We investigate the intrinsic spin Hall effect in a quantum well semiconductor doped with magnetic impurities, as a means to manipulate the carriers' spin. Using a simple Hamiltonian with Rashba spin-orbit coupling and exchange interactions, we analytically compute the spin Hall conductivity. It is demonstrated that using the appropriate order of limits, one recovers the intrinsic universal value. Numerical computations on a tight-binding model, in the weak disorder regime, confirm that the spin Hall effect is preserved in the presence of magnetic impurities. The optical spin conductivity shows large sample to sample fluctuations in the low frequency region. As a consequence, for weak disorder, the static spin conductivity is found to follow a wide Gaussian distribution with its mean value near the intrinsic clean value.
We present a semi-automated computer-assisted method to generate and calculate diagrams in the disorder averaging approach to disordered 2D conductors with intrinsic spin-orbit interaction (SOI). As an application, we calculate the effect of the SOI on the charge conductivity for disordered 2D systems and rings in the presence of Rashba and Dresselhaus SOI. In an infinite-size 2D system, anisotropic corrections to the conductivity tensor arise due to phase-coherence and the interplay of Rashba and Dresselhaus SOI. The effect is more pronounced in the quasi-onedimensional case, where the conduc…
We study the quantum corrections to the conductivity of the two-dimensional disordered interacting electron system in the diffusive regime due to inelastic scattering off rare magnetic impurities. We focus on the case of very different g-factors for electrons and magnetic impurities. Within the Born approximation for the inelastic scattering off magnetic impurities we find additional temperature-dependent corrections to the conductivity of the Altshuler-Aronov type. Our results demonstrate that the low temperature transport in interacting disordered electron systems with rare magnetic impuriti…
We describe an efficient numerical approach to calculate the longitudinal and transverse Kubo conductivities of large systems using Bastin's formulation. We expand the Green's functions in terms of Chebyshev polynomials and compute the conductivity tensor for any temperature and chemical potential in a single step. To illustrate the power and generality of the approach, we calculate the conductivity tensor for the quantum Hall effect in disordered graphene and analyze the effect of the disorder in a Chern insulator in Haldane's model on a honeycomb lattice.
We calculate the thermal conductivity of interacting electrons in disordered metals. In our analysis we point out that the interaction affects thermal transport through two distinct mechanims, associated with quantum interference corrections and energy exchange of the quasi particles with the electromagnetic environment, respectively. The latter is seen to lead to a violation of the Wiedemann-Franz law. Our theory predicts a strong enhancement of the Lorenz ratio $κ/σT$ over the value which is predicted by the Wiedemann-Franz law, when the electrons encounter a large environmental impedance.
We show that the conductivity tensor of a disordered two-dimensional electron gas becomes anisotropic in the presence of both Rashba and Dresselhaus spin-orbit interactions (SOI). This anisotropy is a mesoscopic effect and vanishes with vanishing charge dephasing time. Using a diagrammatic approach including zero, one, and two-loop diagrams, we show that a consistent calculation needs to go beyond a Boltzmann equation approach. In the absence of charge dephasing and for zero frequency, a finite anisotropy σ_{xy} e^2/lhpf arises even for infinitesimal SOI.
The spin-Hall conductivity of a disordered 2D electron gas has been calculated for an arbitrary spin-orbit interaction. We have found that in the diffusive regime of electron transport, in accordance with previous calculations, the dc spin-Hall conductivity of a homogeneous system turns to zero due to impurity scattering when the spin-orbit coupling is represented only by the Rashba interaction. However, when the Dresselhaus interaction is taken into account, the spin-Hall current is not zero. We also considered the spin-Hall currents induced by an inhomogeneous electric field. It is shown tha…
For the two-dimensional ideal electron gas with the Rashba spin-orbit interaction and in the presence of non-magnetic short-ranged potential impurities the spin-Hall conductivity $σ_{sH}$ is found by direct microscopic calculation. Within the semiclassical approximation $\hbar/τ\ll ε_F$ and $Δ\ll ε_F$ the value of $σ_{sH}$ is zero for arbitrary ratio of the spin-orbit splitting $Δ$ and the inverse elastic scattering time $1/τ$.
We study conductivity of strongly disordered amorphous antimony films under high bias voltages. We observe non-linear current-voltage characteristic, where the conductivity value at zero bias is one of two distinct values, being determined by the sign of previously applied voltage. Relaxation curves demonstrate high stability of these conductivity values on a large timescale. Investigations of the antimony film structure allows to determine the percolation character of electron transport in strongly disordered films. We connect the memory effect in conductivity with modification of the percola…
We study the effect of disordered ripples on the conductivity of monolayer graphene flakes. We calculate the relaxation times and the Boltzmann conductivities associated with two mechanisms. First, we study the conductivity correction due to an external in-plane magnetic field $B_\parallel$. Due to the irregular local curvature found at graphene sheets deposited over a substrate, $B_\parallel$ can be mapped into an effective random magnetic field perpendicular to the graphene surface. Second, we study the electron momentum relaxation due to intrinsic pseudo magnetic fields originated from defo…
We consider the spin-Hall current in a disordered two-dimensional electron gas in the presence of Rashba spin-orbit interaction. We derive a generalized Kubo-Greenwood formula for the spin-Hall conductivity $σ$ and evaluate it in an systematic way using standard diagrammatic techniques for disordered systems. We find that in the diffusive regime both Boltzmann and the weak localization contributions to $σ$ are of the same order and vanish in the zero frequency limit. We show that the uniform spin current is given by the total time derivative of the magnetization from which we can conclude that…
Parity anomaly is a quantum mechanical effect that the parity symmetry in a two-dimensional classical action is failed to be restored in any regularization of the full quantum theory and is characterized by a half-quantized Hall conductivity. Here we propose a scheme to explore the experimental signature from parity anomaly in the measurement of optical Hall conductivity, in which the optical Hall conductivity is nearly half quantized for a proper range of frequency. The behaviors of optical Hall conductivity are studied for several models, which reveal the appearance of half-quantized Hall co…
The spin Hall conductivity is usually calculated for periodic solids, while the knowledge of local contributions from inhomogeneous regions like interfaces or surfaces would be desirable. In this work we derive a local expression for the spin Hall conductivity of two-dimensional systems in analogy to the local anomalous Hall conductivity discussed recently. Our approach is applicable to heterogeneous fully bounded systems and nanoribbons with both open and periodic boundary conditions.
We calculate fluctuation corrections to the longitudinal conductivity of disordered superconductors subject to an external magnetic field. We derive analytic expressions that are valid in the entire metallic part of the temperature-magnetic field phase diagram as long as the effect of the magnetic field on the spin degrees of freedom of the electrons may be neglected. Our calculations are based on a kinetic equation approach. For the special case of superconducting films and wires in parallel magnetic fields we perform a detailed comparison with results that were previously obtained with diagr…
Using the recursive Green's function method, we study the problem of electron transport in a disordered single-layer graphene sheet. The conductivity is of order $e^2/h$ and its dependence on the carrier density has a scaling form that is controlled solely by the disorder strength and the ratio between the sample size and the correlation length of the disorder potential. The shot noise Fano factor is shown to have a narrow dip near the neutrality point for weak disorder and to develop a nearly doping independent behavior at strong disorder. Our results are in good qualitative and quantitative …
We consider a disordered topological insulator thin film placed on the top of a ferromagnetic insulator with a perpendicular exchange field $M$ and subjected to a perpendicular electric field. The presence of ferromagnetic insulator causes that bottom surface states of the topological insulator thin film become spin polarized and the electric field provides a potential difference $V$ between the two surface states, resulting in breaking of time-reversal and inversion symmetry in the system. Using Kubo formalism and employing the first Born approximation as well as the self-consistent Born appr…
We carry out a throughout study of the topological Hall and topological spin Hall effects in disordered skyrmionic systems: the dimensionless (spin) Hall angles are evaluated across the energy band structure in the multiprobe Landauer-Büttiker formalism and their link to the effective magnetic field emerging from the real space topology of the spin texture is highlighted. We discuss these results for an optimal skyrmion size and for various sizes of the sample and found that the adiabatic approximation still holds for large skyrmions as well as for few atomic size-nanoskyrmions. Finally, we te…
In this paper, we revisit some quantum mechanical aspects related to the Quantum Hall Effect. We consider a Landau type model, paying a special attention to the experimental and geometrical features of Quantum Hall experiments. The resulting formalism is then used to compute explicitely the Hall conductivity from a Kubo formula.
We study the conductivity of disordered zigzag graphene nanoribbons in the incoherent regime by using the Boltzmann equation approach. The band structure of zigzag nanoribbons contains two energy valleys, and each valley has an excess one-way channel. The crucial point is that the numbers of conducting channels for two propagating directions are imbalanced in each valley due to the presence of an excess one-way channel. It was pointed out that as a consequence of this imbalance, a perfectly conducting channel is stabilized in the coherent regime if intervalley scattering is absent. We show tha…
The semi-classical study of the integer Quantum Hall conductivity is investigated for electrons in a bi-periodic potential $V(x,y)$. The Hall conductivity is due to the tunnelling effect and we concentrate our study to potentials having three wells in a periodic cell. A non-zero topological conductivity requires special conditions for the positions, and shapes of the wells. The results are derived analytically and well confirmed by numerical calculations.
A two-layer system coupled via tunneling and with different carrier masses in each layer is investigated in the integer quantum Hall regime. Striking deviations of the one-layer Hall conductivity from the usual quantization are found, if resonance between Landau levels of different layers is achieved. The appearance of negative jumps in the Hall conductivity is also predicted under suitable conditions. The results obtained are robust against disorder in the system.
We report on the formation of critical states in disordered graphene, at the origin of variable and unconventional transport properties in the quantum Hall regime, such as a zero-energy Hall conductance plateau in the absence of an energy bandgap and Landau level degeneracy breaking. By using efficient real-space transport methodologies, we compute both the dissipative and Hall conductivities of large size graphene sheets with random distribution of model single and double vacancies. By analyzing the scaling of transport coefficients with defect density, system size and magnetic length, we elu…
A static electric field generates circulating currents at the surfaces of a magnetoelectric insulator. The anomalous Hall part of the surface conductivity tensor describing such bound currents can change by multiples of $e^2/h$ depending on the insulating surface preparation, and a bulk calculation does not fix its quantized part. To resolve this ambiguity, we develop a formalism for calculating the full surface anomalous Hall conductivity in a slab geometry. We identify a Berry-curvature term, closely related to the expression for the bulk anomalous Hall conductivity, whose value can change b…
The object of the present work is to study the quantum Hall effect through its symmetries and topological aspects. We consider the model of an electron moving in a two-dimensional lattice in the presence of applied in-plain electric field and perpendicular magnetic field. We refer to this as the two dimensional electric-magnetic Bloch problem (EMB). The Hall conductivity quatizations beyond the linear response approximation is analyzed.
The edge Hall conductivity is shown to be an integer multiple of $e^2/h$ which is almost surely independent of the choice of the disordered configuration. Its equality to the bulk Hall conductivity given by the Kubo-Chern formula follows from K-theoretic arguments. This leads to quantization of the Hall conductance for any redistribution of the current in the sample. It is argued that in experiments at most a few percent of the total current can be carried by edge states.
The temperature ($T$) and frequency ($ω$) dependent conductivity of weakly disordered Luttinger liquids is calculated in a systematic way both by perturbation theory and from a finite temperature renormalization group (RG) treatment to leading order in the disorder strength. Whereas perturbation theory results in $ω/T$ scaling of the conductivity such scaling is violated in the RG traetment. We also determine the non-linear field dependence of the conductivity, whose power law scaling is different from that of temperature and frequency dependence.
In most cases, to observe quantized Hall plateaux, an external magnetic field is applied in intrinsic magnetic topological insulators $\mathrm{MnBi_2Te_4}$. Nevertheless, whether the nonzero Chern number ($C\neq 0$) phase is a quantum anomalous Hall (QAH) state, or a quantum Hall (QH) state, or a mixing state of both is still a puzzle, especially for the recently observed $C=2$ phase [Deng \textit{et al}., Science \textbf{367}, 895 (2020)]. In this Letter, we propose a physical picture based on the Anderson localization to understand the observed Hall plateaux in disordered $\mathrm{MnBi_2Te_4…
We derive a formula for the thermal conductivity tensor of a ballistic phonon Hall model. It is found that, although the diagonal elements of the conductivity tensor diverge to infinite, the off-diagonal elements are finite,antisymmetric, and odd in magnetic field. The off-diagonal elements are non-zero only if the dynamic matrix of the phonon system breaks mirror reflection symmetry. The results are obtained without perturbative assumptions about the spin-phonon interactions.
Graphene has an unusual low-energy band structure with four chiral bands and half-quantized and quantized Hall effects that have recently attracted theoretical and experimental attention. We study the Fermi energy and disorder dependence of its spin Hall conductivity. In the metallic regime we find that vertex corrections enhance the intrinsic spin Hall conductivity and that skew scattering can lead to its values that exceed the quantized ones expected when the chemical potential is inside the spin-orbit induced energy gap. We predict that large spin Hall conductivities will be observable in g…
Inertial effects play an important role in classical mechanics but have been largely overlooked in quantum mechanics. Nevertheless, the analogy between inertial forces on mass particles and electromagnetic forces on charged particles is not new. In this paper, we consider a rotating non-interacting planar two-dimensional electron gas with a perpendicular uniform magnetic field and investigate the effects of the rotation in the Hall conductivi
Dynamical scaling analysis is theoretically performed for the ac (optical) Hall conductivity $σ_{xy}(\varepsilon_F,ω)$ as a function of Fermi energy $\varepsilon_F$ and frequency $ω$ for the two-dimensional electron gas and for graphene. In both systems, results based on exact diagonalization show that $σ_{xy}(\varepsilon_F,ω)$ displays a well-defined dynamical scaling, for which the dynamical critical exponent as well as the localization exponent are fitted and plugged in. A crossover from the dc-like bahavior to the ac regime is identified. The dynamical scaling analysis has enabled us to qu…
We study intrinsic DC spin and valley Hall conductivity in doped ferromagnetic silicene in the presence of an electric filed applied perpendicular to silicene sheet. By calculating its energy spectrum and wavefunction and by making use of Kubo formalism, we obtain a general relation for the transverse Hall conductivity which can be used to obtain spin- and valley-Hall conductivity. Our results, in the zero limit of the exchange field, reduces to the previous results. Furthermore we discuss electrically tunable spin and valley polarized transport in ferromagnetic silicene and obtain the necessa…
We report on robust features of the longitudinal conductivity ($σ_{xx}$) of the graphene zero-energy Landau level in presence of disorder and varying magnetic fields. By mixing an Anderson disorder potential with a low density of sublattice impurities, the transition from metallic to insulating states is theoretically explored as a function of Landau-level splitting, using highly efficient real-space methods to compute the Kubo conductivities (both $σ_{xx}$ and Hall $σ_{xy}$). As long as valley-degeneracy is maintained, the obtained critical conductivity $σ_{xx}\simeq 1.4 e^{2}/h$ is robust up…
We investigate the electric and thermal transport properties in a disordered Weyl ferromagnet on an equal footing by using the Keldysh formalism in curved spacetime. In particular, we calculate the anomalous thermal Hall conductivity, which consists of the Kubo formula and the heat magnetization, without relying on the Wiedemann-Franz law. We take nonmagnetic impurities into account within the self-consistent $T$-matrix approximation and reproduce the Wiedemann-Franz law for the extrinsic Fermi-surface and intrinsic Fermi-sea terms, respectively. This is the first step towards a unified theory…
We calculate the spin-Hall conductivity for a two-dimensional electron gas within the self-consistent Born approximation, varying the strength and type of disorder. In the weak disorder limit we find both analytically and numerically a vanishing spin-Hall conductivity even when we allow a momentum dependent scattering. Separating the reactive from the disspative current response, we find the universal value $σ^R_{sH} = e/8 π$ for the reactive response, which cancels however with the dissipative part $σ^D_{sH} = -e/8 π$.
The electron--electron scattering does not affect the electrical current in Galilean--invariant systems. We show that nevertheless electron--electron collisions may contribute to the electric resistivity of systems with parabolic spectrum provided that they have multiply connected Fermi surface and there is an additional mechanism of scattering. To this end, we calculate the resistivity of a two-dimensional electron gas with two filled transverse subbands in a presence of electron--electron and impurity scattering. Though the collisions between the electrons do not directly affect the current …
The dynamical transport properties near the integer quantum Hall transition are investigated at zero temperature by means of the Dirac fermion approach. These properties have been studied experimentally at low frequency omega and low temperature near the nu=1 filling factor Hall transition, with the observation of an anusual broadening and an overall increase of the longitudinal conductivity Re sigma_{xx} as a function of omega. We find in our approach that, unlike for normal metals, the longitudinal conductivity increases as the frequency increases, whilst the width Delta B (or Delta nu) of t…
High frequency (hf) conductivity in the form $σ^{hf} = σ_1^{hf} - iσ_2^{hf}$ was obtained from the measurement of Surface Acoustic Waves (SAW) attenuation and velocity (f=30 MHz) in GaAs/AlGaAs heterostructures ($n=1.3-7\cdot 10^{11}cm^{-2}$). It has been shown that in the Integer Quantum Hall Effect (IQHE) regime for all the samples at magnetic fields corresponding to the middle of the Hall plateaus and T=1.5 K, $σ_1 / σ_2 =0.14 \pm 0.03$. The ratio $σ_1 / σ_2=0.15$ points the case when the high-frequency hopping conductivity mechanism (electronic transition between the localized states forme…
We establish topological nature of Hall conductivity of graphene and other $Z_2$ crystals in 2D and 3D in the presence of inhomogeneous perturbations. To this end the lattice Weyl-Wigner formalism is employed. The non-uniform mechanical stress is considered, along with spatially varying magnetic field. The relation of the obtained topological invariant to level counting is clarified.
It has been shown recently that the universal dc spin conductivity of two-dimensional electrons with a Rashba spin-orbit interaction is canceled by vertex corrections in a weak scattering regime. We prove that the zero bulk spin conductivity is an intrinsic property of the free-electron Hamiltonian and scattering is merely a tool to reveal this property in terms of the diagrammatic technique. When Zeeman energy is neglected, the zero dc conductivity persists in a magnetic field. Spin conductivity increases resonantly at the cyclotron frequency and then decays towards the universal value.
We study theoretically the anomalous Hall effect due to the vector spin chirality carried by the local spins in the $s$-$d$ model. We will show that the vector spin chirality indeed induces local Hall effect in the presence of the electron spin polarization, while the global Hall effect vanishes if electron transport is homogeneous. This anomalous Hall effect can be interpreted in terms of the rotational component of the spin current associated with the vector chirality.
The gapless energy band spectra make the structures based on graphene and graphene bilayers with the population inversion created by optical or injection pumping to be promising media for the interband terahertz (THz) lasing. However, a strong intraband absorption at THz frequencies still poses a challenge for efficient THz lasing. In this paper, we show that in the pumped graphene bilayer structures, the indirect interband radiative transitions accompanied by scattering of carriers caused by disorder can provide a substantial negative contribution to the THz conductivity (together with the di…
We study the linear Hall response of 2D ballistic system on inhomogeneous magnetic field. We establish that in classical limit the Hall conductivity response on local magnetic field is quantized in units of $α_H \equiv \frac{e^3}{2 π^2 \hbar c}$. The quantized value depends on the point where the field is applied, this dependence being irregular in chaotic billiards. The phenomenon allows for direct tracing of special electron trajectories that belong to fractal repellor. We discuss how quantum effects smooth the quantization.
We report on a novel non-linear electric field effect in the conductivity of disordered conductors. We find that an electric field gives rise to dephasing in the particle-hole channel, which depresses the interference effects due to disorder and interaction and leads to a non-linear conductivity. This non-linear effect introduces a field dependent temperature scale $T_E$ and provides a microscopic mechanism for electric field scaling at the metal-insulator transition. We also study the magnetic field dependence of the non-linear conductivity and suggest possible ways to experimentally verify o…
Electron spin-momentum locking together with background magnetic textures can significantly alter the electron transport properties. We theoretically investigate the electron transport at the interface between a topological insulator and a magnetic insulator with magnetic skyrmions on the top. In contrast to the conventional topological Hall effect in normal metals, the skyrmions yield an additional contribution to the anomalous Hall conductivity even in the absence of in-plane magnetic texture, arising from the phase factor characteristic to Dirac electrons acquired at skyrmion boundary.
We consider the effect of electron-electron interactions on the current-in-plane (CIP) conductivity and exchange coupling energy of a disordered metallic magnetic multilayer. We analyze its dependence on the value of ferromagnetic splitting of conducting electrons and ferromagnetic layers relative magnetizations orientation. We show that contribution to the CIP conductivity and exchange coupling energy as a periodic function of the angle of magnetizations relative orientation experience $ 2π\to π$ transition depending on the characteristic energies: ferromagnetic splitting of the conducting el…
We present a unified understanding of the experimentally observed minimal dc conductivity in weakly disordered graphene. Firstly, based on linear response theory, we reveal that randomness or disorder inevitably induces momentum dependent corrections to the electron self-energy function, which naturally yields a sample-dependent minimal conductivity. Taking the long-ranged Gaussian and Coulomb potentials as examples, we derive the momentum dependent self-energy function within the Born approximation, and further validate it via numerical simulations using the large-scale Lanczos algorithm. The…
A disordered array of metal grains with large and random intergrain conductances is studied within the one-loop accuracy renormalization group approach. While at low level of disorder the dependence of conductivity on log T is nonuniversal (it depends on details of the array's geometry), for strong disorder this dependence is described by a universal nonlinear function, which depends only on the array's dimensionality. In two dimensions this function is found numerically. The dimensional crossover in granular films is discussed.
The mesoscopic fluctuations of the Density of electronic States (DoS) and of the conductivity of two- and three- dimensional lattices with randomly distributed substitutional impurities are studied. Correlations of the levels lying above (or below) the Fermi surface, in addition to the correlations of the levels lying on opposite sides of the Fermi surface, take place at half filling due to nesting. The Bragg reflections mediate to increase static fluctuations of the conductivity in the middle of the band which change the distribution function of the conductivity at half- filling.
We study quantum corrections to conductivity in a 2D system with a smooth random potential and strong spin-orbit splitting of the spectrum. We show that the interference correction is positive and down to the very low temperature can exceed the negative correction related to electron-electron interactions. We discuss this result in the context of the problem of the metal-insulator transition in Si-MOSFET structures.
We study thermal conductivity in the disordered two-dimensional electron liquid in the presence of long-range Coulomb interactions. We describe a microscopic analysis of the problem using as a starting point the partition function defined on the Keldysh contour. We extend the Renormalization Group (RG) analysis developed for thermal transport in the disordered Fermi liquid, and include scattering processes induced by the long-range Coulomb interaction in the sub-temperature energy range. For the thermal conductivity, unlike for the electric conductivity, these scattering processes yield a loga…
We present the electrical transport characteristics of a batch of nano-crystalline diamond films of varying nitrogen concentrations and explain the conduction mechanism by the disordered quasi-superlattice model applied to semiconductor heterostructures. Synthesized by the hot filament chemical vapour deposition technique, the degree of structural disorder in the films, confirmed from Raman spectroscopy, is found to be controllable, resulting in the transition of conduction mechanism from localized and activated to the metallic conduction regime. Hence through high field magneto-resistance mea…
We study a relativistic two-dimensional electron gas in the presence of a uniform external magnetic field and a random static scalar potential. We compute, in first order perturbation theory, the averages of the charge density and of the transverse conductivity for a white-noise potential, and show that, within this treatment, their quantized values are modified by the disorder.
