Dimensionality Analysis

MatID can be used to find out the dimensionality of an atomistic geometry. It should be noted that MatID does not consider the shape or size of a structure when assigning dimensionality, but instead looks at the number of dimensions in which the structure is connected to itself through periodic boundary conditions. This definition is often relevant for atomistic simulations with periodic boundary conditions, but may not be useful in other contexts where e.g. the shape is the deciding factor.

To determine the dimensionality of a system, MatID uses a modified version of the topological scaling algorithm (TSA)¹. The algorithm is based on analyzing the size scaling of atomic clusters when going from the original system to a bigger supercell of the same system. With TSA, the dimensionality $D$ is given by

D=\begin{cases} n_\text{pbc}-\log_n (N_{n}) \text{, when}~n_\text{pbc} \neq 0 \\ 0\text{, when}~n_\text{pbc} = 0 \end{cases}

where $N_n$ is the number of clusters in a supercell that is repeated $n$ times in each periodic direction and $n_\mathrm{pbc}$ is the number of periodic dimensions. For the clustering we use the Density-Based Spatial Clustering of Applications with Noise (DBSCAN)² data clustering algorithm. The advantage of this algorithm is that it does not require an initial guess for the number of clusters and it can find arbitrarily shaped clusters. The clustering requires that we define a metric for the distance between atoms. We use the following metric:

d_{ij} = \lvert \vec{R}_i - \vec{R}_j \rvert^{\text{MIC}} - r_i - r_j

where $\vec{R}_i$ and $\vec{R}_j$ are the cartesian positions of atom $i$ and $j$ , respectively, and $r_i$ and $r_j$ are their radii. The radii definition can be changed and defaults to covalent radii³. It is important to notice that in this metric the distances always follow the minimum image convention (MIC), i.e. the distance is calculated between two closest periodic neighbours. By using the distance to the closest periodic neighbour we obtain the correct clusters regardless of what shape of cell is used in the original simulation.

The clustering uses two parameters: the minimum cluster size $n_\mathrm{min}$ and the neighbourhood radius $\epsilon$ . We set $n_\mathrm{min}$ to 1 to allow clusters consisting of even single atoms and $\epsilon$ defaults to 3.5 Å. At present, a system, in which there is more than one cluster in the original non-repeated system ( $N_1 \gt 1$ ), is classified as unknown. Such a case corresponds to systems with multiple components that are spatially separated, such as a molecule far above a surface, low density gases, widely spaced clusters in vacuum, etc.

The following code illustrates how dimensionality detection can be performed for any atomistic structure with MatID.

from matid.geometry import get_dimensionality

from ase.build import molecule
from ase.build import nanotube
from ase.build import mx2
from ase.build import bulk

# Here we create one example of each dimensionality class
zero_d = molecule("H2O", vacuum=5)
one_d = nanotube(6, 0, length=4, vacuum=5)
two_d = mx2(vacuum=5)
three_d = bulk("NaCl", "rocksalt", a=5.64)

# In order to make the dimensionality detection interesting, we add periodic
# boundary conditions. This is more realistic as not that many electronic
# structure codes support anything else than full periodic boundary conditions,
# and thus the dimensionality information is typically not available.
zero_d.set_pbc(True)
one_d.set_pbc(True)
two_d.set_pbc(True)
three_d.set_pbc(True)

# Here we perform the dimensionality detection with clustering threshold epsilon
epsilon = 3.5
dim0 = get_dimensionality(zero_d, epsilon)
dim1 = get_dimensionality(one_d, epsilon)
dim2 = get_dimensionality(two_d, epsilon)
dim3 = get_dimensionality(three_d, epsilon)

# Printing out the results
print(dim0)
print(dim1)
print(dim2)
print(dim3)

If you wish to get the dimensionality of a Cluster object, then you can directly use the Cluster.get_dimensionality() method, which is essentially a shortcut to the matid.geometry.get_dimensionality() function.

Ashton, M., Paul, J., Sinnott, S. B. & Hennig, R. G. Topology-scaling identification of layered solids and stable exfoliated 2d materials. Phys. Rev. Lett. 118, 106101 (2017) ↩
Ester, M., Kriegel, H.-P., Sander, J. & Xu, X. A density-based algorithm for discovering clusters in large spatial databases with noise. KDD’96 Proceedings of the Second International Conference on Knowledge Discovery and Data Mining 226–231 (1996). ↩
Cordero, B. et al. Covalent radii revisited. Dalton Trans. 2832–2838 (2008) ↩

Dimensionality Analysis

Footnotes​

Footnotes