Self-consistent analysis of doping effect for magnetic ordering in stacked-kagome Weyl system

We theoretically study the carrier doping effect for magnetism in the stacked-kagome system ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$ based on an effective model and the Hartree-Fock method. We show the electron filling and temperature dependencies of the magnetic order parameter. The perpendicular ferromagnetic ordering is suppressed by hole doping, whereas undoped ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$ shows a magnetic Weyl semimetal state. Additionally, in the electron-doped regime, we find a noncollinear antiferromagnetic ordering. Especially, in the noncollinear antiferromagnetic state, by considering a certain spin-orbit coupling, the finite orbital magnetization and the anomalous Hall conductivity are obtained.

Figure 1

Possible phases in doped ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$. In undoped ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$, Weyl semimetal phase with perpendicular ferromagnetic ordering appears. In hole-doped ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$, the ferromagnetic ordering is suppressed and the system becomes paramagnetic. In electron-doped ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$, a noncollinear antiferromagnetic ordering appears.

Figure 2

(a) Crystal structure of ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$. Co forms the kagome lattice network and sandwiches two layers of triangle lattice formed by Sn and S, respectively. (b) Anticipated electronic structures of undoped ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$. A dashed box indicates the limited orbitals in our effective model. The total electron number per unit cell is $n_{\text{e}}=3$. (c) Electronic structure when the total electron number is $n_{\text{e}}=2$. In experimental situation, one hole is doped by substituting Co with Fe, in each unit cell. (d) Electronic structure when the total electron number per unit cell is $n_{\text{e}}=4$. In experimental situation, one electron is doped by substituting Co with Ni, in each unit cell.

Figure 3

Color maps for (a) $z$ component of magnetization $m^z$ per unit cell in units of ${\mu }_{\mathrm{B}}$, (b) in-plane component of magnetization, and (c) $z$ component of vector spin chirality with respect to the filling factor of dopants and temperature. In case $\mathrm{\Delta }{n}_{\text{e}}\sim 0.0$, (a) ferromagnetic ordering with $m^z\sim 0.9$ appears. $m^z$ decreases as $\mathrm{\Delta }{n}_{\mathrm{e}}$ deviates from $\mathrm{\Delta }{n}_{\text{e}}\sim 0$. When $\mathrm{\Delta }{n}_{\text{e}}\sim +1.0$, (a) $z$ component of magnetization diminishes, while (b) in-plane component of magnetization and (c) $z$ component of the vector spin chirality become finite, indicating noncollinear antiferromagnetic state. Electronic band structures of (d) paramagnetic state, (e) perpendicular ferromagnetic state, and (f) noncollinear antiferromagnetic state obtained by the Hartree-Fock method. In (d) $\mathrm{\Delta }{n}_{\mathrm{e}}=-1$, system is paramagnetic and the chemical potential is close to the band gap. In (e) $\mathrm{\Delta }{n}_{\mathrm{e}}=0$, system is ferromagnetic, the chemical potential is located near the local minimum of the spin majority band, corresponding to the Weyl points, and near the gap of the spin minority band. In (f) noncollinear antiferromagnetic state, the electronic band dispersion around the $L$ point remains almost unchanged, compared to that in the ferromagnetic state.

Figure 4

(a) Orbital magnetizations and (b) anomalous Hall conductivity for $t_{\mathrm{KM}}^z=0.0$, $0.1t_1$, and $0.2t_1$, as a function of magnetization angle $\theta$ depicted in the inset of (a).