the Coulomb blockade for superconductor-quantum-dot-superconductor system, calculated using the self consistent field (SCF) method, with E = 1 . eV g =0 . U = 1 . C d /C E = 0 .
5, Γ L = Γ R = 0 .
01 ∆. 21igure 6: Zero temperature Number of electrons- eV d / ∆ for superconductor-quantum-dot-superconductor sys-tem, calculated using the self consistent field (SCF) method, with E = 1 . eV g = 0 . U = 1 . C d /C E = 0 .
5, Γ L = Γ R = 0 .
01 ∆. 22igure 7: Espectral density of the quantum dot- ω for superconductor-quantum-dot-superconductor system,calculated using the self consistent field (SCF) method, with E = 1 . eV g = 0 . eV d = 6 . U = 0 .
01 ∆, C d /C E = 0 .
5, Γ L = Γ R = 0 .
05 ∆. 23igure 8: Espectral density of the quantum dot- ω for superconductor-quantum-dot-superconductor system,calculated using the self consistent field (SCF) method, with E = 1 . eV g = 0 . eV d = 8 . U = 1 . C d /C E = 0 .
5, Γ L = Γ R = 0 .
20 ∆. 24igure 9: Zero temperature I − V characteristics for superconductor-quantum-dot-superconductor system forvarious values of the coupling Γ, calculated using the self consistent field (SCF) method, with E = 1 . eV g = 0 . U = 0 . C d /C E = 0 .
5, Γ L = Γ R .Figure 10: Zero temperature I − V characteristics for superconductor-quantum-dot-superconductor system forvarious values of the coupling Γ, calculated using the self consistent field (SCF) method, with E = 1 . eV g = 0 . U = 0 .
05 ∆, C d /C E = 0 .
5, Γ L = 4Γ R .Figure 11: Zero temperature I − V characteristics for superconductor-quantum-dot-superconductor system forvarious values of the coupling Γ, calculated using the self consistent field (SCF) method, with E = 1 . eV g = 0 . U = 0 .
05 ∆, C d /C E = 0 .
5, Γ R = 4Γ L .Figure 12: Zero temperature I − V characteristics for superconductor-quantum-dot-superconductor system forvarious values of the coupling Γ, calculated using the self consistent field (SCF) method, with E = 1 . eV g = 0 . U = 1 . C d /C E = 0 .
5, Γ L = Γ R . 25igure 13: Zero temperature I − V characteristics for superconductor-quantum-dot-superconductor system forvarious values of the coupling Γ, calculated using the self consistent field (SCF) method, with E = 1 . eV g = 0 . U = 1 . C d /C E = 0 .
5, Γ L = 4Γ R .Figure 14: Zero temperature I − V characteristics for superconductor-quantum-dot-superconductor system forvarious values of the coupling Γ, calculated using the self consistent field (SCF) method, with E = 1 . eV g = 0 . U = 1 . C d /C E = 0 .
5, Γ R = 4Γ L .relevance values of the coupling and capacitances and the implication for experiments is discussed. This wasdone in the weak coupling regime and for ∆ (cid:29) Γ L,R (cid:39) U . A second case when ∆ (cid:39) U (cid:29) Γ L,R also in theweak coupling regime was analyzed. Admittedly, our model of a Hybrid system S/QD/S possess potentiallyphysical extensions. One important missed point is dephasing. This physical effect due to scattering of transportelectrons can be incorporated in the self energy phenomenologically ([45, 46]), or in a stochastic fashion ([47]).Another point is to consider a QD with many energy levels and within the self consistent scheme, to considerthe strong and intermediate regimes and many body correlations due to different kinds of electron-electroninteraction. Here we have to notice that is it not just to scale the level spacing by the charging energy [48]. Itis a genuine many body problem. But the most important missed point was correlations. As pointed out byDatta (chapter III [4]), there has been much effort in order to find a suitable SCF that considers correlations.For example, to modify Equation (78) to consider occupancies probabilities. As discussed in the introduction,Kang [16] and Meir et al. [20] finds a solution for the QD Green function that contains this type of correlation.In other words, one could go to scheme where a more accurate Green function for the QD is used together witha multi-electron picture with associated Master equation. These would mean to use the Anderson model witha Coulomb interaction U that is obtained from a SCF. We want to check if this point of view is correct. Workin this direction is in progress. 26igure 15: The contour C K = C K − ∪ C K + runs on the real axis, but for clarity its two branches C K − and C K + are shown slightly away from the real axis. The contour C K runs from t and return to t . In general, any Green-Keldysh function of two operators A ( t ) and B ( t ) function is given by:G A,B ( t, t (cid:48) ) = − i (cid:104) T K A ( t ) B ( t (cid:48) ) (cid:105) (89)where the operator T K acts on the Keldysh contour shown in Figure 11. A Heaviside function on the Keldyshcontour is given by: Θ ( t, t (cid:48) ) ≡ Θ ( t − t (cid:48) ) , t ∈ C K − , t (cid:48) ∈ C K − ;0 , t ∈ C K − , t (cid:48) ∈ C K + ;1 , t ∈ C K + , t (cid:48) ∈ C K − ;Θ ( t (cid:48) − t ) , t ∈ C K + , t (cid:48) ∈ C K + . (90)whereas be derivative of the Heaviside function on the Keldysh contour is given by: δ ( t, t (cid:48) ) ≡ ∂ Θ ( t, t (cid:48) ) ∂t = − ∂ Θ ( t, t (cid:48) ) ∂t (cid:48) = δ ( t − t (cid:48) ) , t ∈ C K − , t (cid:48) ∈ C K − ;0 , t ∈ C K − , t (cid:48) ∈ C K + ;0 , t ∈ C K + , t (cid:48) ∈ C K − ; − δ ( t − t (cid:48) ) , t ∈ C K + , t (cid:48) ∈ C K + . (91)In general a function F ( t, t ) defined on the Keldsyh contour is given by:F ( t, t (cid:48) ) ≡ Θ ( t, t (cid:48) ) F > ( t, t (cid:48) ) + Θ ( t (cid:48) , t ) F < ( t, t (cid:48) ) (92)where F > ( t, t (cid:48) ) is the so called greater component (greater Keldysh-Green function) and F < ( t, t (cid:48) ) is the lessercomponent (lesser Keldysh-Green function) of F ( t, t (cid:48) ).Directly calculations can be carried out, in this way by deriving by t or t (cid:48) and using Heisenbergequation of motion for the time evolution of the operators, A ( t ) = exp( iHt ) A exp( − iHt ), ∂ A ( t ) ∂t = − i [ A ( t ) , H ] , (93)ones obtains: i ∂ G ( t, t (cid:48) ) ∂t = δ ( t, t (cid:48) ) (cid:10) [ A ( t ) , B ( t )] ∓ (cid:11) − i (cid:104) T K [ A ( t ) , H ] B ( t (cid:48) ) (cid:105) , (94)27 i ∂ G ( t, t (cid:48) ) ∂t = δ ( t, t (cid:48) ) (cid:10) [ A ( t ) , B ( t )] ∓ (cid:11) + i (cid:104) T K A ( t ) [ B ( t (cid:48) ) , H ] (cid:105) . (95)If there is not an applied potential, i.e., if V s = V d = 0, (see Figure 2), the whole system is at equilibrium, onecan use the usual commutator Green Functions, or equivalently, the Matsubara or Temperature-Green Functionto quantify correlations functions. In that case, Equation (89) depends only on the time difference ( t − t (cid:48) ).However, for times t > t , once the potential difference has been applied, the simple dependence on the timedifferences not longer holds which signalizes a non equilibrium situation. In that case the Keldysh methodapplies. We encounter two cases (Equations (28) and (50)), where the general strategy to find an integral expressionfor the Keldyhs-Green Functions is the following: 1) one first considers the equation of motion for for theKeldysh-Green function of each systems (two leads and the QD) (Equation (27) and Equation (46)). 2) Theresulting equation of motion in each case has a delta function as inhomogeneity. 3) The systems are connectedin at t , we have as a result two coupled equations of motion (Equations (20) and (49)). 4) These equations areconverted into an equivalent integral equation on the Keldysh contour. 5) Straightforwardly derivation od theintegral equations results in the differential equation of motion. In this Appendix we explain how we evaluate the some important convolutions used to calculate lesser Keldysh-Green functions [49, 50]. In general a function given as a convolution on the Keldysh contour poses the definitiongiven in Appendix A. One encounter situations where a Keldysh-Green function is given by a convolution oftwo other functions: P ( t, t (cid:48) ) = (cid:90) C K d (cid:48)(cid:48) F ( t, t (cid:48)(cid:48) ) G ( t (cid:48)(cid:48) , t (cid:48) ) . However, for evaluation of quantities like for example the current, one needsP < ( t, t (cid:48) ) = (cid:90) C K d (cid:48)(cid:48) F ( t, t (cid:48)(cid:48) ) G ( t (cid:48)(cid:48) , t (cid:48) ) | t< CK t (cid:48) . One sees that, the relative position of t and t (cid:48) divides the contour in three regions of integration: 1) t (cid:48)(cid:48) < C K t ,2) t < C K t (cid:48)(cid:48) < t (cid:48) and 3) t (cid:48)(cid:48) > C K t (cid:48) . This traduces into the following integrals:P < ( t, t (cid:48) ) = (cid:90) tt d t (cid:48)(cid:48) F > ( t, t (cid:48)(cid:48) ) G < ( t (cid:48)(cid:48) , t (cid:48) ) | t (cid:48)(cid:48) < CK t + (cid:90) t (cid:48) t d t (cid:48)(cid:48) F < ( t, t (cid:48)(cid:48) ) G < ( t (cid:48)(cid:48) , t (cid:48) ) | t< CK t (cid:48)(cid:48)
10 Appendix D
Below we proceed to evaluate the unperturbed Green’s functions g η(cid:126)kσ and f η(cid:126)kσ . To achieve this, it is necessaryto introduce the chemical potential shift in each superconductor, H η = H η − µ η N η , so that a η(cid:126)kσ ( t ) ≡ e i H η t a η(cid:126)kσ e − i H η t = e i H η t (cid:16) e i µ η t a η(cid:126)kσ e − i µ η t (cid:17) e − i H η t = e − i µ η t (cid:16) e i H η t a η(cid:126)kσ e − i H η t (cid:17) → e − i µ η t a η(cid:126)kσ ( t ) , due to e i H η t a η(cid:126)kσ e − i H η t = a η(cid:126)kσ e − iE η(cid:126)k t , e i N η t a η(cid:126)kσ e − i N η t = a η(cid:126)kσ e − i µ η(cid:126)k t , N η , H η ] = 0. Consequently, the first unperturbed retarded Green function g (r) η(cid:126)kσ ( t, t (cid:48) ) is given byg (r) η(cid:126)kσ ( t, t (cid:48) ) = − iΘ ( t − t (cid:48) ) (cid:68)(cid:110) e − i µ η t a η(cid:126)kσ ( t ) , e i µ η t (cid:48) a † η(cid:126)kσ ( t (cid:48) ) (cid:111)(cid:69) = − iΘ ( t − t (cid:48) ) e − i µ η ( t − t (cid:48) ) × (cid:68)(cid:110) u η(cid:126)k γ η(cid:126)kσ ( t ) + σv η(cid:126)k γ † η(cid:126)kσ ( t ) , u η(cid:126)k γ † η(cid:126)kσ ( t (cid:48) ) + σv η(cid:126)k γ η − (cid:126)k − σ ( t (cid:48) ) (cid:111)(cid:69) = − iΘ ( t − t (cid:48) ) e − i µ η ( t − t (cid:48) ) (cid:104) u η(cid:126)k e − iE η(cid:126)k ( t − t (cid:48) ) + v η(cid:126)k e iE η(cid:126)k ( t − t (cid:48) ) (cid:105) , = − iΘ ( t − t (cid:48) ) (cid:104) u η(cid:126)k e − i ( E η(cid:126)k + µ η )( t − t (cid:48) ) + v η(cid:126)k e i ( E η(cid:126)k − µ η )( t − t (cid:48) ) (cid:105) . where the fermion operators γ † η(cid:126)kσ , γ η(cid:126)kσ create and annihilate the “Bogoliubov quasi-particles” and σ = ↑ = 1 ↓ = − a η(cid:126)kσ ( t ) = u η(cid:126)k γ η(cid:126)kσ ( t ) + σv η(cid:126)k γ † η(cid:126)kσ ( t ) ,a † η(cid:126)kσ ( t (cid:48) ) = u η(cid:126)k γ † η(cid:126)kσ ( t (cid:48) ) + σv η(cid:126)k γ η − (cid:126)k − σ ( t (cid:48) ) . Applying the Fourier transformations to g (r) η(cid:126)kσ ( t, t (cid:48) )g (r) η(cid:126)k ( t, t (cid:48) ) = (cid:90) ∞−∞ d ω π e − i ω ( t − t (cid:48) )g (r) η(cid:126)k ( ω ) , with g (r) η(cid:126)k ( ω ) = u η(cid:126)k ω − E η(cid:126)k − µ η + i0 + + v η(cid:126)k ω + E η(cid:126)k − µ η + i0 + . therefore we have V η (cid:88) (cid:126)k g (r) η(cid:126)k ( ω ) = 12 V η (cid:88) (cid:126)k (cid:32) ω − E η(cid:126)k − µ η + i0 + + 1 ω + E η(cid:126)k − µ η + i0 + (cid:33) = V η P (cid:88) (cid:126)k ω − µ η ( ω − µ η ) − E η(cid:126)k −
12 i πV η (cid:88) (cid:126)k (cid:104) δ (cid:16) ω − µ η − E η(cid:126)k (cid:17) + δ (cid:16) ω − µ η + E η(cid:126)k (cid:17)(cid:105) = − N η (0) V η P (cid:90) ∞−∞ d ξ ω − µ η ξ + ∆ − ( ω − µ ) −
12 i πV η (cid:88) (cid:126)k δ (cid:16) | ω − µ η | − E η(cid:126)k (cid:17) . FinallyΣ (r) ( ω ) ≡ V η (cid:88) (cid:126)k g (r) η(cid:126)k ( ω ) = − Γ η (cid:20) ω − µ η ∆ η ζ (∆ η , ω − µ η ) + i ζ ( ω − µ η , ∆ η ) (cid:21) . where Γ η = πN η ( ω − µ η ) V η ≈ πN η (0) V η , ζ ( ω, ω (cid:48) ) ≡ Θ ( | ω | − | ω (cid:48) | ) | ω |√ ω − ω (cid:48) . η = πN η ( ω − µ η ) V η ≈ πN η (0) V η . are the coupling constants between the leads and the quantum dot inthe wide band limit (WBL). N η (0) is the density of states at the η Fermi level and f ( ω ) is the Fermi-Diracdistribution function.The WBL means that the width of the electronic energy bands of the leads are the largest energy.The density of states in the contacts vary on a scale of Fermi energy. These scales are of order 1 −
10 eV (cid:0) ∼ − k (cid:1) which are much larger than the energies involved in the quantum dot ∼ meV ∼
10 K.Furthermore, Γ η ( ω − µ η ) varies slowly with ω − µ η and the prefactor D η ( ω − µ η ) varies in the range of wideband and changes ω − µ η on the average of | V η(cid:126)k | occur on the order of meV. Therefore, we ignore the dependenceof ω − µ η in Γ η ( ω − µ η ). The WBL establish that an electron in the dot decays in an continuum of states of theleads and is sufficient condition for the existence of a stationary state, as has been shown rigourously in [44].
11 Appendix E g <η(cid:126)kσ ( t, t (cid:48) ) = i (cid:68)(cid:110) e i µ η t (cid:48) a † η(cid:126)kσ ( t (cid:48) ) , e − i µ η t a η(cid:126)kσ ( t ) (cid:111)(cid:69) = ie i µ η ( t − t (cid:48) ) (cid:68)(cid:104) u η(cid:126)k γ † η(cid:126)kσ ( t (cid:48) ) + σv η(cid:126)k γ η − (cid:126)k − σ ( t (cid:48) ) (cid:105) × (cid:104) u η(cid:126)k γ η(cid:126)kσ ( t ) + σv η(cid:126)k γ † η − (cid:126)k − σ ( t ) (cid:105)(cid:69) = i (cid:104) u η(cid:126)k e − i ( E η(cid:126)k + µ η )( t − t (cid:48) )f (cid:16) E η(cid:126)k (cid:17) + v η(cid:126)k e i ( E η(cid:126)k − µ η )( t − t (cid:48) )f (cid:16) − E η(cid:126)k (cid:17)(cid:105) . Applying the Fourier transformationsg <η(cid:126)kσ ( t, t (cid:48) ) = (cid:90) ∞−∞ d ω π e − i ω ( t − t (cid:48) )g <η(cid:126)k ( ω ) , with g <η(cid:126)kσ ( ω ) = 2 π i (cid:104) u η(cid:126)k δ (cid:16) ω − E η(cid:126)k − µ η (cid:17) + v η(cid:126)k δ (cid:16) ω + E η(cid:126)k − µ η (cid:17)(cid:105) f ( ω − µ η ) . therefore we have V η (cid:88) (cid:126)k g <η(cid:126)kσ ( ω ) = 2 π i V η (cid:88) (cid:126)k (cid:104) u η(cid:126)k δ (cid:16) ω − E η(cid:126)k − µ η (cid:17) + v η(cid:126)k δ (cid:16) ω + E η(cid:126)k − µ η (cid:17)(cid:105) × f ( ω − µ η ) , = 2 π i V η (cid:88) (cid:126)k δ (cid:16) | ω − µ η | − E η(cid:126)k (cid:17) f ( ω − µ η ) . Finally,Σ < ( ω ) ≡ V η (cid:88) (cid:126)k g <η(cid:126)k ( ω ) = 2iΓ η ζ ( ω − µ η , ∆ η ) f ( ω − µ η ) . f (r) η(cid:126)kσ ( t, t (cid:48) ) = − iΘ ( t − t (cid:48) ) (cid:68)(cid:110) e i µ η t a † η − (cid:126)k − σ ( t ) , e i µ η t (cid:48) a † η(cid:126)kσ ( t (cid:48) ) (cid:111)(cid:69) = − iΘ ( t + t (cid:48) ) e i µ η ( t + t (cid:48) ) × (cid:68)(cid:110) u η(cid:126)k γ † η − (cid:126)k − σ ( t ) − σv η(cid:126)k γ η(cid:126)kσ ( t ) , u η(cid:126)k γ † η(cid:126)kσ ( t (cid:48) ) + σv η(cid:126)k γ η − (cid:126)k − σ ( t (cid:48) ) (cid:111)(cid:69) = − iΘ ( t − t (cid:48) ) e µ η t e − i µ η ( t − t (cid:48) ) σv η(cid:126)k u η(cid:126)k (cid:104) e iE η(cid:126)k ( t − t (cid:48) ) − e − iE η(cid:126)k ( t − t (cid:48) ) (cid:105) = − iΘ ( t − t (cid:48) ) σ ∆ η η(cid:126)k (cid:104) e i ( E η(cid:126)k − µ η )( t − t (cid:48) ) − e − i ( E η(cid:126)k + µ η )( t − t (cid:48) ) (cid:105) . Applying the Fourier transformations V η (cid:88) (cid:126)k f (r) η(cid:126)kσ ( t, t (cid:48) ) = (cid:90) ∞−∞ e µ η t σ d ω π e − i ω ( t − t (cid:48) )Ξ (r) η(cid:126)k ( ω ) , with V η (cid:88) (cid:126)k Ξ (r) η(cid:126)k ( ω ) = ∆ η η(cid:126)k (cid:34) ω + E η(cid:126)k − µ η + i0 + − ω − E η(cid:126)k − µ η + i0 + (cid:35) = V η (cid:88) (cid:126)k ∆ η η(cid:126)k (cid:34) ω + E η(cid:126)k − µ η + i0 + − ω − E η(cid:126)k − µ η + i0 + (cid:35) = V η ∆ η (cid:88) (cid:126)k P 1 ξ η(cid:126)k + ∆ η − ( ω − µ η ) + i π ω − µ η ) δ (cid:16) | ω − µ η | − E η(cid:126)k (cid:17) = N η (0) V η ∆ η × P (cid:90) ∞−∞ d ξ ξ + ∆ − ( ω − µ ) + i πV η ω − µ η ) (cid:88) (cid:126)k δ (cid:16) | ω − µ η | − E η(cid:126)k (cid:17) . Finally,Ξ (r) η(cid:126)k ( ω ) = Γ η (cid:20) ζ (∆ η , ω − µ η ) + i ∆ η ω − µ η ζ ( ω − µ η , ∆ η ) (cid:21) .
13 Appendix G f <η(cid:126)kσ ( t, t (cid:48) ) = i (cid:68)(cid:110) e i µ η t (cid:48) a † η(cid:126)kσ ( t (cid:48) ) , e i µ η t a † η − (cid:126)k − σ ( t ) (cid:111)(cid:69) = ie i µ η ( t + t (cid:48) ) (cid:68)(cid:104) u η(cid:126)k γ † η(cid:126)kσ ( t (cid:48) ) + σv η(cid:126)k γ η − (cid:126)k − σ ( t (cid:48) ) (cid:105) × (cid:104) u η(cid:126)k γ † η − (cid:126)k − σ ( t ) − σv η(cid:126)k γ η(cid:126)kσ ( t ) (cid:105)(cid:69) = ie µ η t e − i µ η ( t − t (cid:48) ) (cid:104) σu η(cid:126)k v η(cid:126)k e i ( E η(cid:126)k )( t − t (cid:48) )f (cid:16) − E η(cid:126)k (cid:17) − σv η(cid:126)k u η(cid:126)k e − i ( E η(cid:126)k )( t − t (cid:48) )f (cid:16) E η(cid:126)k (cid:17)(cid:105) = ie µ η t σ ∆ η η(cid:126)k (cid:104) e i ( E η(cid:126)k − µ η )( t − t (cid:48) )f (cid:16) − E η(cid:126)k (cid:17) + e − i ( E η(cid:126)k + µ η )( t − t (cid:48) )f (cid:16) E η(cid:126)k (cid:17)(cid:105) . V η (cid:88) (cid:126)k f <η(cid:126)kσ ( t, t (cid:48) ) = (cid:90) ∞−∞ e µ η t σ d ω π e − i ω ( t − t (cid:48) )Ξ <η(cid:126)k ( ω ) , (102)with Ξ <η(cid:126)k ( ω ) = V η (cid:88) (cid:126)k π iE η(cid:126)k (cid:104) δ (cid:16) ω + E η(cid:126)k − µ η (cid:17) f (cid:16) − E η(cid:126)k (cid:17) − δ (cid:16) ω − E η(cid:126)k − µ η (cid:17) f (cid:16) E η(cid:126)k (cid:17)(cid:105) = V η (cid:88) (cid:126)k π iE η(cid:126)k (cid:104) δ (cid:16) ω + E η(cid:126)k − µ η (cid:17) − δ (cid:16) ω − E η(cid:126)k − µ η (cid:17)(cid:105) f ( ω − µ η )= − η ω − µ η (cid:34) πV η (cid:88) η δ (cid:16) | ω − µ η | − E η(cid:126)k (cid:17)(cid:35) f ( ω − µ η ) . Finally,Ξ <η(cid:126)k ( ω ) = − η ∆ η ω − µ η ζ ( ω − µ η , ∆ η ) f ( ω − µ η ) . References [1] L. P. Kouwenhofen, D. G. Austing and S. Tarucha, “Few-electron quantum dots”,
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