---
title: 'Penta-Graphene: Pentagonal 2D Carbon Allotrope'
url: https://www.emergentmind.com/topics/penta-graphene
type: topic
---

# Penta-Graphene: Pentagonal 2D Carbon Allotrope

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 \(sp^2\)- and \(sp^3\)-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 [1509.04512][2508.00704].

## 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 \(sp^3\)-like carbons and four \(sp^2\)-like carbons, with a \(C1:C2\) ratio of \(1:2\), lattice constants \(a=b=3.64\ \text{\AA}\), and a thickness of about \(1.2\ \text{\AA}\) [1509.04512]. In symmetry language, the structure is reported as \(p\text{-}421m\) in one band-structure study, while other works describe the lattice as \(P\overline{4}2_1m\) or \(P\bar{4}2_1mm\) and emphasize its nonsymmorphic character [1509.04512][1712.07544][1905.07381].

The mixed bonding is central to essentially all subsequent discussions. The \(sp^3\)-like atoms buckle above and below the layer, whereas the \(sp^2\)-like atoms occupy three-coordinated environments. One optimized structure gives a single bond \(C1{-}C2\) of \(1.55\ \text{\AA}\) and a double bond \(C2{-}C2\) of \(1.34\ \text{\AA}\) [1509.04512]. A more detailed relaxed geometry study reports in-plane distances \(d_1=1.34\) Å, \(d_2=1.54\) Å, \(d_3=1.55\) Å, \(d_4=1.33\) Å, \(d_5=1.50\) Å, \(d_6=1.37\) Å, \(d_7=1.41\) Å, \(d_8=1.45\) Å, \(d_9=1.51\) Å, and \(d_{10}=1.51\) Å, out-of-plane distances \(h_1=0.039\) Å, \(h_2=0.050\) Å, \(h_3=0.047\) Å, and bond angles \(\theta_1=135.01^\circ\), \(\theta_2=139.21^\circ\), and \(\theta_3=137.63^\circ\) [2007.12224].

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 [1509.08651][2401.05429][2105.08797]. 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-\(2\ \text{eV}\) range: \(2.22\) eV with LDA-Teter, \(2.21\) eV with LDA-Hehin-Lundqvist, \(2.29\) eV with GGA-RPBE, \(2.25\) eV with GGA-Z. Wu, \(2.22\) eV with GGA-C09x, \(2.34\) eV with GGA-HTCH147, and \(2.36\) eV with GGA-HTCH407 [1509.04512]. Other DFT-level reports give a quasi-direct band gap of about \(2.35\) eV, an indirect gap of about \(2.32\) eV at the DFT-GGA level, a pristine value of about \(2.4\) eV, and \(2.21\) eV at PBE versus \(3.28\) eV with HSE06 [2007.12224][2401.05429][2001.06062][2210.12697].

The indirect-gap character is described in some detail. One study places the valence-band maximum on the \(T-X\) path and the conduction-band minimum on the \(M-I\) path, with the gap lying between the 12th and 13th bands because the six-atom unit cell contains 24 valence electrons [1509.04512]. The same work notes that a sub-VBM on the \(M-I\) path is only about \(0.01\ \text{eV}\) below the true VBM, which motivates the widely used description of the material as quasi-direct [1509.04512].

Quasiparticle corrections substantially enlarge the gap. One-shot \(G_0W_0\) calculations give quasi-direct gaps of \(4.10\), \(4.14\), and \(4.28\ \text{eV}\), while a later many-body study reports a quasi-direct \(G_0W_0\) gap of about \(5.35\ \text{eV}\) [1509.04512][2401.05429]. In the latter work, Bethe-Salpeter calculations show that electron-hole interaction red-shifts the absorption spectrum relative to \(G_0W_0\)-RPA, with the optical response dominated by the first bound exciton at about \(2.46\ \text{eV}\) and a reported exciton binding energy of about \(3.07\ \text{eV}\) [2401.05429].

| Level or context | Reported gap | Character |
|---|---:|---|
| DFT-LDA/GGA | \(2.21\)–\(2.36\ \text{eV}\) | indirect |
| DFT-GGA/PBE or related pristine values | \(2.21\)–\(2.40\ \text{eV}\) | indirect or quasi-direct |
| HSE06 | \(3.28\ \text{eV}\) | indirect |
| \(G_0W_0\) | \(4.10\)–\(5.35\ \text{eV}\) | 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 \(\Gamma\), and a van Hove singularity \(D(E)\sim 1/\sqrt{E}\) near the band edge [2210.12697]. This feature is tied mainly to the \(sp^2\)-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 \(\pi\)-orbital model based only on the \(sp^2\)-hybridized carbon atoms reproduces the two highest valence bands, while the two lowest conduction bands are improved by integrating out the \(sp^3\) carbons and introducing energy-dependent hopping together with a Hubbard onsite interaction and assisted hopping [1509.08651]. A separate 24-orbital Slater–Koster model with 16 fitted parameters reproduces both the band structure and the linear optical response and extends naturally to nanoribbons [1712.07544]. In the four-band treatment, optical absorption at the \(\Gamma\)-point is predicted to be isotropic and as large as \(24\%\), whereas away from \(\Gamma\) the absorption becomes strongly anisotropic with respect to linear polarization [1509.08651].

## 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 \(C_{11}=277.5\ \text{GPa}\cdot\text{nm}\), \(C_{12}=-26.7\ \text{GPa}\cdot\text{nm}\), \(C_{11}+C_{12}=250.8\ \text{GPa}\cdot\text{nm}\), a Young’s modulus of about \(274.95\ \text{GPa}\cdot\text{nm}\), and a Poisson’s ratio \(\nu=-0.096\) [1703.03789]. Comparison values quoted in the same work include \(Y=263.8\ \text{GPa}\cdot\text{nm}\), \(\nu=-0.068\), thickness \(1.20\ \text{\AA}\) from earlier DFT, and \(Y=257.6\ \text{GPa}\cdot\text{nm}\), \(\nu=-0.096\), thickness \(1.23\ \text{\AA}\) from the authors’ own DFT calculations [1703.03789].

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 \(20\%\) strain before fracture, with DFT ultimate strains of \(19.5\%\) for uniaxial loading and \(23\%\) for biaxial loading, and ultimate tensile strengths of about \(\sim 38\ \text{GPa}\cdot\text{nm}\) for R0 uniaxial, \(\sim 29\ \text{GPa}\cdot\text{nm}\) for R45 uniaxial, and \(\sim 52\ \text{GPa}\cdot\text{nm}\) for biaxial loading [1703.03789]. Fracture is accompanied by the formation of 7-, 8-, and 11-membered rings together with carbon chains of polyyne-like character [1703.03789].

Reactive molecular dynamics under elevated temperature shows marked degradation of these properties. Non-equilibrium MD with ReaxFF reports Young’s modulus values from \(222.80 \pm 1.95\ \text{GPa}\cdot\text{nm}\) at \(200\ \text{K}\) to \(154.76 \pm 3.81\ \text{GPa}\cdot\text{nm}\) at \(1000\ \text{K}\), ultimate tensile stress from \(35.88 \pm 0.19\ \text{GPa}\cdot\text{nm}\) at \(200\ \text{K}\) to \(11.83 \pm 0.51\ \text{GPa}\cdot\text{nm}\) at \(1000\ \text{K}\), and critical strain from \(0.22\) at \(200\ \text{K}\) to \(0.06\) at \(1000\ \text{K}\) [2105.08797]. The same study reports that the material largely preserves its non-coplanar pentagonal structure from \(10\ \text{K}\) to \(600\ \text{K}\), begins losing symmetry around \(900\ \text{K}\), and at \(2000\ \text{K}\) reconstructs into graphene islands, large porous regions, small 1D carbon chains, and negatively curved layers [2105.08797].

These results coexist with earlier statements that penta-graphene is mechanically and dynamically stable up to \(\sim 1000\ \text{K}\) and has a quasi-direct gap of about \(3.25\ \text{eV}\) [1604.03201]. A plausible implication is that “stability” 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 \(sp^3\)-hybridized site and PG@B at an \(sp^2\)-hybridized site [2007.12224]. The \(sp^3\)-vacancy remains comparatively open and retains more nonbonding-electron character, whereas the \(sp^2\)-vacancy reconstructs more strongly and suppresses nonbonding density [2007.12224]. Adsorption curves fitted with the Improved Lennard-Jones potential give the following \(\epsilon\) and \(r_m\) values:

| Configuration | \(\epsilon\) (eV) | \(r_m\) (\(\text{\AA}\)) |
|---|---:|---:|
| PG/\(\mathrm{O_2}\)-H | \(0.39\) | \(2.99\) |
| PG/\(\mathrm{O_2}\)-V | \(0.35\) | \(3.55\) |
| PG@A/\(\mathrm{O_2}\)-H | \(2.08\) | \(1.72\) |
| PG@A/\(\mathrm{O_2}\)-V | \(0.82\) | \(2.03\) |
| PG@B/\(\mathrm{O_2}\)-H | \(0.46\) | \(2.29\) |
| PG@B/\(\mathrm{O_2}\)-V | \(-0.53\) | \(2.55\) |

The standout case is PG@A/\(\mathrm{O_2}\)-H, which is described as chemisorption and has the largest adsorption energy, at least twice that of the other systems considered [2007.12224]. The same work reports short recovery times on the order of picoseconds for the stronger-binding PG@A cases, including \(\tau=2.31\ \text{ps}\) for PG@A/\(\mathrm{O_2}\)-H and about \(1.5\ \text{ps}\) for PG@A/\(\mathrm{O_2}\)-V [2007.12224]. Pristine penta-graphene shows only weak perturbation under \(\mathrm{O_2}\) adsorption, including roughly \(-1.1\ \text{eV}\) 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 [2007.12224].

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 \(sp^2\)-like or \(sp^3\)-like [2001.06062]. For N at \(sp^3\)-like sites, one defect line reduces the gap from \(2.4\ \text{eV}\) to \(1.5\ \text{eV}\), two to three lines produce semimetallic behavior, and additional lines produce metallic behavior [2001.06062]. For N at \(sp^2\)-like sites, even numbers of defect lines remain semiconducting with gaps decreasing to about \(0.6\ \text{eV}\) for six lines, whereas odd numbers of defect lines are semimetallic with very small DOS near the Fermi level [2001.06062]. The same study notes that \(sp^2\)-site doping preserves structural stability better than \(sp^3\)-site doping, based on cohesive-energy trends [2001.06062].

Bilayer penta-graphene adds another layer of site selectivity. In substitutionally doped bilayers, boron produces magnetic moments of about \(1.00\ \mu_B\) for B-\(sp^2\)-in, \(0.93\ \mu_B\) for B-\(sp^3\), and \(1.00\ \mu_B\) for B-\(sp^2\)-out, while oxygen produces no magnetization in O-\(sp^2\)-in or O-\(sp^2\)-out but \(2.00\ \mu_B\) in O-\(sp^3\) [2007.10941]. The pristine bilayer retains an almost indirect band gap of \(2.3\ \text{eV}\), B doping reduces the gap to about \(2.0\ \text{eV}\), and O doping gives gaps around \(2.1\)–\(2.2\ \text{eV}\) depending on site [2007.10941].

Hydrogenation defines a particularly important derivative. First-principles lattice-dynamics calculations report a room-temperature thermal conductivity of \(350\ \text{W/mK}\) for penta-graphene and \(615\ \text{W/mK}\) for hydrogenated penta-graphene, a \(76\%\) increase, whereas hydrogenation of graphene reduces thermal conductivity from \(3590\) to \(1328\ \text{W/mK}\), a \(63\%\) reduction [1604.03201]. The microscopic explanation given is weaker bond anharmonicity in hydrogenated penta-graphene despite an increased phonon scattering phase space [1604.03201].

## 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 \((8,8)\) penta-graphene nanotube, the elastic regime extends to \(\varepsilon \approx 0.062\), the yield stress is about \(56\ \text{GPa}\), plastic deformation continues up to about \(\varepsilon \approx 0.61\), the failure strain is about \(0.61\), the failure strength about \(90\ \text{GPa}\), and the Young’s modulus about \(931\ \text{GPa}\) from stress-strain fitting [1709.08330]. 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 \(1100\ \text{K}\) [1709.08330].

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 \(3.415\ \text{eV}\) in AMS and \(3.433\ \text{eV}\) in VASP [2406.13096]. Adsorption of one Li atom induces a semiconductor-to-metal transition with a formation energy of \(-0.220\ \text{eV}\), and a perpendicular electric field lowers the Li migration barrier from \(0.272\ \text{eV}\) at zero field to \(0.102\ \text{eV}\) at \(2\ \text{V/nm}\) [2406.13096]. The corresponding diffusion coefficient rises from \(3.2\times 10^{-6}\ \text{cm}^2/\text{s}\) at zero field to \(2.3\times 10^{-3}\ \text{cm}^2/\text{s}\) at \(2\ \text{V/nm}\), about \(719\times\) faster than the zero-field case and about \(521\times\) higher than commercial graphitic carbon layers according to that study [2406.13096].

Penta-graphene also serves as the structural parent of a wider family of “penta-materials” 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-PC\(_2\) is a substitutional derivative that becomes a spin-orbit Dirac-node metal, and magnetic penta-MnC\(_2\) is reported as a topological insulator with Chern numbers \(C_x=1\) and \(C_y=-1\) [1905.07381]. The organizing principle in that work is filling-enforced metallicity or topology within the penta-graphene space group [1905.07381].

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 \(1\ \mu_B/\text{hole}\) for hole densities from \(5.60\times10^{14}\ \text{cm}^{-2}\) to \(1.28\times10^{15}\ \text{cm}^{-2}\), and Curie temperatures above room temperature for hole densities from about \(7.36\times10^{14}\ \text{cm}^{-2}\) to \(1.37\times10^{15}\ \text{cm}^{-2}\), peaking at about \(972\ \text{K}\) at \(1.28\times10^{15}\ \text{cm}^{-2}\) [2210.12697]. The same study predicts a sequence \(\text{d-HM} \leftrightarrow \text{b-SC} \leftrightarrow \text{u-HM}\) under gating, together with type-I and type-II Weyl cones and a hybrid quasi-Weyl nodal loop under suitable strain [2210.12697].

A more recent extension is fully three-dimensional. Three 3D allotropes derived from biaxially strained and compressed penta-graphene layers, 3D-PG-\(\alpha\), 3D-PG-\(\beta\), and 3D-PG-\(\gamma\), are reported as dynamically and thermally stable semiconductors with indirect gaps of \(0.91\), \(2.67\), and \(1.76\ \text{eV}\), respectively, and strong mechanical and optical anisotropy [2509.10191]. These structures retain the pentagon-derived mixed \(sp^2/sp^3\) framework while introducing interlayer C–C bonds [2509.10191].

## 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 \(sp^2\) site there is a peak near \(285\ \text{eV}\) associated with \(\pi\)-bonding states, plus higher-energy features around \(293\), \(298\), and \(303\ \text{eV}\), whereas the four-coordinated \(sp^3\) site lacks the low-energy \(\pi\) peak [2508.00704]. The averaged C K-edge spectrum is described as three plateaus at \(283\text{–}288\ \text{eV}\), \(290\text{–}300\ \text{eV}\), and \(>305\ \text{eV}\), with pronounced polarization anisotropy [2508.00704]. Hydrogenation and hydroxylation suppress the low-energy plateau, and Si substitution introduces Si K-edge structures in the \(1840\text{–}1860\ \text{eV}\) range [2508.00704]. 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 \(20\%\) 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 [1703.09071]. 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 [1703.03789]. 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-graphene’s 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, \(d_{36}=-0.065\ \text{pm/V}\), and strain engineering changes it only modestly to \(-0.067\ \text{pm/V}\) at \(-2\%\) biaxial strain or \(-0.074\ \text{pm/V}\) at \(-4\%\) uniaxial strain [1911.03050]. A Janus derivative CCB, obtained by replacing one atomic layer, remains dynamically and mechanically stable, semiconducting with an indirect gap of \(0.66\ \text{eV}\), and exhibits much larger out-of-plane coefficients \(d_{31}=-0.505\ \text{pm/V}\) and \(d_{32}=0.273\ \text{pm/V}\), together with room-temperature electron mobility along \(y\) of \(8865.23\ \mathrm{cm^2V^{-1}s^{-1}}\) [1911.03050]. 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-\(\kappa\) 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 [1604.03201][1905.07381][2406.13096]. The experimentally unresolved status of the phase, together with the diversity of predicted transformations and fingerprints, remains the central context for current research.

Source: https://www.emergentmind.com/topics/penta-graphene