Aluminene as highly hole‐doped graphene

In order to study the stability of aluminene, we have carried out phonon dispersion calculations for all the geometric configurations. The fully optimized geometric structures and their corresponding phonon dispersion spectra are given in figure 1. Our calculations predict that planar aluminene with space group $P6/{mmm}$ forms a stable structure since it contains only positive frequencies for all the vibrational modes (figure 1(a)). On the other hand, the remaining geometric configurations possess structural instabilities in the transverse acoustic modes due to the presence of imaginary frequencies (shown as negative values). Cohesive energy for planar aluminene is estimated to be −1.956 eV and the negative value indicates that this is a bound system. Our calculations yield 4.486 and 2.590 Å for lattice constant and Al-Al bond length for the planar structure, respectively. After the geometry optimization the buckled (puckered) structure has become the AB (AA) bilayer of the triangular lattice (figures 1(b) and (c)). The relative displacement between the A and B layers is along the (1/3, 2/3, 0) direction. Since aluminene in triangular, buckled and puckered configurations is not stable, as observed from their phonon spectra, we do not consider them for further analysis.

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We have also carried out phonon dispersion calculations for the planar, buckled, puckered and triangular configurations made up of other group-III elements (boron, gallium and indium). Our results show that there are no stable configurations.

We present the results of the electronic band structure, density of states (DOS) and partial DOS (PDOS) for planar aluminene in figure 2(a). Aluminene behaves as a metallic system due to the partial occupancies in the σ as well as π bands. We wish to note that the other honeycomb monolayer materials studied so far are either semimetal or semiconductor. At the high symmetric K point in the Brillouin zone, two Dirac cones occur at energies 1.618 eV and $-4.274$ eV. The former is due to the π bonds purely made up of the p_z orbital and the latter is due to σ bonds which has strong contribution from the s orbital. At the Fermi energy, there exist two σ bands and we call them the ${\sigma }_{1}$ and ${\sigma }_{2}$ bands as shown in figure 3(a). The ${\sigma }_{2}$ band is purely made up of p_x and p_y orbitals, while the mixing of the s orbital is not negligible in the ${\sigma }_{1}$ band.

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The electronic band structure of aluminene closely resembles that of graphene [26], but the locations of the Dirac points in these two systems are different. In the case of graphene, the Dirac point lies exactly at the Fermi level where the bands due to the bonding orbitals (σ and π) are completely filled and those of anti-bonding orbitals are completely empty (${\sigma }^{*}$ and ${\pi }^{*}$). On the other hand, in aluminene, the bands due to the bonding orbitals (σ and π) are not completely filled and thus the Dirac point lies about 1.618 eV above the Fermi level. This is due to the fact that the number of valence electrons in Al (trivalent) is smaller than that of C (tetravalent). As a result, aluminene can be regarded as a highly hole-doped graphene. However, the important difference is that the Fermi energy is highly shifted in aluminene. Here, we wish to point out a remarkable feature of aluminene with respect to van Hove singularities. The van Hove singularity of the σ band exists very close to the Fermi energy at $E=-0.212$ eV. It may be possible to tune the Fermi energy around the vanHove singularity and to make the electrical conductivity very large by applying gate voltage.

To understand the nature of bonding in aluminene, we have calculated the valence charge density (figure 2(b)). Except the hollow regions of hexagon, the charge is nearly uniformly extended over the aluminene plane which contributes to the conductivity. We have also calculated the difference between the charge densities of system and its atomic constituents (figure 2(c)). It clearly indicates the presence of covalent bonds between Al atoms in aluminene.

The transport properties of a metallic system are governed by the Fermi surface. The calculated Fermi surface for aluminene is given in figure 3(b). It consists of three closed loops, corresponding to the π, ${\sigma }_{1}$ and ${\sigma }_{2}$ bands. The momentum vectors at which the bands cross the Fermi level along high symmetric k-points are shown in figure 3(a). Magnitudes of momentum vectors along the Γ-K and Γ-M directions for the ${\sigma }_{1}$ and π bands are equal (${k}_{1}={k}_{2}$ and ${k}_{5}={k}_{6}$). Furthermore, Fermi surfaces for these two bands are circular in shape around the Γ point. Thus, the electrons in these bands behave as free-particles. However, the magnitudes of momentum vectors along the Γ-K and Γ-M directions for the ${\sigma }_{2}$ band are not equal (${k}_{3}\ne {k}_{4}$). The Fermi surface for this band is highly hexagonally warped. We comment on these Fermi surfaces further when we analyze the Dirac theory as given below.

The electronic configuration of aluminium is [Ne] 3s²3p¹. We construct an 8-band tight-binding model including the 4 orbitals: 3s, 3p_x, 3p_y and 3p_z. The tight-binding Hamiltonian is given by

$$\begin{eqnarray}&&H=\displaystyle \sum _{{ij}}\displaystyle \sum _{\alpha \beta }{t}_{{ij}}^{\alpha \beta }{c}_{i\alpha }^{\dagger }{c}_{j\beta },\end{eqnarray} \tag{ 1 }$$

where ${t}_{{ij}}^{\alpha \beta }$ denotes the nearest-neighbor transfer integral between the α orbital at the site i and the β orbital at the site j, and ${c}_{i\alpha }$ (${c}_{i\alpha }^{\dagger }$) annihilates (creates) an electron in the α orbital at the i site. We have derived the six independent Slater–Koster parameters (${\varepsilon }_{p}$, ${\varepsilon }_{s}$, ${V}_{{pp}\pi }$, ${V}_{{pp}\sigma }$, ${V}_{{ss}\sigma }$ and ${V}_{{sp}\sigma }$) by fitting the DFT-based band structure at the Γ and K points, and present them in table 1. The tight-binding model reproduces the band structure near the Fermi energy well, as shown in figure 2(a). More details about the tight-binding model are given in appendix.

Table 1.  The Slater–Koster parameters are obtained by fitting the DFT-based electronic band structure at the Γ and K points.

Structure	${\varepsilon }_{p}$	${\varepsilon }_{s}$	${V}_{{pp}\pi }$	${V}_{{pp}\sigma }$	${V}_{{ss}\sigma }$	${V}_{{sp}\sigma }$
Aluminene	1.618	−1.969	−0.815	−1.313	−1.596	−1.737
Graphene [27]	0	−8.370	−3.070	−6.050	−5.729	−5.618

We proceed to construct the Dirac theory to describe the band structure near the Fermi energy. First, the Fermi surface of the ${\sigma }_{2}$ band is highly anisotropic (see figure 3(b)). Hexagonal warping occurs due to the hexagonal symmetry at the Γ point. It is well described by the hexagonal warped Dirac Hamiltonian, which was originally proposed to describe the Dirac fermion on the surface of the 3D topological insulator [28],

$$\begin{eqnarray}&&H={\hslash }v({k}_{x}{\sigma }_{y}-{k}_{y}{\sigma }_{x})+\gamma ({k}_{+}^{3}+{k}_{-}^{3}){\sigma }_{z}+{\varepsilon }_{p},\end{eqnarray} \tag{ 2 }$$

where $v=0.678a/{\hslash }$ is the velocity of Dirac fermion, $\gamma =0.237{a}^{3}$ describes the hexagonal warping effects, ${\varepsilon }_{p}$ is the energy shift of the Dirac point, ${k}_{\pm }={k}_{x}\pm {\rm{i}}{k}_{y}$ and ${\boldsymbol{\sigma }}$ denotes the Pauli matrix. The energy is given by

$$\begin{eqnarray}&&E={\varepsilon }_{p}\pm \sqrt{{({\hslash }{vk})}^{2}+{\gamma }^{2}{({k}_{x}^{3}-3{k}_{x}{k}_{y}^{2})}^{2}}.\end{eqnarray} \tag{ 3 }$$

It explains well the hexagonal warped Fermi surface shown in figure 3(b), where the values of the Fermi momenta ${k}_{3}=0.269\times (2\pi /a)$ along the K direction and ${k}_{4}=0.380\times (2\pi /a)$ along the M direction are different (${k}_{4}\gt {k}_{3}$).

We note that the low-energy Dirac theory is different from that of the minimal model of graphene made of the π bond, where the contribution from the σ bonds is negligible at the Fermi level. On the other hand, in aluminene, the σ bonds play a significant role to determine the electronic properties around the Fermi level. This difference arises because the Fermi energy in aluminene is much lower as compared to that in graphene. Namely, the Dirac Hamiltonian equation (2) is the continuum model of the σ bands at the Γ point. Hence, the theory of aluminene is different from the theory of graphene which is governed by the π band at the K point.

Next, the ${\sigma }_{1}$ band yields a circular shaped Fermi surface with ${k}_{1}={k}_{2}=0.129\times (2\pi /a)$. The ${\sigma }_{1}$ band is described by the Hamiltonian equation (2) with $v=1.996a/{\hslash }$ and $\gamma =0$.

Third, the π band also possesses a circular shaped Fermi surface with ${k}_{5}={k}_{6}=0.375\times (2\pi /a)$. Around the Γ point, the Hamiltonian can be reduced to

$$\begin{eqnarray}&&H=(3-{({ak})}^{2}){V}_{{pp}\pi }{\sigma }_{x}+{\varepsilon }_{p}.\end{eqnarray} \tag{ 4 }$$

We get the following energy for the above Hamiltonian,

$$\begin{eqnarray}&&E={\varepsilon }_{p}\pm (3-{({ak})}^{2}/4){V}_{{pp}\pi },\end{eqnarray} \tag{ 5 }$$

leading to a circularly shaped Fermi surface with the radius

$$\begin{eqnarray}&&k=\sqrt{3+{\varepsilon }_{p}/{V}_{{pp}\pi }}/a=0.355\times (2\pi /a).\end{eqnarray} \tag{ 6 }$$

It is in good agreement with $k=0.375\times (2\pi /a)$ obtained from the first-principles calculations.

The Hamiltonian for the π bonds is given by

$$\begin{eqnarray}H=\left(\begin{array}{cc}{\varepsilon }_{p} & {h}_{{AB}}\\ {h}_{{AB}}^{\dagger } & {\varepsilon }_{p}\end{array}\right),\end{eqnarray} \tag{ A.2 }$$

with

$$\begin{eqnarray}&&{h}_{{AB}}={V}_{{pp}\pi }\left[2{{\rm{e}}}^{-{\rm{i}}{{ak}}_{x}/\sqrt{3}}\mathrm{cos}{{ak}}_{y}+{{\rm{e}}}^{2{\rm{i}}{{k}}_{x}/\sqrt{3}}\right],\end{eqnarray} \tag{ A.3 }$$

which is the same as the tight-binding model of graphene with only the p_z orbitals being included.

By explicitly diagonalizing the Hamiltonian, we obtain the energies ${E}_{\pi }^{K}$ at the K point and ${E}_{\pi }^{\Gamma }$ the Γ point as

$$\begin{eqnarray}&&{E}_{\pi }^{K}={\varepsilon }_{p},\quad {E}_{\pi }^{\Gamma }={\varepsilon }_{p}+3{V}_{{pp}\pi }.\end{eqnarray} \tag{ A.4 }$$

By solving the energies at the K and the Γ, we get

$$\begin{eqnarray}&&{\varepsilon }_{p}={E}_{\pi }^{K},\quad {V}_{{pp}\pi }=({E}_{\pi }^{\Gamma }-{E}_{\pi }^{K})/3.\end{eqnarray} \tag{ A.5 }$$

By comparing the energies of π at these high symmetric points, which are obtained from the DFT (shown in table A1 njp516911t2), we find the two Slater–Koster parameters (${\varepsilon }_{p}$ and ${V}_{{pp}\pi }$) and the values of these two parameters are given in table 1 of the main text.

Table A1.  The energy obtained by the first-principles calculations at the Γ and K points.

Structure	${E}_{\pi }^{K}$	${E}_{\pi }^{\Gamma }$	${E}_{\sigma }^{K}$	${E}_{\sigma 1}^{\Gamma }$	${E}_{\sigma 2}^{\Gamma }$	${E}_{\sigma 3}^{\Gamma }$
Aluminene	1.618	−0.828	−4.274	−6.757	0.871	2.819

For the σ bond, we take the following basis, ${\psi }_{A,{p}_{x}},{\psi }_{A,{p}_{y}},{\psi }_{A,s},{\psi }_{B,{p}_{x}},{\psi }_{B,{p}_{y}},{\psi }_{B,s}$. The Hamiltonian for the σ bonds has the form

$$\begin{eqnarray}H=\left(\begin{array}{cc}{H}_{{AA}} & {H}_{{AB}}\\ {H}_{{AB}}^{\dagger } & {H}_{{BB}}\end{array}\right),\end{eqnarray} \tag{ A.6 }$$

with

$$\begin{eqnarray}{H}_{{AA}}={H}_{{BB}}=\left(\begin{array}{ccc}{\varepsilon }_{p} & 0 & 0\\ 0 & {\varepsilon }_{p} & 0\\ 0 & 0 & {\varepsilon }_{s}\end{array}\right),\end{eqnarray} \tag{ A.7 }$$

and

$$\begin{eqnarray}{H}_{{AB}}=\left(\begin{array}{ccc}{h}_{{xx}} & {h}_{{xy}} & {h}_{{xs}}\\ {h}_{{xy}} & {h}_{{yy}} & {h}_{{ys}}\\ -{h}_{{xs}} & -{h}_{{ys}} & {h}_{{ss}}\end{array}\right),\end{eqnarray} \tag{ A.8 }$$

where the matrix elements are given by

$$\begin{eqnarray}{h}_{{xx}} & = & \displaystyle \frac{1}{2}(-{V}_{{pp}\sigma }{+3{V}_{{pp}\pi }){\rm{e}}}^{-{\rm{i}}{{ak}}_{x}/\sqrt{3}}\mathrm{cos}\displaystyle \frac{{{ak}}_{y}}{2}{-{V}_{{pp}\sigma }{\rm{e}}}^{2{\rm{i}}{{ak}}_{x}/\sqrt{3}},\\ {h}_{{xy}} & = & i\displaystyle \frac{\sqrt{3}}{2}({V}_{{pp}\sigma }+{V}_{{pp}\pi }){e}^{-{\rm{i}}{{ak}}_{x}/\sqrt{3}}\mathrm{sin}{{ak}}_{y},\\ {h}_{{yy}} & = & \displaystyle \frac{1}{2}{(-3{V}_{{pp}\sigma }+{V}_{{pp}\pi }){\rm{e}}}^{-{\rm{i}}{{ak}}_{x}/\sqrt{3}}\mathrm{cos}{{ak}}_{y}+{V}_{{pp}\pi }{{\rm{e}}}^{2{\rm{i}}{{ak}}_{x}/\sqrt{3}},\\ {h}_{{ss}} & = & {V}_{{ss}\sigma }\left[2{{\rm{e}}}^{-{\rm{i}}{{ak}}_{x}/\sqrt{3}}\mathrm{cos}{{ak}}_{y}+{{\rm{e}}}^{2{\rm{i}}{{ak}}_{x}/\sqrt{3}}\right],\\ {h}_{{sx}} & = & {V}_{{sp}\sigma }\left[{{\rm{e}}}^{-{\rm{i}}{{ak}}_{x}/\sqrt{3}}\mathrm{cos}{{ak}}_{y}-{{\rm{e}}}^{2{\rm{i}}{{k}}_{x}/\sqrt{3}}\right],\\ {h}_{{sp}} & = & -{\rm{i}}\sqrt{3}{V}_{{sp}\sigma }{{\rm{e}}}^{-{\rm{i}}{{ak}}_{x}/\sqrt{3}}\mathrm{sin}{{ak}}_{y}.\end{eqnarray} \tag{ A.9 }$$

We can determine the energy at the Γ point as

$$\begin{eqnarray}&&{E}_{\sigma 1}^{\Gamma }={\varepsilon }_{s}+3{V}_{{ss}\sigma },\qquad {E}_{\sigma 2}^{\Gamma }={\varepsilon }_{p}-\displaystyle \frac{3}{2}\left({V}_{{pp}\pi }-{V}_{{pp}\sigma }\right),\qquad {E}_{\sigma 3}^{\Gamma }={\varepsilon }_{s}-3{V}_{{ss}\sigma }\end{eqnarray} \tag{ A.10 }$$

and at the K point as

$$\begin{eqnarray}&&{E}_{\sigma }^{K}=\displaystyle \frac{1}{2}\left({\varepsilon }_{s}+{\varepsilon }_{p}-\sqrt{{\left({\varepsilon }_{p}-{\varepsilon }_{s}\right)}^{2}+18{V}_{{sp}\sigma }^{2}}\right).\end{eqnarray} \tag{ A.11 }$$

By solving the energies at the K and the Γ, we obtain

$$\begin{eqnarray}{\varepsilon }_{s} & = & ({E}_{\sigma 1}^{\Gamma }+{E}_{\sigma 3}^{\Gamma })/2,\qquad {V}_{{pp}\sigma }=({E}_{\pi }^{\Gamma }+2{E}_{\sigma 2}^{\Gamma })/3-{E}_{\pi }^{K},\\ {V}_{{ss}\sigma } & = & ({E}_{\sigma 1}^{\Gamma }-{E}_{\sigma 3}^{\Gamma })/6,\qquad {V}_{{sp}\sigma }=-\sqrt{({E}_{\pi }^{K}-{E}_{\sigma }^{K})({E}_{\sigma 1}^{\Gamma }+{E}_{\sigma 2}^{\Gamma }-2{E}_{\sigma }^{K})}/3.\end{eqnarray} \tag{ A.12 }$$

Again, we find the above-mentioned four Slater–Koster parameters by comparing the energies obtained from the DFT calculations (shown in table A1), and the results are summarized in table 1 of the main text.

It is enough to take into account only the p_x and p_y orbitals to describe the ${\sigma }_{2}$ bands. The Hamiltonian for this case is given by

$$\begin{eqnarray}{H}_{{AA}}={H}_{{BB}}=\left(\begin{array}{cc}{\varepsilon }_{p} & 0\\ 0 & {\varepsilon }_{p}\end{array}\right),\end{eqnarray} \tag{ A.13 }$$

and

$$\begin{eqnarray}{H}_{{AB}}=\left(\begin{array}{cc}{h}_{{xx}} & {h}_{{xy}}\\ {h}_{{xy}} & {h}_{{yy}}\end{array}\right).\end{eqnarray} \tag{ A.14 }$$

with the matrix elements given in equation (A.9). We call it the p_x–p_y model.