Funct Materials 2010

Converse magnetoelectricity in asymmetric magnetoelectric structures G.S. Radchenko, M.G. Radchenko* Institute of Physics, Southern Federal University, 194 Stachki Ave., 344090 Rostov-on-Don, Russia Pedagogical Institute, Southern Federal University, 33 Bolshaya Sadovaya St., 344022 Rostov-on-Don, Russia * Rostov Branch, Moscow State Technical University of Civil Aviation, 262b Sholokhova St., 344009 Rostov-on-Don, Russia Received November 30, 2010 A new computational approach is developed to analyze the asymmetric magnetoelectric structures. The possibility to enhance considerably the composite main effective parameters is found. The acoustic and electromagnetic vibrations in such structures are described comprehensively. Развит новый вычислительный подход к анализу асимметричных магнитоэлектрических структур. Обнаружена возможность значительного усиления основных эффективных параметров композита. Приведено полное описание акустических и электромагнитных колебаний в таких структурах. 1. Introduction The rapid development of modern electronics requires novel materials with useful properties. The promising magnetoelectric (ME) heterogeneous materials possessing converse ME effect (CMEF) are among materials of promise for practical applications. CMEF is the magnetization of a sample under external electric ield. This offers an excellent potential for coil-free electromagnets free of eddy-current existence, devices for sensitivity measurements and for development of ME memory devices and sensors. In literature, the CMEF is a rather new direction and there are some publications aimed at that matter [1-10]. The structures considered in the present work and in [1, 3-5, 7, 8, 10, 11] are layered heterogeneous structures containing combination of highly magnetostrictive and piezoactive substances. The applied external electric ield creates mechanical straining of the piezoelectric plate and then this strain is transferred to magnetic phase, causing a magnetic lux therein due to the magnetostriction. Then the created lux is registered by a testing coil [1, 3, 4, 7, 8, 10], two Helmholtz coils [5], one coil above the vibrating magnet [9], or vibrating sample magnetometer in heterostructures [2]. In the region where the applied electric ield is of resonance frequency, the response is increased drastically and this provides a way for CMEF to be practically essential. However, almost all these publications are aimed at the symmetric strictures, there piezo- and magnetostrictive layers are of similar length. However, in reality, the lengths may be very different, and no attempts to analyze this difference theoretically were made prior to this Functional Materials, 17, 3, 2010 371 G.S. Radchenko, M.G. Radchenko / Converse magnetoelectricity in asymmetric ... work. The converse ME susceptibility is poorly analyzed in literature, too. This has stimulated the present research. 2. Theoretical analysis and results Let us consider an asymmetric layered structure with different lengths of the components. The basic equations [11] for strain tensor component S1 and electric and magnetic induction vectors component D3 and B1 as functions of the x coordinate (X1 direction) can be written as follows (1). p p p * S1p = s11 T1 ( x ) + d31 E3 p p p * D3p ( x ) = d31 T1 ( x ) + e33 E3 (1) m m m bias B1m ( x ) = q11 T1 ( x ) + m11 H1 m m m bias S1m ( x ) = s11 T1 ( x ) + q11 H1 . Here s11p, d31p and ε33p are the elastic compliance, piezoelectric coeficient, and dielectric permittivity of piezoelectric, respectivegly; s11m, q11m and μ33m are the elastic compliance, magnetostrictive coeficient and magnetic permeability of ferrite, respectively. H1bias is the applied bias magnetic ield; E3*, the electric ield equal to the applied one. The mechanical coupling between the phases is supposed to be ideal. Using the basic equation governing acoustic oscillations [12] and boundary conditions [11, 12], we get the expression for the strain vector (2) ux ( x ) = p p m p * m m p m bias d31 h s11 L E3 + q11 h s11 L H1 sin (kx ). æ æ kLp ö÷ æ kLm ö÷ö÷ çç p p m ç ç p m m ÷÷ + h L s cos ç ÷÷ k çh L s11 cos çç çç 2 ÷÷÷÷÷÷ 11 ççè çè 2 ÷÷ø è øø (2) The expression (2) is derived for the irst time and deines completely the strain of the platelike sample due to converse ME interaction and different lengths of the plates. It takes into account the inertia forces and back moving forces of both the phases. Then we place the obtained result to the third equation of the set (1). The arising magnetic induction we obtain by averaging it. The expression for the effective ME susceptibility, taking the above-mentioned average into account, is given by (3). * a13 æ Lm ö÷ p p m p m ÷÷ d31 h s11 L q11 sin çççk çè 2 ÷÷ø . =2 æ æ kLm ö÷ö÷ æ kLp ö÷ ç ç p m m m çç p p m ÷ ÷ ÷ kL çh L s11 cos çç ÷÷ ÷ + h L s11 cos çç ççè çè 2 ÷÷øø÷÷ çè 2 ÷ø÷ (3) Fig. 1. Theoretical (3) frequency dependence of the L-T real and imaginary parts of ME susceptibility of PZT-Terfenol compound. The parameters for theory are: s11p = 15,3·10–12 m2/N, d31p = –175 pC/N; s11m = 45,4·10–12 m2/N; q11m = 4500 м/А; m m11 = 2, 2 × 10-6 Tm/A; Lm = 14 mm, Lp = 16 mm, m h = 1,2 mm, hp = 2 mm, c = 8000 rad/s, ρm = 9200 kg/m. 372 Functional Materials, 17, 3, 2010 G.S. Radchenko, M.G. Radchenko / Converse magnetoelectricity in asymmetric ... Fig. 2. Theoretical (4) frequency dependence of the real and imaginary parts of effective dielectric permittivity ε33* of PZT-Terfenol compound. The parameters for theory are the same as in Fig. 1. Figure 3. Theoretical (5) frequency dependence of the transversal real and imaginary parts of piezoelectric coeficient d31* of PZT-Terfenol compound. The parameters for theory are the same as in Fig. 1. The lengths of the plates being taken to be the same, our formula turns into (6) from [4] and the model is analogous to [11]. To obtain the longitudinal coeficient α33*, we have to substitute q31m for q11m. It is usually much smaller than transversal due to the inluence of demagnetization ields created by the metal electrode surface currents. The attenuation can be regarded as a complex angular frequency ω = 2πf+iχ. We can also determine the effective dielectric response (4) which has another resonance frequency. æ Lp ö÷ 2 ççk p p m 2 ÷ sin d h s p 11 çç 2 ÷÷÷ 31 æ L/2 ö÷ d (4) ç 31 è ø 1 p * ÷÷ / ¶E * = e p D x dx + 2 . e33 = ¶ ççç ( ) 3 ò 3 33 p ÷÷÷ æ æ m öö æ kLp ö÷ çè L -L / 2 s11 ø ç p çç p p m ÷÷ + hm Lm s p cos çç kL ÷÷÷÷÷÷ ks11 ççh L s11 cos ççç çç 2 ÷÷÷÷ 11 ÷ çè è øø è 2 ÷ø ( ) ( ) The dielectric response ε33* consists of frequency-independent part (rigidly ixed sample) and resonance dynamic part, which at f→0 transforms ε33* into the low-frequency response at constant stress. It is impossible to divide the piezoelectric and magnetoelectric contributions to the strain (both phases vibrate in the same manner). Thus, to calculate the effective converse piezoresponse, we need to consider the strains in both the phases according to the average from the total strain. The effective piezoelectric coeficient is determined by formula (5) æ æ p öö æ Lm ö÷ p p m çç p ÷÷ + Lm sin ççk L ÷÷÷÷÷÷ d31 h s11 çL sin çççk (5) çç 2 ÷÷÷÷ ççè çè 2 ÷÷ø è øø * d31 = 2 . æ æ kLp ö÷ æ kLm ö÷ö÷ ç ç p m çç p p m m m ÷÷ + h L s cos ç ÷÷ kL çh L s11 cos çç çç 2 ÷÷÷÷÷÷ 11 ççè çè 2 ÷÷ø è øø The resonance frequencies are determined by assuming the denominators in (3-5) to zero and are dependent on the elastic properties of both the phases and their volume densities and all the dimensions. If the oscillations of the external electric ield are at the resonance frequencies, the sharp increase of induced magnetic induction (3), dynamic polarization (4), and mechanical vibrations (5) will occur in the sample. The physical nature of resonance frequency independence of other integral composite parameters in this work and in [4], in contrast to [11], consists in what follows. The magnetic ield and magnetic induction play opposite parts in the electromagnetic interactions in comparison with electric ield and induction. This results, for example, in the absence of the average induced magnetic Functional Materials, 17, 3, 2010 373 G.S. Radchenko, M.G. Radchenko / Converse magnetoelectricity in asymmetric ... ield H inside the sample due to the converse effect. The ME voltage coeficient is determined in [11] under open electric chain conditions (average D is equal to zero) as E/H comprises ields which play physically opposite roles. Therefore, αE depends upon most of the integral parameters. The attenuation can be regarded as a complex angular frequency ω = ω’+iχ. Here χ is the attenuation parameter. So we also take into account the attenuation as it was done in [11] for direct effect and in [4] for converse one. The resonant enhancements of the most important physical constants are shown in Figs. 1-3. All the considered effective constants show a large resonance enhancement (both in real and imaginary parts). At low frequencies, the ME and other constants are much lower and practically independent of the frequency. 3. Conclusion So, in this work we have investigated and analyzed the converse magnetoelectric effect within the frames of resonance theory. The considerable resonance enhancement of basic effective parameter in plate-like ME composites is shown. These results can help in designing of ME devices, magnetostrictive and piezoelectric transducers, actuators, and other technical devices. References 1. Y. M. Jia, S. W. Or, H. L. W. Chan et al., Appl. Phys. Lett. ,88, 242902 (2006). 2. W. Eerenstein, M. Wiora, J. L. Prieto et al., Nature Mater. Lett., 6, 348 (2007). 3. C. Popov, H. Chang, P. M. Record et al., J. Electroceram, 20, 53 (2008). 4. Y. K. Fetisov, V. M. Petrov, G. Srinivasan, J. Mater. Res., 22, 2074 (2007). 5. J. P. Zhou, W. Zhao, Y.-Y. Guo et al., J. Appl. 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