Magnon thermal mean free path in yttrium iron garnet

The magnetothermal properties of monocrystalline yttrium iron garnet (YIG) are reported. The magnon contribution to both the thermal conductivity and specific heat at low temperatures has been determined by measuring these properties under an applied magnetic field, which allows us to freeze the magnon modes and isolate the phonon contribution relative to the zero-field behavior. These results are interpreted within the framework of a simple kinetic gas model for magnon heat conduction that allows us to estimate the magnon thermal mean free path, i.e., the inelastic scattering length scale for thermally driven bulk magnons. We observe this parameter to reach as high as approximately 100 μm at 2 K. It tracks the acoustic phonon thermal mean free path closely and decreases rapidly as the temperature is increased. This relatively short length scale suggests that magnon modes at thermal energies in YIG are not solely or directly responsible for coherent macroscale thermal spin transport (e.g., in the spin Seebeck effect) at high temperatures. Instead, these results support a growing consensus that subthermal magnons, i.e., those at energies below about 30 ± 10 K, are important for spin transport in YIG at all temperatures. These results also emphasize that magnon effects should be considered wavelength dependent, and that magnon-magnon interactions may be just as important for thermal spin transport as magnon-phonon scattering. This, in turn, has implications for understanding the characteristic temperature and length scales involved in spin caloritronic phenomena.

Figure 1

Temperature (top frame) and magnetic field (bottom frame) dependence of the thermal conductivity ${\kappa }_{xxx}$ in the presence of a magnetic field parallel to the direction of the temperature gradient. The temperature dependence is shown in the absence of field, then in an external magnetic field of 70 kOe. In the bottom frame, the magnetic field dependence of ${\kappa }_{xxx}$ shows a clear saturation at 2 K for applied fields of $H$ > 15 kOe. At 8 K the saturation is not quite achieved even at 70 kOe, yet the saturation value is only about 2% lower than the value reached. For $T$ < 10 K, the high-field value is interpreted as arising from phonon conduction alone, while the zero-field value is interpreted as the sum of independent phonon and magnon contributions, such that the difference is the magnon thermal conductivity.

Figure 2

Temperature (top frame) and magnetic field (bottom frame) dependence of the transverse thermal conductivity ${\kappa }_{xxz}$ in the presence of a magnetic field perpendicular to the direction of the temperature gradient. The description and conclusions are the same as for ${\kappa }_{xxx}$ in Fig. 1.

Figure 3

The specific heat $C$ of YIG vs temperature (a) under constant pressure at zero magnetic field and in the presence of a 70 kOe applied external field. Inset shows the data plotted as $C{T}^{-3/2}$ vs $T^{3/2}$, from which the slope and intercept provide information about the phonon and magnon contributions, respectively. The magnetic field dependence of $C$ is shown in frame (b) at two temperatures; as was the case for ${\kappa }_{xxx}$ and ${\kappa }_{xxz}$, complete saturation is achieved at 2 K, but not quite at 8 K. Again, the value of $C$ in a 70 kOe applied field is interpreted to be due to phonons $\left(C_P\right)$, while the difference between the value in zero field and $C_P$ is interpreted to be due to magnons $\left(C_M\right)$ below 10 K: These are shown in (c) as a function of temperature. The dashed line is the value calculated from a simple Heisenberg model for the magnon specific heat equation (8), assuming a parabolic dispersion for magnons with a characteristic value of $D$ = 47 K. The phonon specific heat is fitted in frame (d) to a Debye model (full red line) with the additional contribution of an Einstein mode (dashed blue line).

Figure 4

Magnon and phonon thermal mean free paths, as described by Eq. (3). The open symbols describe the results obtained from ${\kappa }_{xxz}$ (transverse) and the full symbols those from ${\kappa }_{xxx}$ (longitudinal) measurements, respectively. This analysis may be extended to higher temperatures if the experiment can be repeated at higher magnetic fields $(H$ > 15 T), which may shed insight into the temperature dependence of magnon scattering within the phonon umklapp regime $(T$ > 20 K).