Perturbation theory for two-dimensional hydrodynamic plasmons
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Perturbation theory for two-dimensional hydrodynamic plasmons
Aleksandr S. Petrov ∗ and Dmitry Svintsov Laboratory of 2D Materials’ Optoelectronics, Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia
Perturbation theory is an indispensable tool in quantum mechanics and electrodynamics thathandles weak effects on particle motion or fields. However, its extension to plasmons involvingcomplex motion of both particles and fields remained challenging. We show that this challenge can bemastered if electron motion obeys the laws of hydrodynamics, as recently confirmed in experimentswith ultra-clean heterostructures. We present a unified approach to evaluate corrections to plasmonspectra induced by carrier drift, magnetic field, Berry curvature, scattering, and viscosity. Asa first application, we study the stability of direct current in confined two-dimensional electronsystems against self-excitation of plasmons. We show that arbitrarily weak current in the absenceof dissipation is unstable provided the structure lacks mirror symmetry. As a second application,we indicate that in extended periodic systems – plasmonic crystals – carrier drift induces anomalousDoppler shift, which can be both below and higher than its value in uniform systems. Finally, weexactly evaluate the effect of Berry curvature on spectra of edge plasmons and demonstrate thenon-reciprocity induced by anomalous velocity.
I. INTRODUCTION AND OUTLINE
In quantum mechanics, there is a limited number ofpotential landscapes that allow exact solutions for en-ergy spectra and wave functions. Fortunately, weak po-tentials of arbitrary form can be handled with perturba-tion theory (PT)[1]. Only several decades after success inquantum mechanics, PT was formulated for classical elec-trodynamics [2, 3]. Currently, electrodynamic PT repre-sents an indispensable tool for analysis of non-uniformlaser cavities and inhomogeneous waveguides [4].Simplicity of perturbation theory in electrodynamicsstems from the fact that state of the field is characterizedby two vectors, E and H . Waves propagating near con-ductive surfaces – plasmons – involve not only oscillationsof field but of charge carriers as well. The state of carriersis characterized by distribution function generally havingan infinite number of harmonics in momentum space. Forthis reason, state of plasmon is more complex and for-mulation of plasmonic PT represents a challenging task.Its solution promises a unified approach for treatment ofvarious perturbations on plasma resonances in metal andsemiconductor nanostructures, including magnetic fields,electric currents, electron scattering, and others. Previ-ous attempts to construct PT for plasmonic structuresrequired the synthesis of auxiliary equations of motionfor polarization and velocity fields in materials that pro-vide a necessary form of dielectric function [5].In this paper, we show that formulation of simple PTfor plasmon eigen frequencies and field distributions ispossible when charge carriers in conductors obey thelaws of hydrodynamics. While being a common approx-imation for analysis of carriers motion for nearly a cen-tury [6], the true hydrodynamic phenomena in solidswere demonstrated only recently with advent of high-mobility two-dimensional heterostructures [7–10]. The ∗ [email protected] reason is that the prequisite of hydrodynamics is thedominance of carrier-carrier momentum-conserving col-lisions over all other collisions (electron-impurity andelectron-phonon) [11]. Under these conditions, only threeharmonics of distribution function survive, and the stateof charge carriers is characterized only by three variables:density, velocity, and temperature.Electric charge of electron fluid plays the central role inour theory. It leads to existence of plasma modes settingthe largest frequency scale in the problem (compared,e.g., to the plasmon decay time). As a result, the unper-turbed dynamic matrix is Hermitean, which simplifiesthe formulation of PT. Such simplicity is lacking in PTfor neutral incompressible fluids which motion is stronglyaffected by viscous dissipation [12, 13].Having constructed the PT for hydrodynamic plas-mons, we apply it to physical systems where hydro-dynamics was originally observed, namely, to the two-dimensional electronic systems (2DES). As a first ex-ample, we study the stability of direct electric cur-rent against the excitation of plasmons in confined2DES [keeping in mind a field-effect transistor (FET)shown in 1A]. We succeed to relate the current-inducedgrowth/decay rate of plasmon and its steady-state fielddistribution in 2DES with arbitrary gating geometry andarbitrary contact boundary conditions. On one hand,this relation aggregates the outcome of preceding excur-sive studies of plasma instabilities in 2d electron systeminitiated by Dyakonov and Shur [14–18]. On the otherhand, it shows that structural asymmetry is a necessaryand (in the absence of dissipation) sufficient conditionfor self-excitation of plasmons. This resolves the long-standing experimental puzzle about the relation betweenstructural asymmetry and strength of plasmon-assistedterahertz emission from FETs [19–21].The second example is devoted to drift effects in plas-monic crystals (Fig. 1B). Existing theories [22–24] predictthe instability excitation only at high velocities (higherthan plasma wave velocity) in these structures. Wedemonstrate that low-drift instabilities are indeed pro- V g V d V g V d gate drain s o u r c e
2d channel y xz A B FIG. 1. Schematics of 2DES realizations. (A) Bounded 2DES,namely partly-gated field-effect transistor [16]. (B) Plasmoniccrystal hibited due to energy conservation (Sec. IV); still, plas-monic crystals can be used in resonant photodetectionproviding higher-than Doppler frequency shift in a cer-tain range of parameters.The third example addresses the nature of edge modesin anomalous Hall materials hosting ’anomalous’ carriervelocity [25]. We find that this velocity causes non-reciprocity of the system in the absence of external mag-netic field , introducing the frequency splitting of right-and left-propagating edge modes. The same qualitativeresult was obtained earlier [26, 27] in restrictive quasi-local electrostatics [28]; PT allows us to refrain from thisapproximation and correctly specify the splitting magni-tude.
II. PERTURBATION THEORY FORELECTRON HYDRODYNAMICS
The confinement of a 2DES to a characteristic length L leads to the emergence of collective modes (plasmamodes) with frequencies ω p ∼ [ n e /mεL ] / , where e > n is the electron density, m is the effective mass, and ε is the background dielec-tric constant. These modes have been extensively studiedsince the pioneering works of Stern [29] and Chaplik [30]and host a variety of phenomena when exposed to exter-nal influence, e.g. magnetic field [31] or carrier drift [32].Interestingly, many of these effects can be incorporatedinside a single theoretical shell of plasmonic perturbationtheory (PT), which we develop in this Section.The formulation of plasmonic PT is possible if oscillat-ing charge carriers obeys the laws of hydrodynamics. Hy-drodynamic approach grants a straightforward formula-tion of operator eigenvalue problem on 2d plasmon eigen-frequencies, allowing the isolation of drift, magnetic field,viscosity and pressure operators — which is unachiev- able in other transport regimes. This property enablesus to construct an analogy of quantum-mechanical per-turbation theory with respect to these operators; suchtreatment is possible when the corrections to the eigen-frequency are small as compared to the frequency itself.That is why the perturbative approach is much simpler inthe case of a charged fluid rather than in conventional hy-drodynamics [12]: the former possesses large zero-orderfrequency ω p that generally outweighs the corrections,whereas the latter deals with much less frequencies and isthus laden with dissipation terms in the zero-order prob-lem.The formal inequalities for the perturbative treatmentto be possible are { u /L, ω c , τ − p , ν/L } ≪ ω p , where ν isthe kinematic viscosity and ω c = eB/m is the cyclotronfrequency ( B denotes the magnetic inductance). Takingthe realistic parameters u ≃ m/s, L ≃ µ m, τ p ≃ − s − , B = 0 . T and estimating the viscosity as ν ≃ v τ ee / ≃
250 cm /s [10], we see that the inequalitiesare fulfilled for ω p / π ≃ ∂ t N + ∂ i J i = 0; (1) ∂ t J i + ∂ j P ij = F i /m − J i /τ p , (2)where t denotes time, N is the electron density, J i = N U i is the current, U is the drift velocity, F is the Lorentzforce, and P ij is the stress tensor of the electron fluid(we neglect the bulk viscosity and pressure terms, whichcan be easily restored if needed): F = e N ∇ ϕ + ω c [ˆ z , U ]; (3) P ij = N U i U j − ηm ( ∂ j U i + ∂ i U j − δ ij ∂ k U k ) , (4)where ˆ z is the unitary vector in the direction perpendicu-lar to 2DEG (see Fig. 1), η denotes dynamic viscosity and δ ij is the Kronecker delta, { i, j, k } = 1 ,
2. The set (1)-(4)is completed by the expression for the electric potential ϕ determined by the 2DEG surroundings (consider Fig. 1)through the electrostatic Green function G ( r , r ′ ): ϕ ( r ) = ϕ ext ( r ) − e G [ N ] , (5)where G [ f ] = R d r ′ G ( r , r ′ ) f ( r ′ ) is the self-consistentfield, the r -vector lies in the 2DEG plane, the contribu-tion ϕ ext ( r ) is fixed at the contacts by the voltage sourceand the integration is performed over the whole 2DEG.The following analysis is based on linearization N ( r , t ) = n ( r ) + n ( r ) e − i Ω t , U ( r , t ) = u ( r ) + u ( r ) e − i Ω t and reformulation of (1)-(5) as an operator eigenvalueproblem:( ˆΩ + ˆ V drift + ˆ V sc + ˆ V visc + ˆ V mag ) Φ = Ω Φ . (6)Above, we have introduced the ”three-component wavefunction” Φ = { n, u } T describing density and velocityvariations in plasma modes. The unperturbed motion isdescribed by the ’hydrodynamic’ operator:ˆΩ = − i (cid:18) ∇ [ n ( r ) · ] e m ∇G [ · ] 0 (cid:19) , (7)and we consider the steady carrier drift, magnetic field,scattering, and viscosity as small perturbations given bythe operatorsˆ V drift = − i (cid:18) ∇ ( u · ) 00 ( · , ∇ ) u + ( u , ∇ ) · (cid:19) ; (8)ˆ V visc = i (cid:18) η ∆ (cid:19) ; ˆ V sc = − iτ − p (cid:18) (cid:19) ; (9)ˆ V mag = − iω c (cid:18) e ij (cid:19) ; (10)where e ij is the two-dimensional absolutely antisymmet-ric tensor, i, j = 1 , δ Ω λ = h Φ λ | ˆ V | Φ λ i , where λ enumer-ates the plasmon modes. However, this step is prematureuntil the inner product is specified. Apparently, a stan-dard definition h Φ λ | Φ λ ′ i = R dx [ n ∗ λ n λ ′ + u ∗ λ u λ ′ ] fails: itdoes not ensure that matrix ˆΩ is Hermitean. We resolvethis issue by reformulating the initial problem: we applythe Hamilton operator ˆ H of a charged fluid to Eq.(6) andobtain a generalized eigenvalue problem:ˆ H ( ˆΩ + ˆ V drift + ˆ V sc + ˆ V visc + ˆ V mag ) Φ = Ω ˆ H Φ , (11)ˆ H = (cid:18) e /m G [ · ] 00 ˆ I n ( r ) (cid:19) . (12)Here, the dynamic matrix ˆ H ˆΩ is Hermitean (i.e. h Φ | ˆ H ˆΩ | Φ i = h Φ | ˆ H ˆΩ | Φ i ) as well as the Hamilto-nian ˆ H ; this fact can be shown explicitly by evaluatingthe corresponding matrix elements. Hence, we are nowable to apply the standard perturbative expansion thatleads to the following expression for the first correctionto the eigenfrequency: δ Ω λ = − i h Φ λ | ˆ H ( ˆ V drift + ˆ V sc + ˆ V visc + ˆ V mag ) | Φ λ ih Φ λ | ˆ H | Φ λ i . (13)We stress that the whole procedure does not requireany additional boundary conditions (BC) apart from thetwo natural ones: (1) the Green function vanishes at theelectrodes and (2) J y = 0 at the 2DES edges (no car-rier leakage). In such a way, our analysis holds for 2DESwith arbitrary BCs, which are hardly known in real ex-perimental setups.The constructed perturbation theory is a powerful toolthat can be used to uncover the underlying principles ofmany plasmonic phenomena in 2DES. We apply its for-malism to examine the drift-originated plasmonic effectsin bounded systems (Sec. III) and PCs (Sec. IV), con-summating the article with a specific example of ChiralBerry plasmon (Sec. V).
III. CURRENT-DRIVEN INSTABILITIES INBOUNDED SYSTEMS
Direct current passing in confined 2DES can supply en-ergy to plasmon modes and lead to their self-excitation(plasma instability). The first example of such condi-tions was deduced by Dyakonov and Shur who foundan instability under source grounded and drain held atfixed current[14]. Later, another geometries and bound-ary conditions were realized, including loaded drain [33],partly gated FETs [15, 16], 2DES edges [18], and Corbinodiscs [17]. This search for instabilities has been excursiveand there was no general understanding whether a givenFET structure supports an instability or not.From the prospective of perturbation theory, the op-erator of electron drift ˆ V dr in bounded 2DES is non-Hermitean. Therefore, the eigen frequencies of driftingplasmons are generally complex, which implies plasmonamplitude growth/decay with time. The eigenfrequen-cies may remain real only under specific symmetry con-straints which we are to obtain with the developed PT.We restrict our consideration to one-dimensional oscil-lations of 2d electrons assuming the 2DES to be uniformin the y -direction. Evaluating the matrix elements inEq. (13), we find the correction to plasmon frequency inconfined 2DES induced by a combined action of directcurrent, scattering, and viscosity: δ Ω λ = i j [ K λ (0) − K λ ( L )] − Q loss | Π λ | , (14)where K ( x ) = m | u λ ( x ) | / e L Z d x d x ′ n ∗ λ ( x ) G ( x, x ′ ) n λ ( x ′ ) (15)is the potential energy of interacting charge density fluc-tuations in a 2DES of length L , and Q loss = 12 L Z d x (cid:8) Re σ | E λ | + η | ∂ x u λ | (cid:9) + ηu ∗ λ ∂ x u λ (cid:12)(cid:12)(cid:12)(cid:12) L (16)is the energy loss due to viscous friction and Ohm dissi-pation, where σ = ie n /m (Ω + i/τ p ) is the Drude con-ductivity. The last ’viscous-boundary’ term is also dissi-pative in systems with time-reverse symmetry; if the lat-ter is broken, shear viscosity will contribute to frequencyshift as well.From Eq. (14) we readily observe that plasmon growthrate Im δ Ω λ originates from the excess of kinetic energyentering the mode at the source over the energy lost atthe drain. A particular example is the DS instability,where K (0) − K ( L ) is nonzero due to inequivalent sourceand drain contacts.Apart from the energy interpretation, Eq. (14) im-mediately indicates that only zero-order plasma modeswithout definite parity can be excited by a weak exter-nal drift. Indeed, the even/odd mode profiles require u ( L ) = u (0), which forces the gain term in Eq. (14)vanish. Therefore, FETs with mirror symmetry of sourceand drain do not support unstable modes, but asym-metric structures generally do (in the absence of dissipa-tion). The origin of asymmetry can be arbitrary: eitherasymmetric placement of gates, or asymmetric loading ofsource and drain, or non-uniform carrier density in thechannel, or all of them.We can go beyond the symmetry constraints and spec-ify the requirements on 2DES aiming to maximize theplasmon growth rate. This is equivalent to maximizationof u (0) − u ( L ). The velocity is proportional to electricfield and inverse carrier density. Thus, a 2DES with highfield and small carrier density at the source and low fieldand high density at the drain would be most suitable forturbulence. The simplest realization of this scheme isa partly-gated field-effect transistor (FET) with a shortungated depleted region at the source and long, gatedand enriched region at the drain (see inset on Fig. 2).To prove this hypothesis, we develop a toy model forplasma oscillations in a partly-gated FET. The contactsof such structure are connected to voltage sources, whichis a typical experimental situation, and variation of thegate-source bias allows to form a carrier density step inthe channel ( n + − n junction). We numerically solvethe governing equations in the absence of scattering (Ap-pendix A) and plot in Fig. 2 the drift-induced correctionsto plasmon eigenfrequencies for a set of structures withvarying junction location x and density modulation fac-tor. In accordance with our expectations, the highestgrowth rate (point 1) is achieved for a structure with ashort, depleted, ungated source region. If we swap thecontacts and thus change the signs of u and δ Ω λ , themagnitude of a new maximum (point 2) would be pre-dictably lower due to the gate screening. In addition,Fig. 2 shows that the source regions should not be tooshort, or else the structure would approach the symmet-ric limits of open ( x <
0) or fully-gated ( x >
1) FETswith uniform density that are not subject to instabilities.Experimental data supports our findings. Thus, thefirst plasmonic THz sources [19] were symmetric andprovided broadband radiation only at 4 K whereas fur-ther implication of asymmetric partly-gated FETs [21]enabled the observation of resonant room-temperatureemission due to efficient plasmon-to-drift coupling. Thisfact apparently shows the need for asymmetry. Moreover,the latter experiment demonstrated that the thresholdcurrent is significantly smaller for a depleted source re-gion, which is in agreement with our analysis.However, this depletion should be kept in properbounds. Indeed, a sharp density step not only increasesthe source field, but also causes highly non-uniform dis-tribution of plasma wave velocity in the channel. Thisnon-uniformity promotes viscous losses, which are pro-portional to the velocity gradient, and at some degree ofasymmetry the viscosity takes over gain. I n s t a b ili t y g r o w t h r a t e , δ Ω L u / Gate length/channel length V d V g u n u = 0.3 n g n g n g n g n g n g n g (2) ( )1 n u n g FIG. 2. Calculated instability growth rates for the first plas-mon mode in a partly-gated FET (shown on the inset) withdifferent gate lengths and carrier densities. The carrier den-sity distribution is taken to be n ( x ) = n + ( n − n )[1 + e ( x − x ) /w j ] − , w j = L/
50 is the junction width. The growthrates are normalized by u /L , where u is the drift velocityat the drain (kept the same for all curves), the density at thedrain n is also fixed. The instability benefits if the drift isdirected from low- into high-density region and is especiallypronounced if the low-density region is short and ungated IV. DRIFT-INDUCED PHENOMENA INPLASMONIC CRYSTALS
It turns out that drift-induced phenomena in PCs iscompletely different from that in bounded 2DES. Thereason is that in periodic systems the drift operator isHermitean due to translational invariance and thus con-serves energy. Still, plasma turbulence in PCs can emerge— but only at high drift velocities, at least in the depletedregions [22–24]. In that way, moderate carrier drift inPCs does not affect the stability of plasma modes; in-stead, it changes their spectrum.In order to determine this spectrum change, we ap-ply the constructed perturbation theory with severalamendments. To be more specific, after the lineariza-tion procedure we exploit a usual Bloch decompositionfor the unknown functions and arrive at the same ex-pressions (7)-(10), but with a modified derivative opera-tor ∂ x → D x = ( ∂ x + iq B ) (where q B is the Bloch vector)and the Green function: G P C ( R , R ′ ) → G cell ( r , r ′ ) = n =+ ∞ X n = −∞ G P C ( r , r ′ + ˆ x nL ) e iq B ( x ′ − x ) , (17)where r and r ′ lie within a unit cell of length L . Theunknowns n, u, ϕ should now be understood as periodicparts of the corresponding Bloch functions.These remarks allow us to apply Eq. (13) to drift-originated phenomena in PCs. We arrive at: δ Ω λ Ω λ = j R L d x Im( mu λ D x u ∗ λ ) R L d x Re( − en λ ϕ ∗ λ ) , (18)As expected, the correction is real and corresponds to thezero-mode Doppler shift.Significant shifts can be useful in resonant photodetec-tion exploiting the plasmonic drag effect [34]. Indeed,in a typical PC a normally incident light excites plasmawave packets around q B = 0, where the group velocitydistribution is symmetrical in the absence of drift. Fora sufficient detection, however, one needs a substantialasymmetry in this distribution, which can be introducedvia carrier drift. The latter breaks the dispersion curvesymmetry, and the greater the Doppler shift, the greaterthe group velocity difference.From the first glance, however, PCs may seem not ap-plicable to photodetection: Eq. (18) implies that δ Ω λ turns to zero in the standing-wave limit q B = 0 due toIm( mu λ D x u ∗ λ ) = 0. Nevertheless, they are. Indeed, thelinear correction (18) vanishes in the framework of non-degenerate perturbation theory. But if a PC hosts twoclose modes, we shall use the degenerate theory and inthis case we obtainΩ ± = 12 (cid:26) Ω + Ω ± q (Ω − Ω ) + 4 | V | (cid:27) , (19)where V = h Φ | ˆ H ˆ V dr | Φ ih Φ | ˆ H | Φ i h Φ | ˆ H | Φ i .Fig. 3 illustrates our findings. On the left panel weplot the calculated Doppler shifts for 3rd and 4th modesthat exist in a range of fully-gated PCs (see inset on theupper right panel) with a density step inside the unit cell.The shifts are normalized by the expected shift valueΩ2 πsN/L = ¯ u /N = u L /L + n /n N L/L , (20)where s is plasma wave velocity under the first gate, L and L are the gate lengths, L + L = L , n and n denote carrier densities, and N enumerates the pairs ofmodes; in our case N = 2. The upper right panel showsΩ( u ) dependencies for the first four modes in a PC with L = 0 . L and n = 0 . n . In full accordance withEq. (19), we observe parabolic spectrum at very low ve-locities that transforms into linear spectrum when theperturbation magnitude exceeds the degeneracy contri-bution. Hence, the linear part slope is determined by thenon-diagonal matrix element and in certain parameterrange leads to higher-than Doppler shift.Actually, there exists another way to break the sym-metry constraint Im( mu λ D x u ∗ λ ) = 0. Indeed, magneticfield, being included in the ˆΩ-operator, efficiently entan-gles the real and imaginary parts of the zero-order wavefunctions. Thus, the plasma modes become coupled to Density contrast n n / L e n g t h r a ! o LL / Normalized Doppler shi" - - L+L n n n F r e qu e n c y Ω π L / s Dri ! u /s FIG. 3. (Left and lower right panels) Normalized Dopplershift for 3rd and 4th modes in fully-gated PCs with a densitystep inside a unit cell. (Upper right panel) Ω( u ) dependen-cies for a PC with L = 0 . L and n = 0 . n . Their charac-ter coincides with the predictions of degenerate perturbationtheory Eq. (19). Inset: Scheme of a fully-gated PC drift in the first order (their stability is not affected as asˆ V mag is Hermitean). A more detailed discussion of thisinfluence will be given elsewhere. V. CHIRAL BERRY PLASMONS
Apart from treating the perturbations that directlyenter the equations of motion (1)-(5), our theory alsoaccounts for the boundary condition perturbations. Inparticular, this property can be useful for the descrip-tion of Chiral Berry plasmons (CBPs) — a new type ofedge plasmonic states in 2D materials with nonzero Berryflux [25]. CBPs arise as follows. Nonzero Berry flux pro-vides an ’anomalous’ contribution J a to the total particlecurrent J tot = J + J a , where J = N U is the usual currentdensity. The anomalous current does not affect the gov-erning HD equations (due to ∇ J a = 0), but changes theboundary condition on the 2DEG edge from the usual J ⊥ = 0 to J tot, ⊥ = 0. Song and Rudner [26], andlater Zhang and Vignale [27], found this BC change tosplit the frequencies of right- and left-propagating plas-mons, which were therefore named Chiral Berry plas-mons. However, they did that in a restrictive quasi-local electrostatic approximation [28] that does not, inparticular, reproduce the logarithmic divergence of edgemagnetoplasmon group velocity at short wave vectors instrong magnetic fields [35]. This divergence stems fromthe 2d Coulomb potential divergence, which is violatedin the quasi-local approximation. As we demonstrate be-low, our perturbation theory, which exploits precise elec-trostatics, reproduces the CBP frequency gap for smallBerry flux and corresponding anomalous current.We consider a semi-infinite 2DEG with nonzero Berryflux F confined to a { x > , y } half-plane and requirethe potential and electric field continuity at the x = 0boundary along with J tot,x | x =+0 = 0. The latter BC isthe only difference that distinguishes the CBP problemfrom the edge plasmon problem solved by Volkov andMikhailov [35], who obtained the edge plasmon frequencyΩ = ω d / √ . ω d = p πe n | q | /m , q is the wavevector in the y -direction. Replacing J in Eqs. (1), (2)with J tot and using ∇ J a = 0, we arrive at the followingproblem: ( ˆΩ + ˆ V a )Φ = ΩΦ , (21)where ˆ V a = Ω e F ~ n (cid:18) ∇ a G q [ · ] 0 (cid:19) (22)is the ’anomalous’ perturbation operator induced by the(dimensionless) Berry flux F , ∇ a = ˆ x iq + ˆ y ∂ x , G q [ f ] =2 R ∞ d x ′ K ( q | x − x ′ | ) f ( x ′ ), K ( x ) is the modified Besselfunction of the second kind, q is the wave vector in the y -direction.Thus, we transformed the BC perturbation into amore convenient form. Now the formulation of the zero-order problem (21) coincides with the formulation of edgeplasmon problem [35]. Taking Volkov-Mikhailov modeprofiles, we evaluate the Berry-flux-induced correction h Φ | ˆ H ˆ V a | Φ i / h Φ | ˆ H | Φ i and calculate the gap betweentwo branches of CBP: ~ ∆Ω( q ) = 3 . F e | q | (23)versus the Song-Rudner result ~ ∆Ω SR ( q ) = 8 √ π/ F e | q | ≈ . F e | q | . (24)Thus, our calculations qualitatively approve the ap-proximate Song-Rudner solution and downshift the gapwidth by 10%. This change is a minor one and grants allthe non-reciprocal implementations of CBPs discussedin [26]. VI. DISCUSSION AND CONCLUSION
The developed plasmonic PT has the same functional-ity as PT in quantum mechanics: given the exact solutionof unperturbed problem, we can accurately find correc-tions to eigen-frequency under small perturbations. Un-fortunately, exact solutions in 2d plasmonics are unique,among non-trivial cases to mention the edge modes [35]and plasma oscillations in gated 2DES with infinite con-ductive walls [36]. At the same time, the unperturbedproblem of plasma oscillations due to self-consistent elec-tric field is readily solved with commercial electromag-netic simulators, and the resulting field profiles can besupplied to our perturbation theory.The assumption of hydrodynamic transport used inderivation limits the frequencies ω below the inverseelectron-electron scattering time τ − ee . This may lookrestrictive as τ − ee is order of 1 THz at room tempera-ture [37] and scales as T . Most experimental obser-vations of plasmons correspond to the opposite ballis-tic limit ωτ ee ≫ c to infinity. This is justified for typical 2d plasmonsonce their frequency ω lies below the light cone ω = cq ∼ c/L [40]. Renouncement of this assumption immediatelyleads to radiative plasmon damping and non-Hermiticityof unperturbed problem. Formulation of PT in this caseis also possible [41] but requires dealing with divergingfields far away from 2DES.The presented examples were related to first-order ordegenerate perturbation theory. Higher-order correctionscan also be derived and are important when first-order ef-fects are absent by symmetry (such as Doppler shift in thecenter of plasmon Brillouin zone). Another non-trivialapplication of higher-order corrections is the analysis ofweak steady-state plasma turbulence for direct currentslightly exceeding the threshold [13, 42]. So far, the so-lution of such problems in 2DES was achieved with nu-merical simulations [43] or limited to model systems [44].The general analysis is possible with the developed PTand will be reported elsewhere.In conclusion, we have developed the perturbationtheory for two-dimensional hydrodynamic plasmons anddemonstrated its utility on several examples. We havederived a constitutive relation between current-inducedgrowth rate of plasmon and its steady-state field distri-bution. This expression clarifies the role of structuralasymmetry for efficient excitation of plasmons by directcurrent. In periodic systems – plasmonic crystals – thecurrent does not lead to instabilities in the first order, butdoes induce Doppler shift which can be both above andbelow the conventional value. Finally, we have shown thecapability of PT to handle boundary condition pertur-bations, and obtained an exact result for left-right edgemode splitting due to Berry curvature. ACKNOWLEDGEMENTS
The authors thank M. S. Shur and D. Bandurinfor valuable discussions. The work was supported byprojects 18-37-20058 and 16-37-60110 of the RussianFoundation for Basic Research. A. S. P. acknowledgesthe support of grant 18-37-00206 of the Russian Foun-dation for Basic Research and grant 18-1-5-66-1 of theBasis Foundation.
Appendix A: Numerical method
In order to obtain the results shown in Fig. 2 we applya standard spectral numerical method to the system ofhydrodynamic equations (1-5) with Chebyshev polyno-mials of the first kind T i taken as the basis functions. Tobe more concrete, after writing the linearized Eqs. (1),(2) in dimensionless units ( ξ = 2 x/L −
1, ˜ n = n/n (0),˜ u = u/s (0), ω = Ω L/s (0) where s (0) = e n (0) L/m )we substitute the Chebyshev expansions in the form { ˜ n, ˜ u } = N X i =0 C { n,u } i T i ( ξ ) (A1)and project the system on each of the polynomials T i ( ξ ) , i = 0 ..N . After these manipulations we arrive atthe matrix equation: (cid:18) ˆ M (1) ˆ M (2) ˆ M (3) ˆ M (4) (cid:19) (cid:18) C ni C ui (cid:19) = iω (cid:18) C ni C vi (cid:19) , (A2)where ˆ M (1) ij = ˆ M (4) ij = t ij h T j | w ( ξ ) ∂ ξ | u T i i , ˆ M (2) ij = t ij h T j | w ( ξ ) ∂ ξ | n T i i , ˆ M (3) ij = t ij h T j | w ( ξ ) ∂ ξ | Z − dξ ′ G ( ξ, ξ ′ ) T i ( ξ ′ ) i ,t ij = ( /π, i = 0 , /π, otherwise , and w ( ξ ) = (1 − ξ ) − / is the weight function; { i, j } =0 ..N for all the matrices.Implying boundary conditions of fixed charge densityat the contacts ˜ n ( −
1) = ˜ n (1) = 0, we obtain C nN = − N/ − X i =0 C n i , (A3) C nN − = − N/ − X i =0 C n i +1 , (A4) where N is supposed to be even. We use these expressionsto eliminate C nN and C nN − from the system (A2) and, inorder to keep the matrix dimensions, truncate the firstthree matrices.One may face computational difficulties while evaluat-ing matrix elements ˆ M (3) ij as the Green function is singu-lar on the diagonal ξ = ξ ′ . To overcome this, we decom-pose the Green function into singular ( G s ) and regular( G r ) parts: G = ( G − G s ) + G s ≡ G r + G s , (A5)where G s = ln ( ξ − ξ ′ ) ( ξ + ξ ′ − ( ξ + ξ ′ + 2) (A6)accounts for the singularity provided by the charge itselfas well as by the two nearest mirror images in the elec-trodes. The regular integral is then calculated numeri-cally while the singular one can be significantly simplifiedvia transition to the complex plane.The Green function of a partly-gated structure is givenby G P G ( ξ, ξ ′ ) = ln (cid:20) ( α − α ′ ) + ( β + β ′ ) ( α − α ′ ) + ( β − β ′ ) (cid:21) , (A7)where α ′ + iβ ′ = z ′ , z ′ = cos ψ r tanh π ( d + iξ ′ )2 L + tan ψ (A8)and ψ = πL g / L .The values of the first correction obtained by thedescribed procedure and by the perturbation theory[Eq.(14)] fully coincide at small drift velocities. [1] E. Schrdinger, Quantisierung als eigenwertproblem,Annalen der Physik , 437.[2] H. A. Bethe and J. Schwinger, Perturbation theory forcavities (Massachusetts Institute of Technology, Radia-tion Laboratory, 1943).[3] R. Waldron, Perturbation theory of resonant cavities,Proceedings of the IEE-Part C: Monographs , 272(1960).[4] R. E. Collin, Field theory of guided waves, 2nd edition(IEEE Press, 1991) Chap. 5.[5] A. Raman and S. Fan, Perturbation the- ory for plasmonic modulation and sensing,Phys. Rev. B , 205131 (2011).[6] F. Bloch, Bremsverm¨ogen von atomen mit mehreren elek-tronen, Zeitschrift f¨ur Physik A Hadrons and Nuclei ,363 (1933).[7] M. J. M. de Jong and L. W. Molenkamp, Hy-drodynamic electron flow in high-mobility wires,Phys. Rev. B , 13389 (1995).[8] D. A. Bandurin, I. Torre, R. K. Kumar, M. Ben Shalom,A. Tomadin, A. Principi, G. H. Auton, E. Khes-tanova, K. S. Novoselov, I. V. Grigorieva, L. A. Pono- marenko, A. K. Geim, and M. Polini, Negative local re-sistance caused by viscous electron backflow in graphene,Science , 1055 (2016).[9] J. Crossno, J. K. Shi, K. Wang, X. Liu, A. Harzheim,A. Lucas, S. Sachdev, P. Kim, T. Taniguchi, K. Watan-abe, T. A. Ohki, and K. C. Fong, Observation of the diracfluid and the breakdown of the wiedemann-franz law ingraphene, Science , 1058 (2016).[10] R. K. Kumar, D. Bandurin, F. Pellegrino, Y. Cao,A. Principi, H. Guo, G. Auton, M. B. Shalom, L. A.Ponomarenko, G. Falkovich, et al. , Superballistic flowof viscous electron fluid through graphene constrictions,Nature Physics , 1182 (2017).[11] R. Gurzhi, Hydrodynamic effects in solids at low temper-ature, Physics-Uspekhi , 255 (1968).[12] D. Joseph and D. Sattinger, Bifurcatingtime periodic solutions and their stability,Archive for Rational Mechanics and Analysis , 79 (1972).[13] Y. Kuramoto, Chemical oscillations, waves, and turbu-lence (Courier Corporation, 2003).[14] M. Dyakonov and M. Shur, Shallow water analogy for aballistic field effect transistor: New mechanism of plasmawave generation by dc current, Phys. Rev. Lett. , 2465(1993).[15] V. Ryzhii, A. Satou, and M. S. Shur, Transit-time mecha-nism of plasma instability in high electron mobility tran-sistors, Phys. Status Solidi (a) (2005).[16] A. S. Petrov, D. Svintsov, M. Rudenko, V. Ryzhii,and M. S. Shur, Plasma instability of 2d electrons ina field effect transistor with a partly gated channel,Int. Jour. High Speed Electron. Syst. , 1640015 (2016).[17] O. Sydoruk, R. Syms, and L. Solymar,Plasma oscillations and terahertz instability infield-effect transistors with corbino geometry,Appl. Phys. Lett. , 263504 (2010).[18] M. Dyakonov, Boundary instability of a two-dimensionalelectron fluid, Semiconductors , 984 (2008).[19] W. Knap, J. Lusakowski, T. Parenty, S. Bollaert,A. Cappy, V. Popov, and M. Shur, Terahertz emissionby plasma waves in 60 nm gate high electron mobilitytransistors, Appl. Phys. Lett. , 2331 (2004).[20] N. Dyakonova, A. El Fatimy, J. Lusakowski, W. Knap,M. I. Dyakonov, M.-A. Poisson, E. Morvan, S. Bol-laert, A. Shchepetov, Y. Roelens, C. Gaquiere,D. Theron, and A. Cappy, Room-temperature tera-hertz emission from nanometer field-effect transistors,Applied Physics Letters , 141906 (2006).[21] A. El Fatimy, N. Dyakonova, Y. Meziani, T. Ot-suji, W. Knap, S. Vandenbrouk, K. Madjour,D. Th´eron, C. Gaquiere, M. Poisson, et al. , Al-gan/gan high electron mobility transistors as avoltage-tunable room temperature terahertz sources,J. Appl.Phys. , 024504 (2010).[22] A. Chaplik, Absorption and emission of electro-magnetic waves by two-dimensional plasmons,Surf. Sci. Rep. , 289 (1985).[23] S. Mikhailov, Plasma instability and amplification ofelectromagnetic waves in low-dimensional electron sys-tems, Phys. Rev. B , 1517 (1998).[24] V. Y. Kachorovskii and M. Shur, Current-induced terahertz oscillations in plasmonic crystal,Appl. Phys. Lett. , 232108 (2012).[25] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effectson electronic properties, Reviews of modern physics , 1959 (2010).[26] J. C. Song and M. S. Rudner, Chiral plasmons withoutmagnetic field, Proceedings of the National Academy ofSciences , 201519086 (2016).[27] S. S.-L. Zhang and G. Vignale, Chiral surface and edgeplasmons in ferromagnetic conductors, Physical ReviewB , 224408 (2018).[28] A. L. Fetter, Edge magnetoplasmons in a bounded two-dimensional electron fluid, Physical Review B , 7676(1985).[29] F. Stern, Polarizability of a two-dimensional electron gas,Phys. Rev. Lett. , 546 (1967).[30] A. Chaplik, Possible crystallization of charge carriers inlow-density inversion layers, Soviet Journal of Experi-mental and Theoretical Physics , 395 (1972).[31] D. B. Mast, A. J. Dahm, and A. L. Fetter, Observationof bulk and edge magnetoplasmons in a two-dimensionalelectron fluid, Phys. Rev. Lett. , 1706 (1985).[32] L. C. Silleabhin, H. P. Hughes, A. C.Churchill, D. A. Ritchie, M. P. Grimshaw, andG. A. C. Jones, Raman studies of plasmonmodes in a drifting twodimensional electron gas,Journal of Applied Physics , 1701 (1994).[33] M. Cheremisin and G. Samsonidze, Dyakonov-shur insta-bility in a ballistic field-effect transistor with a spatiallynonuniform channel, Semiconductors , 578 (1999).[34] V. V. Popov, Terahertz rectification by periodic two-dimensional electron plasma, Applied Physics Letters , 253504 (2013).[35] V. Volkov and S. Mikhailov, Edge magnetoplasmons: lowfrequency weakly damped excitations in inhomogeneoustwo-dimensional electron systems, Sov. Phys. JETP ,1639 (1988).[36] D. Svintsov, Exact solution for driven os-cillations in plasmonic field-effect transistors,Phys. Rev. Applied , 024037 (2018).[37] D. Svintsov, Hydrodynamic-to-ballistic crossover in diracmaterials, Phys. Rev. B , 121405 (2018).[38] S. J. Allen, D. C. Tsui, and R. A. Logan, Observation ofthe two-dimensional plasmon in silicon inversion layers,Phys. Rev. Lett. , 980 (1977).[39] P. L. Bhatnagar, E. P. Gross, and M. Krook, A modelfor collision processes in gases. i. small amplitude pro-cesses in charged and neutral one-component systems,Phys. Rev. , 511 (1954).[40] I. V. Kukushkin, J. H. Smet, S. A. Mikhailov,D. V. Kulakovskii, K. von Klitzing, andW. Wegscheider, Observation of retardation ef-fects in the spectrum of two-dimensional plasmons,Phys. Rev. Lett. , 156801 (2003).[41] T. Weiss, M. Mesch, M. Sch¨aferling, H. Giessen,W. Langbein, and E. A. Muljarov, From dark tobright: First-order perturbation theory with ana-lytical mode normalization for plasmonic nanoan-tenna arrays applied to refractive index sensing,Phys. Rev. Lett. , 237401 (2016).[42] L. D. Landau, On the problem of turbulence, in C.R.Acad. Sci. URSS , Vol. 44 (1944) pp. 339–349.[43] A. Gabbana, M. Polini, S. Succi, R. Tripiccione, andF. Pellegrino, Prospects for the detection of electronicpreturbulence in graphene, Physical Review Letters ,236602 (2018).[44] A. Dmitriev, A. Furman, and V. Y. Kachorovskii, Non-linear theory of the current instability in a ballistic field- effect transistor, Physical Review B54