Electric field control of magnetic susceptibility in laminate magnetostrictive/piezoelectric composites: Phase-field simulation and theoretical model

Electric field control of magnetic susceptibility in laminate magnetostrictive/piezoelectric composites promises to create a new class of magnetoelectric elements, voltage tunable inductors. To elucidate the underlying mechanism of electric field modulated magnetic susceptibility at the domain level, phase-field modeling, and computer simulation are employed to systematically study the laminate magnetoelectric composites of Terfenol-D and PZT, where polycrystalline Terfenol-D can provide a giant magnetoelectric coupling that is important for high-tunability voltage tunable inductors. The simulations focus on the interplay between magnetocrystalline anisotropy and stress-induced anisotropy that is induced by electric field and reveal three regimes of magnetic susceptibility behaviors: constant (regime I), fast-varying (regime II), and reciprocal linear (regime III), where regimes II and III can give rise to a high tunability. Such three regimes are attributed to different magnetization distribution and evolution mechanisms that are modulated by the stress-induced anisotropy. To further characterize the electric field control of magnetic susceptibility behaviors, a general theoretical model of laminate magnetoelectric (ME) composites based on polycrystalline magnetostrictive materials is developed, which reproduces the three regimes of susceptibility behaviors for polycrystalline Terfenol-D material. The general theoretical model for this specific system can also be extended to other laminate polycrystalline ME composites.

Figure 1

Grain structure and phase morphology of laminate ME composite. Blue color represents magnetostrictive phase and red color represents piezoelectric phase.

Figure 2

In-plane stress $\sigma$ in Terfenol-D layer induced by bias strain $\varepsilon$ in PZT layer, and resultant domain structures of PZT and Terfenol-D layers at points A and B, which correspond to zero bias strain ($\varepsilon =0$) and zero induced stress ($\sigma =0$), respectively. Black and white arrows represent polarization and magnetization vectors, respectively. To distinguish the composite structure, polarization domain structure in PZT layer is visualized by color map with red, green, blue (RGB) components proportional to $P_x$, $P_y$, $P_z$, and magnetic domain structure in Terfenol-D layer is visualized by color contour proportional to $M_z$.

Figure 3

(a) Simulated magnetic susceptibility χ as a function of electric field $E$ (or induced stress $\sigma$) for Terfenol-D/PZT ME composites under zero internal bias strain ($\varepsilon =0$). Red line represents linear fitting curve with slope $d\chi /dE=-0.087\mathrm{cm}/\mathrm{kV}$. Magnetic domain structures of Terfenol-D layer under (b) $E=0\mathrm{kV}/\mathrm{cm}$ and (c) $E=40\mathrm{kV}/\mathrm{cm}$. Black arrows represent magnetization vectors. Magnetic domain structures are visualized by color map with RGB components proportional to $M_x$, $M_y$, $M_z$.

Figure 4

(a) Simulated magnetic susceptibility χ as a function of electric field $E$ (or induced stress $\sigma$) for Terfenol-D/PZT ME composites under zero internal bias stress ($\sigma =0$). Red line represents linear fitting curve with slope $d\chi /dE=0.037\mathrm{cm}/\mathrm{kV}$. Blue dashed lines represent the linear fitting in regimes I and II. Magnetic domain structure of Terfenol-D layer under (b) $E=0\mathrm{kV}/\mathrm{cm}$ and (c) $E=40\mathrm{kV}/\mathrm{cm}$.

Figure 5

(a) Simulated susceptibility χ at $E=0$ as a function of bias strain $\varepsilon$ from $-2.4$ to $1.5\times {10}^{-3}$. Three susceptibility regimes (I, II, and III) are divided by two critical bias strain values, ${\varepsilon }_{c1}\approx -1.6\times {10}^{-3}$ and ${\varepsilon }_{c2}\approx -0.6\times {10}^{-3}$. (b) The linear fitting slope $d\chi /dE$ as a function of bias strain $\varepsilon$ in the three regimes.

Figure 6

Magnetic domain structures of Terfenol-D layer under representative bias strain of (a) $\varepsilon =-2.4\times {10}^{-3}$, (b) $-0.6\times {10}^{-3}$, and (c) $1.5\times {10}^{-3}$.

Figure 7

(a) Simulated magnetic susceptibility χ as a function of electric field $E$ (or induced stress $\sigma$) for Terfenol-D/PZT ME composites under internal bias strain $\varepsilon =-1.2\times {10}^{-3}$. Red line represents linear fitting curve with slope $d\chi /dE=0.120\mathrm{cm}/\mathrm{kV}$. Magnetic domain structures of Terfenol-D layer under (b) $E=0\mathrm{kV}/\mathrm{cm}$ and (c) $E=40\mathrm{kV}/\mathrm{cm}$.

Figure 8

(a) Schematic illustration of laminate ME composite and coordinate system. Calculated (b) susceptibility $\chi$ and (c) ${\chi }^{-1}$ as a function of stress $\sigma$ based on the theoretical model for polycrystalline magnetostrictive materials.

Figure 9

(a) Simulated magnetic susceptibility χ as a function of induced stress $\sigma$. Three susceptibility regimes (I, II, and III) are defined by two critical induced stress values, ${\sigma }_{c1}\approx -3\mathrm{MPa}$ and ${\sigma }_{c2}\approx -20\mathrm{MPa}$. (b) Plot of ${\chi }^{-1}$ as a function of induced stress σ. Red line represents linear fitting in regime III: ${\chi }_{\mathrm{III}}^{-1}=-0.0035\sigma +0.011$.