Journal of Magnetism and Magnetic Materials 452 (2018) 30–34

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Journal of Magnetism and Magnetic Materials

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Magnon condensation and spin superfluidity ⇑ Yury M. Bunkov a, Vladimir L. Safonov b,c, a Kazan Federal University, Kremlevskaya 18, 420008 Kazan, Russia b Mag and Bio Dynamics, Inc., Granbury, TX 76049, USA c Physical Science Department, Tarrant County College, South Campus, Fort Worth, TX 76119, USA article info abstract

Article history: We consider the Bose-Einstein condensation (BEC) of quasi-equilibrium magnons which leads to spin Received 24 August 2017 superfluidity, the coherent quantum transfer of magnetization in magnetic material. The critical condi- Received in revised form 12 November 2017 tions for excited magnon density in ferro- and antiferromagnets, bulk and thin films, are estimated Accepted 7 December 2017 and discussed. It was demonstrated that only the highly populated region of the spectrum is responsible Available online 8 December 2017 for the emergence of any BEC. This finding substantially simplifies the BEC theoretical analysis and is surely to be used for simulations. It is shown that the conditions of magnon BEC in the perpendicular Keywords: magnetized YIG thin film is fulfillied at small angle, when signals are treated as excited spin waves. Bose-Einstein condensation We also predict that the magnon BEC should occur in the antiferromagnetic hematite at room tempera- Magnons YIG ture at much lower excited magnon density compared to that of ferromagnetic YIG. Bogoliubov’s theory Hematite of Bose-Einstein condensate is generalized to the case of multi-particle interactions. The six-magnon repulsive interaction may be responsible for the BEC stability in ferro- and antiferromagnets where the four-magnon interaction is attractive. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction interesting phenomenon for both fundamental and applied studies. It should be emphasized that the main paradigm of magnetic Spin deviations from the magnetic order in a magnetic material dynamics, the Landau-Lifshitz-Gilbert equation, does not contain (ferromagnet, antiferromagnet or ferrites) are manifested by spin complete information about the Bose-Einstein condensate of mag- waves and their quanta, magnons. Magnons are quasiparticles nons. BEC is the principal result of quantum statistics and for mag- which represent a very useful quantum theoretical tool to describe nons it can exist at room and even higher temperatures. various dynamic and thermodynamic processes in magnets in For the first time the existence of quasi-equilibrium Bose con- terms of magnon gas. Since magnons have magnetic moments, densate was demonstrated in the experiment with nuclear mag- the external alternating magnetic field can excite extra magnons nons in the superfluid antiferromagnetic liquid crystal 3He-B in and increase the disorder in the magnetic system. However, in cer- 1984 [3]. The theoretical explanation of this phenomenon [4] tain conditions, the increase of magnon density leads to a new was developed on the basis of global Ginzburg-Landau energy state, so-called, magnon condensate, in which a macroscopic num- potential. A similar approach was later developed to explain the ber of magnons forms a coherent quantum state (see, e.g., [1,2]). atomic BEC [5]. In the experiments with an antiferromagnetic This macroscopic state can significantly change the properties of 3He-B, the following phenomena were observed: a) transport of magnon gas, its dynamics and transport. An example is the phe- magnetization by spin supercurrent between two cells with mag- nomenon of quasi-equilibrium Bose-Einstein condensation (BEC) non BEC; b) phase-slip processes at the critical current; c) spin cur- of excited magnons on the bottom of their spectrum as a single- rent Josephson effect; d) spin current vortex formation; d) particle long-range coherent state of quantum liquid. This state Goldstone modes of magnon BEC oscillations. Comprehensive generate an uniform long-lived precession of spins formed by reviews of these studies can be found in Refs.[6–8]. Currently mag- quantum specificity of the magnon gas when the magnon density non BEC found in different magnetic systems: i) in antiferromag- exceeds certain critical value. The spatial gradients of this state netic superfluid 3He-A [9,10]; ii) in in-plane magnetized yttrium exhibit a spin superfluidity, the non-potential transport of iron garnet Y3Fe5O12 (YIG) film (with two minima in the magnon deflected magnetization. The spin superfluidity is an extremely spectrum) [11,12] and in normally magnetized YIG film [13]; iii)

in antiferromagnets MnCO3 and CsMnF3 with Suhl-Nakamura ⇑ Corresponding author. indirect nuclear spin-spin interaction [14–16]. An explanation of E-mail address: [email protected] (V.L. Safonov). analogy between the atomic and magnon BEC is given in Ref. [17]. https://doi.org/10.1016/j.jmmm.2017.12.029 0304-8853/Ó 2017 Elsevier B.V. All rights reserved. Y.M. Bunkov, V.L. Safonov / Journal of Magnetism and Magnetic Materials 452 (2018) 30–34 31

= A microscopic theory of quasi-equilibrium magnon BEC was h2 N 2 3 p4=3 T j~ ; j~ 3:65: 6 developed in Refs. [18–21] (‘‘KS theory”). It was predicted that BEC ’ 0 0 ¼ 1=3 ’ ð Þ kBm V s 2 the external strong pumping of magnons leads to a rapid growth of magnon density and saturation. This state can be considered We see that the only difference between Eqs. (4 and (6) is a % in terms of weakly non-ideal gas of ‘‘dressed” magnons in a ther- slightly different ( 10 ) numerical factor. modynamic quasi-equilibrium with an effective chemical potential The fact that the high population Eq. (5) is dominant does not l and effective temperature T. The dressed magnon energy is mean that the BEC phenomenon is a classical one. The criterion ð0Þ ð0Þ of classical Maxwell-Boltzmann statistics exp ðÞl=k T 1 in this defined by e ¼ e þ de , where e ¼ hx is the energy spectrum B k k k k k case can be written as (see, e.g., [27]): of bare magnons and dek is the energy shift due to magnon gas "#Z nonlinearities. Magnon-magnon scattering processes retain the 3 1 V s ek d k total number of dressed magnons in the system and hold their dis- exp ðÞ¼l=kBT exp 1; ð7Þ N k T p 3 tribution function of the form B ð2 Þ 1 or e l k : ! nk ¼ exp 1 ð1Þ 3=2 kBT p2 N k3 N 2 h ; T ¼ 1 ð8Þ The instability at l ¼ min ek in the quasi-equilibrium magnetic V s V s mkBT system is an analog of BEC phenomenon for the bottom dressed k magnons. The distribution (1) seems to underlie the phenomenon where T is the thermal de Broglie wavelength. Substituting BEC of spin superfluidity, since it nullifies the integral of four-magnon temperature Eqs. (6) into (8), we obtain the opposite relation: : > collisions 2 26 1, which obviously corresponds to a degenerate bose gas. Z ð4Þ 3 3 3 2 I fnkg/ d k1d k2d k3jjU4ðk; k1; k2; k3Þ 3. BEC of magnons

ðnk þ 1Þðnk1 þ 1Þnk3 nk4 nknk1 ðnk2 þ 1Þðnk3 þ 1Þ Now let us consider a Bose-Einstein condensation of so-called, d e e e e D ‘‘dressed” magnons as an instability in the externally pumped ð k þ k1 k2 k3 Þ ðk þ k1 k2 k3Þ quasi-equilibrium magnon gas. The total number of magnons ð2Þ Nðl; TÞ is equal to the number of thermal magnons Nð0; TÞ at a and thus this energy loss channel vanishes. given temperature T and the number of magnons Np created by KS theory qualitatively explained the parallel pumping experi- external pumping. So far as the energy shift of dressed magnons ments ([22,23] (YIG at room temperature) and [24] (nuclear mag- is usually much less than the energy of bare magnons ð0Þ ð0Þ nons in CsMnF3), where the accumulation of magnons at the de e e e k min k , for simplicity we can approximate k ’ k . bottom of the spin wave spectrum was observed. One and a half decade later, purposeful experiment [11] directly demonstrated 3.1. BEC in a ferromagnet BEC of quasi equilibrium magnons in the thin film of YIG. Subse- quent experimental studies have shown qualitative correspon- Consider a ferromagnet with the quadratic spectrum (we dence with the predicted distribution of excited magnons [25] neglect details of the dipole-dipole interactions): and agreement with the BEC under noisy pumping [26]. e e e 2: In this paper we analyze critical conditions of quasi-equilibrium k ¼ 0 þ exðÞak ð9Þ magnon BEC in ferro- and antiferromagnets, bulk and thin films, Here eex is the exchange interaction constant and a is the ele- and evaluate the possibilities of their experimental achievements. mentary cell linear size. The quasi-equilibrium BEC will be mainly determined by pumping if the number of pumped magnons is

2. BEC of bose particles much greater than the thermal magnon number Np Nð0; TÞ.In this case we obtain an analog of Eq. (4): Let us first briefly discuss BEC of real bose particles. Their distri- e 2=3 e 2= 2 ex 3 Np bution is defined by Eq. (1), where k ¼ðhkÞ 2m is the kinetic TBEC ¼ j0 a ; ð10Þ energy of particle with the wave vector k and mass m. The total kB V s number of bosons in the system is or, Z 3 = d k 2e N 2 3 Nðl; TÞ¼N ¼ V n ; ð3Þ j~ ex 3 p s k p 3 TBEC ’ 0 a ð11Þ ð2 Þ kB V s where V s is the volume of the system. For the critical condition in the high-population approximation. l ¼ min ek, from (3) follows well-known formula for the BEC critical The above formula, however, does not work for a BEC estimate temperature versus the density of bosons: if Np K Nð0; TÞ. Using a high-population approximation, we write 2 2=3 h N 2p Np ¼ Nðl; TÞNð0; TÞ T j ; j 3:31: 4 BEC ¼ 0 0 ¼ 2=3 ’ ð Þ Z e 2 kBm V s f 3 T 2 kBT kBT k dk ’ V s ; ð12Þ e e l e 2p2 It is interesting to note that the BEC is formed mainly by bosons 0 k k with high populations when Eq. (1) can be written as and obtain at l ¼ e0

1=2 kBT e : Np kBTBEC nk ’ ð5Þ ’ 0 ; ek l p 3 3=2 V s 4 a eex = Let us prove it by direct calculation. Substituting the high tem- e e 1 2 N p ex ex 3 p : perature population (5) into Eq. (3) and cutting the upper integral TBEC ’4 a ð13Þ kB e0 V s limit by the thermal energy eT ’ kBT, one obtains: Forbidden

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