Symmetry constraints on the elastoresistivity tensor

The elastoresistivity tensor $m_{ij,kl}$ characterizes changes in a material's resistivity due to strain. As a fourth-rank tensor, elastoresistivity can be a uniquely useful probe of the symmetries and character of the electronic state of a solid. We present a symmetry analysis of $m_{ij,kl}$ (both in the presence and absence of a magnetic field) based on the crystalline point group, focusing for pedagogic purposes on the $D_{4h}$ point group (of relevance to several materials of current interest). We also discuss the relation between $m_{ij,kl}$ and various thermodynamic susceptibilities, particularly where they are sensitive to critical fluctuations proximate to a critical point at which a point-group symmetry is spontaneously broken.

Figure 1

Schematic illustration of an arbitrary shearless strain in $D_{4h}$ as decomposed in terms of three irreducible components: $\left(\begin{array}{ccc}{\epsilon }_{xx}& 0& 0\\ 0& {\epsilon }_{yy}& 0\\ 0& 0& {\epsilon }_{zz}\end{array}\right)={\epsilon }_{{}_{A_{1g,1}}}\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)+{\epsilon }_{{}_{A_{1g,2}}}\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right)+{\epsilon }_{{}_{B_{1g}}}\left(\begin{array}{ccc}1& \hfill 0& 0\\ 0& \hfill -1& 0\\ 0& \hfill 0& 0\end{array}\right)$, where ${\epsilon }_{{}_{A_{1g,1}}}=\frac{1}{2}({\epsilon }_{xx}+{\epsilon }_{yy}),{\epsilon }_{{}_{A_{1g,2}}}={\epsilon }_{zz}$, and ${\epsilon }_{{}_{B_{1g}}}=\frac{1}{2}({\epsilon }_{xx}-{\epsilon }_{yy})$. The gray parallelepiped represents a crystalline sample; the yellow regions represent electrical contacts used for transport measurements.