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Effect of external mechanical stress on the domain structure of Pb(Zr0.35Ti0.65)O3 thin films
Solid State Communications 135 (2005) 703–706 www.elsevier.com/locate/ssc
Effect of external mechanical stress on the domain structure of Pb(Zr0.35Ti0.65)O3 thin films Li-Ben Lib,*, Xiu-Mei Wua, Xiao-Mei Lua, Jing-Song Zhua a
National Laboratory of Solid State Microstructures, Department of Physics, Nanjing University, Nanjing 210093, People’s Republic of China b Department of Mathematics and Physics, Henan University of Science and Technology, Luoyang 471003, People’s Republic of China Received 2 May 2005; accepted 6 June 2005 by H. Takayama Available online 21 June 2005
Abstract Landau–Devonshire theory is used to investigate the effect of external mechanical stress (or strain) on domain structure and the remanent polarization of Pb(Zr0.35Ti0.65)O3 (PZT) thin films, by considering the competition between the external stress and the intrinsic stresses. A set of intrinsic stress functions in PZT film is obtained by solving elastic mechanical equations. At room temperature, the intrinsic stresses may lead to an alternate a/c/a/c domain structure. While an external in-plane tensile stress increases the Gibbs free energy of the c-phase and decreases that of a-phase. Parts of the c-domains turn to a-domains, so as to reduce the remanent polarization of the PZT films. q 2005 Elsevier Ltd. All rights reserved. PACS: 77.55.Cf; 62.40.Ci; 77.80.Dj; 77.22.Ej Keywords: A. Ferroelectric; A. Film; D. Stress; C. Domain structure; D. Polarization
1. Introduction Pb(ZrxTi1Kx)O3 (PZT) thin films can be used in nonvolatile memories due to their high remanent polarization [1]. The remanent polarization of ferroelectric films is controlled by the domain structure of the material, especially by the volume fraction of the domains with outof-plane polarization (c-domain). An external stress (or strain) can alter the domain direction or lead to the ferroelastic domain wall motion [2,3]. Kelman [2] et al. explained the influence of the external strain on the normalized polarization by a strain accommodation mechanism. Actually, enormous intrinsic stresses can exist in thin films when one material is deposited on another, resulting from difference in crystal lattice parameters and thermal expansion behavior between the film and the underlying * Corresponding author. Tel.: C86 37964908599; fax: C86 37964810223. E-mail address: [email protected] (L.-B. Li).
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2005.06.008
substrate or arising from defects formed during film deposition [4]. Theoretical [5] and Experimental [6] results showed that the intrinsic stresses can also affect the domain structure and the properties of ferroelectric films. Kanno et al. [7] have measured and calculated the influence of the total stress on the remanent polarization in PZT film with 3000 nm thickness. They considered the intrinsic stress (residual stress) being a constant in the whole film. Generally speaking, the intrinsic stress is uneven in the film [8,9], decaying from the interface to the inner of the film. The main goal of this paper is to investigate the influence of the competition between the external stress and the intrinsic stress on the domain structure in PZT films. The calculated result agrees with the experiments. 2. Theory Consider a two-dimensional system that a PZT film with thickness D is grown on a thick substrate. A Cartesian coordinate system is defined that the axes x and z is parallel
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L.-B. Li et al. / Solid State Communications 135 (2005) 703–706
and perpendicular to the interface between the film and the substrate, respectively. The origin is located at the interface. PZT has a prototype phase of symmetry Pm3m and transforms to low temperature ferroelectric phase upon cooling. The Gibbs free energy for two-dimensional PZT film is expressed as [8]: G Z a1 ðP2x C P2z Þ C a11 ðP4x C P4z Þ C a12 P2x P2z C a111 ðP6x C P6z Þ C a112 ðP2z C P2x ÞP2z P2x K Q11 ðsxx P2x
C szz P2z Þ K Q12 ðsxx P2z
C szz P2x Þ
1 K Q44 sxz Pz Px K s11 ðs2xx C s2zz Þ K s12 sxx szz 2 1 K s44 s2xz C G0 2
(1)
where Pi and sij are the polarization and stress; ai, aij, aijk are the dielectric stiffness coefficients at constant stress; sij are the elastic compliances at constant polarization; and Qij are the electrostrictive constant. The dielectric stiffness a1 is given by the Curie–Weiss law, a1Z(TKT0/230C), where T0 and C are the Curie–Weiss temperature and constant of bulk ferroelectric, respectively, and 30 is the permittivity of free space. G0 is the Gibbs free energy in the paraelectric state. There are two possible low-temperature phases in PZT film for the two-dimensional system introduced above: a-phase for Pxs0 and PzZ0; c-phase for PxZ0 and Pzs0. Minimizing Eq. (1) with respect to polarization, the spontaneous polarization for the two low-temperature phases can be obtained. Thus for a-phase sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ka11 C a211 K 3a111 ½a1 K ðQ11 sxx C Q12 szz Þ Px Z 3a111 (2) Ga Z G0 C ½a1 K ðQ11 sxx C Q12 szz ÞP2x C a11 P4x C a111 P6x for c-phase sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ka11 C a211 K 3a111 ½a1 K ðQ12 sxx C Q11 szz Þ Pz Z 3a111 (3) Gc Z G0 C ½a1 K ðQ12 sxx C Q11 szz ÞP2z C a11 P4z C a111 P6z If an external in-plane stress, se, is applied to the film along x-axis, the corresponding strain can be computed from s Z s11 se
(4)
Intrinsic stresses can exist in thin films when one material is deposited on another. Consider the case that at the paraelectric phase the interface between the film and the substrate is ‘strained incommensurate’, i.e. in a periodic array of misfit dislocation on the interface, one lattice losses
one-to-one correspondence between the unit cells of the film and the substrate [10]. The film is thus constructed periodically along the interface. With an extended Frenkel–Kontorova model [11,12], the traction acting on the film from the substrate is described by a shear stress at the interface [13]: 2p sizx Z s0 sin x ; z Z0 (5) Ls where LsZbfbs/jbfKbsj, s0ZGfjbfKbsj/2pd in which bs and bf are the lattice constants of the film and substrate, respectively; Gf is the elastic modulus of PZT film; d the distance of the atomic plane between the substrate and the film; Ls is the length of a periodic array. The distribution of stresses in the film can be obtained by solving elastic mechanical equations [14,15]: sixx Z
v2 f i v2 f i v2 f ; V4 f Z 0 ; s Z ; s Z K zz zx vzvx vz2 vx2
(6)
Further more, we assume that the shear stress at the top of the film is very small and is determined only by the stress at the interface. A possible distribution of stresses in the film is 8 sixx Z Ks0 cosð2px=Ls ÞexpðK2pz=Ls Þ > > < (7) sizz Z s0 cosð2px=Ls ÞexpðK2pz=Ls Þ Z Ksixx > > : i sxz Z s0 sinð2px=Ls ÞexpðK2pz=Ls Þ The magnitude of stress decays with exponential law from the interface to the top of the film, which agrees with the Kim’s result [8]. In-plane compressive stress (sixx ! 0) and tensile stress (sixx O 0) may coexist in the ferroelectric film due to the factor cos(2px/Ls). The feature of the intrinsic stresses, sixx ZKsizz , means that the elastic distortion of the crystal lattice may result in two primary stresses along two directions perpendicular to each other. A unit cell is elongated along x as well as compressed along z, vice versa.
3. Results and discussions We apply the above theories to Pb(Zr0.35Ti0.65)O3 films grown on Si substrate to determine how 908 domain wall motion in response to applied stress would affect the domain structure and the polarization of the film. Part of parameters are taken from Ref. [16]: Q11Z0.065 m4/C2, Q12ZK 0.032 m4/C2, s11Z6.785!10K12 m2/N, s12ZK2.5!10K12 m2/N. Other parameters are fitted from the parameters in Ref. [16]: a1Z1.396(TK423.9)!105 m/F, a11ZK2.274! 10 7 m5/C 2F, a 12Z9.994!10 7 m5/C2F, a 111Z1.549! 108 m9/C4F, a112Z2.231!108 m9/C4F, a123ZK8.0! 109 m9/C4F. The phase which corresponds to minimum G is the equilibrium thermodynamic states of the PZT film. The stress–phase diagrams at 25 and 450 8C are calculated from Eqs. (2) and (3), and plotted in Fig. 1(a) and (b),
L.-B. Li et al. / Solid State Communications 135 (2005) 703–706
Fig. 1. Stress–phsase diagram for (a) TZ25 8C and (b) TZ450 8C.
respectively. At room temperature, the region in the film with sxxOszz is in favor of forming a-phase. And that with sxx!szz is in favor of forming c-phase. At 450 8C, which is higher than the transition temperature of bulk PZT (435.8 8C), ferroelectric and paraelectric phase (p-phase) may coexist in the film as shown in Fig. 1(b). The cause is that PZT is displacive ferroelectrics. The lattice distortion resulting from the stress is helpful in stabilizing ferroelectric state in PZT films. When an external in-plane tensile stress (se) is put on the film, the primary stress along x direction is sixx C se . Let s Z 2sixx C se
(8)
We take that s0Z0.5 GPa, LsZ1600 nm, DZ300 nm. The distribution of s for seZ0, 0.05, 0.1 and 0.25 GPa in a periodic array at room temperature is plotted in Fig. 2. The region in the film with s!0 (white color) is in favor of forming c-phase, that with sO0 (grey color) in favor of forming a-phase, as just described. Without external stress (seZ0), in-plane tensile intrinsic stress is in favor of forming a-phase and a compressive one in favor of forming c-phase. Therefore, the intrinsic stress may result in alternate a/c/a/c domain structure along the interface due to the coexistence of tensile and compressive stresses. This kind of domain structure was observed in PbTiO3 films grown on KTaO3 substrate [15]. The region with a-phase expands as the external tensile stress increases. Contrarily, c-phase enlarges its region as an external compressive stress
705
Fig. 2. The distribution of s in a periodic array for seZ0, 0.05, 0.1 and 0.25 GPa. Without external stress, a- and c-phase coexist equally in a periodic array, while the region with a-phase expands as the external tensile stress increases.
is put on the PZT film. The conclusion is in qualitative agreement with the experiment by Lu¨ et al. [3]. Actually, an external tensile in-plane stress elongates the unit cell in the PZT film along x direction. If the effect of the external tensile stress on a unit cell is greater than that of the intrinsic compressive stress, the direction of spontaneous polarization turns from z to x direction. To compare our theory quantitatively with experiment, we consider the turning of c-domain in PZT film under an external tensile in-plane stress. Take Ls Z
2p 3:925 !10K3 K 2:325 !10K6 D
(9)
in a nano meter unit, which was fitted from the experimental values of the average phase transition temperature in PZT films by Kim et al. [8]. The parameter s0 can be estimated as 80p2/Ls (GPa) from Eq. (7) and the average stress in PZT films given by Kelman [17]. Elastic distortion of the crystal lattice and the domain turning take place when external strain is applied on PZT film. The two effects have influence on the polarization of the film. The normalized (average) polarization, P, on external strain, s, are computed from Eqs. (3), (4) and (7) and plotted in Fig. 3 for DZ250 nm (6 line) and 400 nm (7 line). The P–s curve can be divided into two parts according to its slope. When s is small, the decrease of P is resulted mainly from elastic distortion of the crystal lattice. When s is large enough, domain turning dominates in
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L.-B. Li et al. / Solid State Communications 135 (2005) 703–706
turns to a-domain, which leads to a notable decrease of the polarization. The theoretical result agrees with experiment.
References
Fig. 3. Dependence of the normalized polarization P on the external strain s. The normalized polarization deceases with the increase of the external in-plane strain.
PZT film and leads to a notable decrease of P. The theoretical result is in well agreement with the experiment by Kelman [2] (line for 250 nm and line for 400 nm) as shown in Fig. 3.
4. Conclusion Misfit dislocation at the interface between the substrate and the film may result in a set of sin-modulated intrinsic stresses. Under the action of the intrinsic stresses, the region with (in-plane) tensile stress is in favor of forming a-phase and compressive stress in favor of forming c-phase. c-Phase losses its stability under in-plane external tensile stress. If the external tensile stress on the lattice is greater than two times of the intrinsic in-plane compressive stress, c-domain
[1] J.F. Scott, C.A.P. De Araujo, Science 246 (1989) 1400. [2] M.B. Kelman, P.C. Mclntyre, B.C. Hendrix, S.M. Bilodeau, J.F. Roeder, J. Appl. Phys. 93 (2003) 9231. [3] X. Lu¨, X. Zhang, J. Zhu, Z. Liu, Y. Wang, Ferroelectrics 252 (2001) 517. [4] S.B. Desu, J. Electrochem. Soc. 140 (1993) 2981. [5] N.A. Pertsev, A.G. Zembilgotov, A.K. Tagantsev, Phys. Rev. Lett. 80 (1998) 1988. [6] K.J. Choi, M. Biegalski, Y.L. Li, A. Sharan, J. Schubert, et al., Science 306 (2004) 1005. [7] I. Kanno, Y. Yokoyama, H. Kotera, Phys. Rev. B 69 (2004) 064103. [8] H.J. Kim, S.H. Oh, H.M. Jang, Appl. Phys. Lett. 75 (1999) 3195. [9] L. Roitburd, Phys. Stat. Sol. A 37 (1976) 329. [10] S. Little, K. Zangwill, Phys. Rev. B 49 (23) (1994) 16659. [11] J. Frenkel, T. Kontorova, J. Phys. (USSR) 1 (1939) 137. [12] C. Kittle, Introduction to Solid State Physics, fifth ed., Wiley, New York, 1976. [13] L. Li, J. Zhang, Z. Yin, M. Zhang, Phys. Lett. A 321 (2004) 67. [14] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, third ed., McGraw-Hill, New York, 1970. [15] B.S. Kwak, A. Erbil, J.D. Budai, M.F. Chisholm, L.A. Boatner, B.J. Wilkens, Phys. Rev. B 49 (21) (1994) 14865. [16] A. Amin, R.E. Newnham, L.E. Cross, Phys. Rev. B 34 (1986) 1595. [17] M.B. Kelman, P.C. Mclntyre, A. Gruveman, B.C. Hendrix, S.M. Bilodeau, J.F. Roeder, J. Appl. Phys. 94 (2003) 5210.
Effect of external mechanical stress on the domain structure of Pb(Zr0.35Ti0.65)O3 thin films Li-Ben Lib,*, Xiu-Mei Wua, Xiao-Mei Lua, Jing-Song Zhua a
National Laboratory of Solid State Microstructures, Department of Physics, Nanjing University, Nanjing 210093, People’s Republic of China b Department of Mathematics and Physics, Henan University of Science and Technology, Luoyang 471003, People’s Republic of China Received 2 May 2005; accepted 6 June 2005 by H. Takayama Available online 21 June 2005
Abstract Landau–Devonshire theory is used to investigate the effect of external mechanical stress (or strain) on domain structure and the remanent polarization of Pb(Zr0.35Ti0.65)O3 (PZT) thin films, by considering the competition between the external stress and the intrinsic stresses. A set of intrinsic stress functions in PZT film is obtained by solving elastic mechanical equations. At room temperature, the intrinsic stresses may lead to an alternate a/c/a/c domain structure. While an external in-plane tensile stress increases the Gibbs free energy of the c-phase and decreases that of a-phase. Parts of the c-domains turn to a-domains, so as to reduce the remanent polarization of the PZT films. q 2005 Elsevier Ltd. All rights reserved. PACS: 77.55.Cf; 62.40.Ci; 77.80.Dj; 77.22.Ej Keywords: A. Ferroelectric; A. Film; D. Stress; C. Domain structure; D. Polarization
1. Introduction Pb(ZrxTi1Kx)O3 (PZT) thin films can be used in nonvolatile memories due to their high remanent polarization [1]. The remanent polarization of ferroelectric films is controlled by the domain structure of the material, especially by the volume fraction of the domains with outof-plane polarization (c-domain). An external stress (or strain) can alter the domain direction or lead to the ferroelastic domain wall motion [2,3]. Kelman [2] et al. explained the influence of the external strain on the normalized polarization by a strain accommodation mechanism. Actually, enormous intrinsic stresses can exist in thin films when one material is deposited on another, resulting from difference in crystal lattice parameters and thermal expansion behavior between the film and the underlying * Corresponding author. Tel.: C86 37964908599; fax: C86 37964810223. E-mail address: [email protected] (L.-B. Li).
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2005.06.008
substrate or arising from defects formed during film deposition [4]. Theoretical [5] and Experimental [6] results showed that the intrinsic stresses can also affect the domain structure and the properties of ferroelectric films. Kanno et al. [7] have measured and calculated the influence of the total stress on the remanent polarization in PZT film with 3000 nm thickness. They considered the intrinsic stress (residual stress) being a constant in the whole film. Generally speaking, the intrinsic stress is uneven in the film [8,9], decaying from the interface to the inner of the film. The main goal of this paper is to investigate the influence of the competition between the external stress and the intrinsic stress on the domain structure in PZT films. The calculated result agrees with the experiments. 2. Theory Consider a two-dimensional system that a PZT film with thickness D is grown on a thick substrate. A Cartesian coordinate system is defined that the axes x and z is parallel
704
L.-B. Li et al. / Solid State Communications 135 (2005) 703–706
and perpendicular to the interface between the film and the substrate, respectively. The origin is located at the interface. PZT has a prototype phase of symmetry Pm3m and transforms to low temperature ferroelectric phase upon cooling. The Gibbs free energy for two-dimensional PZT film is expressed as [8]: G Z a1 ðP2x C P2z Þ C a11 ðP4x C P4z Þ C a12 P2x P2z C a111 ðP6x C P6z Þ C a112 ðP2z C P2x ÞP2z P2x K Q11 ðsxx P2x
C szz P2z Þ K Q12 ðsxx P2z
C szz P2x Þ
1 K Q44 sxz Pz Px K s11 ðs2xx C s2zz Þ K s12 sxx szz 2 1 K s44 s2xz C G0 2
(1)
where Pi and sij are the polarization and stress; ai, aij, aijk are the dielectric stiffness coefficients at constant stress; sij are the elastic compliances at constant polarization; and Qij are the electrostrictive constant. The dielectric stiffness a1 is given by the Curie–Weiss law, a1Z(TKT0/230C), where T0 and C are the Curie–Weiss temperature and constant of bulk ferroelectric, respectively, and 30 is the permittivity of free space. G0 is the Gibbs free energy in the paraelectric state. There are two possible low-temperature phases in PZT film for the two-dimensional system introduced above: a-phase for Pxs0 and PzZ0; c-phase for PxZ0 and Pzs0. Minimizing Eq. (1) with respect to polarization, the spontaneous polarization for the two low-temperature phases can be obtained. Thus for a-phase sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ka11 C a211 K 3a111 ½a1 K ðQ11 sxx C Q12 szz Þ Px Z 3a111 (2) Ga Z G0 C ½a1 K ðQ11 sxx C Q12 szz ÞP2x C a11 P4x C a111 P6x for c-phase sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ka11 C a211 K 3a111 ½a1 K ðQ12 sxx C Q11 szz Þ Pz Z 3a111 (3) Gc Z G0 C ½a1 K ðQ12 sxx C Q11 szz ÞP2z C a11 P4z C a111 P6z If an external in-plane stress, se, is applied to the film along x-axis, the corresponding strain can be computed from s Z s11 se
(4)
Intrinsic stresses can exist in thin films when one material is deposited on another. Consider the case that at the paraelectric phase the interface between the film and the substrate is ‘strained incommensurate’, i.e. in a periodic array of misfit dislocation on the interface, one lattice losses
one-to-one correspondence between the unit cells of the film and the substrate [10]. The film is thus constructed periodically along the interface. With an extended Frenkel–Kontorova model [11,12], the traction acting on the film from the substrate is described by a shear stress at the interface [13]: 2p sizx Z s0 sin x ; z Z0 (5) Ls where LsZbfbs/jbfKbsj, s0ZGfjbfKbsj/2pd in which bs and bf are the lattice constants of the film and substrate, respectively; Gf is the elastic modulus of PZT film; d the distance of the atomic plane between the substrate and the film; Ls is the length of a periodic array. The distribution of stresses in the film can be obtained by solving elastic mechanical equations [14,15]: sixx Z
v2 f i v2 f i v2 f ; V4 f Z 0 ; s Z ; s Z K zz zx vzvx vz2 vx2
(6)
Further more, we assume that the shear stress at the top of the film is very small and is determined only by the stress at the interface. A possible distribution of stresses in the film is 8 sixx Z Ks0 cosð2px=Ls ÞexpðK2pz=Ls Þ > > < (7) sizz Z s0 cosð2px=Ls ÞexpðK2pz=Ls Þ Z Ksixx > > : i sxz Z s0 sinð2px=Ls ÞexpðK2pz=Ls Þ The magnitude of stress decays with exponential law from the interface to the top of the film, which agrees with the Kim’s result [8]. In-plane compressive stress (sixx ! 0) and tensile stress (sixx O 0) may coexist in the ferroelectric film due to the factor cos(2px/Ls). The feature of the intrinsic stresses, sixx ZKsizz , means that the elastic distortion of the crystal lattice may result in two primary stresses along two directions perpendicular to each other. A unit cell is elongated along x as well as compressed along z, vice versa.
3. Results and discussions We apply the above theories to Pb(Zr0.35Ti0.65)O3 films grown on Si substrate to determine how 908 domain wall motion in response to applied stress would affect the domain structure and the polarization of the film. Part of parameters are taken from Ref. [16]: Q11Z0.065 m4/C2, Q12ZK 0.032 m4/C2, s11Z6.785!10K12 m2/N, s12ZK2.5!10K12 m2/N. Other parameters are fitted from the parameters in Ref. [16]: a1Z1.396(TK423.9)!105 m/F, a11ZK2.274! 10 7 m5/C 2F, a 12Z9.994!10 7 m5/C2F, a 111Z1.549! 108 m9/C4F, a112Z2.231!108 m9/C4F, a123ZK8.0! 109 m9/C4F. The phase which corresponds to minimum G is the equilibrium thermodynamic states of the PZT film. The stress–phase diagrams at 25 and 450 8C are calculated from Eqs. (2) and (3), and plotted in Fig. 1(a) and (b),
L.-B. Li et al. / Solid State Communications 135 (2005) 703–706
Fig. 1. Stress–phsase diagram for (a) TZ25 8C and (b) TZ450 8C.
respectively. At room temperature, the region in the film with sxxOszz is in favor of forming a-phase. And that with sxx!szz is in favor of forming c-phase. At 450 8C, which is higher than the transition temperature of bulk PZT (435.8 8C), ferroelectric and paraelectric phase (p-phase) may coexist in the film as shown in Fig. 1(b). The cause is that PZT is displacive ferroelectrics. The lattice distortion resulting from the stress is helpful in stabilizing ferroelectric state in PZT films. When an external in-plane tensile stress (se) is put on the film, the primary stress along x direction is sixx C se . Let s Z 2sixx C se
(8)
We take that s0Z0.5 GPa, LsZ1600 nm, DZ300 nm. The distribution of s for seZ0, 0.05, 0.1 and 0.25 GPa in a periodic array at room temperature is plotted in Fig. 2. The region in the film with s!0 (white color) is in favor of forming c-phase, that with sO0 (grey color) in favor of forming a-phase, as just described. Without external stress (seZ0), in-plane tensile intrinsic stress is in favor of forming a-phase and a compressive one in favor of forming c-phase. Therefore, the intrinsic stress may result in alternate a/c/a/c domain structure along the interface due to the coexistence of tensile and compressive stresses. This kind of domain structure was observed in PbTiO3 films grown on KTaO3 substrate [15]. The region with a-phase expands as the external tensile stress increases. Contrarily, c-phase enlarges its region as an external compressive stress
705
Fig. 2. The distribution of s in a periodic array for seZ0, 0.05, 0.1 and 0.25 GPa. Without external stress, a- and c-phase coexist equally in a periodic array, while the region with a-phase expands as the external tensile stress increases.
is put on the PZT film. The conclusion is in qualitative agreement with the experiment by Lu¨ et al. [3]. Actually, an external tensile in-plane stress elongates the unit cell in the PZT film along x direction. If the effect of the external tensile stress on a unit cell is greater than that of the intrinsic compressive stress, the direction of spontaneous polarization turns from z to x direction. To compare our theory quantitatively with experiment, we consider the turning of c-domain in PZT film under an external tensile in-plane stress. Take Ls Z
2p 3:925 !10K3 K 2:325 !10K6 D
(9)
in a nano meter unit, which was fitted from the experimental values of the average phase transition temperature in PZT films by Kim et al. [8]. The parameter s0 can be estimated as 80p2/Ls (GPa) from Eq. (7) and the average stress in PZT films given by Kelman [17]. Elastic distortion of the crystal lattice and the domain turning take place when external strain is applied on PZT film. The two effects have influence on the polarization of the film. The normalized (average) polarization, P, on external strain, s, are computed from Eqs. (3), (4) and (7) and plotted in Fig. 3 for DZ250 nm (6 line) and 400 nm (7 line). The P–s curve can be divided into two parts according to its slope. When s is small, the decrease of P is resulted mainly from elastic distortion of the crystal lattice. When s is large enough, domain turning dominates in
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L.-B. Li et al. / Solid State Communications 135 (2005) 703–706
turns to a-domain, which leads to a notable decrease of the polarization. The theoretical result agrees with experiment.
References
Fig. 3. Dependence of the normalized polarization P on the external strain s. The normalized polarization deceases with the increase of the external in-plane strain.
PZT film and leads to a notable decrease of P. The theoretical result is in well agreement with the experiment by Kelman [2] (line for 250 nm and line for 400 nm) as shown in Fig. 3.
4. Conclusion Misfit dislocation at the interface between the substrate and the film may result in a set of sin-modulated intrinsic stresses. Under the action of the intrinsic stresses, the region with (in-plane) tensile stress is in favor of forming a-phase and compressive stress in favor of forming c-phase. c-Phase losses its stability under in-plane external tensile stress. If the external tensile stress on the lattice is greater than two times of the intrinsic in-plane compressive stress, c-domain
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