Tunable auxetic properties in group-IV monochalcogenide monolayers

Negative Poisson's ratio (NPR) is a counterintuitive material elastic constant that can lead to many unusual auxetic properties. Here, using first-principles calculations, we report tunable negative Poisson's ratio in the out-of-plane direction in group-IV monochalcogenide monolayers $MX$ ($M=\text{Sn}$, Ge and $X=\text{S}$, Se). SnSe, GeS, and SnS monolayers have intrinsic NPR ${\nu }_{zx}$ ranging from $-0.004$ to $-0.210$ in armchair ($x$) tension, whereas GeSe monolayer possesses a much larger NPR ${\nu }_{zy}$ of $-0.433$ in zigzag ($y$) tension. Our analysis attributes the NPR effects to the relative position of $M$ and $X$ in the puckered structure and the smaller bending stiffness of $M\text{−}X\text{−}M$ bond angle. We further established the correlation between electronic structures of materials and their crystal structures. It allows us to fine tune GeSe structure via electron doping, leading to a reversible and continuous change of ${\nu }_{zy}$ from $-0.821$ up to 0.895. We also demonstrate the concept of strain engineering GeSe monolayer to switch its Poisson's ratio ${\nu }_{zx}$ between two different values: 0.583 and $-0.433$. Our in-depth study provides not only fundamental knowledge but also practical routes for designing 2D smart materials with tunable negative Poisson's ratio, which are desirable for smart devices at small scale.

Figure 1

Crystal structures of group-IV monochalcogenide ($MX$) monolayers. The atoms are distributed on four atomic layers. The thickness of these monolayers is defined as the vertical distance between the top and the bottom layers. From the side view, for SnS, SnSe, and GeS, group-IV atoms ($M$) are on the two outer layers, whereas for GeSe, chalcogen atoms ($X$) are on the two outer layers.

Figure 2

The correlations between applied strain along the $x$ ($y$) direction and two resultant transverse strains along the $y$ ($x$) or $z$ direction.

Figure 3

(a) The motion of two atoms in the top two atomic layers under the strains in $x$ or $y$ direction. The black curves represent the $z$ coordinates change of group-IV elements, and red curves represent the $z$ coordinates change of chalcogens. The basis point (zero in $z$ coordinates) is selected at the middle height of puckered crystal structures. (b) The black and red arrows depict the motion direction of the $M$ and $X$ atoms under tensile strains of ${\varepsilon }_x$ or ${\varepsilon }_y$. The blue boxes show the strain conditions of these 2D materials. Note that stretching along the $y$ direction is equivalent to contraction along the $x$ direction because of Poisson's effect.

Figure 4

(a) Projected electronic density of states (PDOS) for $MX$ (GeS) from DFT calculation. The analysis in main text suggests that the $M$ and $X$ atoms undergo $s{p}^3$ hybridization. For each atom, three of the half-filled $s{p}^3$ orbitals form the covalent bonds connecting the $M$ and $X$ atoms. For both $M$ and $X$, there is a lone-pair orbital (the shaded region). (b) Charge density of the lone-pair orbital on $MX$ (GeS). The size of the lone-pair orbitals affects the tetrahedral bond angle ${\theta }_M$ and ${\theta }_X$. (c) The relative magnitude of ${\theta }_M$ and ${\theta }_X$ leads to the three different puckered structures.

Figure 5

Tunable Poisson's ratio of monolayer GeSe via electron doping engineering. (a) DOS (density of states) of GeSe with zero and $0.050e^{-}$/atom electron doping, respectively. The shaded region shows that the excess electrons located at the energy levels ranging from $-0.15$ to 0 eV (relative to Fermi level), which corresponds to the range from LUMO to $\text{LUMO}+0.15$ eV in the zero electron doping case. (b) The distribution of doped excess electrons. (c) Electron doping leads to almost constant of ${\theta }_M$ and significant increase of ${\theta }_X$. The crossover takes places at about $0.040e^{-}$/atom. (d) Poisson's ratio ${\nu }_{zx}$ (${\nu }_{zy}$) as a function of electron doping. They change sign from positive (negative) into negative (positive) arising from the crossover of the two angles in (c). See details in main text.

Figure 6

Tunable Poisson's ratio ${\nu }_{zx}$ of monolayer GeSe via strain engineering in the $x$ direction. (a) The $z$ coordinates change of atom Ge (black line) and atom Se (red line), suggesting that the $z$ coordinates of two atoms on the top two layers will be the same at ${\varepsilon }_x=6\%$. (b) The transverse strain ${\varepsilon }_z$ when GeSe is subjected to a strain ${\varepsilon }_x$ in the range from 0 to 0.14. Poisson's ratio ${\nu }_{zx}$ changes from a positive value 0.583 to a negative value $-0.433$. (c) Cartoons to show the structural change of GeSe under different ${\varepsilon }_x$ strains. There is a transition at ${\varepsilon }_x=6\%$, corresponding to the intersection point of two lines in (a). The competition of bond angles ${\theta }_M$ and ${\theta }_X$ determines the sign of Poisson's ratio. See details in main text.