Nonlinear converse magnetoelectric effects in a

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Nov 20, 2018 - The nature of the DME in composites under excitation by an ac magnetic field .... mental at f1 ¼f0 and amplitude A1 corresponding to the linear.
Nonlinear converse magnetoelectric effects in a ferromagnetic-piezoelectric bilayer L. Y. Fetisov, D. V. Chashin, D. A. Burdin, D. V. Saveliev, N. A. Ekonomov, G. Srinivasan, and Y. K. Fetisov

Citation: Appl. Phys. Lett. 113, 212903 (2018); doi: 10.1063/1.5054584 View online: https://doi.org/10.1063/1.5054584 View Table of Contents: http://aip.scitation.org/toc/apl/113/21 Published by the American Institute of Physics

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APPLIED PHYSICS LETTERS 113, 212903 (2018)

Nonlinear converse magnetoelectric effects in a ferromagnetic-piezoelectric bilayer L. Y. Fetisov,1 D. V. Chashin,1 D. A. Burdin,1 D. V. Saveliev,1 N. A. Ekonomov,1 G. Srinivasan,2,a) and Y. K. Fetisov1,a) 1

MIREA-Russian Technological University, Moscow 119454, Russia Physics Department, Oakland University, Rochester, Michigan 48309, USA

2

(Received 31 August 2018; accepted 3 November 2018; published online 20 November 2018) The strain mediated nonlinear converse magnetoelectric effect (CME) is investigated in a bilayer of an amorphous ferromagnet FeBSiC and piezoelectric lead zirconate titanate (PZT). The magnetic response of the sample to an AC electric field (e) applied to PZT at an acoustic resonance frequency of 76 kHz was measured with a coil wound around the bilayer. With an increase in the amplitude of e over the range 0–250 V/cm, the variation in amplitude of the first and the second harmonics of the induced voltage due to the variation in the magnetic induction B was measured for DC bias magnetic field H ¼ 0–80 Oe. The coefficients of the linear and nonlinear converse ME effects were 5.5 G cm/V and 1.9  102 G cm2/V2, respectively. The nonlinearity of the CME arises due to the nonlinear dependence of the magnetic induction on the stress. A theoretical model for the nonlinear CME is discussed. Published by AIP Publishing. https://doi.org/10.1063/1.5054584

The magnetoelectric (ME) effects in composites containing mechanically coupled ferromagnetic (FM) and piezoelectric (PE) layers have been studied extensively in recent years because of their potential for use in magnetic field sensors, radio frequency signal processing devices, and information storage technologies.1–3 The direct-ME effect (DME) in the composites arises due to combination of the magnetostriction of the FM layer and the piezoelectric effect in the PE layer and manifests as a change in the polarization P under an external magnetic field H. The converse-ME effect (CME) arises as a result of a combination of the inverse piezoelectric effect in the PE layer and the elastomagnetic effect (inverse magnetostriction) in the FM layer and is measured as a change in the magnetic induction B under an electric field E. The nature of the DME in composites under excitation by an ac magnetic field in a variety of ferroic phases and geometries has been reported so far. Ferromagnets with high magnetostriction k, such as Ni, Co, FeGa, Terfenol, FeBSiC, and ferrites, and PE materials including lead zirconate titanate (PZT) and lead magnesium niobate-lead titanate (PMN-PT) with a high ratio of piezoelectric coupling coefficient to dielectric permittivity d/e, were used to achieve strong DME.4,5 In 2006, the effect of frequency doubling was observed for DME in a layered sample with nickel zinc ferrite and PZT.6 It was shown that the nonlinearity arises from the nonlinear dependence of k on the bias magnetic field H. In subsequent years, generation of higher harmonics in the DME response,7–10 mixing of the frequencies of alternating magnetic fields,11–13 nonlinear conversion of magnetic noise,14 and bistability15 were observed in various composites. The characteristics of nonlinear effects were also studied in detail as a function of the frequency and amplitude of the excitation field and dc magnetic field.16,17 The linear converse ME effect (CME) excited by an ac electric field e was also investigated in several composites.18–24 a)

Authors to whom correspondence should be addressed: [email protected] and [email protected]

0003-6951/2018/113(21)/212903/4/$30.00

The strength of the CME is shown to increase with increasing d for the PE layer and piezomagnetic coefficient q of the FM layer and also depends linearly on the amplitude of e. A theory allowing calculation of the linear CME characteristics was developed.25 However, nonlinear phenomena in CME have not been reported so far although it is well known that B in ferromagnets depends nonlinearly on the mechanical stress.26,27 One can therefore expect nonlinearities in the CME effects in the composite structures. This work is on the low-frequency nonlinear CME effects in a composite with a FM alloy and PZT when it is excited by an ac electric field at frequencies corresponding to an acoustic resonance mode in the sample. The CME response of the sample was measured as an induced voltage in a coil that contained the composite. Results of measurements of linear and nonlinear CME coefficients are presented and a model is proposed that explains the occurrence of nonlinear CME. The CME effect was studied in a FM-PE bilayer schematically shown in Fig. 1. The FM layer with in-plane dimensions

FIG. 1. Frequency dependence of the induced voltage u generated in a coil due to converse magnetoelectric effect (CME) in a composite of Metglas and PZT placed inside a pick up coil. An AC electric field of amplitude e ¼ 25 V/cm and frequency f was applied across PZT. An in-plane bias magnetic field H ¼ 14 Oe was applied parallel to the length of the composite. The inset shows the schematics of the ferromagnetic-piezoelectric structure of ferromagnetic (FM) alloy and piezoelectric (PE) PZT.

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of 20 mm  7 mm and thickness of am ¼ 20 lm was an amorphous ferromagnet FeBSiC (Metglas 2605S3A) with saturation magnetostriction kS  21  106 in HS  100 Oe. The PE layer with dimensions of 20 mm  7 mm and thickness ap ¼ 0.2 mm was a polycrystalline Pb0.52Zr0.48TiO3 (PZT) with 2 lm thick electrodes that was poled perpendicular to the plane and the piezoelectric coupling coefficient d31 ¼ 175 pm/N. The FM and PZT layers were bonded with a quick dry adhesive of thickness 2 lm. The composite was placed inside a solenoid 27 mm in length and 20 mm in diameter, containing N1 ¼ 800 turns of 0.18 mm diameter Cu-wire and was used to record changes in the magnetic induction B in the composite due to the CME effect. A DC bias field H ¼ 0–100 Oe directed along the long axis of the structure was produced by Helmholtz coils 20 cm in diameter. The field was measured with a (LakeShore 421) Gaussmeter with an accuracy of 0.1 Oe. A harmonic voltage U cos(2pft) with an amplitude up to 5 V and frequency f ¼ 0–200 kHz was applied across PZT and created an electric field ecos(2pft) with an amplitude e up to 250 V/cm. The induced voltage u(t) in the coil and its frequency spectrum A(f) were recorded using an (Tektronics TDS 3032B) oscilloscope. These measurements were performed as a function of e, f, and H-values. Figure 1 shows the measured f dependence of u for e ¼ 25 V/cm and H ¼ 14 Oe. One observes a peak near f0 ¼ 76 kHz with an amplitude of um ¼ 0.9 V and quality factor Q  15. The peak corresponds to the excitation of the fundamental mode of acoustic oscillations along the length of the composite. One can estimate theffi resonance qffiffiffiffiffiffiffiffiffiffiffiffiffiffi frequency that is given by f0 ¼ ð1=2lÞ Yef =qef , where Yef ¼ ðYm am þ Yp ap Þ=ðam þ ap Þ and qef ¼ ðqm am þ qp ap Þ= ðam þ ap Þ are the effective Young’s modulus and the density of the bilayer, respectively, with m and p representing the magnetostrictive and piezoelectric phases, respectively. For the following values of the parameters Ym ¼ 10.6  1010 N/m2 and Yp ¼ 7  1010 N/m2, qm ¼ 7.3  103 kg/m3, qp ¼ 7.7  103 kg/m3, and l ¼ 20 mm, one obtains fo ¼ 70 kHz. The discrepancy between the estimated and measured values of fo could be attributed to the fact that the formula is true for very thin strips of FM and PE layers and it does not take into account the finite thickness. The fine structure seen in the peak in Fig. 1 is due to different lengths of FM and PE layers with the FM layer slightly shorter (19.5 mm) than the length of the PZT layer. The CME effects under high amplitude excitations are considered next. Figure 2 shows the frequency spectrum of u with the application of e ¼ 250 V/cm at f0 ¼ 76 kHz and H ¼ 14 Oe. The spectrum contains two components; the fundamental at f1 ¼ f0 and amplitude A1 corresponding to the linear CME effect. Second harmonic with f2 ¼ 152 kHz and amplitude A2 demonstrates a nonlinear frequency doubling due to CME which was not reported previously for any composite structures. Figure 3(a) shows the dependences of A1 and A2 on the bias magnetic field H for f0 ¼ 76 kHz and e ¼ 125 V/ cm. The amplitude A1 ¼ 0 for zero bias field increases with increasing H, reaches a maximum at H1  11.5 Oe, and then decreases gradually as the FM layer becomes magnetically saturated. The amplitude A2 is also zero in the absence of the

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FIG. 2. Frequency spectrum for induced voltage u due to the converse ME effect in the Metglas-PZT bilayer for e ¼ 125 V/cm and H ¼ 14 Oe.

field H, then increases to a maximum at H2  18 Oe, and then gradually decreases to zero. Figure 3(b) shows measured A1 and A2 as a function of the electric field amplitude e for f0 ¼ 76 kHz at H ¼ 14 Oe. It is seen that the amplitude A1 increases linearly, A1  e, for up to e  120 V/cm, and then tends to saturation. The amplitude A2 increases quadratically with e, A2  e2 for fields up to 120 V/cm, and then its growth slows down. The figure also shows the best linear and quadratic fits to the data, with A1 ¼ 0.03e and A2 ¼ 0.0001e2. Thus, the e dependences of the amplitudes of the harmonics agree with the theory is discussed next. In order to describe the nonlinear CME effects, we use the approach developed in Ref. 16. First, we consider CME under bias field H and a weak DC field E and then we proceed to the excitation with e. The bilayer is assumed to be in the (1, 2) plane so that axis 1 is directed along the length and axis 3 is directed perpendicular to the sample plane. The thickness of the FM layer is am, and the thickness of the layer PE is ap, the width is w, and the length l  am, ap, w. The field H is along its length and E is perpendicular to the plane. Then the problem is reduced to a one-dimensional case. The deformation of the FM layer caused by H is described by k(H). We also assume that E is too small to consider any nonlinearity of CME. Then, the static

FIG. 3. Dependence of the amplitude A1 of the fundamental mode and the second harmonic A2 on (a) the bias magnetic field H for e ¼ 250 V/cm; and (b) the excitation field amplitude e for H ¼ 14 Oe and f ¼ 76 kHz. The circles are the data. The lines in (a) are guide to the eye and dashed lines in (b) are linear and quadratic fittings to the data.

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mechanical deformations S, mechanical stresses T, and H, B, and E are related by28 Sp1 ¼ sp11 Tp1 þ d31 E;

Sm1 ¼ sm 11 Tm1 þ kðHÞ;

B ¼ B0 ðHÞ þ bðH; Tm1 Þ:

(1)

p In Eq. (1), sm 11 ¼ 1=Ym and s11 ¼ 1=Yp are the material compliance coefficients and B0(H) is the magnetic induction at Tm1 ¼ 0. The nonlinearity of CME arises from the term b (H, Tm1), which describes the elasto-magnetic effect (inverse magnetostriction), i.e., change in b in the FM layer under mechanical stress. The condition for the continuity of deformations at the interface of the layers and the condition for the equilibrium of the structure along axis 1 have the form

Sm1 ¼ Sp1 ;

Tp1 ap  Tm1 am ¼ 0:

(2)

Solving Eqs. (1) and (2), we obtain the following for the mechanical stress in the FM layer Tm1 ¼

kðHÞ  d31 E : p sm 11 þ s11 ðam =ap Þ

(3)

It is seen from Eq. (3) that the stress of the FM layer contains terms caused by k(H) and E   p TH ¼ kðHÞ= sm and 11 þ s11 ðam =ap Þ m  p (4) TE ¼ d31 E= s11 þ s11 ðam =ap Þ : Now, we consider the application of AC field e (f). Expanding b(Tm1) in a Taylor series to the second order terms near TH, we obtain for B bð1Þ d31 e cosð2pftÞ þ sp11 ðam =ap Þ  2 bð2Þ d31 þ e2 cos 2 ð2pftÞ þ    : (5) p 2 sm þ s ða =a Þ p 11 11 m

Bm ¼ B0 ðHÞ þ bðH; TH Þ 

sm 11

2 jTH is the secHere, bð1Þ ¼ @b=@Tm1 jTH and bð2Þ ¼ @ 2 b=@Tm1 ond derivative. After transformations and rearrangement of the terms, we obtain from Eq. (5)

dependence of A1 and A2 for higher e-field is due to transfer of energy to higher harmonics. In order to establish the form of the function b(H, Tm1), the static elasto-magnetic characteristics of the bilayer were measured when it was stretched by an external force. The sample was placed inside a vertically positioned coil, 25 mm in length with a cross-section of S ¼ 16 mm2, containing N2 ¼ 1000 turns of a 0.18 mm diameter wire. A dc field H ¼ 0–100 Oe was applied parallel to the coil axis. The upper end of the sample was rigidly fixed, and a load weight of m ¼ 1–300 g was suspended to the free end. This allowed creating tensile stress Tm1 ¼ 0–2.6 MPa in the Metglas layer. The inductance L of the coil was measured using a RLCmeter with an accuracy of 0.01%. The inductance of empty coil was L0 ¼ 1.9 mH and it increased to L ¼ 16.5 mH when a sample was placed inside the coil. The B-value in the FM layer was determined from L as described below. The coil inductance L0 ¼ n2 V and for coil with the sample L ¼ l  ðR0 =RÞ  n2 V, where l is the differential magnetic permeability of the FM layer, R0/R is the coil filling factor, R0 is the cross-section area of the FM layer, R is the crosssection area of the coil, and n is the number of turns per unit length. Combining the two, we obtain ðH

ðH R LðH; Tm1 ÞdH: BðH; Tm1 Þ ¼ lðH; Tm1 ÞdH ¼ R0 L0 o

(7)

0

It follows from Eqs. (1) and (7), that by measuring the dependence of L on Tm1 for different H, one can find the function B(H, Tm1), and then calculate its derivatives of b with respect to Tm1. Figure 4(a) shows, as an example, the dependence of jB–B0j on the applied stress Tm1 for three values of H. Each field H corresponds to its own B0. The value of Tm1 is calculated taking into account the thicknesses and Young’s moduli of the layers of the structure. It can be seen that B changes approximately linearly with increasing stress and

d31 e cosð2pftÞ p sm þ s 11 11 ðam =ap Þ  2 1 d31 e2 þ bð2Þ m s11 þ sp11 ðam =ap Þ 4  2 bð2Þ d31 þ e2 cosð4pftÞ þ    : p 4 sm þ s ða =a 0 m p 11 11 (6)

B ¼ B0 ðHÞ þ bðH; TH Þ þ bð1Þ

The first two terms in Eq. (6) describe the DC terms, the third term describes the linear CME effect, and the last term describes the nonlinear CME. The model thus well describes the linear dependence of the 1-st harmonic amplitude and quadratic dependence of the 2-d harmonic amplitude on the amplitude, in agreement with the results in Fig. 3(b) that shows the linear and nonlinear dependences for e < 120 V/cm. For large fields, the expansion of magnetic induction in a Taylor series is no longer applicable and the weak

FIG. 4. (a) Dependence of the change B-B0 in magnetic induction in the Metglas layer on the applied stress Tm1 at various H. B0 is the induction at Tm1 ¼ 0. Dashed lines are linear fittings to the data. (b) The derivative b(1) estimated from the magnetic induction B and the stress Tm1 for the ferromagnetic layer as a function of H. Values of b(1) were obtained from data as in Fig. 4(a).

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the slope which is the derivative b(1) ¼ @B/@Tm1 is small in weak fields H ¼ 0.6 Oe, then increases with H, and reaches a maximum at H ¼ 10.6 Oe. This is followed by a decrease in b(1) with increasing field to H ¼ 34.5 Oe and finally tends to zero value when the FM layer is saturated. Figure 4(b) shows the dependence of the derivative b(1) on H and resembles the dependence of A1(H) in Fig. 3(a). The maximum value of the derivative b(1)  670 G/MPa. Hence it follows that the elasto-magnetic characteristic of the FM layer actually determines the amplitude of the first harmonic and the form of H dependence of the CME. It was experimentally verified that the field H1 corresponding to the maximum of the converse linear ME effect, within the accuracy of the measurement, coincided with the field Hm  11 Oe corresponding to the maximum of the direct ME effect. Next, we estimate the linear and nonlinear CME coeffið1Þ ð2Þ cients, aB ¼ dB1 =e and aB ¼ dB2 =e2 , where dB1 and dB2 are the amplitudes of variation in magnetic induction for the 1st and 2nd harmonics, respectively. We find the values of dB using the Faraday’s law of electromagnetic induction A ¼ N1  dB  R0  2pf0 . Here, A is the amplitude of the harmonic generated by the measuring coil. For area R0, we take the cross section of the FM film of 0.14 mm2, since the induction of the field varies only within the film. For parameters corresponding to the data of Fig. 3(b) and e ¼ 100 V/cm, we obtain dB1 ¼ 550 G and dB2 ¼ 187 G. The coefficients of ð1Þ ð2Þ CME are aB ¼ 5.5 G cm/V and aB ¼ 1.9  102 G cm2/ 2 V , respectively. The CME coefficients for the Metglas-PZT bilayer are significantly higher than 0.27 G cm/V for PZT-Ni,19 but smaller than reported value of 30 G cm/V for the MetglasPMN-PT structure.22 Our coefficient for the Metglas-PZT structure is 4 orders of magnitude higher than the CME coefficient of 3  106 G cm2/V2 for the Metglas-electrostrictor structure, where the second harmonic is generated due to the nonlinearity of the electrostriction in the PE layer.29 The coefficient for the linear CME can be estimated using Eq. (7) and experimentally found values b(1)  670 G/MPa and Q ¼ 15 and one obtains dB  750 G and dB/e  7.5 G cm/V. Thus, the calculated efficiency of the linear converse ME effect is in satisfactory agreement with the measured value of 5.5 G cm/V. Finally, it is worth mentioning that the model discussed here did not consider the influence of the epoxy layer thickness on the CME effects. Although it is only 2 lm in thickness, it will weaken the strength of CME due to less than perfect strain transfer at the interface. In conclusion, the generation of the first and second harmonics is studied for the converse ME effect in a ferromagnetic alloy-PZT composite with an ac electric field at the acoustic resonance frequency. It is shown that the nonlinearity in CME arises from the nonlinear dependence of the magnetic induction of the FM layer of the structure on its mechanical stress. A theory is developed that qualitatively describes the dependence of the harmonic amplitudes on e and on the dc bias field H. The nonlinearity of the

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converse ME effect should be taken into account when developing sensors and signal processing devices based on ME composites. The work at MIREA was supported by the Russian Science Foundation, Project No. 17-12-01435. Efforts at Oakland University were supported by Grants for the DARPAMATRIX program and the NSF (No. DMR-1808892). 1

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