Colossal tunability in high frequency magnetoelectric voltage tunable inductors

The electrical modulation of magnetization through the magnetoelectric effect provides a great opportunity for developing a new generation of tunable electrical components. Magnetoelectric voltage tunable inductors (VTIs) are designed to maximize the electric field control of permeability. In order to meet the need for power electronics, VTIs operating at high frequency with large tunability and low loss are required. Here we demonstrate magnetoelectric VTIs that exhibit remarkable high inductance tunability of over 750% up to 10 MHz, completely covering the frequency range of state-of-the-art power electronics. This breakthrough is achieved based on a concept of magnetocrystalline anisotropy (MCA) cancellation, predicted in a solid solution of nickel ferrite and cobalt ferrite through first-principles calculations. Phase field model simulations are employed to observe the domain-level strain-mediated coupling between magnetization and polarization. The model reveals small MCA facilitates the magnetic domain rotation, resulting in larger permeability sensitivity and inductance tunability.

Scale bars in all images represent 10 μm.

Supplementary Note 1: First Principles Calculations
All calculations presented in this work were based on the projector-augmented wave PAW method which is implemented in the Vienna ab initio simulation package (VASP). The GGA+U with PBE functional were used as it was suggested by Ref. 1. A Cut off energy 500 eV and k-space grid of 5×5×3 are used for all calculations for good convergence. For GGA+U calculation, we used U eff =U−J=3 eV for Fe, Co and Ni. 1 We use the pseudo-potential contributing 15 valence electrons per Co(3p 6 4s 2 3d 7 ), 10 valence electrons per Ni(4s 2 3d 8 ), 14 valence electrons per Fe(3p 6 4s 2 3d 6 ), and 6 valence electrons per O(2s 2 2p 4 ).
CFO and NFO are inverse spinel structure (space group Fd3¯m, general formula AB 2 X 4 ). The trivalent cations occupy all A sites as well as 50% of the B sites whereas the remaining 50% of the B sites are occupied by the divalent cations. In the inverse spinel structure, Fd3¯m symmetry is preserved suggesting that the distribution of divalent and trivalent cations on the B sites is completely random. To enable our calculations for random distribution, we considered a unit cell with twice the size of the primitive unit cell. In particular, unit cells with Imma, P4m2 and P4 1 22 symmetries are considered in the calculation as suggested by Ref. 2. The crystal with Imma symmetry is shown in Fig.3 (b). During the geometric relaxation process, we fixed Co/Ni and Fe site, varies O positions during relaxation.
We first determined the equivalent lattice structure of CFO/NFO by relaxation based on GGA+U within collinear calculations. After that, we determined the CFO/NFO geometries under symmetry breaking strain. Based on the relaxed structures, we performed non-collinear calculations to calculate magnetic anisotropy and magnetostriction.
To calculate magnetic anisotropy and magnetostriction, we model the total energy by following model.
(1) where C 11 , C 12 , and C 44 are elastic constants and B 0 , B 1 , and B 2 are magnetoelastic coupling constants. To determine the three independent cubic elastic constants C 11 , C 12 , and C 44 and cubic magnetoelastic coupling constants B 1 , and B 2 by distorting the equilibrium crystal structure in three different ways: (1) hydrostatic strain which satisfies ε xx = ε yy = ε zz (isotropic volume expansion), (2) biaxial strain ε xx = ε yy = -0.5ε zz and (3) applying a volume-conserving shear strain ε ij . The magnetostriction coefficients for a cubic system are then calculated by: where λ α is the pure volume magnetostriction coefficient, λ 100 and λ 111 are the magnetostriction coefficients along [100] and [111]. In particular, we choose biaxial strain ε xx = ε yy = -0.5ε zz =ε, and , , in order to calculate the magnetostriction λ 100 . For this kind of strain, the total energies of the systems are written as (6) To calculate B 1 and C 11 -C 12 , we used (7) ( ) ) ( ) (   In the phase field model, the ME composite system can be described by field variables of magnetization M(r), polarization P(r), and free charge density r(r). The total system free energy under externally applied magnetic field H ex and electric field E ex is 4 : 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.
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, where the grain rotation matrix field R ij (r) describes the grain structure and crystallographic orientation of individual grains. The phase field h(r) describes magnetostrictive phase by h=0 and ferroelectric phase by h=1. The two-phase morphology of the ME composite is illustrated in Fig. 5(d), where the blue and red color represents the magnetostrictive ferrite and ferroelectric PZT phase, respectively. In the local coordinate system, f M (M) is formulated as the MCA energy 5 : where m=M/M is the magnetization direction, and f E (P) is formulated by the Landau-Ginzburg-Devonshire (LGD) polynomial energy 6 : where g is the gyromagnetic ratio and a the damping parameter for magnetization evolution, and L is kinetic coefficient for polarization evolution. The evolution of free charge density field is governed by charge conservation and microscopic Ohm's law 7 : where is the current density field, describes the electrical conductivity distribution in the ME composite, and is the local electric field given by 2  2  4  4  4  2 2  2 2  2 2  E  1  1  2  3  1 1  1  2  3  1 2  1 2  2 3  3 1   6  6  6  4  2  2  4  2  2  4  2  2  111  1  2  3  112  1  2  3  2  3  1  3  1