Construction of Negatively Curved Cubic Carbon Crystals via Standard Realizations

Abstract

In Tagami et al. (Carbon 76:266–274, 2014), we constructed physically stable sp2 negatively curved cubic carbon structures which reticulate a Schwarz P-like surface. The method for constructing such crystal structures is based on the notion of the standard realization of abstract crystal lattices. In this paper, we expound on the mathematical method to construct such crystal structures.

Dedicate to Yumiko Naito

Appendix: Indexes of Single Wall Nanotubes

We summarize the geometric structure of single wall nanotubes (SWNTs). Mathematically, a SWNT is considered as the fundamental region of the $\mathbb{Z}$-action on the standard realization of the hexagonal lattice. In the followings, we explain the geometric structures of SWNTs.

First, we define

$$\displaystyle{\begin{array}{ll} &\boldsymbol{v}_{0} = (0,0),\quad \boldsymbol{v}_{1} = (-\sqrt{3}/2,1/2),\quad \boldsymbol{v}_{2} = (\sqrt{3},1/2),\quad \boldsymbol{v}_{3} = (0,-1), \\ &\boldsymbol{a}_{1} = \boldsymbol{v}_{2} -\boldsymbol{v}_{1} = (\sqrt{3},0),\quad \boldsymbol{a}_{2} = \boldsymbol{v}_{3} -\boldsymbol{v}_{1} = (\sqrt{3}/2,-3/2), \end{array} }$$

then the graph X ₀ = (V ₀, E ₀), $V _{0} =\{ \boldsymbol{v}_{i}\}_{i=0}^{3}$, $E_{0} =\{ (\boldsymbol{v}_{0},\boldsymbol{v}_{i})\}_{i=1}^{3}$ is the fundamental region of the hexagonal lattice, and $\{\boldsymbol{a}_{1},\boldsymbol{a}_{2}\}$ is the basis of the parallel transformations (see Fig. 7). Here, the angle between $\boldsymbol{a}_{1}$ and $\boldsymbol{a}_{2}$ is π∕3. By using the basis, we may write all vertices $\boldsymbol{v}$ of the hexagonal lattice as

$$\displaystyle{\boldsymbol{v} =\alpha _{1}\boldsymbol{a}_{1} +\alpha _{2}\boldsymbol{a}_{2},\quad (\alpha _{1},\alpha _{2}) \in \mathbb{Z} \times \mathbb{Z},\quad \text{or}\quad (\alpha _{1},\alpha _{2}) \in (\mathbb{Z} - 1/3) \times (\mathbb{Z} - 1/3).}$$

Fig. 7

Configuration of the single-wall carbon nanotube, with $\boldsymbol{c} = (6,3)$ and $\boldsymbol{t} = (4,-5)$

Definition

The vector $\boldsymbol{c} = c_{1}\boldsymbol{a}_{1} + c_{2}\boldsymbol{a}_{2}$ is called the chiral vector and (c ₁, c ₂) is called the chiral index if and only if the SWNT is constructed from the hexagonal lattice by identifying $\boldsymbol{x}$ and $\boldsymbol{x} + \boldsymbol{c}$.

The vector $\boldsymbol{t} = t_{1}\boldsymbol{a}_{1} + t_{2}\boldsymbol{a}_{2}$, where $(t_{1},t_{2}) = ((c_{1} + 2c_{2})/d(c),-(2c_{1} + c_{2})/d(\boldsymbol{c}))$, and $d(\boldsymbol{c}) =\gcd (c_{1} + 2c_{2},2c_{1} + c_{2})$, is called the lattice vector of the SWNT with chiral index $\boldsymbol{c} = (c_{1},c_{2})$ (see Fig. 7).

The chiral vector indicates the direction of the circle of the SWNT, and is the period vector of the $\mathbb{Z}$-action on the hexagonal lattice. The chiral vector is orthogonal to the lattice vector, and the lattice vector is the minimum period of the hexagonal lattice along the tube axis of the SWNT. The circumference $L(\boldsymbol{c})$ of the SWNT is derived from the chiral index by $L(\boldsymbol{c}) = \vert \boldsymbol{c}\vert = \sqrt{3(c_{1 }^{2 } + c_{2 }^{2 } + c_{1 } c_{2 } )}.$  The electronic properties of single wall carbon nanotubes depends on the chiral index (see [38]).

Next, let us consider finite length single wall nanotubes. More precisely, consider a finite length SWNT, which terminates at the vertices (atoms) with $\boldsymbol{x}_{0} = \boldsymbol{0}$ and $\boldsymbol{x}_{1} =\alpha _{1}\boldsymbol{a}_{1} +\alpha _{2}\boldsymbol{a}_{2}$, and characterize the length of this SWNT by α ₁ and α ₂. In the followings, $\mathop{\mathrm{SWNT}}\limits (\boldsymbol{c},\alpha _{1}\boldsymbol{a}_{1} +\alpha _{2}\boldsymbol{a}_{2})$ denotes the SWNT which is terminated by $\boldsymbol{0}$ and $\alpha _{1}\boldsymbol{a}_{1} +\alpha _{2}\boldsymbol{a}_{2}$ and whose chiral vector is $\boldsymbol{c}$. It is easy to show that the length of $\mathop{\mathrm{SWNT}}\limits (\boldsymbol{c},\boldsymbol{x})$ is $\left \vert \langle \boldsymbol{x},\boldsymbol{e}_{t}\rangle \right \vert$, where $\boldsymbol{e}_{t} = \boldsymbol{t}/\vert \boldsymbol{t}\vert$. In [39], we define the length index $\ell(\boldsymbol{c},\boldsymbol{x})$ of $\mathop{\mathrm{SWNT}}\limits (\boldsymbol{c},\boldsymbol{x})$ as

$$\displaystyle{\ell(\boldsymbol{c},\boldsymbol{x}) = \left \vert \langle \boldsymbol{x},\boldsymbol{e}_{t}\rangle \right \vert /\sqrt{3} = \frac{\sqrt{3}\vert c_{2}\alpha _{1} - c_{1}\alpha _{2}\vert } {2\sqrt{c_{1 }^{2 } + c_{2 }^{2 } + c_{1 } c_{2}}},\quad \boldsymbol{x} =\alpha _{1}\boldsymbol{a}_{1} +\alpha _{2}\boldsymbol{a}_{2}.}$$

The length index corresponds to how many hexagons are arranged in the direction of the tube axis.

Now, we consider $\mathop{\mathrm{SWNT}}\limits (\boldsymbol{c},\boldsymbol{t})$, which has the canonical length for the given chiral index $\boldsymbol{c}$. By using the definition of $\boldsymbol{t}$, we obtain the length index $\ell(\boldsymbol{c},\boldsymbol{t})$ of $\mathop{\mathrm{SWNT}}\limits (\boldsymbol{c},\boldsymbol{t})$ as

$$\displaystyle{\ell(\boldsymbol{c},\boldsymbol{x}) = \frac{\sqrt{3(c_{1 }^{2 } + c_{2 }^{2 } + c_{1 } c_{2 } )}} {d(\boldsymbol{c})} = \frac{L(\boldsymbol{c})} {d(\boldsymbol{c})},}$$

and its area as

$$\displaystyle{\mathop{\mathrm{Area}}\limits (\mathop{\mathrm{SWNT}}\limits (\boldsymbol{c},\boldsymbol{t})) = \sqrt{3}\ell(\boldsymbol{c},\boldsymbol{t})L(\boldsymbol{c}) = \frac{\sqrt{3}L(\boldsymbol{c})} {d(\boldsymbol{c})}.}$$

Finally, we calculate the number of hexagons in the fundamental region of $\mathop{\mathrm{SWNT}}\limits (\boldsymbol{c},\boldsymbol{t})$, which is the fundamental region of action generated by the lattice $\{\boldsymbol{c},\boldsymbol{t}\}$. Since all hexagons in the lattice are congruent and their volume are $3\sqrt{3}/2$, the fundamental region contains F hexagons, where

$$\displaystyle{ F = \frac{2L(\boldsymbol{c})^{2}} {3d(\boldsymbol{c})}. }$$

(11)

Combining ( 2) and ( 11), we obtain the number of vertices V, of edges E, and of faces F in Table 1.

Note about figures Part of Figs. 6 and 1a are gray scale version of figures published in [1]. Figures 1a, b, 4 and 6c, d of 6-1-1-P are also published in [40].