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arXiv:1001.0011v2 [cond-mat.mes-hall] 16 Apr 2010Guided plasmons in graphene p-njunctions |
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E. G. Mishchenko,1A. V. Shytov∗,1and P. G. Silvestrov2 |
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1Department of Physics and Astronomy, University of Utah, Sa lt Lake City, Utah 84112, USA |
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2Theoretische Physik III, Ruhr-Universit¨ at Bochum, 44780 Bochum, Germany |
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Spatial separation of electrons and holes in graphene gives rise to existence of plasmon waves |
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confined to the boundary region. Theory of such guided plasmo n modes within hydrodynamics of |
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electron-hole liquid is developed. For plasmon wavelength s smaller than the size of charged domains |
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plasmon dispersion is found to be ω∝q1/4. Frequency, velocity and direction of propagation of |
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guided plasmon modes can be easily controlled by external el ectric field. In the presence of magnetic |
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field spectrum of additional gapless magnetoplasmon excita tions is obtained. Our findings indicate |
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that graphene is a promising material for nanoplasmonics. |
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PACS numbers: 73.23.-b, 72.30.+q |
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Introduction . Breakthrough progress in synthesis and |
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characterization has made graphene [2] a promising ob- |
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ject for nanoelectronics. Operation of graphene-based |
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transistors [3] and other components would rely on the |
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propertiesofits single-particle excitations–electronsand |
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holes. However, one can also envisage a completely dif- |
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ferent set of applications which employ collective excita- |
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tions, such as plasmons. Currently, plasmon excitations |
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in metallic structures are a subject of nanoplasmonics, a |
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new field which has emerged at the confluence of optics |
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and condensed matter physics with one of the aims be- |
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ing the developing of plasmon-enhanced high resolution |
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near-field imaging methods [4, 5]. Another objective is |
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possible utilization of plasmons in integrated optical cir- |
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cuits. However, perspectives of graphene for nanoplas- |
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monics are largely unexplored since plasmon modes of |
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graphene flakes have not been addressed so far. As our |
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results indicate a great amount of control over graphene |
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plasmon properties makes it a very promising material |
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for applications. |
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Fundamentally, the spectrum of collective chargeoscil- |
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lations reflects the long-rangenature of Coulomb interac- |
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tion. In conventional two dimensional systems, such as |
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those created in semiconducting heterostructures, plas- |
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mons are gapless, ω2(q) = 2πe2nq/m∗, withnandm∗ |
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being electron density and effective mass, respectively |
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[6]. Such oscillations can be treated hydrodynamically. |
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In clean graphene at zero temperature the plasmon fre- |
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quency,ω2∝ |EF|, vanishes with decreasing the doping |
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levelEF. It has been argued [7] that the interaction be- |
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tweenelectronsandholesinthefinalstatecanmodifythe |
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response functions of Dirac fermions and open up a pos- |
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sibility for the propagation of charge oscillations at low |
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frequencies ω < qv, wherevis electron velocity. Still, hy- |
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drodynamic( ω > qv)analogofconventionalplasmonsre- |
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mains absent unless either temperature is non-zero [8] or |
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graphene is driven away from the charge neutrality point |
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by doping or gating [9]. Expectedly, in both cases plas- |
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mon spectrum has the conventional form, ω(q)∝q1/2. |
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In the present paper we investigate spectra of hydro- |
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dynamic plasmons in spatially inhomogeneous grapheneflakes. Realistic graphene samples are typically subject |
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to disorder potential and mechanical strain [10] that lead |
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totheformationofchargedelectronandholepuddles[11] |
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with boundaries between nandpregions being the lines |
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ofzerochemicalpotential. Moreover,controlled p-njunc- |
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tions can be made with the help of metallic gates [12]. |
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Alsop-njunctions can be created by applying electric |
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field within the plane of a graphene flake, see Fig. 1a. |
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The field separates electrons and holes spatially in a way |
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that allows control of both the amount of induced charge |
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(and thus plasmon frequency) and spatial orientation of |
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the junction (the direction of plasmon propagation). |
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b)2d 2d |
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Ea) |
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0n n |
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p p |
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FIG.1: Twotypesofgraphene p-njunctions: a)field-induced, |
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b) gate-induced. Dot-dashed line indicates boundary betwe en |
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electron and hole regions and, correspondingly, the direct ion |
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of plasmon propagation. In case of field-induced junction it |
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is controlled by the direction of external electric field E0. |
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Below, we demonstrate that such p-njunctions can |
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guide plasmons. We show the existence of charge oscil- |
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lations which are localized at the junction and have the |
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amplitude decaying with the distance to the junction. |
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For wavelengths shorter than the width of the charged |
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domains, we find the plasmon spectrum of the form, |
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ω2 |
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n(q) =αne2v |
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¯h/radicalbigg |
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q|ρ′ |
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0| |
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e, (1) |
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whereρ′ |
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0is the gradient of equilibrium charge density |
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at the junction, vis electron velocity, and n= 0,1,2,...2 |
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enumerates the solutions. The lowest mode has α0= |
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4√ |
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2πΓ(3/4)/Γ(1/4)≈3.39. |
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Below we derive this result and discuss plasmon prop- |
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erties for the two types of p-njunctions: electric field |
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controlled and gate controlled, as shown in Fig. 1. |
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Hydrodynamics of charge density oscillations. We uti- |
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lize the hydrodynamic approach to describe the motion |
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of charged Dirac fermions. The rate of change of electric |
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current density Jdue to dynamic electric field Efollows |
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from the usual intra-band Drude conductivity with the |
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corresponding density of states [13], |
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˙J(r,t) =e2 |
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π¯h2|µ(r)|E(r,t), (2) |
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determined by the local value of chemical potential µ(r) |
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as measured from the Dirac point (positive for electrons |
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and negative for holes). Electric current is related to the |
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variation of charge density δρby means of the continuity |
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equation, |
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δ˙ρ(r,t)+∇·J(r,t) = 0. (3) |
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Finally, the variation of charge density produces electric |
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field according to the Coulomb law [14], |
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E(r,t) =−∇/integraldisplay |
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d2r′δρ(r′,t) |
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|r−r′|. (4) |
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Equations (2)-(4) give a closed system for plasmon exci- |
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tations in graphene flakes. We apply it to a p-njunction |
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created in a strip infinite along the y-axis (direction of |
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plasmon propagation). Using the Fourier representation, |
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δρ(r,t) =δρ(x)exp(iqy−iωt), and eliminating Eand |
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Jwe arrive at the equation for the oscillating part of |
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electron density, |
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ω2δρ(x)+2e2v√π¯h/braceleftBigg |
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d |
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dx/radicalbigg |
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|ρ0(x)| |
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ed |
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dx−q2/radicalbigg |
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|ρ0(x)| |
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e/bracerightBigg |
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×/integraldisplayd |
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−ddx′δρ(x′)K0(|q||x−x′|) = 0,(5) |
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HereK0is the modified Bessel function and 2 dis |
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the width of graphene flake. Within the Thomas- |
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Fermi approximation equilibrium charge density ρ0(x) |
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is related to the chemical potential via ρ0(x) = |
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−sgn(µ)eµ2(x)/π¯h2v2(electron charge is taken to be |
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−e). This follows from the condition that the electro- |
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chemical potential µ(x)−eφ(x) is constant throughout |
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the system. The solutions of Eq. (5) will now be consid- |
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ered for large and small plasmon momenta separately. |
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Short wavelength, q≫1/d. In this case the decay |
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of plasmon density δρ(x) occurs over a distance much |
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smaller than the width of the system and the limits |
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of integration in Eq. (5) can be extended to infinity. |
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Assuming (cf. Eq. (11) below) the linear dependence, |
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ρ0(x) =ρ′ |
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0x, we observe that the integro-differentialequation (5) acquires obvious scaling property. Intro- |
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ducing the variable ξ=qxwe arrive at the plasmon |
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spectrum in the form (1), with dimensionless constants |
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αndetermined from the eigenvalue problem: |
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−2√π/parenleftbiggd |
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dξ/radicalbig |
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|ξ|d |
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dξ−/radicalbig |
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|ξ|/parenrightbigg |
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×/integraldisplay∞ |
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−∞dξ′δρ(n)(ξ′)K0(|ξ−ξ′|) =αnδρ(n)(ξ).(6) |
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Interestingly, this integro-differential equation allows a |
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complete analytic solution, though the detailed analysis |
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is beyond the scope of this paper. Our main findings |
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are as follows. Solutions are enumerated by n= 0,1,2... |
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with even/odd numbers corresponding to even/odd den- |
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sity profile, δρ(n)(−ξ) = (−1)nδρ(n)(ξ). Surprisingly, |
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eigenvalues are doubly-degenerate and given by |
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α2n=α02n+1 |
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4n+1·3·7··(4n−1) |
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1·5··(4n−3), α2n+1=α2n.(7) |
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At large distances all modes have exponential depen- |
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dence,δρ(n)(ξ)∼e−|ξ|, while at |ξ| ≪1 even and |
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odd solutions exhibit different behavior, δρ(even)∼1− |
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const/radicalbig |
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|ξ|andδρ(odd)∼sign(ξ)//radicalbig |
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|ξ|. The first pair |
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of solutions (belonging to the lowest eigenvalue α0) in |
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the Fourier representation δρ(n)(k) =/integraltext |
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dξδρ(n)(ξ)eikξ |
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acquires a simple form: |
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δρ(0)(k)∝1 |
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(1+k2)3/4, δρ(1)(k)∝k |
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(1+k2)3/4.(8) |
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Long wavelength, q≪1/d. In contrast to the above |
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result (1) plasmon spectrum at small qis sensitive to a |
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specific realization of the p-njunction. We address the |
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long-wavelength behavior of plasmons in field controlled |
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junctions. We expect this case to be of more interest, |
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in addition it allows a more complete description. Be- |
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fore analyzing plasmons in this structure, we discuss the |
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equilibrium density profile. As shown in Fig. 1a the flake |
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of width 2 dis placed in external electric field E0applied |
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along the x-direction. The equilibrium density distribu- |
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tionρ(x) is found from, |
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E0x+sgn(x)¯hv |
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e/radicalbiggπ |
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e|ρ0(x)|+2/integraldisplayd |
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0dx′ρ0(x′)lnx+x′ |
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|x−x′|= 0, |
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(9) |
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where it is used that ρ0(x) =−ρ0(−x). Prior to solv- |
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ing Eq. (9) it is instructive to analyze validity of the |
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semiclassical approach. The first condition implies that |
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the change of the electron wavelength is smooth on the |
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scale of itself, d/dx(¯hv/µ)≪1. Estimating µ(x)∼eE0x |
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we obtain that the distance to the p-njunction line |
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(x= 0) should exceed the characteristic electric field |
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lengthlE=/radicalbig |
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e/E0≪x. The second condition requires |
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that the electron wavelength is small compared with the |
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width of the system, d≫¯hv/µ. Noting that in graphene3 |
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¯hv∼e2we can rewrite this second condition simply as |
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lE≪d. Thus, the Thomas-Fermi equation (9) for the |
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equilibrium charge density and the hydrodynamic equa- |
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tion (5) for its variation are applicable as long as |
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lE≪d, q≪1/lE. (10) |
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However, the ratio of qand 1/dcan be arbitrary. For a |
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moderate external electric field ∼104V/m the value of |
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electric length lE∼0.4µm and the first of the conditions |
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(10) is satisfied easily for micron-sized samples. |
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AnalyticsolutionofEq.(9)ispossiblewhenthe second |
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term is small, in which case the charge density is [15] |
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ρ0(x) =E0x√ |
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d2−x2. (11) |
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Substituting this expression back into Eq. (9) we ob- |
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serve that the second term is indeed negligible as long |
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asx≫l2 |
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E/d. This is assured whenever the condi- |
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tions (10) are satisfied. It is also worth pointing out |
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that Eq. (11) justifies the linear approximation for the |
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charge density used in deriving Eq. (1) for q≫1/d, with |
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ρ′ |
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0/e= 1/(l2 |
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Ed). |
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We now turn to the analysis of plasma oscillations |
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propagating on top of the density distribution, Eq. (11). |
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For small plasmon momenta, q≪1/d, electric field ex- |
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tends beyond the width of the flake and the equation (5) |
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needs to be supplemented with the boundary condition, |
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which ensures that electric field (and thus the current) |
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vanishes at the edges, x=±d: |
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P/integraldisplayd |
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−ddxδρ(x) |
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x±d= 0. (12) |
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The spectrum of the lowest symmetric mode can be most |
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easily found by integrating Eq. (5) across the width of |
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the flake. The first term in the brackets will then van- |
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ish exactly due to the boundary condition (12). The |
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remaining integral can now be calculated to the log- |
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arithmic accuracy with the help of the approximation |
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K0(q|x−x′|) =−lnq|x−x′|: |
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/integraldisplayd |
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−ddx/radicalbigg |
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|ρ0(x)| |
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eln(q|x−x′|)≈2dΓ2(3/4) |
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lE√πln(qd). |
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(13) |
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Eqs. (5) and (13) combine to give the equation, [ ω2− |
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ω2 |
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0(q)]/integraltextd |
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−ddxδρ(x) = 0, that yields the dispersion of the |
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gapless symmetric plasmon, |
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ω2 |
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0(q) = Γ2(3/4)4e2vd |
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π¯hlEq2ln(1/qd),(14) |
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reminiscent of the plasmon spectrum in quasi-one- |
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dimensional wires, The remaining modes, n≥1, are |
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gapped. For these modes/integraltextd |
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−ddxδρ(x) = 0 and simple |
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procedure of integrating Eq. (5) over the width of theflake is not useful. Instead, the equation for the n-th fre- |
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quency gap can be obtained by setting q= 0 in Eq. (5). |
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We observe that |
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ω2 |
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n(0) =βne2v |
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¯hlEd, (15) |
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whereβnare the eigenvalues of the equation, |
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2√πd |
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dξ/radicalbig |
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|ξ| |
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(1−ξ2)1/4/integraldisplay1 |
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−1dξ′δρ(n)(ξ′) |
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ξ−ξ′=βnδρ(n)(ξ).(16) |
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The zeroth mode β0= 0, see Eq. (14), is found ana- |
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lytically: δρ(0)∝1//radicalbig |
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1−ξ2. It describes charge dis- |
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tribution in the strip in response to a (uniform along |
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xdirection and smooth along y-direction) change of its |
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chemical potential [16]. Other solutions of Eq. (16) are |
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found numerically: |
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β1= 1.41, β2= 6.49, β3= 6.75,... (17) |
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With increasing nthe eigenmodes of integro-differential |
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equation (16) oscillate faster, but in generaldo not follow |
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the oscillation theorem familiar from quantum mechan- |
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ics. In particular, the solutions with n= 0 andn= 3 are |
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even while n= 1,n= 2 are odd [17]. |
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Finally, we mention the case of a gate-controlled p-n |
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junction, Fig.1b. Theequilibriumdensityprofileislinear |
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nearx= 0 and saturates for large |x|[18]. Eq. (1) is still |
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applicable for q >1/d. In the limit q <1/done should |
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take into account the screening of long-range Coulomb |
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interaction by metallic gates. In this case the logarithm |
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in the spectrum of the gapless plasmon disappears, and |
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the lowest mode Eq. (14) becomes sound-like. |
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Magnetoplasmons. If external magnetic field Bis ap- |
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plied perpendicularly to the plane of graphene the plas- |
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mon spectra acquire new modes. The equation of motion |
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(2) should now be modified to include the Lorentz force, |
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˙J(r,t) =e2 |
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π¯h2|µ(x)|E(r,t)−ev2 |
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cµ(x)J×B.(18) |
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The relative coefficient between electric and magnetic |
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terms in this equation follows from the expression for |
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the Lorentz force acting on a single particle. The last |
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term has opposite sign for electrons and holes. Note that |
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the frequency of cyclotron motion ωB(x) =ev2B/cµ(x) |
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in graphene p-njunctions is position-dependent. The |
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remaining equations (3)-(4) are intact in the presence of |
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magnetic field. The boundary condition requires now the |
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vanishing of the normal component of electric current at |
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the boundary, rather than simply vanishing of the elec- |
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tric field, as in Eq. (12). Eliminating JandEwe arrive |
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at the generalization of equation (5), |
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δρ(x)+2e2 |
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π/braceleftbigg |
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q2Z −q |
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ω(ωBZ)′−d |
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dxZd |
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dx/bracerightbigg |
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×/integraldisplayd |
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−ddx′δρ(x′)K0(|q||x−x′|) = 0,(19)4 |
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whereZ(x) =|µ(x)|/(ω2 |
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B(x)−ω2). |
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The most interesting effect described by Eq. (19) is |
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the appearance of a set of new modes, chiral magne- |
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toplasmons, similar to those considered in Ref. [19] for |
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conventional 2D electron systems with smooth bound- |
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aries. To find their dispersion in strong magnetic fields, |
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whenω≪ωB(x) (the exact condition is given below), |
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one should retain only the second term in Eq. (19). |
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Noticing that ( ωBZ)′=πl2 |
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Bρ′ |
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0(x)/e=πl2 |
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B/(l2 |
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Ed), where |
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lB=/radicalbig |
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¯hc/eBis the magnetic length, we arrive at the |
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integral equation |
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−2c |
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Bq |
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ωdρ0(x) |
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dx/integraldisplayd |
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−ddx′δρ(x′)K0(|q||x−x′|) =δρ(x).(20) |
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SinceK0is positive, propagation of magnetoplasmons |
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withq >0is quenched, indicative oftheir chiral property |
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[20]. As seen from Eq. (20), the plasmon density δρ(x) is |
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concentratedwhere ρ′ |
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0(x) isthestrongest. Thederivative |
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of the charge density in field-induced junctions (11) fea- |
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tures strong singularitynearthe edges of the flake. Thus, |
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low-frequency magnetoplasmon spectrum is strongly de- |
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pendent on the microscopic regularization of this singu- |
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lar behavior and is, therefore, beyond the scope of the |
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Thomas-Fermi approximation used throughout this pa- |
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per. |
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Thegate-induced junctions, however, allow a rather |
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simple analytical description of these modes if we ap- |
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proximate that ρ′ |
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0(x) =e/l2 |
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Edfor|x| ≤dandρ′ |
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0(x) = 0 |
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for|x|> d. The oscillating density δρ(x) then vanishes |
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for|x|> d. The solution inside the strip, |x| ≤d, can |
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be easily found for q≫1/d, where one can assume the |
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range of integration in Eq. (20) to be infinite. The eigen- |
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functions of Eq. (20) are simply given by sin[ q⊥(x+d)], |
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with the values of q⊥=πn/2ddetermined from the con- |
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dition,δρq(±d) = 0. The spectrum of magnetoplasmons |
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is then found to be, |
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ωn(q) =−2πe2l2 |
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B |
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¯hl2 |
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Edq/radicalbig |
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q2+π2n2/4d2, n= 1,2...(21) |
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The magnetoplasmon spectrum (21) is derived under |
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the assumption that magnetic field is strong, ωB(d)≫ω, |
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which implies that lB≪lE. In order to neglect the first |
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and third terms in the brackets in Eq. (19) one has to |
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ensure that q≪(lE/lB)4/d. This condition might turn |
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out to be more orless restrictivethan the hydrodynamics |
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condition q≪1/lE, depending on the particular value of |
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the ratio lB/lE. Note that the smallness of this ratio is |
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not in contradiction to the non-quantized description of |
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electron motion in magnetic filed. The latter is valid as |
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long as the filling factor is large, eEd≫ωB(d), which |
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means that lB≫l2 |
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E/d. For magnetic field ∼1T, and |
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lB∼25nm, using the estimate below Eq. (10) that lE∼ |
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400nm we conclude that the width of the flake should |
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exceedd >10µm. The magnetoplasmon modes (21) are |
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∼(lB/lE)2slowerthan electrons. Note that these modesare undamped since single-particle excitations cannot be |
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induced at frequencies below cyclotron frequency ωB. |
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Conclusions . Graphene p-njunctions are among the |
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most simple and promising applications of this material. |
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Single-electron properties of p-njunctions have been ex- |
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tensively studied. In the present paper we investigated |
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their collective excitations both with and without mag- |
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netic field. We anticipate that plasmon modes will be |
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crucial for the optical response of graphene nanostruc- |
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tures and realistic samples containing electron-hole pud- |
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dles. High degree of experimental control should make |
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them of special interest to nanoplasmonics and electron- |
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ics. Among the most promising applications of plasmons |
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inp-njunctions we envisage a possibility of a “plasmon |
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transistor” [4]. In particular, by simply switching the |
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direction of electric field from across the flake to along |
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it (and back) the propagation of plasmons can be facil- |
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itated (or prevented). In addition, as follows from the |
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above Eqs. (1), (11), the plasmon velocity can be con- |
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trolled with simple change in the magnitude of electric |
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field. This is in a sharp contrast to plasmons in metal- |
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lic nanostructures, whose spectra are typically fixed once |
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devices are fabricated. |
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Acknowledgments. Useful discussions with M. Raikh |
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and O. Starykh are gratefully acknowledged. This |
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work was supported by DOE, Grant No. DE-FG02- |
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06ER46313. P.G.S. was supported by the SFB TR 12. |
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[*] Present address: School of Physics, University of Exete r, |
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EX4 4QL, U.K. |
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[2] M. Wilson, Phys. Today 59, No. 1, 21 (2006). |
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[3] A.K. Geim and K.S. Novoselov, Nature Mater. 6, 183 |
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(2007). |
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[4] H.A. Atwater, Sci. Am. 296, 56 (2007). |
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[5] S.A. Maier, Plasmonics: Fundamentals and Applications |
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(Springer, New York, 2007). |
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[6] F. Stern, Phys. Rev. Lett. 18, 546 (1967). |
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[7] S. Gangadharaiah, A.M. Farid, and E.G. Mishchenko, |
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Phys. Rev. Lett. 100, 166802 (2008). |
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[8] O. Vafek, Phys. Rev. Lett. 97, 266406 (2006). |
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[9] E.H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 |
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(2007). |
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[10] A.H. Castro Neto et al.,Rev. Mod. Phys. 81, 109 (2009). |
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[11] J. Martin et al.,Nature Physics 4, 144 (2008). |
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[12] J.R. Williams, L. DiCarlo, and C.M. Marcus, Science |
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317, 638 (2007). |
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[13] Rigorous derivation of Eq. (2) is based on the “rela- |
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tivistic” stress energy-momentum tensor, see L.D. Lan- |
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dau and E. M. Lifshitz, Fluid Mechanics , Butterworth- |
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Heinemann, Oxford (1987), Ch. 15; M. Mueller, L. Fritz, |
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S. Sachdev, and J. Schmalian, arXiv:0810.3657. |
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[14] In the case of gate controlled junctions the image charg es |
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induced at the gates should be included into Eq. (4). |
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[15] T.A. Sedrakyan, E.G. Mishchenko, and M.E. Raikh, |
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Phys. Rev. B 74, 235423 (2006). |
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[16] P.G. SilvestrovandK.B. Efetov, Phys.Rev.B 77, 155436 |
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(2008).5 |
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[17] In addition even and odd solutions with n >0 have dif- |
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ferent singular behavior at |ξ| ≪1:δρ(even)∼/radicalbig |
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|ξ|, |
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δρ(odd)∼sign(ξ)//radicalbig |
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|ξ|. Atξ→ ±1 all solutions diverge |
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asδρ∼1//radicalbig |
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1−ξ2. |
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[18] L.M. Zhang and M.M. Fogler, Phys. Rev. Lett. 100,116804 (2008). |
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[19] I.L. Aleiner and L.I. Glazman, Phys. Rev. Lett. 72, 2935 |
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(1994). |
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[20] V.A. Volkov and S. A. Mikhailov, JETP Lett. 42, 556 |
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(1985). |