Computational exfoliation of atomically thin one-dimensional materials with application to Majorana bound states

We introduce a computational database with calculated structural, thermodynamic, electronic, magnetic, and optical properties of 820 one-dimensional materials. The materials are systematically selected and exfoliated from experimental databases of crystal structures based on a dimensionality scoring parameter. The database is furthermore expanded by chemical element substitution in the materials. The materials are investigated in both their bulk form and as isolated one-dimensional components. We discuss the methodology behind the database, give an overview of some of the calculated properties, and look at patterns and correlations in the data. The database is furthermore applied in computational screening to identify materials, which could exhibit Majorana bound states.

Figure 1

The $(k_1,k_2)$ values for dimension one for all the materials in ICSD and COD. A clear cluster representing the one-dimensional materials is seen in the upper figure. In the lower figure, the line $s_{1\mathrm{D}}=0.5$ is shown and the materials are colored according to their dimensionalities, where the dimension is determined as the one with the highest scoring parameter. One-dimensional materials are green, while zero-, two-, and three-dimensional materials are shown in blue, yellow, and red, respectively. (The figure is from Ref. [30].)

Figure 2

A simple diagram of the workflow that is used to calculate the structure and properties of 1D materials. This is the workflow used to generate the data in our database. The first part of the workflow considers stability while the other part of our workflow evaluates different properties. A more detailed description of the steps in the workflow can be found in the text.

Figure 3

The ratio between the volumes calculated with PBE-D3 and the experimental ones from ICSD and COD.

Figure 4

The scoring parameter calculated for the bulk materials in the core of the database relaxed with PBE-D3 vs the scoring parameter for the experimentally determined bulk structures.

Figure 5

The structure distance matrix of all materials having three or fewer different elements in the unit cell. The structures have been ordered using the rearrangement clustering method [47] so that the dark squares on the diagonal reveal clusters of similar materials.

Figure 6

A dendrogram of the clusters using single linkage. The inset shows a histogram of all distances in the structure distance matrix. The distances can clearly be separated in two groups with distances smaller or larger than 0.75 $\text{Å}$. In the dendrogram the clusters obtained by using a cutoff distance of 0.75 $\text{Å}$ are shown in different colors.

Figure 7

Distribution of the energies above the convex hull, $\mathrm{\Delta }{H}_{\text{hull}}$ calculated with PBE-D3 for the bulk materials in the core and the shell of the database.

Figure 8

Distribution of above-the-hull energies for the bulk materials in the shell as calculated with PBE-D3. Here, the set of reference materials for the convex hull construction consists of the unary and binary materials from OQMD and the materials in the core of the database.

Figure 9

The distribution of separation energies defined as the difference between the relaxed 3D bulk material and the 1D system per atom calculated using the PBE-D3 functional.

Figure 10

The figure shows calculated exfoliation energies $E_{\text{xf}}$ vs the calculated scoring parameter $s_1$ for all materials in the core of the database. The red points and vertical bars show the average values and standard deviations for points in the intervals [0.5, 0.55], [0.55, 0.60], etc. for $s_1$. The two different measures of exfoliability are clearly correlated for large values of $s_1$.

Figure 11

The energy above convex hull distribution for the extracted 1D materials.

Figure 12

A convex hull plot of a 1D material compound, with the bulk reference structures in green, and the 1D component in orange.

Figure 13

The distribution of energy gaps for all the core and shell 1D materials as calculated with PBE.

Figure 14

Comparison between the PBE calculated band gaps for the bulk systems and the one-dimensional components.

Figure 15

Band structure and electronic density of states for $\mathrm{ISSb}$ for both the bulk system (left figures) and the one-dimensional components (right figures). $\mathrm{ISSb}$ belongs to group 9 in Table 3.

Figure 16

A zoom of the band structure of the one-dimensional component of $\mathrm{HgO}$. The upper figure shows the conduction band and the lower figure the valence band. The magnitude of the $x$ component of the spin is represented by the colors of the circles. Small Rashba splittings are observed in the bands.

Figure 17

A spin-orbit split conduction band in the absence (left) and presence (right) of a $B$ field applied perpendicular to the spin direction. The $B$ field isolates the lower spin band leading to an effective spinless band. The Rashba wave vector and Rasba energies are indicated in the left panel. (Figure adapted from Fig. 5 in Ref. [11]).

Figure 18

Crystal structures of the ITe and SnClI wires.

Figure 19

The band structure around the valence band maximum for the six 1D structures of the database with the largest Rashba splittings (see also Table 5). The color of the bands represent the projection of the spin perpendicular to the wire axis.