Momentum-resolved spin splitting in Mn-doped trivial CdTe and topological HgTe semiconductors

Exchange coupling between localized spins and band or topological states accounts for giant magnetotransport and magneto-optical effects as well as determines spin-spin interactions in magnetic insulators and semiconductors. However, even in archetypical dilute magnetic semiconductors such as ${\mathrm{Cd}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}$ and ${\mathrm{Hg}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}$ the evolution of this coupling with the wave vector $\mathit{k}$ is not understood. For instance, a series of experiments have demonstrated that exchange-induced splitting of magneto-optical spectra of ${\mathrm{Cd}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}$ and ${\mathrm{Zn}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}$ at the $L$ points of the Brillouin zone is, in contradiction to the existing theories, more than one order of magnitude smaller compared to its value at the zone center and can show an unexpected sign of the effective Landé factors, opposite to that found for topological ${\mathrm{Hg}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}$. The origin of these findings we elucidate quantitatively by combining (i) relativistic first-principles density functional calculations with the modified Becke-Johnson exchange-correlation potential; (ii) a tight-binding approach that takes carefully into account $\mathit{k}$ dependence of the potential and kinetic $sp-d$ exchange interactions; (iii) a theory of magnetic circular dichroism (MCD) for $E_1$ and $E_1+{\mathrm{\Delta }}_1$ optical transitions, developed here within the envelope function $k·p$ formalism for the $L$ point of the Brillouin zone in zinc-blende crystals. This combination of methods leads to the conclusion that the physics of MCD at the boundary of the Brillouin zone is strongly affected by the strength of two relativistic effects in particular compounds: (i) the mass-velocity term that controls the distance of the conduction band at the $L$ point to the upper Hubbard $d^6$ band of Mn ions and, thus, a relative magnitude and sign of the exchange splittings in the conduction and valence bands; (ii) the spin-momentum locking by spin-orbit coupling that reduces exchange splitting depending on the orientation of particular $L$ valleys with respect to the magnetization direction.

Figure 1

GGA band structure obtained using the MLWF for ${\mathrm{Cd}}_{0.875}{\mathrm{Mn}}_{0.125}\mathrm{Te}, U_{\text{Mn}}=5$ eV, and assuming ferromagnetic alignment of Mn spins without spin-orbit coupling. A gray (red) dashed line in the left (right) panel depicts the spin-up (-down) channel. The interpolated Cd $s$, Te $p$, and Mn $d$ Wannier bands are shown by a solid green line. Zero energy is set at the valence band top.

Figure 2

GGA band structure of ${\mathrm{Hg}}_{0.875}{\mathrm{Mn}}_{0.125}\mathrm{Te}$ with ferromagnetically aligned Mn spins without spin-orbit coupling and for $U_{\text{Mn}}=5$ eV. Zero energy is set at the valence band top. Bands for spin up and spin down with respect to Mn spins' direction are shown by solid gray and red dashed lines, respectively.

Figure 3

GGA values of effective exchange integrals for the conduction and valence bands $J_c$ in the top panel (red lines) and $J_v$ in the bottom panel (green lines) for ${\mathrm{Cd}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}$ compared to experimental values at the $\mathrm{\Gamma }$ point, determined by exciton magnetospectroscopy, represented by horizontal lines [54]. The experimental effective exchange integral is positive for the conduction band and negative for the valence at the $\mathrm{\Gamma }$ point. The solid, dashed, and dotted lines represent the band spin splitting for entirely spin-polarized Mn ions with concentrations $x=\frac{2}{64}=0.03125,\frac{4}{64}=0.0625$, and $\frac{8}{64}=0.125$, respectively. The computations have been performed neglecting spin-orbit coupling and for $U_{\text{Mn}}=5$ eV.

Figure 4

The values of effective exchange integrals for the conduction and valence bands $J_c$ in the top panel (red lines) and $J_v$ in the bottom panel (green lines) for ${\mathrm{Hg}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}$ compared to experimental values at the $\mathrm{\Gamma }$ point represented by horizontal lines [3, 60]. The sign of the experimental effective exchange integral is negative for the valence band and positive for the conduction band at the $\mathrm{\Gamma }$ point. The solid, dashed, and short-dashed lines represent the spin splitting for $x=\frac{2}{64}=0.03125,\frac{4}{64}=0.0625$, and $\frac{8}{64}=0.125$, respectively. The square, circle, and triangle represent an effective exchange integral of the ${\mathrm{\Gamma }}_6$ band for $x=\frac{2}{64}=0.03125,\frac{4}{64}=0.0625$, and $\frac{8}{64}=0.125$, respectively.

Figure 5

Comparison between the Wannier bands obtained by MBJLDA (solid green line) and our tight-binding model (dashed blue lines) for CdTe (left panel) and HgTe (right panel) taking spin-orbit coupling into account. The bands labeled $L_1, L_2, L_3$, and $L_4$ are described by irreducible representations of the double group $L_6, L_{4,5}, L_6$, and $L_6$, respectively.

Figure 6

Normalized absorption ${\mathcal{A}}_{\iota \phi }^{\pm }$ for ${\sigma }^{+}$ and ${\sigma }^{-}$ polarizations (positive and negative bars, respectively) for optical transitions in the vicinity of the $E_1$ and $E_1+{\mathrm{\Delta }}_1$ gaps at $L$ points in the Brillouin zone for ${\mathrm{Cd}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}$ (left panels) and ${\mathrm{Hg}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}$ (right panels) with $x_{\text{eff}}=0.0625$ and fully polarized Mn spins along $\langle 100\rangle , \langle 110\rangle$, or $\langle 111\rangle$ crystal axis that is also the light propagation direction (upper, middle, and bottom panels, respectively). Arrows show the resulting spectral splittings at $L, \mathrm{\Delta }E$, defined as a distance between the centers of gravity of lines appearing for ${\sigma }^{-}$ and ${\sigma }^{+}$ polarizations, respectively. In the case of ${\mathrm{Cd}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}, \mathrm{\Delta }E$ at $E_1$ is about 30 times smaller than at $E_0$ for the same Mn magnetization. Note that MCD is usually defined as the difference ${\mathcal{A}}^{-}-{\mathcal{A}}^{+}$.