Nonlocal transport and the hydrodynamic shear viscosity in graphene

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Nonlocal transport and the hydrodynamic shear viscosity in graphene

Iacopo Torre, Andrea Tomadin, Andre K. Geim, and Marco Polini

Phys. Rev. B 92, 165433 – Published 30 October 2015

Abstract

Motivated by recent experimental progress in preparing encapsulated graphene sheets with ultrahigh mobilities up to room temperature, we present a theoretical study of dc transport in doped graphene in the hydrodynamic regime. By using the continuity and Navier-Stokes equations, we demonstrate analytically that measurements of nonlocal resistances in multiterminal Hall bar devices can be used to extract the hydrodynamic shear viscosity of the two-dimensional (2D) electron liquid in graphene. We also discuss how to probe the viscosity-dominated hydrodynamic transport regime by scanning probe potentiometry and magnetometry. Our approach enables measurements of the viscosity of any 2D electron liquid in the hydrodynamic transport regime.

Received 3 August 2015

DOI:https://doi.org/10.1103/PhysRevB.92.165433

Authors & Affiliations

Iacopo Torre^1,2,*, Andrea Tomadin³, Andre K. Geim⁴, and Marco Polini²

¹NEST, Scuola Normale Superiore, I-56126 Pisa, Italy
²Istituto Italiano di Tecnologia, Graphene Labs, Via Morego 30, I-16163 Genova, Italy
³NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa, Italy
⁴School of Physics, and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom

^*iacopo.torre@sns.it

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Vol. 92, Iss. 16 — 15 October 2015

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Figure 1
Schematic of the nonlocal transport setup analyzed in this paper. A dc current $I$ is injected (red arrow) into an encapsulated graphene Hall bar of width $W$ . Current injection occurs at a lateral contact located at $x = x_{0}$ and $y = - W / 2$ . The same current is drained (blue arrow) at a contact located at $x = - x_{0}$ and $y = - W / 2$ . Measurements of voltage drops $Δ V$ near the current injection region are sensitive to the kinematic viscosity $ν$ of the two-dimensional massless Dirac fermion liquid. The notion of “vicinity” between voltage probe and current injector is defined by a crucial length scale, i.e., the vorticity diffusion length $D_{ν} = \sqrt{ν τ}$ . Here $τ$ (exceeding 1 ps in high-quality encapsulated devices) represents a phenomenological scattering time due to momentum-non-conserving collisions of a fluid element (and not of single electrons).
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Figure 2
Panel (a) The longitudinal conductivity (21) (in units of $σ_{0})$ is plotted as a function of the ratio $D_{ν} / W$ for different values of the boundary scattering length $l_{b} : l_{b} = \infty$ (thick solid line), $l_{b} = 10 W$ (dashed line), $l_{b} = W$ (dotted line), $l_{b} = 0.1 W$ (dashed-dotted line), and $l_{b} = 0$ (thin solid line). Panel (b) The current-density profile $- e J_{x} (y)$ , normalized by the total current $I / W$ , is plotted as a function of $y$ for $l_{b} = 0$ (no-slip BCs) and different values of $D_{ν} : D_{ν} = 0$ (thick solid line), $D_{ν} = 0.01 W$ (dashed line), and $D_{ν} = 0.1 W$ (dotted line). The case of $D_{ν} ≫ W$ , corresponding to Poiseuille flow, is represented by a thin solid line.
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Figure 3
Steady-state spatial map of the 2D electrical potential $ϕ (r)$ (in units of $ϕ_{0} \equiv I / σ_{0})$ and charge current streamlines $- e J (r)$ in a Hall bar device, such as the one depicted in Fig. 1 with $x_{0} = W$ . Different panels refer to different values of the vorticity diffusion length $D_{ν} : D_{ν} = 0$ [panel (a)], $D_{ν} = 0.5 W$ [panel (b)], and $D_{ν} = W$ [panel (c)]. Whirlpools are clearly seen in the bottom right and bottom left of panels (b) and (c). No whirlpools occur in the absence of viscosity as in panel (a). In each panel, the current streamlines change color from white (high current density) to black (low current density).
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Figure 4
Spatial map of the $\hat{z}$ component of the magnetic field $B_{z} (r, z)$ (in units of $B_{0} \equiv μ_{0} I d / W^{2})$ , generated by the 2D steady-state current pattern $J (r)$ and calculated immediately above the graphene sheet, i.e., for $0 < z ≪ W, D_{ν}$ . These results have been obtained for the same parameters as in Fig. 3. Different panels refer to different values of $D_{ν} : D_{ν} = 0.5 W$ [panel (a)] and $D_{ν} = W$ [panel (b)]. In this figure we have not shown results for $D_{ν} = 0$ : In the absence of viscosity $B_{z} (r, z)$ is identically zero.
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