Frustration and Atomic Ordering in a Monolayer Semiconductor Alloy

Frustrated interactions can lead to short-range ordering arising from incompatible interactions of fundamental physical quantities with the underlying lattice. The simplest example is the triangular lattice of spins with antiferromagnetic interactions, where the nearest-neighbor spin-spin interactions cannot simultaneously be energy minimized. Here we show that engineering frustrated interactions is a possible route for controlling structural and electronic phenomena in semiconductor alloys. Using aberration-corrected scanning transmission electron microscopy in conjunction with density functional theory calculations, we demonstrate atomic ordering in a two-dimensional semiconductor alloy as a result of the competition between geometrical constraints and nearest-neighbor interactions. Statistical analyses uncover the presence of short-range ordering in the lattice. In addition, we show how the induced ordering can be used as another degree of freedom to considerably modify the band gap of monolayer semiconductor alloys.

Figure 1

Monolayer ${\mathrm{Re}}_x{\mathrm{Nb}}_{1-x}{\mathrm{S}}_2$. Top and side view models of monolayer (a) ${\mathrm{ReS}}_2$, (b) ${\mathrm{NbS}}_2$, and (c) ${\mathrm{Re}}_{0.5}{\mathrm{Nb}}_{0.5}{\mathrm{S}}_2$. (d) Total energies of the $1{\mathrm{T}}^{\prime }$ structure with respect to the 1H structure. (e) Compositional formation energy of the lowest-energy phase with respect to the parent TMDs. (f) Computed composition-dependent band gaps for ${\mathrm{Re}}_x{\mathrm{Nb}}_{1-x}{\mathrm{S}}_2$. Filled points indicate a direct gap and energies are reported in $\mathrm{eV}/{\mathrm{MS}}_2$. (g) An unfrustrated system with a square lattice versus (h) a frustrated system with a triangular lattice.

Figure 2

Atomic ordering in monolayer ${\mathrm{Re}}_{0.5}{\mathrm{Nb}}_{0.5}{\mathrm{S}}_2$. (a) An ADF-STEM image of the monolayer ${\mathrm{Re}}_{0.5}{\mathrm{Nb}}_{0.5}{\mathrm{S}}_2$ with the corresponding fast Fourier transform (inset). (b) A higher magnification ADF-STEM image of ${\mathrm{Re}}_{0.5}{\mathrm{Nb}}_{0.5}{\mathrm{S}}_2$ used for statistical analyses and (c) its filtered version. (d) Probability distribution of DNN, (e) spatial correlation functions in the three zigzag directions, and (f) histograms of homoatomic stripe lengths for a $20\times 20$ unit cell portion of the image presented in (b).

Figure 3

Nearest-neighbor-based model describing the preferred distribution of atomic species. (a) DFT total energies of 60 ${\mathrm{Re}}_{0.5}{\mathrm{Nb}}_{0.5}{\mathrm{S}}_2$ configurations with various Re-Nb distributions versus their average $\overline{\mathrm{DNN}}$. Energies are reported in $\mathrm{eV}/{\mathrm{MS}}_2$ and the best fit line is shown. ${\mathrm{R}}^2$ stands for the coefficient of determination, and is close to 1, indicating that the linear fit closely represents the data. (b) $\overline{\mathrm{DNN}}$ versus temperature in the nearest-neighbor model solved by Monte Carlo simulations. A portion of the Nb-Re distribution (c) obtained by the Monte Carlo simulations at $T=1220\mathrm{K}$, (d) extracted from the image in Fig. 2, and (e) for a random configuration, as well as the average probability distribution of DNN and the averaged correlation function for four such configurations. Re: navy; Nb: light violet.

Figure 4

Band gap tunability of ${\mathrm{Re}}_{0.5}{\mathrm{Nb}}_{0.5}{\mathrm{S}}_2$ by ordering and layer number. (a) DFT band gaps versus DFT total energies for 12 experimental and 12 random configurations. Energies are reported in $\mathrm{eV}/{\mathrm{MS}}_2$ and the energy of the reference structure [Fig. 1] is taken as zero. (b) Optical absorption spectra of ${\mathrm{Re}}_{0.5}{\mathrm{Nb}}_{0.5}{\mathrm{S}}_2$ as a function of layer number. The intensity of the spectrum for bulk ${\mathrm{Re}}_{0.5}{\mathrm{Nb}}_{0.5}{\mathrm{S}}_2$ is divided by 10. (c) Experimental and theoretical band gaps versus the ${\mathrm{Re}}_{0.5}{\mathrm{Nb}}_{0.5}{\mathrm{S}}_2$ thickness. For the experimental values, several measurements are performed on each flake and the error bars represent standard deviation. For the theoretical values, the monolayer alloy shown in Fig. 1 is taken as the reference structure, and multiple high-symmetry stacking sequences with lowest energy are used for calculations of band gap for the multilayer alloys. SOI is included.