Coulomb drag induced non-local resistance in double graphene layers

Abstract

We study the effect of Coulomb drag between graphene layers in presence of viscosity term. To do this, we use the simple model of Stokes equations for drift velocities in active and passive layers, known as Pogrebinskii’s approach. The solution to these equations allows us to find the potential distribution, and thus the non-local drag resistance of passive layer. It is shown that in viscous regime the non-local resistance may take negative values, in contrast, the ohmic regime results in positive non-local resistance for all drag strengths. Additionally, we discuss the influence of magnetic field on the non-local drag magnetoresistance.

Introduction

The recent progress in fabrication of two-dimensional (2D) materials and particularly graphene gave rise to the study of electron hydrodynamics, which was difficult to reach before in conventional “dirty” metals^1,2,3,4. The hydrodynamics viscous regime is crucial when the electron-electron scattering mechanism is dominant^3,4. This means that the typical electron-electron scattering length must be the shortest scale compared to other scattering mechanisms, such as phonons, impurities and etc. All these length scales strongly depends on temperature⁵, and the hydrodynamic regime is accessible at intermediate temperatures, which is of the order of hundred Kelvins in graphene monolayer⁶. In contrast, electron impurity mechanism is important at low temperatures whereas electron-phonon scattering dominates at large temperatures⁶. Many surprising experimental result have been demonstrated in hydrodynamical regime⁷. Among them, there are the increase of thermal conductivity and breakdown of the Wiedemann-Franz law in graphene⁸, the increase of electrical conductance in graphene constriction due to superballistic behavior of viscous flow^9,10, the non-local negative resistance in graphene¹¹.

Nowadays, the interest in studying the electron hydrodynamics is growing both experimentally and theoretically^{12,13,14,15,16,17,18}. Particularly, compared to the listed above transport measurements with applied bias or temperature difference, the collective hydrodynamical behavior of electrons was shown using the alternative measurements with the help of quantum spin magnetometer¹⁹. Further, it was theoretically suggested to investigate the hydrodynamics in other novel systems such as 2D anomalous Hall material, fractons, Weyl semimetals in presence of electron-phonon interaction^20,21,22,23. Moreover, in recent past, the Coulomb drag measurements became a powerful tool to investigate properties of 2D graphene based materials^{24,25,26,27,28}

The drag effect allows to investigate the interlayer interaction nature as well as the properties of collective excitations in each interacting layer²⁹. The effect of Coulomb drag was suggested by M. B. Pogrebinskii³⁰, and shortly the idea is as follows: The experimental setup consists of two closely separated conductors, but no charge transfer is allowed between these conductors. In such setup the electric current, which flows in one of the conductors (“active” layer) causes the induced current in other one (“passive” layer), due to Coulomb interaction between charged particles. After Pogrebinskii’s paper there was a huge progress in studying the Coulomb drag effect in different physical systems as such semiconductors and 3D metallic systems, one-dimensional nanowires, zero dimensional nanostructures, quantum point contacts, quantum Hall systems²⁹. Moreover, it was shown that the drag effect can be established in phonon³¹, photon³², ion³³ and some other hybrid systems³⁴. The drag effect was studied experimentally and theoretically for non hydrodynamic regimes in a systems of two parallel graphene layers as well²⁹. It is worth mentioning that one can also consider a bilayer graphene^35,36,37, but in this case the interlayer coupling constant can compete with the pure Coulomb interaction between carriers. The investigation of drag effect in graphene and other 2D systems (such as silicene, germanene, stanene³⁸) is as well essential compared to metallic and semiconductor 3D systems since the low-energy excitations in graphene are described by the linear Dirac spectrum and the Galilean invariance is broken^4,29,39. Concurrently, it was studied theoretically the damping of plasmons in passive layer and Berry phase effects induced by Coulomb drag in 2D graphene layers^40,41.

In this work we study the effect of mutual drag caused by interlayer Coulomb interaction. The system under consideration consists of two identical closely spaced, but electrically isolated two dimensional graphene layers. The electric current $I_1(x)=I\delta (x)$ injected in one of the layers, known as the “active” layer, induces spatial dependent voltage distribution $V_2(x)$ (or associated non-local resistance, $R_{\text {nl,2}}(x)$) in the other, “passive” layer due to interlayer electron-electron interaction (see Fig. 1). The mutual drag effect is taken into account introducing phenomenological relaxation rate $\tau _d=1/\gamma _d$, which generally depends on the form of interlayer interaction, macroscopic nature of “active” and “passive” layers, temperature²⁹. We mainly interested in the influence of viscosity in each layer on drag effect, namely on non-local resistance in passive layer. In order to consider the particular geometry, we mainly follow the steps of Ref.¹¹, where the calculations were performed for pristine graphene in both viscous and ohmic regimes and we as well use the no-slip boundary conditions. We assume that the chemical potentials in both layers larger then the temperature, the viscous regime is reachable in both layers and the interlayer Coulomb effect is weak, thus the contribution of mutual drag on the transport properties of active layer can be considered as a sub-leading correction. The typical values for temperature are $T \simeq 100$ K, the carrier density near charge neutrality is $n \simeq 10^{10} \text {cm}^{-2}$. For these typical values one can show that $\gamma ^{-1} \simeq 0.5$ ps and electron–electorn scattering time is $\gamma ^{-1}_{ee} \simeq 80$ fs. Therefore, hydrodynamics regime is reachable since $\gamma \ll \gamma _{ee}$¹¹. The relaxation rate of drag $\gamma _d$ can be tuned and might be larger or smaller than momentum relaxation time due to impurities, defects and phonons. The rest of the article is organized as follows. We first introduce the Pogrebinskii’s approach, in the spirit of Refs.^29,30. Then, the simple Poiseuille type solution is considered. Further, we provide the qualitative analytical expression for voltage distribution and non-local resistance in passive layer. Additionally, we study the general result for stream function , velocity distribution and electrostatic potential patterns in passive layer numerically. Finally, we present our conclusions and possible future perspectives.

Fig. 1

Sketch of setup. Current I at spatial point $(x,y)=(0,w)$ is injected into active layer. This current induces drag signal in passive layer due to Coulomb interaction between layers.

We assume that in presence of drag effect the hydrodynamical description of transport in each layer is still applicable, namely electron-electron scattering rates in both layers are much faster than those due to impurity and phonon scatterings. We consider the subsonic velocities, namely drift velocities in both layers to be smaller enough then the Fermi velocity, $v_F \sim 10^6$ m/s. It is known that in this so called linear regime, the transport of electrons can be described by an incompressible fluid¹¹. Therefore, in general case the flow of carriers are described by coupled system of Navier-Stokes equations

$$\begin{aligned} {\left\{ \begin{array}{ll} \nabla \cdot \textbf{v}_1=0, \\ (\partial _t+\gamma _1 + \textbf{v}_1 \cdot \nabla ) \textbf{v}_1-\nu _1 \nabla ^2 \textbf{v}_1=\frac{e}{m}\textbf{E}_1-\gamma _d (\textbf{v}_1-\textbf{v}_2)+\textbf{f}_1, \\ \nabla \cdot \textbf{v}_2=0, \\ (\partial _t+\gamma _2 + \textbf{v}_2 \cdot \nabla ) \textbf{v}_2-\nu _2 \nabla ^2 \textbf{v}_2=\frac{e}{m}\textbf{E}_2-\gamma _d (\textbf{v}_2-\textbf{v}_1)+\textbf{f}_2, \end{array}\right. } \end{aligned}$$

(1)

where e is the elementary charge carried by electron, m is the effective mass defined through momentum ($\textbf{p}=m n\textbf{v}$, and n is particle number density), $\textbf{E}_i$ is the electric field distribution in each layer, $\gamma _i$ is the inverse momentum relaxation time due to impurities or phonons within layer, $i=1,2$. The pressure induced forces for each layer are given by $\textbf{f}=\nabla P/mn +(\textbf{v}/mnv^2_F) \partial _t P$. The $\gamma _d$ is the drag inverse time, which describes the mutual friction between the charge carriers in passive and active layers, therefore it couples two independent equations. The exact form of $\gamma _d$ depends on temperature, distance between layers and densities of carriers. We consider the drag rate $\gamma _d$ as known parameter and the scheme of microscopic calculations of it can be found in Refs.²⁹. Non-linear term in drift velocity is an advection term. The additional term on the left hand side describes the viscous nature of electron’s fluid in graphene and $\nu$ is the kinematic viscosity, which is related with dynamical viscosity (or first viscosity coefficient in Navier-Stokes equation) by $\eta =mn \nu$⁴².

For further simplifications, we consider two identical layers and one set $\gamma _1=\gamma _2=\gamma$, similarly $\nu _1=\nu _2=\nu$. Moreover, we are interested in time-independent steady-state case, thus the terms with time derivative in both Eqs. (2) are omitted. Additionally, we neglect the advection term, restricting the problem under consideration to small Reynolds numbers⁴². To remind, we work at degeneracy regime, where the chemical potentials are larger then the temperature and the fluid is incompressible, therefore on the right hand side of Eq. (1), we can omit the forces which are related to gradients of pressures. Thus, in order to study the Coulomb drag effect, we use linearized version of Navier-Stokes equations, known as Stokes equations in hydrodynamics, or the Pogrebinskii’s approach^29,30 with additional viscous term in condensed matter physics. In the question under consideration this is the systems of partial differential equations for drift velocity in each layer

$$\begin{aligned} {\left\{ \begin{array}{ll} \gamma \textbf{v}_1-\nu \nabla ^2 \textbf{v}_1=-\frac{e}{m}\nabla \varphi _1-\gamma _d (\textbf{v}_1-\textbf{v}_2), \\ \gamma \textbf{v}_2-\nu \nabla ^2 \textbf{v}_2=-\frac{e}{m}\nabla \varphi _2-\gamma _d (\textbf{v}_2-\textbf{v}_1), \end{array}\right. } \end{aligned}$$

(2)

where we introduced the two-dimensional electrostatic potential, $\textbf{E}=-\nabla \varphi$. In order to solve Eq.  (2) we consider the strip geometry (see Fig. 1). Apart from that, to simplify the algebra we introduce the “center of mass” and “relative” velocities $\textbf{v}_{\pm }(\textbf{r})=\textbf{v}_1 (\textbf{r}) \pm \textbf{v}_2 (\textbf{r})$ and similarly $\varphi _{\pm }(\textbf{r})=\varphi _{1}(\textbf{r}) \pm \varphi _{2}(\textbf{r})$, $\gamma _{+}=\gamma$ and $\gamma _{-}=\gamma +2\gamma _d$. In terms of new variables we have the following equations

$$\begin{aligned} \gamma _{\pm } \textbf{v}_{\pm }-\nu \nabla ^2 \textbf{v}_{\pm }=-(e/m) \nabla \varphi _{\pm }. \end{aligned}$$

(3)

Below, we use this equation together with boundary conditions to restore the velocity, $\textbf{v}_2$, and potential distribution profile, $\varphi _2$, in passive layer. It is worth mentioning that if $\gamma _1 \ne \gamma _2$ (or/and $\nu _1 \ne \nu _2$), then it is not possible to bring NS equations from Eq. (2) to one equation similar to Eq. (3). Nevertheless, one can consider the general case, but this adds unnecessary cumbersomeness, which does not change the essence of the results. To be precise, in general case the Eq. (2) presents a linear system of two fourth order differential equations in terms of stream functions and can be solved calculating the eigenvalues of characteristic equation and corresponding eigenfunctions.

Fig. 2

The velocity profile in passive layer (see Eq. (7 )). Here $\zeta =w\sqrt{\gamma /\nu }$, $\beta =\gamma _d/\gamma =0.1$ and $v_0=eE/2m\gamma$.

Results

Poiseuille type solution

As an example, let us first consider the Poiseuille type solution. Let assume that the constant electric field is applied in x direction along the strip with width w. Therefore, the Stokes Eqs. (2) takes the form

$$\begin{aligned} {\left\{ \begin{array}{ll} \gamma v_1(y)-\nu \partial ^2_y v_1(y)=eE/m-\gamma _d [v_1(y)-v_2(y)] \\ \gamma v_2(y)-\nu \partial ^2_y v_2(y)=-\gamma _d [v_2(y)-v_1(y)]. \end{array}\right. } \end{aligned}$$

(4)

Consequently, equation for $v_{\pm }=v_1 \pm v_2$ velocities has the form

$$\begin{aligned} \partial ^2_y v_{\pm }(y)-\gamma _{\pm } v_{\pm }(y)/\nu +eE/m\nu =0, \end{aligned}$$

(5)

with boundary conditions $v_1(y=0)=v_1(y=w)=v_2(y=0)=v_2(y=w)=0$. The solution is given by the following structure

$$\begin{aligned} v_{\pm }(y) =\frac{eE}{m\gamma _{\pm }}\left( 1-\frac{ \cosh [\sqrt{\gamma _{\pm }/\nu }(y-w/2)]}{\cosh [\sqrt{\gamma _{\pm }/\nu }w/2]} \right) , \end{aligned}$$

(6)

and the velocity profile in passive layer is given by

$$\begin{aligned} v_2(y)=[v_{+}(y)-v_{-}(y)]/2. \end{aligned}$$

(7)

The velocity profile is plotted at Fig. 2. In viscous regime, $\zeta \ll 1$, at finite $\beta$ the velocity profile has the form $v_2(y)=(eE\gamma _d w^3 y/24m\nu ^2)(1-2y^2/w^2+y^3/w^3)$. The net current can be calculated as the integral with respect to width variable, $I_2=en \int ^w_0 v_2(y) dy$. After performing the integration of Eq. (7) one arrives to “Drude” like relation, $I_2=(ne^2 \tau /m) w E$, where the effective scattering time is given by the renormalized inverse momentum relaxation, $\tau =\mathscr {A}/\gamma$ and the dimensionless factor has the form

$$\begin{aligned} \mathscr {A}=\frac{\beta }{1+2\beta }-\frac{\tanh (\zeta /2)}{\zeta }+\frac{\tanh (\zeta \sqrt{1+2\beta }/2)}{\zeta (1+2\beta )^{3/2}}, \end{aligned}$$

(8)

where $\zeta =w\sqrt{\gamma /\nu }$, $\beta =\gamma _d/\gamma$. One can check that at $\gamma _d \rightarrow 0$ ($\beta \rightarrow 0$), the net current is equals to zero, thus there is no drag effect. In the viscous regime, $\zeta \ll 1$, and finite $\beta$, the effective relaxation time is $\tau \simeq \gamma _d w^4/120 \nu ^2$. It is worth mentioning, that the effective relaxation time in active layer without back-action from passive layer is given by $\tau _{*}=w^2/12\nu$. This means that the viscous effect from passive layer is smaller, since it is “sub-leading” effect arising through the coupling, $\gamma _d=1/\tau _d$ in Eqs. (2).

Non-local resistance in passive layer

In this section we proceed to consider the Stokes Eqs. (2) and (3) for strip geometry $0<y<w$ with no-slip boundary conditions. In order to solve the Eq. (3) we introduce the stream function $\textbf{v}_{i}=\{-\partial _y \psi _{i}(x,y), \partial _x \psi _{i}(x,y)\}$, where $i=1,2$. This can be done since liquids in both layers are assumed to be incompressible. Now, in terms of “±” variables the stream function obtained from Eq. (3) takes the form

$$\begin{aligned} \gamma _{\pm } \Delta \psi _{\pm }(x,y)-\nu \Delta \Delta \psi _{\pm }(x,y)=0. \end{aligned}$$

(9)

This equation is similar to the one obtained for single layer in Ref.¹¹. Since we consider the strip geometry one can apply the Fourier transformation in x spatial coordinate, $\psi _{\pm }(x,y) \propto \int dk \psi _{k,\pm }(y)e^{ikx}$ and arrive to fourth order differential equation in mixed momentum and spatial representation

$$\begin{aligned} \partial ^4_y \psi _{k,\pm }(y)-(k^2+q^2_{\pm }) \partial ^2_y\psi _{k,\pm }(y)+k^2q^2_{\pm }\psi _{k,\pm }(y)=0, \end{aligned}$$

(10)

where $q^2_{\pm }=k^2+\gamma _{\pm }/\nu$. The solution to this equation is given by the sum of four terms

$$\begin{aligned} \psi _{k,\pm }(y)=\frac{I/en}{ik} \left[ C_{\pm ,1}e^{-ky}+C_{\pm ,2}e^{ky}+C_{\pm ,3}e^{q_{\pm }y}+C_{\pm ,4}e^{-q_{\pm } y}\right] \end{aligned}$$

(11)

To find unknown coefficients, the solution is supplemented with no-slip boundary conditions

$$\begin{aligned}&\psi _{k, \pm }(y)|_{y=0}=\psi _{k, \pm }(y)|_{y=w}=\frac{I/en}{ik}, \end{aligned}$$

(12)

$$\begin{aligned}&\partial _y \psi _{k,\pm }(y)|_{y=0}=\partial _y \psi _{k,\pm }(y)|_{y=w}=0. \end{aligned}$$

(13)

In Eq. (12), I is the inlet and outlet current in active layer at points $y=0$ and $y=w$, thus the y component of velocity in these points defined by delta peak, namely $v_{1y}(x,y)|_{y=0}=v_{1y}(x,y)|_{y=w}=I\delta (x)/en$. The second equation is derived from condition on x component of velocity $v_{1x}(x,y)|_{y=0}=v_{1x}(x,y)|_{y=w}=0$. In passive layer we set $v_{2y}(x,y)|_{y=0}=v_{2y}(x,y)|_{y=w}=v_{2x}(x,y)|_{y=0}=v_{2x}(x,y)|_{y=w}=0$. Substituting the Eq. (11) into boundary conditions, Eq. (12) we find the coefficients

$$\begin{aligned} C_{\pm ,1}&=\frac{e^{kw}\left( e^{q_{\pm }w}-1\right) q_{\pm }}{\left( k-q_{\pm }\right) \left( 1-e^{(k+q_{\pm })w}\right) +\left( k+q_{\pm }\right) \left( e^{q_{\pm }w}-e^{kw}\right) }, \nonumber \\ C_{\pm ,2}&=\frac{\left( e^{q_{\pm }w}-1\right) q_{\pm }}{\left( k-q_{\pm }\right) \left( 1-e^{(k+q_{\pm })w}\right) +\left( k+q_{\pm }\right) \left( e^{q_{\pm }w}-e^{kw}\right) }, \nonumber \\ C_{\pm ,3}&=\frac{-\left( e^{kw}-1\right) k}{\left( k-q_{\pm }\right) \left( 1-e^{(k+q_{\pm })w}\right) +\left( k+q_{\pm }\right) \left( e^{q_{\pm }w}-e^{kw}\right) }, \nonumber \\ C_{\pm ,4}&=\frac{-e^{q_{\pm }w}\left( e^{kw}-1\right) k}{\left( k-q_{\pm }\right) \left( 1-e^{(k+q_{\pm })w}\right) +\left( k+q_{\pm }\right) \left( e^{q_{\pm }w}-e^{kw}\right) }. \end{aligned}$$

(14)

The non-local resistance in passive layer $V_2(x)/I=[\varphi _2(x,w)-\varphi _2(x,0)]/I$ can be obtained substituting Eq. (14) into Eq. (11) and then into Eq. (3), since the velocity is defined by the derivatives of stream function. It is given by the following integral relation

$$\begin{aligned} R_\mathrm{{nl},2}(x)=\frac{\eta }{(enw)^2} \int \left[ f(z,\epsilon _{+})-f(z,\epsilon _{-})\right] e^{i\frac{x}{w}z}\frac{dz}{2\pi }, \end{aligned}$$

(15)

where we have introduced the dimensionless integral variable $z=kw$. The function in the integrand has the form

$$\begin{aligned}&f(z,\epsilon _{\pm })=\epsilon _{\pm } \mathscr {N}(z, \epsilon _{\pm })/z\mathscr {D}(z,\epsilon _{\pm }), \nonumber \\&\mathscr {N}(z, \epsilon _{\pm })=(e^{\sqrt{z^2+\epsilon _{\pm }}}-1)(e^{z}-1)\sqrt{z^2+\epsilon _{\pm }}, \nonumber \\&\mathscr {D}(z, \epsilon _{\pm })=(\sqrt{z^2+\epsilon _{\pm }}+z)(e^{\sqrt{z^2+\epsilon _{\pm }}}-e^{z})+(\sqrt{z^2+\epsilon _{\pm }}-z)(e^{\sqrt{z^2+\epsilon _{\pm }}+z}-1) \end{aligned}$$

(16)

where $\epsilon _{\pm }=(enw)^2 \rho _{\pm }/\eta$ is the dimensionless parameter, which specifies the crossover from ohmic to viscous regime, and $\rho _{\pm }=\gamma _{\pm } m/e^2 n$ is the “resistivity”. One can straightforwardly check that there is no induced voltage $V_{2}(x)=0$ for $\gamma _d \rightarrow 0$, since $\epsilon _{+}=\epsilon _{-}$ and thus the difference $f(z,\epsilon _{+})-f(z,\epsilon _{-})=0$ vanishes.

Fig. 3

Asymptotics of $f_1(z)$ function at $|z| \gg 1$ (red) and at $|z| \ll 1$ (blue).

First, we investigate the strong viscous regime, namely $\eta \rightarrow \infty$, thus $\epsilon _{\pm } \propto \rho _{\pm }/\eta \rightarrow 0$. In this case one can expand the function in the integrand, Eq. (16) with respect to $\epsilon _{\pm }$. Namely $f(z, \epsilon _{\pm }) \approx f_0(z)+f_1(z) \epsilon _{\pm }$, where

$$\begin{aligned} f_0(z)&=\frac{2z\tanh (z/2)\sinh (z)}{z+\sinh (z)}, \nonumber \\ f_1(z)&=\frac{z^2\sinh (z)+3\left[ \cosh (z)-1\right] \left[ z+\sinh (z)\right] }{2z \left[ z+\sinh (z)\right] ^2}. \end{aligned}$$

(17)

One can check that the voltage distribution is an even function with respect to x, according to spatial invariance. It is worth mentioning as well that for the single layer the main contribution comes from the first term, $f_0(z)$¹¹. However, the non-local voltage distribution in passive layer is given only by the term $f_1(z)$, since the subtraction in Eq. (15) cancels out the contribution of leading term $f_0(z)$. Introducing the non-local resistance as $R_{\text {nl,2}}(x)=V_2(x)/I$ we get the following integral

$$\begin{aligned} R_{\text {nl,2}}(x)=(-2 \rho _d/\pi )\int _0^{\infty } f_1(z) \cos (xz/w) dz. \end{aligned}$$

(18)

The function $f_1(z)$ in the integrand has a maximum value 1/2 at $z=0$ and it goes to zero at $|z| \rightarrow \infty$ as $\propto 1/|z|$ (See Fig. 3). To make a progress analytically, we consider the asymptotics of Eq. (18) for small $a/w \ll |x|/w \ll 1$ and large $|x|/w \gg 1 \gg a/w$ distances, where a is the source/drain contact’s size. At small distances the main contribution to integral, Eq. (18) comes from $z \gg 1$, and one can estimate the order of integral as follows

$$\begin{aligned} R_{\text {nl,2}}(x)/\rho _d \sim (-3/\pi ) \left[ \log (w/|x|)-\varvec{\gamma }\right] , \end{aligned}$$

(19)

where $\varvec{\gamma } \approx 0.5772$, which is an Euler constant. We keep this constant since it might be the same order as logarithm. Therefore, the non-local resistance in the leading order is negative. At large distances, the main contribution comes from $z \ll 1$ and the non-local resistance demonstrates a weak power-law behavior $R_{\text {nl,2}}(x)/\rho _d \propto w/\pi |x|$.

In the strong ohmic regime, $\eta \rightarrow 0$, but $\epsilon _{\pm } \propto \rho _{\pm }/\eta \rightarrow \infty$, one can set $q_{\pm } w \rightarrow \infty$, thus the coefficients which contributes for non-local resistance in passive layer convert to $C_{\pm ,1}(k) \rightarrow (e^{-kw}+1)^{-1}$ and $C_{\pm ,2}(k) \rightarrow C_{\pm ,1}(-k)$ and the result is given by the integral

$$\begin{aligned} R_{\text {nl,2}}(x)/\rho _d \sim (4/\pi )\int _0^{\infty } \frac{dz}{z}\tanh (z/2) \cos (xz/w) =(4/\pi )\log |\coth (\pi x/2w)| \end{aligned}$$

(20)

Fig. 4

Weak drag regime, $\alpha =\rho _d/\rho =0.01$. The non-local resistance defined in the Eq. (21) for passive layer is plotted in arbitrary units versus the dimensionless distance x/w.

Fig. 5

Weak drag regime, $\alpha =\rho _d/\rho =0.01$. The non-local resistance defined in the Eq. (21) for passive layer is plotted in arbitrary units versus the dimensionless distance x/w.

Additionally, there might be another viscous dominated regime, where $\epsilon \rightarrow 0$ at $\rho \rightarrow 0$, but $\epsilon _{\text {d}}$ is finite. However, this regime is difficult to investigate analytically, thus we present the numerical plots. In order to do this, one can rewrite Eq. (15) for non-local resistance as follows

$$\begin{aligned} R_{\textrm{nl},2}(x)/\rho&= \int _0^{\infty } \frac{dz}{\pi } \Phi (z,\alpha , \epsilon ) \cos (xz/w),\nonumber \\ \Phi (z,\alpha , \epsilon )&=\frac{\mathscr {N}(z,\epsilon )}{z\mathscr {D}(z,\epsilon )}-\frac{(1+2\alpha )\mathscr {N}(z,(1+2\alpha )\epsilon )}{z\mathscr {D}(z,(1+2\alpha )\epsilon )}, \end{aligned}$$

(21)

where the dimensionless variable $\alpha =\rho _d/\rho$ can be regarded as a Coulomb drag strength. One can again see that for a fixed $\rho$ at $\alpha \rightarrow 0$, the drag effect disappears, therefore the Coulomb drag induced non-local resistance loses it’s meaning. The Coulomb drag induced non-local resistance $R_{\text {nl,2}}(x)/\rho _d$, as a function of positive x/w is shown at Figs. 4 and  5. The non-local resistance is the even function of spatial coordinate x, thus it’s behavior for negative x can be restored easily.

Velocity stream lines and potential map

The stream function in passive layer is given by $\psi _2(x,y)=[\psi _{+}(x,y)-\psi _{-}(x,y)]/2$. This can be calculated by inverting the Fourier transformation of Eq. (11). After some algebra the stream function in passive layer takes form

$$\begin{aligned} \psi _2(x,y)&=\frac{I}{2en} \sum _{\sigma =\pm }\int \frac{dk}{2\pi }\frac{e^{ikx}}{ik} \sigma M(k,q_{\sigma }), \nonumber \\ M(k,q_{\sigma })&=\frac{q_{\sigma }\sinh \left[ \frac{q_{\sigma }w}{2} \right] \cosh \left[ k(y-w/2)\right] -k\sinh \left[ \frac{kw}{2} \right] \cosh \left[ q_{\sigma }(y-w/2)\right] }{q_{\sigma }\cosh \left[ \frac{kw}{2}\right] \sinh \left[ \frac{q_{\sigma }w}{2} \right] - k\cosh \left[ \frac{q_{\sigma }w}{2}\right] \sinh \left[ \frac{kw}{2} \right] }. \end{aligned}$$

(22)

The velocity vector field in the passive layer is given by the derivatives of stream function, $\psi _2(x,y)$ multiplied by two orthogonal unit vectors, $\textbf{e}_x$ and $\textbf{e}_y$. Namely

$$\begin{aligned} \textbf{v}_2(\textbf{r})=[-\partial _y \psi _2(x,y)]\textbf{e}_x + [\partial _x \psi _2(x,y)]\textbf{e}_y. \end{aligned}$$

(23)

The velocity stream lines in passive layer for viscous and ohmic regimes are shown at Figs. 6, 7 and 8, 9. The potential map in passive layer can be obtained from Eq. (3). Substituting the solution for stream function, Eq. (11) one can show that only the terms proportional to $\exp (\pm ky)$ contribute to potential in passive layer

$$\begin{aligned} \varphi _2(x,y)=\frac{1}{2}\left[ \varphi _{+}(x,y)-\varphi _{-}(x,y)\right] =\frac{I}{2}\sum _{\sigma =\pm } \int \frac{dk}{2\pi } \frac{e^{ikx}}{k} \sigma \rho _{\sigma } \left[ C_{\sigma ,2}e^{ky}-C_{\sigma ,1}e^{-ky}\right] , \end{aligned}$$

(24)

where the momentum dependent coefficients $C_{\pm ,1}$ and $C_{\pm ,2}$ are given in Eq. (14). It is worth mentioning that the boundary conditions on electrostatic potential as well given by Eq. (3), $\nabla \varphi _2=(-\rho e n +(\eta /en) \nabla ^2)([-\partial _y \psi _2(x,y)]\textbf{e}_x + [\partial _x \psi _2(x,y)]\textbf{e}_y)$, at $y=0,w$, which is known as Neumann boundary condition. The potential map in passive layer for viscous and ohmic regimes are shown at Figs. 10, 11, 12 and 13.

Fig. 6

Weak drag viscous regime, $\alpha =\rho _d/\rho =0.01$ and $\epsilon =(enw)^2 \rho /\eta =0.01$. Velocity stream lines defined in the Eq. (23) for passive layer are plotted.

Fig. 7

Weak drag ohmic regime, $\alpha =\rho _d/\rho =0.01$ and $\epsilon =(enw)^2 \rho /\eta =100$. Velocity stream lines defined in the Eq. (23) for passive layer are plotted.

Fig. 8

Strong drag viscous regime, $\alpha =\rho _d/\rho =100$ and $\epsilon =(enw)^2 \rho /\eta =0.01$. Velocity stream lines defined in the Eq. (23) for passive layer are plotted.

Fig. 9

Strong drag ohmic regime, $\alpha =\rho _d/\rho =100$ and $\epsilon =(enw)^2 \rho /\eta =100$. Velocity stream lines defined in the Eq. (23) for passive layer are plotted.

Fig. 10

Weak drag viscous regime, $\alpha =\rho _d/\rho =0.01$ and $\epsilon =(enw)^2 \rho /\eta =0.01$. Potential distribution defined in the Eq. (24) is plotted.

Fig. 11

Weak drag ohmic regime, $\alpha =\rho _d/\rho =0.01$ and $\epsilon =(enw)^2 \rho /\eta =100$. Potential distribution defined in the Eq. (24) for passive layer is plotted.

Fig. 12

Strong drag viscous regime, $\alpha =\rho _d/\rho =100$ and $\epsilon =(enw)^2 \rho /\eta =0.01$. Potential distribution defined in the Eq. (24) for passive layer is plotted.

Fig. 13

Strong drag viscous regime, $\alpha =\rho _d/\rho =100$ and $\epsilon =(enw)^2 \rho /\eta =100$. Potential distribution defined in the Eq. (24) for passive layer is plotted.

Discussion

To conclude, we studied the non-local resistance, velocity stream lines and potential in passive layer induced by effect of Coulomb drag between graphene layers in presence of viscosity term. To do this, we applied the Stokes equations for drift velocities in active and passive layers. We demonstrated that in viscous regime, the non-local resistance changes the sign and thus might take negative values. However, the non-local resistance is positive in the ohmic regime (see Figs. 4, 5). Apart from that, we have studied the stream lines of drift velocity and potential map (see Figs 6,7, 8, 9, 10, 11, 12 and 13). At strong Coulomb drag regime, when $\alpha =\rho _d/\rho \gg 1$, the stream lines of velocity reproduce almost the same behavior as in the active layer since electrons in the passive layer are dragged by electrons in the active layer. Since there is no source and drain in passive layer, the stream lines form loops and particularly vortexes in the strong hydrodynamics regime, $\epsilon =(enw)^2 \rho /\eta \ll 1$. Additionally, due to the absence of drain and source one can see the whirpools at strong Coulomb and Ohmic regime close to the origin line $x=0$. These whirpools are related with the electrons backflow and formation of loops. The moderate Coulomb drag regime, when $\alpha =\rho _d/\rho \sim 1$ is shown in Supplementary materials. Qualitatively, this case is similar to strong drag regime. The fabrication of double layer systems and experimental detection of the Coulomb drag response is not an easy task. Despite to this fact, the Coulomb drag effect in double layers might be an alternative way to distinguish the ohmic and hydrodynamic regimes compared to monolayer graphene, where hydrodynamic regime was recently investigated using an electric response to applied DC bias, a light-induced photoresistance and a sensitive magnetometry based on nitrogen vacancy centers in diamond^43,44. Coulomb drag response as well might be fruitful especially to see additional quantum corrections to hydrodynamic regime. For instance, this can be done by the studying of quantum corrections to hydrodynamic regime both with and without magnetic fields in context of weak (anti-) localization effects⁴⁵.

Let us say now a few words on non-local resistance in presence of magnetic field. In order to study the behavior of non-local magnetoresistance, we consider the case of classical magnetic fields, namely the cyclotron radius is much larger than the collision mean free length. The constant magnetic field is perpendicular to the active and passive layers, $\textbf{B}=B \textbf{e}_z$, and thus the right hand sides of Eq. (2) acquire additional term, $\omega _c [\textbf{v}_{i} \times \textbf{z}]$, where $\omega _c=eB/mc$ is the cyclotron frequency. Introducing the stream function $\textbf{v}=\textbf{z} \times \nabla \psi (\textbf{r})$ and applying the curl operators on Eqs. (2) and (3) we arrive to the same equation for stream function as in Eq. (9). This is due to the fact that the curl of Lorentz force is equal to zero at constant magnetic field, i.e. $\nabla \times [\textbf{v}_i \times \textbf{B}]=\textbf{v}_i(\nabla \cdot \textbf{B})-\textbf{B}(\nabla \cdot \textbf{v}_i)+(\textbf{B} \cdot \nabla ) \textbf{v}_i-(\textbf{v}_i \cdot \nabla )\textbf{B}=0$, since electron liquids in both 2D layers are assumed to be incompressible, $\nabla \cdot \textbf{v}_i=0$. Moreover, one can show that the constant magnetic field does not change the no-slip boundary conditions⁴⁶. It is worth mentioning that this is true as well for no-stress and mixed boundary conditions⁴⁶. The presence of constant perpendicular magnetic field does not affect on the differential equation for stream function. However, the constant magnetic field changes the potential distribution and it appears as an additional shift for each layer, namely $\varphi _i(B; x,y)=\varphi _i(B=0; x,y)-B\psi _i(B=0; x,y)$, where $i=1,2$ corresponds to active and passive layer and $\varphi _i(B=0; x,y)$ and $\psi _i(B=0; x,y)$ are given in Eqs. (2) and (3) respectively in absence of magnetic field. Further, introducing the voltage distribution at finite magnetic field, $V_2(B; x)=\varphi _2(B; x, w)-\varphi _2(B; x, w)$ and the associated non-local magnetoresistance in passive layer, $R_{\textrm{nl},2} (B; x)= R_{\textrm{nl},2} (B=0; x)-(B/I) \Psi (x)$, where $\Psi (x)=\psi _2(B=0; x,w)-\psi _2(B=0; x,0)$ we obtain that $\Psi (x)$ exactly equals to zero, since this is the difference of stream functions on the edges of strip.

Finally, as a future perspectives it is appropriate to investigate the viscous and the ohmic regimes for Coulomb drag induced electron transport in double graphene layers in presence of time-dependent external perturbation. Compared to our situation, in presence of time-dependent external perturbation the no-slip and no-stress boundary conditions strongly influence differently to the transport of electrons in both hydrodynamic and ohmic regimes⁴⁷.