Lagrangian approach in spin-oscillations problem
Lagrangian of electronic liquid in magneto-inhomogeneous micro-conductor has been constructed. A corresponding Euler-Lagrange equation has been solved. It was shown that the described system has eigenmodes of spin polarization and total electric current oscillations. The suggested approach permits t...
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Інститут фізики конденсованих систем НАН України
2014
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| Назва видання: | Condensed Matter Physics |
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| Цитувати: | Lagrangian approach in spin-oscillations problem / P.V. Pyshkin, A.I. Kopeliovich, A.V. Yanovsky // Condensed Matter Physics. — 2014. — Т. 17, № 4. — С. 43801: 1–4. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1534572019-06-15T01:25:49Z Lagrangian approach in spin-oscillations problem Pyshkin, P.V. Kopeliovich, A.I. Yanovsky, A.V. Lagrangian of electronic liquid in magneto-inhomogeneous micro-conductor has been constructed. A corresponding Euler-Lagrange equation has been solved. It was shown that the described system has eigenmodes of spin polarization and total electric current oscillations. The suggested approach permits to study the spin dynamics in an open-circuit which contains capacitance and/or inductivity. Побудовано Лагранжiан електронної рiдини в магнiтно-неоднорiдному мiкропровiднику. Розв’язано вiдповiдне рiвняння Ейлера-Лагранжа. Показано, що описана система має власнi моди спiнової поляризацiї i осциляцiї сумарного електричного струму. Запропонований пiдхiд дозволяє вивчати спiнову динамiку у вiдкритому контурi, який мiстить ємнiсть i/чи iндуктивнiсть. 2014 Article Lagrangian approach in spin-oscillations problem / P.V. Pyshkin, A.I. Kopeliovich, A.V. Yanovsky // Condensed Matter Physics. — 2014. — Т. 17, № 4. — С. 43801: 1–4. — Бібліогр.: 7 назв. — англ. 1607-324X arXiv:1312.4138 DOI:10.5488/CMP.17.43801 PACS: 81.05.Xj, 75.70.Cn, 75.85.+t http://dspace.nbuv.gov.ua/handle/123456789/153457 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Lagrangian of electronic liquid in magneto-inhomogeneous micro-conductor has been constructed. A corresponding Euler-Lagrange equation has been solved. It was shown that the described system has eigenmodes of spin polarization and total electric current oscillations. The suggested approach permits to study the spin dynamics in an open-circuit which contains capacitance and/or inductivity. |
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Pyshkin, P.V. Kopeliovich, A.I. Yanovsky, A.V. |
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Pyshkin, P.V. Kopeliovich, A.I. Yanovsky, A.V. Lagrangian approach in spin-oscillations problem Condensed Matter Physics |
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Pyshkin, P.V. Kopeliovich, A.I. Yanovsky, A.V. |
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Pyshkin, P.V. |
| title |
Lagrangian approach in spin-oscillations problem |
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Lagrangian approach in spin-oscillations problem |
| title_full |
Lagrangian approach in spin-oscillations problem |
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Lagrangian approach in spin-oscillations problem |
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Lagrangian approach in spin-oscillations problem |
| title_sort |
lagrangian approach in spin-oscillations problem |
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Інститут фізики конденсованих систем НАН України |
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2014 |
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Lagrangian approach in spin-oscillations problem / P.V. Pyshkin, A.I. Kopeliovich, A.V. Yanovsky // Condensed Matter Physics. — 2014. — Т. 17, № 4. — С. 43801: 1–4. — Бібліогр.: 7 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT pyshkinpv lagrangianapproachinspinoscillationsproblem AT kopeliovichai lagrangianapproachinspinoscillationsproblem AT yanovskyav lagrangianapproachinspinoscillationsproblem |
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2025-07-14T04:37:01Z |
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2025-07-14T04:37:01Z |
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1837595699124895744 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 4, 43801: 1–4
DOI: 10.5488/CMP.17.43801
http://www.icmp.lviv.ua/journal
Lagrangian approach in spin-oscillations problem
P.V. Pyshkin, A.I. Kopeliovich, A.V. Yanovsky
B.Verkin Institute for Low Temperature Physics and Engineering
of the National Academy of Sciences of Ukraine, 47 Lenin Ave., 61103 Kharkiv, Ukraine
Received June 27, 2014, in final form July 30, 2014
Lagrangian of electronic liquid in magneto-inhomogeneous micro-conductor has been constructed. A corre-
sponding Euler-Lagrange equation has been solved. It was shown that the described system has eigenmodes
of spin polarization and total electric current oscillations. The suggested approach permits to study the spin
dynamics in an open-circuit which contains capacitance and/or inductivity.
Key words: spin transport, spintronics
PACS: 81.05.Xj, 75.70.Cn, 75.85.+t
At present, spintronics [1] is one of the most active areas of investigation in solid state physics. This
interest is due, above all, to practical applications of spin-transport effects in modern microelectronics
(for example, applications of the giant magnetoresistance [2] phenomenon). In most publications on spin
transport research, a stationary situation is assumed, for instance, the transmission of direct current
through the magnetic inhomogeneous layered conductor [3]. But specific spin-transport effects are also
possible in a non-stationary regime. For example, in [4], AC spin-valve has been investigated. Another
example is the effect of the “spin pendulum” [5]: the oscillations of the total current and the spin polariza-
tion in an inhomogeneous magnetic closed conductor provided the hydrodynamic transport of electrons.
The consideration in [5] is based on the solutions of the spin hydrodynamics equations, which in turn
were obtained using a standard Chapman-Enskog method of quasi-classical kinetic equation solution. A
description of “spin-pendulum” oscillations from more general principles is of independent interest as
well as it can contribute to a better understanding of spin dynamics in solid state systems. In this article,
we have developed classical Lagrange approach to the “spin pendulum” oscillations of a spin electron
liquid. It allows us to study the effects of capacitance and inductivity on the “spin pendulum” oscillations.
The system which is investigated is a small-size conductor (closed or open circuit). We suppose that
the length of the conductor is much bigger than the square root of its cross section and all physical values
depend only on the distance x along the conductor (cross section of the conductor s does not depend
on x). Let us consider a magnetically inhomogeneous conductor, when the value of magnetization of
the conductor depends on x. That is, in terms of a spin of conducting electrons, the spin polarization
of an electron density in such a conductor depends only on x in the equilibrium state: P (x) = [n↑(x)−
n↓(x)]/[n↑(x)+n↓(x)], where P is a spin polarization and n↑(↓)(x) are equilibrium densities of electrons
with spin “up” and “down” (figure 1).
In particular, such a system can be realized by a spatial inhomogeneous doping with magnetic im-
purities and by placing the conductor in a magnetic field. Here, we consider the magnetization direc-
tion that is collinear along the conductor. The behavior of the conducting electrons can be described by
several macroscopic parameters: v(x, t) — the drift velocity which we consider is common for spin-up
and spin-down electrons (this is due to the hydrodynamic transport regime which we also consider),
n↑(↓)(x, t) = n0↑(↓)(x)+δn↑(↓)(x, t) are electron densities [δn↑(↓)(x, t) is non equilibrium perturbation of
electron densities], µ↑(↓)(x, t) is chemical potential for spin-up and spin-down electrons. We can express
µ↑(↓)(x, t) as a sum of the equilibrium value µ0 and perturbation δµ↑(↓)(x, t) of the equilibrium chem-
ical potential. We suppose that all sizes of the system are more than electronic screening length, and
the characteristic frequency is less than the plasmonic frequency. Thus, we can assume that the condi-
tion of electro-neutrality in the conductor is: δn↑(x, t)+δn↓(x, t) = 0. In the case of small perturbation
© P.V. Pyshkin, A.I. Kopeliovich, A.V. Yanovsky, 2014 43801-1
http://dx.doi.org/10.5488/CMP.17.43801
http://www.icmp.lviv.ua/journal
P.V. Pyshkin, A.I. Kopeliovich, A.V. Yanovsky
x
0
s
Figure 1. Magnetically inhomogeneous closed conductor (“spin pendulum”) with a constant cross section
s. Various sized black arrows denote spatial inhomogeneity of equilibrium spin polarization.
(|δµ↑(↓)(x, t)|≪µ0), the electro-neutrality condition can be expressed as follows:
δµ↑N↑+δµ↓N↓ = 0, (1)
where N↑(↓) are equilibrium densities of states at Fermi level.
Neglecting the spin-flip processes, we can write the continuity equation for electrons of each spin σ
∂δnσ
∂t
+
∂(n0σv)
∂x
= 0. (2)
Let us describe the system we study by a classical Lagrange function L = T −U [6], where T is the kinetic
energy and U is the potential energy of conducting electrons. The kinetic energy is connected with the
drift of electrons:
T =
∫
V
n
mv2
2
dV =
s j 2
2
∫
W
m
n(x)
dx ,
where j = nv is total electronic current (due to electro-neutrality ∂ j /∂x = 0), m is an electron mass, V is
the total volume of the conductor, n = n↑ +n↓, s = Const is the cross section of the conductor, W is the
length of the conductor. The potential energy U is connected with the non-equilibrium spin density i.e.,
deviation from the equilibrium value of chemical potential δµσ:
U =
∫
V
µ0+δµ↑
∫
µ0
εN↑dε+
µ0+δµ↓
∫
µ0
εN↓dε
dV , (3)
where ε is electron energy. In the main approximation, N↑(↓) does not depend on energy ε (but it can
depend on the coordinate x). There is an explanatory drawing in figure 2 which contains schematic il-
lustrations of equilibrium (a) and non-equilibrium (b) spin states which correspond to potential energy.
Figure 2. Illustration of the origin of potential energy, which is connected with non-equilibrium spins.
Grey filled area corresponds to the occupied electron states (at zero temperature).
43801-2
Lagrangian approach in spin-oscillations problem
x
0
s
C
Figure 3. Modification of “spin pendulum” — magnetically inhomogeneous conductor with capacitor
plates which can collect electric charge. Inductivity of the conductor is taken into account.
Using electro-neutrality condition (1) and by integration (3) over energy we can transform (3) to the fol-
lowing expression:
U =
s
2
∫
[
N↑δµ
2
↑ +N↓δµ
2
↓
]
dx. (4)
Using the fact that a drift velocity v = j /n is common for two spins, we can transform (2) to:
∂δnσ
∂t
+ j
∂
∂x
(n0σ
n
)
= 0. (5)
It is easy to integrate (5) over time (note that n0σ and n does not depend on time):
δµσ(x, t) =−
1
Nσ(x)
[
∂
∂x
n0σ(x)
n(x)
]
t
∫
j (t ′)dt ′. (6)
Based on the form of (6), let us introduce a generic variable:
q(t)=
t
∫
j (t ′)dt ′, q̇(t)= j (t).
Now we can write a Lagrange function in terms of q(t):
L =
sq̇2
2
∫
m
n
dx −
sq2
2
∫
1
N∗
(
d
dx
n0↑
n
)2
dx. (7)
Using a common Lagrange equation
d
dt
∂L
∂q̇
=
∂L
∂q
(8)
and substituting (7) into (8), we obtain a harmonic oscillator equation with frequency ω
q̈ +ω2q = 0, ω2 =
[
∫
1
N∗
(
d
dx
n0↑
n
)2
dx
][
∫
m
n
dx
]−1
, (9)
where N∗−1 = N−1
↑
+ N−1
↓
. Expression (9) which determines spin oscillations frequency is the same as
obtained in [5]. Notice that our approach in this work does not require the closure of the conductor as in
[5]. Of course, a closed conductor is one of the methodologies to satisfy the electro-neutrality condition
(1), but now we can investigate a more general situation when we have a circuit which consists of a
magnetically inhomogeneous part, capacity and inductor (figure 3). Electro-neutrality of the open-circuit
is reached due to the presence of the capacity.
To take into account the presence of the capacity and inductor in the magneto-inhomogeneous con-
ductor we must add terms into the Lagrangian. The kinetic energy should be supplemented by a term rel-
evant to the inductor energy e2L(sq̇)2/2c2, while potential energy should be supplemented by the term
43801-3
P.V. Pyshkin, A.I. Kopeliovich, A.V. Yanovsky
relevant to the capacity energy e2(sq)2/2C , where L and C are inductivity and capacitance of the conduc-
tor, e is electron charge, c is speed of light. Using (8) with the new Lagrangian we obtain the frequency of
oscillations of our system:
ω2
=
[∫
1
N∗
(
d
dx
n0↑
n
)2
dx +
e2s
2C
][∫
m
n
dx +
e2sL
2c2
]−1
. (10)
It is easy to see that if we neglect the spin inhomogeneous and inertial terms, the expression (10) grades
into the well known Thomson’s oscillation formula for LC circuit.
As can be seen from above, we propose and describe a new device — an “extended” oscillatory cir-
cuit, whose frequency depends on magnetic properties of the conductor. It can be seen from the result
that an external magnetic field can be used to vary the oscillatory frequency of the device (by chang-
ing spatial distribution of the equilibrium spin density n↑(↓)). Due to the presence of a capacity in the
device, it can be easily used as a part of external circuit by a capacitive coupling. The above described
resonant properties will appear as a peak of the impedance dependence on the frequency [4, 7]. It can
be seen from (4) that the potential energy is proportional to a square of the perturbation of a chemical
potential (non-equilibrium additions to spin densities) and to the volume ∆V in which non-equilibrium
densities are enclosed U ∝ δµ2
σ∆V . Thus, spin diffusion (which increases ∆V , decreases δµσ and does
not change δµσ∆V ) must be considered as a dissipative process. Therefore, we emphasize that the ex-
pressions (9) and (10) are valid when νrelax ≪ω≪ νee,νp, where νrelax is the biggest efficient frequency
of relaxation processes (frequency of collisions with a momentum loss, frequency of spin-flip processes,
characteristic diffusion frequency νeel 2
ee/l 2
tr, where lee is electron free path length, ltr is characteristic
length of equilibrium spatial spin inhomogeneity), νee is the frequency of electron-electron collisions,
νp is the plasmonic frequency.
Acknowledgements
This work was supported by the grant of the NAS of Ukraine #4/13–N.
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Метод Лагранжiану в проблемi спiнових осциляцiй
П.В. Пишкiн, О.I.Копелiович, А.В. Яновський
Фiзико-технiчний iнститут низьких температур iменi Б.I. Вєркiна НАН України,
просп. Ленiна, 47, 61103 Харкiв, Україна
Побудовано Лагранжiан електронної рiдини в магнiтно-неоднорiдному мiкропровiднику. Розв’язано вiд-
повiдне рiвняння Ейлера-Лагранжа. Показано, що описана система має власнi моди спiнової поляризацiї
i осциляцiї сумарного електричного струму. Запропонований пiдхiд дозволяє вивчати спiнову динамiку у
вiдкритому контурi, який мiстить ємнiсть i/чи iндуктивнiсть.
Ключовi слова: перенос спiну, спiнтронiка
43801-4
http://dx.doi.org/10.1103/RevModPhys.76.323
http://dx.doi.org/10.1103/PhysRevLett.61.2472
http://dx.doi.org/10.1103/PhysRevB.48.7099
http://dx.doi.org/10.1103/PhysRevLett.107.176604
http://dx.doi.org/10.1103/PhysRevB.73.153204
http://dx.doi.org/10.1063/1.3536341
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