Biphenylene-Based Networks

• Biphenylene-based networks are two-dimensional lattices built from nonbenzenoid biphenylene motifs that tile the plane with four-, six-, and eight-membered rings.
• They exhibit diverse electronic properties, including metallicity, Dirac fermionic states, and topological phases, which are highly sensitive to strain, functionalization, and correlation effects.
• Their versatile structure enables controllable modifications for applications in energy storage, thermoelectrics, gas adsorption, and interfacial engineering.

Searching arXiv for papers on biphenylene-based networks and related derivatives. Search query: biphenylene network arXiv recent topological fluorination heterostructure battery thermoelectric Biphenylene-based networks are two-dimensional lattices built from the non-benzenoid biphenylene motif and, in their most studied realizations, tile the plane with four-, six-, and eight-membered rings rather than the purely benzenoid topology of graphene. The experimentally realized carbon biphenylene network (BPN) has become the reference member of this class, and subsequent work has extended the same structural idea to functionalized carbon sheets, heterostructures, graphyne derivatives, and isoelectronic or chemically analogous lattices based on BCN, group-III nitrides, SiC, and MoS$_2$ (Bafekry et al., 2021, Paupitz et al., 2 Sep 2025).

1. Structural archetype and bonding

The canonical carbon BPN is a planar sp$^2$-hybridized sheet with a rectangular primitive cell containing six C atoms, lattice constants $a=3.75$ Å and $b=4.52$ Å, and no out-of-plane buckling. Its tessellation consists of tetragons, hexagons, and octagons, with representative bond lengths $d_1=1.45$ Å in the four-membered rings, $d_2=1.40$ Å in the six-membered rings, and $d_3=1.44$ Å in the eight-membered rings; the corresponding bond angles are $90^\circ$, $109^\circ$, and $125^\circ$. Electron-localization-function maps place red lobes directly between neighboring carbons and give $^2$0 in the C–C bonding regions, indicating strong localization of electron pairs on all ring edges (Bafekry et al., 2021).

Closely related descriptions use the orthorhombic $^2$1 setting, with primitive vectors $^2$2 Å and $^2$3 Å and a six-site cell in which atoms 1–4 form a square ring and atoms 5–6 form a dimer. In tight-binding analyses of the $^2$4-electron network, the six inequivalent sites are often labeled $^2$5–$^2$6, and the low-energy physics is organized by the rectangular Bravais lattice, mirror operations, inversion, and $^2$7 rotations (Son et al., 2022, Koizumi et al., 2024).

This structural motif generalizes readily. The review literature places both BPN and graphenylene within the broader class of biphenylene-based two-dimensional carbon networks, while later work explicitly transplants the same ring topology into BCN, III-nitride, SiC, MoS$^2$8, and graphyne-derived sheets (Paupitz et al., 2 Sep 2025, Sargin et al., 25 Dec 2025).

2. Electronic structure, optical anisotropy, and correlation effects

Within first-principles treatments based on DFT+PBE geometry and HSE06-refined bands, the carbon BPN monolayer is metallic: valence and conduction bands overlap at $^2$9, the total DOS at $a=3.75$0 is nonzero, and the low-energy carriers are dominated by $a=3.75$1 orbitals, especially from one inequivalent carbon labeled C$a=3.75$2. A tilted Dirac-cone feature appears slightly above $a=3.75$3 along a high-symmetry line (Bafekry et al., 2021).

A more detailed low-energy analysis identifies a zone-center saddle point, a van Hove singularity at $a=3.75$4, and peculiar type-II Dirac fermionic states. In the minimal tight-binding description,

$a=3.75$5

with $a=3.75$6 eV, $a=3.75$7 eV, and $a=3.75$8 eV, the type-II state lies close to a transition to type-I, and moderate uniaxial strain can merge a pair of Dirac points with the zone-center saddle point, producing concurrent Lifshitz transitions of the van Hove singularity and Dirac-pair annihilation (Son et al., 2022).

The description of pristine BPN is not uniform across methodologies. In fully relaxed DFT+$a=3.75$9+$b=4.52$0 calculations with extended Hubbard interactions, the monolayer becomes a non-magnetic semiconductor with an indirect gap of “a few $b=4.52$1–$b=4.52$2 meV,” whereas the fully relaxed bilayer remains a non-magnetic metal. The same study emphasizes that including inter-site $b=4.52$3 in structural relaxation changes lattice constants by $b=4.52$4–$b=4.52$5, increases interlayer spacing by $b=4.52$6 Å, and restores the non-magnetic ground state when unperturbed (Kim, 2024). This suggests that the apparent metallic-versus-gapped character near $b=4.52$7 is highly sensitive to the level of correlation treatment and to whether the lattice is relaxed within that correlated framework.

Optically, BPN is strongly anisotropic. The dielectric tensor $b=4.52$8 gives $b=4.52$9, $d_1=1.45$0, and $d_1=1.45$1. The imaginary part is Drude-like in the $d_1=1.45$2 direction, has a transparent window for $d_1=1.45$3 from $d_1=1.45$4 to $d_1=1.45$5 eV, and vanishes for $d_1=1.45$6 from $d_1=1.45$7 to $d_1=1.45$8 eV. Accordingly, $d_1=1.45$9 and $d_2=1.40$0 show strong in-plane absorption peaks from $d_2=1.40$1 to $d_2=1.40$2 eV, while $d_2=1.40$3 is zero up to $d_2=1.40$4 eV, indicating metallic in-plane response and out-of-plane optical insulation (Bafekry et al., 2021).

3. Topological phases and boundary-state physics

The topological analysis of BPN is naturally formulated in a six-band $d_2=1.40$5-electron model with intracell hopping $d_2=1.40$6 and intercell hopping $d_2=1.40$7. For $d_2=1.40$8, the bulk spectrum is metallic, with two flat bands at $d_2=1.40$9 and four tilted Dirac crossings along $d_3=1.44$0–$d_3=1.44$1. On inversion-symmetric one-dimensional cuts of the Brillouin zone, the Zak phase is quantized to $d_3=1.44$2 or $d_3=1.44$3, and the total phase

$d_3=1.44$4

predicts edge states in the corresponding gap (Koizumi et al., 2024).

Tuning $d_3=1.44$5 drives topological phase transitions at the critical ratios $d_3=1.44$6, $d_3=1.44$7, and $d_3=1.44$8. Between these values, band inversions change parity eigenvalues and cause $d_3=1.44$9 to jump by $90^\circ$0. The resulting bulk–edge correspondence is sharpened by a Wannier-center picture: the number of topological edge states equals the number of Wannier centers cut by the chosen termination. In the $90^\circ$1 regime, the occupied $90^\circ$2 bands form elementary band representations $90^\circ$3, corresponding to Wannier orbitals centered at the two inequivalent bond midpoints, so zigzag and armchair terminations cut different numbers of bond-center orbitals and therefore host different numbers of edge states (Koizumi et al., 2024).

The same formalism predicts higher-order topology. In a finite diamond-shaped nanoflake with $90^\circ$4, choosing corners that cut through the relevant Wannier orbitals yields six in-gap corner modes: two at the left–right corners and four at the top–bottom corners. Their wavefunctions decay exponentially as $90^\circ$5, and chiral symmetry pins them near $90^\circ$6 in the $90^\circ$7-model (Koizumi et al., 2024).

Related topological structures also appear in continuum-inspired effective models. A domain boundary generated by shifting half the lattice by $90^\circ$8 along $90^\circ$9 exchanges the Wannier center by $109^\circ$0, changes the Zak phase by $109^\circ$1, and supports topological grain-boundary states connecting the two Dirac points across the gap, provided mirror symmetry across the boundary is preserved (Son et al., 2022).

Experimental access to this boundary physics has so far been clearest in analog platforms. In a photonic BPN lattice written into a nonlinear SBN crystal by a continuous-wave 488 nm laser, both zigzag and armchair topological in-gap edge states were observed. The armchair case is especially notable because, unlike graphene, BPN has Dirac points at $109^\circ$2; therefore an in-gap state need not be pinned to zero energy or forced onto a single sublattice, and stable armchair edge states become possible (Zhong et al., 11 Mar 2025).

4. Mechanical response, fracture pathways, and thermal stability

The planar BPN monolayer is dynamically stable at $109^\circ$3: phonon band dispersions computed by the small-displacement PHONOPY method show no imaginary frequencies anywhere in the Brillouin zone (Bafekry et al., 2021). In the harmonic approximation, its elastic constants satisfy the Born stability criteria for a two-dimensional rectangular lattice. Using

$109^\circ$4

the reported in-plane Young’s modulus is $109^\circ$5 TPa, the shear modulus $109^\circ$6 TPa, and the Poisson ratio $109^\circ$7; the value $109^\circ$8 was taken to indicate brittle behavior (Bafekry et al., 2021).

Reactive molecular dynamics paints a more direction-dependent and strongly nonlinear picture. For pristine BPN under uniaxial tension, the initial slopes give $109^\circ$9 GPa and $125^\circ$0 GPa, with fracture strains $125^\circ$1 and $125^\circ$2, and ultimate strengths $125^\circ$3 GPa and $125^\circ$4 GPa. Stress–strain curves show an initial elastic regime followed by one or more inelastic plateaus, onset of fracture with linear-atomic-chain formation, and final failure. Under $125^\circ$5, the lattice passes through four distinct pre-fracture morphologies, including elongation of eight-membered rings, formation of 11-atom pores by new E–G bonds, local graphitization patches, and finally LAC-mediated rupture; under $125^\circ$6, a pronounced graphene-like rearrangement precedes pore coalescence and LAC formation (Júnior et al., 2021).

Pre-existing nanocracks modify strengths and critical strains but do not create new fracture morphologies. A horizontal crack parallel to the loading direction can increase $125^\circ$7 and $125^\circ$8 in the $125^\circ$9-direction by allowing bond re-formation across the gap, whereas a vertical crack perpendicular to the load concentrates stress near the crack tip and softens the response in the $^2$00-direction (Júnior et al., 2021).

Thermally, heating-ramp MD from $^2$01 K to $^2$02 K identifies three regimes in $^2$03: an intact lattice from $^2$04 to $^2$05 K, morphology change and graphitization onset from $^2$06 to $^2$07 K, and a gas-like LAC/atom phase above $^2$08 K. A sharp peak in $^2$09 at $^2$10 K marks the solid-to-liquid transition (Júnior et al., 2021).

5. Functionalization, adsorption, disorder, and interfacial engineering

Periodic fluorination has emerged as a controlled route for reshaping the BPN band topology. In the fluorinated series $^2$11, $^2$12, $^2$13, and $^2$14, fluorine adatoms act by manipulating destructive-interference conditions and mirror symmetries. The sequence of phases is explicit: preserving both mirrors and one compact localized state yields type-II Dirac behavior in pristine BPN; breaking $^2$15 while retaining one interference channel gives a gapped type-II phase for $^2$16 and $^2$17; removing all stripe-CLS paths while preserving $^2$18 produces type-I Dirac cones in $^2$19; and preserving both mirrors while removing all CLS paths generates a nodal-line semimetal in $^2$20 (Mo et al., 2024).

Transport calculations on fluorinated and disordered BPN extend this picture into nonequilibrium response. In pristine and ordered fluorinated systems, negative differential resistance and a bias-induced inversion of the preferred transport direction were found. For leads based on $^2$21, armchair current shows NDR above $^2$22 V, peaking near $^2$23 at $^2$24 V and falling to $^2$25 at $^2$26 V, while correlated disorder suppresses the NDR and drives the system toward approximately Ohmic behavior. At high fluorine coverage, the armchair current becomes nonmonotonic with adatom concentration because correlated quasi-linear fluor conformations promote armchair-oriented C-$^2$27 transport channels and suppress zigzag transport (Sousa et al., 16 Jun 2026).

Gas adsorption reveals another facet of chemical tunability. In pristine BPN, CO$^2$28 physisorbs in the eight-membered rings with adsorption energies $^2$29 eV for vertical orientation and $^2$30 eV for horizontal orientation, and the charge transfer remains negligible at $^2$31. Vacancy engineering increases the binding strongly: vertical adsorption at single-atom vacancies remains physisorption-like with $^2$32 eV or $^2$33 eV, while horizontal adsorption becomes chemisorptive with $^2$34 eV or $^2$35 eV and substantial charge transfer $^2$36 or $^2$37. Recovery times estimated by $^2$38 are $^2$39 s and $^2$40 s for pristine physisorption, indicating rapid desorption but weak electronic selectivity (Lima et al., 2023).

Magnetic and mechanical properties can also be engineered interfacially. In a BPN/YIG(111) heterostructure, first-principles calculations with DFT+$^2$41+vdW predict strong $^2$42 hybridization, non-homogeneous electron transfer of $^2$43, and an induced carbon moment of $^2$44 per C at equilibrium, with spin splitting $^2$45 meV near $^2$46. Reducing the vdW gap enhances the splitting to $^2$47 meV and the induced moment to $^2$48 (López-Alcalá et al., 2024).

Fe adsorption produces a different interfacial regime. In monolayer BPN, the average adsorption energy becomes more negative with increasing Fe coverage and reaches its most stable value near $^2$49 eV per Fe at Fe/C $^2$50. In bilayer BPN, the preferred site is the center of the interlayer four-membered ring with $^2$51 eV. In-plane elastic moduli remain close to those of pristine BPN, but interlayer Fe adsorption changes the out-of-plane stiffness dramatically: the pristine bilayer has $^2$52 GPa, whereas Fe/C $^2$53 raises it to $^2$54 GPa. Electrical conductivity remains anisotropic and on the order of $^2$55 S/m at $^2$56 K, with $^2$57 typically $^2$58–$^2$59 larger than $^2$60 (Zhang et al., 21 May 2026).

6. Derived biphenylene lattices and compositional variants

The BPN topology has been exported to a wide range of chemically distinct monolayers. Some remain metallic or nearly metallic; others become narrow-gap or wide-gap semiconductors, topological platforms, or thermoelectrics.

Network	Representative property	Source
$^2$61-BCN-BPN	$^2$62 eV; Li capacity $^2$63 mAh/g	(Parida, 2023)
BPN-AlN / BPN-GaN	$^2$64 / $^2$65 eV; strong UV absorption near $^2$66 eV	(Lima et al., 2023)
InN-BPN	$^2$67-type $^2$68 along zigzag at $^2$69 K	(Sargin et al., 25 Dec 2025)
SiC-biphenylene	$^2$70 eV; first bright exciton at $^2$71 eV	(Singh et al., 2024)
MoS$^2$72-BPN	$^2$73; zigzag NDC	(Sargın, 9 Jun 2026)
$^2$74-BPNGY	spin-polarized semiconducting state with $^2$75 eV	(Rêgo et al., 2024)

The $^2$76-BCN biphenylene monolayer is the most stable among six isoelectronic BCN-BPN isomers, with $^2$77 eV/atom, $^2$78 eV/atom, no imaginary phonon modes up to $^2$79 THz, and retention of planarity in AIMD at $^2$80 K and $^2$81 K for $^2$82 ps. Its electrochemical performance is governed by strong alkali adsorption and low diffusion barriers: $^2$83 eV at the H site, with $^2$84 eV for Li, $^2$85 eV for Na, and $^2$86 eV for K, leading to theoretical capacities of $^2$87, $^2$88, and $^2$89 mAh/g, respectively (Parida, 2023).

Group-III nitride BPNs shift the family decisively into semiconducting and thermoelectric territory. In the four-material set BN-BPN, AlN-BPN, GaN-BPN, and InN-BPN, all structures are dynamically stable, with no imaginary phonon modes, and at $^2$90 K the ballistic phonon thermal conductance decreases monotonically from BN to InN. The valence-band maxima are dominated by N $^2$91 orbitals, conduction-band minima by B $^2$92 or the group-III $^2$93 orbitals, and the nearly dispersionless valence-band region between $^2$94 and $^2$95 strongly enhances $^2$96-type transmission. InN-BPN then attains the best performance, with $^2$97-type $^2$98 along the zigzag direction at $^2$99 K (Sargin et al., 25 Dec 2025). A separate DFT study of BPN-AlN and BPN-GaN reported indirect $a=3.75$00 gaps of $a=3.75$01 and $a=3.75$02 eV at HSE06, maximum Young’s moduli of $a=3.75$03 and $a=3.75$04 GPa, and strong UV absorption peaks at approximately $a=3.75$05 eV (Lima et al., 2023).

SiC-biphenylene introduces strong many-body effects. The monolayer is dynamically and thermally stable, with a melting point of approximately $a=3.75$06 K, an HSE06 direct gap of $a=3.75$07 eV at $a=3.75$08, and a $a=3.75$09 quasiparticle gap of $a=3.75$10 eV. Solving the Bethe–Salpeter equation yields the first optically active exciton at $a=3.75$11 eV with binding energy $a=3.75$12 eV and effective Bohr radius $a=3.75$13 Å, corresponding to a Frenkel-type exciton (Singh et al., 2024).

MoS$a=3.75$14 integrated into the biphenylene configuration exhibits a different form of anisotropy. The room-temperature phonon thermal conductances are $a=3.75$15 nW/K and $a=3.75$16 nW/K along the armchair and zigzag directions, the first $a=3.75$17 peaks are $a=3.75$18 and $a=3.75$19, and the $a=3.75$20–$a=3.75$21 curves differ qualitatively: armchair current increases steadily with bias, whereas zigzag transport shows intrinsic negative differential conductance up to $a=3.75$22 V, producing $a=3.75$23 at $a=3.75$24 V (Sargın, 9 Jun 2026).

The graphyne derivative $a=3.75$25-BPNGY retains the rectangular Bravais lattice of BPN but inserts acetylenic bridges between every original sp$a=3.75$26–sp$a=3.75$27 bond. The non-spin-polarized state is metallic with two nearly flat frontier bands straddling $a=3.75$28, while the spin-polarized ground state is antiferromagnetic with zero net moment and opens a direct gap of about $a=3.75$29 eV. Its elastic constants are highly anisotropic, giving $a=3.75$30 N/m and $a=3.75$31 N/m, with very large Poisson ratios $a=3.75$32 and $a=3.75$33 (Rêgo et al., 2024).

7. Synthesis, experimental status, and conceptual scope

The first extended BPN monolayer was synthesized on Au(111) by interpolymer dehydrofluorination (“HF-zipping”) of octafluorinated biphenylene precursors under ultra-high vacuum, with annealing at approximately $a=3.75$34. STM and noncontact AFM identified the 4–6–8 ring network, and Raman spectroscopy showed characteristic C–C stretch modes near $a=3.75$35 (Paupitz et al., 2 Sep 2025). The first-principles literature on monolayer BPN was explicitly motivated by this synthesis result (Bafekry et al., 2021).

The broader term “biphenylene-based networks” therefore denotes not only the experimentally realized carbon BPN but also a topology-centered research program in which nonbenzenoid ring tilings are used as a platform for metallicity, Dirac physics, strain-driven Lifshitz transitions, higher-order topology, anisotropic fracture, gas adsorption, spin proximity, energy storage, and thermoelectric transport. A plausible implication is that the unifying variable across the family is less the elemental composition than the preservation, controlled distortion, or selective functionalization of the square–hexagon–octagon framework.

One recurring misconception is that BPN should behave as a simple graphene variant. The published record does not support that simplification. Graphene lacks stable armchair edge states without breaking time-reversal symmetry, whereas photonic BPN supports both zigzag and armchair topological edge states (Zhong et al., 11 Mar 2025). Graphene is uniformly hexagonal, whereas BPN hosts square, hexagonal, and octagonal rings and can show a zone-center saddle point and type-II Dirac fermions (Son et al., 2022). Even within BPN itself, metallicity, semimetallicity, small-gap behavior, and magnetic ordering depend on symmetry, strain, hole doping, fluorination pattern, interfacial coupling, and the treatment of extended Hubbard interactions (Kim, 2024).

Within carbon allotropes more broadly, graphenylene remains an important comparator: it is another biphenylene-based two-dimensional carbon network, but unlike BPN it is porous, hexagonal, and semiconducting, with a direct band gap of approximately $a=3.75$36 eV in PBE calculations (Paupitz et al., 2 Sep 2025). This contrast underscores the range already present inside the biphenylene-based category itself.

Taken together, biphenylene-based networks are best understood as a structurally defined but electronically diverse class of low-dimensional materials whose salient properties arise from the interplay between nonbenzenoid topology, anisotropic $a=3.75$37-electron connectivity, and tunable boundary or chemical perturbations.