content_image/420102.jpg content_image/420105.jpg
where Z ( x ) = | µ ( x ) | / ( ω 2 $_{B}$( x ) − ω $^{2}$) . 

The most interesting effect described by Eq. (19) is the appearance of a set of new modes, chiral magne- toplasmons, similar to those considered in Ref. [19] for conventional 2D electron systems with smooth bound- aries. To find their dispersion in strong magnetic fields, when ω ≪ ω$_{B}$ ( x ) (the exact condition is given below), one should retain only the second term in Eq. (19). Noticing that ( ω$_{B}$Z ) ′ = πl 2 $_{B}$ρ ′ $_{0}$( x ) /e = πl 2 $_{B}$/ ( l 2 $_{E}$d ), where l$_{B}$ = √ ¯ hc/eB is the magnetic length, we arrive at the integral equation 

<img src='content_image/420102.jpg'>

Since K$_{0}$ is positive, propagation of magnetoplasmons with q > 0 is quenched, indicative of their chiral property [20]. As seen from Eq. (20), the plasmon density δρ ( x ) is concentrated where ρ ′ $_{0}$( x ) is the strongest. The derivative of the charge density in field-induced junctions (11) fea- tures strong singularity near the edges of the flake. Thus, low-frequency magnetoplasmon spectrum is strongly de- pendent on the microscopic regularization of this singu- lar behavior and is, therefore, beyond the scope of the Thomas-Fermi approximation used throughout this pa- per. 

The gate-induced junctions, however, allow a rather simple analytical description of these modes if we ap- proximate that ρ ′ $_{0}$( x ) = e/l 2 $_{E}$d for | x | ≤ d and ρ ′ $_{0}$( x ) = 0 for | x | > d . The oscillating density δρ ( x ) then vanishes for x > d . The solution inside the strip, | x | ≤ d , can be easily found for q ≫ 1 /d , where one can assume the range of integration in Eq. (20) to be infinite. The eigen- functions of Eq. (20) are simply given by sin [ q$_{⊥}$ ( x + d )], with the values of q$_{⊥}$ = πn/ 2 d determined from the con- dition, δρ$_{q}$ ( ± d ) = 0. The spectrum of magnetoplasmons is then found to be, 

<img src='content_image/420105.jpg'>

the assumption that magnetic field is strong, ω$_{B}$ ( d ) ≫ ω , which implies that l$_{B}$ ≪ l$_{E}$ . In order to neglect the first and third terms in the brackets in Eq. (19) one has to ensure that q ≪ ( l$_{E}$/l$_{B}$ ) $^{4}$/d . This condition might turn out to be more or less restrictive than the hydrodynamics condition q ≪ 1 /l$_{E}$ , depending on the particular value of the ratio l$_{B}$/l$_{E}$ . Note that the smallness of this ratio is not in contradiction to the non-quantized description of electron motion in magnetic filed. The latter is valid as long as the filling factor is large, eEd ≫ ω$_{B}$ ( d ), which means that l$_{B}$ ≫ l 2 $_{E}$/d . For magnetic field ∼ 1T, and l$_{B}$ ∼ 25nm, using the estimate below Eq. (10) that l$_{E}$ ∼ 400nm we conclude that the width of the flake should exceed d > 10 µ m. The magnetoplasmon modes (21) are ∼ ( l$_{B}$/l$_{E}$ ) 2 slower than electrons. Note that these modes 

are undamped since single-particle excitations cannot be induced at frequencies below cyclotron frequency ω$_{B}$ . 

Conclusions . Graphene p - n junctions are among the most simple and promising applications of this material. Single-electron properties of p - n junctions have been ex- tensively studied. In the present paper we investigated their collective excitations both with and without mag- netic field. We anticipate that plasmon modes will be crucial for the optical response of graphene nanostruc- tures and realistic samples containing electron-hole pud- dles. High degree of experimental control should make them of special interest to nanoplasmonics and electron- ics. Among the most promising applications of plasmons in p - n junctions we envisage a possibility of a “plasmon transistor” [4]. In particular, by simply switching the direction of electric field from across the flake to along it (and back) the propagation of plasmons can be facil- itated (or prevented). In addition, as follows from the above Eqs. (1), (11), the plasmon velocity can be con- trolled with simple change in the magnitude of electric field. This is in a sharp contrast to plasmons in metal- lic nanostructures, whose spectra are typically fixed once devices are fabricated. 

Acknowledgments. Useful discussions with M. Raikh and O. Starykh are gratefully acknowledged. This work was supported by DOE, Grant No. DE-FG02- 06ER46313. P.G.S. was supported by the SFB TR 12. 

[*] Present address: School of Physics, University of Exeter, EX4 4QL, U.K. 

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[13] Rigorous derivation of Eq. (2) is based on the “rela- tivistic” stress energy-momentum tensor, see L.D. Lan- dau and E. M. Lifshitz, Fluid Mechanics , Butterworth- Heinemann, Oxford (1987), Ch. 15; M. Mueller, L. Fritz, S. Sachdev, and J. Schmalian, arXiv:0810.3657. 

[14] In the case of gate controlled junctions the image charges induced at the gates should be included into Eq. (4). 

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