Monolayer Amorphous Carbon: Structure & Properties

• Monolayer amorphous carbon is a one-atom-thick carbon membrane lacking long-range crystalline order, featuring 5-, 6-, 7-, and 8-membered rings.
• It is synthesized via laser-assisted CVD and modeled using techniques like kinetic Monte Carlo and melt–quench simulations, revealing a crystallite-containing random network.
• MAC exhibits unique electronic localization, thermal insulation, and mechanical toughness due to heterogeneous bond distributions and disorder-induced stress redistribution.

Monolayer amorphous carbon (MAC) is a free-standing, continuous, stable, one-atom-thick carbon membrane that lacks long-range crystalline order and is topologically distinct from both graphene and disordered graphene. In the experimentally characterized form, it is a threefold-coordinated carbon network containing five-, six-, seven-, and eight-membered rings, with atomic-resolution imaging showing broad bond-length and bond-angle distributions and randomly oriented, strained $\sim 1$ nm hexagonal crystallites embedded in a nonperiodic network rather than a pure Zachariasen continuous random network (Toh et al., 2021). Subsequent work has treated MAC as a model 2D amorphous solid whose disorder can be described structurally, mechanically, electronically, vibrationally, and catalytically with unusual directness because the monolayer geometry permits atom-by-atom structural analysis (Zhang et al., 2021).

1. Structural identity and atomic topology

MAC is defined as a pure carbon monolayer with no long-range periodicity, but it is not merely graphene with incidental defects. The experimentally resolved network is threefold coordinated, predominantly $sp^2$-bonded, and contains a wide distribution of bond lengths, bond angles, and 5-, 6-, 7-, and 8-membered rings. Direct imaging reported projected bond lengths from $0.9$ to $1.8$ $\text{\AA}$ and bond angles from $90^\circ$ to $150^\circ$, far broader than in graphene, while diffraction and Fourier transforms showed only diffuse halos rather than crystalline spots (Toh et al., 2021).

A central structural conclusion is that MAC is amorphous but not structurally homogeneous. Atomic-resolution TEM and STEM identified $\sim 1$ nm hexagonal crystallites that are heavily strained and randomly oriented, embedded within a disordered network of non-hexagonal rings. This motivated the interpretation that the experimentally synthesized material is better described by a crystallite-containing amorphous network than by a pure Zachariasen continuous random network (Toh et al., 2021).

Later kinetic Monte Carlo analysis sharpened that point. For elemental monolayer amorphous carbon, the favored as-formed structure was argued to be a crystallite-containing Z-CRN, because hexagons nucleate rapidly and organize into distorted graphene-like nanocrystallites embedded in a threefold-coordinated random network. In that view, recently fabricated MAC is a metastable, kinetically arrested stage between a relaxed disordered carbon network and polycrystalline graphene, and the old CRN-versus-crystallite debate remains relevant in two dimensions (Zhang et al., 2021).

2. Synthesis, atomistic generation, and disorder descriptors

The first free-standing MAC was synthesized by laser-assisted chemical vapour deposition. On Cu, the reported process used a chamber base pressure of $10^{-6}$ mbar, methane at $10^{-2}$ mbar, a KrF excimer laser at $sp^2$0 nm with fluence $sp^2$1 and repetition rate $sp^2$2 Hz, plus a $sp^2$3 kHz pulsed-DC plasma at $sp^2$4 W. Growth occurred in under $sp^2$5 min over an exposed area of $sp^2$6, with substrate temperatures between $sp^2$7 and $sp^2$8; above $sp^2$9 the product evolved toward nanocrystalline graphene and above $0.9$0 toward polycrystalline graphene (Toh et al., 2021).

Theoretical structures have been generated by several non-equivalent workflows. One route begins from randomly distributed carbon atoms in a plane, relaxes the bonded network, and then evolves it by bond-rotation events in a kinetic Monte Carlo annealing scheme biased toward lower energy, producing a crystallite-containing random network (Zhang et al., 2021). A second route starts from defective graphene and applies melt–quench amorphization with ReaxFF, including heating to $0.9$1 K, equilibration, staged cooling through $0.9$2 K, and final quenching to $0.9$3 K, yielding a buckled network dominated by 5-, 6-, and 7-membered rings and mostly $0.9$4 plus substantial $0.9$5 carbon (S et al., 8 May 2026). A third route compares crystalline-to-amorphous and random-to-crystalline kinetic Monte Carlo pathways, spanning Zachariasen-type CRNs and nanocrystallites embedded in random networks (Kurt et al., 31 May 2026).

Because “degree of disorder” is not uniquely specified by one experimental observable, several structural descriptors have been proposed. One is the crystallite area fraction

$0.9$6

where $0.9$7 is the number of rings in crystallites and $0.9$8 the number in the continuous random network; for MAC, pre-fracture structure and mechanical response were found to collapse strongly when compared at fixed $0.9$9 (Zhang et al., 2023). Another is the local bond-order parameter $1.8$0, used to quantify local hexagonal order and relate topology to thermal transport (Kurt et al., 31 May 2026). A third is the pair correlation function $1.8$1, which was used as the central descriptor in SPRamNet; in that framework, the informative part of the MAC pair statistics was found mainly at $1.8$2, yet remained sufficient to predict electronic and thermal properties across a disorder continuum (Cheng et al., 2024).

3. Electronic structure, optical response, and localization

The electronic characterization of MAC is not methodologically uniform. Direct transport on free-standing MAC showed that it is insulating, with room-temperature sheet resistance of order $1.8$3, an increase to about $1.8$4 at $1.8$5 K, and an Arrhenius activation energy of $1.8$6 meV. Optical measurements on the same material yielded a band edge near $1.8$7 eV, a Tauc gap of $1.8$8 eV, and photoluminescence centered at $1.8$9 nm ($\text{\AA}$0 eV), while DFT on an experimentally motivated model showed many states near the nominal Fermi level but near-zero transmission there because the corresponding wavefunctions localize on non-hexagonal-ring regions (Toh et al., 2021).

By contrast, SCC-DFTB calculations on a 610-atom experimentally motivated sheet classified MAC as metallic, with a density-of-states peak at the Fermi level, no Dirac cone, and broadband optical activity from infrared to ultraviolet. In that model the strongest absorption peak was near $\text{\AA}$1 eV, reflectivity lay in the range $\text{\AA}$2–$\text{\AA}$3 in the infrared and $\text{\AA}$4–$\text{\AA}$5 in the ultraviolet, and the optical response was described as compatible with ultraviolet-filter functionality (Tromer et al., 2020).

Low-temperature STM/STS and atomistic tight binding have added a different picture: disorder in MAC localizes much of the low-energy spectrum, yet a critical-like state persists near the band centre $\text{\AA}$6. The measured critical energy was reported at $\text{\AA}$7, with localization-length exponent $\text{\AA}$8, multifractal strength $\text{\AA}$9 experimentally and $90^\circ$0 numerically, and agreement with the scaling identity $90^\circ$1 only near the band centre. The same study interpreted MAC as the first strictly 2D amorphous electronic system showing Anderson criticality driven purely by topological disorder (Sk et al., 14 May 2026).

At smaller scales, coherent transport through MAC nano-fragments has been analyzed with a Pariser–Parr–Pople Hamiltonian and Landauer theory. Ensemble calculations on 413 junctions found that states near $90^\circ$2 inherit partial characteristics of graphene-like surface states, while band-edge states are more compactly localized in the interior and conduct poorly. Constructive quantum interference between frontier orbitals was found to be a common feature, giving finite transmission even in the nominal gap region (Gastellu et al., 2021). This suggests that device-scale insulating behavior, local spectroscopy, and nanoscale coherent transmission are probing different disorder realizations and different transport regimes.

4. Mechanical response and fracture mechanics

Experimentally, free-standing MAC is mechanically robust despite its amorphous topology. AFM indentation on suspended membranes gave a 2D elastic stiffness of $90^\circ$3 and a breaking strength of $90^\circ$4, more than half the strength of monocrystalline graphene. The same study reported that MAC deforms to a high breaking strength without crack propagation from the point of fracture, and repeated loading showed irreversible flattening and stiffening consistent with a buckled membrane that plastically rearranges under load (Toh et al., 2021).

Reactive molecular dynamics on the same 610-atom structural model compared MAC directly with pristine graphene. In that finite-sheet simulation, MAC had $90^\circ$5, $90^\circ$6, $90^\circ$7, and $90^\circ$8, versus $90^\circ$9, $150^\circ$0, $150^\circ$1, and $150^\circ$2 for pristine graphene. Graphene showed essentially one elastic stage followed by abrupt brittle fracture, whereas MAC passed through several stress-drop stages linked to localized bond rupture and structural reconstruction; complete failure occurred at about $150^\circ$3 fs in MAC versus $150^\circ$4 fs in graphene under the same loading protocol (2002.04682).

A more general structural-mechanical framework was later developed using DFT-accurate machine-learning potentials and kinetic Monte Carlo. There the key order parameter was $150^\circ$5, the areal fraction occupied by crystallites within the continuous random network. Samples with the same $150^\circ$6 but different crystallite sizes and arrangements had nearly identical radial distribution functions, bond-length distributions, bond-angle distributions, and pre-fracture stress–strain curves. With increasing $150^\circ$7, the critical strain showed a downward trend and the critical stress an upward trend. Cracks propagated mainly through the CRN region in meandering paths and were toughened by crystallite-induced arrest, deflection, and bridging (Zhang et al., 2023).

These results converge on a common mechanistic picture. MAC is weaker in peak stress than graphene, but its topological disorder frustrates straight cleavage, concentrates stress heterogeneously, and creates multiple stress-release channels. A plausible implication is that MAC’s unusual toughness is tied less to high intrinsic bond strength than to the way the amorphous network redistributes damage.

5. Thermal stability, vibrational structure, and heat transport

Thermal stability has been addressed mostly through heating-ramp molecular dynamics, and the reported “melting” temperatures are model-dependent. For free-standing MAC, a linear heating ramp from $150^\circ$8 K to $150^\circ$9 K over $\sim 1$0 ns yielded a transition temperature of $\sim 1$1 K, compared with $\sim 1$2 K for pristine graphene under the same AIREBO protocol. The authors explicitly noted that under these conditions the transition may be better interpreted as sublimation into a gas-like carbon phase rather than strict thermodynamic melting (2002.04682).

Heat transport in MAC is exceptionally suppressed relative to graphene. Reverse nonequilibrium molecular dynamics with optimized Tersoff interactions reported an in-plane thermal conductivity of $\sim 1$3 at $\sim 1$4 K, about two orders of magnitude smaller than crystalline graphene, with weak temperature dependence up to $\sim 1$5 K. Vibrational analysis classified the excitations in terms of participation ratio and showed broadened vibrational density of states, strong localization at low frequency, and an unusual extended high-frequency $\sim 1$6 mode at $\sim 1$7–$\sim 1$8 ($\sim 1$9–$10^{-6}$0), attributed to the $10^{-6}$1 carbon network and broken $10^{-6}$2 symmetry (Zhang et al., 2021).

A later fully quantum Green-function treatment found still lower thermal conductivities, depending on topology and amorphization level. At room temperature, predicted values ranged from $10^{-6}$3 to $10^{-6}$4, with stronger disorder producing lower conductivity and classical statistics overestimating the result by roughly a factor of two even at room temperature. In that work, $10^{-6}$5 served as the principal structural disorder metric, and vibrational transport in MAC was decomposed into propagons below roughly $10^{-6}$6, diffusons over the intermediate band, and locons above a mobility edge near $10^{-6}$7 (Kurt et al., 31 May 2026).

Taken together, these studies imply that MAC is both thermally stable on the scale of several thousand kelvin and thermally insulating in-plane on the scale of single-digit to low-double-digit $10^{-6}$8. The latter is not a contradiction: strong covalent bonding preserves structural integrity, while topological disorder, buckling, and quantum statistics suppress heat transport.

6. Derived structures, functionalization, and emerging functionalities

MAC has also been used as a precursor and platform for derived amorphous nanocarbons. Reactive molecular dynamics on amorphous carbon nanotubes and nanoscrolls obtained by rolling the MAC monolayer found that these structures remain very stiff but are markedly weaker than crystalline CNT and CNS analogues. Reported values were $10^{-6}$9, $10^{-2}$0, and $10^{-2}$1 for the amorphous nanotube, and $10^{-2}$2, $10^{-2}$3, and $10^{-2}$4 for the amorphous nanoscroll; the main conceptual conclusion was that in the amorphous derivatives, inherited structural disorder dominates over topology, unlike in the pristine tube-versus-scroll comparison (Júnior et al., 2020).

Substitutional nitrogen creates a second materials family, MAC@N. DFTB simulations found that nitrogen-doped monolayer amorphous carbon remains stable up to $10^{-2}$5 nitrogen incorporation, beyond which the lattice becomes unstable. Pristine MAC and MAC@N were both reported as metallic in that study, but only pristine MAC retained a Dirac-like cone. At $10^{-2}$6 N, the Young’s modulus increased slightly from $10^{-2}$7 to $10^{-2}$8; optically, MAC absorbed mainly in the ultraviolet, while MAC@N shifted activity into the infrared and visible. Thermally, MAC was reported to melt at approximately $10^{-2}$9 K in the abstract, whereas MAC@N lost structural integrity over the range $sp^2$00–$sp^2$01 K depending on nitrogen content (Santos et al., 2024).

MAC has also emerged as a noncrystalline topological platform. In a driven tight-binding model on Voronoi-generated threefold-coordinated amorphous carbon, circularly polarized laser driving induced Floquet edge modes in both the $sp^2$02-gap and the $sp^2$03-gap. The analysis used an energy- and space-resolved topological marker based on the spectral localizer and concluded that local threefold coordination, rather than long-range crystallinity, is the key structural ingredient for the driven topological phase (Ghosh et al., 13 Aug 2025).

Catalytically, disorder in MAC generates a broad distribution of hydrogen adsorption free energies for HER. DFT on 30 local environments found $sp^2$04 values from $sp^2$05 eV to $sp^2$06 eV, while a fine-tuned MACE model extended the full-surface distribution to $sp^2$07 eV through $sp^2$08 eV across about 1183 sites. Approximately $sp^2$09 of sites had $sp^2$10 eV. Feature analysis identified 7-membered rings, curvature, ripple height, and bond variation as activity-enhancing motifs, whereas graphene-like hexagonal order correlated with worse HER performance (S et al., 8 May 2026).

Finally, stacked MAC sheets have been proposed as an ultrathin dielectric platform. In multilayer amorphous carbon described as stacked monolayer amorphous carbon, the thinnest directly grown continuous layer was $sp^2$11 nm and assigned as 1L. That system showed $sp^2$12, dielectric strength $sp^2$13–$sp^2$14, and Cu-ion diffusion time to failure $sp^2$15 s even for a single layer, with the low permittivity attributed to the lack of long-range order, intrinsic 2D nature, $sp^2$16 character, and low density (Toh et al., 29 Jun 2026).

MAC therefore occupies an unusual position within carbon materials science. It is simultaneously a synthesized 2D amorphous membrane, a structurally resolved test case for the CRN-versus-crystallite question, an insulating and localization-dominated electronic system in experiment, a non-brittle atomically thin network, a thermally robust but ultralow-conductivity monolayer, and a tunable platform for rolled geometries, doping, driven topology, catalysis, and atomically thin dielectric design. The remaining controversies—especially over the precise structural taxonomy and the relation between local spectroscopy, coherent transport, and macroscopic electrical measurements—are themselves part of what makes MAC scientifically consequential.