Colossal enhancement of the Seebeck coefficient in FeSb$_{2}$ driven by nearly ballistic phonons

An unusually large $S$ of $-45$ mV/K (at 10 K) was discovered in FeSb$_{2}$ single crystal, which prompted extensive investigations into its physical origin [A. Bentien $et$ $al$., EPL 80, 17008 (2007).] This compound has a small energy gap $\Delta \sim 5$ meV, which may be caused by strong correlations of Fe 3d-electrons, as observed with Kondo insulators, and the colossally large $S$ may be attributed to this unique band structure near the Fermi energy. However, the exceptional value of $S$ has not been clearly explained by electron correlations, suggesting an additional contribution such as the non-equilibrium phonon-drag effect [H. Takahashi $et$ $al$., JPSJ 80, 054708 (2011)., H. Takahashi $et$ $al$., PRB 84, 205215 (2011)., and H. Takahashi $et$ $al$., PRB 88, 165205 (2013).]. Here, we report on the direct investigation of this effect by measuring the transport properties of three samples with cross sections ranging from $250 \times 245$ $\mu$m$^{2}$ to $80\times 160$ $\mu$m$^{2}$. $S$ and $\kappa$ show a significant size effect, indicating that nearly ballistic phonons, which have a long mean free path relative to the sample dimensions, are responsible for the colossal $S$.