Functional dependence of the Hall viscosity-induced transverse voltage in two-dimensional Fermi liquids

TL;DR

This paper investigates how Hall viscosity influences the Hall voltage in two-dimensional Fermi liquids, providing a method to distinguish and measure Hall viscosity effects through specific channel geometries and voltage measurements.

Contribution

The study introduces a channel geometry that uniquely separates Hall viscosity and Lorentz force contributions, deriving their functional dependencies on external parameters for the first time.

Findings

Hall viscous contribution can dominate Lorentz signal up to tens of millitesla in GaAs.

The ratio of Hall viscous to Lorentz force is negative and varies with width, slip-length, density, and mean free path.

Total Hall electric field is parabolic, with the offset indicating Hall viscosity.

Abstract

The breaking of parity and time-reversal symmetry in two-dimensional Fermi liquids gives rise to non-dissipative transport features characterized by the Hall viscosity. In magnetic fields, the Hall viscous force directly competes with the Lorentz force, since both mechanisms contribute to the Hall voltage. In this work, we present a channel geometry that allows us to uniquely distinguish these two contributions and derive, for the first time, their functional dependency on all external parameters. We show that the ratio of the Hall viscous to the Lorentz force contribution is negative and that its modulus decreases with increasing width, slip-length and carrier density, while it increases with the electron-electron mean free path of our channel. In typical materials such as GaAs the Hall viscous contribution can dominate the Lorentz signal up to a few tens of millitesla until the total Hall voltage vanishes and subsequently is overcome by the Lorentz contribution. Moreover, we prove that the total Hall electric field is parabolic due to Lorentz effects, whereas the offset of this parabola is characterized by the Hall viscosity. Hence, our results pave the way to measure and identify the Hall viscosity via both global and local voltage measurements.

Figures

Equations

$$\mathbf{v}=v_{\mathrm{x}}(y){\mathbf{e}}_{\mathrm{x}},\mu ={\mu }_0+\Delta \mu ,T=T_0=\mathrm{constant},$$

$$\eta {\partial }_{\mathrm{y}}^2{v}_{\mathrm{x}}$$

$${\partial }_{\mathrm{y}}p$$

$$\eta =\frac{{\eta }_0}{1+(2l_{\mathrm{ee}}/r_{\mathrm{c}}{)}^2},{\eta }_{\mathrm{H}}=\frac{2\mathrm{sgn}(B){\eta }_0{l}_{\mathrm{ee}}/r_{\mathrm{c}}}{1+(2l_{\mathrm{ee}}/r_{\mathrm{c}}{)}^2},$$

$$\Delta V_{\mathrm{tot}}$$

$$v_{\mathrm{x}}(-l_{\mathrm{s}})=v_{\mathrm{x}}(W+l_{\mathrm{s}})$$

$$\Delta V_{\mathrm{tot}}$$

$$\displaystyle\times\left(A_{1}\sinh\!\left(Wl_{\rm G}^{-1}\right)+A_{2}\left[\cosh\!\left(Wl_{\rm G}^{-1}\right)\!-\!1\right]\right)\!-\!BW\Big{]},$$

$$\Delta V_{{\eta }_{\mathrm{H}}}$$

$$\Delta V_B$$

$$\frac{|\Delta V_{{\eta }_{{}_{\mathrm{H}}}}|}{|\Delta V_{\mathrm{B}}|}=\frac{l_{\mathrm{ee}}^2}{\frac{1}{6}{W}^2+l_{\mathrm{s}}W+l_{\mathrm{s}}^2}+2\frac{l_{\mathrm{ee}}}{l_{\mathrm{imp}}}.$$

$$\Delta V_{{\eta }_{\mathrm{H}}}$$

$$\Delta V_B=-B{\int }_0^W\mathrm{d}y{v}_{\mathrm{x}}(y)$$

$$\Delta V_{{\eta }_{\mathrm{H}}}$$

$$\Delta V_B$$

$$E_{\mathrm{y}}(0)=E_{\mathrm{x}}\left[\frac{2l_{\mathrm{s}}(l_{\mathrm{s}}+W)}{\mathrm{sgn}(B)r_{\mathrm{c}}{l}_{\mathrm{ee}}}-\left(1-\frac{2l_{\mathrm{s}}(l_{\mathrm{s}}+W)}{l_{\mathrm{ee}}{l}_{\mathrm{imp}}}\right)\frac{{\eta }_{\mathrm{H}}}{\eta }\right].$$

$$\left({\partial }_t+\mathbf{v}\cdot \nabla \right)\rho$$

$$m_{\mathrm{eff}}\rho \left({\partial }_t+\mathbf{v}\cdot \nabla \right)\mathbf{v}$$

$$+\mathrm{e}\rho (\mathbf{E}+\mathbf{v}\times \mathbf{B})-\frac{{\rho }_0{v}_{\mathrm{F}}{m}_{\mathrm{eff}}}{l_{\mathrm{imp}}}\mathbf{v}.$$

$$\mu (y)$$

$$T(y)$$

$$\rho (y)={\rho }_0+\frac{\partial {\rho }_0}{\partial {\mu }_0}\Delta \mu +\frac{\partial {\rho }_0}{\partial T_0}\Delta T,$$

$$p(y)=p_0+\frac{\partial p_0}{\partial {\mu }_0}\Delta \mu +\frac{\partial p_0}{\partial T}\Delta T=p_0+{\rho }_0\Delta \mu +s_0\Delta T,$$

$${\partial }_{y}p={\rho }_0{\mu }_0{\partial }_y\left(\frac{\Delta \mu }{{\mu }_0}\right)+s_0{T}_0{\partial }_y\left(\frac{\Delta T}{T_0}\right).$$

$$\rho (y)={\rho }_0+\frac{\partial {\rho }_0}{\partial {\mu }_0}\Delta \mu (y)\wedge p(y)=p_0+{\rho }_0\Delta \mu (y).$$

$$\eta {\partial }_{\mathrm{y}}^2{v}_{\mathrm{x}}(y)$$

$${\partial }_{\mathrm{y}}p$$

$$\eta {\partial }_{\mathrm{y}}^2{v}_{\mathrm{x}}(y)=\left({\rho }_0+\frac{\partial {\rho }_0}{\partial \mu }\Delta \mu \right)\mathrm{e}{E}_{\mathrm{x}}+\frac{{\rho }_0{v}_{\mathrm{F}}{m}_{\mathrm{eff}}}{l_{\mathrm{imp}}}{v}_{\mathrm{x}}(y),$$

$${\partial }_{\mathrm{y}}p={\rho }_0{\partial }_y\Delta \mu =\left(\mathrm{e}B{\rho }_0-{\eta }_{\mathrm{H}}{\partial }_y^2\right)v_{\mathrm{x}}(y).$$

$$v_{\mathrm{x}}(-l_{\mathrm{s}})=v_{\mathrm{x}}(W+l_{\mathrm{s}})$$

$$\displaystyle\Delta\mu\big{|}_{y=0}+\ \Delta\mu\big{|}_{y=W}\ \$$

$$B\to ϵB,{\eta }_{\mathrm{H}}\to ϵ{\eta }_{\mathrm{H}},\Delta \mu \to ϵ\Delta \mu ,v_{\mathrm{x}}=v_{\mathrm{x}}^0+ϵv_{\mathrm{x}}^1.$$

$$\eta {\partial }_{\mathrm{y}}^2{v}_{\mathrm{x}}^0(y)-\frac{m_{\mathrm{eff}}{v}_{\mathrm{F}}{\rho }_0}{l_{\mathrm{imp}}}{v}_{\mathrm{x}}^0(y)$$

$$\left(\mathrm{e}B{\rho }_0-{\eta }_H{\partial }_{\mathrm{y}}^2\right)v_{\mathrm{x}}^0(y)$$

$$\eta {\partial }_{\mathrm{y}}^2{v}_{\mathrm{x}}^1(y)-\frac{m_{\mathrm{eff}}{v}_{\mathrm{F}}{\rho }_0}{l_{\mathrm{imp}}}{v}_{\mathrm{x}}^1(y)$$

$$v_{\mathrm{x}}^0(y)$$

$$\Delta \mu (y)$$

Full text

Introduction. The idea of describing electrons in solids via hydrodynamics goes back to the discovery of the Gurzhi effect in (Al)GaAs quantum wires Molenkamp and de Jong (1994a, b); de Jong and Molenkamp (1995). Recently, hydrodynamic transport has received renewed attention due to the accessibility of the hydrodynamic regime in modern materials Lucas and Fong (2018); Bandurin et al. (2016); Krishna Kumar et al. (2017); Moll et al. (2016), even beyond the Fermi liquid regime Gusev et al. (2018, 2018); Levin et al. (2018). In particular, two-dimensional systems that violate parity invariance are of special interest, since they exhibit novel non-dissipative hydrodynamic transport coefficients, such as the Hall viscosity $\eta_{\mathrm{H}}$ Avron et al. (1995); Hoyos (2014); Alekseev (2016); Link et al. (2018); Pena-Benitez et al. (2019); Copetti and Landsteiner (2019). Recent experiments in graphene have shown that $\eta_{\rm H}$ may be of the same order of magnitude as the shear viscosity $\eta$ and therefore is expected to significantly affect the fluid transport Berdyugin et al. (2019). However, current theoretical and experimental works Delacretaz and Gromov (2017); Pellegrino et al. (2017); Holder et al. (2019); Sulpizio et al. do not provide a quantitative answer to the functional dependency of Hall viscous effects on all parameters describing the system, including a finite slip- $l_{\mathrm{s}}$ and impurity length $l_{\mathrm{imp}}$.

To address this open issue, we predict in this work the Hall viscous contribution $\Delta V_{\eta_{\mathrm{H}}}$ to the total Hall voltage $\Delta V_{\mathrm{tot}}=\Delta V_{\eta_{\mathrm{H}}}+\Delta V_{B}$ measured across a two-dimensional channel in the hydrodynamic regime. Here, $\Delta V_{B}$ is the Lorentz force contribution. Up to first order in the electric field, we derive the complete functional dependency of $\Delta V_{\eta_{\mathrm{H}}}$ and $\Delta V_{B}$ on all external parameters as a function of the transverse channel coordinate. In particular, we evaluate the dependence of both voltage contributions on $l_{\mathrm{imp}}$, $l_{\mathrm{s}}$, the wire width $W$, the equilibrium carrier density $\rho_{0}$ as well as on the magnetic field $B$. This allows to distinguish $\Delta V_{\eta_{\rm H}}$ from $\Delta V_{B}$ via measurements on two-dimensional Hall setups with varying $W$ and $\rho_{0}$, such as shown in Fig. 1. Most remarkably, we derive that the two voltage contributions differ in sign and that the ratio $|\Delta V_{\eta_{\rm H}}|/|\Delta V_{B}|$ decreases with increasing $|B|$. Hence, the total Hall voltage cancels at a certain critical magnetic field $B_{\mathrm{c}}$, which is a smoking gun feature of Hall viscous transport. Additionally, we show that local measurements of the Hall voltage are suitable to identify the Hall viscosity. In particular, we analytically derive the parabolic form in the transverse channel coordinate of the total Hall electric field, as was recently measured in Ref. Sulpizio et al. . While the curvature of this parabola is mainly defined by the Lorentz contribution, its offset is explicitly characterized by $\Delta V_{\eta_{\mathrm{H}}}$ and hence by the Hall viscosity.

For both, weak and strong magnetic fields as well as for clean systems our approach is analytical, whereas for intermediate field regimes we numerically solve the Navier-Stokes equations. In particular, we demonstrate that $\Delta V_{B}$ is proportional to the velocity profile integrated over the channel width, and therefore proportional to the total fluid momentum. In contrast, $\Delta V_{\eta_{\mathrm{H}}}$ exclusively depends on the gradient of the velocity profile at the boundaries of the system. Based on this, we analytically prove that for small fields and clean systems with Poiseuille-like (parabolic) velocity profile the ratio $\Delta V_{\eta_{\mathrm{H}}}/\Delta V_{B}$ is determined completely by the interplay of length scales defining the system. While the modulus of this ratio increases with the transverse electron-electron mean free path $l_{\mathrm{ee}}$, it decreases with $W,l_{\mathrm{s}}$ and $l_{\mathrm{imp}}$. Since $l_{\mathrm{ee}}$ and $l_{\mathrm{imp}}$ are density dependent, the ratio acquires a density dependence, as well. In the absence of impurities $\Delta V_{\eta_{\rm H}}/\Delta V_{B}\!=\!\mathcal{O}(\rho_{0}^{3})$, whereas for weak impurity strength there exists an additional $\mathcal{O}(\rho_{0}^{2})$ contribution.

Hence, it is possible to achieve $|\Delta V_{\eta_{\mathrm{H}}}|/|\Delta V_{B}|\!\gg\!1$ by tuning the width and density of the sample. For $\rho_{0}\!=\!9.1\times 10^{11}\mathrm{cm}^{-1}$ and $W\!=\!3\mu$m, such a regime is for instance realized in GaAs for $|B|\!\leq\!20$mT. Beyond the weak field limit, we show that increasing $|B|$ strongly reduces $|\Delta V_{\eta_{\mathrm{H}}}|/|\Delta V_{B}|$, due to the suppression of $\eta$ and $\eta_{\mathrm{H}}$ Alekseev (2016). For large magnetic fields, for example in GaAs for $|B|\!\gtrsim\!0.5$T, impurities dominate the transport, which causes an Ohmic (flat) fluid profile with vanishing $\Delta V_{\eta_{\mathrm{H}}}/\Delta V_{B}$. The critical field $|B_{\mathrm{c}}|$, at which $\Delta V_{\eta_{\mathrm{H}}}/\Delta V_{B}\!=\!-1$ and the total Hall voltage $\Delta V_{\mathrm{tot}}=0$, increases with decreasing $W,l_{\mathrm{s}},l_{\mathrm{imp}}$ and $l_{\mathrm{ee}}^{-1}$. In particular, for GaAs we find $|B_{\mathrm{c}}|\!\simeq\!\mathcal{O}(10\mathrm{mT})$, which makes our predictions experimentally verifiable.

Model. Our setup is shown in Fig. 1. We analyze the hydrodynamic flow of non-relativistic electrons along a two-dimensional channel under the combined influence of a DC electric field $\mathbf{E}=-E_{\mathrm{x}}{\bf e}_{\mathrm{x}}$ and an out-of-plane magnetic field $\mathbf{B}=-B{\bf e}_{\mathrm{z}}$ 111In what follows, we assume that the screening length in our material is much shorter than the width of our channel. Hence, the electric field sourced by the redistribution of charge carriers in our fluid can be neglected.. To justify our hydrodynamic approach, we assume that $l_{\mathrm{ee}}$ is the shortest length scale in our system Erdmenger et al. (2018). In particular, it is smaller than the cyclotron radius $r_{\mathrm{c}}\!=\!m_{\mathrm{eff}}v_{\mathrm{F}}/|\mathrm{e}B|$, which is defined in terms of the effective electron mass $m_{\mathrm{eff}}$, the electron charge $\mathrm{e}<0$ and the Fermi velocity $v_{\mathrm{F}}$. Additionally, we assume that our system is incompressible and has translational invariance in longitudinal, as well as vanishing current in transversal direction. These assumptions lead to the hydrodynamic variables 222The charge conservation equation $\partial_{t}\rho+\nabla\cdot(\rho\mathbf{v})=0$ is trivially satisfied due to our assumption of a steady and translational invariant flow $\mathbf{v}=v_{\mathrm{x}}(y)\mathbf{e}_{\mathrm{x}}$.

[TABLE]

where $\Delta\mu$ is the local chemical potential fluctuation around the global equilibrium value $\mu_{0}$ 333In general, one could also allow for a temperature gradient $T\!\to\!T\!+\!\Delta T$. However, we show in Ref. sup that this contribution is negligibly small for Fermi liquids. In addition, since we restrict ourselves to $T=\mathcal{O}(1{\rm K})$, phonons are negligible in our approach Molenkamp and de Jong (1994a, b).. For this ansatz, the dynamical equations are defined by the non-relativistic Navier-Stokes equations for incompressible fluids Landau and Lifshitz (1987); Romatschke (2010); Luciano Rezzolla (2013); sup . In the framework of linear response theory with respect to the external electric field $E_{\mathrm{x}}$, these equations are given by 444Let us emphasize that we took into account the term $\propto\!\Delta\mu E_{\mathrm{x}}$, even though it is higher order in $E_{\mathrm{x}}$ and therefore does not affect our linear response results. We did so to explicitly see that the Lorentz and the Hall-viscosity induced force influences the velocity profile in the expected way (cf. Ref. [29]).

[TABLE]

Here, $p\!=\!p_{0}\!+\!\rho_{0}\Delta\mu$ defines the pressure in terms of $\Delta\mu$. Let us emphasize that the incompressibility condition for our unidirectional fluid flow, $\nabla\cdot\mathbf{v}\!=\!0$, solely implies a constant density along the $\mathbf{e}_{\mathrm{x}}$-direction. In particular, $v_{\mathrm{y}}=0$ allows for density fluctuations satisfying $\partial_{\mathrm{y}}\rho\propto\partial_{\mathrm{y}}\Delta\mu\neq 0$. In two dimensions, the dynamics of incompressible, non-relativistic fluids is entirely captured by the shear and Hall viscosities Scaffidi et al. (2017); Alekseev (2016)

[TABLE]

where $\eta_{0}\!=\!m_{\mathrm{eff}}\rho_{0}v_{\mathrm{F}}l_{\rm ee}/4$ is the shear viscosity at zero magnetic field. The main goal of this work is to derive the total Hall voltage

[TABLE]

up to first order in $E_{\mathrm{x}}$, and to separate Hall viscous and Lorentz force contributions therein. Therefore, we solve Eqs. (2) and (3) under the general boundary conditions

[TABLE]

The slip length $l_{\mathrm{s}}$ characterizes the velocity profile at the boundaries, as shown in Fig. 1. For $l_{\mathrm{s}}\!=\!0$, the fluid flow is Poiseuille-like (parabolic), while $l_{\mathrm{s}}\!\to\!\infty$ defines the diffusive regime with a flat $v_{\mathrm{x}}(y)$ 555 Notice that we do not use standard Robin type boundary conditions to define the slip length in our system Kiselev and Schmalian (2019). This relies on the fact that our set of boundary conditions naturally describes the case of a finite drift velocity.In our supplemental material we prove that in our channel geometry, Robin-type boundary conditions are equivalent with our choice of boundary conditions, Eq.(6)..

Hall Response. The total Hall voltage $\Delta V_{\mathrm{tot}}\!=\!\Delta V_{\eta_{\mathrm{H}}}\!+\Delta V_{B}$ consists of two different terms, namely the Hall viscous and the Lorentz force contributions. To analytically derive the functional dependence of these building blocks, we first restrict ourselves to weak magnetic fields, defined by $l_{\mathrm{G}}/r_{\mathrm{c}}\!\ll\!1$. The Gurzhi length $l_{\rm G}\!=\!\sqrt{l_{\rm imp}\eta/(\rho_{0}m_{\rm eff}v_{\rm F})}$ quantifies the relative strength of impurity to shear effects. While the flow is Poiseuille-like for $l_{\rm G}/W\!\gg\!1$, it becomes Ohmic for $l_{\rm G}/W\!\ll\!1$. The assumption of weak magnetic fields allows us to expand the velocity profile and hence the Navier-Stokes equations in powers of $B$.

With this ansatz, the linearized Navier-Stokes Eqs. (2)-(3) predict the total Hall voltage sup

[TABLE]

where the functions $A_{1,2}(l_{\mathrm{s}},W)$ are explicitly defined in Ref. sup . For $W/l_{\rm G}\ll 1$, corresponding to dominant shear effects, Eq. (7) implies up to second order in $l_{\mathrm{G}}^{-1}$

[TABLE]

Hence, for weak magnetic fields, this implies the ratio

[TABLE]

Thus, for small magnetic fields and $W/l_{\mathrm{G}}\!\ll\!1$, this ratio is completely determined by the characteristic length scales of the system. Its modulus decreases as a function of $W$, $l_{\mathrm{s}}$ and $l_{\mathrm{imp}}$, whereas it increases with $l_{\mathrm{ee}}$. While the hydrodynamic assumption $l_{\mathrm{ee}}\!\ll\!l_{\mathrm{imp}}$ strongly suppresses the second term in Eq. (9) 666 In this limit, according to Eq. (Functional dependence of the Hall viscosity-induced transverse voltage in two-dimensional Fermi liquids), $\Delta V_{\eta_{\mathrm{H}}}$ becomes insensitive to the concrete form of the boundary conditions., the first term experimentally can be much larger than one Gusev et al. (2018, 2018); Levin et al. (2018). For very small $l_{\mathrm{s}}$ and $W$ in comparison to $l_{\mathrm{ee}}$, engineered e.g. in Ref. Moessner et al. (2019), the Hall viscous contribution can strongly dominate the Lorentz signal. In addition, Eq. (Functional dependence of the Hall viscosity-induced transverse voltage in two-dimensional Fermi liquids) provides the density dependence of each voltage contribution. For temperatures much smaller than the Fermi energy, we obtain the dependence Giuliani and Quinn (1982); Czycholl (2016)

[TABLE]

where $n_{\mathrm{imp}}$ is the impurity concentration and $f_{1,2,3}$ are density independent functions, given in Ref. sup . Explicitly, Eq. (10) predicts $\Delta V_{\eta_{\mathrm{{}_{H}}}}\!/\Delta V_{\rm B}\!=\!\mathcal{O}_{\mathrm{clean}}(\rho_{0}^{3})+\mathcal{O}_{\mathrm{imp}}(\rho_{0}^{2})$. Hence, in the weak field limit and for $W/l_{\mathrm{G}}\!\ll\!1$, the Hall viscous contribution becomes strongly enhanced in comparison to the Lorentz force signal as the carrier density increases. Summarizing, the distinct dependence of $\Delta V_{\eta_{\mathrm{H}}}$ and $\Delta V_{B}$ on $\rho_{0},W,l_{\mathrm{s}},l_{\mathrm{imp}}$ and $l_{\mathrm{ee}}$ allows to experimentally distinguish these two contributions in the limit of weak magnetic fields and clean systems. Beyond this limit, the Hall viscous and Lorentz force contributions to $\Delta V_{\mathrm{tot}}$ need to be evaluated numerically. By solving Eqs. (2) and (3) for the velocity profile, we obtain

[TABLE]

In what follows, we numerically investigate the dependence of these voltage contributions on $B,W,l_{\mathrm{s}},l_{\mathrm{imp}}$ and $l_{\mathrm{ee}}$. Notice, that while $\Delta V_{B}$ is proportional to the integrated value of $v_{\mathrm{x}}(y)$, which characterizes the overall fluid momentum, $\Delta V_{\eta_{\mathrm{H}}}$ is totally determined by the gradient of the velocity profile at the boundaries of the channel. In Fig. 2, we plot the velocity profile of a general Fermi liquid for different magnetic fields. With increasing $|B|$ the fluid is accelerated due to the corresponding suppression of $\eta$ (cf. Eq. (4)). According to Eq. (11), this leads to an enhancement of $|\Delta V_{B}|$. For large fields, $\eta$ vanishes and solely impurity scattering is responsible for momentum relaxation, leading to an Ohmic velocity profile sup .

The dependence of $\Delta V_{\eta_{\mathrm{H}}}$ on the magnetic field is more subtle. Figure 2 shows that for weak magnetic fields, the gradient $|\partial_{\mathrm{y}}v_{\mathrm{x}}(y)|_{y=0,W}|$ increases as a function of $|B|$. According to Eq. (11), this corresponds to an enhancement of $|\Delta V_{\eta_{\mathrm{H}}}|$. As discussed above, impurity scattering causes an Ohmic (flat) velocity profile for large magnetic fields. This decreases $|\partial_{\mathrm{y}}v_{\mathrm{x}}(y)|_{y=0,W}|$ and therefore reduces $|\Delta V_{\eta_{\mathrm{H}}}|$. Altogether, this implies that systems in which Hall viscous effects dominate the weak magnetic field regime are eventually always driven to $|\Delta V_{\eta_{\rm H}}|/|\Delta V_{\rm B}|\!\ll\!1$. The transition from Hall viscous to Lorentz force dominated transport occurs for $\Delta V_{\eta_{\rm H}}/\Delta V_{\rm B}\!=\!-1$, where $\Delta V_{\mathrm{tot}}=0$. This happens at a certain critical magnetic field $B_{\mathrm{c}}$, which strongly depends on $W$, $l_{\mathrm{s}}$ and $l_{\mathrm{ee}}$.

This is shown in Fig. LABEL:fig:vhall, which displays $|\Delta V_{\eta_{\rm H}}|/|\Delta V_{\rm B}|$ as a function of $W,l_{\mathrm{s}},l_{\mathrm{ee}}$ and $l_{\mathrm{imp}}$ 101010Note that the system lies within the range of validity of the hydrodynamic regime, even though $l_{\rm ee}\lesssim W$ Sulpizio et al. .. According to Eq. (11), the dependence of this ratio on these parameters can be explained by analyzing the corresponding velocity profiles, depicted in Fig. 2. As $l_{\rm s}$ increases, the gradient $|\partial_{\mathrm{y}}v_{\mathrm{x}}(y)|_{y=0,W}|$ decreases, leading to a reduction of $|\Delta V_{\eta_{\rm H}}|$. In contrast, since the overall fluid momentum is enhanced, $|\Delta V_{B}|$ increases as a function of the slip length. Therefore, $|\Delta V_{\eta_{\rm H}}|/|\Delta V_{\rm B}|$ decreases with $l_{\mathrm{s}}$, as illustrated in Fig. LABEL:fig:vhall. Increasing the channel width leads to the same result, since it also increases the overall fluid momentum and at the same time decreases $|\partial_{\mathrm{y}}v_{\mathrm{x}}(y)|_{y=0,W}|$. In contrast, $|\Delta V_{\eta_{\rm H}}|/|\Delta V_{\rm B}|$ increases with the electron-electron mean free path. According to Eq. (4), the viscosities $\eta$, $|\eta_{\mathrm{H}}|$ as well as the ratio $|\eta_{\mathrm{H}}|/\eta$ increase as a function of $l_{\mathrm{ee}}$ as long as $l_{\mathrm{ee}}\!\ll\!r_{\mathrm{c}}$, implied by the hydrodynamic assupmtion. This reduces the overall fluid momentum but increases $|\Delta V_{\eta_{\rm H}}|$. Last but not least, Fig. LABEL:fig:vhall shows that since $l_{\mathrm{ee}}\ll l_{\mathrm{imp}}$, the impurity length does not significantly change $\Delta V_{\eta_{\rm H}}/\Delta V_{\rm B}$.

To demonstrate the experimental relevance of our predictions, Fig. 4 finally shows $|\Delta V_{\eta_{\rm H}}|$, $|\Delta V_{\rm B}|$ and $|\Delta V_{\rm tot}|$ in GaAs as a function of the magnetic field Gusev et al. (2018, 2018)1010footnotemark: 10. Since in this material the slip length is highly dependent on the etching technique, we plot the corresponding curves for $l_{\rm s}=0,0.5,1.0{\rm\mu m}$ 777Private communication with Prof. Laurens Molenkamp, University of Wuerzburg. For $|B|\!\ll\!|B_{\mathrm{c}}|$, the ratio $|\Delta V_{\eta_{\rm H}}|/|\Delta V_{\rm B}|\gg 1$, giving rise to a Hall viscosity dominated transport regime. As indicated by Eq. (9) and confirmed by Fig. LABEL:fig:vhall, this property is strongly pronounced if the slip length decreases. With the exception of $l_{\mathrm{ee}}$, it is possible to evaluate all parameters in Eqs. ​​(2)-(4) precisely. Therefore, measuring $\Delta V_{\eta_{\rm H}}$ and $\Delta V_{\rm B}$ with increasing $|B|$, enables us to determine the electron-electron mean free path in this sample by fitting theoretical and experimental data. In general, this procedure works for any Fermi liquid and therefore is a powerful tool to evaluate the electron interaction strength. For $B\!=\!B_{\mathrm{c}}$, the two voltage contributions are equal in magnitude but opposite in sign, leading to $\Delta V_{\mathrm{tot}}=0$. Experimentally, this smoking gun feature can be used to prove our Hall viscous magnetotransport theory. For $|B|\!\gg\!|B_{\mathrm{c}}|$, Fig. 4 shows that transport is dominated by the Lorentz signal. Eventually, this leads to an Ohmic, impurity driven velocity profile sup . Our present approach does not incorporate the formation of Landau levels which in GaAs occurs beyond the applicability of hydrodynamics for $|B|\!\gtrsim\!1$T. Finally, we remark that the Hall voltage we predict depends non-linearly on the magnetic field strength. Since ballistic transport predicts $\Delta V_{\rm B}\propto B$, this non-linear behavior is a direct consequence of the hydrodynamic regime.

Local Hall response. So far, we focused on global voltage measurements across the entire channel. In addition, Eq. (11) predicts the local voltage contributions:

[TABLE]

Hence, the total Hall voltage $\Delta V_{\mathrm{tot}}$ scales as $y^{3}$, whereas the associated Hall field $E_{\rm y}\!=\!-\mathrm{d}\Delta V_{\rm tot}/\mathrm{d}y$ has parabolic structure. For large $l_{\mathrm{imp}}$, the curvature of this parabola, $\kappa\!=\!-4E_{\mathrm{x}}\left[\mathrm{sgn}(B)/r_{\mathrm{c}}+\eta_{\mathrm{H}}/(\eta l_{\mathrm{imp}})\right]\!/l_{\mathrm{ee}}$, is mainly characterized by the Lorentz contribution. In contrast, the Hall viscous signal significantly defines the offset of this parabola,

[TABLE]

For $l_{s}\!\ll\!l_{\mathrm{ee}}\!\ll\!l_{\mathrm{imp}}$, this relation becomes $E_{\mathrm{y}}(0)\!\simeq\!E_{\mathrm{x}}\eta_{\mathrm{H}}/\eta$. Recently, the authors of Ref. Sulpizio et al. measured the local Hall voltage in the hydrodynamic regime of graphene, particularly satisfying $l_{s}\!\simeq\!l_{\mathrm{ee}}\!\ll\!W$. They explicitly reproduced the parabolic form of the Hall field, as well as that $|\kappa|$ decreases with increasing $l_{\mathrm{ee}}$.

Conclusions. Our results provide new insights into Hall viscous effects of two-dimensional non-relativistic electron fluids in external magnetic fields. We present a setup that allows to distinguish unambiguously between Hall viscous and Lorentz force contributions to the total Hall voltage. In particular, we prove that the ratio of Hall viscous to Lorentz force induced voltage contributions is negative and that in clean systems its modulus decreases with the width, slip-length and carrier density, while it increases with the electron-electron mean free path of our setup. Additionally, we show that this identification is possible by measuring the Hall voltage locally. In particular, we prove that the total Hall field is parabolic in the transverse channel coordinate, as it was recently measured in Ref. Sulpizio et al. . While the curvature of this parabola mainly originates from Lorentz effects, its offset is characterized by the Hall viscosity. For typical Fermi liquids such as GaAs, we find that the Hall viscous signal dominates over the Lorentz force contribution up to a critical magnetic field which is on the order of a few tens of millitesla. At this field, the total Hall voltage vanishes, clearly indicating Hall viscous transport. Hence, our predictions pave the way to experimentally identify Hall viscous effects and explicitly demonstrate how to determine the electron-electron interaction strength. Possible future directions of our approach include two-dimensional massive Dirac systems, where parity and time-reversal symmetry are already broken for vanishing magnetic fields Böttcher et al. (2019) and a torsional Hall viscous term is present Hoyos (2014); Hughes et al. (2013).

Supplemental material

Supplemental material: “Functional dependence of the Hall viscosity-induced transverse voltage in two-dimensional Fermi liquids"

Ioannis Matthaiakakis These three authors contributed equally to this work and are listed alphabetically.

David Rodríguez Fernández These three authors contributed equally to this work and are listed alphabetically.

Christian Tutschku These three authors contributed equally to this work and are listed alphabetically.

Ewelina M. Hankiewicz

Johanna Erdmenger

René Meyer

Equations of motion. In what follows, we derive the equations of motion for a charged two-dimensional Fermi liquid confined to a channel geometry, in the framework of linear response theory. In particular, we study fluid dynamics in presence of a DC electric field $\mathbf{E}$ and an out-of-plane magnetic field $\mathbf{B}$. For such a fluid, the equations of motion are given by charge continuity as well as by momentum conservation (Navier-Stokes):

[TABLE]

In our analysis, we consider a steady, hydrodynamic flow of electrons, which is translationally invariant along the $\mathbf{e}_{\rm x}$-direction. Moreover, we assume a vanishing current in the $\mathbf{e}_{\rm y}$-direction, implying that the velocity profile takes the form $\mathbf{v}=v_{\rm x}(y)\mathbf{e}_{\rm x}$. To obtain an inhomogeneous, non-trivial velocity profile, our system needs to deviate from global thermal equilibrium. However, in order to be able to linearize Eqs. (S1) and (S2), we assume that variations of the chemical potential and temperature are small compared to their global equilibrium values

[TABLE]

For typical Fermi liquids, such as GaAs, $\mu_{0}={\cal O}(50{\rm~{}meV})$ whereas $T_{0}={\cal O}(1{\rm~{}K})$. Note, that due to time and translational invariance in $\mathbf{e}_{\rm x}$-direction, $\Delta\mu$ and $\Delta T$ solely vary in the $\mathbf{e}_{\rm y}$-direction. In terms of $\mu(y)$ and $T(y)$, pressure and density fluctuations in our system are given by

[TABLE]

where $s_{0}$ is the equilibrium entropy of our fluid. Under the assumption of a steady and unidirectional fluid flow with $v_{\mathrm{y}}=0$, Eq. (S1) reduces to $\partial_{\rm x}\rho=0$. This enforces a constant density along the fluid flow, whereas it allows for density fluctuations in $\mathrm{\mathbf{e}}_{\mathrm{y}}$-direction. Explicitly, that is compatible with our incompressibility condition $\nabla\cdot\mathbf{v}\!=\!0$. Together, Eqs. (S1)-(S4) dictate the dynamics of $v_{\rm x}(y)$, $\Delta\mu(y)$ and $\Delta T(y)$. For our system the gradient of temperature is negligible compared to the gradient of the chemical potential. To see this explicitly, we write

[TABLE]

Due to the assumptions in Eq. (S3), the dimensionless gradients $\partial_{y}(\Delta\mu/\mu_{0})$ and $\partial_{y}(\Delta T/T_{0})$ are of the same order of magnitude. As a consequence, the relative strength of the chemical potential contribution to the temperature contribution in Eq. (S5) is given by ${\cal U}=\rho_{0}\mu_{0}/s_{0}T_{0}$. If ${\cal U}\gg 1$, the chemical potential gradient dominates over temperature fluctuations and vice versa. For typical Fermi liquids, such as GaAs, we find ${\cal U}\simeq 10^{16}$. Hence, in those samples the gradient of temperature is clearly negligible. Thus, we assume $T(y)=T_{0}$ and define

[TABLE]

Substituting this into Eq. (S2), leads to

[TABLE]

Let us emphasize that we did not include the electric field sourced by our inhomogeneous density distribution $\rho(y)$ in Eqs. (S7) and (S8). This is justified by the assumption that the screening length of our material is much shorter than the width of our channel. Hence, the corresponding electric field is expected to be much smaller than $E_{\rm x}$, and therefore can be neglected.

So far, Eqs. (S7) and (S8) still deviate from Eqs. (2) and (3) of our main text, due to the presence of additional terms $\propto\!\Delta\mu(y)v_{\rm x}(y)$. These terms induce non-linear corrections to our physical observables in terms of the electric field $E_{\rm x}$. In the framework of linear response theory, we are therefore allowed to drop them without loss of generality. In particular, this leads to Eqs. (2) and (3) of our main text:

[TABLE]

Notice, that we took into account the term $\propto\!\Delta\mu(y)E_{\rm x}$, even though it also scales non-linearly in $E_{\mathrm{x}}$ and, therefore, does not affect our phyiscal observables. However, we kept this term because we want to explicitly check that the Lorentz and the Hall-viscosity induced force influences the velocity profile in the expected way. Namely, we expect $v_{\mathrm{x}}(y)$ to deviate from its symmetric Poiseuille form, corresponding to a fluid flow which concentrates towards one of the sides of the channel, depending on the sign of the magnetic field. The simplest way to see this deviation is to take into account the $\Delta\mu E_{\mathrm{x}}$ term since it only couples to the first order correction of the velocity profile and decouples from the equations for $v^{(0)}_{\mathrm{x}}(y)$.

For our setup, we supplement Eqs. (S9) and (S10) with general boundary conditions of the form

[TABLE]

Below, in section Robin Boundary Conditions, we prove that this choice is equivalent to standard Robin type boundary conditions, as defined in Ref. Kiselev and Schmalian (2019). In what follows, we provide an explicit solution of the Navier-Stokes Eqs. (S9)-(S10) under the boundary condition (S11) for weak and for strong magnetic fields, since in this regime the physical properties of the analytic solutions become most apparent.

Weak magnetic field regime. The weak magnetic field limit $r_{\mathrm{c}}\gg l_{\rm G}$ allows for the expansion of the velocity profile and hence of the Navier-Stokes equations in powers of $B$. Technically, this expansion is achieved by introducing the dimensionless parameter $\epsilon\ll 1$, satisfying

[TABLE]

In particular, this assumes that to first order the system responds linearly in terms of the magnetic field. With this ansatz the Navier-Stokes Eqs. (S9)-(S10) reduce to

[TABLE]

To find an explicit solution of this set of equations we first determine $v^{0}_{\rm x}(y)$ by solving Eq. (S13), which is the zero-field Poiseuille flow equation in the presence of impurities. Once we have obtained $v^{0}_{\rm x}(y)$, we calculate $\Delta\mu(y)$ by integrating Eq. (S14). Substituting this quantity in Eq. (S15) finally allows us to derive the first order velocity correction $v^{1}_{\rm x}(y)$. Explicitly, we find

[TABLE]

where for clarity we defined

[TABLE]

Before we explicitly present our result for $v^{1}_{\rm x}(y)$, we want to emphasize that Eq. (S17) predicts the total Hall voltage $\Delta V_{\rm tot}=-[\Delta\mu(W)\!-\!\Delta\mu(0)]/\mathrm{e}$, measured across the sample. To derive Eq. (8) of the main text, we need to expand the hyperbolic functions in Eqs. (S18)-(S20) up to third order in $W/l_{\rm G}$. As a result, we find

[TABLE]

In Eq. (10) of the main text, we have illustrated the density dependence of these voltage contributions,

[TABLE]

However, so far we did not specify the functions $f_{1,2,3}$. This will be the purpose of the following paragraph. The linear, impurity independent scaling of $\Delta V_{\eta_{\rm H}}$ is defined as the $l_{\mathrm{imp}}\!\rightarrow\!\infty$ limit of Eq. (S21). The density dependence of this term is given by (cf. Eq. (4) of our main text)

[TABLE]

Here, according to Refs. Novikov (2006); Giuliani and Quinn (1982); Alekseev (2016); Qian and Vignale (2005), we introduced the second momentum relaxation time

[TABLE]

where $k_{\mathrm{B}}$ is the Boltzmann constant and $F_{\pi}$ is a geometric factor, characterizing electron-electron scattering amplitudes. Since density variations do not significantly change the ln${}^{2}(\mu/k_{\mathrm{B}}T)$ terms in the Fermi liquid regime $\mu\!\gg\!\mathrm{k}_{\mathrm{B}}T$, we treat $f_{1}$ as density independent function. Moreover, the impurity contribution to $\Delta V_{\eta_{\rm H}}$ is given by

[TABLE]

Here, we considered point like impurities with concentration $n_{\mathrm{imp}}$, scattering strength $\nu_{0}$ and inverse momentum relaxation time $\tau_{\mathrm{imp}}^{{}_{-1}}\!=\!v_{\mathrm{F}}l_{\mathrm{imp}}^{{}_{-1}}\!=\!m_{\mathrm{eff}}\nu_{0}^{2}\hbar^{-3}\rho_{0}n_{\mathrm{imp}}$ Czycholl (2016). In the same manner, Eq. (S22) evolves to

[TABLE]

After having clarified how to disentangle $\Delta V_{\eta_{\rm H}}$ and $\Delta V_{B}$ in terms of their width and density dependence, let us proceed in presenting our solution for $v^{1}_{\rm x}(y)$. Therefore, we substitute Eq. (S17) into Eq. (S15), which leads to the first order velocity correction

[TABLE]

Here, for clarity, we defined the following functions

[TABLE]

Notice, that for $\eta_{\rm H}\neq 0$, the first order correction $v_{\mathrm{x}}^{1}(y)$ breaks the reflection symmetry of the entire velocity profile with respect to $y=W/2$, as expected. Explicitly this is caused by the linear terms in powers of $y$ within Eq. (S28). Moreover, since $C_{1,2},\,\lambda_{1,2}$ and $A_{1,2,3,4}$ are non-linear in $E_{\rm x}$ and $l_{\rm imp}$, Eq. (S28) implies that the first order correction $v^{1}_{\rm x}\ll v^{0}_{\rm x}$ in the linear response regime.

Strong magnetic field regime. We now move on to the discussion of strong magnetic fields, implicitly defined by $r_{\mathrm{c}}\ll l_{\mathrm{G}}$. In this limit, $\eta$ and $\eta_{\mathrm{H}}$ tend to zero (cf. Eq. (4) of our main text), which strongly simplifies Eqs. (S9) and (S10). In particular, these equations yield

[TABLE]

Here, we dropped the term $\propto\!\!\Delta\mu\,E_{\rm x}$ to make explicit predictions for our linear response theory. In particular, the solution of Eq. (S32) reads:

[TABLE]

Hence, the Hall voltage solely depends on $l_{\mathrm{imp}}$ and $W$ in the limit of strong magnetic fields.

Robin Boundary Conditions. In what follows, we prove that our set of boundary conditions in Eq. (S11) is equivalent to Robin type boundary conditions, satisfying

[TABLE]

Here $u^{0}_{\rm x}(y)$ defines the Robin velocity profile and $l_{R}$ is the corresponding slip length. As a solution of the Navier-Stokes Eqs. (S13)-(S15), $u^{0}_{\rm x}(y)$ is given by

[TABLE]

The amplitudes $C_{1}$ and $C_{2}$ are defined to be:

[TABLE]

In order to prove the equivalence between the two types of boundary conditions, it is sufficient to show that the velocity profiles $u^{0}_{\rm x}(y)$ and $v^{0}_{\rm x}(y)$ match each other. According to Eq. (S35), the Robin slip length is defined to be the normalized value of the inverse first derivative of the velocity profile at the boundaries. Hence, let us define the effective Robin slip length for our velocity profile

[TABLE]

This quantity allows us to derive the equivalence of the two types of boundary conditions if we can prove that

[TABLE]

For $l_{\mathrm{G}}/r_{\mathrm{c}}\ll 1$, Eq. (S16) defines $v^{0}_{\rm x}(y)$ analytically. In this limit, Eq. (S38) implies for $l_{R}=l_{R}^{\mathrm{eff}}$:

[TABLE]

In fact, this mapping identifies the velocity profiles $u^{0}_{\rm x}(y)$ and $v^{0}_{\rm x}(y)$, since it maps Eq. (S35) onto Eq. (S16). In particular, the Robin type equivalents to the Hall viscous and the Lorentz force contribution to the total Hall voltage in Eqs. (S21) and (S22) are given by

[TABLE]

Since Eq. (S21) and Eq.(S22) are derived under the assumption $W/l_{\rm G}\ll 1$, Eq.(Supplemental material) is obtained by inserting the Robin slip length, given by Eq. (Supplemental material) in the same limit, $l_{R}\simeq l_{\rm s}+l^{2}_{\rm s}/W$. Beyond the weak magnetic field regime Eq. (S39) is still justified. Since in this limit $l_{\mathrm{G}}/r_{\mathrm{c}}\gtrsim 1$, this can not be proven analytically. Instead, we confirmed the validity of Eq. (S39) by numerically evaluating the Robin slip length in Eq. (S38). As soon as the cyclotron radius $r_{\mathrm{c}}\ll W$, one enters the strong magnetic field regime. While in this limit the Robin slip length diverges, our slip length remains constant by definition. Therefore, we can not justify Eq. (S39) for strong magnetic fields. However, in this limit the equivalence of both types of boundary conditions is given trivially. Since the fluid layer interaction decreases as function of $r_{\mathrm{c}}$, all our results become slip length and therefore boundary condition independent for $r_{\mathrm{c}}\ll W$ (cf. Eq. (S33)).