Blochfuncties

Bloch functions are a fundamental concept in solid-state physics, describing the wave functions of electrons in a periodic potential, such as that found in a crystalline solid. These functions are named after Felix Bloch, who introduced them in 1929.

The defining characteristic of a Bloch function is its periodicity. Specifically, a Bloch function $\psi_{\mathbf{k}}(\mathbf{r})$ can be written as the product of a plane wave $e^{i\mathbf{k} \cdot \mathbf{r}}$ and a periodic function $u_{\mathbf{k}}(\mathbf{r})$ that has the same periodicity as the crystal lattice. Mathematically, this is expressed as $\psi_{\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} u_{\mathbf{k}}(\mathbf{r})$, where $\mathbf{k}$ is the wave vector, and $u_{\mathbf{k}}(\mathbf{r} + \mathbf{R}) = u_{\mathbf{k}}(\mathbf{r})$ for any lattice vector $\mathbf{R}$.

The wave vector $\mathbf{k}$ plays a crucial role in characterizing the electronic states. It is often considered within the first Brillouin zone, a fundamental region in reciprocal space. The energy of an electron is a function of $\mathbf{k}$, leading to the formation of energy bands, which are characteristic features of the electronic structure of solids. The properties of Bloch functions, such as their localization and momentum, are directly linked to the material's electrical and optical properties.