Penta-Graphene: Pentagonal 2D Carbon Allotrope
- Penta-graphene is a two-dimensional carbon allotrope featuring a pentagonal (Cairo-tessellation) structure with buckled geometry and mixed sp2/sp3 hybridization.
- Its electronic characteristics show a tunable band gapāfrom ~2.2 eV at the DFT level up to ~5.35 eV with GW correctionsāand unique optical excitations linked to its frontier bands.
- Mechanical tests reveal auxetic behavior with high elasticity under strain (up to ~20%) and diverse responses under defect engineering, doping, and functionalization.
Penta-graphene is a theoretically proposed two-dimensional carbon allotrope built entirely from pentagons in a Cairo-tessellation-like network, rather than from the hexagons of graphene. Across the literature it is described as a buckled or noncoplanar sheet with mixed - and -hybridized carbon atoms, a finite electronic band gap, unusual mechanical behavior including auxeticity, and a broad sensitivity to defects, functionalization, strain, and reduced dimensionality. It is also treated as experimentally elusive, which has made first-principles, tight-binding, and reactive-dynamics studies central to its characterization (Einollahzadeh et al., 2015, Pedrielli et al., 1 Aug 2025).
1. Atomic structure and crystallographic description
Penta-graphene is commonly described as a pentagon-only carbon network with a buckled geometry and mixed coordination. Several studies use a six-atom unit cell containing two -like carbons and four -like carbons, with a ratio of $1:2$, lattice constants , and a thickness of about (Einollahzadeh et al., 2015). In symmetry language, the structure is reported as in one band-structure study, while other works describe the lattice as or 0 and emphasize its nonsymmorphic character (Einollahzadeh et al., 2015, Bravo et al., 2017, Bravo et al., 2019).
The mixed bonding is central to essentially all subsequent discussions. The 1-like atoms buckle above and below the layer, whereas the 2-like atoms occupy three-coordinated environments. One optimized structure gives a single bond 3 of 4 and a double bond 5 of 6 (Einollahzadeh et al., 2015). A more detailed relaxed geometry study reports in-plane distances 7 Ć , 8 Ć , 9 Ć , 0 Ć , 1 Ć , 2 Ć , 3 Ć , 4 Ć , 5 Ć , and 6 Ć , out-of-plane distances 7 Ć , 8 Ć , 9 Ć , and bond angles 0, 1, and 2 (Lima et al., 2020).
Descriptions of the unit cell are not fully uniform across the provided literature. Several electronic-structure and tight-binding papers use six carbon atoms per unit cell, whereas one mechanical study states that the unit cell shown contains five carbon atoms (Stauber et al., 2015, Minaie et al., 2024, BrandĆ£o et al., 2021). This suggests differing cell conventions or structural representations across studies rather than a single universally adopted notation.
2. Electronic structure, many-body corrections, and optical response
Pristine penta-graphene is consistently treated as a semiconductor, but the reported gap depends strongly on the level of theory. DFT studies place the gap near the low-3 range: 4 eV with LDA-Teter, 5 eV with LDA-Hehin-Lundqvist, 6 eV with GGA-RPBE, 7 eV with GGA-Z. Wu, 8 eV with GGA-C09x, 9 eV with GGA-HTCH147, and 0 eV with GGA-HTCH407 (Einollahzadeh et al., 2015). Other DFT-level reports give a quasi-direct band gap of about 1 eV, an indirect gap of about 2 eV at the DFT-GGA level, a pristine value of about 3 eV, and 4 eV at PBE versus 5 eV with HSE06 (Lima et al., 2020, Minaie et al., 2024, Santos et al., 2020, Jia et al., 2022).
The indirect-gap character is described in some detail. One study places the valence-band maximum on the 6 path and the conduction-band minimum on the 7 path, with the gap lying between the 12th and 13th bands because the six-atom unit cell contains 24 valence electrons (Einollahzadeh et al., 2015). The same work notes that a sub-VBM on the 8 path is only about 9 below the true VBM, which motivates the widely used description of the material as quasi-direct (Einollahzadeh et al., 2015).
Quasiparticle corrections substantially enlarge the gap. One-shot $1:2$0 calculations give quasi-direct gaps of $1:2$1, $1:2$2, and $1:2$3, while a later many-body study reports a quasi-direct $1:2$4 gap of about $1:2$5 (Einollahzadeh et al., 2015, Minaie et al., 2024). In the latter work, Bethe-Salpeter calculations show that electron-hole interaction red-shifts the absorption spectrum relative to $1:2$6-RPA, with the optical response dominated by the first bound exciton at about $1:2$7 and a reported exciton binding energy of about $1:2$8 (Minaie et al., 2024).
| Level or context | Reported gap | Character |
|---|---|---|
| DFT-LDA/GGA | $1:2$9ā0 | indirect |
| DFT-GGA/PBE or related pristine values | 1ā2 | indirect or quasi-direct |
| HSE06 | 3 | indirect |
| 4 | 5ā6 | quasi-direct |
Beyond the gap magnitude, the frontier-band topology is unusual. One recent study identifies a tetragonal Mexican-hat valence-band edge with a shallow inverted shape, a local minimum at 7, and a van Hove singularity 8 near the band edge (Jia et al., 2022). This feature is tied mainly to the 9-like C2 atoms and becomes central in later discussions of doping-induced magnetism and Weyl states.
Low-energy effective modeling has proceeded along two main routes. A four-band 0-orbital model based only on the 1-hybridized carbon atoms reproduces the two highest valence bands, while the two lowest conduction bands are improved by integrating out the 2 carbons and introducing energy-dependent hopping together with a Hubbard onsite interaction and assisted hopping (Stauber et al., 2015). A separate 24-orbital SlaterāKoster model with 16 fitted parameters reproduces both the band structure and the linear optical response and extends naturally to nanoribbons (Bravo et al., 2017). In the four-band treatment, optical absorption at the 3-point is predicted to be isotropic and as large as 4, whereas away from 5 the absorption becomes strongly anisotropic with respect to linear polarization (Stauber et al., 2015).
3. Mechanical response, fracture, and thermal reconstruction
The mechanical literature treats penta-graphene as a buckled auxetic membrane with a large elastic response. A DFT-based study reports 6, 7, 8, a Youngās modulus of about 9, and a Poissonās ratio 0 (Sousa et al., 2017). Comparison values quoted in the same work include 1, 2, thickness 3 from earlier DFT, and 4, 5, thickness 6 from the authorsā own DFT calculations (Sousa et al., 2017).
Under tensile loading, the membrane is described as following two regimes: a linear elastic regime at small strain and a plastic regime involving bond rearrangement and re-hybridization. One combined DFT/MD study states that penta-graphene membranes can hold up to about 7 strain before fracture, with DFT ultimate strains of 8 for uniaxial loading and 9 for biaxial loading, and ultimate tensile strengths of about 0 for R0 uniaxial, 1 for R45 uniaxial, and 2 for biaxial loading (Sousa et al., 2017). Fracture is accompanied by the formation of 7-, 8-, and 11-membered rings together with carbon chains of polyyne-like character (Sousa et al., 2017).
Reactive molecular dynamics under elevated temperature shows marked degradation of these properties. Non-equilibrium MD with ReaxFF reports Youngās modulus values from 3 at 4 to 5 at 6, ultimate tensile stress from 7 at 8 to 9 at 00, and critical strain from 01 at 02 to 03 at 04 (BrandĆ£o et al., 2021). The same study reports that the material largely preserves its non-coplanar pentagonal structure from 05 to 06, begins losing symmetry around 07, and at 08 reconstructs into graphene islands, large porous regions, small 1D carbon chains, and negatively curved layers (BrandĆ£o et al., 2021).
These results coexist with earlier statements that penta-graphene is mechanically and dynamically stable up to 09 and has a quasi-direct gap of about 10 (Wu et al., 2016). A plausible implication is that āstabilityā is being used in multiple senses across the literature, including phonon stability, persistence of the ideal topology under MD, and preservation of strength under thermal disorder.
4. Defects, adsorption, doping, and functionalization
Defects strongly reshape the local chemistry of penta-graphene. In the specific case of oxygen adsorption on defective lattices, two monovacancies have been compared: PG@A at an 11-hybridized site and PG@B at an 12-hybridized site (Lima et al., 2020). The 13-vacancy remains comparatively open and retains more nonbonding-electron character, whereas the 14-vacancy reconstructs more strongly and suppresses nonbonding density (Lima et al., 2020). Adsorption curves fitted with the Improved Lennard-Jones potential give the following 15 and 16 values:
| Configuration | 17 (eV) | 18 (19) |
|---|---|---|
| PG/20-H | 21 | 22 |
| PG/23-V | 24 | 25 |
| PG@A/26-H | 27 | 28 |
| PG@A/29-V | 30 | 31 |
| PG@B/32-H | 33 | 34 |
| PG@B/35-V | 36 | 37 |
The standout case is PG@A/38-H, which is described as chemisorption and has the largest adsorption energy, at least twice that of the other systems considered (Lima et al., 2020). The same work reports short recovery times on the order of picoseconds for the stronger-binding PG@A cases, including 39 for PG@A/40-H and about 41 for PG@A/42-V (Lima et al., 2020). Pristine penta-graphene shows only weak perturbation under 43 adsorption, including roughly 44 shifts of valence and conduction bands but essentially no band-gap change, whereas PG@A exhibits orientation-dependent band-structure changes and flat midgap states (Lima et al., 2020).
Electronic tuning by engineered line defects is similarly rich. Substitutional N or Si arranged as 1 to 7 defect lines can drive penta-graphene between semiconductor, semimetallic, and metallic behavior depending on dopant species and whether the target site is 45-like or 46-like (Santos et al., 2020). For N at 47-like sites, one defect line reduces the gap from 48 to 49, two to three lines produce semimetallic behavior, and additional lines produce metallic behavior (Santos et al., 2020). For N at 50-like sites, even numbers of defect lines remain semiconducting with gaps decreasing to about 51 for six lines, whereas odd numbers of defect lines are semimetallic with very small DOS near the Fermi level (Santos et al., 2020). The same study notes that 52-site doping preserves structural stability better than 53-site doping, based on cohesive-energy trends (Santos et al., 2020).
Bilayer penta-graphene adds another layer of site selectivity. In substitutionally doped bilayers, boron produces magnetic moments of about 54 for B-55-in, 56 for B-57, and 58 for B-59-out, while oxygen produces no magnetization in O-60-in or O-61-out but 62 in O-63 (Santos et al., 2020). The pristine bilayer retains an almost indirect band gap of 64, B doping reduces the gap to about 65, and O doping gives gaps around 66ā67 depending on site (Santos et al., 2020).
Hydrogenation defines a particularly important derivative. First-principles lattice-dynamics calculations report a room-temperature thermal conductivity of 68 for penta-graphene and 69 for hydrogenated penta-graphene, a 70 increase, whereas hydrogenation of graphene reduces thermal conductivity from 71 to 72, a 73 reduction (Wu et al., 2016). The microscopic explanation given is weaker bond anharmonicity in hydrogenated penta-graphene despite an increased phonon scattering phase space (Wu et al., 2016).
5. Derived nanostructures, symmetry engineering, and emergent phases
Reduced-dimensional and symmetry-preserving derivatives substantially extend the penta-graphene concept. Rolled penta-graphene nanotubes inherit the pentagonal precursor but display mechanical behavior unlike conventional carbon nanotubes. For a representative 74 penta-graphene nanotube, the elastic regime extends to 75, the yield stress is about 76, plastic deformation continues up to about 77, the failure strain is about 78, the failure strength about 79, and the Youngās modulus about 80 from stress-strain fitting (Chen et al., 2017). The plasticity is tied to an irreversible pentagon-to-polygon transformation in which hexagons become the dominant motif, and it is reported to be largely independent of diameter, strain rate, and temperature up to about 81 (Chen et al., 2017).
Nanoribbons expose a different regime. A hydrogen-passivated penta-graphene nanoribbon with nine sawtooth carbon chains is dynamically stable and shows an indirect band gap of 82 in AMS and 83 in VASP (Tran et al., 2024). Adsorption of one Li atom induces a semiconductor-to-metal transition with a formation energy of 84, and a perpendicular electric field lowers the Li migration barrier from 85 at zero field to 86 at 87 (Tran et al., 2024). The corresponding diffusion coefficient rises from 88 at zero field to 89 at 90, about 91 faster than the zero-field case and about 92 higher than commercial graphitic carbon layers according to that study (Tran et al., 2024).
Penta-graphene also serves as the structural parent of a wider family of āpenta-materialsā that retain the same nonsymmorphic symmetry while changing electron count through adsorption or substitution. Li- and Na-adsorbed derivatives are reported as nodal-line metals with a continuum of Dirac points around the perimeter of the Brillouin zone, penta-PC93 is a substitutional derivative that becomes a spin-orbit Dirac-node metal, and magnetic penta-MnC94 is reported as a topological insulator with Chern numbers 95 and 96 (Bravo et al., 2019). The organizing principle in that work is filling-enforced metallicity or topology within the penta-graphene space group (Bravo et al., 2019).
Directly on penta-graphene itself, modest hole doping exploits the tetragonal Mexican-hat valence band and associated van Hove singularity. First-principles calculations report ferromagnetism with magnetic moment up to 97 for hole densities from 98 to 99, and Curie temperatures above room temperature for hole densities from about 00 to 01, peaking at about 02 at 03 (Jia et al., 2022). The same study predicts a sequence 04 under gating, together with type-I and type-II Weyl cones and a hybrid quasi-Weyl nodal loop under suitable strain (Jia et al., 2022).
A more recent extension is fully three-dimensional. Three 3D allotropes derived from biaxially strained and compressed penta-graphene layers, 3D-PG-05, 3D-PG-06, and 3D-PG-07, are reported as dynamically and thermally stable semiconductors with indirect gaps of 08, 09, and 10, respectively, and strong mechanical and optical anisotropy (FĆ©lix et al., 12 Sep 2025). These structures retain the pentagon-derived mixed 11 framework while introducing interlayer CāC bonds (FĆ©lix et al., 12 Sep 2025).
6. Spectroscopic identification, transformations, and research status
A recurring issue in the literature is that penta-graphene is treated as theoretically predicted but experimentally elusive. One consequence is the search for diagnostic spectroscopies. Ab initio XANES calculations identify two inequivalent carbon-site fingerprints in pristine penta-graphene: for the three-coordinated 12 site there is a peak near 13 associated with 14-bonding states, plus higher-energy features around 15, 16, and 17, whereas the four-coordinated 18 site lacks the low-energy 19 peak (Pedrielli et al., 1 Aug 2025). The averaged C K-edge spectrum is described as three plateaus at 20, 21, and 22, with pronounced polarization anisotropy (Pedrielli et al., 1 Aug 2025). Hydrogenation and hydroxylation suppress the low-energy plateau, and Si substitution introduces Si K-edge structures in the 23 range (Pedrielli et al., 1 Aug 2025). This establishes XANES as a proposed route for experimental identification.
The question of structural fate under load remains contested. One DFT study reports that penta-graphene undergoes a sudden global transformation at about 24 uniaxial strain into planar biphenylene, accompanied by a sharp drop in energy and stress; the final biphenylene phase is described as metallic, energetically lower than penta-graphene, and dynamically, mechanically, thermally, and electronically stable (Rahaman et al., 2017). By contrast, a separate DFT/ReaxFF fracture study concludes that a mechanically induced transition from penta-graphene to graphene is unlikely, reports no hexagons during tensile fracture, and attributes discrepancies with earlier transformation claims in part to the choice of ReaxFF parameters (Sousa et al., 2017). A plausible reading of these results is that pathway sensitivity, model selection, and the distinction between biphenylene formation and graphene formation are all central to the ongoing interpretation of penta-grapheneās metastability.
The same pattern appears in electromechanical derivatives. Pure penta-graphene, described there as the CCC monolayer, has only a very small pure out-of-plane piezoelectric response, 25, and strain engineering changes it only modestly to 26 at 27 biaxial strain or 28 at 29 uniaxial strain (Guo, 2019). A Janus derivative CCB, obtained by replacing one atomic layer, remains dynamically and mechanically stable, semiconducting with an indirect gap of 30, and exhibits much larger out-of-plane coefficients 31 and 32, together with room-temperature electron mobility along 33 of 34 (Guo, 2019). This suggests that, within the penta-graphene family, asymmetry engineering is more effective than strain alone for producing strong vertical piezoelectric functionality.
Taken together, the literature presents penta-graphene as a pentagonal, mixed-hybridization carbon platform with semiconducting electronic structure, strong method dependence in its quasiparticle gap, nontrivial fracture chemistry, marked sensitivity to vacancies and substitution, and an unusual capacity to generate chemically or symmetry engineered descendants ranging from high-35 thermal derivatives to nodal-line metals, half-metals, Weyl states, piezoelectric Janus layers, nanoribbon battery anodes, nanotubes with topology-driven plasticity, and strain-generated 3D allotropes (Wu et al., 2016, Bravo et al., 2019, Tran et al., 2024). The experimentally unresolved status of the phase, together with the diversity of predicted transformations and fingerprints, remains the central context for current research.