Shift-current response as a probe of quantum geometry and electron-electron interactions in twisted bilayer graphene

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Shift-current response as a probe of quantum geometry and electron-electron interactions in twisted bilayer graphene

Swati Chaudhary, Cyprian Lewandowski, and Gil Refael

Phys. Rev. Research 4, 013164 – Published 28 February 2022

Abstract

Moiré materials, and in particular twisted bilayer graphene (TBG), exhibit a range of fascinating phenomena that emerge from the interplay of band topology and interactions. We show that the nonlinear second-order photoresponse is an appealing probe of this rich interplay. A dominant part of the photoresponse is the shift current, which is determined by the geometry of the electronic wave functions and carrier properties and thus becomes strongly modified by electron-electron interactions. We analyze its dependence on the twist angle and doping and investigate the role of interactions. In the absence of interactions, the response of the system is dictated by two energy scales: (i) the mean energy of direct transitions between the hole and electron flat bands and (ii) the gap between flat and dispersive bands. Including electron-electron interactions both enhances the response at the noninteracting characteristic frequencies and produces new resonances. We attribute these changes to the filling-dependent band renormalization in TBG. Our results highlight the connection between nontrivial geometric properties of TBG and its optical response, as well as demonstrate how optical probes can access the role of interactions in moiré materials.

Received 2 August 2021
Revised 10 January 2022
Accepted 27 January 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.013164

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)

Geometric & topological phases Optical conductivity Photoconductivity Photovoltaic effect

Graphene Two-dimensional electron system

Bloch wave theory Electron-correlation calculations Mean field theory Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Swati Chaudhary^{1,2,3,4,5,*,†}, Cyprian Lewandowski ^1,2,*,‡, and Gil Refael^1,2

¹Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
²Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
³Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
⁴Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA
⁵Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

^*These authors contributed equally to this work.
^†swati.chaudhary@austin.utexas.edu
^‡cyprian@caltech.edu

Article Text

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Issue

Vol. 4, Iss. 1 — February - April 2022

Subject Areas

Condensed Matter Physics

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Images

Figure 1
Interaction-induced modifications to band structure and shift-current response of twisted bilayer graphene. (a) Band structure of the noninteracting (Non-int) model presented in Eq. (2), (b) band structure that includes Hartree corrections at different fillings $ν$ , (c) and (d) contributions to the second-order conductivity $σ_{x x}^{y} (0, ω, - ω)$ from flat-to-flat band transitions for the noninteracting case and for the interacting case with Hartree corrections, and (e) and (f) contributions to the second-order conductivity $σ_{x x}^{y} (0, ω, - ω)$ from flat-to-dispersive band transitions for the noninteracting case and for the interacting case with Hartree corrections. These Hartree corrections flatten both the flat and dispersive bands significantly as the filling is increased. This results in an enhanced second-order response and also gives rise to a second peak in the flat-to-dispersive contribution. As a consequence of a velocity gauge, as explained in the text, there is an apparent $\propto 1 / ω^{2}$ divergence as $ω \to 0$ .
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Figure 2
Band structure and shift-current response of twisted bilayer graphene near the magic angle. (a) Band structure of noninteracting twisted bilayer graphene for a twist angle $θ = 0 . 8^{\circ}$ and a sublattice offset $Δ = 5$ meV on both layers. Here, FF and FD denote flat-to-flat and flat-to-dispersive band transitions, respectively. (b) FF contribution to second-order conductivity as a function of frequency shown in units of average gap between two flat bands. The behavior of this average gap with twist angle $θ$ is shown in the inset. Note that the van Hove singularity in our model occurs near $ν \approx \pm 1.9$ and thus the peak value of $σ_{x x y}$ is significantly larger for fillings close to this value. (c) FD contribution to shift-current conductivity scaled by $ɛ_{f d}^{2} / L_{M}$ (see main text for justification) as a function of frequency in units of the band gap between flat and dispersive bands denoted by $ɛ_{f d}$ . Dependence of this energy gap $ɛ_{f d}$ and the moiré-length-dependent parameter $ɛ_{f d}^{2} / L_{M}$ is shown in the insets. Here the same filling to color assignment is made as in Fig. 1. (d) and (e) The top row shows the conductivity at different fillings, and the bottom row shows the $k$ -space profile of the integrand $\sum_{m, n} f_{m n} R_{m n}^{x x y} δ (ω - ɛ_{m n})$ contributing to the second-order conductivity at the frequency corresponding to the dashed line for flat-to-flat and flat-to-dispersive transitions shown in the top row, respectively.
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Figure 3
Interaction-induced modifications of band structure, second-order conductivity, shift vector, and interband Berry connection for flat-to-flat band transitions. (a) Band structure with Hartree corrections. All curves are shifted to the same energy at $κ^{'}$ . Note how as filling increases (decreases), the electron (hole) flat band flattens and the hole (electron) band broadens. (b) FF contribution to shift-current conductivity for the noninteracting case (upper panel) and the interacting case (bottom panel). Electron-electron band flattening increases the overall magnitude of the response and narrows the resonance in frequency. (c) $k$ -space profiles of the shift vector $S_{m n}^{y x}$ in units of the lattice constant of the monolayer graphene lattice, interband Berry connection magnitude square, $| A_{m n}^{x} |^{2}$ , the integrand $R_{m n}^{x x y}$ , and energy contours for $ɛ_{f d}$ in $k$ space for the transition between two flat bands at four different fillings used to calculate shift-current response in Eq. (10). It is worth noticing that the shift vector can be orders of magnitude larger than the lattice constant $a$ . This is expected as the Berry connection is roughly of the order of the lattice constant of the moiré lattice.
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Figure 4
Interaction-induced modification of band structure, second-order conductivity, shift vector, and interband Berry connection for flat-to-dispersive band transitions. (a) Band structure with Hartree corrections showing flat and dispersive bands, (b) FD contribution to shift-current conductivity for the noninteracting case, and (c) FD contribution when Hartree corrections are included. Note the appearance of additional peaks in (c) as compared with (b). (d)–(g) Shift vector $S_{m n}^{y x}$ , interband Berry connection magnitude square, $| A_{m n}^{x} |^{2}$ , the integrand $R_{m n}^{x x y}$ , and energy contours for $ɛ_{f d}$ in $k$ space for the four FD transitions, where (d) and (e) represent transitions between the hole flat band and hole dispersive bands and (f) and (g) describe transitions between the electron flat band and the electron dispersive bands.
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Figure 5
Comparison between the momentum space profile of the FF contribution of shift-current conductivity for the noninteracting and interacting case at peak frequencies for different filling factors. The flat-to-flat band transition contribution to the peak of second-order conductivity from different $k$ points within the mini BZ at different filling factors for the (a) noninteracting model and (b) interacting model with Hartree corrections. In each set of panels (consisting of an upper and a lower panel), the upper panel shows the variation of the shift-current conductivity with frequency at a given filling, and the lower panel shows the $k$ -space profile of the shift-current integrand from Eq. (10) for the flat-to-flat band transitions at frequencies corresponding to the dashed line in the upper panel. We notice a significant increase in the contribution from the regions near the $μ$ point which mainly arises from the band-flattening effect of interactions.
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Figure 6
Comparison between the momentum space profile of the FD contribution of shift-current conductivity for the noninteracting and interacting case at peak frequencies for different filling factors. The flat-to-dispersive band transition contribution to the peak of second-order conductivity from different $k$ points within the mini BZ at different filling factors for the (a) noninteracting model and (b) interacting model with Hartree corrections. In each set of panels (consisting of an upper and a lower panel), the upper panel shows the variation of the shift-current conductivity with frequency at a given filling, and the lower panel shows the $k$ -space profile of the shift-current integrand from Eq. (10) for the flat-to-dispersive band transitions at frequencies corresponding to the dashed line in the upper panel. We notice a significant increase in the contribution from the regions near the $μ$ point which mainly arises from the band-flattening effect of interactions.
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