Moir\'e fractional Chern insulators. II. First-principles calculations and continuum models of rhombohedral graphene superlattices

Editors' Suggestion

Moiré fractional Chern insulators. II. First-principles calculations and continuum models of rhombohedral graphene superlattices

Jonah Herzog-Arbeitman, Yuzhi Wang, Jiaxuan Liu, Pok Man Tam, Ziyue Qi, Yujin Jia, Dmitri K. Efetov, Oskar Vafek, Nicolas Regnault, Hongming Weng, Quansheng Wu, B. Andrei Bernevig, and Jiabin Yu

Phys. Rev. B 109, 205122 – Published 7 May 2024

Abstract

The experimental discovery of fractional Chern insulators (FCIs) in rhombohedral pentalayer graphene twisted on hexagonal boron nitride (hBN) has preceded theoretical prediction. Supported by large-scale first-principles relaxation calculations at the experimental twist angle of $0 . 77^{\circ}$ , we obtain an accurate continuum model of $n = 3, 4, 5, 6, 7$ layer rhombohedral graphene-hBN moiré systems. Focusing on the pentalayer case, we analytically explain the robust $| C | = 0, 5$ Chern numbers seen in the low-energy single-particle bands and their flattening with displacement field, making use of a minimal two-flavor continuum Hamiltonian derived from the full model. We then predict nonzero valley Chern numbers at the $ν = - 4, 0$ insulators observed in experiment. Our analysis makes clear the importance of displacement field and the moiré potential in producing localized “heavy fermion” charge density in the top valence band, in addition to the nearly free conduction band. Lastly, we study doubly aligned devices as additional platforms for moiré FCIs with higher Chern number bands.

44 More

Received 4 December 2023
Revised 13 February 2024
Accepted 13 February 2024

DOI:https://doi.org/10.1103/PhysRevB.109.205122

Physics Subject Headings (PhySH)

Graphene Twisted heterostructures

Density functional theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jonah Herzog-Arbeitman ^1,*, Yuzhi Wang^2,3,*, Jiaxuan Liu^2,3,*, Pok Man Tam^4,*, Ziyue Qi ^2,3,*, Yujin Jia^2,3, Dmitri K. Efetov^5,6, Oskar Vafek^7,8, Nicolas Regnault^1,9, Hongming Weng^2,3,10, Quansheng Wu^2,3,†, B. Andrei Bernevig^1,11,12,‡, and Jiabin Yu^1,§

¹Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
²Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
³University of Chinese Academy of Sciences, Beijing 100049, China
⁴Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
⁵Faculty of Physics, Ludwig-Maximilians-University Munich, 80799 Munich, Germany
⁶Munich Center for Quantum Science and Technology (MCQST), Ludwig-Maximilians-University Munich, 80799 Munich, Germany
⁷National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA
⁸Department of Physics, Florida State University, Tallahassee, Florida 32306, USA
⁹Laboratoire de Physique de l'cole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, 75005 Paris, France
¹⁰Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
¹¹Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain
¹²IKERBASQUE, Basque Foundation for Science, Bilbao, Spain

^*These authors contributed equally to this work.
^†quansheng.wu@iphy.ac.cn
^‡bernevig@princeton.edu
^§jiabinyu@princeton.edu

Moiré fractional Chern insulators. I. First-principles calculations and continuum models of twisted bilayer ${MoTe}_{2}$

Yujin Jia, Jiabin Yu, Jiaxuan Liu, Jonah Herzog-Arbeitman, Ziyue Qi, Hanqi Pi, Nicolas Regnault, Hongming Weng, B. Andrei Bernevig, and Quansheng Wu

Phys. Rev. B 109, 205121 (2024)

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 109, Iss. 20 — 15 May 2024

Reuse & Permissions

Access Options

Article part of CHORUS

Accepted manuscript will be available starting 7 May 2025.

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) Crystalline and moiré BZs for rhombohedral graphene (blue) and hBN (red). The $K$ points of both materials are labeled, and the blue disks highlight the first shell of moiré reciprocal lattices around $K_{G}$ . The $q_{1}$ vector and its $C_{3}$ partners are shown in orange about the moiré $Γ_{M}$ point. (b) Real space lattice for seven moiré unit cells.
Reuse & Permissions
Figure 2
(a) Two different stacking configurations for R5G/hBN. In the $ξ = 1$ setup, nitrogen (N) atoms align with carbon $C_{A}$ of the closest graphene layer ( ${Gr}_{0}$ ) in the AA region; in the $ξ = 0$ setup, boron (B) atoms align with $C_{A}$ of ${Gr}_{0}$ in the AA region. $E$ indicates the applied electrical field (with arrow labeling the positive direction) and ISP indicates the direction internal symmetrical polarization due to the different chemical environment of outer and inner atoms in the graphene. (b) shows different regions in the moiré pattern for each stacking configuration. In the AA region, both boron and nitrogen are aligned with carbon atoms in lowest layer (denoted ${Gr}_{0}$ ). In AB regions, only boron atoms are aligned with carbon atoms in ${Gr}_{0}$ . In BA region, only nitrogen atoms are aligned with the carbon atoms in ${Gr}_{0}$ . (c) and (d) show the interlayer distance between hBN and ${Gr}_{0}$ for $ξ = 1$ and $ξ = 0$ respectively. (e) and (f) show the intralayer displacement of ${Gr}_{0}$ for $ξ = 1$ and $ξ = 0$ respectively.
Reuse & Permissions
Figure 3
Relaxation of R5G/hBN for $ξ = 1$ . (a) The maximum in-plane displacement vector magnitude of each layer, which decays rapidly each layer away from the hBN. (b) Unlike the intralayer relaxation, the out-of-plane displacement, which measures the displacement along the $z$ direction, remains constant. (c) Depiction of the relaxed pentalayer graphene on hBN (gray) with the out-of-plane relaxation increased by a factor of 20 for visibility. We see that the profile of the out-of-plane relaxation remains throughout the device so that the interlayer distances between neighboring graphene layers stay roughly constant.
Reuse & Permissions
Figure 4
Comparison between rigid SK and relaxed $DFT + SK$ band structures for $0 . 76715^{\circ}$ twist R5G/hBN in the $ξ = 0$ and $ξ = 1$ configurations under different external electric field strengths, depicted with orange-dotted lines for rigid structures and black-dotted lines for relaxed structures. (a) $E = 0$ and $R 5 G / {hBN}_{ξ_{b} = 1}$ , (b) $E = 0$ and $R 5 G / {hBN}_{ξ_{b} = 0}$ , (c) $E = 5 mV / Å$ and $R 5 G / {hBN}_{ξ_{b} = 1}$ , (d) $E = 5 mV / Å$ and $R 5 G / {hBN}_{ξ_{b} = 0}$ , respectively. Here, both K and K' valley bands are included.
Reuse & Permissions
Figure 5
Band structures of R5G/hBN in the $ξ = 1$ configuration at twist angle $θ = 0 . 76715^{\circ}$ for interlayer potential energy differences $V = - 16.65, - 6.66, 0, 6.66, 16.65 meV$ in $K$ valley. The black dots are from the first-principles calculation, and the lines are from the $10 \times 10$ continuum model with (blue solid) and without (orange dashed) moiré potential. For $V = 16.65 meV$ , it is clear that the conduction bands are almost free from the moiré. In fact, the average overlap of the projectors onto the lowest conduction band with and without moiré is $〈 Tr [P_{c} (k) P_{c}^{(0)} (k)] 〉 = 0.999$ , with the average computed over region of the mBZ where the lowest conduction band has a least 0.5 meV direct gap to nearby bands such that the projectors are well defined. The highest valence band, on the other hand, responds strongly to the moiré potential.
Reuse & Permissions
Figure 6
Pristine R5G. (a) The unit cell of rhombohedrally stacked pentalayer graphene is shown with a depiction of the sublattice-polarized chiral states $ψ_{A} (k)$ and $ψ_{B} (\bar{k})$ that make up the low-energy Hilbert space (other unit cells shown faded). They are bound to the outermost orbitals of the unit cell and decay exponentially within the crystal. The in-plane coupling is $v_{F} k$ shown with a dotted line, and $t_{1}, v_{3}, v_{4}$ are the out-of-plane couplings. Note that $V$ is the interlayer potential energy difference of electrons due to the applied field. The arrows show the positive directions of $V$ and $V_{ISP}$ . (b) The spectrum (solid lines) of the low-energy states $\pm E (| k |)$ is shown in the limit $v_{3} = v_{4} = 0$ at two values of $V$ . At $V = 0$ (blue), the bands are monotonically increasing, but $V = 20 meV$ (black), they are nonmonotonic. Dashed lines show analytical approximations (see Appendix pp1-s2) capturing these features.
Reuse & Permissions
Figure 7
Evolution of topology in pentalayer on hBN in the graphene $K$ valley (per spin). (a) Pristine R5G has a Berry curvature monopole corresponding to $C = \pm \frac{5}{2}$ due to the five Dirac cones. (b) Adding displacement field only will split the bands at charge neutrality, and the Chern number will change by 5 going through the gap. The valence bands carry $C = - \frac{5}{2}$ and the conduction bands carry $C = \frac{5}{2}$ . Note that including both valleys causes the total Chern number at charge neutrality to vanish. (c) Adding the moiré potential opens small gaps and isolates the lowest conduction band with $C = 5$ and highest valence band with $C = 0$ . The gap at charge neutrality is not closed, and still carries a nontrivial valley Chern number.
Reuse & Permissions
Figure 8
Comparison of equal-amplitude $T (r)$ (blue) and one-orbital-only coupling in Eq. (17) (red) for $V = 0$ in (a) and $V = 10 meV$ in (b) for $ξ = 1$ in the $K$ valley. We conclude that only the moiré coupling to the outer orbital, which is sublattice polarized due to the low-energy holomorphic basis, plays a role in the band structure. In both plots we have shifted the chemical potential to the minimum of the conduction band and boosted both valleys to the moiré $Γ_{M}$ point.
Reuse & Permissions
Figure 9
Phase diagrams for the lowest conduction band in pentalayer graphene on a single hBN substrate in the $ξ = 0$ stacking configuration. (a) $Δ_{ind, ν = 0}$ denotes the indirect gap below the conduction band at filling $ν = 0$ and (b) $Δ_{ind, ν = 4}$ denotes the indirect gap above the conduction band at filling $ν = 4$ (including spin/valley degeneracy). (c) $Δ_{cond}$ is the direct gap around the conduction band, from which the topological phase transition is visible. Chern number of each phase is indicated. (d) $W_{cond}$ denotes the bandwidth. The dashed line refers to the experimental twist angle at $θ = 0 . 77^{\circ}$ .
Reuse & Permissions
Figure 10
Band structure of the effective model for pentalayer graphene on a single hBN in the $ξ = 1$ stacking. The red-solid (orange dashed) lines show the band structure of the effective $2 \times 2$ model in the graphene $K$ valley with fitted parameters (with parameters perturbatively derived from the $2 n \times 2 n$ continuum model). Black dots are the first-principles calculations. The match to the low-energy bands is very good.
Reuse & Permissions
Figure 11
Topology, $C_{3}$ irreps, and quantum geometry at $V = 3 meV$ for R5G/hBN at $θ = 0 . 76715^{\circ}$ . The Chern number of the lowest conduction band is $C = 5$ consistent with the $C_{3}$ irreps for stacking $ξ = 1$ in (a) and $ξ = 0$ in (b). The lowest bands show the irreps of a trivial atomic limit $p_{x} - i p_{y}$ orbital at the origin in (a) and $p_{x} - i p_{y}$ at the moiré AB site in (b). Here $ω = e^{2 π i / 3}$ denotes the $C_{3}$ eigenvalues at the high-symmetry points. (c) and (e) show the Berry curvature distribution for the lowest conduction band and for $ξ = 1$ and $ξ = 0$ respectively, which is extremely peaked due to the small gaps (clipped regions are shown in black). (d) and (f) show the Fubini-Study metric of the highest valence band for $ξ = 1$ and $ξ = 0$ respectively, which is large despite the triviality of the bands. The integrated Fubini-Study metric $G$ [48, 49, 50] lower bounds the Wannier spread. In (c) through (f), the center of BZ is the $K_{G}$ point.
Reuse & Permissions
Figure 12
Wilson loop $W (k_{1}) = e^{i ϑ (k_{1})} = exp i \int d k_{2} A_{2} (k)$ at $V = - 20 meV$ for $ξ = 1$ , confirming the triviality of the band.
Reuse & Permissions
Figure 13
Quantum geometry of the lowest conduction band at $V = - 20 meV$ for $ξ = 1$ in (a) and (b) and $ξ = 0$ in (c) and (d). We find that the Chern number of the bands vanishes, but the quantum geometry of the bands remains nontrivial as measured by the integrated Fubini-Study metric $G$ [49].
Reuse & Permissions
Figure 14
We plot the band structure at $V = 20 meV$ for $ξ = 1$ (a) and $ξ = 0$ (c). In (b) and (d) we compute the real-space density profile of the highest valence band for $ξ = 1$ and $ξ = 0$ , respectively, which matches the Wannier center derived from the irreps.
Reuse & Permissions
Figure 15
Phase diagrams of hBN/R5G/hBN with stacking orientation $(ξ_{b}, ξ_{t}) = (1, 1)$ in (a) and $(ξ_{b}, ξ_{t}) = (1, 0)$ in (b). Panel (i) shows the indirect gap at filling $ν = 0$ (at charge neutrality), (ii) shows the indirect gap at $ν = 4$ , (iii) shows the minimal direct gap around the lowest conduction band, and (iv) shows the bandwidth of the lowest conduction band. Chern number of the lowest conduction band is indicated in panel (iii) where boundaries of topologically distinct phases can be seen as the direct gap closes. The dashed line is a reference to the twist angle at $θ = 0 . 77^{\circ}$ .
Reuse & Permissions
Figure 16
Topology and $C_{3}$ irreps for hBN/R5G/hBN. (a) shows the band structure for stacking orientation $(ξ_{b}, ξ_{t}) = (1, 1)$ where there is a $\sim 1 meV$ indirect gap at filling $ν = 4$ for the lowest conduction band with Chern number $C = - 1$ . (b) shows the band structure for stacking orientation $(ξ_{b}, ξ_{t}) = (1, 0)$ with a $C = - 4$ lowest conduction band. (c)–(f) show the Berry curvature distribution for the bands labeled in (a) and (b).
Reuse & Permissions
Figure 17
Dispersion $\pm E (k)$ of the $10 \times 10$ pentalayer Hamiltonian is shown in black, in comparison with the analytical energies of the projected model of Eq. (A15) (blue) at $V = 10 meV$ . The red curve is the asymptotic approximation $\sqrt{{(2 V)}^{2} + | v_{F} {k |}^{10} / t_{1}^{8} - 4 V^{2} {| v_{F} k |}^{2} / t_{1}^{2}}$ for $θ = 0 . 767^{\circ}$ .
Reuse & Permissions
Figure 18
3D schematic of the pentalayer graphene (R5G) from the side (a) and the top (b). The unit cell consists of 10 carbon atoms and is connected by a line to guide the eye. Red and blue are used to color the A and B sublattices in each layer.
Reuse & Permissions
Figure 19
(a) Depiction of the moiré Brillouin zone (BZ) for three different commensurate moiré configurations [see Eq. (3)]. Depending on $m - n mod 3$ , with $(m, n) \in Z^{2}$ labeling the commensurate configuration, the graphene $K$ point (at $K_{G}$ ) is either folded onto the moiré $K_{M}, Γ_{M}$ or $K_{M}^{'}$ point. (b) Depiction of the continuum model moiré BZ in which we focus on the degrees of freedom centered around the $K$ graphene valley. The high-symmetry points are labeled as ${\tilde{Γ}}_{M}, {\tilde{K}}_{M}$ , and ${\tilde{K}}_{M}^{'}$ to signify our convention of always boosting the graphene $K_{G}$ onto ${\tilde{Γ}}_{M}$ . The three moiré reciprocal points circled in red are considered in the tripod model analysis in Appendix pp1-s5 for deducing the $C_{3}$ eigenvalue at ${\tilde{K}}_{M}$ .
Reuse & Permissions
Figure 20
Rigid R5G/hBN and hBN/R5G/hBN lattice structures with zero twist. (a) and (b) are pentalayer graphene stack on hBN, (a) with a carbon atom strictly colinear with a boron atom in bottom layer along z axis, (b) with a carbon atom strictly colinear with a nitrogen atom in bottom layer along z axis. (c) and (d) are pentalayer graphene between two hBN layers. (a) and (b) are connected by applying $C_{2}$ rotation (with axis along z axis) on hBN only, (d) has an approximate inversion symmetry while (c) breaks the inversion symmetry strongly.
Reuse & Permissions
Figure 21
Relaxation results of $0.767^{\circ} R 5 G / {hBN}_{ξ_{b} = 1}$ . (a) Rigid structure of $R 5 G / {hBN}_{ξ_{b} = 1}$ for $ξ = 1$ . (b)–(f) show the intralayer displacement of each layer. (g)–(k) show the interlayer distance of each layer. The ${Gr}_{0}$ layer is the lowest layer, which lies on the hBN substrate. The intralayer displacement is the in-plane displacement from rigid position to relaxed position of an atom in the corresponding layer. The interlayer distance indicates the distance between the corresponding layer and the layer under it.
Reuse & Permissions
Figure 22
Relaxation results of $0.767^{\circ} R 5 G / {hBN}_{ξ_{b} = 0}$ . (a) Structure of $R 5 G / {hBN}_{ξ_{b} = 0}$ . (b)–(f) Intralayer displacement of each layer. (g)–(k) Interlayer distance of each layer. The ${Gr}_{0}$ layer is the lowest layer, which lies on the hBN substrate.
Reuse & Permissions
Figure 23
Relaxation results of $0.767^{\circ} {hBN}_{ξ_{t} = 1} / R 5 G / {hBN}_{ξ_{b} = 1}$ . (a) Structure of ${hBN}_{ξ_{t} = 1} / R 5 G / {hBN}_{ξ_{b} = 1}$ . (b)–(f) Intralayer displacement of each layer. (g)–(k) Interlayer distance of each layer. The ${Gr}_{0}$ layer is the lowest, and ${Gr}_{4}$ is the highest layer. The ${Gr}_{0}$ and ${Gr}_{4}$ layers are near hBN substrates.
Reuse & Permissions
Figure 24
Relaxation results of $0.767^{\circ} {hBN}_{ξ_{t} = 0} / R 5 G / {hBN}_{ξ_{b} = 1}$ . (a) Structure of ${hBN}_{ξ_{t} = 0} / R 5 G / {hBN}_{ξ_{b} = 1}$ . (b)–(f) Intralayer displacement of each layer. (g)–(h) Interlayer distance of each layer. The ${Gr}_{0}$ layer is the lowest layer, and ${Gr}_{4}$ is the highest layer. The ${Gr}_{0}$ and ${Gr}_{4}$ layers are near hBN substrates.
Reuse & Permissions
Figure 25
Comparison of the tight-binding Hamiltonian (red circles) and ab initio DFT (black lines) band structure calculations along the $Γ − K − M$ path for ABC-stacked graphene. (a) Illustration of the $γ_{2}$ bond correction in the SK tight-binding model. Panels (b) and (c) for trilayer, (d) and (e) for tetralayer, (f) and (g) for pentalayer, (h) and (i) for hexalayer, and (j) and (k) for septalayer graphene depict band structures near the $K$ point within two energy windows, from −0.1 eV to 0.1 eV and from −1 eV to 1 eV, using a set value of $V_{pp π}^{0} = - 2.81 eV$ and an internal symmetrical polarization (ISP) of 5 mV/Å.
Reuse & Permissions
Figure 26
Comparison of band structures between rigid and relaxed structures of R3G and $0 . 76715^{\circ}$ twisted-angle setting ISP = 5 mV/Å. E is an applied electrical field and ISP is an internal symmetrical polarization due to the different chemical environment of outer and inner atoms in trilayer graphene. The positive direction of E and ISP are shown in Fig. 2. The structures corresponding to the first row to the fourth row are $R 3 G / {hBN}_{ξ_{b} = 1}, {hBN}_{ξ_{t} = 1} / R 3 G / {hBN}_{ξ_{b} = 1}, R 3 G / {hBN}_{ξ_{b} = 0}$ , and ${hBN}_{ξ_{t} = 0} / R 3 G / {hBN}_{ξ_{b} = 1}$ respectively, and the applied electrical fields corresponding to the first column to the fifth column are −5, −2, 0, 2, and 5 mV/Å respectively. The band structures are depicted with orange dotted lines for rigid structures and black dotted lines for relaxed structures. Here, both $K$ valley and $K^{'}$ valley bands are included.
Reuse & Permissions
Figure 27
Comparison of band structures between rigid and relaxed structures of R4G and $0 . 76715^{\circ}$ twisted-angle setting ISP = 5 mV/Å. E is an applied electrical field and ISP is an internal symmetrical polarization due to the different chemical environment of outer and inner atoms in tetralayer graphene. The positive direction of E and ISP are shown in Fig. 2. The structures corresponding to the first row to the fourth row are $R 4 G / {hBN}_{ξ_{b} = 1}, {hBN}_{ξ_{t} = 1} / R 4 G / {hBN}_{ξ_{b} = 1}, R 4 G / {hBN}_{ξ_{b} = 0}$ , and ${hBN}_{ξ_{t} = 0} / R 4 G / {hBN}_{ξ_{b} = 1}$ respectively, and the applied electrical fields corresponding to the first column to the fifth column are −5, −2, 0, 2, and 5 mV/Å respectively. The band structures are depicted with orange dotted lines for rigid structures and black dotted lines for relaxed structures. Here, both $K$ valley and $K^{'}$ valley bands are included.
Reuse & Permissions
Figure 28
Comparison of band structures between rigid and relaxed structures of R5G and $0 . 76715^{\circ}$ twisted angle setting ISP = 5 mV/Å. E is an applied electrical field and ISP is an internal symmetrical polarization due to the different chemical environment of outer and inner atoms in pentalayer graphene. The positive direction of E and ISP are shown in Fig. 2. The structures corresponding to the first row to the fourth row are $R 5 G / {hBN}_{ξ_{b} = 1}, {hBN}_{ξ_{t} = 1} / R 5 G / {hBN}_{ξ_{b} = 1}, R 5 G / {hBN}_{ξ_{b} = 0}$ , and ${hBN}_{ξ_{t} = 0} / R 5 G / {hBN}_{ξ_{b} = 1}$ respectively, and the applied electrical fields corresponding to the first column to the fifth column are −5, −2, 0, 2, and 5 mV/Å respectively. The band structures are depicted with orange dotted lines for rigid structures and black dotted lines for relaxed structures. Here, both $K$ valley and $K^{'}$ valley bands are included.
Reuse & Permissions
Figure 29
Comparison of band structures between rigid and relaxed structures of R6G and $0 . 76715^{\circ}$ twisted angle setting ISP = 5 mV/Å. E is an applied electrical field and ISP is an internal symmetrical polarization due to the different chemical environment of outer and inner atoms in hexalayer graphene. The positive direction of E and ISP are shown in Fig. 2. The structures corresponding to the first row to the fourth row are $R 6 G / {hBN}_{ξ_{b} = 1}, {hBN}_{ξ_{t} = 1} / R 6 G / {hBN}_{ξ_{b} = 1}, R 6 G / {hBN}_{ξ_{b} = 0}$ , and ${hBN}_{ξ_{t} = 0} / R 6 G / {hBN}_{ξ_{b} = 1}$ respectively, and the applied electrical fields corresponding to the first column to the fifth column are −5, −2, 0, 2, and 5 mV/Å respectively. The band structures are depicted with orange dotted lines for rigid structures and black dotted lines for relaxed structures. Here, both $K$ valley and $K^{'}$ valley bands are included.
Reuse & Permissions
Figure 30
Comparison of band structures between rigid and relaxed structures of R7G and $0 . 76715^{\circ}$ twisted angle setting ISP = 5 mV/Å. E is an applied electrical field and ISP is an internal symmetrical polarization due to the different chemical environment of outer and inner atoms in septalayer graphene. The positive direction of E and ISP are shown in Fig. 2. The structures corresponding to the first row to the fourth row are $R 7 G / {hBN}_{ξ_{b} = 1}, {hBN}_{ξ_{t} = 1} / R 7 G / {hBN}_{ξ_{b} = 1}, R 7 G / {hBN}_{ξ_{b} = 0}$ , and ${hBN}_{ξ_{t} = 0} / R 7 G / {hBN}_{ξ_{b} = 1}$ respectively, and the applied electrical fields corresponding to the first column to the fifth column are −5, −2, 0, 2, and 5 mV/Å respectively. The band structures are depicted with orange dotted lines for rigid structures and black dotted lines for relaxed structures. Here, both $K$ valley and $K^{'}$ valley bands are included.
Reuse & Permissions
Figure 31
Valley operator for single layer graphene. (a) shows the valley operator in real space for one unit cell; (b) shows the Haldane's NNN hopping term with local magnetic flux and (c) shows the corresponding NNN hopping term with valley flux; (d) shows the average value of valley operator $〈 ψ (k) | V_{z}^{G r} (k) | ψ (k) 〉$ in high-symmetry point path of reciprocal space. (e) is the energy band of SLG follow the same path as (d), weighted by $〈 ψ (k) | V_{z}^{G r} (k) | ψ (k) 〉$ . (f) is the flow diagram of valley resolved bands calculation, this flow is applicable for both SLG and hBN- $R n G$ .
Reuse & Permissions
Figure 32
Tight-binding band structures of R3G and $0 . 76715^{\circ}$ twist angle setting ISP = 5 meV/Å. E is an applied electrical field, and ISP is an internal symmetrical polarization due to the different chemical environment of outer and inner atoms in trilayer graphene. The positive direction of E and ISP are shown in Fig. 2. The structures corresponding to the first row to the fourth row are $R 3 G / {hBN}_{ξ_{b} = 1}, {hBN}_{ξ_{t} = 1} / R 3 G / {hBN}_{ξ_{b} = 1}, R 3 G / {hBN}_{ξ_{b} = 0}$ , and ${hBN}_{ξ_{t} = 0} / R 3 G / {hBN}_{ξ_{b} = 1}$ respectively, and the applied electrical fields corresponding to the first column to the fifth column are −5, −2, 0, 2, and 5 mV/Å respectively. The band structures are depicted with blue dotted lines for $K$ valley and red dotted lines for $K^{'}$ valley.
Reuse & Permissions
Figure 33
Tight-binding band structures of R4G and $0 . 76715^{\circ}$ twisted angle setting ISP = 5 mV/Å. E is an applied electrical field and ISP is an internal symmetrical polarization due to the different chemical environment of outer and inner atoms in tetralayer graphene. The positive direction of E and ISP are shown in Fig. 2. The structures corresponding to the first row to the fourth row are $R 4 G / {hBN}_{ξ_{b} = 1}, {hBN}_{ξ_{t} = 1} / R 4 G / {hBN}_{ξ_{b} = 1}, R 4 G / {hBN}_{ξ_{b} = 0}$ , and ${hBN}_{ξ_{t} = 0} / R 4 G / {hBN}_{ξ_{b} = 1}$ respectively, and the applied electrical fields corresponding to the first column to the fifth column are −5, −2, 0, 2, and 5 mV/Å respectively. The band structures are depicted with blue dotted lines for $K$ valley and red dotted lines for $K^{'}$ valley.
Reuse & Permissions
Figure 34
Tight-binding band structures of R5G and $0 . 76715^{\circ}$ twisted angle setting ISP = 5 mV/Å. E is an applied electrical field and ISP is an internal symmetrical polarization due to the different chemical environment of outer and inner atoms in pentalayer graphene. The positive direction of E and ISP are shown in Fig. 2. The structures corresponding to the first row to the fourth row are $R 5 G / {hBN}_{ξ_{b} = 1}, {hBN}_{ξ_{t} = 1} / R 5 G / {hBN}_{ξ_{b} = 1}, R 5 G / {hBN}_{ξ_{b} = 0}$ , and ${hBN}_{ξ_{t} = 0} / R 5 G / {hBN}_{ξ_{b} = 1}$ respectively and the applied electrical fields corresponding to the first column to the fifth column are −5, −2, 0, 2, and 5 mV/Å respectively. The band structures are depicted with blue dotted lines for $K$ valley and red dotted lines for $K^{'}$ valley.
Reuse & Permissions
Figure 35
Tight-binding band structures of R6G and $0 . 76715^{\circ}$ twisted angle setting ISP = 5 mV/Å. E is an applied electrical field and ISP is an internal symmetrical polarization due to the different chemical environment of outer and inner atoms in hexalayer graphene. The positive direction of E and ISP are shown in Fig. 2. The structures corresponding to the first row to the fourth row are $R 6 G / {hBN}_{ξ_{b} = 1}, {hBN}_{ξ_{t} = 1} / R 6 G / {hBN}_{ξ_{b} = 1}, R 6 G / {hBN}_{ξ_{b} = 0}$ , and ${hBN}_{ξ_{t} = 0} / R 6 G / {hBN}_{ξ_{b} = 1}$ respectively, and the applied electrical fields corresponding to the first column to the fifth column are −5, −2, 0, 2, and 5 mV/Å respectively. The band structures are depicted with blue dotted lines for $K$ valley and red dotted lines for $K^{'}$ valley.
Reuse & Permissions
Figure 36
Tight-binding band structures of R7G and $0 . 76715^{\circ}$ twisted angle setting ISP = 5 mV/Å. E is an applied electrical field and ISP is an internal symmetrical polarization due to the different chemical environment of outer and inner atoms in septalayer graphene. The positive direction of E and ISP are shown in Fig. 2. The structures corresponding to the first row to the fourth row are $R 7 G / {hBN}_{ξ_{b} = 1}, {hBN}_{ξ_{t} = 1} / R 7 G / {hBN}_{ξ_{b} = 1}, R 7 G / {hBN}_{ξ_{b} = 0}$ , and ${hBN}_{ξ_{t} = 0} / R 7 G / {hBN}_{ξ_{b} = 1}$ respectively, and the applied electrical fields corresponding to the first column to the fifth column are −5, −2, 0, 2, and 5 mV/Å respectively. The band structures are depicted with blue dotted lines for $K$ valley and red dotted lines for $K^{'}$ valley.
Reuse & Permissions
Figure 37
The comparison between the $DFT + SK$ bands (black) and the bands from the $2 n \times 2 n$ continuum model (blue lines) in Eq. (A45) for $n = 3$ . The parameter values are listed in Table 1 and Table 3 of main text.
Reuse & Permissions
Figure 38
The comparison between the $DFT + SK$ bands (black) and the bands from the $2 n \times 2 n$ continuum model (blue lines) in Eq. (A45) for $n = 4$ . The parameter values are listed in Table 1 and Table 3 of main text.
Reuse & Permissions
Figure 39
The comparison between the $DFT + SK$ bands (black) and the bands from the $2 n \times 2 n$ continuum model (blue lines) in Eq. (A45) for $n = 5$ . The parameter values are listed in Tables 1 and 3 of main text.
Reuse & Permissions
Figure 40
The comparison between the $DFT + SK$ bands (black) and the bands from the $2 n \times 2 n$ continuum model (blue lines) in Eq. (A45) for $n = 6$ . The parameter values are listed in Table 1 and Table 3 of main text.
Reuse & Permissions
Figure 41
The comparison between the $DFT + SK$ bands (black) and the bands from the $2 n \times 2 n$ continuum model (blue lines) in Eq. (A45) for $n = 7$ . The parameter values are listed in Table 1 and Table 3 of main text.
Reuse & Permissions
Figure 42
Phase diagrams of R3G-hBN superlattices of various stacking configurations: (a) $ξ = 1$ , (b) $ξ = 0$ , (c) $(ξ_{b}, ξ_{t}) = (1, 1)$ , and (d) $(ξ_{b}, ξ_{t}) = (1, 0)$ . Panel (i) shows the single-particle indirect gap $Δ_{ind, ν = - 4}$ at filling $ν = - 4$ ; (ii) shows the indirect gap $Δ_{ind, ν = 0}$ at filling $ν = 0$ ; (iii) shows the indirect gap $Δ_{ind, ν = 4}$ at filling $ν = 4$ ; (iv) shows the minimal direct gap $Δ_{val}$ around the highest valence band in one valley; (v) shows the bandwidth $W_{val}$ of the highest valence band; (vi) shows the minimal direct gap $Δ_{cond}$ around the lowest conduction band in one valley; (vii) shows the bandwidth $W_{cond}$ of the lowest conduction band. Chern numbers of the highest valence band and the lowest conduction band in $K$ valley are indicated on panel (iv) and panel (vi), respectively, where boundaries of topologically distinct phases can be seen as the direct gap closes.
Reuse & Permissions
Figure 43
Phase diagrams of R4G-hBN superlattices of various stacking configurations: (a) $ξ = 1$ , (b) $ξ = 0$ , (c) $(ξ_{b}, ξ_{t}) = (1, 1)$ , and (d) $(ξ_{b}, ξ_{t}) = (1, 0)$ . Panel (i) shows the single-particle indirect gap $Δ_{ind, ν = - 4}$ at filling $ν = - 4$ ; (ii) shows the indirect gap $Δ_{ind, ν = 0}$ at filling $ν = 0$ ; (iii) shows the indirect gap $Δ_{ind, ν = 4}$ at filling $ν = 4$ ; (iv) shows the minimal direct gap $Δ_{val}$ around the highest valence band in one valley; (v) shows the bandwidth $W_{val}$ of the highest valence band; (vi) shows the minimal direct gap $Δ_{cond}$ around the lowest conduction band in one valley; (vii) shows the bandwidth $W_{cond}$ of the lowest conduction band. Chern numbers of the highest valence band and the lowest conduction band in $K$ valley are indicated on panel (iv) and panel (vi), respectively, where boundaries of topologically distinct phases can be seen as the direct gap closes.
Reuse & Permissions
Figure 44
Phase diagrams of R5G-hBN superlattices of various stacking configurations: (a) $ξ = 1$ , (b) $ξ = 0$ , (c) $(ξ_{b}, ξ_{t}) = (1, 1)$ , and (d) $(ξ_{b}, ξ_{t}) = (1, 0)$ . Panel (i) shows the single-particle indirect gap $Δ_{ind, ν = - 4}$ at filling $ν = - 4$ ; (ii) shows the indirect gap $Δ_{ind, ν = 0}$ at filling $ν = 0$ ; (iii) shows the indirect gap $Δ_{ind, ν = 4}$ at filling $ν = 4$ ; (iv) shows the minimal direct gap $Δ_{val}$ around the highest valence band in one valley; (v) shows the bandwidth $W_{val}$ of the highest valence band; (vi) shows the minimal direct gap $Δ_{cond}$ around the lowest conduction band in one valley; (vii) shows the bandwidth $W_{cond}$ of the lowest conduction band. Chern numbers of the highest valence band and the lowest conduction band in $K$ valley are indicated on panel (iv) and panel (vi), respectively, where boundaries of topologically distinct phases can be seen as the direct gap closes.
Reuse & Permissions
Figure 45
Phase diagrams of R6G-hBN superlattices of various stacking configurations: (a) $ξ = 1$ , (b) $ξ = 0$ , (c) $(ξ_{b}, ξ_{t}) = (1, 1)$ , and (d) $(ξ_{b}, ξ_{t}) = (1, 0)$ . Panel (i) shows the single-particle indirect gap $Δ_{ind, ν = - 4}$ at filling $ν = - 4$ ; (ii) shows the indirect gap $Δ_{ind, ν = 0}$ at filling $ν = 0$ ; (iii) shows the indirect gap $Δ_{ind, ν = 4}$ at filling $ν = 4$ ; (iv) shows the minimal direct gap $Δ_{val}$ around the highest valence band in one valley; (v) shows the bandwidth $W_{val}$ of the highest valence band; (vi) shows the minimal direct gap $Δ_{cond}$ around the lowest conduction band in one valley; (vii) shows the bandwidth $W_{cond}$ of the lowest conduction band. Chern numbers of the highest valence band and the lowest conduction band in $K$ valley are indicated on panel (iv) and panel (vi), respectively, where boundaries of topologically distinct phases can be seen as the direct gap closes.
Reuse & Permissions
Figure 46
Phase diagrams of R7G-hBN superlattices of various stacking configurations: (a) $ξ = 1$ , (b) $ξ = 0$ , (c) $(ξ_{b}, ξ_{t}) = (1, 1)$ , and (d) $(ξ_{b}, ξ_{t}) = (1, 0)$ . Panel (i) shows the single-particle indirect gap $Δ_{ind, ν = - 4}$ at filling $ν = - 4$ ; (ii) shows the indirect gap $Δ_{ind, ν = 0}$ at filling $ν = 0$ ; (iii) shows the indirect gap $Δ_{ind, ν = 4}$ at filling $ν = 4$ ; (iv) shows the minimal direct gap $Δ_{val}$ around the highest valence band in one valley; (v) shows the bandwidth $W_{val}$ of the highest valence band; (vi) shows the minimal direct gap $Δ_{cond}$ around the lowest conduction band in one valley; (vii) shows the bandwidth $W_{cond}$ of the lowest conduction band. Chern numbers of the highest valence band and the lowest conduction band in $K$ valley are indicated on panel (iv) and panel (vi), respectively, where boundaries of topologically distinct phases can be seen as the direct gap closes.
Reuse & Permissions
Figure 47
The comparison between the $DFT + SK$ bands (black) and the bands from the $2 \times 2$ effective continuum model (red lines) in Eq. (A54) for $n = 3$ . The parameter values are listed in Table 2 and Table 4 of main text.
Reuse & Permissions
Figure 48
The comparison between the $DFT + SK$ bands (black) and the bands from the $2 \times 2$ effective continuum model (red lines) in Eq. (A54) for $n = 4$ . The parameter values are listed in Table 2 and Table 4 of main text.
Reuse & Permissions
Figure 49
The comparison between the $DFT + SK$ bands (black) and the bands from the $2 \times 2$ effective continuum model (red lines) in Eq. (A54) for $n = 5$ . The parameter values are listed in Table 2 and Table 4 of main text.
Reuse & Permissions
Figure 50
The comparison between the $DFT + SK$ bands (black) and the bands from the $2 \times 2$ effective continuum model (red lines) in Eq. (A54) for $n = 6$ . The parameter values are listed in Table 2 and Table 4 of main text.
Reuse & Permissions
Figure 51
The comparison between the $DFT + SK$ bands (black) and the bands from the $2 \times 2$ effective continuum model (red lines) in Eq. (A54) for $n = 7$ . The parameter values are listed in Table 2 and Table 4 of main text.
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics

Moiré fractional Chern insulators. II. First-principles calculations and continuum models of rhombohedral graphene superlattices

Jonah Herzog-Arbeitman, Yuzhi Wang, Jiaxuan Liu, Pok Man Tam, Ziyue Qi, Yujin Jia, Dmitri K. Efetov, Oskar Vafek, Nicolas Regnault, Hongming Weng, Quansheng Wu, B. Andrei Bernevig, and Jiabin Yu

Phys. Rev. B 109, 205122 – Published 7 May 2024

Abstract

Physics Subject Headings (PhySH)

Authors & Affiliations

See Also

Moiré fractional Chern insulators. I. First-principles calculations and continuum models of twisted bilayer ${MoTe}_{2}$

Yujin Jia, Jiabin Yu, Jiaxuan Liu, Jonah Herzog-Arbeitman, Ziyue Qi, Hanqi Pi, Nicolas Regnault, Hongming Weng, B. Andrei Bernevig, and Quansheng Wu

Phys. Rev. B 109, 205121 (2024)

Article Text (Subscription Required)

References (Subscription Required)

Issue

Access Options

Physical Review B

covering condensed matter and materials physics

Moiré fractional Chern insulators. II. First-principles calculations and continuum models of rhombohedral graphene superlattices

Jonah Herzog-Arbeitman, Yuzhi Wang, Jiaxuan Liu, Pok Man Tam, Ziyue Qi, Yujin Jia, Dmitri K. Efetov, Oskar Vafek, Nicolas Regnault, Hongming Weng, Quansheng Wu, B. Andrei Bernevig, and Jiabin Yu

Phys. Rev. B 109, 205122 – Published 7 May 2024

Abstract

Physics Subject Headings (PhySH)

Authors & Affiliations

See Also

Moiré fractional Chern insulators. I. First-principles calculations and continuum models of twisted bilayer MoTe2

Yujin Jia, Jiabin Yu, Jiaxuan Liu, Jonah Herzog-Arbeitman, Ziyue Qi, Hanqi Pi, Nicolas Regnault, Hongming Weng, B. Andrei Bernevig, and Quansheng Wu

Phys. Rev. B 109, 205121 (2024)

Article Text (Subscription Required)

References (Subscription Required)

Issue

Access Options

Authorization Required

Other Options

Download & Share

Images

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

Figure 13

Figure 14

Figure 15

Figure 16

Figure 17

Figure 18

Figure 19

Figure 20

Figure 21

Figure 22

Figure 23

Figure 24

Figure 25

Figure 26

Figure 27

Figure 28

Figure 29

Figure 30

Figure 31

Figure 32

Figure 33

Figure 34

Figure 35

Figure 36

Figure 37

Figure 38

Figure 39

Figure 40

Figure 41

Figure 42

Figure 43

Figure 44

Figure 45

Figure 46

Figure 47

Figure 48

Figure 49

Figure 50

Figure 51

Log In

Search

Article Lookup

Paste a citation or DOI

Enter a citation

Moiré fractional Chern insulators. I. First-principles calculations and continuum models of twisted bilayer ${MoTe}_{2}$