Room temperature and low-field resonant enhancement of spin Seebeck effect in partially compensated magnets

Resonant enhancement of spin Seebeck effect (SSE) due to phonons was recently discovered in Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{3}$$\end{document}3Fe\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{5}$$\end{document}5O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{12}$$\end{document}12 (YIG). This effect is explained by hybridization between the magnon and phonon dispersions. However, this effect was observed at low temperatures and high magnetic fields, limiting the scope for applications. Here we report observation of phonon-resonant enhancement of SSE at room temperature and low magnetic field. We observe in Lu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{2}$$\end{document}2BiFe\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{4}$$\end{document}4GaO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{12}$$\end{document}12 an enhancement 700% greater than that in a YIG film and at very low magnetic fields around 10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{-1}$$\end{document}−1 T, almost one order of magnitude lower than that of YIG. The result can be explained by the change in the magnon dispersion induced by magnetic compensation due to the presence of non-magnetic ion substitutions. Our study provides a way to tune the magnon response in a crystal by chemical doping, with potential applications for spintronic devices.


SUPPLEMENTARY NOTE 1: STRUCTURAL CHARACTERIZATION
The structural quality of the Lu 2 BiFe 4 GaO 12 (BiGa:LuIG) film used in the present study was confirmed by X-ray diffraction (XRD) and high-resolution transmission electron microscopy (TEM) measurements. The structural characterization of the LPE-grown YIG used for comparison has been described elsewhere 1 . Supplementary Figure 1a shows the XRD results of the 2θ − ω scan measured in the symmetric configuration around the (004) peak of the Gd 3 Ga 5 O 12 (GGG) substrate at 2θ ∼ 28.83 • . It can be clearly seen that the BiGa:LuIG film peak is at slightly larger angles than that of the GGG, indicating the lower lattice parameter of the film, as expected: a GGG = 1.2383 nm, a LuIG = 1.2284 nm [red line shows expected 2θ value of (004) peak for bulk Lu 3 Fe 5 O 12 (LuIG)]. The Bi-substitution increases the lattice parameter of LuIG to a value closer to that of GGG 2 and due to the good lattice matching with the GGG substrate we cannot fully resolve the thin film peak.
To further check the structural quality of the sample, we also performed symmetric and an- We have also performed high resolution TEM (HR-TEM) (Supplementary Figures 1g, 1h) and HR-BF-STEM (Supplementary Figures 1i, 1j) measurements. If we now take a closer look at the GGG/BiGa:LuIG interface, shown in Supplementary Figures 1h and 1j, we can see that the BiGa:LuIG grows epitaxially on GGG with a sharp, smooth interface, perfect lattice matching and absence of defects. In fact, the film and substrate are nearly indistinguishable at the interface. This is fully consistent with the XRD analysis.
:  resulting in a non-zero magnetic moment. obtained from the magnetization hysteresis loops at each temperature and its comparison to the magnetization of YIG (red symbols) (data for YIG below room temperature (dark red) and above room temperature (light red) were extracted from previous studies on YIG film grown by liquid phase epitaxy in T. Kikkawa et al. 3 and K. Uchida et al. 9 , respectively). f Detail of the low magnetic field dependence of the magnetization (left axis) and SSE (right axis) measured at 300 K.

B. Comparison between magnetization and SSE at low magnetic fields
It has been previously shown that the competition between bulk and surface anisotropies in YIG can result in pronounced differences in the magnetic field dependence behavior of the magnetization and the SSE at low fields. 5,10-12 This effect is quite pronounced for YIG slabs and gradually decreases as the YIG thickness is reduced, being absent for films with thickness lower than 5 µm 10 . Here, in order to confirm the absence of this effect in the BiGa:LuIG(4µm) film, we have performed precise measurements of the SSE and magnetization at low magnetic fields. Supplementary Figure 2f shows the comparison between the magnetic field response of the magnetization and SSE measured at room temperature, we can clearly see that the magnetic field dependence of the SSE and magnetization closely follow each other upon approaching the saturation magnetization, therefore ruling out differences in the magnetic field response due to the competition between surface and bulk anisotropies in the SSE at low magnetic fields.   18,19 , which express the spin creation and annihilation operators in terms of boson operators that create or destroy magnons, and are given by: and then if we consider the Fourier transforms (and their inverse) for the magnon operators , and the definition of the delta Kronecker function: δ k,k = 1 N a,d N a,d i e i(k−k )· r i , we obtain the expression below: and which can be further simplified by defining γ k = γ −k = 1/z i δ e ikδ , with z i = z ad or z da the number of nearest neighbors for octahedral and tetrahedral sites, respectively. Then the Hamiltonian of the ferrimagnet can be expressed as: where the first two terms in the right side describe the ground state energy 20 and the third term describes the magnon excitation and is given by: Now from the above Hamiltonian, if we consider the Heisenberg equations of motion for the magnon operators a k and b † k (i.e. ih(da k /dt) = [a k , H k ]) and assuming exp(−iω m t) time dependence for a k and b † k , we obtain the secular equation which directly yields the ω m dispersion, as follows (for S a = S d = S): which after re-grouping inside the [] 1/2 term, it becomes: if we assume cubic symmetry for simplicity, and ka << 1 the above expression can be simplified to: the + and -solutions of the above magnon dispersion represent the acoustic and optical magnon branches respectively. For the problem we are now concerned, we only need to consider the acoustic dispersion branch (+ solution).
To consider the effect of the number of magnetic ions on each of the sites, we can use the expressions: N a = λN and N d = µN , where N refer to the number of Fe 3+ ions per unit volume and λ (µ) correspond to the ion density in the octahedral (tetrahedral) sites.
Introducing this notation into the dispersion described previously, we obtain: this expression is similar to Eq. 2 of the main text with the external magnetic field B 0 = µ 0 H and the gyromagnetic ratio γ = gµ B /h. The effect of non-magnetic ion substitution can be accounted for by introducing the following relations 21 : where x and y are the concentration of non-magnetic ion substitutions per unit formula in octahedral (a) and tetrahedral (d) sites, respectively.  The magnon-polaron peaks in SSE have been explained by the larger magnon-phonon hybridization over k-space, when the condition for tangential touching between magnon and phonon dispersions is met. Therefore, we want to evaluate the degree of magnon-phonon overlap over k-space as a function of magnetic compensation (non-magnetic substitutions) at the touching condition. The touch condition is met when ω m = ω p and ∂ωm ∂k = ∂ωp ∂k , with ω m and ω p being the frequency of magnon and phonon dispersions, respectively. In order to evaluate the degree of overlap, we can expand the magnon dispersion as a k power series around the touching point k 0 (see Fig. 5), given by the formula: At the tangential touching point k 0 , both magnon and phonon dispersions have the same velocity (i.e. their first derivatives are the same: ∂ωm ∂k k=k 0 = ∂ωp ∂k k=k 0 = c p , where c p is the phonon velocity). Therefore, in order to evaluate the degree of magnon-phonon overlap in k-space, we need to consider the quadratic component, which represents the deviation from the linear behaviour at the touching point and it should decrease in order to increase the magnon-phonon overlap.
Before evaluating the power expansion series, let us re-express the magnon dispersion obtained in Eq. 12 as: where we have introduced the following notation: ω z = γµ 0 H is the Zeeman gap induced by the external magnetic field, A = J ad S(z da µ+z ad λ)

2h
, B = µ−λ µλ , C = 4a 2 3λµ and D = A cp 2 C. From the tangential touching condition ( ∂ωm ∂k = ∂ωp ∂k = c p ), we can obtain the value of k 0 at the touching point: Using the above expression we can evaluate the magnitude of the quadratic component at k 0 , which should be minimized in order to obtain the largest overlap over k-space. This can be obtained from the second order component of the Taylor expansion around k 0 , which gives the expression for the quadratic term: with the parameters A, B, C, D as defined previously. Now, we can obtain the dependence of the quadratic component as a function of the tetrahedral site substitution, y (see Supplementary Figure 6). We can see that, counter-intuitively, the k 2 component gradually increases with the tetrahedral site substitution y (proportional to Ga doping). Therefore the relatively larger magnitude of the magnon-polaron peaks in BiGa:LuIG (compared to YIG) cannot be explained by an increased overlap over k-space. This is clearly illustrated in Supplementary Figure 7, showing the comparison of the magnon and phonon dispersions at the touching condition for BiGa:LuIG and YIG, where we can see that the overlap over k-space is actually larger in the case of YIG.