Emergent altermagnetism at surfaces of antiferromagnets: full symmetry classification and material identification

TL;DR

This work introduces a symmetry-based framework to realize two-dimensional altermagnetism at surfaces of bulk collinear antiferromagnets by terminating the material in ways that break bulk spin degeneracy while preserving spin-only inversion. Central to the approach is the surface spin space group (sSSG) and the surface Laue group, which enable a complete classification of altermagnetic surface states into SAM ($d$-, $g$-, or $i$-wave) and SD-SAM variants. A systematic MAGNDATA screening identifies over 140 AFMs with at least one altermagnetic surface, and a minimal Lieb-lattice-inspired model demonstrates the mechanism of surface-induced $d$-wave altermagnetism. Ab initio studies of NaMnP ($d$-wave SAM) and FeGe$_2$ ($g$-wave SAM) confirm the predicted surface spin-split textures. The results offer a scalable path to 2D altermagnetism without exfoliation, reconcile surface-sensitive experiments with bulk order, and open avenues for interface spintronics, magneto-optics, and topological phenomena at AFM boundaries.

Abstract

We demonstrate the emergence of altermagnetism at the surfaces of antiferromagnets, vastly expanding the number of material candidates with altermagnetic characteristics and establishing a route to two-dimensional altermagnetism through surface-induced symmetry breaking. We do so by developing a surface spin group formalism that fully classifies all surface magnetic states and identifies altermagnetic surface spin groups that can arise at the surfaces of antiferromagnets. We use this formalism to identify over 140 antiferromagnetic entries from the MAGNDATA database with at least one altermagnetic surface, often times with multiple such surfaces in the same material. We illustrate this emergent phenomenon in a realistic Lieb lattice-based minimal model and present ab initio calculations on two representative material candidates, NaMnP and FeGe$_2$, exhibiting $d$-wave and $g$-wave surface altermagnetism, respectively. Our theory naturally resolves the contradiction of recent experimental reports of $d$-wave ARPES measurements on metallic Lieb lattice compounds that have been shown to be antiferromagnetic in the bulk. Hence, we establish a new paradigm for generating two-dimensional altermagnetism by functionalizing the abundant material class of collinear antiferromagnets as viable platforms for controlled surface altermagnetism, creating natural materials for future hybrid device implementation.

Figures (4)

• Figure 1: Minimal model inspired by the layered Lieb lattice compounds. (a) Schematic of the crystal structure for a generic layered Lieb lattice system with the hoppings of the tight-binding model in Eq. \ref{['toy_hamiltonian']} indicated. (b) Kramers' degenarate bulk bands of the minimal model with periodic boundary conditions in all three spatial directions. (c) Spin-resolved surface spectral density at the $(001)$ surface showing altermagnetic splitting emerging from the surface-induced symmetry breaking. Red and blue colors depict spectral weight in the spin-up and down channels, respectively. The mixed color signifies the degeneracy of both channels in their spectral density. (d) Isoenergy cut of the difference between the spin-up and down channel of the surface spectral density in the two-dimensional surface Brillouin zone at $0.8/J$ [see dashed yellow line in (c)], exhibiting the $d$-wave pattern of the spin-splitting. Parameters in units of $J$ read: $t_1 = 0.2, t_{2,a}=0.4, t_{2,b} = 0, u = 1.2,J = 1, v = 0.4$.
• Figure 2: Pie chart summarizing the results of the surface altermagnetism screening of the MAGNDATA database, consisting of 755 total entries that are 3D AFMs. The inner pie chart shows the share of materials that have altermagnetic surfaces, i.e., SAM or SD-SAM surfaces. The outer pie chart shows the shares of the respective surface spin Laue groups. Notice that it is common among the materials with altermagnetic surfaces to have multiple different surfaces with SAM/SD-SAM surface spin Laue groups. The "Other" category encompasses the surface spin Laue groups $^1\bar{3}^2m$, $^26/^2m$, $^14/^1m^2m^2m$, and $^24/^1m$. The total number of surfaces for the respective surface spin Laue group is given in Tab. \ref{['tab:tabsummary']}.
• Figure 3: Candidate material NaMnP for a $\mathcal{PT}$ AFM that exhibits $d$-wave altermagnetic surface states. (a) Crystal unit cell showing compensated collinear magnetic ordering of Mn moments. Two different termination planes (001) and (1$\bar{1}$0) are indicated in green and brown. Surface spin group analysis predicts $d$-wave altermagnetic surface states for (001) and (1$\bar{1}$0), respectively, but with different surface spin Laue groups. (b) Spin-polarized bulk electronic bands calculated without spin-orbit coupling (SOC), showing Kramers' spin degeneracy. Inset highlights how the $\mathcal{PT}$ symmetry connects the two sublattices in the bulk. (c) Spin-up and (d) spin-down channel spectral function ($\mathcal{G}(E,\mathbf{k})$) without SOC of the (001) Mn-terminated surface, indicated with the green plane in (a), showing both bulk and surface states along the path $\bar{\mathrm{\Gamma}}$ (0.0,0.0)-$\bar{\mathrm{X}}$ (0.5,0.0), indicated by a yellow line in panel (e) in the the surface Brillouin zone. (e) The spin-polarized spectral function difference ($\Delta \mathcal{G}(E,\mathbf{k})=\mathcal{G}_{\uparrow}(E,\mathbf{k})-\mathcal{G}_{\downarrow}(E,\mathbf{k})$) at $E_{F} - 0.5$ eV for (001) termination exhibiting $d$-wave nature with two nodal planes, and rotational symmetry. Surface Brillouin zone indicated with a black dashed square. (f) The spectral function difference for the two spin channels calculated for (1$\bar{1}$0) termination at $E_{F} - 0.5$ eV, with the axes system rotated such that $k_{z'}$ parallel to the surface normal, exhibiting $d$-wave nature with two nodal surfaces without rotational symmetry, and with surface Brillouin zone indicated with black dashed rectangle.
• Figure 4: Candidate material $\text{FeGe}_2$ for a $\mathcal{PT}$ AFM that exhibits $g$-wave altermagnetic surface states. (a) Left panel: Top view of the crystal unit cell showing antiparallel magnetic ordering of Fe moments, indicating the [$C_{2\perp}||m_x|t_b$] spin transposing symmetry, where $t_b=(0,\frac{1}{2},0)$. Right panel: Terminated bulk structure, showing the glide $m_x$ plane. (b) Spin-polarized Kramers' degenerate bulk dispersion calculated without spin-orbit coupling (SOC). The $\mathcal{PT}$ symmetric magnetic sublattices are shown in the inset. Spectral function ($\mathcal{G}(E,\mathbf{k})$) of the (001) Fe-terminated surface [shown with yellow plane in the inset of (b)] without SOC, showing both bulk and surface states for spin-up (c) and spin-down (d) channels, respectively. The spin splitting near the Fermi energy is $\sim$0.08 eV along the path $\bar{\mathrm{\Gamma}}$ (0.0,0.0)-$\bar{\mathrm{P}}$ (0.5,0.25), indicated by a dashed yellow line in panel (e) in the surface Brillouin zone (dashed black square). (e) The spin-polarized spectral function difference ($\Delta \mathcal{G}(E,\mathbf{k})=\mathcal{G}_{\uparrow}(E,\mathbf{k})-\mathcal{G}_{\downarrow}(E,\mathbf{k})$) at the Fermi energy for (001) termination exhibits $g$-wave nature with four nodal planes. (f) The spectral function difference for two spin channels calculated for (100) termination [shown with orange plane in the inset of (b)], indicating an uncompensated ferromagnetic nature in agreement with symmetry prediction.