Semiclassical Boltzmann Equation Research Articles

Phonon-mediated ultrafast energy- and momentum-resolved hole dynamics in monolayer black phosphorus.

The electron-phonon scattering plays a crucial role in determining the electronic, transport, optical, and thermal properties of materials. Here, we employ a non-Markovian stochastic Schrödinger equation (NMSSE) in momentum space, together with abinitio calculations for energy bands and electron-phonon interactions, to reveal the phonon-mediated ultrafast hole relaxation dynamics in the valence bands of monolayer black phosphorus. Our numerical simulations show that the hole can initially remain in the high-energy valence bands for more than 100fs due to the weak interband scatterings, and its energy relaxation follows single-exponential decay toward the valence band maximum after scattering into low-energy valence bands. The total relaxation time of holes is much longer than that of electrons in the conduction band. This suggests that harnessing the excess energy of holes may be more effective than that of electrons. Compared to the semiclassical Boltzmann equation based on a hopping model, the NMSSE highlights the persistence of quantum coherence for a long time, which significantly impacts the relaxation dynamics. These findings complement the understanding of hot carrier relaxation dynamics in two-dimensional materials and may offer novel insights into harnessing hole energy in photocatalysis.

Optimal control of a semiclassical Boltzmann equation for charge transport in graphene

An ensemble optimal control problem governed by a semiclassical space-homogeneous Boltzmann equation for charge transport in graphene is formulated and analyzed.The control mechanism is an external time-dependent electric field having the purpose of driving the material density of electrons in graphene to follow a desired trajectory. For this purpose, an ensemble cost functional with quadratic H1 costs is considered.Well-posedness of the control problem is proved, and the characterization of optimal controls by an optimality system is discussed. This system is approximated by a discontinuous Galerkin scheme and solved using a nonlinear conjugate gradient method. Results of numerical experiments are presented that demonstrate the ability of the proposed control framework to design control fields.

Hydrodynamic Navier-Stokes equations in two-dimensional systems with Rashba spin-orbit coupling

We study a two-dimensional (2D) electron system with a linear spectrum in the presence of Rashba spin-orbit (RSO) coupling in the hydrodynamic regime. We derive a semiclassical Boltzmann equation with a collision integral due to Coulomb interactions on the basis of the eigenstates of the system with RSO coupling. Using the local equilibrium distribution functions, we obtain a generalized hydrodynamic Navier–Stokes equation for electronic systems with RSO coupling. In particular, we discuss the influence of the spin-orbit coupling on the viscosity and the enthalpy of the system and present some of its observable effects in hydrodynamic transport.

The Transport and Optical Characteristics of a Metal Exposed to High-Density Energy Fluxes in Compressed and Expanded States of Matter

This article presents a theoretical study of the optical and transport properties of metals. Iron, as an example, was used to discuss, through a theoretical description, the peculiarities of these properties in the compressed and expanded states under the influence of high-density energy fluxes. By solving the semi-classical Boltzmann equation for conduction electrons for a broad range of densities and temperatures, the expressions of electrical conductivity, electronic thermal conductivity, and thermoelectric coefficient calculations were derived. The real and imaginary parts of the iron permittivity and the energy absorption coefficient for the first and second harmonics of Nd:YAG laser radiation were obtained. The calculation peculiarities of the metal’s optical characteristics of matter in an expanded state in a broad range of densities and temperatures were considered. The analysis of the obtained results shows their agreement with the theoretical description for cases of ideal non-degenerate and dense degenerate electron plasmas. It is shown that the behavior of the electrical conductivity and optical characteristics in the critical and supercritical regions of density and temperature are in agreement with the known experimental results.

Electrochemical transport in Dirac nodal-line semimetals

Nodal-line semimetals are topological phases where the conduction and the valence bands cross each other along one-dimensional lines in the Brillouin zone, which are symmetry protected by either spatial symmetries or time-reversal symmetry. In particular, nodal lines protected by the combined symmetry exhibits the parity anomaly of 2D Dirac fermions. In this letter, we study the electrochemical transport in Dirac nodal-line semimetals by using the semiclassical Boltzmann equation approach. We derive a general formula for the topological current that includes both the Berry curvature and the orbital magnetic moment. We first evaluate the electrochemical current by introducing a small mass term (which could be induced by inversion-breaking uniaxial strain, pressure, or an external electric field) and apply it to the hexagonal pnictide CaAgP. The electrochemical current vanishes in the zero-mass limit. Introducing a tilting term that does not spoil symmetry that protects the nodal ring, we obtain a finite electrochemical current in the zero-mass limit, which can be regarded as a direct consequence of the parity anomaly. We show that the parity-anomaly–induced electrochemical transport is also present at nonzero temperatures.

DMRG analysis of magnetic order in the zigzag edges of hexagonal CrN nanoribbons

We investigate the finite-temperature magnetic order at the edges of hexagonal CrN nanoribbons by using density functional theory combined with the density matrix renormalization group (DMRG) method. Moreover, the spin-dependent transport in nanoribbons is calculated within the semiclassical Boltzmann transport theory. We find that the zigzag edges have lower energy with respect to armchair edges. The zigzag edges of CrN nanoribbon show half-metallic electronic character, which is the same as for the two-dimensional (2D) monolayer. The localized electronic states on the zigzag edges reduce the electronic band gap energy for spin-down electrons. The ab initio electronic results are mapped into an effective 1D Heisenberg spin model up to the next-nearest-neighbor exchange interaction term. For zigzag ribbons, the nearest-neighbor and next-nearest-neighbor magnetic exchange are around 10--12 and $\ensuremath{-}2$ to 0 $\text{meV}/\text{Cr}\phantom{\rule{4.pt}{0ex}}\text{atom}$, respectively. The finite spin correlation length in 1D nanoribbons drops sharply to zero with temperature. The absence of long-range spin correlations at the edges is a practical drawback for future room temperature 2D spintronic devices. The maximally localized Wannier functions are used for band interpolation and spin-dependent transport calculations by using the semiclassical Boltzmann equation. We show that the zigzag edges of CrN are a perfect spin filter under both electron and hole doping.

Disorder-induced anomalous Hall effect in type-I Weyl metals: Connection between the Kubo-Streda formula in the spin and chiral basis

We study the anomalous Hall effect (AHE) in tilted Weyl metals with weak Gaussian disorder under the Kubo-Streda formalism in this work. To separate the three different contributions, namely the intrinsic, side jump and skew scattering contribution, it is usually considered necessary to go to the eigenstate (chiral) basis of the Kubo-Streda formula. However, it is more straight-forward to compute the total Hall current in the spin basis. For the reason, we develop a systematic and transparent scheme to separate the three different contributions in the spin basis for relativistic systems by building a one-to-one correspondence between the Feynman diagrams of the different mechanisms in the chiral basis and the products of the symmetric and anti-symmetric part of the polarization operator in the spin basis. We obtain the three contributions of the AHE in tilted Weyl metals by this scheme and found that the side jump contribution exceeds both the intrinsic and skew scattering contribution for the low-energy effective Hamiltonian. We compared the anomalous Hall current obtained from our scheme with the results from the semi-classical Boltzmann equation approach under the relaxation time approximation and found that the results from the two approaches agree with each other in the leading order of the tilting velocity.

Engineering Transistorlike Optical Gain in Two-Dimensional Materials with Berry Curvature Dipoles.

Transistors are key elements of electronic circuits as they enable, for example, the isolation or amplification of voltage signals. While conventional transistors are point-type (lumped-element) devices, it may be interesting to realize a distributed transistor-type optical response in a bulk material. Here, we show that low-symmetry two-dimensional metallic systems may be the ideal solution to implement such a distributed-transistor response. To this end, we use the semiclassical Boltzmann equation approach to characterize the optical conductivity of a two-dimensional material under a static electric bias. Similar to the nonlinear Hall effect, the linear electro-optic (EO) response depends on the Berry curvature dipole and can lead to nonreciprocal optical interactions. Most interestingly, our analysis uncovers a novel non-Hermitian linear EO effect that can lead to optical gain and to a distributed transistor response. We study a possible realization based on strained bilayer graphene. Our analysis reveals that the optical gain for incident light transmitted through the biased system depends on the light polarization, and can be quite large, especially for multilayer configurations.

Fundamental distinction between intrinsic and extrinsic nonlinear thermal Hall effects

We theoretically investigated the fundamental distinction between intrinsic and extrinsic nonlinear thermal Hall effect in the presence of disorder at the second-order response to the temperature gradient in terms of the semi-classical Boltzmann equation. We found that, at low temperatures, the intrinsic contribution of the nonlinear thermal Hall conductivity is proportional to the square of temperature, whereas the extrinsic contributions (side-jump and skew-scattering) are independent of temperature. This distinct dependency on temperature provide a new approach to readily distinguish the intrinsic and extrinsic contributions. Specifically, we analysed the nonlinear thermal Hall effect for a tilted two-dimensional massive Dirac material. In particular, we showed that when the Fermi energy is located at the Dirac point, the signal is solely from the intrinsic mechanism; when the Fermi energy is higher, the extrinsic contributions are dominant, which are uncovered to be two to three orders of magnitude larger than the intrinsic contribution.

Anisotropic and tunable optical conductivity of a two-dimensional semi-Dirac system in the presence of elliptically polarized radiation

We investigate the effect of ellipticity ratio of the polarized radiation field on optoelectronic properties of a two-dimensional (2D) semi-Dirac (SD) system. The optical conductivity is calculated within the energy balance equation approach derived from the semiclassical Boltzmann equation. We find that there exists the anisotropic optical absorption induced via both the intra- and interband electronic transition channels in the perpendicular $xx$ and $yy$ directions. Furthermore, we examine the effects of the ellipticity ratio, the temperature, the carrier density, and the band-gap parameter on the optical conductivity of the 2D SD system placed in transverse and vertical directions, respectively. It is shown that the ellipticity ratio, temperature, carrier density, and band-gap parameter can play the important roles in tuning the strength, peak position, and shape of the optical conductivity spectrum. The results obtained from this study indicate that the 2D SD system can be a promising anisotropic and tunable optical and optoelectronic material for applications in innovative 2D optical and optoelectronic devices, which are active in the infrared and terahertz bandwidths.

Weyl–Wigner description of massless Dirac plasmas: ab initio quantum plasmonics for monolayer graphene

We derive, from first principles and using the Weyl–Wigner formalism, a fully quantum kinetic model describing the dynamics in phase space of Dirac electrons in single-layer graphene. In the limit ℏ → 0, we recover the well-known semiclassical Boltzmann equation, widely used in graphene plasmonics. The polarizability function is calculated and, as a benchmark, we retrieve the result based on the random-phase approximation. By keeping all orders in ℏ, we use the newly derived kinetic equation to construct a fluid model for macroscopic variables written in the pseudospin space. As we show, the novel ℏ-dependent terms can be written as corrections to the average current and pressure tensor. Upon linearization of the fluid equations, we obtain a quantum correction to the plasmon dispersion relation, of order ℏ 2, akin to the Bohm term of quantum plasmas. In addition, the average variables provide a way to examine the value of the effective hydrodynamic mass of the carriers. For the latter, we find a relation in which Drude’s mass is multiplied by the square of a velocity-dependent, Lorentz-like factor, with the speed of light replaced by the Fermi velocity, a feature stemming from the quasi-relativistic nature of the Dirac fermions.

Spin accumulation in metallic thin films induced by electronic impurity scattering

In order to explore the spin accumulation, evaluating the spin galvanic and spin Hall effect, we utilize the semi-classical Boltzmann equation based on input from the relativistic Korringa-Kohn-Rostoker Green's function method, within the density functional theory. We calculate the spin accumulation including multiple contributions, especially skew-scattering (scattering-in term) and compare this to three different approximations, which include the isotropic and anisotropic relaxation time approximation. For heavy metals, with strong intrinsic spin-orbit coupling, we find that almost all the effects are captured within the anisotropic relaxation time approximation. On the other hand, in light metals the contributions from the vertex corrections (scattering-in term) are comparable to the induced effect in anisotropic relaxation time approximation. We put a particular focus on the influence of the atomic character of the substitutional impurities on the spin accumulation as well as the dependence on the impurity position. As impurities will break space inversion symmetry of the thin film, this will give rise to both symmetric and antisymmetric contributions to the spin accumulation. In general, we find the impurities at the surface generate the largest efficiency of charge-to-spin conversion in case of the spin accumulation. Comparing our results to existing experimental findings for Pt we find a good agreement.

Spatiotemporal characteristics of spin transport in compensated metals with electron–hole exchange interaction

We develop a theory describing spatiotemporal behavior of spin transport in two-band metals by postulating a spin–exchange interaction between electrons and holes. Starting with the semiclassical Boltzmann equation, we derive a system of coupled diffusion equations and solve them analytically under steady-state conditions. The solutions reveal two types of electron–hole coupled-spin transport modes: a dissipative mode and a nondissipative mode with an infinite spin diffusion length. The two modes are the manifestations of two types of spin coupling channels. Besides the exchange interaction, we incorporate into our derivation the relaxation caused by the spin–orbit interaction to show how it affects the spin transport characteristics of the two modes.

Quantum kinetics of anomalous and nonlinear Hall effects in topological semimetals

We present a systematic microscopic derivation of the semiclassical Boltzmann equation for band structures with the finite Berry curvature based on Keldysh technique of nonequilibrium systems. In the analysis, an AC electrical driving field is kept up to quadratic order, and both cases of small and large frequencies corresponding to intra- and interband transitions are considered. In particular, this formulation is suitable for the study of nonlinear Hall effect and photogalvanic phenomena. The role of impurity scattering is carefully addressed. Specifically, in addition to previously studied side-jump and skew-scattering processes, quantum interference diffractive contributions are now explicitly incorporated within the developed framework. This theory is applied to multifold fermions in topological semimetals, for which the generic formula for the skew scattering rate from the Pancharatnam phase is obtained along with the corresponding anomalous Hall conductivity.

Quarkonium semiclassical transport in quark-gluon plasma: factorization and quantum correction

We study quarkonium transport in the quark-gluon plasma by using the potential nonrelativistic QCD (pNRQCD) effective field theory and the framework of open quantum systems. We argue that the coupling between quarkonium and the thermal bath is weak using separation of scales, so the initial density matrix of the total system factorizes and the time evolution of the subsystem is Markovian. We derive the semiclassical Boltzmann equation for quarkonium by applying a Wigner transform to the Lindblad equation and carrying out a semiclassical expansion. We resum relevant interactions to all orders in the coupling constant at leading power of the nonrelativistic and multipole expansions. The derivation is valid for both weakly coupled and strongly coupled quark-gluon plasmas. We find reaction rates in the transport equation factorize into a quarkonium dipole transition function and a chromoelectric gluon distribution function. For the differential reaction rate, the definition of the momentum dependent chromoelectric gluon distribution function involves staple-shaped Wilson lines. For the inclusive reaction rate, the Wilson lines collapse into a straight line along the real time axis and the distribution becomes momentum independent. The relation between the two Wilson lines is analogous to the relation between the Wilson lines appearing in the gluon parton distribution function (PDF) and the gluon transverse momentum dependent parton distribution function (TMDPDF). The centrality dependence of the quarkonium nuclear modification factor measured by experiments probes the momentum independent distribution while the transverse momentum dependence and measurements of the azimuthal angular anisotropy may be able to probe the momentum dependent one. We discuss one way to indirectly constrain the quarkonium in-medium real potential by using the factorization formula and lattice calculations. The leading quantum correction to the semiclassical transport equation of quarkonium is also worked out. The study can be easily generalized to quarkonium transport in cold nuclear matter, which is relevant for quarkonium production in eA collisions in the future Electron-Ion Collider.

Tensor monopoles and the negative magnetoresistance effect in optical lattices

We propose that a kind of four-dimensional (4D) Hamiltonians, which host tensor monopoles related to quantum metric tensor in even dimensions, can be simulated by ultracold atoms in the optical lattices. The topological properties and bulk-boundary correspondence of tensor monopoles are investigated in detail. By fixing the momentum along one of the dimensions, it can be reduced to an effective three-dimensional model manifesting with a nontrivial chiral insulator phase. Using the semiclassical Boltzmann equation, we calculate the longitudinal resistance against the magnetic field $B$ and find the negative relative magnetoresistance effect of approximately $ -B^{2} $ dependence when a hyperplane cuts through the tensor monopoles in the parameter space. We also propose an experimental scheme to realize this 4D Hamiltonian by introducing an external cyclical parameter in a 3D optical lattice. Moreover, we show that the quantum metric tensor and Berry curvature can be detected by applying an external drive in the optical lattices.

Band-Gap Sensitived Seebeck Effect in Heavy Group-IV Monolayers

We have systematically investigated the spin- and valley-dependent Seebeck effect in irradiated heavy group-IV monolayers including silicene, germanene, and stanene by means of semi-classical Boltzmann equation in diffusive regime. Due to the interplay of strong intrinsic spin-orbit coupling, perpendicular electric field, and off-resonant light field, the temperature-driven spin and valley conductivity can be controllable effectively. Except for numerical results, the amplitude of Seebeck coefficient is analyzed and found to be in direct proportion to the band gap at high temperature. It supplies a flexible method to modulate the Seebeck effect. In addition, we have compared the thermoelectric transport properties of different group-IV materials and found that the figure of merit shows great enhancement from silicene to stanene. These findings are excepted to provide a platform for the heavy group-IV materials in future spin–valley thermal and energy-saving devices.

Comparing Kinetic and MEP Model of Charge Transport in Graphene

Graphene has attracted the attention of several researchers because of its peculiar features. In particular, the study of charge transport in graphene is challenging for future electron devices. Usually, the physical description of electron flow in graphene given by the semiclassical Boltzmann equation is considered to be a good one. However, due to the computational complexity, its use in simulation tools is not practical and, as already done for traditional semiconductors such as Si or GaAs, simpler models are warranted. Here we will assess the validity of a class of hydrodynamical models based on the maximum entropy principle (MEP), by comparing, in the case of suspended monolayer graphene, the direct solution of the semiclassical Boltzmann equation for electrons, obtained by employing a discontinuous Galerkin approach, with the MEP distribution function. A reasonable agreement is observed.

Equilibrium Wigner Function for Fermions and Bosons in the Case of a General Energy Dispersion Relation.

The approach based on the Wigner function is considered as a viable model of quantum transport which allows, in analogy with the semiclassical Boltzmann equation, to restore a description in the phase-space. A crucial point is the determination of the Wigner function at the equilibrium which stems from the equilibrium density function. The latter is obtained by a constrained maximization of the entropy whose formulation in a quantum context is a controversial issue. The standard expression due to Von Neumann, although it looks a natural generalization of the classical Boltzmann one, presents two important drawbacks: it is conserved under unitary evolution time operators, and therefore cannot take into account irreversibility; it does not include neither the Bose nor the Fermi statistics. Recently a diagonal form of the quantum entropy, which incorporates also the correct statistics, has been proposed in Snoke et al. (2012) and Polkovnikov (2011). Here, by adopting such a form of entropy, with an approach based on the Bloch equation, the general condition that must be satisfied by the equilibrium Wigner function is obtained for general energy dispersion relations, both for fermions and bosons. Exact solutions are found in particular cases. They represent a modulation of the solution in the non degenerate situation.

Magnetotransport of Weyl semimetals with tilted Dirac cones

Weyl semimetals (WSM) exhibit chiral anomaly in their magnetotransport due to broken conservation laws. Here, we analyze the magnetotransport of WSM in the presence of the time-reversal symmetry-breaking tilt parameter. The analytical expression for the magnetoconductivity is derived in the small tilt limit using the semiclassical Boltzmann equation. We predict a planar Hall current which flows transverse to the electric field and in the plane containing magnetic and electric fields and scales linearly with the tilt parameter. A tilt-induced transverse conductivity is also present in the case where the electric and magnetic fields are parallel to each other, a scenario where the conventional Hall current completely vanishes.