Emerging spintronic and valleytronic phenomena in noncentrosymmetric variants of the Kane-Mele $X_4{Y}_2{Z}_6$ materials family ($X$ = Pt, Pd, Ni; $Y$ = Hg, Zn, Cd; $Z$ = S, Se, Te)

The realization of the Kane-Mele model in Xene solids faces challenges due to weak spin-orbit coupling (SOC). Nevertheless, the recently discovered $X_4{Y}_2{Z}_6$ family ($X$=Pt, Pd, Ni; $Y$=Hg, Zn, Cd; and $Z$=S, Se, Te) offers a promising opportunity with larger SOC. However, the presence of centrosymmetry in this family hinders the achievement of several cross-coupling phenomena based on spintronics and valleytronics. This study explores a noncentrosymmetric version of the Kane-Mele family, $X_{4}Y{Y}^{\prime }{Z}_6$ and $X_{4}Y{Y}^{\prime }{\left(Z{Z}^{\prime }\right)}_3$, comprising over 16 experimentally accessible members. The results reveal intertwined phenomena involving topology, spin, and valley degrees of freedom, including the quantum valley/spin Hall effect, spin-valley locking, and spin-valley selective optical transitions. Additionally, Rashba coupling coexists with Ising spin splitting, enabling valley spin valve functionality and out-of-plane spontaneous electric polarization. Quantum valley Hall kink states can be achieved on the domain walls between these noncentrosymmetric monolayers due to opposite spin-valley Berry curvatures. External factors, like electric fields and strain, induce various topological phase transitions. This study lays the foundation for exploring spin-valley physics in low-dimensional topological materials with noncentrosymmetry.

Figure 1

Continuum model: (a)–(c) Schematic representation illustrating the evolution of Berry curvature and the topological phase transition for different strengths of $U$ and ${\lambda }_{\mathrm{so}}$ in the $X_4{Y}_2{Z}_6$ family. (d)–(f) Interplay between ${\lambda }_{\mathrm{so}}$ and $U$ under the condition ${\lambda }_{\mathrm{so}}>{\lambda }_R$. (g)–(i) Same as (d)–(f), but for the case ${\lambda }_{\mathrm{so}}<{\lambda }_R$. QSHI, QVHI, and DSM represent the states of the quantum spin Hall insulator, quantum valley Hall insulator, and Dirac semimetal, respectively. (j) Topological phase diagram as a function of $U$ and ${\lambda }_R$ in the presence of Kane-Mele SOC ${\lambda }_{\mathrm{so}}\sim 81.2$ meV. The spin and valley Chern numbers $({\mathcal{C}}_{\mathrm{s}},{\mathcal{C}}_{\mathrm{v}})$ are also labeled.

Figure 2

Exploiting centrosymmetry in the Kane-Mele $X_4{Y}_2{Z}_6$ family: Crystal structures of (a) ${\mathrm{Pt}}_4{\mathrm{Hg}}_2{\mathrm{Se}}_6$, (b) ${\mathrm{Pt}}_4{\mathrm{Hg}}_2{\mathrm{Se}}_3{(\mathrm{S}/\mathrm{Te})}_3$, (c) ${\mathrm{Pt}}_4\mathrm{Hg}(\mathrm{Cd}/\mathrm{Zn}){\mathrm{Se}}_6$, and (d) ${\mathrm{Pt}}_2{(\mathrm{Pd}/\mathrm{Ni})}_2{\mathrm{Hg}}_2{\mathrm{Se}}_6$. The corresponding point groups are also displayed, where Cs (NCs) denotes centrosymmetry (noncentrosymmetry). (e)–(h) The corresponding charge density profiles. The homogeneity and inhomogeneity in the charge density profile can be observed in (e) and (h) and (f) and (g), reflecting the existence and nonexistence of centrosymmetry, respectively.

Figure 3

The phonon band dispersion of various noncentrosymmetric Kane-Male $X_{4}Y{Y}^{\prime }{Z}_6/X_4{Y}_2{\left(Z{Z}^{\prime }\right)}_3$ monolayers, demonstrating the absence of soft modes. The absence of soft modes signifies both the dynamic stability and experimental accessibility of the proposed monolayers.

Figure 4

(a), (c), (e), (g), (i), and (k) Non-SOC and (b), (d), (f), (h), (j), and (l) the corresponding SOC-included $S_z$-resolved energy spectrum for various noncentrosymmetric Kane-Male $X_{4}Y{Y}^{\prime }{Z}_6/X_4{Y}_2{\left(Z{Z}^{\prime }\right)}_3$ monolayers. The valley-dependent spin polarization is evident. (m) Constant energy contour maps calculated for the top valence bands (right) and the lower conduction bands (left). The triangular and starlike contours reflect the high trigonal distortion around the Fermi level in these proposed monolayers. (n) The size of the momentum-dependent spin splitting in the top valence and lower conduction bands in our proposed monolayers.

Figure 5

Top: Berry curvature profiles in momentum space for various Kane-Male ${\mathrm{Pt}}_{4}Y{Y}^{\prime }{Z}_6$ monolayers, with opposite Berry curvatures in valleys $K$ and $K^{\prime }$, signifying the quantum valley Hall effect. Middle: Corresponding circular dichroism profiles, indicating valley-dependent selective optical band transitions and signifying valley selective optical conductivity. Bottom: Corresponding surface state profiles.

Figure 6

(a) and (b) Spin-texture distribution over the momentum space for the ${\mathrm{Pt}}_4{\mathrm{Hg}}_2{\left(\mathrm{SeTe}\right)}_3$ monolayer. Opposite $S_z$-resolved spin textures in valleys $K$ and $K^{\prime }$ can be noticed. $S_x, S_y$, and $S_z$ spin-resolved constant energy contour profiles at (c) the top valence bands and (d) the lower conduction bands for the ${\mathrm{Pt}}_4{\mathrm{Hg}}_2{\left(\mathrm{SeTe}\right)}_3$ layer. (e) $S_x$-, $S_y$-, and $S_z$-resolved constant-energy contour maps at the bottom of the lower conduction band for the ${\mathrm{Pt}}_4{\mathrm{Hg}}_2{\left(\mathrm{STe}\right)}_3$ layer. The dominant contribution of the in-plane spin component ($S_x, S_y$) signifies the presence of Rashba-type spin splitting around the $\mathrm{\Gamma }$ point.

Figure 7

(a) SOC-included energy spectrum and (b) corresponding surface state profile of the (${\mathrm{PtPd})}_2{\mathrm{Hg}}_2{\mathrm{Se}}_6$ layer. The presence of a Dirac point signifies the robust topology in the (${\mathrm{PtPd})}_2{\mathrm{Hg}}_2{\mathrm{Se}}_6$ layer. The SOC-included energy spectrum with (c) $+E_z=0.3\text{V}/\text{Å}$ and (d) $-E_z=0.3\text{V}/\AA$. The interchange of spin polarization between valleys $K$ and $K^{\prime }$ makes the (${\mathrm{PtPd})}_2{\mathrm{Hg}}_2{\mathrm{Se}}_6$-type layer useful for the valley spin valve effect.

Figure 8

Top: Evolution of the $S_z$-resolved energy spectrum of the ${\mathrm{Pt}}_4{\mathrm{Hg}}_2{\left(\mathrm{SeTe}\right)}_3$ layer under various amounts of stretching. Band gap closing and reopening occur as the stretching increases, signifying a topological phase transition from QVH to QSH and again to the QVH state. Bottom: Corresponding Berry curvature, surface states, and Wannier charge center profiles.

Figure 9

Evolution of the $S_z$-resolved band structure for the (a)–(c) ${\mathrm{Pt}}_4{\mathrm{Hg}}_2{\left(\mathrm{SSe}\right)}_3$, (d)–(f) ${\mathrm{Pt}}_4{\mathrm{Hg}}_2{\left(\mathrm{STe}\right)}_3$, and (g)–(i) ${\mathrm{Pt}}_4{\mathrm{Hg}}_2{\left(\mathrm{SeTe}\right)}_3$ layers under the electric field orientations $E_z<0$ left), $E_z=0$ (middle), and $E_z>0$ (right). The strength of the electric field was chosen to be 0.3, 0.35, and 0.2 $\text{eV}/\mathring{\text{A}}$ in (a)–(f), (g). and (i). respectively. The sizes of the global band gap are highlighted in each plot.

Figure 10

(a) Realization of a quantum valley Hall kink (QVHK) state at the interface between the ${\mathrm{Pt}}_4{\mathrm{Hg}}_2{\left(\mathrm{SSe}\right)}_3$ and ${\mathrm{Pt}}_4{\mathrm{Hg}}_2{\left(\mathrm{TeSe}\right)}_3$ monolayers. (b) Realization of quantum spin-valley Hall kink (QSVHK) states at the interface between ${\mathrm{Pt}}_4{\mathrm{Hg}}_2{\left(\mathrm{TeSe}\right)}_3$ and ${\mathrm{Pt}}_4{\mathrm{Hg}}_2{\mathrm{Se}}_6$.