🔬 condensed matter

Local moment magnon spectrum in conduction electron tunnelling

This paper investigates how dynamic scattering of conduction electrons from local moment magnon excitations, in addition to static magnetic order, modifies the surface tunnelling spectrum of an antiferromagnetic dual system by introducing magnon sidebands.

Original authors: Peter Thalmeier, Alireza Akbari

Published 2026-07-08

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CC BY 4.0

Original authors: Peter Thalmeier, Alireza Akbari

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a bustling city (the material) where two types of residents live side-by-side: conduction electrons, who are like fast-moving commuters zipping through the streets, and local magnetic moments, who are like a group of stationary guards standing in a strict, organized formation.

In this paper, the authors investigate what happens when a tiny probe (like a Scanning Tunneling Microscope, or STM) tries to "listen" to the commuters to see what the city looks like. Usually, the guards (the magnetic moments) are invisible to the commuters' traffic reports because the guards don't move. However, the authors discovered that the guards' presence and their tiny, jittery movements actually change the traffic report in surprising ways.

Here is a breakdown of their findings using simple analogies:

1. The Static Effect: The "Mirror Wall"

First, the authors looked at what happens when the guards stand perfectly still in an organized pattern (Antiferromagnetic order).

The Analogy: Imagine the commuters are running on a track. Suddenly, a giant, invisible mirror wall appears in the middle of the track, forcing the runners to bounce back and forth. This changes the shape of the track and the speed of the runners.
The Result: Even though the guards themselves aren't moving, their organized "stillness" forces the commuters to reorganize their paths. This creates a "ghost signal" in the traffic report. If you look at the data, you see extra peaks (called satellites) that correspond to the guards' formation pattern, even though the guards aren't the ones being measured directly. This part was already known from previous studies.

2. The Dynamic Effect: The "Jittery Guards"

The main new discovery in this paper is what happens when the guards aren't perfectly still. They vibrate, wiggle, and send out ripples of energy called magnons (think of these as sound waves or ripples in a pond).

The Analogy: Now, imagine the guards are jittering and throwing small pebbles (magnons) into the path of the commuters. When a commuter hits a pebble, they get bumped, slow down, or speed up.
The Result: These bumps create a "self-energy" for the commuters. It's like the commuters are wearing heavy backpacks or running on a bumpy road because of the guards' vibrations. This changes the traffic report in two specific ways:
1. The Main Peak Shrinks: The main signal of the commuters gets slightly weaker because their energy is being scattered.
2. Sidebands Appear: This is the big discovery. Just as a singer's voice creates echoes, the commuters now show up in the data with "echoes" or sidebands. These are extra, faint signals appearing at slightly higher and lower energies than the main signal. These sidebands are the direct fingerprint of the guards' vibrations (the magnons).

3. Why This Matters

The authors argue that if you want to understand the "traffic report" (the STM spectrum) of these materials, you can't just look at the static arrangement of the guards. You have to account for their jittery movements.

The "Ghost" Signal: The paper shows that the "ghost signals" (the satellites at the magnetic wave vector) aren't just caused by the guards standing still; they are also heavily influenced by the guards' vibrations.
Reading the Ripples: By looking closely at these "sidebands" in the data, scientists could theoretically figure out the specific "song" (spectrum) the guards are singing (their magnon spectrum), even though the guards themselves are not the ones being directly scanned.

Summary

Think of the STM experiment as listening to a busy street.

Old View: The street layout changes because of a static fence (the magnetic order).
New View (This Paper): The street layout changes and the cars are constantly getting bumped by invisible, vibrating obstacles (magnons). These bumps create "echoes" in the sound of the traffic. The authors calculated exactly how these echoes look, proving that the "jitter" of the magnetic moments leaves a clear, measurable fingerprint on the electrons passing by.

Important Note: The authors emphasize that this is a theoretical study using a simplified model. They are not claiming this has been used in a specific medical device or commercial product yet; they are simply explaining the physics of how these two systems interact so that future experiments can be interpreted correctly.

Technical Summary: Local Moment Magnon Spectrum in Conduction Electron Tunnelling

Problem Statement
Scanning Tunneling Microscopy (STM) experiments on materials with dual electronic character, such as GdRu $_2$ Si $_2$ , have demonstrated that the magnetic order of localized $4f$ moments can be imaged in the conduction electron tunnelling spectrum. Previous theoretical work established that this occurs via the static reconstruction of conduction bands driven by the on-site exchange coupling to the molecular field of the local moments. This results in anomalous spectral satellites at the magnetic ordering wave vector $\pm Q$ , even though the localized moments themselves do not directly contribute to the charge current. However, it remains an open question whether the dynamics of the local moment order—specifically magnon excitations—leave a discernible fingerprint in the STM tunnelling conductance. This work investigates the extent to which the scattering of conduction electrons from local moment magnons modifies the tunnelling spectrum beyond the effects of static magnetic order.

Methodology
The authors employ a simplified theoretical model consisting of a dual system: localized moments with antiferromagnetic (AF) order and a single tight-binding conduction band. The system is defined by:

Static Order: Commensurate easy-axis AF order with wave vector $Q = (\pi, \pi)$ on a 2D square lattice.
Interactions: An on-site exchange interaction ( $I_{ex}$ $I_{e x}$ ) couples the localized spins to conduction electron spins. This interaction is treated in two parts:
- A static molecular field term that reconstructs the conduction bands (opening magnetic gaps).
- A dynamic term describing the scattering of conduction electrons from magnon excitations.
Magnon Model: The localized moments are treated using a Holstein-Primakoff representation for an $xxz$-type Heisenberg model, allowing for the derivation of the magnon dispersion $\omega_q$ and the electron-magnon interaction vertex.
Formalism: The authors calculate the real-space Green's functions for the conduction electrons. The dynamic effects are incorporated via the electron self-energy ( $\Sigma$ ) derived to second order in the electron-magnon coupling constant ( $I_0$ ) using perturbation theory. The tunnelling conductance is proportional to the local density of states (DOS), which is obtained from the imaginary part of the Green's function.

Key Contributions and Results
The paper derives explicit expressions for the renormalized quasiparticle dispersion and the spectral functions, distinguishing between the "normal" spectral function (at $q=0$ ) and the "anomalous" spectral function (at $q=\pm Q$ ).

Self-Energy and Renormalization: The interaction with magnons introduces a complex self-energy $\Sigma(k, \omega)$ . The real part of this self-energy leads to a renormalization (shift) of the quasiparticle band energies, while the imaginary part determines the quasiparticle lifetime (broadening). The authors find that the sign of the real part at the M-point changes as the exchange coupling strength increases, and the broadening is most prominent at the $\Gamma$ and M points where magnon energies are low.
Magnon Sidebands: The primary result is the appearance of "magnon sidebands" in the tunnelling spectrum. The dynamic scattering processes (virtual magnon emission and absorption) shift spectral weight away from the main quasiparticle peaks toward lower and higher energies.
- These sidebands manifest as distinct features in both the normal ( $\rho_0$ ) and anomalous ( $\rho_{AF}$ ) spectral functions.
- The shape and position of these sidebands are determined by the magnon density of states (DOS) and the momentum-dependent coherence factors of the electron-magnon interaction.
Dependence on Anisotropy: The study demonstrates that increasing the exchange anisotropy parameter ( $\Delta$ ) shifts the magnon spectrum to higher energies. This shift suppresses the virtual magnon excitations available at the Fermi level, thereby reducing the magnitude of the self-energy corrections and diminishing the visibility of the sideband features.
Comparison of Static vs. Dynamic Effects: While static band reconstruction is necessary to generate the anomalous satellites at $\pm Q$ , the inclusion of dynamic magnon scattering significantly modifies the lineshape of both the normal and anomalous spectra. The dynamic effects reduce the height of the main spectral peaks and introduce the aforementioned sidebands.

Significance and Claims
The authors conclude that the static description of magnetic order is insufficient for a complete interpretation of STM spectra in exchange-coupled dual systems. They assert that the dynamic magnon self-energy leaves a "visible imprint" in the tunnelling spectrum. Specifically:

The appearance of magnon sidebands provides a potential mechanism to extract features of the AF surface magnon spectrum from surface charge tunnelling experiments.
These dynamic effects are predicted to be more accessible experimentally in the anomalous part of the spectrum (at $\pm Q$ ) than in the normal part ( $q=0$ ), as the background noise associated with long-wavelength inhomogeneities is absent at the finite AF wave vector.
The work serves as a principal theoretical study, establishing that any interpretation of experimental STM spectra in such systems must necessarily include contributions from conduction electron-magnon scattering to accurately describe the spectral weight distribution and lineshapes.

1. The Static Effect: The "Mirror Wall"

2. The Dynamic Effect: The "Jittery Guards"

3. Why This Matters

Summary

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