Fluorinated Biphenylene Network

• Fluorinated biphenylene network is a 2D carbon allotrope modified by periodic fluorination, enabling diverse phases including type-I/II Dirac cones and nodal lines through symmetry control.
• Advanced tight-binding and DFT calculations reveal that specific fluorination patterns alter mirror symmetries and compact localized states, directly impacting band flatness and Dirac point gaps.
• Experimental methods adapted from graphene synthesis, like CF₄ plasma exposure and STM manipulation, offer a pathway to engineer tunable electronic properties in FBPN for novel device applications.

The fluorinated biphenylene network (FBPN) denotes a family of two-dimensional carbon allotropes derived from the biphenylene network (BPN) through periodic adsorption of fluorine atoms. FBPNs exhibit a broad spectrum of nodal semimetallic phases, including gapped and gapless Dirac cones (type I, type II), and symmetry-protected nodal lines, arising from tunable disruption or preservation of mirror symmetries and compact localized states (CLS) intrinsic to BPN’s lattice geometry. These properties emerge under precise stoichiometric fluorination patterns and can be rationalized by tight-binding models and confirmed by density functional theory (DFT) calculations (Mo et al., 2024).

1. Geometry, Symmetry, and Electronic Structure of Pristine Biphenylene Network

BPN consists of planar carbon sheets whose primitive cell is built from four- and eight-membered rings arranged in a motif known as biphenylene, interlinked by hexagonal rings. The structural parameters are lattice constants $a \approx 6.40~\text{Å}$, $b \approx 4.94~\text{Å}$, forming a monoclinic cell frequently recast in an orthorhombic primitive cell. The crystalline symmetry group is Pmmm ($D_{2h}$), and the lattice supports two orthogonal mirror planes: $M_x$ (reflection $x \to -x$) and $M_y$ (reflection $y \to -y$). Time-reversal symmetry $(\Theta)$ is also preserved.

Low-energy electronic states predominantly derive from C $p_z$ orbitals. The minimal tight-binding Hamiltonian, constructed on a basis of six $p_z$ orbitals per unit cell (sites $A$–$F$), includes hoppings along square chains ($t_{sx}$, $t_{sy}$), along $y$-chains ($t_y$), within four-membered rings ($t_d$), long-range diagonals ($t_p$), intra-square bonds ($t_s$), plus an on-site shift $E_0$. Representative values:

$$\{ t_{sx}, t_{sy}, t_y, t_d, t_s, t_p, E_0 \} = \{ -3.0,\ -2.7,\ -2.7,\ -2.8,\ -0.7,\ -0.3,\ -0.5 \}\,\text{eV}$$

Projection onto high-symmetry lines yields strictly flat bands associated with stripe-CLS stabilized by destructive interference on linking carbon sites. Each stripe-CLS is a compact localized state along every second square-pair chain, retaining zero amplitude on certain linking sites due to interference. Flat-band eigenvalues on lines $k_x = 0\ (\Gamma$–Y$)$ and $k_x = \pi\ (\mathrm{X}$–S$)$ are:

$$E_{F1} = -t_p - t_s + t_{sx} - t_{sy} + E_0\ E_{F2} = -t_p - t_s - t_{sx} + t_{sy} + E_0$$

BPN thus hosts nearly flat bands ($F_1$, $F_2$) and, crucially, type-II Dirac points at their crossings with dispersive bands along $\Gamma$–Y.

2. Emergence and Classification of Dirac and Nodal Phases

At high-symmetry points along $\Gamma$–Y, the flat $F_2$ band (mirror eigenvalue $M_x = -1$) crosses a dispersive parabolic band ($M_x = +1$), forming type-II Dirac nodes. The absence of hybridization is dictated by the orthogonality imposed by the mirror operation. The effective Hamiltonian near such a node takes the form:

$$H_{2\times2}(q_x, q_y) \simeq w q_y \sigma_0 + v_x q_x \sigma_x + v_y q_y \sigma_y$$

with $|w| > v_y$, globally tilting the Dirac cone (type-II).

DFT calculations yield tilt ratios $\eta = |w|/v_y \approx 1.2$–$1.4$, confirming the type-II character. The Fermi surface comprises electron and hole pockets in this regime. The manipulation of band topology derives fundamentally from control of mirror symmetries and the persistence or destruction of compact localized states under symmetry-breaking perturbations.

3. Periodic Fluorination Patterns: Symmetry Breaking and Phase Control

Fluorine adsorption on BPN preferentially occurs atop carbon atoms within the square rings, with computed binding energies $\sim3.2$–$3.5~\text{eV}$ per F atom for square sites. FBPNs are indexed by their stoichiometry C$_6$F$_x$ ($x = 0.5, 1.0, 1.5, 2.0$), with each F–C bond exhibiting sp$^3$ character, inducing symmetry reduction or preservation according to adsorption motif.

The four prototypical fluorination patterns induce the following changes:

System	Mirror Symmetry	Emergent Phase
C$_6$F$_{0.5}$	$M_y$ only	Massive type-II Dirac, residual flat bands
C$_6$F$_{1.0}$	$M_y$ only	Massive type-II Dirac, residual flat bands
C$_6$F$_{1.5}$	$M_y$ only	Type-I Dirac (along $\Gamma$–X), gapped elsewhere, no flat bands
C$_6$F$_{2.0}$	$M_x$, $M_y$	Nodal-line semimetal (closed ring)

In each case, selective preservation or breaking of $M_x$ and $M_y$ determines whether flat bands and gapless nodal points/lines are symmetry-protected or gapped.

4. Band Structure Evolution under Fluorination

C$_6$F$_{0.5}$: One F per Square

• $M_x$ is broken (by the off-chain fluorine), $M_y$ is preserved.
• Flat band $F_2$ survives along $\Gamma$–Y; the type-II Dirac crossing on $\Gamma$–Y is gapped (massive Dirac), with DFT gap $\Delta_{DP_1}\approx40$ meV (upper Dirac point), $\Delta_{DP_2}\lesssim5$ meV (lower).
• Along $\Gamma$–X, crossing remains type-I Dirac.

C$_6$F$_{1.0}$: Two F on Opposite Squares

• $M_x$ broken; $M_y$ preserved.
• Both Dirac points along $\Gamma$–Y acquire small gaps ($\Delta_{DP}\approx25$ meV), flat bands persist (stripe-CLS unaffected).

C$_6$F$_{1.5}$: Three Squares Fluorinated

• $M_x$ broken; $M_y$ preserved.
• Adsorption interrupts all stripe-CLS paths—no flat bands remain.
• Along $\Gamma$–X, M$_y$ continues to protect a gapless type-I Dirac dispersion; elsewhere gaps open.

C$_6$F$_{2.0}$: All Four Squares Fluorinated

• Both $M_x$ and $M_y$ preserved (full inversion-symmetric arrangement).
• Flat bands lost; bands cross along both $\Gamma$–X and $\Gamma$–Y, forming a closed nodal line around $\Gamma$ (radius $k_\text{line}\approx0.15\,(2\pi/a)$).
• The nodal line carries a quantized Berry phase $\pi$ on loops linked with the line.

5. Symmetry, Destructive Interference, and Topological Invariants

The existence of flat bands, Dirac nodes, and nodal lines is tightly linked to the preservation of specific mirror symmetries and the ability to construct compact localized states free from fluorinated sites. A stripe-CLS remains an exact eigenstate (yielding a flat band) only if none of the CLS’s destructive-interference sites coincides with a fluorinated atom. Once a protective mirror symmetry is broken (e.g., $M_x$ in C$_6$F$_{0.5}$), any previously symmetry-protected band crossing becomes gapped.

For the nodal line in C$_6$F$_{2.0}$, the topological stability is ensured by both $M_x$, $M_y$; the enclosed Berry phase is $\pi$. Each type-I Dirac point in C$_6$F$_{1.5}$ also possesses a quantized Berry phase.

6. Quantitative Characteristics

Binding Energies (DFT)

• C$_6$F$_{0.5}$: $3.45$ eV/F
• C$_6$F$_{1.0}$: $3.51$ eV/F
• C$_6$F$_{1.5}$: $3.48$ eV/F
• C$_6$F$_{2.0}$: $3.50$ eV/F

Dirac Cone Tilts

• Pristine BPN $\eta=|w|/v_y\approx1.2$–$1.4$

Gap Magnitudes

• C$_6$F$_{0.5}$: $\Delta_{DP_1} \approx 40$ meV, $\Delta_{DP_2} \lesssim 5$ meV
• C$_6$F$_{1.0}$: $\Delta_{DP} \approx 25$ meV

Nodal-Line Radius (C$_6$F$_{2.0}$)

• $k_\text{line} \approx 0.15\,(2\pi/a)$

Topological Invariants

• Nodal line: Berry phase $\pi$ on any loop reporting the line
• Type-I Dirac: $\pi$ Berry phase around the node

7. Synthesis, Experimental Probes, and Applications

Fluorination methods established for graphene—such as CF$_4$ plasma exposure or XeF$_2$ dosing—are directly applicable to BPN. On-surface synthesis of BPN via molecular precursors has been demonstrated, characterized by STM and XPS. Precision control over adatom placement, potentially down to the single-atom level, has precedent in STM manipulation experiments (e.g., H or CO on Cu/graphene).

Distinct semimetal phases are expected to exhibit divergent transport and spectroscopic signatures:

• Type-II Dirac phases: anisotropic magnetoresistance, electron-hole coexistence
• Type-I Dirac semimetals: high carrier mobility, linear magneto-optics
• Nodal-line phases: drumhead surface states, non-conventional plasmonic behavior

FBPN thus constitutes a tunable 2D platform for systematic exploration of relativistic quasiparticle phenomena and symmetry-protected topological phases as a function of atomic-scale patterning (Mo et al., 2024).