Spin-resolved quasiparticle interference patterns on altermagnets via non-spin-resolved scanning tunneling microscopy

TL;DR

The paper addresses identifying spin-split Fermi surfaces in altermagnets, materials with zero net magnetization. It develops a minimal four-site tight-binding model on a Lieb-like lattice with tunable AFM and altermagnetic phases and computes impurity-induced LDOS using the retarded Green's function $G(\omega)=[\omega+i\eta-H]^{-1}$. By placing impurities on distinct magnetic sublattices and Fourier transforming the LDOS variations to obtain FT-LDOS, the authors show that non-spin-resolved STM can yield effectively spin-resolved QPI maps via spin-dependent LDOS scattering, revealing a $d$-wave spin-split Fermi surface. This approach provides a practical experimental route to identifying altermagnetic order in candidate materials without spin-polarized STM by comparing QPI from impurities on different sublattices.

Abstract

We investigate quasiparticle interference on an altermagnetic Lieb-like lattice and show how a non-spin-polarized scanning tunneling microscopy measurement can yield effectively spin-resolved information. Within a four-site tight-binding model, which can be tuned between an antiferromagnetic and a Lieb-type altermagnetic state, we introduce on-site impurities at distinct sublattice sites and compute the real space local density of states (LDOS) via a Green's function approach. A Fourier transformation of the impurity-induced LDOS yields the characteristic $d$-wave spin-split Fermi surface contours of the altermagnetic phase. Notably, by choosing which sublattice the impurity is placed upon, we show that the scattering amplitudes effectively encode spin-dependent contrasts: Impurities on one of the magnetic sublattices highlights predominantly spin-up contributions along one crystallographic direction, while impurities on the other one favor the complementary spin-down channel and orientation.

Figures (4)

• Figure 1: (a) Two-dimensional altermagnetic crystal lattice with two sites hosting opposite magnetic moments (up/red and down/blue) and two non-magnetic sites (white) in a checkerboard pattern. Itinerant electrons couple to the local moments through an exchange term $\pm J\sigma$, with $\sigma=\pm1$ for spin-up and spin-down carriers respectively. The values $\epsilon_{2/3}$ represent on-site energies for the non-magnetic sites and $t$ the hopping strength between nearest-neighbor bonds. By tuning $\epsilon_{2}$ and $\epsilon_3$, the system can be tuned between an antiferromagnetic (AFM) state with $\epsilon_{2}=\epsilon_{3}$ and an altermagnetic (AM) state with $\epsilon_{2}\neq\epsilon_{3}$. (b)Band structure of the model along the high-symmetry-points for the parameters $(\epsilon_2=\epsilon_3=1)$ in black and $(\epsilon_2=0,\epsilon_3=1)$ and $(\epsilon_2=1,\epsilon_3\to\infty)$ in red (spin up) and blue (spin down). In the case of $\epsilon_2\neq\epsilon_3$, the altermagnetic $d$-wave spin splitting becomes visible along the $\Gamma\to X$ and $\Gamma\to Y$ paths. (c) Fermi surface of the model.
• Figure 2: Real space Friedel oscillations in the charge LDOS for a single on-site impurity (red cross) placed on each of the four unit-cell sites (columns 2-5) with the lattice model tuned through three magnetic regimes (rows) at the Fermi level ($\omega=0$). Column 1 shows the undisturbed system. (b-e): AFM state ($\epsilon_{2}=\epsilon_{3}$). (g-j): 4-site AM state (finite $\epsilon_{2}\neq\epsilon_{3})$. (l-o): Lieb lattice limit ($\epsilon_3\to\infty$).
• Figure 3: Clean spin-up LDOS and impurity-induced change in the spin-up LDOS, $\Delta\rho_{\uparrow} = \rho_{\uparrow,\text{impure}}-\rho_{\uparrow,\text{clean}}$, for an impurity (red cross) on each of the four unit-cell sites (columns) with the lattice model tuned through three magnetic regimes (rows) at the Fermi level ($\omega=0$). (a-d): AFM state ($\epsilon_{2}=\epsilon_{3}$). (e-h): 4-site AM state (finite $\epsilon_{2}\neq\epsilon_{3}$). (i-l): Lieb lattice limit ($\epsilon_3\to\infty$).
• Figure 4: Absolute value of the Fourier-transformed LDOS (power spectrum) at the Fermi level ($\omega=0$) for a single impurity placed on each of the four unit-cell sites (columns) with the lattice model tuned through three magnetic regimes (rows). The intensity gives the spin summed power spectrum, while red and blue denote spin up and spin down contributions, respectively. (a-d): AFM state ($\epsilon_{2}=\epsilon_{3}$). (e-h): 4-site AM state (finite $\epsilon_2\neq\epsilon_3$). (i-l): Lieb lattice limit ($\epsilon_3\to\infty$).