Re-distribution (condensation) of magnons in a ferromagnet under pumping
Last years several experiments were performed, which studied the Bose-Einstein condensation of quasi-particles, in particular, magnons in magnetically ordered systems. Recently the Bose-Einstein condensation of magnons was observed at room temperatures in a ferromagnetic film. The theory of the cond...
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irk-123456789-77172017-05-21T15:59:14Z Re-distribution (condensation) of magnons in a ferromagnet under pumping Zvyagin, A.A. Last years several experiments were performed, which studied the Bose-Einstein condensation of quasi-particles, in particular, magnons in magnetically ordered systems. Recently the Bose-Einstein condensation of magnons was observed at room temperatures in a ferromagnetic film. The theory of the condensation (re-distribution) of magnons under the conditions of pumping, which explains many features of that experiment, is presented. The use of the term "Bose-Einstein condensation of magnons" is discussed. 2007 Article Re-distribution (condensation) of magnons in a ferromagnet under pumping / A.A. Zvyagin // Физика низких температур. — 2007. — Т. 33, № 11. — С. 1248-1252. — Бібліогр.: 11 назв. — англ. 0132-6414 PACS: 03.75.Nt; 75.45.+j; 76.50.+g http://dspace.nbuv.gov.ua/handle/123456789/7717 en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Last years several experiments were performed, which studied the Bose-Einstein condensation of quasi-particles, in particular, magnons in magnetically ordered systems. Recently the Bose-Einstein condensation of magnons was observed at room temperatures in a ferromagnetic film. The theory of the condensation (re-distribution) of magnons under the conditions of pumping, which explains many features of that experiment, is presented. The use of the term "Bose-Einstein condensation of magnons" is discussed. |
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Zvyagin, A.A. |
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Zvyagin, A.A. Re-distribution (condensation) of magnons in a ferromagnet under pumping |
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Zvyagin, A.A. |
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Zvyagin, A.A. |
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Re-distribution (condensation) of magnons in a ferromagnet under pumping |
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Re-distribution (condensation) of magnons in a ferromagnet under pumping |
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Re-distribution (condensation) of magnons in a ferromagnet under pumping |
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Re-distribution (condensation) of magnons in a ferromagnet under pumping |
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Re-distribution (condensation) of magnons in a ferromagnet under pumping |
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re-distribution (condensation) of magnons in a ferromagnet under pumping |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Re-distribution (condensation) of magnons in a ferromagnet under pumping / A.A. Zvyagin // Физика низких температур. — 2007. — Т. 33, № 11. — С. 1248-1252. — Бібліогр.: 11 назв. — англ. |
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AT zvyaginaa redistributioncondensationofmagnonsinaferromagnetunderpumping |
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Fizika Nizkikh Temperatur, 2007, v. 33, No. 11, p. 1248–1252
Re-distribution (condensation) of magnons
in a ferromagnet under pumping
A.A. Zvyagin
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: zvyagin@ilt.kharkov.ua
Received April 20, 2007
Last years several experiments were performed, which studied the Bose-Einstein condensation of
quasi-particles, in particular, magnons in magnetically ordered systems. Recently the Bose-Einstein con-
densation of magnons was observed at room temperatures in a ferromagnetic film. The theory of the conden-
sation (re-distribution) of magnons under the conditions of pumping, which explains many features of that
experiment, is presented. The use of the term «Bose-Einstein condensation of magnons» is discussed.
PACS: 03.75.Nt Other Bose-Einstein condensation phenomena;
75.45.+j Macroscopic quantum phenomena in magnetic systems;
76.50.+g Ferromagnetic, antiferromagnetic, and ferrimagnetic resonances; spin-wave resonance.
Keywords: condensation of magnons, parametric pumping.
The interest to the Bose-Einstein condensation (BEC)
of bosonic particles and quasi-particles has been consider-
ably increased during last years. That quantum mechanical
phenomenon from the pure theoretical problem became
one of the most attractive things in the modern physics,
mostly due to recent experiments on dilute ultra-cold gases
of atoms [1–3]. The theory of that phenomenon (BEC) for
particles is well understood by now. On the other hand, the
phenomenon of BEC for quasi-particles causes many dis-
cussions, and very different opinions were expressed about
its nature. Among others, the BEC of magnons (magnetic
quasi-particles, quanta of low-energy excitations of mag-
netic systems) has attracted much attention of physicists
during last years [4–8]. For example, in the recent letter
Demokritov et al. [9] reported the observation of BEC of
magnons at room temperatures. In this study the theory,
which theoretically explains many features of the experi-
ment [9] is presented.
The observation of BEC of magnons under the condi-
tions of the so-called parametric pumping of magnons in
thin films of the yttrium-iron-garnet (YIG) was reported
in [9]. Actually, the authors have studied experimentally
how the spectral density of magnons �( )E , defined as
�( )E � D E n E( ) ( ) (where D E( ) is the density of states of
magnons, and n E( ) is the statistical distribution function
of magnons, E is the energy of magnons), is affected by
the pumping by means of the external magnetic rf field.
For that purpose they used the Brillouin light scattering
(BLS) spectroscopy. Under the conditions of the experi-
ment [9] YIG can be considered as a ferromagnet (as
usual for such kind of experiments, where only the lowest
branch of magnetic excitations of YIG was studied). For
ferromagnets it is commonly accepted that at low temper-
atures (and/or at their low densities) magnons satisfy the
Bose-Einstein statistics, and, hence, n E( ) is the Bose-Ein-
stein distribution function. In the experiment [9] the au-
thors considered the dependence of n, D, and � on the fre-
quency � rather than on the energy E (using the basic
quantum mechanical connection between the energy and
the frequency E h� �, where h � 2�� is the Planck’s con-
stant).
The parametric pumping is usually performed in the
parallel geometry, for which the polarization of the rf
field is collinear to the direction of the time-independent
magnetic field H. The Hamiltonian, which describes a
ferromagnet, can be written as � � �� �0 1( )t , where
�0 � � � � � �� � � � �
n r
n n r z n
z
n r
z
n
x
n r
x
n
y
n r
y
J J S S B S S S S
,
[ ( )]S S
� �g H SB n
z . (1)
It is supposed that the external magnetic field is directed
along the z-axis. In this expression S n
x y z, , are operators of
© A.A. Zvyagin, 2007
projections of the spin S in the nth site of the crystal, g is
the g-factor, B is the Bohr’s magneton, J
0 is the isotro-
pic ferromagnetic coupling, J z and B are the coupling con-
stants of the spin Hamiltonian of the system, related to
magnetically anisotropic interactions. J z denotes the mag-
netic anisotropy of the crystal, parallel to the external mag-
netic field, and B denotes the magnetic anisotropy, per-
pendicular to the direction of the field. In the standard
approach B is determined, e.g., by the orthorhombic mag-
netic anisotropy, and/or by the magnetic dipole-dipole in-
teractions. The latter also gives a contribution to the term
J z (J z can be caused by the uniaxial magnetic anisotropy,
too) [11]. Usually, in standard magnets, including YIG,
one has the condition B J g H Jz B�� , , . In the cubic crys-
tal of YIG, the main contribution to J z and B comes, prob-
ably, from the magnetic dipole-dipole coupling. The sums
are performed over all sites, and r � 0 1 2, , , ...determines ra-
dii of spin-spin interactions. For example, for possible (in
principle) terms, describing a single-ion magnetic aniso-
tropy, one has to put r � 0. Obviously, the magnetically iso-
tropic spin-spin interaction J does not change the direc-
tions of spins of the ferromagnet in the equilibrium.
However, the isotropic spin-spin ferromagnetic coupling is
responsible for the fact, that all spins of the ferromagnet
are parallel to each other in the ground state. For the de-
scription of the low-temperature thermodynamic charac-
teristics of the ferromagnet the spin-wave approximation is
often used [11]. In that approximation one replaces opera-
tors of quantum spins by classical vectors, and looks for a
steady-state (equilibrium) configuration of those vectors.
That steady state is associated with the ground state of the
original quantum spin system. Then small (inhomoge-
neous) deviations of classical vectors from that steady-
state configuration are called spin waves, or, in the quan-
tized form, magnons [11]. The Hamiltonian Eq. (1) in that
approximation can be written as
�0 0� � �E E nk
k
k� , (2)
where E0 is determined by the energy of the lowest energy
equilibrium configuration, Ek (E Ek k� � for the consid-
ered case) and �nk are the energy and the operator of the
number of magnons with quasi-momenta k. Ek depends on
J , J z , B, and H in the standard way [11]. For the consid-
ered ferromagnet the minimal energy of magnons is deter-
mined by the value of the spin gap, which does not depend
on the isotropic ferromagnetic interaction J . When calcu-
lating thermodynamic characteristics of a ferromagnet it is
often useful to replace the summation over quasi-momenta
by the integration over energies �
�k V D E dE0 ( ) ,
where D E( ) is the density of states of magnons, and V0 is
the phase volume. That is why, for example, the low-tem-
perature energy of the ferromagnet in that approximation
is equal to
E E V En E D E dE E V E EdElt � � � �� �0 0 0 0( ) ( ) ( ) ,� (3)
where n E( ) is the Bose-Einstein distribution function. To
remind, in this work, following the letter [9], we use
E h� �, i.e. the integration over energies E are replaced by
integration over frequencies �. It is important to point out
that because of the spin gap, the integration over � (or
over E) is, in fact, performed over the interval � �� m (as
well as � �� max), where h m� is determined by the spin
gap of Ek ( ( ) ( )E h fmk k� �� 2 in our model, where
lim ( ) )k k
�0 0f , and �max is related to the maximal en-
ergy of magnons. This property follows from the density
of states of magnons of the considered ferromagnet D( )� .
The term of the Hamiltonian, describing the effect of the
external rf magnetic field, is given by the expression
�1 0 2( ) cos ( )t g h t SB p n
z
n
� � � �� . (4)
Here h0 and � p are the magnitude and the frequency of
the rf magnetic field, respectively. In this geometry in the
magnon approximation, the resonance takes place if the
frequency of the rf magnetic field is equal to the energy of
two magnons with opposite directions of quasi-momenta
o f ma g n o n s , Ek a n d E�k , i . e . a t h E Ep� � � �k k
( � �2 2E hk � in the considered case).
Several features of the behavior of the BLS spectral
density of magnons were observed in the experiment [9].
Namely, when the power P of the rf pumping field was
weak enough, the observed BLS spectral function was fit-
ted with the help of the same density of states of magnons
and the same Bose-Einstein distribution function of
magnons, which were observed in YIG without pumping.
The intensities of the observed BLS spectra of magnons
(related to the spectral density of magnons) increased
when the power of the rf magnetic field became larger.
The intensity of the BLS spectra increased also with the
time of the action of pumping, until saturation at approxi-
mately 400 500� ns. Before saturation the BLS spectra
were also possible to fit with the help of the above men-
tioned procedure. However, in the saturation regime, it
was already impossible to fit the spectra with the help of
the standard Bose-Einstein distribution function. Demo-
kritov et al. pointed out that to reach the agreement with
the experiment it was necessary to add (by hand, without
derivation) to the distribution function of magnons n( )� a
singularity peak � �� � �( )m , where �m was the frequency
of the maximum of the intensity in the observed BLS
spectra of magnons (which corresponds to the minimal
energy of magnons Ek ). To explain the appearance of the
singularity peak the authors of Ref. [9] supposed that in
the steady-state regime, for t
�400 500 ns, BEC of mag-
nons took place (at room temperature, at which the exper-
Re-distribution (condensation) of magnons in a ferromagnet under pumping
Fizika Nizkikh Temperatur, 2007, v. 33, No. 11 1249
iment was performed). They concluded that the re-distri-
bution of magnons under pumping was related to BEC.
Here we propose the theory, which explains many fea-
tures of the experiment of Demokritov et al. For that pur-
pose we use the approach of the paper [10]. It was shown in
[10] that under the condition of the parametric pumping of
magnons in a ferromagnet the steady-state configuration of
spins (or classical vectors, see above) in the effective Hamil-
tonian, which determines the time evolution of the system un-
der parametric pumping, is changed in resonance. It is very
similar to the procedure, outlined above for the description of
low-temperature thermodynamic characteristics of a ferro-
magnet, but with the application to the effective Hamiltonian,
which describes the low-energy dynamics of the spin system
under pumping. In this work we generalize the approach of
[10]. It is easy to show that out of resonance each spin (or a
classical vector) of that effective Hamiltonian at equilibrium
has the projection to the axis, parallel to the direction of the
external magnetic field S Sz
0 � (other projections are equal to
S x
0 � S / S S
y z
0
2
0
21 2� �( ) ( ) ). On the other hand, in reso-
nance the configuration at equilibrium of spins (vectors) in
the effective Hamiltonian is such that the z-projection of the
effective spin (classical vector) becomes equal to
S
g H h
B g h /h J
z B p
B p z
0
0
2
4
�
�
�
�
�( ( ) )
, (5)
where the small parameter g h /hB p �0 1�� was used. One
can see that at resonance (when the frequency of the rf field
is close to the value � �p E /� k �), the effective position of
equilibrium of spins can be changed dramatically.
All characteristics of any quantum system in mixed
states depend on the density matrix �d . The time evolu-
tion of the latter can be determined by the solution of the
Liouville-like equation [10]
i
t
id
d d� �
�
�
�
�
�
�
�
� � � �
�
� � � �[ , ] ( )� 0 . (6)
Here [., .] denotes the commutator, � is the relaxation
constant (the relaxation is introduced in the simplest
form), and �0 is the density matrix in the equilibrium. It is
worthwhile to suppose, following the paper [10], that the
density matrix in equilibrium is determined by the Hamil-
tonian of the system without pumping � �0 0 0� �( )h ,
�0
0
0
�
�
�
exp ( )
(exp ( ))
�
�
/k T
/k T
B
BTr
, (7)
where k B is the Boltzmann’s constant, and T is the tempera-
ture. Such an assumption is justified, because usually the
magnitude of the pumping field h0 is much smaller than the
value of the stationary field H. At low temperatures we can
replace �0 in Eq. (7) by the approximate Hamiltonian of the
free gas of magnons �0 � E0 �� k k kE n� , see the discus-
sion above. On the other hand, the dynamics of the spin
system under pumping is determined by the effective Hamil-
tonian, mentioned above. We look for the solution of Eq. (6)
using the standard set of unitary transformations [10], and
the resonance approximation.
The solution of the Liouville-like equation manifests
two different regimes, cf. [10]. In the first (dynamical) re-
gime, at t �� �� 1, all characteristics of the system oscillate
with two frequencies: the frequency of the pumping field
� p , and the small frequency of Rabi-like oscillations,
which value is determined in resonance by the magnitude
of the rf field h0. On the other hand, in the second
(steady-state) regime, at t
�� 1, only oscillations with
� p survive. We are interested in the contributions to the
spectral density of magnons, which are proportional to
sin ( )2�� p t , because the main contribution to the density
comes from the term
�
�
�
�
�
�
�
� � � ���1
02 2
t
g h t Sp B p
n
n
z�� ��sin ( ) , (8)
where the line denotes the time averaging
f t A f t dt
A
A
( ) lim ( )�
�
� �1
0
, (9)
and brackets denote the thermal averaging. Following the
approach of the paper [10] and using the small parameter
g h /hB p �0 1�� , we find in the steady-state regime, at
t
�� 1, that the (small) addition to the spectral function of
magnons �� �( ), caused by the pumping, is proportional to
�� � � �
�
�
� �
( ) ( ) ( )
( )
( )
�
�
D n
g h
h
BS
g H
B
p
x
B
0
0 0
2
2 2
2 �
�
, (10)
where
� �
�
2
0
2 0
0
2
2 2 2� � � �
�
�
�
�
�
�
�
�[ ( )] ,g H J S Z h /
g h
h
BSB z
z
p
B
p
x
(11)
Z is the coordination number, and n0( )� is the Bose-Ein-
stein distribution function of magnons without pumping,
which is determined by the Hamiltonian �0 in the spin-
wave approximation. The resonance in the parallel geom-
etry of pumping takes place at � �p � 2 . To show that, we
can write down the Hamiltonian of the system in the
quasi-classical (magnon) approximation:
� � � ��E E n0 k
k
k�
� � ���g h BS
E
t b bB
x
p
��0 0 2
k
k k
k
cos ( ) ( ) ...
† †
H.c. (12)
1250 Fizika Nizkikh Temperatur, 2007, v. 33, No. 11
A.A. Zvyagin
Here E0 does not depend on operators, b
k
†
are the creation
operators of magnons with the quasi-momenta k
( � )n b bk k
†
k� , and H.c. denote, as usually, Hermitian con-
jugated terms. From this formula one immediately sees
that the resonance terms, i.e. those, which give the non-
zero contribution to the linear in g h /EB 0 k response of
the spin system to the rf field, are those, for which Ek �
� h /p� 2, or h h /p� �� 2. That procedure for resonance
terms is similar to the procedure for the obtaining the so-
lution of Eq. (6). First, using the unitary transformation
U i
g h
h
t SB
p
p n
z
n
1
0 2�
�
�
�
�
�
�
�
��exp sin ( )
�
�� , (13)
the time dependence is transferred to the non-diagonal
terms of the Hamiltonian �. Then, we used the small pa-
rameter g h /hB p �0 , and the resonance approximation.
The latter means that we drop all the terms in the obtained
effective Hamiltonian, which determines the spin dynam-
ics, except of those, which give a nonzero contribution in
the linear in g h /hB p �0 response of the system. Other
terms, which depend on g hB 0 (the resonance ones) are
taken into account exactly. Then, the explicit dependence
on time is removed from the effective Hamiltonian, using
the unitary transformation
U i Sp n
z
n
2 � �exp ( )�� . (14)
In the obtained Hamiltonian, which determines the spin
dynamics of the system, we use the quasi-classical ap-
proximation, taking into account the shift of the steady-
state configuration of spins in resonance. And, finally, we
use the unitary transformation
U i /t3 � �exp (
~
)� , (15)
where
~
� is the obtained effective Hamiltonian, which de-
termines the spin dynamics. The time dependence for
�d t( ) is obtained by using the consecutive combination of
the above described procedures. The average value of the
operator �n is obtained after averaging it with �d t T( , ), and
the temperature dependence in this approach is determined
by Eq. (7). Out of resonance, where � �! p /2, the addition
to the spectral function, caused by the pumping, is approxi-
mately zero. In resonance, one may substitute � �p /2 � in
the first term of Eq. (11), see above. Namely that substitu-
tion (justified at resonance) produces the re-distribution of
the spectral density of magnons in Eq. (10). This is why,
the addition �� �( ) in the steady-state regime in resonance is
approximately proportional to � � �( )� m (notice that for
our system in the used approximation one has h m� �
� g HB � �2 0J S Z O Bz
z ( )). Here we used the formula
lim [( ) ] ( )� � �� � � �
� �0
2 2/ x x . Hence, the pumping cau-
ses the coherent condensation of magnons to the state with
the minimal energy h m� . One can see that even if the
half-frequency of the pumping field � p /2 is larger than
�m, but it is in resonance with some of higher-energy
magnons (i.e. 2 2� � �m p� � max), our theory predicts the
re-distribution of the magnon spectral density to the one
with the peak � �� � �( )m near the minimal energy of
magnons. That re-distribution means, that when the photon
of the pumping field is in resonance with spins of the sys-
tem, those spins dynamically change their effective equi-
librium configuration in such a way, that some part of small
deviations of that dynamical equilibrium (magnons) be-
comes re-distributed at the lowest edge of the magnon
spectrum. The physical reason for the re-distribution to the
minimal energy of magnons is due to the fact that in our ap-
proximation (B g H J JB z�� , , ) the k-dependent disper-
sion of the energy of magnons is related to the isotropic
ferromagnetic interaction, which operator commutes with
the z-projection of the total spin of the system (to which the
rf pumping field is connected). It also turns out, that the ad-
dition �� �( ) is increased with the growth of the pumping
power P (because P is proportional to h0
2).
On the other hand, we calculated that for t �� �� 1, in
the dynamical regime, the small contribution to the spec-
tral density of magnons, caused by the pumping, is pro-
portional (at small t, at which one can use the approxima-
tion sin ( )� �t/ t/� �� ) to
�� � � �
�
( ) ( ) ( )
( )
� D n
g h
h
BS
g H
tB
p
x
B
0
0 0
22
�
. (16)
Obviously, in that regime the spectral function of mag-
nons is increased with time (as well as with the power of
the pumping field P). However, in this regime there is no
�-function contribution to the spectral function, and the
latter (even in resonance) is determined by the distribu-
tion function of magnons without pumping. Such a be-
havior is also similar to what was observed in [9].
One can see that the present theory describes many
features of the experiment of Demokritov et al. [9]. First,
in the dynamical regime, the spectral function of magnons
is increased with time, reaching the steady-state regime.
There the pumping renormalizes the distribution of
magnons in such a way, that in resonance some part of
magnons is in the coherent state with the lowest possible
energy (approximately with the energy of the spin gap).
Demokritov et al. called such a process BEC of magnons.
However, one can express doubts about the correctness of
the use of such a term for magnons. It is important to point
out that BEC determines the ground state of bosonic parti-
cles. The situation is drastically different for magnons. It
is well known that for many magnetically ordered sys-
tems (and, moreover, for magnetic systems with the spin
frustration) in most of cases the ground state cannot be
found theoretically exactly [11]. Usually (see above) the
quasi-classical approximation is used, when one replaces
the operators of quantum spins by their classical counter-
Re-distribution (condensation) of magnons in a ferromagnet under pumping
Fizika Nizkikh Temperatur, 2007, v. 33, No. 11 1251
parts (vectors), then the steady-state configuration of
those classical vectors is associated with the ground state
of quantum spins. Small deviations from that steady-state
configuration are called spin waves (or, in the quantized
form, magnons). However, BEC for magnons implies the
renormalization of the ground state: All magnons at
T TBEC� (and, thus, at T � 0 also) are re-distributed to
the BEC state. Then, one faces with the paradoxical situa-
tion: The ground state of the spin system was not deter-
mined exactly, but the use of such a term, BEC of mag-
nons, implies the renormalization of the (unknown!)
ground state of the spin system. Hence, such a use is un-
der question. One can also put similar questions to the dis-
cussion of the results [4–7]. These our doubts correlate
with the recent opinion of Snoke [8]. Notice, however,
that, on the other hand, according to the present theory, it
seems to be clear that magnons in the experiment of
Demokritov et al. were re-distributed by the parametric
pumping to the dynamically induced coherent quantum
state, cf. [8]. Despite the question, whether the Bose-Ein-
stein condensation is the right term to call the observed
phenomena, coherent quantum states of magnons, either
near phase transitions [4–7], or induced by the parametric
pumping [9], are among the most interesting phenomena
of the modern condensed matter physics.
The author dedicates this paper to the memory of his
father, Anatolii Illarionovich Zvyagin.
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A.A. Zvyagin
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