Equilibrium Thermochemistry and Crystallographic Morphology of Manganese Sulfide Nanocrystals

Abstract

Manganese sulfide (MnS) is a p-type magnetic semiconductor whose physicochemical properties are sensitive to nanocrystal (NC) morphology, yet the thermodynamic driving forces governing morphology across MnS polymorphs remain poorly understood. Here, we use density functional theory (DFT) to predict the equilibrium morphologies of rock salt (RS), zinc blende (ZB), and wurtzite (WZ) MnS NCs as a function of the relative chemical potential of sulfur, $Δμ_{S}$. Benchmarking against Heyd$\unicode{x2013}$Scuseria$\unicode{x2013}$Ernzerhof (HSE06) hybrid functional calculations reveals that the r$^2$SCAN meta-generalized gradient approximation reproduces experimental lattice constants and thermochemical reaction energies but underestimates S-terminated polar surface energies by up to a factor of five; applying a Hubbard $U$ correction (r$^2$SCAN+$U$, $U = 2.7$ eV) to the Mn 3d states brings the results into close agreement with HSE06. Using the validated r$^2$SCAN+$U$ framework with the Gibbs$\unicode{x2013}$Wulff theorem, we predict that RS-MnS NCs favor nanocubes across nearly the entire stability window, ZB-MnS NCs transform from rhombic dodecahedra (Mn-rich) to polyhedra with 16 triangular faces (S-rich), and WZ-MnS NCs adopt rod-like morphologies with $Δμ_{S}$-sensitive base truncation. Synthesized RS-MnS NCs confirm the predicted cubic morphology, and high-temperature oxidative solution calorimetry yields an apparent surface energy of 1.15 $\pm$ 0.38 J$\cdot$m$^{-2}$, higher than the theoretical equilibrium value (0.42$\unicode{x2013}$0.43 J$\cdot$m$^{-2}$) due to high-index facet exposure, surface area uncertainty, and non-ideal surface configurations in real samples. This work establishes a framework for predicting the equilibrium morphologies of metal chalcogenide NCs.

Figures (10)

• Figure 1: Schematics illustrating the calculation of (a) the individual surface energy of the (100) facet in rock salt (RS) MnS from a symmetric slab, (b) the combined (0001) and (000$\overline{1}$) surface energies in wurtzite (WZ) MnS from an asymmetric slab, (c) the individual polar surface energy of the ($\overline{1}$1$\overline{1}$)--Mn facet in zinc blende (ZB) MnS from combined slab and wedge models, and (d) the relative surface energy between the (1$\overline{1}$0$\overline{1}$)--S$^2$ and (000$\overline{1}$)--Mn facets in WZ-MnS from combined slab and wedge models. Gray-shaded regions denote atoms fixed (F) during geometry optimization. $E_{\mathrm{slab}}$, $E_{\ce{MnS}}$, and $E_{\ce{S}}$ are the total energies of the slab, bulk MnS per formula unit, and elemental sulfur per atom, respectively. $\sigma = A \cdot \gamma$, where $\gamma$ is the surface energy and $A$ is the surface area of one side of the unit slab. $\Gamma_{\ce{S}} = N_{\ce{Mn}} - N_{\ce{S}}$, where $N_{\ce{Mn}}$ and $N_{\ce{S}}$ are the numbers of Mn and S atoms in the slab, respectively. $\Delta \mu_{\mathrm{S}}$ is the chemical potential of sulfur relative to its standard state. $\theta$ is the dihedral angle between two planes. See the Ab Initio Thermodynamics subsection of Methods for details.
• Figure 2: Percentage deviations of DFT-predicted lattice constants from experimental values for (a) rock salt (RS) MnS, (b) wurtzite (WZ) MnS, (c) zinc blende (ZB) MnS, and (d) pyrite (PY) MnS2. Six exchange-correlation functionals are compared: PBE, PBE-D3+BJ, SCAN, SCAN-rVV10, r$^2$SCAN, and r$^2$SCAN-rVV10. Numerical values are listed in Table S1.
• Figure 3: Comparison of calculated and experimental (a) formation energies for MnCl2, MnI2, rock salt (RS) MnS, and pyrite (PY) MnS2, and (b) reaction energies of RS-MnS with Cl2, iodine (I), and S. In both panels, filled bars correspond to six exchange-correlation functionals (PBE, PBE-D3+BJ, SCAN, SCAN-rVV10, r$^2$SCAN, and r$^2$SCAN-rVV10) and two hybrid-functional benchmarks (HSE06 and HSE06-D3); unfilled green bars denote experimental reaction enthalpies at 298.15 K, and unfilled red bars with red dashed lines extending from their bases indicate experimental values extrapolated to 0 K. (c) Phase diagram of bulk MnS polymorphs as a function of the relative chemical potentials $\Delta \mu_{\ce{Mn}}$ and $\Delta \mu_{\ce{S}}$. Solid lines are derived from formation energies; vertical dash-dotted lines denote polymorph phase boundaries.
• Figure 4: (a) Surface energies of two nonpolar and six polar rock salt (RS) MnS facets calculated using r$^2$SCAN, HSE06, and r$^2$SCAN+$U$ ($U = 2.7$ eV). S and Mn labels following the Miller indices denote the surface termination. (b) Dependence of S-terminated (111) and (131) RS-MnS surface energies on the Hubbard $U$ value in r$^2$SCAN+$U$ calculations. Horizontal dotted lines indicate the corresponding HSE06 surface energies, with values annotated; vertical dashed lines mark the optimal $U$ for each facet (2.9 eV for (111)--S and 2.5 eV for (131)--S) that minimizes the deviation from HSE06. In both panels, polar surface energies are evaluated at $\Delta \mu_{\ce{S}} = 0$ eV.
• Figure 5: Dependence of rock salt (RS) MnS surface energies on the relative chemical potential of sulfur $\Delta \mu_{\ce{S}}$ for (a) low-index facets and (b) the six lowest-energy facets among all $\{h,k,l\} \leq 3$ facets. Solid and dashed lines represent r$^2$SCAN+$U$ and r$^2$SCAN results, respectively. The $\Delta \mu_{\ce{S}}$ range between the two vertical dash-dotted lines ($-2.22$ eV $\leq \Delta \mu_{\ce{S}} \leq -0.10$ eV) indicates the thermodynamically stable region of bulk RS-MnS. Wulff constructions of RS-MnS nanocrystals considering (c) only low-index facets and (d) all facets with Miller indices up to 3. Each Wulff shape corresponds to the $\Delta \mu_{\ce{S}}$ value indicated beneath it. If no specific $\Delta \mu_{\ce{S}}$ value is given, the shape is invariant across the entire stability window. (e) Four representative RS-MnS slab models showing the (010), (100), (131)--S, and (311)--S facets.