Sub-Terahertz/Terahertz Electron Resonances in Hard Ferrimagnets

Introduction

The transition to ultrafast electronics operating at sub-terahertz/terahertz frequencies (sub-THz/THz, 0.1-5 THz) is currently constrained by challenges in fabricating spintronic devices and finding materials with high-frequency functional properties. For the past decade, scientific interest has shifted from ferromagnetic materials (FMs) to antiferromagnets (AFs) and compensated ferrimagnets (CFIMs), mainly motivated by faster spin dynamics timescales reaching hundreds of gigahertz frequencies.

The Challenge

Current approaches to spin current generation face significant limitations:

• Ferromagnetic resonance (FMR) frequencies are generally low (dozens of GHz) and require external magnetic fields up to several Tesla
• Antiferromagnetic resonance (AFMR) operates at sub-THz/THz frequencies but requires:
  • External magnetic fields to remove mode degeneracy
  • Polarized radiation to excite only one chiral mode
  • Complex device configurations

The phenomenon of natural ferromagnetic resonance (NFMR) occurs without external magnetic fields, making it highly prospective for practical spintronics. However, NFMR frequencies are traditionally low (not exceeding a dozen GHz), and there are no proposals in the literature for spintronic devices based on the NFMR effect.

The Solution: Hard Ferrimagnetic Insulators

The key to reaching high NFMR frequencies lies in obtaining large magnetic anisotropy fields ($H_a$) within dielectric materials. Phenomenologically:

where $K_{mca}$ is the magnetocrystalline anisotropy constant and $M_{SV}$ is the volume saturation magnetization.

Researchers investigated cobalt ferrite (CoFe$_2$O$_4$) in the form of nanoparticles and bulk ceramics, synthesized via high-temperature methods.

Materials and Synthesis

Nanoparticle Synthesis

High-quality cobalt ferrite nanoparticles were obtained through high-temperature treatment of Co-Fe-Si-O xerogel:

1. Stoichiometric Fe(NO$_3$)$_3$$\cdot$9H$_2$O and CoCO$_3$ dissolved in water-alcohol solution
2. Tetraethoxysilane (TEOS) added to obtain 20 wt% CoFe$_2$O$_4$ within CoFe$_2$O$_4$/SiO$_2$ composite
3. Thermal treatment at 900-1200 °C with final annealing for 3 hours
4. Silica matrix removed by NaOH treatment

Ceramic Synthesis

Stoichiometric CoCO$_3$ and Fe$_2$O$_3$ were:

1. Mixed and pressed into pellets
2. Heated to 1350 °C for 2 hours
3. Quenched, ground, re-pressed, and annealed again

Key Findings

Single-Domain State and Magnetic Properties

Both nanoparticles and ceramics in single-domain state show broad hysteresis loops due to high magnetic anisotropy fields. The materials exhibit pronounced hard magnetic properties.

Record-Breaking NFMR Frequencies

For the first time, natural ferromagnetic resonance frequencies higher than 0.30 THz were registered:

The ceramic sample demonstrates the highest-known NFMR frequency of 0.35 THz, a record-breaking achievement.

Terahertz Absorption

The samples possess intensive resonance absorption at frequencies higher than 0.20 THz in zero external magnetic fields, making them attractive as isolation media in sub-THz/THz bands.

Theoretical Model

A model based on the Landau-Lifshitz equation was developed to explain the magnetodynamic properties. Key insights:

Two Resonance Modes in Ferrimagnets

Two-sublattice ferrimagnetic materials exhibit two resonance modes:

1. NFMR mode (right-handed) - frequencies in GHz to sub-THz range depending on $H_a$
2. Exchange mode (EF, left-handed) - frequencies in THz range

For soft ferrimagnets ($H_a \ll H_E$):

• NFMR mode in GHz band
• EF mode in THz band

For hard ferrimagnets ($H_a \approx 10-100\%$ of $H_E$):

• Both frequencies lie in sub-THz/THz bands

Spin Current Generation

The spin current can be expressed as:

Modeling reveals critical advantages of ferrimagnets over antiferromagnets:

Key Advantages of Hard Ferrimagnets

1. No chiral modes - pure spin current can be induced by unpolarized radiation even in zero external magnetic fields
2. Much higher magnetic susceptibilities - spin currents 2-4 orders of magnitude higher than AFMR throughout the entire range of anisotropy fields and frequencies
3. Essential spin current from EF mode - even at zero anisotropy field

Comparison with Other Hard Magnetic Insulators

Despite cobalt ferrite being magnetically hard only below 200 K, it is characterized by much higher absorption coefficients and NFMR frequencies compared to other hard magnetic insulators.

Applications in Ultrafast Electronics

1. Electromagnetic Isolation

Due to resonant absorption in sub-THz/THz range, CoFe$_2$O$_4$ and other hard magnetic insulators are attractive as isolation media, both in magnetic field and in its absence.

2. Spin-Wave Transport

Hard ferrimagnetic materials are appropriate for ultrafast short-range spin-wave transport in spintronic nanodevices. The large damping factor, while unsuitable for long-distance transport, is acceptable for nanoscale applications.

3. Pure Spin Current Induction

The most promising application is induction of pure spin current for:

• Detection of high-frequency electromagnetic radiation
• Development of ultrafast electronics
• THz polarizer design (due to different polarization of resonance modes)

Why Ferrimagnets Outperform Antiferromagnets

FMR Limitations

• Frequencies do not exceed dozens of GHz
• Requires external magnetic field for spin current induction
• At $H_{app} = 0$, no magnetization precession occurs

AFMR Limitations

• Two chiral modes are degenerate at $H_{app} = 0$
• Resulting spin current vanishes due to antiparallel angular moments
• Requires either:
  • Polarized radiation to excite only one mode
  • External magnetic field ($H_{app} > 0$) to remove degeneracy
• Bulky equipment needed for high magnetic fields makes practical use impractical

Ferrimagnet Advantages

• No external magnetic field required - NFMR occurs naturally
• Unpolarized radiation sufficient - no chiral mode degeneracy
• Orders of magnitude higher spin currents - due to higher magnetic susceptibility
• Compact device integration - no bulky magnet installations needed

Optimal Material Form for Device Integration

The most appropriate form for electronic device integration is a textured thin film:

• Minimal distribution of easy magnetization axis relative to electromagnetic wave k-vector
• Significantly narrower resonance line
• Epitaxial growth can induce crystal structure distortions, increasing magnetocrystalline anisotropy and resonant frequencies

Alternatively, particles oriented in a magnetic field can be used - easier to fabricate and high continuity is not required for such applications.

Conclusions

This work demonstrates several groundbreaking achievements:

1. First observation of electron resonances in cobalt ferrite materials
2. Record NFMR frequency of 0.35 THz for ceramic samples below 50 K
3. Proof of concept for hard ferrimagnets as candidates for ultrafast electronics integration

The principal advantage of hard ferrimagnets over antiferromagnets is that spin-pumping devices can operate without external magnetic fields while providing much higher spin currents over the entire range of resonance frequencies.

These findings represent an important step toward developing practical THz electronics based on natural ferromagnetic resonance in hard magnetic insulators.

Future Directions

The scientific community should focus on:

1. Methods of increasing magnetic anisotropy in known materials
2. Searching for new materials with high anisotropy fields and saturation magnetization
3. Developing textured thin films for optimal device integration
4. Optimizing damping factors through microstructure and composition control

This article is based on research published in Materials Today (2023): “Sub-terahertz/terahertz electron resonances in hard ferrimagnets” by Evgeny A. Gorbachev, Miroslav V. Soshnikov, Liudmila N. Alyabyeva, Ekaterina S. Kozlyakova, Anastasia S. Fortuna, Asmaa Ahmed, Roman D. Svetogorov, and Lev A. Trusov.