Higher-than-ballistic conduction of viscous flows Haoyu Guoa, Ekin Ilsevena, Gregory Falkovichb,c, and Leonid S. Levitova,1

aDepartment of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139; bDepartment of Physics, Weizmann Institute of Science, Rehovot 76100, Israel; and cInstitute for Information Transmission Problems, Moscow 127994, Russia

Edited by Allan H. MacDonald, The University of Texas at Austin, Austin, TX, and approved January 24, 2017 (received for review July 25, 2016) Strongly interacting can move in a neatly coordinated Conveniently, both regimes are accessible in a single constric- way, reminiscent of the movement of viscous fluids. Here, we tion, because transport is expected to be viscous at elevated tem- show that in viscous flows, interactions facilitate transport, allow- peratures and ballistic at T = 0. The crossover temperature can ing conductance to exceed the fundamental Landauer’s ballistic be estimated in terms of the ee scattering mean free path as limit Gball. The effect is particularly striking for the flow through l (T )/w = π2/16 ≈ 0.62. [2] a viscous point contact, a constriction exhibiting the quantum ee mechanical ballistic transport at T = 0 but governed by electron This relation is found by setting Rvis = Rball and expressing viscos- 1 hydrodynamics at elevated temperatures. We develop a theory ity as η = νnm = 4 vF leenm, where m is the carrier mass and the of the ballistic-to-viscous crossover using an approach based on kinetic viscosity ν is estimated in Eq. S21. The condition Eq. 2 quasi-hydrodynamic variables. Conductance is found to obey an can be readily met in micron-size junctions. additive relation G = Gball + Gvis, where the viscous contribution Several effects of electron interactions on transport in con- Gvis dominates over Gball in the hydrodynamic limit. The super- strictions were discussed recently. Refs. 14 and 15 study junc- ballistic, low-dissipation transport is a generic feature of viscous tions with spatially varying electron density and, using the electronics. time-dependent current density functional theory, predict a sup- pression of conductance. A hydrodynamic picture of this effect electron hydrodynamics | graphene | strongly correlated systems was established in ref. 16. In contrast, here we study junctions in which, in the absence of applied current, the carrier density ree electron flow through constrictions in metals is often is approximately position-independent. This situation was ana- Fregarded as an ultimate high-conduction charge transfer lyzed in ref. 27 perturbatively in the ee scattering rate, finding a mechanism (1–5). Can conductance ever exceed the ballistic limit conductance enhancement that resembles our results. value? Here we show that superballistic conduction is possi- The relation Eq. 1 points to a simple way to measure viscos- ble for strongly interacting systems in which electron movement ity by the conventional transport techniques. Precision measure- resembles that of viscous fluids. Electron fluids are predicted ments of viscosity in fluids date as far back as the 19th century to occur in quantum-critical systems and in high-mobility con- (28). They relied, in particular, on measuring resistance of a vis- ductors, so long as momentum-conserving electron–electron (ee) cous fluid discharged through a narrow channel or an orifice, scattering dominates over other scattering processes (6–9). Vis- a direct analog of our constriction geometry. Further, viscosity- cous electron flows feature a host of novel transport behaviors induced electric conduction has a well-known counterpart in the (10–22). Signatures of such flows have been observed in ultra- kinetics of classical gases, where momentum exchange between clean GaAs, graphene, and ultrapure PdCoO2 (23–26). atoms results in a slower momentum loss and a lower resistance We will see that electrons in a viscous flow can achieve of gas flow. Viscous effects are responsible, in particular, for a through cooperation what they cannot accomplish individually. dramatic drop in the hydrodynamic resistance upon a transition As a result, resistance and dissipation of a viscous flow can be from Knudsen to Poiseuille regime. For a viscous flow through markedly smaller than that for the free- transport. As a simplest realization, we discuss viscous point contact (VPC), Significance where correlations act as a ”lubricant” facilitating the flow. The reduction in resistance arises due to the streaming effect Free electron flows through constrictions in metals are often illustrated in Fig. 1, wherein electron currents bundle up to regarded as an ultimate high-conduction charge transfer. We form streams that bypass the boundaries, where momentum loss predict that electron fluids can flow with a resistance that occurs. This surprising behavior is in a clear departure from the is much smaller than the fundamental quantum mechani- common view that regards electron interactions as an impedi- cal ballistic limit for nanoscale electronics. The “superballis- ment for transport. tic” low-dissipation transport is particularly striking for the A simplest VPC is a 2D constriction pictured in Fig. 1A. The flow through a viscous point contact, a constriction exhibit- interaction effects dominate in constrictions of width w exceed- ing the quantum mechanical ballistic transport at zero tem- ing the carrier collision mean free path l (and much greater ee perature but governed by viscous electron hydrodynamics than the Fermi wavelength λF ). The VPC conductance, evalu- at a higher temperature. Unlike other mechanisms of low- ated in the absence of impurity scattering, scales as a square of dissipation transport, for example, , the vis- the width w and inversely with the electron viscosity η, cous electron flows can be realized at elevated temperatures, πn2e2w 2 granting a new route for the low-power electronics research. G (w) = , w  l , [1] vis 32η ee Author contributions: H.G., G.F., and L.S.L. designed research; H.G., E.I., G.F., and L.S.L. where n and e are the carrier density and charge. In the opposite performed research; and H.G., G.F., and L.S.L. wrote the paper. limit, lee  w, the ballistic free-fermion model (1, 5) predicts the The authors declare no conflict of interest. 2 conductance Gball = 2e /h N , where N ≈ 2w/λF is the number This article is a PNAS Direct Submission. of Landauer’s open transmission channels. The conductance Gvis 1To whom correspondence should be addressed. Email: [email protected]. grows with width faster than Gball. Therefore, for large enough This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. w, viscous transport yields G values above the ballistic bound. 1073/pnas.1612181114/-/DCSupplemental.

3068–3073 | PNAS | March 21, 2017 | vol. 114 | no. 12 www.pnas.org/cgi/doi/10.1073/pnas.1612181114 Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 in Estimating diffusion tion. momentum of with time form similar the constriction using the way across this in modeled be and form mass Drude-like a values in conduc- the electron them compare for can limit we ballistic Indeed, quantum tion. the overcome to cle resistance. the lowering thus and aver- scatterers on from away collectively, surface, staying flow age particles scatterer makes particle, the exchange per rate Momentum aver- loss to spatial momentum the the a normal down slows reduce which by component scatterer elucidated velocity a be near age can collisions scatterers Particle to argument: proximity in sions time ballistic For path. free is time distance is transfer a tum by spaced scatterers phe- Gibbs the out smooth to used was filter nomenon. space Fourier A current. total u tal. et Guo as regime long dynamic so values limit ballistic the below regime, viscous the are in semicircle used a to regime constriction ballistic Eq. the the in in constant evolves distribution a distribution The from Current values. mean-free-path (B) collision Eq. bound- carrier value. resistance, different the for limit the avoiding ballistic allowing stream, the and narrow below occurs a dissipation forms where den- Current aries the to streamlines. proportional of is magnitude sity Velocity constriction. a through flow 1. Fig. h icst-nue rpi eitnecnb sda vehi- a as used be can resistance in drop viscosity-induced The colli- particle fast from originating correlations peculiar The lutaigteitrcinidcdsraigefc.Parameters effect. streaming interaction-induced the illustrating 10, τ urn temie bak n oeta oo a o viscous for map color potential and (black) streamlines Current (A) 0 L = = τ 3w L/v sasial oetmrlxto ie Eq. time. relaxation momentum suitable a as τ ` 0 τ and . T  w ∼ ν where , τ = & 0 L h icu time viscous the L, b = 2 = 1 4 l /ν ee w v 10 . F τ /v ≈ l 5 v = ee v T F L h itiuin r omlzdt unit to normalized are distributions The . eseta Eq. that see we , w R 2 steflgttm costeconstric- the across time flight the as stemlvlct and velocity thermal is /v 2 = /ν T m h yia ieo momen- of time typical the L, whereas , hra,fra da a,this gas, ideal an for whereas, `, /ne τ 2 τ τ smc ogrta the than longer much is R with , & vis R 1 τ and ball 0 rdcsresistance predicts .. ntehydro- the in i.e., , m a epti a in put be can R stecarrier the as ball ` stemean the is yputting by m odrop to 1, v v 1 ⊥ can v /L. ⊥ , φ Eq. tion plane function unknown Eq. Namely, resulting the hcns,adpstoe nthe in positioned and thickness, field electric width nal of strip ideal-metal an for lem interval the outside zero where func- the determine therefore not must is We that tions slit. potential a the yield inside would constant distribution current generic a Eq. relation potential-current the on focus function influence the where current slit, different the to in due elements contributions of superposition a as plane at in component of current the that ensures metry of ratio any scales for length VPC the provides through transport model for This solution same. closed-form the a are harmon- distribution nonconserved all momentum a for of rates use ics in relaxation we chosen the that operator crossover, way collision the a ee such tackle simplified a to with Here, equation task. kinetic nontrivial a is over frcaiy ednt Dcodntsb aia etr) Poten- letters). capital by coordinates tial 3D denote we clarity, (for all for the at ρ source current Eq. point-like from a obtained to edge, due half-plane a in potential the obeys that flow electron Eq. (29). low-Reynolds in equation resistance a Stokes VPC of the model of the derivation using simple a with start We Regime Hydrodynamic The terms interpo- the which model, limits, in viscous microscopic and ballistic a the between from lates derived dependence, This cteig u nlssrle nasmer ruetand con- the argument model symmetry We 1A Fig. a problem. in electrostatic striction on auxiliary relies an or invokes impurity analysis to Our due resistivity scattering. ohmic describes term ond Here +0 0 = rcal,rte hnpoiigaslto oorpolm the problem, our to solution a providing than rather Crucially, cross- ballistic-to-viscous the at behavior the Understanding ouino hsitga qainsc that such equation integral this of solution A y y y Φ (x sarsl,tequantities the result, a As . 0 = hra ohtecomponent the both whereas j 3D φ(r 3) Here (30). y − = ) (x j (x ≥ hsosrainalw st rt h oeta nthe in potential the write to us allows observation This . (X w 5 2 ) ) ) 0 and as − − < steeeti potential, electric the is , stecurrent the is deas edge Y w β 2 y x 2 φ(x 5 φ(x φ(x ) < > Z < utb rae sa nerleuto o an for equation integral an as treated be must (η sahroi ucin osato h ti and strip the on constant function, harmonic a is −∞ w , E w ∞ β 0 2 , X ∇ y , y PNAS saslit a as 0 G half-plane. = y j a eotie rma3 lcrsai prob- electrostatic 3D a from obtained be can and = ) φ ) dx 2 = (x ) < ball +0 − π aihso h line the on vanishes efcnitnl,i a htesrsthat ensures that way a in self-consistently, G λ 0 eoigptnilvle ttehalf- the at values potential Denoting ). ( w (x 2η Z l en  ˆ 2 h ti stknt eifiie zero infinite, be to taken is strip The x. and VPC (ne ee | [− 4 − y (x ) , = ) rdcigasml diierelation additive simple a predicting , w − ac 1 2017 21, March 2 2 ihn-lpbudr odtosand conditions boundary no-slip with w 2 opnn,wihi nt nieand inside finite is which component, w ) = n,wtotls fgnrlt,we generality, of loss without and, , 2 − G w 2 dx 2 Y , φ(x = ρ)v(r j G vis 5 x w < 2 (x 0 0 = ball 0 epciey dominate. respectively, , Y R(x eeyhlst oei.Indeed, it. pose to helps merely ]. x j + , 0 R(x x ) j y 0 = < η x + , i ) and w 0) n h potential the and y stevsoiy n h sec- the and viscosity, the is − w G 2 =+0 , lcdi nfr exter- uniform a in placed ne 2 −∞ ln uhthat such plane , y vis x | y φ = ) + 0 ∇φ(r). . , ecnwieterela- the write can we , 0 = o.114 vol. y j aihwti h slit the within vanish (x y y < )j w β ( 0 = sa vnfunction even an is x The . ( (x Z −  2 y +y 2 φ j 0 −x x < ), l (x +0 nieteslit. the inside | ee 2 0 ) 2 y ∞ − 0 2 o 12 no. (x and ) ) −y → i ) φ 0) describes vanishes w r odd are 2  |  , sym- 3069 [5] [3] [7] [4] l [6] ee 1 ,

PHYSICS D ∓iθ E behaving asymptotically as −E0X . It is easily checked that the f0 = hf (θ)iθ, f±1 = e f (θ) , [13] 3D electrostatic problem translates to the 2D viscous problem as θ H dθ 3D, Y = 0 : X σ(X ) Ex (X ) where we introduced notation h...iθ = ... 2π . The quantities f0 ↓ ↓ ↓ . [8] and f±1, conserved in the ee collisions, represent the zero modes β I I 2D, y = +0 : x − ∂j /∂x φ (x) of ee. For suitably chosen ee, the task of solving the kinetic equa- 2 +0 tion in a relatively complicated constriction geometry is reduced to analyzing a self-consistency equation for the variables f0 and This mapping transforms Coulomb’s charge–field relation f±1. We will derive a linear integral equation for these quan- between the electric field at Y = 0 and the surface charge den- 0 0 tities and solve it to obtain the current density, potential, and R ∞ σ(X )dX sity, Ex (X ) = 2 −∞ X −X 0 , into the 2D viscous relation in conductance. Eq. 6. Potential Φ3D , obtained through a textbook application To facilitate the analysis, we model Iee by choosing a single of conformal mapping, then equals relaxation rate for all nonconserved harmonics,

r 2 2 w X Φ3D (X , Y ) = − Re λ ζ − , ζ = X + iY . [9] Iee(f ) = −γ(f − Pf ), P = |mi hm|, [14] 4 m=0,±1

Eq. 9 describes the net contribution of the external field E0 and the charges σ(X ) induced on the strip. The field component where γ represents the ee collision rate, with lee = v/γ, and w w Ex (X ) = −∂X Φ3D vanishes on the strip − 2 < x < 2 and equals P is a projector in the space of angular harmonics of f (θ) λ far outside. We can therefore identify λ with V /2 in the viscous that selects the harmonics conserved in ee collisions. Here problem (Fig. 1A). we introduced Dirac notation for f (θ) with the inner product 9 H dθ ¯ Charge density on the strip, found from with the help of hf1|f2i = 2π f1(θ)f2(θ). Namely, Gauss’s law, σ(X ) = √ λX , under the mapping Eq. 8, 2π w2/4−X 2 I 0 imθ X dθ im(θ−θ0) 0 gives a semicircle current distribution, hθ|mi = e , Pf (θ) = e f (θ ). 2π r m=0,±1 w λ w 2  w  j (|x| < ) = − x 2, j |x| > = 0. [10] 2 πβ 4 2 As in quantum theory, the Dirac notation proves to be a useful The potential map in Fig. 1A is then obtained by plugging this bookkeeping tool to account, on equal footing, for the distribu- result into Eq. 5. The flow streamlines are obtained from a sim- tion function position and wavenumber dependence, as well as ilar relation for the stream function (see ref. 30). Evaluating the the angle dependence. w R 2 2 To simplify our analysis, we replace the constriction geometry current I = − w j (x)dx = λw /8β and setting λ = V /2 yields 2 by that of a full plane, with a part of the line y = 0 made impene- 2 R = V /I = 16β/w , which is Eq. 1. The inverse-square scaling trable through a suitable choice of Ibd(f ). Scattering by disorder −2 −1 R ∝ w is distinct from the w scaling found in the ballis- at the actual boundary conserves f0 but not f±1. We can therefore tic free-fermion regime. The scaling, as well as the lower-than- model momentum loss due to collisions at the boundary using ballistic R values, can serve as a hallmark of a viscous flow. Potential, inferred from the 2D/3D correspondence, is  w 0, |x| < 0  2  V |x| w Ibd(f ) = −α(x)P f , α(x) = w , [15]  p sgn y |x| > bδ(y), |x| ≥ φ(x) = 2 x 2 − w 2/4 2 , [11] 2 y=±0 w  0 |x| < 2 where P 0 is a projector defined in a manner similar to P, pro- f m = ±1 α(x) where sgn y corresponds to the upper and lower sides, jecting on the harmonics . The term describes y = 0 y = ±0. Potential grows toward the slit, diverging at the end momentum relaxation on the line , equal to zero within the w slit and to b outside. The parameter b > 0 with the dimension of points x = ± 2 . This interesting behavior, representing an up- converting DC current transformer, arises due to the electric velocity, introduced for mathematical convenience, describes a field pointing against the current near the viscous fluid edge as partially transparent boundary. An impenetrable no-slip bound- discussed in ref. 29. ary, which corresponds to the situation of interest, can be mod- eled by taking the limit b → ∞. Crossover to the Ballistic Regime We will analyze the flow induced by a current applied along y Our next goal is to develop a theory of the ballistic-to-viscous the direction, described by a distribution, crossover for a constriction. Because we are interested in the lin- (0) (0) ear response, we use the kinetic equation linearized in deviations f (θ, x) = f (θ) + δf (θ, x), f (θ) = 2j sin θ. [16] of particle distribution from the equilibrium Fermi step (assum- E ing kB T  EF ), (0) (0) Here, f and δf , which we will also write as f and |δf i, (∂t + v∇x) f (θ, x, t) = Iee(f ) + Ibd(f ), [12] represent a uniform current-carrying state and its distortion due to scattering at the y = 0 boundary. The quantity j is the current θ where is the angle parameterizing particle momentum at density. Once found, the spatial distribution f (θ, x) will allow us I I the 2D Fermi surface. Here, ee and bd describe momentum- to determine the resulting potential and resistance. conserving carrier collisions and momentum-nonconserving scat- The kinetic equation, Eq. 12, reads tering at the boundary, respectively. In the presence of momentum-conserving collisions, trans- 0 ˆ port is succinctly described by “quasi-hydrodynamic variables” [∂t + K + α(x)P ]|f i = 0, K = v∇ + γ1 − γP [17] defined as deviations in the average particle density and momen- tum from local equilibrium (31). These quantities can be (from now on, we suppress the coordinate and angle dependence expressed as angular harmonics of the distribution f (θ, x, t), of f and switch to the Dirac notation). Plugging f = f (0) + δf , we

3070 | www.pnas.org/cgi/doi/10.1073/pnas.1612181114 Guo et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 ingvsmti lmns(here elements matrix gives all tion which in transport describing including harmonics, function Green’s auxiliary an is qaini bandb cigo ohsdswt h operator the with sides both on + acting 1 by obtained is equation matrix ana- First, steps: two 2D. in into proceed to convenient Eq. more lyze is it 2D, in operator integral the Eq. inverting by principle, in found, the on projected defined we Here, tity 3 Eq. P in quantities the the projecting on by variables hydrodynamic Eq. operator, in an term as scattering the treat we representation, momentum where as solution operator formal a write We state. steady u tal. et Guo where of terms in result the expressing series, a the resummed we Here, the sinc where Eq. rewrite n efrigatdosbtsrihfradmti inversion, matrix straightforward but obtain we tedious a performing and matrix the but that note ing We channels. “active” of number finite Lippmann–Schwinger as function Green’s actual γ the expanding by sinhβ denote we where projector the absorbed we ness, and P 3 × esatwt finding with start We quantity The quasi- for equation integral closed-form a derive we Next, h matrix The

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  |k (k (k , g ac 1 2017 21, March and X ± opnnsvns nthe on vanish components Z κ sa nerleuto,replacing equation, integral an as γ (0) dk |v n 1 1 2π √ D −∞ −2κ . ) ) + 1 ∞ D 2 −i 2 E ssmls oudrtn using understand to simplest is , 0 2κ . , ... k G , and α ss κ ˜ ial,mliligby multiplying Finally, . 2 γκ γκ γκ D D dk √ 1 k 2π 0s sarsl eoti block a obtain we result a as 25; 2 2 (κ + nEq. in κ −k sa ievleof eigenvalue an is R 0c , cc 2κ D v πκ 4 4 2 2 2 | κ n t eainwt the with relation its and 29, r d in odd are + s (k (k G R 2πδ 0 ss ˜ 1,2 i 1 g + 2 1 (| 1 k (k y = 0 ) 0 ) 0 )cot 1) + k | = eobtain we 25, |0i, (k 0 = 1 2 = |v |+1i−|−1i , o.114 vol. ) k γ D −2κ v 2  k → −i so pure a of is |c √ ss πµδ ). 2κ 1,2 2i (k and i, γκ γκ γκ γ √ 1 k Lδ 1 2 , −1 κ 2 = ) 1 R (k 2κ κ 4 4 2 ) 2 | k n therefore and + k   κ R ,0 = nti basis, this In . ), 1 2 o 12 no. , π . , + = 2 |s p v y D         i . |s 2π D κ 0 = L y |±1i , n we and 1 2 o the for i −1 n −y → | + form, [25] [24] [27] [30] [28] [26] [29] line. we , 3071 and κ k G to 2 2 ˜ k 1 .

PHYSICS Z 0 ˜ 0  −1  0  f0(x) = − dx G0s (x, x ) µ − Dss g (x ) . [31]

√ ˜ −i 2k2 Plugging in G0s (k) = 2 2 , Fourier-transforming, and carry- v(k1 +k2 ) R ik2y ik2 ing out the k2 integral by the residue method, dk2e 2 2 = k1 +k2 −πe−|yk2|sgn y, we obtain

Z sgn y dk1 ik1x−|k1y| −1 f0(x) = √ e [Dss (k1)gk − 2πµδ(k1)]. [32] 2v 2π 1

The resulting distributions, shown in Fig. 2, are step-like. At µ large y, the µ-term dominates, giving f0(|x|  w) ≈ − √ sgn y. √ 2v 2 Therefore, the step height equals v µ regardless of the parame- ter values used. This relation provides a route to evaluate resistance. Namely, because of charge neutrality, the density f0 obtained from a noninteracting model translates directly into potential distribu- 1 tion φ(x) = f0(x), where ν0 is the density of states. Divid- eν0 √ ing the potential difference V = 2µ by the total current eν0v I = R dxg(x) hev sin θ|si = √ev g 2 k1=0 yields a simple expression for resistance,

A Fig. 2. Potential distribution induced by a unit current through a constric- tion (A) at the crossover, lee ≈ w, and (B) in the viscous regime, lee  w. The spikes at the constriction edges in B are a signature of a hydrodynamic behavior (Eq. 11 and accompanying text). Plotted is particle density devia- tion from equilibrium, f0(x), which is proportional to potential (see discus- sion preceding Eq. 33). Parameters used are (i) γ = v/w and (ii) γ = 15v/w; other parameter values are the same as in Fig. 1B.

that is, putting the problem on a cylinder of circumference L, we see that the values G˜ ss (k1, k2) vanish for k1 = 0 and any k2. This observation implies that the quantity Dss (k1) also vanishes for k1 = 0, and thus the operator D does not have an inverse. In this case, caution must be exercised when multiplying by D −1. Namely, the quantities D −1 |f i are defined modulo a null vec- tor of D, which is the k1 = 0 mode with an unspecified coeffi- B cient, represented by the µ-term. We note, parenthetically, that discretization has no impact on the values Dss (k1 =6 0) given in Eqs. 27 and 28. We obtain current distribution by solving, numerically, Eq. 29, discretized as in Eq. 30, and subsequently Fourier-transforming gk to position space (Supporting Information). A large value b = 105v was used to ensure that current vanishes outside the w w interval [− 2 , 2 ]. The resulting distribution, shown in Fig. 1B, features interesting evolution under varying γ: Flat at small γ, the distribution gradually bulges out as γ increases, peak- w ing at x = 0 and dropping to zero near x = ± 2 . In the limit γ  v/w, it evolves into a semicircle coinciding with the hydro- dynamic result, Eq. 10. Current suppression near the constriction edges is in agreement with the streaming picture discussed in the Introduction. The solution on the line y = 0 can now be used to deter- mine the solution in the bulk. For example, to obtain the den- Fig. 3. (A) The resistance R, Eq. 33, plotted vs. γ. Upon rescaling R → Rw, γ → γw, all of the curves collapse on one curve, confirming that the sity f0(x), we project the relation Eq. 20 on m = 0 harmonic, tak- (0) ˜ only relevant parameter is the ratio w/lee = wγ/v.(B) Scaled conductance ing into account that both f and αf are of an |si form. This G = 1/(Rw) vs. γw. All curves collapse onto a single straight line, which can procedure yields an expression for the 2D density of the form −1 be fitted with (0.694 + 0.378γw)ρ∗ . This dependence matches Eq. 3 which R 0 ˜ 0 0 0 −1 6 f0(x) = − dx G0s (x, x )α(x )g(x ), with x being a 2D coordi- corresponds to (2/π + π/8γw)ρ∗ . Parameters used are b = 10 v, the num- nate and −∞ < x 0 < ∞. To avoid handling the b → ∞ limit in α, ber of sampling points within the constriction ≈160, and the length unit = 1 . we write this relation, using Eq. 29, as w0 30 L

3072 | www.pnas.org/cgi/doi/10.1073/pnas.1612181114 Guo et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 5 o ,VgaeG iVnr 21)Vsoscretost h eitneo nano- of resistance the to corrections Viscous (2011) M Ventra Di G, DFT-LDA Vignale the D, to corrections Roy Dynamical (2005) 15. M Ventra Di G, Vignale M, Zwolak N, Sai flow. graphene 14. in regimes Preturbulent (2011) S Succi HJ, Herrmann M, Mendoza 13. 6 nre V iesnS,Sia 21)Hdoyai ecito ftasotin transport of description Hydrodynamic (2011) B Spivak SA, Kivelson AV, Andreev 16. M 12. M J, graphene. Schmalian in L, scaling Fritz critical 11. Quantum (2007) J Schmalian DE, Sheehy 10. u tal. et Guo value the plugging by obtained Eq. be can dependence Spatial conductance Landauer value. collisionless the with coincides result This limit the In eepc httedpnec on dependence grounds, the general that on expect First, we unexpected. some and expected some [− where e.g., states of density 2D where efidafml fcre htalclas noecre(hsuni- (this curve one on collapse all that curves of family a find we ratio distribution, step-like a by Eq. integral limit collisionless tion. as decreases 3A Fig. independent. Ohm .DmeK ahe 19)Nneotmeauetasotna unu critical quantum near transport Nonzero-temperature (1997) S effect. Sachdev size dc K, the for Damle model Electron-fluid 9. (1991) R Jaggi (1981) LP 8. Pitaevskii EM, Lifshitz 7. temperature]. low at solids in effects contacts. [Hydrodynamic (1968) point RN Gurzhi Quantum (1996) 6. 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(|x k sse rmtepoint the from seen is ee =0 tan = ) √ hsRvB Rev Phys ned ltigtersae quantity rescaled the plotting Indeed, . µ o rpee,w find we graphene), for | 2v b = 29 > γ h oa urn is current total the ∞, → o h tpheight. step the for R nrae,ie,crircliin nac conduc- enhance collisions carrier i.e., increases, R w un noa leri qain hc ssolved is which equation, algebraic an into turns 56:8714–8733. hsC Phys J f −1 g 0 )= /2) = γ R (x 78:085416. 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Rw 69:816–820. π s iilNauk Fiziol Usp 2µ o hsJETP Phys Sov v hsRvLett Rev Phys hsRvLett Rev Phys v g n the and , . (x vs. ) [35] [34] [36] [33] 49(7): into γ Phys Phys w µ- R , 1 eisnI 17)Budr odtosi hnnhydrodynamics. (2014) phonon FJ in Blatt conditions in Boundary 32. current (1977) and IB potential Levinson of distributions 31. spatial Linking (2016) L Levitov G, nonlocal Falkovich negative and 30. vortices current viscosity, Electron (2016) G Falkovich L, liquids. Levitov of viscosity 29. of measurements of accuracy The (1971) RS Marvin ballistic wide 28. in scattering electron-electron of Effects (2008) the OS hydro- Ayvazyan of for KE, Evidence Nagaev breakdown (2016) AP the 27. Mackenzie and B, Schmidt fluid N, Nandi Dirac P, Kushwaha the PJW, Moll of Observation 26. (2016) al. back- et electron viscous J, by Crossno caused resistance 25. local Negative (2016) high-mobility al. et in DA, flow Bandurin electron Hydrodynamic 24. (1985) LW Molenkamp MJM, Jong de 23. 2 ua ,CosoJ ogK,KmP ahe 21)Tasoti inhomogeneous in Transport (2016) S Sachdev P, Kim KC, Fong J, Crossno A, Lucas 22. 2D the of viscosities shear Ferreir and A, Cortijo Bulk (2016) 21. M Polini M, Carrega G, Vignale Sch A, Principi M, 20. Titov IV, Gornyi elec- quantum BN, 2D for Narozhny viscometer disk 19. Corbino (2014) M Polini G, properties Vignale Electromagnetic A, Tomadin (2014) D Marel 18. der van D, Valentinis J, Zaanen D, Forcella 17. ues to proportional tion with linearly scales conductance, term for rule all addition for simple a identical into is at which offset by line, positive described straight a The with 3B. line Fig. straight perfect nearly plotting a and quantity this inverting ably, h usa cec onainPoet1-2029(oG.F.). and G.F.); (to (to 14-22-00259 882 Project Seed Grant Foundation Foundation MIT-Israel Science Science Russian MISTI Israeli the the (L.S.L.); G.F.); and Foundation (L.S.L. Science Fund Binational L.S.L.); US-Israel (to Institute W911NF-13-D-0001 the The Materials Contract through under Army Office Quantum Nanotechnologies, US Research Soldier the Army for by Integrated US support the partial for and L.S.L.); Laboratory (to Research Center 1231319 Award the NSF under (CIQM) of support acknowledge ACKNOWLEDGMENTS. value. exceeds collisionless conductance the collisions, by suppressed scat- being than conduc- other Rather for to rule Eq. due addition tance, loss relaxation the momentum Furthermore, momentum in mechanisms. impact tering result only themselves, can by ee but not, momentum-conserving for do question that of collisions indepen- out The certainly independent. is mutually depend dence are transport they cannot unless affecting probabilities summed rule factors scattering be individual This the because if (32). other, each valid solids on not many course, in of scat- a observed is, different an as of also mandates presence mechanisms, is our the that tering in It of rule resistivity for best grounds. Matthiessen’s behavior the the simple additive from to on departure and, anticipated stark surprise not a is as knowledge, comes crossover 3B. viscous Fig. in data the to fit best a values from the with agreement large at holds only versality h diiebhvo fcnutnea h ballistic-to- the at conductance of behavior additive The icu lcrnc.arXiv:1607.00986. electronics. viscous graphene. in resistance A Sect Stand microcontacts. PdCoO in flow electron dynamic graphene. in law Wiedemann-Franz graphene. in flow wires. unu rtclflisadi h ia udi graphene. in fluid Dirac the in and fluids critical quantum crystals. Dirac in fields gauge tic sheet. graphene doped a in liquid electron transport. Linear-response graphene: liquids. tron fluids. charged viscous of a 1 G 2 = hsRvB Rev Phys ball /π ecie a describes ecie tiig“niMthesn behavior: “anti-Matthiessen” striking a describes 3, hsRvLett Rev Phys 75A(6):535–540. ρ atise’ Rule Matthiessen’s hsRvLett Rev Phys and ∗ sY adtie ,Vzein A 21)Hl icst rmelas- from viscosity Hall (2015) MAH Vozmediano K, Landsteiner Y, os ´ 51:13389–13402. /(Rw Science a PNAS 2 w a Phys Nat etakM enkvfrueu icsin and discussions useful for Reznikov M. thank We γ = = ) hsRvB Rev Phys whereas , 351:1055–1058. 113:235901. htsae as scales that γ epciey hsfidn si good in is finding This respectively. 8, π/ 101:216807. idpnetblitccnrbto that contribution ballistic -independent | a 2 1 hsRvLett Rev Phys 12:672–676. ac 1 2017 21, March . MGa-il e York). New (McGraw-Hill, + Science cf. b, a Science 1 90:035143. hsRvB Rev Phys a 0 = G 2 t ,Mri D(05 yrdnmc in Hydrodynamics (2015) AD Mirlin M, utt ¨ γ i.S1 Fig. vis 351(6277):1061–1064. w hsRvB Rev Phys .694 351(6277):1058–1061. ecie icu contribu- viscous a describes hsdpnec translates dependence This . 115:177202. w 91:035414. 2 and h w em il val- yield terms two the ; .Scn,qieremark- quite Second, ). | 1/(Rw o.114 vol. 93:125410. G a hsRvB Rev Phys 2 = 0 = ) G vs. | . ball 378 JETP o 12 no. w γ 93(7):075426. + w e alBur Natl Res J γ aus is values, 46:165–172. G obtained efind we , 0 = vis | The . see ; 3073

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