# Evolution of elastic moduli through a two-dimensional structural transformation

## TL;DR

This paper presents a method to estimate elastic moduli softening in 2D materials using an elastic energy landscape, highlighting quantum effects and phase stability in SnO monolayers.

## Contribution

It introduces a conceptual procedure to estimate elastic moduli from a 2D elastic energy landscape, considering quantum effects and phase stability in soft matter.

## Key findings

- Elastic moduli soften near mechanical instability at finite temperature.
- Escape times from energy landscape wells increase exponentially with barrier height.
- Arguments against additional lattice transformations due to high energy costs.

## Abstract

We use a classical analytical and separable elastic energy landscape describing SnO monolayers to estimate the softening of elastic moduli through a mechanical instability occurring at finite temperature in this material. Although not strictly applicable to this material due to its low energy barrier $J$ that leads to a quantum paraelastic phase, the present exercise is relevant as it establishes a conceptual procedure to estimate such moduli straight from a two-dimensional elastic energy landscape. As additional support for the existence of a quantum paraelastic phase, we carry a qualitative WKB analysis to estimate escape times from an individual well on the landscape; escape times increase exponentially with the height of the barrier $J$. We also provide arguments against an additional transformation onto a planar lattice due to its high energy cost. These results continue to establish a case for the usefulness of soft matter concepts in two-dimensional materials, and of the potential lurking of quantum effects into soft matter.