Quantum and classical ripples in graphene

TL;DR

This study uses atomistic path-integral Monte Carlo simulations to explore the quantum effects on thermal ripples in graphene at low temperatures, revealing a classical-quantum crossover and scale-dependent roughness.

Contribution

It provides a realistic quantitative description of quantum ripples in graphene, showing how quantum and classical fluctuations coexist and evolve with temperature and scale.

Findings

Quantum regime is never fully attained at finite temperatures.

Zero-temperature quantum graphene is flatter at large scales but rougher at short scales.

Angular fluctuations can be described by two Gaussian distributions, indicating classical and quantum contributions.

Abstract

Thermal ripples of graphene are well understood at room temperature, but their quantum counterparts at low temperatures are still in need of a realistic quantitative description. Here we present atomistic path-integral Monte Carlo simulations of freestanding graphene, which show upon cooling a striking classical-quantum evolution of height and angular fluctuations. The crossover takes place at ever-decreasing temperatures for ever-increasing wavelengths so that a completely quantum regime is never attained. Zero-temperature quantum graphene is flatter and smoother than classical at large scales, yet rougher at short scales. The angular fluctuation distribution of the normals can be quantitatively described by coexistence of two Gaussians, one classical strongly T-dependent and one quantum about $2^{\circ}$ wide, of zero-point character. The quantum evolution of ripple-induced height and angular spread should be observable in electron diffraction in graphene and other two-dimensional materials like MoS$_2$, bilayer graphene, boron nitride, etc.