Unlocking Doping Effects on Altermagnetism in MnTe: Emergence of Quasi-altermagnetism

Abstract

Governed by specific symmetries, altermagnetism is an emerging field in condensed matter physics, characterized by unique spin-splitting of the bands in the momentum space co-existing with the compensated magnetization as in antiferromagnets. As crystals can have tailored and unintended defects, it is important to gain insights on how altermagnets are affected by the defects-driven symmetry-breaking which, in turn, can build promising perspectives on potential applications. In this study, considering the widely investigated MnTe as a prototype altermagnet, defects are introduced through substitutional doping to create a large configuration space of spin space groups. With the aid of density functional theory calculations, symmetry analysis, and model studies in this configuration space, we demonstrate the generic presence of spin-split of the antiferromagnetic bands in the momentum space. This is indicative of a wider class of quasi-altermagnetic materials, augmenting the set of ideal altermagnetic systems. Furthermore, we show that while pristine MnTe does not show anomalous Hall conductivity (AHC) with out-of-plane magnetization, suitable doping can be carried out to obtain finite and varied AHC. Our predictions of quasi-altermagnetism and doping-driven tailored AHC have the potential to open up as-yet-unexplored directions in this developing field.

Figures (11)

• Figure 1: Geometric structure and band structure of unit cell and supercell of pristine MnTe. The $2 \times 2 \times 2$ supercell of MnTe is shown in panel (a), with the unit cell highlighted in purple. The corresponding Brillouin zone for the unit cell and the supercell are shown in panel (b), where the outer and inner zones correspond to unit cell and supercell of MnTe, respectively. As expected, enlarging the real-space crystal reduces the size of the Brillouin zone in reciprocal space. However, since the supercell is formed by doubling the unit cell along all three crystallographic directions, the Brillouin zone retains the same shape, and the high-symmetry $k$-points of the unit cell can be mapped onto those of the supercell. Panels (c) and (d) show the band structures for the unit cell and the supercell, respectively. These two band structures can be connected by mapping the $k$-points of their respective Brillouin zones. In both cases, spin-split bands -- characteristic of the altermagnetic behaviour -- are clearly visible along the $L$–$\Gamma$–$L'$ path.
• Figure 2: Altermagnetism in MnTe supercell with single non-magnetic atom substitution. The spin-polarized DOS of the 2 $\times$ 2 $\times$ 2 supercell of MnTe with one Te atom substituted by Se is shown in panel (a), where the blue and red colours represent the spin-up and spin-down channels, respectively. The DOS confirms a magnetically compensated state. Panel (b) shows the constant energy surfaces for the spin-up and spin-down channels (blue and red, respectively) at $E=E_F-0.85$ eV, illustrating the spin-splitting in momentum space. Panels (c)–(e) display the spin-resolved electronic band structures of MnTe supercells with a single Te atom substituted by Se, Sb, and I, respectively. All these structures exhibit momentum-dependent spin-splitting along the $L$–$\Gamma$–$L'$ direction. Panels (f) and (g) illustrate symmetry operations connecting opposite spin sublattices in a single Te-substituted MnTe supercell. The mirror plane perpendicular to the $z$-axis (M$_z$), which connects opposite spin sublattices, is present in panel (f). The structures shown in panels (f) and (g) are related by a six-fold roto-inversion symmetry operation ($S_{6z}$), which consists of a 60° rotation followed by inversion, with the dopant atom serving as the inversion center. Atoms from each sublattice are labelled in planes adjacent to the dopant atom in (f), and their new positions after the $S_{6z}$ operation are shown in panel (g). These features from DFT calculations and symmetry analysis indicate the persistence of $g$-wave altermagnetism in MnTe upon substitution of a single Te atom.
• Figure 3: Altermagnetic and quasi-altermagnetic states in MnTe for different configurations with a pair of Se substitutions. Substituting a pair of Te atoms with Se atoms in a 2$\times$2$\times$2 MnTe supercell yields a system with 12.5% doping. This specific substitution results in 120 unique configurations, distributed among five distinct MSGs. The top panel [(a) and (b)] illustrates representative configurations from each MSG, categorized into two classes:(a) those exhibiting ideal altermagnetism, and (b) those displaying quasi-altermagnetic characteristics. The corresponding electronic band structures for these two classes are presented in panels (c) and (d), respectively. The valence and conduction band edges are shown in the insets, where the characteristic features of both altermagnetic and quasi-altermagnetic states are highlighted. The constant energy surfaces for a pair of spin-up (blue) and spin-down channels (red) at $E=E_F-0.7$ eV for altermagnets and quasi-altermagnets are shown in panels (e) and (f), respectively. Constant energy surfaces of altermagnets illustrate the six-fold roto-inversion relation of opposite channels, whereas a distortion is visible in that of quasi-altermagnets. Panel (g) provides a pie chart summarizing the distribution of the two-atom-doped configurations across the five MSGs. The grey, green, and yellow segments in the pie chart represent configurations that exhibit ideal altermagnetism, collectively accounting for approximately 46.66% of the total configurations.
• Figure 4: Electronic band structures for MnTe supercell with Sb and I pair substitution. Panels (a) and (b) show the band structures of the MnTe supercell in which a pair of non-magnetic atoms have been substituted with Sb, corresponding to different MSGs. Panels (c) and (d) present the band structures for the MnTe supercell with the same substitution by I. The band structures in (a) and (c) exhibit a perfect altermagnetic character, whereas this feature is absent in (b) and (d). Note the shift of the Fermi level due to non-isovalent doping.
• Figure 5: Schematic illustrations defining quasi-altermagnetism and its distinctness w.r.t. ferromagnetism, antiferromagnetism, and altermagnetism. The spin-split occurs in ferromagnetism due to a Zeeman like field. The rest of the configurations have sublattices, denoted here as A and B. The symmetry connections among these sublattices are expressed. The inversion symmetry in antiferromagnet introduces sub-band degeneracy, which is lifted in altermagnet due to the breakdown of the inversion symmetry. The equal but opposite momentum-dependent spin split mirroring a nodal plane in altermagnets is due to perfect rotational symmetries. Lack of this symmetry blurs the nodal plane and results in unequal splitting, which gives rise to quasi-altermagnetism. The $k$-dependent energy relations for each of the configurations are expressed. Here, $k_\perp$ denotes the perpendicular distance from the nodal plane.