Theory of finite-temperature two-dimensional structural transformations in group-IV monochalcogenide monolayers

One account of two-dimensional (2D) structural transformations in 2D ferroelectrics predicts an evolution from a structure with ${\mathrm{Pnm}2}_1$ symmetry into a structure with square P4/nmm symmetry and is consistent with experimental evidence, while another argues for a transformation into a structure with rectangular Pnmm symmetry. An analysis of the assumptions made in these models is provided here, and six fundamental results concerning these transformations are contributed as follows: (i) Softened phonon modes produce rotational modes in these materials. (ii) The transformation to a structure with P4/nmm symmetry occurs at the lowest critical temperature $T_c$. (iii) The hypothesis that one unidirectional optical vibrational mode underpins the 2D transformation is unwarranted. (iv) Being successively more constrained, a succession of critical temperatures $\left(T_c<T_c^{\prime }<T_c^{″}\right)$ occurs in going from molecular dynamics calculations with the NPT and NVT ensembles onto models with unidirectional oscillations. (v) The choice of exchange-correlation functional impacts the estimate of the critical temperature. (vi) Crucially, the correct physical picture of these transformations is one in which rotational modes confer a topological character to the 2D transformation via the proliferation of vortices.

Figure 1

(a) Ferroelectric rectangular ground-state u.c. (with $\mathrm{Pnm}{2}_1$ group symmetry) of a SnSe ML. The Sn (Se) atom is gray (orange) [or black (gray) in grayscale]; $\mathrm{\Delta }{\alpha }_0>0$. (b) Paraelectric square (P4/nmm) phase in which $a_1=a_2=a_s,d_{3,s}=d_{2,s},{\alpha }_1={\alpha }_2={\alpha }_3={\alpha }_{1,s},{\delta }_{x,s}={\delta }_{y,s}=0$, and $\mathrm{\Delta }{\alpha }_s=0$. (c) Paraelectric rectangular (Pnmm) phase in which $a_1$ and $a_2$ retain ground-state magnitudes (so $\mathrm{\Delta }{\alpha }_r=\mathrm{\Delta }{\alpha }_0)$, $d_{3,r}=d_{2,r},{\alpha }_1={\alpha }_3={\alpha }_{1,r}\ne {\alpha }_{2,r},{\delta }_{x,r}={\delta }_{y,r}=0$. The energy $(J_s$ or $J_r)$ in turning from structure (a) into (b) or (c) is indicated, too. (Reproduced from Ref. [14].)

Figure 2

(a) Phonon dispersion for the SnSe ML at its ground state $({\mathrm{Pnm}2}_1$ symmetry). Eigenfrequencies $\nu$ are listed in ascending order, according to their magnitudes around the $\mathrm{\Gamma }$ point. (b) Rotational signatures of degenerate modes 7 and 9 at the $M$ point for all momentum directions are emphasized by red quadratic fits. [(c) and (d)] Phonon dispersion in the P4/nmm and Pnmm u.c., respectively [gray curves are those in subplot (a); they help identifying softened phonons]. [(e) and (f)] Identification of softened phonons in P4/nmm and Pnmm phases at the $\mathrm{\Gamma }$ or $M$ point, respectively. Note that soft (optical) mode 7 is not even unidirectional at the $M$ point, in contradiction to assumption UOV 3. Bounding rectangles identify degenerate eigenfrequencies.

Figure 3

(a) Top and side views of a 2D monolayer in a periodic cell calculation. The structure within the central supercell is shown in golden (dark gray) color, and atoms in neighboring supercells are seen in a lighter color. The circles represent a sphere with a cutoff radius after which interatomic forces die off; there is sufficient vacuum to disconnect periodic images along the out-of-plane $\left(z\right)$ direction. (b) Top and side views of a 2D monolayer with vacuum around the edges of the finite-size flake.

Figure 4

Critical temperatures for a SnSe ML according to different ensembles/models (columns) and two XC functionals (rows). Once an ensemble/model and XC functional are chosen (separate panels), the thermal evolution of (i) $d_1,d_2$, and $d_3$; (ii) ${\theta }_x$ and ${\theta }_y$; (iii) ${\alpha }_1,{\alpha }_2$, and ${\alpha }_3$; (iv) $\mathrm{\Delta }\alpha$; (v) $h$; and (vi) $\sigma$ are listed. Orange (gray) boxes seen toward the far left in all subplots indicate $T$ at which the ferroelectric to paraelectric 2D transformation has been reached. $T_c,T_c^{\prime }$, and $T_c^{″}$ as well as $J_s$ and $J_r$ are strongly dependent on XC functionals and increase as structural constraints are added.

Figure 5

Violation of the angle-covariant approximation in ab initio molecular dynamics calculations, irrespective of ensemble. The angle-covariant approximation (hypothesis UOV 4 in Ref. [12]) holds if $\sqrt{|{\theta }_{x,1}^2-{\theta }_{x,2}^2|}=0$.

Figure 6

Distribution of ${\delta }_x$ and ${\delta }_y$ at finite $T$ within the NPT and NVT ensembles. The black dot indicates ${\delta }_{x,0}$, and the white dot [yellow (light gray) curve] the average (standard deviation) of $\left({\delta }_x,{\delta }_y\right)$ for $T$ as indicated; ${\delta }_x$ coalesces to zero at a smaller $T$ on the NPT ensemble. The near radial symmetry of ${\delta }_x$ and ${\delta }_y$ at finite $T$ evidences rotational motion.

Figure 7

(a) Vorticities are computed within four-point closed loops. (b) Snapshot of the projections $({\delta }_x,{\delta }_y)$ at a given time $t$ for a calculation within the NPT ensemble at $T<T_c$. (c) Similar plot for $T>T_c$. (d) Distribution of vortex lifetimes for the NPT ensemble. (e) The number of vortices increases at $T_c\left(T_c^{\prime }\right)$ and eventually saturates in the NPT (NVT) ensemble.

Figure 8

(a) Thermal evolution of the intrinsic dipole moment $P$, using the parameters provided in Ref. [12] based on the PBE XC functional (choice UOV 5). (b) Specific heat as obtained with that model. $T_c$ agrees with the value reported in that work when using their fitting parameters, thus validating our Monte Carlo results.

Figure 9

Energy landscape as a function of ${\theta }_{1,x}$ and ${\theta }_{2,x}$ in the rectangular structure with zero-temperature lattice parameters $a_{1,x}$ and $a_{2,x}$, using the (a) vdW-DF-cx and (b) PBE XC functionals. Plots to the right indicate the mean field coupling of neighboring dipoles without subtracting a strain contribution (solid black) or when subtracting it (dashed red curve). Use of the solid black curve would increase $T_c^{″}$ even further, and the red (dashed) curve was employed in computing $T_c^{″}$.

Figure 10

$J_r$ is underestimated in Ref. [12] (they report 39.6 K/u.c. while its correct value is 117.9 K/u.c.), as determined using exactly the same numerical tool but well-converged inputs. See Ref. [14] for more details, and Ref. [25] for another work with lattice parameters in exact agreement with ours. This numerical inaccuracy impacts the magnitude of $T_c^{″}$ estimated in Ref. [12].