Nonlinear Conductivity Tensor Research Articles

Nonlinear spin Hall effect in PT -symmetric collinear magnets

We theoretically investigate a nonlinear spin Hall effect in $\mathcal{PT}$-symmetric antiferromagnetic metals, which serve as an efficient spin current generator. We elucidate that an emergent spin-dependent Berry curvature dipole is a microscopic origin of the nonlinear spin Hall effect, which becomes nonzero with neither relativistic spin-orbit coupling, uniform magnetization, nor spin-split band structure. By analyzing a microscopic antiferromagnetic model without spin-orbit coupling for an intuitive understanding of the phenomena, we elucidate essential hopping processes and a condition to enhance the nonlinear spin Hall conductivity. We also provide a complete table to include useful correspondence among the N\'eel vector, odd-parity multipoles, nonlinear spin conductivity tensor, and candidate materials in all the $\mathcal{PT}$-symmetric black-and-white magnetic point groups.

Quantum theory of the nonlinear Hall effect

The nonlinear Hall effect is an unconventional response, in which a voltage can be driven by two perpendicular currents in the Hall-bar measurement. Unprecedented in the family of the Hall effects, it can survive time-reversal symmetry but is sensitive to the breaking of discrete and crystal symmetries. It is a quantum transport phenomenon that has deep connection with the Berry curvature. However, a full quantum description is still absent. Here we construct a quantum theory of the nonlinear Hall effect by using the diagrammatic technique. Quite different from nonlinear optics, nearly all the diagrams account for the disorder effects, which play decisive role in the electronic transport. After including the disorder contributions in terms of the Feynman diagrams, the total nonlinear Hall conductivity is enhanced but its sign remains unchanged for the 2D tilted Dirac model, compared to the one with only the Berry curvature contribution. We discuss the symmetry of the nonlinear conductivity tensor and predict a pure disorder-induced nonlinear Hall effect for point groups C3, C3h, C3v, D3h, D3 in 2D, and T, Td, C3h, D3h in 3D. This work will be helpful for explorations of the topological physics beyond the linear regime.

Quantized electrochemical transport in Weyl semimetals

We show that under the effect of an external electric field and a gradient of chemical potential, a topological electric current can be induced in Weyl semimetals without inversion and mirror symmetries. We derive analytic expressions for the nonlinear conductivity tensor and show that it is nearly quantized for small tilting when the Fermi levels are close to the Weyl nodes. When the van Hove point is much larger than the largest Fermi level, the band structure is described by two linearly dispersing Weyl fermions with opposite chirality. In this case, the electrochemical response is fully quantized in terms of fundamental constants and the scattering time, and it can be used to measure directly the topological charge of Weyl points. We show that the electrochemical chiral current may be derived from an electromagnetic action similar to axion electrodynamics, where the position-dependent chiral Fermi level plays the role of the axion field. This posits our results as a direct consequence of the chiral anomaly.

Nonlinear quantum optical properties of graphene

We present a semiclassical theory of the linear and nonlinear optical response of graphene. The emphasis is placed on the nonlinear optical response of graphene from the standpoint of the underlying chiral symmetry. The Bloch quasiparticles in the low-energy limit around the degeneracy points are dominantly chiral. It is shown that this chiral behavior in conjunction with scale invariance in graphene around the Dirac points results in the strong nonlinear optical response. Explicit expressions for the linear and nonlinear conductivity tensors are derived based on semiconductor Bloch equations (SBEs). The linear terms agree with the result of Kubo formulation. The three main additive mechanisms contribute in the nonlinear optical response of graphene: pure intraband, pure interband and the interplay between them. For each contribution, an explicit response function is derived. The Kerr-type nonlinearity of graphene is then numerically studied and it is demonstrated that the nonlinear refractive index of graphene can be tuned and enhanced by applying a gate voltage. It is also discussed that a strong Kerr nonlinearity can be achieved in a gated graphene monolayer. However, this nonlinearity is accompanied with a significant amount of absorption loss.

Rigorous calculation of nonlinear parameters in graphene-comprising waveguides

We describe a rigorous formalism for the calculation of the nonlinear parameter of arbitrary three-dimensional nanophotonic graphene-comprising waveguides. Graphene is naturally implemented as a zero-thickness conductive sheet, modeled solely by complex linear and nonlinear surface conductivity tensors, whose values are extracted from theoretical models. This representation is compared to the more commonly employed equivalent bulk-medium representation and is found superior. We numerically calculate the nonlinear parameters of several optical waveguide archetypes overlaid with infinite graphene monolayers, including silicon-wire and plasmonic metal-slot and metal-stripe configurations. The metal-slot configuration offers the most promising performance for Kerr-type nonlinear applications. Finally, we apply the same formalism to probe the potential of graphene nanoribbon waveguide nonlinearity in the terahertz band.

Quantum theory of third-harmonic generation in graphene

A quantum theory of the third-harmonic generation in graphene is presented. An analytical formula for the nonlinear conductivity tensor $\sigma^{(3)}_{\alpha\beta\gamma\delta}(\omega,\omega,\omega)$ is derived. Resonant maxima of the third harmonic are shown to exist at low frequencies $\omega\ll E_F/\hbar$, as well as around the frequency $\omega=2E_F/\hbar$, where $E_F$ is the Fermi energy in graphene. At the input power of a CO$_2$ laser ($\lambda\approx 10$ $\mu$m) of about 1 MW/cm$^2$ the output power of the third-harmonic ($\lambda\approx 3.3$ $\mu$m) is expected to be $\simeq 50$ W/cm$^2$.

A method for the identification of nonlinear components of the thermal conductivity tensor for anisotropic materials

A method is proposed for the numerical solution of coefficient inverse problems on nonlinear heat transfer in anisotropic materials used for heat shielding in the aerodynamic heating of hypersonic craft (HC). The main characteristics of anisotropic heat-shielding materials at high temperatures are the components of a nonlinear thermal conductivity tensor that are to be restored based on the results of the experimental measurements of the temperature in space-time nodes.

Kinetic theory of nonlinear waves interaction in relativistic plasmas

On a basis of the relativistic Vlasov–Maxwell equations for an electron unmagnetized collisionless plasma, a three-dimensional analysis of the nonlinear mixing of electromagnetic and longitudinal waves is performed. Using a conducting relativistic plasma model two waves generate a nonlinear current which is the source of a longitudinal plasma wave described by a nonlinear differential equation and by a nonlinear conductivity tensor. A dielectric relativistic plasma model is used to explain the nonlinear excitation of a transverse wave, as an effect of the nonlinear polarization, and described by a nonlinear differential equation and by a nonlinear susceptibility tensor.

Conventional and torque magnetometry on disc-shaped high-T c superconductor samples : I. A steady-state dynamical theory

A nonlinear, macroscopic electrodynamic theory is given to explain the irreversible part of the magnetic momentum m irr and the torque momentum Γ irr observed in conventional (CM) and torque magnetometry (TM) on HTSC discs and thin films with oblique external time-dependent magnetic field H 0( t) and off-plane rotation of H 0( t), respectively. This theory relates m irr and Γ irr with the nonlinear conductivity tensor σ of the mixed state, and with microstructure-dependent material parameters entering σ, which govern the thermally activated vortex motion under anisotropic pinning. In the steady state, m irr and Γ irr is understood to result from a current j excited by an “applied” electrical field E 0 induced by the external inductance B 0 ( t) = H 0( t). The limiting case of discs with large aspect ratio and thickness much smaller than the nonlinear skin depth is studied in detail. It is shown that, except of some singular cases, j and m irr becomes strictly azimuthal and axial, respectively, while | δB| ⪡ | B 0| where B = B o + δB is the total inductance. The boundary value problem is solved in a self-consistent approximation making use of the strong sublinear characteristic of j versus the total electrical field E under CM and TM conditions, which allows to replace σ by an effective pseudo-linear quantity ḡs governed by the dominating azimuthal E -mode. Formulae or m irr and Γ irr are given, showing, in spite of the simple configuration of E and B , a complicated nonlinear dependence upon the components of σ. To interpret CM and TM experiments, a phenomenological description of σ( E, B ) is presented based upon symmetry arguments, as well as theoretical and experimental facts, which is valid for anistropic and layered HTSC, including the isotropic, and the anisotropic intrinsic and twin-boundary pinning. The possibility to determine σ and related material parameters from CM and TM results is discussed in detail.

Nonlinear response of electron-phonon interaction inn-type nondegenerate piezoelectric semiconductors

The nonlinear response of the electron-phonon interaction in nondegenerate piezoelectric semiconductors such asn-type InSb in the presence of a dc magnetic fieldB directed along the propagation of acoustic waves has been studied by using a quantum mechanical treatment. The effect of electron scattering in solids has been taken into consideration, so the electron relaxation time cannot be neglected. The nonlinear nature of the energy band is corrected for, using the Heisenberg equation of motion. It is found that the nonlinear response is proportional to a nonlinear longitudinal conductivity tensor τ zzz when acoustic waves propagate parallel to the [111] direction for both piezoelectric and deformation-potential couplings. Numerical calculations for τ zzz of n-type InSb at low temperatures are presented. Results show that the nonlinear response decreases rapidly with the sound frequency, and decreases slowly with the dc magnetic field and temperature. Therefore, the electron relaxation time and the nonparabolicity of energy bands play important roles for the electron-phonon interaction due to the piezoelectric and deformation-potential couplings in the microwave region.

Random-phase-approximation study of the response function describing optical second-harmonic generation from a metal selvedge.

Optical second-harmonic generation in centrosymmetric metals stems from the breaking of inversion symmetry in the selvedge and from nonlocal effects. In the present work a random-phase-approximation description of the nonlocal nonlinear conductivity tensor of an inhomogeneous jellium is given. Main emphasis is devoted to a study of the electric dipole response from the selvedge, and to the magnetic dipole and electric quadrupole responses from the profile region and the bulk. A detailed analysis of the tensor symmetry schemes associated with the p\ensuremath{\cdot}A and A\ensuremath{\cdot}A interactions is presented. Dipole and quadrupole transitions are identified, and the role of direct and indirect quantum processes is discussed. On the basis of the so-called infinite-barrier model, a few numerical results are given.

Nonlinear wave coupling in a warm plasma in the fluid approximation

The general expression for nonlinear coupling between plasma modes is obtained. The nonlinear conductivity tensor is then calculated by means of the two-fluid plasma description taking into account the thermal pressure effects.

Symmetry of the complete second-order nonlinear conductivity tensor for an unmagentized relativistic turbulent plasma

Symmetry of the complete second-order nonlinear conductivity tensor for an unmagentized relativistic turbulent plasma

Symmetry of the complete second-order nonlinear conductivity tensor for an unmagnetized relativistic turbulent plasma

A new exact symmetry is proved for the complete second-order nonlinear conductivity tensor of an unmagnetized relativistic turbulent plasma. The symmetry is not limited to principal parts. If Cerenkov resonance is ignored, the new symmetry reduces to the well-known symmetry related to the Manley–Rowe relations, crossing symmetry, and nondissipation of the principal part of the nonlinear current. Also, a new utilitarian representation for the complete tensor is obtained in which all derivatives are removed and the pole structure is clearly exhibited.

Exact symmetry of the second-order nonlinear conductivity for a relativistic turbulent plasma

The second-order nonlinear conductivity tensor for a relativistic turbulent plasma is shown to have an exact symmetry which, neglecting resonant wave-particle interaction, becomes the widely known symmetry related to the nondissipative nature of the nonlinear current.

Theory of Non-linear Longitudinal Kerr Magneto-optic Effect in a Ferromagnetic Metallic Thin Film

A theory of the non-linear longitudinal Kerr magneto-optic effect, previously developed for bulk materials [1], is now applied to a magnetic thin film. The material model is based on the classical equation of motion for a free electron with a finite relaxation frequency under the action of a Lorentz force. A second harmonic current density, dependent upon three non-linear conductivity tensors, reduces to results obtained by Jha [2] in the limit of vanishing ferromagnetic state. The second harmonic reflection coefficients are approximated and are given in terms of the first harmonic reflection coefficients as derived by Smith [3].

Boltzmann-equation approach to nonlinear acoustoelectric interactions in piezoelectric semiconductors

Ultrasonic second-harmonic generation in piezoelectric semiconductors is investigated using the classical Boltzmann-equation approach with the ansatz of a constant relaxation time. The flux at the second-harmonic frequency is calculated in terms of the flux in the fundamental and the linear and nonlinear conductivity tensors. It is found that when $\mathrm{ql}\ensuremath{\ll}1$, our results for second-harmonic generation reduce to those of Conwell and Ganguly, while when $\mathrm{ql}\ensuremath{\gg}1$ they reduce to those of Wu and Spector. The existence of a separate regime when $\mathrm{ql}\ensuremath{\gg}1$ and $\ensuremath{\omega}\ensuremath{\tau}\ensuremath{\ll}1$ as found by Nakamura is shown not to occur.

Nonlinear conductivity tensor in graded mixed semiconductors

A mixed semiconductor crystal with graded composition (i.e., a heterojunction) is characterized by a position-dependent effective mass, which leads to an additional force acting on free carriers. Taking this force into account, as well as the carrier-concentration gradient, the Boltzmann equation is solved in the presence of two oscillating electric fields of arbitrary frequencies, and a general expression for the second-order conductivity tensor is obtained. This tensor is linear in the effective-mass gradient and in the carrier-concentration gradient, and independent of carrier statistics. Limiting cases $\ensuremath{\omega}\ensuremath{\tau}\ensuremath{\gg}1$ and $\ensuremath{\omega}\ensuremath{\tau}\ensuremath{\ll}1$ are discussed in detail. For $\ensuremath{\omega}\ensuremath{\tau}\ensuremath{\gg}1$, the expected second-harmonic generation associated with the mass gradient is at least comparable with that observed in homogeneous semiconductors. For $\ensuremath{\omega}\ensuremath{\tau}\ensuremath{\ll}1$, nonlinear effects are much stronger than in a homogeneous material in this frequency range. Measurement of the nonlinear current offers a method of directly determining the effective-mass gradient.

Ultrasonic-harmonic generation in bismuth

Second-harmonic generation of ultrasound in a semimetal like bismuth is investigated using a quantum-mechanical treatment which is valid at high frequencies and in strong magnetic fields. It is shown that the amplitude of the second harmonic can be expressed in terms of the fundamental using the linear and nonlinear conductivity tensors for ultrasonic waves propagating parallel to a dc magnetic field. At first we assume that the energy-band structure of bismuth is the Cohen nonellipsoidal nonparabolic (NENP) model. It is found that the magnitude of the second-harmonic generation oscillates with the dc magnetic field. Some suboscillations with the dc magnetic field are also observed. These suboscillations occur owing to the spin-splitting effect of the Landau levels in bismuth. When the sound frequency increases, these oscillations will be diminished due to the breakdown of the screening. If the energy-band structure is assumed to be the ellipsoidal nonparabolic (ENP) model, the oscillations and suboscillations can also be observed. However, the suboscillations will be diminished very rapidly with the sound frequency increasing. In fact, these suboscillations are washed out at the region of high frequencies. When the energy-band structure is assumed to be the ellipsoidal parabolic (EP) model, no suboscillations can be observed. It is also shown that the second-harmonic generation increases with increasing the sound frequency for the NENP model. The second-harmonic generation will peak in the neighborhood of $\ensuremath{\omega}=2\ifmmode\times\else\texttimes\fi{}{10}^{11}$ rad/sec owing to the effect of highly anisotropic Fermi surfaces in bismuth. However, no peaks occur at the high-frequency region for the ENP or EP models.

Magnetoacoustic effects of electrical conductivity tensors for nonellipsoidal nonparabolic band structure

Using time-independent perturbation theory, the energy eigenvalue equation for a free-electron gas in the presence of a dc magnetic field $\stackrel{\ensuremath{\rightarrow}}{B}$ is solved. The energy-band structure applied here is supposed to be the Cohen nonellipsoidal nonparabolic model. Then the longitudinal linear and nonlinear conductivity tensors are calculated by using the quantum-mechanical treatment which is valid in the high-frequency region. It is found that these conductivity tensors for ultrasound propagating parallel to the dc magnetic field depend upon the magnetic field and sound frequency. Some oscillations and discontinuities in these conductivity tensors are observed. This can be explained by the fact that the linear and nonlinear conductivity tensors have the logarithmic singularities of a purely quantum origin related to the degeneracy of the electron gas. We also present some numerical calculations of the absorption coefficient and change in sound velocity in bismuth. It is found that the absorption coefficient and change in sound velocity oscillate with the dc magnetic field and sound frequency when the ultrasound propagates parallel to the magnetic field.