# Review of Magnetoelectric Effects on Coaxial Fibers of Ferrites and Ferroelectrics
## 2 Theory  
In this section, we review the results of theoretical estimates on the strengths of ME interactions in the coaxial fibers of spinel and hexagonal ferrites and ferroelectrics. Nickel ferrite was considered in the models due to its large magnetostriction \(\lambda\) and piezomagnetic coefficient q = d\(\lambda\)/dH [41]. Although cobalt ferrite has a large \(\lambda\), the piezomagnetic coefficient q is small due to slow variation in \(\lambda\) under H and is yet to be considered for modeling. In another report, fibers with hexagonal ferrites were considered [47]. Ferrimagnetic hexagonal ferrites contain cubic spinel (S-) blocks and two types of hexagonal blocks, T and R, in their crystal structure. Depending on the arrangement of S, T, and/or R blocks, they are classified into several categories including M-, W-, Y-, U-types, etc. Most of them have either a uniaxial or a planar magneto-crystalline anisotropy field, H\({}_{\rm a}\)[46]. The choice for the ferroelectric phase in the fibers was either PZT or barium titanate (BTO), since their composites with ferrites are found to be free of any impurity phases. Both BTO and PZT have a large piezoelectric coefficient d. The important magnetic parameters, saturation magnetization 4\(\pi\)M\({}_{\rm s}\), \(\lambda\), and q for spinel ferrites and hexagonal ferrites and ferroelectric order parameters, d, and permittivity \(\varepsilon\) for PZT and BTO, used for theoretical estimates, are given in Table 1. Estimates of strengths of direct ME effects in terms of the low-frequencyMEVC for core-shell fibers of NFO-BTO [41] and Y- and W-type hexagonal ferrites and PZT [47] are considered next. The theories developed are for the direct ME effect on a fiber with a length that is much greater than its radius. It is assumed that the fiber is subjected to a static magnetic field H and an AC magnetic field \(\delta\)H, both along the length of the fiber (direction 3). The resulting piezomagnetic deformation in the ferrite when transferred to the ferroelectric phase will produce an induced electric field \(\delta\)E. Figure 1 shows the estimated \(\text{MEVC}=\delta\text{E}/\delta\)H as a function of the ratio of the radius of the core to the radius of the fiber [41]. Figure 1a is for a fiber with a BTO core and an NFO shell (sample A). As \(\text{r}_{\text{p}}=\text{core radius}/\text{shell radius}\) increases from zero (i.e., in the absence of a BTO core), the MEVC increases from zero, is expected to reach a maximum value of 1.6 V/cm Oe for \(\text{r}_{\text{p}}=\) 0.33, and then decreases to zero for \(\text{r}_{\text{p}}=\) 1 when the fiber is BTO and does not contain a ferrite shell. Similar estimates for a fiber with an NFO core and a BTO shell (sample B) are shown in Figure 1b. A maximum \(\text{MEVC}\)\(\sim\)1.6 V/cm Oe is expected for this case when the radius ratio is 0.95. The dependence of \(\text{MEVC}\) in Figure 1 is significantly different from similar estimates for bulk or 2-2 laminate (layered) composites in which a maximum \(\text{MEVC}\) is predicted for composites with an equal volume fraction or thickness of the two phases [8]. Figure 2 shows the results of similar modeling for fibers with W- or Y-type hexagonal ferrites and PZT [47]. The material parameters in Table 1 are used for theoretical values of \(\text{MEVC}\) as a function of the ratio of the PZT core to the ferrite shell.