The existence/absence of Dirac cones in graphynes

We first analyze the bonding character and electronic structure using MLWFs, which may serve as ideal building blocks in models for the electronic structure of nanostructures [26]. In contrast to graphene, there are both sp and sp² hybridized carbon atoms in graphyne, resulting in different bonds [27, 28]. Figure 1 summarizes the MLWF analysis for α graphyne, as a typical example of various graphynes. Each vertex atom exhibits sp² hybridization with neighboring atoms to form three σ bonds (figure 1(a)), which correspond to deep valence bands. Because of the tetravalent nature of C, the p_z orbital of vertex atoms forms a localized π orbital (figure 1(b)), which constitutes the main component of bands near the Fermi level. The triple bonds in acetylenic linkages are represented by three MLWFs (figure 1(c)), lying between two sp C atoms that are distinguished by a 120° rotation. It is therefore straightforward to express the Hamiltonian in the basis of the MLWFs from figures 1(a)–(c) and then to re-calculate the band structure, i.e., to make the Wannier interpolation with restricted basis. A comparison of the direct first-principles results and the Wannier interpolation is shown in figure 1(d), where we see excellent agreement. The massless bands near the Fermi level are reproduced quite well, vindicating the choice of MLWFs in the analysis.

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As described above, when expressed using a basis of selected MLWFs, the Hamiltonian reproduces first-principles results quite accurately. Because of the strong localization of MLWFs, the matrix elements of two MLWFs decay rapidly with increasing distance between them, and by neglecting long-distance hopping terms, a TB model can be constructed [29]. A graphyne model obtained in this way contains a large number of MLWFs. However, we can adopt further approximations to reduce the TB Hamiltonian and facilitate the analysis. Some MLWFs have deep energy levels: the σ bonds for example (figure 1(a)) are characterized by a very low on-site energy (∼12 eV below the Fermi level), so they can be safely cast. An additional approximation is derived from the observation that the main contributions to bands near the Fermi level come from the p_z orbitals of vertex atoms. This implies that an ideal construction involves an effective TB Hamiltonian containing only vertex p_z orbitals. However, in graphynes, electrons from one p_z orbital can hop to another p_z orbital either directly or through other MLWFs (such as found in triple bonds). Between p_z orbitals separated by acetylenic linkages, direct hopping of electrons is negligible, and indirect hopping is dominant. To construct the effective Hamiltonian that considers this indirect hopping process, we made use of the downfolding method, as briefly explained below.

The downfolding method is also known as a matrix condensation or Shur complement. It is an effective way to derive a few-band Hamiltonian from first-principles results with 'complete' MLWF basis [30, 31]. The method yields a minimal basis set from a large set of MLWFs by integrating out the other degrees of freedom. In the example of graphynes, their MLWFs may be divided into two parts: the vertex p_z orbitals, {Ψ^π}, which form the main components in the spectrum near the Fermi level; and the remainder of MLWFs, {Ψ^Tri}, which are mainly the triple-bond orbitals. The Schrödinger equation takes the form

$$\begin{equation} \mathbf{H}_{\pi,\pi}|\Psi^{\pi}\rangle + \mathbf{H}_{\pi,{\rm Tri}}|\Psi^{\rm Tri}\rangle = E |\Psi^{\pi}\rangle, \end{equation} \tag{ 1 }$$

$$\begin{equation} \mathbf{H}_{{\rm Tri},{\rm Tri}}|\Psi^{\rm Tri}\rangle + \mathbf{H}_{{\rm Tri},\pi}|\Psi^{\pi}\rangle = E |\Psi^{\rm Tri}\rangle, \end{equation} \tag{ 2 }$$

where H_{π(Tri),π(Tri)} are the blocks of the Hamiltonian matrix and E is the eigenenergy. One can solve for |Ψ^Tri〉 in (2)

$$\begin{equation} |\Psi^{\rm Tri}\rangle = - (\mathbf{H}_{{\rm Tri},{\rm Tri}}-E)^{-1} \mathbf{H}_{{\rm Tri},\pi}|\Psi^{\pi}\rangle \end{equation} \tag{ 3 }$$

and substitute the result into (1), yielding an equation for |Ψ^π〉:

$$\begin{equation} [ \mathbf{H}_{\pi,\pi} - \mathbf{H}_{\pi,{\rm Tri}} (\mathbf{H}_{{\rm Tri},{\rm Tri}}-E)^{-1} \mathbf{H}_{{\rm Tri},\pi} ] |\Psi^{\pi}\rangle = E |\Psi^{\pi}\rangle. \end{equation} \tag{ 4 }$$

The contents inside the square brackets can be regarded as an E-dependent Hamiltonian, which has to be determined self-consistently. Since we are only interested in the spectrum near the Fermi level and the on-site energy of {Ψ^Tri} (the diagonal elements of H_Tri,Tri) lies far from the spectrum of interest, (4) can be approximately written as

$$\begin{equation} [ \mathbf{H}_{\pi,\pi} - \mathbf{H}_{\pi,{\rm Tri}} (\mathbf{H}_{{\rm Tri},{\rm Tri}}-E_{\rm F})^{-1} \mathbf{H}_{{\rm Tri},\pi} ] |\Psi^{\pi}\rangle = E |\Psi^{\pi}\rangle, \end{equation}\noindent \tag{ 5 }$$

where E_F is the Fermi level. Thus, we obtain an effective Hamiltonian for the minimal orbitals of {Ψ^π} (assuming that E_F = 0)

$$\begin{equation} \tilde{\mathbf{H}}= \mathbf{H}_{\pi,\pi} - \mathbf{H}_{\pi,{\rm Tri}} (\mathbf{H}_{{\rm Tri},{\rm Tri}})^{-1} \mathbf{H}_{{\rm Tri},\pi}. \end{equation}\noindent \tag{ 6 }$$

Based on the first-principles calculation of α graphyne, the blocks of the original Hamiltonian matrix containing two vertex atoms and an acetylenic linkage are given by

$$\begin{eqnarray} \mathbf{H}_{\pi,\pi} & = & \left(\begin{array}{@{}cc@{}} e_0 & t_0 \\ t_0 & e_0 \end{array} \right), \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} \mathbf{H}_{{\rm Tri},{\rm Tri}} & = & \left(\begin{array}{@{}ccc@{}} e_3 & -t_3 & -t_3 \\ -t_3 & e_3 & -t_3 \\ -t_3 & -t_3 & e_3 \end{array} \right), \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} \mathbf{H}_{\pi,{\rm Tri}} & = & \left(\begin{array}{@{}ccc@{}} t & -t & 0 \\ t & -t & 0 \end{array} \right), \end{eqnarray} \tag{ 9 }$$

where e₀ = −2.66, e₃ = −6.17, t₀ = 0.32, t₃ = 3.55, and t = 1.11 (all in eV). The third triple-bond MLWF (see figure 1(c)) is perpendicular to p_z orbitals, so the hopping terms between them are zero, as shown in (9). The effective Hamiltonian for two vertex p_z orbitals separated by an acetylenic linkage can be denoted as

$$\begin{equation} \tilde{\mathbf{H}}= \left(\begin{array}{@{}cc@{}} \tilde{e}_0 & \tilde{t}_0 \\ \tilde{t}_0 & \tilde{e}_0 \end{array} \right). \end{equation} \tag{ 10 }$$

Then the effective hopping between neighboring p_z orbitals via an acetylenic linkage, $\tilde {t}_0$, can be deduced analytically from (6)–(9) to be

$$\begin{equation} \tilde{t}_0=t_0 -\frac{2t^2}{e_3+t_3}. \end{equation}\noindent \tag{ 11 }$$

For α graphyne, it was determined that $\tilde {t}_0=1.25$ eV. It is noted that, in comparison with the approach adopted by Kim and Choi [14], the exact value of the effective hopping parameter can be determined from first-principles calculations in the current scheme as described. Our approach also differs from what was adopted by Liu et al [15] where on-site energy was supposed to be zero at all C atoms. After this downfolding process, the effective Hamiltonian for α graphyne is almost identical to that of graphene except that the hopping parameter takes different values. This sheds light on the facts that the band structure of α graphyne is very similar to that of graphene and that their Dirac points are located at the same K and K' points as graphene [10].

Since the major effect of acetylenic linkages is to mediate the effective hopping terms between vertex p_z orbitals as described above, all graphynes may be described in a unified scheme in which p_z orbitals reside at vertices of a honeycomb lattice, and electrons hop from one vertex to its neighboring vertices as described by two kinds of hopping terms: one is for the hoppings between vertices separated by acetylenic linkages, denoted as $\tilde {t}_0$ as described above; and the other is for direct hopping between vertices that are connected by single bonds, which we therefore denote with t_d. Different combinations of such hopping terms describe different graphynes, giving rise to the existence or absence of Dirac cones.

As was previously discovered [10, 14, 15, 20], β graphyne is a semimetal with Dirac cones, while γ graphyne is a semiconductor (figure 2). Both β and γ graphynes (and even α graphyne) can be described by a common model composed of a $\sqrt {3}\times \sqrt {3}R30^{\circ }$ supercell and a hopping pattern as shown in figure 2(d). In this hopping pattern, 1/3 of hoppings belong to one class (t_r, represented by red edges in the figure) and 2/3 of hoppings belong to the other class (t_b, represented by blue edges). For α graphyne, both red and blue edges represent indirect hopping ($\tilde {t}_0$). For β graphyne, red edges represent direct hopping (t_d), while blue edges represent indirect hopping ($\tilde {t}_0$) (figures 2(c), (d)), where the situation is reversed for γ graphyne (figures 2(d), (e)). From our analysis in appendix A, the criterion for the existence of Dirac cones in the common model of figure 2(d) is given by the satisfaction of either of the following:

$$\begin{equation} \frac{t_{\rm r}}{t_{\rm b}}=1 \end{equation} \tag{ 12 }$$

or

$$\begin{equation} -2\leqslant \frac{t_{\rm r}}{t_{\rm b}} \leqslant -1. \end{equation} \tag{ 13 }$$

The indirect and direct hopping parameters derived from first-principles calculations for various graphynes are summarized in table 1 together with the original elements in MLWFs. It is determined that t_r/t_b = 1 for α graphyne and ${t_{\mathrm { r}}}/{t_{\rm b}}=t_{\rm d}/\tilde {t}_0{=}-1.79/1.52$ for β graphyne, which satisfy (12) and (13), respectively. For γ graphyne, however, ${t_{\mathrm { r}}}/{t_{\rm b}}=\tilde {t}_0/t_{\rm d}{=}-1.74/2.11$, satisfying neither (12) nor (13). This explains why Dirac cones are found to exist in α and β graphynes, but not in γ graphyne. It is interesting that, although the effective hopping parameters obtained by Liu et al [15] are different from ours (because different TB models and renormalization procedures were used), the application of the current criterion on their data also arrives at the same conclusion. In addition, the analytic solution predicts that the Dirac cones are located at the Γ–M lines (see appendix A), consistent with the first-principles calculations in figures 1 and 2 (noting that the K and K' points for α graphyne fold to the Γ point if the same supercell with six vertex atoms were adopted).

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Table 1. The TB Hamiltonian matrix elements obtained by MLWFs and the downfolding method (in units of eV).

	e0	t0	t	e3	t3	td	$\tilde {t}_0$
α graphyne	−2.66	0.32	1.11	−6.17	3.55	–	1.25
β graphyne	−2.18	0.36	1.22	−6.14	3.57	−1.79	1.52
γ graphyne	−1.66	0.45	1.30	−6.19	3.59	−2.11	1.74
6,6,12-graphyne	−1.53	0.48	1.26	−6.02	3.56	−2.05	1.76

Another interesting example is found in 6,6,12-graphyne. It does not have a hexagonal symmetry, but it also possesses Dirac cones in the band structure [10]. According to the effective TB model derived previously, 6,6,12-graphyne (figure 3(a)) should be equivalent to a system on a honeycomb lattice with rectangular supercells containing eight vertices (figure 3(b)). In other words, although 6,6,12-graphyne has a rectangular symmetry, its TB topology is equivalent to that of graphene. The resemblance between 6,6,12-graphyne and graphene is also demonstrated by comparing their band structures where the same supercell is adopted in graphene (figure 3(c) and (d)). The Dirac cones in graphene are folded to locate at the Γ–X' line due to the band-folding under the supercell. 6,6,12-graphyne has similar Dirac cones at the Γ–X' line, reflecting an inherent relationship with the underlying honeycomb topology. 6,6,12-graphyne has extra Dirac cones at the M–X line. Empirical calculations using various t_r and t_b values suggest that Dirac cones will not appear at the M–X line in this system when t_r and t_b have the same sign. So the existence of Dirac cones at the M–X line originates from the opposite sign of t_r and t_b. The construction of an effective TB model using the downfolding method also provides clues to the underlying mechanism of the self-doping effect in 6,6,12-graphyne [10]. Using (6)–(9), the effective on-site energy $\tilde {e}_0$ in (10) can be solved, and when the vertex atom is connected by one acetylenic linkage, it is found to be

$$\begin{equation} \tilde{e}_0=e_0 -\frac{2t^2}{e_3+t_3}. \end{equation}\noindent \tag{ 14 }$$

Furthermore, when vertex atoms have different numbers of acetylenic linkages, as is the case for 6,6,12-graphyne, the resulting on-site energies will be different, which leads to the self-doping effect.

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Aside from acetylenic linkages, other chemical groups can also provide effective hopping scenarios between neighboring p_z orbitals. Therefore, it comes as no surprise that a graphyne containing –B ≡ N– linkages has been recently shown to also possess Dirac cones in its band structure [13].

With the understanding that all graphynes have an equivalent honeycomb topology wherein Dirac cones arise from an adequate combination of hopping terms, one may expect that Dirac cones exist in many types of graphynes, not just in specific examples. Here we predict the existence of Dirac cones in two additional graphynes (figure 4).

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The first of these is 14,14,14-graphyne, which has a rhombic lattice with two vertex atoms in a unit cell as shown in figure 4(a). Each vertex atom is connected to its three nearest neighbors via one direct hopping (t_d) and two indirect hoppings through an acetylenic linkage ($\tilde {t}_0$). Consequently, it is equivalent to graphene that is subject to a uniaxial strain along the armchair direction [6]. According to the criterion in the smallest honeycomb unit cell [32], if the three hopping terms can form a triangle, Dirac cones should exist. For 14,14,14-graphyne considered here, the values of $\tilde {t}_0$ and t_d are close to those in β and γ graphynes (see table 1), so we have $2|\tilde {t}_0|>|t_{\mathrm { d}}|$, satisfying the triangular criterion. Therefore, 14,14,14-graphyne is a semimetal with Dirac cones (but is not metallic as claimed previously [9]), which was verified by first-principles calculations in figures 4(b)–(d).

The second type is 14,14,18-graphyne, whose atomic structure is very similar to 14,14,14-graphyne, but with the difference being that one direct hopping in the unit cell is replaced by an acetylenic linkage (see figure 4(e)). This graphyne has rectangular symmetry. Unlike the case of 14,14,14-graphyne, the rectangular unit cell of 14,14,18-graphyne with its four vertex atoms cannot be reduced into a smaller unit (i.e. one with two vertex atoms). First-principles calculations on the band structure indicate that 14,14,18-graphyne also possesses Dirac cones (figure 4(f)). In the vicinity of Dirac cones (±0.3 eV from the Fermi level), the valence and conduction bands exhibit linear dispersion in both the k_x and k_y directions with different slopes, namely 5.0 and 4.1 eV Å, respectively. The anisotropy of Dirac cones is related to the rectangular symmetry, just as with 6,6,12-graphyne [10]. Since 14,14,18-graphyne contains only four vertex atoms in a unit cell, the analyses become much easier. We performed an analytic study (see appendix B) and obtained an expression for the location of Dirac cones and the Fermi velocity, which confirms that Dirac cones locate at the X–M lines and that the Fermi velocity is direction dependent.