We study anomalous Hall conductivity ($sigma$$_{rm AHC}$) and electronic band structures of Si-substituted Mn$_{2}$CoAl (Mn$_{2}$CoAl$_{1-x}$Si$_{x}$). First-principles calculations reveal that the electronic band structure is like a spin-gapless system even after substituting a quaternary element of Si for Al up to $x = $0.2 in Mn$_{2}$CoAl$_{1-x}$Si$_{x}$. This means that the Si substitution enables the Fermi level shift without largely changing the electronic structures in Mn$_{2}$CoAl. By using molecular beam epitaxy (MBE) techniques, Mn$_{2}$CoAl$_{1-x}$Si$_{x}$ epitaxial films can be grown, leading to the systematic control of $x$ (0 $le$ $x$ $le$ 0.3). In addition to the electrical conductivity, the values of $sigma$$_{rm AHC}$ for the Mn$_{2}$CoAl$_{1-x}$Si$_{x}$ films are similar to those in Mn$_{2}$CoAl films shown in previous reports. We note that a very small $sigma$$_{rm AHC}$ of $sim$ 1.1 S/cm is obtained for $x =$ 0.225 and the sign of $sigma$$_{rm AHC}$ is changed from positive to negative at around $x =$ 0.25. We discuss the origin of the sign reversal of $sigma$$_{rm AHC}$ as a consequence of the Fermi level shift in MCA. Considering the presence of the structural disorder in the Mn$_{2}$CoAl$_{1-x}$Si$_{x}$ films, we can conclude that the small value and sign reversal of $sigma$$_{rm AHC}$ are not related to the characteristics of spin-gapless semiconductors.
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