Voltage characteristics of hydrodynamic Dirac electron nozzles with supersonic flow

Abstract

In clean Dirac electron systems such as graphene, electron-electron interactions can dominate over other relaxation mechanisms such as phonon or impurity scattering. In this limit, collective electron dynamics can be described by hydrodynamic equations. The prerequisites for electron hydrodynamics have already been fulfilled in experiments, and signatures of hydrodynamic flow have been identified in transport measurements. Here, we derive the pressure-driven hydrodynamic flow profile across a de Laval nozzle profile for Dirac electrons in the subsonic and supersonic regimes. Based on this, we resolve the local voltage characteristics, which provide clear signatures of supersonic hydrodynamic flow. In particular, we identify two distinct features in the experimentally measurable potential profile: a pronounced asymmetry of the local voltage profile on opposite sides of the nozzle, and a sharp differential resistance signature induced by an electron shock wave on the exit side of the nozzle.

Figures (11)

• Figure 1: A graphene-based de Laval nozzle that is connected to two leads, over which a bias voltage $U_{{\mathrm{L}}{\mathrm{R}}}$ is applied, and a noninvasive probe that locally measures the voltage difference with respect to the left lead $U_\mathrm{p}$. The lines with arrows indicate the laminar hydrodynamic flow of charge carriers through the nozzle. The lumped-element model for resolving the voltage characteristics, with Ohmic and hydrodynamic de Laval sections between two leads, is schematically depicted (see text for details).
• Figure 2: (a) The relation between pressure and cross section for the hydrodynamic flow through a two-dimensional nozzle. Three flow profiles are indicated: subsonic flow with the pressure reaching pressure $P_{\textrm{t}}$ at the throat with cross section $A_{\textrm{t}}$ and returning to the initial pressure $P_{\mathrm{L}}$ (blue line A, back and forth), critical flow that reaches the critical pressure $P_*$ and the speed of sound at the throat before returning to the initial pressure (yellow line B, back and forth), and supersonic flow with supersonic flow speeds between the throat and the shock front at cross section $A_{\textrm{sf}}$ (line B, C & D), where there is a pressure jump $\Delta P$ and a speed drop $\Delta V$ (brown dashed line). The ideal supersonic flow profile is realized for $P_{\mathrm{R}}=0$ (line B, C & E). (b),(c) The (b) flow speed and (c) pressure profiles as a function of the position along the nozzle are shown for the flow profiles indicated in (a) matching the corresponding labels and colors. We consider a two-dimensional nozzle with length $L$ and width profile given by $A(x) = A_{\textrm{t}}/[1-(2x/L)^2]$ here, such that the leads are infinitely wide: $A(\pm L/2) = +\infty$.
• Figure 3: (a),(b) The chemical potential and temperature for vanishing flow speed at the nozzle exit to the right of the shock front in the supersonic regime, in the limit regimes with (a) $T_{{\mathrm{L}},{\mathrm{R}}} \gg \mu_{{\mathrm{L}},{\mathrm{R}}}$ and (b) $T_{{\mathrm{L}},{\mathrm{R}}} \ll \mu_{{\mathrm{L}},{\mathrm{R}}}$.
• Figure 4: Voltage characteristics of a graphene-based de Laval nozzle with Ohmic leads, as presented in Fig. \ref{['fig:setup']}, in the subsonic ($V < v_\mathrm{s}$) and supersonic ($V > v_\mathrm{s}$) flow regimes. (a) Cross-sectional profile of the de Laval nozzle. (b)-(d) The normalized (a) probe voltage, (c) differential resistance, and (d) its spatial derivative as a function of the position of the probe along the along the nozzle. The position of the shock front for the supersonic solution is indicated with a vertical dashed line. We consider graphene with $v =10^6\,\text{m/s}$, carrier density $n_\mathrm{2D} = 10^{11}\,\text{cm$^{-2}$}$, mobility $\mu_\mathrm{mob} = 10 000 \, \text{cm$^2$/(V$\cdot$s)}$, and a transport geometry with $5\,\text{\textmu m}$-long, $1\,\text{\textmu m}$-wide Ohmic sections and a $1\,\text{\textmu m}$-long de Laval nozzle that is $0.2\,\text{\textmu m}$ wide at the throat (see Appendix \ref{['subsec:nozzle-Ohmic']} for details).
• Figure A1: Two-dimensional nozzle geometry with local coordinate system that aligns with the direction of the flow.