On chaotic regimes of conductivity behavior in the tight-binding approximation

TL;DR

The study addresses the probability of detecting chaotic conductivity regimes in metals with tight-binding dispersion under strong magnetic fields, focusing on the regime $\omega_B \tau \gg 1$ with $\omega_B = \frac{eB}{m^{*}c}$. It analyzes semiclassical electron dynamics on the Fermi surface via angular diagrams on $\mathbb{S}^2$, distinguishing stable open trajectories within Stability Zones and chaotic trajectories of Tsarev/Dynnikov types, and estimates the likelihood of observing chaos for cubic-lattice dispersions. The main result is that for simple cubic and body-centered cubic lattices the leading-order width of the chaotic interval is zero, while for face-centered cubic lattices the width is nonzero but very small, requiring external tuning of $\epsilon_F$ to observe chaos. This work guides experimental searches for chaotic conductivity by linking Fermi-surface topology and tight-binding models to measurable transport anisotropies in high-field regimes.

Abstract

We investigate the probability of detecting the most nontrivial conductivity behavior regimes in metals whose electron spectrum is described by the tight-binding approximation. These regimes are associated with the emergence of highly complex electron trajectories on the Fermi surface and correspond to a nontrivial (scaling) behavior of the conductivity tensor in strong magnetic fields. The geometry of such trajectories, as well as the corresponding conductivity regimes, have been well studied theoretically; however, they have not yet been observed experimentally. The results of our study allow us, in particular, to estimate the probability of their occurrence and to indicate the conditions for their possible detection for a wide class of conductors.

Figures (17)

• Figure 1: Trajectories of system (\ref{['MFSyst']}) in the $\, {\bf p}$ - space.
• Figure 2: Closed (a) and open periodic trajectories (b) of the system (\ref{['MFSyst']}) on the Fermi surface.
• Figure 3: The form of a stable open trajectory of system (\ref{['MFSyst']}) in a plane orthogonal to $\, {\bf B} \,$ (schematically).
• Figure 4: The angular diagram of a fixed dispersion relation $\, \epsilon ({\bf p}) \,$ (schematically).
• Figure 5: The form of Dynnikov's chaotic trajectory (schematically).