Kinetic instability and bose condensation of nonequilibrium magnons

ELSEVIER

Journal of Magnetism and Magnetic Materials 132 (1994) 180-184

Kinetic instability and bose condensation of nonequilibrium magnons G.A. Melkov

a, V.L. Safonov

by*, A.Yu

Taranenko

a, S.V. Sholom

a

aDepartment of Radrophysrcs, Kleb State Umvers@, 252127 KU%, Ukrame b Institute of Molecular Physzcs, Russian Research Center “Kurchatov Institute’: 123182 Moscow, Russia (Received 31 August 1993)

Abstract Several models m which a microwave radlatlon of the frequency w1 arises from the bottom of spm-wave spectrum under condltlon of parametric resonance of magnons of the frequency w,/2 are consldered Weak radiation of the frequency o2 = wp + o1 has been detected m thm ferrite film This radiation can be treated m favour of both the theory of kinetic mstablhty and the thermodynamic theory of parametric resonance of magnons Posslblhty of Bose condensation of nonequlhbrmm magnons IS discussed

1. Introduction

Spm waves (and their quanta magnons) serve as a very smtable image for a descrlptlon of different perturbations m magneto-ordered crystals The most powerful tool for the spm wave research 1s sure to be their parametric excltatlon by means of microwave field In the present paper we shall consider experlmental and theoretlcal aspects of noneqmhbrmm magnons system under intense pump We shall mamly concern phenomena m ferromagnet as the simple model for iron yttrium garnet (YIG) Lavrmenko et al [l] predlcted and detected experimentally a new phenomenon m YIG at room temperature microwave radlatlon from the bottom of spm-wave spectrum This radlatlon

* Correspondmg

author

arises when the parallel pump microwave amphtude h exceeds a certam value h, which 1s it, (6-10) hth, where h,, IS the parametric excltatlon threshold amplitude The radlatlon frequency o1 approximately equals the mmlmum frequency w, of the spm waves The latter 1s defined by the followmg expression ~*=y(H,-K

M,)

(1)

Here H, 1s the external magnetic field, N, 1s the demagnetlzatlon factor of the sample, M, 1s the magnetization and y 1s the gyromagnetlc ratio Later Krutsenko et al [2] have detected an analogous phenomenon at helmm temperature The radiation from the bottom of spin-wave spectrum was also observed at arbitrary pumpmg [3l and m the case of nonlmear ferromagnetic resonance [4] The appearance of noisy radiation from the bottom of spin-wave spectrum means that the occupation number of magnons with the fre-

0304-8853/94/$07 00 0 1994 Elsevler Science B V All nghts reserved SSDI 0304-8853(93)E0609-G

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quency w1 and wave vector k, = 0 substantially exceeds their thermal level Presently three theoretical explanations for this magnon accumulation are known The first 1s so-called ‘kinetic’ mstablllty which was proposed and developed m [l&7] on the base of S theory [8] The second 1s the thermodynamic theory of parametric resonance of magnons [9-111 The third 1s the Bose condensation of nonequlhbrmm magnons [ 12-141 Kmetlc mstablhty arises when m the sample a highly non-equlhbrmm dlstrlbutlon of spm waves exists m the phase space Such a dlstrlbutlon can be created, for mstance, by means of parallel pumping of the frequency wp The presence of superheated domains m the spm-wave spectrum leads to energy flows to the new, previously equlhbrmm, parts of the spectrum These flows, m the case of four-magnon nonlmeanty, are directed to the spm waves with frequencies o1 and w2 and wave vectors k, and k, w,+w,=w,/2+0,/2, k, +k,=k’,+k;,

w,
= Ik;l

(2)

=k,,

where k, 1s the wave vector of parametrically excited spm waves (PSW) According to the theory of kmetlc mstablhty, the effective damping y(k,) of spm wave of the frequency w1 should dlmmlsh as y(k1) = Yo(k1) - +,

N,j2/@,

(3)

Here y&k) 1s the equlhbrmm damping, Tp is the four-magnon interaction coefficient, N, 1s the number of parametric magnons, up 1s the group velocity of PSW From (3) follows that d N, is greater than the value N,, at which y(k,) = 0, then the amplitude of spm waves of the frequency o1 will grow exponentially According to [1,5--71, w1 1: w, and k, --f 0 It 1s obvious that spm waves with small wave vectors can be transformed mto microwave radiation from the sample The dlstrlbutlon function for excited magnons beyond the kinetic mstablhty 1s shown m Fig 1 To the present time the concept of kinetic mstablhty was widely used for interpretation of expenments with highly pumped spm waves [15-M] However there are several contradlctlons For example, m the experiment [18] It was expected

181

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N.J-Fig 1 Spectrum and dlstrlbutlon waves beyond the kmetlc mstablhty

function [6]

of excited

spm

that the threshold h$) a y(k,) of an additional microwave field of the frequency w, = 20, should dlmmlsh m the vlcmlty of kinetic mstablhty But the threshold h$,) increased and did not influence the frequency of radiation The thermodynamic approach m a rotating coordinate frame has been developed m the theory of parametric resonance of magnons m order to describe strongly driven spin-wave systems According to the theory [9-111 all forced magnetic vibrations can be divided mto two parts The first corresponds to the magnons coherent with the pump field The second 1s the noisy deviations bemg described by an equlhbrmm quasi particle gas with a certain temperature T and chemical potential p Dlstrlbutlon function of magnons m a saturated state (when the excited system does not absorb the microwave power) 1s

PSW

L L_ 4

(-442

CJ

Fig 2 Dlstrlbutlon of excited spm waves m accordance with thermodynamic theory of parametric resonance of magnons

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shown m Fig 2 Accumulation of low-frequency magnons m this model arises due to speclflclty of occupation numbers at w + o1 N, = {exp[( ho - p)/k,T]

- 1)-l,

Note that m this case there 1s no necessity at all for reahzatlon of Eqs (2) Govorkov and Tulm [19] studled parametric pumpmg of nuclear magnons m antlferromagnet CsMnF, m which the Eqs (2) are not valid They detected a radlatlon from the bottom of nuclear spm-wave spectrum at h/hth = 10 So this fact testifies m favour of apphcablhty of thermodynamic theory of parametric resonance of magnons [9,10] Thermodynamic reasons are also used m an Idea of Bose condensation of noneqmhbrmm magnons Parametrically pumped spm waves bemg concentrated m a narrow frequency range play for the other points of spectrum a role of incoherent pump by means of four-magnon scattermg processes As a result a quaslequlhbrmm dlstrlbutlon function of the form (4) 1s settled Effective chemical potential increases from I_L= 0 m absence of external pump to I_L= tiw, at some critical pump power Then a coherent magnon state appears on the bottom of spm-wave spectrum and the correspondmg coherent radiation arises In the present paper we made an attempt to fmd out an accumulation of magnons with the frequency w2 > 0,/2 The substantial grouth of the magnon population m this region may be treated m favour of the theory of kinetic mstabllItY

2. Expenment The above magnon accumulation can be detected by the correspondmg microwave radlatlon However a great value of wavevector k, prevents such a detection In order to avoid this prevention, the experiment has been carried out with tangentially magnetized YIG film m which the frequency w,/2 1s less than the upper (at k + 0) spin-wave boundary frequency w I (see Fig 3)

0

1

k

Fig 3 Spectrum of backward volume spm waves (quahtatlvely) for the tangentially magnetized ferrite film Sohd lines correspond to branches with different dlstrlbutlon of magnetlzatlon m transverse plane Pomts mdlcate waves excited beyond the kmetlc mstablhty

Parametric mstablhty excites PSW of the frequency w,/2 and wave vector k, - lo5 cm-’ In the case of kinetic mstablhty PSW create waves with w1 and k, - lo3 cm-‘, and o2 and k, - lo5 can be cm -’ The radlatlon from (02, k&waves detected as a result of then two-magnon scattermg on lmperfectlons of the sample This leads to an excitation of new waves with the same frequency, but with the lesser value of wave vector (m Fig 3 these are the waves of the frequency w2 with the indexes n,, n2, n3 at k + 0) Such a scattering has been already consldered m Ref [17] for studying PSW with k - lo5 cm-‘> Film of 29 9 Frn thickness under parallel mlcrowave pumping of the frequency w,/2~ = 9 52 GHz, was studied at room temperature Parametric excltatlon m the film was produced by means of an open dielectric cavity with TM,,, oscillation mode The radlatlon induced m the wire antenna an electromagnetic slgnal which was detected by means of a microwave receiver of - lo-l4 W sensitivity We have detected the radiation of w2 from the film at the limit sensltlvlty of the apparatus The results of the experiment are given m Fig 4 At every field H, there are two radiated frequencies, one with wI near the bottom of spin-wave

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f,GHz

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3. Dlscussion

6

Fig 4 The magnetic field dependence for frequencies of electromagnetic radlatlon f = o/2a, (1) w,, (2) o = tip - W, The circles are the experlmental data

spectrum, and the other with w2 near the frequency w = wP - w, as it follows from the theory of kinetic mstablhty According to this theory the intensity of radiation of w2 must be 2-3 orders weaker than that of w, So, It has been observed when the pump power was 3-6 dB greater than the critical power of kinetic mstablhty Note that the spectrum of irradiation of the frequency w1 1s several times wider than that of frequency w2 it 1s about 400 MHz at h/hth = 7 Thus, the results of our experiment can be explained m the framework of the theory of klnetlc mstablhty However one can propose a quite different and natural interpretation of the origin of radlatlon with the frequency w2 The corresponding nonlinear process 1s shown m Fig 5 It means that two uniform osclllatlons with frequencies wP and w1 being mixed on magnetic nonlmearlties give a harmonic o2 = wP + w1 Such a process can take place at any mechanism of accumulation of magnons with w1 2: w,

From the above analysis follows that m order to choose an adequate model of an orlgm of radiation from the bottom of the spin-wave spectrum we need additional mformatlon For example, there 1s still an open question about bottom magnons damping which, m prmclple, can be determined by means of a transverse microwave magnetic field One of the most promlsmg ways of studymg the orlgm of the radlatlon from the spin-wave spectrum bottom appears to he m the use of modulation method of mvestlgatlon of parametric magnons (see e g Ref [20]) So with the help of this method it became possible to determine the nature of mstablhty of the statlonary state of parametrically pumped nuclear magnons [21] It should be also mentioned that one and the same spin-wave Hamlltoman with four-magnon nonlinear@ is the basis of all models considered here We can regard these models as approximate solutions of a nonequlhbrmm many-body problem So, experimental and theoretical research m this field could enrich our understandmg m the physics of highly driven systems The special interest between the discussed questions 1s certainly to have a posslblhty of Bose condensation of nonequlhbrmm magnons This phenomenon 1s very attractive both from applied standpoint (we shall obtain a kmd of transformer from noise radiation mto coherent radiation) and from fundamental standpomt The critical magnon density 1s defined by the followmg equation n, = /{exp[ r‘z(wk- w,)/k,T,]

- l}-’ dk

/(2r)3Y where T, 1s the thermal In order to estimate the spin-wave spectrum = w0 + w~,(u~)~, where quency and a isthe cell Rg 5 Process of mIcrowave uradlatlon frequency wz by means of three-magnon

(wavy hne) anmhllatlon

of the

(5) bath temperature Eq (5) one can consider m the simplest form We o,, 1s the exchange frelinear size Then we have

n,=O 06a-3(k,T,,/hm,,)3'2,

(6)

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or for YIG (where a = lo-’ cm, hw,,/k,

of Magnetwn and Magnetic Materials 132 (1994) 180-184

= 50 JS)

nC[cm-“1 = 1 7 10”(T,[K])3’2 So, we can expect Bose condensation of magnons at low (helium) temperatures as far as the values n - 101* cmP3 are typical for strongly pumped spm waves In order to prevent superheating of the system, the low-frequency magnons should be pumped only One more transverse microwave field of the frequency w, can play the role of ‘crystahzatlon nucleus’ for the condensatmg magnons In prmclple, the experiment may be supplemented by extra coohng of the spm wave system, for example, by means of adiabatic reduction of external magnetic field Thus, the above analysis shows that Bose condensation of noneqmhbrmm magnons m YIG can be attamed by means of purposeful experiments

Acknowledgement

The authors wish to thank Dr A V Andrlenko for a helpful discussion

References [l] AV Lavrmenko, V S L’vov, GA Melkov and V B Cherepanov, Zh Eksp Teor FIZ 81 (1981) 1022 [Sov Phys JETP 54 (1981) 5421 [2] I V Krutsenko, A V Lavrmenko and G A Melkov, FIZ Nlzk Temp 9 (1983) 1289 [Sov Phys Low Temp Phys 9 (1983) 6641

[3] A V Lavrmenko,

GA Melkov and A Yu Taranenko, FIZ Tverd Tela 26 (1984) 1499 [Sov Phys Sohd State 26 (1984) No 51 Zh Eksp Teor Rz [41 GA Melkov and A Yu Taranenko, 91 (1986) 1007 [Sov Phys JETP 64 (1986) 5921 Zh Eksp Teor FIZ 81 [51 V S L’vov and V B Cherepanov, (1981) 1406 [Sov Phys JETP 54 (1981) 7461 Kmetlcs of strongly excited magnons, 161 V B Cherepanov, D SC Thesis, Novoslblrsk, 1986 [71 V S L’vov, Nonlinear Spin Waves (Nauka, Moscow, 1987, m Russian), Nonlinear Dynamics and Kinetics of Magnons (Springer Series Nonlinear Dynamics, 1991) Bl V E Zakharov, V S L’vov and S S Starobmets, Usp FIZ Nauk 114 (1974) 609 [Sov Phys Usp 17 (1975) 8961 191Yu D Kalafati and VL Safonov, J Phys France 50 (1989) 1157 DOI Yu D Kalafatl and V L Safonov, Zh Eksp Teor FIZ 95 (1989) 2009 [Sov Phys JETP 68 (1989) 11621 1111V L Safonov, Physica A 188 (1992) 675 [=I Yu D Kalafatl and V L Safonov, Pls’ma Zh Eksp FIZ 50 (1989) 135 [JETP Lett 50 (1989) 1491 [131 Yu D Kalafatl and V L Safonov, Zh Eksp Teor FIZ 100 (1991) 1511 [Sov Phys JETP 73 (1991) 8361 [141 Yu D Kalafatl and V L Safonov, JMMM 123 (1993) 184 Zh Eksp Teor D51 A Yu Taranenko and V B Cherepanov, Flz 95 (1989) 1810 [Sov Phys JETP 68 (1989) 10461 1161V S Lutovmov, G A Melkov, A Yu Taranenko and V B Cherepanov, Zh Eksp Teor FIZ 95 (1989) 760 [Sov Phys JETP 68 (1989) 4321 [171 GA Melkov and S V Sholom, Zh Eksp Teor FIZ 99 (1991) 610 [Sov Phys JETP 72 (1991) 3411 1181A V Lavrmenko, G A Melkov and Fal’kovlch, Zh Eksp Teor FIZ 87 (1984) 205 [Sov Phys JETP 60 (1984) 1183 [191 S A Govorkov and VA Tuhn, Zh Eksp Teor FIZ 95 (1989) 1398 [Sov Phys JETP 68 (1989) 8071 V L Safonov and A Yu Yakubovskn, ml A V Andrlenko, Zh Eksp Teor FIZ 93 (1987) 907 [Sov Phys JETP 66 (1987) 5111 [21] AV Andrlenko, VI Ozhogm, LV Podd’yakov, V L Safonov and A Yu Yakubovsku, Zh Eksp Teor Flz 94 (1988) 251 [Sov Phys JETP 67 (1988) 1411