1. Unit cells

The basic building block of the atomic structures modeled in the atomistic solvers of QTCAD is a crystallographic unit cell. It is assumed that all atoms in an atomic structure (excluding surface atoms) are tetrahedrally bonded. The diamond and zincblende crystal structures are two of the most common tetrahedrally-bonded crystal structures. In particular, silicon and germanium, two materials of particular interest in the context of atomistic simulations of quantum dots, have a diamond crystal structure. More generally, ignoring strain, silicon–germanium alloys have a zincblende crystal structure.

QTCAD provides a class to model the unit cells of diamond and zincblende crystal structures: UnitCellZincblende. There are two standard unit cells used to describe these crystal structures [AM76]:

• the Wigner–Seitz unit cell, which is rhombohedral in shape and contains two atoms;

• the conventional unit cell, which is cubic in shape and contains eight atoms.

Both of these unit cells are stored in UnitCellZincblende objects. We characterize such an object with the following parameters:

• its lattice constant \(a\), namely the side length of its conventional unit cell;

• the chemical species of the two atoms that are contained in the Wigner–Seitz unit cell.

In the following, we define the Wigner–Seitz and conventional unit cells, and we describe the corresponding Brillouin zone.

Wigner–Seitz unit cell

In the Wigner–Seitz unit cell, the primitive lattice vectors are given by:

(1.31)\[ \begin{align}\begin{aligned}\mathbf{W}_1 &= \frac{a}{2} \left(0, 1, 1\right)\,,\\\mathbf{W}_2 &= \frac{a}{2} \left(1, 0, 1\right)\,,\\\mathbf{W}_3 &= \frac{a}{2} \left(1, 1, 0\right)\,.\end{aligned}\end{align} \]

The two atoms of the Wigner–Seitz unit cell are located at the following positions:

(1.32)\[ \begin{align}\begin{aligned}\mathbf{w}_1 &= \left(0, 0, 0\right)\,,\\\mathbf{w}_2 &= \frac{a}{4} \left(1, 1, 1\right)\,.\end{aligned}\end{align} \]

The Wigner–Seitz unit cell may be visualized using the visualize method, and is shown below.

Fig. 1.1 Wigner–Seitz unit cell of an ordered \(\mathrm{Si}_{0.5}\mathrm{Ge}_{0.5}\) alloy produced by QTCAD.

Conventional unit cell

In the conventional unit cell, the primitive lattice vectors are given by:

(1.33)\[ \begin{align}\begin{aligned}\mathbf{C}_1 &= a \left(1, 0, 0\right)\,,\\\mathbf{C}_2 &= a \left(0, 1, 0\right)\,,\\\mathbf{C}_3 &= a \left(0, 0, 1\right)\,.\end{aligned}\end{align} \]

The eight atoms of the conventional unit cell are located at the following positions:

(1.34)\[ \begin{align}\begin{aligned}\mathbf{c}_1 &= \left(0, 0, 0\right)\,,\\\mathbf{c}_2 &= \frac{a}{4} \left(0, 2, 2\right)\,,\\\mathbf{c}_3 &= \frac{a}{4} \left(2, 0, 2\right)\,,\\\mathbf{c}_4 &= \frac{a}{4} \left(2, 2, 0\right)\,,\\\mathbf{c}_5 &= \frac{a}{4} \left(1, 1, 1\right)\,,\\\mathbf{c}_6 &= \frac{a}{4} \left(1, 3, 3\right)\,,\\\mathbf{c}_7 &= \frac{a}{4} \left(3, 1, 3\right)\,,\\\mathbf{c}_8 &= \frac{a}{4} \left(3, 3, 1\right)\,.\end{aligned}\end{align} \]

Furthermore, the atoms located at positions \(\mathbf{c}_{1,2,3,4}\) (\(\mathbf{c}_{5,6,7,8}\)) are of the same chemical species as the atom located at position \(\mathbf{w}_1\) (\(\mathbf{w}_2\)) in the Wigner–Seitz unit cell.

The conventional unit cell may be visualized using the visualize method with the wigner_seitz argument set to False, and is shown below.

Fig. 1.2 Conventional unit cell of an ordered \(\mathrm{Si}_{0.5}\mathrm{Ge}_{0.5}\) alloy produced by QTCAD.

Brillouin zone

In QTCAD, the high-symmetry points of the Brillouin zone of the zincblende crystal structure are defined as follows:

(1.35)\[ \begin{align}\begin{aligned}\Gamma &= \left(0, 0, 0\right)\,,\\X &= \frac{\pi}{a} \left(2, 0, 0\right)\,,\\W &= \frac{\pi}{a} \left(2, 0, 1\right)\,,\\L &= \frac{\pi}{a} \left(1, 1, 1\right)\,,\\K &= \frac{\pi}{a} \left(\frac{3}{2}, 0, \frac{3}{2}\right)\,,\\U &= \frac{\pi}{a} \left(2, \frac{1}{2}, \frac{1}{2}\right)\,.\end{aligned}\end{align} \]

The coordinates of these high-symmetry points are of particular relevance in the context of bandstructure calculations, wherein electronic bands are computed on lines joining the high-symmetry points. Note that by symmetry of the Brillouin zone, the bands at the points \(K\) and \(U\) are equal to each other. However, the bands in the vicinity of these two points are not equal to each other. Bandstructures may be computed and visualized using the plot_bandstructure method. The bandstructure is computed within the tight-binding model described in Tight-binding Schrödinger solver.