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Inequivalence of direct and converse magnetoelectric coupling at electromechanical resonance Gaojian Wu, Tianxiang Nan, Ru Zhang, Ning Zhang, Shandong Li et al. Citation: Appl. Phys. Lett. 103, 182905 (2013); doi: 10.1063/1.4827875 View online: http://dx.doi.org/10.1063/1.4827875 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v103/i18 Published by the AIP Publishing LLC. Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors APPLIED PHYSICS LETTERS 103, 182905 (2013) Inequivalence of direct and converse magnetoelectric coupling at electromechanical resonance Gaojian Wu,1,2,3,a) Tianxiang Nan,2,a) Ru Zhang,1,3 Ning Zhang,3 Shandong Li,4 and Nian X. Sun2,b) 1 Department of Applied Physics, Nanjing University of Technology, Nanjing 210009, People’s Republic of China 2 Electrical and Computer Engineering Department, Northeastern University, Boston, Massachusetts 02115, USA 3 Magnetoelectronic Lab, Nanjing Normal University, Nanjing 210097, People’s Republic of China 4 College of Physics Science, Qingdao University, Qingdao 266071, People’s Republic of China (Received 28 August 2013; accepted 13 October 2013; published online 30 October 2013) Resonant direct and converse magnetoelectric (ME) effects have been investigated experimentally and theoretically in FeGa/PZT/FeGa sandwich laminate composites under the same electric and magnetic bias conditions. Resonant direct ME effect (DME) occurs at antiresonance frequency while resonant converse ME effect (CME) occurs at resonance frequency. The antiresonance and resonance frequencies have close but different values under identical bias conditions. The magnitudes of resonant effective ME coefficients for direct and converse ME effects are also not equal. A model was developed to describe the frequency response of DME and CME in laminate C 2013 AIP Publishing LLC. composite, which was in good agreement with experimental results. V [http://dx.doi.org/10.1063/1.4827875] Strong magnetoelectric (ME) coupling can be achieved in multiferroic materials, which can be categorized as direct ME effect (DME, a change of polarization in an external magnetic field) and converse ME effect (CME, a change of magnetization in an external electric field).1 An ever increasing amount of interest has been devoted to investigations on DME and CME effects because of their fundamental science and potential applications in devices, such as sensors, transducers, and actuators.1–7 Compared to single phase multiferroic materials, multiferroic laminates composed of piezoelectric and magnetostrictive phases have shown much stronger ME effect due to product property.4–11 A sharp increase in the magnitude of ME coupling coefficient has been observed at the electromechanical resonance (EMR) for both DME and CME effects, which provides great opportunities for applications.12–16 However, the relationship between DME and CME, especially at the EMR, is controversial and confusing. The first serious issue is about the resonance peak position of DME and CME. The results in Refs. 15–19 demonstrated that the resonance peak for DME and CME appears at the same EMR frequency while other researchers reported that resonant DME and CME effects were observed at two very close but significantly different frequencies, i.e., antiresonance frequency (fa ) for DME and resonance frequency (fr ) for CME, respectively.13,20–24 Moreover, the origin of the discrepancy of antiresonance and resonance frequencies is not very clear. Cho et al. explained the difference between the two resonance frequencies by combining piezoelectric constitutive equations with resonance boundary conditions, a) G. Wu and T. Nan contributed equally to this work. Author to whom correspondence should be addressed. Electronic mail: [email protected]. b) 0003-6951/2013/103(18)/182905/5/$30.00 believing that fr is obtained when impedance z of piezoelectric phase is minimized while fa is obtained when z is maximized.13 On the theory side, Filippov et al.20,21 and Bichurin et al.22 derived the different resonance conditions for DME and CME effects using the same constitutive equations of the materials. Another issue is on the magnitude of resonant DME and CME coefficients. Usually, DME and CME coefficients are defined as aE ¼ dE=dH and aB ¼ dB=dE, respectively, where E, H, and B are electric field, magnetic field, and magnetic induction, respectively. It is evident that these two coefficients are not equivalent according to unit dimension analysis. In order to compare the magnitude between DME and CME, effective DME and CME coefficients defined as ad ¼ dp=dH and ac ¼ l0 dm=dE, respectively, were introduced, where p and m are effective electrical and magnetic dipole moments of the entire system under equivalent conditions of electric and magnetic bias, respectively. Consequently, the equivalence between DME and CME effects in single phase, as well as in laminate composite at low frequency, was verified theoretically and experimentally.23,25,26 It is of physical and technical interest to investigate if this equivalence still exists at the EMR. In this work, we systematically measured the frequency response of DME and CME effect in FeGa/PZT/FeGa sandwich at the same bias conditions of different dc electric and magnetic fields. The experimental results confirm that antiresonance frequency for DME and resonance frequency for CME have close but different values, and the resonant DME and CME are not equivalent in magnitude, demonstrating the inequivalence of resonant DME and CME effects. At the same time, antiresonance and resonance frequencies and resonant DME and CME coefficients show similar variation tendency with dc bias magnetic or electric fields, indicating 103, 182905-1 C 2013 AIP Publishing LLC V 182905-2 Wu et al. Appl. Phys. Lett. 103, 182905 (2013) the consistency between resonant DME and CME effects. In addition, based on analyzing the transfer relationship between stress and strain, as well as electric field and magnetic field, new expressions for frequency response of ME effects have been derived by adopting different sets of constitutive equations of the materials for DME and CME effects. The theoretical results are in good agreement with the experimental results. The ME laminate composite used for measurement consists of a thickness direction polarized 25 mm 4 mm 0.279 mm lead zirconate titanate (PZT) (PSI-5H4E, from PIEZO Systems, Inc.) slab whose faces are sandwiched between two polycrystalline 0.290 mm-thick layers of magnetostrictive Fe82Ga18 (Galfenol, from ETREMA Products, Inc.). Fe82Ga18 was selected as the ferromagnetic phase due to its large magnetostriction up to 400 ppm and relatively low optimum bias field.27 The much larger length than the width intensifies the dominating vibration mode along the length direction. An electromagnetic copper coil of 70 turns was wound around this structure to pick up variations in magnetic induction or apply an ac magnetic field. The dc bias magnetic field generated by an external coil was applied along the length direction. For DME effect measurement, a harmonic voltage from the lock-in amplifier (Zurich Instruments, UHF-DEV2031) with frequency f ¼ 1–100 kHz and magnitude U ¼ 100 mV was applied to the coil and excited an ac magnetic field. The dc bias voltage was applied on the sample along thickness direction using high voltage amplifier. The generated ac voltage between the electrodes of PZT was measured by the lock-in amplifier with input impedance of 1 MX. For CME measurement, the same harmonic voltage signal and dc bias electric voltage were applied to the electrodes of PZT simultaneously using a bias tee (Picosecond Pulse Labs 5530B). The variation of magnetic induction results in a voltage in the coil, which was measured by the lock-in amplifier. Consider a layered ME composite consisting of a piezoelectric plate sandwiched between two magnetostrictive plates with thickness t, width w, and length L. The piezoelectric phase is polarized along its thickness direction with electrodes on its top and bottom surfaces. The plate thickness and width are assumed to be much smaller than its length, i.e., t L, w L, so the stress components of T2 and T3 may be ignored, and only nonzero T1 is taken into account. For DME effect, the dc bias and ac magnetic fields are applied along the length direction. The induced strain in the magnetostrictive component due to piezomagnetic effect is transferred to piezoelectric component by elastic coupling. Then the generated stress results in polarization and electric field between the two electrodes in the piezoelectric phase. During this process, strain and electric displacement are independent variables, and stress and electric field are dependent variables, so the fourth set of constitutive equations are lef f ¼ adopted for the piezoelectric phase. Since the magnetic induction B in the magnetostrictive phase along length direction should satisfy the divergence-free condition, the third set of constitutive equations for the magnetostrictive phase are adopted p p p T 1 ¼ p cD 11 S1 h31 D3 ; (1) E3 ¼ p h31 p S1 þ D3 =eS33 ; (2) m S1 ¼ m sB11 m T 1 þ m g11 B1 ; (3) H1 ¼ m g11 m T 1 þ B1 =lT11 ; (4) where superscripts p and m denote piezoelectric and magnetostrictive phases, respectively; T1 is stress component; S1 is strain component; cD 11 is elastic coefficient under open-circuit condition; p h31 is piezoelectric coefficient defined as p h31 ¼ @ p T 1 =@D3 ; eS33 is permittivity under constant strain; E3 and D3 are electric field and electric displacement; m sB11 is compliance coefficient under opencircuit condition; m g11 is piezomagnetic coefficient defined as m g11 ¼ @ m S1 =@B1 ; and lT11 is permeability under constant stress. In the EMR region, the equation of motion for the medium has the following form:22 q @ p T ij @ m T ij @ 2 ui ¼ v þ ð1 vÞ ; @t2 @xj @xj (5) is where v is volume fraction of piezoelectric phase and q average density of the composite. Expressing the stress components in terms of strain from Eqs. (1) and (3), substituting these expressions into Eq. (5), and using the open-circuit condition D3 ¼ 0, we can get the differential equation for displacement ux . The solution to this equation can be obtained by taking into account the stressfree boundary condition at both ends. Then substituting the displacement expression into Eq. (2) and averaging the electric field over the sample length, the effective DME coefficient is obtained as ad ¼ 2vð1vÞp d 31 m d 11 lef f p3 k1 L ¼ Vtan ; T B D m p H3 k1 Ll11 ðv s11 þð1vÞ s11 Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðp svD þ m1v where k1 ¼ x q Þ1 , sB 11 m 11 p T d31 ¼ p h31 p sD 11 e33 , (6) m d 11 g11 lT11 , ¼ x is the angular frequency, V is volume of the composite, and lef f is effective permeability that can be found from Eq. (4). Expressing m T 1 from Eq. (3), substituting it into Eq. (4), and then averaging over the sample length, the effective permeability can be obtained as m B T k1 Lðvm sB11 þ ð1 vÞp sD 11 Þ s11 l11 : k L m sB þ lT m g2 Þ 2tan 1 lT ð1 vÞp sD m g 2 k1 Lðvm sB11 þ ð1 vÞp sD Þð 11 11 11 11 11 11 2 11 (7) 182905-3 Wu et al. Appl. Phys. Lett. 103, 182905 (2013) According to Eq. (6), the resonance condition for DME effect is found as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2n þ 1 1 v 1v þ m B ; (8) fa ¼ p sD 2L q s11 11 where a sharp increase in the DME coefficient is observed at the antiresonance frequency fa . In the case of CME effect, ac electric field is applied to the piezoelectric phase along thickness direction, and the dc bias magnetic field is applied along length direction of the laminate. The strain generated in the piezoelectric phase due to applied ac electric field by converse piezoelectric effect is transferred to magnetostrictive phase by elastic coupling, resulting in magnetic induction B by piezomagnetic effect. During this process, stress and electric field are independent variables, strain and electric displacement are dependent variables for piezoelectric phase, and so the first set of constitutive equations are adopted. For the magnetostrictive phase, stress and magnetic induction are dependent variables, and the second kind of constitutive equations are used p S1 ¼ p sE11 p T 1 þ p d 31 E3 ; (9) D3 ¼ p d31 p T 1 þ eT33 E3 ; (10) m m T 1 ¼ m cH 11 S1 e11 H1 ; (11) B1 ¼ m e11 m S1 þ lS11 H1 ; (12) m where p sE11 is compliance coefficient under short-circuit condition; p d31 is piezoelectric coefficient defined as p d31 ¼ @D3 =@ p T 1 ; eT33 is permittivity under constant stress; m cH 11 is compliance coefficient under short-circuit condition; m e11 is piezomagnetic coefficient defined as m e11 ¼ @H1 =@ m T 1 ; and lS11 is permeability under constant strain. Since the input impedance of the lock-in is much larger than the coil resistivity, the open-circuit condition H1 ¼ 0 is valid. Then with similar derivation as that for DME effect, the effective CME coefficient is obtained as m 2vð1 vÞp d31 m d 11 k2 L ; (13) ¼ ac ¼ p E Vtan E3 k2 Lðvm sH 2 þ ð1 vÞ s Þ 11 11 where m d11 ¼ m e11 m sH 11 , k2 ¼ x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðp svE þ 1v q . One can p sH Þ 11 11 find that the resonance condition for CME effect is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2n þ 1 1 v 1v þ p H ; fr ¼ p sE11 2L q s11 (14) where a sharp increase occurs at the resonance frequency fr . Using Eqs. (6) and (13), we can discuss the performance of DME and CME at low frequency and EMR. In the case of p E low frequency region, the difference between p sD 11 and s11 , m B m H s11 and s11 under different boundary conditions is negligikL ble, and tan kL 2 2 is satisfied, so Eqs. (6) and (13) can be reduced to ac or ad ¼ vð1 vÞp d 31 m d 11 V: p 11 þ ð1 vÞ s11 vm s (15) Equation (15) indicates that equivalence between DME and CME at low frequency can be expected and has been theoretically and experimentally proved.23,25 For the case at EMR, p E m B m H the difference between p sD 11 and s11 , s11 and s11 cannot be ignored, resulting in the prominent difference between fa and fr . Moreover, the magnitude of resonant effective DME and CME coefficients cannot be exactly equivalent in spite of the same order, as can be seen from Eqs. (6) and (13). First, we measured frequency dependence of DME and CME effect at increasing dc bias magnetic field and fixed dc bias voltage. Fig. 1 shows the bias magnetic field dependence of fa and fr and corresponding resonant ME coefficients at fixed increasing bias voltage 0, 100, 200, 300, and 400 V. As shown in Figs. 1(a) and 1(b), at each same bias condition, there is distinct difference between fa and fr . These two frequencies are very close, but fa is a little larger than fr , which is consistent with observations by other groups.13,24 In addition, it can be seen clearly from Figs. 1(c) and 1(d) that resonant effective ME coefficients ad and ac are also not equal in magnitude, as expected from Eqs. (6) and (13). This observation is not consistent with the results for single phase or twophase composites at low frequency.23,25,26 The measured ad and ac are calculated according to ad ¼ ac ¼ dp uout eVp ; ¼ dH1 tnI l0 dm tuout ¼ Vm ; 2pfNAuin dE where uout is induced output voltage in the piezoelectric phase or the search coil, n is the number of coil turns per meter, I is current in the coil due to applied voltage, e is permittivity of piezoelectric phase, f is applied ac electric field frequency, N is the number of coil turns, A is cross section area of the coil, uin is input voltage onto the piezoelectric phase, and Vp and Vm are the volume of piezoelectric phase and magnetostrictive phase, respectively. It should be pointed out that the electric field dependence of permittivity e of piezoelectric phase has been taken into account in the calculation. The relative permittivity with bias voltage was obtained by measuring capacitor versus voltage curve, which exhibits a butterfly characteristic, resulting from the ferroelectric property of PZT. Measured permittivity increases from 3800 to 4500 with increasing E field, then experiences an abrupt drop to 2954 around Uc 150 V, and keeps the low value until U ¼ 400 V. When the electric field is decreased back from this value, the permittivity continuously increases. Now we apply our model to the experimental results. Fig. 2 shows the comparison of calculated frequency response of ad and ac using Eqs. (6) and (13) to the experimental results for FeGa/PZT/FeGa sandwich composite under optimum bias magnetic field Hbias ¼ 180 Oe and zero bias electric field. The material parameters used for numerical calculation are listed as follows:28 p sE11 ¼ 1:61 1011 m2 =N, p d 31 ¼ 320 1012 C=N, p q ¼ 7800 kg=m3 , m m H s11 ¼ 1:65 1011 m2 =N, d 11 ¼ 10 e33 =e0 ¼ 3800, 9 m 3 10 m=A, q ¼ 7870 kg=m , l11 =l0 ¼ 70, and l ¼ 25mm. The circular frequency x is represented by complex quantity xð1 i=QÞ to take into account the energy loss, where Q is quality factor, determined from the line-width the 182905-4 Wu et al. Appl. Phys. Lett. 103, 182905 (2013) FIG. 1. Bias magnetic field H dependence of (a) antiresonance frequency fa ; (b) resonance frequency fr ; (c) resonant ad ; and (d) resonant ac at fixed increasing bias voltage 0, 100, 200, 300, 400 V. experimental curve. One observes good agreement between theory and data. The discrepancy between calculated and experimental resonance frequencies is mainly due to DE effect,29 namely, the shift of antiresonance and resonance frequencies with applied dc bias magnetic field, to be discussed in the following sections. The amplitude discrepancy for CME is attributed to the drawback of the experimental method,23 where the search coil is wound around the entire sample rather than magnetostrictive phase, resulting in lower measured magnetic induction than that actually induced in magnetostrictive phase. Even taking into account this factor, resonant DME and CME would not be expected to be equivalent in magnitude, as indicated in Eqs. (6) and (13). On the other hand, as can be seen from Figs. 1(a) and 1(b), fa and fr show similar variation tendency with bias magnetic field. With increasing bias magnetic field, both fa and fr decrease to a minimum value at around 180 Oe and then rise FIG. 2. Comparison of theoretical and experimental frequency response of effective DME and CME coefficients with Hbias ¼ 180 Oe and zero bias electric field. to a saturated value at bias magnetic field up to 400 Oe at each fixed bias voltage. It is interesting that resonant DME and CME exhibit a tendency with bias magnetic field in opposition to fa and fr , as shown in Figs. 1(c) and 1(d). The change of fa and fr , as well as opposite change of resonant ME effects with increasing bias magnetic field, can be explained in terms of motion of domain walls.30 That is, with bias magnetic field increasing to optimum value, deformation contribution from non-180 domain wall motion reaches maximum, leading to maximized compliance and magnetostrictive strain and, consequently, lowest fa and fr and strongest resonant ME coefficients. Further increasing bias magnetic field the motion of non-180 domain-wall will be suppressed due to the interaction with applied magnetic field, resulting in a decrease in compliance and magnetostrictive strain. Accordingly fa and fr increase, and resonant DME and CME decrease. The similar variation tendency of fa and fr as well as resonant ME effects indicate that bias magnetic field affects DME and CME at EMR in the same manner. As indicated in Fig. 1, bias electric field also has significant influence on resonant DME and CME. In order to further investigate the influence of bias voltage on resonant DME and CME, the frequency responses of DME and CME were measured with bias voltage variation loop at a fixed bias magnetic field 120 Oe. As shown in Figs. 3(a) and 3(c), the bias voltage dependence of fa and fr shows similar standard “butterfly” curve, which is attributed to the widely observed butterfly curves of piezoelectric strain and electric field due to ferroelectric behavior of PZT.31 Figs. 3(b) and 3(d) present the corresponding resonant ME coefficients variation with bias voltage, which also exhibits similar butterfly variation tendency. It is worth noting that resonant ME effect also shows nearly opposite variation tendency with bias voltage to resonance frequency, similar with the case for bias 182905-5 Wu et al. Appl. Phys. Lett. 103, 182905 (2013) FIG. 3. Bias voltage dependence of: (a) fa ; (b) fr ; (c) resonant ad ; and (d) resonant ac at fixed bias magnetic field of 120 Oe. magnetic field. In Figs. 3(a) and 3(c), the lowest resonant frequencies occur at 150 V corresponding to the electric coercive field which is due to variation of the compliance coefficient of PZT with bias voltage while in Figs. 3(b) and 3(d) the maximum ME coefficients occur at 50 V which is possibly due to the combined effect of compliance coefficient, piezoelectric coefficient, and permittivity of PZT with bias voltage, as indicated in Eqs. (6) and (13). The similar variation of fa and fr as well as resonant DME and CME coefficients indicate that the bias electric field affects DME and CME in the same manner. For a laminated composite, we have measured the frequency response of DME and CME coefficients at the same bias magnetic and voltage conditions. The antiresonance frequency for DME and resonance frequency for CME have very close but distinguishable values, showing similar variation tendency with bias magnetic and electric fields. 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