Corbino Disk Viscometer for 2D Quantum Electron Liquids

Andrea Tomadin, Giovanni Vignale, and Marco Polini

Phys. Rev. Lett. 113, 235901 – Published 2 December 2014

Abstract

The shear viscosity of a variety of strongly interacting quantum fluids, ranging from ultracold atomic Fermi gases to quark-gluon plasmas, can be accurately measured. On the contrary, no experimental data exist, to the best of our knowledge, on the shear viscosity of two-dimensional quantum electron liquids hosted in a solid-state matrix. In this work we propose a Corbino disk device, which allows a determination of the viscosity of a quantum electron liquid from the dc potential difference that arises between the inner and the outer edge of the disk in response to an oscillating magnetic flux.

Received 11 January 2014

DOI:https://doi.org/10.1103/PhysRevLett.113.235901

Authors & Affiliations

Andrea Tomadin^1,*, Giovanni Vignale², and Marco Polini¹

¹NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa, Italy
²Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA

^*andrea.tomadin@sns.it

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Vol. 113, Iss. 23 — 5 December 2014

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Images

Figure 1
A viscometer for 2D electron liquids. The light gray surface represents the 2D electron system, which is shaped into a Corbino disk geometry. The intermediate green region represents a dielectric layer of thickness $d$ , which separates the 2D electron system from a back gate (dark gray region). The internal hole of the Corbino disk is threaded by an oscillating magnetic flux $Φ (t) = Φ_{0} \cos (Ω t)$ , which induces an azimuthal electric field $E_{\hat{θ}}$ oscillating at the same frequency $Ω$ . The magnitude of the azimuthal electric field decreases as one goes from the inner to the outer rim. The inner rim of the disk is grounded, while the outer rim is free to adjust its voltage. A dc electrical potential energy difference $Δ U$ appears between the inner and the outer rim in response to the oscillating magnetic flux. The quantity $Δ U$ sensitively depends on the shear viscosity of the 2D electron fluid and the frequency $Ω$ of the magnetic flux.
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Figure 2
Radial profile of the azimuthal component of the velocity—in units of $v_{in}$ as defined in Eq. (4)—at $t = 0$ , obtained by solving Eq. (15) with boundary conditions (12) (main panel) and (13) (inset). The results are obtained with $r_{out} / r_{in} = 10.0$ , and for several values of the dimensionless parameter $ζ$ defined in Eq. (16): $ζ = 0.2$ (dashed line), 0.5 (dotted line), and 1.0 (dash-dotted line). The value of the viscosity in the different solutions increases as shown by the arrow. The solid line corresponds to the analytical result (4), which holds at $ζ = 0$ .
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Figure 3
The dc potential energy difference $Δ U$ between the outer and inner rims of the Corbino disk as a function of the frequency $f = Ω / (2 π)$ of the oscillating magnetic flux $Φ (t)$ (main panel) and the dimensionless variable $Ω τ$ (inset). Results in this plot have been obtained with the parameters as in Eq. (6). In the main panel $Ω τ = 10^{3}$ . In the inset $f = 10 MHz$ . Different curves correspond to different values of $ν$ : $1.0 {cm}^{2} / s$ (dashed line), $5.0 {cm}^{2} / s$ (dotted line), and $15.0 {cm}^{2} / s$ (dash-dotted line). The latter value corresponds to a simple order-of-magnitude estimate of the kinematic viscosity [16], i.e., $ν \sim ℏ / m$ . The solid line represents the viscous-free analytical result in Eq. (5).
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Figure 4
Same as in Fig. 2 but for the radial profile of the vorticity $ω_{z}^{(1)} (r, t)$ (in units of $v_{in} / r_{in}$ ) at time $t = 0$ . The inset shows the position $L_{\max}$ of the maximum of $v_{θ}^{(1)} (r, t = 0)$ , as displayed in Fig. 2, for several values of the vorticity penetration depth $L_{η}$ (in units of $r_{in}$ ). The dashed line corresponds to the expected relation $L_{\max} = L_{η}$ .
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