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Now that we’ve discussed the physics underlying a microwave isolator, we must inves- tigate the physics underlying the magnetoelectrical tuning mechanism. To understand the magnetoelectric effect as a product property of magnetostriction and piezoelectric- ity, it is beneficial to understand those effects separately.

1.1.4.1 Magnetostriction

Magnetostriction is defined as the deformation of a body in response to a change in its magnetization. More specifically, the magnetostritive coefficientsNijkl relate strainSij

to the square of the magnetization MkMl [21]

SI =NIJMkMl. (1.32)

Magnetostriction, therefore, should not be mistaken with piezomagnetism — the latter being defined as linear relationship between strain and magnetization.

All magnetic materials exhibit magnetostriction to some degree. Empirical evi- dence, has shown that, in most cases, the second order effect of magnetostriction is far larger than the first order effect of piezomagnetism [11]. That being said, a pseudo- piezomagnetic response can be elicited by applying a static bias field which exceeds the driving field, as depicted in the grey region in Fig. 1.10(b).

a) b) Saturation Rotation Domains x’₃ H’₃ Linear Regime + -e m + + +

Figure 1.10: Origin And Behavior of Magnetostriction - a) Schematic of a 4f

charge density rotating in a magnetic field. After Ref. [11]b) Schematic of the magnetic field vs. strain behavior for a magnetostrictive material. After Ref. [21]

The magnetoelastic coupling — the coupling of a magnetic order to an elastic order — is the tendency of neighboring ions to shift their positions in response to the rotation of the magnetic moment and its rigidly attached anisotropic charge cloud, as can be seen in Fig. 1.10(a). This results in the anisotropic change in length of the crystal due to the application of a magnetic field which we call magnetostriction [11].

Quantitatively, magnetostriction is a change in elastic energy associated with a rotation of the magnetic moment. Or, conversely, a change in magnetic anisotropy en- ergy due to a specified strain. Induced strain is related to the rotation of the magnetic moment, and thus, both saturate at some large field, resulting in the saturation magne- tostriction (λs) and saturation magnetization (Ms) respectively. Materials with large

magnetostriction also exhibit the reciprocal effect: a large change in magnetization due to induced strain [11]. The induced strain is mainly caused by rotation of the magnetic dipoles rather than domain wall motion.

The strain in any direction can be evaluated from the following expression ∆l l =λ= 3 2λ100 α₁2β₁2+α₂2β₂2+α2₃β₃2−1 3 + 3λ111(α1α2β1β2+α1α3β1β3+α2α3β2β3) (1.33)

where (α1, α2, α3) specify the direction of the magnetization relative to the principal

axes. In the case of a cubic crystal, the principal axesZ1,Z2, Z3 are along the cube

edges [100], [010] and [001]. A second set of direction cosines (β1, β2, β3) specify the

directionZ₃0 in which the strainx0₃₃is measured. The magnetostrictive coefficientsλ100

andλ111 are the saturation strains measured along the specified crystal directions [21]

given by λ100= 2h1 3 (1.34) and λ111= h2 3 (1.35) with h1 = (N11−N12)Ms2 (1.36) and h2 =N44Ms2. (1.37)

We have now established a mechanism which can induce a change in magnetic anisotropy energy by imposing a specified strain. As discussed in Section 1.1.2, the Kittel equation determines the FMR frequency and depends on a magnetic shape anisotropy term. Thus, we can influence this term by imposing a strain on a mag- netostrictive material. By using a magnetostrictive material as the magnetic core of our microwave device, we can then, by imposing a strain, tune the FMR frequency of the device. In magnetoelectric composites, the strain is imposed by a piezoelec- tric element over which a voltage is applied. Understanding the underlying effect of piezoelectricty thus becomes important.

1.1.4.2 Piezoelectricity

The direct piezoelectric effect refers to the linear coupling between mechanical stress and electric polarization [21] as

Pi=diJTJ. (1.38)

The converse piezoelectric effect refers to the linear coupling between mechanical strain and applied electric field.

SJ =dJ iEi. (1.39)

The notation conventions used are explained in Appendix B.2.

Of the materials already discovered to be piezoelectric, the complex oxides, and more specifically the perovskite piezoelectrics, have the largest piezoelectric coefficients. The perovskite crystal system is characterized by the general formula ABO₃, where A and B are cationic metals. A depiction of the perovskite structure is shown in Fig. 1.11(a). Generally perovskites can be characterized by BO₆ octahedra surrounded by octahedrally coordinated A atoms. The overall stoichiometry becomes ABO₃.

PbTiO₃ Symmetry 4mm Pb O Ti Z3 Z₁ P3=d33X3 d33≈120 pC/N + + – – _P3=d31X1 d31≈50 pC/N – – + + P3=d15X5 d15≈300 pC/N – – + + (a) (b) (c) (d)

Figure 1.11: Structure-Property Relations for the Intrinsic Piezoelectric Effect in PbTiO3- In the unstressed state there is an electric dipole associated with the off-center

shift of the titanium atom. Under stress, this dipole can be increased (d33), decreased (d31),

or tilted (d51). After Ref. [21]

While there are many oxide crystals that display a piezoelectric response, the best piezoelectrics tend to also be ferroelectric [10]. A ferroelectric is an insulating system with two or more discrete stable states of different nonzero electric polarization. A ferroelectric system can switch between states under influence of an applied electric field. This changes the relative energy of the states through the coupling of the field to the polarization [22]. Each of these states is crystallographically equivalent. In

ferroelectrics, these orientation states can be visualized as a double energy well sym- metrically positioned around the centrosymmetric cation position. The applied electric field transitions the ion through the energy barrier, to one of the other orientation states. This changes the strain state of the system. It is this double well that leads to the high piezoelectric coefficients found in ferroelectric materials [10].

1.1.4.3 Magnetoelectric Effect

The magnetoelectric effect is defined as the occurrence of a polarization in response to a magnetic field, or conversely, the occurrence of a magnetization in response to an electric field. Although this effect has been observed in single phase materials, only composites systems have shown ME voltages at room temperature of significant mag- nitude for practical purposes. Thus we only consider composite systems consisting of a magnetostrictive and a piezoelectric phase. One then arrives at the ME suscepti-

bility α = δP/δH by taking the product of piezomagnetic deformation δz/δH with

piezoelectric charge generation δQ/δz. Thus, an effective magnetic field is induced in the magnetostrictive phase by imposing a voltage-induced strain via the converse piezoelectric effect in the piezoelectric phase.

In ME materials, the induced polarizationP is related to the applied magnetic field

H as [23]

Pi =αijHj (1.40)

and the induced magnetizationM to the applied electric fieldE as

µ0Mi =αjiEj (1.41)

where α is the second rank ME-susceptibility tensor. Another important quantity is the ME voltage coefficient

αE =δE/δH (1.42)

which is related to α by

α=0rαE (1.43)

whereris the relative permittivity of the material. MostlyαEis provided in V·cm−1Oe−1.

These equations are valid for a magnetoelectric composite, regardless of the con- nectivity of its phases. There are a handful of ways to interconnect the phases of a

two-part composite [24]. Not all connectivity schemes are equally practical, or effective. We will focus on the laminate composites (with connectivity 2-2) for their proven high ME coefficients and ease of production.

1.1.4.4 Magnetoelectric Laminate Composites

A bilayer or multilayer configuration for ME composites has several advantages over bulk composites and other composite connectivities.

1. The loss of polarization in bulk composites due to leakage currents can be over- come in layered structures

2. It is generally easier to apply large electric fields to thin film samples rather than bulk crystal or ceramic samples, because these require relatively smaller bias voltages

3. The piezoelectric phase can be poled to enhance the ME coupling

4. It is also possible to vary the poling and applied field directions to achieve maxi- mum ME coupling

5. Different composites might experience line broadening effects for increased piezo- electric phase ratios

6. The planar thin-film geometry is more likely to be compatible with conventional deposition techniques employed in the IC industry compared to bulk and semi- monolithic alternatives

In order to obtain a better understanding as to what physical quantities are involved

in ME coupling, the theory provided by Bichurin et al. [25] for low-frequency ME

coupling, is considered. These authors consider a bilayer in the (1, 2) plane as can be seen in Fig. 1.12. The system is biased in a magnetic fieldH on which an AC magnetic fieldδH is superimposed. This gives rise to a pseudo-piezomagnetic effect (See Section 1.1.4.1) leading to an AC electric fieldδE across the piezoelectric phase. Bichurinet al.

then estimate the ME voltage coefficient by solving the elastostatic and electrostatic equations.

1

Piezoelectric phase Magnetostrictive phase

2 3

Figure 1.12: Piezoelectric-Magnetostrictive Composite- A bilayer of magnetostric- tive and piezoelectric phases in the (1,2) plane with its thickness along axis 3. The bilayer is assumed to be poled with an electric field along axis 3. A bias magnetic field H and an AC magnetic field δH result in an AC electric field δE across the piezoelectric layer. Adapted from Ref. [26]

The value for α33, with both the magnetic and electric field oriented out-of-plane, is

comparatively smaller. In fact, the maximum value forα31is five times larger than the

maximum value forα33. Other orientations are not discussed due to their comparatively

low ME coefficients. The ME voltage coefficientα31 is given by [25]:

α31= δE3 δH1 = −t(1−t)( m_q 11+mq21)pd31 p_ε 33(ms12+ms11)t+pε33(ps11+ps12)(1−t)−2(pd31)2(1−t) (1.44)

where m and p denote magnetostrictive and piezoelectric phases, respectively. The

piezoelectric and piezomagnetic coupling coefficients are denoted by d and q respec- tively. The compliance coefficient is denoted by s, is the permittivity at constant stress, µ is the tensor permeability and pt and mt are the thicknesses of the magne- tostrictive and the piezoelectrics respectively. Lastly,t=pt/(pt+mt) is the fractional thickness for the piezoelectric layer. Bichurinet al.show that, in the case of a NFO-PZT bilayer, α31 is maximized for t= 0.4−0.6 [25].

Although this not a general result, it is reasonable to assume, based on eq. (1.44), that for equal compliances for the piezoelectric and magnetostrictive phases, the opti- mal fractional thickness will lie around 0.5. It so happens that most candidate materials for ME bilayers (electroceramics, magnetostrictive ceramics and metals) have compli- ances in the range 4–12 1012 m2/N [27]. In general, maximizing the piezoelectric and magnetostrictive coefficients will maximize the magnetoelectric coupling.

In the area of magnetoelectric composites for microwave devices, the ME voltage is often deduced by the induced shift in FMR frequency. The induced strain in the

magnetostrictive material results in a shift in FMR frequency which can be measured as a function of applied voltage over the piezoelectric layer, see eq. (1.45).

ωF M R=γ q

(Hk+HDC+ ∆Hef f)(4πMs+Hk+HDC+ ∆Hef f) (1.45)

Magnetoelectric composite materials consisting of both magnetic and ferroelec- tric phases with strong magnetoelectric coupling have led to many different classes of devices, one being, electrostatically tunable microwave devices [28–32]. These het- erostructures have consisted of either piezoelectric substrates with thin polycrystalline ferro-/ferrimagnetic films or ferrite slabs epoxy bonded onto piezoelectric substrates.

Now that we have gone through the relevant theory and some of the cutting edge applications, we are ready to more accurately define the research goal.