Supplementary Material
Magnetoelectric effect in shear-mode Pb(Zr,Ti)O3/NdFeB composite
cantilever
Jinchi Han,1 Jun Hu,1,a) Zhongxu Wang,1 Shan X. Wang,1,2 and Jinliang He1
1
The State Key Lab of Power System, Department of Electrical Engineering,
Tsinghua University, Beijing, 100084, China
2
Center for Magnetic Nanotechnology, Stanford University, 450 Serra Mall,
Stanford, CA 94305, USA
Derivation of the quasi-static model
The shear-mode response of individual piezoelectric unimorph is shown in Fig. S1.


Fig. S1. Torsional vibration of individual piezoelectric unimorph.
The shear-mode piezoelectric constitutive equation is given as
D1  d15t5  11E1  d15

s55E p
 11E1
(1)
where D1, E1 are the electric displacement, electric field along 1-direction,
respectively, t5 is the shear stress, α is the shear strain, and d15, 11 , s55E p are the shear
piezoelectric coefficient, dielectric constant, elastic compliant coefficient of the
1
piezoelectric layer.
The torque Tm generated from the interaction of magnetic moment of permanent
magnet and the external AC magnetic field is given by
Tm  m  Be  MVBe sin 
(2)
where m, M, V are the magnetic moment, magnetization, and volume of the
permanent magnet, respectively, Be is the flux density of the external magnetic field,
and θ is the angle between the orientations of magnetization of the permanent magnet
and the external magnetic field.
The angular displacement β resulting from the induced torque Tm equals to

Tm L
D
(3)
where L is the active length of piezoelectric cantilever, and D is the torsional rigidity
of the overall shear-mode cantilever. The torsional rigidity can be derived based on
the torsional rigidity of shear-mode plate, i.e., D0  G0 h0 w03 3 , where G0, w0, h0 are the
shear modulus, width and thickness of the plate. According to the proposed
shear-mode cantilever, the torsional rigidity is expressed as
D
(2Gn hn  G p hp )((2b  2d )3  8d 3 )
3

8G p bhp (b 2  3bd  3d 2 )
3
(4)
where b is the width of individual piezo-unimorph, 2d is the distance between the two
piezo-unimorphs, Gp, hp are the shear modulus, thickness of the piezoelectric layer,
respectively, Gn and hn are the shear modulus, thickness of the nylon plastic layer,
respectively.
Therefore, the angular displacement of the overall cantilever is calculated as
2

3Tm L
8Gpbhp (b2  3bd  3d 2 )
(5)
The overall cantilever can be seen as the integration of many shear-mode cantilevers
each based on two symmetrical unimorphs seperated with a distance 2dn and each
unimorh has dimensions of L×d(dn)×(hp+2hn). These cantilever components have the
same angular displacement but different shear strains, and thus different induced
electric-field strengths. Substituting Eq. (3), relationship between the anuglar
displacement β and shear strain α(dn), i.e.,  (d n  d(d n ))   (d n )l , and the
open-circuit boundary condition D1=0, the induced electric field of a cantilever
element (two piezo-unimorphs spaced 2dn) is
E1 (dn )  
d15 (dn )
d  (d n  d(d n ))
3d15Tm (d n  d(d n ))
  15

E
E
11s55 p
11s55 p L
811bhp (b2  3bd  3d 2 )
(5)
The charge output from the cantilever element is
dQ(dn )  E1 (d n )hp dC (d n )  
3d15Tm dn L
d(d n )
8bhp (b2  3bd  3d 2 )
(6)
Therefore, the overall charge induced by the whole ME cantilever is integrated as
Q
bd
d
dQ(d n )  
3d15Tm L(b  2d )
16hp (b2  3bd  3d 2 )
(7)
The average quasi-static ME voltage of individual piezo-unimorph is
U ME ( static ) 
3d15Tm (b  2d )
Q

C
1611b(b 2  3bd  3d 2 )
(8)
Derivation of the dynamic model
Under static or quasi-static condition, the torque generated by the interaction of
magnet and static external magnetic field is constant, and the shear stress distribution
in different cross-section is therefore consistent. However, it is not the case under
3
dynamic torque excitation. Regarding a shear-mode ME cantilever element with a
length of dz, a difference dT exists between the torques applied on the two surfaces,
the dynamic equation of torsional vibration is given by
 2  ( z, t )
 I  2  ( z, t )
1  2  ( z, t )




t 2
D
t 2
c 2 t 2
(9)
where β(z,t) is the angular displacement, ρ is the equivalent mass density of the
cantilever, c is the wave velocity of torsional vibration, expressed as
c  D I
(10)
where I is the torsional moment of inertia, approxinmately calculated as
I
2b(b 2  3bd  3d 2 )(hp  2hn )
3
(11)
Therefore, the wave velocity of torsional vibration equals
c
4G p hp (  p hp   n hn )
(12)
where ρn is the mass density of nylon layer and Gp, ρp are shear modulus and mass
density of the piezoelectric layer, respectively. It indicates the velocity c is only
influenced by the mass fraction of the piezoelectric layer in the unimorph.
The general solution of Eq. (9) is


 ( z, t )   Af cos(

c
z )  B f sin(


z )  eit
c 
(13)
Considering the boundary condition at the clamped end, the coefficient Af equals zero.
According to the free end, the boundary condition is expressed as
Tm  T ( z, t ) z  L  I m0
 2  ( z, t )
t 2 z  L
(14)
Equivalent as,
4
Tm(peak)eit  D
 ( z, t )
 2  ( z, t )
 I m0
z z  L
t 2 z  L
(15)
where I m0 is the rotational inertia of the tip mass, calculated as
I m0   r 2 dm 
V
 ml
hw(h 2  w2 )
12
(16)
where ρm, l, w, h are the mass density, length, width, and height of the magnet,
respectively.
Combining Eqs. (13), (15) and (16), the coefficient Bf can be calculated as
Bf 
Tm (peak)
(17)
 l



 m hw(h 2  w2 )sin( L)  D cos( L)
12
c
c
c
2
Therefore, the torsional angular is
 ( z, t ) 

Tm (peak) sin(

2
 ml
12
hw(h  w ) sin(
2
2
12Tm (peak)  IL
c

c
2

z)
L)  D


c
cos(

c
eit
L)
sin(



c
z)
12  IL
 mlhw(h  w ) D cos( L)  L tan( L) 
c
c
c
c
 mlhw(h 2  w2 )
2


eit
(18)
The resonance frequency corresponds to the solution of the Eq. (19).



12  IL
L tan( L) 
0
c
c
 mlhw(h 2  w2 )
(19)
The induced electric field in a cantilever element based on two piezo-unimorphs
(spaced 2dn) each with dimensions of L×(hp+2hn)×d(dn) is expressed as
5
E (d n , z, t )  
d15 [  ( z  dz , t )   ( z , t )](d n  d(d n ))
11s55E dz
d15Tm (peak) (d n  d(d n ))eit



[sin( ( z  dz ))  sin( z )]
 L



c
c
11s55E dz[ 2 m hw(h 2  w2 ) sin( l )  D cos( l )]
12
c
c
c
it
d15Tm (peak) (d n  d(d n ))e



 cos( z )
 L



c
11s55E [ 2 m hw(h 2  w2 ) sin( l )  D cos( l )] c
12
c
c
c
(20)
The induced charge is therefore calculated as
dQ(t )  U (d n , z, t )C (dS )  E (d n , z , t )hpC (d n , z )


cos( z )eit
c
c

dzd(d n )

L



E
2
2
2
m
s55 [
hw(h  w )sin( l )  D cos( l )]
12
c
c
c
d15Tm (peak) d n
(21)
The overall charge induced from the ME cantilever can be integrated as
Q(t )  
L
0


bd
d

d15Tm (peak) d n
s [
E
55
2
ml
12

c
hw(h  w ) sin(
2
2
6d15Tm (peak)b(b  2d ) sin(
E
s55
[ 2  mlhw(h 2  w2 ) sin(

c

cos(

c

c
c
z )eit
L)  D

c
cos(
L)
L)  12 D

c
cos(

c

c
dzd( d n )
L)]
(22)
eit
L)]
Therefore, the average ME voltage between electrodes of individual piezo-unimorph
is
U (t )  Q(t ) C  
6d15Tm (peak) (b  2d )hp sin(

c
L)



11s55E L[ 2  mlhw(h 2  w2 ) sin( L)  12 D cos( L)]
c


6d15Tm (peak) (b  2d )hp tan(

c
11s55E  mlhw(h 2  w2 )c
L)
6d15Tm (peak) (b  2d )hp  I tan(
11s55E D

c

c
 mlhw(h 2  w2 )

1


c
12  IL
 L tan( L) 
c
c
 mlhw(h 2  w2 )
L)

c


1
eit
eit
12  IL
 L tan( L) 
c
c
 mlhw(h 2  w2 )
(23)
eit
6
The dynamic model can be degraded to the static model while the angular frequency
ω→0, which helps unify the models under different operating conditions. The
approximate value of the induced ME voltage is

6d15Tm (peak) (b  2d )hp  I tan( L)
1
c 
lim U (t )  lim 
eit
 0
 0



12

IL
11s55E  mlhw(h 2  w2 ) D
[ L tan( L) 
]
c
c
c
 mlhw(h 2  w2 )


d15Tm (peak) (b  2d )hp
211s55E D
3d15Tm (peak) (b  2d )
1611b(b 2  3bd  3d 2 )
(24)
It observes that the limit values of the induced voltage is consistent with that in Eq.
(8), which means the static model and dynamic model can be synthesized and the
magnetoelectric coupling is thus appropriately expressed.
7