Anisotropy-assisted magnon condensation in ferromagnetic thin films

We theoretically demonstrate that adding an easy-axis magnetic anisotropy facilitates magnon condensation in thin yttrium iron garnet (YIG) films. Dipolar interactions in a quasi-equilibrium state stabilize room-temperature magnon condensation in YIG. Even though the out-of-plane easy-axis anisotropy generally competes with the dipolar interactions, we show that adding such magnetic anisotropy may even assist the generation of the magnon condensate electrically via the spin transfer torque mechanism. We use analytical calculations and micromagnetic simulations to illustrate this effect. Our results may explain the recent experiment on Bi-doped YIG and open a pathway toward applying current-driven magnon condensation in quantum spintronics.

The study of magnon condensation is not only interesting from an academic point of view but also of great importance in various areas of emerging quantum technology and applied spintronics [15,[30][31][32][33]. Therefore, it is crucial to clarify the intricate microscopic mechanisms at play and to present theoretical proposals to electrically control the generation of magnon condensate.
At low magnon densities, the interaction between magnons is weak, and they behave as free quasiparticles.But when the magnon population increases, as in magnon condensation experiments, the interactions between magnons become stronger and more crucial.Moreover, nonlinear magnon interactions facilitate the quasi-thermalization process of injected nonequilibrium magnons.A stable and steady quasi-equilibrium magnon condensation requires an effective repulsive interaction between injected magnon quasiparticles.It is known that in a system mainly influenced by Heisenberg exchange interactions, interaction between magnons is attractive.However, it was shown that dipolar interactions may assist in the generation of a metastable double-degenerate magnon condensate in YIG [12][13][14][34][35][36][37][38][39][40][41][42].
Recently, it was theoretically shown that the quasithermalization time of magnon condensation is reduced in confined nanoscopic systems [43].It was also demonstrated that the lateral confinement in YIG thin films enhances the dipolar interaction along the propagation direction of magnons and causes a deeper band depth, i.e., the difference between the ferromagnetic resonance (FMR) and magnon band minima.Increasing the lifetime of the magnon condensate was attributed to this enhancement of the band depth [43].
In another recent achievement in magnon condensation experiments, Divinsky et al. [22] found evidence of condensation of magnons by spin-transfer torque mechanism.They introduced a small perpendicular magnetocrystalline anisotropy (PMA) through bismuth doping in the thin film of YIG, while the magnetic ground state still resides within the plane.This discovery opens a route toward electrical control of magnon condensation.
However, the interplay between the dipolar interactions, which was previously shown to be essential for the stability and thermalization of magnon condensation, and the counteracting out-of-plane easy-axis magnetic anisotropy is so far uncharted.In this Letter, we analyze the stability of condensate magnons in the presence of a PMA in YIG.We present simulations within the Landau-Lifshitz-Gilbert framework [44][45][46] that support our analytical calculations.
Model-.We consider a thin ferromagnetic film in the y − z plane to model YIG.The magnetic moments are directed along the z direction by an in-plane external magnetic field of strength H 0 .The magnetic potential energy of the film contains contributions from the isotropic exchange interaction H ex , Zeeman interaction H Z , dipolar interaction H dip , and additionally a PMA energy H an in the x direction, normal to the film plane.YIG has a weak in-plane easy-axis that can be neglected compared to the other energy scales in the system.The total spin Hamiltonian of the system reads, The PMA energy is given by, where K an > 0 is the easy-axis energy, ℏS j is the vector of spin operator at site j, and ℏ is the reduced Planck constant.Details of the Hamiltonian can be found in the Supplemental Material (SM) [47].The Holstein-Primakoff spin-boson transformation [48] allows us to express the spin Hamiltonian in terms of magnon creation and annihilation operators.The amplitude of the effective spin per unit cell in YIG at room temperature is large S ≈ 14.3 ≫ 1, [39,49,50], and thus we can expand the spin Hamiltonian in the inverse powers of the spin S, which is equivalent to the semiclassical regime.Up to the lowest order in nonlinear terms, the magnon Hamiltonian H of a YIG thin film can be expressed as the sum of two components: H 2 and H 4 .The former represents a noninteracting magnon gas comprising quadratic magnon operators.The latter, on the other hand, constitutes nonlinear magnon interactions characterized by quartic magnon operators; see SM for details [47].Note that three-magnon interactions are forbidden in our geometry by the conservation laws [51] Magnon dispersion of YIG with a finite PMA-.The magnon dispersion in YIG is well known and has been studied extensively in both experimental and theoretical works [2,52,53].Magnons traveling in the direction of the external magnetic field have the lowest energy.These so-called backward volume magnetostatic (BVM) magnons have a dispersion with double degenerate minima at finite wavevectors q z = ±Q.When pumping magnons into the thin film, the magnons may thermalize and eventually form a condensate state in these two degenerate minima with opposite wavevectors.
The noninteracting magnon Hamiltonian and the dispersion of BVM magnons, along the z direction, in the presence of a finite PMA reads, where ĉ † qz (ĉ qz ) are magnon creation (annihilation) operators, which are Bogoliubov bosons [47], and Here, D ex = J ex Sa 2 is the exchange constant, J ex is the Heisenberg exchange coupling and M S = γℏS/a 3 is the saturation magnetization, where γ = 1.2 × 10 −5 eV Oe −1 is the gyromagnetic ratio, and a = 12.376 Å is the lattice constant of YIG.The form factor f q = (1 − e −|qz|Lx )/(|q z |L x ) stems from dipolar interactions in a thin magnetic film with thickness L x [54,55].Figure 1 shows the effect of PMA on the magnon dispersion of YIG.PMA decreases the FMR frequency ω qz=0 , in addition to a more significant decrease in the magnon band minima ω qz=±Q .Therefore the band depth ∆ω = ω qz=0 − ω qz=±Q is increased.The position of the band minima at q z = ±Q is also shifted to larger momenta.In addition, the curvature of the minima increases as a function of the anisotropy strength.Above a critical PMA, K c2 an , the magnetic ground state is destabilized and the in-plane magnetic state becomes outof-plane.We are interested in the regime in which the magnetic ground state remains in the plane, and thus the effective saturation magnetization is positive The effect of PMA on magnon dispersion resembles the effect of confinement in the magnon spectra of YIG.In Ref. 43, it was shown that transverse confinement in a YIG thin film leads to an increase of the FMR frequency, the band depth, as well as shifting the band minima to higher momenta while the magnon band gap at the band minima is also increased.It was shown that this change of the spectrum in confined systems increases the magnon condensate lifetime.Therefore, we expect PMA to increase the magnon condensate lifetime and assist in the generation of magnon condensation.
Nonlinear magnon interactions in the presence of PMA-.To check the stability of condensate magnons at low energy, we should show that the interactions do not destabilize and destroy magnon condensation.To achieve this goal, we turn on the magnon interaction between the condensate magnons at q z = ±Q.This magnon interaction consists of intra-and inter-band contributions, , where

.). (5b)
The intraband magnon interaction, parametrized by A, preserves magnon number.However, the interband magnon interaction includes both a magnon conserving contribution, parametrized by B, and nonconserving contributions, parametrized by C and D, see SM [47].The interaction amplitudes are given by Here, N is the total number of spin sites.The dimensionless parameters α 1 , α 2 , and α 3 are related to the Bogoliubov transformation coefficients, listed in SM [47].
An off-diagonal long-rage order characterizes the condensation state.The condensate state is a macroscopic occupation of the ground state and can be represented by a classical complex field.Therefore, to analyze the stability of the magnon condensate, we perform Madelung's transform ĉ±Q → N ±Q e iϕ ±Q , in which the macroscopic condensate magnon state is described with a coherent phase ϕ ±Q and a population number N ±Q [39,40].The total number of condensed magnons is is set in the system by an external magnon pumping mechanism and is a constant.We also define the total phase as Φ = ϕ +Q + ϕ −Q .Finally, the macroscopic four-magnon interaction energy of condensed magnons is expressed as , This expression is similar to the one recently obtained without PMA [56], but the interaction amplitudes, Eq.

FIG. 2.
The analytical nonlinear interaction energy of magnon condensate state, Eq. ( 7), as a function of the PMA strength.N and Nc are the total spins and condensate magnons, respectively.K c 1 an represents the critical value of the PMA at which the sign of nonlinear interaction energy is changed.On the other hand, K c 2 an corresponds to the critical value of PMA at which the in-plane magnetic ground state becomes unstable.We set Lx = 50 nm and H0 = 1 kOe.K sim an = 0.5 µeV denotes the PMA used in our micromagnetic simulations.
We now look at the total interaction energy and amplitudes of condensate magnons in more detail.Figure 2 shows the effective interaction potential of condensate magnons as a function of the PMA.In a critical PMA strength, K c1 an , the sign of the interaction changes.This means that below K c1 an , the interaction reduces the total energy of condensate magnons while above K c1 an the interaction increases its energy.This critical anisotropy is well below the critical magnetic anisotropy strength K c2 an that destabilizes the in-plane magnetic ground state.In the following, we consider a PMA strength below the critical anisotropy K an < K c1 an .The interacting potential energy of the condensate magnons, Eq. ( 7), has five extrema, ∂V 4 (δ i , Φ i ) = 0, at, (8b) δ i = 0 indicates condensate states with symmetric magnon populations in the two magnon band minima while δ i ̸ = 0 represents states with nonsymmetrical  We plot the magnon interaction energy V4/N 2 c , Eq. ( 7), as a function of the film thickness Lx and the magnetic field strength H0, applied along the z direction.Different states are labeled based on different extrema listed in Eq. ( 8).The dashed black lines indicate the boundaries between the different condensate phases, Eq. ( 8).We set Kan = 0.5 µeV in (b).magnon populations.Whether any of these extrema represents the actual minimum of the interacting potential energy, i.e., ∂ 2 V 4 (δ i , Φ i ) > 0, depends on the system parameters.Finding these minima allows us to construct the phase diagram for magnon condensate.
Phase diagram for magnon condensate-.Now, we theoretically explore the (meta)stability of the magnon condensate as a function of the film thickness L x and the magnetic field strength H 0 , applied along the z direction in the plane, using the YIG spin parameters, see Table I.We characterize a (meta)stable state of a magnon condensate in the phase diagram as one that minimizes the interaction potential V 4 while satisfying the condition V 4 < 0, ensuring a reduction in the total magnon condensate energy at q z = ±Q through interactions.
First, we present the phase diagram for magnon condensation in YIG, in the absence of PMA, in Fig. 3a.The thinner films are expected to have a symmetric distribution of magnons between the two magnon band minima, the state with δ 1 = 0, and only thicker films with larger applied magnetic fields tend to have nonsymmetric magnon populations, the state with δ 4 ̸ = 0.This phase diagram is in agreement with previous studies [39,56].
Next, we add a PMA, with strength K an = 0.5 µeV, and plot the phase diagram of the magnon condensate in Fig. 3b for different thicknesses.Compared to the case without PMA, we see that the condensate magnons can only be stabilized for thinner films, and within our material parameters, we do not have any metastable condensation above 90nm since the sign of the total interaction energy becomes positive.In addition, we have a richer phase diagram in the presence of PMA.PMA tends to push the magnon condensate within our material parameters toward a more nonsymmetric population distribution between the two magnon band minima, states with δ 4 ̸ = 0 and δ 5 ̸ = 0. Since both minima are degenerate, there is an oscillation of magnon population between these two minima.In very thin films, less than 30 nm, we may have a symmetric condensate magnon state, δ 1 = 0, in our system.
This phase diagram shows that in the presence of a PMA, magnon condensate can be still survived as a metastable state.In addition, as we discussed earlier, a PMA increases the band depth and reduces the curvature of noninteracting magnon dispersion, see Fig. 1, leading to an enhancement of the condensate magnon lifetime.Thus, we expect that introducing a small PMA into a thin film of YIG facilitates the magnon condensation process.It is worth mentioning that the stabilizing condensated magnons in thinner films with finite PMA, is not a real problematic issue since the injection of magnons into the system by electrical means is more efficient in thin films.
Micromagnetic simulation of magnon condensate-.To validate our theoretical predictions and illustrate the facilitation of magnon condensate formation by incorporating a PMA, we conducted a series of micromagnetic simulations.Simulations were performed using MuMax 3 [57], which solves the semiclassical Landau-Lifshitz-Gilbert (LLG) equation that describes magnetization's precessional motion; see SM [47].In the limit of large S, it is important to note that we enter the semiclassical regime and hence the LLG equation may effectively capture and accurately describe the spin dynamics of the system.In our ferromagnetic thin film simulation, magnons are excited via spin-transfer torque at zero temperature, eliminating thermal magnons.The nonequilibrium magnons in the film are the result of injection of spin current across the sample surface [22].Optimal spin torque strength ensures that the magnon population reaches the critical density required for forming condensed magnons; see SM for simulation details [47].By introducing spin torque into the system, we excite magnons with varying wavevectors and frequencies, as illustrated in Figs.4(a) and 4(c).A portion of these nonequilibrium magnons undergoes thermalization through nonlinear magnon-magnon interactions, leading to the establishment of a stable and quasi-equilibrium state of condensed magnons located at the minima of the magnon band spectra, ±Q, as depicted in Figs.4(b) and 4(d).This sharp peak in the number of magnons at the band minima is a signature of magnon condensate.
The numerical simulations confirm the supportive role of PMA in the condensation process.First, there is a reduction in the threshold of spin-transfer torque necessary to inject the critical magnon density into the system, enabling the system to attain said critical magnon density even at lower torque amplitudes.Second, the final condensate magnons in the presence of the PMA are more localized around the band minima than in the absence of PMA.Simulations also indicate that PMA shifts the population of condensate magnons from a symmetric distribution between two band minima to a nonsymmetric distribution, Fig. 4.This agrees with the analytical phase diagram in Fig. 3(b).
Summary and concluding remarks-.Dipolar interactions are assumed to be relevant to stabilizing the magnon condensate within YIG.The presence of a PMA is expected to counteract dipolar interactions.In this Letter, we show that even at intermediate strengths of the PMA field, a magnon condensate state can exist as a metastable state.We note that the anisotropy increases the band depth and curvature of the magnon dispersion.These adjustments to the magnon spectrum are expected to facilitate magnon condensate formation.From the calculations of effective magnon-magnon interactions and minimizing the interaction potential at the band minima, we find the magnon condensate phase diagram.We demonstrate that the inclusion of PMA results in a magnon condensate with a more intricate phase diagram compared to when PMA is absent.A finite PMA has the tendency to drive the magnon condensate towards a nonsymmetric magnon population at band minima in thinner films and lower magnetic fields, as compared to the absence of PMA.Micromagnetic simulations within the LLG framework confirm our analytical results and analyses.

FIG. 1 .
FIG.1.The analytical dispersion of noninteracting BVM magnons in a YIG thin film for various PMA strengths, Eq. (3b).The inset shows the depth of the magnon band minima as a function of the PMA strength.We set the thickness Lx = 50 nm and the magnetic field in the z direction H0 = 1 kOe.

FIG. 3 .
FIG.3.The theoretical phase diagram of the condensate magnons in the absence (a) and presence (b) of PMA.We plot the magnon interaction energy V4/N 2 c , Eq. (7), as a function of the film thickness Lx and the magnetic field strength H0, applied along the z direction.Different states are labeled based on different extrema listed in Eq. (8).The dashed black lines indicate the boundaries between the different condensate phases, Eq. (8).We set Kan = 0.5 µeV in (b).

FIG. 4 .
FIG. 4.Micromagnetic simulation of nonequilibrium magnon distribution injected by spin torque mechanism for a YIG thin film with a thickness of Lx = 50 nm and lateral sizes of Ly = Lz = 5 µm,in the presence of an external magnetic field along the z direction H0 = 1 kOe.(a) and (b) show magnon distributions of the initial nonequilibrium injected magnons and final quasi-equilibrium magnon condensate steady state, respectively, when Kan = 0. (c) and (d) show magnon distributions of initial nonequilibrium excited magnons and final quasi-equilibrium magnon condensate steady state, respectively, when Kan = 0.5 µeV.The dotted line indicates the analytical dispersion relation of noninteracting magnons, Eq. 3b.Because of magnon-magnon interactions, the simulated magnon dispersion has a nonlinear spectral shift compared to the analytical noninteracting magnon dispersion.Although the duration of magnon pumping by spin-transfer torque is the same in the absence or presence of the PMA, the critical torque amplitude is lower in the presence of PMA.

TABLE I :
The material parameters used in the micromagnetic simulations.