We propose an efficient basis expansion for electron orbitals to describe real-time electron dynamics in crystalline solids. Although a conventional grid representation in the three-dimensional Cartesian coordinates works robustly, it requires a large amount of computational resources. To reduce computational costs, we consider basis expansion methods employing eigenstates of the ground state Hamiltonian with a truncation. We have found that adding occupied eigenstates of nearby $k$ points to the truncated basis functions composed of eigenstates of the original $k$ point is crucially important. We demonstrate the usefulness of the method for linear and nonlinear electron dynamics calculations in crystalline ${\mathrm{SiO}}_{2}$ .We propose an efficient basis expansion for electron orbitals to describe real-time electron dynamics in crystalline solids. Although a conventional grid representation in the three-dimensional Cartesian coordinates works robustly, it requires a large amount of computational resources. To reduce computational costs, we consider basis expansion methods employing eigenstates of the ground state Hamiltonian with a truncation. We have found that adding occupied eigenstates of nearby $k$ points to the truncated basis functions composed of eigenstates of the original $k$ point is crucially important. We demonstrate the usefulness of the method for linear and nonlinear electron dynamics calculations in crystalline ${\mathrm{SiO}}_{2}$.

arXiv:1404.5118Physical Review BJun 30, 201416 citations

Electron Interactions in Solids

Assa Auerbach

Graduate texts in contemporary physicsJan 1, 19941 citation

Wave and Uncertainty Properties of Electrons in Crystalline Solids

Wlodek Zawadzki

Herein, relations between particle and wave properties for charge carriers in periodic potentials of crystalline metals and semiconductors are derived. The particle aspects of electrons in periodic potentials are considered using properties of quasimomentum (QM), while the wave aspects are described using wave packets of Bloch waves. The two aspects are combined in the derivation of QM–wavelength relations for energy bands of arbitrary nonparabolicity and nonsphericity. An effective mass relating electron QM to its average velocity for spherical energy bands is defined and used to calculate energy dependences of wavelengths for electrons in narrow‐gap semiconductors, graphene, and surface states of topological insulators. An uncertainty relation between electron QM and its spatial coordinate in periodic potentials is derived. It is emphasized that the described properties apply to the average (not instantaneous) electron behavior. Analogies and differences between the wave and uncertainty properties of electrons in crystalline solids and those in vacuum are traced. An interference experiment with semiconductor nanostructures is proposed to verify the theoretical predictions and demonstrate to what extent the wave properties of electrons in solids are distinct from those in vacuum.Herein, relations between particle and wave properties for charge carriers in periodic potentials of crystalline metals and semiconductors are derived. The particle aspects of electrons in periodic potentials are considered using properties of quasimomentum (QM), while the wave aspects are described using wave packets of Bloch waves. The two aspects are combined in the derivation of QM–wavelength relations for energy bands of arbitrary nonparabolicity and nonsphericity. An effective mass relating electron QM to its average velocity for spherical energy bands is defined and used to calculate energy dependences of wavelengths for electrons in narrow‐gap semiconductors, graphene, and surface states of topological insulators. An uncertainty relation between electron QM and its spatial coordinate in periodic potentials is derived. It is emphasized that the described properties apply to the average (not instantaneous) electron behavior. Analogies and differences between the wave and uncertainty properties of electrons in crystalline solids and those in vacuum are traced. An interference experiment with semiconductor nanostructures is proposed to verify the theoretical predictions and demonstrate to what extent the wave properties of electrons in solids are distinct from those in vacuum.

arXiv:1903.07124physica status solidi (b)Feb 26, 20201 citation

k

points to the truncated basis functions composed of eigenstates of the original

k

point is crucially important. We demonstrate the usefulness of the method for linear and nonlinear electron dynamics calculations in crystalline

{\mathrm{SiO}}_{2}