TODD-Graphene: Porous 2D Carbon Allotrope

• TODD-Graphene is a 2D porous carbon allotrope defined by a 3-8-10-12 ring network that exhibits strong anisotropic mechanical and electronic characteristics.
• It is computationally validated via density functional theory and molecular dynamics to possess metallic behavior, robust thermal stability, and distinct optical responses.
• The structure facilitates lithium adsorption and diffusion, making it a promising candidate for lithium-ion battery anode applications.

Searching arXiv for TODD-Graphene and closely related papers to ground the article in the current literature. TODD-Graphene (TODD-G) is a two-dimensional planar carbon allotrope introduced as a porous carbon network composed of interconnected 3-, 8-, 10-, and 12-membered rings and investigated primarily through density functional theory, ab initio molecular dynamics, and classical reactive molecular dynamics simulations (Santos et al., 2023). In the reported first-principles characterization, TODD-G is described as intrinsically metallic, structurally porous, and mechanically anisotropic, with calculated dynamical, thermal, and mechanical stability and with lithium adsorption and diffusion characteristics that motivate its consideration as an anode material for lithium-ion batteries (Santos et al., 2023). Subsequent work extended the “TODD-G” label to structurally distinct allotropes—$\alpha$-, $\beta$-, and $\gamma$-TODD-G—whose ring topologies, hybridization patterns, and anisotropies differ from the original 3-8-10-12 network (Lima et al., 7 Aug 2025).

1. Definition and structural identity

TODD-Graphene was introduced as a novel 2D planar carbon allotrope with a porous structure composed of 3-8-10-12 carbon rings (Santos et al., 2023). The reported structure is orthorhombic, with space group P1, and its unit cell contains 14 carbon atoms with lattice constants $a = 7.03\ \text{Å}$ and $b = 6.54\ \text{Å}$ (Santos et al., 2023). Bond distances vary between $1.368\ \text{Å}$ and $1.435\ \text{Å}$, and the planar density is given as $0.30\ \text{atom/Å}^2$, compared in the source with graphene ($0.38\ \text{atom/Å}^2$), graphdiyne ($0.23\ \text{atom/Å}^2$), and graphyne ($\beta$0) (Santos et al., 2023).

The defining geometric feature of TODD-G is its extensive porous network built from large 8-, 10-, and 12-membered rings (Santos et al., 2023). In the original study, this topology is directly connected to two proposed functional consequences: periodically distributed pores that facilitate ion transport and an enlarged set of adsorption sites for Li atoms (Santos et al., 2023). This suggests that the structural identity of TODD-G is not merely topological but also application-oriented, since pore geometry, ring size distribution, and planarity are treated as central determinants of electrochemical behavior.

A later paper broadened the nomenclature by introducing $\beta$1-, $\beta$2-, and $\beta$3-TODD-G as three additional 2D planar carbon allotropes (Lima et al., 7 Aug 2025). These are not variants of the original 3-8-10-12 framework in a narrow crystallographic sense; rather, they are distinct networks composed of 3-8-12-16, 3-8-12-16, and 3-4-8-12 interconnected carbon rings with $\beta$4 hybridization, respectively (Lima et al., 7 Aug 2025). A common misconception is therefore to treat “TODD-G” as a single uniquely defined lattice. The literature instead supports two usages: the original TODD-G denotes the 3-8-10-12 porous allotrope (Santos et al., 2023), whereas the later family designation encompasses multiple planar carbon phases with different ring connectivities and anisotropies (Lima et al., 7 Aug 2025).

2. Stability and energetics

The original TODD-G study reports a calculated formation energy of $\beta$5, noted to be higher, i.e. less negative, than graphene’s $\beta$6, but still consistent with stable carbon allotropes (Santos et al., 2023). Dynamical stability is supported by phonon dispersion calculations showing no imaginary frequencies, with the highest phonon frequency reaching $\beta$7, which the source attributes to stiff bonds, particularly in the fused 3-8-membered rings (Santos et al., 2023).

Thermal stability was assessed using both ab initio and classical reactive dynamics (Santos et al., 2023). In AIMD at $\beta$8 for $\beta$9, the total energy per atom remained stable and no bond breaking or network reconstruction was observed (Santos et al., 2023). In ReaxFF simulations with a heating ramp from $\gamma$0 up to $\gamma$1, the structure was reported to remain intact up to $\gamma$2 with no reconstructions, while the melting point was estimated to be approximately $\gamma$3, above which the network decomposed into linear atomic chains and clusters (Santos et al., 2023). The paper’s abstract highlights the same conclusion in condensed form, stating that classical reactive MD simulations suggested structural integrity with no bond reconstructions at $\gamma$4 (Santos et al., 2023).

Mechanical stability was evaluated from LDA elastic constants. The reported values are $\gamma$5, $\gamma$6, $\gamma$7, and $\gamma$8, and the structure is said to satisfy the Born-Huang stability criteria for orthorhombic crystals (Santos et al., 2023). The directional Young’s modulus is highly anisotropic, with $\gamma$9 in the $a = 7.03\ \text{Å}$0-direction and $a = 7.03\ \text{Å}$1 in the $a = 7.03\ \text{Å}$2-direction, while the maximum Poisson’s ratio is reported as $a = 7.03\ \text{Å}$3 (Santos et al., 2023). Relative to graphene, whose modulus is cited in the source as approximately $a = 7.03\ \text{Å}$4, TODD-G is mechanically softer yet still stable (Santos et al., 2023).

The later $a = 7.03\ \text{Å}$5-, $a = 7.03\ \text{Å}$6-, and $a = 7.03\ \text{Å}$7-TODD-G study likewise reports planarity, no imaginary phonon frequencies, and AIMD stability at $a = 7.03\ \text{Å}$8 for $a = 7.03\ \text{Å}$9 for all three phases (Lima et al., 7 Aug 2025). Their cohesive energies are listed as $b = 6.54\ \text{Å}$0, $b = 6.54\ \text{Å}$1, and $b = 6.54\ \text{Å}$2 for $b = 6.54\ \text{Å}$3-, $b = 6.54\ \text{Å}$4-, and $b = 6.54\ \text{Å}$5-TODD-G, respectively (Lima et al., 7 Aug 2025). This suggests that stability, as used in the TODD-G literature, is established through a common triad of energetic, phononic, and finite-temperature criteria rather than through a single descriptor.

3. Electronic structure and optical response

In the original report, both DFT-PBE and HSE06 calculations indicate metallic character for TODD-G (Santos et al., 2023). The HSE06 calculation is described as opening only a minute gap of approximately $b = 6.54\ \text{Å}$6, while the band dispersion is reported to be anisotropic, metallic along Q-Z and semiconducting along other directions (Santos et al., 2023). The density of states near the Fermi level is dominated by $b = 6.54\ \text{Å}$7-states, mainly $b = 6.54\ \text{Å}$8, which the study interprets as evidence of robust $b = 6.54\ \text{Å}$9-conjugation and metallicity; a Dirac point slightly above the Fermi level is also reported (Santos et al., 2023).

These features are significant because the original battery-oriented interpretation depends not only on porosity and Li adsorption but also on a conducting host lattice (Santos et al., 2023). The paper explicitly links the metallic framework and significant density of states at the Fermi level to high conductivity (Santos et al., 2023). Strain engineering is treated more conservatively: the band structure is said not to be readily tunable by strain, and any strain-induced gap opening remains minor, with a maximum of approximately $1.368\ \text{Å}$0 under tensile strain along the $1.368\ \text{Å}$1-direction (Santos et al., 2023).

The same study also reports anisotropic optical properties (Santos et al., 2023). The refractive index is described as anisotropic and birefringent in the infrared, with strong visible and ultraviolet activity; the reflectivity is very low, below $1.368\ \text{Å}$2 in the visible and UV; and the absorption coefficient reaches $1.368\ \text{Å}$3, with the first main absorption peak at approximately $1.368\ \text{Å}$4, contrasted in the source with graphene’s approximately $1.368\ \text{Å}$5 (Santos et al., 2023). These statements place TODD-G within the broader literature on 2D carbon allotropes whose utility is evaluated simultaneously in electronic, optical, and electrochemical terms.

The $1.368\ \text{Å}$6-, $1.368\ \text{Å}$7-, and $1.368\ \text{Å}$8-TODD-G paper generalizes this electronic picture by reporting metallic behavior in all three phases, with Dirac-like features and tilted Dirac cones that suggest anisotropic charge transport (Lima et al., 7 Aug 2025). The projected density of states near the Fermi level is again dominated by carbon $1.368\ \text{Å}$9-orbitals (Lima et al., 7 Aug 2025). Optical differentiation is more pronounced across the later family: $1.435\ \text{Å}$0-TODD-G is reported to show strong infrared absorption, whereas $1.435\ \text{Å}$1- and $1.435\ \text{Å}$2-TODD-G mainly absorb in the visible and ultraviolet ranges (Lima et al., 7 Aug 2025). A plausible implication is that the TODD-G designation increasingly marks a class of porous metallic carbon sheets whose phenomenology is controlled by the interplay between ring topology and anisotropy.

4. Charge transport and carrier mobility

The original TODD-G paper reports calculated charge carrier mobilities for both electrons and holes and states that the values surpass those of graphene (Santos et al., 2023). The numerical values given in the summary are $1.435\ \text{Å}$3 and $1.435\ \text{Å}$4 for electron mobility along the $1.435\ \text{Å}$5- and $1.435\ \text{Å}$6-directions, respectively, and $1.435\ \text{Å}$7 and $1.435\ \text{Å}$8 for hole mobility along the same directions (Santos et al., 2023). For comparison, the source cites graphene’s reported electron mobility as up to $1.435\ \text{Å}$9 (Santos et al., 2023).

The strong anisotropy of these transport values is attributed in the source to the porous and topologically complex 3-8-10-12 ring network (Santos et al., 2023). This is consistent with the anisotropic band dispersion and with the highly directional elastic response. The literature therefore presents TODD-G as a case in which porosity does not preclude high mobility; rather, the specific porous topology is argued to preserve a metallic $0.30\ \text{atom/Å}^2$0-conjugated pathway while inducing marked directional dependence (Santos et al., 2023).

In the later $0.30\ \text{atom/Å}^2$1-, $0.30\ \text{atom/Å}^2$2-, and $0.30\ \text{atom/Å}^2$3-TODD-G work, tilted Dirac cones and anisotropic carrier group velocities are emphasized instead of explicit mobility values (Lima et al., 7 Aug 2025). $0.30\ \text{atom/Å}^2$4-TODD-G is described as exhibiting pronounced tilted Dirac cones, while $0.30\ \text{atom/Å}^2$5-TODD-G features nearly ideal, symmetric Dirac cones with slight tilt (Lima et al., 7 Aug 2025). This suggests an emerging internal differentiation within the TODD-G family between transport anisotropy arising from open $0.30\ \text{atom/Å}^2$6 networks and transport isotropy associated with more symmetric ring arrangements.

5. Lithium adsorption, diffusion, and battery relevance

The primary application context for the original TODD-G paper is lithium-ion battery anodes (Santos et al., 2023). The preferred Li adsorption site is reported to be a hollow site centered on a 10-membered ring, identified using UMC combined with AIMD (Santos et al., 2023). The adsorption energy is defined as

$0.30\ \text{atom/Å}^2$7

and the final Li adsorption energy is given as $0.30\ \text{atom/Å}^2$8, with a final adsorption height of $0.30\ \text{atom/Å}^2$9 above the TODD-G plane (Santos et al., 2023). During AIMD at $0.38\ \text{atom/Å}^2$0, Li is reported to migrate rapidly between pores (Santos et al., 2023).

Three Li migration paths were examined: 12-membered ring to 12-membered ring, 12-membered ring to 10-membered ring, and 12-membered ring to 8-membered ring (Santos et al., 2023). The corresponding diffusion barriers are reported as $0.38\ \text{atom/Å}^2$1, $0.38\ \text{atom/Å}^2$2, and $0.38\ \text{atom/Å}^2$3, respectively (Santos et al., 2023). The average diffusion barrier is given as approximately $0.38\ \text{atom/Å}^2$4, with the paper noting a slight discrepancy in the range $0.38\ \text{atom/Å}^2$5–$0.38\ \text{atom/Å}^2$6 depending on rounding (Santos et al., 2023). The abstract states a low average diffusion barrier of $0.38\ \text{atom/Å}^2$7 (Santos et al., 2023). Compared with the values cited in the source for graphene ($0.38\ \text{atom/Å}^2$8), popgraphene ($0.38\ \text{atom/Å}^2$9), $0.23\ \text{atom/Å}^2$0 ($0.23\ \text{atom/Å}^2$1), biphenylene network ($0.23\ \text{atom/Å}^2$2), and phagraphene ($0.23\ \text{atom/Å}^2$3), TODD-G is positioned as intermediate: slower than graphene but substantially more favorable than some other porous 2D carbons (Santos et al., 2023).

The original paper repeatedly refers to TODD-G’s high theoretical Li storage capacity, but the extracted summary explicitly notes that no actual numerical value was provided in the extracted content (Santos et al., 2023). Accordingly, only a qualified statement is warranted: the study associates high theoretical capacity with extensive porosity and the presence of large 8-, 10-, and 12-membered pores (Santos et al., 2023). This suggests that the battery relevance of TODD-G rests on a composite rationale—porosity, metallicity, and acceptable diffusion barriers—rather than on a single benchmark figure.

The paper’s conclusion synthesizes these claims in direct application terms, stating that TODD-G exhibits a low average diffusion barrier of about $0.23\ \text{atom/Å}^2$4 and a metallic framework boasting excellent conductivity, emerging as a promising anode material for lithium-ion batteries, while classical reactive MD simulations suggest structural integrity with no bond reconstructions at $0.23\ \text{atom/Å}^2$5 (Santos et al., 2023). The term “promising” here is part of the source’s own evaluative language.

6. Relation to later TODD-G allotropes and scope of the term

The later paper on $0.23\ \text{atom/Å}^2$6-, $0.23\ \text{atom/Å}^2$7-, and $0.23\ \text{atom/Å}^2$8-TODD-G expands the TODD-G label from a single porous $0.23\ \text{atom/Å}^2$9-dominated 3-8-10-12 carbon sheet to a family of three additional 2D planar carbon allotropes with distinct ring motifs and mixed $\beta$00 hybridization (Lima et al., 7 Aug 2025). $\beta$01-TODD-G and $\beta$02-TODD-G are both built from 3-, 8-, 12-, and 16-membered rings, while $\beta$03-TODD-G is built from 3-, 4-, 8-, and 12-membered rings (Lima et al., 7 Aug 2025). Their reported lattice constants are $\beta$04, $\beta$05 for $\beta$06; $\beta$07, $\beta$08 for $\beta$09; and $\beta$10, $\beta$11 for $\beta$12 (Lima et al., 7 Aug 2025).

The mechanical characterization of these later phases emphasizes tunable anisotropy. $\beta$13-TODD-G is described as strongly anisotropic, $\beta$14-TODD-G as moderately anisotropic, and $\beta$15-TODD-G as nearly isotropic (Lima et al., 7 Aug 2025). All three are metallic, all display Dirac-like band crossings, and their optical responses differ substantially, especially in the infrared versus visible/UV ranges (Lima et al., 7 Aug 2025). What is absent from the provided summary of the 2025 paper is a parallel lithium-storage analysis of the type carried out for the original TODD-G. Therefore, the battery-oriented identity of TODD-G remains tied primarily to the 2023 3-8-10-12 allotrope (Santos et al., 2023), while the later work situates the name within a broader taxonomy of porous metallic carbon monolayers (Lima et al., 7 Aug 2025).

A further source of possible confusion is the unrelated use of “Todd” in other mathematical and high-energy contexts, such as Todd classes and Todd polynomials (Jiang, 2020, Nagy et al., 2024). These papers concern hyperkähler geometry and asymptotic charges at null infinity, respectively, and are not connected to the carbon allotrope literature despite lexical overlap. In the materials-science context, TODD-G refers specifically to the graphene-like carbon allotropes described above (Santos et al., 2023, Lima et al., 7 Aug 2025).

7. Position within the 2D carbon allotrope literature

Within the literature summarized here, TODD-G is situated among porous 2D carbon allotropes investigated for combined electronic and electrochemical performance (Santos et al., 2023). The original study compares its planar density with graphene, graphdiyne, and graphyne; its Li adsorption energy with popgraphene and net-$\beta$16; and its diffusion barriers with graphene, popgraphene, $\beta$17, biphenylene network, and phagraphene (Santos et al., 2023). These comparisons frame TODD-G as part of a design space in which ring topology, pore size, and bonding network are systematically traded against stability, conductivity, and ion kinetics.

Several broad conclusions follow from the reported data. First, TODD-G is presented as an example of a porous metallic carbon monolayer whose porosity does not eliminate electronic conductivity (Santos et al., 2023). Second, its anisotropy appears across multiple observables—band dispersion, carrier mobility, elastic response, and optics—indicating that structural anisotropy is a governing principle rather than an incidental feature (Santos et al., 2023). Third, the extension to $\beta$18-, $\beta$19-, and $\beta$20-TODD-G indicates that the underlying design motif is generative: varying ring connectivity and $\beta$21 content preserves planarity and metallicity while tuning anisotropy and optical response (Lima et al., 7 Aug 2025).

At the same time, the current literature remains largely computational in the material provided here. The claims are based on first-principles calculations, AIMD, and classical reactive MD rather than experimental synthesis or electrochemical testing (Santos et al., 2023, Lima et al., 7 Aug 2025). A plausible implication is that the near-term significance of TODD-G lies as much in carbon-allotrope design methodology as in any single predicted application. In that sense, TODD-G occupies a dual role: a specifically defined 3-8-10-12 porous graphene analogue with reported lithium-ion anode potential (Santos et al., 2023), and a broader family label for later metallic porous 2D carbon phases with tunable anisotropy and Dirac-like electronic structure (Lima et al., 7 Aug 2025).