Quantum Paraelastic Two-Dimensional Materials

We study the elastic energy landscape of two-dimensional tin oxide (SnO) monolayers and demonstrate a transition temperature of $T_c=8.5\pm 1.8\mathrm{K}$ using ab initio molecular dynamics (MD) that is close to the value of the elastic energy barrier $J$ derived from $T=0\mathrm{K}$ density functional theory calculations. The power spectra of the velocity autocorrelation throughout the MD evolution permit identifying soft phonon modes likely responsible for the structural transformation. The mean atomic displacements obtained from a Bose-Einstein occupation of the phonon modes suggest the existence of a quantum paraelastic phase that could be tuned with charge doping: SnO monolayers could be 2D quantum paraelastic materials with a charge-tunable quantum phase transition.

Figure 1

(a) Elastic energy landscape for the SnO monolayer under zero doping. (b) Unit cells at the energy minimum (structures $A$ and $B$) and for the square structure at point $C$. Structural order parameters are shown. (c) Energy cuts through the black and red dashed lines shown in subplot (a).

Figure 2

(a) Thermal evolution of lattice constants from MD (data) and our model (solid trend lines). $T_c=8.5\pm 1.8\mathrm{K}$. (b),(c) Mean values and expectation values of $X$ and $Y$. (d),(e) Thermal evolution of interatomic distances and angles defined in Fig. 1. (f) Thickness dependence on $T$; the inset is a structural snapshot during the thermal evolution.

Figure 3

PSD resolved over chemical element and spatial direction for (a) $T<T_c$ and (b) $T>T_c$. (c) Phonon dispersion for $T<T_c$ (left, mode resolved) and $T>T_c$ (right). The continuous trend was obtained at $T=0$ using standard methods and is a guide to the eye. At least one phonon line above 10 THz at the $\mathrm{\Gamma }$ point on the left subplot (vertical arrow) appears to have gone soft to accumulate near zero frequency. Diagonal arrows showcase lack of symmetry when $T<T_c$.

Figure 4

(a) Energy due to phonons according to Bose-Einstein statistics, and to the classical occupation linear in $T$. (b)–(d) Atomic displacements of Sn and O atoms, calculated with bosonic occupations (solid lines) and classical occupations. (e) Cuts $U(X,0)$, $U(X_0,Y)$, and $U(0,Y)$ show the evolution of the elastic energy landscape and the tuning of $J$ with charge doping.

Figure 5

(a) Electronic band structure of a neutral SnO monolayer. Blue (red) shadowed regions point to locations of reciprocal space from which electrons are added (removed). (b) Electron doping occurs around the $\mathrm{\Gamma }$ point but (c) hole doping first happens along a shallow pocket along the $\mathrm{\Gamma }-Y$ line. Units in (b) and (c) are $|e|/\mathrm{u}.\mathrm{c}$. The Brillouin zone boundaries in (c) reflect the change of $a_1$ and $a_2$ upon doping.