Free energy barrier and thermal-quantum behavior of sliding bilayer graphene

TL;DR

The study addresses how thermal and quantum fluctuations renormalize the free-energy barrier governing sliding between graphene layers. It employs the stochastic self-consistent harmonic approximation (SSCHA), augmented by an interpolation technique, to compute temperature-dependent free energies at both the equilibrium and saddle-point configurations, including unstable states. The results show the barrier drops from $V_{\mathrm{SP}}=1.24$ meV/atom (fixed nuclei) to about $0.80$ meV/atom at $T=100$ K (≈$35\%$ reduction), remains near this value up to ~500 K, and further decreases to ~ $0.64$ meV/atom at $T=1000$ K, with quantum effects negligible above ~100 K and interlayer shear modes driving the reduction. This leads to a corrected maximum shear stress of $0.13$ GPa in agreement with experiments, demonstrating that thermal effects must be included in free-energy barrier calculations and providing a general framework for studying thermally activated sliding in other macroscopic layered systems.

Abstract

In multilayer graphene, the stacking order of the layers plays a crucial role in the electronic properties and the manifestation of superconductivity. By applying shear stress, it is possible to induce sliding between different layers, altering the stacking order. Here, focusing on bilayer graphene, we analyze how ionic fluctuations alter the free energy barrier between different stacking equilibria. We calculate the free energy barrier through the state-of-the-art self-consistent harmonic approximation, which can be evaluated at unstable configurations. We find that above 100 K there is a large reduction of the barrier of more than 30% due to thermal vibrations, which significantly improves the agreement between previous first-principles theoretical work and experiments in a single graphite crystal. As the temperature increases, the barrier remains nearly constant up to around 500 K, with a more pronounced decrease only at higher temperatures. Our approach is general and paves the way for systematically accounting for thermal effects in free energy barriers of other macroscopic systems.

Figures (5)

• Figure 1: Energy profile of bilayer graphene when moving one layer relative to the other along the bond direction. The lower layer is fixed in position A. Configuration AA (one layer on top of each other) is the least favorable configuration. There are two minima of the same energy, AB and AC, which correspond to the two stable configurations. To go from one minima to the other one applying shear stress, the upper layer has to overcome the barrier $V_\mathrm{SP}$, which has a value of 1.24 meV/atom relative to the minima. The diagrams below illustrate the different configurations, with the upper layer (darker blue) moving relative to the lower one in position A (lighter blue).
• Figure 2: Barrier as a function of temperature. It is reduced from 1.24 meV/atom, the value at fixed nuclei (V$_\mathrm{SP}$ in Fig. \ref{['fig:profile1D']}), to about 0.8 meV/atom, corresponding to a large reduction of about 35% due to phonons. The barrier does not change much up to about 500 K, but the value at 1000 K is about 20% smaller relative to the value at 100 K. The lines are a guide to the eye, as in the other figures of this work. The standard quantum SSCHA result (blue) and its classical limit (red) give almost identical results, showing quantum effects are negligible above 100 K.
• Figure 3: Bilayer graphene phonon dispersion. The inset shows a zoom in close to $\Gamma$ to better visualize the doubly degenerate shear modes S at $\Gamma$, and the breathing mode B. Due to the high speed velocity in graphene, the region of the shear mode in the BZ is small (a fraction of $\Gamma$M), and is overrepresented in small supercell calculations. The breathing mode, on the other hand, is flat, so nearby sampling points are not necessary to converge the free energy.
• Figure 4: Shear frequency as a function of temperature using SSCHA (blue circles) and QHA (red crosses). There is no experimental data for the temperature dependence of the shear frequency as a function of temperature for bilayer graphene. Experimental values correspond to bulk (blue crosses) and folded 2+2 graphene (orange circles), which are corrected by the factor of a nearest neighbor modelHe2016 to compare to the bilayer case (and rigidly shifted by -5 cm$^{-1}$ to facilitate the visual comparison). The agreement with the temperature dependence of the bulk values is good.
• Figure 5: (a) Graphite out-of-plane coefficient of TE as a function of temperature using SSCHA and QHA. PIMD calculationsHerrero2020 and experimental data Bailey1970 are also included. PIMD overall agrees well, while QHA and SSCHA (with the potential of Ref. Wen2018) underestimates the TE. (b) Graphite in-plane coefficient of TE using SSCHA and QHA. QHA again underestimates the dependence at low temperatures, and also at high temperatures. On the other hand, PIMD gives a dependence that is surprisingly very far from experimental values. SSCHA gives excellent results both at low and high temperatures, considering also the disagreement within experimental data.