Finite-temperature spintronic transport through Kondo quantum dots: Numerical renormalization group study

In the early 1970s, Wilson developed the concept of a fully nonperturbative renormalization group transformation. When applied to the Kondo problem, this numerical renormalization group (NRG) method gave for the first time the full crossover from the high-temperature phase of a free spin to the low-temperature phase of a completely screened spin. The NRG method was later generalized to a variety of quantum impurity problems. The purpose of this review is to give a brief introduction to the NRG method, including some guidelines for calculating physical quantities, and to survey the development of the NRG method and its various applications over the last 30 years. These applications include variants of the original Kondo problem such as the non-Fermi-liquid behavior in the two-channel Kondo model, dissipative quantum systems such as the spin-boson model, and lattice systems in the framework of the dynamical mean-field theory.

We investigate the effects of spin-polarized leads on the Kondo physics of a quantum dot using the numerical renormalization group method. Our study demonstrates in an unambiguous way that the Kondo effect is not necessarily suppressed by the lead polarization: While the Kondo effect is quenched for the asymmetric Anderson model, it survives even for finite polarizations in the regime where charge fluctuations are negligible. We propose the linear tunneling magnetoresistance as an experimental signature of these behaviors. We also report on the influence of spin-flip processes.

We study the Kondo model --a magnetic impurity coupled to a one dimensional wire via exchange coupling-- by using Wilson's numerical renormalization group (NRG) technique. By applying an approach similar to which was used to compute the two impurity problem we managed to improve the bad spatial resolution of the numerical renormalization group method. In this way we have calculated the impurity spin - conduction electron spin correlation function which is a measure of the Kondo compensation cloud whose existence has been a long standing problem in solid state physics. We also present results on the temperature dependence of the Kondo correlations.

Using numerical renormalization group method the temperature dependences of the magnetic susceptibility $\chi(T)$ and specific heat $C(T)$ are obtained in the single-impurity Anderson model with inclusion of the $d-f$ Coulomb interaction. It is shown that the exciton effects owing to this interaction can change considerably the dependence $C(T)$ in comparison with the standard Anderson model at not too low temperatures, whereas the dependence $\chi (T)$ remains universal. The renormalization of the effective hybridization parameter and $f$-level position, which is connected with the $d-f$ interaction, is calculated, a satisfactory agreement with the Hartree-Fock approximation being derived.

A superconductor (SC) in proximity to a ferromagnetic insulator (FMI) is predicted to exhibit mixed singlet and triplet correlations. The magnetic proximity effect of FMI spin splits the energy of Bogoliubov excitations and leads to a spin polarization at the surface for superconducting films that are thinner than the superconducting coherence length. In this work, we study manifestations of these phenomena in the properties of a magnetic impurity coupled via Kondo coupling to this FMI/SC system. Using the numerical renormalization group (NRG) method, we compute the properties of the ground state and low-lying excited states of a model that incorporates the Kondo interaction and a Ruderman-Kittel-Kasuya-Yosida (RKKY)-like interaction with the surface spin polarization. Our main finding is an energy splitting of the lowest even fermion-parity states caused by the proximity to the FMI. As the Kondo coupling increases, the splitting grows and saturates to a universal value equal to twice the exchange field of the FMI. We introduce a two-site model that can be solved analytically and provides a qualitative understanding of this and other NRG results. In addition, using perturbation theory, we demonstrate that the mechanism behind the splitting involves the RKKY field and the triplet correlations of the spin-split superconductor. A scaling analysis combined with NRG shows that the splitting can be written as a single-parameter scaling function of the ratio of the Kondo temperature and the superconducting gap, which is also numerically obtained. Published by the American Physical Society 2024

We study the spin-resolved transport properties of T-shaped double quantum dots coupled to ferromagnetic leads. Using the numerical renormalization group method, we calculate the linear conductance and the spin polarization of the current for various model parameters and at different temperatures. We show that an effective exchange field due to the presence of ferromagnets results in different conditions for Fano destructive interference in each spin channel. This spin dependence of the Fano effect leads to perfect spin polarization, the sign of which can be changed by tuning the dots' levels. Large spin polarization occurs due to Coulomb correlations in the dot, which is not directly coupled to the leads, while finite correlations in the directly-coupled dot can further enhance this effect. Moreover, we complement accurate numerical results with a simple qualitative explanation based on analytical expressions for the zero-temperature conductance. The proposed device provides a prospective example of an electrically-controlled, fully spin-polarized current source, which operates without an external magnetic field.

We investigate the spin-related currents and tunnel magnetoresistance through a quantum dot, which is side-coupled with a Majorana fermion zero mode and two thermal-driven ferromagnetic electrodes. It is found that the interplay of Majorana fermion and electrodes’ spin polarization can induce a nonlinear thermal-bias spin current. This interplay also decreases the total magnitude of spin or charge current, in either parallel or antiparallel configuration. In addition, a thermal-driven negative tunnel magnetoresistance is found, which is an unique feature to characterize Majorana fermion. With large temperature difference, a step phenomenon is observed in gate tuned spin-up current. When the coupling between quantum dot and topological superconductor is strong enough, this step will evolve into a linear relation, revealing Majorana fermion’s robustness.

An impurity four-level Kondo model, in which an ion is tunneling among 4-stable points and interacting with surrounding conduction electrons, is investigated using both perturbative and numerical renormalization group methods. The results of numerical renormalization group studies show that it is possible to construct the ground state wavefunction including the excited ion states if we take into account the interaction between the conduction electrons and the ion. The resultant effective mass of quasiparticles is moderately enhanced. This result offers a good explanation for the enhanced and magnetically robust Sommerfeld coefficient observed in SmOs$_4$Sb$_{12}$, some other filled-skutterudites, and clathrate compounds.

The thermoelectric properties of strongly correlated quantum dots, described\nby a single level Anderson model coupled to conduction electron leads, is\ninvestigated using Wilson's numerical renormalization group method. We\ncalculate the electronic contribution, $K_{\\rm e}$, to the thermal conductance,\nthe thermopower, $S$, and the electrical conductance, $G$, of a quantum dot as\na function of both temperature, $T$, and gate voltage, ${\\rm v}_g$, for strong,\nintermediate and weak Coulomb correlations, $U$, on the dot. For strong\ncorrelations and in the Kondo regime, we find that the thermopower exhibits two\nsign changes, at temperatures $T_{1}({\\rm v}_g)$ and $T_{2}({\\rm v}_g)$ with\n$T_{1}< T_{2}$. Such sign changes in $S(T)$ are particularly sensitive\nsignatures of strong correlations and Kondo physics. The relevance of this to\nrecent thermopower measurements of Kondo correlated quantum dots is discussed.\nWe discuss the figure of merit, power factor and the degree of violation of the\nWiedemann-Franz law in quantum dots. The extent of temperature scaling in the\nthermopower and thermal conductance of quantum dots in the Kondo regime is also\nassessed.\n

We investigate the Josephson effect through a two-level quantum dot with an exchange coupling between two dot electrons. We compute the superconducting phase relationship and construct the phase diagram in the superconducting gap--exchange coupling plane in the regime of the singlet-triplet transition driven by the exchange coupling. In our study two configurations for the dot-lead coupling are considered: one where effectively only one channel couples to the dot, and the other where the two dot orbitals have opposite parities. Perturbative analysis in the weak-coupling limit reveals that the system experiences transitions from 0 to $\pi$ (negative critical current) behavior, depending on the parity of the orbitals and the spin correlation between dot electrons. The strong coupling regime is tackled with the numerical renormalization group method, which first characterizes the Kondo correlations due to the dot-lead coupling and the exchange coupling in the absence of superconductivity. In the presence of superconductivity, many-body correlations such as two-stage Kondo effect compete with the superconductivity and the comparison between the gap and the relevant Kondo temperature scales allows to predict a rich variety of phase diagrams for the ground state of the system and for the Josephson current. Numerical calculations predicts that our system can exhibit Kondo-driven 0-$\pi$-0 or $\pi$-0-$\pi$ double transitions and, more interestingly, that if proper conditions are met a Kondo-assisted $\pi$-junction can arise, which is contrary to a common belief that the Kondo effect opens a resonant level and makes the 0-junction. Our predictions could be probed experimentally for a buckminster fullerene sandwiched between two superconductors.

We investigate f-electron charge susceptibility in a two-impurity Anderson model on the basis of Wilson's numerical renormalization group method. The f-electron charge susceptibility diverges logarithmically at the critical point of this model when conduction-electron bands exhibit particle–hole symmetry. Although the critical point disappears without the particle–hole symmetry, the f-electron charge fluctuation is much more enhanced near the crossover regime between the Kondo–Yosida singlet and intersite spin-singlet states than that in the single-impurity case. This result shows that charge fluctuations are enhanced owing to the competition between intersite and Kondo–Yosida spin singlets. A possible scenario for the enhanced residual resistivity near the region where the Kondo temperature becomes comparable with the Néel temperatures under pressure in some heavy-fermion compounds is proposed.

The numerical renormalization group method is used to investigate zero-temperature phasetransitions in quantum impurity systems, in particular in the particle–hole symmetricsoft-gap Anderson model. The model displays two stable phases whose fixed points can bebuilt up of non-interacting single-particle states. In contrast, the quantum phase transitionsturn out to be described by interacting fixed points, and their excitations cannot bedescribed in terms of free particles. We show that the structure of the many-bodyspectrum of these critical fixed points can be understood using renormalizedperturbation theory close to certain values of the bath exponents which play the role ofcritical dimensions. Contact is made with perturbative renormalization groupcalculations for the soft-gap Anderson and Kondo models. A complete description ofthe quantum critical many-particle spectra is achieved using suitable marginaloperators; technically this can be understood as epsilon-expansion for full many-bodyspectra.

We propose a new approximation scheme within the equation of motion approach (EOM)to spin polarized transport through a quantum dot coupled to ferromagneticleads. It has some advantages over a standard EOM technique widely used inthe literature, in particular when we are interested in spin polarized quantities.Namely, it gives values of the dot spin polarization which are closer to the onesobtained within the numerical renormalization group (NRG) than the standardEOM approach. While restoring the Kondo effect, the spin polarization vanishesand the transport becomes unpolarized, in agreement with NRG and real timediagrammatic calculations. The standard EOM procedure gives non-zero values of the spinpolarization, and the transport is still spin polarized. Both approximations give the samecorrect splitting of the Kondo peaks due to ferromagnetism in the electrodes.

A quantum theory of spin-dependent electronic transport through a single Mn2+-ion-doped II-VI quantum dot weakly coupled to ferromagnetic (FM) leads is presented. The electron-Mn-ion exchange interaction in the quantum dot strongly enhances the spin polarization of the current and the tunneling magnetoresistance (TMR) and yields a highly spin-polarized current even when the FM source is not 100% polarized. Both the spin polarization and the TMR depend strongly on the bias voltage VSD. In the presence of an external magnetic field, the TMR oscillates with VSD. The results open the way to voltage-controlled spin filters and magnetic sensors.

In this thesis, antiferromagnetic and superconducting order in the Kondo-lattice model is investigated by means of the dynamical mean-field theory (DMFT) in combination with the numerical renormalization group (NRG) method. First, these numerical methods are introduced and a special emphasis is put on the inclusion of superconducting order by employing the Nambu formalism. In the next chapter, the Kondo-lattice model as a paradigmatic model for heavy-fermion system is introduced. The analysis of the paramagnetic state at zero temperature reviews the description of the electronic system as a Fermi liquid of heavy quasiparticles. Afterwards ordered phases are investigated. In the antiferromagnetically ordered phase at zero temperature and half filling, a Néel state is found to be stable for small Kondo interactions. A peculiar feature arises in the density of states: A “spin resonance” is observed at energies which scale with the Kondo coupling. These resonances are the more pronounced, the higher the conduction band electrons are polarized – hence they are quite likely related to spin fluctuations. As a main result of the thesis, a superconducting state at zero temperature is found to be stable in a large parameter regime within the DMFT+NRG approach. The pair correlations are the most pronounced in the region of the parameter space, where AF order breaks down. Although only even-frequency singlet s-wave superconductivity can be treated within the utilized approach, the superconducting state is actually found to be closely related to local spin fluctuations, which act as retarded “glue” for s-wave pairing of heavy quasiparticles. This is a novel mechanism for superconductivity in HF systems. Similar to the antiferromagnetic case, resonances are observed in the density of states at energies related to the spin-fluctuation spectrum. These resonances are interpreted by means of a static mean-field description as a separation of the flat quasiparticle bands from the strongly dispersive part of the bandstructure. Such a structure is also observed in the spectral functions obtained by DMFT. Therefore, the conclusion is drawn that the pairing appears among the heavy quasiparticles. This conclusion is underpinned by sign changes of the dynamic gap function at the very frequencies, where the resonances respectively the band splitting is observed. In addition, the observed rule that the critical temperature for the superconducting state is generally lower than the coherence scale, indicates that the heavy-fermion state is a prerequisite for pairing in the Kondo-lattice model. Moreover, the ratio of the gapwidth and critical temperature is strongly enhanced over the universal BCS ratio. This fact is associated with a non-phononic origin of the attractive interaction. In summary, the stable superconducting state in the Kondo-lattice model is identified to be highly unconventional.