Current distribution in a slit connecting two graphene half-planes
CCurrent distribution in a slit connecting two graphene half-planes
Sergey S. Pershoguba , , Andrea F. Young , and Leonid I. Glazman ; Department of Physics, Yale University, New Haven, CT 06520, USA Department of Physics and Astronomy, University of New Hampshire, Durham, New Hampshire 03824, USA and Department of Physics, University of California, Santa Barbara, CA 93106 (Dated: September 7, 2020)We investigate the joint effect of viscous and Ohmic dissipation on electric current flow througha slit in a barrier dividing a graphene sheet in two. In the case of the no-slip boundary condition,we find that the competition between the viscous and Ohmic types of the charge flow results inthe evolution of the current density profile from a concave to convex shape. We provide a detailedanalysis of the evolution and identify favorable conditions to observe it in experiment. In contrast,in the case of the no-stress boundary condition, there is no qualitative difference between the currentprofiles in the Ohmic and viscous limits. The dichotomy between the behavior corresponding todistinct boundary conditions could be tested experimentally.
I. INTRODUCTION
Recent years have seen a revival of interest in theidea that charge transport in solids under some con-ditions is best described by hydrodynamic flow of anelectron liquid. Graphene provides an ideal platformfor observing hydrodynamic effects due to the extremelylong electron mean free path for impurity scattering .In constrained geometries, viscous electron flow differsfrom both the Ohmic and ballistic transport regimes.The simplest manifestations of that difference are seenin the conductance: it exceeds the ballistic limit for aslit connecting two conducting half-planes , and maybecome negative for certain configurations of contactsalong the edge of a conducting stripe . These mani-festations are fairly insensitive to the type of boundarycondition for the electron liquid flowing around obsta-cles. For example, the conductance of a slit in the hy-drodynamic regime exceeds the ballistic limit, regardlessthe liquid “sticking” to the boundary or “sliding” alongit.Sticking to or sliding along the boundary corresponds,respectively, to the no-slip or no-stress boundary condi-tions for the electron liquid. There is no consensus in theliterature (see Ref. [2] vs [8]) regarding which bound-ary condition is appropriate for graphene. Theoreticalwork [13] discussed the relation of the hydrodynamicboundary conditions to the microscopic conditions forelectron scattering off the boundary.Recently, spatially resolved experimental techniqueshave made it possible to investigate the velocity distri-bution in the electron flow , giving direct informa-tion about the boundary conditions for hydrodynamiccharge carriers. That motivates us to investigate the-oretically the effect of boundary conditions and of theOhmic losses in the bulk on the on the velocity distri-bution. We focus on the electron flow through a slit, seeFig. 1(a).Our main finding is that the velocity profile may al-low one to unambiguously determine the type of bound-ary conditions as well as to identify the viscous regime.We also elucidate the domain for the sample parameters (the slit width, charge carrier density, and temperature)favoring the hydrodynamic regime.We start with a brief review in Sec. II of thecontinuous-medium (hydrodynamic) equations whichaccount for the electron viscosity and Ohmic losses. Inthe same Section, we identify the width of the bound-ary layer defined by the competition between the viscousand Ohmic terms in the hydrodynamic equations. Thecomparison of the limiting cases where either the vis-cous or Ohmic term dominates allows us to conclude inSec. III that in the case of no-stress boundary conditionit may be hard to distinguish in an experiment betweenthe Ohmic and viscous electronic flows. In contrast,for the no-slip boundary condition, we notice a qualita-tive feature: the current density profile is concave andconvex in the Ohmic and viscous limits, respectively.In practice, the viscous term in the dynamic equationfor the electron liquid coexists with the Ohmic term. InSec. IV, we study the crossover between the two regimescontrolled by a single dimensionless parameter, the ra-tio of the slit width to the width of the boundary layerintroduced in Sec. II. We find the current density pro-file numerically at any value of this control parameterand present a simplified model allowing for an analyticalsolution, which agrees well with the numerical results.The control parameter may be varied in situ by chang-ing the electron density and temperature. We identifythe domain of parameters favoring the hydrodynamicregime of electron flow and map out the crossover linesseparating the Ohmic, viscous and ballistic regimes fromeach other in Sec. V. We conclude in Sec. VI. II. HYDRODYNAMIC DESCRIPTION OFELECTRONIC FLOW IN GRAPHENE.
In this section, we set up hydrodynamic equationsand briefly discuss their applicability. Following previ-ous literature , the electronic flow in graphenemay be described, at low applied bias, by the linearizedStokes equation in two dimensions r = ( x, y ):[ η ∆ − ( ne ) ρ ] v ( r ) = ne ∇ φ ( r ) . (1) a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Here, φ ( r ) and n are the electric potential and elec-tronic density; η and ρ are the viscosity coefficient andthe electric resistivity, respectively. It is assumed thatthe velocity v ( r ) of the electronic fluid is small, so thehigher-order in v terms are dropped (see discussion inRef. [2]). In addition, the stationary continuity equa-tion for current density j = ne v is used:0 = ∇ · j ( r ) = ne ∇ · v ( r ) , (2)where, in the last equality, we assumed that the elec-tronic liquid is incompressible at hydrodynamic lengthscales, i.e., n ( r ) = const.We intend to solve Eqs. (1) and (2) for the “slit”geometry. To be more specific, we assume that thegraphene sheet is divided by the opaque (for electrons)barrier with a slit of finite width 2 w as illustrated inEq. 1(a). For the purposes of analytical calculations,we assume that the barrier is infinitely thin.The specifics of the boundary conditions imposed bythe barrier is crucial for determining the profile of theflow. In the microscopic approach, the pioneering workby Fuchs discussed two types of boundary conditionsfor electrons: (i) the diffuse and (ii) specular scattering.In the phenomenological hydrodynamic approach, theboundary conditions on each side of the impenetrablebarrier may be formulated in a concise form, v y | | x | >w, y → = 0 ,v x | | x | >w, y → = λ ( ∇ y v x ) | | x | >w, y → . (3)The first of these two equations states that the normalcomponent of the velocity vanishes at the barrier. Thesecond equation states that the tangential velocity atthe boundary is proportional to the viscous stress. Theparameter λ allows to interpolate between the no-slip( λ = 0) and no-stress ( λ = ∞ ) boundary conditions.There is no consensus in the literature (see Ref. [2] vs[8]) regarding which boundary condition is appropriatefor graphene. Recent theoretical work [13] discusseda relation between the microscopic and hydrodynamicboundary conditions.By inspecting the left-hand-side of Eq. (1), it is in-structive to define the parameter l = 1 ne (cid:114) ηρ , (4)which has units of length. Comparison of l with thegeometric scale of the problem w allows us to definethe two regimes in which (i) the Ohmic term dominates( l (cid:28) w ), or (ii) the viscous term dominates ( l (cid:29) w ). Wediscuss the current distribution in these limiting casesin the following Section. Then, in Sec. IV, we discussthe crossover between the two limits. (a)(b)FIG. 1. (a) Schematic representation of the electric currentflow in graphene through a slit of finite width 2 w . The scaleof a graphene lattice is artificially enlarged for visualization.The distribution of current within the slit (i.e. at | x | < w and y = 0) may allow to distinguish between the viscous/non-viscous regimes as well as clarify the role of the boundaryconditions. (b) The current distribution within the slit inthe Ohmic (9), viscous no-slip (18), and viscous no-stress(17) cases. To plot them simultaneously, we set the commonnormalization constant v c = I/πnew , which corresponds tofixed total current I . III. CURRENT DISTRIBUTION IN THELIMITING CASES.A. Current distribution in the Ohmic limit( l/w → ). As a warm up, we consider the Ohmic limit l/w → ∝ η ) term in Eq. (1).In order to resolve the continuity Eq. (2), we introducethe stream function v ( r ) = [ ˆ z × ∇ ψ ( r )]. Then, Eq. (1)reduces to [ ˆ z × ∇ ψ ( r )] = − neρ ∇ φ ( r ) . (5)We seek a solution of Eq. (5) with the normal compo-nent of velocity vanishing at the wall. That bound-ary condition amounts to ψ being constant at thetwo sides of the barrier, i.e. ψ ( r ) | x>w,y → = 0 and ψ ( r ) | x< − w,y → = ψ . The constant ψ is related to thetotal current I flowing through the slit, ψ = − I/ne .Equation (5) may be interpreted as the Cauchy-Riemann condition for an analytical function of a com-plex variable z = x + iy : f ( z ) = φ ( r ) neρ + i ψ ( r ) . (6)Then, it is practical to perform a confor-mal transformation to a new variable z =ln (cid:2)(cid:0) z + √ z − w (cid:1) /w (cid:3) , in which the complicatedslit geometry (see Fig. 1(a)) transforms into a horizon-tal stripe, i.e. −∞ < x < ∞ and 0 < y < π . In thelatter geometry, the boundary conditions at the edgesof the stripe become Im f ( z ) | z → x + iπ = ψ = − I/ne and Im f ( z ) | z → x + i = 0. It is straightforward tofind the function satisfying that boundary condition: f ( z ) = − z I/neπ . So, in the original variable z , wehave f ( z ) = − Iπne ln (cid:34) z + √ z − w w (cid:35) . (7)The functions φ and ψ may be read off from Eq. (7)using Eq. (6). Few comments about the solution (7)are in order. (i) The potential is logarithmically large φ ( r ) = neρ Re f ( z ) ∼ ( Iρ/π ) ln r/w at r → ∞ . Physi-cally, it corresponds to a logarithmically large resistance R ∼ ( ρ/π ) ln L/w , where L is the size of the system.(ii) Using that ψ ( r ) = Im f ( z ) and definition of ψ ( r ),one may evaluate the velocity (cid:18) v x ( r ) v y ( r ) (cid:19) = Iπne (cid:18)
Re[1 / √ z − w ] − Im[1 / √ z − w ] (cid:19) . (8)Within the slit, the flow has only the ˆ y component, v y | Ohmic | x |
0. For that, let us establishthe boundary condition satisfied by f ( z ) on the realaxis, i.e, at z → x + i
0. It is convenient to set theelectric potential φ ( r ), such that φ ( r ) | r →∞ , π>ϕ> = 0and φ ( r ) | r →∞ , π>ϕ>π = V , where V is the applied biasand ϕ is the polar angle of vector r . Then, by invokingthe symmetry of the problem, the electric potential isconstant within the slit, i.e., φ ( r ) | | x | We summarize the results of the current section byplotting the velocity distributions in the Ohmic (9),viscous no-stress (17), and viscous no-slip (18) limitsin Fig. 1(b). Observe that both the Ohmic (9) andviscous no-stress (17) distributions have an integrable v y ∝ / √ x ± w singularity at the edges of the slit.Physically, that divergence stems from the requirementto accommodate the non-vanishing flow along the im-penetrable boundary. The profiles of velocity for theOhmic (9) and viscous no-stress (17) limits appear sim-ilar qualitatively. Therefore, it would be challenging toexperimentally distinguish the two limits.In contrast, the velocity profile (18) in the case of theno-slip boundary conditions is a convex function with amaximum at the center of the interval ( − w, w ). It is sig-nificantly different from the concave velocity profile in case of the Ohmic flow. Once the Ohmic ( ∝ ρ ) and vis-cous ( ∝ η ) terms become of comparable strength, i.e., l/w ∼ 1, the solutions (9) and (18) corresponding tothe limiting cases are not applicable, and we expect acrossover between the concave and convex velocity dis-tributions across the slit ( | x | ≤ w ). In the next section,we develop a method of integral equation to describethat crossover. IV. CROSSOVER BETWEEN THE OHMICAND NO-SLIP VISCOUS LIMITS ( l/w ∼ ).A. Integral equation In the spirit of Refs. [5,7], we find the solution of the“point-source” (ps) problem ψ ps ( x, y ) = (cid:90) ∞−∞ dk x πi k x e ik x x q − | k x | (cid:104) q e − y | k x | − | k x | e − y q (cid:105) ,q = (cid:112) k x + l − , (19)where the parameter l , defined in Eq. (4), measuresthe relative strength of the viscous and Ohmic terms.Equation (19) solves Eqs. (1) and (2) for arbitrary η and ρ with no-slip boundary condition and a “point-source” current at the boundary y = 0. In otherwords, it satisfies v x | y → +0 = −∇ y ψ ps | y → +0 = 0 and v y | y → +0 = ∇ x ψ ps | y → +0 = δ ( x ). One may view Eq. (19)as a Green’s function allowing to relate ψ ( x, y ) in theplane to the velocity v ( x ) within a finite-width slit: ψ ( x, y ) = (cid:90) w − w dx (cid:48) ψ ps ( x − x (cid:48) , y ) v ( x (cid:48) ) . (20)For clarity, the components v x and v y stand for the ve-locity at arbitrary r , whereas v ( x ) ≡ v y ( x, y ) | y → de-notes the velocity distribution within the slit. Natu-rally, ψ ( x, y ) satisfies the correct boundary conditionsat y = 0 as well as the condition on the total current at r → ∞ . In addition, the velocity distribution must sat-isfy the symmetry condition that y = 0 is the inflectionpoint for v x , which amounts to ∇ y ψ (cid:12)(cid:12) | x | 0. In order to ex-pose that singularity, we re-write the rational functionof the integrand in Eq. (22) as: K ( x ) =lim δ → (cid:90) ∞ dt e − t δ l sin (cid:16) t xl (cid:17) (cid:20) t + 32 − t + √ t + 1) (cid:21) . We may explicitly evaluate the integrals correspondingto the first two terms in the square brackets and retainthe last term in K reg ( x ): K ( x ) = − l x + 32 x + K reg ( x ) , (23) K reg ( x ) = − (cid:90) ∞ dt l sin ( t ( x/l ))( t + √ t + 1) . The first two terms in Eq. (23) are singular, and, corre-spondingly, the integral (21) is understood in the senseof Cauchy’s principal value. In contrast, the integralin K reg ( x ) converges well and, so, the regularizing ex-ponent is dropped. It has the following asymptotes: K reg ( x ) = ( x/l ) ln( l/ | x | ) and K reg ( x ) = − / x +2 l /x + O ( l /x ) at x/l (cid:28) x/l (cid:29) 1, respec-tively. B. Limiting cases Let us demonstrate that the limiting cases are consis-tent with the integral equation approach. First, considerthe Ohmic limit l → 0, in which case the kernel (23) be-comes K ( x ) = x . Then, it is straightforward to checkthat the Ohmic velocity profile (9) satisfies the integralequation (21): − (cid:90) w − w dx (cid:48) x − x (cid:48) (cid:20) v c √ − x (cid:48) (cid:21) = 0 . (24)In the opposite strongly viscous case l → ∞ , the kernelbehaves as K ( x ) = − l /x . One may show that thevelocity profile (18) satisfies the corresponding integralequation (21), − (cid:90) w − w dx (cid:48) − l ( x − x (cid:48) ) (cid:104) v c (cid:112) − x (cid:48) (cid:105) = 0 , (25)and the boundary condition v ( ± w ) = 0 at the edges ofthe slit. The consideration above prompts the followinginterpretation of the singular terms in kernel (23). Thetwo terms ∝ /x and ∝ l /x correspond to the Ohmicand viscous parts of the kernel, respectively. C. Numerical solution Equation (23) is conveniently split in singular ( ∝ /x and ∝ l /x ) as well as non-singular K reg ( x ) terms. The FIG. 2. Normalized velocity profile through the slit eval-uated for the no-slip boundary condition. We present re-sults ranging from the strongly viscous l/w (cid:29) l/w (cid:28) l/w (cid:39) . 5. The numeri-cal and (approximate) analytical (32) curves are shown withsolid and dashed lines, respectively. strategy is to simplify the singular terms by analyticalmethods, whereas the non-singular term may be treatednumerically.We proceed by substituting the kernel (23) in Eq. (21)and recognize that the viscous term ( ∝ l /x ) may bewritten via a second derivative: − l d dx (cid:20) − (cid:90) w − w dx (cid:48) v ( x (cid:48) ) x − x (cid:48) (cid:21) + 32 (cid:20) − (cid:90) w − w dx (cid:48) v ( x (cid:48) ) x − x (cid:48) (cid:21) + (cid:90) w − w dx (cid:48) K reg ( x − x (cid:48) ) v ( x (cid:48) ) = 0 . (26)In order to tackle this integro-differential equation, weemploy the Chebyshev polynomials of both first T n ( x )and second U n ( x ) kinds. They are tailored for a prob-lem on a finite interval. We expand the velocity profilein series v ( x ) = v c (cid:112) − ( x/w ) ∞ (cid:88) n =0 c n T n ( x/w ) , (27)where v c = Iπnew denotes the characteristic value of ve-locity. The summation is carried over the polynomialsof even order, which are even functions of x , thus cor-responding to the symmetry of the problem. The valueof the first coefficient c = 1 is fixed by the constraint (cid:82) w − w dx v ( x ) = I/ne , whereas c n are unknown for n ≥ − (cid:90) w − w dx (cid:48) v ( x (cid:48) ) x − x (cid:48) = − v c ∞ (cid:88) n =1 c n π U n − ( x/w ) , (28)where we used Eq. (18.17.42) of Ref. [26]. The last termin Eq. (26) may also be presented as a linear combina-tion of U n ( x ) [see Eq. (A6)]. Therefore, by relying onthe orthogonality of the polynomials U n ( x ), Eq. (26)reduces to an infinite system of linear equations onthe coefficients ( c , c , c , . . . ) [see Eq. (A9)]. In addi-tion, given Eq. (27) and the property T n ( ± 1) = 1,the boundary condition v ( ± w ) = 0 leads to the condi-tion (cid:80) n ≥ c n = − c = − c n = 0for n > N , renders a finite system of linear equationsamenable to a numerical solution. The elements of thatmatrix depend on the parameter l/w , allowing us to in-vestigate the crossover between the Ohmic and viscousflows. The evaluated coefficients c n are then substitutedin Eq. (27) thereby producing the velocity profile.In Fig. 2, we present the result of the numerical proce-dure outlined above for the parameters ranging from thestrongly viscous l/w (cid:29) l/w (cid:28) l/w (cid:29) 1, the velocityprofile is a convex function with a single maximum at x = 0. With decrease of l/w (i.e. with the decreaseof η ), the profile further flattens at the center until thesecond derivative of velocity vanishes at x = 0 for somecritical value of parameter l/w (cid:39) . 5. The two shallowmaxima appear in the vicinity of x = 0 for l/w < . l/w , the two maxima sharpenand drift towards the edges of the slit as the velocityprofile approaches Eq. (9) evaluated in the Ohmic limit. D. Analytical interpolation between the viscousand Ohmic limits We recall that the distribution of the velocity v ( x ) inthe two limits can be obtained from an integral equationwith the kernel truncated to the corresponding singularterm [see Eqs. (24) and (25)]. Next, we note that theboundary values v ( − w ) = v ( w ) = 0 would be enforcedby the stronger singularity of the viscous − l /x partof the kernel (23) at any l , even if l (cid:28) w and the Ohmicterm dominates everywhere except the vicinity of theends of the slit. Therefore, it is clear that the qualitativebehavior of v ( y ) should be captured by a solution of theintegral equation Eq. (26) with an omitted part K reg .The resulting equation, − l d dx (cid:20) − (cid:90) w − w dx (cid:48) v ( x (cid:48) ) x − x (cid:48) (cid:21) + 32 (cid:20) − (cid:90) w − w dx (cid:48) v ( x (cid:48) ) x − x (cid:48) (cid:21) = 0(29)can be solved analytically. Remarkably, this solutionprovides one with an excellent fit to the numerical re-sults in a broad range of the ratios w/l which includesthe crossover between the concave and convex profilesof v ( x ).We view Eq. (29) as a second-order differential equa- tion. When solving it, we pick the odd in x solution, − (cid:90) w − w dx (cid:48) v ( x (cid:48) ) x − x (cid:48) = C sinh (cid:32) x √ l (cid:33) , (30)where the constant C will be determined below. In orderto invert Eq. (30), we expand both the left- and right-hand sides of Eq. (30) in Chebyshev polynomials U n ( x ).For the left-hand side, we use Eq. (28). For the right-hand side, we evaluate an expansionsinh (cid:32) √ x l (cid:33) = 8 l √ w ∞ (cid:88) n =1 n I n (cid:32) √ w l (cid:33) U n − (cid:16) xw (cid:17) , (31)where I n ( x ) are the modified Bessel functions. Thereby,the left- and right-hand sides of Eq. (30) are pre-sented as series in orthogonal U n − ( x ) polynomials.So, the expansion coefficients c n may be read off: c n = − C n I n (cid:16) √ w l (cid:17) for n > 0. Recall that the coefficient c = 1 is determined by fixing the total current. So, weobtain the analytical expression for velocity v ( x ) = (32) v c (cid:112) − ( x/w ) (cid:34) − C ∞ (cid:88) n =1 n I n (cid:32) √ w l (cid:33) T n (cid:16) xw (cid:17)(cid:35) . The remaining constant C is determined from theboundary condition v ( ± w ) = 0, producing C − = ∞ (cid:88) n =1 n I n (cid:32) w √ l (cid:33) . (33)For comparison, we superpose the numerical curves withanalytical result (32) in Fig. 2. As expected, the ana-lytical and numerical curves agree perfectly at l/w (cid:29) | x | ≤ w . It is remarkable that at l/w (cid:39) l/w (cid:28) 1, the analytical curves give a verygood approximation to the numerical results in that en-tire range. Our rationalization of such a good agreementthat it is the competition between the singular terms inthe kernel ( ∝ l /x and ∝ /x ) that determines thevelocity profile v ( x ) through the slit. The regular term K reg ( x ) is subdominant and may only slightly renormal-ize the relative strength of the singular terms. Thereforethe extrapolation by means of Eqs. (32) and (33) pro-vides a convenient way for a quantitative comparison ofexperimental results with theory predictions. V. CONDITIONS FOR EXPERIMENTALOBSERVATION OF THE OHMIC-TO-VISCOUSFLOW CROSSOVER In experimental setting, the slit width 2 w is fixedwithin a specific device. One may examine the effect of FIG. 3. Diagram of different transport regimes in the ( T, l )plane. The lengths and temperature are normalized by val-ues given in Eq. (36). The lines corresponding to the trans-port mean free path l tr (red) and the mean-free path forthe electron-electron scattering l ee (blue), have distinct scal-ing with temperature [see Eq. (37)]. Their geometric mean,shown in green, determines the Ohmic-to-viscous crossoverline [see Eqs. (38) and (39)]. Lowering of a temperature atfixed electron density corresponds to a motion along somehorizontal (dashed) line with vertical coordinate represent-ing the fixed slit width 2 w . Its intersection with the threecurves determines three temperatures: T , T , and T . At T > T , the flow through the slit is in the Ohmic regime. At T = T , the crossover to the viscous regime, discussed in thiswork, occurs. At T < T , the notion of local conductivitybecomes inapplicable, but the viscous flow regime persists;at point T , the viscous-to-ballistic crossover occurs [8]. temperature T and electron density n variation on thecurrent density distribution within a slit. In this section,we address two questions which arise in that context:(i) what is the optimal width 2 w for the observation ofcrossover, and (ii) what are the temperature and elec-tron density at which the crossover is likely to occur.Apart from technological constraints limiting the long-scale homogeneity of a sample, additional considerationsfor choosing w come from a remarkably long electrontransport mean-free path l tr at low temperatures . Thetemperature dependence l tr ( T ) comes from the electronscattering off phonons. Upon lowering the temperature,the increase of l tr saturates at some value l tr (0) ∼ µ mdue to the residual scattering off impurities .The sample homogeneity requirement favors smallervalues of w , so in the following we assume w (cid:28) l tr (0)and account only for the phonon contribution to l tr .Furthermore, considering the temperature dependenceof l tr , we focus on T above the Bloch-Gr¨uneisen tem-perature , l tr ( T ) = 4 (cid:126) v F v ph ρ M √ πD T √ n . (34)Here ρ M , v ph , and D are, respectively, the mass density,phonon velocity, and deformation potential in graphene,and v F is the Fermi velocity of the charge carriers;hereinafter T is measured in units of energy. The vis-cosity is proportional to the electron mean free path l ee with respect to the electron-electron scattering : η = νnm = (1 / v F l ee nm ; here n is the charge carriersdensity, and m = p F /v F is the mass conventionally re-lated to the Fermi momentum p F and velocity v F (forreference, we also introduced here the kinematic viscos-ity ν used instead of η in some works ). The mean freepath l ee = α (cid:126) v F p F /T is also temperature-dependent.We may re-write l ee in terms of n instead of p F , l ee ( T ) = √ πα (cid:126) v F T √ n ; (35)the interaction constant α = e / ( (cid:126) v F (cid:15) ) depends on thedielectric constant (cid:15) of the environment (in re-writing,we accounted for the valley and spin degeneracy). Itis convenient to parametrize l tr ( T ) and l ee ( T ) by tem-perature T ee − tr ( n ) at which the two lengths equal eachother, l tr ( T ee − tr ) = l ee ( T ee − tr ) ≡ l ee − tr ( n ), and by thatlength ( l ee − tr ): T ee − tr ( n ) = πα D ρ m v ph n ; l ee − tr ( n ) = √ πα (cid:126) v F T ee − tr ( n ) √ n. (36)With these notations, we find l tr ( T ) = l ee − tr T ee − tr T ; l ee ( T ) = l ee − tr (cid:18) T ee − tr T (cid:19) . (37)The temperature-dependent scattering lengths l tr ( T )and l ee ( T ) are plotted in Fig. 3 in units defined byEq. (36).As shown in Sec. IV, the competition between theviscous and Ohmic terms defines the width l of theboundary layer for the spatial distribution of the cur-rent density [see Eq. (4)]. Using the Drude formula forresistivity, ρ = mv F / ( ne l tr ), and the expression for vis-cosity, η = (1 / v F l ee nm , we may conveniently express l in terms of l tr ( T ) and l ee ( T ): l = 12 (cid:112) l tr ( T ) l ee ( T ) = 12 l ee − tr (cid:18) T ee − tr T (cid:19) / . (38)For the current flow through a slit, the applicability ofthe hydrodynamic description requires that the width ofthe slit exceeds the electron-electron scattering length,i.e. 2 w (cid:38) l ee , while using the notion of resistivity relieson 2 w (cid:38) l tr . Under these conditions, we found theOhmic-to-viscous crossover to occur at w ≈ l . Werewrite this condition using Eq. (38) as2 w = 2 l ee − tr (cid:18) T ee − tr T (cid:19) / . (39)Here, we multiply by 2 the left- and right-hand sides ofEq. (39) in order to display it on par with l ee and l tr inFig. 3.Figure 3 sets the stage for determining the range ofthe slit widths 2 w most favorable for observing theviscous flow, and the temperature of the Ohmic-to-viscous crossover at a given value of 2 w . At 2 w > (1 / l ee − tr ( n ), the crossover from Ohmic regime to vis-cous flow occurs when the slit width 2 w exceeds themean free paths l tr and l ee , justifying the hydrodynamicdescription of electron liquid. This type of crossover isconsidered in detail in this work. One may see fromFig. 3 that a slit of width 2 w (cid:46) l ee − tr ( n ) is the mostfavorable for observing this type of crossover. Furtherreduction of temperature makes scattering off phononsirrelevant, once l tr exceeds the slit width. At even lowertemperatures, the viscous flow gives way to ballistic elec-tron propagation .The temperature of the Ohmic-to-viscous crossoverincreases with the decrease of 2 w . At 2 w =(1 / l ee − tr ( n ) the crossover temperature is 4 T ee − tr , seeEq. (39). (The corresponding point is slightly off theplot in Fig. 3.) At 2 w < (1 / l ee − tr ( n ) the crossover toviscous flow occurs upon lowering the temperature, once l tr ( T ) exceeds the slit width. This type of crossover isnot considered in this work; however, it is clear that theconcave-to-convex transition would occur in the case ofno-slip boundary conditions, while the current flow pro-file would remain concave in the case of no-stress bound-ary condition [cf. Eqs. (17) and (18)].The temperature domain for the viscous flow is alsoconstrained from below (see Fig. 3): the charge car-rier transport enters the ballistic regime once both l ee ( T ) and l tr ( T ) exceed 2 w . Neglecting the electrondiffraction, which occurs on the length scale of theFermi wavelength 2 π (cid:126) /p F , one finds a flat distribution( v y | | x | The goal of this work is to identify the favorableconditions for observing the viscous electron flow ingraphene and to facilitate an accurate measurement ofthe density profile of the current constrained by the de-vice geometry. We find the slit geometry promising asit creates large gradients of electric potential and rapidspatial variations of electron velocity near the edges ofthe wall cut by the slit. It may help gaining informa-tion about the boundary conditions for the electron flowfrom the local-probe measurements .In the case of Ohmic flow, the divergent electric fieldcauses 1 / √ x singularities of the current density at theedges of the slit [see Eq. (9)]. We establish that the ve-locity in the viscous flow with no-stress boundary con-dition also results in 1 / √ x divergence at the edges [seeEq. (17)]. It qualitatively resembles the velocity pro-file in the Ohmic limit, making it difficult to distinguishbetween the two types of flow in an experiment. Incontrast, the velocity profile in a viscous flow with theno-slip boundary condition is significantly different fromthe Ohmic limit: it is convex in the former and concavein the latter case.At a fixed electron density n , the electron transportmean free path l tr depends on temperature due to theelectron scattering off phonons; resistivity ρ is inverselyproportional to l tr . The viscosity η of electron liquidis controlled by the electron-electron scattering and isa function of temperature as well. The competitionbetween the viscous and Ohmic flows determines thewidth l of the boundary layer in the electron liquidmoving around an obstacle [see Eq. (4)]; l is propor-tional to (cid:112) η/ρ and also is a function of temperature.The crossover from Ohmic to viscous flow upon low-ering the temperature occurs once l tr or l exceeds thewidth 2 w of the slit. The former case was alluded to inRef. [8]. Our work investigates the details of Ohmic-to-viscous crossover in the latter case (interplay between l and w ). We develop a method based on a solution ofthe integral equation (21), which depends on the param-eter l and describes the crossover. We find an efficientnumerical scheme to solve that equation and establishthat the crossover occurs at l/w (cid:39) . 5. In addition,by dropping certain term in the kernel K ( x ) of the in-tegral equation and solving it analytically, we producea convenient extrapolation formula [see Eq. (32)]. Thecrossover is marked by the change in the current profilefrom concave to a convex one.The profile evolves slowly with the ratio l/w and israther flat at l/w = 0 . . That the-ory informed the experiment which, in turn, indicatedthat the boundary condition falls in between the no-slip and no-stress limits. Theory also indicated thatthe crossover between the hydrodynamic and ballisticregimes is quite broad for the channel geometry. Inaddition, for the ballistic regime the kinetic approach predicted a robust spike of the Hall field in the middleof the channel, if exactly two cyclotron orbits fit intothe channel’s width. This beautiful observation is rem-iniscent of the physics of Gantmakher-Kaner effect .Works [30] and [16] provide a strong motivation to ex-tend the kinetic theory, with an account for the effectof magnetic field, to a slit geometry. ACKNOWLEDGMENTS We thank A. Bleszynski Jayich, M. Goldstein,Z. Raines, and J. Zang for useful discussions. The workis supported by NSF DMR Grant No. 2002275 (LG)and by NSF DMR Grant No. 1810544 (AY). R. N. Gurzhi, “Hydrodynamic effects in solids at low tem-perature,” Sov. Phys. Usp. , 255 (1968). I. 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Yacoby, and R. L. Walsworth, “Imagingviscous flow of the Dirac fluid in graphene,” Nature ,537 (2020). Here we use that ψ is defined up to a constant, so we mayarbitrary shift it for our convenience. L. D. Landau and E. M. Lifshitz, Fluid Mechanics:Volume 6 (Course of Theoretical Physics) , 2nd ed.(Butterworth-Heinemann, 1987). It is instructive to view that conformal transformationas a sequence of two mappings, z = g ( f ( z )). The firstone, ˜ z = f ( z ) = (cid:0) z + √ z − w (cid:1) /w , is the inverse to theJoukowsky transform, and it maps the slit geometry ontothe upper half-plane (i.e., ˜ z = ˜ x + i ˜ y with ˜ y > z = g (˜ z ) = ln ˜ z transforms the upper-halfplane into the horizontal stripe (i.e. z = x + iy with π > y > M. A. Lavrentiev and B. V. Shabat, Methods of the The-ory of Functions of Complex Variable , 4th ed. (Nauka,Moscow, 1973). From Eq. (2) of Ref. 7, we extract the large- r behavior ofvelocity v ( r ) = I r πner (cid:0) − cos 2 θ (cid:1) . 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B , 245305 (2019). ´E. A. Kaner and V. F. Gantmakher, “Anomalous penetra-tion of eletromagnetic field in a metal and radiofrequencysize effects,” Sov. Phys. Usp. , 81 (1968). H. Prodinger, “Representing derivatives of Chebyshevpolynomials by Chebyshev polynomials and related ques-tions,” Open Math. , 1156 (2017). Appendix A: Details on numerical solution ofEq. (26). In this Appendix, we provide the details of a numer-ical solution of the integral Eq. (21). We rely on theChebyshev polynomials of both first T n and second U n kind, which are well suited for solving (differential orintegral) equations on a finite interval.(i) Let us treat the principal value integral appearingin Eq. (26). We substitute the expansion (27) in thatintegral and, using Eq. (18.17.42) of Ref. [26], obtain − (cid:90) w − w dx (cid:48) v ( x (cid:48) ) x − x (cid:48) = − v c ∞ (cid:88) n =1 c n π U n − ( x/w ) , (A1)where U m are the Chebyshev polynomials of secondkind. In addition, we express the second derivative of the Chebyshev polynomial U m − using polynomials oflesser degrees d U m − ( x/w ) dx = (A2)= (cid:26) w (cid:80) m − n =1 n ( m − n ) U n − ( x/w ) , m ≥ , , m = 1 . (ii) Let us treat the last term in Eq. (21). The goal isto expand that term in series of U m − ( x/w ). We recallthe definition of K reg ( x ) in Eq. (23), and, using parityof v ( x ) under x → − x , drop odd terms in the integrand (cid:90) w − w dx (cid:48) K reg ( x − x (cid:48) ) v ( x (cid:48) ) (A3)= − (cid:90) w − w dx (cid:48) v ( x (cid:48) ) (cid:90) ∞ dt l sin ( t ( x − x (cid:48) ) /l )( t + √ t + 1) = − (cid:90) w − w dx (cid:48) v ( x (cid:48) ) (cid:90) ∞ dt l sin ( t x/l ) cos ( t x (cid:48) /l )( t + √ t + 1) It allows to treat the x and x (cid:48) parts independently. Wesubstitute the expansion (27) and integrate over x (cid:48) usingthe identity (cid:90) w − w dx (cid:48) cos( tx (cid:48) /l ) T m ( x (cid:48) /w ) (cid:112) w − x (cid:48) = ( − m J m ( tw/l ) . (A4)Further, we expandsin( tx/l ) = 4 lt w ∞ (cid:88) n =1 ( − n +1 n J n ( tw/l ) U n − ( x/w ) . (A5)Equations (A4) and (A5) allow to cast Eq. (A3) in aconcise form (cid:90) w − w dx (cid:48) K reg ( x − x (cid:48) ) v ( x (cid:48) ) (A6)= v c ∞ (cid:88) n =1 m =0 U n − (cid:16) xw (cid:17) K nmreg c m ,K nmreg = ( − m + n πn (cid:90) ∞ dt J m ( tw/l ) J n ( tw/l ) t ( t + √ t + 1) . The integrals in K nmreg are evaluated numerically.(iii) Equations (A1), (A2) and (A6) allow to writeEq. (26) in the form ∞ (cid:88) n =1 U n − ( x/w ) (cid:40) K n reg + ∞ (cid:88) m =1 (cid:20) π l w n ( m − n ) θ mn − π δ nm + K nmreg (cid:21) c m (cid:41) = 0 , (A7)1where the notation θ mn = (cid:26) , m > n, , m ≤ n, (A8)was introduced for simplicity. For reference, the three terms in the square brackets of the latter equation correspondto the three respective terms in Eq. (26). Using the orthogonality of the Chebyshev polynomials U n − ( x/w ), thesystem of linear equations is read-off from Eq. (A7) ∞ (cid:88) m =1 (cid:20) π l w n ( m − n ) θ mn − π δ nm + K nmreg (cid:21) c m = − K n reg , for n = 1 , , . . . . (A9)We supplement it with the boundary condition v ( ± w ) = 0, which, given expansion (27) and c = 1, translates into ∞ (cid:88) n =1 c n = − . (A10)Equations (A9) and (A10) comprise the infinite system of linear equations for the expansion coefficients C =( c , c , . . . ). We solve it numerically by truncating, i.e. by setting c n = 0 for n > Nn > N