Lattice and Orbital-Resolved Fermiology of Metallenes

TL;DR

This work provides a systematic, element-resolved map of Fermiology for 270 monolayers of metallenes across six lattice geometries, uncovering how lattice type fixes Fermi-line topology while buckling reshapes crossings and pocket formation. By introducing four orbital- and contour-based metrics ($N_{ m{cross}}/L$, $F_{ m{flat}}$, $A_{ m{line}}$, $G$) and the global pocketness score $
P$, the authors present a compact framework to predict transport-relevant features and guide ARPES and SdH experiments. The findings show that lattice geometry primarily governs loop shape and placement, with element-driven orbital character determining how bands populate $E_F$, enabling targeted design for isotropic vs. anisotropic transport, central pockets vs. peripheral loops, and tunable near-$E_F$ flatness. This baseline model, while rooted in freestanding monolayers, provides actionable screening guidelines for substrate-coupled metallenes in catalysis, plasmonics, and orbitronics, and lays the groundwork for high-throughput explorations incorporating environmental effects. Overall, the pocketness framework offers a practical route to engineer Fermi-surface features in 2D metals for pocket-driven device concepts.

Abstract

Atomically thin metallenes have emerged as a new member of the two-dimensional (2D) materials family. Recent experimental realization of metallenes in the Ångström limit has further intensified interest in this class of 2D materials. However, achieving sub-atomic insight into them demands the most detailed and systematic characterization of their electronic structure. Such understanding is essential for the rational design and exploitation of their properties in plasmonics, catalysis, and quantum optics. Existing electronic-structure studies are either scattered or focus on a few selected systems, and a comprehensive view of their band structures and Fermi surfaces remains missing. Here, we address this gap by studying 45 elemental metallenes in six monolayer lattices (honeycomb, square, hexagonal, and their buckled forms) using density-functional theory. We found that lattice type primarily fixes the shape and radial placement of the Fermi-lines, while out-of-plane buckling introduces controlled modifications: it shortens long straight Fermi-line segments, and occasionally creates, removes, or merges small Fermi-line pockets. The electronic configuration determines which orbital type dominates the Fermi level. We summarized Fermiology using a single score for each element, termed pocketness, derived from four descriptors that combine element properties (symmetry, coordination) with electronic characteristics (dispersion, Fermi-surface topology). This score enables targeted angle-resolved photoemission spectroscopy (ARPES) tests, controlled Lifshitz transitions, and provides a predictive basis for transport and device applications.

Figures (5)

• Figure 1: (a) Structural and Fermiology metric schematics. Top: hexagonal (hex), honeycomb (hc) and square (sq) metallenes with computational unit cells (black quadrilateral) and a side view illustrating out-of-plane buckling. Bottom: schematics defining the four fermiology metrics: $N_{\mathrm{cross}}/L$ (number of band crossings at $E_F$ per path length), $F_{\mathrm{flat}}$ (fraction of the near-$E_F$ path spent on flat segments), $A_{\mathrm{line}}$ (degree of Fermi-line isotropy, low for nearly isotropic ($\Gamma$-centred loops, high for edge-aligned segments), and $G$ ($\Gamma$-centricity, i.e. weight of Fermi lines near $\Gamma$ versus the zone edge). The grey dashed hexagon schematically marks the first Brillouin zone. Representative band structures and Fermiologies of metallenes. (b) Mg in honeycomb (hc) and buckled honeycomb (bhc) lattices, (c) Al in square (sq) and buckled square (bsq) lattices, and (d) Ir in hexagonal (hex) and buckled hexagonal (bhex) lattices. For each case, the left panel shows the band structure along the high-symmetry path with the Fermi level fixed at $E_F = 0$; bands are colored by orbital type projection as implemented in QuantumATK smidstrup2019quantumatk (blue ($s$), orange ($p$), red ($d$)). The right panel shows the corresponding Fermi-line map in the first Brillouin zone (FBZ) at $E_F$, here the color encodes the Fermi velocity ($v_F$). The red dot marks the $\Gamma$-point at the FBZ center, and black dots on the FBZ boundary indicate high-symmetry points: edge centers are $M$ for hc/bhc and $X/Y$ for sq/bsq/hex/bhex, while zone corners are $K$ for hc/bhc and $C$ for sq/bsq/hex/bhex. For each band-structure and Fermi-line plot, the values of $N_{\mathrm{cross}}/L$ (blue) and $F_{\mathrm{flat}}$ (red), $A_{\mathrm{line}}$ (green) and $G$ (purple) are shown. The complete set of bands and Fermi maps for all metallenes and lattices is in the SI.
• Figure 2: Violin plots show the distribution of $\langle v_F\rangle$ (m/s) for elemental metallenes grouped by valence family. (a) $s$-led (Groups 1–2), (b) closed-$d$ (Groups 11–12), (c) $p$-led (Groups 13–15), (d) early-$d$ (Groups 3–6), and (e) late-$d$ (Groups 7–10). In each panel, the six lattices (hc, sq, hex, bhex, bsq, bhc) appear on the horizontal axis. Individual symbols indicate single metallenes, the central diamond marks the median, and the vertical bar shows the interquartile range for that lattice and family.
• Figure 3: (a) Orbital-resolved band structure and fermiology metrics vs lattice. For each of the six lattices, the panels show lattice-wise medians of $N_{\mathrm{cross}}/L$, $F_{\mathrm{flat}}$, $A_{\mathrm{line}}$, and $G$ for $s$-led (Group 1-2), closed$d$ (Group 11-12), $p$-led (Group 13-15), early-$d$-led (Group 3-6), and late-$d$-led (Group 7-10) metallenes; error bars denote the interquartile range. (b) Lattice-resolved statistics of the four metrics. From top to bottom, boxplots show the distributions of $N_{\mathrm{cross}}/L$, $F_{\mathrm{flat}}$, $A_{\mathrm{line}}$, and $G$ for all elemental metallenes in the six lattices. For each lattice, the grey box and whiskers indicate the interquartile range and overall spread of the metric, while the black dot and connecting line mark the lattice-wise median, highlighting how coordination and buckling systematically reshape multiband connectivity, band flatness, Fermi-line anisotropy, and $\Gamma$-centricity. Together, the trends reveal how Fermi-level crossing density, band flatness, Fermi-line anisotropy, and $\Gamma$-centricity depend jointly on lattice geometry and dominant orbital character.
• Figure 4: Correlation between metrics across all metallenes. (a-f) Scatter plots show the pairwise relationships between the four metrics $N_{\mathrm{cross}}/L$, $F_{\mathrm{flat}}$, $A_{\mathrm{line}}$, and $G$. Each point corresponds to a single metallene and is colored by dominant orbital character: $s$-led (Group 1-2, 11-12) (blue), $p$-led (Group 13-15) (orange), and $d$-led (Group 3-10) (red). The plots reveal how multiband connectivity, band flatness, Fermi-line anisotropy, and $\Gamma$-centricity co-vary and how $s$-, $p$-, and $d$-dominated metallenes occupy distinct regions of the correlation space.
• Figure 5: (a) Identifying metallenes on the basis of $\mathcal{P}$, a metrics-weighted per-element score for locally flat bands with fewer $E_F$ crossings and nearly isotropic Fermi contours which sits near $\Gamma$ (details in SI). Low-$\mathcal{P}$ values cluster among the alkali and coinage metallenes together with a few light $p$-led and 3$d$ metallenes, the majority of the remaining main-group and transition metallenes have a moderate value of $\mathcal{P}$, whereas high-$\mathcal{P}$ are of alkaline-earth and early/late $d$-metallenes. (b) Schematic band structures and Fermi-line topologies representative of low- and high-$\mathcal{P}$ behavior. Low $\mathcal{P}$ shows many steep crossings of $E_F$ and Fermi-line contours whose radius changes a lot and remain close to the FBZ edges, while high $\mathcal{P}$ shows fewer, flatter bands near $E_F$ and compact, nearly isotropic $\Gamma$-centred loops.