Kekul\'e spirals and charge transfer cascades in twisted symmetric trilayer graphene

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Kekulé spirals and charge transfer cascades in twisted symmetric trilayer graphene

Ziwei Wang, Yves H. Kwan, Glenn Wagner, Nick Bultinck, Steven H. Simon, and S. A. Parameswaran

Phys. Rev. B 109, L201119 – Published 16 May 2024

Abstract

We study the phase diagram of magic-angle twisted symmetric trilayer graphene in the presence of uniaxial heterostrain and interlayer displacement field. For experimentally reasonable strain, our mean-field analysis finds robust Kekulé spiral order whose doping-dependent ordering vector is incommensurate with the moiré superlattice, consistent with recent scanning tunneling microscopy experiments, and paralleling the behavior of closely related twisted bilayer graphene (TBG) systems. Strikingly, we identify a possibility absent in TBG: the existence of commensurate Kekulé spiral order even at zero strain for experimentally realistic values of the interlayer potential in a trilayer. Our studies also reveal a complex pattern of charge transfer between weakly and strongly dispersive bands in strained trilayer samples as the density is tuned by electrostatic gating, that can be understood intuitively in terms of the “cascades” in the compressibility of magic-angle TBG.

Received 2 February 2024
Accepted 3 May 2024

DOI:https://doi.org/10.1103/PhysRevB.109.L201119

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Strain

Graphene Twisted bilayer graphene

Hartree-Fock methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Ziwei Wang ¹, Yves H. Kwan², Glenn Wagner³, Nick Bultinck^1,4, Steven H. Simon¹, and S. A. Parameswaran¹

¹Rudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford OX1 3PU, United Kingdom
²Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
³Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
⁴Department of Physics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium

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Vol. 109, Iss. 20 — 15 May 2024

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Figure 1
Twisted symmetric trilayer graphene. (a) The large hexagons denote graphene Brillouin zones (BZs) of individual layers and the small hexagon denotes the moiré BZ for valley $K$ . (b) Without interlayer potential, TSTG decomposes into a TBG sector (red) and a graphene sector (blue) with opposite mirror eigenvalues. (c) A finite interlayer potential (100 meV) breaks mirror symmetry and hybridizes the two sectors. The color represents the relative weight of the wave function on the zero-field TBG and graphene sectors. (d), (e) TBG sector conduction band at zero interlayer potential without and with strain ( $ε = 0.1 %$ ), respectively. Strain breaks $C_{3}$ symmetry, unpinning the Dirac points, and increases the bandwidth. The corresponding band structures in valley $K^{'}$ can be found using time-reversal symmetry.
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Figure 2
Phase diagrams as a function of strain and interlayer potential at integer fillings $ν$ . We have performed $10 \times 10$ self-consistent Hartree-Fock calculations, scanning over all possible Kekulé spiral vectors $q$ . $Δ E$ is the energy difference per unit cell between the $q = 0$ state with the overall energy minimum, and may be compared with temperature of a typical experiment in the field, which is of the order of 0.1 meV. The acronyms are K-IVC for Kramers-intervalley coherence, VP for valley polarization, (C/I)KS for (commensurate/incommensurate) Kekulé spiral, SM for (compensated) semimetal. An asterisk (*) denotes states that break $C_{2} T$ symmetry. The (C/I)KS*(VP) region in $ν = 3$ consists of Kekulé spiral states with possible finite valley polarization. The red lines on the zero-strain axis indicate commensurate Kekulé spirals. The shaded regions have a finite charge gap. For $ν = 1$ , we have shown the phases and charge gaps of the two spin sectors separately, where $ν_{s}$ denotes the filling of each spin sector. As $Δ E$ cannot be separately defined for each spin sector, the data are simply duplicated in the two plots of $ν = 1$ .
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Figure 3
“Charge transfer plateaus” in TSTG. We plot the fillings of TBG and graphene sectors from $10 \times 10$ self-consistent HF, interpolated to $60 \times 60$ , as a function of total fillings. We notice plateaus at integer TBG fillings, which are due to a correlated charge gap in the TBG sector. In the inset, we additionally show the total unsigned Fermi volumes of all bands (“Fermi volume”) as a measure of the metallicity of the system. We notice that, due to nontrivial charge transfer between the two sectors, minima in the Fermi volume need not occur exactly at total integer fillings.
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Figure 4
(a) Doping-dependent $q_{Kekulé}$ vectors, found from $10 \times 10$ HF, of a system with $ε = - 0.12 %$ and $φ = 87^{\circ}$ , consistent with the experimental parameters in Ref. [25]. The hexagon [as well as the ones in (b) and (c)] represents the moiré Brillouin zone (mBZ), and $q_{Kekulé}$ is only defined modulo mBZ. We set $Δ V = 0$ as a displacement field in the single-gate geometry is weak, and use a single-gate screened Coulomb interaction $V (q) = (e^{2} / 2 ε_{0} ε_{r} q) (1 - e^{- 2 q d})$ with screening length $d = 25$ nm and relative permittivity $ε_{r} = 20$ . We take the middle-layer graphene to be rotated counterclockwise, as in experiment but opposite to our usual convention. (b), (c) $q_{Kekulé}$ dependence of the energies of IKS states for $ν = - 2$ and $ν = - 2.5$ , respectively, where $Δ E = E (q_{Kekulé}) - E_{min}$ .
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Physical Review B

covering condensed matter and materials physics

Kekulé spirals and charge transfer cascades in twisted symmetric trilayer graphene

Ziwei Wang, Yves H. Kwan, Glenn Wagner, Nick Bultinck, Steven H. Simon, and S. A. Parameswaran

Phys. Rev. B 109, L201119 – Published 16 May 2024

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Physical Review B

covering condensed matter and materials physics

Kekulé spirals and charge transfer cascades in twisted symmetric trilayer graphene

Ziwei Wang, Yves H. Kwan, Glenn Wagner, Nick Bultinck, Steven H. Simon, and S. A. Parameswaran

Phys. Rev. B 109, L201119 – Published 16 May 2024

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