Geometric Control of Universal Hydrodynamic Flow in a Two-Dimensional Electron Fluid

Fluid dynamics is one of the cornerstones of modern physics and has recently found applications in the transport of electrons in solids. In most solids, electron transport is dominated by extrinsic factors, such as sample geometry and scattering from impurities. However, in the hydrodynamic regime, Coulomb interactions transform the electron motion from independent particles to the collective motion of a viscous “electron fluid.” The fluid viscosity is an intrinsic property of the electron system, determined solely by the electron-electron interactions. Resolving the universal intrinsic viscosity is challenging, as it affects the resistance only through interactions with the sample boundaries, whose roughness not only is unknown but also varies from device to device. Here, we eliminate all unknown parameters by fabricating samples with smooth sidewalls to achieve the perfect slip boundary condition, which has been elusive in both molecular fluids and electronic systems. We engineer the device geometry to create viscous dissipation and reveal the true intrinsic hydrodynamic properties of a 2D system. We observe a clear transition from ballistic to hydrodynamic electron motion, driven by both temperature and magnetic field. We directly measure the viscosity and electron-electron scattering lifetime (the Fermi quasiparticle lifetime) over a wide temperature range without fitting parameters and show they have a strong dependence on electron density that cannot be explained by conventional theories based on the random phase approximation.

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The equations describing fluid flow are universal, depending only on the fluid viscosity. In practice, fluid flow is affected by interactions with solid boundaries, giving rise to complex physics that depends on the nature of the boundaries and on ambient conditions. We overcome this boundary problem by creating a fluid of electrons in a pure semiconductor channel with perfectly smooth walls—the first realization of a viscous fluid in perfectly slippery channels. By tuning the shape of the channel, we study the intrinsic, universal properties of the electron fluid and observe some unexpected behavior.

At low temperatures, electrons in a pure solid can behave collectively as a viscous fluid, instead of individually bouncing around off impurities and thermal vibrations, as they commonly do. The fluid viscosity is a universal and intrinsic property of the electron system and should be independent of the sample details. However, the nature of the sample boundaries varies between experiments and is hard to quantify, resulting in sample-dependent viscous flow.

We eliminate the boundary problem by developing devices with smooth walls and then deliberately create viscous flow by engineering obstacles into the boundaries. We observe a clear transition to hydrodynamic electron motion, driven by decreasing temperature (which is expected) and by increasing magnetic field (which is unexpected). Furthermore, the absence of any fitting parameters and precision of the measurement technique reveals an unexpected deviation from existing theoretical models.

The engineering of smooth boundaries provides a new approach to study many-body electron systems in a variety of different material systems over a wide temperature range and opens new possibilities for viscous electronics.

Figure 1

(a),(b) 3D schematics showing (a) a conventional modulation-doped $\mathrm{GaAs}/{\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ heterostructure. The conduction channel is patterned using chemical etching, which causes rough sidewalls, while random surface charge on the sidewalls creates additional disorder. (b) An accumulation-mode $\mathrm{GaAs}/{\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ device. The channel is defined by a metal top gate and kept away from etched sidewalls and surface charge. (c)–(f) Theoretical simulations (see the Appendix pp4) of the power dissipation density ($W/m^2$)under the perfect slip boundary condition (width $W=2.5\mu \mathrm{m}$, length $L=25\mu \mathrm{m}$) in the linear response regime (Reynolds number much smaller than 1). The color scale represents the magnitude of the dissipation, and white lines show the electron flow streamlines. (c) Power dissipation density of a straight channel. The dissipation is purely Ohmic, as viscous contribution vanishes with perfect slip boundaries. (d) Ohmic power dissipation density of a crenellated channel (crenellation size $l_{\mathrm{cren}}=1\mu \mathrm{m}$) with zero viscosity, i.e., nonviscous electron transport. The Ohmic resistance of the crenellated channel is smaller than the straight channel. (e) Viscous power dissipation density of the crenellated channel with a viscosity of $\nu =1.15\times {10}^{-2}{\mathrm{m}}^2/\mathrm{s}$. The viscous power dissipation density concentrates around the regions where the streamlines deform the most to form slow whirlpools in the crenellations [19]. The viscous power dissipation vanishes near the boundaries due to the perfect slip boundary condition. (f) Total power dissipation density of the crenellated channel for a viscous electron flow summing up both Ohmic and viscous contributions.

Figure 2

(a) Image of the accumulation-mode GaAs heterostructure device containing a high-mobility 2DEG. The gate-defined channel contains two sections: a $2.5\text{−}\mu \mathrm{m}$-wide and $25\text{−}\mu \mathrm{m}$-long straight channel labeled “$S$” and a crenellated channel “$C$” of the same dimensions with crenellations of $l_{\mathrm{cren}}=1\mu \mathrm{m}$. (b) Key length scales and transport regimes in the system at $n=2.45\times {10}^{11}{\mathrm{cm}}^{-2}$. $l_{\mathrm{mfp}}$ is the momentum-relaxing mean free path due to electron-phonon ($l_{\text{phonon}}$) and electron-impurity ($l_{\mathrm{imp}}$) scattering. $l_{\mathrm{cren}}$ is the size and spacing of the square crenellations. ${\lambda }_F$ is the Fermi wavelength. $l_{ee}$ is the theoretically calculated electron-electron scattering length. The shaded blue area indicates the gradual crossover from hydrodynamic to ballistic transport regimes around $T\sim 10\mathrm{K}$ as the smallest length scale in the system varies between $l_{\mathrm{cren}}$ and $l_{ee}$. (c),(d) Experimentally measured resistance of both straight and crenellated channels $R_{\text{straight}}$ and $R_{\mathrm{cren}}$, respectively, as a function of the temperature for three electron densities $n=2.45$, 1.78, and $1.45\times {10}^{11}{\mathrm{cm}}^{-2}$, respectively. The resistances of two straight channels with different widths $W=2.5$ and $5\mu \mathrm{m}$ but the same length-to-width ratio are shown by the red solid circles and gray squares and are indistinguishable. The solid red lines are fits to $R_{\text{straight}}$ including both electron-impurity and electron-phonon scattering with the shaded area presenting the uncertainty. The resistance of the crenellated channel $C$ is shown both at $B=0$ (dark blue empty circles) and with a small out-of-plane magnetic field $B=0.1\mathrm{T}$ (light blue empty circles).

Figure 3

(a) Electron-electron scattering length $l_{ee}$ and the viscous effective scattering length $l_{\mathrm{eff}}$ as a function of the temperature for an electron density of $n=2.45\times {10}^{11}{\mathrm{cm}}^{-2}$. The solid red line is the parameter-free theoretical calculation of $l_{ee}$ for a Fermi liquid using RPA and Boltzmann theory (Appendix pp6). The dashed black line is the calculated $l_{\mathrm{eff}}$ using Eq. (2). The black squares and red filled circles are the values of $l_{\mathrm{eff}}$ and $l_{ee}$, respectively, extracted from the measured resistance of the crenellated channel. The error bars are due to the uncertainty in the Ohmic component of resistance. (b),(c) High-temperature magnetoresistance of the crenellated and straight channels for $n=2.45\times {10}^{11}{\mathrm{cm}}^{-2}$. The straight channel (c) shows no magnetoresistance as expected for smooth sidewalls, whereas the crenellated channel (b) shows a strong negative magnetoresistance due to viscous effects. The dashed lines in (b) are theoretical magnetoresistance curves, calculated using Eq. (3) and the $l_{ee}$ measured in (a). (d) Density dependence of $l_{ee}$ at $T=20\mathrm{K}$. Values extracted from the experiment (symbols, with the error margin shown by the shading) are substantially lower than the low $T$ theory of Giuliani and Quinn [29] (as expected) but show a stronger density dependence than predicted theoretically from the more complete RPA (black dashed line) and Hubbard calculations (blue dash-dotted line).

Figure 4

(a) The resistances in the straight devices with dimensions $W=2.5$ and $5.0\mu \mathrm{m}$ and $L=25$ and $50\mu \mathrm{m}$, respectively, as a function of the temperature. The dotted lines are the high and low fits (see Table 2). (b) Electron-impurity mean free time and corresponding impurity limited mobility (at $T=0.25\mathrm{K}$) with respect to number density. A volumetric distribution of $n_i=1.13\times {10}^{15}{\mathrm{cm}}^{-3}$ impurities yields the dotted line based on Eqs. (b1) and (b2) and explains the observed data shown by filled circles.

Figure 5

Extracted $l_{ee}$ and the viscous effective length $l_{\mathrm{eff}}=4\nu /v_F$ as a function of the temperature for three different number densities: (a) 1.45, (b) 1.78, and (c) 2.45 in units of ${10}^{11}{\mathrm{cm}}^{-2}$. The error bars are the $l_{ee}$ and the viscous effective length $l_{\mathrm{eff}}=v_F/(4\nu )$ extracted from resistance data by solving Navier-Stokes equation (e1) and using the formula in Eq. (g1) that captures size effects. The uncertainty is due to the possible variation of impurity concentration as discussed in Appendixes pp1 and pp4. Dotted orange lines are the theoretical calculation of the $l_{ee}$ using RPA and Boltzmann theory, with no free parameters, as outlined in Appendix pp6. The dotted black line shows the viscous effective length we calculate by using theoretical $l_{ee}$ and Eq. (g1).

Figure 6

(a) The scattering probability at the Fermi surface, $P({\epsilon }_k=E_F)$, and (b) the dimensionless inverse viscosity $⟨P⟩=\hslash /(2m^{*}\nu )$, and, equivalently, $⟨P⟩=2/(k_F{l}_{ee})$, as a function of normalized temperature $T/T_F$ for different values of interaction parameter $r_s=\kappa /(\sqrt{2}{k}_F)$ in the three different approximations in Eq. (g2), namely, contact interaction, RPA, and HA. As $r_s$ decreases, the trend $\mathrm{RPA}\to \mathrm{HA}$ is clearly visible. At low temperatures, only the Fermi surface contributes to the average; hence, $⟨P⟩\to P$ or $l_{ee}\to v_F{\tau }_{ee}$.

Figure 7

Normalized polarization function with respect to momentum $q$, for different temperatures.

Figure 8

(a) The measured $l_{ee}$ shown by error bars, compared to the theory with the Hubbard local field correction (dotted orange lines) described in Appendix pp6 and the measured effective length $l_{\mathrm{eff}}$ that contains low-$T$ ballistic corrections, compared to the theory with the Hubbard local field correction (dotted black lines). See similar plots in Fig. 3, where the dotted theory curves are based on RPA. The density dependence of $l_{ee}$ at (b) $T=20\mathrm{K}$ and (c) $T=40\mathrm{K}$ in the Hubbard and RPA approximations compared to the experimentally measured values shown by the blue solid line. The shaded area depicts the uncertainty due to the possible variation of impurity concentration as discussed in Appendixes pp1 and pp4. The correlation effects captured by HA but missed in RPA account for an approximately 50% increase in the interaction length yet do not explain the strong density dependence observed in the experimental data.

Figure 9

(a) The charge contour plot over the plane of 2DEG for the highest gate voltage $V=1.6\mathrm{V}$, where the number density deep inside the channel (along $y=0$) is $n=2.45\times {10}^{11}{\mathrm{cm}}^{-2}$. (b) The electrical shape of the channel near the crenellation for three different densities, compared to the top gate marked by the dotted black line. The boundary of the channel is assumed to be the contour on which the density is half its maximum value. (c) The comparison of $l_{ee}$ at $T=20\mathrm{K}$ extracted by assuming boundaries of the top gate and the softened boundaries in (a).

Figure 10

Magnetoresistance of (a) the crenellated channel after removing the zero-field resistance of the straight channel $R_{\text{straight}}(B=0)$ and (b) the straight channel after removing $R_{\text{straight}}(B=0)$ at $n=2.45\times {10}^{11}{\mathrm{cm}}^{-2}$ for different temperatures. (c) Resistance of the crenellated channel as a function of the temperature at different magnetic fields. Dashed (dotted) lines are straight line guides for low (high) temperatures depicting the change in slope.

Figure 11

Comparison of our $l_{ee}$ measurements as a function of electron number density $n$ with that of the recent experiment by Gupta et al. [27] and the parameter-free calculations of $l_{ee}$ as described in Appendix pp6 (a) at $T=13\mathrm{K}$ (our data are obtained at $T=14\mathrm{K}$) and (b) $T=28\mathrm{K}$ (our data are obtained at $T=30\mathrm{K}$).