# Wave and uncertainty properties of electrons in crystalline solids

## TL;DR

This paper explores the wave and uncertainty characteristics of electrons in crystalline solids, deriving relations between their particle and wave properties, and applying these to various materials including semiconductors and topological insulators.

## Contribution

It introduces a unified approach to relate electron quasimomentum and wavelength in nonparabolic, nonspherical energy bands, and derives an uncertainty relation specific to electrons in periodic potentials.

## Key findings

- Derived QM-wavelength relations for arbitrary energy band shapes.
- Calculated energy-dependent wavelengths for electrons in various materials.
- Established an uncertainty relation between quasimomentum and position in crystals.

## Abstract

Relations between particle and wave properties for charge carriers in periodic potentials of crystalline metals and semiconductors are derived. The particle aspects of electrons and holes in periodic potentials are considered using properties of quasimomentum (QM), while the wave aspects are described employing 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 quasimomentum 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 between the wave and uncertainty properties of electrons in crystalline solids and those in vacuum are traced.