Emergence of multiple quasi-ferromagnetic magnon modes induced by strong magnetoelastic coupling in $TmFeO_3$ single crystal

This study demonstrates that strong magnetoelastic coupling in $TmFeO_3$ single crystals induces the emergence of multiple hybridized magnon modes during spin-reorientation phase transitions, driven by nonuniform spin-wave excitations within the intermediate magnetic domain structure.

The Gist

Imagine a crystal of Thulium Orthoferrite (TmFeO₃) not as a hard, static rock, but as a giant, microscopic dance floor filled with billions of tiny magnetic dancers (the atoms' spins).

In this paper, scientists are watching how these dancers move when they change the temperature or apply a magnetic field. They discovered something surprising: when the dancers are in the middle of a "rearrangement," they don't just move as one big group. Instead, they suddenly start performing multiple, distinct dance routines at the same time, creating a complex symphony of vibrations that wasn't there before.

Here is the breakdown of the discovery using simple analogies:

1. The Three Dance Styles (The Magnetic Phases)

The magnetic dancers in this crystal have three main ways they like to stand and move, depending on the temperature:

• The "Cold" Style (Γ2 phase): At low temperatures, the dancers face one specific direction.
• The "Hot" Style (Γ4 phase): At high temperatures, they all turn to face a completely different direction.
• The "In-Between" Style (Γ24 phase): In the middle, they are in a state of flux. They are slowly turning from the cold style to the hot style. This is the Spin-Reorientation Phase Transition.

2. The Experiment: Shaking the Floor

The researchers used microwaves (like a very gentle, high-speed shaking of the dance floor) to see how the dancers responded. They measured this shaking at different temperatures and with different magnetic "pushes."

Usually, when you shake a magnetic crystal, you expect to hear one main sound (a single vibration frequency). This is like a choir singing one note. In the "Cold" and "Hot" phases, that's exactly what happened.

3. The Surprise: The "Choir" Splits into Many Voices

But when the dancers were in the "In-Between" phase, something weird happened. Instead of one note, the researchers heard multiple distinct notes appearing at once, separated by small gaps in pitch.

It's as if a single choir suddenly split into several smaller groups, each singing a slightly different note (separated by about 0.5 to 2 GHz), all at the same time.

4. Why Did This Happen? The "Rubber Band" Effect

The paper explains this using a concept called Magnetoelastic Coupling. Think of the magnetic dancers as being connected to the floor by rubber bands (the crystal lattice).

• The Connection: When the dancers move, they stretch the rubber bands. When the rubber bands snap back, they pull the dancers. They are tightly linked.
• The Transition: When the dancers are in the middle of turning (the transition phase), the floor itself gets a bit "squishy" or unstable. The rubber bands get loose and stretchy.
• The Result: Because the floor is squishy, the dancers can't just move as one big block. The crystal breaks up into tiny neighborhoods (domains). In some neighborhoods, the dancers face slightly left; in others, slightly right.
• The Hybrid Dance: Because the rubber bands (elasticity) and the dancers (magnetism) are so tightly linked, the "neighborhoods" start vibrating against each other. This creates new, hybrid vibrations. The magnetic movement and the physical stretching of the crystal get mixed together, creating these extra "notes" in the sound.

5. The "Softening" Effect

The researchers also noticed that as the dancers approached the moment of turning, the main vibration slowed down (got "softer"), almost stopping before picking up speed again. This is like a swing slowing down at the very top of its arc before swinging back. The fact that it didn't stop completely (it had a "finite gap") proved that the rubber bands (the crystal structure) were holding it back, confirming the strong link between the magnetism and the physical shape of the crystal.

Why Does This Matter?

This discovery is like finding a new musical instrument.

• Tunability: Because these extra "notes" appear only when the crystal is in this specific, unstable state, scientists can control them by simply changing the temperature or the magnetic field.
• Future Tech: This could lead to new types of magnonic devices (computers that use magnetic waves instead of electricity). Imagine being able to tune a computer chip to generate multiple signals at once just by warming it up slightly or turning a magnet.

In short: The scientists found that when a magnetic crystal is in the middle of changing its mind, the connection between its magnetic spins and its physical structure gets so strong that it creates a whole new set of vibrations, turning a simple hum into a complex, tunable chord.

Technical Summary

1. Problem Statement

Rare-earth orthoferrites ($RFeO_3$) are known for their complex magnetic properties, including weak ferromagnetism, multiferroicity, and strong magnetoelastic coupling. A key feature of these materials is the Spin-Reorientation Phase Transition (SRPT), where the magnetic sublattices rotate between different symmetry configurations (denoted as $\Gamma_2$, $\Gamma_{24}$, and $\Gamma_4$) driven by the interplay between $Fe^{3+}$ and $R^{3+}$ spins.

While previous studies have identified the softening of the quasi-ferromagnetic (q-FM) resonance mode near SRPT points using THz pulse excitation, significant limitations existed:

• Thermal Artifacts: Intense THz pulses cause local heating, hindering precise temperature control required to study the narrow SRPT region.
• Field Limitations: Existing microwave studies were often restricted to low frequencies or limited magnetic field ranges, failing to fully explore the combined influence of temperature and external magnetic fields.
• Unexplained Phenomena: The emergence of multiple magnon modes in the intermediate phase and their specific coupling to lattice dynamics (phonons) were not systematically understood.

The authors aimed to systematically investigate the temperature- and magnetic-field-dependent magnetization dynamics of Thulium Orthoferrite ($TmFeO_3$) across the SRPT region using low-power microwave excitation to avoid heating artifacts and to resolve the origin of multiple hybridized magnon modes.

2. Methodology

The study utilized broadband microwave absorption spectroscopy on a high-quality $(001)$-oriented $TmFeO_3$ single crystal grown via the optical floating-zone technique. The experiments covered a frequency range up to 87.5 GHz using two complementary configurations to ensure comprehensive coverage:

• Low-Frequency Regime (Up to 40 GHz):

  • Technique: Flip-chip Ferromagnetic Resonance (FMR) using a Coplanar Waveguide (CPW).
  • Setup: The sample was mounted on a CPW inside a cryostat with a superconducting magnet (up to 8 T).
  • Geometry: DC magnetic field ($H_{DC}$) applied along the $c$-axis; microwave magnetic field ($h_{rf}$) along the $b$-axis.
  • Data Acquisition: Transmission parameter ($S_{21}$) recorded as a function of frequency and magnetic field, processed using the derivative-divide method to enhance resonance visibility.

• High-Frequency Regime (70.5 – 87.5 GHz):

  • Technique: Rectangular waveguide (WR-12) based FMR.
  • Setup: Sample placed inside a WR-12 waveguide compatible with the cryostat.
  • Geometry: $H_{DC}$ along the $c$-axis; $h_{rf}$ within the $ab$-plane.
  • Instrumentation: Vector Network Analyzer (VNA) combined with frequency extenders and a synthesizer for harmonic mixing.

The study systematically varied Temperature (spanning the SRPT region, approx. 80 K – 95 K) and Magnetic Field (0 – 8 T) to map the evolution of resonance modes.

3. Key Contributions

• Comprehensive Phase Mapping: Provided a detailed map of the q-FM resonance modes across the entire SRPT region ($\Gamma_2 \to \Gamma_{24} \to \Gamma_4$) under varying temperature and magnetic fields, overcoming the thermal limitations of THz studies.
• Discovery of Multiple Modes: Identified the emergence of multiple distinct q-FM magnon modes specifically within the intermediate $\Gamma_{24}$ phase, separated by frequency gaps of 0.5–2 GHz.
• Mechanism Elucidation: Attributed these additional modes to nonuniform spin-wave excitations arising from the periodic magnetic domain structure in the $\Gamma_{24}$ phase, which hybridize with acoustic phonons due to strong magnetoelastic coupling.
• Quantification of Coupling: Demonstrated that the finite magnon gap observed at transition points is a direct signature of strong magnetoelastic coupling, preventing the mode frequency from reaching zero (softening).

4. Key Results

• Mode Softening and Critical Fields:

  • The primary q-FM mode exhibits characteristic softening (frequency reduction) at the critical fields ($H_c$) corresponding to the $\Gamma_2 \to \Gamma_{24}$ and $\Gamma_{24} \to \Gamma_4$ transitions.
  • Unlike ideal Goldstone modes, the frequency does not drop to zero at the transition; instead, a finite magnon gap remains (lowest frequency ~9 GHz at $H_c$), confirming strong coupling to acoustic phonons.
  • The critical field $H_c$ decreases linearly with increasing temperature, consistent with theoretical predictions.

• Emergence of Multiple Modes in $\Gamma_{24}$:

  • In the intermediate $\Gamma_{24}$ phase (84 K < T < 94 K), the spectrum reveals multiple q-FM modes (a "comb" of frequencies) rather than a single uniform mode.
  • These modes are separated by ~0.5–2 GHz and exhibit similar field/temperature dependence.
  • Outside the $\Gamma_{24}$ phase (in pure $\Gamma_2$ or $\Gamma_4$), only a single dominant mode with strong field dependence is observed; the additional modes vanish or become negligible.

• Domain Structure and Hybridization:

  • The authors propose that the $\Gamma_{24}$ phase supports a periodic magnetic domain structure (four possible domain orientations) due to lower symmetry.
  • The periodicity of these domains (estimated ~0.5 $\mu$m) allows a uniform microwave field to excite nonuniform spin-wave modes.
  • Magnetoelastic Coupling: The strong interaction between these nonuniform magnetic textures and acoustic phonons (elastic strains) leads to hybridized magnon-phonon modes. The domain walls, possessing both magnetic and elastic components, act as the source of these additional excitations.

• Exclusion of Alternative Mechanisms:

  • While Suhl instability (parametric down-conversion) was considered, it was deemed less likely due to the absence of a clear power threshold characteristic of such nonlinear processes. The domain structure explanation fits the data more robustly.

5. Significance

• New Platform for Magnonics: The study establishes $TmFeO_3$ as a natural platform for generating tunable, hybridized magnon modes. The ability to control the number and frequency of these modes via temperature and magnetic field offers new avenues for magnonic devices.
• Insight into Spin-Lattice Coupling: The work provides direct experimental evidence of how strong magnetoelastic coupling modifies the magnon spectrum, specifically preventing mode softening to zero and creating finite gaps.
• Domain Engineering: It highlights the critical role of magnetic domain textures in the intermediate phase of SRPT, suggesting that domain engineering could be used to manipulate spin-wave propagation and hybridization in rare-earth orthoferrites.
• Methodological Advance: The successful application of broadband, low-power microwave spectroscopy sets a standard for studying delicate phase transitions without the thermal artifacts associated with THz excitation.

In conclusion, the paper demonstrates that the interplay between magnetic domain structures and strong magnetoelastic coupling in the intermediate phase of $TmFeO_3$ leads to the generation of multiple, tunable quasi-ferromagnetic magnon modes, opening new possibilities for controlling spin dynamics in quantum materials.