Bachelorthesis: Calculation of the magnetic properties of quarternary ThMn$_{12}$-type compounds with Zr as a substitution for Nd

This bachelor thesis employs density functional theory calculations to investigate the intrinsic and magnetic properties of Nd-substituted ThMn$_{12}$-type compounds with Zr and Ti and to identify promising Nd-poor quaternary materials as potential alternatives to rare-earth magnets.

The Gist

The Big Picture: Finding a Cheaper, More Eco-Friendly Magnet

Imagine the modern world as a giant machine powered by electricity. For this machine to function efficiently (as in electric cars, wind turbines, and hard drives), we need powerful magnets. Currently, the "gold standard" material for these magnets is a compound called Nd-Fe-B (Neodymium-Iron-Boron).

Think of Neodymium (Nd) as the VIP guest at a party. He makes the magnet incredibly strong, but he is expensive, hard to find, and almost everyone relies on a single country (China) to supply him. This creates a bottleneck in the supply chain, similar to a single narrow bridge that everyone wants to cross.

The goal of this research is to find a substitute for this VIP guest. The author, Nico Yannik Merkt, proposes using Zirconium (Zr). Zirconium is like the reliable, affordable neighbor who is abundant, cheaper, and easier to obtain. The question is: If we swap the VIP for the neighbor, will the party (the magnet) still work just as well?

The Problem: The "Unstable House"

The specific type of magnetic structure under investigation is called ThMn12 (or the "1:12 phase").

• The Blueprint: Imagine a house built according to a specific blueprint where you have 1 rare-earth atom (the VIP) and 12 iron atoms.
• The Problem: If you try to build this house using only the VIP (Neodymium) and the 12 iron atoms, the house is unstable. It is like trying to build a skyscraper on a sand foundation; it collapses.
• The Solution: To keep the house standing, you need a "stabilizer." In this case, the researchers use Titanium (Ti). Think of Titanium as the steel beams you insert into the frame so the house doesn't fall apart.

The Experiment: A Virtual Construction Site

Since building these magnets in a real laboratory is expensive and time-consuming, the author used supercomputers to simulate the construction. This is called Density Functional Theory (DFT).

• The Simulation: Instead of mixing chemicals in a beaker, the computer calculates how the atoms "sense" each other. It asks: "If I insert Zirconium here, will the house stand? How strong will the magnetic pull be?"

What the Computer Found

The paper goes through several "what-if" scenarios to see how swapping Neodymium for Zirconium affects the magnet's performance. Here are the key findings:

1. Stability (Will the house stand?)

• Pure Neodymium: Without help, the house is unstable.
• Pure Zirconium: Surprisingly, a house made entirely of Zirconium and iron is stable.
• The Mixture (50/50): When they mixed half Neodymium and half Zirconium, the house was a bit wobbly. They had to add more "steel beams" (Titanium) to keep it stable.
• Conclusion: You can replace Neodymium with Zirconium, but you must choose the recipe carefully to keep the structure stable.

2. Strength (How strong is the magnet?)

• The Trade-off: The VIP (Neodymium) is naturally very magnetic. The neighbor (Zirconium) is less magnetic.
• The Result: When they swapped Neodymium for Zirconium, the magnet became slightly weaker. It is like replacing a super-strong rope with a slightly thinner one. Nevertheless, the magnet is still very strong – strong enough to be useful.
• The "Energy Product": This is a measure of how much energy the magnet can store. The new Zirconium-based magnets achieved very high values, beating some older magnet types and coming close to the current champion (Nd-Fe-B).

3. Heat Resistance (The Curie Temperature)

• Magnets lose their strength when they get too hot. The "Curie temperature" is the point where the magnet gives up and stops working.
• The Finding: The new Zirconium magnets can withstand heat almost as well as the Neodymium magnets. They will not melt or lose their strength in a hot electric motor.

4. Directional Dependence (The "One-Way Street")

• A good permanent magnet must be "hard" to be demagnetized. It must hold its direction, like a one-way street.
• The Finding: The Zirconium magnets are very good at holding their direction. In fact, in some calculations, the Zirconium magnets were even better at holding their direction than the Neodymium magnets.

The Verdict: Is it a Winner?

The paper concludes that Zirconium is a very promising substitute for Neodymium.

• The Advantages: It is cheaper, more abundant, and less critical for supply chains. The resulting magnets are stable and exhibit excellent magnetic properties.
• The Disadvantages: The magnets are slightly weaker than pure Neodymium magnets, and they currently fall just short of being "perfectly hard" magnets (they are "semi-hard").
• The Future: The author suggests that with a little more fine-tuning (such as adding nitrogen or adjusting the recipe), these Zirconium magnets could become a real alternative to the expensive Neodymium magnets we use today.

In short: The author used a computer to demonstrate that we can build a strong, stable magnet using the cheap, abundant neighbor (Zirconium) instead of the expensive VIP (Neodymium). It is not quite as strong as the VIP version, but close enough to potentially revolutionize how we manufacture magnets for electric cars and green energy.

Technical Summary

1. Problem Statement and Motivation

The global demand for high-performance permanent magnets is driven by renewable energy technologies and electric vehicles. Currently, the industry relies heavily on Nd$_2$Fe$_{14}$B magnets. However, Neodymium (Nd) and other Rare-Earth (RE) elements are considered critical resources due to monopolies in the supply chain (mainly China) and dwindling reserves.

• The Challenge: Development of "RE-lean" or RE-free magnets that maintain high magnetic performance without relying on critical elements.
• The Specific Gap: While the ThMn$_{12}$-type (1:12) structure (RFe$_{12}$) exhibits a lower RE content than Nd$_2$Fe$_{14}$B, pure RFe$_{12}$ phases are thermodynamically unstable. They require stabilization (usually by Titanium, Ti) and often suffer from lower Curie temperatures or anisotropy compared to Nd$_2$Fe$_{14}$B.
• The Goal: This work investigates Zirconium (Zr) as a replacement for Nd in the 1:12 phase. Zr is abundant, low-cost, and non-critical. The study aims to determine whether Zr-substituted quaternary (Zr,Nd)Fe$_{12-y}$Ti$_y$ and quinary (Zr,Nd)Fe$_{10}$CoTi compounds can serve as viable, high-performance alternatives to Nd-Fe-B magnets.

2. Methodology

The research employs ab-initio computational methods based on Density Functional Theory (DFT) to predict intrinsic magnetic properties.

• Software and Framework:
  • VASP (Vienna Ab-initio Simulation Package): Used for structural relaxation and calculations of total energies, magnetic moments, and magnetocrystalline anisotropy.
  • AkaiKKR (MACHIKANEYAMA): Used to calculate Curie temperatures ($T_C$) via the Korringa-Kohn-Rostoker (KKR) Green's function method.
• Theoretical Approaches:
  • Exchange-Correlation Functionals: Primarily the Generalized Gradient Approximation (GGA-PBE). For the magnetocrystalline anisotropy energy (MAE), the Local Spin Density Approximation (LSDA) was additionally employed, as it often yields more accurate anisotropy constants for these systems.
  • Treatment of f-Electrons: Nd 4f-electrons were treated using the DFT+U method (Dudarev approach, $U = 6$ eV) to correct self-interaction errors and strong electron correlations not correctly captured by standard DFT.
  • Pseudopotentials: Projector Augmented-Wave (PAW) method.
• Performed Calculations:
  • Phase Stability: Calculation of formation energies and solution enthalpies to identify stable configurations of Ti and Co substitutions at Fe sites, as well as Zr at RE sites.
  • Magnetic Properties: Total magnetic moment ($m_{tot}$), saturation magnetization ($\mu_0 M_S$), and maximum energy product ($|BH|_{max}$).
  • Anisotropy: Calculation of magnetocrystalline anisotropy energy (MAE) by rotating magnetization directions ([001] to [010]) to derive the anisotropy constant ($K_1$) and anisotropy field ($H_a$).
  • Curie Temperature ($T_C$): Calculated using the Mean Field Approximation (MFA) and the Local Moment Disorder (LMD) model to account for thermal fluctuations.
  • Hardness Factor ($\kappa$): Calculated to determine if the material qualifies as a hard magnet ($\kappa > 1$).

3. Main Contributions

1. Systematic Zr Substitution Analysis: This work provides the first comprehensive theoretical investigation of Zr as a replacement for Nd in the 1:12 ThMn$_{12}$ structure, going beyond previous studies focused on Y or Ce.
2. Stabilization Mechanisms: It is confirmed that while pure (Zr,Nd)Fe$_{12}$ is unstable, the addition of Ti (at the 8i site) and Co (at the 8f/8j sites) stabilizes the phase. It is found that Zr substitution allows for a reduction in Ti concentration compared to pure Nd-based analogs.
3. Methodological Validation: The study validates the use of LSDA for calculating anisotropy in these systems and finds that GGA+U often underestimates $K_1$ for Nd-containing compounds, whereas LSDA yields results closer to experimental data.
4. Identification of Promising Candidates: The identification of (Zr,Nd)Fe$_{11}$Ti and (Zr,Nd)Fe$_{10}$CoTi as promising candidates that balance cost (reduced Nd) and performance.

4. Main Results

A. Stability and Structure

• Ti Stabilization: Ti preferentially occupies the 8i crystallographic site. A concentration of ~7.7 at.% Ti (2 atoms per 26-atom supercell) is sufficient to stabilize the (Zr,Nd)Fe$_{12}$ phase.
• Co Substitution: Co prefers the 8f and 8j sites. While Co slightly improves magnetic moments, it has a lower stabilizing effect than Ti.
• Lattice Parameters: Zr substitution leads to a reduction in cell volume compared to pure Nd compounds due to the smaller atomic radius of Zr, following the trend: $Nd_2Fe_{24} > ZrNdFe_{24} > Zr_2Fe_{24}$.

B. Magnetic Moments and Saturation Magnetization ($\mu_0 M_S$)

• Influence of Zr: Replacing Nd with Zr reduces the total magnetic moment, as Zr exhibits a lower magnetic moment (-0.47 $\mu_B$) than Nd (-0.27 $\mu_B$ in GGA, but -3.3 $\mu_B$ when f-electrons are included via DFT+U).
• Calculated Values:
  • NdFe$_{11}$Ti: $\mu_0 M_S \approx 1.69$ T (DFT+U), consistent with experimental ranges (1.38–1.70 T).
  • (Zr,Nd)Fe$_{11}$Ti: $\mu_0 M_S \approx 1.62$ T.
  • ZrFe$_{11}$Ti: $\mu_0 M_S \approx 1.51$ T.
• Conclusion: Although Zr substitution slightly lowers $M_S$, the values remain high enough for practical applications.

C. Maximum Energy Product ($|BH|_{max}$)

• Performance: The quaternary compound (Zr,Nd)Fe$_{11.5}$Ti$_{0.5}$ shows a theoretical $|BH|_{max}$ of ~600 kJ/m³, exceeding the theoretical limit of Nd$_2$Fe$_{14}$B (~512 kJ/m³).
• Stable Phase: The stable (Zr,Nd)Fe$_{11}$Ti compound yields ~510–525 kJ/m³, comparable to Nd$_2$Fe$_{14}$B and significantly higher than Sm$_2$Co$_{17}$ (294 kJ/m³).
• Co Effect: Co substitution slightly increases $M_S$ but only marginally decreases $|BH|_{max}$ due to volume changes.

D. Curie Temperature ($T_C$)

• Trend: $T_C$ decreases with increasing Ti content but increases with Co substitution.
• Values:
  • NdFe$_{11}$Ti: $T_C \approx 652$ K (LMD) / 792 K (FM).
  • (Zr,Nd)Fe$_{11}$Ti: $T_C \approx 652$ K (LMD) / 783 K (FM).
  • Comparison: These values are higher than those of Nd$_2$Fe$_{14}$B ($T_C \approx 588$ K), making them suitable for higher-temperature applications.
• Note: The FM approximation overestimates $T_C$; LMD values are considered more realistic but still indicate a viable operating temperature.

E. Magnetocrystalline Anisotropy ($K_1$) and Hardness Factor ($\kappa$)

• Anisotropy Constant ($K_1$): Zr substitution significantly increases $K_1$.
  • NdFe$_{11}$Ti: $K_1 \approx 1.68$ MJ/m³ (LSDA).
  • (Zr,Nd)Fe$_{11}$Ti: $K_1 \approx 2.41$ MJ/m³ (LSDA).
  • ZrFe$_{11}$Ti: $K_1 \approx 2.47$ MJ/m³ (LSDA).
  • Significance: Zr enhances anisotropy, a crucial property for coercivity.
• Hardness Factor ($\kappa$):
  • (Zr,Nd)Fe$_{11}$Ti yields $\kappa \approx 0.97$ (LSDA).
  • This classifies the material as semi-hard ($\kappa < 1$), whereas Nd$_2$Fe$_{14}$B is hard ($\kappa \approx 1.54$).
  • The value lies very close to the threshold for hard magnets, suggesting that minor modifications (e.g., nitriding) could bring it into the hard magnet range.

5. Significance and Outlook

• Strategic Significance: This work demonstrates that Zirconium is a viable, low-cost, and abundant replacement for Neodymium in ThMn$_{12}$-type magnets without compromising critical magnetic properties such as saturation magnetization or Curie temperature.
• Performance Potential: The calculated properties suggest that (Zr,Nd)Fe$_{11}$Ti-based magnets could compete with Nd-Fe-B in terms of energy product and thermal stability while offering a safer supply chain.
• Future Directions:
  • Experimental Verification: Theoretical predictions require synthesis (e.g., via Selective Laser Melting) and experimental validation.
  • Optimization: Since the hardness factor is slightly below 1.0, future work should investigate nitriding (formation of R-Fe-N phases) or further fine-tuning of Co/Ti to achieve $\kappa > 1.0$.
  • Extension: The methodology can be applied to other RE-lean structures (2:14, 2:17 phases) or other abundant elements (e.g., La, Ce) to further reduce dependence on critical rare earths.

In summary, the bachelor's thesis successfully identifies (Zr,Nd)Fe$_{12-y}$Ti$_y$ compounds as promising candidates for the next generation of sustainable, high-performance permanent magnets.