Formation of Bose–Einstein magnon condensate via dipolar and exchange thermalization channels
Thermalization of a parametrically driven magnon gas leading to the formation of a Bose–Einstein condensate at the bottom of a spin-wave spectrum was studied by time- and wavevector-resolved Brillouin light scattering spectroscopy. Two distinct channels of the thermalization process related on dip...
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irk-123456789-1280852018-01-13T14:01:08Z Formation of Bose–Einstein magnon condensate via dipolar and exchange thermalization channels Bozhko, D.A. Clausen, P. Chumak, A.V. Kobljanskyj, Yu.V. Hillebrands, B. Serga, A.A. К 80-летию уравнения Ландау–Лифшица Thermalization of a parametrically driven magnon gas leading to the formation of a Bose–Einstein condensate at the bottom of a spin-wave spectrum was studied by time- and wavevector-resolved Brillouin light scattering spectroscopy. Two distinct channels of the thermalization process related on dipolar and exchange parts of a magnon gas spectrum are clearly determined. It has been found that the magnon population in these thermalization channels strongly depends on applied microwave pumping power. The observed magnon redistribution between the channels is caused by the downward frequency shift of the magnon gas spectrum due to the decrease of the saturation magnetization in the course of injection of parametrically pumped magnons. 2015 Article Formation of Bose–Einstein magnon condensate via dipolar and exchange thermalization channels / D.A. Bozhko, P. Clausen, A.V. Chumak, Yu.V. Kobljanskyj, B. Hillebrands, A.A. Serga // Физика низких температур. — 2015. — Т. 41, № 10. — С. 1024–1029. — Бібліогр.: 30 назв. — англ. 0132-6414 PACS: 05.30.Jp, 75.30.Ds, 75.70.–i http://dspace.nbuv.gov.ua/handle/123456789/128085 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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К 80-летию уравнения Ландау–Лифшица К 80-летию уравнения Ландау–Лифшица |
| spellingShingle |
К 80-летию уравнения Ландау–Лифшица К 80-летию уравнения Ландау–Лифшица Bozhko, D.A. Clausen, P. Chumak, A.V. Kobljanskyj, Yu.V. Hillebrands, B. Serga, A.A. Formation of Bose–Einstein magnon condensate via dipolar and exchange thermalization channels Физика низких температур |
| description |
Thermalization of a parametrically driven magnon gas leading to the formation of a Bose–Einstein condensate
at the bottom of a spin-wave spectrum was studied by time- and wavevector-resolved Brillouin light scattering
spectroscopy. Two distinct channels of the thermalization process related on dipolar and exchange parts of a
magnon gas spectrum are clearly determined. It has been found that the magnon population in these
thermalization channels strongly depends on applied microwave pumping power. The observed magnon redistribution
between the channels is caused by the downward frequency shift of the magnon gas spectrum due to the
decrease of the saturation magnetization in the course of injection of parametrically pumped magnons. |
| format |
Article |
| author |
Bozhko, D.A. Clausen, P. Chumak, A.V. Kobljanskyj, Yu.V. Hillebrands, B. Serga, A.A. |
| author_facet |
Bozhko, D.A. Clausen, P. Chumak, A.V. Kobljanskyj, Yu.V. Hillebrands, B. Serga, A.A. |
| author_sort |
Bozhko, D.A. |
| title |
Formation of Bose–Einstein magnon condensate via dipolar and exchange thermalization channels |
| title_short |
Formation of Bose–Einstein magnon condensate via dipolar and exchange thermalization channels |
| title_full |
Formation of Bose–Einstein magnon condensate via dipolar and exchange thermalization channels |
| title_fullStr |
Formation of Bose–Einstein magnon condensate via dipolar and exchange thermalization channels |
| title_full_unstemmed |
Formation of Bose–Einstein magnon condensate via dipolar and exchange thermalization channels |
| title_sort |
formation of bose–einstein magnon condensate via dipolar and exchange thermalization channels |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2015 |
| topic_facet |
К 80-летию уравнения Ландау–Лифшица |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/128085 |
| citation_txt |
Formation of Bose–Einstein magnon condensate via dipolar and exchange thermalization channels / D.A. Bozhko, P. Clausen, A.V. Chumak, Yu.V. Kobljanskyj, B. Hillebrands, A.A. Serga // Физика низких температур. — 2015. — Т. 41, № 10. — С. 1024–1029. — Бібліогр.: 30 назв. — англ. |
| series |
Физика низких температур |
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2025-07-09T08:23:44Z |
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| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10, pp. 1024–1029
Formation of Bose–Einstein magnon condensate
via dipolar and exchange thermalization channels
D.A. Bozhko1,2, P. Clausen1, A.V. Chumak1, Yu.V. Kobljanskyj3,
B. Hillebrands1, and A.A. Serga1
1Fachbereich Physik and Landesforschungszentrum OPTIMAS,
Technische Universität Kaiserslautern, Kaiserslautern 67663, Germany
2Graduate School Materials Science in Mainz, 47 Gottlieb-Daimler-Straße, Kaiserslautern 67663, Germany
E-mail: bozhko@physik.uni-kl.de
3Faculty of Radiophysics, Electronics and Computer Systems, Taras Shevchenko National University of Kyiv
Kyiv 01601, Ukraine
Received April 17, 2015, published online August 25, 2015
Thermalization of a parametrically driven magnon gas leading to the formation of a Bose–Einstein conden-
sate at the bottom of a spin-wave spectrum was studied by time- and wavevector-resolved Brillouin light scatter-
ing spectroscopy. Two distinct channels of the thermalization process related on dipolar and exchange parts of a
magnon gas spectrum are clearly determined. It has been found that the magnon population in these
thermalization channels strongly depends on applied microwave pumping power. The observed magnon redistri-
bution between the channels is caused by the downward frequency shift of the magnon gas spectrum due to the
decrease of the saturation magnetization in the course of injection of parametrically pumped magnons.
PACS: 05.30.Jp Boson systems;
75.30.Ds Spin waves;
75.70.–i Magnetic properties of thin films, surfaces, and interfaces.
Keywords: magnon, thermalization, Bose–Einstein condensate.
In ferromagnetic materials, atoms having unpaired elec-
trons act as individual magnets. Since these atomic mag-
nets orient in the same direction due to the exchange inter-
action, a macroscopic magnetic moment M appears. In the
presence of an external bias magnetic field H0 this magnet-
ic moment tends to be oriented along the field direction.
When the equilibrium position of the magnetization is dis-
turbed, M starts to precess around the equilibrium position.
Such precession is described by the classical Landau–
Lifshitz equation of magnetization dynamics [1]:
eff eff2= ( ),d
dt M
γλ
−γ × − × ×
M M H M M H (1)
where γ is the gyromagnetic ratio and λ is a dissipation
parameter. An effective internal magnetic field effH in-
cludes various terms:
3 2
eff 0= ( ) G( , ) ( ) ...
sV
t dr
M
η′ ′ ′+ + ⋅ + ∇ +
γ∫H H h r r M r M
(2)
where G( , )′r r is a tensor Green's function, η is a spin
stiffness constant, and sM is a saturation magnetization at
a given temperature. The first and the second terms in
Eq. (2) represent external static and dynamic magnetic
fields, respectively. The third term describes an effective
dipole–dipole interaction field and the fourth term repre-
sents an effective field of the exchange interaction. Equa-
tion (2) can also contain the effective fields mel =H
mel / ,U= −δ δM = / ,a aU−δ δH M and others, which de-
pend on magneto-elastic interaction energy mel ,U energy
of magnetic anisotropy ,aU etc. [2].
In the approximation of a small precession angle, the
solution of Eq. (1) with the effective field given by Eq. (2)
and proper boundary conditions represents a manifold of
spatially non-uniform magnetic excitations — waves of
magnetization precession usually called spin waves,—
whose dispersion characteristics are significantly non-
linear and anisotropic in relation to the direction of the bias
magnetic field H0. Figure 1 shows an example of calcula-
tion of the first 47 thickness spin-wave modes performed
© D.A. Bozhko, P. Clausen, A.V. Chumak, Yu.V. Kobljanskyj, B. Hillebrands, and A.A. Serga, 2015
mailto:bozhko@physik.uni-kl.de
Formation of Bose-Einstein magnon condensate via dipolar and exchange thermalization channels
using the method described in Ref. 3 for two specific prop-
agation directions in an in-plane magnetized ferrimagnetic
film of yttrium iron garnet (Y3Fe5O12, YIG). Taking into
account in the calculation the magneto-elastic field melH
leads to the hybridization of all spin-wave modes with the
clockwise-polarized phonon branches [4].
As the behavior of the transversal spin-wave branch
family 0( )⊥q H is explicitly determined by the exchange
interaction their frequencies increase proportionally to
2 ,qη where q is an in-plane wavevector. The same ex-
change-dependent mechanism (see the fourth term in
Eq. (2)) is responsible for the increase in the frequency of
the high thickness modes ( 1)n ≥ having harmonic dynam-
ic magnetization profiles over the film thickness. At the
same time, the behavior of the branches from the longitu-
dinal family 0( || )q H is completely different due to the
influence of dynamic demagnetizing fields originating
from the dipolar magnetic interaction. For example, in the
exchange-free case the frequency of the fundamental spin-
wave mode = 0n (blue curves in Fig. 1) drops monoton-
ically from the ferromagnetic resonance frequency
0 0= ( 4 )FMR zf H H Mγ + π down to 0=Hf Hγ when |q|
increases from nearly zero to infinity. Here zM is a pro-
jection of the precessing magnetic moment M on the di-
rection z of the static magnetization. However, existence
of the exchange interaction leads to the appearance of two
global energy minima at finite wavevector values, for the
given experimental conditions q = ±4·104 rad/cm as it is
shown in Fig. 1.
In terms of the second quantization, a spin-wave en-
semble of a magnetic sample can be treated as a gas of
weakly interacting quasi-particles called magnons. Each
magnon corresponds to the spatially delocalized spin-flip
and possesses a magnetic moment of two Bohr magnetons
2 .Bµ Magnons have an integer spin and, therefore, obey
the Bose–Einstein statistics. In the thermal equilibrium the
number of magnons depends on the sample temperature
and, thus, is not fixed. In such a case the chemical potential
of the magnon gas is zero. However, in 1991 it was pre-
dicted by Kalafati and Safonov that external injection of
magnons can increase the chemical potential up to the bot-
tom of the spin-wave spectrum [5]. These results in the
Bose–Einstein magnon condensation: the spontaneous ap-
pearance of a coherent state at the global energy minima of
the spin-wave spectrum (see Fig. 1).
The most effective way to reach the necessary population
of a magnon gas is the parallel parametric pumping, where
an alternating magnetic pumping field ( )th acts along the
direction of the bias magnetic field H0 [7,8]. This technique
is widely used to generate [9,10], amplify [11–13], and re-
store [14,15] spin-wave signals in macro- and micro-sized
magnetic structures. In 2006, this technique was successfully
used for the experimental observation of the Bose–Einstein
condensate (BEC) of magnons at room temperature [16].
This discovery has attracted the common interest to the
physics of the parametrically driven magnon gases, which is
now under active theoretical and experimental investigation.
Shortly after the first observation, Bugrij and Loktev showed
that the condensation of magnons has no restrictions on the
temperature of the magnon gas [17]. Later, Rezende [18]
presented extended theoretical analyses of the process.
Nowadays, the time dependent behavior of the magnon gas
in the phase-energy space [19–22], the temperature of the
magnon gas [23], and the role of different scattering mecha-
nisms [24] in the process of the thermalization of the inject-
ted magnons are in the focus of attention.
However, the exact mechanism of the thermalization of
the parametrically-injected magnons is not clarified yet. In
particular, a question about the role of dipolar (dipolar
channel of the thermalization) and exchange (exchange
channel) magnons in the thermalization process is still not
answered. Furthermore, it is expected that specific contri-
butions of the different thermalization channels depend on
the spectral position of the initially pumped magnon group,
which can vary with change in the pumping power. The
mechanism behind this phenomenon is following: with
increase in the number n of the excited magnons the mag-
netization component zM is reduced [2]:
= .z s BM M g− µn (3)
Fig. 1. (Color online) The magnon spectrum calculated in dipole-
exchange approximation for the first 47 thickness modes of an in-
plane magnetized single crystal yttrium iron garnet film of
6.7 µm thickness. The external bias field H0 = 1735 Oe, stiffness
constant η = 9.15·10–2 cm2·s–1, saturation magnetization 4πMs =
= 1750 G, the total field of cubic crystallographic and uniaxial
out off plane anisotropies Ha = –70 Oe. Red arrows indicates the
magnon injection by parametric pumping. The initially pumped
magnon group is marked by two violet dots. Green dots indicates
position of the magnon Bose–Einstein condensate.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10 1025
D.A. Bozhko, P. Clausen, A.V. Chumak, Yu.V. Kobljanskyj, B. Hillebrands, and A.A. Serga
Here g is the g-factor of an electron. Such a reduction may
lead to the frequency shift of the magnon spectrum [6] sig-
nificantly influencing the process of spin-wave parametric
excitation and affecting further thermalization dynamics.
Here, we provide experimental insight into the evolu-
tion of a magnon gas affected by the four-magnon scatte-
ring process in the presence of an external pumping field
of different magnitudes. We show that the magnons initial-
ly pumped to the low wavenumber region of the perpen-
dicular spin-wave branch (magnon wavevector 0 )⊥q H
scatter into the lowest energy states located at the longitu-
dinal branch 0( || )q H passing through both dipolar and
exchange parts of the spectrum. With increase in the
pumping power the initially pumped magnons occupy the
large wavenumber region of the transverse spin-wave
branch and transition to the bottom occurs mainly through
the exchange region of spectrum. Such a thermalization
can be followed by the formation of a magnon Bose–
Einstein condensate.
The measurements were performed using a low-
damping YIG film by means of a combined microwave
and Brillouin light scattering (BLS) setup schematically
shown in Fig. 2. The YIG film of 6.7 µm thickness with a
saturation magnetization 4πMs = 1750 G was grown in the
(111) crystallographic plane on a gallium gadolinium gar-
net substrate by liquid phase epitaxy. The YIG-film was
in-plane magnetized along the 110〈 〉 axis. In this case, the
crystallographic anisotropy fields can be easily considered
in the calculation of the magnon spectrum shown in Fig. 1
[25]. The sample with lateral dimensions of 2·10 mm2 was
placed on top of a 50 µm wide microstrip resonator, which
was used to induce the pumping Oersted field.
The resonator was driven by microwave pulses at a carri-
er frequency of = /2 = 13.74 GHzp pf ω π with peak pow-
ers pP ranging from 1.25 W to 40 W. The chosen pump
pulse duration of 100 ns was sufficiently long to observe the
response of the spin-wave system even at the lowest applied
pumping powers. A pulse repetition time of 12.5 µs was
chosen to give the system enough time to relax to the ground
state and to prevent microwave-heating effects.
The magnon dynamics was analyzed by means of
time- and wavevector-resolved BLS spectroscopy [26]
with a time resolution of 1 ns. Both the sample and the
magnetic system are mounted on a rotating stage in order
to be able to change the angle Θq|| between the YIG sam-
ple and the probing laser beam holding the magnetization
conditions constant. By varying this angle, spin waves
with different wavevectors parallel to the external mag-
netic field 0H can be resolved. A three-dimensional piv-
otal lever utilizing a small xyz-moving stage (not shown
in Fig. 2) allows for a precise positioning of the sample.
The magnetic system comprises two NdFeB permanent
magnets embedded in an iron yoke. The permanent mag-
nets ensure long-term stability of the bias magnetic field
in combination with a sufficiently large field strength.
Iron shunts are used to tune the magnetic field strength
inside the pole gap. For the automated fine control of the
bias magnetic field a small electric coil in combination
with a Hall sensor is installed. As a result, the magnetic
field strength can be adjusted in the range from 1200 to
2100 Oe with an accuracy better than 0.5 Oe. The prob-
ing light beam of 532 nm wavelength and 32 mW power
is generated by a single-mode solid-state laser. The inci-
dent light is redirected by a beam-splitter cube and fo-
cused by a precise aspherical objective onto the sample.
The diameter of the focal point is about 25 µm. The same
objective collects the inelastically scattered light. A pin-
hole placed behind the objective and the beam-splitter
cube assures that only the light coming directly through
the middle of the objective can pass to the interferometer.
It increases the sensitivity of the optical setup to the small
declination of the scattered light, and consequently im-
proves the wavevector resolution. For a proper focusing
and spatial stabilization the setup is equipped with a CCD
camera and a white light source (not shown in Fig. 2)
installed behind the beam-splitter cube and enabling di-
rect observation of the sample. Described setup allowed
us to resolve wide longitudinal wavenumber range up to
15·104 rad/cm. The wavevector resolution was set to
2·103 rad/cm by a pinhole in the path to the interferome-
ter. In the reported experiment, the frequency resolution
of the BLS setup is 100 MHz.
In course of the parallel pumping process the photons of
the microwave pumping field split into magnon pairs with
opposite wavevectors at half of the pumping frequency as
Fig. 2. (Color online) Geometry of the experiment. The setup
allows to vary the probing laser beam incidence angle Θq|| allow-
ing for wavevector resolution. The YIG sample is placed on top
of a microstrip resonator which is fabricated on top of an alumina
substrate. Microwave pumping pulses are applied to the
microstrip in order to drive the spin-wave system. The probing
laser beam is directed through a beam splitter and focused by
objective on to the YIG sample. The inelastically scattered light
is redirected to the Fabry–Pérot interferometer. An additional
pinhole is used to increase the wavenumber resolution.
1026 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10
Formation of Bose-Einstein magnon condensate via dipolar and exchange thermalization channels
illustrated in Fig. 1. The strength of the bias field
0 = 1735H Oe has been chosen to allow for a pumping of
the magnon pairs slightly above the FMR frequency. In
this case, firstly, the parallel pumping achieves its highest
efficiency because the magnons are pumped to the trans-
versal spectral branch, which has the lowest threshold of
the parametric instability [6]. Secondly, no kinetic instabil-
ity process, which corresponds to a one-step scattering of
the parametric magnons to the bottom of the spin-wave
spectrum [27] and which may disturb the formation of the
magnon BEC, is allowed due to energy and momentum
conservation [28].
Let us address now the formation of the magnon BEC
and specifically a question regarding the influence of the
injected magnon quantity (i.e., different microwave pump-
ing powers Pp) on the transition of thermalized magnons to
the bottom of the spectrum. Figure 3 represents a wave-
number-resolved BLS intensity maps of the time evolution
of gaseous magnons with q || H0 within the frequency range
from 4.7 GHz to 5.7 GHz, which includes the bottom of the
magnon spectrum. The spectra are normalized to their max-
imum. The intensity scale is logarithmic, with low intensi-
ties displayed in blue and high displayed in red. Edges of the
microwave pulse are marked with vertical white dot lines at
0 and 100 ns. A dark gap, which is visible in Fig. 3(a)–(d) in
all four measured spectra at wavevector of 7.9·104 rad/cm,
corresponds to the area of hybridization between the funda-
mental spin-wave mode and the clockwise-polarized phonon
branch [4]. The formation of the magnon BECs is clearly
visible at 4·104 rad/cm in Fig. 3(a)–(c) after switching off
the microwave pump pulse. However, during the action of
the pump pulse the formation of the magnon BEC is pre-
vented by the extremely high effective temperature of the
parametrically driven magnon gas [23]. After the end of the
pumping, the magnon temperature rapidly decreases due to
the evaporative supercooling process [23] and magnons
form the Bose–Einstein condensate. It is worth to note that
the magnon BEC formation is not observed in Fig. 3(d),
because the microwave pumping power does not exceed the
threshold of the magnon Bose–Einstein condensation.
In the case of the highest pumping power Pp = 40 W (see
Fig. 3(a)), the first magnons appear at the bottom of the
Fig. 3. (Color online) Dependence of the density of thermalized magnons near the bottom of the spin-wave spectrum on the pumping
power Pp, W: 40 (a), 10 (b), 2.5 (c), 1.25 (d). Vertical dotted lines mark the moments when the microwave pumping pulse was switched
on and off. Dashed horizontal lines indicate the position of the global energy minima at |q| = 4·104 rad/cm in the spin-wave spectrum.
Red curves in panels (a)–(d) schematically show the time dependent transition paths of dipolar and exchange thermalized magnons to
the bottom of the spin-wave spectrum. The formation of the magnon BEC at the global energy minima can be observed in the panels
(a)–(c). The gap in the magnon density distribution is visible in all panels around the dashed-dotted lines indicating the magnon–phonon
hybridization area (see also Fig. 1).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10 1027
D.A. Bozhko, P. Clausen, A.V. Chumak, Yu.V. Kobljanskyj, B. Hillebrands, and A.A. Serga
spin-wave spectrum already 30 ns after the pump pulse in
the entire wavevector range of (1–15)·104 rad/cm. The short
transition time is a consequence of the high rate of magnon-
scattering processes due to the large population of the prima-
ry group of magnons. The four-magnon scattering processes,
which are caused by the exchange interaction (see the fourth
term in Eq. (2)) constitutes the main mechanism for the
magnons gas thermalization. This process conserves the
number of magnons in the gas and leads to the redistribution
of magnons in a phase space. The center of the density dis-
tribution of the initially thermalized magnons lies in the ex-
change region of the spectrum 7·104 rad/cm. In the same
time the magnon density in a range of (1–4)·104 rad/cm is
relatively weak (take, please, into account the used loga-
rithmic intensity scale). In the case of the lower micro-
wave pumping power Pp = 10 W shown in Fig. 3(b), the
first magnons appear after 50 ns delay time in the dipo-
lar thermalization channel in the wavenumber range of
(1–5)·104 rad/cm and only starting from 75 ns the ex-
change channel is populated (the corresponding therma-
lization channels are indicated with red dashed guide-
lines for the eye in Fig. 3(a)–(d)). In Fig. 3(c) (pumping
power Pp = 2.5 W), the first magnons appear only after
switching off the pump pulse with 105 ns delay — the
smaller density of the initially injected magnons results
in a lower four-magnon scattering rate and causes, thus,
the longer termalization time. As previous, the first mag-
nons appears in the dipolar spectral region (∼ 2·104 rad/cm).
In Fig. 3(d) the pumping power is decreased to Pp =
= 1.25 W. In this case, the densities of thermalized magnons
are very weak but still detectable. The two distinct areas of
thermalization, which are separated by the energy minimum
at a wavevector of 4·104 rad/cm, are well recognizable. The
first magnons, which appear in the dipolar dominated region
in the wavevector range of (1–4)·104 rad/cm, are visible
beginning from 175 ns, after the pump pulse is switched off.
In the wavevector range (4–15)·104 rad/cm belonging to the
exchange-dominated spectral area, the magnon density rises
to a detectable level starting from 175 ns. The observed dy-
namics can be summarized in the following way. One can
admit, that the relative time shift of the dipolar therma-
lization channel practically does not depend on applied mi-
crowave power. In the same time, thermalized magnons
appear in the exchange spectral area with a delay that
strongly depends on the pumping power. This delay shortens
with increasing of the pumped magnons density.
The explanation of observed thermalization behavior is
following. As it was already mentioned, the strength of the
external magnetic field H0 was chosen in such a way that in
the case of the highest pump power the parametric magnons
were injected into the exchange spectral area just above the
FMR frequency. It is necessary to note that in this case the
spin-wave spectrum is already shifted to lower frequencies
by a huge amount of the injected magnons reducing the
magnetization Mz in accordance with Eq. (3). For a small
quantity of the pumped magnons this shift is less pro-
nounced and the spectral position of the magnon injection is
located at or even below the FMR frequency. As a result, at
the beginning of the pumping process, the parametric
magnons are injected into the long-wavelength dipolar area
[29] in all experimental cases presented in Fig. 3. This ex-
plains why in Fig. 3(a)–(d) the dipolar thermalization area
always gets populated in the same way — as the magnon
density increases during the action of the pumping pulse, the
spectrum shifts down and the pumping point moves to the
exchange area of the spectrum. As a result no more magnons
are injected into the dipolar area. Thus, the resulting popu-
lation of this thermalization channel does not change sig-
nificantly with the increase in the pumping power from
Pp = 2.5 W to 40 W.
Another situation occurs in the exchange thermalization
region. With increase of the pumping power, magnon den-
sity increases faster, so this region starts to be populated
earlier. Moreover, since the magnon number increases dur-
ing the pumping process, injection points continuously
shifts towards the higher wavevectors. This leads to the
wide spreading of thermalized magnons in the phase space
(see exchange areas in Fig. 3(a)–(c)), and thus enhances
efficiency of the magnon gas thermalization. In addition,
as it is shown in Ref. 30, at high pumping powers a second
group of short-wavelength magnons is excited at a half of
the pumping frequency significantly increasing the popula-
tion of the exchange thermalization channel.
Worth to mention, that the injection of magnons below
the FMR frequency and subsequent thermalization has al-
ready been investigated experimentally in 2008 by Demidov
et al. [20] and theoretically in [24]. The pumping conditions
were set to inject magnons far below (∼0.5 GHz) the FMR
frequency, ensuring the efficient population of the dipolar
region of the spectrum. However, an accessible wavevector
range in the experiment presented in Ref. 20 was not wide
enough to probe the exchange region of the spectrum.
In conclusion, we investigated the thermalization of the
parametrically driven magnon gas and subsequent magnon
Bose–Einstein formation in a phase space. Our experi-
mental findings suggest two channels of the thermalization
process, which pass through dipolar and exchange areas of
the spin-wave spectrum. During the pumping process, the
increasing magnon density causes shift of the spectrum due
to the change in the saturation magnetization. This effect
leads to the corresponding shift towards higher wave-
vectors in the position of parametrically injected magnon
pairs. It has been shown, that the dipolar thermalization
channel doesn't change its qualitative behavior with in-
crease of the pumping power. In the same time, the ex-
change channel becomes dominating at high pumping
powers.
Financial support by the DFG within the SFB/TR49 and
Graduate School Materials Science in Mainz is gratefully
acknowledged.
1028 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10
Formation of Bose-Einstein magnon condensate via dipolar and exchange thermalization channels
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