Peculiar width dependence of the electronic properties of carbon nanoribbons

Nanoribbons (nanographite ribbons) are carbon systems analogous to carbon nanotubes. We characterize a wide class of nanoribbons by a set of two integers $⟨p,q⟩$, and then define the width $w$ in terms of $p$ and $q$. Electronic properties are explored for this class of nanoribbons. Zigzag (armchair) nanoribbons have similar electronic properties to armchair (zigzag) nanotubes. The band gap structure of nanoribbons exhibits a valley structure with streamlike sequences of metallic or almost metallic nanoribbons. These sequences correspond to equiwidth curves indexed by $w$. We reveal a peculiar dependence of the electronic property of nanoribbons on the width $w$.

Figure 1

(a) A typical structure of nanoribbons. A solid circle stands for a carbon atom with one $\pi$ electron, while an open circle for a different atom such as a hydrogen. A closed area represents a unit cell. It is possible to regard the lattice made of solid circles as a part of a honeycomb lattice. (b) A nanoribbon is constructed from a chain of $m$ connected carbon hexagons, as depicted in dark gray, and by translating this chain by the translational vector $\mathit{T}=\pm q\mathit{a}+\mathit{b}$ many times, as depicted in light gray, where $q<m$. A nanoribbon is indexed by a set of two integers $⟨p,q⟩$ with $p=m-q$. Here we have taken $m=4$, $q=2$, $p=2$.

Figure 2

The geometric configuration of (a) the polyacene series having zigzag edges, (b) the polyphenanthrene series having armchair edges, and (c) the polyperylene series having armchair edges.

Figure 3

The band structure of nanoribbons. The horizontal axis is the crystal momentum $k$, $-\pi <k⩽\pi$, while the vertical axis is the energy $\epsilon$, $-3\mid t\mid ⩽\epsilon ⩽3\mid t\mid$ with $\mid t\mid =3.033\mathrm{eV}$. The band structure depends strongly on the index $q$ for fixed $p$, but depends on the index $p$ only weakly for fixed $q$.

Figure 4

The band gap structure of nanoribbons. (a) The horizontal and vertical axes represent the indices $p$ and $q$, respectively. Magnitudes of band gaps are represented by gray squares. Dark (light) gray squares represent small (large) gap semiconductors. Especially, metallic states are represented by black squares. (b) A bird’s eye view. The vertical axis is the energy gap $\Delta$ in units of $\mid t\mid =3.033\mathrm{eV}$. Nanoribbons make a valley structure with stream like sequences of metallic points in the $pq$ plane.

Figure 5

The band gaps $\Delta$ in unit of $\mid t\mid =3.033\mathrm{eV}$ as a function of the width $w$. The band gap behaves inversely to $w$. Band gaps take local minima almost at the same values of the width $w$ for any $q$.

Figure 6

Illustration of metallic points, sequences, and equiwidth curves. Metallic points are denoted by solid circles. Solid curves represent sequences of MAM points, while dotted curves represent the points $⟨p,q⟩$ possessing the same width. The $n\text{th}$ sequence is tangent to the equiwidth curve with $w=n$ at $q=1$. These two curves become almost identical for sufficiently wide nanoribbons.

Figure 7

The density of state (DOS) of the ⟨1,0⟩ nanoribbon. The horizontal axis is the energy $\epsilon$. There are many van Hove singularities because of one-dimensional structure of nanoribbon.

Figure 8

Plot of van Hove singularities in the $w\text{−}\epsilon$ plane. For a given $⟨p,q⟩$ nanoribbon, we calculate the width $w$ and the energies $\epsilon$ at which van Hove singularities develop. We have plotted the points $(w,\epsilon )$ for $q=1,2,3,4$ and for all $p$ in the region $w⩽3$. A stripe pattern is manifest.

Figure 9

Illustration of edge carbons and bulk carbons. A gray circle denotes an edge carbon connected by two carbons and a hydrogen. A black circle denotes a bulk carbon connected by three carbons. The energy and the transfer energy associated with edge carbons are set to be ${\epsilon }_{\text{edge}}$ and $t_{\text{edge}}$, respectively. Hydrogen atoms are omitted in this figure.

Figure 10

(a) The original band structure of the ⟨5,0⟩ nanoribbon. $\left({\mathrm{a}}^{\prime }\right)$ The band structure of the nanoribbon with the choice of $t_{\text{edge}}=0.7t_{\text{bulk}}$. The difference is hardly recognizable.

Figure 11

(a) and (b) The original band structures of the ⟨5,0⟩ and ⟨2,1⟩ nanoribbons. $\left({\mathrm{a}}^{\prime }\right)$ and $\left({\mathrm{b}}^{\prime }\right)$ The band structures of the nanoribbon with the choice of ${\epsilon }_{\text{edge}}={\epsilon }_{\text{bulk}}+0.3t$. The inversion symmetry is broken and the Fermi energy is slightly moved from ${\epsilon }_{\text{bulk}}=0$.

Figure 12

The band gap structure of nanoribbons with the edge correction ${\epsilon }_{\text{edge}}={\epsilon }_{\text{bulk}}+0.05t$ and $t_{\text{edge}}=0.99t_{\text{bulk}}$. See Fig. 4 for other details. Nanoribbons make a valley structure with streamlike sequences of almost metallic points in the $pq$ plane. Zigzag nano-ribbons remain gapless. Armchair nanoribbons are most strongly affected by edge corrections.

Figure 13

The band gap structure of nanoribbons with the edge correction ${\epsilon }_{\text{edge}}={\epsilon }_{\text{bulk}}+0.10t$ and $t_{\text{edge}}=0.95t_{\text{bulk}}$. See Fig. 4 for other details. Nanoribbons make a valley structure with streamlike sequences of almost metallic points in the $pq$ plane. Zigzag nano-ribbons remain gapless. Armchair nanoribbons are most strongly affected by edge corrections.