Topological Transformations in Hyperuniform Pentagonal 2D Materials Induced by Stone-Wales Defects

We now describe our numerical procedure for introducing SW defects in an initially perfect P5 tiling and use PtP₂ as an example for the corresponding 2D materials. Note that the general methodology discussed here are readily applicable for other 2D pentagonal AB₂ materials. As shown in Fig. 1c, we apply the flipping operation to the P-P bonds, i.e., by fixing the center of bond and rotation it by $\pi/2$ around the center, to create SW defects. We define the SW defect concentration $p$ as

where $N_{tot}$ is the total number of available P-P bonds for rotation and $N_{rot}$ is the number of rotated bonds. Once the topological defects are introduced, we use molecular static simulations to relax the system to a minimal-energy state before collecting structural statistics and perform additional simulations for physical properties. The P-P bonds in the original P5 tiling (and subsequent disordered tilings) assume two perpendicular orientations, which are distinguished with red and blue colors respectively (see Figs. 2a and 2b).

P5-C46 Transformation. We first restrict the rotation operation to P-P bonds with the same orientation. A convenient way to understand this process is to consider that a prescribed fraction $p$ of the fundamental cells containing 4 pentagons of the P5 tiling are randomly selected, and the P-P bonds in the middle of the cells are flipped to create SW defects. This process is repeated for all $p\in[0,1]$, and the resulting material configurations are shown in Fig. 2a. Fig. 2c shows the fractions of different types of polygons in the material resulted from the transformation as a function of $p$. Since one can only have three types of polygons, i.e., rhombus, pentagon and hexagon for any $p$ value, the fractions of the three types add up to unity. It can be clearly seen the fractions of rhombi $\phi_{4}$ and hexagons $\phi_{6}$ increase linearly with $p$, i.e., $\phi_{4}=\phi_{6}=0.5p$; while the fraction of pentagons $\phi_{5}$ decreases linearly with $p$, i.e., $\phi_{5}=1-(\phi_{4}+\phi_{6})=1-p$. In addition, the rhombus-hexagon network percolates at p = 0.5 (see Fig. 2a). This can be easily obtained by mapping the percolation of fundamental cells containing rhombi and hexagons to the site percolation process on triangular lattice.

The P5 and C46 crystals at the two extreme concentrations $p=0$ and $p=1$ are periodic structures, and are also hyperuniform Torquato and Stillinger (2003); Torquato (2018), i.e., the (normalized) infinite-wavelength density fluctuations in these systems are completely suppressed. This is manifested in their structure factor $S(k)$ as $\lim_{k\rightarrow 0}S(k)=0$, as shown in Fig. 3(e). Here $S(k)$ is defined as

where $\tilde{h}(k)$ is the Fourier transform of the total correlation function $h(r)=g_{2}(r)-1$, and $g_{2}(r)$ is the pair correlation function, $\rho$ is the number density of the system. Note that this definition implies that the forward scattering contribution to the diffraction pattern is omitted Torquato and Stillinger (2003); Torquato (2018). Moreover, the P5 and C46 crystals satisfy a stronger condition called stealthy hyperuniformity Torquato and Stillinger (2003); Torquato et al. (2015); Torquato (2018), for which $S(k)=0$ for a range of wavenumbers $0\leq k<K$, where $K$ in the crystalline cases is the location of the first Bragg peak.

The intermediate configurations are random tilings consisting of two types of building blocks of varying concentrations, with each type corresponding to a particular orientation of the P-P bonds in the middle of the fundamental cell. They are neither crystals or quasicrystals, but since the SW defects do not induce any distortion of the fundamental cell, and the number density is strictly preserved on the fundamental cell level, we expect that hyperuniformity in the P5 or C46 crystal is preserved in these random tilings. This is verified by the computed $S(k)$ of these structures as $S(k)$ continues to decrease to extremely small values as $k$ decreases, as shown in Fig. 3(e). However, we note that the stealthiness of the P5 or C46 crystal is lost in these intermediates structures, i.e., $S(k)$ is not strictly zero at small, yet finite wavenumbers. Nonetheless, the topological transformation in pentagonal 2D materials is a novel class of hyperuniformity-preserving operations that have not been identified before Chen et al. (2020); Kim and Torquato (2018). The locations of the Bragg peaks also shift as $p$ varies, reflecting the increasing fractions of rhombi and hexagons in the system. Interestingly, the intermediate structures are tiling variants of the P5 or C46 crystal within the 2D plane, and in this regard can be viewed as 2D analogs of disordered Barlow packings Middlemas et al. (2019), which are stacking variants of fcc or hcp crystals in three dimensions.

We note that transformation is a phenomenological analogy of anti-ferromagnetic to ferromagnetic transition in the two-state Ising system, with the two distinct orientations of P-P bonds respectively corresponding to the two spin polarization directions. The global nematic order parameter $S$ associated with the P-P bonds, which is the analog of the net magnetization for Ising systems, is simply given by

where $n_{\parallel}$ and $n_{\perp}$ are the number of P-P bonds along two orthogonal directions, and $N_{b}$ is the total number of P-P bonds in the system. It is clear that $S(p)=p$ for this transformation, which is also verified from the numerical data shown in Fig. 3c. Fig. 3a shows the P-P bond orientation correlation function, i.e.,

where ${\bf s}({\bf x})$ is the unitary direction vector of a P-P bond located at ${\bf x}$. The strong oscillations in $C(r)$ with $p=0$ indicates the periodic alternation of bond orientations in the P5 structure, while $C(r)=1$ with $p=1$ indicates perfect alignment of all bonds in the C46 structure. The gradual building up of long-range orientation correlation as $p$ increases can be clearly seen from the evolution of $C(r)$.

P5-R456 Transformation. We now allow all of the P-P bonds to rotate, but still require that each bond can only be rotated once. The fractions of different polygons as a function of $p$ are shown in Fig. 2d and the representative configurations associated with different $p$ are shown in Fig. 2b. Similar to the P5-C46 transformation, the initial increase of $p$ from 0 leads to an rapid increase of $\phi_{4}$ and $\phi_{6}$ and decrease of $\phi_{5}$. In the low-$p$ limit, the defects are well separated and independent from one another, and thus, the same linear relations between $\phi_{n}$ ($n=4,5,6$) and $p$ as in the P5-R456 transformation also hold. As $p$ further increases, the neighboring perpendicular bond of a rotated bond might also be rotated, which transforms a rhombus/hexagon back to a pentagon. This leads to slower growth of $\phi_{4}$ and $\phi_{6}$ (and equivalently, slower decay of $\phi_{5}$). Therefore, the P5-R456 transform can be considered to be mainly controlled by the orientational decorrelation of the neighboring P-P bonds and the maximal $\phi_{4}$ and $\phi_{6}$ are achieved when the “most decorrelated” state is reached at $p_{c}=0.5$, i.e., 50% of the bonds are randomly flipped (see Fig. 2b). This is the analog of the anti-ferromagnetic to paramagnetic transition in Ising systems. As $p$ further increases beyond 0.5, more rhombi and hexagons are converted back to pentagons, leading to deceasing $\phi_{4}$ and $\phi_{6}$. Finally, at $p=1$, all of the P-P bonds have been rotated exactly once and the tiling transforms back to P5, albeit the entire system is rotated by 90 degree with respect to the original P5 tiling.

The aforementioned process is also manifested in $S$ and $C(r)$, respectively shown in Figs. 3c and 3b. Since random bond flipping on average preserves the number of bonds with either orientation, $S(p)$ remains close to 0 for all $p\in(0,1)$, with largest fluctuations occurring around $p=0.5$ for finite systems. In the infinite-system limit, $S(p)$ converges to exactly 0 for all $p$ as statistical fluctuations vanish. On the other hand, the strong oscillations in $C(r)$ associated with the P5 structure are clearly diminished as $p$ increases due to the induced disorder or decorrelation of P-P bond orientations. The most random state characterized by an almost flat $C(r)\approx 0.5$ (i.e., due to equal amount of parallel and perpendicular bond pairs at all separation distances) is achieved at $p=0.5$. Further increasing $p$ gradually removes disorder, leading to perfect P5 at $p=1$.

Similar to the P5-C46 pathway, the intermediate structures in the P5-R456 pathway are neither crystals nor quasicrystals, and yet they preserve hyperuniformity of the P5 crystal. Specifically, their corresponding $S(k)$ approaches zero as $k$ approaches 0, as shown in Fig. 3(f). The structure factor also shows diminished and shifted peaks as $p$ increase from 0 to 0.5, a common signature of increasing disorder. Moreover, these intermediate structures can be viewed as random tiling variants of P5 (or C46) crystal consisting of finite types of building blocks, with each type corresponding to a particular combination of orientations of the P-P bonds in the fundamental cell.

Figure 3(d) shows the energies $\Delta E$ required to generate SW defects of different $p$ values for the two transition pathways. We can see that, for the P5-R456 pathway, the required energy gradually increases with increasing $p$, which reach the maximum near $p=0.5$ and decreases with further increase in $p$. On the other hand, $\Delta E$ associated with the P5-C46 pathway increases almost linearly with $p\in[0,1]$. These trends of $\Delta E$ are completely consistent with the observed structural changes during the two types of topological transformations. We note that the largest $\Delta E$ is much less than 1.0 eV per atom for the full spectrum of structures obtainable from the transformation.

Fig. 4a and 4b compare the band structures of the 2D materials derived from the P5 and C46 structures. As reported in Ref. Liu and Zhuang (2018), the P5 structure is semiconducting with a direct band gap of 0.52 eV. By contrast, the C46 structure is metallic with Dirac-cone-like dispersion in the conduction bands near the X point. This dispersion is caused by the presence of the hexagonal six-membered rings in the C46 structure. The metallic nature is also shared by the intermediate structures with p values lying between 0 and 1. For example, Fig. 4c shows the density of states of the two PtP₂ structures with the same p (p = 0.5) from the two different transformation pathways. Both density of states show sizable magnitudes at the Fermi level and their overall shapes are similar in the entire energy window, consistent with their close energies at $p$ = 0.5 as shown in Fig. 4d.

Due to the metallic properties of the C46 structure, we envision a potential application of forming P5-C46 heterostructure as a Schottky barrier. Fig. 4d illustrates the average local potentials a carrier experiences in the out-of-plane direction of single-layer P5 and C46 structures. We can see that the quantum wells are nearly identical in spite of the two different atomic structures. We extract the conduction band minimum (CBM; -5.37 eV) and valence band maximum (VBM; -5.89 eV) energy levels of the P5 structure and the Fermi level (i.e. -5.46 eV or 5.46 eV as the work function) of the C46 structure by aligning these three energy levels with the vacuum level set to zero for the two structures. We find that the Fermi level of the C46 structure lies between the CBM and VBM of the P5 structure with the distances from the CBM and VBM being 0.09 and 0.43 eV, respectively.

In summary, we have studied the effects of the Stone-Wales defects on the structure and electronic properties of a novel class of single-layer $AB_{2}$ pyrite materials derived from the pentagonal Cairo tiling. Unlike the SW defects in hexagonal 2D materials which cause distortions, the defects in pentagonal 2D materials preserve the shape and symmetry of the fundamental cell of P5 tiling and are associated with a minimal energy cost. We discovered two distinct transformation pathways encompassing a rich spectrum of disordered 2D materials composed of a mixture of rhombus, pentagon and hexagon rings, including the ordered P5 structure and C46 structure as the two extremes. The intermediate structures along the two pathways are neither crystals nor quasicrystals, and yet they preserve hyperuniformity of the P5 or C46 tiling. These random tilings can be viewed as 2D analogs of disordered Barlow packings in three dimensions.

The resulting 2D materials possess metallic properties, making them promising candidates for forming Schottky barriers with the semiconducting P5 material. The semiconductor-to-metal transition in these pentagonal 2D materials as disorder is introduced into the system are similar to the insulator-to-metal transition observed in disordered hyperuniform silica Zheng et al. (2020), and suggest a potential broad linkage between disordered hyperuniformity (DHU) and enhanced electron transport in low-dimensional quantum materials, which is complementary to the conventional wisdom of landmark “Anderson localization” Anderson (1958) that disorder generally diminishes transport. In future works, we will continue to explore this novel concept of disordered hyperuniform quantum materials (DHQM), i.e., disordered hyperuniform atomic-scale low-dimensional materials with nontrivial quantum characteristics Zheng et al. (2020); Chen et al. (2020)), which could potentially have broad implications in many fields of condensed-matter physics and materials science.