Leitfähigkeitstensors

Leitfähigkeitstensors, also known as conductivity tensors, are mathematical objects used in physics and materials science to describe how readily a material conducts electric current. In many materials, electrical conductivity is not uniform in all directions. This anisotropy means that the current density might not be parallel to the applied electric field. The conductivity tensor accounts for this directional dependence.

A conductivity tensor is typically a second-rank tensor, represented as a 3x3 matrix in three-dimensional space. For a material with conductivity tensor $\sigma$, the relationship between current density $\mathbf{j}$ and electric field $\mathbf{E}$ is given by $\mathbf{j} = \sigma \mathbf{E}$. In component form, this is $j_i = \sum_{j} \sigma_{ij} E_j$. The diagonal elements $\sigma_{ii}$ represent conductivity along the principal axes, while the off-diagonal elements $\sigma_{ij}$ (where $i \neq j$) describe the coupling between components of the electric field and current density in different directions.

In isotropic materials, where conductivity is the same in all directions, the conductivity tensor is simply a scalar multiple of the identity matrix, meaning $\sigma_{ij} = \sigma \delta_{ij}$, where $\sigma$ is a scalar conductivity and $\delta_{ij}$ is the Kronecker delta. Many crystalline materials, such as certain semiconductors and anisotropic conductors, exhibit non-diagonal conductivity tensors. The properties of the conductivity tensor are closely related to the crystal structure and the nature of charge carriers within the material. Analyzing the conductivity tensor can reveal important information about the material's electronic band structure and transport phenomena.