Download Supplementary Material for Large and electric field tunable

Supplementary Material for
Large and electric field tunable superelasticity in BaTiO3
crystals predicted by an incremental domain switching criterion
Y. W. Li1, X.B. Ren3, F.X. Li1,4,a), H.S. Luo5, D.N. Fang2,b)
1State
Key Lab for Turbulence and Complex Systems, College of Engineering, Peking University,
Beijing, 100871, China
2Department
3Materials
of Aeronautics & Astronautics, College of Engineering, Peking University, Beijing,
100871, China
Physics Group, National Institute for Materials Science, Tsukuba, 305-0047, Ibaraki,
Japan
4HEDPS,
Center for Applied Physics and Technologies, Peking University, Beijing, China
5Shanghai
Institute of Ceramics, Chinese Academy of Sciences, Shanghai, 200050, China
*E-mail: [email protected]
a)
b)
Corresponding author, Email: [email protected]
Co-corresponding author, Email: [email protected]
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P-E hysteresis loop and butterfly curve
Fig S1. P-E hysteresis loop and butterfly curve of the poled BaTiO3 crystal cube
After poling, the D-E hysteresis loop and butterfly loop of the poled BaTiO3 crystal were tested to
check if the poling is complete, as shown in Fig. S1. The remnant polarization after poling is
measured to be 25.5μC/cm2 which is very close to the saturated value of 26μC/cm2, indicating that
little reverse domain switching occurs after poling and the obtained poled sample is nearly the
single-domain state as that in Fig.1a (in the paper). The very small strain variations (about 0.01%)
during bi-polar electric loading implies that little 900 domain switching was involved in the
polarization reversal.
Testing Setup
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Fig S2. Testing setup for electromechanical loading of BaTiO3 crystal cubes to realize the large
superelasticity.
Fig.S2 shows the testing setup for electromechanical loading of BaTiO3 crystal cubes to realize
the large superelasticity. The specimen is immersed into silicon oil during testing to prevent
charge leakage. The electric loading is provided by a high-voltage amplifier controlled by a
functional generator. Compressive stress is applied by a Shimadzu testing machine and a spherical
hinge is used to avoid any bias compression. Alumina blocks were used to insulate the specimen
from the loading equipment. Brass plates of 0.3 mm-thick were pasted on the alumina blocks as
electrodes for electric loading. Two strain gauges, with the dimension of 3*3mm2, were glued on
the opposite 5*5mm2 faces to measure the longitudinal strain during compression. A 2μF
capacitor is connected in series with the testing specimen and an electrometer with high-input
resistance is connected in parallel with the capacitor to detect the polarization change.
Domain switching process and a proposed domain switching criterion
The compression depolarization process in Fig.2a and Fig.2b of the paper can be divided into
three stages. At stage I when the stress is less than 1.8MPa, both the strain and polarization
responses are almost linear, i.e., no domain switching occurs. At stage II when the compression is
above 1.8MPa, both the strain and polarization show intensive nonlinearities indicating domain
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switching occurs and the initial coercive stress is thus determined to be 1.8MPa. At this stage, the
strain and polarization first increase quickly with the stress with a slightly hardening effect until at
about 6MPa when the hardening effect is more severe (seen in Fig.2b). When the stress reaches
above 13MPa, i.e., stage III in Fig.2a and 2b, the strain again increases almost linearly with the
stress and the polarization keeps almost constant, which implies that domain switching saturates
at this stage. The maximum strain and polarization at 15MPa is 1.01% and 24.5 μC/cm2,
respectively, among which about 0.99% is the switching strain and 24.3 μC/cm2 is the switching
polarization. This indicates that about 95% “c” domains can be switched to be “a” domains, and
another 5% “c” domains may be frozen by the space charge or internal bias field and cannot
switch under compression loading.
During stress unloading, the polarization still keeps constantly at 24.5 μC/cm2 and the strain
decreases almost linearly with the decreasing stress only showing slightly nonlinearities below
5MPa, indicating that little back domain-switching occurs. After unloading, the remnant strain is
0.92%, slightly smaller than the maximum switching strain of 0.99%. During one stress
loading/unloading cycle, the dissipated energy density is measured to be 36 kJ/m3.
Similarly, the re-poling process can also be divided into three stages. At stageⅰwhen the electric
field (E3) is below 70V/mm, there is almost no strain response and the polarization response is
nearly linear, i.e., no domain switching occurs. At stage ⅱ when E3>70V/mm, both the strain and
polarization increases rapidly, indicating that 900 domain switching occurs at this stage. The
polarization saturates at about 200V/mm while the strain does not saturate until the field reaches
400V/mm. The maximum electric-field-induced-strain is 0.91%, among which 0.9% is the
switching strain, very close to the remnant strain of 0.92% in Fig.2a. The remnant polarization
upon removing the electric field is about 27.5μC/cm2, slightly larger than that of 24.5μC/cm2 after
compression loading in Fig.2b, implying that a small amount of 1800 switching occurs during
electric re-poling which contributes about 3μC/cm2 of polarization. The dissipated energy density
during the electric loading/unloading cycle is about 28 kJ/m3, among which about 26 kJ/m3 is
associated with 900 domain switching, obviously lower than that of 36 kJ/m3 during compression
loading/unloading. This confirms that considerable amount of strain energies were stored in the
900 domain walls after compression depolarization and it will be released to assist the 900 domain
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switching during electric re-poling.
The proposed quasi-static 900 domain switching criterion under uni-axial electromechanical
loading is:
D
 ( 33 S 0  E3 Ps )  f  Wsto   U 90
( f )  f
(S1)
Where  33 is the applied compressive stress along the poling direction, S 0  c / a  1 is the
spontaneous strain for the tetragonal crystal. E 3 is the applied electric field, Ps is the
D
( f ) is the energy barrier (per unit volume) for 900 domain
spontaneous polarization; U 90
switching which is a function of the volume fraction f of switched domains. The “+” and “-“ in
front of the parentheses are for the 900 domain switching during compression depolarization and
electric re-poling, respectively. Wsto is the stored strain energy (per unit volume) in the 900
domain walls, which is dependent of the domain configurations and Wsto .is the increment. Here
we take the volume fractions of the “a” domains as f1 , then when f1  0 , there is no 900
domain wall at all and Wsto  0 . When f1 reaches the maximum value, the domain
configuration turns to be close to that in Fig.1b, i.e., most the 900 domain walls are “a-a” domain
walls and the strain energy density of the 900 domain walls reaches the maximum.
During compression depolarization without dc bias field, Eq.(S1) reduced to be:
 33 S 0  f  Wsto  U 90D ( f )  f
(S2)
By integrating of Eq.(S2) from f  0 to f  0.95 and assuming that during compression
unloading, no domain switching occurs (which is close to the case in Fig.2a that little domain
switching occurs), we get
0.95
W  Wsto 
U
D
90
( f )df
0
5
(S3)
Where W is the loop area of the stress-strain curves during compression loading/unloading, as
shown in Fig.2a (in the paper).
During the re-poling process after compression loading, Eq.(S1) become:
D
E3 Ps  f  Wsto  U 90
( f )  f
(S4)
Similarly, by integrating of Eq.(S4) from f  0 to f  0.95 and assuming that during electric
field unloading, no domain switching occurs (which is close to the case in Fig.2c (in the paper)
that little domain switching occurs), we get
0.95
WE  Wsto 
U
D
90
( f )df
(S5)
0
Where WE is the loop area of the electric field-polarization curves during electric
loading/unloading, as shown in Fig.2d (in the paper).
0.95
It is reasonable to assume that the energy barriers for
900
domain switching, i.e.,
U
D
90
( f )df ,
0
are the same for domain switching caused by stress and electric field. Then, from Eq.(S3) and
Eq.(S5), we get the maximum stored elastic energy density in domain walls
Wsto  (W  WE ) / 2  5 kJ/m 3
(S6)
0.95
U
D
90
( f )df  31 kJ/m3
(S7)
0
D
If we further assume that the U 90
increases linearly with the volume fraction of the switched
domains (close to the cases in Fig.2a and 2c in the paper) and Wela increases linearly with the
volume fraction of the ”a” domains, i.e., f1 , then
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Wsto ( f1 )  5.26 f1 kJ/m3
f1 [ 0 , 0 . 9 5 )
(S8)
Obviously the energy barrier for initial 900 domain switching can be determined from the coercive
stress in Fig.2a (in the paper), i.e.,
U 90D (0)  1.8MPa *1.04%  5.26kJ/m3  13.46 kJ/m3
(S9)
As to the volume fraction dependent energy barrier for 900 domain switching, during compression
loading (Fig.2a in the paper), we have
U 90D ( f1 )  (1.8  4.04 f1 )MPa *1.04%  5.26 kJ/m 3  (13.46  42 f1 )kJ/m3
(S10)
And during the electric re-poling process after compression (Fig.2d in the paper), we have
U 90D ( f1 )  (184  164 f1 ) V/mm*26C/cm 2  5.26 kJ/m 3  (53.3  43 f1 ) kJ/m3 (S11)
Using Eqs.S1, S9-S11, the domain switching curves in Fig.2 (in the paper) can then be estimated
and compared with experimental data, as shown in the following Fig.S3. The good fitness
between the predictions and the experimental results indicate that the proposed domain switching
criterion for 900 domain switching in Eq.(S1) is effective.
Fig.S3 Comparison of the predicted domain switching curves and the experimental results. (a)
stress-strain curves; (b) stress-depolarization curve; (c) electric field induced strain curve; (d)
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electric field induced polarization curve. (the black lines and the red lines represent the
experimental curves and the predicted curves, respectively)
Compression depolarization with a dc bias field of 800V/mm
Fig.S4 Measured stress-strain curve and stress-polarization curve of BaTiO3 crystals under
compression loading/unloading with a dc bias field of 800V/mm.
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