Effect of the magnon dispersion on the longitudinal spin Seebeck effect in yttrium iron garnets

We study the temperature dependence of the longitudinal spin Seebeck effect (LSSE) in an yttrium iron garnet ${\mathrm{Y}}_3\mathrm{F}{\mathrm{e}}_5{\mathrm{O}}_{12}$ (YIG)/Pt system for samples of different thicknesses. In this system, the thermal spin torque is magnon driven. The LSSE signal peaks at a specific temperature that depends on the YIG sample thickness. We also observe freeze-out of the LSSE signal at high magnetic fields, which we attribute to the opening of an energy gap in the magnon dispersion. We observe partial freeze-out of the LSSE signal even at room temperature, where $k_{B}T$ is much larger than the gap. This suggests that a subset of the magnon population with an energy below $k_B{T}_C \left(T_C\sim 40\mathrm{K}\right)$ contributes disproportionately to the LSSE; at temperatures above $T_C$, we label these magnons subthermal magnons. The $T$ dependence of the LSSE at temperatures below the maximum is interpreted in terms of an empirical model that ascribes most of the temperature dependence to that of the thermally driven magnon flux, which is related to the details of the magnon dispersion.

Figure 1

Experimental setup for LSSE measurement. (a) Experimental setup on a liquid helium cryostat used for both sample A and B. (b) Schematic illustration of the measurement geometry for sample A. The same geometry applies to sample B except that the GGG substrate is absent in sample B. The sample dimensions are not drawn to scale.

Figure 2

Low-field LSSE measurement. (a–c) Transverse voltage $V_y$ as a function of magnetic field $H_x$ (a,b), and $\mathrm{\Delta }{V}_y$ versus applied temperature difference $\mathrm{\Delta }{T}_z$ (c) for sample A. (d–f) $V_y$ as a function of $H_x$ (d,e), and $\mathrm{\Delta }{V}_y$ versus $\mathrm{\Delta }{T}_z$ (f) for sample B.

Figure 3

High-field LSSE measurement on sample B. (a,b) $V_y/\mathrm{\Delta }{T}_z$ as a function of $H_x$ measured up to $H_x=\pm 70\mathrm{kOe}$ (“high field”) at $T_{\mathrm{avg}}=296\mathrm{K}$ (a) and at $T_{\mathrm{avg}}=10\mathrm{K}$ (b). (c,d) Magnification of low-field $(H_x=\pm 3\mathrm{kOe})$ results at $T_{\mathrm{avg}}=296\mathrm{K}$ (c) and at $T_{\mathrm{avg}}=10\mathrm{K}$ (d).

Figure 4

(a) $T$ dependence of $\mathrm{\Delta }{S}_y/S_{y,max}$ for sample B. Here, $\mathrm{\Delta }{S}_y/S_{y,max}\equiv \left(S_{y,max}-S_{y,70\mathrm{kOe}}\right)/S_{y,max}$ with $S_{y,max}$ and $S_{y,70\mathrm{kOe}}$ defined as in the text. $\mathrm{\Delta }{S}_y/S_{y,max}=1$ indicates the fully suppressed $S_y$ at $H=\pm 70\mathrm{kOe}$ (i.e., $S_{y,70\mathrm{kOe}}=0$). The inset shows the same data extended to 300 K in log scale. $T_{\mathrm{avg}}$ denotes the average sample temperature. (b) Schematic diagram of magnon dispersion of YIG below 0.6 THz (30 K) per Ref. [27]. The red arrow indicates opening of a magnon gap equivalent to $\sim 9\mathrm{K}$ when $H=\pm 70\mathrm{kOe}$ is applied. The red dashed lines mark the temperature $(\sim 15\mathrm{K})$ below which significant increase in $\mathrm{\Delta }{S}_y/S_{y,max}$ starts to occur.

Figure 5

$T$ dependence of the spin Seebeck coefficient $S_y$ for the 4-μm film sample A (green diamond) and the bulk sample B (red circle). The dashed line represents a power law for the bulk sample, and the full line represents the values calculated from the model [Eq. (9))] in the text for the thin-film sample, renormalized to go through the data.