The creation of temperature variations in magnetization, and hence in the frequencies of the magnon spectrum in laser-heated regions of magnetic films, is an important method for studying Bose–Einstein condensation of magnons, magnon supercurrents, Bogoliubov waves, and similar phenomena. In our study, we demonstrate analytically, numerically, and experimentally that, in addition to the magnetization variations, it is necessary to consider the connected variations of the demagnetizing field. In the case of a heat-induced local minimum of the saturation magnetization, the combination of these two effects results in a local increase in the minimum frequency value of the magnon dispersion at which the Bose–Einstein condensate emerges. As a result, a magnon supercurrent directed away from the hot region is formed.

The phenomenon of Bose–Einstein condensation, predicted by Einstein^{1} for an ideal gas and subsequently by Fröhlich^{2} for quanta of collective excitations, has been attracting the attention of the scientific community for a long time. Such attention is warranted not only by the universality and physical depth of this phenomenon but also by such practically significant consequences as coherency, superfluidity, and superconductivity. In overpopulated gases of excitons,^{3} magnons,^{4,5} photons,^{6} polaritons,^{7} etc., the Bose–Einstein condensate (BEC) manifests itself as a spontaneous occurrence of macroscopic coherent oscillations at the lowest frequency of the spectrum.^{8} For magnons in a magnetic insulator such as yttrium iron garnet (YIG),^{9,10} this condensation can be achieved even at room temperature,^{5} which is relevant for practical applications.^{11–17} The same applies to magnon supercurrents—a collective motion of condensed magnons driven by a phase gradient $ \u2207 \phi $ of the BEC wave function $ \psi ( x , t )$.^{18,19} At present, the dynamics of magnon condensates and supercurrents remains intriguing and not fully understood as it can be affected by the spatial distribution of magnetization *M,*^{18–20} variations in the bias magnetic field $ H ext$,^{21} and by various nonlinear effects.^{22,23}

Here, for a tangentially magnetized magnetic film, we show theoretically and experimentally that a spatially localized decrease in the saturation magnetization $ M s ( r , T )$ induced by local optical heating leads to a local increase in the BEC frequency, which is consistent with the observed supercurrent direction and opens broad possibilities for controlling transport in magnon condensates. As discussed in the following, this effect is caused by the demagnetizing field generated by the local variations in $ M ( r , T )$.

To provide a qualitative description of the expected phenomena, let us consider the geometry, shown in Fig. 1(a). It consists of an unbounded magnetic plate with the surface in the plane $ ( x \u0302 , y \u0302 )$ placed in a tangential magnetic field $ H ext$, which is aligned with the $ x \u0302$ axis. The magnetization $ M plate$ is also parallel to $ x \u0302$. For simplicity, let us assume that the plate hosts an ellipsoid of revolution (spheroid) around $ z \u0302$, whose axis ratio $ R = c / a$ is a parameter of the problem. The magnetization of this spheroid $ M sph ( T )$ is smaller than $ M plate$. To find the intrinsic magnetic fields in the plate and in the spheroid, $ H int plate$ and $ H int sph$, respectively, we use the continuity condition across a surface for the orthogonal component of the magnetic flux density ** B**. We obtain that in the plate $ H int plate = H ext$.

*N*is the demagnetization factor that varies with the value of

_{x}*R*. The three factors

*N*( $ j = x , y , z$) satisfy the sum rule $ N x + N y + N z = 1$. In our geometry, we conclude that

_{j}*N*varies between 1/2 and 0 [see inset in Fig. 1(c)].

_{x}^{25}

*η*is the nonuniform exchange constant,

*M*is the magnetization of the medium,

*θ*is the angle between

**and**

*k***, and**

*M**γ*is the gyromagnetic ratio.

In the following, we describe the effect of temperature modification of the magnon spectrum using the example of two typical frequencies with *θ* = 0 and $ \theta = \pi / 2$ and $ k \u2192 0$. These frequencies are easily available for the experimental study.

^{17}

*N*= 0. The critical value of

_{x}*N*at which $ \omega \u22a5 sph = \omega \u22a5 plate$ is as follows:

_{x}Let us examine the thermal modification of the magnon frequencies under more realistic conditions. For this goal, we conduct micro-magnetic simulations of the internal magnetic field in a tangentially magnetized YIG plate for a given magnetization profile using the open-source GPU-based software MuMax 3.10.^{27} The chosen geometry is determined by a bias magnetic flux density *B* of 1300 G ( $ B SI \u2009 units$ = 130 mT) and a film thickness of 2.1 *μ*m. The spatial distribution of magnetization was chosen as a cylindrical well with a Gaussian profile and depth $ \Delta M$, as shown in Fig. 1(b). Thus, the model accounts for the magnetization gradient in the plane of the plate, assuming uniform magnetization along its thickness. The simulation results in a profile $ H int ( x )$ for the whole sample, which can be recalculated into a profile $ \omega | | ( x )$ using a relationship $ \Delta \omega | | ( x ) = \gamma \Delta H int ( x )$.

The obtained frequency profiles for the magnetization wells of different diameters and fixed depths $ 4 \pi \Delta M \u2243 400$ G are shown in Fig. 1(c).^{24} Indeed, we see that the frequency $ \omega \u2225 ( y )$ increases in the hot-spot region, and this effect becomes more pronounced (even at constant $ \Delta M$) as the hot-spot diameter decreases.

*w*to the plate thickness

*d*. One sees that the numerical dependence $ N x eff ( w / d )$ is in good quantitative agreement with the analytical dependence $ N x ( a / b )$

^{28}shown by the solid line. It means that $ H int ( 0 )$ at the center of our magnetization profile is well approximated by $ H int sph$ for a spheroid with $ \Delta M = \Delta M ( 0 )$ and $ a / b = w / d$. This opens the possibility of analytically finding the magnon frequency profiles of hot spots in magnetic films without numerical modeling.

To clarify the dynamics of the magnon supercurrent in the vicinity of the hot spot, we numerically solved the Gross–Pitaevskii equation. Since this supercurrent is highest in the direction perpendicular to the magnetic field,^{18} we can, in first approximation, limit ourselves to the one-dimensional equation

^{15,22}with wave vectors $ k = + k 0$ and $ k = \u2212 k 0$ in the two frequency minima of the magnon spectrum of the tangentially magnetized magnetic film. The dispersion coefficient $ \omega \u2032 \u2032 y y = \u2202 2 \omega ( k ) / ( \u2202 k y ) 2$ calculated at $ k = \xb1 k 0$ is inversely proportional to the effective mass of condensed magnons. The frequency profile $ \Omega ( y ) = \Omega max \u2009 exp \u2009 ( \u2212 y 2 / ( 2 \delta 2 ) )$ plays the role of an external potential.

As seen in Fig. 2, our solution demonstrates a supercurrent propagating outward from the region of the elevated BEC frequency and, hence, decreased magnetization. Further steps toward understanding the magnon supercurrent dynamics require consideration of the nonlinear terms of the two-dimensional Gross–Pitaevskii equation, in particular, those related to the static demagnetization field,^{23} which in the case of supercurrents varies with the spatial distribution of the BEC density. This problem is beyond the scope of this paper.

In order to experimentally verify the theoretical results, we used a dedicated optical system [see Fig. 3(a)] that allows us to create thermal profiles of various forms and various sizes, in particular, to produce a hot spot with a diameter down to 2 *μ*m.^{29} Desired thermal patterns were generated by phase-based wavefront modulation^{30,31} of the heating laser in combination with Fourier optics. The *Cobolt Twist* laser source with a wavelength of 457 nm is directed to a spatial light modulator, which imprints a spatial distribution of phase shifts. The intermediate image of the modified laser wavefront, visible after a lens, is focused on the sample via a microscope objective. Depending on the focal plane, we obtain a spot diameter between 2 and 9 *μ*m. Spatial resolution is obtained by moving the thermal pattern over the sample surface.

The magnon density spectrum $ N ( \omega )$ was measured by means of Brillouin light scattering (BLS) spectroscopy.^{32–34} This spectroscopy is based on the process of inelastic scattering of an incident photon by a magnon. The intensity of the inelastically scattered light is proportional to the density of magnons, whose frequency corresponds to the measured frequency shift of the photons. The probing laser source is a *Coherent Verdi* laser operating at a wavelength of 532 nm. The frequency of the scattered light was analyzed using the tandem multi-pass Fabry–Pérot interferometer. To reduce the heating of the sample by the probing laser source, we pulse the laser with an acousto-optic modulator.

The described micro-BLS system was integrated with an optical heating system in one experimental setup as shown in Fig. 3(a). The setup was controlled using the *thaTEC:OS* automation framework, and data evaluation was performed using Python libraries such as *PyThat*^{35} and *xarray.*^{36}

First of all, we measured the relative number of thermal magnons $ N ( \omega )$ as a function of their frequency in the center of the hot spot.^{29} The resulting function $ N ( \omega )$ is presented in Fig. 3(b) in comparison to the dispersion curves of the fundamental dipole-exchange magnon mode shown in Fig. 3(c). $ N ( \omega )$ is proportional to the density of magnon states $ D ( \omega ) \u221d [ d \omega ( k ) / d k ] \u2212 1$ and has two clear peaks near $ \omega | |$ and $ \omega \u22a5$, where $ D ( \omega )$ formally goes to infinity.

It can be seen that heating of the investigated region leads to a decrease in the frequency of the upper peak of the magnon density and an increase in the frequency of the peak at the bottom of the magnon spectrum. This behavior is in perfect agreement with our expectations. Following our calculations presented in Fig. 1(c), the shift of these frequencies by a few hundred MHz indicates a strong localized heating of the film reaching more than 120 °C.^{18,30,44}

To reveal the effect of thermally created magnetic inhomogeneity on the magnon Bose–Einstein condensate (BEC), we equipped our setup with a specialized microwave circuit. It was used to create a dense magnon population, allowing us to reach the threshold of BEC formation. Therefore, a microstrip resonator with a width of 50 *μ*m and a length of 3.5 mm is placed under the YIG sample tangentially magnetized by a 1510 Oe field ( $ B SI \u2009 units$ = 151 mT). Being driven by external microwave pulses, this microstrip induces a pumping magnetic field parallel to the external field $ H ext$ and along the equilibrium direction of the magnetization ** M**. Thus, the geometry of parallel parametric pumping is fulfilled.

^{26}The energy transfer from the electromagnetic to the spin system happens in the form of a microwave photon with a wavenumber close to zero, which decays into two magnons with half the pumping frequency $ \omega p = 12.705 \u2009 GHz$ and opposite wave vectors.

^{37–39}Due to four-magnon scattering processes, parametrically pumped magnons thermalize in the lower region of the spectrum and form a BEC at its bottom at $ \omega | |$ [see Fig. 3(c)] when the threshold density is reached.

^{5,40}

Figure 4 shows the frequency-spatial magnon distribution on the longitudinal axis of the microstrip around the hot spot in the center of the pumping zone at different moments of time *t*. After the start of pumping [see Fig. 4(a)], parametric magnons injected at $ \omega \u22a5 = \omega p / 2 =$ 6.352 GHz move through the step-by-step Kolmogorov–Zakharov scattering cascade to $ \omega | |$.^{41} Although, at this time, the expected decrease in $ \omega \u22a5$ is already noticeable in the hot region, nothing can be said about the behavior of $ \omega | |$ yet, since the thermalizing magnons have not yet reached this frequency.

After some time, [see Figs. 4(b) and 4(c)] the magnons fill the entire frequency region between $ \omega \u22a5$ and $ \omega | |$, concentrating at the bottom of the spectrum. Due to changes in the parametric pumping conditions in the hot spot, the density of gaseous magnons here is somewhat lower than in the surrounding areas. This is not the case for the near-bottom magnons, where the upward shift of $ \omega | |$ to about 80 MHz becomes clearly visible. The frequency shift profile can be well approximated by a Gaussian curve with 10 *μ*m full width at half maximum.

Of particular interest is the dynamics of the spatial distribution of near-bottom magnons. One can see from Figs. 4(b)–4(d) that during the pumping action and some time after its termination at *t* = 0, the hot spot is surrounded by areas of increased magnon density. This phenomenon finds a natural interpretation in the dynamics of the magnon supercurrents flowing out of the region of increased frequency.^{19,20} The strength of the supercurrent depends on the phase gradient of the BEC wave function and, hence, on the BEC frequency gradient.^{18} Due to the decrease in this gradient with distance from the heating region, the supercurrent decreases, and magnons accumulate due to the “bottle-neck” effect when the inflow of quasiparticles exceeds their outflow. The decrease in the BEC density caused by its outflow from the hot region is compensated by the condensation of the parametrically overpopulated magnon gas. Such compensation ceases after the pumping is turned off, which leads to the formation of a deep BEC density dip in the hot spot [see Figs. 4(d)–4(f)]. The formation of this dip is further intensified by the growth of the supercurrent due to the increasing BEC coherence after the termination of the disturbing effect of pumping.^{42,43} Soon, due to the decrease in supercurrents owing to the depletion of the magnon condensate in the region of maximum heating, the magnon density humps disappear as well [see Figs. 4(e) and 4(f)]. However, the spatial redistribution of the condensed magnons leads to the fact that during the entire time after pumping is turned off, the BEC density outside the hot spot remains higher than in the absence of heating.

In summary, we see that an increase in the lower frequency limit of the magnon spectrum due to the influence of demagnetization fields in a locally heated region causes supercurrents to flow out of the heated area. At the same time, such an increase depends on the ratio of the diameter of this region and the film thickness, which opens up opportunities for controlling magnon supercurrents in thermal landscapes. It can also be expected that in sub-micrometer-thick YIG films, where the lower frequency of the dipole-exchange magnon spectrum lies significantly above $ \omega | |$, and, therefore, strongly depends on the magnetization change, local heating can lead to an inversion of the supercurrent direction.

This study was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), TRR 173-268565370 Spin+X (Project B04). V.S.L. was in part supported by NSF-BSF Grant No. 2020765.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Matthias R. Schweizer:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Franziska Kühn:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Victor S. L'vov:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Anna Pomyalov:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Georg von Freymann:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Burkard Hillebrands:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Alexander A. Serga:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

## REFERENCES

*Albert Einstein: Akademie-vorträge, sitzungsberichte Der Preußischen akademie der wissenschaften 1914–1932*

^{3}He

*Magnonics*

^{−1}) in contrast to $\n4\n\pi \nM\n\u2243\n1750$ G ( $\n\nM\n\nSI\n\u2009\nunits\n=$ 140 kA m

^{−1}) in the ambient plate.

^{18,30,44}Although the decrease of

*M*may seem excessive, it will be shown in the following that it leads to a frequency shift comparable to the experimental results.

*N*=

_{x}*N*. In the case of a very elongated spheroid,

_{y}*N*tends to zero, and thus $\n\nN\nx\n=\n\nN\ny\n\u2192\n1\n/\n2$. Similarly, for a very flattened spheroid, $\n\nN\nz\n\u2192\n1$, and $\n\nN\nx\n=\n\nN\ny\n=\n0$.

_{z}*Magnetization Oscillations and Waves*

*Festkörperprobleme*

*μ*W, magnon accumulation at $\n\n\omega \n\n|\n|$ began when $\n\nP\np$ was increased by 20 dB, and the results presented in Fig. 4 were obtained at a pump power exceeding $\n\nP\np\n\nthr$ by 36dB (5W). Due to the relatively large size of the microstrip, and hence the area of magnon BEC formation, compared to the wavelength of condensed magnons $\n\lambda \n=\n2\n\pi \n/\nk\n\u2243\n1$

*μ*m, we can talk about the formation of a spatially homogeneous BEC along the microstrip.