Energy Bands In Solids Research Articles

Berry curvature dipole generation and helicity-to-spin conversion at symmetry-mismatched heterointerfaces.

The Berry curvature dipole (BCD) is a key parameter that describes the geometric nature of energy bands in solids. It defines the dipole-like distribution of Berry curvature in the band structure and plays a key role in emergent nonlinear phenomena. The theoretical rationale is that the BCD can be generated at certain symmetry-mismatched van der Waals heterointerfaces even though each material has no BCD in its band structure. However, experimental confirmation of such a BCD induced via breaking of the interfacial symmetry remains elusive. Here we demonstrate a universal strategy for BCD generation and observe BCD-induced gate-tunable spin-polarized photocurrent at WSe2/SiP interfaces. Although the rotational symmetry of each material prohibits the generation of spin photocurrent under normal incidence of light, we surprisingly observe a direction-selective spin photocurrent at the WSe2/SiP heterointerface with a twist angle of 0°, whose amplitude is electrically tunable with the BCD magnitude. Our results highlight a BCD-spin-valley correlation and provide a universal approach for engineering the geometric features of twisted heterointerfaces.

Movable but not removable band degeneracies in a symmorphic crystal

Crossings of energy bands in solids that are not pinned at symmetry points in the Brillouin zone and yet cannot be removed by perturbations are thought to be conditioned on the presence of a nonsymmorphic symmetry. In this Letter we show that such band crossings can actually appear also in a symmorphic crystal. A study of a class of tight-binding multiband one-dimensional lattice models of spinful electrons reveals that chiral, time-reversal and site-mirror symmetries are suffcient to produce such movable but not removable band degeneracies.

Michelson interferometry of high-order harmonic generation in solids

For the high-order harmonic generation in solids, we find a distinct and clean interference pattern in the high-energy part of the spectrum which can be interpreted as a Michelson interferometer of the Bloch electron. Our results are achieved by a numerical solution to the time-dependent Schrödinger equation of the quasi-electron in solids and can be explained by an analytical model based on the principle of the Michelson interferometry. The present study deepens our understanding of the HHG mechanism in crystalline materials and may find potential applications in imaging of the dispersion relation or topological structure of the energy bands in solids.

Similarities and differences between high-harmonic generation in atoms and solids

The basic features of high-harmonic generation in dilute atomic gases and solid-state systems are analyzed using a microscopic many-body theory. Whereas it is sufficient for atomic gases to include the ground state and the ionization continuum, the semiconductor approach has to deal with the relevant interband polarizations and intraband currents. For both systems, a closed set of Bloch-type equations is derived, which has to be solved numerically. As common features, the emission of odd-order harmonics, the development of a plateau, and a high-frequency cutoff are observed if the solid system is modeled as a two-band system. However, already the addition of a third coupled band effectively breaks the inversion symmetry and leads to the appearance of even harmonic orders with comparable strength. In all cases, frequency- and time-resolved studies reveal a fundamentally different timing in the emission dynamics of the individual frequency components in atomic and solid-state systems, which can be attributed to the presence of energy bands in solids.

Symmetric coordinates in solids: magnetic Bloch oscillations

There has recently been a revival of the Bloch theory of energy bands in solids. This revival was caused, on one hand, by the discovery of topological insulators and the discovery of graphene, and, on the other end, by a very efficient new technique that was developed for creating artificial solids. These are the cold atoms in optical lattices. Last year geometric phases were measured in energy bands of cold atoms in a one-dimensional optical lattice by using Bloch oscillations. These phases are related to the Wyckoff positions, or the symmetry centers in the Bravais lattice. In this lecture a theoretical frame is developed for magnetic Bloch oscillations, meaning oscillations in the presence of a magnetic field. The theory is based on the kq-representation and the symmetric coordinates in solids. It is shown that for a Bloch electron in a magnetic field the orbit quasi-center is a conserved quantity. This is similar to the conservation of the quasi-momentum for an electron in a periodic potential. When an electric field is turned on, the orbit quasi-center oscillates in a similar way to the Bloch oscillations in the absence of a magnetic field. But there is a difference because the magnetic Brillouin zone is different. It depends on the strength of the magnetic field. An analogy is drawn between Bloch oscillations and magnetic Bloch oscillations. By using the magnetic translations it is indicated that a magnetic Wannier-Stark ladder appears in the spectrum of a Bloch electron in crossed magnetic and electric fields. The geometric phases for magnetic Bloch oscillations should be magnetic field dependent.

Allowed wavevectors under the application of incommensurate periodic boundary conditions

While the energy bands of solids are often thought of as continuous functions of the wavevector, k, they are in fact discrete functions, due to the periodic boundary conditions applied over a finite number of primitive cells. The traditional approach enforces periodicity over a volume containing Ni primitive unit cells along the direction of the primitive lattice vector ai. While this method yields a simple formula for the allowed k, it can be problematic computer programs for lattices such as face-centred cubic (FCC) where the boundary faces of the primitive cell are not orthogonal. The fact that k is discrete is of critical importance for supercell calculations since they include only a finite number of unit cells, which determines the number of wavevectors, and have a given geometry, which determines their spacing. Rectangular supercells, with the faces orthogonal to the Cartesian axes, are computationally simplest but are not commensurate with the FCC unit cell, so that the traditional approach for determining the allowed k-values is no longer useful. Here, we present a simple method for finding the allowed k-values when periodic boundary conditions are applied over a rectangular supercell, answering the question in both its practical and pedagogical aspects.

Electronic dynamic Hubbard model: Exact diagonalization study

A model to describe electronic correlations in energy bands is considered. The model is a generalization of the conventional Hubbard model that allows for the fact that the wave function for two electrons occupying the same Wannier orbital is different from the product of single-electron wave functions. We diagonalize the Hamiltonian exactly on a four-site cluster and study its properties as a function of band filling. The quasiparticle weight is found to decrease and the quasiparticle effective mass to increase as the electronic band filling increases, and spectral weight in one- and two-particle spectral functions is transferred from low to high frequencies as the band filling increases. Quasiparticles at the Fermi energy are found to be more ``dressed'' when the Fermi level is in the upper half of the band (hole carriers) than when it is in the lower half of the band (electron carriers). The effective interaction between carriers is found to be strongly dependent on band filling becoming less repulsive as the band filling increases and attractive near the top of the band in certain parameter ranges. The effective interaction is most attractive when the single-hole carriers are most heavily dressed, and in the parameter regime where the effective interaction is attractive, hole carriers are found to ``undress,'' hence become more like electrons, when they pair. It is proposed that these are generic properties of electronic energy bands in solids that reflect a fundamental electron-hole asymmetry of condensed matter. The relation of these results to the understanding of superconductivity in solids is discussed.

CONSTRUCTION OF A DISPERSION RELATION FROM AN ARBITRARY DENSITY OF STATES

The dispersion relations of energy bands in solids are characterized by their density of states, but a given density of states may originate from various band structures. We show how a spherically symmetric dispersion can be constructed for any one-band density of states. This method is applied to one-, two- and three-dimensional systems. It also serves to establish that any one-band spectrum with finite bandwidth can be obtained from a properly scaled dispersion relation in the limit of infinite dimensions.

Simulation of Ultrafast Optical Transitions using Genetic Algorithm

The purpose of this paper is to investigate the possibility of application of the genetic algorithm to quantum control of electronic transitions between energy bands in solids. In particular, the hole transitions between valence bands induced by ultrashort (femtosecond duration) electric eld pulse will be considered. Examples are presented to illustrate the effciency of the algorithm in this case.

Energy Spectra in Dual-Periodic Multilayer Structures

Transmission spectra have been calculated for the dual-periodic multilayers, by means of the transfer matrix method using a model of a sequential square-well type variation of refractive index modulated by a sinusoidal function with a variable periodicity relative to that of the basic multilayer. The calculated energy diagrams are plotted against the phase shift parameter of the latter period. These diagrams exhibit a remarkable feature showing butterfly-like patterns, similar to those in the electron energy diagrams for solid crystals in uniform magnetic fields. It is suggested that the one-dimensional photonic bands in the dual-periodic multilayers are an optical analogue of the electronic energy bands of solids in the quantum magnetic regime.

FERMI SURFACE MAPPING BY PHOTOEMISSION

Angle-resolved ultraviolet photoelectron spectroscopy (ARUPS) data are usually measured spectrum by spectrum at various emission angles or photon energies in order to observe the dispersion of energy bands in solids and on their surfaces. In these lecture notes an alternative experimental procedure is described which yields a direct mapping of constant energy surfaces within the band structure, and specifically of the Fermi surface. This approach appears very promising, in particular when applied to magnetic systems and systems with narrow bands. Fermi surfaces of surface states are seen in direct relation to the underlying bulk Fermi surface.

Correlation energy calculations for infinite systems

AbstractA survey is given of ab initio calculations for solids and polymers, which start from an SCF calculation and treat correlations by using local operators. The theory is formulated by employing a projection method. As regards applications, special attention is paid to elemental semiconductors and to polymers like polyethylene and polyacetylene. With the help of reduced Hamiltonians, insight can be gained into various correlation effects. This is demonstrated by a discussion of the problem of dimerization in polyacetylene. Finally, a formalism is described that allows for treating the effects of correlations on energy bands of solids. It is shown that qualitative new aspects of correlations enter when a quasiparticle in, e.g., a semiconductor is considered. The theory is applied to a calculation of the energy bands of silicon.

Photon/polariton modes in a magnetic/nonmagnetic layered structure with magnetization antiparallel to each other (abstract)

We have calculated the photon/polariton spectrum of the surface and bulk waves in a magnetic/nonmagnetic layered structure where the magnetization in subsequent magnetic layers is antiparallel to each other. The magnetic layers are magnetized to saturation by a uniaxial anisotropy field parallel to the layer surfaces and the wave propagation direction is assumed to be perpendicular to the anisotropy axis (the Voigt configuration). Similar to electronic energy bands in solids the dispersion relations of the bulk modes form dispersion bands in a 1D superlattice of the layered structure. Surface modes form dispersion curves in between the bulk bands. The results are compared with those associated with the case where the magnetization of the layered structure is aligned parallel to each other by an external magnetic field.1,2 Polariton/photon modes are excited for both magnetization configurations. However, under ordinary conditions we find both forward and backward wave propagation dispersions of the surface waves only for the antiparallel magnetization case. A new type of surface propagating waves is also found whose appearance is unique to the Kittel, rather than the Voigt configuration.

Distortion of Energy Surfaces by a Strain Field

Deformation of energy bands in solids in the presence of a strain field is discussed from a symmetry point of view. The very particular case of the conduction band of silicon in an orthorhombic strain is examined in some detail. The distortion of the energy surface in this case gives rise to an additional contribution to piezoresistance and predicts the correct sign and magnitude of the piezoresistance coefficient 1144, which has been predicted to vanish by the conventional electron transfer theory.

Biorthogonal sets for energy bands in solids

A biorthogonal set is constructed for a basis of a band representation. A simple fact is used that any band representation can be induced from a one-dimensional representation of a point group (the latter is not necessarily an isotropy group of a Wyckoff position). It is shown that the biorthogonal set has the same symmetry structure as the original basis of the band representation. Symmetry relations are established for the matrix connecting the biorthogonal set with the functions of the original basis.

Berry's phase for energy bands in solids.

Berry's phase is defined for the dynamics of electrons in periodic solids and an explicit formula is derived for it. Because of the special torus topology of the Brillouin zone a nonzero Berry phase is shown to exist in a one-dimensional parameter space. Symmetry of the Bloch functions in the Brillouin zone leads to the quantization of Berry's phase. A connection is established between the latter and the Wyckoff positions in the crystal in the framework of band representations of space groups. Berry's phase can therefore be used for labeling energy bands in solids.

Symmetry and analyticity of energy bands in solids.

We show that pairs of simple energy bands exist that have identical symmetries for their Bloch functions, but different symmetries for the corresponding Wannier functions. For finite crystals such pairs of bands belong to equivalent band representations. In the case of infinite crystals, we use the analyticity of simple bands to prove the inequivalence of their band representations. This means that the energies of such pairs of bands cannot coincide despite the identical symmetries of their Bloch functions.

Effects of Deformation-Potential Coupling on Acoustic Second-Harmonic Generation in n-Type InSb

Acoustic second-harmonic generation has been studied in n-type InSb with a dc magnetic field \\vecB directed along the acoustic waves propagating. We are interested in both high- and low-frequency regions, the electron relaxation time due to the scattering in solids is taken into account. The dominant electron-phonon interaction in solids is via deformation-potential coupling. It is shown that the ratio of the acoustic intensity in the second harmonic to the square of the acoustic intensity in the fundamental decreases rapidly and then increases slowly with the phonon frequency. A comparison between the deformation-potential coupling and piezoelectric coupling has been made. It is also found that the second-harmonic generation can be influenced by changing the dc magnetic field due to the nonlinear nature of energy bands in solids.

Two classes of composite energy bands in solids.

Composite energy bands are subdivided into two classes. Class-I bands are shown to decompose into a set of simple bands. This is achieved by replacement of the Bloch functions (labeled by the energy) by basis functions that are labeled by the symmetry centers of the band. Because of their decomposition into simple bands, class-I bands have uniquely defined Wannier functions with an exponential falloff. Symmetry arguments are used to prove that Bloch functions of any composite band are necessarily discontinuous.

Energy bands in solids: bonding, energy levels and orbitals

Abstract Qualitative aspects of semiempirical energy band calculations are surveyed with the purpose of highlighting the key features of the electronic structures of some insulators, semiconductors and metallic systems. The nodal structure of the electronic orbitals is stressed as the common underlying theme in such effects as: the Peierls distortions of metallic systems; the changing electron counts (due to charge transfer or different d-electron counts); the high-pressure transformations of solids. The range of applicability of semiempirical non-self-consistent band theories is assessed. The discussion is illustrated by several examples.