Cubic magnetic anisotropy in $B$20 magnets: Interplay of anisotropy and magnetic order in Fe$_{1-x}$Co$_{x}$Si

This study systematically quantifies the cubic magnetocrystalline anisotropy in MnSi and Fe$_{1-x}$Co$_{x}$Si single crystals, revealing that specific low cobalt concentrations in Fe$_{1-x}$Co$_{x}$Si generate sufficient anisotropy to controllably stabilize a low-temperature skyrmion lattice, thereby identifying it as the first chiral metallic system where such a phase is induced by cubic anisotropy.

The Gist

Imagine a group of tiny, invisible magnets (atoms) inside a piece of metal. In most materials, these magnets just line up in a straight row, like soldiers marching in formation. But in a special family of metals called B20 magnets (like MnSi and Fe₁₋ₓCoₓSi), these tiny magnets are a bit more playful. They twist and turn into a spiral, like a corkscrew or a DNA strand. This twisting is called a helimagnetic state.

Recently, scientists discovered that under the right conditions, these spirals can organize themselves into a grid of tiny, swirling tornadoes called Skyrmions. Think of a Skyrmion as a stable, microscopic whirlpool in a river of magnetism. These are exciting because they could be the future of super-fast, super-small computer memory.

However, there's a catch. These Skyrmion whirlpools are usually only stable in a very specific, narrow "sweet spot" of temperature and magnetic field, usually right near the point where the material starts to get magnetic. Scientists wanted to know: Can we make these Skyrmions stable at much lower temperatures, far away from that sweet spot?

In the past, they found this was possible in an insulator (a material that doesn't conduct electricity) called Cu₂OSeO₃, thanks to a hidden force called cubic anisotropy.

The Problem: The "Invisible Hand"

Cubic anisotropy is a fancy term for a rule that says, "It's easier for the magnets to point in certain directions (like the corners of a cube) than others." In most of these metals, this rule is very weak—so weak that it's like trying to hear a whisper in a hurricane. It's usually ignored because the main forces (the "hurricane") are so much stronger.

Because this "whisper" is so quiet, it's incredibly hard to measure. It's like trying to weigh a feather while standing on a trampoline; the trampoline's movement (the main magnetic forces) drowns out the feather's weight.

The Experiment: Tuning the Radio

The researchers in this paper decided to investigate a tunable alloy: Iron-Cobalt-Silicon (Fe₁₋ₓCoₓSi).

• Imagine this material as a radio dial. By changing the amount of Cobalt (Co) in the mix (turning the dial), they could change the material's properties without building a new machine.
• They tested samples with different amounts of Cobalt, from very little to quite a lot.

They used a super-sensitive scale (a SQUID magnetometer) to measure how much energy it took to rotate the magnetic "whisper" (anisotropy) in different directions. They had to be very clever to filter out the noise and find the true signal.

The Discovery: Finding the Sweet Spots

Here is what they found, using a simple analogy:

1. The "Goldilocks" Zones: They discovered that the strength of the "whisper" (anisotropy) changes wildly depending on how much Cobalt is in the mix.

   • Low Cobalt: The whisper is loud and clear. The magnets really want to point in specific directions.
   • Medium Cobalt: The whisper disappears. The magnets don't care which way they point; it's all the same.
   • High Cobalt: The whisper comes back, but it's much quieter than before.

2. The Big Reveal: The researchers realized that the "whisper" is strongest right at the edges of the magnetic world—where the material is just starting to become magnetic or just about to stop being magnetic.

   • Specifically, for a sample with about 15% Cobalt, the "whisper" is strong enough to act like a safety net.

Why This Matters: The "Low-Temperature Skyrmion"

In materials like Cu₂OSeO₃, this strong "whisper" (anisotropy) acts like a magnetic anchor. It pins the Skyrmion whirlpools in place, allowing them to survive even when the temperature drops way down, far below the usual "sweet spot."

The paper concludes that Fe₁₋ₓCoₓSi with low Cobalt (around 10-15%) is the first metallic system (a material that conducts electricity) where this "anchoring" effect is strong enough to potentially create a Low-Temperature Skyrmion phase.

The Takeaway

Think of it like this:

• Skyrmions are delicate soap bubbles. Usually, they pop if the temperature gets too cold.
• Cubic Anisotropy is a bubble wand that holds the bubble together.
• For a long time, we thought this wand was too weak to hold the bubble in metals.
• This paper shows that by tuning the recipe (adding just the right amount of Cobalt), we can make the wand strong enough to keep the bubbles stable in the cold.

This is a huge step forward because it suggests we might be able to build new types of computer memory that use these stable, tiny magnetic whirlpools, which could be faster and more efficient than what we have today. The researchers have essentially found the "recipe" to make these magnetic tornadoes last longer and survive in colder conditions.

Technical Summary

1. Problem Statement

The B20 family of chiral magnets, specifically MnSi and the doped semiconductor Fe$_{1-x}$Co$_x$Si, is characterized by a generic magnetic phase diagram dominated by isotropic exchange and Dzyaloshinskii-Moriya (DM) interactions. While these interactions determine the existence of helical ground states and high-temperature skyrmion lattices (HTS), cubic magnetocrystalline anisotropy is a weaker energy scale that plays a crucial, often overlooked role.

• The Gap: In insulating chiral magnets like Cu$_2$OSeO$_3$, cubic anisotropy has been shown to stabilize a distinct low-temperature skyrmion phase (LTS) far below the ordering temperature. However, it remains unclear if such a phase exists in metallic B20 systems (MnSi and Fe$_{1-x}$Co$_x$Si).
• The Challenge: Measuring cubic anisotropy in these systems is difficult because it is weak (lowest in the energy hierarchy) and exhibits strong temperature dependence. Furthermore, standard magnetization measurements can be confounded by non-collinear spin textures (helicoids, skyrmions) and crossover effects between easy and hard axes.
• The Goal: To systematically quantify the cubic anisotropy in Fe$_{1-x}$Co$_x$Si across the entire magnetic doping range ($0.08 \le x \le 0.70$) and determine if specific concentrations allow for the stabilization of an LTS phase.

2. Methodology

The authors employed a multi-faceted approach combining experimental measurements, phenomenological fitting, and theoretical simulation:

• Sample Preparation: High-quality single crystals of MnSi and Fe$_{1-x}$Co$_x$Si (with $x = 0.08, 0.15, 0.20, 0.30, 0.50, 0.60, 0.65, 0.70$) were grown using the tri-arc Czochralski method. Samples were cut into cubes aligned with the $\langle 100 \rangle$ crystal axes.
• Angle-Resolved Magnetization:
  • Measurements were performed using a rotatable sample holder in a SQUID magnetometer at $T = 5$ K and $10$ K.
  • The magnetic field was rotated relative to the crystal lattice (parameterized by angles $\theta$ and $\phi$) to map the magnetization energy landscape.
  • Correction: Geometric offsets and demagnetization effects were rigorously calculated and subtracted to isolate intrinsic anisotropy.
• Data Analysis & Modeling:
  • Magnetization Energy: The anisotropy energy $E(\hat{m})$ was calculated by integrating the magnetization curves ($E = \mu_0 \int H \cdot dM$).
  • Phenomenological Fitting: The data was fitted to a cubic anisotropy expansion: $E_{cub} = K_0 + K_1(m_1^2m_2^2 + m_2^2m_3^2 + m_1^2m_3^2)$. The fourth-order constant $K_1$ was extracted as the primary measure of anisotropy.
  • Dimensionless Anisotropy ($k_c$): To compare with theory, the authors calculated the unitless anisotropy constant $k_c = K_1 / (2\mu_0 H_{c2} M_s)$, where $H_{c2}$ is the critical field for field polarization and $M_s$ is the saturation magnetization.
  • Theoretical Cross-Validation: A second method was used to determine $k_c$ by analyzing the angular dependence of the critical field $H_{c2}$ and comparing it to numerical simulations based on the phenomenological Dzyaloshinskii model.

3. Key Results

A. Evolution of Cubic Anisotropy ($K_1$)

The study revealed a non-monotonic dependence of cubic anisotropy on Co concentration ($x$):

1. MnSi ($x=0$): $K_1$ is negative, making the $\langle 111 \rangle$ directions the easy axes (consistent with neutron scattering data).
2. Low-Doping Regime ($x \le 0.30$): $K_1$ becomes positive, switching the easy axes to $\langle 100 \rangle$. The anisotropy strength peaks around $x \approx 0.15$.
3. Intermediate Doping ($x \approx 0.50$): The cubic anisotropy vanishes ($K_1 \approx 0$) within experimental resolution. The system exhibits four-fold symmetry consistent with zero anisotropy.
4. High-Doping Regime ($x \ge 0.60$): $K_1$ recovers to positive values but is roughly one order of magnitude weaker than in the low-doping regime.

B. Decoupling of Order and Anisotropy

The ordering temperature ($T_{HM}$) and the anisotropy constant ($K_1$) do not track each other. While $T_{HM}$ peaks near $x \approx 0.35$, the anisotropy is strongest at the phase boundaries (low and high $x$). This suggests that magnetic order and cubic anisotropy are governed by different energy scales and mechanisms (e.g., band-filling effects in itinerant magnets).

C. Identification of Low-Temperature Skyrmion Candidates

By calculating the dimensionless anisotropy $k_c$ and comparing it to a theoretical threshold ($k_c \approx 0.039$) required for LTS stability:

• Candidates: The samples with $x = 0.08$ and $x = 0.15$ exhibit $k_c$ values exceeding the critical threshold.
• Best Candidate: Fe$_{0.85}$Co$_{0.15}$Si is identified as the most promising platform. It combines a sufficiently high ordering temperature ($T_{HM}$) and critical field ($H_{c2}$) with strong relative anisotropy, making the experimental detection of an LTS phase feasible.
• High-Doping Limit: While $k_c$ is also high for $x \ge 0.60$, the extremely low $H_{c2}$ fields and vanishing DM interaction make the experimental realization of an LTS phase unlikely in this regime.

D. Methodological Validation

The study validated two independent methods for determining anisotropy:

1. Integration of magnetization curves ($E(\hat{m})$).
2. Analysis of the angular dependence of $H_{c2}$ combined with numerical simulations.
   Both methods yielded consistent results for most samples, confirming that magnetization energy integration is a viable technique for itinerant systems, provided crossover effects are accounted for.

4. Significance and Conclusions

• First Metallic LTS Candidate: This work proposes Fe$_{1-x}$Co$_x$Si (specifically $x \approx 0.15$) as the first chiral metallic system where a low-temperature skyrmion phase could be controllably stabilized by cubic anisotropy. This contrasts with Cu$_2$OSeO$_3$, which is an insulator.
• Tunability: The study highlights the unique advantage of the Fe$_{1-x}$Co$_x$Si system: the ability to tune anisotropy and magnetic order independently by varying Co concentration, offering a "knob" to explore the interplay between topology and anisotropy.
• Theoretical Insight: The findings confirm that while anisotropy is weak compared to exchange interactions, it becomes dominant relative to other energy scales near phase boundaries (where magnetic order emerges or vanishes). This relative strength is the key factor enabling the stabilization of secondary skyrmion phases.
• Experimental Impact: The identification of specific doping levels ($x=0.08, 0.15$) provides clear targets for future transport and neutron scattering experiments aimed at discovering and characterizing the elusive low-temperature skyrmion lattice in metallic chiral magnets.