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Structures of group-15 elemental solids from an effective boundary theory

Ashland Knowles, R. Ganesh

Abstract

We present an effective description for the crystal structures of pnictogen elemental solids. In these materials, each atom contains three valence electrons in $p$ orbitals. They are shared between neighbouring atoms to form valence bonds. We propose a trivalent network model on the simple cubic lattice. As a generalization of a dimer model, we impose a constraint that three dimers must touch every site. We argue that intra-orbital Coulomb repulsion prohibits the formation of two adjacent, parallel dimers. This leads to a tripod-like local configuration at every site. More importantly, it forces every line of the cubic lattice to have alternating dimers and blanks. There is no dynamics as dimers cannot be locally rearranged. A bulk-boundary mapping emerges whereby bonds in the interior are fully described by Ising variables on three bounding planes -- a simple example of holography that may be realized in real materials. To describe the energetics of bonding, we formulate a minimal model in terms of boundary Ising spins. Symmetries reduce the problem to that of three identical, independent, two-dimensional Ising models. An antiferromagnetic Ising-ground-state corresponds to the A7 structure seen in antimony and grey arsenic. An antiferromagnetic phase within a bilayer describes the structure of phosphorene. By stacking such bilayers, we obtain the A17 structure of black phosphorus. The stripe phase of the Ising models describes the cubic gauche structure of nitrogen. As a testable signature, we demonstrate that single impurities will induce long-ranged domain walls.

Structures of group-15 elemental solids from an effective boundary theory

Abstract

We present an effective description for the crystal structures of pnictogen elemental solids. In these materials, each atom contains three valence electrons in orbitals. They are shared between neighbouring atoms to form valence bonds. We propose a trivalent network model on the simple cubic lattice. As a generalization of a dimer model, we impose a constraint that three dimers must touch every site. We argue that intra-orbital Coulomb repulsion prohibits the formation of two adjacent, parallel dimers. This leads to a tripod-like local configuration at every site. More importantly, it forces every line of the cubic lattice to have alternating dimers and blanks. There is no dynamics as dimers cannot be locally rearranged. A bulk-boundary mapping emerges whereby bonds in the interior are fully described by Ising variables on three bounding planes -- a simple example of holography that may be realized in real materials. To describe the energetics of bonding, we formulate a minimal model in terms of boundary Ising spins. Symmetries reduce the problem to that of three identical, independent, two-dimensional Ising models. An antiferromagnetic Ising-ground-state corresponds to the A7 structure seen in antimony and grey arsenic. An antiferromagnetic phase within a bilayer describes the structure of phosphorene. By stacking such bilayers, we obtain the A17 structure of black phosphorus. The stripe phase of the Ising models describes the cubic gauche structure of nitrogen. As a testable signature, we demonstrate that single impurities will induce long-ranged domain walls.

Paper Structure

This paper contains 10 sections, 2 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Bonding and Configurations of Dimers
  3. Boundary Theory
  4. Network configurations
  5. Energy in the model
  6. Phases
  7. Black Phosphorus
  8. Impurity textures
  9. Discussion
  10. Unique Structures in the Stripe Phase

Figures (11)

  • Figure 1: A simple cubic lattice viewed along the body diagonal of a cube. Three $p$ orbitals are shown at each site, with $p_x$ orbitals oriented along the $x$-axis, $p_y$ orbitals along the $y$-axis, and $p_z$ orbitals along the $z$-axis.
  • Figure 2: Overlaps between $p$ orbitals. The $p_z$ orbitals in green are on neighboring sites, but their overlap is weak since the line connecting them is not along the $z$-axis. They can, at best, form a weak $\pi$ bond. A $p_z$ orbital and its neighbouring $p_y$ orbital have zero overlap due to symmetry. The $p_y$ orbitals in red are also on neighboring sites. They have a strong overlap, since the line connecting them is along the $y$-axis. They can form a strong $\sigma$ bond. When there is an electron in each of the two overlapping $p_y$ orbitals, they form a covalent bond which we represent with a dimer, as shown on the right.
  • Figure 3: Eight possible orientations of a tripod at each site. The tripod consists of three dimers attached to the site at the centre.
  • Figure 4: a) A simple cubic lattice with open boundaries. Boundary variables are defined on square grids on three bounding surfaces. Each boundary variable determines dimer placements along a line that is directed into the cube. The $xy$ plane is colored green and has variables $\{\sigma\}$, the $yz$ plane is colored blue and has variables $\{\chi\}$, and the $zx$ plane is colored red with variables $\{\eta\}$. Boundary variables are Ising-like, taking values $+1$ or $-1$. b) A plane of dimers with $\chi$ and $\eta$ variables at the boundaries. Dimers fixed by the $\chi$ variables (shown in blue) are perpendicular to the $yz$ plane. Similarly, dimers fixed by $\eta$'s (shown in red) are perpendicular to the $zx$ plane. The same convention is used to highlight dimers in (c) and (d). c) A plane of dimers with $\sigma$ and $\eta$ variables at the boundaries. d) A plane of dimers with $\sigma$ and $\chi$ variables at the boundaries.
  • Figure 5: a) Perpendicular lines on the cubic lattice. Each line contains an alternating sequence of dimers and blanks. Dimers are shown in red, blue and green on a $5 \times 5\times 5$ section of a large cubic lattice. Shaded circles mark the ends where the line intersects the boundary of the $5\times 5\times 5$ section. The sides of the cube that are nearest to the viewer are transparent and are shaded in red, green, and blue. b) A plane from (a) that contains the line of red dimers is highlighted. The line of green dimers from (a) intersects this plane and is perpendicular to it. A mirror reflection about this plane leaves the red dimers unchanged, but shifts the green dimers.
  • ...and 6 more figures