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.
Arguably, carbon is the most versatile element being capable to form various stable structures with graphene1,2 being its most prominent two-dimensional allotrope out of which most carbon structures can be built such as fullerene3, carbon-nanotubes4 or multi-layer graphene5 and graphite.
Recently, a new allotrope was proposed which cannot be composed from a graphene sheet: penta-graphene6. It entirely consists of carbon atoms forming pentagons within the Cairo tiling and it remains almost flat by adding to its sp2-hybridised carbon atoms also sp3-hybridised carbon atoms, arranging them in three parallel horizontal planes separated by approximately half an Angstrom. Penta-graphene, therefore, only experiences a buckling on the atomic scale of a unit cell and would represent another example of a two-dimensional semiconductor. Also other pentagonal monolayer crystals of boron nitride and silver azide have recently been investigated7.
Even though penta-graphene has not been synthesised experimentally, the theoretical analysis point out many intriguing properties which are worth discussing in more detail. Most strikingly in terms of applications is probably the large predicted band-gap of 3.25 eV which makes it a potential candidate for blue absorption/emission. Nevertheless, the optical properties such as absorption due to a linearly polarised light field have not been addressed, yet.
The objective of this work is two-fold. First, we will discuss the possibility of a simple tight-binding description to model the valence and conduction bands closest to the neutrality point. Our model will only include the π-orbitals of the sp2-hybridised carbon atoms for which an analytical solution is possible. This is reminiscent to widely-used tight-binding models for graphene8,9,10 and carbon-nanotubes11,12. Second, using the analytical approximation, we will also determine the absorption of linearly polarized light via Fermi’s Golden Rule.
The paper is organised as follows. In Sec. II, we discuss the general structure of penta-gaphene. In Sec. III, we introduce the effective 4-band model and present its analytical approximation. In Sec. IV, we discuss the absorption at the high-symmetry points of the Brillouin zone and close with a summary and conclusions. In the supplementary information, we point out the importance of correlation effects in order to justify the parameters of the extended tight-binding model.
The regular Cairo pentagonal patterning is characterized by four bonds forming the pentagon, three of which have the distance a and one has the distance . The unit cell consists of six carbon atoms and is defined by the two lattice vectors and . The length of the quadratic unit cell is obtained from first-principle studies to be 3.64 Å6, which translates into a = 1.49 Å and b = 1.09 Å.
The real distances as obtained from first-principle calculations turn out to be slightly different, i.e., C1 − C2 = 1.55 Å and b = C2 − C2 = 1.34 Å, where there are two C1-atoms and four C2-atoms denoting the carbon atoms with sp3 and sp2-hybridization, respectively. Furthermore, the atoms are arranged in three different horizontal planes, of which the C1-atoms form the central plane and two of the four C2-atoms the upper and the lower plane, respectively. The total distance between the C1 and C2-atomic horizontal planes is h = 0.6 Å which yields the projected 2D-distance a = C1 − C2(projected) = 1.43 Å and b = C2 − C2 = 1.34 Å.
The distorted Cairo pentagonal patterning is shown on the left hand side of Fig. 1 where the C1 and C2-atoms are represented by red and black dots, respectively. The black horizontal and vertical bonds of length b connect the C2-atoms whereas the red bonds of the projected length a connect the C1 with the C2-atoms. There is a slight difference of 1% between the two bonds connecting the C1 − C2 atoms that form a pentagon, but this will be neglected. We also assume that the projected distance C1 − C2(projected) is the distance a of the regular Cairo patterning. The unit cell is denoted by the shaded square and consists of two C1-atoms and four C2-atoms.
Let us finally comment on the symmetry group. The three-dimensional lattice possesses the S4 point group and D2d full space group. The latter includes the following symmetry elements: one C2 axis along the direction perpendicular to the a1 − a2 plane, two C2′ axes perpendicular to C2, two dihedral planes σd bisecting the angles formed by pairs of the C2′ axes and two improper S4 axes. For the strictly two-dimensional lattice, the symmetry elements is doubled going from D2d to D4h full space group.
Density Functional Theory calculations
We calculate the band structure of pentagraphene using the VASP code13,14, based on density functional theory (DFT). All calculations were done using the Projector Augmented Wave potentials15 and employing the Perdew-Burke-Ernzerhof flavour of the generalized gradient approximation for the exchange-correlation functional16. Plane waves with a energy cut-off of 500 eV were employed to describe the valence electrons (2s2 2p2) of the C atoms. The employed Brillouin zone for the relaxation calculations is 9 × 9 × 1 in the Monkhorst-Pack scheme17, while high accurate electronic calculations to estimate the relative C1/C2 weight were performed using 11 × 11 × 1. The band structure of free standing pentagraphene layer was calculated for a fixed lattice constants of a = b = 3.64 Å6, and including a vacuum distance of around 20 Å. In all calculations the atomic positions are relaxed till forces are less than 0.015 e-/Å.
We find slightly larger atomic distances than the one cited in the previous subsection which were taken from ref. 6. The obtained band structure shows the expected narrowing of the band gap, common to all local and semi-local exchange-correlation approximations, which is corrected via a rigid shift of the conduction band to mimic the obtained direct Γ-Γ band gap using hybrid functionals6. After the rigid shift, the band dispersion shown in Fig. 2 as red stars agrees remarkably well to the more costly hybrid functionals.
In order to obtain more insight of how the atomic and orbital nature influences the electronic structure we plot on the right hand side of Fig. 2 the projected density of states (PDOS) of the valence bands (VBs) and conduction bands (CBs). We observe that the band gap after correction is of the order of 3.5 eV. For both the VBs and CBs there is a noticeable difference between the total and the s-orbital, evidencing that the highest contribution comes from the p orbital, whose contribution is above 94 and 90% respectively for the VB and CB. Similarly we can discuss the atomic contribution using the C1/C2 ratio which is less than 9% and 15% respectively for the VBs and CBs. We thus conclude that a four band model is an adequate approximation.
Despite the useful comparison obtained using the PDOS, the projected values should be considered qualitatively since the associated numerical errors are noticeable. These errors can be estimated as the difference between the real band charge, which is exactly 2 electrons, and the projected charge of the VBs and CBs, which is 1.5 and 1.35 electrons respectively.
Tight-binding and analytical approach
Our goal is to introduce a tight-binding model that only considers the four C2-atoms where the atoms A and B form the vertical dimer and the atoms C and D the horizontal dimers, see right hand side of Fig. 1. Both dimers are coupled by the hopping matrix element t0 which connects the two π-orbitals. We will set this to the typical value t0 = 2.7 eV5. The other hopping matrix elements involve hopping processes between the two dimers t and next-nearest hopping processes between the same dimers t′. To simplify our model, we will set t = t′ and determine its value from a fit to the DFT-band structure. As we will show, this model can also be well described analytically. But first, we will discuss the full tight-binding model including the six atoms of the unit cell and 4 orbitals within the Slater-Koster formalism18.
We start our tight-binding description with the full model that will include all six atoms of the unit cell and all four orbitals of the second atomic energy level, i.e., 2s, 2px, 2py, 2pz. The hopping parameters are chosen to be the one of graphene scaled by the corresponding distance and taken from ref. 19: Vssσ = −5.34 eV, Vspσ = 6.40 eV, Vppσ = 7.65 eV, and Vppπ = −2.80 eV. The atomic energy levels of the sp2-hybridized C2-atoms are εs = −2.85 eV, , ; the ones of the sp3-hybridized C1-atoms are chosen as εs = −4.05 eV, εp = 2 eV. The 24 × 24 Hamiltonian is obtained following the Slater-Koster parametrization including only nearest-neighbor hopping18.
In Fig. 2, the lowest energy bands obtained from the full tight-binding model are shown (black lines) and compared to the DFT-results (red stars). Qualitatively, the tight-binding can reproduce the lowest conduction bands and the highest valence bands. Still, quantitatively there are several disagreements: i) the energy gap is not accurately reproduced; ii) the splitting of the two conduction as well as of the two valence bands is not small; iii) bands further away from the Fermi level become worse and they can at most only be described qualitatively.
The above analysis does not contain any fitting parameter. Nevertheless, the quantitative disagreement suggests other terms to become important. In fact, in the case of the effective four-band model, we will be only able to adequately describe the bands by including an onsite interaction as well as an assisted hopping term. This shall be discussed in the following. Also the fitting parameters should be improved and extracted from first-principle calculations for penta-graphene, thus avoiding the use of parameters obtained for graphene.
Effective four-band model
As outlined above, we will build up an effective tight-binding model by only considering lattice sites with an unbounded π-electron. There are thus four atoms in the unit cell and direct hopping (t0) within the unit cell is only between the dimers A, B and C, D, respectively. All other hopping processes involve intermediate lattice sites with sp3-hybridisation and we assume that this hopping is the same (t = t′) and considerably less than the direct π − π hopping. The Hamiltonian is thus given by with:
where with σx, σy the Pauli-matrices and c1/2 = cos(k · a1/2), s1/2 = sin(k · a1/2). We further set E0 = 0. The coupling between the two dimers is given by
In Fig. 3, we compare the resulting band structure (black lines) to the full model obtained from DFT-calculations (red stars). Whereas the two valence bands (VBs) approximately agree with the exact first-principle band structure, the two conduction bands (CBs) do not show the large splitting in the Γ − X and Γ − M direction and seem to be inverted. Still, we want to emphasize that by setting t0 = 2.7 eV (which is the typical tight-binding hopping parameter of π-bands5) only one fitting parameter is involved by choosing t/t0 = 0.056. Further, we applied a rigid energy shift of E0 = 2.3 eV. Also, the basic features of the band structure are reproduced, i.e., an (no) energy separation of the valence (conduction) bands at the Γ-point and a two-fold degeneracy along the X − M direction.
The conduction bands can be considerably improved within our four-band model by introducing an energy-dependent hopping parameter. This term is formally obtained by including an effective pz-orbital of the sp3-hybridized C1-atoms which are then projected out, see appendix. The projected 4 × 4 Hamiltonian is again given by Eq. (1) with where the effective energy difference of the pz-atomic orbitals and . With , EX = −3.75 eV and an additional rigid energy shift , the eigenenergies of the conduction band are then obtained self-consistently to yield the upper blue lines of Fig. 3.
In order to justify the above parameters, it is necessary to include an onsite interaction at the C2-atoms and assisted hopping terms, see SI. Neglecting these correlations, the onsite energy shift yields EX = 2 eV and the two valence bands of the original four-band model with constant hopping amplitude t = 0.056 t0 are almost perfectly reproduced by the four-band model with energy-dependent hopping parameter of . Additionally, we introduced an energy shift of . This can be seen in Fig. 3 where the lower blue lines are almost on top of the black lines.
Our analysis indicates that correlation effects are important in order to reproduce the band-structure of the conduction band obtained from DFT-calculations and that electron-electron interactions are effectively screened in the valence band. The change in and the different constant energy shifts and further suggest an assisted hopping amplitude which depends on the occupation number of the C2-atoms20,21.
The inclusion of both correlation terms yields a consistent picture when choosing the Hubbard-interaction U ≈ 10 eV and the assisted hopping term W ≈ 2 eV. The assisted hopping term may further lead to a superconducting condensate when pumping electrons into the conduction bands20,21,22,23.
The eigenenergies of the bands around the neutrality point can be well approximated by the (extended) 4 × 4 model, reproducing correctly the degeneracy of the VBs and CBs along the X − M-direction and the band splitting along the other directions with a (zero) gap in the VBs (CBs). Nevertheless, to also obtain orthonormal eigenvectors, we will continue the discussion by approximating the energy-dependent hopping parameter by the constant hopping parameter t also for the conduction bands. In the following, we set t0 = 1 and assume . A first approximation is thus given by neglecting the inter-dimer coupling h12. Comparing this approximation with the exact numerical solution, we see that the conduction bands are well described by neglecting h12. For the eigenvalues, we then obtain with
However, the valence band experiences a splitting that cannot be account for by simply setting h12 = 0. Let thus U denote the unitary transformation which diagonalizes the Hamiltonian with h12 = 0 and lets only consider the upper left and lower right 2 × 2-matrix of U†HU that connects the two valence and conduction band states, respectively. Keeping now only terms of these matrices that are linear in t yields the linear approximation for and the following eigenvalues for the valence band:
The corresponding eigenvectors need to be transformed by U to obtain the approximate eigenstates of the original Hamiltonian. We thus end up with the following set of 4-dimensional eigenvectors:
with and .
The above approximation matches excellently the numerically exact band structure for , see Fig. 4.
Penta-graphene displays a band gap of 3.25 eV which makes it a potential candidate for blue absorption/emission. In order to discuss the optical response of the system, we will obtain the coupling Hamiltonian via the Peierls substitution in k-space. Nevertheless, for systems with various atoms in the unit cell, it is crucial to represent the Hamiltonian within a basis that distinguishes the relative phase of the atoms within the same unit cell24. The above representation has thus to be modified by using the k-states of the following basis:
Placing the origin of the unit cell in the middle of the AB-dimer and choosing it in the direction of the y-axis, we have δA = (0, −b/2), δB = (0, b/2), δC = (b+, 0), and δD = (b+ + b, 0) where . For the regular Cairo patterning, we have ; the unit-cell vectors are and with .
With respect to the new basis, the 2 × 2 Hamiltonians of Eq. (1) now read
The eigenvectors read
Fermi’s “golden rule”
We are interested in the system response at small fields and therefore expand the Hamiltonian up to terms linear in k. With (minimal coupling), we obtain the following coupling Hamiltonian:
with , and
where the velocities ħv = bt0 and .
We can now apply Fermi’s “golden rule” to calculate the absorption due to an incoming electric field E(t) = −∂tA. The incoming energy flux of a propagating sinusoidal linearly polarized electromagnetic plane wave of a fixed frequency is given by and the absorbed energy flux Wa = ηħω with η the transition rate. Since the momentum is conserved in the absorption process, only transitions from (k, v) to (k, c) are allowed. The total transition rate is then obtained by summing over all initial states k which yields:
where the sum goes over the first Brillouin zone. For a system with Nc = N2 unit cells, the possible k-values thus read:
with and .
Only considering the leading term in t0, we can neglect v12 and obtain for the matrix-elements:
By interchanging kx ↔ ky, we have φ+ ↔ φ−, indicating that the absorption displays the underlying four-fold symmetry within our approximation (degenerate conduction band). But we will neglect φ± in the following since it would yield another correction of order t. We can thus write the transition rate as
One measure for absorption is the joint density of states (DOS):
where the 2 comes from the two-fold degeneracy of the conduction band.
The DOS of the (degenerate) conduction band ρc is also given by Eq. (19) by setting and neglecting the summation over m. With , we can obtain an analytical expression:
where K(x) is the complete elliptic function. At the band-edges, we thus have a constant density of state of . At the band-center, there is a van-Hove singularity at the X-point with a logarithmically diverging density of states .
The DOS of the valence band (setting εc = 0) can also be expressed in closed form, but already this and especially the joint density of states are quite complicated analytical function and we will not pursue this any further. Instead, we will expand the dispersion around the high-symmetry points and calculate the optical absorption which should be particularly high at van-Hove singularities.
The optical absorption is defined by the ratio of the absorbed and the incoming energy flux, . The expansion around the Γ-point (Γ = 0) yields an isotropic absorption for both transitions from the two (m = ±)-valence bands. These are related to the transition energies ħω+ = 2t0 and ħω− = 2t0 + 8t, respectively. To leading order in t0/t, we get:
The numerical values are obtained from (b/a)2 ≈ 0.88 and t0/t ≈ 18 in units of the universal absorption of suspended graphene with the fine-structure constant25,26. This universal absorption is also present in InAs-quantum wells27 and other systems28. But here, the expression depends on material constants and is thus non-universal. It is also substantially higher with . This is remarkable having in mind that the maximal absorption of suspended two-dimensional materials is 50% and to our knowledge the highest value of a suspended two-dimensional system29. However, at such high absorption the simple “golden rule”-approach cannot be trusted anymore and more accurate calculations based on first principles are needed. In the Supplementary Information, we report on DFT-results for the optical conductivity which reaches absorption values up to .
In this work, we have investigated the possibility of an analytical description of penta-graphene. We were able to adequately approximate the two highest valence and lowest conduction bands within a simple four band model, but to match the conduction bands an energy dependent hopping element was necessary. To justify the parameters of the effective model, the inclusion of an onsite energy at the sp2-hybridized C2-atoms was necessary. This interaction should be screened for the valence band and we also expect an assisted hopping term in the effective model which might lead to a superconducting condensate at low temperatures when pumping electrons into the conduction bands20,21,22,23. A possible extension of our approach would include multi-orbitals in the spirit of the Weaire-Thorpe (WT) model30,31.
Using a constant hopping element between dimers, we were able to present an analytical solution for the eigenvalues as well as for the eigenvectors and calculated the optical absorption within this approximation. For small field strengths and for transitions around the Γ-point, this yielded an isotropic absorption of up to 24% which is remarkably large compared to usual two-dimensional materials such as graphene28. This suggests the interpretation that the quasi-isolated sp2-dimers are capable to act as antennas which efficiently couple to the incoming radiation.
It would be interesting to further investigate the influence of correlation effects on the band-structure as well as on the optical properties. Also the inclusion of electron-hole interaction and excitonic resonances are likely to change the absorption close to the band edges. Moreover, ferromagnetism is expected due to quasi-flat bands32. These issues shall be discussed in the future.
How to cite this article: Stauber, T. et al. Tight-binding approach to penta-graphene. Sci. Rep. 6, 22672; doi: 10.1038/srep22672 (2016).
We thank Félix Yndurain and Paul Wenk for useful discussions. This work has been supported by Spain’s MINECO under grants FIS2013-48048-P and FIS2014-57432-P, by the Comunidad de Madrid under grant S2013/MIT-3007 MAD2D-CM, and by Deutsche Forschungsgemeinschaft via GRK 1570. JIB thanks the ERC starting Investigator Award, grant #239739 STEMOX.
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