Anomalous Hall effect in three ferromagnetic compounds: Eu Fe 4 Sb 12 , Yb 14 Mn Sb 11 , and Eu 8 Ga 16 Ge 30

The Hall resistivity $({\ensuremath{\rho}}_{xy})$, resistivity $({\ensuremath{\rho}}_{xx})$, and magnetization of three metallic ferromagnets are investigated as a function of magnetic field and temperature. The three ferromagnets, $\mathrm{Eu}{\mathrm{Fe}}_{4}{\mathrm{Sb}}_{12}$ $({T}_{c}\ensuremath{\approx}84\phantom{\rule{0.3em}{0ex}}\mathrm{K})$, ${\mathrm{Yb}}_{14}\mathrm{Mn}{\mathrm{Sb}}_{11}$ $({T}_{c}\ensuremath{\approx}53\phantom{\rule{0.3em}{0ex}}\mathrm{K})$, and ${\mathrm{Eu}}_{8}{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ $({T}_{c}\ensuremath{\approx}36\phantom{\rule{0.3em}{0ex}}\mathrm{K})$ are Zintl compounds with carrier concentrations between $1\ifmmode\times\else\texttimes\fi{}{10}^{21}$ and $3.5\ifmmode\times\else\texttimes\fi{}{10}^{21}\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$. The relative decrease in ${\ensuremath{\rho}}_{xx}$ below ${T}_{c}$ $[{\ensuremath{\rho}}_{xx}({T}_{c})∕{\ensuremath{\rho}}_{xx}(2\phantom{\rule{0.3em}{0ex}}\mathrm{K})]$ is 28, 6.5, and 1.3 for $\mathrm{Eu}{\mathrm{Fe}}_{4}{\mathrm{Sb}}_{12}$, ${\mathrm{Yb}}_{14}\mathrm{Mn}{\mathrm{Sb}}_{11}$, and ${\mathrm{Eu}}_{8}{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$, respectively. The low carrier concentrations coupled with low magnetic anisotropies allow a relatively clean separation between the anomalous $({\ensuremath{\rho}}_{xy}^{\ensuremath{'}})$, and normal contributions to the measured Hall resistivity. For each compound the anomalous contribution in the zero field limit is fit to $a{\ensuremath{\rho}}_{xx}+{\ensuremath{\sigma}}_{xy}{\ensuremath{\rho}}_{xx}^{2}$ for temperatures $Tl{T}_{c}$. At $T=0$ the anomalous Hall conductivity ${\ensuremath{\sigma}}_{xy}^{0}$, is $\ensuremath{-}220\ifmmode\pm\else\textpm\fi{}5\phantom{\rule{0.3em}{0ex}}({\ensuremath{\Omega}}^{\ensuremath{-}1}\phantom{\rule{0.2em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1})$, $\ensuremath{-}14.7\ifmmode\pm\else\textpm\fi{}1\phantom{\rule{0.3em}{0ex}}({\ensuremath{\Omega}}^{\ensuremath{-}1}\phantom{\rule{0.2em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1})$, and $28\ifmmode\pm\else\textpm\fi{}3\phantom{\rule{0.3em}{0ex}}({\ensuremath{\Omega}}^{\ensuremath{-}1}\phantom{\rule{0.2em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1})$ for $\mathrm{Eu}{\mathrm{Fe}}_{4}{\mathrm{Sb}}_{12}$, ${\mathrm{Yb}}_{14}\mathrm{Mn}{\mathrm{Sb}}_{11}$, and ${\mathrm{Eu}}_{8}{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$, respectively, and is independent of temperature for $Tl{T}_{c}$ if the change in spontaneous magnetization (order parameter) with temperature is taken into account. These data are consistent with recent theories [T. Jungwirth, Q. Niu, and A. H. MacDonald, Phys. Rev. Lett. 88, 207208 (2002); Y. Yao, L. Kleinman, A. H. MacDonald, J. Sinova, T. Jungwirth, D.-S. Wang, E. Wang, and Q. Niu, Phys. Rev. Lett. 92, 037204 (2004); I. V. Solovyev, Phys. Rev. B 67, 174406 (2003); Z. Fang, N. Nagosa, K. S. Takahashi, A. Asamitsu, R. Mathieu, T. Ogasawara, H. Yamada, M. Kawasaki, Y. Tokura, and K. Terakura, Science 302, 92 (2003); and Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa, and Y. Tokura, Science 291, 2573 (2001)] of the anomalous Hall effect that suggest that even for stochiometric ferromagnetic crystals, such as those studied in this article, the intrinsic Hall conductivity is finite at $T=0$, and is a ground state property that can be calculated from the electronic structure.