Kelvin-Mach wake in a two-dimensional Fermi sea
KKelvin-Mach wake in a two-dimensional Fermi sea
Eugene B. Kolomeisky and Joseph P. Straley Department of Physics, University of Virginia, P. O. Box 400714, Charlottesville, Virginia 22904-4714, USA Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055, USA (Dated: September 26, 2018)The dispersion law for plasma oscillations in a two-dimensional electron gas in the hydrodynamicapproximation interpolates between Ω ∝ √ q and Ω ∝ q dependences as the wave vector q increases.As a result, downstream of a charged impurity in the presence of a uniform supersonic electriccurrent flow, a wake pattern of induced charge density and potential is formed whose geometry iscontrolled by the Mach number M . For 1 < M (cid:54) √ M > √
2. These wakes also trail an external chargetraveling supersonically a fixed distance away from the electron gas.
PACS numbers: 72.80.Vp, 52.35.Hr, 47.35.-i, 47.40 Ki
An object uniformly moving relative to a medium givesrise to a series of effects ranging from formation of a Machshockwave cone behind a supersonic projectile [1] andCherenkov radiation emitted by a rapidly moving charge[2], to creation of wakes on water surface by ships [3].One feature these effects have in common is that the in-teraction between the object and the medium triggers thecoherent emission of collective excitations of the mediumwhich combine constructively to form the wake [4]. Herewe describe the coherence effect wherein plasma wavesemitted by a two-dimensional (2 d ) electron gas form awake pattern resembling both Mach and ship wakes [3].Hereafter we speak of the electron gas; the theory for thegas of holes is the same.The starting point of our analysis is an expression forthe dynamical dielectric function of the 2 d electron gas inthe hydrodynamic approximation which, neglecting theeffects of dissipation and retardation, is given by [5, 6]: (cid:15) ( ω, q ) = ω − Ω ( q ) ω − s q (1)where ω is the frequency, Ω( q ) is the frequency of plasmaoscillations as a function of the wave vector q ,Ω ( q ) = gq + s q (2)and q = | q | . This description encompasses systems rang-ing from those whose electrons obey parabolic [5, 6] tolinear (graphene) dispersion laws [7]. The material pa-rameters g = 2 πne v F /κζ ( n ) (a characteristic acceler-ation) and s = v F ( ∂p/∂ε ) / (the speed of sound) aredetermined by the equilibrium electron number density n , the equation of state in the neutral limit (entering viathe density dependence of the chemical potential ζ ( n )and the energy density dependence of the pressure p ( ε )),the background dielectric constant κ , and the limiting(Fermi) velocity v F [8].In the long-wavelength limit q → ( q ) = gq [3] (where in this context g is thefree-fall acceleration), an observation due to Dyakonovand Shur [9]. Since plasma oscillations are classical innature [8], a series of effects analogous to classical waveson water are then expected in electron layers.One of the most familiar manifestations of the Ω( q → ∝ √ q dispersion law in fluid mechanics is the Kelvinwake that trails a traveling pressure source: the 39 ◦ an-gle of the wake is independent of the source velocityand has a characteristic ”feathered” pattern [3]. Onethen might infer that an external charge traveling non-relativistically a fixed distance away from the plane ofthe electron system disturbs the latter in the form of an”electron” Kelvin wake. Such a conclusion was recentlymade in the literature in the context of doped graphene[10]; it is misleading because it overlooks crucial deviationfrom the strict √ q dispersion law. The same criticism ap-plies to a conjecture that stationary Kelvin wake shouldbe formed downstream of a defect in the 2 d electron gasin the presence of a current [11]. A wake is formed be-hind a moving source whenever there is a mode whosephase velocity matches the speed of the source (the pre-cise statement is given by Eq.(9) below). For a strictly √ q dispersion law such a mode can always be found nomatter what the speed of the source. However, the spec-trum of plasma oscillations (2) deviates from the √ q law,and the phase velocity Ω /q is always above the speed ofthe sound s, which is thus the critical velocity for wakeformation in 2 d electron systems. If the acceleration g inEq.(2) were zero, the wake pattern would resemble thatformed behind a supersonic projectile, with a wake angledetermined by the Mach number M = v/s [1], where v is the speed of the source. For finite Mach number onewould then expect a pattern sharing features of both theKelvin and Mach wakes, hereafter called the Kelvin-Machwake.In an earlier study, Fetter [6] has analyzed various as-pects of the electromagnetic response of an electron layer a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y to a moving charge. However, the problem was solved inthe Fourier representation, and the real space pattern ofthe induced charge and potential were not addressed.Our goal is to solve for the geometry of the wake in-duced by the moving charge. This is done by focusing onthe case when the external charge is in the plane of theelectron system. Since only the relative motion of thecharge and the medium matters, in practice this situa-tion can be realized by subjecting an electron layer withan embedded Coulomb impurity to a supersonic currentflow.Supersonic flows are experimentally accessible, as wewill now show.*The speed of sound s is less than the Fermi velocity v F but typically has the same order of magnitude.*For a parabolic dispersion law it can be estimatedas s (cid:39) v F (cid:39) c ( m / m ) a B √ n where m is the elec-tron mass in vacuum and a B is the Bohr radius. For m /m = 10 and n = 10 cm − the speed of sound canbe estimated as s (cid:39) cm/s . This large value can beattained at low temperature, where the mobility can beas large as 10 cm / ( V · s ) [5]. The required electric fieldwould be 10 V /cm , which is five orders of magnitudesmaller than the dielectric breakdown field of the
SiO insulating layer common to various practical realizationsof electron layer systems [5].*Similarly, in graphene (a linear dispersion material)the Fermi velocity is two orders of magnitude smallerthan the speed of light but the electron mobility is of theorder 10 cm / ( V · s ) [12] which translates into a 10 V /cm electric field needed to propel graphene’s electrons pastthe speed of sound. There exists direct experimental ev-idence [13] that the saturation velocity in graphene on
SiO above room temperature exceeds 3 × cm/s atlow carrier density while the intrinsic graphene satura-tion velocity could be more than twice the quoted value.*The ratio d = s /g (the Debye screening length of theelectron gas [5]) sets the length scale of the effects to bediscussed. It is of the order 10 − cm (and weakly dopingdependent) in materials with parabolic dispersion law [5]and of the order κ/ √ n in graphene [12].*There is a further advantage of studying graphenerather than the electron layers of the past [5]. Chargedimpurities can be embedded into graphene in a controlledmanner [14] and high-resolution non-invasive imagingof charge currents in graphene structures [15] can beemployed to directly observe the electron Kelvin-Machwake; in other systems the formation of the wake canonly be inferred indirectly from the onset of non-zerowave resistance.We will be studying the electromagnetic response of anelectron layer to an external potential ϕ ext ( r , t ), where r is the position within the layer and t is the time; thedependence on these quantities is in respose to an ex-ternal charge (number) density n ext ( r , t ). Their Fouriertransforms are related by the Coulomb law ϕ ext ( ω, q ) = 2 πen ext ( ω, q ) /κq [5, 6]. According to the linear responsetheory, the Fourier components of the induced density n in ( ω, q ) and induced potential ϕ in ( ω, q ) are given by n in ( ω, q ) = (cid:20) (cid:15) ( ω, q ) − (cid:21) n ext ( ω, q ) = gqn ext ( ω, q ) ω − Ω ( q ) (3) ϕ in ( ω, q ) = (cid:20) (cid:15) ( ω, q ) − (cid:21) ϕ ext ( ω, q ) = 2 πegκ n ext ( ω, q ) ω − Ω ( q )(4)Inverting the Fourier transforms we find the electromag-netic response in the direct space and time representation n in ( r , t ) = g (cid:90) d qdω (2 π ) qn ext ( ω, q ) e i ( q · r − ωt ) ( ω + i − Ω ( q ) (5) ϕ in ( r , t ) = 2 πegκ (cid:90) d qdω (2 π ) n ext ( ω, q ) e i ( q · r − ωt ) ( ω + i − Ω ( q ) (6)where ω in the denominators of the integrands is en-dowed with infinitesimally small positive imaginary part( ω → ω + i
0) to guarantee analyticity of the integrandsin the upper half-plane of complex ω [2]. A unit externalcharge moving with constant velocity v within the layeris described by n ext ( r , t ) = δ ( r − v t ) whose Fourier trans-form is n ext ( ω, q ) = 2 πδ ( ω − q · v ). Substituting this intoEqs.(5) and (6) and changing the frame of reference tothat of the charge, r − v t → r , we find n in ( r ) = g (cid:90) d q (2 π ) qe i q · r ( q · v + i − Ω ( q ) (7) ϕ in ( r ) = 2 πegκ (cid:90) d q (2 π ) e i q · r ( q · v + i − Ω ( q ) (8)This is the electrodynamic response of the electron layerhaving an initially uniform flow velocity − v to a pointCoulomb impurity of unit charge fixed at the origin or,equivalently, to a traveling charge in the co-moving ref-erence frame.For v = 0 Eqs.(7) and (8) describe the static screeningresponse of the electron layer to a point charge [5, 6].Slow motion ( v < s ) brings anisotropy to the responsebut no other qualitative changes occur because the de-nominators of the integrands in Eqs.(7) and (8) cannotvanish for q real; the + i v exceeds the speed of sound, the denominators ofthe integrands in Eqs.(7) and (8) can vanish; integrals (7)and (8) are dominated by the real wave vectors q givenby the solutions to Ω( q ) = ± q · v (9)The response pattern is now qualitatively different andthe presence of the + i g and s (2), and velocity of the source v :(i) For s = 0 (the Kelvin wake) the only parameterhaving dimensions of length that can be formed out of g and v is the characteristic length scale of the wake, λ = v /g . Measuring the length in units of λ , density inunits of 1 /λ , and potential in units of e/κλ eliminates allthe parameters from the problem. Thus all Kelvin wakesare geometrically similar. While this argument does notsupply the value of the wake angle, it does predict thatit is independent of v and g .(ii) For s (cid:54) = 0 (the Kelvin-Mach wake) two independentlength scales can be formed out of the parameters of theproblem: λ = v /g and d = s /g (the Debye screeninglength). Their ratio, λ/d = v /s = M , is the squareof the Mach number; once this is fixed, either λ or d may be used to characterize the length scale of the wake.Measuring the length in units of d , density in units of1 /d ) and the potential in units of e/κd eliminates allthe parameters from the problem except for the Machnumber. Thus all the Kelvin-Mach wakes of the sameMach number M = v/s are geometrically similar.Even though the Fourier integrals (7) and (8) cannotbe computed in closed form, the geometry of the wakepattern can be inferred with the help of Kelvin’s methodof stationary phase [3]. The idea is that when the phasefactor f = q · r in the integrands in (7) and (8) variesrapidly with q , the exponentials are highly oscillatoryso that contributions from various elements d q canceleach other; this is the case of destructive interferencewith almost zero net result. This cancelation, however,will not occur for the wavelengths for which f is station-ary with respect to q (which is additionally restrictedby the Cherenkov-Landau condition (9)); this is the caseof constructive interference. Since the integrands of theinduced charge (7) and potential (8) differ by a smoothfactor of q , the two wake patterns have the same geome-try.Let us choose the positive x direction along the veloc-ity vector v and measure length in units of the Debyescreening length d = s /g . The trace of the externalcharge divides the plane into two regions related to oneanother by reflection; without the loss of generality wecan focus on the y > q x,y >
0. Then the phase f = q · r is given by f = (cid:110) M − q y + 1 + (cid:2) M − M q y (cid:3) / (cid:111) / ( M − √ x + q y y (10)where instead of q x we substituted the positive solution ofthe Cherenkov-Landau equation (9) corresponding to theplasma spectrum (2). Direct inspection of Eq.(10) showsthat the condition of stationary phase df /dq y = 0 canonly be satisfied for x < z = [1 + 4( M − M q y ] / (cid:62) f = ( z − / M ( M − / (cid:20) ( z − M ) / ( M − / ( z − / x + y (cid:21) (11)The condition of stationary phase f (cid:48) ( z ) = 0 now becomes − yx = 1( M − / ( z − / ( z + M ) z ( z − M ) / (12)Since the phase f is constant along the wavefront,Eqs.(11) and (12) can be solved relative to x and y togive the equation for the wavefront in parametric form: x ( z ) = 2 f ( M − M z ( z − M ) / ( z + 1) / (13) y ( z ) = − f ( M − / M ( z + M )( z − / ( z + 1) / (14)We now see that internal consistency of the argumentrequires the phase to be negative, f < r from the source which in the original units oflength means r (cid:29) d = s /g .To put the consequences of Eqs.(12)-(14) into perspec-tive we begin with the Kelvin case s = 0 which corre-sponds to M = ∞ . In this limit Eq.(12) simplifies to − yx = ( z − / √ z (15)whose right-hand side vanishes at z = 1, z → ∞ andreaches a maximum value of 1 / √ − y/x = 0, two solutions for 0 < − y/x < / √ − y/x = 1 / √
2, and none for − y/x > / √
2. The angle between the wake edges is2 arctan(1 / √ ≈ ◦ which is Kelvin’s classic result[3].In order to take the Kelvin s = 0 limit in Eqs.(13)and (14) we temporarily restore original units of length, -10 0-55 A BC
FIG. 1. (Color online) Wavefronts of the Kelvin wake, Eqs.(16), with the source at the origin traveling to the right. Thewake is confined within the shaded light blue 39 ◦ wedge; theunit of length is λ = v /g . -10 0-1010 A BC
FIG. 2. (Color online) Wavefronts of the Kelvin-Mach wakefor 1 < M (cid:54) √
2, Eqs. (13) and (14), with an external chargeat the origin traveling to the right. The wake consists of trans-verse wavefronts confined within shaded light green Mach sec-tor of angle 2 arctan( M − − / , the unit of length is theDebye screening length d = s /g , and M = 1 . ( x, y ) → ( x, y ) /d = ( g/s )( x, y ) followed by selecting λ = v /g as a new unit of length with the result x ( z ) = 2 √ f z ( z + 1) / , y ( z ) = − f ( z − / ( z + 1) / (16)A series of these wavefronts is shown in Figure 1 where forthe purpose of illustration we chose f = − π ( l +1 / , l =0 , ,
2; the y <
ABC connecting the edges of the pattern acrossthe central line y = 0 and the diverging wavefronts AO and CO connecting the source at the origin to the edgesof the pattern [3]. The two wavefronts meet at A and C at the edges of the pattern.For M finite the right-hand side of Eq.(12) vanishesat z = 1 and approaches ( M − − / as z → ∞ ; theintermediate behavior depends on the Mach number:(i) When 1 < M (cid:54) √ z . Thus the -200 -100 0-5050 A BC DE
FIG. 3. (Color online) Same as in Figure 2 for
M > √ M = 4was employed to produce the drawing. equation of stationary phase (12) has one (transverse)solution for 0 (cid:54) − y/x < ( M − − / and none for − y/x (cid:62) ( M − − / . Therefore the angle of the wakeis 2 arctan( M − − / which is Mach’s classic result[1]. A series of wavefronts (13) and (14) employing thesame choice for the phase f as in Figure 1 is shown inFigure 2. The wake consists of transverse wavefronts ABC connecting the edges of the pattern. We stressthat in view of the dispersion relation (2) the wake is not the classic Mach wake; the wavefronts of the lat-ter, y/x = ± ( M − − / , coincide with its geometricalboundary [1].(ii) When M > √ M + 1) / / (2 M − / at z =(2 M − / ( M − (cid:54) − y/x < ( M − − / , two (transverse and diverging) solutionsfor ( M − − / (cid:54) − y/x < ( M + 1) / / (2 M − / coalescing at − y/x = ( M + 1) / / (2 M − / , andnone for − y/x > ( M + 1) / / (2 M − / . The wakepattern shown in Figure 3 is confined within a sector ofangle φ ( M ) = 2 arctan ( M + 1) / (2 M − / (17)that is wider than Mach’s. In addition to the trans-verse wavefronts ABC connecting the edges of the pat-tern, the diverging wavefronts AD and CE are also foundoutside the Mach (light green) sector; the region withtwo types of wavefronts present is shaded light blue. Incontrast to the Kelvin wake (Figure 1), divergent wave-fronts connect the edges of the Kelvin-Mach wake tothe boundaries of the Mach sector, a consequence of the z → ∞ limit of Eqs.(12)-(14). As M increases, the Machsector becomes narrow, closing as M → ∞ while thewake angle (17) decreases approaching Kelvin’s limit of φ ( ∞ ) = 2 arctan(1 / √ M > √ d electron sys-tems which we hope will be observed in future experi-ments.We thank G. Rousseaux and M. I. Dyakonov for in-forming us of Refs.[4, 11], and M.I. Dyakonov and E. Y.Andrei for valuable comments. [1] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Perg-amon, Oxford, 1987), Section 82.[2] L. D. Landau and E. M. Lifshitz,
Electrodynamics ofContinuous Media (Pergamon, Oxford, 1984), Sections82 and 115.[3] H. Lamb,
Hydrodynamics (6th ed., Cambridge UniversityPress, 1975), Chapter IX.[4] For a review of different types of wakes see I. Carusottoand G. Rousseaux, in