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E0x+sgn(x)¯hv
e/radicalbiggπ
e|ρ0(x)|+2/integraldisplayd
0dx′ρ0(x′)lnx+x′
|x−x′|= 0,
(9)
where it is used that ρ0(x) =−ρ0(−x). Prior to solv-
ing Eq. (9) it is instructive to analyze validity of the
semiclassical approach. The first condition implies that
the change of the electron wavelength is smooth on the
scale of itself, d/dx(¯hv/µ)≪1. Estimating µ(x)∼eE0x
we obtain that the distance to the p-njunction line
(x= 0) should exceed the characteristic electric field
lengthlE=/radicalbig
e/E0≪x. The second condition requires
that the electron wavelength is small compared with the
width of the system, d≫¯hv/µ. Noting that in graphene3
¯hv∼e2we can rewrite this second condition simply as
lE≪d. Thus, the Thomas-Fermi equation (9) for the
equilibrium charge density and the hydrodynamic equa-
tion (5) for its variation are applicable as long as
lE≪d, q≪1/lE. (10)
However, the ratio of qand 1/dcan be arbitrary. For a
moderate external electric field ∼104V/m the value of
electric length lE∼0.4µm and the first of the conditions
(10) is satisfied easily for micron-sized samples.
AnalyticsolutionofEq.(9)ispossiblewhenthe second
term is small, in which case the charge density is [15]
ρ0(x) =E0x√
d2−x2. (11)
Substituting this expression back into Eq. (9) we ob-
serve that the second term is indeed negligible as long
asx≫l2
E/d. This is assured whenever the condi-
tions (10) are satisfied. It is also worth pointing out
that Eq. (11) justifies the linear approximation for the
charge density used in deriving Eq. (1) for q≫1/d, with
ρ′
0/e= 1/(l2
Ed).
We now turn to the analysis of plasma oscillations
propagating on top of the density distribution, Eq. (11).
For small plasmon momenta, q≪1/d, electric field ex-
tends beyond the width of the flake and the equation (5)
needs to be supplemented with the boundary condition,
which ensures that electric field (and thus the current)
vanishes at the edges, x=±d:
P/integraldisplayd
−ddxδρ(x)
x±d= 0. (12)
The spectrum of the lowest symmetric mode can be most
easily found by integrating Eq. (5) across the width of
the flake. The first term in the brackets will then van-
ish exactly due to the boundary condition (12). The
remaining integral can now be calculated to the log-
arithmic accuracy with the help of the approximation
K0(q|x−x′|) =−lnq|x−x′|:
/integraldisplayd
−ddx/radicalbigg
|ρ0(x)|
eln(q|x−x′|)≈2dΓ2(3/4)
lE√πln(qd).
(13)
Eqs. (5) and (13) combine to give the equation, [ ω2−
ω2
0(q)]/integraltextd
−ddxδρ(x) = 0, that yields the dispersion of the
gapless symmetric plasmon,
ω2
0(q) = Γ2(3/4)4e2vd
π¯hlEq2ln(1/qd),(14)
reminiscent of the plasmon spectrum in quasi-one-
dimensional wires, The remaining modes, n≥1, are
gapped. For these modes/integraltextd
−ddxδρ(x) = 0 and simple
procedure of integrating Eq. (5) over the width of theflake is not useful. Instead, the equation for the n-th fre-
quency gap can be obtained by setting q= 0 in Eq. (5).
We observe that
ω2
n(0) =βne2v
¯hlEd, (15)
whereβnare the eigenvalues of the equation,
2√πd
dξ/radicalbig
|ξ|
(1−ξ2)1/4/integraldisplay1
−1dξ′δρ(n)(ξ′)
ξ−ξ′=βnδρ(n)(ξ).(16)
The zeroth mode β0= 0, see Eq. (14), is found ana-
lytically: δρ(0)∝1//radicalbig
1−ξ2. It describes charge dis-
tribution in the strip in response to a (uniform along
xdirection and smooth along y-direction) change of its
chemical potential [16]. Other solutions of Eq. (16) are
found numerically:
β1= 1.41, β2= 6.49, β3= 6.75,... (17)
With increasing nthe eigenmodes of integro-differential
equation (16) oscillate faster, but in generaldo not follow
the oscillation theorem familiar from quantum mechan-
ics. In particular, the solutions with n= 0 andn= 3 are