Ultra-complex conductivity diagrams in the nearly free electron approximation

Technical Summary: Ultra-Complex Conductivity Diagrams in the Nearly Free Electron Approximation

Problem Statement
This paper investigates the conditions under which "ultra-complex" conductivity diagrams emerge in normal metals with cubic symmetry when subjected to strong magnetic fields. In the limit of large mean free paths and strong magnetic fields ($\omega_B \tau \gg 1$), electron transport is governed by the geometry of electron trajectories on the Fermi surface ($S_F$). While simple Fermi surfaces yield simple conductivity behaviors (dominated by closed trajectories or simple periodic open trajectories), sufficiently complex Fermi surfaces (topological rank 3) can support "stable open trajectories" confined to specific angular domains (Stability Zones).

The central problem addressed is the existence and location of "Type B" angular diagrams. These diagrams occur when the Fermi level ($\epsilon_F$) falls within a specific, narrow energy interval $[\epsilon^B_1, \epsilon^B_2]$. In this regime, the angular diagram contains an infinite number of Stability Zones ($\Omega_\alpha$) separating regions of electron-type and hole-type Hall conductivity. The accumulation points of these zones correspond to "chaotic" (Dynnikov) trajectories, leading to highly complex anisotropic conductivity behavior. The paper seeks to estimate the position and width of this interval $[\epsilon^B_1, \epsilon^B_2]$ within the conduction band for metals described by the nearly free electron approximation.

Methodology
The authors employ a topological and geometric analysis of the electron quasi-momentum evolution system $\dot{p} = \frac{e}{c} [\nabla \epsilon(p) \times B]$ on the Fermi surface. The methodology relies on the following steps:

1. Topological Classification: The study utilizes the classification of trajectories on the Fermi surface, distinguishing between closed, periodic open, and stable open trajectories. Stable open trajectories exist within "Stability Zones" ($W_\alpha$) on the unit sphere of magnetic field directions ($S^2$).
2. Definition of the Critical Interval: The interval $[\epsilon^B_1, \epsilon^B_2]$ is defined as the energy range where the angular diagram contains regions of both electron and hole Hall conductivity separated by chains of Stability Zones. Mathematically, this interval is bounded by:
   $$\epsilon^B_1 = \min_{S^2} \tilde{\epsilon}_2(n), \quad \epsilon^B_2 = \max_{S^2} \tilde{\epsilon}_1(n)$$
   where $\tilde{\epsilon}_1(n)$ and $\tilde{\epsilon}_2(n)$ are continuous extensions of the energy boundaries for open trajectories.
3. Approximation Strategy: To estimate these bounds without solving the full complex geometry for all directions, the authors focus on the boundaries of the largest, symmetric Stability Zones ($W_\alpha$). Specifically, they analyze the "symmetric" points $P$ and $Q$ on the boundary of a primary zone $W_1$.
4. Geometric Construction: The boundaries of the Stability Zones correspond to the simultaneous disappearance of two cylinders of closed trajectories (one electron-type, one hole-type) that separate the carriers of open trajectories. The authors calculate the energy $\tilde{\epsilon}_0(n)$ at which these cylinders vanish for specific symmetric directions.
5. Model Application: The method is applied to three specific lattice types under the nearly free electron approximation (where $\epsilon(p) \approx p^2$ near band minima):
   • Simple Cubic (SC)
   • Face-Centered Cubic (FCC)
   • Body-Centered Cubic (BCC)

Key Contributions and Results
The paper provides quantitative estimates for the position and width of the ultra-complex conductivity interval $[\epsilon^B_1, \epsilon^B_2]$ for the first conduction band in cubic crystals.

• Simple Cubic Lattice:

  • The first conduction band spans $\epsilon \in [0, 3]$.
  • The ultra-complex interval is estimated as $[\epsilon^B_1, \epsilon^B_2] \simeq [1.2, 1.22]$.
  • This corresponds to an energy $\approx 0.4 \epsilon_{max}$ from the band bottom.
  • The width of the interval is approximately 0.7% of the total conduction band width.

• Face-Centered Cubic Lattice:

  • The first conduction band spans $\epsilon \in [0, 13/9]$.
  • The ultra-complex interval is estimated as $[\epsilon^B_1, \epsilon^B_2] \simeq [0.817, 0.820]$.
  • This lies at a distance of $\approx 0.57 \epsilon_{max}$ from the band bottom.
  • The width is approximately 0.2% of the band width.

• Body-Centered Cubic Lattice:

  • The first conduction band spans $\epsilon \in [0, 1]$.
  • The ultra-complex interval is estimated as $[\epsilon^B_1, \epsilon^B_2] \simeq [0.5153, 0.5161]$.
  • This lies at a distance of $\approx 0.515 \epsilon_{max}$ from the band bottom.
  • The width is approximately 0.1% of the band width.

In all three cases, the interval is found to be extremely narrow. The calculations involve determining the radius $r$ of the intersection circles of the Fermi surface spheres at the specific symmetric points where the topological structure of the trajectories changes (i.e., where the separating cylinders of closed trajectories vanish).

Significance and Claims
The paper claims that the emergence of ultra-complex conductivity diagrams is a rare phenomenon in the nearly free electron approximation for cubic metals. The primary finding is that the required Fermi level must fall into very narrow energy intervals (less than 1% of the band width) to observe these diagrams.

The authors attribute this narrowness to two factors:

1. The high symmetry of the cubic crystal lattices.
2. The simplest analytical form of the dispersion relations $\epsilon(p)$ used in the nearly free electron approximation.

The paper contrasts these results with the tight-binding approximation, noting that for the simple cubic lattice, the tight-binding model predicts a zero-width interval in the leading order, whereas the nearly free electron model yields a non-zero (though still very small) width.

The authors conclude that detecting these diagrams in real experiments is difficult due to the low probability of the Fermi level naturally falling within these narrow intervals. They suggest that observing such phenomena may require external influences to shift the Fermi level within the conduction band. The results serve as a theoretical estimate for the search for ultra-complex conductivity diagrams and highlight the sensitivity of such complex topological features to the specific details of the dispersion relation and crystal symmetry.