Pressure and electric field effects on piezoelectric responses of KNbO3 · 2013. 1. 18. · Pressure and electric field effects on piezoelectric responses of KNbO 3 Linyun Liang,1,a)

Pressure and electric field effects on piezoelectric responses of KNbO3Linyun Liang, Y. L. Li, Fei Xue, and Long-Qing Chen Citation: J. Appl. Phys. 112, 064106 (2012); doi: 10.1063/1.4752418 View online: http://dx.doi.org/10.1063/1.4752418 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v112/i6 Published by the American Institute of Physics. Related ArticlesPositive effective Q12 electrostrictive coefficient in perovskites J. Appl. Phys. 112, 094106 (2012) Large and temperature-independent piezoelectric response in Pb(Mg1/3Nb2/3)O3-BaTiO3-PbTiO3 Appl. Phys. Lett. 101, 192901 (2012) Piezoelectric nonlinearity and frequency dispersion of the direct piezoelectric response of BiFeO3 ceramics J. Appl. Phys. 112, 064114 (2012) Structure and properties of La-modified Na0.5Bi0.5TiO3 at ambient and elevated temperatures J. Appl. Phys. 112, 054111 (2012) Modeling of dielectric and piezoelectric response of 1-3 type piezocomposites J. Appl. Phys. 112, 044107 (2012) Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Pressure and electric field effects on piezoelectric responses of KNbO3Linyun Liang,1,a) Y. L. Li,2 Fei Xue,1 and Long-Qing Chen11Department of Materials Science & Engineering, The Pennsylvania State University,Pennsylvania 16802, USA2Pacific Northwest National Laboratory, Richland, Washington 99352, USA

(Received 15 May 2012; accepted 15 August 2012; published online 18 September 2012)

The dielectric and piezoelectric properties of a KNbO3 single crystal under applied hydrostatic

pressure and positive bias electric field are investigated using phenomenological Landau-Ginzburg-

Devonshire thermodynamic theory. It is shown that the hydrostatic pressure effect on the dielectric

and piezoelectric properties is similar to temperature, suggesting a common underlying mechanism

for the piezoelectric anisotropy and its enhancement. The stable phase diagram of KNbO3 as a

function of temperature and positive bias electric field is constructed. The maximum piezoelectric

coefficient do�33 varying with temperature and electric field is calculated. VC 2012 American Instituteof Physics. [http://dx.doi.org/10.1063/1.4752418]

Much attention has been paid to environmental preser-

vation worldwide in recent years. In the research field of

piezoelectric ceramics, there is an increasing strong

demand to develop alternative lead-free piezoelectric mate-

rials against PZT based compounds, which are being

widely used in various fields as the most important piezo-

electric materials. Potassium niobate, KNbO3 is one of the

important candidates of lead-free piezoelectric materials

since the single crystals show a large electromechanical

coupling factor and the high Curie point.1–4 Although the

KNbO3 has good piezoelectricity, piezoelectric coefficients

of KNbO3 are much lower than those of PZT based materi-

als. Various experimental approaches have been explored

to improve the piezoelectric responses including applied

external fields,4,5 chemical etching,3 domain engineering,6

etc. However, so far, no significant improvement has been

achieved.

In understanding of mechanisms of the piezoelectric

coupling in ferroelectric piezoelectrics, most of the pro-

gresses have been centered on the discovery of the

enhanced piezoelectric responses along nonpolar directions.

Large piezoelectric responses along nonpolar crystallo-

graphic directions were reported in both complex relaxors7–

9 and simple perovskites3,10 in the past years. The origin of

such enhancement of piezoelectric response in perovskites

has been explained using a polarization rotation mechanism

based on first-principle calculations11–13 and free-energy

flattening mechanism based on a phenomenological thermo-

dynamic analysis.14,15 Based on the thermodynamic analy-

sis, the effects of electric field, composition, stress and

temperature on the piezoelectric properties of BaTiO3,

PbTiO3, and PZT perovskite crystals have been examined

using the Landau-Ginzburg-Devonshire (LGD) thermody-

namic theory.16–20 It is shown that the application of exter-

nal fields enhances piezoelectric coefficients along the

nonpolar directions, which is related to the phase transitions

that are accompanied by huge shear piezoelectric coeffi-

cients, resulting an enhancement of longitudinal piezoelec-

tric coefficient d�33ðh;/;wÞ. For the tetragonal BaTiO3single crystal, a higher piezoelectric response was observed

along no-polar [111].10 While in the orthorhombic phase,

the highest piezoelectric responses with d33 over 500 pC/Nwere observed when an electric field was applied along

[001] no-polar direction.21 It is well known that KNbO3crystals exhibit a series of transitions similar to BaTiO3with cooling from Curie temperature but the stable phase is

orthorhombic at room temperature. It is expected that

KNbO3 crystals may have similar enhancement of piezo-

electric response under external fields. Room temperature

piezoelectric coefficients of KNbO3 have been well deter-

mined experimentally in the earlier studies.22–25 Wada

et al.4 investigated the piezoelectric properties of KNbO3crystals along the polar [110]c direction and [001]c of engi-

neered domain direction. In their following work, they have

obtained all piezoelectric coefficients based on the single-

domain crystals.6 In our previous work, we calculated a set

of piezoelectric coefficients as a function of crystallographic

orientation and temperature,26 and also the piezoelectric

coefficients along the applied electric field directions.27

However, theoretical analysis of the crystallographic orien-

tation dependence of piezoelectric coefficients of KNbO3single crystals under the application of electric fields and

hydrostatic pressure has not been studied. Thus, the objec-

tive of this paper is to study the hydrostatic pressure and

electric field effect on the dielectric and piezoelectric

responses of single-domain KNbO3 crystals using the LGD

theory.

Under external pressure and electric fields, the LGD free

energy can be written as28a)Author to whom correspondence should be addressed. Electronic mail:

[email protected].

0021-8979/2012/112(6)/064106/6/$30.00 VC 2012 American Institute of Physics112, 064106-1

JOURNAL OF APPLIED PHYSICS 112, 064106 (2012)


http://dx.doi.org/10.1063/1.4752418http://dx.doi.org/10.1063/1.4752418

fLGDðPi; r;EiÞ ¼ a1ðP21 þ P22 þ P23Þ þ a11ðP41 þ P42 þ P43Þ þ a12ðP21P22 þ P22P23 þ P21P23Þþ a111ðP61 þ P62 þ P63Þ þ a112½P41ðP22 þ P23Þ þ P42ðP21 þ P23Þ þ P43ðP21 þ P22Þ�þ a123P21P22P23 þ a1111ðP81 þ P82 þ P83Þ þ a1112½P61ðP22 þ P23Þ þ P62ðP21 þ P23Þ þ P63ðP21 þ P22Þ�þ a1122ðP41P42 þ P42P43 þ P41P43Þ þ a1123ðP41P22P23 þ P42P23P21 þ P43P21P22Þ

� 12

s11ðr21 þ r22 þ r23Þ � s12ðr1r2 þ r2r3 þ r3r1Þ �1

2s44ðr24 þ r25 þ r26Þ

�Q11ðr1P21 þ r2P22 þ r3P23Þ � Q12½r1ðP22 þ P23Þ þ r2ðP21 þ P23Þ þ r3ðP21 þ P22Þ��Q44½r4P2P3 þ r5P1P3 þ r6P1P2Þ� � E1P1 � E2P2 � E3P3;

(1)

where a with subscript index represents expansion coeffi-cient under zero stress, sij is elastic compliance at constantpolarization Pi, Qij is electrostrictive coefficient, ri and Eiare stress and electric field. The values of aij, sij, and Qij aretaken form Ref. 29.

We use the superscript ‘p’ with p¼ c, t, o, r to indicatethe physical quantities measured in the cubic, tetragonal,

orthorhombic, and rhombohedral crystallographic coordinate

systems, respectively. The dielectric stiffness coefficient vcijis obtained via vcij ¼ e0@2fLGD=@Pi@Pj.

28 The dielectric sus-

ceptibility gcij can be determined from the reciprocal of vcij

using gcij ¼ Aji=D; where Aji and D are the cofactor and deter-minant of the vcij matrix. For the stable orthorhombic phaseat room temperature, we rotate the cubic coordinate system

into a new coordinate system defined by ½100�o ¼ ½100�c,½010�o ¼ ½01�1�c, and ½001�o ¼ ½011�c. The dielectric suscepti-bility goij in the new coordinate system can be simplyobtained by goij ¼ 1=voij due to the diagonalized matrix ofdielectric stiffness coefficient voij. The relationship of thedielectric stiffness coefficients between the old and new

coordinate systems is given by, vo11 ¼ vc11, vo22 ¼ vc33 � vc23,vo33 ¼ vc33 þ vc23, and vo12 ¼ vo13 ¼ vo23 ¼ 0.

28 Therefore, the

piezoelectric coefficients in the orthorhombic phase depend-

ing on the dielectric susceptibilities and polarizations in the

rotated system are given by

do15 ¼ e0go11ðri;EiÞQ44Po3ðri;EiÞ;do24 ¼ 2e0go22ðri;EiÞ½Q11 � Q12�Po3ðri;EiÞ;do31 ¼ 2e0go33ðri;EiÞQ12Po3ðri;EiÞ;

do32 ¼1

2e0g

o33ðri;EiÞ½2Q11 þ 2Q12 � Q44�Po3ðri;EiÞ;

do33 ¼1

2e0g

o33ðri;EiÞ½2Q11 þ 2Q12 þ Q44�Po3ðri;EiÞ; (2)

where goij is related to the dielectric coefficient eoij by a rela-

tion of goij � 1þ goij ¼ eoij.The orientation dependence of the longitudinal piezo-

electric coefficient do�33 in the orthorhombic KNbO3 crystalunder hydrostatic pressure and electric field can be calcu-

lated by26

do�33ðh;/Þ ¼ ðdo15ðri;EiÞ þ do31ðri;EiÞÞcosh sin2h sin2/

þ�

do24ðri;EiÞ þ do32ðri;EiÞ�

cosh sin2h cos2/

þ do33ðri;EiÞcos3h; (3)

where the Euler angle / describes the rotation around the[001]o¼ [011]c axis, and h describes the rotation away from

the [001]o axis around the rotated [100]o axis. / ¼ p=2 andh ¼ arcsinð1=

ffiffiffi3pÞ correspond to the coordinates associated

with the rhombohedral phase while / ¼ 0 and h ¼ �p=4 areassociated with the tetragonal phase. The other set of piezo-

electric coefficients in different ferroelectric phases under

the applied external fields can be obtained in a similar way.

We first consider the effects of hydrostatic pressure on

the dielectric and piezoelectric responses of orthorhombic

KNbO3. The hydrostatic pressure is expressed by

r1 ¼ r2 ¼ r3 ¼ �r; r4 ¼ r5 ¼ r6 ¼ 0. It alters the phasetransition temperature. A phase diagram as a function of

hydrostatic pressure r and temperature T has been con-structed elsewhere.29 Based on this thermodynamic phase

diagram, the pressure dependencies of dielectric susceptibil-

ities and piezoelectric coefficients at two chosen tempera-

tures T¼ 25 �C and 100 �C for the orthorhombic phase arepresented in Fig. 1. It should be noted that the applied pres-

sure range is different for the two different temperatures

since the phase transition-pressure is different based on the

thermodynamic diagram. It is seen that the dielectric suscep-

tibilities go11 and go22 which are perpendicular to the polariza-

tion direction [001]o. are sensitive to the pressure. go11decreases and go22 increases with the pressure. g

o33 is rela-

tively insensitive to the pressure. Correspondingly, we can

easily see from Eq. (2) that the piezoelectric coefficients do15(/ go11) and do24(/ go22) change significantly with the pressurewhile do31, d

o32, and d

o33(/ go33) change slightly (the value of

do31 is close to do32 but not plotted in Fig. 2 for the sake of

clarity). The large enhancement of the shear piezoelectric

coefficients do24 is ascribed to the ferroelectric phase transi-tion from the applied hydrostatic pressure. Their change with

the pressure will result in strong anisotropy of do�33. In orderto illustrate the crystallographic orientation dependence of

do�33, the do�33 surface as a function of pressure at room temper-

ature is plotted in Fig. 2. We analyzed the piezoelectric

responses on the (010)o and (100)o planes which are corre-

sponding to / ¼ 90� and / ¼ 0�, respectively. At room tem-perature, the piezoelectric coefficient do�33 as a function ofangle h with various pressures in different planes is given inFig. 3. It is seen that when / ¼ 90�, the maximum of do�33decreases with increasing pressure and its corresponding

angle hmax reduces rapidly from 52.0� at 0 GPa to 14.6� at6 GPa. The hydrostatic pressure tends to rotate the direction

of maximum do�33 into the [001]o direction. On the other hand,

for / ¼ 0�, do�33 increases significantly while hmax increasesslightly with pressure. For example, do�33max is 100.9 pC/N

064106-2 Liang et al. J. Appl. Phys. 112, 064106 (2012)


without applied pressure and then increases to 366.2 pC/N at

6 GPa. do�33max for / ¼ 0� is nearly nine times larger than thatof / ¼ 90� at 6 GPa. The largest do�33max always lies in the(100)o plane (/ ¼ 0�) in the orthorhombic stable phase.

As seen from Eq. (3) and Fig. 1, do�33 is dominated by do24

when close to the orthorhombic-tetragonal phase transition

temperatures. The shear piezoelectric coefficients do15 anddo24 are related to the permittivities perpendicular to thedirection of the spontaneous polarization axis [001]o. It is

known that the significant enhancement of do15 and do24 is

directly related to the presence of the orthorhombic-

rhombohedral or tetragonal-orthorhombic phase transitions.

Therefore, the largest do�33 along (100)o plane at high pressure

implies that do24 dominate do�33 and the tetragonal-

orthorhombic phase transition has a stronger effect on do�33.The pressure dependencies of the dielectric susceptibil-

ity for the orthorhombic phase can be understood from the

pressure dependence of its free energy profile (Fig. 4). The

free energy of the orthorhombic phase at a given pressure as

a function of polarization can be obtained through assuming

the polarization P1 ¼ 0 and P2 ¼ P3 ¼ffiffiffi2p

PS. The flatteningof the free energy profile implies an increase of the system’s

FIG. 2. At room temperature, the orientation dependence of piezoelectric coefficient do�33 under three selected pressures, (a) r¼ 0 GPa; (b) r¼ 3 GPa; (c)r¼ 6 GPa. Angles /max and hmax at which the maximum of do�33 occurs are indicated for each pressure. The three coordinate axes correspond toxo1 ¼ do�33 sinh cos/, xo2 ¼ do�33 sinh sin/, xo3 ¼ do�33 cosh, respectively. The numerical values marked on the axes have unit pC/N. The piezoelectric coefficientsare symmetry with respective to the plane of xo3 ¼ 0, so only the upper half of the coordinate space is shown.

FIG. 1. The hydrostatic pressure depend-

ence of (a) dielectric susceptibilities and

(b) piezoelectric coefficients of KNbO3at 25 �C (solid line) and 100 �C (dashedline).

FIG. 3. Room temperature piezoelectric

coefficient do�33 in the orthorhombicKNbO3 as a function of angle h atvarious pressures in different planes, (a)

/ ¼ 0� and (b) / ¼ 90�.



dielectric susceptibility, and thus the increase of its piezo-

electric response, i.e., a dielectric softening of the crystal

along the polar directions. It is noted that the flattening of

free energy under a pressure becomes more prominent near a

phase transition.

Next, we calculate the piezoelectric responses of

KNbO3 crystal under applied electric field. In order to find

the stable ferroelectric phase with the applied electric field, a

phase diagram as a function of temperature and electric field

is constructed and given in Fig. 5.

From the temperature-electric field phase diagrams, it is

seen that the applied electric field could induce ferroelectric

phases other than the regular tetragonal phase (P1¼P2¼ 0,P3> 0), orthorhombic phase (P1¼ 0, P2¼P3> 0), andrhombohedral phase (P1¼P2¼P3> 0). For the sake of sim-plicity, here we only focus on the dielectric susceptibilities

goij and piezoelectric coefficients doij at the chosen two tem-

peratures 200 �C and �45 �C for an applied field along thepolar direction (E[011]

c) where there is only the regular ortho-

rhombic phase. The calculated dielectric susceptibilities and

piezoelectric coefficients are shown in Fig. 6. It is easily

seen that a positive bias field reduces the dielectric suscepti-

bilities and piezoelectric coefficients. Of particular interest

are the temperature regimes close to two adjacent phase tran-

sitions at the small electric field, i.e., from the orthorhombic

phase to high-temperature phase a [defined by P3 > P2;P1 ¼ 0 in Fig. 5(b)] and from the orthorhombic phase tolow-temperature phase b [defined by P2 ¼ P3 > P1 > 0 inFig. 5(b)]. For example, at 200 �C close to the orthorhombic-a phase transition, the positive bias field reduces go22, andconsequently, leads to a low do24 [see Fig. 6 and Eq. (2)]. Ifthe temperature decreases to �45 �C which is close to theorthorhombic-b phase transition at small applied electricfield E[011]

c, a similar trend for go11 was observed but with alow do15 [see Fig. 7 and Eq. (2)]. In generally speaking, athigh temperatures, do24 is relatively large, while at lower tem-peratures, do15 becomes larger than d

o24 [Fig. 6(b)]. It is known

that the significant enhancement of do24 and do15 near the tran-

sition points is directly related to the orthorhombic-

tetragonal and orthorhombic-rhombohedral phase transitions,

respectively, which involves a polarization rotation from the

½011�c cubic (½001�o orthorhombic) axis toward the ½001�c

axis and from ½011�c cubic (½001�o orthorhombic) axis towardthe ½111�c cubic axis. However, our calculating results showthat the temperature dependence of the shear piezoelectric

coefficients is significantly reduced by the electric field along

the polar direction [Fig. 6(b)]. The temperature effects on the

piezoelectric coefficients are diminished under the applied

electric field. As a result, it is not easy to transform KNbO3crystal from the orthorhombic phase to either tetragonal or

rhombohedral phase by changing the temperature. The result

is consistent with the temperature-electric field phase dia-

gram [Fig. 5(b)] that the transition temperatures for the

orthorhombic-a and orthorhombic-b phase transitionincrease and decrease under an electric field, respectively.

FIG. 4. A calculated free energy as a function of polarization (Ps) under var-ious selected pressures in the orthorhombic phase at room temperature.

FIG. 5. Phase diagram as a function of temperature and applied electric

field (a) E ¼ ð0; 0;E0Þ so that E½001�c ¼ E0; (b) E ¼ ð0;E0;E0Þ so thatE½011�c ¼

ffiffiffi2p

E0; (c) E ¼ ðE0;E0;E0Þ so that E½111�c ¼ffiffiffi3p

E0.



In order to illustrate the crystallographic orientation de-

pendence of do�33, the do�33 surface under the given electric

fields of 0 and 35 MV/m at temperature 200 �C is plotted inFig. 7. The angle corresponding to the maximum do�33 isnearly independent of the electric field.

Figure 8 shows the orientation dependence of piezoelec-

tric coefficients do�33ðh;/Þ [see Eq. (3)] under the electricfields of 0 and 35 MV/m at temperature 200 �C and �45 �C.At high temperature, the positive bias field decreases the

maximum do�33 from 278.1 (without any electric field) to120.1 pC/N for / ¼ 0�. It can be easily seen from Eq. (3)that the decrease of do�33 along the (100)

o plane is primarily

due to the decrease of do24. On the other hand, at �45 �C, thepositive bias field reduces do�33 to 92.0 pC/N for / ¼ 90�which is larger than that at / ¼ 0�(65.8 pC/N), implying thatthe maximum do�33 is along the (010)

o plane. Compared with

its value at higher temperature, do�33 is dominated by do15 in

the lower temperature regime. It is consistent with the results

of the shear piezoelectric coefficients do24 and do15 which are

always larger than others at both high and low temperatures

under any bias electric field. Thus, the piezoelectric

responses are determined by the rotation direction of the

polarization resulted by both temperature changes and exter-

nal electric fields.

FIG. 7. The orientation dependence of

the piezoelectric coefficient do�33 for twoselected positive bias electric fields at

200 �C, (a) E¼ 0 MV/m; (b) E¼ 35 MV/m. The three coordinate axes correspond

to xo1 ¼ do�33sinhcos/, xo2 ¼ do�33sinhsin/,xo3 ¼ do�33cosh, respectively. The numeri-cal values marked on the axes have unit

pC/N. The piezoelectric coefficients are

symmetry with respective to the plane of

xo3 ¼ 0, so only the upper half of thecoordinate space is shown.

FIG. 6. Calculated electric field depend-

ence of (a) the dielectric susceptibility

goij and (b) the piezoelectric coefficientsdoij for single-domain KNbO3 crystal at200 �C (solid lines) and �45 �C (dashedlines).

FIG. 8. Piezoelectric coefficient do�33 in the orthorhombic KNbO3 as a function of angle h under various applied electric fields E[011]c at temperature (a) 200�C

and (b) �45 �C. Only the piezoelectric coefficients in the angle h between 0� and 90� are shown.



In summary, we have studied the effects of hydrostatic

pressure and electric field (E[011]c) on the piezoelectricresponses of single-domain KNbO3 single crystals using the

Landau-Ginzburg-Devonshire thermodynamic theory. The

effect of hydrostatic pressure on the piezoelectric response is

very similar to that of temperature if we assume a ferroelec-

tric phase transition is in the first-order. The phase diagram

as a function of electric field and temperature is constructed

based on the thermodynamic calculations. The applied elec-

tric fields parallel to the polar direction reduce the dielectric

and piezoelectric responses of KNbO3 crystals. The maxi-

mum of the piezoelectric coefficient do�33 is determined by therotation of the polarization resulted by both temperatures

and external electric fields. To further calculate the piezo-

electric responses of KNbO3 sintered ceramics under the

applied external fields, we have to extend the current thermo-

dynamic calculations to phase field simulations, in which the

effect of polycrystal microstructures and multi-domains can

be included.

The work is partially supported by NSF under ECCS-

0708759 and DMR-1006541. The computer simulations

were carried out on the LION and Cyberstar clusters at the

Pennsylvania State University supported in part by NSF

Major Research Instrumentation Program through Grant No.

OCI-0821527.

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