The high-order cyclotron modes in Fermi liquid of Q2D layered conductors
The propagation of electromagnetic waves in layered conductors at the presence of an external magnetic field is investigated theoretically. At certain orientations of a magnetic field concerning the layers of the conductor the collisionless absorption is absent and weakly damping collective modes...
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Інститут фізики конденсованих систем НАН України
2005
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| Цитувати: | The high-order cyclotron modes in Fermi liquid of Q2D layered conductors / O.V. Kirichenko, V.G. Peschansky, D.I. Stepanenko // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 835–844. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1210582017-06-14T03:04:54Z The high-order cyclotron modes in Fermi liquid of Q2D layered conductors Kirichenko, O.V. Peschansky, V.G. Stepanenko, D.I. The propagation of electromagnetic waves in layered conductors at the presence of an external magnetic field is investigated theoretically. At certain orientations of a magnetic field concerning the layers of the conductor the collisionless absorption is absent and weakly damping collective modes can propagate even under the strong spatial dispersion. In a short-wave limit the existence of electromagnetic waves with frequencies near the cyclotron resonances is possible at an arbitrary orientation of the wave vector with respect to an external magnetic field. We have obtained the spectrum of waves with the frequencies near the cyclotron resonances of high order with regard to the Fermi-liquid interaction of the electrons. Теоретично досліджено поширення електромагнітних хвиль у шаруватих провідниках у присутності зовнішнього магнітного поля. При деяких орієнтаціях магнітного поля щодо шарів провідника беззіткнене поглинання відсутнє і поширення слабозгасаючих колективних мод є можливим навіть в умовах сильної просторової дисперсії. У короткохвильовій межі можливе існування електромагнітних хвиль з частотами біля резонансів при довільній орієнтації хвильового вектора щодо зовнішнього магнітного поля. Знайдено спектр хвиль з частотами біля кратних циклотронних резонансів з врахуванням Фермі-рідинної взаємодії електронів провідності. 2005 Article The high-order cyclotron modes in Fermi liquid of Q2D layered conductors / O.V. Kirichenko, V.G. Peschansky, D.I. Stepanenko // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 835–844. — Бібліогр.: 14 назв. — англ. 1607-324X PACS: 72.15.Nj, 72.30.+q DOI:10.5488/CMP.8.4.835 http://dspace.nbuv.gov.ua/handle/123456789/121058 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
The propagation of electromagnetic waves in layered conductors at the
presence of an external magnetic field is investigated theoretically. At certain
orientations of a magnetic field concerning the layers of the conductor
the collisionless absorption is absent and weakly damping collective modes
can propagate even under the strong spatial dispersion. In a short-wave
limit the existence of electromagnetic waves with frequencies near the cyclotron
resonances is possible at an arbitrary orientation of the wave vector
with respect to an external magnetic field. We have obtained the spectrum
of waves with the frequencies near the cyclotron resonances of high order
with regard to the Fermi-liquid interaction of the electrons. |
| format |
Article |
| author |
Kirichenko, O.V. Peschansky, V.G. Stepanenko, D.I. |
| spellingShingle |
Kirichenko, O.V. Peschansky, V.G. Stepanenko, D.I. The high-order cyclotron modes in Fermi liquid of Q2D layered conductors Condensed Matter Physics |
| author_facet |
Kirichenko, O.V. Peschansky, V.G. Stepanenko, D.I. |
| author_sort |
Kirichenko, O.V. |
| title |
The high-order cyclotron modes in Fermi liquid of Q2D layered conductors |
| title_short |
The high-order cyclotron modes in Fermi liquid of Q2D layered conductors |
| title_full |
The high-order cyclotron modes in Fermi liquid of Q2D layered conductors |
| title_fullStr |
The high-order cyclotron modes in Fermi liquid of Q2D layered conductors |
| title_full_unstemmed |
The high-order cyclotron modes in Fermi liquid of Q2D layered conductors |
| title_sort |
high-order cyclotron modes in fermi liquid of q2d layered conductors |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121058 |
| citation_txt |
The high-order cyclotron modes in Fermi liquid of Q2D layered conductors / O.V. Kirichenko, V.G. Peschansky, D.I. Stepanenko // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 835–844. — Бібліогр.: 14 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT kirichenkoov thehighordercyclotronmodesinfermiliquidofq2dlayeredconductors AT peschanskyvg thehighordercyclotronmodesinfermiliquidofq2dlayeredconductors AT stepanenkodi thehighordercyclotronmodesinfermiliquidofq2dlayeredconductors AT kirichenkoov highordercyclotronmodesinfermiliquidofq2dlayeredconductors AT peschanskyvg highordercyclotronmodesinfermiliquidofq2dlayeredconductors AT stepanenkodi highordercyclotronmodesinfermiliquidofq2dlayeredconductors |
| first_indexed |
2025-07-08T19:07:11Z |
| last_indexed |
2025-07-08T19:07:11Z |
| _version_ |
1837106867996721152 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 4(44), pp. 835–844
The high-order cyclotron modes in
Fermi liquid of Q2D layered conductors
O.V.Kirichenko, V.G.Peschansky, D.I.Stepanenko
B.I.Verkin Institute for Low Temperature Physics and Engineering,
National Academy of Sciences of the Ukraine,
47 Lenin ave, Kharkiv, 61103, Ukraine
Received July 18, 2005
The propagation of electromagnetic waves in layered conductors at the
presence of an external magnetic field is investigated theoretically. At cer-
tain orientations of a magnetic field concerning the layers of the conductor
the collisionless absorption is absent and weakly damping collective modes
can propagate even under the strong spatial dispersion. In a short-wave
limit the existence of electromagnetic waves with frequencies near the cy-
clotron resonances is possible at an arbitrary orientation of the wave vector
with respect to an external magnetic field. We have obtained the spectrum
of waves with the frequencies near the cyclotron resonances of high order
with regard to the Fermi-liquid interaction of the electrons.
Key words: layered conductors, Fermi liquid, cyclotron waves
PACS: 72.15.Nj, 72.30.+q
Experimental studies of galvanomagnetic phenomena in a large family of tetrathi-
afulvalene based ion-radical salts of the form (BEDT − TTF)2X (X stands for a set
of various anions) indicate that these layered compounds possess the metal type elec-
trical conductivity. This permits to describe electron processes in such conductors
based on the well-developed concept of quasiparticles, analogous to conduction elec-
trons in metals. Observation of Shubnikov-de Haas magnetoresistance oscillations
prove that the approximation of free path time τ in these layered conductors can be
sufficient for charge carriers to manifest their dynamic properties. Their cyclotron
frequency Ω may significantly exceed τ−1(see, for example, [1–3] and citations there-
in).
The sharp anisotropy of the conductivity of layered conductors is apparently
connected with the anisotropy of the velocities on the Fermi surface and restricts
the choice of suitable models for the Fermi surface. The model of the Fermi sur-
face in the form of a weakly corrugated cylinder is in a good agreement with the
experimental investigations of many layered conductors. The results of calculations
based on this model are in complete agreement with the experimentally observed
c© O.V.Kirichenko, V.G.Peschansky, D.I.Stepanenko 835
O.V.Kirichenko, V.G.Peschansky, D.I.Stepanenko
Shubnikov-de Haas oscillations in the tetrathiafulvalene salts (BEDT − TTF)2JBr2
and (BEDT − TTF)2J3.
The charge carrier energy ε(p) in layered conductors weakly depends on the
momentum projection pz = pn on the normal n to the layers. The electron energy
spectrum can be represented as the Fourier series with respect to the variable pz/p0
with rapidly decreasing coefficients. Confining ourselves to the zeroth and first har-
monics and neglecting the anisotropy in the layer-plane we can write the dispersion
law as follows:
ε (p) =
p2
x + p2
y
2m
− ε0 cos
pz
p0
. (1)
Here ε0 = ηvFp0, v2
F ≡ 2εF/m , m is the effective mass in the layer-plane, ~/p0 is the
distance between the layers, ~ is the Planck constant, εF is the Fermi energy. The
parameter η characterizes the anisotropy of the charge carrier energy spectrum. In
the absence of a magnetic field the ratio of the conductivity across the layers to the
in-plane conductivity is about η2.
The experimental studies of the propagation of electromagnetic waves in the
layered conductors has also attracted the attention of the researches. Several pa-
pers appeared, in which the results of the observation of cyclotron resonance in
α − (BEDT − TTF)2KHg(SCN) and in (BEDT − TTF)2ReO4(H2O)) have been re-
ported [4–6].
The kinetic phenomena in alternating fields contain rich information about the
electron energy spectrum and relaxation properties of charge carriers. In the presence
of a high external magnetic field H0 the other types of weakly attenuating electro-
magnetic waves (helicoidal, magnetohydrodynamic and cyclotron waves) occur. The
spectrum of their high frequency branches depends essentially on the constants of
Fermi-liquid interaction. The results of the studies of the wave processes in con-
ventional metals with regard to the Fermi liquid interaction of charge carriers, are
reported in the monograph [7]. Herein below we consider the propagation of elec-
tromagnetic waves in a Fermi liquid of the layered conductors placed in an external
magnetic field. The specific features of the quasi-two dimensional electron energy
spectrum of a layered conductor manifest themselves in its kinetic properties. The
velocity vH of the charge carriers drift along the magnetic field direction gets zero
for some values of the angle ϑ between H0 = (H0 sin ϑ, 0, H0 cos ϑ) and the normal
n to the layers. This is the condition when there is no collisionless absorption of the
electromagnetic wave energy by electrons and weak damping waves can propagate
even under the strong spatial dispersion [8]. In a short-wave limit the existence of
electromagnetic waves with frequencies near the cyclotron resonance is possible at
an arbitrary orientation of the wave vector k with respect to H0. We analyze the
spectrum of cyclotron waves under the strong spatial dispersion with regard to the
Fermi-liquid interaction.
Kinetic properties of the system of fermions should be described by means of
the kinetic equation for the density matrix ρ̂. In the quasiclassical case when ~Ω .
T � ηεF the quantization of the charge carriers energy in the magnetic field does
not essentially affect the magnetization M (T is the temperature). Under these
836
The high-order cyclotron modes
conditions the density matrix %̂ can be presented as an operator in the space of spin
variables and as a function depending on coordinates and momentum.
The interaction between electrons results in the correction to their energy
δε̂(p, r, t) = Trσ
′
∫
d3p
′
(2π~)3
Λ(p, σ̂,p
′
, σ̂
′
)δρ̂(p, r, σ̂
′
, t) (2)
which can be described with the aid of the correlation function [9,10]
Λ(p, σ̂,p
′
, σ̂
′
) = L(p,p
′
) + S(p,p
′
)σ̂σ̂
′
, (3)
where δρ̂ is the nonequilibrium correction to the density matrix, σ are Pauli ma-
trices. The second term on the right-hand part of (3) corresponds to the exchange
interaction between electrons.
The alternating electric E and magnetic H∼ fields produced by the electric cur-
rent
j (r, t) = eTrσ
∫
d3p
(2π~)3 ρ̂(p, r, σ̂, t)
∂ε̂
∂p
+ cµ0rot Trσ
∫
d3p
(2π~)3
σ̂ρ̂(p, r, σ̂, t) (4)
should be determined from the Maxwell equations
rot H∼ =
4π
c
j +
1
c
∂E
∂t
, rot E = −
1
c
∂H∼
∂t
, (5)
where e is the electron charge, c is the velocity of light.
Instead of the density matrix ρ̂, it will be more convenient to use the distribution
function f (r,p, t) = Trσρ̂ and the spin density g(r,p, t) = Trσ(σ̂ρ̂). The function
g(r,p, t) and the second term on the right-hand part of (4) describe paramagnetic
spin waves predicted by Silin [11] and observed in isotropic metals by Dunifer and
Schultz [12]. The properties of spin waves are determined by the magnetic suscepti-
bility tensor χik(ω, k). For the frequencies, that do not coincide with the frequency
of eigen-oscillations of the spin density, the components χik(ω, k) are of the order
of the static paramagnetic susceptibility χ0 ' µ2
0ν(εF) ∼ 10−6 (ν(εF) is the density
of states at the Fermi level). For this reason we neglect the spin magnetism of the
media, when considering the electromagnetic modes arising from the oscillations of
the electron distribution function.
Let us present the distribution function f in the form f (r,p, t) = f0 (ε) −
eΨ(r,p, t)E · ∂f0/∂ε, where f0(ε) is the equilibrium Fermi function. The nonequi-
librium correction Ψ(r,p, t) satisfies the following kinetic equation
∂Ψ
∂t
+
(
v
∂
∂r
+
e
c
(v × H0)
∂
∂p
)
Φ = v + Icoll(Φ). (6)
Here
Φ ≡ Ψ + 〈LΨ〉 , 〈LΨ〉 ≡
∫
2d3p′
(2π~)3
(
−
∂f0 (ε′)
∂ε′
)
L (p,p′)Ψ (r,p′, t) ,
Icoll is the collision integral.
837
O.V.Kirichenko, V.G.Peschansky, D.I.Stepanenko
The wave process can be regarded as harmonic and the coordinate and time
dependencies of the fields E,H∼ and of the function Ψ can be represented as
exp(ikr − iωt). Regardless of the weak current of spin magnetization in (4), the
current density in a conductor can be written as follows:
ji = e2
∫
2d3p
(2π~)3
(
−
∂f0
∂ε
)
viΦjEj = σij(ω,k)Ej , (7)
where σij(ω,k) is the electrical conductivity tensor, v = ∂ε(p)/∂p. Substituting
E,H∼ in the form of a harmonic wave into the Maxwell equations (5) we obtain the
dispersion equation
det
[
k2δij − kikj −
ω2
c2
εij(ω,k)
]
= 0, (8)
which determines the spectrum ω(k) for the electromagnetic field oscillations. Here
εij(ω,k) = δij + 4πiω−1 σij(ω,k) is the dielectric tensor. We shall use the reference
frame where the wave vector k = (k sinφ, 0, k cosφ) is oriented in the xz plane. For
frequencies much less than the plasma frequency ωp the first term in the expression
for εij can be neglected.
In the lowest order approximation about the small parameter η the function
L(p,p′) does not depend on pH = (pH0)/H0 and can be presented as
L (p,p′) =
∞
∑
n=−∞
Ln (ε) ein(ϕ−ϕ′). (9)
We have chosen the integrals of motion of an electron in a magnetic field ε, pH and
the phase ϕ = Ωt1 at its orbit in the magnetic field as variables in the p-space. Here
t1 = (c/eH0)
∫
dl/v⊥ is the time of motion along the trajectory ε(p) = εF, pH =
const, dl =
(
dp2
x + dp2
y
)1/2
, v⊥ is the velocity component orthogonal to the external
magnetic field [13]. Due to the symmetry of the function Λ(p, σ̂,p
′
, σ̂
′
) with respect
to its arguments, the coefficients in (9) satisfy the condition Ln = L−n. Allowing for
next-order terms of the expansion for the correlation function about η does not lead
to the noticeable correction of the results.
Expanding the functions Φ and Ψ into a Fourier series with respect to ϕ and
equalizing coefficients at exp (inϕ) in the equality Φ = Ψ + 〈LΨ〉, we obtain
Ψ = Φ −
∞
∑
n=−∞
λnΦ̄
(n)
einϕ. (10)
Here
Φ̄
(n)
=
1
(2π)2
∫ 2π
0
dϕe−inϕ
∫ π
−π
dβΦ (εF, β) ≡
〈
e−inϕΦ
〉
β,ϕ
(11)
is the Fourier component for the function 〈Φ〉β, β = pH/p0 cos ϑ, 〈· · · 〉β = 1/2π ×
∫ π
−π
dβ · · ·, λn = L∼
n /(1 + L∼
n ), L∼
n = ν(εF)Ln. Substituting (10) into the equation
838
The high-order cyclotron modes
(6) and using the equality ∂/∂t1 = (e/c)(v ×H0)∂/∂p we find the kinetic equation
for the perturbation of the renormalized distribution function for electrons with the
quasi-two-dimensional dispersion law [14]:
∂Φ
∂ϕ
−
i
Ω
(ω − kv)Φ =
v
Ω
−
iω
Ω
∞
∑
n=−∞
λnΦ̄n (ε) einϕ +
1
Ω
Icoll (Φ) . (12)
The collision integral in the τ−approximation can be written as Icoll (Φ) = −τ−1Φ
(τ is the relaxation time for the momentum ). Below we consider the wave processes
in the range of frequencies satisfying the condition
ωp � ω � τ−1, (13)
where the asymptotic expression for the spectrum of collective modes does not essen-
tially depend on the concrete form of the collision integral. Here ωp = (4πe2n0/m)1/2
is plasma frequency, n0 is charge carrier density.
In the first order in η, the components of electron velocity are easily found by
means of standard method of nonlinear mechanics
vx(t1) = v(0)
x (t1) + v(1)
x (t1), v(0)
x (t1) = v⊥ cos Ω(β)t1, v(0)
y (t1) = −v⊥ sin Ω(β)t1,
v(1)
x (t1) = ηvF tan ϑ J0(α) sin β − ηvF tan ϑ
∞
∑
n=2
Jn (α) sin (β − nπ/2)
n2 − 1
cos nΩ(β)t1,
vz(t1) = ηvF sin (β − α cos Ω(β)t1) .
(14)
Here
Ω(β) = Ω0
(
1 +
1
2
η tan ϑ J1(α) cos β
)
is cyclotron frequency of quasi-particles with the energy spectrum (1) in the magnetic
field H0 = (H0 sin ϑ,0, H0 cos ϑ), Ω0 = (|e|H0/mc) cos ϑ, α = (mvF/p0) tan ϑ, Jn(α)
is the Bessel function, the initial phase is chosen so that vy(0) = 0,
v⊥ = vF
(
1 −
v
(1)
x (0)
vF
+
ηp0
mvF
cos(β − α)
)
is the amplitude of the first harmonic of vx(t).
After straightforward calculations the equation (12) can be transformed into the
form
Φ =
∫ ϕ
−∞
dϕ′ exp
(
i
Ω
∫ ϕ
ϕ′
dϕ′′ (ω̃ − kv(ϕ′′, pH))
)
(
v
Ω
− i
ω
Ω
∞
∑
p=−∞
λpΦ̄
(p)
eipϕ′
)
,
(15)
where ω̃ = ω + iτ−1, in the collisionless limit (τ → ∞) we may set ω̃ = ω + i0.
After multiplying the equation (15) by e−inϕ and then integrating with respect to
β and ϕ, we obtain the infinitesimal set of linear equations for the Fourier coefficients
839
O.V.Kirichenko, V.G.Peschansky, D.I.Stepanenko
Φ̄
(p)
of the function 〈Φ〉β ≡ 1/(2π)
∫ π
−π
dβΦ(εF, β, ϕ)
∑
p
=
∞
∑
p=−∞
(
δnp − λp
ω
Ω
〈fn,p (β)〉β
)
Φ̄
(p)
=
1
Ω
〈
1
2π
∫ 2π
0
∫ 2π
0
dϕdϕ1v (β, ϕ − ϕ1)e
−inϕ+iR(ϕ,ϕ1)
1 − e2πiR(2π,2π)
〉
β
, (16)
fn,p (β) =
1
2πi
∫ 2π
0
∫ 2π
0
dϕdϕ1e
i(p−n)ϕ−ipϕ1+iR(ϕ,ϕ1)
1 − e2πiR(2π,2π)
. (17)
Here R (ϕ, ϕ1) ≡ Ω−1
∫ ϕ
ϕ−ϕ1
dϕ′ (ω̃ − kv(β, ϕ′)), δnp, is the Kroneker symbol. The
dependence of the cyclotron frequency on pH should be taken into account in the
expression kxvx/Ω in the exponent when ηkvF & Ω.The coefficients of the Fourier
series for the smooth function L∼(p,p′) significantly decrease with their number
increasing, so we shall restrict ourselves to taking into account a finite number of
the terms in the right part of (16).
The parameter η2 is usually of the order of 10−3−10−5. If the inequality (kr0)
−3×
(ηωpvF/Ωc)2 � 1, (r0 = vF/Ω ) is satisfied, the components of electrical conductivity
tensor σzα, α = x, y, proportional to η, and the component σzz ∼ η2 in the dispersion
equation can be neglected. Substituting the Fourier series expansion of Φi into (7)
we obtain
σxx =
1
2
e2vFν(εF)
(
Φ̄(1)
x + Φ̄(−1)
x
)
, σyy = −
1
2i
e2vFν(εF)
(
Φ̄(1)
y − Φ̄(−1)
y
)
,
σxy =
1
2
e2vFν(εF)
(
Φ̄(1)
y + Φ̄(−1)
y
)
=
1
2i
e2vFν(εF)
(
Φ̄(1)
x − Φ̄(−1)
x
)
. (18)
Here ν(εF) = mp0/π~
3 is the density of states of electrons at the Fermi level. Equati-
ons (8), (16), (18) describe the eigenmodes of the electromagnetic field in the Fermi
liquid.
Under the strong spatial dispersion kr0 � 1, ηkr0 ∼ 1 the integrals about ϕ, ϕ1
in the formula (16) can be calculated by means of the stationary phase method,
and if ω � kr0, the stationary points are determined from the equations vx(ϕ) =
0, vx(ϕ − ϕ1) = 0. It is easy to see that the maximum component of the tensor
σij is σyy which is proportional to (kr0)
−1. The expansion in power series of the
components σxα, α = x, y, z begins with the terms of higher order in (kr0)
−1. In
the basic approximation in small parameters (kr0)
−1 and η we obtain from (8) the
following dispersion equation
k2c2
ω2
=
4πi
ω
σyy . (19)
840
The high-order cyclotron modes
The asymptotic expression for fn,p(β) for n, p � kr0 takes the form
fn,p(β) =
1
kxr0
cot
π
Ω
(ω̃ − 〈kv〉ϕ) cos
π
2
(n − p)
+
sin
(
R
(
π
2
, π
)
− π
Ω
〈kv〉ϕ + π
2
(n + p)
)
sin π
Ω
(ω̃ − 〈kv〉ϕ)
. (20)
Outside the domain of values of ω,k satisfying the condition
|ω − nΩ0| > max |〈kv〉ϕ|, (21)
the integrands in the formula (16) have a pole and after integration over pH the
dispersion equation gets an imaginary part responsible for the strong absorption of
the wave.
After averaging the component of electron velocity over the period of motion
along the cyclotron orbit, we obtain
〈kv〉ϕ = kvB = ηvFJ0(α)(kx tan ϑ + kz) sin β. (22)
For the directions of H0 when α is equal to one of the roots αi = (mvF/p0) tan ϑi of
the equation J0(α) = 0, the average 〈kv〉ϕ ∼ η2 is of the second order in η and there
is no collisionless absorption. Having determined Φ̄
(±1)
using (16), we can find the
conductivity tensor components. Within the framework of the model which allows
for the zeroth and first Fourier harmonics of the Landau function
L(p,p′) = L0 + 2L1 cos(ϕ − ϕ′), (23)
the dispersion equation becomes
1 − λ0
ω
Ω0
1
ξ0
(
cot
πω̃
Ω0
+
〈sin R (ϑi)〉β
sin πω̃
Ω0
)
= 2
ω
Ω0
1
ξ0
(
(
ωpvF
Ω0cξ0
)2
− λ1
)
×
− cot
πω̃
Ω0
+
〈sin R (ϑi)〉β
sin πω̃
Ω0
+
λ0
ξ0
ω
Ω0
(
cos2 πω̃
Ω0
− (〈sin R (ϑi)〉
2
β+〈cos R (ϑi)〉
2
β)
)
sin2 πω̃
Ω0
.
(24)
Here
R(ϑi) =
π/2
∫
−π/2
kv(ϕ)
Ω(β)
dϕ = 2
kxv⊥
Ω(β)
− πη
kzvF
Ω0
H0(αi) cos β
+ η
kxvF
2Ω0
tan ϑi cos β
∞
∑
n=1
J2n+1 (αi)
n(n + 1)(2n + 1)
, (25)
841
O.V.Kirichenko, V.G.Peschansky, D.I.Stepanenko
H0(α) = (2/π)
π/2
∫
0
dϕ sin(α cos ϕ) is the Struve function, ξ0 = kxr0.
The equation (24) has the solution in the region of the resonance
ω = nΩ0 + 4ω, 0 < |4ω| < Ω0, n = 1, 2, 3 . . . , (26)
where nΩ0 is the frequency corresponding to the cyclotron resonance of n order. In
quasi-isotropic metals the similar types of waves take place only when k is perpen-
dicular to the direction of the magnetic field.
In the case when (kr0)
−3 (ωpvF/Ωc)2 � 1, which is usually realized in the conduc-
tors whose charge carriers density is about one electron per atom, the left-hand part
of the equation (24) may be neglected. If the inequality 1− | 〈sin R(ϑi)〉β |� ξ−1
0 is
satisfied, the spectrum for the cyclotron waves can be found in the analytical form
4ω = 4ω0 + 4ω1,
4ω0 = Ω0
(
1
2
−
(−1)n
π
arcsin 〈sin R (ϑi)〉β
)
,
4ω1 =
Ω0λ0
πξ0
(
n +
4ω0
Ω0
)
〈cos R(ϑi)〉
2
β)
1 − 〈sin R(ϑi)〉
2
β
. (27)
In the other limited case (kr0)
−3 (ωpvF/Ωc)2 � 1, the solution of (24) can be
represented in the form (26) with:
4ω =
nΩ0
πξ0
(
a±
√
a2 + 2 (γ − λ1) λ0
(
1 − 〈sin R(ϑi)〉
2
β − 〈cos R(ϑi)〉
2
β
)
)
, (28)
a =
1
2
[
λ0− 2 (γ − λ1) + (−1)n (λ0 + 2 (γ − λ1)) 〈sin R(ϑi)〉β
]
, γ =
(
ωpvF
Ω0cξ0
)2
.
If ηkvF � Ω0, the spatial dispersion along the z-direction is not essential, R(ϑi) =
2kxr0 and formulas (27),(28) transform into the expressions obtained in [14] for the
case when the wave vector k = (k, 0, 0) is orthogonal to an external magnetic field.
842
The high-order cyclotron modes
References
1. Wosnitza J., Springer Tracts in Mod. Phys., 1996, 136, 1.
2. Kartsovnik M.V., Laukhin V.N., J. Phys. I., 1996, 6, 1753.
3. Singleton J., Rep. Prog. in Phys., 2000, 63, 1111.
4. DemishevS.V., SemenoA.V., Sluchanko N.E., Samarin N.A., Pis’ma Zh. Eksp. Teor.
Fiz., 1995, 61, 299 (in Russian); JETP Lett. 1995, 61, 313.
5. Demishev S.V., Semeno A.V., Sluchanko N.E., et al., Phys. Rev. B, 1996, 53, 12794.
6. Demishev S.V., Semeno A.V., Sluchanko N.E., et al., Zh. Eksp. Teor. Fiz., 1997, 111,
979 (in Russian); JETP, 1997, 84, 540.
7. Platzman P.M., Wolf P.A. Waves and Interactions in Solid State Plasma. Academic,
New York, 1973.
8. Kirichenko O.V., Peschansky V.G., Stepanenko D.I., Zh. Eksp. Teor. Fiz., 2004, 126,
1435 (in Russian); JETP, 2004, 99, 1253.
9. Landau L.D., Zh. Eksp. Teor. Fiz., 1956, 30, 1058 (in Russian); JETP, 1956, 3, 920.
10. Silin V.P., Zh. Eksp. Teor. Fiz., 1957, 33, 495 (in Russian); JETP, 1958, 6, 387.
11. Silin V.P., Zh. Eksp. Teor. Fiz., 1958, 35, 1243 (in Russian); JETP, 1959, 8, 870.
12. Schultz S., Dunifer G., Phys. Rev. Lett., 1967, 18, 283.
13. Abrikosov A.A. Fundamentals of the Theory of Metals. North-Holland, Amsterdam,
1988.
14. Kirichenko O.V., Peschansky V.G., Stepanenko D.I., Phys. Rev. B, 2005, 71, 045304.
843
O.V.Kirichenko, V.G.Peschansky, D.I.Stepanenko
Високочастотні циклотронні моди у Фермі рідині
Q2D шаруватих провідників
О.В.Кириченко, В.Г.Піщанський, Д.І.Степаненко
Фізико-Технічний Інститут Низьких температур ім. Б.І.Веркіна НАН
України, Україна, 61164, м. Харків, пр. Леніна, 47
Отримано 18 липня 2005 р.
Теоретично досліджено поширення електромагнітних хвиль у шару-
ватих провідниках у присутності зовнішнього магнітного поля. При
деяких орієнтаціях магнітного поля щодо шарів провідника беззітк-
нене поглинання відсутнє і поширення слабозгасаючих колективних
мод є можливим навіть в умовах сильної просторової дисперсії. У
короткохвильовій межі можливе існування електромагнітних хвиль
з частотами біля резонансів при довільній орієнтації хвильового
вектора щодо зовнішнього магнітного поля. Знайдено спектр хвиль
з частотами біля кратних циклотронних резонансів з врахуванням
Фермі-рідинної взаємодії електронів провідності.
Ключові слова: шаруваті провідники, Фермі-рідина, циклотронні
хвилі
PACS: 72.15.Nj, 72.30.+q
844
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