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# Strain and the optoelectronic properties of nonplanar phosphorene monolayers

Edited by Monica Olvera de la Cruz, Northwestern University, Evanston, IL, and approved March 31, 2015 (received for review January 11, 2015)

## Significance

Phosphorene is a new 2D atomic material, and we document a drastic reduction of its electronic gap when under a conical shape. Furthermore, geometry determines the properties of 2D materials, and we introduce discrete differential geometry to study them. This geometry arises from particle/atomic positions; it is not based on a parametric continuum, and it applies across broad disciplinary lines.

## Abstract

Lattice kirigami, ultralight metamaterials, polydisperse aggregates, ceramic nanolattices, and 2D atomic materials share an inherent structural discreteness, and their material properties evolve with their shape. To exemplify the intimate relation among material properties and the local geometry, we explore the properties of phosphorene––a new 2D atomic material––in a conical structure, and document a decrease of the semiconducting gap that is directly linked to its nonplanar shape. This geometrical effect occurs regardless of phosphorene allotrope considered, and it provides a unique optical vehicle to single out local structural defects on this 2D material. We also classify other 2D atomic materials in terms of their crystalline unit cells, and propose means to obtain the local geometry directly from their diverse 2D structures while bypassing common descriptions of shape that are based from a parametric continuum.

Two-dimensional materials (1⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–20) are discrete surfaces that are embedded on a 3D space. Graphene (1, 2) develops an effective Dirac-like dispersion on the sublattice degree of freedom and other 2D atomic materials exhibit remarkable plasmonic, polariton, and spin behaviors too (18⇓–20).

The properties of 2D materials are influenced by their local geometry (12⇓⇓⇓⇓–17, 21⇓⇓⇓⇓⇓⇓⇓–29), making a discussion of the shape of 2D lattices a timely and fundamental endeavor (24, 30⇓–32). A dedicated discussion of the shape of 2D materials is given here within the context of nets. Nets are discrete surfaces made from vertices and edges, with vertices given by particle/atomic positions (31⇓⇓–34). The discrete geometry that originates from these material nets is richer than its smooth counterpart because the net preserves the structural information of the 2D lattice completely, yielding exact descriptions of shape that remain accurate as the lattice is subject to arbitrarily large structural deformations (15, 16), as the particle/atomic lattice becomes the net itself.

A discrete geometry helps address how strain influences chemistry (35), how energy landscapes (36⇓–38) correlate to nonplanar shapes, and it provides the basis for a lattice gauge theory for effective Dirac fermions on deformed graphene (39, 40). In continuing with this program, the optical and electronic properties of phosphorene cones will be linked to their geometry in the present work. At variance with 2D crystalline soft materials that acquire topological defects while conforming to nonplanar shapes (12⇓⇓–15), the materials considered here have strong chemical bonds that inhibit plastic deformations for strain larger than 10% (41).

The study is structured as follows: We build conical structures of black and blue phosphorene, determine their local shape, and link this shape to the magnitude of their semiconducting gap. It is clear that a discrete geometry applies for arbitrary 2D materials.

## Results and Discussion

Phosphorene (8, 9) has many allotropes that are either semiconducting or metallic depending on their 2D atomistic structure (42⇓–44). The most studied phase, black phosphorene (Fig. 1*A*), has a semiconducting gap that is tunable with the number of layers, and by in-plane strain (26, 27, 45, 46). Theoretical studies of defects on planar phosphorene indicate that dislocation lines do not induce localized electronic states (47), and algorithms to tile arbitrary planar 2D phosphorene patterns have been proposed as well (44). Unit cells of planar monolayers of black and blue phosphorene are displayed in Fig. 1.

Four invariants from the metric (*g*) and curvature (*k*) tensors determine the local geometry of a 2D manifold (24, 30):*H* is the mean curvature, and *K* is the Gaussian curvature.

An infinite crystal can be built from these unit cells, and the geometry of such ideal planar structure can be described by *τ* is equal to *g*) and Det(*g*) will be functions of the lattice constant, but here they are normalized with respect to their values in the reference crystalline structures seen in Fig. 1, to enable direct comparisons of the metric among black and blue phosphorene monolayers.

Now, structural defects can induce nonzero curvature and strain (35), and may also be the culprits for the chemical degradation of layered phosphorene. To study the consequences of shape on the optical and electronic properties of phosphorene monolayers, we create finite-size conical black and blue phosphorene monolayers, characterize the atomistic geometry, and investigate the influence of shape on their semiconducting gap. [We have proposed that hexagonal boron nitride could slow the degradation process while allowing for a local characterization of phosphorene allotropes (48), and a study of chemical reactivity of nonplanar phosphorene will be presented elsewhere.]

Phosphorene cones are built from finite disk-like planar structures that have hydrogen-passivated edges (Fig. 2). The black phosphorene cone seen in Fig. 2*A* is created as follows: We remove an angular segment—that subtends a *A* are joined afterward to create a disclination line. Atoms are placed in positions dictated by an initial (analytical) conical structure, and there is a full structural optimization via molecular dynamics at the ab initio level to relieve structural forces throughout (*Methods*).

Blue phosphorene has a (buckled) honeycomb structure reminiscent of graphene, so the conical structure seen to the left of Fig. 2*B* was generated by removing an angular segment subtending a

The subplots displayed to the right in Fig. 2 *A* and *B* show the local discrete geometry at individual atoms (Eq. **1**; see details in *Methods*). For each allotrope, the data are arranged into three rows that indicate the geometry of the planar structure at sublayer

There is strain induced by the dislocation line on the planar black phosphorene structure, as indicated by the color variation on the *H* plot in Fig. 2*A*. The planar blue phosphorene sample does not have any dislocation line, and for that reason the metric shows zero strain [*B*).

The black phosphorene conical structure seen in Fig. 2*A* has the following features: A compression near its apex, as displayed by the white color on the metric invariants; this compression is not radial-symmetric. An asymmetric elongation is visible toward the edges. In addition, there is a radially asymmetric nonzero curvature; the observed asymmetry reflects the presence of the dislocation/disclination axis, which provides the structure with an enhanced structural rigidity. This rigidity is confirmed by the ratio *A*).

The blue phosphorene cone has a more apparent radial symmetry, except for the disclination line that is created by the conical structure, as reflected on the metric invariants and on *H*. There is compression (elongation) at the apex, and elongation (compression) along the disclination line in sublayer

Fig. 2 indicates that the blue phosphorene cone acquires the largest curvatures of these two cones. This is so because the angular segment removed from the planar blue phosphorene sample has a comparatively larger value of *ϕ*. One notes the rather smooth curvature at the apex on the blue phosphorene conical structure after the structural optimization.

The change of *τ* with respect to

The main point from Fig. 2 is the strain induced by curvature. The strain pattern observed in that figure is far more complex than those reported before for planar phosphorene (26, 27, 45, 46), and the discrete geometry captures the strain pattern with the precision given by actual atomic positions. We will describe the tools that lead to this geometry based from atoms later on (*Methods*), but we describe the effect of shape on the material properties of these cones first.

Black and blue phosphorene monolayers are both semiconducting 2D materials with a direct bandgap, and we investigate how this semiconducting gap evolves with their shape. The semiconducting gap *Methods*).

We determine size effects on the semiconducting gap on planar structures first: The magnitude of the gap

The conical structures have fewer atoms than their parent planar structures once the angular segments seen in Fig. 2 *A* and *B* are removed. We learned in the previous paragraph that the electronic gap increases as the number of atoms is decreased on planar structures. Following that argument, one may expect the semiconducting gap for the conical structures to be larger than the one observed on their parent planar structures.

However, instead of increasing with the removal of atoms, the gap decreases dramatically on the conical structures, going down to 0.84 (1.70) eV for the black (blue) phosphorene cone, as indicated by the open red symbols, and the tilted arrows in Fig. 3: Shape influences phosphorene’s material properties. A practical consequence of this result is that optical probes could provide a measure of the local shape of phosphorene. The relation between the semiconducting gap and a nonplanar shape has been observed in transition metal dichalcogenide monolayers recently (25), and we establish here that structural defects on semiconducting 2D materials create a similar effect. Let us next address the reason for such reduction of the semiconducting gap.

The semiconducting gap is a global material property. On finite samples it is possible to project the electronic density onto individual atoms to learn about the spatial localization of the electronic states below and above the Fermi level, whose energy difference is equal to the electronic gap. Those states are known as the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), and they were spin-degenerate in all of the samples we studied. Additionally, the *n*th electronic orbital below (above) the HOMO (LUMO) is labeled HOMO

The HOMO and LUMO states cover the entire planar black and blue phosphorene structures (Fig. 4) and are thus delocalized. On the other hand, the conical samples have orbitals below and above the Fermi energy that display a certain amount of localization.

Such effect is most visible for the HOMO and HOMO*B*. Thus, unlike a dislocation line (47) the pentagon defect that is responsible for the curvature of the blue conical structure is localizing the HOMO state within an

Given the localization observed on some of these orbitals, we can further ask: At a given atomic position, what is the first orbital with a nonzero density at such location below and above the Fermi energy? This question can be rephrased in terms of the difference in energy among the first orbitals below and above the Fermi level that have a nonzero probability density at a given atomic location: In the third column in Fig. 4 *A* and *B* we display such energy difference *i*. There exists a clear correlation among

To end this work, we must state that the discrete geometry used to tell the shape of the phosphorene conical structures applies to other 2D atomic materials with varied unit cells, some of which are listed here: (*i*) Regular honeycomb lattices [graphene (2, 39, 40, 49) and hexagonal boron nitride (4)]. (*ii*) Low-buckled honeycomb hexagonal lattices [silicene and germanene (5); we established that freestanding stanene (50) is not the structural ground state]. (*iii*) “High-buckled” hexagonal close-packed bilayers of bismuth (51), tin, and lead (52). (*iv*) Thin trigonal–prismatic transition-metal dichalcogenides (53, 54). (*v*) Materials with buckled square unit cells––quad-graphs (31)––such as AlP (7).

Structures (*i*)–(*iv*) are equilateral triangular lattices with a basis, for which individual planes represent regular equilateral triangular nets; and structure (*v*) realizes a regular quad-graph.

Low-buckled structures (silicene, germanene, blue phosphorus), hexagonal close-packed bilayers (bismuth, tin, and lead), and black phosphorus have two parallel sublayers, *S*_{1} and *S*_{2}. Transition-metal dichalcogenides and AlP have three parallel sublayers *S*_{1}–*S*_{3}. The discrete geometry clearly stands for other 2D materials that do not form strong directional bonds (12, 14) as well.

## Concluding Remarks

Shape is a fundamental handle to tune the properties of 2D materials, and the discrete geometry provides the most accurate description of 2D material nets. This geometry was showcased on nonplanar phosphorene allotropes for which the electronic gap decreases by 20% with respect to its value on planar structures. The discrete geometry can thus be used to correlate large structural deformations to intended functionalities on 2D materials with arbitrary shapes.

## Methods

### Creation of Conical Structures.

Consider structures having atomic positions at planar disks; these positions can be parameterized by *ϕ* the angular segments being removed from these planar structures, the range of the angular variable

### Calculation of the Electronic Gap of Phosphorene Samples.

We obtain *i*. We call *i*, and report

### A Discrete Geometry Based on Triangulations at Atomic Positions.

There is an extrinsic and continuum (Euclidean) geometry in which material objects exist. These material objects are made out of atoms that take specific locations to generate their own intrinsic shape.

The intrinsic shape of 2D atomic materials can be idealized as a continuum: The basic assumption of continuum mechanics is that a continuum shape is justified at length scales *l* much larger than interatomic distances that are characterized by a lattice parameter

Consider three directed edges

Now consider the change in orientation among normals *l* and highlighted by dashed lines. The dual edge is defined by **1**, **2**, and **3** and

This way, the discrete metric tensor takes the following form on the basis given by dual edges*:

with

The discrete curvature tensor has an identical structure:

The parentheses **2** and **3** become 3

Topological defects (exemplified by a pentagon seen by a dot-dash line in Fig. 5) break translation symmetry, making it impossible to recover a crystalline structure by means of elastic deformations. However, Eqs. **2** and **3** provide a geometry along topological defects seamlessly still, as seen in the geometry provided in Fig. 2.

The computation of the distribution of the thickness *τ* in both conical structures was performed by finding for each atom in the lower sublayer *τ* of the structure at that specific atom.

## Acknowledgments

This work was supported by the Arkansas Biosciences Institute (S.B.-L.) and by NSF EFRI-1433311 (to H.T.). Calculations were performed at Arkansas’ Razor II and Texas Advanced Computing Center’s Stampede (NSF-XSEDE, TG-PHY090002).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: sbarraza{at}uark.edu.

Author contributions: H.T., E.O.H., A.A.P.S., and S.B.-L. designed research; M.M., K.U., H.T., E.O.H., A.A.P.S., and S.B.-L. performed research; M.M., K.U., H.T., E.O.H., A.A.P.S., and S.B.-L. analyzed data; and S.B.-L. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

↵*Weischedel C, Tuganov A, Hermansson T, Linn J, Wardetzky M (2012) Construction of discrete shell models by geometric finite differences, The 2nd Joint Conference on Multibody System Dynamics, May 29–June 1, 2012, Stuttgart, Germany.

Freely available online through the PNAS open access option.

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