Nature - USA (2020-01-16)

Nature | Vol 577 | 16 January 2020 | 353

octahedral perovskites^25 ,^26. The light absorption coefficient of the
a.c.-poled sample is found to be almost zero at wavelengths longer
than 400 nm, whereas the absorption coefficient of the d.c.-poled
sample remains large and decreases monotonically with increasing
wavelength (Extended Data Fig. 6).
The birefringence of the a.c.-poled crystals is approximately one
order of magnitude higher than that of their d.c.-poled counterparts
(Fig. 3c). This difference is associated with the different domain struc-
tures. The principle axis of the optical indicatrices of the domains
on both sides of a 71° domain wall are perpendicular to each other
on the (001) plane, resulting in cancellation of the birefringence as
light travels through a 71° domain wall (see Methods section ‘PLM’).
For a.c.-poled crystals, however, the birefringence is approximately
equal to that of the intrinsic value of a single-domain rhombohedral

PMN-28PT crystal because of the substantially decreased number of

71° domain walls.

The a.c.-poled crystals also exhibit a 30% enhancement in d 33 over

the d.c.-poled crystals. This phenomenon was observed in many

a.c.-poled relaxor–PT crystals^27 –^30. To understand the mechanism for

the enhanced piezoelectricity, we conducted phase-field simulations

to calculate the piezoelectricity of PMN-28PT with different 71° domain

sizes, as shown in Fig. 4a. The variation of polar vectors around a 71°

domain wall is depicted in Fig. 4b. For an ideal rhombohedral domain,

the polar vectors are along the ⟨111⟩ directions, and the angle between

the polar vector and the [011] direction is 35.3°. However, the presence

of 71° domain walls causes the polar vectors to rotate towards the [011]

direction to minimize the polarization gradient and elastic energies

associated with the polarization/strain discontinuities around the

500 1,000 1,500 2,000 2,500

0

20

40

60

80

100

Visible light

Transmittance (%)

Wavelength (nm)

a.c.-poled (0.5 mm)

a.c.-poled (1.8 mm)

d.c.-poled (0.5 mm)

d.c.-poled (1.8 mm)

Surface reection loss

abc

0

0.005

0.010

0.015

0.020

0.025

Birefringence

Birefringence Piezoelectric coefcient

d.c.-poled a.c.-poled

0

480

960

1,440

1,920

2,400

Piezoelectric coefcient (pC N

–1)

a.c.-poled d.c.-poled

Fig. 3 | Comparison of the properties of a.c.-poled and conventional
d.c.-poled [001]-oriented PMN-28PT crystals. a, Photograph of a.c- and
d.c.-poled PMN-28PT crystals. The thicknesses of the crystals are 0.5 mm and
1.8 mm, respectively. b, Light transmittance of the a.c.- and d.c.-poled PMN-
28PT crystals. c, d 33 and birefringence of the a.c.- and d.c.-poled PMN-28PT

crystals. Ten a.c.-poled and ten d.c.-poled samples are used for the

characterization of d 33 , whereas the birefringence is determined from seven

points of the a.c.- and d.c.-poled samples. The red graded arrow indicates the

increase in d 33 and birefringence for PMN-PT crystals after a.c. poling. The error

bars in c present the standard deviation of the corresponding data.

0.0 0.1 0.2 0.3 0.4

0

0.5

1.0

1.5

2.0

2.5

3.0

Change in free energy

density (a.u.)

Pz (μC cm–2)

D 71 = 64 nm

D 71 = 128 nm

D 71 = 256 nm

D 71 = Inf.

a

d e

b

D 71

D 109

D 71 = Inf. D 71 = 256 nm D 71 = 128 nm D 71 = 64 nm

E

c

f

[111] [111]

[111] [111]

Polar directions for different colours

[010]

[001]

[100]

Angle (°)

15 20 25 30 35

[011]

[100]

0 100 200 300 400 500

0

0.05

0.10

0.15

0.20

0.25

Px

Py

Average polarization (C m

–1)

D 71 (nm)

Pz

0

5

10

15

20

Rotation angle from [111]

(º)pc

1,000

1,300

1,600

1,900

Inf.

256

128

64

1,850

1,650

1,610

1,560

D 71 (nm)

3,000

4,000

5,000

6,000

Inf.

256

128

64

5,500

4,890

4,700

4,480

H^33

/H^0

D 71 (nm)

(^7171)
[011][011]
d^33
(pC N
–1)
Fig. 4 | Phase-f ield simulations of the effect of domain size on the
polarization, free-energy density and properties of the PMN-28PT crystal.
a, Two-dimensional schematic of a [001]-poled PMN-28PT with various 71°
domain widths (D 71 ). ‘Inf.’, no 71° domain walls are present in the system. The
polar directions for different colours are depicted on the right. b, Schematic of
polar vectors around a 71° domain wall (marked by the green line in a). The
colours of the arrows represent the angles between each polar vector and the
[011] direction. D 71 in this schematic is 64 nm. c, The average magnitude of the
spontaneous polarization components Px ([100]), Py ([010]) and Pz ([001]) of a
ferroelectric domain as a function of D 71. The red line describes the angle
between the overall polarization and the [111] direction as a function of D 71.
d, The average free-energy density of the system with respect to Pz. The free-
energy density at the stable state (that is, ∆P = 0) is selected as the reference.
e, Dielectric permittivity (ε 33 /ε 0 ) as a function of D 71. f, d 33 as a function of D 71.