Getting Started

Getting Started¶

This page describes the structure of a TiBi project and explains how to create, modify, and interpret its core components — systems, unit cells, sites, and states. It also explains how to build a tight-binding Hamiltonian that can be used for calculating various quantities of interest.

Project Structure¶

In TiBi, a single project can contain multiple systems. A system consists of a unit cell with sites (atoms). Each site can have an arbitrary number of states (orbitals).

Example

Graphene (unit cell)
- A (site, sublattice A)
  - pz (state, orbital)
- B (site, sublattice B)
  - pz (state, orbital)
BN (unit cell)
- B (site, boron)
  - pz (state, orbital)
- N (site, nitrogen)
  - pz (state, orbital)

The orbitals couple to each other via hopping terms. Having multiple systems within a project makes it easy to compare their properties, such as the band structures of two graphene bilayers with different transverse electric fields.

Creating a System¶

To create a system, you’ll define basis vectors, add sites to the unit cell, and attach states to each site. To start, click the + Add Unit Cell button in the project layout view. Once a new unit cell has been added, it is possible to rename it by double clicking on it in the project layout view. Press Enter to confirm the new name.

Basis Vectors¶

By default, unit cells start as cubic with no periodicity (i.e., 0D). This means that the unit cell is not repeated along any of its basis vectors. The dimensionality can be set between 0 and 3 by clicking the appropriate radio button. The system’s dimensionality determines how the unit cell is repeated and how the Brillouin zone is computed. For \(n\)-dimensional systems, the unit cell is repeated along the first \(n\) basis vectors. You can edit each basis vector using the input fields next to it. Press Enter to apply changes.

For notational consistency, there are certain restrictions on the basis vectors depending on the system dimensionality:

0D: Basis vectors are orthogonal, with the first one pointing in the \(x\) direction, second in the \(y\) direction, and the third in the \(z\) direction.
1D: Vector 1 is periodic. Basis vectors are orthogonal, with the first one pointing in the \(x\) direction, second in the \(y\) direction, and the third in the \(z\) direction.
2D: Vectors 1 and 2 are periodic. The periodic vectors lie in the \(xy\) plane; the third vector points along the \(z\) direction.
3D: Vectors 1, 2, and 3 are periodic. There is no restriction on their directions or magnitudes.

To visualize the lattice even in the absence of any sites, click the Toggle wireframe button from the toolbar. Depending on the dimensionality, it is also possible to change the number of unit cells to be shown along each periodic vector.

As the basis vectors and the dimensionality are modified, the Brillouin zone is updated. The Brillouin zone plot shows the edges and the high-symmetry points of the zone that are used for the band calculation:

0D: the Brillouin zone contains a single Γ point
1D: the Brillouin zone contains three points (Γ and two points at either end of the zone)
2D: the high-symmetry points are the Γ point, vertices, and edge midpoints
3D: the high-symmetry points are the Γ point, vertices, edge midpoints, and face centers

Tip

After defining a new unit cell, click the Reduce Unit Cell button. This step ensures that the basis is "most orthogonal", making sure that the algorithm for determining the Brillouin zone functions correctly. For example, if one were to pick \(\mathbf{v}_1\) = (1,0,0) and \(\mathbf{v}_2\) = (1,1,0), the reduction would set \(\mathbf{v}_2\) = (0,1,0).

If the volume of the unit cell is zero, the Brillouin zone is not shown.

Sites¶

Sites can be added to a selected unit cell by clicking the "+" icon next to its name in the project layout view. The sites can be renamed the same was as the unit cells. In addition to changing a site's name, it is also possible to modify its radius and color. These two properties are entirely cosmetic and their effect can be seen in the system view.

The position of each site inside the unit cell is given by the fractional coordinates \(c_1\), \(c_2\), and \(c_3\), ranging from 0 to 1. As with the color and radius, the position is, strictly speaking, also a cosmetic feature since the coupling between different orbitals is determined by hopping elements, not the physical distance.

States¶

States are added to sites by clicking the "+" icon next to the selected site's name. The states can also be renamed, just as unit cells and sites.

Note

States don’t have an explicit energy parameter. Instead, the diagonal elements of the Hamiltonian (i.e., state energies) are defined as hopping terms from a state to itself. Therefore, the state energies are added when constructing the hopping matrix.

Tip

States, Sites, and Unit Cells can be deleted by either clicking the trash bin icon when the item or selected, or by pressing Backspace/Del keys.

Hoppings¶

After assembling the sites and states of the system, it is possible to introduce couplings between the states using the Hopping tab of the central panel.

Coupling Matrix¶

The Hamiltonian of the system with \(s\) states can be represented by a \(s \times s\) matrix. To build the Hamiltonian, TiBi provides a \(s\times s\) button matrix, where each button contains the couplings between pairs of states. If the button is white, the two states do not couple directly. Pink color indicates that the states are coupled, but the couping is not Hermitian (i.e., the transpose partner of the button does not have the appropriate couplings). Finally, if the button is blue, the coupling is Hermitian. In this case, the transpose partner of the button will also be blue.

Hopping Table¶

Clicking on a button in the coupling matrix brings up a table with 5 columns. Each table row corresponds to a single coupling between two states. The first three columns give the separation between the unit cells containing the two states in terms of the three basis vectors. The remaining two columns are the real and imaginary parts of the hopping term. The suggested workflow for adding the hoppings is as follows:

After bringing up the table, click on the New Hopping button once or more to add hoppings between the selected states.
Fill out the table with the appropriate values.
Click the Save Hoppings button to update the coupling matrix.
The newly-added hoppings will likely be non-Hermitian, indicated by the pink color of the hopping button. To make the Hamiltonian Hermitian, either right click on the hopping button currently being edited and select Set transpose element or click on the transpose button and select Get transpose element.

Tip

Clicking on the hopping buttons shows the existing hoppings in the 3D system plot making it easier to spot mistakes in the displacement columns of the hopping table.

Note

There are two details one should be aware of:

If multiple hopping elements in the same table have the same displacement triplet, clicking Save will collect them into a single element.
If a non-periodic vector has a non-zero entry in the displacement column, clicking Save will set it to zero.