Holey Graphene: Engineered 2D Carbon Material

• Holey graphene is an engineered 2D carbon material featuring periodic atomic-scale holes that modify its band gap and transport properties.
• Precise synthesis methods, including bottom-up organic synthesis and lithography, create uniform pore patterns that tailor its electronic and optical responses.
• The material offers advanced applications in electronics, spintronics, photovoltaics, and energy storage through tunable quantum phases and enhanced anisotropic properties.

Holey graphene (HG) denotes a class of engineered two-dimensional (2D) carbon nanomaterials based on graphene sheets in which an array of atomic-scale holes or vacancies is introduced in a periodic or quasi-periodic manner. Unlike pristine graphene, which is a zero-gap semimetal with exceptional carrier mobility but limited band-gap tunability, holey graphene leverages the deliberate introduction of nanoscale pores—via bottom-up synthesis or top-down nanofabrication—to tailor its electronic, optical, magnetic, and thermal properties. The resulting materials can exhibit tunable direct band gaps, flat bands, enhanced electron correlation effects, topological phases, and anisotropic transport, enabling a broad range of device concepts in electronics, optoelectronics, spintronics, nanophotonics, and energy storage.

1. Structural Design and Classification

Holey graphene is realized by engineering periodic or aperiodic vacancies in the otherwise robust sp² carbon network of graphene. These vacancies may be produced by atomically precise bottom-up organic synthesis (yielding uniform pores and edge terminations), patterned lithography and etching, or post-growth chemical modification. The resulting superstructures are typically classified by:

• Hole size and periodicity: The periodic unit cell is defined by $n$ times the graphene lattice constant. For example, a supercell with hole periodicity $n$ has a rectangular or hexagonal Brillouin zone, with $n$ governing the folding of Dirac points.
• Chemical composition: The pores may be stabilized solely by carbon–hydrogen edge terminations (pristine HG) or by heteroatom incorporation, resulting in structures such as C$_2$N (“nitrogenated holey graphene”), C$_2$P, and C$_2$As (Yagmurcukardes et al., 2016, Mortazavi et al., 2017, Xu et al., 2017).
• Morphology: Single-layer sheets with uniform hole patterns (periodic honeycomb, rectangular, random) and multi-layer or porous networks.

Atomic-level bottom-up synthesis has yielded HG with precise control over the shape and size of the holes and edge passivation, resulting in unit cells with lattice constants (e.g., $a = 32.383$ Å, $b = 8.583$ Å) and C–C bond lengths of 1.38–1.43 Å. In nitrogenated variants, C–N bonds are ~1.35 Å with bond angles modified from pristine graphene’s 120° due to the presence of holes and heteroatoms (Xu et al., 2017).

2. Electronic Structure and Band Gap Engineering

The introduction of ordered holes transforms the semimetallic band structure of pristine graphene, enabling band gap opening and the creation of flat bands. The mechanism is rooted in quantum confinement, symmetry breaking, and sublattice imbalance:

• Direct Band Gap Formation: DFT calculations have shown that monolayer HG can have a direct band gap: 0.65 eV (PBE), 0.95 eV (HSE06), which agrees with experimental values (Singh et al., 2020). Nitrogenated C$_2$N exhibits a gap of 1.66–2.47 eV (GGA/HSE06) (Yagmurcukardes et al., 2016, Xu et al., 2017). The gap can be tuned by strain (between 1.3–1.7 eV under ±6% uniaxial/biaxial strain), layer thickness (gap decreases for bulk/multilayer), and pore topology.
• Flat Bands and Compact Localized States: Periodic holes with $n \equiv 0 \mod 3$ retain folded Dirac points at $\Gamma$; for $n \equiv \pm1 \mod 3$ a gap opens along $\Gamma$–X due to sublattice and inversion symmetry breaking (Espinosa-Champo et al., 2023). This induces flat bands with states localized near hole edges (CLS), underpinning strong electron–electron interaction and correlation.
• Low-Energy Hamiltonians: Near-folded Dirac points, HG’s dispersion is described by effective $\alpha$–$\mathcal{T}_3$ Hamiltonians:

$$H_0(\alpha) = \hbar v_F \mathbf{S}_\xi(\alpha) \cdot \vec{k},$$

with pseudospin matrices mixing conventional graphene (when $\alpha=0$) and an additional hub site at finite $\alpha$, reproducing the three-band structure (two dispersive bands and one flat).

• Transport and Mobility: Introduction of holes and heteroatoms increases effective mass but provides band gap tunability, with hole and electron mobilities in the range $10^3$–$10^4$ cm$^2$V$^{-1}$s$^{-1}$ (lower than pristine graphene, but suited for transistor operation and improved on/off ratio) (Singh et al., 2020).

3. Magnetic and Correlated Electronic Phases

Deliberate vacancy engineering in HG leads to phenomena inaccessible in pristine graphene:

• Flat-Band Superconductivity: Periodic sublattice imbalance induces flat bands near $E_F$ as predicted by Lieb’s theorem; these bands localize on the majority sublattice (Sousa et al., 2021, Espinosa-Champo et al., 2023). The enhanced density of states favors superconductivity even for moderate coupling $V$, with the order parameter peaking on the majority sublattice in mean-field Bogoliubov–de Gennes calculations.
• Altermagnetism via Atomic Manipulation: Adsorption of non-magnetic $sp$ impurities (B, S) at specific bridges between carbon rings modulates local electronic structure and triggers superexchange. The AFM energy gain $\Delta E = E_{\text{AFM}} - E_{\text{FM}}$ stabilizes a collinear compensated AFM (altermagnetic) state with a Néel temperature up to 210 K (for B adsorption on holey graphyne; the strategy translates directly to HG) (Li et al., 7 Apr 2025).
• Tunability by Defect Engineering: Creation of N or P single-atom vacancies in C$_2$N or C$_2$P induces metallicity, while double-H impurities create local magnetic moments of 1 $\mu_B$ per defect, tunable for spintronic applications (Yagmurcukardes et al., 2016).

4. Optical and Plasmonic Properties

HG is distinguished by its strongly modified optical response and plasmonic characteristics:

• Anisotropic Optical Conductivity: The introduction of holes and the breaking of bipartite and inversion symmetry lead to direction-dependent optical conductivity. Tensor components $\sigma_{xx}(\omega)$ and $\sigma_{yy}(\omega)$ differ significantly, especially near flat band transitions, for instance with pronounced $\sigma_{yy}$ response at energies linked to the X and M points (Espinosa-Champo et al., 30 Jul 2025).
• Strong Visible Absorption and Excitonic Effects: First-principles GW+BSE calculations show HG supports strong and broad absorption across the visible spectrum, with the first excitonic peak at 1.28 eV and a weak binding energy of 80 meV, suitable for photovoltaic and photonic applications (Singh et al., 2020).
• Hyperbolic Plasmons: Optical anisotropy produces hyperbolic plasmon bands—characterized by isofrequency contours $$k_y = \pm k_x \sqrt{|\Im[\sigma_{xx}]/\Im[\sigma_{yy}]|}$$ when $\Im[\sigma_{xx}]\Im[\sigma_{yy}] < 0$—allowing subwavelength confinement, slow-light propagation, and strong field localization. The flat plasmonic bands observed in large-hole configurations are signatures of the underlying flat electronic bands (Espinosa-Champo et al., 30 Jul 2025).
• Enhanced Sensing via Graphene-Coated Holey Films: Graphene-coated gold holey films exhibit surface plasmon resonance enhancement from strong coupling between graphene charge carriers and metal plasmons. A key result is a $\sim33\%$ larger plasmon resonance wavelength shift upon molecule adsorption (e.g., ethanol) compared to bare gold, with sensitivity controlled by gold-induced Fermi energy shifts in graphene (Reckinger et al., 2013).

5. Mechanical, Thermal, and Energetic Characteristics

Holey graphene systems present mechanical and thermal behaviors distinct from pristine graphene:

• Elastic Modulus and Tensile Strength: Nitrogenated HG has an elastic modulus of 335–400 GPa at room temperature–DFT and MD simulations (Mortazavi et al., 2017). Tensile strengths up to 60 GPa are observed, decreasing with temperature.
• Thermal Conductivity: The intrinsic room-temperature thermal conductivity of NHG (C$_2$N) is 64.8 W/m-K, with a phonon mean free path of 34.0 nm—orders of magnitude lower than graphene—due to periodic holes and heteroatom scattering (Mortazavi et al., 2017).
• Stability and Cohesive Energies: Cohesive energy trends confirm that C$_2$N ($E_{\text{coh}}=7.64$ eV/atom) is more robust than C$_2$P or C$_2$As, as both the electronegativity of the heteroelement and the degree of network "holeyness" decrease (Yagmurcukardes et al., 2016).
• Mechanical Flexibility: These materials withstand moderate biaxial strain and display mechanical flexibility for application in flexible optoelectronic and nanoelectronic devices (Yagmurcukardes et al., 2016). Stiffness and Poisson ratios vary by composition: C$_2$N (C ≈ 9.27 eV/Å$^2$, $\nu$ ≈ 0.26) > C$_2$P > C$_2$As.

6. Energy Storage, Functionalization, and Applications

The periodic holes in HG render it suitable for chemical functionalization, energy storage, and device integration:

• Hydrogen Storage: Li-decorated holey graphyne achieves a gravimetric hydrogen capacity of 12.8 wt% (vs. US DOE’s 9 wt% target), with an average adsorption energy of –0.22 eV/H$_2$ and high site uniformity, exceeding capacities in holey graphene (Gao et al., 2020). Scandium-decorated HGY exhibits reversible hydrogen storage up to 9.8 wt% via Kubas interactions, a desorption temperature $T_a$ of 464 K, and no Sc clustering due to high migration barriers (Mahamiya et al., 2022).
• Band Engineering and Doping: Hole and electron mobilities, gap size, and electronic behavior (metal–semiconductor transitions) can be tuned by chemical doping, type and concentration of defects, pore geometry, and external strain fields (Kamal et al., 2015, Yagmurcukardes et al., 2016, Xu et al., 2017).
• Optoelectronics and Thermoelectrics: HG displays a high Seebeck coefficient (thermopower up to 1662.59 μV/K) and a room-temperature $ZT_e$ of 1.13, with low reflectivity (<20%), making it suitable for thermoelectric, photovoltaic, and transparent conductive applications (Singh et al., 2020).
• Topological States and Quantum Devices: Insights from topologically nontrivial carbon allotropes (e.g., holey graphyne) are directly translatable to HG (Jiang et al., 2023). By engineering the hole pattern and symmetry, higher-order topological phases (Berry phase γ = π) and helical or corner-localized edge modes (protected by $\mathbb{Z}_2$ invariants in hydrogenated films) can be realized, with utility for robust electronic/quantum computation.
• Spintronic Devices: Atomic manipulation strategies (adsorption of specific non-magnetic impurities bridging carbon rings) allow for robust altermagnetic states with possible room-temperature operation, enabling efficient spin-polarized current generation without stray fields (Li et al., 7 Apr 2025).

7. Summary of Key Mechanisms and Design Principles

The fundamental behaviors of holey graphene derive from a synergy between geometric control, electronic structure engineering, defect and impurity manipulation, and symmetry considerations. The following table outlines central design variables and their impact:

Design Parameter	Primary Effect	Representative Value/Observation
Hole periodicity ($n$)	Dirac cone folding, bandgap opening	$n \bmod 3$ governs gap location
Sublattice imbalance	Flat band emergence	Flat bands at $E_F$ (Espinosa-Champo et al., 2023)
Heteroatom type	Bandgap size, stiffness, flexibility	C$_2$N: $E_g$ up to 2.47 eV
Edge passivation	Electronic/magnetic states	Magnetic moments with 2H doping
Impurity adsorption	Altermagnetism, superexchange	B, S bridges, $T_N$ up to 210 K
Doping/functionalization	Fermi level tuning, transport	Li, Sc, Au, Bi, Sb, defects

Through deterministic patterning of holes, chemical functionalization, and atomic-scale control, holey graphene provides a tunable material platform for exploring strong electron correlation, emergent topological phases, and advanced light–matter interactions, with pathways toward scalable fabrication and integration in future solid-state devices.