Imaging flat band electron hydrodynamics in biased bilayer graphene

Abstract

Hydrodynamic electron transport arises when carrier kinetics are dominated by interelectron collisions rather than the relaxation of momentum out of the electron system. In recent years, signatures of electron hydrodynamics have been reported in graphene devices owing to the low disorder and weak electron-phonon coupling. However, these experiments have been performed in regimes where the carrier mass is light, and the electron-electron collision length--though smaller than corresponding lengths for phonon or impurity scattering--remains large in absolute terms, typically several hundred nanometers. This restricts hydrodynamic transport phenomena to large length scales, limiting miniaturization of devices based on hydrodynamic flow. The advent of dual-gated rhombohedral graphene multilayers introduces a new route toward enhanced hydrodynamic behavior via their large--and tunable--effective mass. Here, we employ a scanning superconducting magnetic sensor to image local current flow in dual-gated bilayer graphene. Exploiting a sample geometry sensitive to both laminar and vortical flow, we identify three distinct transport regimes--ballistic, hydrodynamic, and diffusive--across the full phase space spanned by carrier density and displacement field. The strongest hydrodynamic transport is observed in the flat band regime, where fitting our results to a unified Boltzmann transport model reveals the electron-electron scattering length to be comparable to the Fermi wavelength of ~50 nm. High-current measurements, meanwhile, reveal striking nonlinearities in the flow pattern. Our results pave the way for miniaturized electronic devices based on linear and nonlinear electron hydrodynamics.

Figures (13)

• Figure 1: Tunable effective mass in dual-gated R2G.a, Schematic view of a dual-gated R2G device. Two graphite gates, separated from the R2G flake by hexagonal boron nitride (hBN) dielectrics, together control the carrier density $n_\mathrm{e}$ and the applied displacement field $D$. b, Single-particle band structure of R2G near the Brillouin zone corner calculated within a tight-binding modelJung_Accurate_2014. Different curves correspond to different values of interlayer potential difference $\Delta_U\propto D$. c, Density of states effective mass $m^*$ as a function of $n_\mathrm{e}$ and $\Delta_U$. d, Illustration of the experimental geometry, showing nSOT sensor, current streamlines (red and blue), and flux lines associated with the fringe magnetic field (black).
• Figure 1: Sample geometry.a, Optical image of dual-gated R2G after patterning. Scale bar: 10. b, Zoomed-in atomic force microscopy (AFM) image of the active region. Scale bar: 1.
• Figure 1: Current-driven vortex deformation. Evolution of current streamlines at $n_\mathrm{e}=\qty{-0.25e12}{cm^{-2}}$ and $D=\qty{0}{V/nm}$ upon increasing $I_0$ from 27 to 100. Scale bar: 1. 1 red line = $0.05I_0$, 1 solid blue line = $0.005I_0$. Additional dashed streamlines serve as guides to the eye.
• Figure 2: Contrasting microscopic transport regimes.a, Reconstructed current density $J_y(x,y)$ at $n_\mathrm{e}=\qty{0e12}{cm^{-2}}$ and $D=\qty{0}{V/nm}$, where R2G is a charge-neutral gapless semimetal. Scale bar: 1. $J_y$ is normalized by $J_0\equiv I_0/w$, the average current density in the channel. b, Current streamlines at the same $n_\mathrm{e}$ and $D$ as panel a. c, $J_y(x,y)$ for $n_\mathrm{e}=\qty{-1.3e12}{cm^{-2}}$ and $D=\qty{0.39}{V/nm}$. d, Current streamlines at the same $n_\mathrm{e}$ and $D$ as panel c. e, $J_y(x,y)$ for $n_\mathrm{e}=\qty{-0.4e12}{cm^{-2}}$ and $D=\qty{0.39}{V/nm}$. f, Current streamlines at the same $n_\mathrm{e}$ and $D$ as panel e. g, $J_y(x)/J_0$ averaged between measurements at $y=\pm\qty{2}{\micro\meter}$ (see inset) for the data in panels a, c, and e as marked. h, Normalized transverse current $I_x(x)/I_0$ (see inset and Eq. \ref{['eq2']} of main text) corresponding to the data in panels b, d, and f as marked. i, Combined root mean square deviation (RMSD) between experimental data at $n_\mathrm{e}=\qty{0e12}{cm^{-2}}$ and $D=\qty{0}{V/nm}$ and the Boltzmann model parameterized by $\ell_\mathrm{mr}$ and $\ell_\mathrm{ee}$. The RMSD is normalized by the minimum value, indicated by the dot and corresponding to the best fit. Contours of 1.1, 1.3, and 1.5 delineate phenomenological confidence intervals. j, RMSD for $n_\mathrm{e}=\qty{-1.3e12}{cm^{-2}}$ and $D=\qty{0.39}{V/nm}$. k, RMSD for $n_\mathrm{e}=\qty{-0.4e12}{cm^{-2}}$ and $D=\qty{0.39}{V/nm}$. l, Theoretical current streamlines for $\ell_\mathrm{mr}=\qty{600}{nm}$ and $\ell_\mathrm{ee}=\qty{161}{nm}$, corresponding to the best fit in panel i. m, Theoretical current streamlines for $\ell_\mathrm{mr}=\qty{100}{\micro\meter}$ and $\ell_\mathrm{ee}=\qty{4.2}{\micro\meter}$, corresponding to the best fit in panel j. n, Theoretical current streamlines for $\ell_\mathrm{mr}=\qty{18.2}{\micro\meter}$ and $\ell_\mathrm{ee}=\qty{29}{nm}$, corresponding to the best fit in panel k. o, $J_y(x)/J_0$ at $y=\pm\qty{2}{\micro\meter}$ for the best fits described in panels l, m and n.
• Figure 2: Magnetic field, current density, and current streamline plots for the data in Fig. \ref{['fig:fig2']}.a, Fringe magnetic field signal $B(x,y)$ measured at $n_\mathrm{e}=\qty{0e12}{cm^{-2}}$ and $D=\qty{0}{V/nm}$. Scale bar: 1. $B(x,y)$ is normalized by $B_\mathrm{range}\equiv B_{\max}-B_{\min}$. b, Reconstructed current density $J_x(x,y)$ at the same $n_\mathrm{e}$ and $D$ as panel a. $J_x$ is normalized by $J_0\equiv I_0/w$. c,d, Reproduction of Fig. \ref{['fig:fig2']}a,b for comparison. e, $B(x,y)$ at $n_\mathrm{e}=\qty{-1.3e12}{cm^{-2}}$ and $D=\qty{0.39}{V/nm}$. f, $J_x(x,y)$ at the same $n_\mathrm{e}$ and $D$ as panel e. g,h, Reproduction of Fig. \ref{['fig:fig2']}c,d for comparison. i, $B(x,y)$ at $n_\mathrm{e}=\qty{-0.4e12}{cm^{-2}}$ and $D=\qty{0.39}{V/nm}$. j, $J_x(x,y)$ at the same $n_\mathrm{e}$ and $D$ as panel i. k,l, Reproduction of Fig. \ref{['fig:fig2']}e,f for comparison.