Puckered Honeycomb Lattice: Properties & Applications

• The puckered honeycomb lattice is a 2D atomic structure characterized by hexagonal motifs formed by two vertically displaced sublayers, breaking planar symmetry.
• It exhibits tunable electronic band structures, anisotropic mechanical responses, and nontrivial topological edge states driven by buckling and sublattice coordination.
• Applications span advanced field-effect transistors, spintronic devices, and acoustic metamaterials, highlighting its versatility in materials science.

A puckered honeycomb lattice is a two-dimensional (2D) atomic or engineered structure in which hexagons are realized by two sublayers of atoms connected via sp³-like threefold coordination. The "puckering" refers to the vertical displacement between the sublayers, breaking the planar symmetry of a flat honeycomb and imparting in-plane anisotropy as well as distinctive mechanical, electronic, magnetic, and topological properties. This geometry underpins the atomic structures of monolayer black phosphorus (phosphorene), α-antimonene, puckered arsenene, various oxide perovskites, and a class of metamaterials and phononic or acoustic lattices.

1. Crystal Geometry and Atomic Structure

The archetype of a puckered honeycomb lattice is seen in 2D group-V systems such as phosphorene, α-antimonene (α-Sb), and puckered arsenene. The unit cell is orthorhombic, containing four atoms, with two atoms in an upper sublayer and two in a lower sublayer. The vertical separation ("buckling height" or "puckering height," Δz or h) typically ranges from 0.16–0.28 nm, depending on the material. Key parameters for α-Sb (DFT-GGA-PBE) are:

• Lattice constants:
  • a (zigzag) = 4.36 Å, b (armchair) = 4.74 Å
• Sb–Sb bond lengths:
  • in-plane ≈ 2.87 Å, out-of-plane ≈ 2.90 Å
• Bond angles: θ₁ ≈ 97°, θ₂ ≈ 102°
• Buckling height: Δz ≈ 1.6 Å
• Atomic coordinates (Cartesian): Sb₁: (0,0,–Δz/2); Sb₂: (a/2, b/2, –Δz/2); Sb₃: (0, b/2, +Δz/2); Sb₄: (a/2, 0, +Δz/2)

For puckered arsenene, analogous geometry is observed, with a = 3.677 Å, b = 4.765 Å, and bond angles θ₁ ≈ 100.8°, θ₂ ≈ 94.6° (Shi et al., 2020, Kamal et al., 2014).

Distinct from planar (D₆h) and buckled (β-type) honeycombs, the puckered motif lacks sixfold rotation symmetry, resulting in a mirror and twofold axis only. This symmetry breaking fundamentally influences physical responses.

2. Growth, Kinetics, and Stability

Van der Waals epitaxy (e.g., for α-Sb) demonstrates a kinetics-limited two-step growth pathway. Initially, Sb adatoms form a metastable distorted-hexagonal (dH) half-layer at T ≈ 350 K and ≈0.5 monolayer coverage, which is observable as 4.9 Å-high corrugated islands. Upon annealing or local tip-induced heating, an activation barrier ΔE‡ ≈ 0.241 eV/atom (computed via cNEB) is surpassed, yielding conversion to the thermodynamically stable puckered honeycomb (full α-Sb) with total thickness of 6.5 Å.

STM confirms the structural evolution and persistent kinetic barriers (absence of tip-induced flips). These barriers ensure that low-temperature or kinetics-controlled sample growth "traps" defect-poor, large-area α-Sb, whereas direct high-temperature growth results in immediate conversion but more frequent defects. The two-step mechanism is summarized below:

Step	Structural Phase	Height (Å)	ΔE (vs. α-Sb)	Comments
1. Deposition	dH half-layer	4.9	+59 meV/atom	Metastable, all atoms in one sublayer
2. Annealing	full puckered α-Sb	6.5	0	Puckered, two sublayers, lowest energy

(Shi et al., 2020, Shi et al., 2019)

3. Electronic, Magnetic, and Topological Properties

Puckered honeycomb lattices exhibit strong electronic anisotropy, tunable bandgaps, and nontrivial topological phases.

Semimetallicity and Band Structure:

Monolayer α-Sb displays a highly anisotropic, linearly dispersing Fermi surface with vₓ ≈ 5.2×10⁵ m/s, v_y ≈ 2.8×10⁵ m/s, and low effective masses mₓₓ ≈ 0.03 m_e, m_yy ≈ 0.05 m_e, explained by DFT and corroborated by STM QPI and transport (Shi et al., 2019). Puckered arsenene is an indirect-gap semiconductor (E_g ≈ 0.83 eV, zero strain). Application of tensile strain along a₁ yields a continuous indirect-to-direct gap transition at ≈1% strain, and gap closure at ≈6% strain; the gap is tunable at ≈–10 meV/% strain (Kamal et al., 2014).

Edge Physics and Anisotropic Band Topology:

The intrinsic anisotropy splits the three nearest neighbor hoppings (tₐ ≠ t_b ≠ t_c), admitting new edge terminations—skewed-zigzag (sZZ) and skewed-armchair (sAC). Unlike normal zigzag or armchair edges, skewed terminations realize dual edge spectra: sZZ ribbons are always semiconducting (no flat bands), while sAC ribbons host double-degenerate quasi-flat bands at the Fermi level (metallic unless gapped by infinitesimal out-of-plane field E_z). Application of E_z = 25 mV/Å (d ≈ 2.1 Å) opens a gap E_g ≈ Δ, enabling field-effect transistor operation based on topological edge states (Grujić et al., 2015).

Edge Type	Normal Edge (nZZ/nAC)	Skewed Edge (sZZ/sAC)
Zigzag	QFB edge mode	Gapped
Armchair	Gapped	QFB edge mode (metallic)

Higher-Order Topological Insulator Phases:

In acoustic metamaterial analogs, the puckered phosphorene geometry with p_{x,y}-orbital bands supports higher-order topology: flat armchair edge states and in-gap corner modes, directly observed by pump–probe measurements. The key mechanism is the obstruction of s-like Wannier centers at bond midpoints, which, when cut by sample boundaries or corners, leave charge imbalance and thus topological boundary states. Chiral symmetry pins these modes to midgap energy (Wu et al., 10 May 2024).

4. Mechanical and Metamaterial Realizations

Puckered honeycomb lattices admit mechanical auxetic behavior in 2D metamaterials engineered via rigid sticks and angular/dihedral springs. Analytical and numerical modeling demonstrates independently tunable in-plane (membranal Poisson, ν^m) and out-of-plane (bending Poisson, ν^b) coefficients. The continuum limit yields, for the out-of-plane response:

$$\nu^{(b)} = -\frac{4\sqrt{3}k^{\Theta} + 3\tau_0}{4\sqrt{3}k^{\Theta} - 3\tau_0}$$

where k^{\Theta} is the dihedral spring stiffness and τ₀ is wedge self-stress. By tuning the ratio ξ = –τ₀/k^{\Theta}, ν^{(b)} can be varied continuously between –1 (extreme synclastic) and +1 (extreme anticlastic), with ν^{(m)} fixed at 1. Full-field simulations confirm the accuracy and tunability of the mechanical response even in modest system sizes (Davini et al., 2017).

5. Magnetic Properties and Buckled Perovskite Oxides

Transition-metal oxides such as Sr₃CaOs₂O₉ realize a "buckled" honeycomb lattice of metallic cations (Os), in which alternate layering produces a puckering height h ≈ 0.28 Å and buckling angle θ ≈ 3–4°, departing only mildly from the ideal planar geometry. This modest out-of-plane modulation splits the Os sublattice into two (Os1, Os2). The resulting Heisenberg spin network features strong intraplanar (J₁ ≈ 8.3 meV) and substantial interlayer (J₂ ≈ 3.3 meV) antiferromagnetic couplings. The presence of J₂, enabled by puckering, lifts geometric frustration, yielding high-T_N = 385 K antiferromagnetic order, as opposed to the quasi-2D, low-T ordering in flat honeycomb analogues. Comparison with Sr₃CaRu₂O₉ (h < 0.2 Å, J₁ ≈ 5 meV, T_N = 200 K) emphasizes the significant role of puckering in tuning magnetic dimensionality and exchange (Thakur et al., 2022).

6. Symmetry, Wannier Analysis, and Fundamental Distinctions

The reduction in symmetry compared to planar (D₆h) or buckled (β-type) honeycombs is critical. In puckered lattices:

• The unit cell is doubled (four atoms), and the Bravais lattice becomes orthorhombic.
• Puckering leads to distinct bond angles and lengths, breaking C₃ and sixfold symmetry; only mirrors and twofold axes survive.
• This symmetry lowering yields strongly direction-dependent phonon, electronic, and topological properties.
• Wannier orbital analysis in both quantum and classical models (acoustic lattices) places maximally localized centers at bond midpoints (Wyckoff positions 2a, 2b, 4g), with the presence/absence of edge/corner state dictated by which Wanniers are cut in a given sample shape.

The exact locus of the "puckered honeycomb" concept is thus at the intersection of crystal symmetry lowering, strong in-plane anisotropy, multi-orbital band topologies, and sensitivity to boundary and external fields—properties absent in more symmetric honeycomb models (Shi et al., 2020, Grujić et al., 2015, Kamal et al., 2014, Wu et al., 10 May 2024).

7. Applications and Emerging Directions

Puckered honeycomb systems are fertile ground for electronic, optoelectronic, and spintronic device engineering:

• α-antimonene and puckered arsenene offer robust high-mobility, anisotropic metallic/semiconducting conduction, with Fermi surface and bandgap controlled by strain or ribbon orientation, directly relevant for channel materials in FETs and IR optoelectronics (Kamal et al., 2014, Shi et al., 2019).
• sAC nanoribbons in phosphorene and α-Sb serve as the platform for topological FET switching: zero-gap metallic at E_z=0, gapped insulating at E_z≠0 (Grujić et al., 2015).
• Acoustic and photonic metamaterials based on the puckered geometry realize higher-order topological boundary modes, including robust, symmetry-protected corner charges and flat edge bands, with unambiguous experimental detection (Wu et al., 10 May 2024).
• Puckered oxide perovskites such as Sr₃CaOs₂O₉ demonstrate that modest out-of-plane structure is sufficient to drive the crossover from low-T 2D to robust 3D magnetism (Thakur et al., 2022).
• Theoretical and simulation advances in mechanical metamaterials show how to disentangle and tune in-plane and out-of-plane auxeticity, expanding the design space for 2D functional materials (Davini et al., 2017).

The puckered honeycomb motif thus provides a unified structural origin for a diverse set of 2D phenomena, with broad prospects for technological exploitation and fundamental symmetry-protected phases.