Thermodynamical Properties and Stability of Crystalline Membranes in the Quantum Regime
- PDF / 876,492 Bytes
- 12 Pages / 612 x 792 pts (letter) Page_size
- 38 Downloads / 209 Views
Thermodynamical Properties and Stability of Crystalline Membranes in the Quantum Regime B. Amorim1, R. Roldán1, E. Cappelluti2, A. Fasolino3, F. Guinea1, and M. I. Katsnelson3 1 Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, E28049 Madrid, Spain 2 Istituto dei Sistemi Complessi, CNR, U.O.S. Sapienza, v. dei Taurini 19, 00185 Roma, Italy 3 Radboud University Nijmegen,Institute for Molecules and Materials, NL-6525AJ Nijmegen, The Netherlands
ABSTRACT We study the thermodynamical properties and lattice dynamics of two-dimensional crystalline membranes, such as graphene and related compounds, in zero temperature limit, where quantum effects are dominant. We find out that, just as in the high temperature classical limit, a fundamental role is played by the anharmonic coupling between in-plane and out-of plane lattice modes, which leads to a strong reconstruction of the dispersion relation of the outof-plane mode. We identify a crossover temperature, T*, bellow which quantum effects are dominant. We estimate that for graphene T* ~ 70 - 90 K. Inclusion of anharmonic effects makes the thermal expansion finite in the thermodynamic limit, and below T* it tends to zero as a power law as T→0 as required by the third law of thermodynamics. The specific heat also goes to zero as a power law as T→0, but with a exponent that differs from the one predicted by the harmonic theory. INTRODUCTION The recent experimental developments in the growth and isolation of single layers of crystalline materials, such as graphene, MoS2, WS2, BN and similar materials, have attracted an enormous interest in the study and understanding of the mechanical and thermodynamical properties of atomically thin, two-dimensional (2D) crystals [1] . One major issue in this field concerns the very stability of these free 2D crystals (for a review see Refs. [2, 3]). At the simple harmonic level, where the out-of-plane (flexural) and the in-plane modes are decoupled, the mean square atomic displacement of both modes diverges. Anharmonic effects, coupling inplane and out-of-plane modes, are commonly thought to stabilize the flat phase of these systems [4]. This can be seen by looking at the Fourier transform of the displacement-displacement correlation functions. For a free flat membrane, the harmonic theory predicts that, at high temperature, the out-of-plane and in-plane correlation functions behave, respectively, as and (where k is the momentum). It was first pointed out in Ref. [5] that, at high temperature, inclusion of anharmonic effects changes the harmonic theory result. For a free membrane in the thermodynamic limit, anharmonic effects change the behavior of the out-of-plane and in-plane correlation functions at small momentum to and , respectively, where and are anomalous exponents, which are related by [6, 7]. The existence of a flat phase with long-range orientational order requires that and [6]. The most reliable theoretical calculations give us [8-10]. Therefore, we see that the anharmonic coupling
between in-plane and out-of-plan
Data Loading...
