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Twisted Bilayer h-BN Flat Bands

1. Introduction

This tutorial calculates the band structure of the twisted hexagonal boron nitride (h-BN) bilayers created in the structure creation tutorial and reads off the width of their top set of flat valence bands, as in the following manuscript.

Manuscript

Lede Xian, Dante M. Kennes, Nicolas Tancogne-Dejean, Massimo Altarelli, and Angel Rubio, "Multiflat Bands and Strong Correlations in Twisted Bilayer Boron Nitride: Doping-Induced Correlated Insulator and Superconductor" Nano Lett. 19, 4934–4940 (2019) DOI: 10.1021/acs.nanolett.9b00986 1

Twisted bilayer h-BN from the manuscript and the structure notebook

1.1. What the manuscript found

The manuscript computes the band structure of unrelaxed twisted bilayers along Γ–K–M of the moiré cell. The top set of valence bands is flat, and its width decreases from 101.4 meV at 7.34° to 1.6 meV at 2.64° (text p. 4936; Fig. 1g). This tutorial computes the 7.34° point.

2. Prerequisites

Run the structure creation tutorial first. Its notebook saves each bilayer and the monolayer into the uploads folder under a name such as h-BN bilayer 7.34deg alpha d3.23, and this notebook loads them back by exactly those names.

3. What is calculated

One job per material runs the Standata "Band Structure" workflow (band_structure.json) on a structure saved by the structure notebook. It chains pw_scf, the self-consistent run on the material's k-grid, and pw_bands, the bands along Γ–K–M of its cell.

structure twist interlayer distance atoms
h-BN bilayer 7.34deg alpha d3.23 60° + 7.34° (α) 3.23 Å 244
h-BN bilayer 7.34deg beta d3.23 7.34° (β) 3.23 Å 244
h-BN monolayer — — 2

The simulation notebook computes the monolayer and α by default; β and the smaller angles are commented entries in MATERIALS.

3.1. The structure is not relaxed

The manuscript's band widths are computed with flat layers 3.23 Å apart (p. 4935–4936). The structure notebook builds the bilayers the same way, and the workflow has no relaxation step.

3.2. Which configuration the twist angle gives

Configurations α and β share one moiré cell and their twist angles differ by 60° (p. 4935). Their unrelaxed band structures are almost identical, and the manuscript does not say which one Fig. 1g uses (p. 4936). With the h-BN from Standata, an angle of 60 + 7.34 in the structure notebook gives α, and 7.34 gives β.

4. Calculation parameters

The manuscript used VASP with the local density approximation (LDA) and projector augmented-wave (PAW) pseudopotentials (Supporting Information, p. 2):

this tutorial manuscript
Code Quantum ESPRESSO VASP
Functional LDA (pz) LDA
Pseudopotentials ultrasoft (GBRV) PAW
Cutoff 40 Ry wavefunction, 200 Ry density 400 eV
Self-consistent k-points Γ (bilayers), 12×12×1 (monolayer) Γ (bilayers), not stated (monolayer)
Band path Γ–K–M, 7 points (bilayers), 61 points (monolayer) Γ–K–M, number of points not stated
Lattice constant 2.509 Ã… (Standata) 2.512 Ã…
Interlayer distance 3.23 Ã… 3.23 Ã…
Vacuum 20 Ã… above the top layer not stated
Spin polarization off not stated
Geometry not relaxed not relaxed

Cell 1.3 sets the DFT parameters:

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MODEL_SUBTYPE = "lda"
FUNCTIONAL = "pz"
PSEUDOPOTENTIAL_TYPE = "us"  # GBRV
ECUTWFC = 40  # Ry
ECUTRHO = 200  # Ry; paper: VASP PAW, 400 eV

MODEL_TAG = f"{FUNCTIONAL}-{PSEUDOPOTENTIAL_TYPE} {ECUTWFC}-{ECUTRHO}Ry"

SCF_UNIT = "pw_scf"
BANDS_UNIT = "pw_bands"

PLOT_ENERGY_RANGE = (-0.5, 0.1)  # eV, relative to the top of the flat bands

4.1. K-point sampling

KPATH and KPATH_MONOLAYER, in cell 1.2, list the vertices of a path and the number of steps from each to the next; the steps values set how many points the bands are computed at. Each entry of MATERIALS names its path under "kpath".

4.2. The band structure path belongs to the moiré cell

The path is Γ–K–M of the moiré cell, as in the manuscript's Fig. 1f and Supporting Information Fig. S2. The manuscript gives the width of "the top of the valence bands (the first set of flat bands)" without saying where in the Brillouin zone it is taken (p. 4936). Here it is the spread of the top valence flat set over the points of KPATH, vertices included. The set starts at the highest occupied band and takes in each band below whose maximum reaches the set's minimum; its separation is the distance from the set's minimum down to the maximum of the next band.

5. Step-by-step instructions

5.1. Create the structures

Run the structure creation notebook. Its default parameters build the 7.34° α and β bilayers and save them as h-BN bilayer 7.34deg alpha d3.23 and h-BN bilayer 7.34deg beta d3.23.

5.2. Open the simulation notebook

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other/materials_designer/specific_examples/interface_bilayer_twisted_commensurate_lattices_boron_nitride_SIMULATION.ipynb

5.3. Select the materials

Cell 1.2 holds the band paths, the materials with the k-grid, path and manuscript value of each, the workflow and the compute settings:

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from datetime import datetime
from mat3ra.ide.compute import QueueName

ORGANIZATION_NAME = None  # set to your organization name (full or partial); otherwise, your default one is used
FOLDER = "./uploads"
# paper: Γ–K–M, number of points not stated; ours evenly spaced (|ΓK| = 2|KM|), 7 points for the moiré cells
KPATH = [
    {"point": "Γ", "steps": 4},
    {"point": "K", "steps": 2},
    {"point": "M", "steps": 1},
]
# paper Fig. 1d: Γ–K–M, number of points not stated; ours 61, evenly spaced
KPATH_MONOLAYER = [
    {"point": "Γ", "steps": 40},
    {"point": "K", "steps": 20},
    {"point": "M", "steps": 1},
]
# Names as saved by the structure notebook, each with the k-grid of its self-consistent run, the path of its bands and
# the paper's value: the monolayer's LDA band gap (eV, SI p. 10) or the width of the bilayer's top valence flat set
# (meV, configuration not stated; 7.34° and 2.64°: text p. 4936; 6.01, 5.08 and 3.89°: read off Fig. 1g). The twisted
# cells use the paper's Γ-only self-consistent run. One job per material; every angle below 7.34° exceeds one 4 h
# single-node job.
MATERIALS = {
    # paper: k-grid not stated; ours a multiple of 3, which keeps K on the mesh
    "h-BN monolayer": {"kgrid": [12, 12, 1], "kpath": KPATH_MONOLAYER, "paper_band_gap": 4.7},
    "h-BN bilayer 7.34deg alpha d3.23": {"kgrid": [1, 1, 1], "kpath": KPATH, "paper_flat_band_width": 101.4},
    # "h-BN bilayer 7.34deg beta d3.23": {"kgrid": [1, 1, 1], "kpath": KPATH, "paper_flat_band_width": 101.4},
    # "h-BN bilayer 6.01deg alpha d3.23": {"kgrid": [1, 1, 1], "kpath": KPATH, "paper_flat_band_width": 54.9},
    # "h-BN bilayer 6.01deg beta d3.23": {"kgrid": [1, 1, 1], "kpath": KPATH, "paper_flat_band_width": 54.9},
    # "h-BN bilayer 5.08deg alpha d3.23": {"kgrid": [1, 1, 1], "kpath": KPATH, "paper_flat_band_width": 31.6},
    # "h-BN bilayer 5.08deg beta d3.23": {"kgrid": [1, 1, 1], "kpath": KPATH, "paper_flat_band_width": 31.6},
    # "h-BN bilayer 3.89deg alpha d3.23": {"kgrid": [1, 1, 1], "kpath": KPATH, "paper_flat_band_width": 11.7},
    # "h-BN bilayer 3.89deg beta d3.23": {"kgrid": [1, 1, 1], "kpath": KPATH, "paper_flat_band_width": 11.7},
    # "h-BN bilayer 2.64deg alpha d3.23": {"kgrid": [1, 1, 1], "kpath": KPATH, "paper_flat_band_width": 1.6},
    # "h-BN bilayer 2.64deg beta d3.23": {"kgrid": [1, 1, 1], "kpath": KPATH, "paper_flat_band_width": 1.6},
}

WORKFLOW_SEARCH_TERM = "band_structure.json"
MY_WORKFLOW_NAME = "Band Structure"
APPLICATION_NAME = "espresso"

CLUSTER_NAME = None  # specify full or partial name i.e. "cluster-001" to select
QUEUE_NAME = QueueName.OR
PPN = 16  # OR allows 16 cores per job
TIME_LIMIT = "04:00:00"

timestamp = datetime.now().strftime("%Y-%m-%d %H:%M")
POLL_INTERVAL = 60  # seconds

Each name in MATERIALS must match a name the structure notebook saved; one job is created per entry. Set ORGANIZATION_NAME to the account's organization.

5.4. Run the notebook

Select Run > Run All. The notebook will authenticate with the platform, load and save the materials, submit the jobs one after another and wait for them. Section 7 reads the band structures, the flat-band widths and the separations of the bilayers and the band gap of the monolayer, and section 8 prints them beside the manuscript's values.

6. Expected results

6.1. Flat-band width and band gap

For each bilayer the last cell prints the number of bands in the top valence flat set and its width along Γ–K–M beside the manuscript's value, and the separation of the set from the bands below; for the monolayer, its band gap along Γ–K–M beside the manuscript's. For the defaults:

quantity this tutorial manuscript
h-BN bilayer 7.34deg alpha d3.23: width of the top valence flat set (2 bands), along Γ–K–M 101.3 meV 101.4 meV (text p. 4936; Fig. 1g)
h-BN bilayer 7.34deg alpha d3.23: separation from the bands below 57.6 meV not given at 7.34°
h-BN monolayer: band gap along Γ–K–M 4.56 eV 4.7 eV (Supporting Information, p. 10)

6.2. Band structure of the monolayer

The manuscript's Fig. 1d beside the monolayer job's band structure as the platform plots it:

Band structure of monolayer h-BN from the manuscript and from the platform

7. Troubleshooting

7.1. Material not found

ValueError: No material named … means the structure notebook has not been run, or a name in MATERIALS differs from the names it saved. Run the structure tutorial first.

8. Interactive JupyterLite notebook

The notebook below calculates the band structure and prints the comparison. Select Run > Run All Cells.

9. References

  1. Lede Xian, Dante M. Kennes, Nicolas Tancogne-Dejean, Massimo Altarelli, and Angel Rubio. Multiflat bands and strong correlations in twisted bilayer boron nitride: doping-induced correlated insulator and superconductor. Nano Lett., 19(8):4934–4940, 2019. URL: https://doi.org/10.1021/acs.nanolett.9b00986. ↩