Arbitrary Band Structure Research Articles

Electronic transport computation in thermoelectric materials: from ab initio scattering rates to nanostructures

Over the last two decades a plethora of new thermoelectric materials, their alloys, and their nanostructures were synfthesized. The ZT figure of merit, which quantifies the thermoelectric efficiency of these materials increased from values of unity to values consistently beyond two across material families. At the same time, the ability to identify and optimize such materials, has stressed the need for advanced numerical tools for computing electronic transport in materials with arbitrary bandstructure complexity, multiple scattering mechanisms, and a large degree of nanostructuring. Many computational methods have been developed, the majority of which utilize the Boltzmann transport equation (BTE) formalism, spanning from fully ab initio to empirical treatment, with varying degree of computational expense and accuracy. In this paper we describe a suitable computational process that we have recently developed specifically for thermoelectric materials. The method consists of three independent software packages that we have developed and: (1) begins from ab initio calculation of the electron–phonon scattering rates, (2) to then be used within a Boltzmann transport simulator, and (3) calculated quantities from the BTE are then passed on to a Monte Carlo simulator to examine electronic transport in highly nanostructured material configurations. The method we describe is computationally significantly advantageous compared to current fully ab initio and existing Monte Carlo methods, but with a similar degree of accuracy, thus making it truly enabling in understanding and assessing thermoelectric transport in complex band, nanostructured materials.

Quantum kinetic theory of nonlinear optical currents: Finite Fermi surface and Fermi sea contributions

The quantum kinetic framework provides a versatile method for investigating the dynamical optical and transport currents of crystalline solids. In this paper, starting from the density-matrix equations of motion, we present a general theoretical path to obtain the nonlinear optical response in an elegant and transparent manner. We devise an extensive kinetic theory that can be applied to materials with arbitrary band structures and captures intraband and interband coherence effects, finite Fermi surfaces, and disorder effects. We present a classification of the nonlinear optical currents arising from the interference of the interband and intraband components of the density matrix with distinct symmetry and quantum geometrical origin for each contribution. In this context, we report the following four primary findings: (i) The Fermi golden rule approach is insufficient to derive the correct expression for the injection current, a shortcoming that we remedy in our theory while associating the injection current with the intraband-interband contribution to the second-order density matrix. (ii) The interband-intraband contribution yields a resonant current that survives irrespective of any symmetry constraint in addition to the well-known anomalous nonlinear current (nonresonant), which requires time-reversal symmetry. (iii) Quite generally, the nonlinear current is significantly enhanced by contributions arising from the finite Fermi surface. (iv) The finite Fermi surface and Fermi sea additionally lead to sizable novel nonlinear effects via contributions we term double resonant and higher-order pole. We investigate such effects in sum frequency and difference frequency generation. As an illustration, we compute the nonlinear response of the topological antiferromagnet CuMnAs and thin film tilted Weyl semimetals as model systems dominated by interband coherence contributions. We find that the nonlinear response of CuMnAs is responsive to the direction of the finite magnetization field and the response of Weyl semimetal to the tilt. In addition, the choice of the polarization angle of the beam is crucial to have a nonlinear current in CuMnAs, while it is not the case for Weyl semimetals.

Geometric approach to fragile topology beyond symmetry indicators

We present a framework to systematically address topological phases when finer partitionings of bands are taken into account, rather than only considering the two subspaces spanned by valence and conduction bands. Focusing on $C_2\mathcal{T}$-symmetric systems that have gained recent attention, for example in the context of layered van-der-Waals graphene heterostructures, we relate these insights to homotopy groups of Grassmannians and flag varieties, which in turn correspond to cohomology classes and Wilson-flow approaches. We furthermore make use of a geometric construction, the so-called Pl\"ucker embedding, to induce windings in the band structure necessary to facilitate non-trivial topology. Specifically, this directly relates to the parametrization of the Grassmannian, which describes partitioning of an arbitrary band structure and is embedded in a better manageable exterior product space. From a physical perspective, our construction encapsulates and elucidates the concepts of fragile topological phases beyond symmetry indicators as well as non-Abelian reciprocal braiding of band nodes that arises when the multiple gaps are taken into account. The adopted geometric viewpoint most importantly culminates in a direct and easily implementable method to construct model Hamiltonians to study such phases, constituting a versatile theoretical tool.

Discrete Kernel Preserving Model for 3D Electron–Optical Phonon Scattering Under Arbitrary Band Structures

In Li et al. (J Sci Comput 62:317–335, 2015), we thoroughly investigated the structure of the kernel space of the discrete one-dimensional (1D) non-polar optical phonon (NPOP)-electron scattering matrix, and proposed a strategy to setup grid points so that the uniqueness of the discrete scattering kernel is preserved. In this paper, we extend the above work to the three dimensional (3D) case, and also investigate the polar optical phonon (POP)-electron case. In numerical discretization, it is important to get a discretization scattering matrix that keeps as many properties of the continuous scattering operator as possible. We prove that for the 3D NPOP-electron scattering, (i) the dimension of the kernel space of the discrete scattering matrix is one and (ii) the equilibrium distribution is constant with respect to the angular coordinates as long as the mesh over the energy interval obeys the rule proposed in Li et al. (2015). For the POP-electron scattering, (i) is also kept under the same condition, but to keep (ii) becomes a challenging task, since generally a simple uniform mesh in the azimuth and polar angular coordinates will not preserve (ii). Based on high degree of symmetry of the Platonic solids and regular pyramids, we propose two conditions and prove they are sufficient to guarantee (ii). Numerical experiments strongly support our theoretical findings.

Two-Particle Collisional Coordinate Shifts and Hydrodynamic Anomalous Hall Effect in Systems without Lorentz Invariance.

We show that electrons undergoing a two-particle collision in a crystal experience a coordinate shift that depends on their single-particle Bloch wave functions and derive a gauge-invariant expression for such a shift, valid for arbitrary band structures and arbitrary two-particle interaction potentials. As an application of the theory, we consider two-particle coordinate shifts for Weyl fermions in space of three spatial dimensions. We demonstrate that such shifts in general contribute to the anomalous Hall conductivity of a clean electron liquid.

Quantitative mappings between symmetry and topology in solids

The study of spatial symmetries was accomplished during the last century and had greatly improved our understanding of the properties of solids. Nowadays, the symmetry data of any crystal can be readily extracted from standard first-principles calculation. On the other hand, the topological data (topological invariants), the defining quantities of nontrivial topological states, are in general considerably difficult to obtain, and this difficulty has critically slowed down the search for topological materials. Here we provide explicit and exhaustive mappings from symmetry data to topological data for arbitrary gapped band structure in the presence of time-reversal symmetry and any one of the 230 space groups. The mappings are completed using the theoretical tools of layer construction and symmetry-based indicators. With these results, finding topological invariants in any given gapped band structure reduces to a simple search in the mapping tables provided.

Microscopic bosonization of band structures: x-ray processes beyond the Fermi edge

Bosonization provides a powerful analytical framework to deal with one-dimensional strongly interacting fermion systems, which makes it a cornerstone in quantum many-body theory. However, this success comes at the expense of using effective infrared parameters, and restricting the description to low energy states near the Fermi level. We propose a radical extension of the bosonization technique that overcomes both limitations, allowing computations with microscopic lattice Hamiltonians, from the Fermi level down to the bottom of the band. The formalism rests on the simple idea of representating the fermion kinetic term in the energy domain, after which it can be expressed in terms of free bosonic degrees of freedom. As a result, one- and two-body fermionic scattering processes generate anharmonic boson−boson interactions, even in the forward channel. We show that up to moderate interaction strengths, these non-linearities can be treated analytically at all energy scales, using the x-ray emission problem as a showcase. In the strong interaction regime, we employ a systematic variational solution of the bosonic theory, and obtain results that agree quantitatively with an exact diagonalization of the original one-particle fermionic model. This provides a proof of the fully microscopic character of bosonization, on all energy scales, for an arbitrary band structure. Besides recovering the known x-ray edge singularity at the emission threshold, we find strong signatures of correlations even at emission frequencies beyond the band bottom.

An arbitrary band structure construction of totally nonnegative matrices with prescribed eigenvalues

The construction of totally nonnegative (TN) matrices with prescribed eigenvalues is an important topic in real-valued nonnegative inverse eigenvalue problems. TN matrices are square matrices whose minors are all nonnegative. Our previous paper (Numer. Algor. 70, 469–484, ??2015) presented a finite-step construction of TN matrices limited to upper or lower Hessenberg forms with prescribed eigenvalues, based on the discrete hungry Toda (dhToda) equation which is derived from the study of integrable systems. Building on our previous paper, we produce the construction of banded TN matrices with an arbitrary number of diagonals in both lower and upper triangular parts and prescribed eigenvalues, involving upper Hessenberg, lower Hessenberg, and dense TN matrices with prescribed eigenvalues. We first prepare an infinite sequence associated with distinct eigenvalues \(\lambda _{1},\lambda _{2},\dots ,\lambda _{m}\) and two integers M and N which determine the upper and lower bandwidths of m-by-m banded matrices, respectively. Both M and N play a key role for achieving our purpose. The study follows similar lines to our previous paper, but is complicated by the introduction of N. We next consider extended Hankel determinants and extended Hadamard polynomials involving elements of the infinite sequence and then derive their relationships. These relationships help us understand banded TN matrices with eigenvalues \(\lambda _{1},\lambda _{2},\dots ,\lambda _{m}\) from the viewpoint of an extension of the dhToda equation. Finally, we propose a finite-step procedure for constructing TN matrices with an arbitrary upper and lower bandwidths and prescribed eigenvalues and also give illustrative examples.

Einstein's 1D Photo Emission and the Von Klitzing Constant.

An attempt is made to present a very easy integration which generates Von Klitzing constant in one hand and Einstein's 1D photo current from quantum wires having arbitrary band structures under the practical conditions of extreme degeneracy and quantum limits on the other hand, the two radically different concepts, depending on the selection of the values of the upper and lower limits of the integral.

Breakdown of the universality of the Kadowaki-Woods Ratio in multi-band metals

We calculate the Kadowaki-Woods ratio (KWR) in Fermi liquids with arbitrary band structures. We find that, contrary to the single band case, the ratio is not generally independent of the effects of electronic correlations (universal). This is very surprising given the experimental findings of a near universal KWR in many multiband metals. We identify a limit where the universality of the ratio, which has been observed experimentally in many strongly correlated electron systems, is recovered. We discuss the KWR in Dirac semimetals in two and three dimensions. In the two-dimensional case we also generalize the KWR to account for the logarithmic factor in the self-energy. In both cases we find that the KWR is independent of correlations, but strongly dependent on the doping of the system: for massless fermions the KWR is proportional to the inverse square of the carrier density, whereas the KWR for systems with massive quasiparticles is proportional to the inverse of the carrier density.

Equivalent-medium theory for metamaterials made by planar electronic materials

Mapping a planar conductor with arbitrary electronic band structure (e.g., linear or parabolic) to a slab of bulk medium exhibiting the same optical responses, we formulated an equivalent-medium approach to study the optical properties of metamaterials (MTMs) made by patterning such a planar conductor, particularly for the graphene system. The theory was applied to study various graphene-based MTMs, with calculation results in one particular system (with no adjustable parameters) agreeing well with available experiments.

Yet Another Hydrodynamic Model with Correlated Parameters for Silicon Devices

In this paper we present a complete hydrodynamic model (HDM) describing the transport of charge carriers in semiconductor devices with arbitrary band structure. The model is appended with advanced physical models fo r almost all the physical parameters of interest in modern Si Devices. These parameters are inter-related v ia the carrier energy relaxation time. An improved analytical model of the carrier heat flu x, on the basis of a fourth mo ment of the Bolt zmann transport equation (BTE), is also presented. In order to resolve the heat evacuation problems in SOI and power devices, the model is coupled with a lattice heat conservation equation. The transport of hot carriers is simu lated, according to the proposed HDM and the results are compared with the conventional data obtained using the drift-diffusion model (DDM) and MC simu lation. We show from the HD simu lation, the effect of the energy relaxat ion time value on the hot carrier transport in general and the breakdown voltage in particu lar, of both MOSFETs and bipolar devices.

Extended Quantum Correction Model Applied to Six-Band ${\bf k}\cdot {\bf p}$ Valence Bands Near Silicon/Oxide Interfaces

The modified local density approximation (MLDA) was developed to describe quantum corrections due to size quantization near a semiconductor-oxide interface for parabolic ellipsoidal bands. In this paper, we propose an extended MLDA model in order to consider arbitrary band structures. The results are compared with the original MLDA model and to self-consistent Poisson-Schrodinger solutions using both constant effective mass and confined six-band Hamiltonians. The extended MLDA model is verified for three major surface orientations and high stress conditions.

Geometric phases and competing orders in two dimensions

We discuss the problem of characterizing ``quantum-disordered'' ground states, obtained upon loss of antiferromagnetic order on general lattices in two spatial dimensions, with arbitrary electronic band structure. A key result is the response in electron bilinears to the skyrmion density in the local antiferromagnetic order, induced by geometric phases. We also discuss the connection to topological terms obtained under situations where the electronic spectrum has a Dirac form.

Fluctuating spin density waves in metals

Recent work has used a U(1) gauge theory to describe the physics of Fermi pockets in the presence of fluctuating spin density wave order. We generalize this theory to an arbitrary band structure and ordering wave vector. The transition to the large Fermi-surface state, without pockets induced by local spin density wave order, is described by embedding the U(1) gauge theory in a SU(2) gauge theory. The phase diagram of the SU(2) gauge theory shows that the onset of spin density wave order in the Fermi liquid occurs either directly, in the framework discussed by Hertz, or via intermediate non-Fermi-liquid phases with Fermi surfaces of fractionalized excitations. We discuss application of our results to the phase diagram of the cuprates.

Influence of quantum confinement on the carrier contribution to the elastic constants in quantum confined heavily doped non-linear optical and optoelectronic materials: simplified theory and the suggestion for experimental determination

In this paper we study the carrier contribution to elastic constants in quantum confined heavily doped non-linear optical compounds on the basis of a newly formulated electron dispersion law taking into account the anisotropies of the effective electron masses and spin orbit splitting constants together with the proper inclusion of the crystal field splitting in the Hamiltonian within the framework of k.p formalism. All the results of heavily doped three, and two models of Kane for heavily doped III-V materials form special cases of our generalized analysis. It has been found, taking different heavily doped quantum confined materials that, the carrier contribution to the elastic constants increases with increase in electron statistics and decrease in film thickness in ladder like manners for all types of quantum confinements with different numerical values which are totally dependent on the energy band constants. The said contribution is greatest in quantum dots and least in quantum wells together with the fact the heavy doping enhances the said contributions for all types of quantum confined materials. We have suggested an experimental method of determining the carrier contribution to the elastic constants in nanostructured materials having arbitrary band structures.

Analysis of nano-scale MOSFET including uniaxial and biaxial strain

In this paper, we focus on uniaxial and biaxial strain technologies, and we also investigate an optimum combination of strain method and channel direction. We linked the first principles band calculation program to the FUJITSU ensemble full band Monte Carlo simulator FALCON directly, which enables to incorporate arbitrary Si band structures such as uniaxial and biaxial strained-Si into device characteristics analysis. We show that the combination of biaxial tensile strain and current for NMOS, and compressive uniaxial strain and channel for PMOS are optimum methods for current enhancement. However, considering technological difficulties and process cost, it is one of the candidate methods to use the combination of uniaxial tensile strain and channel direction for NMOS and that of uniaxial compressive strain and channel direction for PMOS.

Quantum transport for a Bloch electron quasiparticle in an inhomogeneous electric field scattering from a random distribution of impurities: A Wigner approach

The quantum transport equation is derived in terms of the Wigner distribution function for a Bloch electron quasiparticle, that is, a Bloch electron in a single band, interacting with a random, inhomogeneous distribution of impurities, and subject to general homogeneous and inhomogeneous electric fields. The time dependent homogeneous electric field is described through the vector potential gauge. The derivation of the transport equation makes use of a unitary transformation of the Liouville equation based on the interaction picture to a form in which the scattering interaction appears quadratically, and utilizes accelerated Bloch states as basis states; the resulting generalized drift and diffusion terms are obtained exactly for an arbitrary band structure. In taking the limit of slowly varying inhomogeneous electric field and slowly varying scatterer density distribution, a quantum generalization of the Boltzmann-like Wigner transport equation is obtained which includes impurity-related intracollisional field effects in the collision term and a drift term comprising the total force due to both the homogeneous and inhomogeneous fields.

NMR relaxation time in a clean two-band superconductor

We study the spin-lattice relaxation rate of nuclear magnetic resonance in a two-band superconductor. Both conventional and unconventional pairing symmetries for an arbitrary band structure in the clean limit are considered. The importance of the inter-band interference effects is emphasized. The calculations in the conventional case with two isotropic gaps are performed using a two-band generalization of the Eliashberg theory.

NMR relaxation rate in noncentrosymmetric superconductors

The spin-lattice relaxation rate of nuclear magnetic resonance in a clean superconductor without an inversion center is calculated for an arbitrary pairing symmetry and band structure, in the presence of strong spin-orbit coupling.