Tools for defect calculations

This section gives an (incomplete) overview of features in ASE that help in the preparation and analysis of supercell calculations as most commonly employed in the computation of defect properties.

Supercell creation

Background

Defect properties are most commonly investigated in the so-called dilute limit, i.e. under conditions, in which defect-defect interactions are negligible. While alternative approaches in particular embedding techniques exist, the most common approach is to use supercells. To this end, one creates a supercell by a suitable (see below) repetition of the primitive unit cell, after which a defect, e.g., a vacancy or an impurity atom, is inserted. This procedure can be schematically depicted as follows:

The calculation thus corresponds to a periodic arrangement of defects. Accordingly, care must be taken to keep the interactions between defects as small as possible, which generally calls for large supercells. It is furthermore indicated to maximize the defect-defect separation in all directions, which is in principle achieved if the supercell used has a suitable shape. Consider for illustration the following three 2D lattices with identical unit cell area but different lattice symmetry:

In the case of the square lattice, each defect has \(Z_1=4\) nearest neighbors at a distance of \(r_1=a_0\), where \(a_0=\sqrt{A}\) with \(A\) being the unit cell area. By comparison in a rectangular lattice with an aspect ratio of 2:1, the defects are much closer to each other with \(r_1 = 0.5 a_0\) and \(Z_1=2\). The largest defect-defect distance (at constant unit cell area) is obtained for the hexagonal lattice, which also correponds to the most closely packed 2D arrangement. Here, one obtains \(r_1=\sqrt{2}/\sqrt[4]{3}=1.075 a_0\) and \(Z_1=6\). For defect calculation supercells corresponding to hexagonal or square lattices have thus clear advantages. This argument can be extended to 3D: Square lattices in 2D correspond to cubic lattices (supercells) in 3D with \(r_1=a_0\) and \(Z_1=6\). The 3D analogue of the hexagonal 2D lattice are hexagonal and cubic close packed structures, both of which yield \(r_1 = \sqrt{3}/2 a_0\) and \(Z_1=12\).

It is straightforward to construct cubic or face-centered cubic (fcc, cubic closed packed) supercells for cubic materials (including e.g, diamond and zincblende) by using simple repetitions of the conventional or primitive unit cells. For countless materials of lower symmetry the choice of a supercell is, however not necessarily so simple. The algorithm below represents a general solution to this issue.

In the case of semiconductors and insulators with small dielectric constants, defect-defect interactions are particularly pronounced due to the weak screening of long-ranged electrostatic interactions. While various correction schemes have been proposed, the most reliable approach is still finite-size extrapolation using supercells of different size. In this case care must be taken to use a sequence of self-similar supercells in order for the extrapolation to be meaningful. To motivate this statement consider that the leading (monopole-monopole) term \(E_{mp}\), which scales with \(1/r\) and is proportional to the (ionic) dielectric constant \(\epsilon_0\). The \(E_{mp}\) term is geometry dependent and in the case of simple lattices the dependence is easily expressed by the Madelung constant. The geometry dependence implies that different (super)cell shapes fall on different lines when plotting e.g., the formation energy as a function of \(N^{-1/3}\) (equivalent to an effective inverse cell size, \(L^{-1} \propto N^{-1/3}\). For extrapolation one should therefore only use geometrically equivalent cells or at least cells that are as self-similar to each other as possibly, see Fig. 10 in [Erhart] for a very clear example. In this context there is therefore also a particular need for supercells of a particular shape.

Algorithm for finding optimal supercell shapes

The above considerations illustrate the need for a more systematic approach to supercell construction. A simple scheme to construct “optimal” supercells is described in [Erhart]. Optimality here implies that one identifies the supercell that for a given size (number of atoms) most closely approximates the desired shape, most commonly a simple cubic or fcc metric (see above). This approach ensures that the defect separation is large and that the electrostatic interactions exhibit a systematic scaling.

The ideal cubic cell metric for a given volume \(\Omega\) is simply given by \(\Omega^{1/3} \mathbf{I}\), which in general does not satisfy the crystallographic boundary conditions. The \(l_2\)-norm provides a convenient measure of the deviation of any other cell metric from a cubic shape. The optimality measure can thus be defined as

\[\Delta_\text{sc}(\mathbf{h}) = ||\mathbf{h} - \Omega^{1/3} \mathbf{1}||_2,\]

Any cell metric that is compatible with the crystal symmetry can be written in the form

\[\mathbf{h} = \mathbf{P} \mathbf{h}_p\]

where \(\mathbf{P} \in \mathbb{Z}^{3\times3}\) and \(\mathbf{h}_p\) is the primitive cell metric. This approach can be readily generalized to arbitrary target cell metrics. In order to obtain a measure that is size-independent it is furthermore convenient to introduce a normalization, which leads to the expression implemented here, namely

\[\bar{\Delta}(\mathbf{Ph}_p) = ||Q\mathbf{Ph}_p - \mathbf{h}_\text{target}||_2,\]

where \(Q = \left(\det\mathbf{h}_\text{target} \big/ \det\mathbf{h}_p\right)^{1/3}\) is a normalization factor. The matrix \(\mathbf{P}_\text{opt}\) that yields the optimal cell shape for a given cell size can then be obtained by

\[\mathbf{P}_\text{opt} = \underset{\mathbf{P}}{\operatorname{argmin}} \left\{ \bar\Delta\left(\mathbf{Ph}_p\right) | \det\mathbf{P} = N_{uc}\right\},\]

where \(N_{uc}\) defines the size of the supercell in terms of the number of primitive unit cells.

Implementation of algorithm

For illustration consider the following example. First we set up a primitive face-centered cubic (fcc) unit cell, after which we call find_optimal_cell_shape() to obtain a \(\mathbf{P}\) matrix that will enable us to generate a supercell with 32 atoms that is as close as possible to a simple cubic shape:

from ase.build import bulk
from ase.build import find_optimal_cell_shape, get_deviation_from_optimal_cell_shape
import numpy as np
conf = bulk('Au')
P1 = find_optimal_cell_shape(conf.cell, 32, 'sc')

This yields

\[\begin{split}\mathbf{P}_1 = \left(\begin{array}{rrr} -2 & 2 & 2 \\ 2 & -2 & 2 \\ 2 & 2 & -2 \end{array}\right) \quad \mathbf{h}_1 = \left(\begin{array}{ccc} 2 a_0 & 0 & 0 \\ 0 & 2 a_0 & 0 \\ 0 & 0 & 2 a_0 \end{array}\right),\end{split}\]

where \(a_0\) =4.05 Å is the lattice constant. This is indeed the expected outcome as it corresponds to a \(2\times2\times2\) repetition of the conventional (4-atom) unit cell. On the other hand repeating this exercise with:

P2 = find_optimal_cell_shape(conf.cell, 495, 'sc')

yields a less obvious result, namely

\[\begin{split}\mathbf{P}_2 = \left(\begin{array}{rrr} -5 & 5 & 5 \\ 5 & -4 & 5 \\ 5 & 5 & -4 \end{array}\right) \quad \mathbf{h}_2 = a_0 \left(\begin{array}{ccc} 5 & 0 & 0 \\ 0.5 & 5 & 0.5 \\ 0.5 & 0.5 & 5 \end{array}\right),\end{split}\]

which indeed corresponds to a reasonably cubic cell shape. One can also obtain the optimality measure \(\bar{\Delta}\) by executing:

dev1 = get_deviation_from_optimal_cell_shape(np.dot(P1, conf.cell)
dev2 = get_deviation_from_optimal_cell_shape(np.dot(P2, conf.cell)

which yields \(\bar{\Delta}(\mathbf{P}_1)=0\) and \(\bar{\Delta}(\mathbf{P}_2)=0.201\).

Since this procedure requires only knowledge of the cell metric (and not the atomic positions) for standard metrics, e.g., fcc, bcc, and simple cubic one can generate series of shapes that are usable for all structures with the respective metric. For example the \(\mathbf{P}_\text{opt}\) matrices that optimize the shape of a supercell build using a primitive FCC cell are directly applicable to diamond and zincblende lattices.

For convenience the \(\mathbf{P}_\text{opt}\) matrices for the aforementioned lattices have already been generated for \(N_{uc}\leq2000\) and are provided here as dictionaries in json format.

• Transformation of face-centered cubic metric to simple cubic-like shapes: Popt-fcc2sc.json

• Transformation of face-centered cubic metric to face-centered cubic-like shapes: Popt-fcc2fcc.json

• Transformation of body-centered cubic metric to simple cubic-like shapes: Popt-bcc2sc.json

• Transformation of body-centered cubic metric to face-centered cubic-like shapes: Popt-bcc2fcc.json

• Transformation of simple cubic metric to simple cubic-like shapes: Popt-sc2sc.json

• Transformation of simple cubic metric to face-centered cubic-like shapes: Popt-sc2fcc.json

The thus obtained \(\bar{\Delta}\) values are shown as a function of the number of unit cells \(N_{uc}\) in the panel below, which demonstrates that this approach provides access to a large number of supercells with e.g., simple cubic or face-centered cubic shapes that span the range between the “exact” solutions, for which \(\bar{\Delta}=0\). The algorithm is, however, most useful for non-cubic cell shapes, for which finding several reasonably sized cell shapes is more challenging as illustrated for a hexagonal material (LaBr₃) in [Erhart].

Generation of supercell

Once the transformation matrix \(\mathbf{P}\) it is straightforward to generate the actual supercell using e.g., the cut() function. A convenient interface is provided by the make_supercell() function, which is invoked as follows:

from ase.build import bulk
from ase.build import find_optimal_cell_shape
from ase.build import make_supercell
conf = bulk('Au')
P = find_optimal_cell_shape(conf.cell, 495, 'sc')
supercell = make_supercell(conf, P)

[Erhart] (1,2,3)

P. Erhart, B. Sadigh, A. Schleife, and D. Åberg. First-principles study of codoping in lanthanum bromide, Phys. Rev. B, Vol 91, 165206 (2012), doi: 10.1103/PhysRevB.91.165206; Appendix C