High-frequency longitudinal oscillations of the quasi-two-dimensional electron liquid
The specific character of longitudinal collective electromagnetic oscillations in a layered conductor with the quasi-two-dimensional electron energy spectrum has been analyzed.
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | High-frequency longitudinal oscillations of the quasi-two-dimensional electron liquid / V.M. Gokhfeld, O.V. Kirichenko, V.G. Peschansky // Condensed Matter Physics. — 2012. — Т. 15, № 1. — С. 13704: 1-7. — Бібліогр.: 21 назв. — англ. |
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irk-123456789-1201692017-06-12T03:03:38Z High-frequency longitudinal oscillations of the quasi-two-dimensional electron liquid Gokhfeld, V.M. Kirichenko, O.V. Peschansky, V.G. The specific character of longitudinal collective electromagnetic oscillations in a layered conductor with the quasi-two-dimensional electron energy spectrum has been analyzed. Проаналiзовано специфiку поздовжнiх колективних електромагнiтних коливань у шаруватому провiдни-ку з квазi-двовимiрним електронним енергетичним спектром. 2012 Article High-frequency longitudinal oscillations of the quasi-two-dimensional electron liquid / V.M. Gokhfeld, O.V. Kirichenko, V.G. Peschansky // Condensed Matter Physics. — 2012. — Т. 15, № 1. — С. 13704: 1-7. — Бібліогр.: 21 назв. — англ. 1607-324X PACS: 71.20.-r DOI:10.5488/CMP.15.13704 arXiv:1204.5997 http://dspace.nbuv.gov.ua/handle/123456789/120169 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The specific character of longitudinal collective electromagnetic oscillations in a layered conductor with the quasi-two-dimensional electron energy spectrum has been analyzed. |
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Article |
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Gokhfeld, V.M. Kirichenko, O.V. Peschansky, V.G. |
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Gokhfeld, V.M. Kirichenko, O.V. Peschansky, V.G. High-frequency longitudinal oscillations of the quasi-two-dimensional electron liquid Condensed Matter Physics |
| author_facet |
Gokhfeld, V.M. Kirichenko, O.V. Peschansky, V.G. |
| author_sort |
Gokhfeld, V.M. |
| title |
High-frequency longitudinal oscillations of the quasi-two-dimensional electron liquid |
| title_short |
High-frequency longitudinal oscillations of the quasi-two-dimensional electron liquid |
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High-frequency longitudinal oscillations of the quasi-two-dimensional electron liquid |
| title_fullStr |
High-frequency longitudinal oscillations of the quasi-two-dimensional electron liquid |
| title_full_unstemmed |
High-frequency longitudinal oscillations of the quasi-two-dimensional electron liquid |
| title_sort |
high-frequency longitudinal oscillations of the quasi-two-dimensional electron liquid |
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Інститут фізики конденсованих систем НАН України |
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2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/120169 |
| citation_txt |
High-frequency longitudinal oscillations of the quasi-two-dimensional electron liquid / V.M. Gokhfeld, O.V. Kirichenko, V.G. Peschansky // Condensed Matter Physics. — 2012. — Т. 15, № 1. — С. 13704: 1-7. — Бібліогр.: 21 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT gokhfeldvm highfrequencylongitudinaloscillationsofthequasitwodimensionalelectronliquid AT kirichenkoov highfrequencylongitudinaloscillationsofthequasitwodimensionalelectronliquid AT peschanskyvg highfrequencylongitudinaloscillationsofthequasitwodimensionalelectronliquid |
| first_indexed |
2025-07-08T17:21:38Z |
| last_indexed |
2025-07-08T17:21:38Z |
| _version_ |
1837100224251691008 |
| fulltext |
Condensed Matter Physics, 2012, Vol. 15, No 1, 13704: 1–7
DOI: 10.5488/CMP.15.13704
http://www.icmp.lviv.ua/journal
High-frequency longitudinal oscillations
of quasi-two-dimensional electron liquid
V.M. Gokhfeld1, O.V. Kirichenko2, V.G. Peschansky2
1 Donetsk Institute for Physics and Engineering named after O.O. Galkin of the National Academy
of Sciences of Ukraine, Donetsk, Ukraine
2 B.I. Verkin Institute for Low Temperature Physics and Engineering of the National Academy
of Sciences of Ukraine, Kharkiv, Ukraine
Received August 29, 2011, in final form January 30, 2012
The specific character of longitudinal collective electromagnetic oscillations in a layered conductor with the
quasi-two-dimensional electron energy spectrum has been analyzed.
Key words: layered conductors, fermi-liquid, longitudinal oscillations
PACS: 71.20.-r
1. Introduction
Numerous low-dimensional conductors differ by their layered structure and highly anisotropic elec-
trical conductivity. Their in-plane conductivity is much higher than that in perpendicular direction, i.e.
along the OZ -axis. Such an anisotropy is typical of cuprates, e.g. YBaCuO in a nonsuperconducting phase,
transition metal dichalcogenides (NbSe2, TaS2), graphite and its intercalates in particular, as well as a
broad family of tetrathiafulvalene salts of (BEDT-TTF)2I3 type [1–5]. A common feature of many layered
conductors is their weak electron-energy dependence on the momentum projection pz . This is due to
the fact that their interlayer distance a is much higher than the in-plane crystal lattice parameter. Then,
the wave functions of appropriate electrons weakly overlap, and the energy of single-particle charged
excitations may be represented by rapidly converging series
ε(p) =
∞∑
n=0
εn (p⊥)cos
anpz
ħ
, (1.1)
where the functions εn decrease with increasing n and ε1(p⊥) is much less than ε0(p⊥). Here ħ is the
Planck constant, p⊥ = [pn] is the momentum projection onto the layer plane.
In conductors with high charge carriers density allowance for the electron-electron interaction is
of great importance. The system of quasiparticles carrying a charge should be regarded as the Fermi
liquid. According to the Landau-Silin theory [6, 7], the interaction between quasiparticles may be taken
into account in the form of self-consistent field. In this case, the quasiparticle energy depends on the
distribution function n(p,r, t) for the other quasiparticles. As a result, the energy of an electron acquires
the correction δε to the dispersion law (1.1)
δε=
∫
2d3p ′
(2πħ)3
L(p,p′)n1(p′,r, t), (1.2)
where n1 is the correction to the equilibrium Fermi distribution function n0(ε). In quasi-two-dimensio-
nal conductors, the Landau correlation function L(p,p′), as well as the quasiparticle energy in the “gas”
approximation (1.1), depends weakly on pz .
© V.M. Gokhfeld, O.V. Kirichenko, V.G. Peschansky, 2012 13704-1
http://dx.doi.org/10.5488/CMP.15.13704
http://www.icmp.lviv.ua/journal
V.M. Gokhfeld, O.V. Kirichenko, V.G. Peschansky
The specificity of quasi-two-dimensional electron energy spectrum provides the most favorable con-
ditions for a detailed study of the electron-electron Fermi liquid correlations. If the initial dispersion
law (1.1) is unknown, investigation of nonequilibrium processes in the electron system in a stationary
and uniform external field does not provide novel information about electron-electron Fermi liquid cor-
relations. The solution of the inverse problem of restoring the electron energy spectrum from experi-
mental data makes it possible to determine the dependence of the electron energy ε̃(p) = ε(p)+δε(p)
renormalized by the Fermi liquid interaction, at the Fermi surface only.
However, wave processes are very sensitive to the form and magnitude of the Fermi liquid correla-
tions. The gas approximation does not provide for propagation of the spin waves in nonmagnetic metals,
predicted by Silin [8], as well as for a series of Fermi liquid effects. Spin waves and transverse electro-
magnetic waves in the Fermi liquid have been studied in detail not only in metals (see, for example, the
monograph [9, 10]) but also in low-dimensional conductors [11–17].
Here we consider longitudinal oscillations in the quasi-two-dimensional Fermi liquid. Following Vla-
sov [18] and Landau [19] we should solve the set of equations including the Poisson equation
div E = 4πρ′, (1.3)
the continuity equation
∂ρ′
∂t
+div j = 0 (1.4)
and the constitutive equation relating the electric field density J to the electric field E ∼ exp(−iωt) with
allowance for the Fermi liquid correlations.
For the sake of brevity in calculations, we shall confine ourselves to the simplest case where the elec-
tron dispersion law contains only zero and first harmonics and the expression (1.1) takes the following
form:
ε(p) = ε0(p⊥)+ t⊥ cosθ. (1.5)
Here ε0(p⊥) is an arbitrary function, t⊥ is the constant value, θ = apzħ−1.
We consider the case where the electric field and the wave vector k are parallel to the normal to
the layers, i.e. to the hard direction for the electric current. We shall derive the dielectric function with
allowance for the Fermi liquid interaction (FLI), ascertain the dispersion law for longitudinal plasma
oscillations and find out the distribution of the high-frequency electric field in a half-infinite sample. We
shall also consider low-frequency collective excitations in a layered conductor which are possible in the
presence of two bands of the type (1.5) in its electron spectrum.
2. Dielectric function
In order to determine the relation between the current density
j =
∫
2d3p
(2πħ)3
e
∂ε̃
∂p
n(p,r, t) (2.1)
and the electric field of the wave, it is necessary to solve the kinetic equation
∂n
∂t
+
∂n
∂r
dr
dt
+
∂n
∂p
dp
dt
= Ŵcoll{n} . (2.2)
Naturally, the collision operator Ŵcoll{n} vanishes working on the equilibrium Fermi function
n0(ε̃)=
[
1+exp
(
ε̃−µ
T
)]−1
(2.3)
which depends on the energy ε̃ = ε(p)+δε(p,r, t) renormalized by the Fermi liquid correlations. In the
case of a weak perturbation of the system of charge carriers, i.e. when the value of the electric field of
the wave is small, the collision integral is the linear integral operator working on the function
n1(p,r, t) = n(p,r, t)−n0(ε̃)=−Φ(p,r, t)
∂n0(ε̃)
∂ε̃
. (2.4)
13704-2
High-frequency longitudinal oscillations
We shall confine ourselves to the τ-approximation for the collision integral, where Ŵcoll corresponds
to the operator of multiplication nonequilibrium part of the distribution function for conduction elec-
trons by the the frequency of their collisions 1/τ:
Ŵcoll{n} =
1
τ
Φ(p,r, t)
∂n0(ε̃)
∂ε̃
. (2.5)
In the kinetic equation, it should be taken into account that besides the electric field, conduction
electrons experience the force of the self-consistent field of interacting quasiparticles
dp
dt
= F = eE− ∂
∂r
δε(p,r, t). (2.6)
The current density depends only on the nonequilibrium part of the distribution function of charge
carriers
j(r, t) =−e
∫
m∗dε
∫
dθ
∫
dϕ
∂n0(ε)
∂ε
vΦ(ε,θ,ϕ,r, t)
1
2π3ħ2a
. (2.7)
Here m∗ is the cyclotron effective mass, ϕ is the variable of integration over charge carriers states with
constant values of energy and momentum projection on the normal to the layers, v = ∂ε(p)/∂p.
The kinetic equation for the case under consideration takes the form
kvzΦ−ωΨ+ ieEvz = i
Φ
τ
, (2.8)
where Ψ=Φ−δε, the relaxation time τ is supposed to be sufficiently large.
Within the chosen model for energy spectrum (1.5), electron velocity across the layers vz = v1 sin(θ),
with v1 =−at⊥ħ−1, is dependent on θ only andmuch less than the characteristic Fermi velocity vF for the
motion along the layers. Then, in order to find out the dielectric permeability and finally the spectrum of
longitudinal vibrations, the averaging over φ of the nonequilibrium distribution function of the charge
carriers along with the Lanadau-Silin correlation function is sufficient.
In accordance with the symmetry of the problem, the Landau correlation function L(p,p′), which
connects the effective and the real distribution of charge carriers, may be represented in the form
L(θ,θ′) = L0 +2L1 cos(θ−θ′). (2.9)
In this case
Φ(θ) =Ψ(θ)+
π∫
−π
dθ′
2π
L(θ,θ′)Ψ(θ′) (2.10)
and one can easily solve the equations (2.8)–(2.10). In particular, for Ψ0 = 〈Ψ〉/〈1〉, averaging over θ we
have
Ψ0 =
ieE
k
ω̃W
ω+W
[
iτ−1 + ω̃
(
L0 +λω2/k2v2
0
)] , (2.11)
where
W (k,ω) =
〈
kvz
kvz + ω̃
〉
〈1〉−1 = 1− ω̃
√
ω̃2 −k2v2
1
, (2.12)
λ= 2L1
(1+2L1)
, ω̃=ω+ i
τ
, 〈1〉 = 2
(2πħ)3
∫
∂n0
∂ε
d3p .
With regard to (2.7) and (2.8), the dielectric function ǫ= 1+4π〈Ψ〉/ikE takes the form
ǫ(k,ω) = 1+ κ2
k2
ω̃W
ω+W
[
iτ−1 + ω̃
(
L0+λω2
k2v2
0
)] , (2.13)
where
κ2 = 4πe2〈1〉 = 4m2e2
aħ2
13704-3
V.M. Gokhfeld, O.V. Kirichenko, V.G. Peschansky
is the square of the decrement of the static screening in the gas approximation. As is easily seen from
(2.13) and the dispersion equation ǫ(k,ω) = 0, the decrement with allowance for FLI equals κ
(p
1+L0
)−1
.
For the activation frequency of the plasma oscillations we have
ω2
p =
2me2v2
1
aħ2
(1+L1) = η
4πNe2
m
(1+L1) , (2.14)
where η = (v1/vF)2. Due to the smallness of η, the plasma frequency in the quasi-two-dimensional con-
ductor essentially reduces in comparison with the case of a conventional metal. Making use of the for-
mula (2.14) we obtain the dispersion law for longitudinal plasmons:
ω2(k) =
ω2
p
λ
k2
2κ2
−K +λK +
√(
K − k2
2κ2
)2
+λK
k2
κ2
, (2.15)
K (k) = 1+ (1+L0)k2
κ2
.
When the Landau correlation function has only a zero harmonic, the dispersion relation becomes notice-
ably simpler:
ω(k) = ev1
ħ
√
2m
a
κ2 + (1+L0)k2
κ
√
κ2 + (1/2+L0)k2
. (2.16)
3. Distribution of the electric field in a specimen
Having calculated the dielectric function for the infinite specimen, we can solve the boundary prob-
lem for the half-space z Ê 0. Landau [19] was the first to consider the penetration of the high-frequency
longitudinal electric field into the Maxwell plasma. Propagation of the longitudinal waves in a degener-
ated isotropic conductor was studied in [20] where a detailed analysis of the boundary conditions for an
electron distribution function was made. In the layered conductor, the angles of incidence of electrons
on the boundary do not exceed v1/vF ≪ 1, which allows us to suppose that charge carriers are reflected
specularly. The electric field outside the conductor (i.e. between the capacitor plates, one of which is the
specimen) is supposed to be given by E (z É 0) = (0,0,E0 exp(−iωt)). Then, the field distribution in the
specimen is described by the following expression
E (z)=
E0
iπ
∞∫
−∞
dk
kǫ(k,ω)
exp(ikz). (3.1)
The k = 0 pole residue gives the field value in the bulk of the conductor E (∞) which is much less than E0,
if ω≪ωp :
E (∞) = E0
ǫ(k,ω)
, ǫ(0,ω) = 1−
ω2
p
ω(ω̃+ iL1/τ)
. (3.2)
However, at ωτ≫ 1, the field attenuates in a nonmonotonous manner because the integral (3.1) contains
an oscillating component
E1(z) =−4E0F 2
π
∞∫
1
dxx
p
x2 −1exp(ik1xz)
(
x2 −1
) [
2+ (1+L0)F 2x2
]2 +
(
2+L0F 2x2
)2
(3.3)
originating from the branching point k1 = ω̃/v1. At z ≫ v1/|ω̃|, the ratio of the oscillating component to
E (∞) takes the form
E1(z)
E∞
= 1−F 2
p
2π(1+L0F 2/2)2
( v1
ω̃z
)3/2
exp(izω̃/v1). (3.4)
Here F = ω̃/ωp. In the layer of the order of the electron free path lz = v1τ, the field oscillates with the
period 2πv1/ω. In the layered conductor, both scale lengths reduce in comparison with the isotropic
metal, remaining, however, microscopic for non-high frequencies ω and τ−1.
13704-4
High-frequency longitudinal oscillations
For the sake of completeness, the resonance case ω=ωp should be considered. At such high frequen-
cies, the direct stimulation of the monochromatic field is hard to achieve, but the character of the spatial
distribution of the resonance harmonics may become apparent in the pulsed mode, as well as in the
experiments with electron beams.
Near the plasma frequency ωp, the radicals in the expression for the dielectric function may be ex-
panded into a series about small k2, and in the main approximation we have
ǫ(0,ω) = 1−
ω2
p
ω(ω̃+ iL1/τ)
, |ǫ(0,ω)|≪ 1. (3.5)
4. The spectrum of the short-wave plasmons
Consider the case of large values of k in collisionless limit. Strictly speaking, this is the case where the
quasiclassical formula for the conductivity
σ(k,ω) =−ie2
〈
v2
z
kv− ω̃
〉
(4.1)
is inappropriate because it is not invariant with respect to translation k → k + 2π/a. The translation-
invariant dielectric function ǫ(k) = 1+4πiσ(k)/ω can be easily constructed if we note that the magnitude
ħkv is the result of the expansion in small k of the energy difference in the quantum perturbation theory
formulas. In the case of the spectrum (1.5), this difference is
ε
(
p+ħk
2
)
−ε
(
p−ħk
2
)
=ħvz
(
2
a
)
sin
(
k
a
2
)
,
i.e., in (4.1) k should be replaced by Q = (2/a)sin(ka/2). As a result, we have
ǫ(k,ω) = 1+ κ2
Q2
1− ω
√
ω2 −Q2v2
1
. (4.2)
Usually, the application of the quasiclassical approximation for the calculation of the plasmon spectrum
is limited by the condition ħk ≪ pF,ħ/a (see, for example, [21]). In the case under consideration, this
restriction is of no necessity because the dispersion law (1.5) is determined everywhere in the Brillouin
zone. From (4.2) we obtain:
ω(k) =ωp0
√
2
b
b + sin2(ka/2)
√
2b + sin2(ka/2)
, (4.3)
where ωp0 = (ev1/ħ)
p
2m/a, b = (κa)2/4 = ane2/ħ2 i.e., the ratio of the lattice period to the Bohr radius.
As far as Bohr radius (at m ≃ m0) is of the order of 10−8 cm, the parameter b may be very large, especially
in artificial superlattices with great separation between conducting layers. In this case, the spectrum (4.3)
corresponds to a narrow band
ωmax
p
ω0
−1=
b +1
p
b2 +b/2
−1 ≃
{
3/4b (b ≫ 1)p
2/b (b ≪ 1)
(4.4)
and the static screening decrement κ is replaced by
κ=
2
a
arsinh
(p
b
)
. (4.5)
It is easily seen that the results obtained above are valid in an external magnetic field applied across
the layers, because it has no effect on the electron velocity along the normal to the layers for the energy
spectrum under discussion.
13704-5
V.M. Gokhfeld, O.V. Kirichenko, V.G. Peschansky
5. Two-band model
The Fermi surface is supposed to be singly connected. However, the electron structure of “synthetic
metals” is quite complicated. It is probable that there exist two charge carrier groups (for example, elec-
trons and holes) of the type (1.5) with the different values Vzmax : v (2)
1 = B v (1)
1 and B Ê 1. In the case of
two charge carrier groups in an isotropic metal FLI (even in its simplest form L(p,p′) = Lα,β, α,β,= 1,2),
there appears a longitudinal collective mode of the type of zero sound at sufficiently low frequencies,
τ−1 ≪ω≪ωp. Its phase velocity V = uv (2)
1 satisfies the characteristic equation
D(k) =
∑
ζαWα = LW1W2 = ζ1W (Bu)+ζ2W (u)+LW (Bu)W (u) = 0, (5.1)
where
ζα = mα
m1 +m2
, L = ζ1ζ2 (L11 +L22 −L12 −L21)
and W (u) = W (ω̃/kv (2)
1 ) is given by the formula (2.12). The equation (5.1) has a real root (u > 0) only at
sufficiently intensive FLI, namely at
L Ê Lmin(B) =−ζ2/W (B).
In the quasi-two-dimensional metal, the threshold value Lmin is much less than that in the isotropic metal
for the same B , especially near B = 1. In this case Lmin = 0, and the solution of the equation (5.1) is
u(L) = 1+L
p
1+2L
. (5.2)
The wave under consideration corresponds to antiphase partial oscillations of the electron densities
for both charge carrier groups. The elastic force generated by non-equilibrium charge carrier distribu-
tion allows one to detect the electron zero sound by means of the concomitant elastic wave. This effect
was observed in a series of ordinary metals (W, Al, Ga) [21]. It may be described by jointly solving the
electron kinetic equation and the elastic theory equation for the half-space with the given oscillation of
the boundary u0 exp(−iωt). As a result, for the displacement of ions in the k-representation, we obtain
U (k) =
2U0
ik
[
1+
ω2
s2k2(1+R)−ω2
]
, (5.3)
R(k) = ωΛ2〈1〉
ω̃ρs2
ζ1ζ2(1+L)
[
1−W1W2
1+L
D(k)
]
,
where ρ is the mass density, s is the velocity of the longitudinal sound. We assume that s ≪ v1. In this
case, U (k) has two different poles: the pole ks ≃ ω/s is connected to the ordinary sound; the other one,
k1s , is close to the root of the equation (5.1) and describes an extra elastic wave. It is easy to see that in
the degenerated case (v (1)
1 = v (2)
1 = v) we have:
U0s ≃U0
ωΛ2〈1〉
ω̃ρv2
1
ζ1ζ2
2L
1+L
exp
[
iω̃z
p
1+2L
v1(1+L)
]
. (5.4)
This expression is proportional to v−2
1 instead of v−2
F
in an ordinary metal.
6. Conclusions
The analysis given above shows that the electrodynamic characteristics of the layered conductor es-
sentially differ from those of the isotropic metal with the same charge carrier density. In particular, the
activation frequency and the velocity of the longitudinal waves, propagating along the normal to the lay-
ers, decrease essentially. Moreover, the case of quasi-two-dimensional charge carrier spectrum is most
favorable in observing the electron zero sound and in studying the electron phenomena in various mod-
ifications of carbon.
13704-6
High-frequency longitudinal oscillations
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Високочастотнi поздовжнi осциляцiї квазi-двовимiрної
електронної рiдини
В.М. Гохфельд1,2, О.В. Кириченко2, В.Г. Пiщанський2
1 Донецький фiзико-технiчний iнститут iм. А.А. Галкiна, Донецьк, Україна
2 Фiзико-технiчний iнститут низьких температур iм. Б.I. Вєркiна НАН України, Харкiв, Україна
Проаналiзовано специфiку поздовжнiх колективних електромагнiтних коливань у шаруватому провiдни-
ку з квазi-двовимiрним електронним енергетичним спектром.
Ключовi слова: шаруватi провiдники, фермi-рiдина, поздовжнi коливання
13704-7
http://dx.doi.org/10.1088/0034-4885/63/8/201
http://dx.doi.org/10.1021/cr0306891
http://dx.doi.org/10.1063/1.1884422
http://dx.doi.org/10.1063/1.593703
http://dx.doi.org/10.1063/1.1374724
http://dx.doi.org/10.1134/1.1625734
http://dx.doi.org/10.1134/1.1854813
http://dx.doi.org/10.1103/PhysRevB.71.045304
Introduction
Dielectric function
Distribution of the electric field in a specimen
The spectrum of the short-wave plasmons
Two-band model
Conclusions
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