Tight-binding approach to penta-graphene

We introduce an effective tight-binding model to discuss penta-graphene and present an analytical solution. This model only involves the π-orbitals of the sp2-hybridized carbon atoms and reproduces the two highest valence bands. By introducing energy-dependent hopping elements, originating from the elimination of the sp3-hybridized carbon atoms, also the two lowest conduction bands can be well approximated - but only after the inclusion of a Hubbard onsite interaction as well as of assisted hopping terms. The eigenfunctions can be approximated analytically for the effective model without energy-dependent hopping elements and the optical absorption is discussed. We find large isotropic absorption ranging from 7.5% up to 24% for transitions at the Γ-point.

− t C1−C2 ∑ m,n;σ d † m,n;σ (e m,n+1;σ + f m+1,n;σ ) + H.c. , where e and f denote the lower and upper red atom in the unit cell of Fig. 1a) of the main text. The four sp 2 -hybridized atoms are defined in Fig. 1b) of the main text. The onsite Hamiltonian given by with E g = E C1 for g = e, f and E g = E C2 for g = a, b, c, d and n g m,n;σ = g † m,n;σ g m,n;σ for g = a, b, c, d, e, f . We also need to introduce a Hubbard term and an assisted hopping contribution H a = W ∑ m,n;σ n a m,n;σ a † m,n;σ (e m,n;σ + f m,n−1;σ ) + H.c.
+W ∑ m,n;σ n d m,n;σ d † m,n;σ (e m,n+1;σ + f m+1,n;σ ) + H.c. , whereσ is the opposite spin-projection of σ . The interaction terms are most easily treated within the mean-field approximation. For the Hubbard interaction, we set H H ≈ U ∑ m,n;σ g=a,b,c,d n g m,n;σ n g m,n;σ + E U , where the constant energy shift reads E U = −U ∑ n g m,n;↑ n g m,n;↓ with the sum over m, n and g = a, b, c, d. The assisted hopping term is approximated analogously by the following: H a = W ∑ m,n;σ n a m,n;σ a † m,n;σ (e m,n;σ + f m,n−1;σ ) + H.c. + n b m,n;σ b † m,n;σ (e m−1,n;σ + f m,n;σ ) + H.c.
Within this mean-field approximation, the two spin-projections decouple and we will drop the spin-degree of freedom. We are now in the position to reduce the resulting 6-band model to an effective 4-band model by projecting out the C1-atoms. This results in the effective Hamiltonian of Eq. (1) of the main text with t →˜t With an additional shift of E c 0 = 1.15eV, the two conduction bands can be well approximated with the above parameters.
The electronic density of the two valence bands is more located between the C2-atoms than the one of the two conductance band. Also self-energy corrections are usually more dominant for the bands further away of the Fermi level, i.e., in our case the conduction band. The two valence bands are thus calculated by setting U = W = 0. The larger energy shift of E v 0 = 2.61eV needed to fit the data matches well with the predicted value E c − E v ≈ E W . The four bands are shown as blue curves in Fig. 3 of the main text.
We note that the hopping Hamiltonian H 0 has already been discussed in the Supplementary Information S3 of Ref. 3 Here, we showed that it is crucial to also include the onsite, Hubbard and assisted hopping terms in order to reproduce the band structure of the bands close to half-filling.

Optical absorption obtained from first principles
In order to compare the results for the optical absorption obtained using the effective Hamiltonian, we will here perform the calculations using the Wannier90 software 4 on top of the VASP DFT calculations. By this, we can also verify the quality of the fit of the band structure based on our effective Hamiltonian.
The Wannier90-code allows for an effective approach to construct maximally localized Wannier functions (MLWF) from the plane wave expansion employed with the VASP code. 5 Furthermore, within the MLWF description, the Wannier90-code can provide the three-dimensional (3D) optical conductivity using the Kubo-Greenwood equation. In order to obtain the 2D conductivity for a penta-graphene unit cell (6 C atoms) in a 3D vector set, we have to multiply the conductivity by the norm of the perpendicular vector of the unit cell which has been chosen to be 20Å.
The final result is express in units of the universal conductivity σ 0 = π 2 e 2 h , the conductivity of neutral single-layer graphene. The corresponding absorption is then obtained by A 0 = σ 0 ε 0 c = πα, with α the fine-stucture constant. Let us finally note that the results are isotropic and that we will thus not distinguish between the different directions.
The optical conductivity σ (ω) depends on the uniform interpolation k-point mesh parameter of Wannier90. After convergence to a 48x48x1 k-mesh, σ shows a plateau region of value 3σ 0 , with two maxima reaching 4σ 0 , as shown in Fig. 1. The obtained maximum values within the Wannier90 toolkit agree well with those observed in the main text using the tight-binding approach, which related the high absorbance of penta-graphene to its chemical structure.