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Two-dimensional electron gas under the effect of constrained potential and magnetic field in curved space

H. Pahlavani, M. Botshekananfard

TL;DR

This work investigates a two-dimensional electron gas confined to a curved cylindrical surface in a uniform magnetic field, focusing on how surface curvature alters the Landau-like spectrum. Using the constraining-potential (da Costa) framework, it derives an effective Hamiltonian that includes a curvature-dependent geometric potential U = -ħ^2/(8mR^2) and analyzes curvature effects via perturbation theory. It provides two complementary results: first, perturbative energy shifts E_n^1 that modify the Landau levels for finite curvature, and second, a high-field treatment showing a curvature-modulated dispersion E_n = (n+1/2) ħ ω_c sqrt(1 - x_0^2/R^2) - ħ^2/(8mR^2); in the limit R → ∞, the familiar flat-surface Landau levels are recovered. The findings quantify how cylindrical curvature influences magnetotransport and offer benchmarks for curved 2DEG nanostructures and related devices.

Abstract

The effect of the curvature of a cylindrical surface on the energy spectrum for a curved two-dimensional electron gas in a homogeneous magnetic field is considered. The corrections to the energy spectrum are obtained for the first time perturbatively, in contrast to previous works where it was obtained numerically. The dispersion relationship is obtained as a function of curvature radius and the results for curved surface have been compared with the flat surface.

Two-dimensional electron gas under the effect of constrained potential and magnetic field in curved space

TL;DR

This work investigates a two-dimensional electron gas confined to a curved cylindrical surface in a uniform magnetic field, focusing on how surface curvature alters the Landau-like spectrum. Using the constraining-potential (da Costa) framework, it derives an effective Hamiltonian that includes a curvature-dependent geometric potential U = -ħ^2/(8mR^2) and analyzes curvature effects via perturbation theory. It provides two complementary results: first, perturbative energy shifts E_n^1 that modify the Landau levels for finite curvature, and second, a high-field treatment showing a curvature-modulated dispersion E_n = (n+1/2) ħ ω_c sqrt(1 - x_0^2/R^2) - ħ^2/(8mR^2); in the limit R → ∞, the familiar flat-surface Landau levels are recovered. The findings quantify how cylindrical curvature influences magnetotransport and offer benchmarks for curved 2DEG nanostructures and related devices.

Abstract

The effect of the curvature of a cylindrical surface on the energy spectrum for a curved two-dimensional electron gas in a homogeneous magnetic field is considered. The corrections to the energy spectrum are obtained for the first time perturbatively, in contrast to previous works where it was obtained numerically. The dispersion relationship is obtained as a function of curvature radius and the results for curved surface have been compared with the flat surface.
Paper Structure (5 sections, 29 equations, 4 figures)

This paper contains 5 sections, 29 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Hamiltonian of a 2DEG in a magnetic field
  3. The energy spectrum
  4. Curved 2DEGS in high magnetic fields
  5. conclusion

Figures (4)

  • Figure 1: Curved electron gas coordinate system in a homogeneous magnetic field
  • Figure 2: Plot of the energies versus the radius $R$ with different Landau indices $n$, under the effect constrained potential in a curved sample of Eq.(\ref{['eq20']}); here, $e=1.602 \times 10^{-19} C, m=9.109 \times 10^{-31} kg, c=3 \times 10^{8} m/s,\hbar=1.055 \times 10^{-34} J.s, k=1.75 \times 10^{11} C.T/kg, B=1T.$ In the limit $R\rightarrow \infty$ the energies spectrum have asymptotic behavior.
  • Figure 3: A two-dimensional electron gas in a magnetic field
  • Figure 4: Plot of the energies versus the radius $R$ with different Landau indices $n$, in a curved sample of Eq.(\ref{['eq28']}); here, $e=1.602 \times 10^{-19} C, m=9.109 \times 10^{-31} kg, c=3 \times 10^{8} m/s,\hbar=1.055 \times 10^{-34} J.s, k=1.75 \times 10^{11} C.T/kg, B=1T.$ The energy levels have an asymptotic behavior as the radius $R$ increases. In the limit $R\rightarrow \infty$ they become the landau levels which in the planar case are dispersionless.