Theory of the low-temperature longitudinal spin Seebeck effect

Using a simplified microscopic model of coupled spin and lattice excitations in a ferromagnetic insulator we evaluate the magnetic-field dependence of the spin Seebeck effect at low temperatures. The model includes Heisenberg exchange coupling, a harmonic lattice potential, and a pseudodipolar exchange interaction. Our approach goes beyond previous work [Phys. Rev. B 98, 134421 (2018)] in that it does not rely on the a priori assumption of a fast equilibration of the magnon and phonon distributions. Our theory shows that singular features in the magnetic-field dependence of the spin Seebeck effect at low temperatures observed by Kikkawa et al. [Phys. Rev. Lett. 117, 207203 (2016)] are independent of the relative strength of magnon-impurity and phonon-impurity scattering.

Figure 1

Geometry for the longitudinal spin Seebeck effect: A ferromagnetic insulator F of length $L$ (center, gray) is placed between an insulator I (bottom, red) and a normal metal N of thickness $l$ (top, blue), which also act as heat reservoirs held at a temperature difference $\mathrm{\Delta }T$. Via magnon-phonon coupling, the applied temperature gradient leads to a nonequilibrium magnon distribution, which causes the flow of a spin current ${\mathbf{j}}^{\mathrm{s}}$ into the normal metal. The spin current can be measured in the normal metal by means of the inverse spin Hall effect.

Figure 2

(a) Magnon-polaron frequency dispersions: ${\omega }_1$ (blue), ${\omega }_2$ (green), transverse phonon dispersion ${\omega }_3$ (orange, dashed), and ${\omega }_4$ (red). In panel (b) the polaron dispersions for the “critical” applied magnetic fields $B_{\mathrm{t}}=2.6\text{T}$ and $B_{\mathrm{l}}=9.2\text{T}$ are shown. Magnifications of the dispersions: (c) around the crossing of the magnon-polaron mode with the transverse phonon mode and (d) longitudinal phonon mode for different angles $\theta$ defined by $q_x=qcos\theta$.

Figure 3

Relaxation lengths for various scattering mechanisms in a magnetic insulator for temperatures $T=10\mathrm{K}$ (a) and (b) and $T=30\mathrm{K}$ (c) and (d). Panels (a) and (c) are for a magnetic field $B=0.1\mathrm{T}$; panels (b) and (d) are for $B=7\mathrm{T}$. Top and bottom panels show relaxation lengths for phonons and magnons, respectively. Material parameters for YIG are taken from Table 1. The microscopic model for impurity scattering and for the inelastic scattering processes is discussed in Sec. 3c; values for the impurity potential are taken from Table 3, center column. The relaxation lengths for phonon modes are averaged over polarization. The scattering processes shown in the figure are magnon-impurity and phonon-impurity scattering, three-magnon, four-magnon, three-phonon, exchange-based magnon-phonon scattering, and relativistic or dipole-dipole-based inelastic magnon-phonon scattering. These processes are shown schematically in Table 2. The dashed curve shows the length scale ${\lambda }_{\mathrm{imp}}$ for impurity-mediated intermode scattering of magnon polarons.

Figure 4

(a) Schematic picture of the interface reflection of a magnon polaron from branch ${\nu }^{\prime }$, and transmission of a phonon from branch ${\lambda }^{″}$, into a magnon polaron of branch $\nu$ with the respective reflection and transmission coefficients $R_{\nu {\nu }^{\prime }}$ and $T_{\nu {\lambda }^{″}}$. (b) Angle-averaged reflection and transmission coefficients $R_{0,\nu {\nu }^{\prime }}$ and $T_{0,\nu {\lambda }^{″}}$ at the IF interface for scattering into magnon-polaron mode $\nu =4$ as a function of the frequency. (c) Coefficients $R_{0,\nu {\nu }^{\prime }}$ and $T_{0,\nu {\lambda }^{″}}$ at the FN interface for magnon-polaron modes $\nu =1,4$, as well as the probability $P_{0,\nu \mathrm{N}}$ that the magnon-polaron mode $\nu$ at the FN interface is excited by a spinful excitation of the conduction electrons in the normal metal. Panels (d) and (e) show magnifications of the resonant regions in (b). The coefficients shown in panels (b)–(e) are for incident longitudinal-phonon-like or magnon-like modes ${\nu }^{\prime }=1$, 4 (in F) and ${\lambda }^{″}=1$ (in I or N). Coefficients for the transverse-phonon-like incident modes ${\nu }^{\prime }=2$, 3 or ${\lambda }^{\prime }=2$, 3 are approximately zero (not shown). Within the accuracy of the figure, the probability $P_{0,\mathrm{N}\nu }$ that the magnon-polaron mode ${\nu }^{\prime }$ incident on the FN interface excites a spinful excitation of the conduction electrons in N is equal to $P_{0,\nu \mathrm{N}}$. In (f) the transmission amplitude $T_{11}$ is shown as a function of the polar angle $\theta$ and azimuthal angles $\varphi$ for different frequencies $\omega$ in the vicinity of the resonance frequency. Panels (b)–(f) are evaluated for an applied magnetic field $B=7\mathrm{T}$. The nonzero reflection and transmission coefficients at the critical magnetic fields $B_{\mathrm{l}}\approx 9.2\mathrm{T}$ and $B_{\mathrm{t}}\approx 2.6\mathrm{T}$ are shown as a function of the frequency in panels (g) and (h), respectively. In panels (b)–(h) material parameters for YIG are taken from Table 1.

Figure 5

Relaxation lengths ${\lambda }_i\left(\omega \right)$, $i=1,2,3$, of Eq. (90) for the material parameters given in Table 1 and impurity scattering parameters from Table 3 (center column) for the white-noise impurity model (60) with the mean-free paths (61) (a) and for the microscopic model (52), (55) with mean-free paths from Eqs. (57) and (58) (b). The dots indicate the magnon-impurity (blue) and phonon-impurity scattering lengths (green and red).

Figure 6

Isotropic moment for impurity scattering only. We take material parameters from Table 1 and set the magnetic field equal to $B=7\mathrm{T}$, so that the magnon dispersion crosses that of the longitudinal phonons, but not the transverse phonons; see panel (a). To keep the presentation of the frequency dependence of the isotropic moment (44) as simple as possible we use white-noise impurity scattering rates (60) with potentials according to Table 3, center column. Panels (c)–(e) show the isotropic moment ${\psi }_0\left(\omega \right)$ of the distribution function for three choices of the system length $L$ as compared to the relaxation lengths ${\lambda }_i$, as schematically indicated in (b). Panel (c) shows the frequency dependence of the isotropic moment ${\psi }_0\left(\omega \right)$ at the FN interface at $x=L$. Panels (d) and (e) show the dependence on position $x$ for a generic frequency [$\omega /2\pi =1\mathrm{THz}$, panel (d)] and a frequency at the crossing point of the magnon and phonon dispersions [$\omega /2\pi =0.76\mathrm{THz}$, panel (e)]. Panel (f) shows a close-up of the spatial dependences near the FN interface for the longest system length considered.

Figure 7

Frequency-resolved spin current $j_x^{\mathrm{s}}\left(\omega \right)=\hslash {\mathcal{J}}_x^{\mathrm{s}}\left(\omega \right)\mathrm{\Delta }T/T$ at the FN interface (blue, solid) for $B=7\mathrm{T}$ and $T=10\mathrm{K}$ at sample length $L=0.1\mu \mathrm{m}$ (a), $L=0.5\mathrm{mm}$ (b), $L=10\mathrm{mm}$ (c), and $L=1\mathrm{m}$ (d). System parameters are taken from Table 1. Impurity scattering of magnon and phonons is described by the white-noise model (60), with parameter values taken from the center column of Table 3. For comparison, panels (a) and (d) also show the frequency-resolved spin current calculated from the short-length approximation of Eq. (99) and the long-length approximation (97), respectively (red, dashed).

Figure 8

Spin Seebeck coefficient for elastic scattering only. Material parameters are taken from Table 1, and impurity scattering of magnon and phonons is described by the microscopic model of Eqs. (52) and (55) with parameter values taken from the center column of Table 3, unless noted otherwise. Panel (a) shows the spin Seebeck coefficient at $L=10\mu \mathrm{m}$ for temperatures $T=2\mathrm{K}$, $5\mathrm{K}$, $10\mathrm{K}$, and $20\mathrm{K}$ (bottom to top). Panel (b) shows the spin Seebeck coefficient for $T=10\mathrm{K}$ at different system lengths $L=10\mu \mathrm{m}$ (green), $5\mathrm{cm}$ (red), and $10\mathrm{cm}$ (blue). Panel (c) shows $S$ for $B=7\mathrm{T}$ and $B=9.2\mathrm{T}$ as a function of the system length $L$. Panels (d) and (e), which are evaluated at $T=10\mathrm{K}$ and at $L=10\mu \mathrm{m}$ and $L=1\mathrm{cm}$, respectively, show the spin Seebeck coefficient $S$ vs magnetic field $B$ for spin mixing conductivity ${\sigma }_{\uparrow \downarrow }$ equal to 1, ${10}^{-1}$, and ${10}^{-2}$ times the value listed in Table 1, from top to bottom, respectively. Panel (f) shows the spin Seebeck coefficient for $L=10\mu \mathrm{m}$, $T=10\mathrm{K}$, and magnon-phonon coupling parameters $D$ and $D^{\prime }$ a factor 1, 0.5, and 0.1 times the values listed in Table 1, from top to bottom.

Figure 9

Relaxation lengths (a), isotropic distribution function ${\psi }_0\left(\omega \right)$ at the FN interface (b), frequency-resolved spin current $j_x^{\mathrm{s}}\left(\omega \right)=\hslash {\mathcal{J}}_x^{\mathrm{s}}\left(\omega \right)\mathrm{\Delta }T/T$ at $T=10\mathrm{K}$ (c), and spin Seebeck coefficient $S$ vs. magnetic field $B$ (d) for a system in which magnon-impurity scattering is stronger than phonon-impurity scattering. Material parameters are taken from Table 1. Impurity scattering is described by the microscopic model of Eqs. (52) and (55) (a, top panel, and d) and the white-noise model of Eq. (60) (a, bottom panel, and b, c) with parameters taken from the right column in Table 3. Dots in panel (a) indicate the magnon-impurity mean free path (blue) and the phonon-impurity mean free paths (green and red) in the absence of magnon-phonon coupling. The system length $L$ for panels (b)–(d) is indicated by the horizontal lines in panel (a). The spin Seebeck coefficient in panel (d) is evaluated for temperatures $T=2\mathrm{K},5\mathrm{K},10\mathrm{K}$, and $20\mathrm{K}$ (bottom to top).

Figure 10

(a) Frequency-dependent relaxation lengths ${\lambda }_i\left(\omega \right)$, $i=1,2,3$, for the case of elastic scattering only. (b) and (c) Relaxation lengths ${\lambda }_j$, $j=1,...,4(N+1)-1$ in the presence of inelastic scattering, calculated for a frequency grid with $N=200$ frequencies. For clarity, only every $20\mathrm{th}$ relaxation length is shown. In (b) the relaxation lengths are shown vs the strength of the inelastic scattering processes, normalized to the inelastic processes at $10\mathrm{K}$. In (c) the relaxation lengths are shown vs temperature. The black curves represent those relaxation lengths that are divergent in absence of inelastic scattering.

Figure 11

Relaxation lengths (a), (d) as in Fig. 5 with material parameters taken from Table 1. Frequency-resolved isotropic moment ${\psi }_0\left(\omega \right)$ and frequency-resolved spin current $j_x^{\mathrm{s}}\left(\omega \right)=\hslash {\mathcal{J}}_x^{\mathrm{s}}\left(\omega \right)\mathrm{\Delta }T/T$ at the FN interface for $T=10\mathrm{K}$ (b), (e) and $T=30\mathrm{K}$ (c), (f) together with the values obtained from a theory with elastic scattering only (dotted lines). Impurity scattering is modeled by the white-noise model (60) with parameters taken from the center column (panels (a)–(c)) and rightmost column (panels (d)–(e)) in Table 3, corresponding to the cases of higher magnetic and acoustic quality, respectively.

Figure 12

Spin Seebeck coefficient $S$ for systems of higher magnetic quality (upper panels) and higher acoustic quality (lower panels), as a function of magnetic field for $L=10\mathrm{cm}$ and temperatures $T=2\mathrm{K}$, $5\mathrm{K}$, and $10\mathrm{K}$ from bottom to top (a) and for $L=200\mu \mathrm{m}$ and $T=5\mathrm{K}$ and $10\mathrm{K}$ from bottom to top (b). Panel (c) shows $S$ as a function of length $T=10\mathrm{K}$ (red) and $T=5\mathrm{K}$ (blue). Panels (d) and (e) show a comparison to the incoherent theory of Ref. [22] (red, dashed) for $L=10\mu \mathrm{m}$ (d) and $L=10\mathrm{cm}$ (e) and $T=10\mathrm{K}$. In all panels solid curves show results of the full theory, including inelastic scattering, whereas dotted curves include elastic scattering only. Material parameters are taken from Table 1. Impurity scattering of magnons and phonons is described by the microscopic model of Eqs. (52) and (55), with parameter values taken from Table 3.

Figure 13

Spin current $j_x^{\mathrm{s}}$, normalized to the spin current for elastic scattering only at $T=5\mathrm{K}$ and $T=20\mathrm{K}$, $L=200\mu \mathrm{m}$, and $B=7\mathrm{T}$, with or without physical processes as indicated on the left. The physical processes are impurity-mediated intermode scattering (${\lambda }_{\mathrm{imp}}$), inelastic three-phonon scattering (3-pho), inelastic three-magnon scattering (3-mag), inelastic four-magnon scattering (4-mag), inelastic exchange-based magnon-phonon scattering (mag-pho), and relativistic/dipole-based inelastic magnon-phonon scattering (rel). Phonon-impurity and magnon-impurity scattering are included in all cases, using the microscopic model of Eqs. (52) and (55) with values taken from the center and right column of Table 3 [panels (a) and (b), respectively].

Figure 14

Frequency-resolved spin current $j_x^{\mathrm{s}}\left(\omega \right)=\hslash {\mathcal{J}}_x^{\mathrm{s}}\left(\omega \right)\mathrm{\Delta }T/T$ at the FN interface for $T=10\mathrm{K}$ and $B=7\mathrm{T}$ (left) and for $T=10\mathrm{K}$ and $B=9.2\mathrm{T}$ (right), calculated from the short-length prediction (99) based on the ansatz (45) (blue, solid) and the exact result based on Eq. (G2) (red, dashed). System parameters are taken from Table 1. The magnetic field $B=9.2\mathrm{T}$ is the highest magnetic field for which the dispersions of magnons and phonons touch.