Mechanically coupled phase field modeling of ferromagnetics unter thermal fluctuation
Mechanics
Mechanical Properties of Metallic Materials and their Microstructural Origins
Final Report Abstract
1. In summary, we have developed a continuum and thermodynamically consistent constraint-free mechanically-coupled phase field model for the simulation of magnetic domain evolution in ferromagnetic materials under thermal fluctuations at finite temperatures. We take the polar and azimuthal angles as the order parameters and derive the model in a thermodynamic framework which involves a microforce system for the domain microstructure. By representing the thermal fluctuations as white-noise nature and thus random field, we obtain a stochastic evolution equation via the fluctuation-dissipation theorem. We prove that the utilization of Fokker-Planck equation could result in an alternative stochastic evolution equation with two random numbers (three random numbers in the original stochastic evolution equation). 2. The phase-field model containing a stochastic evolution equation is numerically implemented by the Monte Carlo scheme (MCS) based stochastic finite element (SFEM). A deterministic FE problem is solved for a large number of times with different random numbers. And then the average magnetization and the statistical properties are calculated by the Monte Carlo estimator. Benchmark tests on the random walk on a spherical surface, thermal fluctuation induced magnetization switching, and magnetization switching probability as functions of temperature and volume in a spherical single-domain particle show the model can recapture the basic physics of magnetization dynamics under thermal fluctuations. 3. The electric-field control of magnetic state in nanomagnets by the strain-mediated magnetoelectric effect in a nanomagnetic/piezoelectric heterostructure has been studied by phase field simulations. It is found that, depending on the nanomagnet size, the electric-fieldinduced equilibrium magnetic state can be either non- volatile or volatile. For large nanomagnet size, a nonvolatile vortex state can be achieved, which persists after removing the electric field or applying a reverse electric field. For small nanomagnet size, coherent switching occurs, and the electric-field-induced single-domain state is unstable and volatile. For the nanomagnet size between the above two cases, an intermediate magnetic state between single domain and vortex exists. By using the precessional magnetization dynamics in the case of single-domain state and intermediate state which is close to the single-domain state, a 180o switching can be achieved by an electric-field pulse. On the contrary, for the case of vortex state and intermediate state which is close to the vortex state, a 180o switching is impossible. Careful design of the electricfield magnitude, pulse width, and ramp time can result in a 180o switching time of less than 10 ns, which is close to that in the traditional STT-MRAM, MRAM, and DRAM. It is anticipated that the present study provides valuable insight into the design of electric-field control of nonvolatile magnetic states and 180o switching without electric currents for achieving low-power, highspeed, nonvolatile, and highly compact memory devices. 4. The voltage-driven charge-mediated perpendicular and in-plane 180° magnetization switching at 0 K and room temperature (300 K) has been studied by using a multiscale theoretical framework which combines first-principles calculations and temperature-dependent magnetization dynamics. Especially, the influence of thermal fluctuations on the switching dynamics is disclosed. For the epitaxial metal- magnet-insulator (Pt/FePt/MgO) hetero-nanostructure as the model system, it is found from first-principles calculations that the interfacial charges induced by electric fields induce a giant modulation of MAE of the nanomagnet. From the temperature-dependent magnetization dynamics using first-principles results, it is found that both in-plane and perpendicular 180° magnetization switching is possible in the case of suitable epitaxial strain, E pulse width, and E ramp rate. But the temperature effect disturbs the switching behavior and makes the 180° switching as probability events. The E magnitude and pulse width should be carefully designed for a low-error-probability 180° switching at room temperature. Statistical analysis indicates that a fast (around 4 ns) 180° switching of low error probability can be achieved at room temperature. This work not only demonstrates a charge- mediated way for controlling magnetization by voltage, but also inspires the rational design of miniaturized nanoscale spintronic devices where temperature-induced thermal fluctuation plays a critical role. 5. Finite-temperature simulations using the developed stochastic evolution equation are performed to calculate the magnetic reversal, thermal-activation volume 푣, thermal-fluctuations-induced coercivity reduction 퐻푡ℎ and its ratio Δℎ푡ℎ, and coercivity Hc and its temperature coefficient 훽 in a pure Nd2Fe14B and Nd2Fe14B grain with surface defect layer or Dy-rich hard shell. Specifically, the stepwise external field and the step time for calculating the magnetic reversal curves are optimized. It is found that apart from the anisotropy field decreasing with temperature, the thermal fluctuations further reduce Hc by 5–10% and 훽 by 0.02–0.1%/K. The defect layer with strong magnetization (e.g., 1 T) is demonstrated to result in a remarkably increased 푣 (which can be reduced by adding the Dy-rich hard shell) and significantly decreased Hc, while suppressing the influence of thermal fluctuations and thus reducing 퐻푡ℎ and Δℎ푡ℎ. It is also revealed that even though the presence of a Dy-rich hard shell cannot fully cancel out the reduction of coercivity from the defect layer, a 4.5-nm- thick (Nd0.53Dy0.47)2Fe14B shell enhances Hc by 0.5 T and considerably improves the thermal stability. Both Hc and 훽 are found to saturate at a Dy-rich shell thickness of 6–8 nm. An even thicker shell or Dy alloying into the core prior to grain-boundary diffusion is not necessary. The calculation results are useful for the design of highperformance Nd-Fe-B permanent magnets used at high temperatures in terms of microstructure engineering.
Publications
- “Multiscale examination of strain effects in Nd-Fe-B permanent magnets,” Physical Review Applied, 8, 014011, 2017
Min Yi, Hongbin Zhang, Oliver Gutfleisch and Bai-Xiang Xu
(See online at https://doi.org/10.1103/PhysRevApplied.8.014011) - “Voltage-driven charge-mediated fast 180 degree magnetization switching in nanoheterostructure at room temperature,” npj Computational Materials, 3, 38, 2017
Min Yi, Hongbin Zhang and Bai-Xiang Xu
(See online at https://doi.org/10.1038/s41524-017-0043-x) - “Calculating temperature-dependent properties of Nd2Fe14B permanent magnets by atomistic spin model simulations,” Physical Review B, 99, 214409, 2019
Qihua Gong, Min Yi, Richard F. L. Evans, Bai-Xiang Xu and Oliver Gutfleisch
(See online at https://doi.org/10.1103/PhysRevB.99.214409) - “Multiscale simulations toward calculating coercivity of Nd-Fe-B permanent magnets at high temperatures,” Physical Review Materials, 3, 084406, 2019
Qihua Gong, Min Yi and Bai-Xiang Xu
(See online at https://doi.org/10.1103/PhysRevMaterials.3.084406) - “Strain-mediated magnetoelectric effect for the electric-field control of magnetic states in nanomagnets,” Acta Mechanica, 230, 1247-1256, 2019
Min Yi, Bai-Xiang Xu, Ralf Müller and Dietmar Gross
(See online at https://doi.org/10.1007/s00707-017-2029-7) - “Calculating the magnetocaloric effect in second-order-type material by micromagnetic simulations: A case study on Co2B,” Scripta Materialia, 177, 218-222, 2020
Dominik Ohmer, Min Yi, Maximilian Fries, Oliver Gutfleisch and Bai-Xiang Xu
(See online at https://doi.org/10.1016/j.scriptamat.2019.10.039) - “Electric field induced magnetization reversal in magnet/insulator nanoheterostructure,” International Journal of Smart and Nano Materials, 11, 298-309, 2020
Qihua Gong, Min Yi and Bai-Xiang Xu
(See online at https://doi.org/10.1080/19475411.2020.1815132)
