Evidence for spin current driven Bose-Einstein condensation of magnons

The quanta of magnetic excitations – magnons – are known for their unique ability to undergo Bose-Einstein condensation at room temperature. This fascinating phenomenon reveals itself as a spontaneous formation of a coherent state under the influence of incoherent stimuli. Spin currents have been predicted to offer electronic control of Bose-Einstein condensates, but this phenomenon has not been experimentally evidenced up to now. Here we show that current-driven Bose-Einstein condensation can be achieved in nanometer-thick films of magnetic insulators with tailored nonlinearities and minimized magnon interactions. We demonstrate that, above a certain threshold, magnons injected by the spin current overpopulate the lowest-energy level forming a highly coherent spatially extended state. We quantify the chemical potential of the driven magnon gas and show that, at the critical current, it reaches the energy of the lowest magnon level. Our results pave the way for implementation of integrated microscopic quantum magnonic and spintronic devices.

For a long time, the only experimentally proven approach enabling observation of room-temperature Bose-Einstein condensation (BEC) of magnons was the approach based on the utilization of microwave pumping [1][2][3][4][5][6][7][8][9][10] .This approach relies on the injection of low-energy non-equilibrium magnons within a relatively small interval of energies, followed by their thermalization and accumulation in the lowest-energy level.Although this process was shown to result in a formation of the quasi-equilibrium magnon gas, which can be described by a non-zero chemical potential and an effective temperature, it was found to predominantly affect low-energy magnons, which do not fully equilibrate with high-energy ones resulting in a very high effective temperature of the former 5 .Another recently demonstrated approach relies on the dynamic spectral redistribution of magnons in an overheated magnon gas caused by its rapid cooling 11 .Due to its dynamic origin, this approach can only be used to create BEC-like states for a few tens of nanoseconds.
Alternative way to drive magnon gas into a steady quasi-equilibrium state characterized by a non-zero chemical potential [12][13][14][15] can be based on the utilization of dc electric currents converted into pure spin currents by the spin-Hall effect (SHE) 16,17 .In strong contrast to the microwave pumping, the spin torque exerted by the spin current simultaneously affects population of all magnon modes 18,19.Importantly, it hence allows avoiding strongly nonequilibrium transient states of the magnon gas.The possibility to create magnon BEC by using the above driving mechanism has been theoretically predicted long time ago [20][21][22] .
However, up to now, no direct experimental confirmation of the current-driven BEC has been reported.Here we demonstrate that this mechanism can indeed lead to the magnon BEC in a system that satisfies certain requirements.
The most striking and necessary manifestation of the magnon BEC is the spontaneous formation of a coherent dynamical state.However, the observation of such a state alone is insufficient to claim the BEC transition 23 .Indeed, every magnetic oscillator driven by spintransfer effects [24][25][26][27][28][29][30][31] exhibits more or less coherent magnetic oscillations.Although these oscillations are formed spontaneously, the underlying mechanism is not necessarily BEC.
Instead, it can be a strongly non-equilibrium process, similar to that responsible for generation of coherent optical radiation in lasers.Strictly speaking, an assertion of unambiguous BEC identification requires that the magnon gas remains in a (quasi-) equilibrium thermodynamic state: while the emergent coherent dynamics is described in terms of the overpopulation of the lowest-energy magnon level, the population of the rest of the magnon states follows the Bose-Einstein statistics.Additionally, formation of coherent magnetic dynamics via BEC, which is the condensation of particles in the phase space, is inconsistent with the formation of a dynamical bullet 26,32 or droplet [33][34][35] , which represents the condensation of particles in the real space due to their attractive interaction.As a general rule of thumb, the nonlinear interactions between magnons is one of the main factors hindering the magnon BEC 14 .
In relatively thick films of Yttrium Iron Garnet (YIG), where microwave-driven BEC has been evidenced [1][2][3][4][5][6][7][8][9][10] , the magnons at the lowest-energy level exhibit very weak nonlinear interactions resulting a stabilization of BEC by extrinsic effects 10,36 .In contrast, in nanometer-thick YIG films, which can be efficiently driven by spin currents, the energy of the lowest magnon level contains a significant dynamic dipolar contribution, which results in a strong attractive interaction between magnons leading to spatial instabilities and collapses 26,37 .The nonlinearities associated with dipolar effects have also been recently shown to be responsible for the strong scattering of magnons out from the lowest-energy state preventing its overpopulation 38 .
Here we study a current-driven system based on a nanometer-thick Bi-substituted YIG film, in which the out-of-plane magnetic anisotropy has been engineered in order to fully compensate the dipolar demagnetizing field resulting in vanishing nonlinear magnon interactions.We show that, in such a system, the gas of weakly interacting magnons can undergo Bose-Einstein condensation under the influence of pure spin currents.The BEC transition is documented by the observation of the overpopulation of the lowest-energy level resulting in a spatially extended dynamical state exhibiting high temporal and spatial coherence.Additional confirmation of the BEC nature of the observed transition is provided by the measurements of current-dependent density of high-energy magnons.In agreement with the BEC scenario, this density saturates, as the chemical potential of the magnon gas reaches the lowest magnon energy, while the density of lowest-energy magnons continues to grow.This observation indicates that the additionally injected magnons accumulate in the lowest-energy level, which is a clear signature of the BEC transition.
In Figure 1a, we show the schematics of our experiment.The studied system is based on 20-nm thick film of Bi-doped Yttrium Iron Garnet (BiYIG) (Bi1Y2Fe5O12) grown by the pulsed laser deposition on substituted Gallium Gadolinium Garnet (sGGG) substrate 39 .The film is magnetized to saturation by a static in-plane magnetic field H0.It exhibits significant strain-induced perpendicular magnetic anisotropy (PMA) with the effective anisotropy field µ0Ha = 167 mT, which is very close to the saturation magnetization of the film µ0Ms = 175 mT.This leads to the compensation of the effects associated with the dipolar anisotropy and thus results in minimization of nonlinear magnon interactions 38 .The spin system of the BiYIG film is driven out of the equilibrium by spin torque through the injection of the pure spin current Is created by the SHE in the Pt electrode carrying dc electric current I.The injected spin current creates additional magnons within a broad interval of energies 18 , which distribute in the energy space according to the Bose-Einstein distribution with a non-zero chemical potential 12,14 .We experimentally evaluate the population of the magnon states by using the microfocus Brillouin light scattering (BLS) spectroscopy 40 (see Methods for details).The measured signal -the BLS intensity -is proportional to the spectral density of magnons at the position, where the probing laser light is focused (Fig. 1a).By moving the focal spot over the surface of the BiYIG film, we additionally obtain information about the spatial variations of the spectral distribution of magnons.
Figure 1b shows the BLS spectrum recorded at µ0H0=100 mT and I=0, describing magnons, which exist in the magnetic film due to thermal fluctuations at room temperature.Under these conditions, the magnon gas is at thermal equilibrium with the lattice and is characterized by zero chemical potential and room temperature 1 .The recorded spectrum exhibits an asymmetric peak with the maximum at 2.9 GHz with the high-frequency tail extending to about 3.3 GHz and an additional nearly symmetrical peak located at 39 GHz.
Figure 1c shows the dispersion spectrum of magnons calculated according to Ref. 41, which allows identification of the peaks.The low-frequency part of the spectrum corresponds to magnons characterized by a nearly uniform distribution of the dynamic magnetization across the film thickness (see the inset in Fig. 1c).These magnons exhibit an anisotropic dispersion in the plane of the film, as seen from the essentially different dispersion curves corresponding to magnons with the wavevector k oriented parallel (solid curves) and perpendicular (dashed curves) to the direction of the static field.The two dispersion curves merge at k = 0 at the frequency of the uniform ferromagnetic resonance fFMR.The dispersion curve characterizing magnons with k ⊥ H0 shows a monotonous rise, while that for magnons with k || H0 shows non-monotonous behavior resulting in the appearance of a minimum at k = 7.6 μm -1 and the frequency fmin, which is by 0.09 GHz smaller than fFMR.The state corresponding to this minimum is the lowest-energy magnon state, where BEC is expected to take place.
As seen from the comparison of the data of Figs.1b and 1c, the experimentally observed low-frequency peak has the maximum at fFMR.This is due to the wavevectordependent sensitivity of the BLS technique, which maximizes at k = 0 and gradually decreases with the increase of k.Accordingly, the BLS intensity decreases with the increase of the frequency and vanishes at frequencies corresponding to k > kmax, which is mainly determined by the wavevector of the probing light.Note here that, at I = 0, the BLS intensity at fmin is significantly smaller than that at fFMR due to the significantly smaller experimental sensitivity.
The small high-frequency BLS peak in Fig. 1b corresponds to the so-called perpendicular standing-wave (PSW) magnon modes 42,43 , which possess a non-uniform distribution of the dynamic magnetization across the film thickness (see the inset in Fig. 1c).
These modes are characterized by a nearly isotropic dispersion (k || H0 and k ⊥ H0 curves are indistinguishable in Fig. 1c).As will be shown below, although these modes are located at frequencies far above the frequency of the lowest-energy magnon state, their analysis provides important information about the state of the current-driven magnon gas.
We now analyze the modifications of the magnon spectral distribution caused by the injection of the spin current.As seen from the data shown in Fig. 2a, at small currents, the population of all magnon states becomes increasingly enhanced with the maximum effect observed for states at lower frequencies.In particular, the shape of the spectrum recorded at I = 1.2 mA clearly indicates that the population of the states at frequencies close to fmin grows much faster in comparison with that at fFMR, as expected for the chemical potential gradually approaching the minimum energy.Finally, at I > 1.3 mA, a narrow intense peak appears in the spectra (Fig. 2b) at the frequency, which is by about 0.08 GHz below fFMR.This frequency difference is very close to the theoretically estimated value fFMR -fmin = 0.09 GHz.This allows us to conclude that the observed peak corresponds to the overpopulated lowestenergy magnon state, which is a first indication of the onset of BEC in the driven magnon gas.
We emphasize that the frequency of the peak and its spectral width remain nearly constant within the entire range I = 1.3-2.0mA (see Fig. 2c, which shows the normalized color-coded BLS spectra in the current-frequency coordinates), while the peak intensity increases by more than one order of magnitude (Fig. 2d).Note that, in magnetic films without PMA, the frequency is known to exhibit a strong negative shift with the increase of the density of magnons 32,44 .Generally, this means that the local increase of the particle density is energetically favorable, which corresponds to the attractive interaction between them causing local accumulation of particles in the real space followed by a collapse -the phenomenon, which is known to hinder formation of stable Bose-Einstein condensates 45,46 .As follows from the independence of the frequency of the observed peak from its intensity (Figs.2c and 2d), thanks to the effects of PMA, the attractive interactions are well suppressed in our system, enabling the BEC transition.
We then address the spectral coherence of the observed BEC.Because of the limited frequency resolution of the BLS apparatus, it is not possible to directly measure the spectral linewidth of the formed monochromatic oscillations.The fitting of the BLS spectra shown in Fig. 2b with the Gaussian function yields the full width at half maximum of 0.051 GHz.This value is very close to the frequency resolution of the BLS apparatus, which, for the used experimental arrangement, can be estimated as 0.04-0.05GHz.Therefore, we can conclude that the spontaneously formed dynamical state is characterized by a high temporal coherence.
To illustrate that the state is also coherent in the space domain, we perform spectral measurements at different spatial locations within the active area of the electrode.Figures 3a   and 3b show the color-coded BLS spectra recorded by moving the probing spot along the zand x-direction, as schematically shown in the corresponding insets.As seen from these data, the frequency and the linewidth of the BLS peak does not change noticeably across the entire 1×2 μm 2 -large active area, indicating that a spatially extended BEC is formed in the studied system.
Note here that, in contrast to most of the previously studied systems exhibiting currentdriven magnetic oscillations [24][25][26][27][28][29][30][31] , our system does not show transitions to multi-frequency oscillations or mode jumps -behaviors, which are inconsistent with the BEC scenario, where all additional magnons created at large driving stimuli should accumulate in the single, lowest-energy state.The latter condition can be unambiguously proven by analyzing the current-dependent population of a spectral state corresponding to the magnon energy far above the energy of the lowest state.To perform this analysis, we utilize the possibility to experimentally access PSW magnons at the frequency 39 GHz (Fig. 1b).In Figure 4a, we show the current-dependent intensity of the PSW peak together with the same dependence for the lowest magnon state at fmin.First, the data show that the small-current enhancement of the population of the high-frequency PSW states is significantly smaller than that of the state at fmin (note different vertical scales).At I = 1 mA, the enhancement of the BLS intensity of the PSW peak does not exceed 20%, while the intensity at fmin increases by nearly a factor of 10.
This result is consistent with the assertion that the main effect of the spin current on the magnon gas is the increase of its chemical potential [12][13][14][15] .More importantly, at currents where the formation of the coherent state takes place (Fig. 2b), the intensity of the PSW peak saturates, while the BLS intensity at fmin exhibits further strong growth.This result indicates that, in agreement with the BEC scenario, additional magnons created at large currents do not spread over the entire energy space, but instead overpopulate the lowest-energy state.
A key parameter to characterize the BEC transition is the chemical potential that we extract from the acquired data following the approach in Ref. 11.As seen from Fig. 4b, the chemical potential expressed in frequency units μ/h (here h is the Plank constant) monotonically increases at small currents, reaches fmin at I = IC = 1.3 mA, where the formation of the narrow spectral peak is observed (Fig. 2b), and then saturates.This feature allows us to clearly identify the BEC transition.
Finally, we aim at determining the conditions of the BEC formation over a broad range of the static magnetic field H0.The experimental data show that the variation of the static field mainly influences the value of the critical current IC, which is found to increase with the increase of H0.As seen from the data of Fig. 5, the field dependence of IC is nearly linear and extrapolates to a finite value at H0 = 0.This result is expectable, since the efficiency of the magnon creation by the spin torque depends on the magnon relaxation frequency r ω , which can be written as a sum of two terms: the field-independent term 0 r ω originating from the influence of the inhomogeneity of the magnetic film, and the term 0 H αγ , which scales linearly with H0 (here α is the Gilbert damping constant and γ is the gyromagnetic ratio) 28 .
In conclusion, our experimental results provide the direct experimental evidence of roomtemperature BEC in a magnon gas driven by spin currents -the phenomenon, which was theoretically predicted nearly a decade ago and has been actively discussed in the scientific community since then.We show that stable spatially extended Bose-Einstein condensate of magnons can be created in systems, where nonlinear magnon interactions are minimized by using compensation of the dipolar effects by the anisotropy.These findings open new avenues for the studies of magnon BEC by demonstrating an approach enabling creation of magnon BEC without breaking the equilibrium between low-energy and high-energy magnons.The simple and robust driving method based on the utilization of SHE at the micrometer scale enables implementation of a large variety of new experiments.We believe that our findings will strongly move forward the field of magnon thermodynamics and quantum magnetic phenomena in general.

Sample fabrication.
The growth of BiYIG films by pulsed laser deposition (PLD) is realized using stoichiometric BiYIG target.The laser used is a frequency tripled Nd:YAG laser ( λ = 355 nm), of a 2.5 Hz repetition rate and a fluency of about 1J/cm 2 and a substrate temperature of about 500°C.The distance between the target and the substrate is fixed at 44 mm.Prior to the deposition, the substrate is annealed at 700°C under 0.4 mbar of O2.For the growth, the pressure is set at 0.25 mbar O2 pressure.At the end of the growth, the sample is cooled down under 300 mbar of O2.No post annealing is performed.Deposition of Pt is performed using dc magnetron sputtering.Prior to Pt deposition, a slight O2 etch is performed to remove photo-resist residues and promote surface spin-transparency.The Pt electrode is 6 nm thick, its width is reduced to 1 µm over a length of 2 µm.Hence, the current density and the resulting spin-orbit torque is maximum over this 1x2 µm 2 constriction.
Micro-focus BLS measurements.All the measurements were performed at room temperature.The measurements of current-dependent magnon spectral distributions were based on the analysis of the inelastic scattering of laser light from magnons.The probing light with the wavelength of 532 nm and the power of 0.1 mW was produced by a singlefrequency laser possessing the spectral linewidth <10 MHz.The light was focused through the sample substrate (sGGG) into a submicrometer-size diffraction-limited spot by using a 100x corrected microscope objective lens with the numerical aperture of 0.85.The scattered light was collected by the same lens and analysed by a six-pass Fabry-Perot interferometer.
The lateral position of the probing spot was controlled by using a high-resolution customdesigned optical microscope and was actively stabilized with the precision better than 50 nm.

Figure 1 |
Figure 1 | Experimental system (a) Schematics of the experiment.Magnon gas in a BiYIG film with PMA is driven by the injection of the spin current Is created by SHE in the Pt electrode with the active area of 1 x 2 μm 2 .Spectral distribution of magnons is measured by analysing the light inelastically scattered from magnons.Transparent sGGG substrate is not shown for clarity.(b) Normalized BLS spectrum of magnons in the BiYIG film measured at I=0 and µ0H0=100 mT.(c) Calculated dispersion spectrum of magnons in the BiYIG film.Solid curves correspond to magnons with the wavevector k oriented parallel to H0. Dashed curves correspond to magnons with the wavevector k oriented perpendicular to H0. fFMR marks the frequency of the ferromagnetic resonance.fmin marks the frequency of the lowest-energy magnon state.Insets schematically show the distributions of the dynamic magnetization across the film thickness for the fundamental and the PSW magnon mode.

Figure 2 |
Figure 2 | Formation of the Bose-Einstein condensate of magnons.(a) and (b) Representative BLS spectra recorded at different currents in the Pt electrode, as labelled.Dashed lines mark the frequency of the ferromagnetic resonance.Note formation of the narrow intense spectral peak at I=1.3 mA.(c) and (d) Modifications of the formed peak at I≥1.3 mA.(c) Color-coded normalized BLS spectra in the current-frequency coordinates.(d) Current dependence of the peak intensity normalized by its value at I=1.3 mA.Solid line -guide for the eye.The data were obtained at µ0H0=100 mT.

Figure 3 |
Figure 3 | Spatial characterization of the BEC state.(a) and (b) Color-coded normalized BLS spectra recorded by moving the probing spot along the z-and xdirection, as schematically shown in the corresponding insets.The data were obtained at I=1.3 mA and µ0H0=100 mT.

Figure 4 |
Figure 4 | Thermodynamic characteristics of the current-driven magnon gas.(a) Current dependences of the BLS intensity for the PSW peak and that for the lowest magnon state at fmin, as labelled.The data are normalized by the value at I=0.Note different vertical scales for the two dependences.The data were obtained at µ0H0=100 mT.(b) Current dependence of the chemical potential expressed in frequency units.Horizontal dashed line marks the frequency of the lowest-energy magnon state fmin.Vertical dashed line marks the critical current IC, at which the formation of the BEC peak is observed.

Figure 5 |
Figure 5 | Dependence on the magnetic field.Static-field dependence of the critical current IC.Line is the linear fit of the experimental data.