Lev Krainov
Introduction
Bilayer graphene(BLG) has been a material of intensive interest recently. In particular, twisted bilayer graphene(TBLG) has attracted a lot of attention due to presence of flat bands at Fermi energy at the so-called “magic” angle of twisting. Recent ab initio study suggests that dependence of interlayer distance on the local stacking pattern is an important ingredient for the band structure of TBLG. In this study we use DFT to find the interlayer distance for simple bilayer graphene in case of AA and AB(Bernal) stacking.
Figure 1. AA and AB stacking of bilayer graphene.[8]
Interlayer interaction
The cohesion in layered graphitic structures is a combination of long-ranged van der Waals(vdW) and short-ranged orbital overlap contributions. The vdW coupling is missing in the local density LDA and the generalized gradient approximations GGA, so these methods do not adequately describe the interlayer cohesion in graphite.
In this study we use Tkatchenko-Scheffler method[1] to account for vdW interaction. It is phenomenologically approximated by a pairwise interaction potential C₆/r⁶ where the constant C₆ is found from the results of the DFT calculations. Such method of evaluation adds flexibility into an otherwise empirical description. We perform calculations both with and without TS correction and compare the results.
Computational details
Ground state energy computations were performed using DFT plane-wave pseudopotential method implemented in CASTEP[2]. With CASTEP, we use the GGA-PBE as an exchange-correlation functional [3]. We also employ On-the-fly generated (OTFG) ultrasoft pseudopotential was used to describe the interactions of ionic core and valance electrons with a core radius of 1.4Bohr(0.74 Å) [4]. Pseudo atomic calculation is performed for 2s2 2p2 orbitals of C atoms. SCF convergence tolerance was set to 5.0E-7eV/atom. Since plane-wave method requires periodic boundary conditions we make a box of height h=17 Å and a cross section matching unit cell of graphene(a=2.46 Å).
First we establish ECUT and KPOINTS required energy convergence within Etol=0.001eV for AB stacking with interlayer spacing set to 3.35 Å. For KPOINTS Monkhorst-Pack grid[7] was utilized. The usual energy convergence analysis yields ECUT=650eV and KPOINTS 17x17x1. Later we will use the same parameters for AA stacking, assuming that different stacking pattern will not change convergence drastically.
Results
Figure 2 shows energy as a function of interlayer distance for AB BLG. For convenience all energies are offset by E0 = -631.09eV. Without TS correction the minimum is around 4.2-4.4 Å and is quite shallow. TS correction shifts minimum to 3.34-3.37 Å and makes it much more pronounced. Much higher second derivative means that layers are mostly bound by vdW forces. Corrected interlayer distance is in much better agreement with the experimental value of 3.34 Å [5]
Figure 2. Geometry optimization for AB stacked BLG with(blue) and without(orange) TS correction
Figure 3 shows energy as a function of interlayer distance for AA stacked BLG. Energy is minimized at around 3.5Å layer separation. As expected from previous numerical studies[6], the layers are further apart compared to AB BLG. Also, energy of AA stacked BLG is approximately 25meV higher for only a single honeycomb unitcell. This energy difference will scale as a square of the linear size, making AB staking strongly preferred.
Figure 3. Geometry optimization for AA stacked BLG with TS correction.
Conclusion
We have shown that van der Waals interaction is a crucial element of interlayer interaction in bilayer graphene. We have found AB stacked BLG to be closer(3.35Å) and energetically prefered compared to AA stacked one(3.5Å) in accordance with previous studies.
Bibliography
[1] A. Tkatchenko, M. Scheffler, “Accurate molecular van der waals interactions from ground-state electron density and free-atom reference data”, Phys. Rev. Lett. 102, 073005 (2009).
[2] S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson, M. C. Payne, “First principles methods using CASTEP”, Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005)
[3] Perdew, J. P; Burke, K; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865-3868
[4] CASTEP GUIDE, BIOVIA, UK, 2014. URL : http://www.tcm.phy.cam.ac.uk/castep/documentation/WebHelp/content/pdfs/castep.htm.
[5]Y. Baskin and L. Mayer, Phys. Rev. 100, 544 1955.
[6] A.N. Kolmogorov and V.H. Crespi, Phys. Rev. B 71, 235415 2005
[7] Monkhorst H.J., Pack J.D. Special points for Brillouin-zone integrations. Phys Rev B 13,5188 1976
[7]https://phys.org/news/2017-01-scientists-bilayer-graphene.html
