Electronic band structure of twisted bilayer graphene
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Bilayer graphene (BLG) consists of two graphene layers, usually twisted
at an angle with respect to one another. The following types of BLG are
identified with respect to the layer stacking: AA-stacked BLG for twist angle
of 0°, AB-stacked (or Bernal) BLG for twist angle of 30°, and twisted BLG
(tBLG) for twist angles between 0° and 30°; the two layers can also be
shifted with respect to one another. Here, we consider the general case of
twisted BLG but without layer shifting. The twisting of the layers with
respect to the AA-stacked BLG preserves the hexagonal symmetry of the parent
graphene and the Brillouin zone is still hexagonal. Figure 1 shows the atomic
structure of tBLG with twist angle of 21.79°, which has the smallest period
and the smallest number of atoms, equal to 28. The experimentally observed
tBLG normally have much larger period and number of atoms in the unit cell.
The electronic band structure of tBLG can be obtained by using the
tight-binding Slonczewski–Weiss–McClure model, which has been around since
1960. The use of this model is limited to Bernal BLG and cannot be extended
to arbitrary tBLG. A parametrization of the tight-binding model with
distance-depending parameters is found useful for studying transport
phenomena (Trambly de Laissardiere) but does not have the necessary accuracy
for modeling of optical phenomena. On the other hand, the calculation of
microscopic properties within non-orthogonal tight-binding models (Sato) and
ab-initio ones (Latil) demands rapidly increasing computer resources with
increasing the unit cell size and is thus limited only to a few tBLG with
small unit cells.
The problem of solving electron and phonon eigenvalue problems for the
large number of atoms in the unit cell of tBLG can be tackled by taking into
account that the interlayer interaction is much weaker than the intralayer
one. This allows using the quantum-mechanical perturbation theory for
degenerate electronic levels. In particular, the unperturbed wavefunction of
tBLG is chosen as a linear combination of the wavefunctions of the two
isolated layers. Next, the interaction Hamiltonian is considered as a
perturbation. In view of the sparseness of the matrix of the interaction
Hamiltonian, the computational time for solving the electronic eigenvalue
problem increases quasi-linearly with the number of atoms in the unit cell.
For making the problem feasible for relatively large unit cells, we use the
non-orthogonal tight-binding model with model parameters, taken over from
ab-initio studies [1-3] (for details, see [4]).
The electronic band structure of the tBLG in Fig. 1 is shown in Fig. 2. For small energies, the band structure preserves the linear bands, crossing at the Fermi energy. For larger energies, however, the crossing of the bands of the two layers results in a modification of the bands. Namely, the bands split and extrema appear for some of the subbands close to the M point of the Brillouin zone. At the extrema 1, 2, 3, and 4 the electronic density of states in Fig. 2 has singularities (Fig. 3) in the form of spikes (extrema 2 and 3) and kinks (extrema 1 and 4). The density of states of the tBLG is very similar to that of any single-walled carbon nanotube. Led by this similarity, one is tempted to assume that an optical transition would take place for light excitation with energy, matching the separation between the spikes 2 and 3 (Coh, Dresselhaus). Other authors have argued that this might not be the case, because transitions 1-4 and 2-3 are forbidden, while transitions 1-3 and 2-4 are allowed (Wu, Moon). In order to derive the optical transition energy, we calculated the imaginary part of the dielectric function ε2 (Fig. 4). The plot of ε2 shows that strong light absorption indeed takes place for transitions 1-3 and 2-4, while there is no peak, corresponding to a transition 2-3. The presented computational scheme for the electronic band structure of tBLG was applied to a number of tBLG with up to several hundred atoms in the unit cell [4]. The comparison with available theoretical data shows that the proposed approach can successfully tackle the electron eigenvalue problem in the case of large atoms in the unit cell. References: 1. V. N. Popov, New J. Phys. 6 (2004) 1-17. 2. V. N. Popov and L. Henrard, Phys. Rev. B 70 (2004) 115407. 3. V. N. Popov and Ch. van Alsenoy, Phys. Rev. B 90 (2014) 245429. 4. V. N. Popov, J. Raman Spectrosc. 49 (2018) 31. |
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Valentin Popov
July 15, 2017