Relation between phase transition and impurity-polarized clusters in

PHYSICAL REVIEW B 66, 104105 共2002兲

Relation between phase transition and impurity-polarized clusters in Sr1À ␤ Ca␤ TiO3 Lei Zhang,1,2 Wolfgang Kleemann,2 and Wei-Lie Zhong1 1

2

Department of Physics, Shandong University, Jinan, 250100, People’s Republic of China Laboratorium fu¨r Angewandte Physik, Gerhard Mercator Universita¨t Duisburg, D-47048 Duisburg, Germany 共Received 24 April 2002; published 18 September 2002兲

The temperature and concentration dependence of the dielectric response in Sr1-␤ Ca␤ TiO3 共SCT兲 is explained within the framework of the transverse field Ising model. Impurity-polarized clusters induce the appearance of a second-order phase transition, while quenched random fields give rise to smeared dielectric response. When ␤ increases, the dopant-induced internal pressure counteracts the increase of the dipolar moment of the impurity-polarized clusters. This gives rise to a decrease of the peak response in agreement with the experimental observation. DOI: 10.1103/PhysRevB.66.104105

PACS number共s兲: 77.22.Ch, 77.80.Bh

I. INTRODUCTION

Quantum paraelectrics containing impurities,1–7 such as Sr1⫺ ␤ Ca␤ TiO3 共SCT兲 have attracted great attention in the past. It is well known1 that SrTiO3 crystals doped up to 12% with Ca2⫹ show some interesting features. Above a critical concentration ␤ c⫽0.0018, the quantum mechanical critical point, the dielectric constant begins to peak. Upon an increase of the Ca concentration at ␤ r⬎0.016, the peaks in dielectric constant are rounding and decreasing again. SrTiO3 , one of the end members of SCT, is a quantum paraelectric. The onset of its ferroelectric instability is hindered by quantum fluctuations. Within the framework of the transverse Ising model,8 this can be modeled by setting 2⍀ ⬎ZJ, where ⍀ is the tunneling frequency, J is the interaction constant between pseudospins, and Z⫽6 is the coordination number of elementary cells in the cubic lattice.9 It is well known, however, that a ferroelectric phase transition may be induced in SrTiO3 by polar impurities like Ca2⫹ 共Ref. 1兲 or Ba2⫹. 10 This can be mimicked by the transverse Ising model by introducing either reduced values of ⍀ or increased values of J at the impurity sites.11,12 Apart from the divergence of the dielectric susceptibility at finite Curie temperatures, the dependences of the spontaneous polarization on the concentration ␤ and the electric field dependence of the peak susceptibility are reproduced in a satisfactory way. However, some important properties have not yet been explained by the models developed so far. Neither the considerable broadening and flattening of the susceptibility peak nor the virtual temperature independence of its peak position as ␤ ⬎0.02 共Ref.1兲 have found convincing interpretation up to now. It is the purpose of this paper to fill this gap. On the one hand, the idea of introducing quenched random fields1,5 is added to the formalism of the transverse Ising model more consequently than in previous attempts.13 On the other hand, an electrostrictive correction due to the local reduction of the unit cell by doping with Ca2⫹ 共Ref. 14兲 is taken into account. The latter effect is analogous to that of hydrostatic pressure, which is known to decrease the unit cell size and the dielectric constant of SrTiO3 . 15 Thus we extend previous studies on SCT into the intermediate concentration range, up to ␤ ⬇0.1. Based on some experimental results, a 0163-1829/2002/66共10兲/104105共7兲/$20.00

Landau-type expansion and the transverse field Ising model, it is shown that the change of the dielectric behavior in SCT can mainly be attributed to changes of the dipolar moment and the quenched random field.

II. MODEL A. Hamiltonian

Consider a system of dipole impurities, randomly distributed in a crystal. At low temperature, the orientation of every dipole may be considered parallel to the local field being due to adjacent impurity-polarized clusters, and random quenched dipolar fields. Due to the varying sign of the local frozen field, the dipoles are thought to be frozen in different orientations in the limit of strong random fields. The dipolar moments in undistorted cells are referred to as host pseudospins S j . In the following, we define the pseudospin ␴ m as the cluster spin in the impurity-polarized cluster. In order to treat the interactions of the dipolar moments, as well as the quantum mechanical effects in a unified framework, we use the following Hamiltonian

H⫽⫺⍀ 1 ⫺

1

z z J 1␴ m ␴ n -2␮ 2 兺 共 E⫹E mr 兲 ␴ mz 兺i ␴ mx ⫺ 2 兺 mn m

1

z z J 3␴ m S j ⫺⍀ 2 兺 S xi ⫺ 兺 J 2 S zi S zj 兺 2 ij mj i

⫺2 ␮ 1

兺i 共 E⫹E ri 兲 S zi ,

共1兲

x z (␴m ) are the x共z兲 components of the where Sxi (S zi ) and ␴ m host and cluster pseudospins. i and j sum over all host pseudo-spins, and m and n sums over dopant pseudospins. ⍀ 1 and ⍀ 2 are the cluster and host pseudospin tunneling frequencies. J 1 , J 2 , and J 3 are the cluster-cluster, host-host, and host-cluster pseudospin interaction constants. ␮ 1 and ␮ 2 are the effective dipolar moments of the cluster and host r are pseudospins. E stands for the external field, E ri and E m site-dependent intrinsic quenched random fields.

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©2002 The American Physical Society

PHYSICAL REVIEW B 66, 104105 共2002兲

LEI ZHANG, WOLFGANG KLEEMANN, AND WEI-LIE ZHONG

FIG. 1. Schematic sketch of regions 共1兲 and 共2兲 around a central Ca2⫹ dopant ion in Sr1⫺ ␤ Ca␤ TiO3 共see the text兲.

Some experiments have demonstrated that the size of the cluster polarized by an impurity is a few lattice constants.16 –18 In our model, we suppose that the size rc of the cluster is ␰ a, where ␰ is a constant. Thus every cluster includes ␰ 3 distorted cells and one Ca2⫹ impurity. In region 共2兲, the number of the undistorted cells is 1/␤ -␰ 3 . In the following model, we will study the dielectric behavior of SCT for ␰ ⬍ ␤ ⫺1/3. The condition ␰ ⬍ ␤ ⫺1/3 accounts for the fact that the volume of region 共2兲 is nonzero in Fig. 1. Now we discuss the average pseudospin interaction constants of regions 共1兲 and 共2兲. In region 共1兲, the number of pseudospins is ␰ 3 . Their total coordination numbers for cluster-cluster and cluster-host pseudospin interactions are 3( ␰ 3 -␰ 2 ) and 6 ␰ 2 , respectively. Thus the molecular field ¯E a1 in region 共1兲 is ¯E a1 ⫽

B. Basic model

In a real system, the numerical solution of Eq. 共1兲 is very difficult. Thus we introduce some simplifying suppositions. The impurities are assumed to be distributed uniformly in the crystal. In our model, we divide the crystal into M cubic supercells 共henceforth denoted as ’’regions’’兲 with the same size as shown in Fig. 1. Every region has an impuritypolarized cluster at its center 关region 共1兲兴. The surrounding undistorted cells will be referred to as region 共2兲. Every region contains one impurity (Ca2⫹ ). Thus the number of the cells in one region is 1/␤ . The length of the edges of the region is ␤ ⫺1/3a, where a is the lattice constant.

¯E a2 ⫽

兺m

J 2 S zi ⫹

兺i

z J 3␴ m ⫽

兺m

z J 1␴ m ⫹

兺i

J 3 S zi ⫽

z 共 ␰ ⫺1 兲 J 1 具 ␴ m 典 ⫹2J 3 具 S zi 典

␰ ⫹1

共2兲

z where 具 ␴ m 典 z and 具 Si 典 the

is the average value of the spins in region 共1兲, average value of the spins in region 共2兲. In region 共2兲, the number of pseudospins is 1/␤ -␰ 3 . The total coordination number for the host-cluster pseudospin interaction is 6 ␰ 2 . In the internal part of region 共2兲, the total coordination number for host-host pseudospin interactions is 3 ␤ ⫺1 -3 ␤ ⫺2/3-3 ␰ 3 -3 ␰ 2 . On the edge of every region, the host spins have an interaction with the nearest neighboring spins of the other region 共2兲. The total coordination number is 6 ␤ ⫺2/3. Thus the molecular field ¯E a2 in region 共2兲 is

z 共 ␤ ⫺1 ⫺ ␤ ⫺2/3⫺ ␰ 3 ⫺ ␰ 2 兲 J 2 具 S zi 典 ⫹2 ␰ 2 J 3 具 ␴ m 典 ⫹2 ␤ ⫺2/3J 2 具 S zi 典

共3兲

␤ ⫺1 ⫹ ␤ ⫺2/3⫺ ␰ 3 ⫹ ␰ 2

Every cubic region 共2兲 in our model has six neighboring cubic regions 共2兲. 具 S zi 典 is the average spin of the six regions 共2兲. The possibility of one region with a positive frozen field equals that of the region with a negative frozen field. So we suppose the average spin 具 S zi 典 ⫽0.5具 S zi 典 ⫹ ⫹0.5具 S zi 典 ⫺ , where 具 S zi 典 ⫹ ( 具 S zi 典 ⫺ ) is the spin of the region with a positive 共negative兲 frozen field. When E⫽0, we obtain 具 S zi 典 ⫹ ⫽⫺ 具 S zi 典 ⫺ and 具 S zi 典 ⫽0. Thus the interaction among different regions 共2兲 can be ignored in Eq. 共3兲. When a field E is applied to the crystal, 具 S zi 典 will deviate from 0 obviously, and the interaction among different regions must be considered. For SCT,1 some of Ca2⫹ may be located at Ti4⫹ sites. In order to balance the charge misfit, a next-neighboring oxygen can be vacant, V 0 , forming a Ca2⫹ -V 0 neutral center. The Ca2⫹ -V 0 centers set up local frozen fields.1,6 These centers are located in region 共1兲. Thus the random fields E ri in r region 共2兲 are smaller than those in region 共1兲, E m . In regions 共1兲 and 共2兲, we set constant values in the sense of statistical averages

,

r Em ⫽E r ,

共4兲

E ri ⫽ ␤ ␰ 3 E r ,

共5兲

respectively, where E r is a constant. Equation 共5兲 shows that E ri is zero for ␤ ⫽0, and increases linearly with increasing ␤ . According to Fig. 1, the volume of region 共2兲 is zero for ␤ ⫺1/3⫽ ␰ . When ␤ ⫺1/3→ ␰ , Eq. 共5兲 shows that E ri will increase up to E r . According to the average-field approximation, one obtains

具 ␴ mz 典 ⫽

冉 冊

¯E a1 ⫹2 ␮ 1 E⫾2 ␮ 1 E r h1 tanh h1 2kT

共6兲

in region 共1兲, where h 1 ⫽ 冑⍀ 21 ⫹ 关 ¯E a1 ⫹2 ␮ 1 E⫾2 ␮ 1 E r 兴 2 . while

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具 S zi 典 ⫽

冉 冊

¯E a2 ⫹2 ␮ 2 E⫾2 ␮ 2 ␤ ␰ 3 E r h2 tanh h2 2kT

共7兲

PHYSICAL REVIEW B 66, 104105 共2002兲

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1 G 共 X, P 兲 ⫽ s 11共 X 2 ⫹Y 2 ⫹Z 2 兲 ⫹s 12共 Y Z⫹XY ⫹ZX 兲 2 1 1 ⫹ A 共 T Q ⫺T 0 兲 P 2 ⫹ B P 4 2 4 ⫹Q 11Z P 2 ⫹Q 12共 X⫹Y 兲 P 2 ,

共9兲

where X, Y, and Z are normal stress components, P is the polarization, s 11 , s 12 is the elastic compliance, T Q ⫽(⍀ 2 /2k)coth(⍀2/2kT), T 0 ⫽J 20/4k, Q 11 , and Q 12 are electrostrictive coefficients, and A and B are coefficients of the free energy function. J 20 is the interaction constant of the free SrTiO3 crystal, and A⫽k/(N ␮ 22 ). From Eq. 共9兲 the equation of state becomes

冋

E⫽A T Q ⫺

FIG. 2. Temperature dependence of the dielectric constant of Sr1⫺ ␤ Ca␤ TiO3 crystals with ␤ between 0 and 0.12. Solid circles and lines are the experimental Ref. 1 and theoretical results, respectively.

According to Eq. 共10兲, the coupling constant J 2 under a hydrostatic pressure X ⬘ is J 2 ⫽J 20⫺8k 共 Q 11⫹2Q 12兲 X ⬘ /A,

共11兲

where X ⬘ ⫽X⫽Y ⫽Z. For SrTiO3 , the electrostrictive coefficients21 at 4.2 K are Q 11⫽5.3⫻10⫺2 m4 /C2 and Q 12 ⫽⫺8.5⫻10⫺3 m4 /C2 . In order to find values of J 2 in the differently stressed regions 共2兲 using Eq. 共11兲, the local stress component X ⬘ has to be considered. a and c are the lattice constants in the tetragonal phase. The lattice parameter ␥ is a or 3冑a 2 c. The strain x is ⌬ ␥ / ␥ or (s 11⫹2s 12)X. We obtain

in region 共2兲, where h 2 ⫽ 冑⍀ 22 ⫹ 关 ¯E a2 ⫹2 ␮ 2 E⫾2 ␮ 2 ␤ ␰ 3 E r 兴 2 . The average polarization P in the region is z P⫽2 具 ␴ m 典 N ␤ ␰ 3 ␮ 1 ⫹2 具 S zi 典 N 共 1⫺ ␤ ␰ 3 兲 ␮ 2 ,

册

J 20⫺8k 共 Q 11Z⫹Q 12X⫹Q 12Y 兲 /A P⫹B P 3 . 4k 共10兲

共8兲

d␥ dX ⬘

where N is the density of dipolar moments. In our model, the dielectric susceptibility can be obtained by differentiating Eq. 共8兲 with respect to an arbitrarily weak uniform electric field. The possible polarization P is reverse when the frozen field is reverse in different regions. However, a uniform dielectric susceptibility is obtained in different regions.

⫽ 共 s 11⫹2s 12兲 ␥ .

共12兲

For SrTiO3 , Ref. 22 reported s 11⫽5.4⫻10⫺12 m2 /N in a wide range of the low temperature. We can obtain s 12⫽ ⫺8.50⫻10⫺13 m2 /N from Ref. 21. For Sr1⫺ ␤ Ca␤ TiO3 , Ref. 14 gave a linear relation between ␥ and ␤ ;

␥ ⫽3.897⫺0.062␤ 共 Å 兲 , ␤ ⬍0.2

共13兲

From Eqs. 共12兲 and 共13兲, we obtain C. Initial ascertainment of some parameters

In order to evaluate Eq. 共8兲 quantitatively, we need some parameters describing the SCT system explicitly. For SrTiO3 , the dielectric susceptibility in the paraelectric phase is ␹ ⫽ 关 N ␮ 22 /( ⑀ 0 k) 兴 / 关 (⍀ 2 /2)coth(⍀2/2kT)⫺J 2 /4兴 . 9,19 From the experimental curve 1 of Fig. 2, we obtain three fitting parameters, J 2 ⫽138 K, ⍀ 2 ⫽79.8 K and ␮ 2 ⫽1.89e Å. For SCT, an increase of ␤ will decrease the cell volume, while SrTiO3 cells in region 共2兲 are not distorted by impurities. Thus the physical behavior in region 共2兲 can be explained according to a simple cell volume effect of SrTiO3 . Extending the quantum mechanically modified Landau expansion of the free energy20 by elastic and piezoelectric contributions, we obtain

X ⬘⫽

0.062␤ 3.897共 s 11⫹2s 12兲

共14兲

From Eqs. 共9兲 and 共11兲, we know that the interaction constant J 2 can be adjusted by the pressure, and the dipolar moment ␮ 2 and the tunneling frequency ⍀ 2 are constants. According to Eqs. 共10兲, 共11兲, and 共14兲, we obtain the theoretical temperature dependence of the dielectric constant for SrTiO3 with different X ⬘ . In Ref. 15, dielectric measurements showed that dT 0 /dX ⬘ of SrTiO3 was ⫺14 ⫻10⫺8 K/Pa. In our calculation, we find dT 0 /dX ⬘ ⫺12 ⫻10⫺8 K/Pa, in good agreement with the experimental result. Further, from Eqs. 共11兲 and 共14兲, we obtain the ␤ dependence of J 2 in SCT.

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LEI ZHANG, WOLFGANG KLEEMANN, AND WEI-LIE ZHONG

FIG. 3. ␤ dependence of ␮ 1 共solid circles兲, J 3 共open circles兲, and E r 共open triangles兲 in SCT. The solid lines are three fitted curves.

In order to reduce the number of the adjustable parameters when calculating the dielectric constant ␧( ␤ ,T) from Eq. 共8兲, we try to find a relationship between ␮ 1 and J 1 . We consider a pure single crystal involving only one kind of pseudospin with an interaction constant J and a dipolar moment ␮ . We obtain J⫽4 ␮ E g if we consider a relation between the transverse field Ising model and the classical order-disorder model.23 E g is the internal electrostatic field when the dipoles are completely ordered. In classical theory it is known that the electrostatic field originating from a dipole has a linear relationship with the dipolar moment. For our system, this means that the E g in the lattice has a linear relationship with ␮ . Consequently, J has a linear relationship with ␮ 2 . In the following we suppose identical proportionality constants in regions 共1兲 and 共2兲:

␻⫽

J1

␮ 21

⫽

J 20

␮ 22

.

共15兲

This supposition suggests that the arrangement of the reorienting dipoles in impurity-polarized clusters is the same as the arrangement of the dipoles in a free SrTiO3 crystal. III. RESULTS AND DISCUSSIONS

According to Eqs. 共8兲, 共11兲, 共14兲, and 共15兲, we can obtain the temperature dependence of the dielectric constant of SCT by adjusting ␰ , ⍀ 1 , ␮ 1 , J 3 , and E r . The theoretical curves are shown in Fig. 2 along with the experimental results.1 In the calculation, we suppose ␰ ⫽2 and ⍀ 1 ⫽103 K. ␮ 1 , J 3 , and E r are shown in Fig. 3 as functions of ␤ . In Fig. 2, the theoretical dielectric constant of Sr0.998Ca0.002TiO3 is higher than that of SrTiO3 obviously in the range of the low temperature. The experimental circles of Sr0.998Ca0.002TiO3 show a faint maximum. However, the theoretical curve 2 does not evidence the existence of the maximum. In our model, both ⍀ 2 (79.8 K) and ⍀ 1 (103 K) are very high. A high tunneling frequency means a large temperature range dominated by the quantum mechanical stabilization. This is the reason that the present calculation does not exhibit a dielectric peak near 0 K. It is notable that the

FIG. 4. ␤ dependence of the peak temperature T m in SCT. Open circles are the experimental result; the solid line is the fitted theoretical result; the dashed line 1 is the ( ␤ -␤ c ) 1/2 law; the dashed line 2 is the theoretical result omitting the frozen field; the dashed line 3 is the theoretical result with ␮ 1 ⫽2.95e Å, J 3 /k⫽430 K, and E r ⫽0 V/m.

␤ ⫽0.002 is near the quantum mechanical critical point ( ␤ c ⫽0.0018). Dielectric measurements7 showed that a free Sr0.998Ca0.002TiO3 crystal is a quantum paraelectric. Figure 2 shows that a high and sharp dielectric peak exists in Sr0.9925Ca0.0075TiO3 and Sr0.9893Ca0.0107TiO3 . Thus the impurity-polarized cluster induces the appearance of a second-order phase transition in SCT. At higher concentrations as in Sr0.90Ca0.10TiO3 and Sr0.88Ca0.12TiO3 , rather low and broadened dielectric peaks are encountered 共Fig. 2兲. The obvious diffusion in Sr0.90Ca0.10TiO3 and Sr0.88Ca0.12TiO3 is attributed to the existence of the strong frozen field caused by frozen dipoles. For the origin of the frozen field, we can obtain some inspiration from TGS doped with alanine impurities. For TGS, experiments24,25 and theory25 showed that alanine impurities can cause a strong internal bias field in the crystal lattice. It leads into the existence of a diffusive dielectric peak. The physical picture in TGS should be similar to that in SCT. In Fig. 2, the theoretical curve with ␤ ⫽0.0107 is narrower than the actual dielectric peak. Similar observations apply to Sr0.90Ca0.10TiO3 and Sr0.88Ca0.12TiO3 . The difference between the theory and the experiment probably originates from the fact that we ignore the randomness of the impurity distribution. In a real SCT crystal, ␮ 1 and J 3 are two constants, and the density ␳ of impurity-polarized clusters is expected to deviate from its average value in different crystal regions. In the following part, the dashed line 3 in Fig. 4 gives an approximate explanation for the case. Along the dashed line 3, the Curie temperature increases monotonically with increasing ␳ when ␮ 1 and J 3 are constant. Because of the density dependence of the Curie temperature different crystal regions in SCT have phase transitions at different temperatures. This means a diffused phase transition. Both the frozen field and the fluctuating density can cause the diffusion. We can judge the role of the frozen field qualitatively from two factors. At first, let us suppose that the frozen field does not exist in SCT. Thus SCT will show a

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RELATION BETWEEN PHASE TRANSITION AND . . .

high and narrow dielectric peak. A frozen field will decrease the dielectric peak obviously in the whole range of the temperature. On the other hand, a fluctuation of the density ␳ will decrease the peak dielectric constant. However, it will increase the dielectric constant on both sides of the dielectric peak . In Fig. 2, the dielectric constant of Sr0.88Ca0.12TiO3 is very small in the whole range of the temperature. This means that the frozen field in two factors should play a main role for the diffusion. Fig. 3 shows the ␤ dependence of ␮ 1 , J 3 , and E r for SCT. In Fig. 3, both ␮ 1 and J 3 decrease with increasing ␤ . For SCT, the cell size decreases when a Ca ion substitutes for a Sr ion. When the cell volume decreases, the deviation of off-center ions from the center tends to decrease, hence the decrease of the ␮ 1 . J 3 represents the interaction between the moments ␮ 1 and ␮ 2 in a classical model. The decrease of ␮ 1 causes the decrease of J 3 in SCT. In Fig. 3, E r increases with increasing ␤ . The three solid lines in Fig. 3 are the fitted curves of theoretical parameters

␮ 1 ⫽3.7–4.0␤ 0.37共 e Å 兲 ,

共16兲

J 3 /k⫽600–900␤ 0.40 共 K兲 ,

共17兲

E r ⫽0.08⫹60␤ 2.0 共 106 V/m兲 .

共18兲

Equations 共16兲–共18兲 give approximate results on ␮ 1 , J 3 , and E r . According to these equations, we can obtain the ␤ dependence of the temperature T m in SCT 共solid line in Fig. 4兲, as compared to the experimental result 共hollow circles兲,1 while the dashed line 1 indicates the traditional ( ␤ -␤ c) 1/2 law of SCT.1 According to our model, the change of T m in SCT is related to that of ␮ 1 , J 3 , E r and the density ␳ of impuritypolarized clusters. In order to clarify the relation among them, we suppose E r ⫽0 in SCT. Thus we obtain the theoretical result according to Eqs. 共16兲 and 共17兲. It is the broken line 2 of Fig. 4. The difference between the solid line and the broken line 2 shows that the frozen field obviously increases T m . It is notable that the change of T m is more obvious for a large ␤ . It is attributed to the increase of E r with increasing ␤ . According to Eqs. 共16兲 and 共17兲, ␮ 1 and J 3 /k should be 2.95e Å and 430 K, respectively for ␤ ⫽0. Now we suppose that the ␮ 1 (2.95e Å) and J 3 /k(430 K) are independent of ␤ in SCT. Furthermore, we suppose E r ⫽0 in SCT. Thus we obtain a new theoretical curve. It is the dashed line 3 of Fig. 4. In the curve, the T m increases monotonically with increasing ␤ . For some quantum ferroelectrics 共such as Sr1⫺ ␤ Ba␤ TiO3 ), the ( ␤ -␤ c) 1/2 law can describe the experimental behavior well in a wide range. However, it is obvious that the ( ␤ -␤ c) 1/2 law of SCT deviates from the experimental result when ␤ ⬎0.02 in Fig. 4. The deviation is attributed to the obvious decrease of cell volume with increasing ␤ in Sr1⫺ ␤ Ca␤ TiO3 . In Fig. 4, the dashed line 3 is in good agreement with the ( ␤ -␤ c) 1/2 law. Thus the ( ␤ -␤ c) 1/2 law can describe the behavior of quantum ferroelectrics with constant dipolar moments. For Sr1⫺ ␤ Ba␤ TiO3 , the dipolar moment of

z FIG. 5. 共a兲 ␤ dependence of 具 ␴ m 典 and 具 S iz 典 for SCT. 共b兲 ␤ z z dependence of ⳵ 具 ␴ m / ⳵ ␤ and ⳵ S / ⳵ ␤ 典 具 i 典 . The inset in 共b兲 shows the theoretical dielectric constant in the ground-state as a function of ␤ .

impurities has an obvious increase with increasing ␤ . Thus a sustained increase of the T m with increasing ␤ exists in Sr1⫺ ␤ Ba␤ TiO3 . Consequently, the ( ␤ -␤ c) 1/2 law should be able to describe the tendency. For SCT, the obvious decrease of ␮ 1 and J 3 with increasing ␤ leads into that T m is independent of ␤ for 0.02⬍ ␤ ⬍0.10. Thus the ( ␤ -␤ c) 1/2 law deviates from the experimental result in this range. According to Eqs. 共16-18兲, we obtain the ␤ dependence of z 具 ␴ m 典 , 具 S zi 典 , ⳵ 具 ␴ mz 典 / ⳵ ␤ , and ⳵ 具 S zi 典 / ⳵ ␤ , at T⫽0 K 共Fig. 5兲. In z Fig. 5共a兲, 具 ␴ m 典 and 具 S zi 典 increase with increasing ␤ . 具 S zi 典 is z smaller than 具 ␴ m 典 at different ␤ . In Fig. 5共b兲, we find three z different ranges I, II, and III for ⳵ 具 ␴ m 典 / ⳵ ␤ and ⳵ 具 S zi 典 / ⳵ ␤ versus ␤ . In range I ( ␤ ⬍0.001), both quantities are approxiz mately zero. In range II (0.001⬍ ␤ ⬍ ␤ c), ⳵ 具 ␴ m 典 / ⳵ ␤ and z ⳵ 具 S i 典 / ⳵ ␤ increase with increasing ␤ . The theoretical value is ␤ c⫽0.004. It is slightly larger than the experimental value z 共0.0018兲. In range III ( ␤ ⬍ ␤ c), ⳵ 具 ␴ m 典 / ⳵ ␤ and ⳵ 具 S zi 典 / ⳵ ␤ decrease with increasing ␤ . These results are interpreted as follows. In range I of Fig. 5共b兲, the density of impurity-polarized clusters in the crystal is very small. Hence the polarization in the host lattice is approximately zero. Thus the interactions among these clusters can be ignored, since the impurity-polarized clusters are dispersed in the nonpolar host-lattice. In range II, a faint interaction among clusters exists. Thus the increase of the z density of impurity-polarized clusters increases ⳵ 具 ␴ m 典/⳵␤ z and ⳵ 具 S i 典 / ⳵ ␤ quickly. It is a glass state because of the absence of ferroelectricity. In range III, the decrease of

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LEI ZHANG, WOLFGANG KLEEMANN, AND WEI-LIE ZHONG

FIG. 6. Theoretical ground-state hysteresis loop for Sr0.9925Ca0.0075TiO3 . The solid loop 1 and the solid loop 3 are the results including the frozen field; the dashed loop 2 and the dashed loop 4 are the theoretical results ignoring the frozen field.

⳵ 具 ␴ mz 典 / ⳵ ␤ and ⳵ 具 S zi 典 / ⳵ ␤ with increasing ␤ is understandable according to the change of ⳵ T m / ⳵ ␤ . In Fig. 4, T m increases quickly with increasing ␤ near ␤ c . However, the increase slows down with increasing ␤ , and ⳵ T m / ⳵ ␤ is largest near ␤ c . Hence, the decrease of ⳵ 具 ␴ mz 典 / ⳵ ␤ and ⳵ 具 S zi 典 / ⳵ ␤ above ␤ c should originate from the quick decrease of ⳵ T m / ⳵ ␤ mainly. The inset in Fig. 5共b兲 shows the theoretical ground-state dielectric constant as a function of ␤ . In the inset, a maximum appears at ␤ ⫽ ␤ c . When ␤ ⬎ ␤ c , an increasing ferroelectric polarization stabilizes the lattice and decreases the dielectric constant obviously. In a SCT crystal, the magnitude of the frozen field E r can be assumed to be constant. It tends to cause a multi-domain state. On the other hand, the ferroelectric long-range force increases with decreasing temperature, and tends to cause a single-domain state. Now the question arises whether or not the SCT crystal will change from the multidomain state to the single-domain state with decreasing temperature. Figure 6 shows the theoretical ground-state hysteresis loops of region 共1兲 and region 共2兲 for Sr0.9925Ca0.0075TiO3 . While assuming a positive value of E r ⫽0.1⫻106 V/m. z When E⫽0 V/m, both 具 ␴ m 典 and 具 S zi 典 are positive, since the dipoles in the region should be aligned with the positive

z FIG. 7. 共a兲.Temperature dependence of 具 ␴ m 典 and 具 S iz 典 for Sr0.9925Ca0.0075TiO3 . Solid and dashed lines are the results including and ignored, respectively, the contribution of the frozen field. 共b兲 Remanent polarization as a function of T for Sr0.96Ca0.04TiO3 . The parameters of the calculated result 共solid line, left-hand scale兲 come from Eqs. 共16兲, 共17兲, and 共18兲. The experimental result 共open circles, right-hand scale兲 is measured under a field of 5.25 kV/m 共Ref. 23兲.

frozen field E r . When two regions have antiparallel frozen fields, their dipoles will be antiparallel. In a real SCT crystal, it means the existence of the multidomain state. Since the frozen fields exist at all temperatures, a true paraelectric state does not exist in SCT. When we omit the frozen field, the ground-state loops 2 and 4 共Fig. 6兲 are obtained in regions 共1兲 and 共2兲 respectively. According to our calculation, we find that the multidomain state always exists below the Curie point for ␤ ⬎ ␤ c . In fact, the frozen field in SCT is very large. A full proof is that it lowers the dielectric peaks of SCT crystal obviously. Such a frozen field can stabilize the multi-domain state even in the ground state. z Figure 7共a兲 shows the temperature dependence of 具 ␴ m 典 and 具 S zi 典 for Sr0.9925Ca0.0075TiO3 . According to the calculaz tion, 具 ␴ m 典 and 具 S zi 典 decrease with increasing temperature. Under the frozen field, they keep deviating from zero in the whole range of the temperature. For Sr0.9925Ca0.0075TiO3 , 具 ␴ mz 典 decreases from 0.14 to 0.005 when the temperature increases from 0 to 100 K. This means that the impuritypolarized region tends to become faint with increasing temperature. Uwe et. al.17 investigated KTaO3 both pure and doped with Sr and Ca impurities. For the regions polarized

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RELATION BETWEEN PHASE TRANSITION AND . . .

PHYSICAL REVIEW B 66, 104105 共2002兲

by defects, these authors demonstrated the size r c ⫽4a at low temperature and r c ⫽1.3a at T⫽100 K, where r c is the size of these regions and a the lattice constant. The experimental result fully complies with our theoretical result. For T⬎40 K, 具 S zi 典 is so small that it can be ignored. Hence the frozen-field-induced domains in SCT become very faint at high temperatures. Now we suppose that the frozen field is z ignored in the crystal. According to Fig. 6, 具 ␴ m 典 and 具 S zi 典 will disappear simultaneously at 17 K with increasing temperature. The ideal crystal shows a typical second-order phase transition. SCT has no global ferroelectric polarization because of the existence of multidomain states. However, the reversal of domains under an ac field can induce a ’’remanent polarization.’’ We suppose that the remanent polarization is directly proportional to the average polarization in a domain. Thus we obtain the temperature dependence of the remanent polarization for Sr0.96Ca0.04TiO3 according to Eqs. 共16兲–共18兲 关Fig. 7共b兲, solid line兴. Obviously the theoretical curve can describe the experimental result quite well in Fig. 7共b兲.

IV. CONCLUSION

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A detailed model is established to explain the dielectric response in SCT. Apart from the transverse Ising model considered previously11,12 the action of local frozen fields and piezoelectric interaction due to the Ca2⫹ ions is taken into account. In SCT, the impurity-polarized clusters induce the ferroelectric phase, and frozen fields give rise to the appearance of a multidomain state. According to our model, both ␮ 1 and J 3 decrease with increasing Ca content ␤ . The frozen field E r increases with increasing ␤ . In SCT, the observed changes of the smearing of the transition and of T m can be attributed to changes of ␮ 1 , J 3 , and E r .

ACKNOWLEDGMENTS

Thanks are due to the Alexander von Humboldt-Stiftung for a research grant to L. Z.

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