Kondo peak splitting and Kondo dip induced by a local moment

Many features like spin-orbit coupling, bias and magnetic fields applied, and so on, can strongly influence the Kondo effect. One of the consequences is Kondo peak splitting. However, Kondo peak splitting led by a local moment has not been investigated systematically. In this research we study theoretically electronic transport through a single-level quantum dot exchange coupled to a local magnetic moment in the Kondo regime. We focus on the Kondo peak splitting induced by an anisotropic exchange coupling between the quantum dot and the local moment, which shows rich splitting behavior. We consider the cases of a local moment with S = 1/2 and S = 1. The longitudinal (z-component) coupling plays a role of multivalued magnetic fields and the transverse (x, y-components) coupling lifts the degeneracy of the quantum dot, both of which account for the fine Kondo peak splitting structures. The inter-level or intra-level transition processes are identified in detail. Moreover, we find a Kondo dip at the Fermi level under the proper parameters. The possible experimental observations of these theoretical results should deepen our understanding of Kondo physics.

Scientific RepoRts | 5:18021 | DOI: 10.1038/srep18021 We consider theoretically the Kondo peak splitting phenomenon in electronic transport through a single level quantum dot exchange coupled to a local moment, mainly focusing on an anisotropic coupling. Despite the fact that the relevant aspects of the Kondo effect have been widely studied in many models, for example, the T-shaped double quantum dot or two-impurity Kondo problems [27][28][29][30][31][32][33] , the anisotropic exchange coupling between two spin degrees of freedom and its impact on the Kondo effect have not to be explored in detail. One of the challenges of this study is that we have to deal with two anisotropy couplings: the longitudinal anisotropy coupling J and the transverse anisotropy coupling J ⊥ . In the limiting case of strong longitudinal anisotropy coupling, the local moment behaves like a multi-valued quantum magnetic field, and in the isotropic coupling limit, the results show a three-peak Kondo structure consistent with ref. 25. In the general anisotropic case, the degeneracy is lifted and we can identify the entangled states via which Kondo transport occur. The inter-level or intra-level transitions are also identified, which account for the fine Kondo-peak-splitting structures. Moreover, we find a Kondo dip at the Fermi level under proper parameters. To evaluate the dot's density of states and differential conductance, we use a non-equilibrium Hubbard operator Green's function method 34 . To calculate the relevant Hubbard operator Green's functions from the equations of motion, we apply the truncation scheme introduced in ref. 5.

Results
Hamiltonian. We consider a model consisting of a single level (SL) quantum dot exchange coupled to a local magnetic moment (spin S). This model may be realized in some systems, for instance, magnetic adatoms coupled to a magnetic cluster, quantum dots coupled to a localized magnetic impurity, or a CdTe quantum dot doped with a single metal ion [35][36][37][38][39][40][41] In the numerical results presented below, we assume symmetric dot-lead couplings Γ L = Γ R and a symmetrically applied source-drain bias (μ L = V/2, μ R = − V/2). The energy unit is taken as Γ = Γ L + Γ R = 1. Since the presence of the Kondo resonance and its possible splitting play an important role in the transport, it is helpful to first estimate the Kondo temperature T K , which can be calculated from the half width at half maximum of the Kondo resonance [44][45][46] . For the parameters used in Fig. 1, one gets T K /Γ ≈ 0.026. Figures 1 and 2 show the splitting effects of the anisotropy parameters α and β for the minimal non-trivial spin S = 1/2. Figure 1 shows the differential conductance, related energy spectrum, and three transition processes. Figure 1(a) shows the differential conductance with a gradual transverse anisotropy parameter, from α = 0 to α = 1, and fixed longitudinal parameter β = 1, in which we can clearly observe the Kondo peak splitting effects. In Fig. 1(b), we present the energy spectrum of H D and the corresponding eigenstates, which have been calculated in section II.  , and Δ 4 = ± J, which are shown in Fig. 1 Without the local moment (J = 0), the Kondo singlet state transporting through the SL quantum dot manifests itself as a well-known single sharp peak at V = 0 in conductance. For the strong longitudinal anisotropy case (β = 1 and α = 0), the effect of the local moment is equivalent to a magnetic field. The dot's single level α β Ψ = ↑ + ↓ is now coupled with a " Zeeman field" S z and split into two energy levels ↑↓ ( ↓↑ ) and ↓↓ ( ↑↑ ), as seen in Fig. 1(b), at α = 0. This mechanism is similar to magnetic field induced Kondo peak splitting 5 . The Kondo singlet state tunneling through the dot is a virtual spin-flip process, and thus only inter-level transitions are allowed, which explains the two peaks with α = 0 in Fig. 1(a) and why there is no peak at the Fermi surface (V = 0). However, the magnetic field here is a "quantum" one, and it can take two values with ħ/2 and − ħ/2, which are degenerate.
When we turn on the transverse anisotropy parameter α, i.e., the term (J ⊥ /2)s + S − + (J ⊥ /2)s − S + , the degeneracy is lifted. In fact, there are two effects of (J ⊥ /2)s + S − + (J ⊥ /2)s − S + : it i) entangles the states ↑↓ and ↓↑ and ii) lifts the degeneracy of ↑↓ and ↓↑ and splits the dot's energy level. For 0 < α < 1, there are seven peaks observed in Fig. 1(a). This splitting, originating from the dot's energy level splitting, can be explained with energy level differences. When Kondo transport occurs, it can happen via inter-level transitions or intra-level transitions [see schematic representation of three transitions in Fig. 1(c)]. Δ i (i = 1, 2, 3) correspond to six inter-level transitions, and the peak at V = 0 corresponds to intra-level transitions. Thus, there are seven peaks observed. Specifically, when α ≠ 0, three separated states form: induce the Kondo peaks at V = ± Δ 1 . It seems that the position of the peak at V = 0 is the same as for a single-level quantum dot, but the underlying mechanism is different. The former is a two-electron entangled state, while the latter is a one-electron state.
When α = 1, the exchange coupling becomes isotropic. This isotropic interaction induces a three-peak structure. This structure is also observed in spin-orbit coupling 11 , which can be formulated as → ⋅ → J s l SO . When the anisotropic interaction develops into an isotropic one, one observes that the number of Kondo peaks is reduced. This is because the entangled state − 10 becomes degenerate with the fully polarized states [see Fig. 1(b)]. We now focus on the strong transverse anisotropy coupling case (β = 0 and α = 1), which is shown in Fig. 2. In Fig. 2 we present the results with a gradual longitudinal anisotropy parameter, from β = 0 to β = 1, with fixed transverse parameter α = 1. For 0 < β < 1, there are also seven peaks observed, which confirm the result in Fig. 1(a). For β = 0, we observe five Kondo peaks, which are located at V = 0, ± J ⊥ /2 and ± J ⊥ . These five peaks may also be explained by using the language of intra-and inter-level transitions.
Splitting behaviors for S = 1. Up to now, we have studied the conductance diagram of the system as a function of the anisotropy parameters for S = 1/2. However, for a given magnetic cluster or molecule, the local spin usually has the value S > 1/2. Here, we restrict our discussion to the case of a spin S = 1. We will study the behavior of the conductance varying the anisotropy parameters. The goal is to understand how the values of the local spin influence the behavior of the conductance and Kondo peak splittings. Figures 3 and 4 present the differential conductance for integer spin S = 1. As with S = 1/2, we vary one of the anisotropy parameters while the other one remains fixed. As seen in Fig. 3, when β = 1 and 0 < α < 1, seven peaks are still observed. This is because the energy levels are split into three as before, and are enumerated as ε + (1, ± 1/2), ε − (1, ± 1/2), and the fully polarized states ε(↑ , S). However, one can deduce that when S grows greater, energy levels increase, and likewise with energy level differences and Kondo peaks. When β = 1 and α = 0, one has the strong longitudinal anisotropy case. In this situation, the quantum magnetic field S z takes three values, − ħ, ħ, and 0. As a result, at V = ± J there are two peaks corresponding to inter-level transitions and for S z = 0 it is equivalent to a zero-Zeeman-field corresponding to the Kondo peak at V = 0. For the strong transverse anisotropic case (β = 0 and α = 1), we observe five peaks, which are located at = , ± / ⊥ V J 0 2 2 and ± ⊥ J 2 , as seen in Fig. 4. In Figs 1-4, one can observe a Kondo dip at the Fermi surface (V = 0). This dip is a feature of anisotropic coupling. Under proper parameters, the anisotropic coupling will change the sign of Kondo self-energy at V = 0 and induce the Kondo dip. Moreover, one can conclude that the interplay of J , J ⊥ , and temperature can strongly influence the Kondo resonance at the Fermi surface, i.e., the formation or destruction of a Kondo peak (dip) at Fermi surface.

Methods
The non-equilibrium Green's function method has been widely used in the discussion of Kondo effects in quantum dots 5,[47][48][49][50][51] . In order to handle the large spin involved in many problems, we use the Hubbard operator Green's function method 34 , which has been used to solve the transport problems in the linear response, non-linear response, and Kondo regimes [52][53][54] We use the equation of motion method to calculate the retarded Green's function. Although the equation of motion method is an approximate one due to truncation scheme, it was shown that this method can capture qualitatively the Kondo physics 48   where the upper (lower) sign applies for σ = ↑ (↓ ). The averages are defined as P 0m ≡ 〈 X 00 Y mm 〉 , P σm ≡ 〈 X σσ Y mm 〉 . One may notice that in the above equation there is a newly generated Green's function that has the same order as Other newly generated Green's functions are high-order ones that contain just one lead-operator. The equation of motion of these high-order Green's functions will generate higher-order ones containing two lead-operators. Here is an example: We truncate these higher-order Green's functions following the procedure proposed by Meir, Wingreen, and Lee 5 , in which we pull the operator pair of leads out of the Green's functions and regard them as an average. This truncation approximation captures Kondo physics well when the temperature of the system is low. After this truncation, Eq. (5) becomes are useful. In Eqs. (7) and (8)  where Δ 1,2 are defined as In principle, in Eqs. (9)-(12) at the pole positions of the self-energies, the Kondo peaks should manifest themselves in density of states and conductance. However, at the same time, these self-energies are modulated by parameters of the system. For example, Σ ↑ a is a combination of Kondo self-energies: