Download Inequivalence of direct and converse magnetoelectric coupling at electromechanical resonance

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
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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
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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)
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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
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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
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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. The corresponding resonant DME and CME coefficients show similar
variation tendency with bias magnetic field and voltage but
have inequivalent magnitude under the same bias magnetic and
electric conditions. A theoretical model is developed to explain
the inequivalence and consistency between resonant DME and
CME, showing good agreement. This work is of significance
for better understanding the relationship between DME and
CME by demonstrating the inequivalence and consistency of
resonant DME and CME effect in laminate composites.
This work was financially supported by AFRL through
UES FA8650-090-D-5037 and Semiconductor Research
Corporation, National Natural Science Foundation of China
(NSFC) 51328203 and 51132001.
1
W. Eerenstein, M. Wioral, J. L. Prieto, J. F. Scott, and N. D. Mathur,
Nature Mater. 6, 348 (2007).
2
N. A. Spaldin and M. Fiebig, Science 309, 391 (2005).
3
W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature (London) 442, 759 (2006).
4
C. W. Nan, M. I. Bichurin, S. X. Dong, D. Viehland, and G. Srinivasan,
J. Appl. Phys. 103, 031101 (2008).
5
M. Liu, S. D. Li, O. Obi, J. Lou, S. Rand, and N. X. Sun, Appl. Phys. Lett.
98, 222509 (2011).
6
M. Liu, Z. Y. Zhou, T. X. Nan, B. M. Howe, G. J. Brown, and N. X. Sun,
Adv. Mater. 25, 1435 (2013).
7
T. X. Nan, Y. Hui, M. Rinaldi, and N. X. Sun, Sci. Rep. 3, 1985 (2013).
8
J. Ma, J. M. Hu, Z. Li, and C. W. Nan, Adv. Mater. 23, 1062–1087 (2011).
9
S. Priya, R. Islam, S. Dong, and D. Viehland, J. Electroceram. 19, 149 (2007).
10
R. Islam, Y. Ni, A. Khachaturyan, and S. Priya, J. Appl. Phys. 104,
044103 (2008).
11
S. C. Yang, C. S. Park, K. H. Cho, and S. Priya, J. Appl. Phys. 108,
093706 (2010).
12
M. I. Bichurin, D. A. Filippov, V. M. Petrov, V. M. Laletsin, N.
Paddubnaya, and G. Srinivasan, Phys. Rev. B 68, 132408 (2003).
13
K. H. Cho and S. Priya, Appl. Phys. Lett. 98, 232904 (2011).
14
S. Y. Chen, Q. Y. Ye, W. Miao, D. H. Wang, J. G. Wan, J. M. Liu, Y. W.
Du, Z. G. Huang, and S. Q. Zhou, J. Appl. Phys. 107, 09D901 (2010).
15
Y. K. Fetisov, V. M. Petrov, and G. Srinivasan, J. Mater. Res. 22, 2074
(2007).
16
C. Popov, H. Chang, P. M. Record, E. Abraham, R. W. Whatmore, and Z.
Huang, J. Electroceram. 20, 53 (2008).
17
J. G. Wan, J. M. Liu, G. H. Wang, and C. W. Nan, Appl. Phys. Lett. 88,
182502 (2006).
18
Y. Wang, F. Wang, S. W. Or, H. L. W. Chan, X. Zhao, and H. Luo, Appl.
Phys. Lett. 93, 113503 (2008).
19
J. P. Zhou, L. Meng, Z. H. Xia, P. Liu, and G. Liu, Appl. Phys. Lett. 92,
062903 (2008).
20
D. A. Filippov, T. A. Galkina, and G. Srinivasan, Tech. Phys. Lett. 36, 984
(2010).
21
D. A. Filippov, T. A. Galkina, V. M. Laletin, and G. Srinivasan, Phys.
Solid State 53, 1832 (2011).
22
M. I. Bichurin, V. M. Petrov, R. M. Petrov, and S. Priya, Solid State
Phenom. 189, 129 (2012).
23
T. Wu, C. M. Chang, T. K. Chung, and G. Carman, IEEE Trans. Magn.
45(10), 4333 (2009).
24
Z. Wu, Z. H. Xiang, Y. M. Jia, Y. H. Zhang, and H. S. Luo, J. Appl. Phys.
112, 106102 (2012).
25
J. Lou, G. N. Pellegrini, M. Liu, N. D. Mathur, and N. X. Sun, Appl. Phys.
Lett. 100, 102907 (2012).
26
M. Fiebig, J. Phys. D: Appl. Phys. 38, R123 (2005).
27
A. E. Clark, K. B. Hathaway, M. Wun-Fogle, J. B. Restorff, T. A. Lograsso, V.
M. Keppens, G. Petculescu, and R. A. Taylor, J. Appl. Phys. 93, 8621 (2003).
28
The material parameters for PZT were provided by PIEZO Systems, Inc.;
the material parameters for galfenol were from correspondence with
Etrema, Inc. via email.
29
B. Lewis and R. Street, Proc. Phys. Soc. London 72, 604 (1958).
30
S. W. Or, T. Li, and H. L. W. Chan, J. Appl. Phys. 97, 10M308 (2005).
31
S. E. Park and T. R. Shrout, J. Appl. Phys. 82, 1804 (1997).