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Electrode Position Optimization in Magnetoelectric Sensors Based on Magnetostrictive-Piezoelectric Bilayers on Cantilever Substrates Uzzal B. Bala, Matthias C. Krantz, and Martina Gerken, Member, IEEE Abstract—Finite element method (FEM) simulations are performed to investigate the sensitivity to dc magnetic fields of magnetoelectric sensors on cantilever substrates with trenches or weights at different positions. For a simple layered cantilever, a 15% higher signal voltage across the piezoelectric layer is obtained for optimally positioned electrodes and an insulating magnetostrictive material. A further 25% increase in the signal voltage is achieved for a trenched cantilever design with a pick-up region.
I. Introduction
M
agnetoelectric (ME) composites have been employed for room-temperature sensing of picotesla ac magnetic fields [1] and nanotesla dc magnetic fields [2]. In these ME composites, the magnetic field induces a strain in the magnetostrictive component, which is transferred via elastic coupling to a strain in the piezoelectric component, inducing a voltage signal [3]. For the integration of such ME sensors with microelectromechanical systems (MEMS), as well as for the fabrication of sensor arrays, thin-film-based 2–2 composites consisting of a piezoelectric layer and a magnetostrictive layer on a cantilever substrate are of particular interest [4]–[8]. The cantilever substrate prevents clamping. For resonant operation in the first bending mode, significant enhancement in the sensitivity is achieved. Resonant frequencies are from 100 Hz to the kilohertz range and may be adjusted by changing the size of the cantilever or adding a weight at the tip [5], [9], [10]. Cantilever designs with different regions may also be employed to improve the mechanical stability of the sensor or to realize specific signal pick-up regions for better sensitivity [5], [10]. Although the behavior of ME cantilever sensors without weights or pick-up regions may be modeled analytically [4], [11]–[13], a detailed analysis of the behavior for structured cantilevers is only possible numerically. In [10], we simulated the performance of three different types of trenched cantilevers using the finite element method (FEM). By thinning the substrate in Manuscript received September 11, 2013; accepted December 13, 2013. This work was supported by the German Science Foundation (DFG) within the Collaborative Research Centre SFB 855 “Magnetoelectric Composite Materials—Biomagnetic Interfaces of the Future.” The authors are with the Institute of Electrical and Information Engineering at the Christian-Albrechts-Universität zu Kiel, Kiel, Germany (e-mail: [email protected]). DOI http://dx.doi.org/10.1109/TUFFC.2014.2924 0885–3010
the trench region and modifying the electrode design, the resonance frequency was reduced and the ME coefficient was increased by a factor of 9, in good agreement with experimental results. Here, we present a numerical investigation of the optimal cantilever design and electrode positioning for cantilevers without and with a weight at the tip for dc magnetic field detection. As shown schematically in Fig. 1(a), the cantilever sensor is assumed to consist of a silicon cantilever substrate that is clamped rigidly on the bottom left and has two bending regions of lengths L1 and L2 and heights h1 and h2. The silicon substrate is assumed to be conductive and serves as the bottom electrode for the piezoelectric layer (alternatively, a thin metal layer may be deposited on top of the silicon cantilever to serve as the bottom electrode). Because strain-induced charges may vary in the plane of the piezoelectric layer, special attention is given to the influence of magnetostrictive layer conductivity. In the case of a continuous conductive magnetostrictive material serving as the top electrode, all surfaces of the magnetostrictive material exhibit the same electric potential, clamping the surface in contact with the piezoelectric material to a specific electric potential. In contrast, an insulating magnetostrictive material acts as a series capacitance, and the electric potential on the surface of the magnetostrictive material is position-dependent. Thus, an optimal electrode positioning of the second electrode allows for higher ME coefficients. A small, local second electrode probes the local induced electric potential. A larger electrode of a highly conductive material forces a constant electric potential to the top surface. Thus, the cantilever is deformed (via the piezoelectric material) such that the electrode voltage is constant. The resulting constant voltage may be approximated by averaging over the induced electric potential at the corresponding surface for the case of an insulating magnetostrictive material without an electrode. This will be discussed in Section III. This electrode design concept may be transferred to conductive magnetostrictive materials by electrically separating different regions of the magnetostrictive material with small trenches, as depicted in Fig. 1(b). The conductivity of the magnetostrictive material forces a constant electric potential at the interface between the piezoelectric layer and the respective area of the magnetostrictive material. In the following, we calculate the electric potential distribution in the cantilever beam using FEM. From the
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Fig. 1. (a) Schematic of investigated magnetoelectric sensor structure based on an insulating magnetostrictive (MS) layer and a piezoelectric (PE) layer on a cantilever substrate (Sub) with a weight at the tip. The position of electrode 2 is optimized. (b) Schematic of sensor with conductive MS layer structured with trenches for electric insulation.
simulation results assuming an equivalent insulating magnetostrictive material, we derive the optimal positioning of the second electrode as a function of the lengths L1 and L2 and heights h1 and h2. For the case of the conductive magnetostrictive material, these results describe which region of the magnetostrictive material should be used electrically as the pick-up region. The paper is structured as follows. The model for FEM simulations is described in Section II. Section III presents results for unstructured cantilever substrates with a single thickness of h1 = h2. Section IV investigates cantilevers with weights or trenches. Conclusions are given in Section V. II. Model for Finite-Element-Simulations The small-signal behavior of the cantilever magnetoelectric sensors is modeled using linear constitutive material equations (1)–(3) [14]–[16]:
m σ ij = c ijkl εkl − e kijE k − e kij H k (1)
Di = e ijk ε jk + κijE j (2)
m B i = e ijk ε jk + µijH j . (3)
Here, σ and ε are the stress and strain tensors. E, D, H, and B are the electric field, electric flux density, magnetic field, and magnetic flux density. c, e, em, κ, and µ are the stiffness, strain to electric field coupling constant, strain to magnetic field coupling constant, permittivity, and permeability. In this formulation, it is assumed that the direct magnetoelectric effect is negligible in the employed materials. Lead zirconate titanate (PZT) is used as the piezoelectric material and Terfenol-D as the magnetostrictive material with the material parameters given in [15]. For PZT, we set em = 0, and for Terfenol-D, e = 0. The conductivity of Terfenol-D is included in the model by a perfectly conductive boundary condition, as presented in
[15]. For the calculation of an insulating magnetostrictive material, we assume an equivalent material that is insulating, but has otherwise the same properties as Terfenol-D. In the simulation, this is achieved by removing the conductive boundary condition. We use this approach to investigate the influence of conductivity without changing any other parameters. After deriving the optimal pick-up region using this equivalent-material approach, a patterned Terfenol-D region should be included at this position in the real design. The mechanical properties of additional electrode metal layers may be neglected in the simulation because these layers are typically much thinner than other layers. The silicon cantilever substrate is modeled as an isotropic material with a Young’s modulus of 131 GPa, a Poisson’s ratio of 0.27, κ = 12.1 ∙ 8.854 ∙ 10−12 F/m, µ = 4 ∙ π ∙ 10−7 H/m, e = 0, and em = 0. The coupled elastostatic/elastodynamic and electrostatic/magnetostatic equations are solved assuming an ideal interface lamination, such that the deformation of the magnetostrictive material for an applied magnetic field is transferred to the piezoelectric material without slip. As described in [15], we assume that the electric field E and the magnetic field H may be expressed by the scalar potentials V and Vm as E = −∇V and H = −∇Vm. It is assumed that no charge density and conduction currents are present. A system of five differential equations is set up for the five variables u1, u2, u3, V, and Vm, where u is the displacement vector and εij = (ui,j + uj,i)/2:
σ ij, j + f i = ρu i
with i = 1, 2, 3 (4)
∇ ⋅ D = 0 (5)
∇ ⋅ B = 0. (6)
Because we are considering the low-frequency behavior simulated at 0 Hz, the right-hand sides of Newton’s laws (4) are set to zero. Furthermore, it is assumed that no other forces are present and fi = 0. The system of differential equations is solved in 3-D numeric simulations with Comsol Multiphysics (Comsol Inc., Burlington, MA).
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More details on an FEM implementation of the equations may be found in [14], [15]. III. Results for Single Cantilever Substrate Thickness First, we consider the case of cantilever substrates without a weight at the tip (h1 = h2 in Fig. 1) and compare the induced electric potential at the surface for conductive and insulating magnetostrictive materials. The geometric parameters of the cantilever are chosen as depicted in Fig. 1(a) with a total cantilever length of 2.5 mm. An H-field of 400 A/m is applied along the cantilever by setting the magnetic potential on the surfaces at x = 0 to Vm = 0 and at the surfaces at x = 2.5 mm to Vm = 1 A. The bottom electrode 1 is set to an electric potential of 0 V. Fig. 2 plots the resulting electric potential distribution for the case of h1 = h2 = 100 µm. As expected, a constant electric potential is observed at the surface of the conductive magnetostrictive material. For the insulating magnetostrictive material, the electric potential is lower at both ends of the cantilever. The maximum electric potential is with 0.037 V for the insulating magnetostrictive material, 16% higher than the value of 0.032 V for the conductive magnetostrictive material. Thus, by positioning the second electrode between 0.6 mm and 2.0 mm, an approximately 15% higher voltage signal between top and bottom electrode is expected. By covering the complete cantilever surface with the second electrode, a voltage of 0.032 V is obtained, which is exactly the voltage for the conductive magnetostrictive case. The slight variation of the electric potential for the conductive magnetostrictive case is due to the inaccuracy of the numerical solution in fulfilling the perfectly conductive boundary condition. Next, we consider the influence of the silicon cantilever thickness on the induced ME voltage between top and bottom electrode. Fig. 3 plots the results for the conductive and the insulating magnetostrictive case. For an optimal positioning of electrode 2, a higher ME voltage is achieved in case of the insulating compared with the conductive MS material. On the other hand, the induced ME voltage is identical for a conductive MS material and an insulating MS layer with an all-surface electrode 2. The induced voltage is strongly influenced by substrate thickness with a maximum voltage at h1 = h2 = 100 µm. Figs. 4(a) and 4(b) show the strain ε11 and stress σ11 in the xdirection as a function of the position z in the cantilever at half the width and half the length of the cantilever for an insulating magnetostrictive material. Fig. 4(c) shows the electric potential V as a function of the position z in the cantilever. Four different values for the silicon substrate thickness are evaluated. The strain plots in Fig. 4(a) show that the magnetostrictive layer is compressed because of the applied magnetic field, causing a deformation and upwards bending of the cantilever. For an increasing substrate thickness, the strain in the magnetostrictive layer with fixed thickness is reduced because of the larger com-
Fig. 2. (a) Electric potential distribution on surface for conductive magnetostrictive layer; h1 = h2 = 100 µm. (b) Electric potential distribution on surface for insulating magnetostrictive layer; h1 = h2 = 100 µm. (c) Comparison of electric potential on the surface of the magnetostrictive layer along dashed line indicated in (a) and (b).
bined thickness of piezoelectric layer and substrate that must be deformed. The combined effects of strain magnitude and strain slope are responsible for the induced stress and electric potential in the piezoelectric layer, such that for h1 = h2 = 50 µm and h1 = h2 = 200 µm nearly identical voltages are obtained between the top and the bottom electrode. The position of the neutral axis (axis with zero strain) moves from within the piezoelectric layer for h1 = h2 = 20 µm to positions further and further in the substrate for increasing values of h1 = h2. Note that for the static case considered here, the position of the neutral axis is not predominantly determined by the mechanical properties of the cantilever as in the resonant case [4], [5], [13]. For h1 = h2 = 20 µm, strain and stress are positive on the substrate side of the piezoelectric layer and negative on the other side. This causes the drop and subsequent increase in the electric potential observed in Fig. 4(c). The partial voltage
Fig. 3. Induced open-circuit magnetoelectric (ME) voltage between top and bottom electrode as a function of the silicon cantilever substrate thickness h1 = h2. For the case of the insulating magnetostrictive layer the optimal electrode 2 positioning corresponds to placing electrode 2 at the location with the maximum induced voltage on the magnetostrictive (MS) surface. For comparison, the ME voltage is calculated for an electrode 2 covering the complete MS surface for an insulating MS layer.
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Fig. 4. (a) Strain ε11 in the x-direction as a function of the position z in the cantilever at half the length (x = 1.25 mm) and half the width (y = 325 µm) of the cantilever for an insulating magnetostrictive material and four different silicon cantilever substrate thicknesses. (b) Stress σ11 in the x-direction as a function of the position z for the same parameters. Calculated mesh points are connected by lines for easier visualization, resulting in finite slopes at the interfaces. (c) Induced electric potential V as a function of the position z for the same parameters.
cancellation in the piezoelectric layer explains the reduced performance for thin substrates. The optimal value of h1 = h2 = 100 µm derived here is valid for the geometric parameters defined in Fig. 1. For other ratios of geometric parameters, the solution must be
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recomputed to determine the optimal substrate thickness and electrode position. Table I lists the optimal cantilever substrate thickness h1 = h2 = hopt for other active layer thicknesses hPE = hMS with other parameters identical to the ones given in Fig. 1. For thinner active layers, the optimal substrate thickness is thinner as well and (within the 10 µm simulation step resolution) identical to the active layer thickness. Fig. 5(a) plots the electric potential on the surface of an insulating magnetostrictive layer for each of the five optimal cases of Table I. It is observed that the position of the maximum is closer to the fixed end of the cantilever for smaller layer thicknesses; also, a more distinct peak forms in the electric potential. Table I lists the maximum induced voltage and the average induced voltage. The average induced ME voltage scales approximately with the thickness hPE = hMS of the active layers. The enhancement obtainable with electrode 2 placed at the peak electric potential increases for thinner active layers. For layer thicknesses of 20 µm, the voltage increases from an average value of 7 mV for an electrode 2 covering the whole surface to a value of 11 mV, if the electrode is placed only at the peak position. This is an enhancement of 57%. From noise considerations, wider electrodes with a certain minimum capacitance may be required. Thus, we evaluated the positioning of electrode 2, if a 5% drop from the peak ME voltage is permitted. As shown in Fig. 5(b), we consider an electrode covering the complete width of the cantilever and extending from x1 (95% Vmax) to x2 (95% Vmax). Table I lists the calculated ME voltages. A drop in induced voltage is observed for hopt ≤ 60 µm, whereas no voltage drop is observed for larger hopt within the simulation resolution. This difference is caused by the y-direction profile of the induced ME potential. For thick cantilevers, the induced electric potential is nearly independent of the y-coordinate, as observed in Fig. 2(b). For thinner cantilevers, a voltage peak forms close to the fixed end around the center of the cantilever in the y-direction, as shown in Fig. 6(a). For an electrode covering the complete width of the cantilever, as depicted in Fig. 6(b), a reduced voltage is observed, as seen in Fig. 6(c). These results demonstrate that the optimal electrode positioning depends on the cantilever aspect ratios. Because of the linear constitutive laws, a scaling of all cantilever dimen-
TABLE I. Simulated Optimal Cantilever Substrate Thickness h1 = h2 = hopt for Varying Active Layer Thicknesses hPE = hMS With Other Parameters Identical to Those Given in Fig. 1(a). hPE = hMS (µm) 20 40 60 80 100
h1 = h2 = hopt (µm)
Vmax (mV)
Vavg (mV)
x1 (95% Vmax)
20 40 60 80 100
11 19 24 30 37
7 13 20 26 32
20% 20% 20% 20% 21%
L L L L L
x2 (95% Vmax) 24% 28% 34% 46% 80%
L L L L L
Velec (95% Vmax) (mV) 9 17 23 30 37
Simulations were carried out in steps of 10 µm for h1 = h2. Vmax is the maximum voltage from Fig. 5 along the center line shown dashed in Fig. 2(b). Vavg is the average voltage along the center line. Allowing for a drop in voltage by 5% from the maximum in Fig. 5, the optimal electrode 2 should extend from x1 (95%) to x2 (95%) given in units of the length L of the cantilever. Velec(95%) is the simulated ME voltage obtained for this electrode positioning.
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Fig. 5. (a) Electric potential on the surface of an insulating magnetostrictive layer along the dashed line indicated in Fig. 2(b) for five different layer thicknesses hPE = hMS = h1 = h2 with other parameters the same as Fig. 1(a). (b) Optimal placement of electrode 2, allowing for a 5% drop from the maximum voltage Vmax for the case of hPE = hMS = h1 = h2 = 100 µm and length L = 2.5 mm. The electrode extends from x1 (95% Vmax) = 20.8% L = 0.52 mm to x2 (95% Vmax) = 80.4% L = 2.01 mm.
sions and layer thicknesses by an identical factor results in a correspondingly scaled solution. Thus, the optimal electrode positions relative to the cantilever length remain unchanged. For an investigation of the influence of the cantilever width and length on the solution, we refer to [17].
IV. Results for Cantilevers With Pick-up Region
Next, we consider fixed lengths of L1 = 1500 µm and L2 = 500 µm and vary the thicknesses h1 and h2. Figs. 8(a) and 8(b) give an example for a cantilever with h1 = 400 µm and h2 = 100 µm. Here, the pick-up region is positioned at the tip of the cantilever, with a maximum voltage value of 0.041 V. In Fig. 8(c), the resulting maximum induced voltage for a systematic variation of h1 and h2 is given (L1 = 1500 µm and L2 = 500 µm in all cases).
During the fabrication of the cantilever, a trench may be added as a pick-up region while maintaining the mechanical properties in other regions of the cantilever. In this section, the influence of the thickness and length of the weight in such a trenched cantilever is investigated for the case of insulating magnetostrictive layers, and optimal electrode positioning is derived. Again, the static behavior is evaluated. First, a constant silicon substrate thickness of h1 = 100 µm is assumed for the trench with the other parameters given in Fig. 1. The length L2 of the weight and the thickness h2 are varied. Figs. 7(a) and 7(b) show an example of the induced voltage on the surface for a trench length of L1 = 300 µm, a weight length of L2 = 1700 µm and a weight height h2 = 300 µm. Above the trench, a significantly higher induced voltage of 0.042 V is observed compared with 0.024 V above the weight. Thus, the second electrode should only be positioned above the trench. Table II lists the calculated ME voltages for different electrode configurations. For an electrode above the trench, a 50% higher signal is obtained compared with a second electrode covering the complete cantilever. In Fig. 7(c), the resulting maximum induced voltage for a systematic variation of the weight length L2 and weight height h2 is given (h1 = 100 µm in all cases). A maximum induced voltage of 0.047 V is observed for a height h2 = 500 µm and a length L2 = 1800 µm. Thus, the trenched cantilever enables an up to 27% larger signal voltage above the h1 = 100 µm pick-up region compared with the cantilever with h1 = h2 = 100 µm in Fig. 2(b).
Fig. 6. (a) Electric potential distribution on surface of insulating magnetostrictive layer for hPE = hMS = h1 = h2 = 20 µm. (b) Electrode extending from x1 (95% Vmax) = 20.4% L = 0.51 mm to x2 (95% Vmax) = 24.4% L = 0.61 mm. (c) Electric potential distribution on surface with electrode.
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TABLE II. Calculated Magnetoelectric (ME) Voltage for Different Simulation Setups for h1 = 100 µm, h2 = 300 µm, L1 = 300 µm, and L2 = 1700 µm. Conductive boundary condition on top of MS layer
Conductive boundary condition on bottom of MS layer 27 mV
27 mV
41 mV
42 mV
a) 25 mV b) 39 mV c) 25 mV
A conductive boundary condition on top simulates the behavior of an insulating magnetostrictive (MS) material with an electrode on top. A conductive boundary condition on the bottom corresponds to a conductive MS material.
Optimal voltage values are obtained if either h1 or h2 has a thickness of 100 µm. This may be explained by the optimal strain distribution observed in Figs. 3 and 4 for this substrate thickness. Slightly higher maximum signal voltages are expected for the case of a 100-µm h2 region at the tip of the cantilever. In all cases, the second electrode should be positioned in the region with maximum induced voltage. Larger electrodes result in an averaging effect and reduced signal voltages.
Fig. 7. (a) Electric potential distribution on surface for insulating magnetostrictive layer with trench as pick-up region. (b) Electric potential on the surface of the magnetostrictive layer along dashed line indicated in (a). (c) Maximum induced voltage on piezoelectric (PE) surface as a function of h2 and L2 for h1 = 100 µm and L1 = 2 mm − L2.
V. Conclusion We presented numerical simulation results on the optimal design and electrode positioning of cantilever magnetoelectric sensors employing an insulating magnetostrictive material and operated in the static or quasistatic regime. The deformation of the magnetostrictive layer under an applied magnetic field is transferred to the piezoelectric layer and the silicon cantilever substrate. For a structured cantilever, the deformation varies in space, resulting in a position-dependent induced electric potential across the
Fig. 8. (a) Electric potential distribution on surface for insulating magnetostrictive layer with tip as pick-up region. (b) Electric potential on the surface of the magnetostrictive layer along dashed line indicated in (a). (c) Maximum induced voltage on piezoelectric (PE) surface as a function of h1 and h2 for L1 = 1.5 mm and L2 = 0.5 mm.
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piezoelectric layer. Thus, the top electrode should be positioned in the region with maximum induced voltage. The pick-up region should be designed for an optimal positioning of the neutral axis. Compared with a cantilever without a trench and optimal substrate thickness, an additional increase in the signal voltage of approximately 25% is achieved by employing a structured cantilever. The results presented here may be transferred to conductive magnetostrictive materials. In this case, different regions of the magnetostrictive material should be insulated electrically by trenches. Alternatively, the magnetostrictive material may only be deposited in regions providing a high signal voltage. Here, the influence on the excitation mechanism must be considered. References [1] S. X. Dong, J. Y. Zhai, F. Bai, J. F. Li, and D. Viehland, “Pushpull mode magnetostrictive/piezoelectric laminate composite with an enhanced magnetoelectric voltage coefficient,” Appl. Phys. Lett., vol. 87, no. 6, art. no. 062502, 2005. [2] S. X. Dong, J. Y. Zhai, J. F. Li, and D. Viehland, “Small dc magnetic field response of magnetoelectric laminate composites,” Appl. Phys. Lett., vol. 88, no. 8, art. no. 082907, 2006. [3] C. W. Nan, M. I. Bichurin, S. X. Dong, D. Viehland, and G. Srinivasan, “Multiferroic magnetoelectric composites: Historical perspective, status, and future directions,” J. Appl. Phys., vol. 103, no. 3, art. no. 031101, 2008. [4] N. Tiercelin, V. Preobrazhensky, P. Pernod, and A. Ostaschenko, “Enhanced magnetoelectric effect in nanostructured magnetostrictive thin film resonant actuator with field induced spin reorientation transition,” Appl. Phys. Lett., vol. 92, no. 6, art. no. 062904, 2008. [5] P. Zhao, Z. Zhao, D. Hunter, R. Suchoski, C. Gao, S. Mathews, M. Wuttig, and I. Takeuchi, “Fabrication and characterization of allthin-film magnetoelectric sensors,” Appl. Phys. Lett., vol. 94, no. 24, art. no. 243507, 2009. [6] H. Greve, E. Woltermann, H.-J. Quenzer, B. Wagner, and E. Quandt, “Giant magnetoelectric coefficients in (Fe90Co10)78Si12B10-AlN thin film composites,” Appl. Phys. Lett., vol. 96, no. 18, art. no. 182501, 2010. [7] H. Bhaskaran, M. Li, D. Garcia-Sanchez, P. Zhao, I. Takeuchi, and H. X. Tang, “Active microcantilevers based on piezoresistive ferromagnetic thin films,” Appl. Phys. Lett., vol. 98, no. 1, art. no. 013502, 2011. [8] S. Marauska, R. Jahns, H. Greve, E. Quandt, R. Knöchel, and B. Wagner, “MEMS magnetic field sensor based on magnetoelectric composites,” J. Micromech. Microeng., vol. 22, no. 6, art. no. 065024, 2012. [9] H. Greve, E. Woltermann, R. Jahns, S. Marauska, B. Wagner, R. Knöchel, M. Wuttig, and E. Quandt, “Low damping resonant magnetoelectric sensors,” Appl. Phys. Lett., vol. 97, no. 15, art. no. 152503, 2010. [10] R. Jahns, A. Piorra, E. Lage, C. Kirchhof, D. Meyners, J. Gugat, M. Krantz, M. Gerken, R. Knöchel, and E. Quandt, “Giant magnetoelectric effect in thin film composites,” J. Am. Ceram. Soc., vol. 96, no. 6, pp. 1673–1681, 2013. [11] M. I. Bichurin, V. M. Petrov, and G. Srinivasan, “Theory of low-frequency magnetoelectric coupling in magnetostrictive-piezoelectric bilayers,” Phys. Rev. B, vol. 68, no. 5, art. no. 054402, 2003.
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[12] M. Guo and S. X. Dong, “A resonance-bending mode magnetoelectric-coupling equivalent circuit,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 56, no. 11, pp. 2578–2586, 2009. [13] M. Krantz and M. Gerken, “Theory of magnetoelectric effect in multilayer nanocomposites on a substrate: Static bending-mode response,” AIP Adv., vol. 3, no. 2, art. no. 022103, 2013. [14] M. Gerken, “Resonance line shape, strain and electric potential distributions of composite magnetoelectric sensors,” AIP Adv., vol. 3, no. 6, art. no. 062115, 2013. [15] J. F. Blackburn, M. Vopsaroiu, and M. G. Cain, “Verified finite element simulation of multiferroic structures: Solutions for conducting and insulating systems,” J. Appl. Phys., vol. 104, no. 7, art. no. 074104, 2008. [16] D. A. Berlincourt, D. R. Curran, and H. Jaffe, Physical Acoustics, W. P. Mason, Ed., New York, NY: Academic, 1964, pt. 1A. [17] J. L. Gugat, M. C. Krantz, and M. Gerken, “Two-dimensional versus three-dimensional finite element method simulations of cantilever magnetoelectric sensors,” IEEE Trans. Magn, vol. 49, no. 10, pp. 5287–5293, 2013.
Uzzal B. Bala received the B.Sc. degree in electrical and electronic engineering from the Bangladesh Institute of Technology, Khulna, in 1998. He received the M.Sc. degree in electrical communication engineering from the University of Kassel, Germany, in 2004, and the Ph.D. degree in electrical engineering from Leibniz Universität Hannover, Germany, in 2008. From 2008 to 2010, he held a postdoctoral position at the University of Paderborn, Germany. He was a research engineer at Christian-Albrechts-Universität zu Kiel, Germany, from 2010 to 2011. He is now working as a contingent employee from Alten GmbH at Intel Mobile Communications, Munich, Germany.
Matthias C. Krantz received the Diplom, M.S., and Ph.D. degrees in physics from Christian-Albrechts-Universität zu Kiel, Germany, and the University of Utah, Salt Lake City, UT, in 1984 and 1987, respectively. From 1988 to 1993, he held postdoctoral and Staff Engineer positions at IBM, San Jose. In 1994 and 1995, he took a sabbatical at Max Planck Institut Stuttgart, Germany. From 1996 to 2001, he was Systems Engineer for Terascan at KLA-Tencor, San Jose. From 2002 to 2009, he was a Principal Scientist at Carl Zeiss SMT-AG, Germany. Since 2010, he has been with the Integrated Systems and Photonics Group at the Christian-Albrechts-Universität zu Kiel, Germany.
Martina Gerken (S’00–M’03) received the Dipl.-Ing. degree in electrical engineering from the University of Karlsruhe, Germany, in 1998, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in 2003. From 2003 to 2008, she was an Assistant Professor at the University of Karlsruhe, Germany. In 2008, she was appointed as a full Professor of Electrical Engineering and head of the Integrated Systems and Photonics Group at the Christian-AlbrechtsUniversität zu Kiel, Germany.
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Electrode Position Optimization in Magnetoelectric Sensors Based on Magnetostrictive-Piezoelectric Bilayers on Cantilever Substrates Uzzal B. Bala, Matthias C. Krantz, and Martina Gerken, Member, IEEE Abstract—Finite element method (FEM) simulations are performed to investigate the sensitivity to dc magnetic fields of magnetoelectric sensors on cantilever substrates with trenches or weights at different positions. For a simple layered cantilever, a 15% higher signal voltage across the piezoelectric layer is obtained for optimally positioned electrodes and an insulating magnetostrictive material. A further 25% increase in the signal voltage is achieved for a trenched cantilever design with a pick-up region.
I. Introduction
M
agnetoelectric (ME) composites have been employed for room-temperature sensing of picotesla ac magnetic fields [1] and nanotesla dc magnetic fields [2]. In these ME composites, the magnetic field induces a strain in the magnetostrictive component, which is transferred via elastic coupling to a strain in the piezoelectric component, inducing a voltage signal [3]. For the integration of such ME sensors with microelectromechanical systems (MEMS), as well as for the fabrication of sensor arrays, thin-film-based 2–2 composites consisting of a piezoelectric layer and a magnetostrictive layer on a cantilever substrate are of particular interest [4]–[8]. The cantilever substrate prevents clamping. For resonant operation in the first bending mode, significant enhancement in the sensitivity is achieved. Resonant frequencies are from 100 Hz to the kilohertz range and may be adjusted by changing the size of the cantilever or adding a weight at the tip [5], [9], [10]. Cantilever designs with different regions may also be employed to improve the mechanical stability of the sensor or to realize specific signal pick-up regions for better sensitivity [5], [10]. Although the behavior of ME cantilever sensors without weights or pick-up regions may be modeled analytically [4], [11]–[13], a detailed analysis of the behavior for structured cantilevers is only possible numerically. In [10], we simulated the performance of three different types of trenched cantilevers using the finite element method (FEM). By thinning the substrate in Manuscript received September 11, 2013; accepted December 13, 2013. This work was supported by the German Science Foundation (DFG) within the Collaborative Research Centre SFB 855 “Magnetoelectric Composite Materials—Biomagnetic Interfaces of the Future.” The authors are with the Institute of Electrical and Information Engineering at the Christian-Albrechts-Universität zu Kiel, Kiel, Germany (e-mail: [email protected]). DOI http://dx.doi.org/10.1109/TUFFC.2014.2924 0885–3010
the trench region and modifying the electrode design, the resonance frequency was reduced and the ME coefficient was increased by a factor of 9, in good agreement with experimental results. Here, we present a numerical investigation of the optimal cantilever design and electrode positioning for cantilevers without and with a weight at the tip for dc magnetic field detection. As shown schematically in Fig. 1(a), the cantilever sensor is assumed to consist of a silicon cantilever substrate that is clamped rigidly on the bottom left and has two bending regions of lengths L1 and L2 and heights h1 and h2. The silicon substrate is assumed to be conductive and serves as the bottom electrode for the piezoelectric layer (alternatively, a thin metal layer may be deposited on top of the silicon cantilever to serve as the bottom electrode). Because strain-induced charges may vary in the plane of the piezoelectric layer, special attention is given to the influence of magnetostrictive layer conductivity. In the case of a continuous conductive magnetostrictive material serving as the top electrode, all surfaces of the magnetostrictive material exhibit the same electric potential, clamping the surface in contact with the piezoelectric material to a specific electric potential. In contrast, an insulating magnetostrictive material acts as a series capacitance, and the electric potential on the surface of the magnetostrictive material is position-dependent. Thus, an optimal electrode positioning of the second electrode allows for higher ME coefficients. A small, local second electrode probes the local induced electric potential. A larger electrode of a highly conductive material forces a constant electric potential to the top surface. Thus, the cantilever is deformed (via the piezoelectric material) such that the electrode voltage is constant. The resulting constant voltage may be approximated by averaging over the induced electric potential at the corresponding surface for the case of an insulating magnetostrictive material without an electrode. This will be discussed in Section III. This electrode design concept may be transferred to conductive magnetostrictive materials by electrically separating different regions of the magnetostrictive material with small trenches, as depicted in Fig. 1(b). The conductivity of the magnetostrictive material forces a constant electric potential at the interface between the piezoelectric layer and the respective area of the magnetostrictive material. In the following, we calculate the electric potential distribution in the cantilever beam using FEM. From the
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Fig. 1. (a) Schematic of investigated magnetoelectric sensor structure based on an insulating magnetostrictive (MS) layer and a piezoelectric (PE) layer on a cantilever substrate (Sub) with a weight at the tip. The position of electrode 2 is optimized. (b) Schematic of sensor with conductive MS layer structured with trenches for electric insulation.
simulation results assuming an equivalent insulating magnetostrictive material, we derive the optimal positioning of the second electrode as a function of the lengths L1 and L2 and heights h1 and h2. For the case of the conductive magnetostrictive material, these results describe which region of the magnetostrictive material should be used electrically as the pick-up region. The paper is structured as follows. The model for FEM simulations is described in Section II. Section III presents results for unstructured cantilever substrates with a single thickness of h1 = h2. Section IV investigates cantilevers with weights or trenches. Conclusions are given in Section V. II. Model for Finite-Element-Simulations The small-signal behavior of the cantilever magnetoelectric sensors is modeled using linear constitutive material equations (1)–(3) [14]–[16]:
m σ ij = c ijkl εkl − e kijE k − e kij H k (1)
Di = e ijk ε jk + κijE j (2)
m B i = e ijk ε jk + µijH j . (3)
Here, σ and ε are the stress and strain tensors. E, D, H, and B are the electric field, electric flux density, magnetic field, and magnetic flux density. c, e, em, κ, and µ are the stiffness, strain to electric field coupling constant, strain to magnetic field coupling constant, permittivity, and permeability. In this formulation, it is assumed that the direct magnetoelectric effect is negligible in the employed materials. Lead zirconate titanate (PZT) is used as the piezoelectric material and Terfenol-D as the magnetostrictive material with the material parameters given in [15]. For PZT, we set em = 0, and for Terfenol-D, e = 0. The conductivity of Terfenol-D is included in the model by a perfectly conductive boundary condition, as presented in
[15]. For the calculation of an insulating magnetostrictive material, we assume an equivalent material that is insulating, but has otherwise the same properties as Terfenol-D. In the simulation, this is achieved by removing the conductive boundary condition. We use this approach to investigate the influence of conductivity without changing any other parameters. After deriving the optimal pick-up region using this equivalent-material approach, a patterned Terfenol-D region should be included at this position in the real design. The mechanical properties of additional electrode metal layers may be neglected in the simulation because these layers are typically much thinner than other layers. The silicon cantilever substrate is modeled as an isotropic material with a Young’s modulus of 131 GPa, a Poisson’s ratio of 0.27, κ = 12.1 ∙ 8.854 ∙ 10−12 F/m, µ = 4 ∙ π ∙ 10−7 H/m, e = 0, and em = 0. The coupled elastostatic/elastodynamic and electrostatic/magnetostatic equations are solved assuming an ideal interface lamination, such that the deformation of the magnetostrictive material for an applied magnetic field is transferred to the piezoelectric material without slip. As described in [15], we assume that the electric field E and the magnetic field H may be expressed by the scalar potentials V and Vm as E = −∇V and H = −∇Vm. It is assumed that no charge density and conduction currents are present. A system of five differential equations is set up for the five variables u1, u2, u3, V, and Vm, where u is the displacement vector and εij = (ui,j + uj,i)/2:
σ ij, j + f i = ρu i
with i = 1, 2, 3 (4)
∇ ⋅ D = 0 (5)
∇ ⋅ B = 0. (6)
Because we are considering the low-frequency behavior simulated at 0 Hz, the right-hand sides of Newton’s laws (4) are set to zero. Furthermore, it is assumed that no other forces are present and fi = 0. The system of differential equations is solved in 3-D numeric simulations with Comsol Multiphysics (Comsol Inc., Burlington, MA).
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More details on an FEM implementation of the equations may be found in [14], [15]. III. Results for Single Cantilever Substrate Thickness First, we consider the case of cantilever substrates without a weight at the tip (h1 = h2 in Fig. 1) and compare the induced electric potential at the surface for conductive and insulating magnetostrictive materials. The geometric parameters of the cantilever are chosen as depicted in Fig. 1(a) with a total cantilever length of 2.5 mm. An H-field of 400 A/m is applied along the cantilever by setting the magnetic potential on the surfaces at x = 0 to Vm = 0 and at the surfaces at x = 2.5 mm to Vm = 1 A. The bottom electrode 1 is set to an electric potential of 0 V. Fig. 2 plots the resulting electric potential distribution for the case of h1 = h2 = 100 µm. As expected, a constant electric potential is observed at the surface of the conductive magnetostrictive material. For the insulating magnetostrictive material, the electric potential is lower at both ends of the cantilever. The maximum electric potential is with 0.037 V for the insulating magnetostrictive material, 16% higher than the value of 0.032 V for the conductive magnetostrictive material. Thus, by positioning the second electrode between 0.6 mm and 2.0 mm, an approximately 15% higher voltage signal between top and bottom electrode is expected. By covering the complete cantilever surface with the second electrode, a voltage of 0.032 V is obtained, which is exactly the voltage for the conductive magnetostrictive case. The slight variation of the electric potential for the conductive magnetostrictive case is due to the inaccuracy of the numerical solution in fulfilling the perfectly conductive boundary condition. Next, we consider the influence of the silicon cantilever thickness on the induced ME voltage between top and bottom electrode. Fig. 3 plots the results for the conductive and the insulating magnetostrictive case. For an optimal positioning of electrode 2, a higher ME voltage is achieved in case of the insulating compared with the conductive MS material. On the other hand, the induced ME voltage is identical for a conductive MS material and an insulating MS layer with an all-surface electrode 2. The induced voltage is strongly influenced by substrate thickness with a maximum voltage at h1 = h2 = 100 µm. Figs. 4(a) and 4(b) show the strain ε11 and stress σ11 in the xdirection as a function of the position z in the cantilever at half the width and half the length of the cantilever for an insulating magnetostrictive material. Fig. 4(c) shows the electric potential V as a function of the position z in the cantilever. Four different values for the silicon substrate thickness are evaluated. The strain plots in Fig. 4(a) show that the magnetostrictive layer is compressed because of the applied magnetic field, causing a deformation and upwards bending of the cantilever. For an increasing substrate thickness, the strain in the magnetostrictive layer with fixed thickness is reduced because of the larger com-
Fig. 2. (a) Electric potential distribution on surface for conductive magnetostrictive layer; h1 = h2 = 100 µm. (b) Electric potential distribution on surface for insulating magnetostrictive layer; h1 = h2 = 100 µm. (c) Comparison of electric potential on the surface of the magnetostrictive layer along dashed line indicated in (a) and (b).
bined thickness of piezoelectric layer and substrate that must be deformed. The combined effects of strain magnitude and strain slope are responsible for the induced stress and electric potential in the piezoelectric layer, such that for h1 = h2 = 50 µm and h1 = h2 = 200 µm nearly identical voltages are obtained between the top and the bottom electrode. The position of the neutral axis (axis with zero strain) moves from within the piezoelectric layer for h1 = h2 = 20 µm to positions further and further in the substrate for increasing values of h1 = h2. Note that for the static case considered here, the position of the neutral axis is not predominantly determined by the mechanical properties of the cantilever as in the resonant case [4], [5], [13]. For h1 = h2 = 20 µm, strain and stress are positive on the substrate side of the piezoelectric layer and negative on the other side. This causes the drop and subsequent increase in the electric potential observed in Fig. 4(c). The partial voltage
Fig. 3. Induced open-circuit magnetoelectric (ME) voltage between top and bottom electrode as a function of the silicon cantilever substrate thickness h1 = h2. For the case of the insulating magnetostrictive layer the optimal electrode 2 positioning corresponds to placing electrode 2 at the location with the maximum induced voltage on the magnetostrictive (MS) surface. For comparison, the ME voltage is calculated for an electrode 2 covering the complete MS surface for an insulating MS layer.
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Fig. 4. (a) Strain ε11 in the x-direction as a function of the position z in the cantilever at half the length (x = 1.25 mm) and half the width (y = 325 µm) of the cantilever for an insulating magnetostrictive material and four different silicon cantilever substrate thicknesses. (b) Stress σ11 in the x-direction as a function of the position z for the same parameters. Calculated mesh points are connected by lines for easier visualization, resulting in finite slopes at the interfaces. (c) Induced electric potential V as a function of the position z for the same parameters.
cancellation in the piezoelectric layer explains the reduced performance for thin substrates. The optimal value of h1 = h2 = 100 µm derived here is valid for the geometric parameters defined in Fig. 1. For other ratios of geometric parameters, the solution must be
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recomputed to determine the optimal substrate thickness and electrode position. Table I lists the optimal cantilever substrate thickness h1 = h2 = hopt for other active layer thicknesses hPE = hMS with other parameters identical to the ones given in Fig. 1. For thinner active layers, the optimal substrate thickness is thinner as well and (within the 10 µm simulation step resolution) identical to the active layer thickness. Fig. 5(a) plots the electric potential on the surface of an insulating magnetostrictive layer for each of the five optimal cases of Table I. It is observed that the position of the maximum is closer to the fixed end of the cantilever for smaller layer thicknesses; also, a more distinct peak forms in the electric potential. Table I lists the maximum induced voltage and the average induced voltage. The average induced ME voltage scales approximately with the thickness hPE = hMS of the active layers. The enhancement obtainable with electrode 2 placed at the peak electric potential increases for thinner active layers. For layer thicknesses of 20 µm, the voltage increases from an average value of 7 mV for an electrode 2 covering the whole surface to a value of 11 mV, if the electrode is placed only at the peak position. This is an enhancement of 57%. From noise considerations, wider electrodes with a certain minimum capacitance may be required. Thus, we evaluated the positioning of electrode 2, if a 5% drop from the peak ME voltage is permitted. As shown in Fig. 5(b), we consider an electrode covering the complete width of the cantilever and extending from x1 (95% Vmax) to x2 (95% Vmax). Table I lists the calculated ME voltages. A drop in induced voltage is observed for hopt ≤ 60 µm, whereas no voltage drop is observed for larger hopt within the simulation resolution. This difference is caused by the y-direction profile of the induced ME potential. For thick cantilevers, the induced electric potential is nearly independent of the y-coordinate, as observed in Fig. 2(b). For thinner cantilevers, a voltage peak forms close to the fixed end around the center of the cantilever in the y-direction, as shown in Fig. 6(a). For an electrode covering the complete width of the cantilever, as depicted in Fig. 6(b), a reduced voltage is observed, as seen in Fig. 6(c). These results demonstrate that the optimal electrode positioning depends on the cantilever aspect ratios. Because of the linear constitutive laws, a scaling of all cantilever dimen-
TABLE I. Simulated Optimal Cantilever Substrate Thickness h1 = h2 = hopt for Varying Active Layer Thicknesses hPE = hMS With Other Parameters Identical to Those Given in Fig. 1(a). hPE = hMS (µm) 20 40 60 80 100
h1 = h2 = hopt (µm)
Vmax (mV)
Vavg (mV)
x1 (95% Vmax)
20 40 60 80 100
11 19 24 30 37
7 13 20 26 32
20% 20% 20% 20% 21%
L L L L L
x2 (95% Vmax) 24% 28% 34% 46% 80%
L L L L L
Velec (95% Vmax) (mV) 9 17 23 30 37
Simulations were carried out in steps of 10 µm for h1 = h2. Vmax is the maximum voltage from Fig. 5 along the center line shown dashed in Fig. 2(b). Vavg is the average voltage along the center line. Allowing for a drop in voltage by 5% from the maximum in Fig. 5, the optimal electrode 2 should extend from x1 (95%) to x2 (95%) given in units of the length L of the cantilever. Velec(95%) is the simulated ME voltage obtained for this electrode positioning.
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Fig. 5. (a) Electric potential on the surface of an insulating magnetostrictive layer along the dashed line indicated in Fig. 2(b) for five different layer thicknesses hPE = hMS = h1 = h2 with other parameters the same as Fig. 1(a). (b) Optimal placement of electrode 2, allowing for a 5% drop from the maximum voltage Vmax for the case of hPE = hMS = h1 = h2 = 100 µm and length L = 2.5 mm. The electrode extends from x1 (95% Vmax) = 20.8% L = 0.52 mm to x2 (95% Vmax) = 80.4% L = 2.01 mm.
sions and layer thicknesses by an identical factor results in a correspondingly scaled solution. Thus, the optimal electrode positions relative to the cantilever length remain unchanged. For an investigation of the influence of the cantilever width and length on the solution, we refer to [17].
IV. Results for Cantilevers With Pick-up Region
Next, we consider fixed lengths of L1 = 1500 µm and L2 = 500 µm and vary the thicknesses h1 and h2. Figs. 8(a) and 8(b) give an example for a cantilever with h1 = 400 µm and h2 = 100 µm. Here, the pick-up region is positioned at the tip of the cantilever, with a maximum voltage value of 0.041 V. In Fig. 8(c), the resulting maximum induced voltage for a systematic variation of h1 and h2 is given (L1 = 1500 µm and L2 = 500 µm in all cases).
During the fabrication of the cantilever, a trench may be added as a pick-up region while maintaining the mechanical properties in other regions of the cantilever. In this section, the influence of the thickness and length of the weight in such a trenched cantilever is investigated for the case of insulating magnetostrictive layers, and optimal electrode positioning is derived. Again, the static behavior is evaluated. First, a constant silicon substrate thickness of h1 = 100 µm is assumed for the trench with the other parameters given in Fig. 1. The length L2 of the weight and the thickness h2 are varied. Figs. 7(a) and 7(b) show an example of the induced voltage on the surface for a trench length of L1 = 300 µm, a weight length of L2 = 1700 µm and a weight height h2 = 300 µm. Above the trench, a significantly higher induced voltage of 0.042 V is observed compared with 0.024 V above the weight. Thus, the second electrode should only be positioned above the trench. Table II lists the calculated ME voltages for different electrode configurations. For an electrode above the trench, a 50% higher signal is obtained compared with a second electrode covering the complete cantilever. In Fig. 7(c), the resulting maximum induced voltage for a systematic variation of the weight length L2 and weight height h2 is given (h1 = 100 µm in all cases). A maximum induced voltage of 0.047 V is observed for a height h2 = 500 µm and a length L2 = 1800 µm. Thus, the trenched cantilever enables an up to 27% larger signal voltage above the h1 = 100 µm pick-up region compared with the cantilever with h1 = h2 = 100 µm in Fig. 2(b).
Fig. 6. (a) Electric potential distribution on surface of insulating magnetostrictive layer for hPE = hMS = h1 = h2 = 20 µm. (b) Electrode extending from x1 (95% Vmax) = 20.4% L = 0.51 mm to x2 (95% Vmax) = 24.4% L = 0.61 mm. (c) Electric potential distribution on surface with electrode.
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TABLE II. Calculated Magnetoelectric (ME) Voltage for Different Simulation Setups for h1 = 100 µm, h2 = 300 µm, L1 = 300 µm, and L2 = 1700 µm. Conductive boundary condition on top of MS layer
Conductive boundary condition on bottom of MS layer 27 mV
27 mV
41 mV
42 mV
a) 25 mV b) 39 mV c) 25 mV
A conductive boundary condition on top simulates the behavior of an insulating magnetostrictive (MS) material with an electrode on top. A conductive boundary condition on the bottom corresponds to a conductive MS material.
Optimal voltage values are obtained if either h1 or h2 has a thickness of 100 µm. This may be explained by the optimal strain distribution observed in Figs. 3 and 4 for this substrate thickness. Slightly higher maximum signal voltages are expected for the case of a 100-µm h2 region at the tip of the cantilever. In all cases, the second electrode should be positioned in the region with maximum induced voltage. Larger electrodes result in an averaging effect and reduced signal voltages.
Fig. 7. (a) Electric potential distribution on surface for insulating magnetostrictive layer with trench as pick-up region. (b) Electric potential on the surface of the magnetostrictive layer along dashed line indicated in (a). (c) Maximum induced voltage on piezoelectric (PE) surface as a function of h2 and L2 for h1 = 100 µm and L1 = 2 mm − L2.
V. Conclusion We presented numerical simulation results on the optimal design and electrode positioning of cantilever magnetoelectric sensors employing an insulating magnetostrictive material and operated in the static or quasistatic regime. The deformation of the magnetostrictive layer under an applied magnetic field is transferred to the piezoelectric layer and the silicon cantilever substrate. For a structured cantilever, the deformation varies in space, resulting in a position-dependent induced electric potential across the
Fig. 8. (a) Electric potential distribution on surface for insulating magnetostrictive layer with tip as pick-up region. (b) Electric potential on the surface of the magnetostrictive layer along dashed line indicated in (a). (c) Maximum induced voltage on piezoelectric (PE) surface as a function of h1 and h2 for L1 = 1.5 mm and L2 = 0.5 mm.
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piezoelectric layer. Thus, the top electrode should be positioned in the region with maximum induced voltage. The pick-up region should be designed for an optimal positioning of the neutral axis. Compared with a cantilever without a trench and optimal substrate thickness, an additional increase in the signal voltage of approximately 25% is achieved by employing a structured cantilever. The results presented here may be transferred to conductive magnetostrictive materials. In this case, different regions of the magnetostrictive material should be insulated electrically by trenches. Alternatively, the magnetostrictive material may only be deposited in regions providing a high signal voltage. Here, the influence on the excitation mechanism must be considered. References [1] S. X. Dong, J. Y. Zhai, F. Bai, J. F. Li, and D. Viehland, “Pushpull mode magnetostrictive/piezoelectric laminate composite with an enhanced magnetoelectric voltage coefficient,” Appl. Phys. Lett., vol. 87, no. 6, art. no. 062502, 2005. [2] S. X. Dong, J. Y. Zhai, J. F. Li, and D. Viehland, “Small dc magnetic field response of magnetoelectric laminate composites,” Appl. Phys. Lett., vol. 88, no. 8, art. no. 082907, 2006. [3] C. W. Nan, M. I. Bichurin, S. X. Dong, D. Viehland, and G. Srinivasan, “Multiferroic magnetoelectric composites: Historical perspective, status, and future directions,” J. Appl. Phys., vol. 103, no. 3, art. no. 031101, 2008. [4] N. Tiercelin, V. Preobrazhensky, P. Pernod, and A. Ostaschenko, “Enhanced magnetoelectric effect in nanostructured magnetostrictive thin film resonant actuator with field induced spin reorientation transition,” Appl. Phys. Lett., vol. 92, no. 6, art. no. 062904, 2008. [5] P. Zhao, Z. Zhao, D. Hunter, R. Suchoski, C. Gao, S. Mathews, M. Wuttig, and I. Takeuchi, “Fabrication and characterization of allthin-film magnetoelectric sensors,” Appl. Phys. Lett., vol. 94, no. 24, art. no. 243507, 2009. [6] H. Greve, E. Woltermann, H.-J. Quenzer, B. Wagner, and E. Quandt, “Giant magnetoelectric coefficients in (Fe90Co10)78Si12B10-AlN thin film composites,” Appl. Phys. Lett., vol. 96, no. 18, art. no. 182501, 2010. [7] H. Bhaskaran, M. Li, D. Garcia-Sanchez, P. Zhao, I. Takeuchi, and H. X. Tang, “Active microcantilevers based on piezoresistive ferromagnetic thin films,” Appl. Phys. Lett., vol. 98, no. 1, art. no. 013502, 2011. [8] S. Marauska, R. Jahns, H. Greve, E. Quandt, R. Knöchel, and B. Wagner, “MEMS magnetic field sensor based on magnetoelectric composites,” J. Micromech. Microeng., vol. 22, no. 6, art. no. 065024, 2012. [9] H. Greve, E. Woltermann, R. Jahns, S. Marauska, B. Wagner, R. Knöchel, M. Wuttig, and E. Quandt, “Low damping resonant magnetoelectric sensors,” Appl. Phys. Lett., vol. 97, no. 15, art. no. 152503, 2010. [10] R. Jahns, A. Piorra, E. Lage, C. Kirchhof, D. Meyners, J. Gugat, M. Krantz, M. Gerken, R. Knöchel, and E. Quandt, “Giant magnetoelectric effect in thin film composites,” J. Am. Ceram. Soc., vol. 96, no. 6, pp. 1673–1681, 2013. [11] M. I. Bichurin, V. M. Petrov, and G. Srinivasan, “Theory of low-frequency magnetoelectric coupling in magnetostrictive-piezoelectric bilayers,” Phys. Rev. B, vol. 68, no. 5, art. no. 054402, 2003.
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[12] M. Guo and S. X. Dong, “A resonance-bending mode magnetoelectric-coupling equivalent circuit,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 56, no. 11, pp. 2578–2586, 2009. [13] M. Krantz and M. Gerken, “Theory of magnetoelectric effect in multilayer nanocomposites on a substrate: Static bending-mode response,” AIP Adv., vol. 3, no. 2, art. no. 022103, 2013. [14] M. Gerken, “Resonance line shape, strain and electric potential distributions of composite magnetoelectric sensors,” AIP Adv., vol. 3, no. 6, art. no. 062115, 2013. [15] J. F. Blackburn, M. Vopsaroiu, and M. G. Cain, “Verified finite element simulation of multiferroic structures: Solutions for conducting and insulating systems,” J. Appl. Phys., vol. 104, no. 7, art. no. 074104, 2008. [16] D. A. Berlincourt, D. R. Curran, and H. Jaffe, Physical Acoustics, W. P. Mason, Ed., New York, NY: Academic, 1964, pt. 1A. [17] J. L. Gugat, M. C. Krantz, and M. Gerken, “Two-dimensional versus three-dimensional finite element method simulations of cantilever magnetoelectric sensors,” IEEE Trans. Magn, vol. 49, no. 10, pp. 5287–5293, 2013.
Uzzal B. Bala received the B.Sc. degree in electrical and electronic engineering from the Bangladesh Institute of Technology, Khulna, in 1998. He received the M.Sc. degree in electrical communication engineering from the University of Kassel, Germany, in 2004, and the Ph.D. degree in electrical engineering from Leibniz Universität Hannover, Germany, in 2008. From 2008 to 2010, he held a postdoctoral position at the University of Paderborn, Germany. He was a research engineer at Christian-Albrechts-Universität zu Kiel, Germany, from 2010 to 2011. He is now working as a contingent employee from Alten GmbH at Intel Mobile Communications, Munich, Germany.
Matthias C. Krantz received the Diplom, M.S., and Ph.D. degrees in physics from Christian-Albrechts-Universität zu Kiel, Germany, and the University of Utah, Salt Lake City, UT, in 1984 and 1987, respectively. From 1988 to 1993, he held postdoctoral and Staff Engineer positions at IBM, San Jose. In 1994 and 1995, he took a sabbatical at Max Planck Institut Stuttgart, Germany. From 1996 to 2001, he was Systems Engineer for Terascan at KLA-Tencor, San Jose. From 2002 to 2009, he was a Principal Scientist at Carl Zeiss SMT-AG, Germany. Since 2010, he has been with the Integrated Systems and Photonics Group at the Christian-Albrechts-Universität zu Kiel, Germany.
Martina Gerken (S’00–M’03) received the Dipl.-Ing. degree in electrical engineering from the University of Karlsruhe, Germany, in 1998, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in 2003. From 2003 to 2008, she was an Assistant Professor at the University of Karlsruhe, Germany. In 2008, she was appointed as a full Professor of Electrical Engineering and head of the Integrated Systems and Photonics Group at the Christian-AlbrechtsUniversität zu Kiel, Germany.
