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44.4k
2πΓ(3/4)/Γ(1/4)≈3.39.
Below we derive this result and discuss plasmon prop-
erties for the two types of p-njunctions: electric field
controlled and gate controlled, as shown in Fig. 1.
Hydrodynamics of charge density oscillations. We uti-
lize the hydrodynamic approach to describe the motion
of charged Dirac fermions. The rate of change of electric
current density Jdue to dynamic electric field Efollows
from the usual intra-band Drude conductivity with the
corresponding density of states [13],
˙J(r,t) =e2
π¯h2|µ(r)|E(r,t), (2)
determined by the local value of chemical potential µ(r)
as measured from the Dirac point (positive for electrons
and negative for holes). Electric current is related to the
variation of charge density δρby means of the continuity
equation,
δ˙ρ(r,t)+∇·J(r,t) = 0. (3)
Finally, the variation of charge density produces electric
field according to the Coulomb law [14],
E(r,t) =−∇/integraldisplay
d2r′δρ(r′,t)
|r−r′|. (4)
Equations (2)-(4) give a closed system for plasmon exci-
tations in graphene flakes. We apply it to a p-njunction
created in a strip infinite along the y-axis (direction of
plasmon propagation). Using the Fourier representation,
δρ(r,t) =δρ(x)exp(iqy−iωt), and eliminating Eand
Jwe arrive at the equation for the oscillating part of
electron density,
ω2δρ(x)+2e2v√π¯h/braceleftBigg
d
dx/radicalbigg
|ρ0(x)|
ed
dx−q2/radicalbigg
|ρ0(x)|
e/bracerightBigg
×/integraldisplayd
−ddx′δρ(x′)K0(|q||x−x′|) = 0,(5)
HereK0is the modified Bessel function and 2 dis
the width of graphene flake. Within the Thomas-
Fermi approximation equilibrium charge density ρ0(x)
is related to the chemical potential via ρ0(x) =
−sgn(µ)eµ2(x)/π¯h2v2(electron charge is taken to be
−e). This follows from the condition that the electro-
chemical potential µ(x)−eφ(x) is constant throughout
the system. The solutions of Eq. (5) will now be consid-
ered for large and small plasmon momenta separately.
Short wavelength, q≫1/d. In this case the decay
of plasmon density δρ(x) occurs over a distance much
smaller than the width of the system and the limits
of integration in Eq. (5) can be extended to infinity.
Assuming (cf. Eq. (11) below) the linear dependence,
ρ0(x) =ρ′
0x, we observe that the integro-differentialequation (5) acquires obvious scaling property. Intro-
ducing the variable ξ=qxwe arrive at the plasmon
spectrum in the form (1), with dimensionless constants
αndetermined from the eigenvalue problem:
−2√π/parenleftbiggd
dξ/radicalbig
|ξ|d
dξ−/radicalbig
|ξ|/parenrightbigg
×/integraldisplay∞
−∞dξ′δρ(n)(ξ′)K0(|ξ−ξ′|) =αnδρ(n)(ξ).(6)
Interestingly, this integro-differential equation allows a
complete analytic solution, though the detailed analysis
is beyond the scope of this paper. Our main findings
are as follows. Solutions are enumerated by n= 0,1,2...
with even/odd numbers corresponding to even/odd den-
sity profile, δρ(n)(−ξ) = (−1)nδρ(n)(ξ). Surprisingly,
eigenvalues are doubly-degenerate and given by
α2n=α02n+1
4n+1·3·7··(4n−1)
1·5··(4n−3), α2n+1=α2n.(7)
At large distances all modes have exponential depen-
dence,δρ(n)(ξ)∼e−|ξ|, while at |ξ| ≪1 even and
odd solutions exhibit different behavior, δρ(even)∼1−
const/radicalbig
|ξ|andδρ(odd)∼sign(ξ)//radicalbig
|ξ|. The first pair
of solutions (belonging to the lowest eigenvalue α0) in
the Fourier representation δρ(n)(k) =/integraltext
dξδρ(n)(ξ)eikξ
acquires a simple form:
δρ(0)(k)∝1
(1+k2)3/4, δρ(1)(k)∝k
(1+k2)3/4.(8)
Long wavelength, q≪1/d. In contrast to the above
result (1) plasmon spectrum at small qis sensitive to a
specific realization of the p-njunction. We address the
long-wavelength behavior of plasmons in field controlled
junctions. We expect this case to be of more interest,
in addition it allows a more complete description. Be-
fore analyzing plasmons in this structure, we discuss the
equilibrium density profile. As shown in Fig. 1a the flake
of width 2 dis placed in external electric field E0applied
along the x-direction. The equilibrium density distribu-
tionρ(x) is found from,