Interfaces between 2D Materials: h-BN and Graphene.

1. Introduction

This tutorial creates graphene on four layers of hexagonal boron nitride (h-BN) for three stackings and a list of graphene–h-BN distances, following the manuscript below.

Manuscript

Gianluca Giovannetti, Petr A. Khomyakov, Geert Brocks, Paul J. Kelly and Jeroen van den Brink Substrate-induced band gap in graphene on hexagonal boron nitride: Ab initio density functional calculations Physical Review B 76, 073103 (2007) DOI: 10.1103/PhysRevB.76.073103 ¹

The Materials Designer imports the two materials from Standata, and the JupyterLite notebook builds the interfaces.

2. Load and preview materials

First, navigate to Materials Designer and import graphene (2dm-3993) and bulk h-BN (mp-7991) from Standata.

2.1. Launch JupyterLite session

Select the "Advanced > JupyterLite Transformation" menu item to launch the JupyterLite environment, with h-BN as the first material and graphene as the second.

2.2. Open the notebook and set the parameters

Open the interface_2d_2d_boron_nitride_graphene.ipynb notebook. Cell 1.1 sets the parameters:

LATTICE_CONSTANT = 2.445  # Å, graphene LDA (Giovannetti et al. 2007); h-BN is compressed to it
H_BN_LAYERS = 4
H_BN_INTERLAYER_DISTANCE = 3.24  # Å, the paper's LDA value
VACUUM = 15.0  # Å above graphene

# Giovannetti et al. 2007 compute all three stackings at every distance 2.5–3.9 Å; the defaults run
# stacking (c) at three distances around its minimum. Uncomment entries to compute more
# (one job per stacking × distance, ~25 min each on queue D with two cores).
STACKINGS = [
    "c",
    # "a",
    # "b",
]
DISTANCES = [
    3.1, 3.2, 3.3,
    # 2.5, 2.6, 2.7, 2.8, 2.9, 3.0, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9,
]
STACKING_SHIFTS = {"a": 0, "b": 1, "c": -1}  # in units of a/√3 along y
REGISTRY_TOLERANCE = 1e-3  # crystal units

STACKINGS and DISTANCES list the registries and distances to build; the defaults build stacking (c) at 3.1, 3.2 and 3.3 Å, and uncommenting the rest builds the paper's 3 × 15 set.

3. Create the interfaces between h-BN and graphene

3.1. Strain the materials

Both materials are strained in-plane to a = 2.445 Å, and h-BN along c to 3.24 Å between layers:

from mat3ra.made.tools.build_components.operations.core.modifications.strain.helpers import create_strain

film_scale = LATTICE_CONSTANT / film.lattice.a
substrate_scale = LATTICE_CONSTANT / substrate.lattice.a
substrate_c_scale = 2 * H_BN_INTERLAYER_DISTANCE / substrate.lattice.c
film = create_strain(film, strain_matrix=[[film_scale, 0, 0], [0, film_scale, 0], [0, 0, 1]])
substrate = create_strain(
    substrate, strain_matrix=[[substrate_scale, 0, 0], [0, substrate_scale, 0], [0, 0, substrate_c_scale]]
)

3.2. Set the AA' stacking of h-BN

The standata bulk h-BN entry is not AA': its boron atoms sit over hexagon centres of the next layer. The atoms of the upper layer are moved by (1/3, 2/3, 0) in crystal coordinates, so that boron sits over nitrogen in adjacent layers:

from mat3ra.made.tools.analyze.other import get_atom_indices_with_condition_on_coordinates
from mat3ra.made.tools.operations.core.unary import translate_atoms

upper_layer_ids = get_atom_indices_with_condition_on_coordinates(substrate, lambda coordinate: coordinate[2] > 0.5)
substrate = translate_atoms(
    substrate, atom_ids=upper_layer_ids, vector=[1 / 3, 2 / 3, 0], use_cartesian_coordinates=False
)

3.3. Create the slabs

The h-BN slab has four layers, two per bulk cell, and the graphene slab one layer:

from mat3ra.made.tools.build.pristine_structures.two_dimensional.slab import SlabConfiguration, SlabBuilder

substrate_slab_config = SlabConfiguration.from_parameters(
    material_or_dict=substrate,
    miller_indices=(0, 0, 1),
    number_of_layers=H_BN_LAYERS // 2,  # two BN layers per bulk cell
    vacuum=0.0,
)

film_slab_config = SlabConfiguration.from_parameters(
    material_or_dict=film,
    miller_indices=(0, 0, 1),
    number_of_layers=1,
    vacuum=0.0,
)

substrate_slab = SlabBuilder().get_material(substrate_slab_config)
film_slab = SlabBuilder().get_material(film_slab_config)

3.4. Create the interface at each distance and stacking

The loop places graphene on the h-BN slab at each distance, shifts it along y by the registry shift of each stacking, names the interface and prints the measured registry:

import numpy as np
from mat3ra.made.tools.analyze.other import get_average_interlayer_distance
from mat3ra.made.tools.convert.interface_parts_enum import InterfacePartsEnum
from mat3ra.made.tools.helpers import create_interface_zsl_between_slabs as create_zsl_interface_between_slabs
from mat3ra.made.tools.modify import interface_displace_part


def get_registry(interface):
    elements = np.array(interface.basis.elements.values)
    coordinates = np.array(interface.basis.coordinates.values)
    top_layer = (elements != "C") & np.isclose(coordinates[:, 2], coordinates[elements != "C", 2].max())
    registry = []
    for carbon in coordinates[elements == "C"]:
        in_plane_offsets = (coordinates[top_layer, :2] - carbon[:2] + 0.5) % 1 - 0.5
        atoms_below = elements[top_layer][np.all(np.abs(in_plane_offsets) < 1e-3, axis=1)]
        registry.append(atoms_below[0] if len(atoms_below) else "hollow")
    return registry


interfaces = []
for stacking in STACKINGS:
    for distance in DISTANCES:
        # the builder adds the gap to the vacuum above the film
        interface = create_zsl_interface_between_slabs(
            substrate_slab=substrate_slab, film_slab=film_slab, gap=distance, vacuum=VACUUM - distance
        )
        interface = interface_displace_part(
            interface=interface,
            displacement=[0, STACKING_SHIFTS[stacking] * interface.lattice.a / np.sqrt(3), 0],
            use_cartesian_coordinates=True,
        )
        interface.name = f"Gr/hBN ({stacking}) d{distance:.2f}"
        interface.lattice.type = "HEX"
        interfaces.append(interface)
        interlayer_distance = get_average_interlayer_distance(
            interface, InterfacePartsEnum.SUBSTRATE.value, InterfacePartsEnum.FILM.value
        )
        vacuum = interface.lattice.c * (1 - max(coordinate[2] for coordinate in interface.basis.coordinates.values))
        print(f"{interface.name}: {len(interface.basis.elements.ids)} atoms, a = {interface.lattice.a:.4f} Å, "
              f"gamma = {interface.lattice.gamma:.1f}°, distance = {interlayer_distance:.3f} Å, "
              f"vacuum = {vacuum:.2f} Å, C over {' / '.join(get_registry(interface))}")

3.5. Run the notebook

Select Run > Run All.

4. Pass the Material to Materials Designer

The user can pass the materials to the current Materials Designer environment and save them.

Or the user can save or download the materials in Material JSON format or POSCAR format.

The interfaces are named Gr/hBN (<stacking>) d<distance>, for example Gr/hBN (c) d3.10, and the simulation tutorial loads them by these names.

5. Interactive JupyterLite Notebook

The interactive JupyterLite notebook for creating Gr/h-BN interface can be accessed below. To run the notebook, click on the "Run All" button.

⛶ Full Screen

6. References

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1. Gianluca Giovannetti, Petr A. Khomyakov, Geert Brocks, Paul J. Kelly, and Jeroen van den Brink. Substrate-induced band gap in graphene on hexagonal boron nitride: ab initio density functional calculations. Physical Review B, 76:073103, 2007. doi:10.1103/PhysRevB.76.073103. ↩