Kondo Exchange Coupling Research Articles

Additional applications of the Lambert W function to solid state physics

Analytical solutions are always desirable in physics for the sake of mathematical exactness and physical insight. In this study, I obtain closed-form analytical expressions for various physical quantities taken from solid state physics. More precisely, I derive analytical expressions in terms of the Lambert W function for the work function and the field-enhancement factor of a field emitting material from the Fowler–Nordheim equation. Additionally, I derive similar analytical expressions for the localization length and the density of states in amorphous semiconductors from the Mott hopping conductivity. Similarly, I also derive analytical formulae based on the Lambert W function to compute the extrinsic-intrinsic transition temperature in a partially compensated semiconductor and the Kondo exchange coupling constant. All the obtained results are exact and explicit. Moreover, some of them allow a direct determination of some physical quantities of interest compared to their indirect determination from semi-logarithmic experimental plots. The findings of this paper are accessible and suitable for students enrolled in graduate solid state physics courses.

Aubry-André Anderson model: Magnetic impurities coupled to a fractal spectrum

The Anderson model for a magnetic impurity in a one-dimensional quasicrystal is studied using the numerical renormalization group (NRG). The main focus is elucidating the physics at the critical point of the Aubry-Andre (AA) Hamiltonian, which exhibits a fractal spectrum with multifractal wave functions, leading to an AA Anderson (AAA) impurity model with an energy-dependent hybridization function defined through the multifractal local density of states at the impurity site. We first study a class of Anderson impurity models with uniform fractal hybridization functions that the NRG can solve to arbitrarily low temperatures. Below a Kondo scale $T_K$, these models approach a fractal strong-coupling fixed point where impurity thermodynamic properties oscillate with $\log_b T$ about negative average values determined by the fractal dimension of the spectrum. The fractal dimension also enters into a power-law dependence of $T_K$ on the Kondo exchange coupling $J_K$. To treat the AAA model, we combine the NRG with the kernel polynomial method (KPM) to form an efficient approach that can treat hosts without translational symmetry down to a temperature scale set by the KPM expansion order. The aforementioned fractal strong-coupling fixed point is reached by the critical AAA model in a simplified treatment that neglects the wave-function contribution to the hybridization. The temperature-averaged properties are those expected for the numerically determined fractal dimension of $0.5$. At the AA critical point, impurity thermodynamic properties become negative and oscillatory. Under sample-averaging, the mean and median Kondo temperatures exhibit power-law dependences on $J_K$ with exponents characteristic of different fractal dimensions. We attribute these signatures to the impurity probing a distribution of fractal strong-coupling fixed points with decreasing temperature.

Subgap states in superconducting islands

We study an interacting quantum dot in contact with a superconducting island described by the Richardson model with a Coulomb repulsion term controlling the number of electrons on the island. This Hamiltonian admits a compact matrix-product-operator representation and can be efficiently and accurately solved using the density-matrix renormalization group. We systematically explore the effects of the charging energy $E_c$. For $E_c$ comparable to the superconducting gap $\Delta$, the subgap states are stabilized by the combination of Kondo exchange coupling and charge redistribution driven by the Coulomb interaction. The subgap states exist for both even and odd superconductor ground-state occupancy, but with very distinctive excitation spectra in each case. The spectral peaks are not symmetric with respect to the chemical potential and may undergo discontinuous changes as a function of gate voltages.

Spin dependent transmission of nickelocene-Cu contacts probed with shot noise

The current $I$ through nickelocene molecules and its noise are measured with a low temperature scanning tunneling microscope on a Cu(100) substrate. Density functional theory calculations and many-body modeling are used to analyze the data. During contact formation, two types of current evolution are observed, an abrupt jump to contact and a smooth transition. These data along with conductance spectra ($dI/dV$) recorded deep in the contact range are interpreted in terms of a transition from a spin-1 to a spin-1/2 state that is Kondo screened. Many-body calculations show that the smooth transition is also consistent with a renormalization of spin excitations of a spin-1 molecule by Kondo exchange coupling. The shot noise is significantly reduced compared to the Schottky value of $2eI$ but no influence of the Kondo effect or spin excitations are resolved. The noise can be described in the Landauer picture in terms of spin-polarized transmission of $\approx$35% through two degenerate $d_\pi$-orbitals of the Nickelocene molecule.

Tuning the Kondo effect via gating-controlled orbital selection in the LaAlO3/SrTiO3 interfacial d -electron system

The orbital degree of freedom plays a critical role in contemporary condensed-matter physics, with recent discoveries of orbital-dependent many-body effects including electron correlation and superconductivity. Nevertheless, these orbital-dependent many-body effects have mainly been observed in bulk materials, wherein the electrons' orbital attribute is challenging to tailor selectively. Here, by planar tunneling into the two-dimensional electron system (2DES) at the ${\mathrm{LaAlO}}_{3}/{\mathrm{SrTiO}}_{3}$ interface, we observed a Kondo resonance, a model many-body phenomenon. The observation of the Kondo resonance provides an ideal opportunity to quantitatively exploit the emerging interfacial Kondo physics. Furthermore, it was found that the Kondo coupling strength and resonance line shape can be effectively regulated by electrostatic gating and show a sharp transition at the Lifshitz point, where the orbital occupancy of the 2DES changes from only ${d}_{\mathrm{xy}}$ to both ${d}_{\mathrm{xy}}$ and ${d}_{\mathrm{xz}/\mathrm{yz}}$. This unusual behavior is attributed to the orbital selectivity effect in this unique multiple-$d$-band system, wherein the itinerant ${d}_{\mathrm{xy}}$ and ${d}_{\mathrm{xz}/\mathrm{yz}}$ electrons have different orbital symmetries and therefore different Kondo exchange coupling with the localized ${d}_{\mathrm{xy}}$ electrons. The present study may pave the way for manipulating many-body effects in 2D multiband materials via orbital engineering of itinerant electrons.

Superconducting Correlations in the One-Dimensional Kondo Lattice Models under Magnetic Fields

We analyze superconducting correlations in the one-dimensional Kondo lattice models with Ising anisotropy under transverse magnetic fields, using the density matrix renormalization group. For the spin-1/2 local spin model, the Ising anisotropy is introduced by the ferromagnetic Ising interaction between the local spins, while for the spin-1 model, it is taken by the single-ion anisotropy. The magnetic properties under the transverse fields for the spin-1/2 model are very similar to those for the spin-1 model [K. Suzuki and K. Hattori, J. Phys. Soc. Jpn. 88, 024707 (2019).]. For the superconducting correlations, we analyze various Cooper pairs within nearest-neighbor pairs including composite ones between the local spins and the electrons. We find that, for the spin-1/2 model, the superconducting correlations are highly enhanced in the Tomonaga-Luttinger liquid state near the Kondo-plateau phase, where the conduction electrons and the local spins are strongly coupled with a finite spin gap for the Ising axis. This is a clear contrast to the model under the longitudinal magnetic fields, where there are no noticeable superconducting correlations. Competitions between the transverse magnetic field and the Kondo singlet formation lead to this enhanced superconducting correlations. For the spin-1 model, the single-ion anisotropy suppresses the superconducting correlation and there is no noticeable enhancement. We also examine the large Kondo exchange coupling limit. For the moderate ferromagnetic Ising interaction between the local spins, we find that another type of superconducting correlation is enhanced inside the ferromagnetic phase. We discuss a possible relation between our results and reentrant superconductivity in U-based ferromagnetic superconductors under transverse magnetic fields.

Renormalization of single-ion magnetic anisotropy in the absence of the Kondo effect

Inelastic spin flip excitations associated with single-ion magnetic anisotropy of quantum spins, can be strongly renormalized by Kondo exchange coupling to the conduction electrons in the substrate, as shown recently for the case of Co adatoms on CuN$_2$ islands. In this case differential conductance spectra show zero-bias anomalies due to a Kondo effect of the doubly degenerate ground state, and finite-bias step features due to spin flip excitations. Here I consider spin-1 quantum magnets with positive uniaxial anisotropy, where the ground state is non-degenerate and hence the Kondo effect does not take place. Nevertheless the renormalization of inelastic spin excitations due to exchange coupling by hybridization of the quantum spin with the conduction electrons still takes place despite the complete absence of the Kondo effect in the ground state. Additionally, I show that away from particle-hole symmetry, charge fluctuations have a similar effect to Kondo exchange coupling, leading to the renormalization of spin flip excitations. However, in contrast to the renormalization by Kondo exchange, charge fluctuations lead to asymmetric spectra, which for strong charge fluctuations can mimic Fano behavior.

Quantum impurity models for magnetic adsorbates on superconductor surfaces

Magnetic atoms adsorbed on surfaces have a quenched orbital moment while their ground-state spin multiplet is partially split as a consequence of the spin-orbit coupling which, even if intrinsically weak, has a large effect due to the abrupt change of the potential at the surface. Such metal adsorbates should be modelled using quantum impurity models that include the relevant internal degrees of freedom and the interaction terms, in particular the magnetic anisotropy and the Kondo exchange coupling. When adsorbed on superconducting surfaces, these impurities have complex spectra of sub-gap excitations due to magnetic anisotropy splitting and Kondo screening. Both anisotropy splitting and Zeeman splitting due to the external magnetic field are significantly renormalized by the coupling to the substrate electrons. In this work I discuss the quantum-to-classical crossover and the applicability of classical static-local-spin picture for discussing magnetic nanostructures on superconductors.

Effect of assisted hopping on thermopower in an interacting quantum dot

We investigate the electrical conductance and thermopower of a quantum dot tunnel coupled to external leads described by an extension of the Anderson impurity model which takes into account the assisted hopping processes, i.e., the occupancy-dependence of the tunneling amplitudes. We provide analytical understanding based on scaling arguments and the Schrieffer–Wolff transformation, corroborated by detailed numerical calculations using the numerical renormalization group method. The assisted hopping modifies the coupling to the two-particle state, which shifts the Kondo exchange coupling constant and exponentially reduces or enhances the Kondo temperature, breaks the particle-hole symmetry, and strongly affects the thermopower. We discuss the gate-voltage and temperature dependence of the transport properties in various regimes. For a particular value of the assisted hopping parameter we find peculiar discontinuous behaviour in the mixed-valence regime. Near this value, we find very high Seebeck coefficient. We show that, quite generally, the thermopower is a highly sensitive probe of assisted hopping and Kondo correlations.

Magneto-Electric Effect in Three-Dimensional Coupled Zigzag Chains

Stimulated by recent studies of quantum phases with broken local inversion symmetry, we study the magnetoelectric effect in locally noncentrosymmetric metals. We consider three-dimensional (3D) coupled zigzag chains and demonstrate that the antiferromagnetic moment is induced by the electric current through a staggered antisymmetric spin-orbit coupling. This current-induced magnetism is much larger than that in globally noncentrosymmetric metals. We provide an intuitive understanding of the current-induced antiferromagnetic moment by showing the inverse magnetoelectric effect, that is, the ferroic p-wave charge nematic order accompanied by the asymmetric band structure in the antiferromagnetic state. We also examine conduction electrons coupled to localized spins via Kondo exchange coupling and demonstrate a significant enhancement of the magnetoelectric effect. A possible experimental observation of the magnetoelectric effect in metals is discussed, with focus on LnM_2Al_10 compounds, such as NdRu_Al_10 and TbRu_2Al_10.

Inhomogeneous superconducting phases in the frustrated Kondo-Heisenberg chain

We use bosonization and renormalization group methods to determine the ground state phase diagram of a one-dimensional frustrated Kondo-Heisenberg system consisting of a one-dimensional spin-1/2 Luttinger liquid coupled by a Kondo exchange interaction $J_K$ to a frustrated quantum antiferromagnetic Heisenberg chain, with a nearest-neighbor exchange coupling $J_1$ and a next-nearest-neighbor (frustrating) exchange interaction $J_2$. We analyze the interplay of quantum frustration in the antiferromagnetic chain with the Kondo exchange coupling $J_K$ with the Luttinger liquid. We discuss the structure of the phase diagram of this system as a function of the ratios $J_K/J_1$, $J_2/J_1$ and of the parameters of the Luttinger liquid. In particular we discuss in detail the regimes in which a pair-density-wave state may be realized and its relation with the spin correlations in the frustrated antiferromagnetic chain.

Phase diagram of the Holstein-Kondo lattice model at half filling

We study the Kondo lattice model which is modified by the Holstein term, involving both the Kondo exchange coupling and the electron-phonon coupling constants, characterized by $J$ and $g$, respectively. The model is solved by employing the dynamical mean-field theory in conjunction with the exact diagonalization technique. A zero-temperature phase diagram of symmetry unbroken states at half filling is mapped out which exhibits an interplay between the two interactions and accounts for both spin and charge fluctuations. When the Kondo exchange coupling is dominant the system is in the Kondo insulator state. Increasing $g$ for small values of $J$ leads to a Kondo insulator-metal transition. Upon further enhancement of $g$ a transition to the bipolaronic insulating phase takes place. Also a small region with non-Fermi-liquid behavior is found near the Kondo insulator-metal transition.

Magnetic impurities in a correlated electron system with spin-triplet pairing

We consider an exactly solvable two-band model of electrons moving in one dimension and interacting with a δ -function spin-exchange potential. The interaction leads to the formation of spin-triplet bound states of the Cooper type (hard-core bosons that do not condensate and exist at all temperatures). Without destroying the integrability we introduce a finite concentration of mixed-valent impurities with two magnetic configurations of spin S and S− 1 2 , respectively, which hybridize with the conduction states of the host. We derive the Bethe ansatz equations diagonalizing the correlated host with impurities and discuss the ground state properties as a function of magnetic field, the crystalline field splitting of the bands, the concentration of the impurities and the Kondo exchange coupling. In the zero-field ground state the spin-triplet states are effectively free bosons and a threshold band splitting Δ is required to break up a triplet pair. The gap of the unpaired electron excitations is gradually reduced by the band splitting and closes at Δ . For a splitting larger than Δ a fraction of the electrons is spin-polarized. The impurities are antiferromagnetically correlated with each other and the effect of a magnetic field is to suppress these correlations as well as to gradually align the spins of the spin-triplets with the field. Spin- 1 2 impurities always enhance the gap Δ , while with impurities of higher spin the gap decreases if the coupling between the triplets and impurities is weak (small Kondo temperature), but increases otherwise. For small T K the gap is closed at a critical concentration x cr and the two electron bands have different populations. For x > x cr we obtain a zero-magnetization phase with three coexisting components, namely, itinerant ferromagnetism of unpaired electrons, antiferromagnetism of the impurities and superconducting fluctuations of ferromagnetically correlated spin triplets. The critical exponents of correlation functions are also briefly discussed in the conformal field theory limit.

Spin-gapped two-band correlated electron system with magnetic impurities

We consider an integrable model consisting of two parabolic bands of electrons of equal mass interacting via a δ-function exchange potential. The exchange is attractive or repulsive depending on whether the interacting particles are in a spin-singlet or spin-triplet. Without destroying the integrability we introduce a finite concentration of mixed-valent impurities with two magnetic configurations of spin S and (S+ 1 2 ) , respectively, which hybridize with the conduction states of the host. We derive the Bethe ansatz equations diagonalizing the correlated host with impurities and discuss the ground state properties as a function of magnetic field, the crystalline field splitting of the bands, the concentration of impurities and the Kondo exchange coupling. The attractive interaction in the host leads to Cooper-pair-like bound states, and a magnetic field larger than a critical one H c is required to break up the singlet pairs. The spin-gap is gradually reduced by a magnetic field and closes at H c. For H> H c a fraction of the electrons is spin-polarized. The impurities are antiferromagnetically correlated with each other and the effect of a magnetic field is to suppress these correlations, which are totally quenched at the saturation field H s. Depending on the off-resonance shift θ of the impurities the spin-gap may decrease or increase linearly with the concentration of impurities. For small θ and a weak interaction among the electrons the spin-gap is closed at a critical concentration x cr. For x> x F> x cr ferromagnetism is induced, so that three phases coexist, namely itinerant ferromagnetism, ferrimagnetism of the impurity spins and superconducting fluctuations. The constructive interference between antiferromagnetic and superconducting fluctuations for large θ could be related to the spin-compensation of the impurity spins among each other and the dominating forward scattering of the impurities. The critical exponents of correlation functions are also briefly discussed in the conformal field theory limit.

Mixed valence of the integrable magnetic impurity in correlated-electron hosts

In this paper we show that the Bethe ansatz integrable theories for a magnetic impurity embedded in the correlated electron host correspond to the impurity which carries both spin and charge internal degrees of freedom. Therefore the magnetic impurity necessarily reveals either nonmagnetic behavior, the Kondo-like magnetic regime, or the mixed valence regime, depending on the values of the external magnetic field and the applied voltage, which control the band filling and the magnetization of the correlated-electron host. It is shown that several previously published papers contain invalid statements that the integrability demands only the Kondo exchange coupling of the magnetic impurity to the correlated host together with the action of the ``fine-tuned'' scalar static potential at the impurity site. We prove that instead of that static potential the integrability implies the dynamic scalar interaction of the impurity in the charge sector, too, which causes the valence of the impurity to depend on the external parameters and the parameters of the impurity-host coupling. We show how in the Bethe ansatz framework the effects of the dynamic impurity, external boundary potentials, applied to the edges of the open chain, and the effects of the free edges of the chain themselves are clearly separated. The impurity valence, magnetization, susceptibility, and the mesoscopic effects are calculated as functions of the impurity-host coupling, spin of the impurity, external magnetic field, applied voltage, and temperature for several one-dimensional exactly solvable models of highly correlated electrons. Limitations of the Bethe ansatz approach are discussed.

One-dimensional heavy fermion lattice model

We consider a one-dimensional supersymmetric t−J model and introduce a finite concentration of impurities of arbitrary spin S without destroying the integrability. Using the Bethe ansatz equations diagonalizing the correlated host with impurities, we discuss the ground state properties as a function of magnetic field H and the Kondo exchange coupling. While an isolated impurity of spin S>1/2 has a magnetic ground state of effective spin (S−12), a finite concentration introduces a heavy impurity band and yields a singlet ground state with antiferromagnetically correlated impurities. A field H first polarizes the narrow impurity band. The impurities have mixed valent properties induced in part by correlations in the host. Some aspects of the model are related to heavy fermion alloys.

Exact solution for a degenerate Anderson impurity in the limit embedded into a correlated host

We consider the one-dimensional t - J model, which consists of electrons with spin S on a lattice with nearest neighbor hopping t constrained by the excluded multiple occupancy of the lattice sites and spin-exchange J between neighboring sites. The model is integrable at the supersymmetric point, J = t. Without spoiling the integrability we introduce an Anderson-like impurity of spin S (degenerate Anderson model in the limit), which interacts with the correlated conduction states of the host. The lattice model is defined by the scattering matrices via the Quantum Inverse Scattering Method. We discuss the general form of the interaction Hamiltonian between the impurity and the itinerant electrons on the lattice and explicitly construct it in the continuum limit. The discrete Bethe ansatz equations diagonalizing the host with impurity are derived, and the thermodynamic Bethe ansatz equations are obtained using the string hypothesis for arbitrary band filling as a function of temperature and external magnetic field. The properties of the impurity depend on one coupling parameter related to the Kondo exchange coupling. The impurity can localize up to one itinerant electron and has in general mixed valent properties. Groundstate properties of the impurity, such as the energy, valence, magnetic susceptibility and the specific heat coefficient, are discussed. In the integer valent limit the model reduces to a Coqblin-Schrieffer impurity.

Integrable one-dimensional heavy fermion lattice model

We consider a one-dimensional lattice gas of interacting electrons embedding a finite concentration of magnetic impurities. The host electrons propagate with nearest neighbor hopping t, constrained by the excluded multiple occupancy of the lattice sites, and interact with electrons on neighboring sites via spin exchange J and a charge interaction ( t- J model). The host is integrable at the supersymmetric point, J = 2 t, where charges and spin form a graded FFB superalgebra. Without destroying the integrability we introduce a finite concentration of mixed-valent impurities with two magnetic configurations of spin S and (S + 1 2 ) , respectively, which hybridize with the conduction states of the host. We derive the Bethe ansatz equations diagonalizing the correlated host with impurities and discuss the ground-state properties as a function of magnetic field and the Kondo exchange coupling. While for an isolated impurity the ground state is magnetic of effective spin S, a finite concentration of impurities introduces an additional Dirac sea (the impurity band), which gives rise to a singlet ground state. The impurities are antiferromagnetically correlated and frustrated in zero-field. As a function of field first the narrow impurity band is spin polarized. The van Hove singularities of the spindashrapidity bands define critical fields at which the susceptibility diverges. The number of itinerant electrons and the concentration of impurities can be varied continuously, so that the Kondo problem, the supersymmetric t- J model and the antiferromagnetic Heisenberg chain are contained as special limits. The properties of the model are related to heavy fermion alloys and the Kondo lattice.

Coqblin-Schrieffer impurity in a one-dimensional correlated electron lattice

We consider electrons of effective spin S on a chain with nearest-neighbour hopping t constrained by the excluded multiple occupancy of the lattice sites, and spin exchange J between neighbouring sites. The spin space of the model is invariant, and at the supersymmetric point, J = t, the charge and the spin play identical roles forming an integrable SU(2S + 2) superalgebra. Without compromising the integrability we introduce a Coqblin-Schrieffer-like impurity of spin S, which interacts with the correlated conduction states of the host. The discrete Bethe ansatz equations diagonalizing the host with impurity are derived and the ground-state properties of the impurity are studied as a function of the Kondo exchange coupling. An undercompensated impurity of spin S embedded into a chain of interacting spin- electrons is also discussed.

Kondo impurity band in a one-dimensional correlated electron lattice

We consider a system consisting of a one-dimensional lattice gas of electrons with a finite concentration of magnetic impurities. The host electrons propagate with nearest-neighbor hopping $t$, constrained by the excluded multiple occupancy of the lattice sites, and interact with electrons on neighboring sites via spin exchange $J$ and a charge interaction. The host is integrable at the supersymmetric point $J=2t$, where charges and spin form a SU(3) BBB permutation algebra (this differs from the graded FFB superalgebra of the traditional supersymmetric $t\ensuremath{-}J$ model, where B and F stand for boronic and fermionic degree of freedom). Without destroying the integrability, we introduce a finite concentration of impurities of arbitrary spin $S$, which hybridize with the conduction states of the host. We derive the Bethe ansatz equations diagonalizing the correlated host with impurities and discuss the ground-state properties as a function of magnetic field and the Kondo exchange coupling. While an isolated impurity of spin $Sg1/2$ has a magnetic ground state of effective spin $S\ensuremath{-}\frac{1}{2}$, a finite concentration introduces an additional Dirac sea (the impurity band), which gives rise to a singlet ground state. The impurities are antiferromagnetically correlated and frustrated in zero field. As a function of the field, first the narrow impurity band is spin polarized. The Van Hove singularities of the spin-rapidity bands define critical fields at which the susceptibility diverges. The impurities have in general mixed valent properties induced in part by the correlations in the host. Some of the aspects of the model are related to heavy-fermion alloys. A distribution of Kondo temperatures may give rise to non-Fermi-liquid properties.