Electronic and structural properties of Rh- and Pd-based kagome layered shandites from first principles

Abstract

The shandite structure hosts transition metal ions arranged in kagome layers. These layers are stacked rhombohedrally and are interspersed with post-transition metal ions and chalcogens. The electronic states near the Fermi level are dominated by the transition metal $d$-orbitals and feature saddle points near several of the high-symmetry positions of the Brillouin zone, most notably the F and L points. Combining symmetry considerations with ab initio methods, we study the electronic and structural properties of these materials with an emphasis on the connection between electronic saddle points at specific momenta and structural instabilities at these momenta. While the parent compounds studied are all found to be structurally stable under ambient conditions, we show that, in specific compounds, moving the saddle point closer to the Fermi level using either hydrostatic pressure or doping, can induce a structural instability. The importance of the electronic degrees of freedom in driving this instability is supported by the dependence of the frequency of the soft phonon mode on the electronic smearing temperature. Our first-principles calculations show that as the smearing temperature is increased, the compound becomes structurally stable again, indicating that the electron-phonon coupling is playing an important role. Our findings survey the structural properties of a large family of shandite materials and shed light on the role played by saddle points in the electronic structure in driving structural instabilities in rhombohedrally stacked kagome-layered materials.

Figures (15)

• Figure 1: (a) Schematic crystal structure of the shandite compounds. The rhombohedral primitive unit cell is denoted by the light blue arrows, while the hexagonal conventional cell is outlined in black. The kagome layers consist of the transition-metal ions, $M$, in deep blue. The post transition-metal ions, $A$, are green while the chalcogen ions, Ch, are orange. (b) First Brillouin zone for the space group $R\bar{3}m$. The Brillouin zone of the primitive unit cell is denoted in light blue, while the dark gray denotes the Brillouin zone of the conventional cell. The relevant high-symmetry points are highlighted by blue (F), black (L), and red (T) arrows, respectively. The stars of F and L contain three points while there is only a single point in the star of T.
• Figure 2: Characters of the irreps of the little group, C$_{\rm 2h}$, of the F or the L point of $R\bar{3}m$, alongside an illustration of the rotation axes and inversion point with respect to the kagome planes right above the origin at $c=1/6$ and right below the origin at $c=-1/6$. The table shows the characters of the generators. The little group irreps are one-dimensional, while the corresponding space group irreps are three-dimensional since there are three vectors in the star of F or L. The $\pm$ sign indicates the character under inversion, where the inversion center is placed at the origin of the conventional cell [Fig. \ref{['fig:r3m_crystal_and_bz']}(a)] in between two kagome layers.
• Figure 3: Orbitally resolved electronic structures of shandite materials $M_3A_2$S$_2$ for (a)--(e) $M=$Rh and (f)--(j) $M=$Pd. Different colors correspond to different orbital weights. Blue denotes the $d$-orbitals of the $M$ atoms, green the $p$-orbitals of the $A$ atoms while orange shows the $p$-orbitals of the S atoms. The irreducible representations $F_{1,2}^\pm$ and $L_{1,2}^\pm$ of the states closer to the Fermi level are shown with magenta (+) or cyan (-) triangles pointing upwards (1) or downwards (2). The orbitally resolved electronic density of states is attached to the right of each figure.
• Figure 4: Orbitally resolved electronic structures of shandite materials $M_3A_2$Se$_2$ for (a)--(e) $M=$Rh and (f)--(j) $M=$Pd. Different colors correspond to different orbital weights. Blue denotes the $d$-orbitals of the $M$ atoms, green the $p$-orbitals of the $A$ atoms while orange shows the $p$-orbitals of the Se atoms. The irreducible representations $F_{1,2}^\pm$ and $L_{1,2}^\pm$ of the states closer to the Fermi level are shown with magenta (+) or cyan (-) triangles pointing upwards (1) or downwards (2). The orbitally resolved electronic density of states is attached to the right of each figure.
• Figure 5: Some examples of the wavefunctions induced by the $d$-orbitals of the atoms forming the kagome layers in shandites at the high-symmetry points $\Gamma$, T, F and L, and their irreps. While these wavefunctions, shown on a single layer of the conventional cell at $c=1/2$, are obtained from DFT, their forms are consistent with group theory predictions. The corresponding Wyckoff position, 9d, has local symmetry C$_{\rm 2h}$ ($2/m$) so that the d-orbitals split into the $A_g$ (even under $2_{100}$) and $B_g$ (odd under $2_{100}$) irreps. Note that the periodicity of these wavefunctions between different layers are different according to their different wavevectors.