Guided plasmons in graphene p - n junctions E. G. Mishchenko, 1 A. V. Shytov $^{∗}$, 1 and P. G. Silvestrov 2 $^{1}$Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, USA $^{2}$Theoretische Physik III, Ruhr-Universit¨ at Bochum, 44780 Bochum, Germany Spatial separation of electrons and holes in graphene gives rise to existence of plasmon waves confined to the boundary region. Theory of such guided plasmon modes within hydrodynamics of electron-hole liquid is developed. For plasmon wavelengths smaller than the size of charged domains plasmon dispersion is found to be ω ∝ q 1 / $^{4}$. Frequency, velocity and direction of propagation of guided plasmon modes can be easily controlled by external electric field. In the presence of magnetic field spectrum of additional gapless magnetoplasmon excitations is obtained. Our findings indicate that graphene is a promising material for nanoplasmonics. PACS numbers: 73.23.-b, 72.30.+q Introduction . Breakthrough progress in synthesis and characterization has made graphene [2] a promising ob- ject for nanoelectronics. Operation of graphene-based transistors [3] and other components would rely on the properties of its single-particle excitations – electrons and holes. However, one can also envisage a completely dif- ferent set of applications which employ collective excita- tions, such as plasmons. Currently, plasmon excitations in metallic structures are a subject of nanoplasmonics, a new field which has emerged at the confluence of optics and condensed matter physics with one of the aims be- ing the developing of plasmon-enhanced high resolution near-field imaging methods [4, 5]. Another objective is possible utilization of plasmons in integrated optical cir- cuits. However, perspectives of graphene for nanoplas- monics are largely unexplored since plasmon modes of graphene flakes have not been addressed so far. As our results indicate a great amount of control over graphene plasmon properties makes it a very promising material for applications. Fundamentally, the spectrum of collective charge oscil- lations reflects the long-range nature of Coulomb interac- tion. In conventional two dimensional systems, such as those created in semiconducting heterostructures, plas- mons are gapless, ω $^{2}$( q ) = 2 πe $^{2}$nq/m $^{∗}$, with n and m ∗ being electron density and effective mass, respectively [6]. Such oscillations can be treated hydrodynamically. In clean graphene at zero temperature the plasmon fre- quency, ω 2 ∝ | E$_{F}$ | , vanishes with decreasing the doping level E$_{F}$ . It has been argued [7] that the interaction be- tween electrons and holes in the final state can modify the response functions of Dirac fermions and open up a pos- sibility for the propagation of charge oscillations at low frequencies ω < qv , where v is electron velocity. Still, hy- drodynamic ( ω > qv ) analog of conventional plasmons re- mains absent unless either temperature is non-zero [8] or graphene is driven away from the charge neutrality point by doping or gating [9]. Expectedly, in both cases plas- mon spectrum has the conventional form, ω ( q ) ∝ q 1 / $^{2}$. In the present paper we investigate spectra of hydro- dynamic plasmons in spatially inhomogeneous graphene where ρ ′ 0 is the gradient of equilibrium charge density at the junction, v is electron velocity, and n = 0 , 1 , 2 , ... Below, we demonstrate that such p - n junctions can guide plasmons. We show the existence of charge oscil- lations which are localized at the junction and have the amplitude decaying with the distance to the junction. For wavelengths shorter than the width of the charged domains, we find the plasmon spectrum of the form, flakes. Realistic graphene samples are typically subject to disorder potential and mechanical strain [10] that lead to the formation of charged electron and hole puddles [11] with boundaries between n and p regions being the lines of zero chemical potential. Moreover, controlled p - n junc- tions can be made with the help of metallic gates [12]. Also p - n junctions can be created by applying electric field within the plane of a graphene flake, see Fig. 1a. The field separates electrons and holes spatially in a way that allows control of both the amount of induced charge (and thus plasmon frequency) and spatial orientation of the junction (the direction of plasmon propagation). arXiv:1001.0011v2 [cond-mat.mes-hall] 16 Apr 2010 ω 2 $_{n}$( q ) = α$_{n}$ e $^{2}$v ¯ h √ q | ρ ′ 0 | e , (1) b) 2d 2d E a) 0 n n p p FIG. 1: Two types of graphene p - n junctions: a) field-induced, b) gate-induced. Dot-dashed line indicates boundary between electron and hole regions and, correspondingly, the direction of plasmon propagation. In case of field-induced junction it is controlled by the direction of external electric field E$_{0}$ .