Theoretical Model and Computer Simulation of Metglas/PZT Magnetoelectric Composites for Voltage Tunable Inductor Applications

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1 Accepted Manuscript Theoretical Model and Computer Simulation of Metglas/PZT Magnetoelectric Composites for Voltage Tunable Inductor Applications Liwei D. Geng, Yongke Yan, Shashank Priya, Yu U. Wang PII: DOI: S (17) /j.actamat Reference: AM To appear in: Acta Materialia Received Date: 23 May 2017 Revised Date: 13 August 2017 Accepted Date: 15 August 2017 Please cite this article as: Liwei D. Geng, Yongke Yan, Shashank Priya, Yu U. Wang, Theoretical Model and Computer Simulation of Metglas/PZT Magnetoelectric Composites for Voltage Tunable Inductor Applications, Acta Materialia (2017), doi: /j.actamat This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

2 Theoretical Model and Computer Simulation of Metglas/PZT Magnetoelectric Composites for Voltage Tunable Inductor Applications Liwei D. Geng, 1 Yongke Yan, 2 Shashank Priya, 2 and Yu U. Wang 1,a) 1 Department of Materials Science and Engineering, Michigan Technological University, Houghton, MI 49931, USA 2 Center for Energy Harvesting Materials and Systems (CEHMS), Virginia Tech, Blacksburg, VA 24061, USA Control of magnetic permeability through voltage promises to create novel electronic devices, such as voltage tunable inductors. The relationship between the structure and property of voltage tunable inductors comprising of magnetoelectric Metglas/PZT composites and the underlying domain-level mechanisms are investigated using theoretical analysis, computer simulation, and complementary experiments. A theoretical model is developed to analyze the roles of material anisotropy, inductor shape, and stress in controlling the Metglas permeability and its tunability. The analysis reveals key roles played by stress-induced anisotropy and the resultant ground magnetization state, and predicts two stress-dependent regimes of inductance tunability. The theory is validated using systematic experiments. The experimental results are used to determine the material and physical parameters. To further elucidate the underlying domain-level mechanisms responsible for controlling the behavior of voltage tunable inductor, phase field modeling is employed to simulate domain microstructures and magnetic permeability of Metglas/PZT composites under varying voltage. The computational results confirm the two regimes of inductance tunability and the controlling role of stress-induced anisotropy. The findings suggest engineering of internal bias stress as an effective means to optimize the inductance tunability of magnetoelectric Metglas/PZT composites. Keywords: Phase field model; Magnetoelectric composite; Metglas; Multiferroic material; Domain mechanism; Tunable inductor a) Author to whom correspondence should be addressed; electronic mail: wangyu@mtu.edu 1

3 1. Introduction Electric field or voltage control (E- or V-control) of magnetic properties in multiferroic materials via magnetoelectric (ME) coupling effect promises to create novel electronic and spintronic devices with lower power consumption, smaller volume, and faster operation, which has attracted much interest in recent years [1-4]. In multiferroic composite materials consisting of magnetostrictive and piezoelectric phases, the ME effect is mediated by strain that mechanically couples the two constituent phases through elastic interface [5-7]. Through E- or V-control of the piezoelectric strain, the strain is transferred across the interfaces to the magnetostrictive phase in the composite structure, resulting in E- or V-control of the magnetic properties. One of the novel functional devices enabled by such multiferroic ME materials is voltage tunable inductors (VTIs) [8-12]. As one of the fundamental components of electronic circuits, inductors find extensive applications in power converters, control electronics, communication systems, etc. Tunable inductors are required to tune the electronic circuits. The tunability of conventional magnetic inductors is achieved by mechanically moving magnetic cores in the coils, sliding contacts along the coils, or electrically adjusting DC current in the coils, which, however, are usually bulky, slow, noisy and/or energy inefficient. Multiferroic ME VTIs, on the other hand, achieve the tunability through V-control of magnetic permeability of the ME composite cores. Because the VTIs are solid-state devices without moving components and variable voltage is readily available in the circuits, they are more compact, faster, less noisy, and more power efficient. The functionality and performance of multiferroic ME VTIs critically depend on the properties of the constituent materials of magnetostrictive and piezoelectric phases, both of which are ferroic in nature (ferromagnetic and ferroelectric, respectively) with their behaviors sensitive to domain structures [13,14]. The tunable piezoelectric strain of the poled ferroelectric material iscontrolled by applying electric field or voltage, which is transferred to the magnetostrictive material to tune its magnetic permeability. The resulting internal stress as well as the shape of the inductor and the material s inherent magnetic anisotropy have profound effect on the response of magnetic domains, which, in turn, influences the magnetic properties of the inductor materials. In particular, it is the interplay between material anisotropy, shape anisotropy, and stress-induced anisotropy that determines the tunable magnetic permeability. In this work, VTIs made of ME Metglas/PZT composites are investigated by using theoretical analysis and computer simulation. A theoretical model is developed to analyze the roles of material anisotropy, inductor shape, and stress in the tunability of Metglas inductance. The analysis results are compared with and validated by complementary experiments. To further elucidate the underlying domain- 2

4 level mechanisms, phase field modeling [15] is employed to simulate domain structures and magnetic permeability of Metglas/PZT composites under varying voltage. 2. Theoretical Analysis 2.1. Development of Theoretical Model A theoretical model of magnetostrictive inductor material has been proposed [16], which addresses the interplay among magnetocrystalline anisotropy, shape anisotropy, stress-induced anisotropy, and magnetic bias-induced anisotropy. For an inductor core made of Metglas ribbon, since the thickness of ribbon is much smaller than the transverse dimensions of the inductor core (e.g., length and width of plateshaped core, or inner and outer diameters of ring-shaped core), the inductor core can be modeled by the same geometry as illustrated in Figure 1(a). The main difference between plate-shaped and ring-shaped cores is characterized by different demagnetization factor along axis-1 direction (i.e., length direction for plate and circumferential direction for ring), which is non-zero for plate-shaped core of solenoid inductor and zero for ring-shaped core of toroidal inductor (due to closed magnetic flux path in ring-shaped inductor core). In addition to different shape anisotropy energies, other differences can also exist in the energies of magnetic domain structures and strain-mediated ME coupling between Metglas ribbons and PZT layers. Magnetic domains formed in the plate-shaped and ring-shaped ribbon cores may exhibit different domain sizes and thus produce different stray fields and different energies of demagnetization field across the ribbon thickness. This difference is taken into account by treating the demagnetization factor along axis-3 as a fitting parameter that assumes different values in the plate-shaped and ring-shaped inductor cores, as will be discussed in Section 2.2. Since the PZT layers are electrically poled in the thickness direction, the voltage tunable piezoelectric strain in the PZT layers and the induced stress in the Metglas ribbons possess in-plane isotropy in the plate-shaped core and in-plane cylindrical symmetry in the ring-shaped core, thus there is no major difference in the ME coupling between the two inductors; however, a minor difference may result from the different transverse geometries, which is taken into account through a small difference between the two in-plane stress components, as will also be discussed in Section 2.2. The total free energy, F, of the Metglas inductor core consists of contributions from magnetic anisotropy, demagnetizing field, applied stress, and magnetic field: 3

5 1 F K u 1 M N m N m N m s 11 m1 22 m2 0M s H m p m 0 s where is the permeability of free space, m M M is a unit vector along the direction of 0 magnetization vector M, and M is the saturation magnetization. The first term describes the s magnetocrystalline anisotropy energy, where K is the anisotropy constant, and the unit vector p defines u the easy axis of magnetization. It is worth noting that, while amorphous materials in principle do not exhibit magnetocrystalline anisotropy, certain degree of magnetic anisotropy is usually produced during material processing. In particular, annealed Metglas ribbons usually exhibit uniaxial anisotropy [17-20], whose magnitude and easy axis are characterized by K and p, respectively. The second term describes the u s (1) demagnetization energy, where N i is the demagnetization factor along axis-i direction as defined in Figure 1(a). The third term describes the energy of magnetoelastic coupling, where s is the saturation magnetostriction constant, and ij are the stress components. For layered ME Metglas/PZT inductor structure illustrated in Figure 2(a), the piezoelectric layer only induces in-plane biaxial stress ( and ) in the magnetic layer. The last term describes the Zeeman energy in the magnetic field H To apply Eq. (1) to the voltage tunable ME inductors fabricated and measured in the complementary experiments (discussed in Section 2.2), the easy direction p and demagnetization factors N need to be considered. Annealed Metglas 2605SA1 exhibits in-plane uniaxial anisotropy and, according to the experimentally observed formation of stripe magnetic domains across the width of annealed Metglas ribbons [20], the easy axis of magnetization is assumed along axis-2, i.e., p 0,1,0 i. We define parameter K d 2 0M s 2 for convenience in discussion of the demagnetization energies and, using M 110 s 6 A/m for Metglas, M 5 Kd 710 J/m 3. For the magnetic field-induced magnetization component M sm along axis-1, the demagnetization factor N1 is determined by the inductor core geometry, i.e., 1 1 N ~ for the plate-type core of length 14.8 mm width 2.8 mm thickness mm [21] (fabricated in the complementary experiment as analyzed in Section 2.2) and correspond to N1 0 for the ring-type core, which K ~ 700 d1 Kd N1 J/m 3 and 0 J/m 3, respectively. It is worth noting that, compared to the small magnetic anisotropy K 38 J/m 3 of amorphous Metglas [22], the demagnetizing field plays an u 4

6 important role in axis-1 direction of the plate-type core and must be taken into account. Along axis-2 across the ribbon width, the formation of stripe magnetic domains effectively eliminates the demagnetizing field, thus the demagnetization factor N ~ 0 2 is assumed in the analysis. Along axis-3 across the ribbon thickness, however, the demagnetization factor N 3 cannot be neglected because of the much smaller thickness of the ribbon as compared to its transverse dimensions. Nevertheless, formation of magnetic domains also significantly reduces the demagnetizing field in axis-3 direction when magnetization component M M m 3 s 3 is induced by stress. In the following analysis of the complementary experimental data, and K K N in both cases of plate-type and ring-type inductor cores and K K N in Ku d 3 d 3 d1 d 1 the case of plate-type core will be treated as fitting parameters, whose values depend on the material processing and detailed domain microstructures [22]. While these parameters are not investigated in the complementary experiments, their values can be extracted from the measurements by a fitting procedure as discussed here. Taking into account above relevant material parameters, the free energy in Eq. (1) for a Metglas inductor core in external magnetic field applied along axis-1 direction becomes: F K 1m K m K m m m M Hm u 2 d1 1 d 3 3 s s 1 First, consider the initial equilibrium magnetization state without applied magnetic field (i.e., ground state at H 0 ). For simplicity, assume in-plane biaxial compressive stress Then Eq. (2) (2) becomes: F K K m K K K m K 2 2 u d1 1 u d (3) where K 3s 2, and the normalization condition m1 m2 m3 1 has been used to eliminate m2, leaving two independent variables and m. Since K K 1 0 for both plate and ring, the ground m1 3 state with minimum energy is always reached at field. Therefore, Eq. (3) is reduced to: u d m1 0, as expected from the absence of applied magnetic F Km K where K Ku Kd 3 K is an effective magnetic anisotropy constant. It is readily seen that when K 0 (which will be called Regime I for clarity), the ground state corresponds to and the magnetization vector M 3 0, thus (4) m m 0,1,0 lies along the material easy axis (i.e., axis-2), in agreement with the experimental observation of stripe magnetic domains across the width of annealed Metglas ribbons [20]. 5

7 On the other hand, when K 0 (called Regime II), the ground state corresponds to m3 1, thus m 0,0,1 and the magnetization vector M lies along the stress-induced easy axis (i.e., axis-3). The ME inductors in the two ground states exhibit different voltage tunability behaviors, as discussed in the following Regime I and II. A transition between these two regimes occurs at K K K u d 3, which determines the critical stress for such a transition: 2 Ku K c 3 s d 3 K 0, or more explicitly (5) The susceptibility of the ME inductors strongly depends on the stress in Regime II when c. Under magnetic field H applied along axis-1, Eq. (3) becomes: 2 FII Kd1 Kd 3 K m1 0M shm1 Ku Kd 3 2K 3 (6) 2 2 where m 1 m has been used since m2 0 in Regime II. The magnetic field-induced magnetization M 3 1 M m is obtained from the equilibrium condition df dm1 0, which yields 1 s 1. The magnetic susceptibility dm dh is given by: m1 0M sh 2 K Kd1 Kd 3 1 Kd II K K K d1 d 3 At the transition between Regime I and II, K K K, thus K K K. Using above u d 3 II d u d1 5 mentioned values of K 710 J/m 3, K 38 J/m 3, K ~ J/m 3 (for plate) and ~ 0 1 J/m 3 (for d II u 3 4 ring), it is estimated that is of the order of magnitude 10 ~ 10 in Regime II. For a large susceptibility ( 1), the tunability defined in terms of the permeability 1 becomes = essentially, which, according to Eq. (7), is: c c K K K 3 K K II K K 2 K K K K which is a linear function of the compressive stress. d u d 3 s u d 3 u d1 u d1 u d1 II K d (7) (8) The susceptibility of ME inductors only weakly depends on the stress in Regime I when c. In fact, if is assumed, stress would produce no effect on the susceptibility. The weak stress dependence comes from the small difference between the stress components and, which results

8 from the different transverse dimensions of the layered ME inductor structure and thus different strain transfer in the two in-plane directions across the interfaces between PZT and Metglas layers. The magnitude of the compressive stress component in the length direction (axis-1) is expected to be slightly greater than that in the width direction (axis-2). Defining, then under magnetic field H applied along axis-1, Eq. (2) becomes: F K K K m M Hm 2 I u d1 1 0 s 1 s 11 s where K 3s 2, and m2 1 m1 has been used since m3 0 in Regime I. Again, the magnetic field-induced magnetization M M m is obtained from df dm1 0, which yields 1 s 1 m1 0M sh 2 K Ku Kd1. The magnetic susceptibility dm1 dh is given by: Kd I K K K At zero stress, 0, thus K K K, which is the same as at the transition K u d1 I d u d1 between Regime I and II. The tunability is obtained from Eq. (10): 0 K 3 s I K K 2 K K u d1 u d1 which is directly proportional to the stress component difference. I (9) (10) (11) It is worth noting that, for layered ME inductor structures, the stress component difference (which corresponds to twice the in-plane shear stress) is usually much smaller than the biaxial normal stress components, i.e., =, therefore the effect of in Regime II is much smaller than that of and thus has been neglected in above analysis of Regime II for clarity. In doing so, the stress in Regime II should be regarded as the average magnitude of the two normal stress components. The magnetization behaviors in Regime I and II are illustrated in Figure 1(b) Comparison with Complementary Experiments To complement the theoretical analysis, ME inductors are fabricated using Metglas ribbons and PMN-PZT laminates [16]. The layered structure is illustrated in Figure 2(a). Because the thickness (0.5 mm) of PMN-PZT layer is much greater than the thickness (0.023 mm) of Metglas layer, the elastic strain and stress are practically concentrated in the thin Metglas ribbon. Therefore, the biaxial stress in the 7

9 Metglas ribbon can be expressed as a function of the piezoelectric strain d31e produced in the PMN- PZT layer under electric field, i.e., Y 1 Yd E 1, where Y 110 GPa and 0.3 E are the Young s modulus and Poisson s ratio of Metglas, and 31 d pc/n is the piezoelectric constant of PMN-PZT. According to Eq. (8), the tunability corresponds to a linear function at high stress (or high electric field) in Regime II, where the slope with respect to electric field and the intercept at zero electric field are respectively: d II 3sYd 31 de 2 K K 1 II u d1 K K u d 3 E 0 u K K d1 (12a) (12b) Using above material parameters and s for Metglas into Eq. (12), there remain three fitting Ku Kd1 K d 3 parameters, namely,, and, which can be determined from the complementary experiments. In order to do so, both ring-type and plate-type inductors made of the same materials are required. In particular, a three-step fitting procedure is employed: (i) the ring-type inductor has Kd1 0, which allows determination of K u from the fitted slope of ring-type inductor experimental data in Regime II; (ii) the plate-type inductor has the same K u (because the same Metglas material is used), which allows determination of K d1 from the fitted slope of plate-type inductor experimental data in Regime II; (iii) with Ku Kd1 K d 3 and determined, can be determined from the fitted intercepts of the experimental data in Regime II for ring-type and plate-type inductors, respectively. Figure 3 shows the experimental data of the inductance and its tunability as measured from the fabricated ring-type and plate-type Metglas/PMN-PZT VTIs. The Metglas ribbon is mm thick. The plate-type inductor is 14.8 mm long and 2.80 mm wide, and the ring-type inductor has inner and outer diameters of 8 mm and 18 mm, respectively [16], thus their transverse dimensions are much greater than their thickness, satisfying above simplifying conditions assumed in the theoretical model. It is worth noting that the inductance ratio ratio E E 0 LE L 0 of the inductors measured in Figure 3 is equivalent to the permeability of the magnetic materials used in the inductors, thus the tunability can be equivalently defined through either inductance or permeability. As shown in Figure 3, at high electric field (Regime II), the linear fitting yields slope d II de 2.12 and 0.24 for ring-type and plate-type inductors, respectively, and intercept E and 1.34, respectively. Following above discussed three-step fitting II 8

10 procedure, the following material and physical parameters are extracted from the experimental measurements: K 126 u J/m 3 for the Metglas ribbon used to fabricate both ring-type and plate-type inductors, Kd1 988 J/m 3 for plate-type inductor, Kd J/m 3 for the ring-type inductor, and Kd J/m 3 for the plate-type inductor. It is worth noting that, while the extracted value Ku 126 J/m 3 is about 3 times the typical value reported values (for example, Ku 257 K 38 u J/m 3 [17], this value, nevertheless, is within the range of J/m 3 has been reported based on direct measurement [20]). The extracted value Kd1 988 J/m 3 is in good agreement with the estimated value K ~ J/m 3 d for a plate according to its dimensions (14.8 mm 2.8 mm mm) and demagnetization factor in the length direction ( N ~ ) [21]. The extracted values Kd J/m 3 for ring-type inductor and Kd J/m 3 for plate-type inductor are of the same order of magnitude as, while their difference can be K d1 attributed to different magnetic domains (and thus different effective demagnetization factors in the ribbon thickness direction) formed in the ring-type and plate-type inductors under stress. The tunability in Regime I also corresponds to a linear function at low stress (or low electric field). For simplicity, assume k, i.e., the difference in the two normal stress components is proportional to the magnitude of the stress. According to Eq. (11), the slope with respect to electric field is: d I 3sYd 31k de 2 K K 1 As shown in Figure 3, at low electric field (Regime I), the linear fitting yields slope u d1 d I (13) de and for ring-type and plate-type inductors, respectively. Using above extracted material and physical parameters into Eq. (13), the value k 0.11 for the ring-type inductor and k 0.04 for the plate-type inductor are extracted from the experimental measurements. As expected, the value of k is much smaller than 1; in particular, k is negligibly small for plate-type inductor, which is due to the more symmetric geometry in the length and width directions of a rectangular plate than in the circumferential and radius directions of a ring. Using the above extracted parameters,, K and k as well as the relations Ku Kd1 d 3 Y 1 Yd E 1 and k into Eqs. (7), (8), (10) and (11) predicts the inductance 31 L L E ratio and the tunability of the ring-type and plate-type inductors as functions of electric field E 0 in both Regime I and Regime II, as plotted in Figure 3, where the experimental data are also plotted for 9

11 comparison. According to Eq. (5), the transition between Regime I and Regime II occurs at the critical electric field: E c K K 2 u d Yd which gives E 2.6 kv/cm for ring-type inductor and E 5.6 kv/cm for plate-type inductor. These c values agree well with the experimental data. s 31 c (14) As observed in Figure 3, the experimental data exhibit a smooth transition rather than a sharp transition between Regime I and II. In particular, the tunability gradually deviates from the initial low-field linear segment of Regime I and smoothly approaches the high-field linear segment of Regime II. Such deviations from idealized theoretical predictions are attributed to the non-uniformities in the layered ME K inductors, such as spatial variations in, and that are associated with the microstructures of d 3 magnetic domains in Metglas layer and polarization domains in PMN-PZT layer. Such domain-level phenomena have been averaged out in above theoretical analysis, and are investigated in the following computer simulation study. 3. Computer Simulation 3.1. Phase Field Modeling To further elucidate the underlying mechanisms responsible for the VTI behavior, domain-level phase field modeling and computer simulation is employed to investigate the effects of tunable piezoelectric strain on the susceptibility and its tunability in ME Metglas/PZT composites under varying voltage. The phase field model of ME composites developed in our previous work [15] is adopted, which integrates the phase field models of magnetostrictive materials [13] and ferroelectric materials [14] into one unified model to treat domain processes and grain microstructures in polycrystalline composites of magnetostrictive and ferroelectric phases. This model explicitly addresses the domain-level strain-mediated coupling between magnetization and polarization. In the following, the employed phase field model is briefly described, while more detailed description can be found in prior study [15]. In the phase field model, the state of a ME composite is described by field variables of magnetization M(r), polarization P(r), and free charge density (r). The total system free energy under externally applied magnetic field H ex and electric field E ex is [15]: 10

12 1 ij j ij j 2 2 ex ex 3 F fm R M fe R P M Μ E P 0H M E P d r 3 2 d k * 3 n M in P K ijkl ij kl k 2 where f M (R ij M j ) and f E (R ij P j ) are the local free energy density functions of magnetostrictive and ferroelectric phases, respectively. Both M(r) and P(r) are defined in a global coordinate system attached to the ME composite instead of a local coordinate system aligned with <100> axes of the local crystal lattice. The operations R ij M j and R ij P j in the functions f M (R ij M j ) and f E (R ij P j ) transform M(r) and P(r) from the global sample system to the local crystallographic system in each grain. The grain rotation matrix field R ij (r) describes the grain structure and crystallographic orientation of individual grains. The phase field (r) distinguishes magnetostrictive phase (=0) and ferroelectric phase (=1). For amorphous Metglas, the processing-produced uniaxial anisotropy is defined by easy axis unit vector (15) p, and R ij (r)= ij within the magnetostrictive phase. The phase morphology of Metglas/PZT composite and grain structure of PZT phase are illustrated in Figure 2(b). In the local coordinate system, f M (M) is formulated as the uniaxial anisotropy energy [22]: M m p 2 fm K u 1 where m=m/m is the magnetization direction, and f E (P) is formulated by the Landau-Ginzburg-Devonshire (LGD) polynomial energy [23]: P P1 P2 P3 112 P1 P2 P3 P2 P3 P1 P3 P1 P2 f P P P P P P P P P P P P E P P P The two gradient terms in Eq. (15) characterize the energy contributions from the magnetization gradient (exchange energy) and polarization gradient, respectively. The spontaneous strain ε (16) (17) is a function of magnetization M and polarization P, ij ijklmk ml Qijkl Pk Pl, where ijkl and Qijkl are magnetostrictive and electrostrictive coefficient tensors, respectively. The k-space integral terms in Eq. (15) characterize the long-range magnetostatic, electrostatic and elastostatic interaction energies, where 0 and 0 are permeability and permittivity of free space, K ijkl =C ijkl n m C ijmn np C klpq n q, ik =(C ijkl n j n l ) -1, C ijkl is the elastic stiffness tensor, n=k/k, M k, P k, k and ε k are the Fourier transforms of the respective field variables M r, P r, r and ε r. 11

13 M r The evolution of magnetization,t and polarization P r,t are respectively governed by the Landau-Lifshitz-Gilbert equation [22] and the time-dependent Ginzburg-Landau equation [24]: M r, t F F M M M t Mr, t Mr, t P r, t t F L P r, t where and are gyromagnetic ratio and damping parameter, respectively, for magnetization evolution, and L is kinetic coefficient for polarization evolution. The evolution of free charge density field r,t governed by charge conservation and microscopic Ohm s law [25]: r, t t j j r, t E i ik k where j r,t is the current density field, r describes the electrical conductivity distribution in the ME composite, and the local electric field ik Er is: k 3 ex 1 d k Er E 3 i e n P k n 0 2 k ikr (18) (19) is (20) (21) (22) To perform phase field simulation of ME Metglas/PZT inductors, the following material parameters are used: uniaxial anisotropy constant J/m 3 6 K 126, saturation magnetization M A/m, u 6 saturation magnetostriction constant 2710 for Metglas 2605SA1, in agreement with Section 2.2; s and LGD coefficients 1 = m/f, 11 = m 5 /C 2 F, 12 = m 5 /C 2 F, 111 = m 9 /C 4 F, 112 = m 9 /C 4 F, 123 = m 9 /C 4 F, electrostrictive constants Q 11 = m 4 /C 2, Q 12 = m 4 /C 2, Q 44 = m 4 /C 2 for Pb(Zr 0.5 Ti 0.5 )O 3 [23,26]. It is worth noting that, while PMN-PZT is used in the complementary experiments in Section 2.2, a complete set of material parameters is not available for PMN-PZT, thus PZT is considered in the simulation study, whose material parameters have been experimentally determined [23,26]. Consideration of PZT instead of PMN-PZT does not change the general behaviors of Metglas-based ME VTIs, as shown in the following section. s 3.2. Simulation Results 12

14 The computer simulations focus on the magnetic susceptibility of ring-type Metglas/PZT VTI under different E-controlled tunable strain. The demagnetization factor N1 0 for the ring-type inductor core, where the closed magnetic flux path is provided in the simulation by periodic boundary condition in axis- 1 direction as illustrated in Figure 2(b). External electric field is applied on the PZT layer along axis-3, which induces piezoelectric strain that transfers across the layer interface to stress the Metglas layer. Such an E-controlled stress in the Metglas layer will affect the magnetization distribution and magnetic susceptibility in the Metglas layer, leading to tunable inductor behaviors. According to the theoretical analysis in Section 2, two regimes are expected depending on the stress or electric field. To investigate such behaviors, computer simulations are performed under a series of electric field E in the range of 0 to 40 kv/cm. It is worth noting that the electric field is higher in the computer simulations than that in the complementary experiments because PZT considered in the simulations has smaller piezoelectric coefficient than PMN-PZT used in the experiments, thus higher electric field is needed to produce the same tunable strain and stress in the simulations. This, nevertheless, does not change the general behaviors of Metglas-based ME VTIs. The simulation procedure is as follows. In the first step, Metglas/PZT composite with poled PZT layer and Metglas layer of zero internal stress is prepared, which corresponds to the ME Metglas/PMN- PZT VTI fabricated by bonding Metglas ribbon and poled PMN-PZT plate in the complementary experiments. In this step, the PZT layer is poled by applying high electric field along axis-3 and subsequently removing the field, following the same process as in previous work [15]. This poling treatment induces strain and stress in both PZT and Metglas layers. In order to relieve the internal stress, an additional 0 misfit strain Q P P ij ijkl k l is introduced to the PZT layer in the phase field modeling (. denoting spatial average in PZT layer), similar to the technique used in prior work to engineer internal residual stress [27]. In the second step, apply a tuning electric field of given magnitude along axis-3 to induce the piezoelectric strain in the PZT layer and, in turn, the tunable internal stress in the Metglas layer, and wait until polarization and magnetization distributions in the respective layers reach equilibrium. In the third step, a small magnetic field H is applied along axis-1, the induced magnetization response M along axis-1 is simulated, and the susceptibility is evaluated as M H. The same procedure is employed in the simulation cases presented in this section. Figure 4(a) shows the simulated domain structures in poled ferroelectric PZT layer and magnetostrictive Metglas layer of the ME inductor composite under zero electric field, where the poling-induced internal stress has been relieved to zero. In the poled PZT layer, polarizations are reoriented towards axis-3 (poling direction), rendering a macroscopic polar axis and piezoelectricity to the PZT layer. In the Metglas layer, due to the processing-produced small uniaxial 13

15 anisotropy of easy axis along axis-2, stripe domains of magnetizations along positive and negative directions of axis-2 are formed, which are separated by Bloch walls, where Bloch lines are also observed within the domain walls. With such an equilibrium magnetization distribution, under magnetic field along axis-1, the magnetic susceptibility of the Metglas layer is dominated by magnetization rotation rather than domain wall motion. The simulated susceptibility value for this case (E=0, no internal bias stress) is 4700, in good agreement with the theoretical value K K 4987 according to Eq. (10), where d u K 0 at E=0 and Kd1 0 for ring-type inductor. The simulated behaviors of E-controlled ME inductor, i.e., the induced stress, susceptibility and its tunability as a function of the controlling electric field, are shown in Figure 5. As expected, our simulated susceptibility and its tunability also exhibit two linear regimes, namely, Regime I for lower electric field and Regime II for higher electric field. Comparing the simulation results in Figure 5(a,b) with the experimental results in Figure 3(a) for the ring-type inductor, we observe two differences. The first difference is in Regime I: the simulation reveals a constant susceptibility and zero tunability at low electric field as shown in Figure 5(a) and (b), respectively, while the experiment observes decreasing susceptibility and non-zero tunability in Regime I as shown in Figure 3(a). According to Eqs. (10) and (11), Regime I behaviors are determined by the difference in in-plane stress components, In our simulation, 0 due to the idealized condition, while in the experiment due to the non- equivalency in the circumferential and radius directions of a ring, which results in non-zero tunability in Regime I in the experiment. The second difference between the simulation and experiment is the range of electric field, which arises from the different values of d 31 : PZT considered in the simulation has a much smaller d 31 than PMN-PZT used in the experiment. Figure 5(c) plots the induced stress as a function of the electric field, which exhibits a linear relationship d YE E. Using this linear relationship gives pc/n, where the factor 2 accounts for the roughly equal thickness of Metglas and PZT layers modeled in the simulation. This value of d 31 is much smaller than d 31 =420 pc/n of PMN- PZT in the experiment, thus a higher electric field is required in the simulation to achieve the same level of induced stress in Metglas layer as in the experiment. Nevertheless, this difference does not change the general behaviors of Metglas-based ME VTIs, i.e., the two regimes shown in Figure 5(a,b). In Regime II, linear fitting of the simulated tunability behavior in Figure 5(b) yields 0.137E The fitted slope agrees with the predicted slope of d de according to Eq. (12a). According to Eq. (12b), the fitted intercept allows determination of the effective demagnetization 14

16 energy K J/m 3, which is much smaller than the value K J/m 3 determined for ring-type d 3 37 inductor in the experiment. Such a reduced demagnetization energy along axis-3 is associated with the simulated domain structures shown in Figure 6. The interface between the Metglas and PZT layers is along the PZT grain boundaries, as illustrated in Figures 2(b) and 4. The uneven interface causes formation of closure domains with surface cap structures [22] near the interface in Regime II, as shown in Figure 6 (E=10 kv/cm), which significantly reduces the demagnetization field in axis-3 direction. Under E=0 kv/cm, there is no induced stress in the Metglas layer, and the processing-produced small uniaxial anisotropy of easy axis along axis-2 causes formation of stripe domains, where magnetizations are aligned along positive and negative directions of axis-2 and are separated by Bloch walls, as shown in Figures 4(a) and 6 (E=0 kv/cm). This is the domain structure in Regime I. With increasing electric field, the induced stress in the Metglas layer increases as shown in Figure 5(c), the stress-induced anisotropy becomes sufficiently large to overwhelm the uniaxial anisotropy, and the magnetizations start to rotate towards the stress-induced easy axis-3, as shown in Figure 6 (E=10, 12, 40 kv/cm), which are the domain structures in Regime II. In particular, the electric field reaches a critical value of E c =9.4 kv/cm to overcome the energy barrier K K for the magnetizations to gradually rotate from in-plane direction along axis-2 towards out-ofu d 3 plane direction along axis-3, leading to a transition from Regime I to Regime II. The evolution of magnetic domain structure during the regime transition is illustrated in Figure 6. It is worth noting that, as observed in Figure 6, to reduce the effective demagnetization field in axis-3 direction, the magnetizations of narrower in-plane domains are the first to switch into out-of-plane domains, while the wider in-plane domains gradually split into narrower out-of-plane domains. At E12 kv/cm, all magnetizations are aligned along the stress-induced easy axis-3 with the only exception near the interface where closure domains with surface cap structures are still formed to eliminate the stray field, as observed in Figure 6. Also as expected, the larger stress-induced anisotropy at higher electric field leads to narrower magnetic domain walls, as also observed in Figure 6. d As a direct consequence of the above discussed domain process during increasing electric field, the transition from Regime I to Regime II occurs smoothly, as shown in Figure 5(a,b), which is in agreement with the experimentally measured behaviors shown in Figure 3. In particular, the domain process shown in Figure 6 reveals a gradual evolution of magnetizations from in-plane domains in Regime I towards out-ofplane domains in Regime II, and the inhomogeneous domain structure produces spatially non-uniform distribution of demagnetization field in axis-3 direction, which is responsible for the observed smooth transition between Regime I and II. 15

17 Above simulated behavior of tunable ME inductor with controlling electric field E in the range of 0 to 40 kv/cm covers two regimes, as shown in Figure 5. Since the tunability in Regime I is very low in comparison with that in Regime II, it is desirable to operate the tunable ME inductor in Regime II, which requires the controlling electric field to stay always above the critical value E c 10 kv/cm. However, from device application point of view, it is also desired that the tunable ME inductor operates with the controlling electric field in a range starting from 0 kv/cm. This working condition can be achieved by introducing compressive internal bias stress in the Metglas layer, which shifts the critical electric field E c to negative value. This engineering technique is demonstrated in the following computational study. In the simulation, a compressive internal bias stress of magnitude 0 7 MPa is generated in the Metglas layer by introducing a pre-existing in-plane bias strain of magnitude into the PZT layer, like the same technique used in prior work to engineer internal residual stress [27]. According to Figure 5(c), 0 7 MPa corresponds to E=16 kv/cm in the previous case of zero bias stress (i.e., in Regime II), thus E c is effectively shifted to -6 kv/cm, making E=0 kv/cm inside Regime II. Figure 4(b) shows the simulated domain structures in poled ferroelectric PZT layer and magnetostrictive Metglas layer of the ME inductor composite under zero electric field. In particular, the pre-existing internal bias stress aligns the magnetizations along the stress-induced easy axis-3, while near the interface closure domains with surface cap structures are formed to eliminate the stray field, which is the characteristic domain structure of Regime II as discussed above. Application of electric field starting from 0 kv/cm linearly increases the internal compressive stress starting from 7 MPa, as shown in Figure 7(a). It is noticed that the linearly fitted slope of -E curve is slightly different with and without internal bias stress, i.e., d de 0.45 and 0.42 as shown in Figures 7(a) and 5(c), respectively. Such a slight difference is caused by the nonlinear piezoelectric behavior of the PZT layer that is stressed in tension by the Metglas layer with internal compressive bias stress (so that the bilayer as a whole is in self-equilibrium). Figure 7(b,c) shows the susceptibility and its tunability as a function of electric field, which clearly exhibit the behavior of Regime II. In particular, a linear tunability behavior 0.069E is observed in Figure 7(c). It is worth noting that, according to the definition of tunability, the slope of -E curve is proportional to the E0 E / E value ; since 2235 and 4700 at E=0 kv/cm with and without internal bias stress as shown in E 0 Figures 7(b) and 5(a), respectively, the -E curves in the two cases shown in Figures 7(c) and 5(b) exhibit different slopes accordingly. In Regime II, the permeability decreases with increasing electric field and thus increasing internal compressive stress. The decreased permeability is associated with the increasing stability of the magnetic 16

18 domains and the increasing magnetic anisotropy with stress-induced easy axis-3 in Regime II. Figure 8 shows the magnetic domain structures in the Metglas layer with compressive internal bias stress 0 7 MPa under electric field E=0 kv/cm and 40 kv/cm, both being in Regime II. While the well-formed stripe domains of magnetizations aligned along the easy axis-3 do not change significantly with increasing electric field, the increasing stress-induced anisotropy makes the magnetization rotation more difficult and thus decreases the susceptibility. 4. Conclusion The VTIs fabricated from ME Metglas/PZT composites are investigated by theoretical analysis, computer simulation, and complementary experiments. In the theoretical model of Metglas, the roles of material anisotropy, inductor shape anisotropy, and stress-induced anisotropy in the permeability and its tunability are analyzed. The model reveals key role of stress and predicts two stress-dependent regimes of permeability tunability. Based on the piezoelectric behavior of PZT, these two regimes are analyzed in terms of electric field and are compared with complementary experiments, where good agreement is obtained. Such comparison not only validates the theoretical model but also allows determination of relevant material and physical parameters from the experimental measurements, such as anisotropy constant and demagnetization factors. Phase field modeling is employed to perform computer simulations to further investigate the underlying domain-level mechanisms responsible for the VTI behaviors. The simulations confirm the two regimes of permeability tunability and the key role of stress-induced anisotropy. The magnetic domain structures in the two regimes are studied, which are responsible for the different permeability behaviors in the two regimes. Furthermore, the effect of internal bias stress is investigated, which can shift the critical electric field and bring the VTIs into Regime II of high tenability. The established structure-property relationship together with the elucidated domain-level mechanisms show that engineering of internal bias stress is an effective means to optimize the performance of ME Metglas/PZT composite-based VTIs. Acknowledgments Financial support from DARPA MATRIX Program is acknowledged. The parallel computer simulations were performed on XSEDE supercomputers. 17

19 References [1] Y.H. Chu, L.W. Martin, M.B. Holcomb, M. Gajek, S.J. Han, Q. He, N. Balke, Ch.H. Yang, D. Lee, W. Hu, Q. Zhan, P.L. Yang, A. Fraile-Rodríguez, A. Scholl, S.X. Wang, R. Ramesh, Electric-field control of local ferromagnetism using a magnetoelectric multiferroic, Nature Materials 7 (2008) [2] J.T. Heron, M. Trassin, K. Ashraf, M. Gajek, Q. He, S.Y. Yang, D.E. Nikonov, Y.H. Chu, S. Salahuddin, R. Ramesh, Electric-field-induced magnetization reversal in a ferromagnet-multiferroic heterostructure, Phys. Rev. Lett. 107 (2011) [3] T.H.E. Lahtinen, K.J.A. Franke, S. Van Dijken, Electric-field control of magnetic domain wall motion and local magnetization reversal, Scientific Reports 2 (2012) 258. [4] F. Matsukura, Y. Tokura, H. Ohno, Control of magnetism by electric fields, Nature Nanotechnology 10 (2015) [5] C.W. Nan, M.I. Bichurin, S. Dong, D. Viehland, G. Srinivasan, Multiferroic magnetoelectric composites: Historical perspective, status, and future directions, J. Appl. Phys. 103 (2008) [6] C.A.F. Vaz, Electric field control of magnetism in multiferroic heterostructures, J. Phys.: Condens. Matter 24 (2012) [7] T. Taniyama, Electric-field control of magnetism via strain transfer across ferromagnetic/ferroelectric interfaces, J. Phys.: Condens. Matter 27 (2015) [8] J. Lou, D. Reed, M. Liu, N.X. Sun, Electrostatically tunable magnetoelectric inductors with large inductance tunability, Appl. Phys. Lett. 94 (2009) [9] G. Liu, X. Cui, S. Dong, A tunable ring-type magnetoelectric inductor, J. Appl. Phys. 108 (2010) [10] N.X. Sun, G. Srinivasan, Voltage control of magnetism in multiferroic heterostructures and devices, SPIN 2 (2012) [11] H. Lin, J. Lou, Y. Gao, R. Hasegawa, M. Liu, B. Howe, J. Jones, G. Brown, N.X. Sun, Voltage tunable magnetoelectric inductors with improved operational frequency and quality factor for power electronics, IEEE Trans. Magn. 51 (2015) [12] B. Peng, C. Zhang, Y. Yan, M. Liu, Voltage-impulse-induced nonvolatile control of inductance in tunable magnetoelectric inductors, Phys. Rev. Appl. 7 (2017) [13] Y.Y. Huang, Y.M. Jin, Phase field modeling of magnetization processes in growth twinned Terfenol- D crystals, Appl. Phys. Lett. 93 (2008)

20 [14] Y.U. Wang, Field-induced inter-ferroelectric phase transformations and domain mechanisms in highstrain piezoelectric materials: Insights from phase field modeling and simulation, J. Mater. Sci. 44, (2009) [15] F.D. Ma, Y.M. Jin, Y.U. Wang, S.L. Kampe, S. Dong, Phase field modeling and simulation of particulate magnetoelectric composites: Effects of connectivity, conductivity, poling and bias field, Acta Mater. 70 (2014) [16] Y. Yan, L.D. Geng, L. Zhang, X. Gao, S. Gollapudi, H.C. Song, S. Dong, M. Sanghadasa, K. Ngo, Y.U. Wang, S. Priya, Correlation between tunability and anisotropy in magnetoelectric voltage tunable inductor (VTI) (submitted). [17] M.L. Spano, K.B. Hathaway, H.T. Savage, Magnetostriction and magnetic anisotropy of field annealed Metglas 2605 alloys via dc M H loop measurements under stress, J. Appl. Phys. 53 (1982) [18] H.T. Savage, M.L. Spano, Theory and application of highly magnetoelastic Metglas 2605SC, J. Appl. Phys. 53 (1982) [19] G.H. Hayes, W.A. Hines, D.P. Yang, J.I. Budnick, Low field magnetic anisotropy in Metglas 2605CO ribbons, J. Appl. Phys. 57 (1985) [20] M. Ali, Growth and study of magnetostrictive FeSiBC thin films for device applications, Ph.D. dissertation, University of Sheffield (1999). [21] D.X. Chen, E. Pardo, A. Sanchez, Demagnetizing factors of rectangular prisms and ellipsoids, IEEE Trans. Magnetics 38 (2002) [22] A. Hubert, R. Schäfer, Magnetic Domains: The Analysis of Magnetic Microstructures, Springer, Berlin, Germany, [23] A. Amin, M.J. Haun, B. Badger, H. McKinstry, L.E. Cross, A phenomenological Gibbs function for the single cell region of the PbZrO 3 :PbTiO 3 solid solution system, Ferroelectrics 65 (1985) [24] S. Semenovskaya, A.G. Khachaturyan, Development of ferroelectric mixed states in a random field of static defects, J. Appl. Phys. 83 (1998) [25] Y.M. Jin, Phase field modeling of current density distribution and effective electrical conductivity in complex microstructures, Appl. Phys. Lett. 103 (2013) [26] M.J. Haun, Z.Q. Zhuang, E. Furman, S.J. Jang, L.E. Cross, Electrostrictive properties of the lead zirconate titanate solid-solution system, J. Am. Ceram. Soc. 72 (1989) [27] F.D. Ma, Y.M. Jin, Y.U. Wang, S.L. Kampe, S. Dong, Effect of magnetic domain structure on longitudinal and transverse magnetoelectric response of particulate magnetostrictive-piezoelectric composites, Appl. Phys. Lett. 104 (2014)

21 Figure Captions Figure 1. (a) Schematic illustration of inductor core geometry and coordinate system. (b) Magnetization M under magnetic field H applied along axis-1 direction in Regime I (blue) and Regime II (red) that are defined by stress. Figure 2. Layered ME Metglas/PZT inductor structures considered in (a) theoretical analysis and (b) computer simulation where two simulation boxes are displayed with periodic boundary highlighted by dashed line. Figure 3. Comparison of theoretical and experimental results of inductance and tunability of Metglas/PMN- PZT ME VTIs: (a) ring-type and (b) plate-type. Regime I and II are shown in blue and red lines, respectively. Figure 4. Simulated domain structures in poled ferroelectric PZT layer and magnetostrictive Metglas layer of the ME inductor composite under zero electric field: (a) zero internal bias stress and (b) compressive internal bias stress 0 7 MPa in Metglas layer. Domain patterns are visualized by color maps with red, green, blue (RGB) components proportional to M x, M y, M z in Metglas layer and P x, P y, P z in PZT layer, respectively. Figure 5. Simulation results of E-controlled ME inductor with zero internal bias stress: (a) two regimes of susceptibility ; (b) two regimes of tunability ; and (c) induced compressive stress E0 E / E as a function of electric field E. Green dots represent simulated data, and solid lines represent fitting curves. Figure 6. Simulated magnetic domain structures in Metglas layer of E-controlled ME inductor composite under electric field E=0 kv/cm, 10 kv/cm, 12 kv/cm, and 40 kv/cm. Domain patterns are visualized by color maps with RGB components proportional to M x, M y, M z. Figure 7. Simulation results of E-controlled ME inductor with compressive internal bias stress 0 7 MPa: (a) E-controlled compressive stress; (b) susceptibility and (c) tunability in Regime II. Green dots represent simulated data, and solid lines represent fitting curves. E0 E / E 20

22 Figure 8. Simulated magnetic domain structures in Metglas layer of E-controlled ME inductor composite with compressive internal bias stress 0 7 MPa under electric field E=0 kv/cm and 40 kv/cm. Domain patterns are visualized by color maps with RGB components proportional to M x, M y, M z. 21

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