Nonreciprocal spin waves of helical magnetization states in CoFeB/NiFe bilayers

This study investigates nonreciprocal spin waves in CoFeB/NiFe bilayers by extending the dynamic matrix formalism to reveal that interlayer exchange interactions, alongside dipolar forces, drive frequency shifts in helical magnetization states, offering tunable sub-100 nm wavelengths for potential magnonic applications.

The Gist

The Big Picture: A Magnetic Dance Floor

Imagine you have a dance floor made of two different types of wood glued together: a very stiff, heavy wood (CoFeB) and a softer, more flexible wood (NiFe). This is your bilayer.

On this dance floor, there are tiny magnetic dancers (atoms) holding hands. Usually, they all stand in a straight line, facing the same direction. But in this experiment, the researchers apply a magnetic "wind" (an external field) that pushes the dancers on the soft side to twist and turn, while the dancers on the stiff side try to stay straight.

The result? The whole line of dancers forms a helix (like a spiral staircase or a slinky) as you move from the bottom of the floor to the top.

The Problem: The "One-Way" Street

The researchers are studying spin waves. Think of these as ripples or waves that travel through the line of dancers. If you push the dancers at one end, a wave travels to the other end.

In most materials, a wave traveling to the right behaves exactly the same as a wave traveling to the left. It's like a highway where traffic flows the same speed in both directions.

However, in this twisted, helical magnetic state, the researchers discovered something weird: The waves behave differently depending on which way they go.

• A wave going right might have a high pitch (high frequency).
• A wave going left might have a low pitch (low frequency).

This is called non-reciprocity. It's like a magical slide where sliding down one way is fast and fun, but sliding down the other way is slow and bumpy. This is crucial for building "magnonic diodes"—devices that let information flow in only one direction, which is essential for future, ultra-fast, low-energy computers.

The Discovery: Who is Driving the Bus?

For a long time, scientists thought this "one-way" effect was caused mostly by magnetic fields pushing on each other (called dipolar interactions). Imagine the dancers waving their arms and pushing the air; that air pressure affects their neighbors.

This paper says: "Wait a minute, that's not the whole story."

The authors found that the hand-holding between the layers (called interlayer exchange interaction) is actually the main driver of this effect, especially when the dancers are twisted in a spiral.

• The Old View: The waves are different because the air pressure (dipolar fields) is different.
• The New View: The waves are different because the dancers on the soft side are twisted at different angles. When a wave moves right, the dancers have to stretch their hands (exchange interaction) in one way. When it moves left, they stretch them in a completely different way. This stretching changes the speed and pitch of the wave significantly.

The Analogy: The Spiral Staircase

Imagine a spiral staircase where the steps are made of two materials.

1. The Twist: The stairs twist as they go up.
2. The Wave: Imagine a ball rolling down the stairs.
3. The Direction:
   • If the ball rolls clockwise (with the twist), it hits the steps at a shallow angle. It rolls smoothly and fast.
   • If the ball rolls counter-clockwise (against the twist), it hits the steps at a steep angle. It bounces and slows down.

The researchers realized that the "bouncing" (the change in frequency) isn't just because the stairs are uneven (dipolar); it's because the connection between the steps (the exchange interaction) changes depending on the direction of the roll.

Why Does This Matter? (The "Superpower")

The paper shows that by tweaking two things, you can control this effect perfectly:

1. How hard you push the magnetic wind (External Field): This changes how tight the spiral twist is.
2. How thick the soft wood is (NiFe thickness): This changes how much the dancers can twist.

The Result: They found a "sweet spot" where they can create waves that are:

• Tiny: Smaller than 100 nanometers (thinner than a human hair by a factor of 1,000).
• Fast: Huge differences in frequency (pitch) between left and right.

The Takeaway

This paper is like discovering a new rule for a game of marbles.

• Before: We thought the marble's path was only determined by the slope of the hill.
• Now: We realize the texture of the hill and how the marbles are glued together matters just as much.

By understanding this "glue" (exchange interaction) in these twisted magnetic layers, scientists can design better, smaller, and faster devices for the future of computing. They can essentially build a "traffic cop" for magnetic waves, forcing information to go only one way, which is the holy grail for efficient electronics.

Technical Summary

1. Problem Statement

The directional control of spin waves (non-reciprocity) is crucial for developing low-energy magnonic devices such as diodes and logic elements. While non-reciprocity is well-understood in standard geometries (e.g., Damon-Eshbach modes) driven by dynamic dipolar interactions or Dzyaloshinskii-Moriya interaction (DMI), there is a lack of theoretical frameworks explaining non-reciprocity in heterostructured multilayers with spatially varying, non-collinear magnetization profiles (specifically helical states).

Recent literature often attributes frequency shifts in such systems solely to dipolar interactions. However, this explanation is incomplete for systems where interlayer exchange coupling plays a significant role. The authors aim to systematically explain how helical magnetization, magnetization variation, and external fields determine non-reciprocity in spin waves, specifically addressing the missing role of interlayer exchange interactions.

2. Methodology

The authors developed a comprehensive theoretical and analytical approach:

• System Model: They modeled a CoFeB/NiFe bilayer (an exchange spring system) where the magnetization twists layer-by-layer along the thickness, forming a helical equilibrium state. The system is treated as a stack of $N$ sublayers.
• Dynamic Matrix Formalism (DMM): The authors extended the DMM to handle arbitrary non-collinear configurations.
  • Equilibrium: They solved Brown's equation to determine the equilibrium magnetization angles ($\phi_n$) for each sublayer, considering Zeeman, uniaxial anisotropy, and interlayer exchange interactions.
  • Dynamics: They linearized the Landau-Lifshitz (LL) equations around the helical equilibrium state. This resulted in an eigenvalue problem ($\omega \mathbf{m} = \mathbf{N}\mathbf{m}$) where $\mathbf{N}$ is a $2N \times 2N$ dynamic matrix.
  • Interaction Terms: The dynamic matrix $\mathbf{N}$ includes contributions from dipolar, interlayer exchange, intralayer exchange, uniaxial anisotropy, and Zeeman interactions.
• Validation: The analytical model was validated against micromagnetic simulations and experimental data from a similar SmCo/Fe exchange spring system (Jiang et al., 2025).
• Parameters: The study focused on a CoFeB (hard magnetic, 25 nm) / NiFe (soft magnetic, variable thickness) bilayer. They varied the NiFe thickness ($d_{NiFe}$) and applied magnetic field ($H$) to stabilize different helical states and analyze the resulting dispersion relations.

3. Key Contributions

• Extended Theoretical Framework: The primary contribution is the extension of the dynamic matrix formalism to describe spin-wave propagation in multilayers with in-plane helical equilibrium magnetization.
• Identification of Interlayer Exchange Role: The paper challenges the prevailing view that frequency shifts are solely due to dipolar interactions. It demonstrates that interlayer exchange interactions play a "starring role" in the frequency shift, particularly when counterpropagating spin-wave modes have distinct profiles across the bilayer thickness.
• Mechanism of Non-Reciprocity: The authors propose that the frequency shift ($\Delta f = f[k] - f[-k]$) arises from differences in dynamic dipolar and interlayer exchange interactions caused by the asymmetry in spin-wave mode profiles for counterpropagating waves at the same wave vector.
• Tunability: They identified a mechanism to tune non-reciprocity via the "twisting degree" of the helical state (controlled by external field and layer thickness).

4. Key Results

• Frequency Shift Magnitude: The system exhibits large frequency shifts (up to $|\Delta f| \approx 10$ GHz) combined with sub-100 nm spin-wave wavelengths.
• Role of Mode Profiles:
  • Thin NiFe (25 nm): The counterpropagating modes have very similar orbit distributions. Here, the frequency shift is dominated by dipolar interactions (surface charges).
  • Thick NiFe (47 nm): The counterpropagating modes exhibit significantly different orbit distributions (inhomogeneity) across the thickness. In this regime, interlayer exchange interactions dominate the frequency shift, contributing 2–3 orders of magnitude more than dipolar interactions.
• Field Dependence:
  • Below the coercive field ($H < H_c$), the magnetization is homogeneous, and the frequency shift is relatively constant.
  • Above $H_c$, the system enters a helical state. As the magnetic field increases, the "twist" of the magnetization increases, altering the mode profiles and significantly modifying the frequency shift.
  • For the thicker NiFe layer, the frequency shift can be tuned from positive to negative values depending on the field strength and the resulting mode inhomogeneity.
• Mode Hybridization: The study observed mode hybridization in the dispersion relation, particularly in thicker bilayers at zero field, which is linked to the specific symmetry breaking introduced by the CoFeB layer.

5. Significance

• Fundamental Physics: The work corrects the theoretical understanding of non-reciprocity in exchange-spring systems, proving that interlayer exchange is a critical, previously overlooked driver of frequency shifts.
• Magnonic Applications: The ability to achieve large frequency shifts ($\sim$10 GHz) at short wavelengths (sub-100 nm) makes this CoFeB/NiFe system highly relevant for next-generation magnonic devices.
• Device Design: The findings offer a new design parameter for magnonic diodes and logic elements: the non-reciprocity can be actively tuned not just by material choice, but by adjusting the external magnetic field and the thickness of the soft magnetic sublayer to control the helical twisting degree. This provides a versatile platform for reconfigurable magnonic circuits.