Twisted Bismuth Bilayers

• Twisted bismuth bilayers are bismuth-based two-layer systems in which one layer is rotated relative to the other, creating moiré patterns that tune electronic, topological, and superconducting properties.
• Twist angles govern the evolution from semiconducting to metallic behavior by modifying the mexican-hat dispersion and inducing spin splitting without external fields.
• Finite twisted bilayers reveal pronounced edge effects that enhance the density of states and superconductivity, offering a platform for engineered topological and spintronic applications.

Searching arXiv for papers on twisted bismuth bilayers and closely related bismuthene bilayer systems. Twisted bismuth bilayers are bismuth-based two-layer systems in which one layer is rotated relative to the other by a finite twist angle, producing twist-dependent changes in electronic structure, spin texture, vibrational response, and, in some formulations, superconducting and topological behavior. In current arXiv literature, the term spans several distinct realizations: a periodic twisted bilayer bismuthene homostructure built from two identical 2D bismuthene layers in AA stacking (Zullo et al., 2024), finite twisted bilayer flakes derived from the Bi-I (Wyckoff) structure and studied without periodic boundary conditions (Rodríguez et al., 2023, Rodríguez et al., 26 Nov 2025), and a 30° twisted heterostructure formed from a $β$-bismuthene monolayer and a planar bismuthene layer on SiC(0001) (Pelliccia et al., 15 Apr 2026). Across these realizations, the common theme is that the large intrinsic SOC of Bi makes twist a direct handle on relativistic band topology, spin splitting, and low-energy spectral weight.

1. Structural realizations and geometric definitions

The periodic homostructure denoted twisted bilayer bismuthene, or TB-Bi, is defined as a homostructure made by stacking two identical 2D bismuthene layers and rotating one layer relative to the other by a finite twist angle. Its untwisted reference is bilayer bismuthene composed of two buckled Bi honeycomb monolayers stacked in the stable AA configuration; each monolayer has two sublattices in different vertical planes with buckling distance $\Delta_B$, while the bilayer introduces the minimum interatomic distance along $z$ between layers, $\Delta_M$, and the interlayer distance $\Delta_I=\Delta_B+\Delta_M$ (Zullo et al., 2024).

A separate line of work models twisted bismuth bilayers as finite nanosheet pairs derived from the Bi-I crystal structure. One study starts from a $10\times 8\times 2$ supercell and reduces it to two finite bilayers of 120 atoms each, for 240 atoms total, then removes periodic boundary conditions and rotates one bilayer around the perpendicular $z$-axis (Rodríguez et al., 2023). Another constructs two freestanding flakes with 145 atoms each, for 290 atoms total, from a $20\times 16\times 2$ Bi-I supercell and then relaxes the system for twist angles from $0^\circ$ to $30^\circ$ (Rodríguez et al., 26 Nov 2025). A further extension uses two distinct bismuthene polymorphs: a zigzag $\Delta_B$0-bismuthene monolayer placed with a 30° rotation on a planar bismuthene layer stabilized on SiC(0001), producing a commensurate heterostructure rather than a homobilayer (Pelliccia et al., 15 Apr 2026).

System	Structural description	Twist-angle domain
Twisted bilayer bismuthene (TB-Bi)	Two identical 2D bismuthene layers in AA-derived homostructure	Six commensurate angles from $\Delta_B$1 to $\Delta_B$2
Twisted bismuth bilayers (finite TBB)	Two finite bilayers of 120 atoms each from Bi-I	$\Delta_B$3 to $\Delta_B$4
Edge-dependent twisted flakes	Two freestanding flakes of 145 atoms each from Bi-I	$\Delta_B$5 to $\Delta_B$6
Twisted $\Delta_B$7-/flat bismuthene heterostructure	$\Delta_B$8-bismuthene on planar Bi/SiC(0001)	Fixed at $\Delta_B$9

For periodic TB-Bi, the moiré primitive vectors $z$0 and $z$1 are defined from the monolayer primitive vectors, and the commensurate twist angle is obtained from integer indices $z$2 through

$z$3

The moiré lattice constant is

$z$4

so the supercell size grows as the twist angle decreases (Zullo et al., 2024).

2. Untwisted bilayer reference: symmetry, mexican-hat dispersion, and baseline DOS

The untwisted AA bilayer establishes the reference from which twist effects are measured. In the periodic bismuthene bilayer, time-reversal symmetry gives

$z$5

and, when inversion symmetry is also present,

$z$6

Accordingly, the untwisted bilayer has inversion symmetry and strong SOC does not split the bands: Kramers degeneracy remains intact. In contrast, monolayer bismuthene lacks inversion symmetry and shows Rashba-split bands. In the bilayer, SOC shapes the valence-band maximum into a mexican-hat dispersion (Zullo et al., 2024).

Within PBE+SOC, the untwisted bilayer is described as a perfectly compensated semimetal. Along the $z$7 path, the Fermi level intersects the valence band maximum, the conduction band maximum lies along $z$8, and the valence bands show a mexican-hat shape. The dispersion is anisotropic: the paper defines $z$9 for the peak along $\Delta_M$0 and $\Delta_M$1 for the peak along $\Delta_M$2, with

$\Delta_M$3

The peak along $\Delta_M$4 lies lower than that along $\Delta_M$5 in the untwisted bilayer (Zullo et al., 2024).

External electric field provides a controlled symmetry-breaking benchmark. For fields in the range $\Delta_M$6–$\Delta_M$7, inversion symmetry is broken and the spin splitting

$\Delta_M$8

grows approximately linearly with field strength, with $\Delta_M$9. The spin-split bands along $\Delta_I=\Delta_B+\Delta_M$0 shrink slightly in $\Delta_I=\Delta_B+\Delta_M$1-space by about $\Delta_I=\Delta_B+\Delta_M$2 relative to the original $\Delta_I=\Delta_B+\Delta_M$3. The untwisted spin textures show mostly in-plane spin circulation, purely out-of-plane spin around $\Delta_I=\Delta_B+\Delta_M$4 and $\Delta_I=\Delta_B+\Delta_M$5, a vortex-like pattern around $\Delta_I=\Delta_B+\Delta_M$6, and a hexagonal pattern around $\Delta_I=\Delta_B+\Delta_M$7; under a $\Delta_I=\Delta_B+\Delta_M$8 field, helicity swaps between the two top valence bands and the $\Delta_I=\Delta_B+\Delta_M$9-centered texture becomes flower-like with triangular lobes (Zullo et al., 2024).

A distinct untwisted reference appears in the Bi(111) bilayer work based on a 72-atom supercell with periodic boundary conditions and vacuum thicknesses of $10\times 8\times 2$0, $10\times 8\times 2$1, and $10\times 8\times 2$2. There, the electronic density of states is metallic at the Fermi level, and for $10\times 8\times 2$3 and $10\times 8\times 2$4 the eDoS curves are practically identical with $10\times 8\times 2$5 states/atom for the bilayer, compared with $10\times 8\times 2$6 states/atom for the semimetallic Wyckoff phase. The vibrational density of states shows the expected gap associated with the bilayer structure, and the authors estimate $10\times 8\times 2$7 under a BCS-style assumption that the Cooper pairing potential is essentially the same for the different bismuth phases and structures (Hinojosa-Romero et al., 2017).

These baseline results are not identical, but they are not directly inconsistent: they correspond to different structural models and computational setups. A plausible implication is that “twisted bismuth bilayers” should not be treated as a single universal reference system even before twist is introduced.

3. Twist-driven electronic reconstruction in periodic twisted bilayer bismuthene

In the periodic AA-derived homostructure, twist angle acts as an internal control knob that reshapes the SOC-dominated low-energy bands. Large-scale first-principles DFT calculations use VASP, the GGA-PBE exchange-correlation functional, PAW pseudopotentials, explicit SOC, and the rev-vdW-DF2 functional for structural optimization. Two PAW setups are used: a 5-valence-electron Bi $10\times 8\times 2$8 potential and a 15-valence-electron potential including the Bi $10\times 8\times 2$9 shell; the 15-electron PAW is used for geometry optimization, while band structures and DOS are computed using only the $z$0 valence states because the $z$1 levels lie far below the Fermi energy ($z$2) and do not affect low-energy properties. Numerical details include plane-wave cutoffs of $z$3 or $z$4 depending on PAW setup, vacuum spacing of about $z$5, Gaussian smearing of $z$6, convergence criteria of total energy change $z$7 and forces $z$8, and an $z$9 $20\times 16\times 2$0-mesh for the untwisted bilayer scaled appropriately for supercells (Zullo et al., 2024).

The central electronic result is a twist-driven semimetal-to-semiconductor transition followed by a return to metallic behavior as the twist angle is reduced. For $20\times 16\times 2$1 $20\times 16\times 2$2, the indirect gap is $20\times 16\times 2$3; for $20\times 16\times 2$4 $20\times 16\times 2$5, the indirect gap is $20\times 16\times 2$6. In this semiconducting regime, the valence-band maximum lies along $20\times 16\times 2$7 and the conduction-band minimum is at $20\times 16\times 2$8. At $20\times 16\times 2$9 $0^\circ$0, valence and conduction bands overlap, the higher valence branch crosses the Fermi level, and the lower valence branch remains below it; this angle is identified as the critical angle, $0^\circ$1. For $0^\circ$2, $0^\circ$3, and $0^\circ$4, TB-Bi is metallic, and the DOS at the Fermi level per atom is zero for the first two angles, finite and peaking at the critical region, and increasing as twist angle decreases (Zullo et al., 2024).

$0^\circ$5	$0^\circ$6	Electronic regime
$0^\circ$7	$0^\circ$8	Indirect-gap semiconductor, $0^\circ$9 meV
$30^\circ$0	$30^\circ$1	Indirect-gap semiconductor, $30^\circ$2 meV
$30^\circ$3	$30^\circ$4	Critical angle, band overlap
$30^\circ$5	$30^\circ$6	Metallic
$30^\circ$7	$30^\circ$8	Metallic
$30^\circ$9	$\Delta_B$00	Metallic

Twist also modifies the mexican-hat valence bands in a way that is analogous to, but not exhausted by, electric-field splitting. The two highest valence bands keep the MH character, but twisting changes the peak energies, the splitting between the two SOC-related branches, and the positions of the peaks in $\Delta_B$01-space. For large twists, the splitting increases with twist and reaches a maximum near the critical angle; for smaller twists, once metallicity sets in, the splitting decreases again. A key difference from the electric-field case is that twisting changes the reciprocal-space anisotropy so that $\Delta_B$02, and in twisted bilayers the opposite energy ordering emerges: the peak along $\Delta_B$03 is no longer lower than that along $\Delta_B$04 (Zullo et al., 2024).

The authors interpret twist as a pseudo electric field because it breaks the equivalence of the two layers and lifts band degeneracy even without any applied field. At the same time, twist also changes the moiré potential, the Brillouin-zone size, the band anisotropy in momentum space, and the localization of states in AA-like regions. This is why the twisted paired valence bands no longer have spin textures that are exact opposites of each other at every $\Delta_B$05-point, unlike the near mirror-related textures of the untwisted case (Zullo et al., 2024).

4. Superconductivity proposals in finite twisted bismuth bilayers

A separate literature analyzes superconductivity in finite twisted bismuth bilayers within a conventional BCS-like framework rather than through periodic moiré minibands. In the 240-atom finite TBB model, first-principles calculations are performed with DMol$\Delta_B$06 in the Materials Studio suite using a double numerical plus polarization basis, DSPP core treatment, LDA exchange-correlation with the Vosko-Wilk-Nusair functional, a fine integration grid, a real-space cutoff radius of $\Delta_B$07, and only the $\Delta_B$08/T point because the system is non-periodic. The vibrational density of states is obtained by the finite-displacement or frozen-phonon method with displacement amplitude $\Delta_B$09 and no $\Delta_B$10-points (Rodríguez et al., 2023).

In that finite-cluster setting, all studied structures are metallic at the Fermi level: the non-rotated bilayer and every twisted case from $\Delta_B$11 to $\Delta_B$12. The total electronic DOS shows a sharp peak at the Fermi level, which the authors interpret as arising mainly from border effects of the finite nanolayers rather than from the core of the bilayer. When the core atoms only are considered, the sharp peak is strongly reduced or disappears, bringing the result closer to previously reported bismuth and bismuthene DOS profiles. Near the Fermi level, the upper band is dominated by $\Delta_B$13 character with substantial $\Delta_B$14 admixture and very small $\Delta_B$15 contribution (Rodríguez et al., 2023).

The vibrational DOS exhibits a forbidden frequency gap roughly between $\Delta_B$16 meV and $\Delta_B$17 meV; this gap shrinks as twist increases and disappears at $\Delta_B$18, though a minimum near $\Delta_B$19 meV remains. Above $\Delta_B$20, isolated high-frequency phonon modes begin to appear above $\Delta_B$21 meV and reach up to $\Delta_B$22 meV at $\Delta_B$23; these modes are attributed mainly to border effects in the finite bilayer geometry. Within the adopted superconductivity model, the impact of these high-frequency isolated modes on $\Delta_B$24 is relatively small compared with the effect of the electronic DOS (Rodríguez et al., 2023).

The superconductivity estimate assumes conventional BCS-type superconductivity and invariance of the Cooper pairing potential $\Delta_B$25 between crystalline bismuth and twisted bismuth bilayers: $\Delta_B$26

$\Delta_B$27

Using literature values $\Delta_B$28 and $\Delta_B$29, and extracting $\Delta_B$30 for TBB from the vibrational DOS via the Grimvall method, the study predicts a maximum superconducting transition temperature

$\Delta_B$31

at the magic angle of $\Delta_B$32. The overall trend is that $\Delta_B$33 rises to a maximum at $\Delta_B$34, then decreases steadily with twist angle, reaches a minimum near $\Delta_B$35 at approximately $\Delta_B$36, and increases again above $\Delta_B$37; the authors interpret this as mostly electronically driven through twist-induced amplification of $\Delta_B$38 (Rodríguez et al., 2023).

This superconductivity literature should be read with a basic distinction in mind: the finite-TBB calculations do not model an infinite periodic moiré lattice. Their central spectral feature at $\Delta_B$39 is explicitly assigned to border effects, and their $\Delta_B$40 values are model estimates contingent on unchanged pairing potential.

5. Edge heterogeneity, finite flakes, and strongly angle-dependent superconducting enhancement

The edge-dependent finite-flake study makes boundary heterogeneity the central variable rather than a correction. The twisted object is divided into a core, a transition region, and an edge: a crystalline-like core of approximately $\Delta_B$41 diameter and around 10 atomic layers, an edge of about $\Delta_B$42 thickness and roughly 3 atomic layers, and a transition region between them. DFT calculations use DMol3 in Materials Studio, the VWN-LDA functional, a double numerical plus polarization basis, DSPP pseudopotentials, $\Delta_B$43 $\Delta_B$44-point sampling, a $\Delta_B$45 cutoff, BFGS geometry optimization, and harmonic vibrational analysis; vibrational stability is confirmed by the absence of imaginary frequencies above $\Delta_B$46 (Rodríguez et al., 26 Nov 2025).

Structurally, the pair distribution function shows that the core matches a 3.4% compressed Bi-I crystal very closely, with first and second neighbor distances matching the compressed crystalline reference and third and fourth neighbors also close within less than 4% error. The edge strongly deviates from crystalline order, showing broad delocalized peaks, loss of crystalline order, no liquid-bismuth shoulder between $\Delta_B$47 and $\Delta_B$48, and about 3.9% compression relative to amorphous and liquid Bi. For the untwisted flake, the plane-angle distribution in the core has peaks at $\Delta_B$49 and $\Delta_B$50, similar to crystalline Bi-I, while the edge has broad peaks near $\Delta_B$51 and $\Delta_B$52, interpreted as a random network of distorted triangles, squares, and pentagons (Rodríguez et al., 26 Nov 2025).

Electronically, both core and edge remain metallic-like in the sense that $\Delta_B$53, but their twist dependence differs sharply. The core electronic density of states is almost invariant with twist angle and remains semimetal-like across the full angle range, with a broad maximum

$\Delta_B$54

at $\Delta_B$55. The edge shows non-monotonic, oscillatory $\Delta_B$56 with local maxima at $\Delta_B$57, $\Delta_B$58, $\Delta_B$59, and $\Delta_B$60, and the reported maximum edge enhancement reaches about 10 times the perfect-crystalline Bi-I value of

$\Delta_B$61

that is, roughly

$\Delta_B$62

The edge Fermi-level peak is described as resembling a van Hove singularity and is interpreted as boundary-induced localization of states near $\Delta_B$63 (Rodríguez et al., 26 Nov 2025).

The vibrational DOS yields a Debye temperature that varies periodically and non-monotonically with twist angle. For $\Delta_B$64, the maximum is $\Delta_B$65 at $\Delta_B$66 and the minimum is $\Delta_B$67 at $\Delta_B$68. The reported fit is

$\Delta_B$69

with $\Delta_B$70 and $\Delta_B$71 (Rodríguez et al., 26 Nov 2025).

Using the Mata-Valladares approach with angle-dependent $\Delta_B$72 and $\Delta_B$73, and the experimental $\Delta_B$74 of Bi-I stated as $\Delta_B$75, the authors obtain a smooth core superconducting response with a maximum

$\Delta_B$76

at $\Delta_B$77, but a highly oscillatory edge response with maxima at the same angles as the $\Delta_B$78 peaks and a largest predicted value

$\Delta_B$79

at $\Delta_B$80. The paper explicitly interprets twist angle together with edge disorder as a design parameter for engineering enhanced topological properties, electronic localization, and superconductivity in finite twisted bismuth bilayers (Rodríguez et al., 26 Nov 2025).

A common misconception is that twist-angle physics in bismuth bilayers is exhausted by the moiré interior. The finite-flake calculations argue the opposite: in realistic finite systems, the strongest DOS and $\Delta_B$81 enhancements can be edge-dominated rather than bulk-like.

6. Spin-orbit and topological extensions: from spin textures to a QSH bismuthene heterostructure

Within the periodic TB-Bi homostructure, the principal SOC effect of twist is not only band splitting but also strong spin-texture reconstruction. The untwisted bilayer under electric field retains paired valence-band textures that are essentially the same up to spin reversal, whereas in TB-Bi the paired bands no longer have spin textures that are exact opposites of each other at every $\Delta_B$82-point. Together with twist-tunable splitting of the mexican-hat valence states, this makes TB-Bi a platform for twistronics with built-in spin control and twist-dependent spin-texture engineering (Zullo et al., 2024).

A distinct topological extension is provided by the 30° twisted $\Delta_B$83-/flat bismuthene heterostructure on SiC(0001). The planar Bi/SiC layer has lattice constant $\Delta_B$84 and Bi–Bi distance $\Delta_B$85; free-standing $\Delta_B$86-bismuthene has lattice constant $\Delta_B$87, Bi–Bi distance $\Delta_B$88, and buckling height $\Delta_B$89. A 30° rotation produces a commensurate supercell with $\Delta_B$90 and requires only a moderate tensile strain of about 6% on the $\Delta_B$91-bismuthene layer. The optimized interlayer spacing is

$\Delta_B$92

smaller than the sum of the Bi van der Waals radii, indicating stronger-than-van-der-Waals coupling described as partially covalent or metavalent bonding. The structure is dynamically and thermally stable: phonon calculations show no imaginary frequencies, and AIMD simulations find it stable at $\Delta_B$93 with partial disorder only around $\Delta_B$94 (Pelliccia et al., 15 Apr 2026).

The electronic mechanism is an interlayer orbital hybridization between Bi $\Delta_B$95-derived states of the planar and twisted layers. Because of Brillouin-zone folding in the $\Delta_B$96 supercell, the planar layer’s valence-band maximum is mapped from $\Delta_B$97 to $\Delta_B$98, enabling overlap with the $\Delta_B$99-layer conduction states. The heterostructure develops a new direct gap at $z$00,

$z$01

whereas without SOC it becomes metallic and shows a Dirac-like crossing at $z$02. The system lacks inversion symmetry because one side faces the SiC-supported planar layer and the other the twisted $z$03-bismuthene layer; together with strong Bi SOC, this produces Rashba spin splitting absent in the isolated monolayers. The spin-resolved Fermi surface shows a helical in-plane spin texture with opposite spin polarization on the split branches (Pelliccia et al., 15 Apr 2026).

The topological character is established by the $z$04 invariant computed from Wannier charge center evolution,

$z$05

and by the spin Hall conductivity calculated with WannierTools. Within the band gap, the twisted heterostructure has

$z$06

compared with $z$07 for Bi/SiC and $z$08 for isolated $z$09-bismuthene. Sb substitution in the zigzag layer,

$z$10

increases the interlayer distance from $z$11 to $z$12 and reduces the direct gap from $z$13 to $z$14 while preserving nontrivial topology and finite SHC, with reported values of approximately $z$15, $z$16, $z$17, and $z$18 as $z$19 increases (Pelliccia et al., 15 Apr 2026).

Taken together, these results show that twisted bismuth bilayers are not a single phenomenon but a family of SOC-dominated bilayer problems. In periodic homobilayers, twist controls semimetallic, semiconducting, and metallic regimes together with mexican-hat splitting and spin textures. In finite flakes, twist can act through boundary heterogeneity to amplify $z$20 and the superconducting response within a BCS-like model. In mixed-polymorph heterostructures, large-angle twist and broken inversion symmetry can instead stabilize a QSH phase with Rashba splitting and enhanced SHC.