Charge-transfer insulation in twisted bilayer graphene

Louk Rademaker and Paula Mellado

Phys. Rev. B 98, 235158 – Published 26 December 2018

Abstract

We studied the real-space structure of states in twisted bilayer graphene at the magic angle $θ = 1 . 08^{\circ}$ . The flat bands close to charge neutrality are composed of a mix of “ring” and “center” orbitals around the AA stacking region. An effective model with localized orbitals is constructed which necessarily includes more than just the four flat bands. Long-range Coulomb interaction causes a charge transfer at half filling of the flat bands from the center to the ring orbitals. Consequently, the Mott phase is a featureless spin-singlet paramagnet. We estimate the effective Heisenberg coupling that favors the singlet coupling to be $J = 3.3$ K, consistent with experimental values. The superconducting state depends on the nature of the dopants: hole-doping yields $(p + i p)$ -wave, whereas electron-doping yields $(d + i d)$ -wave pairing symmetry.

Received 24 May 2018

DOI:https://doi.org/10.1103/PhysRevB.98.235158

Physics Subject Headings (PhySH)

Superconductivity

Bilayer films Graphene Strongly correlated systems

Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Louk Rademaker^1,2 and Paula Mellado^2,3

¹Department of Theoretical Physics, University of Geneva, 1211 Geneva, Switzerland
²Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5
³School of Engineering and Sciences, Adolfo Ibáñez University, Santiago 7941169, Chile

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Vol. 98, Iss. 23 — 15 December 2018

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Images

Figure 1
The wave functions of the lowest-energy orbitals at $Γ$ (top row) and $K$ (middle row), computed using the tight-binding model of Fig. 3. Shown here is the hexagonal unit cell with the AA stacking at the center and the six corners having alternating AB and BA stacking. The size of the dots is proportional to the wave function modulus squared on each atom; the color represents the phase. The $K$ orbitals have large spectral weight at the center of the AA region. The $Γ$ orbitals, on the other hand, have suppressed weight at the AA center but have nonzero weight in a ring around it. The averaged wave function squared as a function of distance from the AA centers, measured in units of the single-layer graphene lattice parameter $a$ , is shown in the bottom plot. We propose that the insulating phase is characterized by a charge transfer from the center to the ring orbitals due to long-range Coulomb repulsion. Note that there is a nonzero overlap between the $Γ$ and $K$ orbitals. The orbital nature of the flat band smoothly varies from ring to center orbitals as a function of momentum. The qualitative difference between the real-space wave function of the flat bands at $Γ$ and $K$ implies that one needs more than four localized Wannier orbitals to capture the flat bands correctly.
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Figure 2
The enlarged unit cell of twisted bilayer graphene (TBG), with the two different layers shown in red and blue. The new unit cell, spanned by vectors $G_{i}$ , has one sixfold rotation center with AA interlayer stacking and two threefold rotation centers with AB or BA stacking. There are no mirror symmetries: a reflection along one of the dashed lines interchanges the two layers. The corresponding space group of TBG is therefore the double cover of $p 6$ , as opposed to $p 6 m$ for single-layer graphene or bilayers without twist.
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Figure 3
Tight-binding band structure for bilayer graphene with twist angle $θ = 1 . 08^{\circ}$ , using the parameters provided in the text. This model contains 11 164 bands. Four flat bands with a bandwidth $W = 11.25$ meV arise around charge neutrality, as is clearly shown in the bottom panel. At the $K$ point there is a small gap of 13. $8 μ eV$ at the approximate Dirac cones, as shown in the inset.
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Figure 4
An effective eight-band model, where each band is doubly degenerate, describing the essential low-energy physics of the twisted bilayer, Eq. (5). The fitted parameters are $t_{K} = 1.53$ meV, $t_{Γ} = 9.50$ meV, $Δ_{Γ} = 61.60$ meV, and $t^{'} = 4.67$ meV. The full band structure of Fig. 3 is shown in gray.
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Figure 5
Overlap between the eigenstates of the flat band from the full band problem, with ring orbitals in yellow and center orbitals in blue. Note that any filling away from charge neutrality leads to an uneven charge distribution in the unit cell, as is shown in Fig. 6.
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Figure 6
A comparison of the charge distribution within one unit cell in the noninteracting model at half filling of the flat band (top left) versus the charge-transfer scenario (top right). An excess of charge is shown in green, and an absence of charge is shown in red. The charge density per carbon atom as a function of distance from the unit cell center is shown in the bottom panel. In the noninteracting case the Coulomb energy is $E_{int} = 317.6$ meV, whereas in the charge-transfer picture this is reduced to $E_{int} = 103.9$ meV.
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