Kubo Linear Response Research Articles

Electron-phonon correlations inducing excitonic excitations in semimetal and semiconducting materials

The influence of phonons on the low-energy excitonic excitations at zero temperature in the extended Falicov-Kimball model has been investigated. In the framework of the unrestricted Hartree-Fock approximation, a set of self-consistent equations for the excitonic condensate order parameter and a lattice distortion is derived when both electron-phonon coupling and electron-hole Coulomb interaction are treated on an equal footing. The low-energy excitation properties of the excitonic condensate are addressed in signatures of the optical conductivity and the dynamical excitonic susceptibility function. The real part of the optical conductivity is evaluated by the Kubo linear response theory and the imaginary part of the dynamical excitonic susceptibility is found by adapting the random phase approximation. In the semimetal state, one always finds a sharp peak in the optical conductivity spectrum indicating the stability of the excitonic condensation in the BCS type if the correlation between electrons and phonons becomes significant. In contrast, the peak is smeared out on the semiconducting side indicating the stability of the BEC-type excitonic condensate. In this semiconducting side, the sharp peak signature appears and the system turns to the BCS-type excitonic condensation state by increasing the electron-phonon correlations. In either the semimetal or the semiconducting normal state, increasing the electron-phonon correlations always reinforces a low-energy sharp peak in the dynamical excitonic susceptibility spectrum, indicating the existence of the tightly bound excitonic excitations before the condensation state. Specifically, on the semiconducting side, the ``halo'' phase with the preformed excitons exiting outside of the BEC-excitonic condensation state has been specified. The halo phase becomes more recognizable by raising the electron-phonon correlations.

Magnetization Dynamics in FexCo1-x in Presence of Chemical Disorder

In this paper, we present a theoretical formulation of magnetization dynamics in disordered binary alloys, based on the Kubo linear response theory, interfaced with a seamless combination of three approaches: density functional-based tight-binding linear muffin-tin orbitals, generalized recursion and augmented space formalism. We applied this method to study the magnetization dynamics in chemically disordered FexCo1−x (x = 0.2, 0.5, 0.8) alloys. We found that the magnon energies decreased with an increase in Co concentration. Significant magnon softening was observed in Fe20Co80 at the Brillouin zone boundary. Magnon–electron scattering increased with increasing Co content, which in turn modified the hybridization between the Fe and Co atoms. This reduced the exchange energy between the atoms and softened down the magnon energy. The lowest magnon lifetime was found in Fe50Co50, where disorder was at a maximum. This clearly indicated that the damping of magnon energies in FexCo1−x was governed by hybridization between Fe and Co, whereas the magnon lifetime was controlled by disorder configuration. Our atomistic spin dynamics simulations show reasonable agreement with our theoretical approach in magnon dispersion for different alloy compositions.

Excitonic properties in a double-layer graphene

This paper investigates theoretically the excitonic condensation state at zero temperature in a double-layer graphene structure. In the framework of the unrestricted Hartree–Fock approximation, the electron-hole system in the structure described in the two-band electronic model is analyzed and one finds a set of self-consistent equations determining the excitonic order parameter. The optical properties of the excitonic condensation state then are examined in the Kubo linear optical response theory. Our results indicate that in the case of sufficiently large Coulomb interaction, the BEC excitonic condensation state might occur at low electronic excitation density. By turning the external electric field, the superfluid state stabilizes in the BCS-type excitonic condensate. The optical conductivity spectrum also provides us more insight into the excitonic condensation states.

Far-field radiation of biased monolayer PbBiI

We use the Kubo formula and linear response theory to study the radiation of energy and angular momentum from biased monolayer PbBiI. Our results show that a critical bias field Vbc≃0.326 eV emerges for the radiations. For bias fields below Vbc, there is a negligible shift of the threshold frequency at which the spin-dependent energy radiation is maximum, while above which, it significantly moves to the higher frequencies. At all the threshold frequencies, the sign of possible angular momentum radiations is switched. Further, we find that contrary to Planck's law which is valid for the unbiased energy radiation (proportionality to T4), the biased one is proportional to T5. Also, it is possible to confirm the behaviors of the system below and above Vbc via the temperature-dependent angular momentum radiation. This work illuminates/engineers the application of angular momentum degree of freedom in processing data.

Thermal fluctuations in a realistic ionic-crystal model

We investigate the thermal fluctuations of the ionic motions in a Born model of ionic crystals, namely, a model in which the electrons are eliminated, being replaced by suitable effective potentials among the ions. The model is studied in its classical version, computing the Newtonian trajectories of the ions. The general motivation is that, although being an essential ingredient within Green–Kubo linear response theory, thermal fluctuations apparently were not studied systematically by molecular dynamics methods, as was done instead for the approach to equilibrium in the Fermi–Pasta–Ulam problem. The time evolution of the fluctuations is studied in terms of the time-changes of the mode-energies of the system. The stages of the “regression” of the fluctuations are described, from a first stage of strong time-correlations up to a final decorrelation, and a comparison with the process of approach to equilibrium is performed. Finally, the dependence on specific energy is investigated.

Open Quantum System Response from the Hierarchy of Pure States.

The spectral properties of a quantum system are essential when probing theoretical predictions against experimental data. For an open quantum system strongly interacting with its environment, spectral features are challenging to calculate. Here we demonstrate that the stochastic Hierarchy of Pure States (HOPS) approach is well suited to calculate the response of an open quantum system to a, possibly strong, coherent probe driving. For weak driving, where Kubo's linear response theory is applicable, it turns out that the HOPS method is highly efficient since fluctuations inherent to the stochastic dynamics cancel for the response function and, thus, allow us to obtain the susceptibility easily. Our results are in agreement with experimental data for a strongly damped spin system showing that the transition from oscillatory to overdamped motion is also reflected by the transmission spectrum. As a further application we demonstrate that the susceptibility, quantifying the amplitude of the response, as a function of temperature exhibits a maximum which is the hallmark of stochastic resonance. Beyond the linear regime, the exact open system dynamics shows the asymptotic Floquet state. We use the topic of probe driving and response to present the HOPS approach in a novel and self-contained way. This includes the importance sampling scheme which yields the nonlinear HOPS as well as the stochastic treatment of a thermal initial environmental state within the zero temperature formalism. Special attention is given to the exponential representation of the algebraic Ohmic bath correlation function and the truncation condition for the hierarchy.

Chemical sensing with graphene: A quantum field theory perspective

We studied theoretically the effect of a low concentration of adsorbed polar molecules on the optical conductivity of graphene, within the Kubo linear response approximation. Our analysis is based on a continuum model approximation that includes up to next to nearest neighbors in the pristine graphene effective Hamiltonian, thus extending the field-theoretical analysis developed in Refs.[1,2]. Our results show that the conductivity can be expressed in terms of renormalized quasiparticle parameters $\tilde{v}_F$, $\tilde{M}$ and $\tilde{\mu}$ that include the effect of the molecular surface concentration $n_{dip}$ and dipolar moment $\boldsymbol{\mathcal{P}}$, thus providing an analytical model for a graphene-based chemical sensor.

Origins of anisotropic transport in the electrically switchable antiferromagnet Fe1/3NbS2

Recent experiments on the antiferromagnetic intercalated transition metal dichalcogenide $\mathrm{Fe_{1/3}NbS_2}$ have demonstrated reversible resistivity switching by application of orthogonal current pulses below its magnetic ordering temperature, making $\mathrm{Fe_{1/3}NbS_2}$ promising for spintronics applications. Here, we perform density functional theory calculations with Hubbard U corrections of the magnetic order, electronic structure, and transport properties of crystalline $\mathrm{Fe_{1/3}NbS_2}$, clarifying the origin of the different resistance states. The two experimentally proposed antiferromagnetic ground states, corresponding to in-plane stripe and zigzag ordering, are computed to be nearly degenerate. In-plane cross sections of the calculated Fermi surfaces are anisotropic for both magnetic orderings, with the degree of anisotropy sensitive to the Hubbard U value. The in-plane resistance, computed within the Kubo linear response formalism using a constant relaxation time approximation, is also anisotropic, supporting a hypothesis that the current-induced resistance changes are due to a repopulating of AFM domains. Our calculations indicate that the transport anisotropy of $\mathrm{Fe_{1/3}NbS_2}$ in the zigzag phase is reduced relative to stripe, consistent with the relative magnitudes of resistivity changes in experiment. Finally, our calculations reveal the likely directionality of the current-domain response, specifically, which domains are energetically stabilized for a given current direction.

A comparative study on the influence of intra- and inter-valley scattering to Hall conductivity of Dirac semimetals

Within the theoretical framework of Kubo linear response theory as well as the self-consistent Born approximation(SCBA), we perform a comparative study about the Hall conductivities of a Dirac semimetal(DSM) subject to, respectively, the intra- and inter-valley scattering. By calculating the Hall conductivity as a function of Fermi level, we find that when the Fermi level lies in the region around the Dirac node, the Hall conductivities between the two cases, i.e. subject to intra- and inter-valley scattering respectively, do not show appreciable difference, regardless of the scattering strength. As the Fermi level goes away from the Dirac node, the Hall conductivity in the case of inter-valley scattering is smaller than that in the case of intra-valley scattering if the scattering has relatively weak or moderate strength. In contrast, as scattering strength increases further, the Hall conductivity spectra between the two cases look almost identical no matter whether the Fermi level is close to or far away from the Dirac node. The underlying physics of these numerical results is that the Fermi surface has nontrivial contribution to the Hall conductivity in the case of weak or moderate scattering. However, in the case of strong scattering, it becomes much smaller than the contribution arising from all the occupied states. In addition, our study indicates that the cone tilt brings about the effective enhancement of Hall conductivity of DSMs.

Dynamical polarizability of graphene with spatial dispersion

We perform a detailed analysis of electronic polarizability of graphene with different theoretical approaches. From Kubo's linear response formalism, we give a general expression of frequency and wave-vector dependent polarizability within the random phase approximation. Four theoretical approaches have been applied to the single-layer graphene and their differences are on the band-overlap of wavefunctions. By comparing with the \textit{ab initio} calculation, we discuss the validity of methods used in literature. Our results show that the tight-binding method is as good as the time-demanding \textit{ab initio} approach in calculating the polarizability of graphene. Moreover, due to the special Dirac-cone band structure of graphene, the Dirac model reproduces results of the tight-binding method for energy smaller than \SI{3}{\electronvolt}. For doped graphene, the intra-band transitions dominate at low energies and can be described by the Lindhard formula for two-dimensional electron gases. At zero temperature and long-wavelength limit, with the relaxation time approximation, all theoretical methods reduce to a long-wave analytical formula and the intra-band contributions agree to the Drude polarizability of graphene. Effects of electrical doping and temperature are also discussed. This work may provide a solid reference for researches and applications of the screening effect of graphene.

Electronic Structure and Conductivity of a Disordered A1 – xBx Binary Alloy in the Cluster Approach for the Hubbard Model

We propose a method for calculating the electronic band structure of disordered systems with strong electron correlations. Various approaches to the description of electrical conductivity of disordered systems are considered. Calculations are based on determining the one-particle Green function of the system, which is averaged over different configurations of a cluster, on the Boltzmann formalism, and the Kubo linear response theory. As the basic model, we use the Hubbard model for an A1 – xBx binary alloy.

Symmetrized decomposition of the Kubo-Bastin formula

The Smrcka-Streda version of Kubo's linear response formula is widely used in the literature to compute non-equilibrium transport properties of heterostructures. It is particularly useful for the evaluation of intrinsic transport properties associated with the Berry curvature of the Bloch states, such as anomalous and spin Hall currents as well as the damping-like component of the spin-orbit torque. Here, we demonstrate in a very general way that the widely used decomposition of the Kubo-Bastin formula introduced by Smrcka and Streda contains an overlap, which has lead to widespread confusion in the literature regarding the Fermi surface and Fermi sea contributions. To remedy this pathology, we propose a new decomposition of the Kubo-Bastin formula based on the permutation properties of the correlation function and derive a new set of formulas, without an overlap, that provides direct access to the transport effects of interest. We apply these new formulas to selected cases and demonstrate that the Fermi sea and Fermi surface contributions can be uniquely addressed with our symmetrized approach.

Semirealistic tight-binding model for spin-orbit torques

We compute the spin-orbit torque in a transition metal heterostructure using Slater-Koster parameterization in the two-center tight-binding approximation and accounting for d-orbitals only. In this method, the spin-orbit coupling is modeled within Russel-Saunders scheme, which enables us to treat interfacial and bulk spin-orbit transport on equal footing. The two components of the spin-orbit torque, dissipative (damping-like) and reactive (field-like), are computed within Kubo linear response theory. By systematically studying their thickness and angular dependence, we were able to accurately characterize these components beyond the traditional "inverse spin galvanic" and "spin Hall" effects. Whereas the conventional field-like torque is purely interfacial, we unambiguously demonstrate that the conventional the damping-like torque possesses both an interfacial and a bulk contribution. In addition, both field-like and damping-like torques display substantial angular dependence with strikingly different thickness behavior. While the planar contribution of the field-like torque decreases smoothly with the nonmagnetic metal thickness, the planar contribution of the damping-like torque increases dramatically with the nonmagnetic metal thickness. Finally, we investigate the self-torque exerted on the ferromagnet when the spin-orbit coupling of the nonmagnetic metal is turned off. Our results suggest that the spin accumulation that builds up inside the ferromagnet can be large enough to induce magnetic excitations.

Drude conductivity of a granular metal

We present a complete derivation of the analog to Drude conductivity in granular metals using diagrammatic methods. The derivation of Drude conductivity in a homogeneous metal using Kubo linear response theory is somewhat involved, with the evaluation of a singular double integral being required to cancel the diamagnetic response. We first provide a history and careful derivation of this standard result, before analyzing the equivalent situation in the model of a granular metal as a lattice of metallic grains connected by tunneling. A singular triple integral occurs here, and its evaluation gives the result expected from the Einstein relation and Fermi’s golden rule.

Anisotropy of resistivity in hexagonal late transition metals and their alloys

Transport properties of hexagonal transition metals Co, Ru, and Os at finite temperatures are studied by means of ab initio electronic structure techniques and the Kubo linear response theory. An alloy analogy model for a quantitative treatment of the electrical conductivities due to temperature-induced lattice vibrations (phonons) and spin fluctuations is applied with focus on anisotropy induced by the hexagonal structure. The resistivity anisotropy in Co is found opposite to that in Ru and Os, in agreement with existing experimental data. This result is ascribed to the strong itinerant ferromagnetism of Co which leads to profound differences in the electronic structure and conductivities in the majority and minority spin channels. A similar sensitivity to spin polarization is predicted for the anisotropy of residual resistivity in random hexagonal Co-rich Co–Ni and Co–Ni–Fe alloys.

Fermi Arcs and DC Transport in Nanowires of Dirac and Weyl Semimetals

AbstractThe transport properties and electron states in cylinder nanowires of Dirac and Weyl semimetals are studied paying special attention to the structure and properties of the surface Fermi arcs. The latter make the electric charge and current density distributions in nanowires strongly nonuniform as the majority of the charge density is accumulated at the surface. It is found that a Weyl semimetal wire also supports a magnetization current localized mainly at the surface because of the Fermi arcs contribution. By using the Kubo linear response approach, the direct current (DC) conductivity is calculated and it is found that its spatial profile is nontrivial. By explicitly separating the contributions of the surface and bulk states, it is shown that when the electric chemical potential and/or the radius of the wire is small, the electron transport is determined primarily by the Fermi arcs and the electrical conductivity is much higher at the surface than in the bulk. Due to the rise of the surface‐bulk transition rate, the relative contribution of the surface states to the total conductivity gradually diminishes as the chemical potential increases. In addition, the DC conductivity at the surface demonstrates noticeable peaks when the Fermi level crosses energies of the surface states.

Ab initio theory of the spin-dependent conductivity tensor and the spin Hall effect in random alloys

We present an extension of the relativistic electron transport theory for the standard (charge) conductivity tensor of random alloys within the tight-binding linear muffin-tin orbital method to the so-called spin-dependent conductivity tensor, which describes the Kubo linear response of spin currents to external electric fields. The approach is based on effective charge- and spin-current operators, that correspond to intersite electron transport and that are nonrandom, which simplifies the configuration averaging by means of the coherent potential approximation. Special attention is paid to the Fermi sea term of the spin-dependent conductivity tensor, which contains a nonzero incoherent part, in contrast to the standard conductivity tensor. The developed formalism is applied to the spin Hall effect in binary random nonmagnetic alloys, both on a model level and for Pt-based alloys with an fcc structure. We show that the spin Hall conductivity consists of three contributions (one intrinsic and two extrinsic) which exhibit different concentration dependences in the dilute limit of an alloy. Results for selected Pt alloys (Pt-Re, Pt-Ta) lead to the spin Hall angles around 0.2; these sizable values are obtained for compositions that belong to thermodynamically equilibrium phases. These alloys can thus be considered as an alternative to other systems for efficient charge to spin conversion, which are often metastable crystalline or amorphous alloys.

Quantization of the Hall conductivity in the Harper-Hofstadter model

We study the robustness of the quantization of the Hall conductivity in the Harper-Hofstadter model towards the details of the protocol with which a longitudinal uniform driving force $F_x(t)$ is turned on. In the vector potential gauge, through Peierls substitution, this involves the switching-on of complex time-dependent hopping amplitudes $\mathrm{e}^{-\frac{i}{\hbar}\mathcal{A}_x(t)}$ in the $\hat{\mathbf{x}}$-direction such that $\partial_t \mathcal{A}_x(t)=F_x(t)$. The switching-on can be sudden, $F_x(t)=\theta(t) F$, where $F$ is the steady driving force, or more generally smooth $F_x(t)=f(t/t_{0}) F$, where $f(t/t_{0})$ is such that $f(0)=0$ and $f(1)=1$. We investigate how the time-averaged (steady-state) particle current density $j_y$ in the $\hat{\mathbf{y}}$-direction deviates from the quantized value $j_y \, h/F = n$ due to the finite value of $F$ and the details of the switching-on protocol. Exploiting the time-periodicity of the Hamiltonian $\hat{H}(t)$, we use Floquet techniques to study this problem. In this picture the (Kubo) linear response $F\to 0$ regime corresponds to the adiabatic limit for $\hat{H}(t)$. In the case of a sudden quench $j_y \, h/F$ shows $F^2$ corrections to the perfectly quantized limit. When the switching-on is smooth, the result depends on the switch-on time $t_{0}$: for a fixed $t_{0}$ we observe a crossover force $F^*$ between a quadratic regime for $F<F^*$ and a {\em non-analytic} exponential $\mathrm{e}^{-\gamma/|F|}$ for $F>F^*$. The crossover $F^*$ decreases as $t_{0}$ increases, eventually recovering the topological robustness. These effects are in principle amenable to experimental tests in optical lattice cold atomic systems with synthetic gauge fields.

Anomalous Hall effect in dense QCD matter

In this letter, we investigate the anomalous Hall effect in dense QCD matter. We use here the Nambu–Jona Lasinio model as an effective model of QCD. When the dual chiral density wave which is the spatially modulated chiral condensate appears in the medium, it gives rise to two Weyl points to the single-particle energy-spectrum and then the anomalous Hall conductivity becomes nonzero. Then, dense QCD matter is analogous to the Weyl semimetal. The direct calculation of the Hall conductivity by way of Kubo's linear response theory gives the term proportional to the distance between the Weyl points. Unlike the Weyl semimetal, there appears the additional contribution induced by axial anomaly.

Coherence-enhanced phase-dependent dissipation in long SNS Josephson junctions: Revealing Andreev bound state dynamics

One of the best known causes of dissipation in ac driven quantum systems stems from photon absorption. Dissipation can also be caused by the retarded response to the time-dependent excitation, and in general gives insight into the system's relaxation times and mechanisms. We address the dissipation in a mesoscopic normal wire with superconducting contacts, that sustains a supercurrent at zero frequency and that may be expected to remain dissipationless at frequency lower than the superconducting gap. We probe the high frequency linear response of a Normal/Superconductor ring to a time-dependent flux by coupling it to a highly sensitive multimode microwave resonator. Far from being the simple derivative of the current-phase relation, the ring's ac susceptibility also displays a dissipative component whose phase dependence is a signature of the dynamical processes occurring within the Andreev spectrum. We show how dissipation is driven by the competition between the two aforementioned mechanisms. Depending on the relative strength of those contributions, dissipation can be maximal at $\pi$, when the minigap closes, or can be maximal near $ \pi/2$, when the dc supercurrent is maximal. We also find that the dissipative response increases at low temperature and can even exceed the normal state conductance. The results are confronted with predictions of the Kubo linear response and time-dependent Usadel equations. This experiment shows the power of the ac susceptibility measurement of individual hybrid mesoscopic systems in probing in a controlled way the quantum dynamics of ABS. By spanning different physical regimes, our experiments provide a unique access to inelastic scattering and spectroscopy of an isolated quantum coherent system. This technique should be a tool of choice to investigate topological superconductivity and detect the topological protection of edge states.