Twisted Bilayer PdSe2: Moiré in Rectangular Lattices

• Twisted bilayer PdSe2 is a moiré system defined by rotated rectangular-lattice monolayers with strong orbital-selective interlayer coupling.
• At special twist angles, the structure undergoes dimensional collapse to form 1D flat bands and highly anisotropic electronic behavior.
• Its unique properties enable studies of 1D Luttinger-liquid physics, Rashba spin–orbit coupling, and potential topological states.

Searching arXiv for recent and foundational papers on twisted bilayer PdSe2 and PdSe2 interlayer coupling. {"query":"twisted bilayer PdSe2 arXiv", "max_results": 10} {"query":"PdSe2 interlayer coupling arXiv (Ryu et al., 2021)", "max_results": 10} Twisted bilayer PdSe$_2$ is a moiré system formed by rotating two monolayers of PdSe$_2$, a low-symmetry semiconductor with a rectangular Bravais lattice and unusually strong orbital-driven interlayer coupling. In this setting, twist does not merely perturb two weakly coupled van der Waals sheets: the underlying PdSe$_2$ electronic structure already contains a highly dispersive out-of-plane valence band, and at special twist values the moiré pattern becomes effectively one-dimensional, producing directionally localized flat bands, strong spin–orbit coupling, and nontrivial Berry curvature (Ryu et al., 2021, An et al., 19 Jul 2025).

1. Structural setting and crystallographic anisotropy

Monolayer PdSe$_2$ is an air-stable 2D semiconductor with a rectangular Bravais lattice and pronounced in-plane anisotropy. Its primitive lattice vectors are

$$\mathbf{a} = (a,0),\qquad \mathbf{b} = (0,b),\qquad b \neq a.$$

Bulk PdSe$_2$ is a layered, puckered transition-metal dichalcogenide with low symmetry, often compared structurally to black phosphorus. The bulk structure is orthorhombic, contains zigzag chains of Se–Se dimers within each layer, and has two Pd atoms per layer in the crystallographic unit cell. Because the stacking of Se–Se dimers doubles the in-plane periodicity, the crystallographic cell contains two PdSe$_2$ layers per unit cell along $c$, whereas the “reduced” in-plane cell contains one Pd atom and one Se–Se dimer per layer (Ryu et al., 2021).

The local bonding geometry is central to twisted bilayers. Intralayer Se–Se bonding is strong, with an intralayer Se–Se distance of approximately $2.36$ Å, while the interlayer Se–Se separation is approximately $3.75$ Å and remains electronically active. Pd sits in a square-planar coordination environment with four Se ligands in the plane, and the local $_2$0 axis is slightly canted relative to the crystallographic $_2$1 axis. This low-symmetry, puckered framework already departs from the isotropic paradigms of hexagonal moiré materials and provides the geometric basis for twist-induced one-dimensional moiré channels (Ryu et al., 2021, An et al., 19 Jul 2025).

In twisted bilayer PdSe$_2$2, one layer is fixed and the other is rotated by an angle $_2$3 about an axis perpendicular to the plane. Because the lattice is rectangular rather than hexagonal, twist generates moiré structures with reduced symmetry and, at special angles, a dimensional collapse from a 2D superlattice to a 1D array of channels rather than the usual triangular or hexagonal moiré pattern (An et al., 19 Jul 2025).

2. Orbital-selective interlayer coupling in PdSe$_2$4

The electronic starting point for twisted bilayer PdSe$_2$5 is the unusually strong interlayer coupling of the parent material. Angle-resolved photoemission spectroscopy reveals a pronounced dispersive valence band along the out-of-plane $_2$6–$_2$7 direction with a total bandwidth of approximately $_2$8 eV, and this dispersion is explicitly described as “even more dispersive than those along the in-plane directions.” The corresponding $_2$9 coordinate was obtained through the free-electron final-state approximation,

$_2$0

with $_2$1 eV; the photon energy was varied from $_2$2 to $_2$3 eV under ultra-high vacuum below $_2$4 Torr (Ryu et al., 2021).

Projected density of states and X-ray absorption linear dichroism identify a strongly anisotropic orbital structure. The conduction-band minimum consists mainly of Pd in-plane $_2$5 states $_2$6 and Se in-plane $_2$7 states $_2$8, whereas the valence-band maximum is dominated by Pd $_2$9 and Se $_2$0 character. The top valence states therefore derive from out-of-plane orbitals that form the highly dispersive $_2$1–$_2$2 band, while the conduction edge remains predominantly in-plane (Ryu et al., 2021).

The orbital origin of this behavior is tied to an unusual chemical configuration. XPS core levels show Pd $_2$3 at $_2$4 eV and Se $_2$5 at $_2$6 eV, and the line shapes and binding energies are consistent with monovalent Pd and Se states rather than Pd$_2$7. The inferred configuration is therefore Pd$_2$8($_2$9)–(Se$$\mathbf{a} = (a,0),\qquad \mathbf{b} = (0,b),\qquad b \neq a.$$0)$$\mathbf{a} = (a,0),\qquad \mathbf{b} = (0,b),\qquad b \neq a.$$1, not Pd$$\mathbf{a} = (a,0),\qquad \mathbf{b} = (0,b),\qquad b \neq a.$$2($$\mathbf{a} = (a,0),\qquad \mathbf{b} = (0,b),\qquad b \neq a.$$3)–2Se$$\mathbf{a} = (a,0),\qquad \mathbf{b} = (0,b),\qquad b \neq a.$$4. In the square-planar ligand field, the $$\mathbf{a} = (a,0),\qquad \mathbf{b} = (0,b),\qquad b \neq a.$$5 state is pushed highest in energy and remains unoccupied at the conduction edge, while the occupied $$\mathbf{a} = (a,0),\qquad \mathbf{b} = (0,b),\qquad b \neq a.$$6 state sits at the top of the valence band. The dominant interlayer path is Pd–Se$$\mathbf{a} = (a,0),\qquad \mathbf{b} = (0,b),\qquad b \neq a.$$7–Pd, mediated by Se–Se dimer orbitals aligned along the layer normal rather than by direct Pd–Pd overlap (Ryu et al., 2021).

A simple tight-binding description of the out-of-plane valence dispersion is

$$\mathbf{a} = (a,0),\qquad \mathbf{b} = (0,b),\qquad b \neq a.$$8

so a total $$\mathbf{a} = (a,0),\qquad \mathbf{b} = (0,b),\qquad b \neq a.$$9–$_2$0 bandwidth of about $_2$1 eV gives $_2$2 eV. This provides the scale for the interlayer valence-band hopping relevant to twisted structures. A plausible implication is that twisting reconstructs a large, orbital-directional interlayer hopping channel rather than merely modulating a weak van der Waals tunneling amplitude (Ryu et al., 2021).

3. Critical angles and one-dimensional moiré geometry

The geometry of twisted bilayer PdSe$_2$3 is governed by the commensurability condition

$_2$4

where $_2$5 and $_2$6 are the rotated primitive vectors. For rectangular lattices, the key one-dimensional moiré solutions arise from the sign-change class $_2$7, $_2$8, especially the case $_2$9, $_2$0, which preserves a common lattice translation along the diagonal vector

$_2$1

At the associated critical angles,

$_2$2

atomic periodicity is preserved along $_2$3, while the orthogonal direction exhibits a long-range moiré modulation. The resulting structure is a set of 1D moiré stripes or channels rather than a fully 2D moiré superlattice (An et al., 19 Jul 2025).

For PdSe$_2$4, the simplest case is $_2$5, which gives

$_2$6

and a first critical angle

$_2$7

At this angle, the moiré supercell contains a short vector $_2$8 along the channel direction and a much longer vector $_2$9 that defines the 1D moiré wavelength between adjacent channels. The reciprocal-space construction is likewise effectively one-dimensional because $c$0 and $c$1 are collinear and identical, leaving only one independent small moiré reciprocal vector (An et al., 19 Jul 2025).

A distinctive feature of rectangular moiré systems is the non-analytic behavior near a critical angle. As $c$2, the number of atoms per 2D supercell diverges; exactly at the critical angle, it drops because the structure becomes one-dimensional. For twisted bilayer PdSe$c$3, DFT total energy shows a minimum at $c$4, approximately $c$5 meV per atom lower than nearby noncritical angles. The same framework yields an infinite hierarchy of higher critical angles; seven representative critical angles were explicitly identified for PdSe$c$6 using $c$7 values up to $c$8 (An et al., 19 Jul 2025).

4. Electronic structure at the first critical angle

First-principles calculations for twisted bilayer PdSe$c$9 at $2.36$0 show that the 1D moiré geometry is reflected directly in the electronic structure. The system is an indirect-band-gap semiconductor with

$2.36$1

and the low-energy bands near the Fermi level are strongly anisotropic: they are highly dispersive along $2.36$2 and essentially flat along $2.36$3. Along the $2.36$4 and $2.36$5 directions, the bands display what is described as “1D flat-band behavior,” whereas dispersion persists along the channel direction (An et al., 19 Jul 2025).

The strongest flattening occurs in the valence sector. The two highest valence bands, VB1 and VB2, have bandwidths below $2.36$6 meV and are particularly flat near $2.36$7. Higher in the conduction band, $2.36$8 dispersion reappears, but the valence bands retain their one-dimensional flatness over a wide energy range. The resulting low-energy electronic structure is therefore an effectively 1D electronic system embedded in a 2D crystal (An et al., 19 Jul 2025).

Spin–orbit coupling does not remove this 1D character. With SOC included, the bands remain degenerate at $2.36$9 and $3.75$0, and along $3.75$1 and $3.75$2, but significant spin splitting appears elsewhere in the moiré Brillouin zone. The momentum-dependent splitting is interpreted as a strong Rashba-type SOC arising from inversion-symmetry breaking at finite twist together with the relatively heavy Pd and Se atoms (An et al., 19 Jul 2025).

5. Channel localization, Berry curvature, and candidate many-body regimes

Real-space partial charge densities establish that the flat bands are not merely anisotropic in momentum space; they are also localized in real space. For VB1 and VB2, the charge density is strongly localized along the 1D moiré channels, forming 1D “charge wires” that follow the direction of $3.75$3. CB1 and CB2 are also channel-like, but their localization is less extreme, consistent with their somewhat larger dispersion (An et al., 19 Jul 2025).

The topological structure of the valence manifold is encoded in its Berry curvature. With SOC included, the second and third valence bands show Berry-curvature peaks of opposite sign at band crossing or avoided-crossing points; the fourth valence band also has a peak with magnitude comparable to that of the third band but with opposite sign. This pattern is described as indicating strong interband topological connection and topological obstruction, implying that at least a two-band low-energy description is needed for a faithful effective model (An et al., 19 Jul 2025).

The combination of very narrow valence bandwidth, 1D confinement, strong SOC, and nonzero Berry-curvature distribution makes twisted bilayer PdSe$3.75$4 at the first critical angle a candidate platform for several distinct regimes. The work explicitly identifies 1D Luttinger-liquid physics, arrays of coupled wires, and Rashba-wire phenomenology as relevant frameworks; it further notes that proximitized channels could provide the standard ingredients for Majorana zero modes at channel ends. The same anisotropy implies strongly direction-dependent transport and polarized optical response, with conductivity expected to be much larger along the channel direction than across the channels (An et al., 19 Jul 2025).

6. Quantitative parameters and modeling implications

Several quantitative results define the current modeling baseline for twisted bilayer PdSe$3.75$5:

Quantity	Value or statement	Source
Bulk transport gap	$3.75$6 eV	(Ryu et al., 2021)
$3.75$7–$3.75$8 valence bandwidth	$3.75$9 eV	(Ryu et al., 2021)
Estimated interlayer hopping	$_2$00 eV	(Ryu et al., 2021)
ARPES inner potential	$_2$01 eV	(Ryu et al., 2021)
First critical angle	$_2$02	(An et al., 19 Jul 2025)
Twisted-bilayer indirect gap at $_2$03	$_2$04 eV	(An et al., 19 Jul 2025)
Top two valence-band widths	$_2$05 meV	(An et al., 19 Jul 2025)
DFT energy at $_2$06	about $_2$07 meV per atom lower than nearby noncritical angles	(An et al., 19 Jul 2025)

For low-energy modeling, the unfolding analysis of bulk PdSe$_2$08 indicates that the electronic structure largely follows the reduced unit cell containing a single Pd atom and a single Se$_2$09 dimer per layer, while the top valence band is barely influenced by the cell-doubling potential. This supports an effective one-Pd, one-Se$_2$10 basis per layer for the low-energy Hamiltonian. At the same time, the large out-of-plane dispersion and the Pd $_2$11–Se$_2$12–Pd bonding path imply that any realistic twisted-bilayer model must include geometry-dependent interlayer hopping matrices rather than an isotropic “vertical tunneling” term (Ryu et al., 2021).

Taken together, the two arXiv studies place twisted bilayer PdSe$_2$13 in a “beyond-vdW” category. Relative to hexagonal moiré systems, it combines rectangular-lattice critical-angle geometry with orbital-directional interlayer hybridization. This suggests that hole bands, which derive from the strongly 3D valence states, should undergo substantially stronger twist-induced reconstruction than electron bands, whose edge states remain predominantly in-plane. It also explains why twisted bilayer PdSe$_2$14 is treated as a paradigmatic example of moiré physics in twisted rectangular lattices rather than as a minor variant of weakly coupled TMD moiré materials (Ryu et al., 2021, An et al., 19 Jul 2025).