**Strain-Mediated Multiferroics:**

This page provides a tutorial Dr. Domann presented at the 2020 Joint Conference of the IEEE International Frequence Control Symposium and IEEE International Symposium on Applications of Ferroelectrics (https://ifcs-isaf2020.org/). The presentation walks through the fundamentals of modeling strain-mediated magnetoelectric coupling in composite materials made from piezoelectric and magnetostrictive components. As most conference attendees are well versed in piezoelectricity and ferroelectricity, the video primarily focusses on modeling magnetostrictive materials. Descriptions valid at both the mesoscale (nanometers up through 100's of microns), and macroscale are provided.

**Files: **Presentation

**Macroscale Magnetostriction **

Right after 1:53:00 in the video macroscale magnetostriction curves are discussed. On that slide, Dr. Domann highlights that understanding domain-level changes is vital to understanding the measured macroscale response. This document provides a domain-level explanation of the magnetostriction curves. Finally, Dr. Domann made a small error when saying the saturation magnetostriction of the material he's discussing is 3/2 the maximum measured value (250 microstrain). Instead he should have said it was 2/3 that value (again, see the document for why).

**Micromagnetic Simulation Packages**

While both MuMax and OOMMF are mentioned in the presentation, there are numerous open source and commercial simulation tools avaible for use when solving micromagnetics problems. Additionally, a fairly nice overview of the different packages be found in the 2019 review article: Tomorrow's Micromagnetic Simulations in the Journal of Applied Physics.

**Comercial FEA Packages**

Within the last 5 years, a few of the 'standard' commerical finite element packages like COMSOL and ANSYS have started supporting macroscale magnetostriction simulations. Typically these allow you to use either a piezomagnetic linearization, or a simplified Langevin function based model (see here for COMSOL's implementation). While the Langevin based models are intrinsically nonlinear, and appear to share several similarities with micromagnetics, I will caution you that they should also be used with care. While a Langevin model should be derivable from the Boltzmann statistics equations presented near the end of the tutorial, the required integrals only have closed form solutions when all the anisotropy energies are small compared to the thermal energy. In other words, at best the closed form expressions found in Langevin magnetoelasticity are the result of a linearization step, which directly impacts their range of validity. If you look at the zero-field magnetic susceptibility predicted by Langevin magnetoelasticity, you'll find that they become negative (i.e., unstable) if you apply a large enough stress. This is a mathematical instability in the equations due to a linearization, it is not something that is experimentally observed. Additionally, these equations assume that the magnetostriction is proportional to the squared magnetization (i.e., not just the unit vectors). This means that when the magnetization is zero, these equations predict zero magnetostriction. That is in direct contrast with the experimental findings that the delta-E effect tends to be largest at zero field.