Diffusive and hydrodynamic magnetotransport around a density perturbation in a two-dimensional electron gas

TL;DR

This work addresses magnetotransport around a circular density depletion in a 2D electron gas under strong magnetic fields, focusing on a gradual density tail with exponent $\beta>2$. It develops diffusive and hydrodynamic continuum models to compute the electrochemical potential and current patterns, revealing a no-go region whose size grows with the magnetic field and a Landauer resistivity dipole whose orientation is rotated relative to the external field. In diffusion, the no-go radius scales as $R_{\mathrm{no}} \sim \alpha^{-1/\beta} a$ with $\alpha = 1/(\omega_c \tau_{mr})$, and $R_L$ scales as $R_L^2/a^2 = \frac{\Gamma\left(1-\frac{2}{\beta}\right)}{\Gamma\left(1+\frac{2}{\beta}\right)} (\frac{1}{\alpha \beta})^{\frac{2}{\beta}}$, while in the viscous regime the no-go length scales as $R_{\mathrm{no}} = \left(\beta \frac{a^2 \omega_c}{\nu}\right)^{\frac{1}{\beta-2}} a$ with Gurzhi length $l_G = \sqrt{\nu \tau_{mr}}$, and the dipole size becomes $R_L \simeq \frac{\sqrt{2}\, l_G}{\sqrt{\ln(l_G / a)}}$. The analysis predicts spiraling current patterns inside the no-go region and provides measurable signatures for nanoimaging in graphene and related 2D systems, along with limitations of the continuum approach and directions for incorporating kinetic or quantum effects.

Abstract

We study current flow around a circular density depletion in a two-dimensional electron gas in the presence of a strong magnetic field. The depletion is parametrized by a power-law tail with an exponent $β> 2$. We show that current and electrochemical potential are exponentially suppressed inside a surrounding area much larger than the geometric size of the depletion region. The corresponding ``no-go'' radius grows as a certain power of the magnetic field. Residual current and potential exhibit spiraling patterns inside the no-go region. Outside of it, they acquire corrections inversely proportional to the distance, which is known as the Landauer resistivity dipole. The Landauer dipole is rotated by the angle $π(1 - 1 / β)$ with respect to the direction of the average electric field. We also consider the effect of electron viscosity and show that the variation of the no-go radius with magnetic field becomes more rapid if viscosity is large enough. In that regime the size of the Landauer dipole is set by the Gurzhi length, which is much larger than the no-go radius, which is in turn much larger than the geometric size of the depletion. Our results may be useful for interpreting nanoimaging of current distribution in graphene and other two-dimensional systems.

Figures (6)

• Figure 1: (a) Schematic of the experiment that motivated this study. A local probe measures the electrochemical potential produced by a current flowing around a density depletion in a 2D electron gas. In the presence of a magnetic field, the current forms a spiral pattern around the obstacle. (b) Examples of density profiles $n(r)$ studied: abrupt (black line), smooth with a flat insulating core (blue), and smooth everywhere (red line).
• Figure 2: (a) $\Re \mathrm{e}\,\phi(r)$ for $\alpha = 0.01$, $\beta = 3$. The solid line ($r/a \ge 1$ part) is Eq. \ref{['eqn:phi_outer']}, the dashed line is Eq. \ref{['eqn:f_far']}. (b) Same for $\Im \mathrm{m}\, \phi(r)$.
• Figure 3: False color and contour plots of the electrochemical potential $e\Phi(x,y)$ in the diffusive regime for the density profile given by Eq. \ref{['eqn:denprof2']}. Parameters: $\alpha = 0.01$, $\beta=3$.
• Figure 4: A cartoon of the dependence of the no-go length $R_\mathrm{no}$ and the Landauer dipole size $R_L$ on $1 / \alpha$, i.e., the magnetic field. The crossover from the viscous regime (where $R_\mathrm{no} \ll R_L$) to the diffusive regime (where $R_\mathrm{no} \sim R_L$) occurs as the field increases and $R_\mathrm{no}$ (or equivalently, $R_L$) exceeds the Gurzhi length $l_G$.
• Figure 5: Analytical and numerical results for the stream function. The panels (a) and (b) depict, respectively, the real and imaginary parts of $r \delta \psi(r) \equiv r [r + \psi(r)] / R_\mathrm{no}^2$ that approaches $\tilde{\lambda} / R_\mathrm{no}^2 = 51.2 + 22.9i$ at large $r$. The solid line is the numerical solution of Eq. \ref{['eqn:psi_equation2']} with the boundary conditions $\psi(r_\mathrm{min}) = \psi'(r_\mathrm{min}) = 0$, $\psi(r_\mathrm{max}) = -r_\mathrm{max} + \tilde{\lambda} / r_\mathrm{max}$, $\psi'(r_\mathrm{max}) = -1 - \tilde{\lambda} / r_\mathrm{max}^2$. The dotted line is obtained from Eq. \ref{['eqn:psi_Meijer']}; the dashed line (plotted for $r > l_G$) is from Eqs. \ref{['eqn:psi_far_hydro']}, \ref{['eqn:mu_from_c']}, \ref{['eqn:Rdip_hydro_beta3']}. Parameters: $\beta = 3$, $l_G / R_\mathrm{no} = 10$, $r_\mathrm{min} / R_\mathrm{no} = 0.1$, $r_\mathrm{max} / R_\mathrm{no} = 100$.