Magnetic hallmarks of viscous electron flow in graphene

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Magnetic hallmarks of viscous electron flow in graphene

Karina A. Guerrero-Becerra, Francesco M. D. Pellegrino, and Marco Polini

Phys. Rev. B 99, 041407(R) – Published 28 January 2019

Abstract

We propose a protocol to identify spatial hallmarks of viscous electron flow in graphene and other two-dimensional viscous electron fluids. We predict that the profile of the magnetic field generated by hydrodynamic electron currents flowing in confined geometries displays unambiguous features linked to whirlpools and backflow near current injectors. We also show that the same profile sheds light on the nature of the boundary conditions describing friction exerted on the electron fluid by the edges of the sample. Our predictions are within reach of vector magnetometry based on nitrogen-vacancy centers embedded in a diamond slab mounted onto a graphene layer.

Received 25 October 2018

DOI:https://doi.org/10.1103/PhysRevB.99.041407

Physics Subject Headings (PhySH)

Classical transport Color centers Electrical conductivity

Graphene Two-dimensional electron system

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Karina A. Guerrero-Becerra¹, Francesco M. D. Pellegrino^2,3, and Marco Polini^1,4

¹Istituto Italiano di Tecnologia, Graphene Labs, Via Morego 30, I-16163 Genova, Italy
²Dipartimento di Fisica e Astronomia, Università di Catania, Via Santa Sofia 64, I-95123 Catania, Italy
³INFN, Sezione di Catania, I-95123 Catania, Italy
⁴School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom

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Issue

Vol. 99, Iss. 4 — 15 January 2019

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Images

Figure 1
Schematic representation of an NV magnetometry experiment aimed at detecting electron whirlpools in graphene. A graphene rectangular sample (hexagonal lattice) is mounted onto a diamond slab (pale-pink slab) hosting an array of NV centers (zoom). A back gate (yellow) controls the equilibrium electron density in graphene. (Graphene encapsulation in, e.g., hexagonal boron nitride to ensure high electronic quality is not shown.) An optical setup (structure sketched above the diamond slab) prepares the quantum state of the NV centers through a laser beam and accesses its photoluminescence (PL) as a function of the frequency of a microwave excitation (not shown). This setup enables measurements of the magnetic field $B (x, z)$ generated by the current density flowing in graphene and a spatial map of the 2D current density $J (x)$ can be reconstructed from the Cartesian components of $B$ .
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Figure 2
Spatial map of the $\hat{x}$ component, $B_{x} (x, z = d^{'})$ , of the magnetic field (indicated by the color map and in $μ T$ ) generated by $J_{y} (x)$ . The current density is represented by the vector field. Panel (a): Ohmic case, $D_{ν} = 0$ . Panel (b): Viscous case, $D_{ν} = W / 4$ . The injector is located at position $(0, - W / 2)$ . The collector (not shown) is set on the lower edge of the strip, at position $(- 20 W, - W / 2)$ . An electron whirlpool is clearly seen to the right of the injector in the viscous case. The vertical solid and dashed lines denote the horizontal positions where one-dimensional cuts of $B_{x} (x, z = d^{'})$ have been taken; see Fig. 3.
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Figure 3
One-dimensional cuts of the 2D spatial maps reported in Fig. 2. Panels (a) and (c) [(b) and (d)] illustrate $B_{x} (x, z = d^{'})$ (in units of $μ T$ ) [ $J_{y} (x)$ (in units of $A / m$ )] as a function of $y / W$ , evaluated at $x = W / 2$ and $x = 3 W / 2$ , respectively. These horizontal positions have been marked by a solid and a dashed line in Fig. 2, respectively. Solid lines in this plot correspond to the viscous case ( $D_{ν} = W / 4$ ), while dashed lines correspond to the Ohmic case ( $D_{ν} = 0$ ).
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Figure 4
Same as in Fig. 2 but for $B_{y} (x, z = d^{'})$ . As in Fig. 2, panel (a) is for the Ohmic case ( $D_{ν} = 0$ ) while panel (b) refers to the viscous case ( $D_{ν} = W / 4$ ). The vertical and horizontal lines represent the positions where one-dimensional cuts have been taken; see Fig. 5.
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Figure 5
One-dimensional cuts of the 2D spatial maps reported in Fig. 4. Panel (a) [(b)] illustrates $- B_{y} (x, z = d^{'})$ (in units of $μ T$ ) [ $J_{x} (x)$ (in units of $A / m$ )] evaluated at $x = W / 2$ —see solid line in Fig. 4—and as a function of $y / W$ . Panels (c) and (d): Same as in panels (a) and (b) but in this case $- B_{y} (x, z = d^{'})$ and $J_{x} (x)$ are evaluated at $y = - 0.4 W$ —see dashed line in Fig. 4—and shown as functions of $x / W$ . Solid lines in this plot correspond to the viscous case ( $D_{ν} = W / 4$ ), while dashed lines correspond to the Ohmic case ( $D_{ν} = 0$ ).
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Figure 6
The components $B_{y}$ and $B_{z}$ of the magnetic field generated by longitudinal current flow—Eq. (9)—are plotted as functions of $y / W$ in panels (a) and (b), respectively. The current density profile (9) is shown in panel (c). Red solid line: $D_{ν} = 10 W$ and $ℓ_{b} = 0$ ; black dotted line: $D_{ν} = W / 4$ and $ℓ_{b} = 0$ ; black solid line: $ℓ_{b} = \infty$ . (With free-surface boundary conditions the result is independent of $D_{ν}$ .)
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