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even while n= 1,n= 2 are odd [17].
Finally, we mention the case of a gate-controlled p-n
junction, Fig.1b. Theequilibriumdensityprofileislinear
nearx= 0 and saturates for large |x|[18]. Eq. (1) is still
applicable for q >1/d. In the limit q <1/done should
take into account the screening of long-range Coulomb
interaction by metallic gates. In this case the logarithm
in the spectrum of the gapless plasmon disappears, and
the lowest mode Eq. (14) becomes sound-like.
Magnetoplasmons. If external magnetic field Bis ap-
plied perpendicularly to the plane of graphene the plas-
mon spectra acquire new modes. The equation of motion
(2) should now be modified to include the Lorentz force,
˙J(r,t) =e2
π¯h2|µ(x)|E(r,t)−ev2
cµ(x)J×B.(18)
The relative coefficient between electric and magnetic
terms in this equation follows from the expression for
the Lorentz force acting on a single particle. The last
term has opposite sign for electrons and holes. Note that
the frequency of cyclotron motion ωB(x) =ev2B/cµ(x)
in graphene p-njunctions is position-dependent. The
remaining equations (3)-(4) are intact in the presence of
magnetic field. The boundary condition requires now the
vanishing of the normal component of electric current at
the boundary, rather than simply vanishing of the elec-
tric field, as in Eq. (12). Eliminating JandEwe arrive
at the generalization of equation (5),
δρ(x)+2e2
π/braceleftbigg
q2Z −q
ω(ωBZ)′−d
dxZd
dx/bracerightbigg
×/integraldisplayd
−ddx′δρ(x′)K0(|q||x−x′|) = 0,(19)4
whereZ(x) =|µ(x)|/(ω2
B(x)−ω2).
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,
whenω≪ωB(x) (the exact condition is given below),
one should retain only the second term in Eq. (19).
Noticing that ( ωBZ)′=πl2
Bρ′
0(x)/e=πl2
B/(l2
Ed), where
lB=/radicalbig
¯hc/eBis the magnetic length, we arrive at the
integral equation
−2c
Bq
ωdρ0(x)
dx/integraldisplayd
−ddx′δρ(x′)K0(|q||x−x′|) =δρ(x).(20)
SinceK0is positive, propagation of magnetoplasmons
withq >0is quenched, indicative oftheir chiral property
[20]. As seen from Eq. (20), the plasmon density δρ(x) is
concentratedwhere ρ′
0(x) isthestrongest. Thederivative
of the charge density in field-induced junctions (11) fea-
tures strong singularitynearthe 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.
Thegate-induced junctions, however, allow a rather
simple analytical description of these modes if we ap-
proximate that ρ′
0(x) =e/l2
Edfor|x| ≤dandρ′
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/2ddetermined from the con-
dition,δρq(±d) = 0. The spectrum of magnetoplasmons
is then found to be,
ωn(q) =−2πe2l2
B
¯hl2
Edq/radicalbig
q2+π2n2/4d2, n= 1,2...(21)
The magnetoplasmon spectrum (21) is derived under
the assumption that magnetic field is strong, ωB(d)≫ω,
which implies that lB≪lE. In order to neglect the first
and third terms in the brackets in Eq. (19) one has to
ensure that q≪(lE/lB)4/d. This condition might turn
out to be more orless restrictivethan the hydrodynamics
condition q≪1/lE, depending on the particular value of
the ratio lB/lE. 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≫ωB(d), which