Ac Hall Conductivity Research Articles

Valley-Hall alternatively changing conductivity in gapped and strained graphene.

We investigated the alternatively changing (ac) valley-Hall effect in a graphene/h-BN structure under the modulations of a static electric field E0, magnetic field B0, and light field EA1. The proximity to the h-BN film leads to a mass gap and strain-induced pseudopotential for electrons in graphene. Starting from the Boltzmann equation, we derive the ac conductivity tensor σ, including the orbital magnetic moment, Berry curvature, and anisotropic Berry curvature dipole. It is found that under B0 ≠ 0, σ for the two valleys can have different amplitudes and even have the same sign, leading to a net ac Hall conductivity. The ac Hall conductivities and the optical gain can be altered by both the amplitude and the direction of E0. These features can be understood from the changing rate of σ with E0 and B0, which is valley-resolved and varies nonlinearly with the chemical potential.

Origin of the anomalous Hall effect in two-band chiral superconductors

We consider the origin of the anomalous Hall effect in a general model of a clean two-band chiral superconductor. Within the Kubo formalism we derive an analytic expression for the high-frequency ac Hall conductivity valid close to the critical temperature. This expression involves two distinct gauge-invariant time-reversal-odd bilinear (TROB) functions involving the pairing potential and its Hermitian conjugate. We argue that the existence of at least one of these TROBs generically implies a nonzero ac Hall conductivity. The TROBs allow us to clarify the roles of intra- and interband pairing, and provide a straightforward criterion for a superconducting state to exhibit the anomalous Hall effect. We briefly exemplify our results with model calculations for a chiral $p$-wave pairing state in strontium ruthenate and a chiral $d$-wave pairing state on the honeycomb lattice.

Momentum-dependent mass and AC Hall conductivity of quantum anomalous Hall insulators and their relation to the parity anomaly

The Dirac mass of a two-dimensional QAH insulator is directly related to the parity anomaly of planar quantum electrodynamics, as shown initially in Phys. Rev. Lett. 52, 18 (1984). In this work, we connect the additional momentum-dependent Newtonian mass term of a QAH insulator to the parity anomaly. By calculating the effective action, we reveal that the Newtonian mass term acts like a parity-breaking element of a high-energy regularization scheme. As such, it is directly related to the parity anomaly. In addition, the calculation of the effective action allows us to determine the finite frequency correction to the DC Hall conductivity of a QAH insulator. We derive that the leading order AC correction contains a term proportional to the torsional Hall viscosity. This paves the way to measure this non-dissipative transport coefficient via electrical or magneto-optical experiments. Moreover, we prove that the Newtonian mass significantly changes the resonance structure of the AC Hall conductivity in comparison to pure Dirac systems like graphene.

Loop Currents and Anomalous Hall Effect from Time-Reversal Symmetry-Breaking Superconductivity on the Honeycomb Lattice

We study a tight-binding model on the honeycomb lattice of chiral $d$-wave superconductivity that breaks time-reversal symmetry. Due to its nontrivial sublattice structure, we show that it is possible to construct a gauge-invariant time-reversal-odd bilinear of the pairing potential. The existence of this bilinear reflects the sublattice polarization of the pairing state. We show that it generates persistent loop current correlations around each lattice site and opens a topological mass gap at the Dirac points, resembling Haldane's model of the anomalous quantum Hall effect. In addition to the usual chiral $d$-wave edge states, there also exist electron-like edge resonances due to the topological mass gap. We show that the presence of loop-current correlations directly leads to a nonzero intrinsic ac Hall conductivity, which produces the polar Kerr effect without an external magnetic field. Similar results also hold for the nearest-neighbor chiral $p$-wave pairing. We briefly discuss the relevance of our results to superconductivity in twisted bilayer graphene.

Composite fermion Hall conductivity and the half-filled Landau level

We consider the Hall conductivity of composite fermions in the theory of Halperin, Lee, and Read (HLR). We present a fully quantum mechanical numerical calculation that shows, under suitable conditions, the HLR theory exhibits a particle-hole symmetric dc electrical Hall response in the presence of quenched disorder. Remarkably, this response of the HLR theory remains robust even when the disorder range is of the order of the Fermi wavelength. We find that deviations from particle-hole symmetric response can appear in the ac Hall conductivity at frequencies sufficiently large compared to the inverse system size. Our results agree with a recent semi-classical analysis by Wang et al., Phys. Rev. X 7, 031029 (2017) and complement the arguments based on the fully quantum-mechanical model by Kumar et al., Phys. Rev. B 98, 11505 (2018). These results provide further evidence that the HLR theory is compatible with an emergent particle-hole symmetry.

Imaginary part of Hall conductivity in a tilted doped Weyl semimetal with both broken time-reversal and inversion symmetry

We consider a Weyl semimetal (WSM) with finite doping and tilt within a continuum model Hamiltonian with both broken time reversal and inversion symmetry. We calculate the absorptive part of the anomalous AC Hall conductivity as a function of photon energy ($\Omega$) for both type I and type II Weyl semimetal. For a given Weyl node, changing the sign of its chirality or of its tilt changes the sign of its contribution to the absorptive Hall conductivity with no change in magnitude. For a noncentrosymmetric system we find that there are ranges of photon energies for which only the positive or only the negative chirality node contributes to the imaginary (absorptive) part of the Hall conductivity. There are also other photon energies where both chirality contribute and there can be other ranges of $\Omega$ where there is no absorption associated with the AC Hall conductivity in type I and regions where it is instead constant for type II. We comment on implications for the absorption of circular polarized light.

Intrinsic ac anomalous Hall effect of nonsymmorphic chiral superconductors with an application to UPt3

We identify an intrinsic mechanism of the anomalous Hall effect for non-symmorphic chiral superconductors. This mechanism relies on both a nontrivial multi-band chiral superconducting order parameter, which is a mixture of pairings of even and odd angular momentum channels, and a complex normal state inter-sublattice hopping, both of which are consequences of the nonsymmorphic group symmetry of the underlying lattice. We apply this mechanism to the putative chiral superconducting phase of the heavy-fermion superconductor $\mathrm{UPt_3}$ and calculate the anomalous ac Hall conductivity in a simplified two-band model. From the ac Hall conductivity and optical data we estimate the polar Kerr rotation angle and compare it to the measured results for $\mathrm{UPt_3}$ [E. R. Schemm \textit{et al.}, Science \textbf{345},190(2014)].

Anomalous Hall Effect in Type-I Weyl Metals

We study the ac anomalous Hall conductivity σ_{xy}(ω) of a Weyl semimetal with broken time-reversal symmetry. Even in the absence of free carriers these materials exhibit a "universal" anomalous Hall response determined solely by the locations of the Weyl nodes. We show that the free carriers, which are generically present in an undoped Weyl semimetal, give an additional contribution to the ac Hall conductivity. We elucidate the phy146sical mechanism of the effect and develop a microscopic theory of the free carrier contribution to σ_{xy}(ω). The latter can be expressed in terms of a small number of parameters (the electron velocity matrix, the Fermi energy μ, and the "tilt" of the Weyl cone). The resulting σ_{xy}(ω) has resonant features at ω∼2μ which may be used to separate the free carrier response from the filled-band response using, for example, Kerr effect measurements. This may serve as a diagnostic tool to characterize the doping of individual valleys.

Three-band intrinsic Kerr effect in Sr2RuO4

The Kerr effect in the chiral triplet superconductor Sr2RuO4 has proved difficult to explain unless either impurity scattering or multiband superconductivity is included in the theory. In this paper, we concentrate on the analysis of the ac Hall conductivity and the Kerr signal in terms of a realistic three-dimensional three-band model of the Fermi surface. We consider two possible unique observable signatures which could be used to distinguish the three-band model from other possible sources of the Kerr signal. Firstly, we consider in detail the temperature dependence of the Kerr signal. For most frequencies, this varies with temperature as near to Tc while for others, Hebel-Slichter-like enhancement or decrease below Tc are observed. In contrast, the sum-rule integral over all frequencies is monotonic and follows the temperature dependence of . Secondly, we examine the Kerr effect in the presence of a c-axis magnetic field. There is some evidence that in a small c-axis field the order parameter transforms from chiral to a non-chiral ground state. We show that in the non-chiral state, both spin – orbit coupling and a finite B field are necessary for the appearance of a non-zero Hall conductivity and Kerr effect. The transition from chiral to non-chiral pairing state would lead to an observable dramatic change in Kerr signal at some critical magnetic field.

Remarks on the Hall Conductivity in Chiral Superconductors: Weak vs. Strong Coupling Approach

We discuss the calculations of the ac Hall conductivity for superconductors with the time reversal symmetry breaking states. In the weak coupling theories these states show vanishing Hall response in one band models, even though one expects otherwise on symmetry grounds. On the other hand, the strong coupling approach based on the anti-de Sitter conformal eld theory correspondence leads to the non-vanishing Hall conductivity. We discuss the possible reasons of the discrepancy. The weak coupling many orbital theory leading to the Hall conductivity with correct temperature dependence is also brie y presented.

Transport on the surface of a topological insulator

We study theoretically dc and ac transport on the surface of a three-dimensional topological insulator when its time-reversal symmetry is broken. Starting with a Kubo formula, we derive an explicit expression for the dc Hall conductivity, valid for finite temperatures. At zero temperature this expression gives the dc half-quantum Hall conductivity, provided the Fermi level lies in the gap. Corrections when the Fermi level is outside the gap and scattering by impurities are quantified. The longitudinal conductivity is also examined. At finite frequencies, we find a modified Drude term in σxx(ω) and logarithmic, frequency-dependent corrections in σyx(ω). The ac Hall conductivity exhibits a robust logarithmic singularity for excitation energies equal to the gapwidth. For these energies, we also find that the power spectrum, which is pertinent to optical experiments, exhibits drastic increase. The Hall conductivity remains almost unaffected for temperatures up to approximately 300 K.

Intrinsic Optical Dichroism in the 2d Model of Chiral Superconducting State

Recently, we have found a new intrinsic mechanism of dichroism in chiral superconductors. It relies on the existence of many orbital character of the superconducting state as in Sr2RuO4. Using three orbital but a two-dimensional model relevant for superconducting strontium ruthenate, we calculate temperature T and frequency ω dependent ac Hall conductivity σxy(ω,T) for a number of parameters. In particular, we study the changes of σxy due to changes in the light frequency ω, spin–orbit coupling λ and effective interaction parameters Uij between electrons occupying in-plane d2g orbitals. Our calculations qualitatively agree with the measured Kerr rotation angle in Sr2RuO4 and have a potential to describe other superconductors.

Intrinsic Optical Dichroism in the Chiral Superconducting State ofSr2RuO4

We present an analysis of the Hall conductivity σ(xy)(ω,T) in time reversal symmetry breaking states of exotic superconductors. We find that the dichroic signal is nonzero in systems with interband order parameters. This new intrinsic mechanism may explain the Kerr effect observed in strontium ruthenate and possibly other superconductors. We predict coherence factor effects in the temperature dependence of the imaginary part of the ac Hall conductivity Imσ(xy)(ω,T), which can be tested experimentally.

All Optical Measurement Proposed for the Photovoltaic Hall Effect

We propose an all optical way to measure the recently proposed "photovoltaic Hall effect", i.e., a Hall effect induced by a circularly polarized light in the absence of static magnetic fields. This is done in a pump-probe experiment with the Faraday rotation angle being the probe. The Floquet extended Kubo formula for photo-induced optical response is formulated and the ac-Hall conductivity is calculated. We also point out the possibility of observing the effect in two layered graphene, three-dimensional graphite, and more generally in multi-band systems such as materials described by the dp-model.

Anomalous criticality at then=0quantum Hall transition in graphene: The role of disorder preserving chiral symmetry

We investigate numerically whether the chiral symmetry is the sole factor dominating the criticality of the quantum Hall transitions in disordered graphene. When the disorder respects the chiral symmetry, the plateau-to-plateau transition at the $n=0$ Landau level is shown to become anomalous (i.e., step-function-like). Surprisingly, however, the anomaly is robust against the inclusion of the uniform next-nearest neighbor hopping, which degrades the chiral symmetry of the lattice models. We have also shown that the ac (optical) Hall conductivity exhibits a robust plateau structure when the disorder respects the chiral symmetry.

Dynamical scaling analysis of the optical Hall conductivity in the quantum Hall regime

Dynamical scaling analysis is theoretically performed for the ac (optical) Hall conductivity ${\ensuremath{\sigma}}_{xy}({\ensuremath{\epsilon}}_{F},\ensuremath{\omega})$ as a function of Fermi energy ${\ensuremath{\epsilon}}_{F}$ and frequency $\ensuremath{\omega}$ for the two-dimensional electron gas and for graphene. In both systems, results based on exact diagonalization show that ${\ensuremath{\sigma}}_{xy}({\ensuremath{\epsilon}}_{F},\ensuremath{\omega})$ displays a well-defined dynamical scaling, for which the dynamical critical exponent as well as the localization exponent are obtained. A crossover from the dc-like behavior to the ac regime is identified. The dynamical scaling analysis has enabled us to quantify the plateau in the ac Hall conductivity previously obtained and to predict that the plateaux structure in ac is robust enough to be observed in the terahertz regime.

Frequency and temperature dependence of the anomalous ac Hall conductivity in a chiralpx+ipysuperconductor with impurities

We calculate frequency and temperature dependence of the anomalous ac Hall conductivity induced by impurity scattering in a chiral ${p}_{x}+i{p}_{y}$ superconductor, such as ${\text{Sr}}_{2}{\text{RuO}}_{4}$, with spontaneous time-reversal-symmetry breaking in the absence of an external magnetic field. We consider two models of disorder, Gaussian and non-Gaussian, characterized by the second and third moments of the random impurity potential, respectively. Within both models, we find that the anomalous Hall conductivity has a finite real value at zero frequency, exhibits singularities at the threshold of photon absorption across the superconducting gap, and decays as some power of the high frequency $\ensuremath{\omega}$. The Hall conductivity increases linearly with the decrease in temperature below the superconducting transition and saturates at zero temperature. Using our results for the high-frequency Hall conductivity, we estimate the polar Kerr angle for light reflection from the material and compare it with the experimental measurements in ${\text{Sr}}_{2}{\text{RuO}}_{4}$ by Xia et al., Phys. Rev. Lett. 97, 167002 (2006).

Infrared magneto-optical properties of (III,Mn)V ferromagetic semiconductors

We present a theoretical study of the infrared magneto-optical properties of ferromagnetic (III,Mn)V semiconductors. Our analysis combines the kinetic exchange model for (III,Mn)V ferromagnetism with Kubo linear response theory and Born approximation estimates for the effect of disorder on the valence band quasiparticles. We predict a prominent feature in the ac-Hall conductivity at a frequency that varies over the range from 200 to 400 meV, depending on Mn and carrier densities, and is associated with transitions between heavy-hole and light-hole bands. In its zero frequency limit, our Hall conductivity reduces to the $\vec k$-space Berry's phase value predicted by a recent theory of the anomalous Hall effect that is able to account quantitatively for experiment. We compute theoretical estimates for magnetic circular dichroism, Faraday rotation, and Kerr effect parameters as a function of Mn concentration and free carrier density. The mid-infrared response feature is present in each of these magneto-optical effects.

Midinfrared Hall effect in thin-film metals: Probing the Fermi surface anisotropy in Au and Cu

A sensitive mid-infrared (MIR, 900-1100 cm-1, 112-136 meV) photo-elastic polarization modulation technique is used to measure simultaneously Faraday rotation and circular dichroism in thin metal films. These two quantities determine the complex AC Hall conductivity. This novel technique is applied to study Au and Cu thin films at temperatures down to 20 K and magnetic fields up to 8 T. The Hall frequency is consistent with band theory predictions. We report the first measurement of the MIR Hall scattering rate, which is significantly lower than that derived from Drude analysis of zero magnetic field MIR transmission measurements. This difference is qualitatively explained in terms of the anisotropy of the Fermi surface in Au and Cu.

Hall effect near the vortex-glass transition in high-temperature superconductors.

We discuss the Hall effect in the mixed state of type-II superconductors with an emphasis on the regime near the vortex-glass transition. This transition is argued to have a ``hidden'' particle-hole symmetry, and a new exponent is introduced to characterize the irrelevance of the particle-hole asymmetry. This leads to a natural explanation of the scaling of the Hall and Longitudinal resistivities observed in ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7}$. At the transition, the nonlinear Hall voltage is predicted to vary with a universal power of the current, and the linear ac Hall conductivity with a universal power of the frequency.