The origin of ultrahigh piezoelectricity in relaxor

ARTICLE Received 24 Jun 2016 | Accepted 3 Nov 2016 | Published 19 Dec 2016

DOI: 10.1038/ncomms13807

OPEN

The origin of ultrahigh piezoelectricity in relaxor-ferroelectric solid solution crystals Fei Li1,2, Shujun Zhang1,3, Tiannan Yang1, Zhuo Xu2, Nan Zhang2, Gang Liu4,5, Jianjun Wang1, Jianli Wang3, Zhenxiang Cheng3, Zuo-Guang Ye6,2, Jun Luo7, Thomas R. Shrout1 & Long-Qing Chen1

The discovery of ultrahigh piezoelectricity in relaxor-ferroelectric solid solution single crystals is a breakthrough in ferroelectric materials. A key signature of relaxor-ferroelectric solid solutions is the existence of polar nanoregions, a nanoscale inhomogeneity, that coexist with normal ferroelectric domains. Despite two decades of extensive studies, the contribution of polar nanoregions to the underlying piezoelectric properties of relaxor ferroelectrics has yet to be established. Here we quantitatively characterize the contribution of polar nanoregions to the dielectric/piezoelectric responses of relaxor-ferroelectric crystals using a combination of cryogenic experiments and phase-field simulations. The contribution of polar nanoregions to the room-temperature dielectric and piezoelectric properties is in the range of 50–80%. A mesoscale mechanism is proposed to reveal the origin of the high piezoelectricity in relaxor ferroelectrics, where the polar nanoregions aligned in a ferroelectric matrix can facilitate polarization rotation. This mechanism emphasizes the critical role of local structure on the macroscopic properties of ferroelectric materials.

1 Department

of Materials Science and Engineering, Materials Research Institute, Pennsylvania State University, University Park, Pennsylvania 16802, USA. Materials Research Laboratory, Key Laboratory of the Ministry of Education and International Center for Dielectric Research, Xi’an Jiaotong University, Xi’an 710049, China. 3 Institute for Superconducting and Electronic Materials, Australian Institute of Innovative Materials, University of Wollongong, Wollongong, New South Wales 2500, Australia. 4 High Pressure Synergetic Consortium, Geophysical Laboratory, Carnegie Institute of Washington, Argonne, Illinois 60439, USA. 5 Center for High Pressure Science and Technology Advanced Research, Shanghai 201203, China. 6 Department of Chemistry and 4D LABS, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6. 7 TRS Technologies Inc., 2820 E. College Avenue, State College, Pennsylvania 16801, USA. Correspondence and requests for materials should be addressed to S.Z. (email: [email protected]) or to L.-Q.C. (email: [email protected]). 2 Electronic

NATURE COMMUNICATIONS | 7:13807 | DOI: 10.1038/ncomms13807 | www.nature.com/naturecommunications

1

ARTICLE

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13807

0.1

0k –5 k 0

100

200

300

400

0.0

10 k

0.2

0

0.1

–10 k

0

100

0.4

The variation accounts for 50% value of RT

Δ=2,600

2k 1k

0.3 0.1 kHz 1 kHz 0.2 10 kHz 100 kHz

0.1

0 0

50 100 150 200 250 300

0.0

Dielectric permittivity (22/0)

5k

3k

400

500

0.0

25 k 20 k 15 k


10 k

Δ=20,000

f

0.4 0.3 0.1 kHz 1 kHz 10 kHz 100 kHz

0

0.2 0.1

–5 k 0

0.3

10 k 0.2 0

0.1 0

0.5

5k

0.4

100

200

300

400

500

0.0

Temperature (K)

PMN-0.32PT

50 100 150 200 250 300

0.0

6k

0.5

PZN-0.15PT

5k

Δ=4,000

4k


3k 2k 1k

0.4 0.3 0.1 kHz 0.2 1 kHz 10 kHz 100 kHz 0.1

0 0

50 100 150 200 250 300

0.0

Temperature (K)

Temperature (K)

Temperature (K)

h

g 3k

d15 of PMN-0.28PT crystal d24 of PMN-0.32PT crystal d15 of PZN-0.15PT crystal

2k

1k

0 0

50

100 150 200 250 300 Temperature (K)

Di/piezo electricity

Shear piezo. coeff. (pC N–1)

300

e

0.5

PMN-0.28PT

Loss factor


6k

4k

200

0.1 kHz 1 kHz 10 kHz 100 kHz

20 k

Temperature (K)

Temperature (K)

d

0.3

0.5

PZN-0.15PT

Loss factor

0.2

5k

20 k

30 k

Loss factor

0.3

0.4


10 k

30 k

c

0.5

PMN–0.32PT 0.1 kHz 1 kHz 10 kHz 100 kHz

Loss factor

0.4

0.1 kHz 1 kHz 10 kHz 100 kHz

15 k

40 k

Loss factor

20 k

b

0.5

PMN–0.28PT


25 k

Loss factor


a

By considering the diversity and instability of ferroelectric phases at MPBs, a number of qualitative mechanisms have been proposed to elucidate the high piezoelectric activity in relaxor–PT crystals. The corresponding mechanisms include ‘electric field-induced phase transition’6, ‘ease of polarization rotation via a monoclinic phase’13,14, ‘giant electromechanical response as a critical phenomenon’15, ‘adaptive domain structure’16 and so on. However, these mechanisms fail to explain why relaxor-ferroelectric solid solutions exhibit significantly higher piezoelectricity when compared with nonrelaxor-based MPB ferroelectrics, for example, Pb(ZrxTi1 x)O3 crystals (500–1,000 pC N 1 for x around 0.5). To further clarify the origin of ultrahigh piezoelectricity in relaxor–PT crystals, the microstructural characteristics of relaxor ferroelectrics should be surveyed first. Relaxors, for example, PMN, are characterized by cation disorder on the nanoscale17–20,


P

erovskite ferroelectrics (general formula, ABO3) exhibit the highest electromechanical activity among all known piezoelectrics. The enhanced piezoelectric response of perovskite ferroelectrics are generally associated with the longrange cooperative phenomena near morphotropic phase boundaries (MPBs)1–5. In proximity of an MPB, different ferroelectric phases possess similar energies, leading to facilitated variation of polarization and strain under an external stimulus. One of the most remarkable breakthroughs in perovskite ferroelectrics was the discovery of ultrahigh piezoelectricity (d33* ¼ 1,500–2,500 pC N 1) and electromechanical coupling factors (k33*40.9) in domain-engineered relaxor–PbTiO3 (PT) solid solution crystals with MPB compositions, for example, Pb(Mg1/3Nb2/3)O3–PT (PMN–PT) and Pb(Zn1/3Nb2/3)O3–PT (PZN–PT) crystals6–12.

Relaxor-PT crystals Classical ferroelectrics

Room temperature property

Long-range phase transition

This jump is a cornerstone for ultrahigh piezo. of relaxor-PT system

Stage II

Stage I

Temperature

Figure 1 | The temperature dependences of transverse dielectric (e11 and e22) and shear piezoelectric (d15 and d24) properties for single-domain relaxor–PT crystals. (a–c) Dielectric permittivities for the rhombohedral PMN–0.28PT, orthorhombic PMN–0.32PT and tetragonal PZN–0.15PT crystals, respectively. In the PMN–0.28PT crystals, a rhombohedral–tetragonal and a tetragonal–cubic phase transitions exist at 368 and 405 K, respectively. In the PMN–0.32PT crystal, an orthorhombic–tetragonal and a tetragonal–cubic phase transitions occur at 360 and 430 K, respectively. In the PZN–0.15PT crystal, a tetragonal–cubic phase transition exists at 468 K. (d–f) Enlarged low-temperature sections (20–300 K) for a–c, respectively. (g) The measured shear piezoelectric coefficients for the three relaxor–PT crystals as a function of temperature. (h) A schematic plot showing the major finding of this work for the enhancement of dielectric/piezoelectric properties in relaxor–PT crystals (the temperature-dependent piezoelectric/dielectric responses of the classical ferroelectrics is inferred from phenomenological theory and first-principle calculations of Pb(Zr0.5Ti0.5)O3 (refs 50,51)). For comparison, the corresponding temperature dependences of the longitudinal dielectric permittivity e33 of single-domain PMN–0.28PT, PMN–0.32PT and PZN–0.15PT crystals are shown in Supplementary Fig. 1. For orthorhombic crystals, e11 and e22 are two independent tensors. The permittivity e11 versus temperature of the orthorhombic PMN– 0.32PT crystal is given in Supplementary Fig. 2. For the PZN–0.15PT crystal, the temperature dependence of the reciprocal of relative dielectric constant is given in Supplementary Fig. 3, for identifying its relaxor characteristic. 2


ARTICLE


leading to random fields19,20 and local phase fluctuations17,18. These factors lead to a unique characteristic of relaxor-based ferroelectrics in contrast to classical ferroelectrics, that is, the presence of polar nanoregions (PNRs)17–21, which are believed to be responsible for the high dielectric properties of relaxors17,18,22. Therefore, there have been numerous studies to understand the relationship between the high piezoelectric properties and PNRs in relaxor–PT systems23–27. For example, Xu et al.23 proposed that the softening of the transverse acoustic mode was due to the existence of PNRs, while Manley et al.24 further demonstrated that aligning PNR vibrational modes by a poling electric field can enhance the phonon softening. However, there have been no sufficient evidence from piezoelectric/dielectric measurements to substantiate the contribution of PNRs, which highly impedes the understanding of the corresponding mechanism(s) for ultrahigh piezoelectricity in relaxor–PT systems. In this work, we obtain crucial experimental evidences for elucidating the contribution of PNRs to the piezoelectric activity in relaxor–PT crystals through cryogenic measurements, whereupon the PNR’s contribution accounts for 50–80% of the roomtemperature dielectric and piezoelectric properties. In addition, we perform phase-field simulations of dielectric/piezoelectric responses of ferroelectric domains in the presence of PNRs and propose a mesoscale mechanism that successfully explains the ultrahigh dielectric/piezoelectric properties and the corresponding temperature dependence for relaxor–PT crystals. Results Crystal samples. Our investigations focused on the transverse dielectric and shear piezoelectric responses of single-domain relaxor–PT crystals. As listed in Supplementary Table 1, the transverse dielectric and shear piezoelectric properties are significantly larger than their longitudinal counterparts, and thus play a dominant role in the high electromechanical performance of relaxor–PT crystals. It should be noted that the large

kα2

Intensity (a.u.)

50 K

37.5

[111] poled PMN-0.28PT

kα1

38.0

b

38.5

39.0

30.5

Intensity (a.u.)

300 K R3m PMN-0.28PT

100

100 K R3m PMN-0.28PT

5.8


300 K 150 K 50 K

31.0

2 (°)

d

Cryogenic experiments. Figure 1 shows the transverse dielectric permittivity (e11/e0 or e22/e0) versus temperature for single-domain rhombohedral PMN–0.28PT, orthorhombic PMN–0.32PT and tetragonal PZN–0.15PT crystals, respectively, measured over a frequency range of (102B105 Hz). Above room temperature, several phase transitions are found to exist. For example, rhombohedral-to-tetragonal (Trt: 368 K), orthorhombic-to-tetragonal (Tot: 360 K) and tetragonal-to-cubic (Ttc: 468 K) phase transitions are observed in the PMN–0.28PT, PMN–0.32PT and PZN–0.15PT crystals, respectively. These phase transitions destroy the singledomain states of the crystals, and thus our studies are focused on temperatures well below these phase transitions. The temperature-dependent dielectric behaviour of the singledomain phases can be divided into two stages: stage I (T4200 K) and stage II (To200 K). In stage I, the frequency dispersion of the dielectric permittivity is minimal, and the variation of dielectric response versus temperature can be explained by the Landau– Devonshire phenomenological theory, as previously reported30. In stage II, a drastic increase in transverse dielectric permittivity, e11/e0 and/or e22/e0, is observed in all three crystals with increasing temperature, accompanied by dielectric relaxation, that is, frequency dispersion of permittivity and loss maxima. As expected, the shear piezoelectric coefficients d15/d24, which can be expressed as d15/24 ¼ 2Q55/44e11PS, follow similar temperaturedependent behaviour to the dielectric permittivities in stage II (Fig. 1g), since variations in spontaneous polarization PS

c Intensity (a.u.)

300 K 150 K

Intensity (a.u.)

a

longitudinal piezoelectric properties in domain-engineered relaxor–PT crystals originate from the high shear piezoelectric response in the corresponding single-domain state28–30. To demonstrate the generality of our results and the proposed mechanism, we investigated three different types of phases with all of them single-domain crystals: (1) rhombohedral [111]-poled PMN–0.28PT; (2) orthorhombic [011]-poled PMN–0.32PT; and (3) tetragonal [001]-poled PZN–0.15PT crystals.

31.5 2 (°)

32.0

32.5

300 K 150 K 50 K

21.0

110

Synchrotron results Refinement data


c-plane

a-plane

21.5

22.0 2 (°)

22.5

23.0

111

Synchrotron results Refinement data

5.9

6.0

8.4

8.3

10.2

10.3

2 (°)

Figure 2 | X-ray diffraction experiments of relaxor–PT at cryogenic temperatures. (a–c) X-ray diffraction patterns for single-domain crystals: the (111) surface of [111]-poled PMN–0.28PT; the (011) surface of [011]-poled PMN–0.32PT; and the (001) surface of [001]-poled PZN–0.15PT crystals, respectively. For all the three crystals, the X-ray diffraction peaks do not show any anomaly over the temperature range of 50–300 K, indicating the lack of any longrange ferroelectric phase transition at cryogenic temperatures. Note: the peak splitting in PZN–0.15PT is due to an incomplete single-domain state, especially around the surface of the sample49. Because of the high lattice parameter ratio c/a (1.023) and the associated mechanical clamping, a fully single-domain state is very difficult to obtain. (d) A selected synchrotron X-ray diffraction pattern for PMN–0.28PT powders (ground from crystal samples), where the Rietveld-refinement parameters are given in Supplementary Table 2. The synchrotron X-ray diffraction results indicate that the long-range structure of PMN–0.28PT is R3m from 300 down to 100 K. NATURE COMMUNICATIONS | 7:13807 | DOI: 10.1038/ncomms13807 | www.nature.com/naturecommunications

3

ARTICLE


a

b

40

120 46

y (nm)

y (nm)

80

42 0 38

40

Angle (°)

20

–20 34 0

–40 0

40

80

120

x (nm)

58

62

66

70

x (nm)

Figure 3 | Phase-field-simulated microstructures for a [100]-poled PNR-ferroelectric composite at 100 K. (b) The enlarged square area in a, where the polar vectors of every grid are shown. The x and y axes represent the [100] and [010] crystallographic directions, respectively. The unit for x and y axes is nanometre. The colour bar denotes the angle between the polar vector of the grids and the [100] direction.

(Supplementary Fig. 4) and electrostrictive coefficients Q55/44 are minimal in this temperature range31. It should be noted that the drastic increase in transverse dielectric permittivities observed in stage II account for 50–80% of their room-temperature values, as depicted in Fig. 1d–f, hence they play a dominant role in the ultrahigh room-temperature properties of relaxor–PT crystals. The enhancement of dielectric/piezoelectric properties in stage II cannot be explained by the ferroelectric phenomenological theory based on the long-range order parameter (spontaneous polarization) for the following reasons. First, according to phenomenological theory, a significant variation of the dielectric/piezoelectric response is generally associated with a longrange phase transition. However, no phase transition occurs in stage II, as confirmed by cryogenic experiments on the long-range crystalline structure (Fig. 2, Supplementary Figs 5 and 6, Supplementary Table 2 and refs. 32,33), the domain patterns on the micron-scale (Supplementary Fig. 7) and the spontaneous polarization (Supplementary Fig. 4). Second, the dielectric relaxation frequency is found to be in the range of 100–108 Hz (much lower than the typical phonon frequency) at temperatures of 120 and 150 K (Supplementary Figs 8 and 9 and Supplementary Note 1), demonstrating that a considerable portion of the dielectric response is associated with the switching of specific dipoles and/or interface motion. This contribution cannot be simply explained by ferroelectric long-range order. Therefore, it can be concluded that the major difference between relaxor–PT crystals and classical ferroelectrics exists in stage II. In other words, the enhancement of piezoelectricity in stage II can be recognized as a cornerstone for the ultrahigh piezoelectric activity in relaxor–PT crystals when compared with classical ferroelectric crystals, as schematically shown in Fig. 1h. Thus, exploring the mechanism(s) responsible for the dielectric/piezoelectric variation in stage II should enable us to clarify why relaxor–PT crystals exhibit significantly higher dielectric/piezoelectric properties than those of classical ferroelectrics. Phase-field simulations. We use the phase-field method of ferroelectric domain structures34 to simulate the mesoscale microstructure of relaxor–PT crystals to guide the understanding of the mesoscale mechanism(s) underpinning the high dielectric and piezoelectric properties. Phase-field method has been used to model the effects of random defects/fields on ferroelectric domains and domain-switching to simulate the relaxor behaviour35–38. In this work, we consider two important mesoscale characteristics of relaxor–PT: (i) the coexistence of PNRs and long-range ferroelectric domains; and (ii) the symmetry of PNRs being different from that of the long-range ferroelectric domains. For 4

example, the symmetries of PNRs and long-range ferroelectric domains were found to be orthorhombic and rhombohedral in PZN–xPT (xo0.08) crystals, respectively39,40. Thus, relaxor–PT solid solution crystals are treated as PNR-ferroelectric nanocomposites in the phase-field simulations. In the following simulation, a crystal with orthorhombic PNRs and tetragonal ferroelectric domain matrix is selected as a representative model system. The Curie temperature of the PNRs is assumed to be 620 K based on the Burns temperature of relaxor–PT systems18, while the Curie temperature of the tetragonal matrix is the same as the Curie temperature of PZN–0.15PT, that is, 468 K. Below the Curie temperature, we assume there are no ferroelectric phase transition for both the orthorhombic PNRs and/or tetragonal matrix. The spontaneous polarization of the orthorhombic PNRs and tetragonal matrix is set to be 0.40 C m 2 at 300 K. On the basis of neutron-scattering results41,42 and the activation energy of PNR switching (Supplementary Fig. 10 and Supplementary Note 2), the diameter of PNRs is assumed to be 3–5 nm with a random distribution, while the volume fraction of PNRs is set to be 7.5%, at which the tetragonal-like domain structure is maintained (Supplementary Fig. 11). Volume fractions in the range of 5–15% yielded similar results as given in Supplementary Fig. 12. Figure 3 describes the simulated results of a [100]-poled PNRferroelectric composite at 100 K. At temperatures below 100 K, PNRs tend to retain their original orthorhombic state dictated by their Landau energy despite the fact that electrostatic, gradient and elastic energies are significant due to the discontinuity of polarization and strain around the interfaces between PNRs and ferroelectric matrix. It can be seen that only PNRs with polar vectors along [110] or ½110 exist, while PNRs with polar vectors along ½110 or ½110 are absent due to unfavourable electrostatic interaction energy within the tetragonal ferroelectric matrix. The average symmetry of the [100]-poled PNR-ferroelectric composite is still 4mm although there exist PNRs with mm2 symmetry. With increasing temperature, the polar vectors of some PNRs transform to the [100] direction, that is, the polar vector of PNRs becomes ‘collinear’ with the polar direction of the ferroelectric matrix, as shown in Fig. 4a. The reason why PNRs transform to the ‘collinear’ state is that this transformation minimizes the free energy of the entire composite system. In the ‘collinear’ state, although PNRs’ Landau energy is high, the total electrostatic, gradient and elastic energies are low since the discontinuousness of polarization and strain is minimal for the ‘collinear’ state. As temperature increases, the difference in PNR’s Landau energy between orthorhombic and tetragonal states decreases. At relatively high temperatures, therefore, the decrease in the total


ARTICLE


electrostatic, gradient and elastic energies can more than offset the increase in Landau energy arising from the transformation of some orthorhombic PNRs to the tetragonal phase. In general, smaller and more isolated PNRs transform to the ‘collinear’ state at relatively lower temperatures. At temperatures above 350 K, all PNRs are in the ‘collinear’ state as shown in Fig. 4a. Figure 4b presents the calculated transverse and longitudinal (small inset) dielectric permittivities of a PNR-ferroelectric composite with respect to temperature. The transverse dielectric permittivity e11/e0 of the composite is found to be almost the same as that of the matrix for temperatures below 100 K. As the temperature increases from 100 to 200 K, the permittivity e11/e0 of the composite significantly increases from 1,600 to 4,500, while in the temperature range of 200–400 K, the permittivity of the composite is found to be quite stable with respect to temperature. It should be noted here that the simulated piezoelectric coefficient d15 shows a similar temperature dependence to the transverse permittivity e11/e0, as given in Supplementary Figs 15 and 16. In contrast, the longitudinal dielectric permittivity e33/e0 of the composite is found to possess the same value as the permittivity of the matrix, indicating that the impact of PNRs on the single-domain longitudinal dielectric response is minimal. The simulated temperature-dependent dielectric responses are in good agreement with experimental observations for PZN–0.15PT crystals (Fig. 1c and Supplementary Fig. 1c). Let us now discuss the mechanism(s) responsible for the enhanced transverse dielectric permittivity in a PNR-ferroelectric composite. Figure 5a,b display the microstructural evolution and polarization variation of the [100]-poled PNR-ferroelectric composite under a perpendicular [010] E-field, respectively. From Fig. 5a, the enhanced transverse dielectric response can be attributed to two contributions (schematically depicted in Fig. 5c): I—for ‘non-collinear’ PNRs, they may switch from one stable state to another whose polar direction is favoured by the E-field, that is, PNRs along the ½110 direction can be switched to the [110] direction under an [010] E-field (switching may also occur via a metastable state, that is, with the polar vector along the [100] direction); II—for ‘collinear’ PNRs, their polarizations are much easier to rotate under a perpendicular E-field when compared with the matrix, and this rotation will also facilitate the polarizations of the nearby regions to rotate (please see the

a

100 K

150 K

colour variation of the grids near PNRs) to reduce electrostatic, elastic and gradient energies at the interfaces. During this process, the average (macroscopic) polarization rotation under a perpendicular E-field is highly facilitated by PNRs. At 50 K, both contributions I and II disappear, since PNRs are ‘frozen’ in the ferroelectric matrix. Therefore, the dielectric response of the composite is similar to that of the ferroelectric matrix. At 150 K, PNRs become unstable and some of them are in the ‘collinear’ state. Under this condition, both I and II contribute to the enhanced dielectric response. With increasing temperature, more PNRs become ‘collinear’ and consequently contribution II becomes dominant. At 350 K, all the contribution of PNRs comes from II. On the basis of the simulated transverse polarization-electric field (PE) response of the PNR-ferroelectric composites (Fig. 5b), a relatively high hysteresis is observed at 150 K. This is due to the fact that the switching of PNRs (contribution I) needs to overcome specific energy barriers. At both 50 and 350 K, a linear PE response without hysteresis is observed, since contribution I is absent. The linear PE response at 350 K also indicates that contribution II is not accompanied by hysteresis, which is very important for practical piezoelectric and dielectric applications. Contribution II raises the question ‘why are PNRs easy to rotate when they are in the ‘collinear’ state?’ In the PNRferroelectric composite, the transformation of a PNR to a ‘collinear’ state is driven by the reduction in elastic, gradient and electrical energies of the system. To study the properties of PNRs in the ‘collinear’ state, it is reasonable to use a [100] d.c. E-field as the driving force to mimic the situation, where PNRs are allowed to rotate to the [100] direction. Figure 6 shows the Landau energy profile for an orthorhombic PNR under various levels of [100] d.c. E-field. At zero d.c. E-field, the polar vector of the stable state is along h110i for an orthorhombic PNR. With increasing d.c. E-field, the polar vector approaches the [100] direction, and the energy barrier among the stable states decreases, as shown in Fig. 6a,b. The lower energy barrier indicates the ease of PNR switching from one stable state to another under an external stimulus. When the driving force is sufficiently high (that is, [100] d.c. E-field Z180 MV m 1), the polar vector of a PNR is rotated to the [100] direction, analogous to the scenario where PNRs are in the ‘collinear’ state within the ferroelectric matrix. At this condition, as shown in Fig. 6c, the

b

200 K

Become collinear 250 K 300 K

350 K

–20

0

20

Composite Matrix PNR

10 k

10 k

5k 0k 0

5k

200 400 Temperature (K)

600

Collinear PNRs begin to exist

0k –40

15 k

33/0

Dielectric permittivity 11/0

15 k

0

100

40

Angle (°)

200

300

400

500

600

Temperature (K)

Figure 4 | Temperature-dependent microstructure and dielectric response for [100]-poled PNR-ferroelectric composites. (a,b) Temperaturedependent microstructures and dielectric permittivities, respectively. In a, the colour bar denotes the angle between the polar vector of the grids and the [100] direction. Here an example of specific PNRs’ transformation to the ‘collinear’ state is given with increasing temperature from 100 to 150 K, as marked by the black dashed circles. In b, the intrinsic dielectric permittivities of the ferroelectric matrix and the PNR are given for comparison. The error bars represent the s.d.’s and are obtained from five different random distributions of PNRs with a volume fraction of 7.5%. For calculation of the dielectric permittivities, the magnitude and period of the a.c. E-field were 104 Vm 1 and 105 time steps, respectively. The corresponding three-dimensional simulations, given in Supplementary Figs 13 and 14, show similar results to the two-dimensional simulations. NATURE COMMUNICATIONS | 7:13807 | DOI: 10.1038/ncomms13807 | www.nature.com/naturecommunications

5

ARTICLE


a

b

Temperature

0 MV m–1

0.3 MV m–1

1 MV m–1 [010] E-field

0.6 MV m–1

350 K

350 K

250 K

Polarization (C m–2)

250 K

150 K

150 K

ΔP=0.04 C m–2

50 K 50 K

–1.0 –0.5 –40

–20

20

0

40

0.0

0.5

1.0

Electric field (MV m–1)

Angle (°)

c

Contribution II — rotation of collinear PNRs

Contribution I — switching of non-collinear PNRs E

E

E

E

E=0

E=0

[010]

[100]

Figure 5 | Perpendicular E-field-induced microstructure and polarization variations for the [100]-poled PNR-ferroelectric composites. (a) Microstructure evolution for the [100]-poled PNR-ferroelectric composites with applied perpendicular E-field along [010] at various temperatures. The colour bar denotes the angle between the polar vector of the grids and the [100] direction. (b) Simulated transverse PE responses for the PNR-ferroelectric composite, where the amplitude and period of the a.c. electric field are 1 MVm 1 and 105 time steps, respectively (see Supplementary Movies 1–3 for details of E-field-induced microstructure evolution). (c) Schematics for contribution I and contribution II of PNRs to the transverse dielectric response, where the orange regions represent PNRs, and the polar vectors of PNRs and ferroelectric matrix are shown by purple and blue arrows. The corresponding three-dimensional simulation can be found in Supplementary Figs 17 and 18, showing similar results to the two-dimensional simulation.

polarization rotation path is greatly flattened. This flattening explains why ‘collinear’ PNRs are relatively easy to rotate by a perpendicular E-field. It is worth noting that as the d.c. E-field further increases, the polar vector of a PNR will be further stabilized along the [100] direction, making the polarization rotation path steeper as presented in Fig. 6d. According to the above analysis, the relatively stable transverse dielectric permittivity over the temperature range of 200–400 K can also be explained. With increasing temperature, it is expected that the impact of a driving force on PNRs becomes more prominent due to a decrease in the Landau energy barrier between tetragonal and orthorhombic phases. The enhanced impact of the driving force includes the following: (1) stabilizing the ‘collinear’ PNRs in the matrix (as depicted in Fig. 6d), leading to a decrease in the dielectric response; and (2) destabilization of the ‘non-collinear’ PNRs, whereupon they transform to the ‘collinear’ state (as depicted in Fig. 6a–c), leading to an increase of dielectric response. Furthermore, the dielectric response of the ferroelectric matrix increases with increasing temperature. Thus, over the temperature range of 200–400 K, several factors may compete with each other, leading to an overall temperature-stable dielectric response, as simulated in Fig. 4b and experimentally observed in PZN–0.15PT crystals (Fig. 1c). 6

Discussion On the basis of the experimental and phase-field simulated results, the contribution of PNRs to the dielectric and piezoelectric properties in relaxor–PT crystals can be explained on the mesoscale. At room temperature, the high piezoelectric activity of relaxor–PT crystals is mainly associated with the ‘collinear’ PNRs, and not from the switching of ‘non-collinear’ PNRs, thus loss and hysteresis are extremely small at room temperature (Supplementary Figs 19–21). With decreasing temperature, the ‘collinear’ PNRs begin to become isolated from the ferroelectric matrix due to Landau energy dominating the polar direction of PNRs at low temperatures, resulting in a decreased contribution of ‘collinear’ PNRs, but with an increased contribution from PNR switching. Thus, increased loss and hysteresis associated with PNR switching are observed. By further decreasing temperature, all PNRs are isolated from the matrix and then become frozen, consequently contributions to the dielectric and piezoelectric properties are minimal. In addition to the explanation of the ultrahigh dielectric/ piezoelectric response and related temperature dependence, the phase-field modelling results are also consistent with the microstructural observations of relaxor–PT materials41,43–45. Neutron and X-ray diffraction studies have shown that diffuse


ARTICLE


ΔG (MJ m–3)

Increase of [100] d.c. E-field

[100]

a

b

c

0.35

0.4

0.45

0.25 0.3

0.36 0.2

0.43 0.2

] [001

0

0 –0.3

0 [010]

–0.3 – [110]

–40

[100]

–20

0 20 Angle (°)

0.3

–0.2

0

0.2

2.0

0.46 0.44 0.2 0

–0.2

1.0 0

–0.2

0

0.2

0.2 –0.2

–0.2

–40

–20

0 20 Angle (°)

40

–40

–20

0 20 Angle (°)

40

0

0 –0.2

‘Flattened’ energy profile

[110]

40

d

ΔG = 2.5 MJ m–3

–40

–20

0 20 Angle (°)

40

Figure 6 | The variation of Landau energy profile for an orthorhombic PNR under [100] d.c. E-field at 350 K. (a) At the d.c. E-field of 0 MVm 1; (b) at the d.c. E-field of 70 MVm 1; (c) at the d.c. E-field of 180 MV m 1; (d) at the d.c. E-field of 250 MVm 1. The axes in the three-dimensional figures represent the polarizations along the [100], [010] and [001] directions (unit: C m 2). The distance between the point on the figure surface and the origin is the optimized polarization, at which the free energy is minimized for the corresponding direction. The value of energy density is described by the colour and shown in the colour bar, where the minimal energy density is set as the reference, and the scale of energy density is 2.5 MJ m 3 for all energy profiles. To clearly show the energy path for polarization rotation, the corresponding two-dimensional (2D) energy profiles for a ½110–[100]–[110] polarization rotation path are given, where the x axis represents the angle between the polar vector and the [100] direction, and the y axis represents the minimal energy for a specific polar direction. The minimal energy states are represented by the orange balls in the 2D figures.

scattering becomes stronger at lower temperatures41,43, indicating the local symmetry of PNRs deviates more from the macroscopic symmetry, and/or there is an increased volume proportion of ‘non-collinear’ PNRs at lower temperatures. This is consistent with the scenario shown in Fig. 4a. In addition, convergent beam electron diffraction patterns obtained by transmission electron microscope revealed that the local symmetry could be lower (monoclinic or triclinic) than the long-range symmetry in PMN– PT crystals44,45. This is also confirmed in our simulations where the symmetry of some specific grids (local structure) could be totally broken by the interaction between the ferroelectric matrix and the PNRs, as shown in Fig. 3. It is worth contrasting our study with previous phonon studies on the contribution of PNRs. Phonon studies showed that the PNR–phonon interaction can induce a phase instability (softening of the transverse acoustic mode) in relaxor–PTs, thus this phenomenon was thought to play an important role in the ultrahigh piezoelectric response23. In our model, it is proposed that PNRs facilitate polarization rotation and enhance the shear piezoelectric response of the single domain state, as shown in Fig. 5 and Supplementary Fig. 15. This suggests a possible connection between the softening of the TA mode (shear) and the PNRs. In summary, based on combined experimental observations and phase-field simulations, the contributions of PNRs to the dielectric and piezoelectric properties of relaxor–PT crystals were quantified, accounting for 50–80% of the room-temperature values. At room temperature, the high piezoelectricity was attributed to the ‘collinear’ PNRs, which can facilitate polarization rotation and enhance the shear piezoelectric response of a singledomain state. The impact of such local structure on the macroscopic properties is not limited to relaxor–PT systems. It also exists in other material systems with structural fluctuations46–48, such as lead barium niobate Pb1 xBaxNb2O6 (ref. 47) and cubic pyrochlores Bi1.5Zn1.0Nb1.5O7 (ref. 48). Therefore, local structural engineering has the potential to be an effective and

general method for the future design of high-performance functional materials. Methods Sample preparation. The PMN–0.28PT and PMN–0.32PT single crystals were grown by a modified Bridgman method. The PZN–0.15PT crystals were grown by high-temperature flux method. The samples were oriented by real-time Laue X-ray back diffraction. Vacuum-sputtered gold electrodes were applied to the faces of the samples. The PMN–0.28PT, PMN–0.32PT and PZN–0.15PT single crystals were poled along their respective polar directions, that is, [111], [011] and [001]. To obtain single-domain states, a high-temperature poling approach was used49. For the PMN–0.28PT and PMN–0.32PT crystals, the samples were poled at 100 °C with a 1 MV m 1 E-field, and then cooled to room temperature with the E-field kept on. For the PZN–0.15PT crystals, the samples were poled at 250 °C with a 0.2 MV m 1 E-field, subsequently cooled to room temperature under E-field. After poling, the electrodes were removed and re-electroded on the ð110Þ, (100) and (100) faces for the PMN–0.28PT, PMN–0.32PT and PZN–0.15PT crystals, respectively, for allowing determination of the transverse dielectric permittivities and shear piezoelectric coefficients. The single-domain dielectric and piezoelectric coefficients were measured using the standard coordinate systems of single-domain crystals. For rhombohedral, orthorhombic and tetragonal single-domain crystals, the standard coordinate systems are X: ½110 Y: ½112 Z: [111], X: ½011 Y: [100] Z: [011] and X: [100] Y: [010] Z: [001], respectively. Characterization. Dielectric measurements. The temperature dependence of the various dielectric permittivities was measured using an LCR meter (HP4980) being connected to a computer-controlled temperature chamber (high temperature: home-made temperature controller; low temperature: Model 325 Cryogenic Temperature Controller, Lake Shore). Wide-frequency spectra (10 1–105 Hz) of dielectric response were measured at various temperatures by High Performance Modular Measurement System (Novocontrol Technologies). Piezoelectric measurements. Shear piezoelectric coefficients were determined by the resonance method, following the IEEE standard on piezoelectricity, using an HP4294 impedance analyser. X-ray diffraction measurements. X-ray analysis was carried out using an X-ray diffractometer (PANalytical) with Cu Ka radiation with temperature controller. The operation voltage and current were 40 kV and 40 mA, respectively. Synchrotron X-ray diffraction measurements. In situ synchrotron X-ray diffraction experiments were performed at Argonne National Laboratory beam line 11-BM from 300 down to 100 K. The wavelength is 0.414212 Å. The instrument resolution is Dd/d ¼ 0.00017, representing the state-of-the-art d-spacing resolution for diffraction measurements.


7

ARTICLE


Spontaneous polarization. The room-temperature spontaneous polarization was determined by the polarization-electric field loops, which were measured using a modified Sawyer–Tower circuit. To determine the variation of spontaneous polarization with respect to temperature, the pyroelectric current was measured using a KEITHLEY6485 picoammeter connected to a temperature chamber (Delta Design 2300). Phase-field simulation. The temporal evolution of the polarization field and the domain structure evolution are described by the time-dependent Ginzburg–Landau equations @Pi ðr; tÞ dF ; ¼ L @t dPi ðr; tÞ

ði ¼ 1; 2; 3Þ;

ð1Þ

where L is the kinetic coefficient, t the time, F the total free energy of the system and Pi(r, t) the polarization. The total free energy of the system includes the bulk free energy, the elastic energy, the electrostatic energy and the gradient energy: Z F¼ fbulk þ felas þ felec þ fgrad dV ð2Þ V

where V is the system volume of the PNR-ferroelectric composite, fbulk denotes the bulk free energy density, felas the elastic energy density, felec the electrostatic energy density and fgrad the gradient energy density. The detailed expressions and parameters of these energy densities are given in Supplementary Note 3. The impacts of volume fraction of PNRs (Supplementary Fig. 12 and Supplementary Note 4) and thermal noise (Supplementary Figs 22–25 and Supplementary Note 5) were also included and discussed in the phase-field simulations. In the computer simulations, we used two-dimensional 128 128 discrete grid points (three-dimensional: 64 64 64 grid points) and periodic boundary conditions. The grid space in real space is chosen to be Dx ¼ Dy ¼ 1 nm. Data availability. The data corresponding to this study are available from the first author and corresponding authors on request.

References 1. Jaffe, B., Cook, W. R. & Jaffe, H. Piezoelectric Ceramics (Academic, 1971). 2. Ahart, M. et al. Origin of morphotropic phase boundaries in ferroelectrics. Nature 451, 545–548 (2008). 3. Saito, Y. et al. Lead-free piezoceramics. Nature 432, 84–87 (2004). 4. Damjanovic, D. A morphotropic phase boundary system based on polarization rotation and polarization extension. Appl. Phys. Lett. 97, 062906 (2010). 5. Damjanovic, D. Comments on origins of enhanced piezoelectric properties in ferroelectrics. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 56, 1574–1585 (2009). 6. Park, S. E. & Shrout, T. R. Ultrahigh strain and piezoelectric behavior in relaxor based ferroelectric single crystals. J. Appl. Phys. 82, 1804–1811 (1997). 7. Service, R. F. Materials science—shape-changing crystals get shiftier. Science 275, 1878 (1997). 8. Kuwata, J., Uchino, K. & Nomura, S. Dielectric and piezoelectric properties of 0.91Pb(Zn1/3Nb2/3)O3 0.09PbTiO3 single crystals. Jpn J. Appl. Phys. 21, 1289–1302 (1982). 9. Zhang, S. J. & Li, F. High performance ferroelectric relaxor-PbTiO3 single crystals: status and perspective. J. Appl. Phys. 111, 031301 9 (2012). 10. Sun, E. W. & Cao, W. W. Relaxor-based ferroelectric single crystals: growth, domain engineering, characterization and applications. Prog. Mater. Sci. 65, 124–210 (2014). 11. Zhang, S. J. et al. Advantages and challenges of relaxor-PbTiO3 ferroelectric crystals for electroacoustic transducers—a review. Prog. Mater. Sci. 68, 1–66 (2015). 12. Ye, Z.-G. High-performance piezoelectric single crystals of complex perovskite solid solutions. MRS Bull. 34, 277–283 (2009). 13. Fu, H. X. & Cohen, R. E. Polarization rotation mechanism for ultrahigh electromechanical response in single-crystal piezoelectrics. Nature 403, 281–283 (2000). 14. Noheda, B. Structure and high-piezoelectricity in lead oxide solid solutions. Curr. Opin. Solid State Mater. Sci. 6, 27–34 (2002). 15. Kutnjak, Z., Petzelt, J. & Blinc, R. The giant electromechanical response in ferroelectric relaxors as a critical phenomenon. Nature 441, 956–959 (2006). 16. Jin, Y. M., Wang, Y. U., Khachaturyan, A. G., Li, J. F. & Viehland, D. conformal miniaturization of domains with low domain-wall energy: monoclinic ferroelectric states near the morphotropic phase boundaries. Phys. Rev. Lett. 91, 197601 (2003). 17. Cross, L. E. Relaxor ferroelectrics. Ferroelectrics 76, 241–276 (1987). 18. Bokov, A. A. & Ye, Z.-G. Recent progress in relaxor ferroelectrics with perovskite structure. J. Mater. Sci. 41, 31–52 (2006). 19. Kleemann, W. Relaxor ferroelectrics: cluster glass ground state via random fields and random bonds. Phys. Status Solidi B 251, 1993–2002 (2014). 8

20. Westphal, V., Kleemann, W. & Glinchuk, M. D. Diffuse phase transitions and random-field-induced domain states of the ‘‘relaxor’’ ferroelectric PbMg1/3Nb2/3O3. Phys. Rev. Lett. 68, 847–850 (1992). 21. Jeong, I. K. et al. Direct observation of the formation of polar nanoregions in Pb (Mg1/3Nb2/3)O3 using neutron pair distribution function analysis. Phys. Rev. Lett. 94, 147602 (2005). 22. Viehland, D., Jang, S. J., Cross, L. E. & Wuttig, M. Freezing of the polarization fluctuations in lead magnesium niobate relaxors. J. Appl. Phys. 68, 2916–2921 (1990). 23. Xu, G. Y., Wen, J. S., Stock, C. & Gehring, P. M. Phase instability induced by polar nanoregions in a relaxor ferroelectric system. Nat. Mater. 7, 562–566 (2008). 24. Manley, M. E. et al. Giant electromechanical coupling of relaxor ferroelectrics controlled by polar nanoregion vibrations. Sci. Adv. 2, e1501814 (2016). 25. Phelan, D. et al. Role of random electric fields in relaxors. Proc. Natl Acad. Sci. USA 111, 1754–1759 (2014). 26. Pirc, R., Blinc, R. & Vikhnin, R. Effect of polar nanoregions on giant electrostriction and piezoelectricity in relaxor ferroelectrics. Phys. Rev. B 69, 212105 (2004). 27. Farnsworth, S. M., Kisi, E. H. & Carpenter, M. A. Elastic softening and polarization memory in PZN-PT relaxor ferroelectrics. Phys. Rev. B 84, 174124 (2011). 28. Zhang, R., Jiang, B. & Cao, W. Single-domain properties of 0.67Pb(Mg1/3Nb2/3)O3-0.33PbTiO3 single crystals under electric field bias. Appl. Phys. Lett. 82, 787–789 (2003). 29. Damjanovic, D., Budimir, M., Davis, M. & Setter, N. Monodomain versus polydomain piezoelectric response of 0.67Pb(Mg1/3Nb2/3)O3-0.33PbTiO3 single crystals along nonpolar directions. Appl. Phys. Lett. 83, 527–529 (2003). 30. Li, F., Zhang, S. J., Xu, Z., Wei, X. Y. & Shrout, T. R. Critical property in relaxor-PbTiO3 single crystals—shear piezoelectric response. Adv. Funct. Mater. 21, 2118–2128 (2011). 31. Li, F., Jin, L., Xu, Z. & Zhang, S. J. Electrostrictive effect in ferroelectrics: an alternative approach to improve piezoelectricity. Appl. Phys. Rev. 1, 011103 (2014). 32. Noheda, B., Cox, D. E., Shirane, G., Gao, J. & Ye, Z. G. Phase diagram of the ferroelectric relaxor (1 x)PbMg1/3Nb2/3O3-xPbTiO3. Phys. Rev. B 66, 054104 (2002). 33. La-Orauttapong, D. et al. Phase diagram of the relaxor ferroelectric (1 x)Pb(Zn1/3Nb2/3)O3 x PbTiO3. Phys. Rev. B 65, 144101 (2002). 34. Chen, L. Q. Phase-field method of phase transitions/domain structures in ferroelectric thin films: a review. J. Am. Ceram. Soc. 91, 1835–1844 (1998). 35. Semenovskaya, S. & Khachaturyan, A. G. Development of ferroelectric mixed states in a random field of static defects. J. Appl. Phys. 83, 5125–5136 ð1998Þ: 36. Semenovskaya, S. & Khachaturyan, A. G. Ferroelectric transition in a random field: possible relation to relaxor ferroelectrics. Ferroelectrics 206, 157–180 (1998). 37. Wang, D. et al. Phase diagram of polar states in doped ferroelectric systems. Phys. Rev. B 86, 054120 (2012). 38. Wang, S., Yi, M. & Xu, B. X. A phase-field model of relaxor ferroelectrics based on random field theory. Int. J. Solids Struct. 83, 142–153 (2016). 39. Xu, G. Y., Zhong, Z., Hiraka, H. & Shirane, G. Three-dimensional mapping of diffuse scattering in Pb (Zn1/3Nb2/3) O3-xPb TiO3. Phys. Rev. B 70, 174109 (2004). 40. Welberry, T. R., Goossens, D. J. & Gutmann, M. J. Chemical origin of nanoscale polar domains in PbZn1/3 Nb2/3O3. Phys. Rev. B 74, 224108 (2006). 41. Xu, G. Y., Shirane, G., Gopley, J. R. D. & Gehring, P. M. Neutron elastic diffuse scattering study of Pb(Mg1/3Nb2/3)O3. Phys. Rev. B 69, 064112 (2004). 42. Manley, M. E. et al. Phonon localization drives polar nanoregions in a relaxor ferroelectric. Nat. Commun. 5, 3683 (2014). 43. Xu, Z. J. et al. Freezing of the local dynamics in the relaxor ferroelectric [Pb(Zn1/3Nb2/3)O3]0.955[PbTiO3]0.045. Phys. Rev. B 86, 144106 (2012). 44. Kim, K. H., Payne, D. A. & Zuo, J. M. Symmetry of piezoelectric (1 x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 (x ¼ 0.31) single crystal at different length scales in the morphotropic phase boundary region. Phys. Rev. B 86, 184113 (2012). 45. Kim, K. H., Payne, D. A. & Zuo, J. M. Determination of fluctuations in local symmetry and measurement by convergent beam electron diffraction: applications to a relaxor-based ferroelectric crystal after thermal annealing. J. Appl. Cryst. 46, 1331–1337 (2013). 46. Keen, D. A. & Goodwin, A. L. The crystallography of correlated disorder. Nature 521, 303–309 (2015). 47. Guo, R., Bhalla, A. S., Randall, C. A. & Cross, L. E. Dielectric and pyroelectric properties of the morphotropic phase boundary lead barium niobate (PBN) single crystals at low temperature (10–300 K). J. Appl. Phys. 67, 6405–6410 (1990).


ARTICLE


48. Kamba, S. et al. Anomalous broad dielectric relaxation in Bi1.5Zn1.0Nb1.5O7 pyrochlore. Phys. Rev. B 66, 054106 (2002). 49. Li, F., Wang, L. H., Jin, L., Xu, Z. & Zhang, S. J. Achieving single domain relaxorPT crystals by high temperature poling. CrystEngComm 16, 2892–2897 (2014). 50. Haun, M. J., Furman, E. S., Jang, J. & Cross, L. E. Thermodynamic theory of the lead zirconate-titanate solid-solution system, part I-V: theoretical calculations. Ferroelectrics 99, 13–86 (1989). 51. Bellaiche, L., Garcıá, A. & Vanderbilt, D. Finite-temperature properties of Pb(Zr1 xTix)O3 alloys from first principles. Phys. Rev. Lett. 84, 5427 (2000).

Acknowledgements This work was supported by ONR (Grants Nos. N00014-12-1-1043 and N00014-12-11045). F.L. acknowledges the support by the Office of China Postdoctoral Council, the National Natural Science Foundation of China (Grant Nos. 51572214 and 51372196), the Natural Science Foundation of Shaanxi province (2015JQ5135) and the 111 Project (B14040). S.Z. thanks to the support of ONRG (Grant No. N62909-16-1-2126). T.Y. acknowledges the support of National Science Foundation under the Grant Number DMR-1410714. L.-Q.C. is supported by U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-FG0207ER46417. F.L. and G.L. thank Dr Saul Lapidus for the technical supports of synchrotron X-ray diffraction experiments. J.L.W. and Z.C. thank Qingyong Ren for the cryogenic X-ray diffraction measurements. The use of the Advanced Photon Source was supported by the U.S. Department of Energy, Office of Science and Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.

Author contributions The project was conceived and designed by F.L., S.Z., Z.X., T.R.S. and L.-Q.C.; F.L. prepared crystal samples, performed the dielectric and piezoelectric experiments, and the phase-field simulations; T.Y. and J.J.W. assisted in phase-field simulations; Z.-G.Y. assisted in the analysis of experimental dielectric responses and discussed with F.L. on

the concepts of PNRs; G.L. performed synchrotron X-ray diffraction experiments on powders; N.Z. did the Rietveld refinements of the structures. J.L.W. and Z.C. performed the cryogenic X-ray diffraction experiments for crystals; J.L. and S.Z. grew the crystals; F.L., S.Z., Z.-G.Y., T.R.S. and L.-Q.C. prepared the manuscript.

Additional information Supplementary Information accompanies this paper at http://www.nature.com/ naturecommunications Competing financial interests: The authors declare no competing financial interests. Reprints and permission information is available online at http://npg.nature.com/ reprintsandpermissions/ How to cite this article: Li, F. et al. The origin of ultrahigh piezoelectricity in relaxorferroelectric solid solution crystals. Nat. Commun. 7, 13807 doi: 10.1038/ncomms13807 (2016). Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

r The Author(s) 2016


9

0

0.1

-10k 100

200

300

400

500

0.2 5.0k 0.1 0.0

0.0

0

100

800

0.3 0.2

=120

200

Loss

0.4

0.1 kHz 1 kHz 10 kHz

400

0

0.1

0

50

100

150

200

250

300

400

500

10k

0.2

0

100

300

0.0

0.4


0.3

=110

400

0.2

200 0

0.1

0

50

100

150

200

300

400

500

0.0

250

300

0.0

0.5 PZN-0.15PT 0.4

600

0.3

=150

400 0.1 kHz 1 kHz 10 kHz

200

0

0

50

100

150

200

250

300

0.2 0.1 0.0

Temperature (K)

Temperature (K)

Temperature (K)

200

800

0.5

600

0.4

0

0.0

PMN-0.32PT 800

0.6


Temperature (K)

1000

Dielectric permittivity (33/0)


0.5 PMN-0.28PT

600

200

0.8 20k

Temperature (K)

Temperature (K)


0

0.3

1.0 PZN-0.15PT

Loss

0.2

10k


10.0k

30k

Loss

0.3

0.4

15.0k



20k

0.5 PMN-0.32PT

20.0k

Loss

30k

Loss

0.4

40k

25.0k

Loss

PMN-0.28PT



0.5 50k

Supplementary Figure 1. Temperature dependence of the longitudinal dielectric permittivity for relaxor-PT crystals. a, rhombohedral PMN-0.28PT; b, orthorhombic PMN-0.32PT, and c, tetragonal PZN-0.15PT crystals. Figs. d, e and f are enlarged low temperature parts for Figs. a, b and c, respectively. Compared with the transverse dielectric response, the abnormal variation of longitudinal dielectric response at cryogenic temperatures is quite low (in both absolute and relative values). The variation of longitudinal dielectric response only accounts for 15-25% of the respective room temperature values. Furthermore, it is speculated that this variation is not related to the intrinsic property of the longitudinal response, but associated with inadequate poling. For example, in the [001] poled PZN-0.15PT crystal, if there are 4% perpendicular domains ([010], [100], [1̅00], or [01̅0] domains) remained, the transverse dielectric response variation of these domains can account for the change of longitudinal permittivity along [001] direction at low temperature. (4%×4000=160, where 4000 is the variation of transverse permittivity, as shown in Fig. 1f of the main paper)

0.4

6000 0.3

5000 4000 =2900

3000 2000 1000 0

0

50

100

150

200

100 Hz 0.2 200 Hz 500 Hz 1000 Hz 2000 Hz 0.1 5000 Hz 10000 Hz 20000 Hz 0.0 250 300

Loss factor


7000

Temperature (K)

Supplementary Figure 2. The temperature dependence of the transverse dielectric permittivity ε11/ε0 for PMN-0.32PT crystals. 1

2.0

1

-4.0



-4.4

lg(1/-1/m)

1/r 10-4

Curie-Weiss Law

1.5

1.0

0.5

TC 300

350

400

450



1

m



(T  Tm ) γ C =1.5

-4.8

-5.2

-5.6

500

0.6

550

0.8

1.0

1.2

1.4

1.6

1.8

lg(T-Tm)

Temperature (K)

Supplementary Figure 3. Relaxor characteristic of the PZN-0.15PT crystal. a, Temperature dependence of the reciprocal of relative dielectric constant for a [001]-oriented PZN-0.15PT crystal; b, lg(1/ε-1/εm) vs. lg(T-Tm) figure at temperatures above Tm for obtaining parameter γ. γ is the degree of diffuseness. A higher value of γ represents a higher level of relaxor behavior. For classical ferroelectrics, the value of γ is 1, while for relaxor, the value is 2. The measured γ value of the PZN0.15PT crystal is 1.5.

PS (C m-2)

0.5 0.4 0.3

PMN-0.28PT PMN-0.32PT PZN-0.15PT

0.2 0.1 0.0 100

150

200

250

300

Temperature (K) Supplementary Figure 4. The spontaneous polarization with respect to temperature for the PMN0.28PT, PMN-0.32PT and PZN-0.15PT crystals. All the crystals were poled along their respective polar directions.

2

300K Measured Refined Difference 100K Measured Refined Difference

5

10

15

20

2θ (deg.)

25

4.024

0.20

4.023

0.15

4.022

0.10

4.021

0.05

4.020

100

150

200

250

300

90 - (deg.)

a (Angstron)

Supplementary Figure 5. Synchrotron powder XRD patterns for the PMN-0.28PT. a, at 300 K; b, at 100 K.

0.00

Temperature (K)

Supplementary Figure 6. Temperature-dependent lattice parameters for the PMN-0.28PT.

3

Supplementary Figure 7. Polarized light microscopic pictures for [001] oriented PMN-0.28, PMN0.32 and PZN-0.15PT crystals at temperatures of 100 K and 300 K. The domain structures of relaxorPT crystals were observed using a polarizing light microscope (PLM) (Olympus BX51) with a LINKAM heating-cooling stage. The samples were polished to 50-70 μm and annealed at 573 K for domain observations.

30.0k

0.15

Dielectric permittivity 22/0

PMN-0.32PT

PMN-0.32PT

Loss factor

25.0k 20.0k 15.0k 10.0k 5.0k 10-1

120 K 180 K 240 K 100

150 K 210 K 270 K 101

300 K 102

103

0.10

150 K 210 K 270 K

0.05

0.00 10-1

104

120 K 180 K 240 K 300 K

100

101

102

103

104

Frequency (Hz)

Frequency (Hz)

0.15

PZN-0.15PT 5k

Loss factor


6k

4k

3k

2k 100

120 K 180 K 240 K 101

150 K 210 K 270 K 102

103

0.10

PZN-0.15PT 120 K 180 K 240 K 300 K

150 K 210 K 270 K

0.05

300 K 104

Frequency (Hz)

0.00 100

105

101

102

103

104

105

Frequency (Hz)

Supplementary Figure 8. Frequency spectra of transverse dielectric permittivity and loss factor for relaxor-PT crystals. a, dielectric permittivity of PMN-0.32PT; b, loss factor of PMN-0.32PT; c, dielectric permittivity of PZN-0.15PT; d, loss factor of PZN-0.15PT. 4

5.0k PZN-0.15PT @ 120 K

4k



5k

= 1980 = 2700 f1= 46.9 Hz

3k

f2= 4.45107 Hz Measured data Fitting data

2k 100

101

102

103

104

105

PZN-0.15PT @ 150 K

= 3340

4.5k

= 1470 f1= 7.25103 Hz f2= 6.53107 Hz

4.0k

Measured data Fitting data 3.5k 100

106

101

102

Frequency (Hz)

103

104

105

106

Frequency (Hz)



18.0k PMN-0.32PT @ 120 K

14.0k

PMN-0.32PT @ 150 K

16.0k

12.0k

= 4950 10.0k 8.0k

f2= 2.19108 Hz

100

101

= 7780 f1= 48.1 Hz f2= 2.75108 Hz

12.0k

Measured data Fitting data 6.0k 10-1

= 9190

14.0k

= 9640 f1= 1.19 Hz

102

103

104

Measured data Fitting data

10.0k 100

105

101

Frequency (Hz)

102

103

104

105

Frequency (Hz)

Supplementary Figure 9. Measured and fitted transverse dielectric permittivity vs. frequency for PMN-0.32PT and PZN-0.15PT crystals. a, PZN-0.15PT at 120 K; b, PZN-0.15PT at 150 K; c, PMN0.32PT at 120 K; d, PMN-0.32PT at 150 K.

200

800

100 Hz 200 Hz 500 Hz 1 kHz 2 kHz 5 kHz 10 kHz 20 kHz

50

400

50

100

150

200

250

0

300

0

50

Temperature (K)

10

7

4

f0=(1.7~8.7)1012 Hz Ea/kB=1976 77 K 0.010

0.011

0.012

150

200

10

ln(f) (Hz)

ln(f) (Hz)

8

5

100

250

0

300

Temperature (K)

measrued data Fitting

9

200

0.013

9

9

8

8

7

f0=(0.24~1.7)1012 Hz

5

Ea/kB=1861 82 K

4

0.009

0.010

0.011

100

150

200

250

300

0.012


7 6 5

1/Tm (K-1)

1/Tm (K-1)

50

10


6

0

Temperature (K)

ln(f) (Hz)

0

100 Hz 200 Hz 500 Hz 1 kHz 2 kHz 5 kHz 10 kHz 20 kHz

300

100

200

0

6

2 kHz 5 kHz 10 kHz 20 kHz

11''/0

100

600

22''/0

11''/0

150

400 100 Hz 200 Hz 500 Hz 1 kHz

4

f0=(1.0~3.4)1012 Hz Ea/kB=2419 68 K 0.0075 0.0080 0.0085 0.0090 0.0095 0.0100

1/Tm (K-1)

Supplementary Figure 10. Inference of the size of PNRs by Arrhenius law for relaxor-PT crystals. a, b, and c are the temperature dependence of the imaginary part (ε″/ε0) of the transverse dielectric permittivity for PMN-0.28PT, PMN-0.32PT, and PZN-0.15PT crystals, respectively. d, e, and f are the Tms (temperature of maximum ε″/ε0) versus frequency for PMN-0.28PT, PMN-0.32PT and PZN0.15PT crystals, respectively. 5

Supplementary Figure 11. Microstructure of a [110]-poled PNR-ferroelectric composite at 150 K (poled at room temperature, i.e., 300 K), where the x- and y-axes represent the [100] and [010] directions respectively. The scale of this simulation is 512512 nm. The color bar denotes the angle between the polar vector and the [100] direction. Selected orientations are given in the color bar. The average polar directions of the macro-domains are represented by yellow and green arrows.

15.0k

Dielectric permittivit 11/0

Dielectric permittivit 11/0

15.0k

5% 7.5% 9%

10.0k

5.0k

0.0

0

100

200

300

400

500

10.0k

5.0k

0.0

600

9% 12.5% 15%

0

100

200

300

400

500

600

Temperature (K)

Temperature (K)

Supplementary Figure 12. Simulated transverse dielectric permittivity ε11/ε0 versus temperature for the [100]-poled PNR-ferroelectric composites at different volume fractions of the PNRs. a, volume fractions of 5%, 7.5% and 9%; b, volume fractions of 9%, 12.5% and 15%.

6

Supplementary Figure 13. Microstructure of a [001]-poled PNR-ferroelectric composite with respect to temperature. The translucent-blue color represent the polar vectors along [001] direction. The cyan, orange, red and yellow colors represent the polar vector along [011], [101], [01̅1] and [1̅01] directions, respectively. Actually, the exactly tetragonal- and orthorhombic-type polar vectors will not exist due to the interaction between ferroelectric matrix and PNRs. In this figure, the angle deviation of 10o is allowed for plotting different color regions. The x-, y- and z-axes represent the [100], [010] and [001] directions, respectively.

10000

22/0: composite

10000

Dielectric permittivity

Dielectric permittivity

33/0 of composite

11/0: composite

12000

11/0: tetragonal matrix

8000 6000 4000 2000

33/0 of matrix

8000 6000 4000 2000 0

0

0

100

200

300

400

500

600

0

Temperature (K)

100

200

300

400

500

600

Temperature (K)

Supplementary Figure 14. Simulated dielectric permittivites versus temperature for a [001]-poled PNR-ferroelectric composite. a, transverse dielectric permittivies; b, longitudinal dielectric permittivities. The dielectric permittivities ε11/ε0, ε22/ε0, and ε33/ε0 are measured along [100], [010] and [001] direction, respectively.

7

Supplementary Figure 15. Microstructure evolution of a [100]-poled PNR-ferroelectric composite under shear stress σ12. The x- and y-axes represent [100] and [010] directions, respectively. The color bar denotes the angle (unit: degree) between the polar vector and the [100] direction. a, b, c and d are the microstructures under zero stress at 50 K, 150 K, 250 K and 350 K, respectively. e, f, g and h are the microstructures under 35 MPa at 50 K, 150 K, 250 K and 350 K, respectively.

Supplementary Figure 16. The calculated temperature dependence of shear piezoelectric coefficient d15 for a PNR-ferroelectric composite and ferroelectric matrix. The amplitude and period of the applied stress is 10 MPa and 105 time steps, respectively. Compared with dielectric response, piezoelectric response show a slight decrease in the temperature range of 200 K-400 K. This is due to the decrease of spontaneous polarization with increasing temperature, as shown in the inset of this figure.

8

Supplementary Figure 17. The E-field induced microstructure variation for a [001]-poled PNRferroelectric composite at 350 K. a, under zero E-field; b, under a 1 MV m-1 E-field along the [100] direction; c, under a 1 MV m-1 E-field along the [010] direction. The translucent-blue color represents the polar vectors along [001] direction. The cyan and orange colors represent the polar vector along the [011] and [101] directions, respectively. The x-, y- and z-axes represent the [100], [010] and [001] directions, respectively.

Supplementary Figure 18. E-field induced microstructure variation for a [001]-poled PNRferroelectric composite at 150 K. a, under zero E-field; b, under a 1 MV m-1 E-field along the [100] direction; c, under a 1 MV m-1 E-field along the [010] direction. The translucent-blue color represents the polar vectors along [001] direction. The cyan, orange, red and yellow colors represent the polar vector along the [011], [101], [01̅1] and [1̅01] directions, respectively. The x-, y- and z-axes represent the [100], [010] and [001] directions, respectively.

9

120 K

Polarization (10-3C m-2)


4

Loss = 10.5% 2

Permittivity = 3650

0

-2

-4 -1.0

-0.5

0.0

0.5

4

150 K Loss = 2.8%

2

Permittivity = 4350

0

-2

-4 -1.0

1.0

200 K Loss = 0.6%

2

Permittivity = 4500

0

-2

-4 -1.0

-0.5

0.0

0.5

4

-0.5

250 K Loss = 1.0%

2

Permittivity = 4700

0

-2

-4 -1.0

1.0

-0.5

0.0

0.0

0.5

Electric field (kV cm-1)


4




0.5

4

300 K

2

Permittivity = 4950

Loss = 2.4%

0

-2

-4 -1.0

1.0

-0.5

0.0

0.5

1.0




1.0

Supplementary Figure 19. Transverse polarization-electric field (PE) responses for PZN-0.15PT crystals at various temperatures. a, 120 K; b, 150 K; c, 200 K; d, 250 K; e, 300 K. The test frequency is 100 Hz. 8

120 K 4



8

Loss = 14% Permittivity = 11000

0

-4

-8

150 K 4

Loss = 8.4% Permittivity = 14500

0

-4

-0.4

-0.2

0.0

0.2

-8

0.4

-0.4



0

-4

-8

-0.4

-0.2

0.0

0.2


0.4

4


0

-4

-8

0.2

0.4

8

250 K


4

0.0

8

200 K



8

-0.2


-0.4

-0.2

0.0

0.2


0.4

300 K 4


0

-4

-8

-0.4

-0.2

0.0

0.2

0.4


Supplementary Figure 20. Transverse PE responses (22-mode) for PMN-0.32PT crystals at various temperatures. a, 120 K; b, 150 K; c, 200 K; d, 250 K; e, 300 K. The test frequency is 100 Hz.

10

50K 100K 150K 200K 250K 300K 350K

Polarization (C m-2)

0.020 0.015

Temperature increase

0.010 0.005 0.000

-1.0

-0.5

0.0

0.5

1.0


Supplementary Figure 21. Simulated PE of transverse dielectric response for the PNR-ferroelectric composite. The driving field is 1 kV cm-1. The reduced period of ac electric field is 105 time steps. The simulated results are similar to the measured one, as shown in Supplementary Figures 19-20.

Supplementary Figure 22. Simulated domain morphology for the [100]-poled tetragonal matrix at different magnitudes of thermal noise. a, b, and c represent the reduced thermal noise magnitude of 0.1, 1 and 5, respectively. The color bar denotes the angle between the polar vectors and the [100] direction. Temperature is 350 K.

Supplementary Figure 23. Simulated domain morphology for a [100]-poled PNR-ferroelectric composite at 350 K with the reduced thermal noise magnitude of 5. a, b and c are obtained at different time step, being 1000, 2000 and 3000, respectively. The color bar denotes the angle between the polar vectors and the [100] direction.

11

Supplementary Figure 24. Simulated domain morphology for a [100]-poled PNR-ferroelectric composite at 150 K, 200 K and 250 K with the reduced thermal noise magnitude of 5. The microstructures are shown at different time steps, being the 1000, 2000 and 3000. The color bar denotes the angle between the polar vectors and the [100] direction.

15.0k



15.0k matrix Magnitude of noise = 0.1 Magnitude of noise = 1 Magnitude of noise = 5

10.0k

10.0k

5.0k

0.0 0

100

200

300

400

matrix Magnitude of noise = 0.1 Magnitude of noise = 1 Magnitude of noise = 5

500

600

5.0k

0.0 0

Temperature (K)

100

200

300

400

500

600

Temperature (K)

Supplementary Figure 25. Simulated temperature dependence of dielectric permittivities for PNRferroelectric composites under different levels of thermal noise. a, the transverse dielectric permittivity ε11/ε0; b, the longitudinal dielectric permittivity ε33/ε0.

12

Supplementary Table 1. Dielectric and piezoelectric coefficients of relaxor-PT and PZT crystals. The data for PZT crystals is from the phenomenological calculation. Data for single domain state

Relaxor crystals

ε33/ε0

ε11/ε0

d33 (pC N-1)

d15 (pC N-1)

PMN-0.28PT(R)*

750

5200

90

2900

PMN-0.32PT (O)*

800

6000 (ε22/ε0~22000)

200

3600 (d24 ~2200)

PZN-0.08PT (R) 1

700

11000

90

6000

PZN-0.15PT (T)*

600

5500

380

900

PMN-0.12PT (T) *

870

12000

520

1500

Pb(Zr0.52Ti0.48)O3

(R)2-3

450

1200

130

1350

Pb(Zr0.48Ti0.52)O3(T)2-3

350

1500

300

500

* data obtained in this work. Supplementary Table 2. Refined structural parameters for the rhombohedral PMN-0.28PT solid solution. The Rietveld refinement was carried out using TOPAS Academic4, and the anisotropic strain broadening function proposed by Stephens5, was incorporated into the refinements. It can be seen that the long-range structure of PMN-0.28PT is R3m over the temperature range of 100-300K. Temp. [K]

R3m

100

125

150

200

250

300

a [Å]

4.02090(1)

4.02111(1)

4.02124(1)

4.02159(1)

4.02194(1)

4.02239(1)

alpha

89.8228(7)

89.8284(7)

89.8352(7)

89.8513(7)

89.8716(7)

89.8942(5)

Δ(Pb)

0.0270(3)

0.0262(2)

0.0260(3)

0.0254(3)

0.0224(4)

0.0209(5)

Δ(Nb/Mg/Ti)

0.0146(3)

0.0142(3)

0.0143(3)

0.0139(4)

0.0122(5)

0.0111(6)

Rwp [%]

11.16

11.17

11.26

11.39

11.33

11.29

Rbragg [%]

1.72

1.56

1.82

2.02

2.02

1.96

13

Supplementary Note 1: Analysis of frequency-dependent transverse dielectric responses for relaxor-PT crystals. The frequency-dependent dielectric permittivity and loss of PMN-0.32PT and PZN-0.15PT crystals are shown in Supplementary Figure 8. In the frequency range of 10-1-105 Hz, frequency dependent behaviors were clearly observed at the temperature below 200 K. According to Debye theory, the relaxational dielectric spectrum can be described by means of the distribution function of the relaxation time g(τ)6:

 ( )      



0

g ( ) d ln( ) 1   2 2

(1)

where g(τ) is normalized by the following equation: 

 g ( )d ln( )  1

(2)

0

In this study, we assume there is a strong Debye relaxation with a constant distribution between certain upper value (τ1) and lower value (τ2). Thus, the g(τ) is assumed to be:

1 / ln( 1 /  2 ) g ( )   0

 2    1   1 ,   2

(3)

In this condition, 𝜀∞ represents the dielectric permittivity at frequency much higher than 1/τ2. Δ𝜀 represents the contribution of the relaxation to the static permittivity. Based on Eqs. 1-3, the frequency dependent dielectric permittivity can be expressed as:

 ( )    

 1 1   2 12 [ln( 1 /  2 )  ln( )] ln( 1 /  2 ) 2 1   2 22

(4)

The dielectric spectra of PMN-0.32PT and PZN-0.15PT crystals are fitted by using Eq. 4 and given in Supplementary Figure 9. It can be seen that the upper frequency limit is around 107-108 for PZN0.15PT and PMN-0.32PT, being much lower than the frequency of phonon modes (1012-1014 Hz). The observed relaxation frequency range (100-108) is similar to that of dipole reorientation. This indicates that some interface motion or dipoles’ switching significantly contributes to the dielectric response at 120 K and 150 K for relaxor-PT crystals.

Supplementary Note 2: Inference of the size of PNRs by Arrhenius law. Supplementary Figure 10 a, b and c show the temperature dependence of the imaginary part (ε″/ε0) of the transverse dielectric permittivity for PMN-0.28PT, PMN-0.32PT and PZN-0.15PT crystals, respectively. Supplementary Figure 10 d, e and f show the temperature of maximum ε″/ε0 versus frequency. It can be seen that ln(f) and 1/Tm follow a linear relationship, which can be expressed by Arrhenius law: 14

f  f 0 exp[  Ea /( k BTm )]

(5)

where f is the measurement frequency, Ea the activation energy, kB the Boltzmann constant, f0 the attempt frequency, and Tm the temperature of maximum ε″/ε0. By fitting Eq. 5, it is found that the activation energy Ea is around 2000kB for PMN-0.28PT, PMN-0.32PT and PZN-0.15PT crystals. Activation energy Ea can be expressed as ΔGV, where ΔG is the energy barrier density for reorientation of a PNR and V the volume of a single PNR. According to Landau-Devonshire formalism of free energy 2,7, the typical values of ΔG are in the range of 105~107 J m-3. Based on this typical value of ΔG, the volume of a PNR (V) is calculated to be in the range of (27.4~274)×10-27 m3. 3

Thus, the size of PNRs ( √𝑉) is inferred to be in the range of 1.4~6.5 nm.

Supplementary Note 3: Framework and numerical simulation of the phase-field model (1) Framework of the phase field model The temporal evolution of a polarization field is described by the Time-Dependent Ginzburg-Landau (TDGL) equation, Pi (r, t ) F  L  i (r, t ) , (i  1,2,3), t Pi (r, t )

(6)

where L is the kinetic coefficient, F, the total free energy of the system, and Pi (r, t ) is the polarization. F / Pi (r, t ) is the thermal dynamic driving force for the spatial and temporal evolution of Pi (r, t ) . i (r, t ) represents the effect of thermal noise, which is a Gaussian random fluctuation satisfying:

 i (r, t )  0 and  i (r, t ) j (r' , t ' )  Tm (r  r' ) (t  t ' )

(7)

where T is the temperature, m represents the amplitude of thermal noise, δij is the Kronecker delta,  (r  r' ) and  (t  t ' ) are the delta function. The bracket denotes an average value. The total free energy of the system includes the bulk free energy, elastic energy, electrostatic energy, and the gradient energy: F   [ f bulk  f elas  f elec  f grad ]dV

(8)

V

where V is the system volume of the PNR-ferroelectric composite, fbulk denotes the Landau bulk free energy density, felas the elastic energy density, felec the electrostatic energy density and fgrad the gradient energy density. The bulk free energy density is expressed by Landau theory, i.e. f bulk  1 ( P12  P22  P32 )  11( P14  P24  P34 )  12 ( P12 P22  P22 P32  P32 P12 )  111( P16  P26  P36 )  112[ P14 ( P22  P32 )  P24 ( P12  P32 )  P34 ( P12  P22 )]  123P12 P22 P32 15

(9)

where α1, α11, α12, α111, α112 and α123 are Landau energy coefficients. The values of these coefficients determine the thermodynamic behaviors of the bulk phases (paraelectric or ferroelectric). In our simulation work, the difference between the ferroelectric matrix and PNRs are reflected in the fbulk, where different Landau coefficients were used for matrix and PNRs, respectively. The gradient energy density, associated with the formation and evolution of domain walls, can be expressed as: 1 f grad  G11 ( P12,1  P22, 2  P32,3 )  G12 ( P1,1 P2, 2  P2, 2 P3,3  P1,1 P3,3 ) 2 1  G44 [( P1, 2  P2,1 ) 2  ( P2,3  P3, 2 ) 2  ( P1,3  P3,1 ) 2 ] 2 1 '  G44 [( P1, 2  P2,1 ) 2  ( P2,3  P3, 2 ) 2  ( P1,3  P3,1 ) 2 ] 2

(10)

where Gij are gradient energy coefficients. Pi,j denote ∂Pi/∂rj. The corresponding elastic energy density can be expressed as: 1 1 f elas  cijkl eij ekl  cijkl ( ij   ij0 )( kl   kl0 ) 2 2

(11)

where cijkl is the elastic stiffness tensor, εij the total strain,  kl0 the electrostrictive stress-free strain, i.e.,  kl0  Qijkl Pk Pl . The electrostatic energy density is given by: 1 f elec   Eiin Pi  Eiex Pi 2

(12)

where Eiin is the E-field induced by the dipole moments in the specimen. The detailed expression of Eiin can be found in Ref. 8. Eiex is an applied external E-field.

(2) Material constants and numerical simulation parameters of PNR-ferroelectric composite A semi-implicit Fourier-spectral method is adopted for numerically solving the TDGL equation9. Based on the experimental data (polarization and dielectric response) and the parameters of typical perovskite ferroelectrics (Pb(Zr,Ti)O3 and BaTiO3)2,10, the following parameters are adopted in the present simulation. For the ferroelectric matrix, the Landau free energy parameters are: , , , 1  2.38(T  468)  105 C 2m2 N 12  3.37  108 C 4m6 N 11  5.56  107 C 4m6 N 9 6 10 9 6 10 8 6 10 111  3.23  10 C m N , 112  1.02  10 C m N , 123  1.21  10 C m N . In the whole temperature range, there is only a Cubic-Tetragonal phase transition at 468 K for the ferroelectric matrix. The spontaneous polarization of matrix is about 0.39 C m-2 at 300 K, being similar to the experimental value of the PZN-0.15PT crystal. At 20 K, the dielectric permittivity ε11/ε0 and ε33/ε0 are 850 and 160 respectively for ferroelectric matrix, being similar to the measured data of PZN0.15PT (At temperature of 20 K and frequency of 100 Hz, the measured ε11/ε0 and ε33/ε0 are 930 and 250, respectively.) For PNRs, the Landau free energy parameters are: 1  2.2(T  620)  105 C 2m2 N , , , , 11  2.67  108 C 4m6 N 111  3.33  109 C 6m10N 12  2.67  108 C 4m6 N 10 6 10 8 6 10 112  3.33  10 C m N , 123  3.0  10 C m N , where T is the temperature in Kelvin. In the whole temperature range, there is only a Cubic-orthorhombic phase transition around 620 K for PNRs. The spontaneous polarization of PNR is 0.41 C m-2 at 300 K without consideration of interactions from the matrix. The elastic constants and electrostrictive coefficients are set to be the same for the matrix 16

D and PNRs: s11D  20  1012 m2 N 1 , s12D  7.5  1012 m2 N 1 , s44  20  1012 m 2 N 1 , Q11  0.089C 2 m 4 , Q12  0.030C 2m4 , Q44  0.034C 2m4 (the elastic constants refer to the data of PZN-0.12PT crystal11 and the electrostrictive coefficients refer to data for PbTiO3 2). The background dielectric permittivity (kb), associated with the contribution of the hard mode to the permittivity12, is set to be 100×ε0. In the computer simulations, we employed 2D 128×128 discrete grid points (3D: 64×64×64 grid points) and periodic boundary conditions. The grid space in real space is chosen to be x  y  z  1nm . The gradient energy coefficients are chosen to be G11 / G110  1.5 , G12 / G110  0 , 11 2 4 13 G44 / G110  G44 / G110  0.75 , where G110  7.04  10 C m N . Based on these parameters, the simulated width of domain walls was found to be 1-2 nm, being consistent with existing experimental measurements14,15. The time step for integration is t / t0  0.01 , where t0  1 / L1 T 300K . For 2D simulation, Pz is set at zero.

Supplementary Note 4: The effect of PNRs volume fraction on the transverse dielectric response for the PNR-ferroelectric composite. The temperature dependent dielectric permittivities were simulated under different PNR volume fractions for a PNR-ferroelectric composite, as shown in Supplementary Figure 12. It was found that the increase of volume fraction of PNRs may bring two effects. First, higher volume fraction of PNRs may result in higher dielectric response when most of the PNRs are unstable or in “collinear” state. Secondly, higher volume fraction of PNRs could result in stronger interaction among PNRs due to the decrease of average distances among PNRs. Thus, PNRs are more stable for higher volume fractions, where higher temperatures are needed to make PNRs unstable and/or in “collinear” state. As shown in Supplementary Figure 12, a composite with higher volume fraction of PNRs shows higher dielectric response at elevated temperature region, but at the cost of temperature stability.

Supplementary Note 5: The effect of thermal fluctuations on the dielectric responses for the PNR-ferroelectric composite. Thermal fluctuation is considered in phase-field simulation, where it is randomly changed for every step. The magnitude of thermal noise is proportional to the temperature, as given in Eq. 7. Supplementary Figure 22 exhibits the impact of thermal noise on the polarization of the tetragonal matrix at 350 K. The maximum angle between polar vector and [100] direction is about 7 o and 20o for the reduced thermal noise magnitude of 1 and 5, respectively. Obviously, the thermal noise with magnitude of 5 is extremely large, which will be used to check the extreme condition in our simulations. Supplementary Figure 23 shows the microstructure of PNR-ferroelectric composite at 350 K with thermal noise magnitude of 5. Due to the impact of thermal noise, the polar vector for specific grids varied at different time. The maximum angle between polar vector and [100] direction is about 30 o. The maximum angle generally comes from the “collinear” PNRs, since they are easy to be rotated compared to the tetragonal matrix. Supplementary Figure 24 shows microstructure of a PNRferroelectric composite at 150-250 K with reduced thermal noise magnitude of 5. It can be seen that the PNRs are not affected too much by thermal fluctuation at different time steps. Supplementary Figure 25 shows the calculated temperature-dependent dielectric permittivities for 17

PNR-ferroelectric composite at different levels of thermal fluctuation. Over all, the temperature dependent properties aren’t affected significantly by thermal noise, though a bit higher fluctuation is observed for the magnitude of 5.

18

Supplementary references 1.

Jin, J., Rajan, K. K. & Lim, L.C. Properties of single domain Pb(Zn1/3Nb2/3)O3-(6-7)%PbTiO3 single crystal. Jpn. J. Appl. Phys. 45, 8744-8747 (2006).

2.

Haun, M. J., Furman, E. S., Jang, J. &. Cross L. E. Thermodynamic theory of the lead zirconate-

3.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

15.

titanate solid-solution system, part I-V: Theoretical calculations. Ferroelectrics 99, 13-86 (1989). Du, X. H., Zheng, J., Belegundu, U., & Uchino, K. Crystal orientation dependence of piezoelectric properties of lead zirconate titanate near the morphotropic phase boundary. Appl. Phys. Let. 72, 2421-2423 (1998). Coelho, A. A. A bound constrained conjugate gradient solution method as applied to crystallographic refinement problems. J. Appl. Cryst. 38, 455-461 (2005). Stephens, P. W. Phenomenological model of anisotropic peak broadening in powder diffraction. J. Appl. Cryst. 32, 281-289 (1999). Jonscher, A. K. Dielectric Relaxation in Solids (London: Chelsea Dielectric, 1983). Bell, A. J. Calculations of dielectric properties from the superparaelectric model of relaxors. J. Phys.: Condens. Matter 5, 8773-8792 (1993). Hu, H. L. & Chen, L. Q. Three-dimensional computer simulation of ferroelectric domain formation. J. Am. Ceram. Soc. 81, 492-500 (1998). Chen, L. Q. & Shen J. Applications of semi-implicit Fourier-spectral method to phase field equations. Comput. Phys. Commun. 108, 147-158 (1998). Chen, L. Q. APPENDIX A–Landau free-energy coefficients. In Physics of Ferroelectrics (pp. 363-372). Springer Berlin Heidelberg (2007). Li, F., et al. Electromechanical properties of tetragonal Pb (In1/2Nb1∕2)O3–Pb (Mg1∕3Nb2∕3) O3– PbTiO3 ferroelectric crystals. J. Appl. Phys. 107, 054107 (2010). Tagantsev, A. K. Landau expansion for ferroelectrics: which variable to use? Ferroelectrics 375, 19-27 (2008). Li, Y. L., Hu, S. Y., Liu, Z. K., & Chen, L. Q. Effect of substrate constraint on the stability and evolution of ferroelectric domain structures in thin films. Acta Mater. 50, 395-411 (2002). Jia, C. L., Mi, S. B., Urban, K., Vrejoiu, I., Alexe, M., & Hesse, D. Atomic-scale study of electric dipoles near charged and uncharged domain walls in ferroelectric films. Nat. Mater. 7, 57-61 (2008). Stemmer, S., Streiffer, S. K., Ernst, F., & Rüuhle, M. Atomistic structure of 90o domain walls in ferroelectric PbTiO3 thin films. Phil. Maga. A 71, 713-724 (1995).

19