Robust topological surface states in skyrmion-host magnets Eu(Ga,Al)4: evidence for dual topology

This study provides conclusive evidence for the dual-topology nature of Eu(Ga,Al)$_4$ magnets by using high-resolution angle-resolved photoemission spectroscopy to directly observe robust topological surface states derived from bulk Dirac nodal lines and their unique coupling to magnetic skyrmions.

The Gist

Imagine you have a magical material that is a master of disguise. It has two different "personalities" or "superpowers" hidden inside it, and for the first time, scientists have caught them both in the act.

This paper is about a family of crystals called Eu(Ga, Al)₄ (think of them as a special alloy of Europium, Gallium, and Aluminum). The researchers discovered that these crystals are what they call a "Dual-Topology" system.

To understand what that means, let's break down the two "superpowers" using simple analogies:

1. The First Superpower: The "Swirling Vortex" (Real-Space Topology)

Imagine a calm pond. If you drop a stone, you get ripples. But in this material, the tiny magnetic spins (the tiny compass needles inside the atoms) don't just point up or down. They swirl around each other like a whirlpool or a tornado.

• The Analogy: Think of a Skyrmion as a microscopic, stable tornado of magnetism. Even if you poke it or push it, it doesn't fall apart; it just keeps spinning. This is "Real-Space Topology." It's a physical shape in the material that is incredibly tough to destroy.

2. The Second Superpower: The "Magic Highway" (Momentum-Space Topology)

Now, imagine the electrons (the tiny particles that carry electricity) moving through this crystal. Usually, they get stuck or bounce around like cars in a traffic jam. But in "Topological" materials, the rules of the road change.

• The Analogy: Think of the electrons as cars on a magic highway that only exists on the surface of the crystal. No matter how bumpy the road gets or how many potholes (impurities) are there, the cars on this highway can't stop or crash. They flow perfectly. This is "Momentum-Space Topology," and the "cars" are called Topological Surface States (TSSs).

The Big Discovery: Finding the "Magic Highway"

For a long time, scientists knew these crystals had the "Swirling Vortex" (Skyrmions). But they weren't sure if the "Magic Highway" (the Surface States) actually existed there. It was like knowing a building had a secret elevator, but never seeing the doors open.

The researchers used a super-powerful camera called ARPES (Angle-Resolved Photoemission Spectroscopy). Think of this camera as a high-speed flash photography that takes pictures of the electrons as they fly off the surface of the crystal.

What they found:

1. The Highway is Real: They took clear pictures of the "Magic Highway" (the Surface States) and proved it connects to a hidden "Drumhead" shape inside the crystal.
2. It's Unbreakable: This highway is incredibly tough.
   • The "Pothole" Test: The surface of the crystal rearranged itself (like a road repaving itself), but the highway stayed perfect.
   • The "Chemical" Test: They looked at two different sides of the crystal (one side covered in Europium, the other in Gallium/Aluminum). Even though the "skin" of the crystal changed, the highway was still there.
   • The "Freeze" Test: When they cooled the crystal down, the magnetic "vortices" (Skyrmions) started spinning. Usually, this messes up electron traffic. But here, the highway stayed open!

The "Ghost Train" Effect

Here is the coolest part. When the magnetic "vortices" started spinning (below a certain cold temperature), the electrons didn't just keep flowing; they started doing a weird dance.

• The Analogy: Imagine a train moving on a track. Suddenly, a "ghost train" appears right next to it, moving in perfect sync but slightly shifted. This is called a Replica Band.
• Why it matters: This ghost train proves that the magnetic "vortices" and the electron "highway" are holding hands. They are talking to each other. The magnetism is directly controlling the electricity.

Why Should We Care?

Usually, to get these two superpowers to work together, you have to glue two different materials together (like a sandwich), which is messy and hard to control.

This paper shows that Eu(Ga, Al)₄ is a single, natural material that has both powers built-in.

• The Future: Because the magnetism and the electricity are so tightly linked, scientists might be able to control the "Skyrmion tornadoes" just by using an electric field (like a remote control), instead of using heavy magnetic fields. This could lead to super-fast, super-efficient computers that use almost no power.

In a Nutshell

Scientists found a crystal that is a double agent. It has a stable magnetic swirl inside it and a super-fast electron highway on its surface. They proved these two things not only exist together but also talk to each other, opening the door to a new era of "smart" electronics that are controlled by both magnetism and electricity in a single, robust package.

Technical Summary

1. Problem Statement

The field of condensed matter physics has established two distinct realms of topology:

• Real-space topology: Characterized by magnetic textures like skyrmions (particle-like spin vortices).
• Momentum-space topology: Characterized by topological surface states (TSSs) arising from band inversions or nodal lines in the electronic structure.

While the interplay between these two realms is predicted to yield novel phenomena (e.g., charged skyrmions, emergent gauge couplings), materials hosting both topologies simultaneously are extremely scarce. The family of helimagnets Eu(Ga$_{1-x}$Al$_x$)$_4$ (specifically EuGa$_2$Al$_2$ and EuAl$_4$) has been a prime candidate for such a "dual-topology" system. Previous studies confirmed their real-space topology (skyrmion phases and topological Hall effects) and predicted momentum-space topology (Dirac nodal lines, DNLs). However, conclusive experimental evidence for the momentum-space topology (specifically the TSSs) had been missing, preventing the definitive identification of this family as a true dual-topology material.

2. Methodology

The authors employed high-resolution Angle-Resolved Photoemission Spectroscopy (ARPES) to directly probe the electronic structure and surface states.

• Samples: EuGa$_2$Al$_2$ ($x=0.5$, $T_N = 19.0$ K) and EuAl$_4$ ($x=1.0$, $T_N = 15.4$ K).
• Technique:
  • Used vacuum ultraviolet (VUV) photons (e.g., $h\nu = 113$ eV, 90 eV) to ensure high surface sensitivity.
  • Mapped the Fermi surface (FS) and band dispersions along high-symmetry cuts ($\bar{\Gamma}-\bar{X}$).
  • Utilized a micro-focused photon beam ($\phi \sim 10$ $\mu$m) to scan across cleaved surfaces, identifying distinct surface terminations (Eu-rich vs. GaAl-rich) based on core-level intensity ratios.
  • Performed measurements both above and below the Néel temperature ($T_N$) to study magnetic ordering effects.
• Theoretical Support:
  • Density Functional Theory (DFT) calculations for bulk band projections.
  • Slab-model calculations to simulate surface states and confirm their origin.
  • Topological invariant calculations (Berry phase and Zak phase) to verify the nontrivial nature of the observed states.

3. Key Contributions and Results

A. Direct Observation of Topological Surface States (TSSs)

• The study directly observed TSSs in both EuGa$_2$Al$_2$ and EuAl$_4$ that originate from bulk Dirac nodal lines (DNLs).
• Fermi Surface Mapping: In EuGa$_2$Al$_2$, the experimental FS revealed three sheets (S1, B1, B3), whereas bulk calculations predicted only two. The additional sheet (S1) and a flat band (S2) at $\sim 0.5$ eV binding energy were identified as surface states because they have no bulk counterparts and extend into the gapped region of the bulk projection.
• Connectivity: The S2 band was found to be connected to the bulk DNL, forming a "drumhead-like" dispersion, a hallmark signature of TSSs in Dirac semimetals.
• Topological Verification:
  • Berry Phase: Calculated to be quantized to $\pi$ for the path encircling the DNL, confirming nontrivial topology.
  • Zak Phase: Calculated to be $\pi$ in the $k$-region outside the DNL but not inside, consistent with the experimental emergence of the S2 band in the large-$k$ region.

B. Exceptional Robustness of TSSs

The authors demonstrated that these TSSs are robust against significant perturbations:

1. Surface Reconstruction: The TSSs persist despite a $2\times1$ surface reconstruction (evidenced by replica bands in the FS).
2. Surface Termination: TSSs were observed on two chemically and structurally distinct surfaces:
   • GaAl-terminated: Exhibits $2\times1$ reconstruction; shows S1 and S2 bands.
   • Eu-terminated: Exhibits $1\times1$ structure; shows a new, nearly flat band (S3) near the $\bar{X}$ point. Slab calculations confirmed S3 is a TSS connected to the bulk DNL.
   • Significance: This proves the TSSs are an intrinsic property of the bulk topology, not an artifact of a specific surface termination.
3. Magnetic Order: The TSSs persist deep into the magnetically ordered phase (below $T_N$).

C. Magneto-Topological Coupling

• Replica Bands: Below $T_N$, the authors observed replica bands shifted by the magnetic ordering vector ($\mathbf{Q}$). This indicates that the magnetic order actively couples to the electronic structure, causing band folding.
• Surface Dependence: This magnetic band folding was pronounced on the Eu-terminated surface but weak or absent on the GaAl-terminated surface. This suggests the coupling is mediated by the local magnetic moments of the surface Eu atoms, highlighting a tunable interaction between magnetism and topology.

D. Generality

The findings were extended to EuAl$_4$ (the Al-end member), where analogous TSSs (S1 and S2) were observed, confirming that the dual-topology nature is a fundamental characteristic of the entire Eu(Ga$_{1-x}$Al$_x$)$_4$ family for $0.5 \le x \le 1.0$.

4. Significance

• Definitive Dual-Topology Material: This work provides the missing experimental evidence to establish Eu(Ga$_{1-x}$Al$_x$)$_4$ as a rare material system where real-space (skyrmions) and momentum-space (TSSs) topologies coexist and are intrinsically coupled.
• Stability: Unlike other DNL semimetals (e.g., ZrSiS) where surface states can be unstable or decoupled from other orders, the TSSs in Eu(Ga$_{1-x}$Al$_x$)$_4$ are remarkably robust against surface reconstruction and magnetic ordering.
• Platform for New Physics: The strong coupling between skyrmions and TSSs opens the door to exploring predicted phenomena such as:
  • Charged Skyrmions: Confinement of TSSs at the skyrmion radius.
  • Electric-Field Control: Potential for dissipationless, electric-field-driven manipulation of skyrmions, moving beyond conventional current-driven methods.
• Tunability: The discovery that the magneto-topological coupling is highly sensitive to surface termination offers a new degree of freedom for controlling topological interactions by engineering the surface environment.

In conclusion, the paper successfully bridges the gap between theoretical predictions and experimental reality for dual-topology systems, providing a robust platform for future research into the interplay of magnetic and electronic topologies.