Weyl semimetal in the rare-earth hexaboride family supporting a pseudonodal surface and a giant anomalous Hall effect

Fast Electrically Switchable Large Gap Quantum Spin Hall States in MGe2Z4

“…This study obtain the SHC $$\sigma _{xy}^z$$ using the Kubo formula [ 70–74 ] . $$$$\begin{equation} \sigma _{xy}^z = \frac{-e^2}{VN_{\bf k}^3\hbar }\sum _{\bf k} \Omega _{xy}^z({\bf k}) \end{equation}$$$$where, $$$$\begin{equation} \Omega _{xy}^z({\bf k}) = \sum _n f(E_{n,{\bf k}}) \Omega _{n,xy}^z({\bf k}) \end{equation}$$$$is the k ‐resolved spin Berry curvature and $$$$\begin{equation} \Omega _{n,xy}^z({\bf k}) = \hbar ^2 \sum _{m\ne n} \frac{-2Im[\bra {n{\bf k}}j_x^z\ket {m{\bf k}} \bra {m{\bf k}}\hat{v}_y\ket {n{\bf k}}]}{(E_{n,{\bf k}}-E_{m,{\bf k}})^2} \end{equation}$$$$<...…”

“…The surface energy spectrum was obtained within the iterative Green's function method. [69] This study obtain the SHC 𝜎 z xy using the Kubo formula [70][71][72][73][74] .…”

Topology and Symmetry in Quantum Materials

“…[2][3][4] The 2D band-touching points forming nodal surfaces have been observed. [123,124] Using first-principles predictive theoretical modeling combined with various experimental probes, many materials have been verified to host a variety of topological semimetallic states. [4] Examples include Dirac states in Na 3 Bi, [89,96,97] Cd 3 As 2 , [98,99] KZnBi, [100] and transition-metal dichalcogenides, [101] Weyl fermions in nonmagnetic TaAs family, Mo x W 1−x Te 2 , [4] LaAlGe [122] and TaIrTe 4 , [125] and the magnetic Heuslers, [119,[126][127][128] Co 3 Sn 2 S 2 , [129] and the (Ce,Pr)Al(Si,Ge) materials.…”

Switchable large-gap quantum spin Hall state in two-dimensional MSi$_2$Z$_4$ materials class

“…Having established the QSH state in 1T -MSi 2 Z 4 monolayers, we now discuss their intrinsic spin Hall conductivity (SHC). We obtain the SHC σ z xy using the Kubo formula [40][41][42][43][44] as:…”