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Dielectric behavior of ferroelectric thin films grown on orthorhombic substrates
Physics Letters A 357 (2006) 485–490 www.elsevier.com/locate/pla
Dielectric behavior of ferroelectric thin films grown on orthorhombic substrates P.F. Liu, J.H. Qiu, Q. Jiang ∗ Department of Physics, Suzhou University, Suzhou 215006, China Received 21 January 2006; received in revised form 8 April 2006; accepted 25 April 2006 Available online 3 May 2006 Communicated by R. Wu
Abstract Using Landau–Devonshire (LD)-type phenomenological model, we investigate the dielectric behaviors of single-domain PT thin films grown on orthorhombic substrates. Effects of in-plane misfit strains and external electric fields are combined on dielectric properties. In our calculation of permittivities, we take account of the susceptibility matrix. Basing on the assumption that in-plane misfit strains um1 = −um2 , we find that in-plane permittivities ε11 and ε22 are symmetrical about the line where in-plane misfit strains are zero. Same is out-plane permittivity ε33 . Anisotropic in-plane misfit strains result in anomalies of permittivities at boundaries of phase transitions. Further calculation reveals that dielectric tunability reach to maximums at boundaries of phase transitions where the same direction polarization component appears or disappears. So the permittivities and tunabilities can be chosen to satisfy different demands by controlling applied fields and changing in-plane misfit strains. © 2006 Elsevier B.V. All rights reserved. PACS: 77.55.+f; 77.22.Ej; 77.84.-s
1. Introduction During the recent years, considerable experimental and theoretical efforts have been devoted to the study of ferroelectric thin film materials that have remarkably ferroelectric, dielectric, piezoelectric, and electro-optic properties. The perovskite-type ferroelectric thin films have dependence of permittivity in an electric field, which make them attractive for application in tunable microwave devices [1,2], non-volatile memory devices [3–5], infrared sensors [6], and electro-optic devices [7–9]. Previous investigations show that properties of ferroelectric films are greatly affected by dimension of grain, stress, composition, etc. As has been known, internal stress originates from misfit of lattice parameters between film and substrate. When substrate lattice parameter is larger than that of film, the tensile stress occurs and strain um is a positive quantity. In contrast, com* Corresponding author.
E-mail address: [email protected] (Q. Jiang). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.04.084
pressive stress appears and the strain um is negative. Basing on LD-type phenomenological model, Pertsev et al. [10] developed a new form of the thermodynamic potential which corresponds to the actual mechanical boundary conditions in order to construct misfit strain–temperature phase diagrams of BaTiO3 (BT) and PbTiO3 (PT) thin films epitaxially grown on cubic substrates. Ban et al. [11–13] took into account the formation of misfit dislocations at the growth temperature and analyzed theoretically dielectric coefficients, tunability, and pyroelectric coefficients of epitaxial Bax Sr1–x O3 (BST) and PbZrx Ti1–x O3 (PZT) thin films on different substrates as a function of film thickness. Moreover, Lang Chen et al. [14] introduced spontaneous strain of film which roots in spontaneous polarization to study non-linear electric field dependence of piezoelectric response in epitaxial ferroelectric PZT thin films. As a conclusion, strain mechanism has an important effect on phase transitions and properties of epitaxial ferroelectric thin films. Although many groups are interested in effects of stress, just those films grown on cubic substrates have been investigated widely, in which in-plane stresses are isotropic. Recently, experiments and theories pay more attention to the
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films of anisotropic in-plane strains which are not grown on cubic substrates [15–21]. In theory, Zembilgotov et al. [16] firstly considered phase diagrams of a single-domain PT film and a Pb0.35 Sr0.65 TiO3 film grown on orthorhombic substrates and predicted the equilibrium polarization states of thin films. Wang and Zhang [17] presented misfit strain–misfit strain and temperature–misfit strain phase diagrams for single-domain BT and PT thin films grown on tetragonal substrates. In experiment, Simon et al. [18,19] investigated the permittivity of BST 60/40 film grown on NdGaO3 by pulsed laser deposition. Bellotti et al. [20,21] studied in-plane anisotropy in BST films and SrTiO3 films. In this Letter, we investigate the permittivities and tunabilities of single-domain PT film deposited on orthorhombic substrates by using the non-linear thermodynamic theory brought forward by Ref. [16]. Compared with usual ways, permittivities are gained by working out the susceptibility matrix. Further, we investigate the effects of external electric fields on permittivities by calculating tunabilities. Taking into consideration that films in devices are usually applied at room temperature (RT = 25 ◦ C) and there are new ferroelectric phases [16] which do not exist on cubic substrates at RT , we take T = 25 ◦ C in our calculation. According to phase diagram of Ref. [16], we assume that value of in-plane misfit strain um1 along [100] direction is equal to value of in-plane misfit strain um2 along [010] direction. And if one in-plane strains is compressive strain, the other must be tensile strain. 2. Theoretical development For single-domain ferroelectric films grown on thick dissimilar substrates, the elastic Gibbs energy function is introduced in Refs. [10,22]. It is well known that in-plane misfit strains are governed by substrate lattice parameters and lattice parameters of film. Because the thickness of substrate is much larger than the film thickness, the substrate is stress free and the internal stresses are concentrated in the film. In this Letter, we consider a single-domain ferroelectric film in the cubic paraelectric phase grown on a dissimilar orthorhombic substrate which induces two different in-plane strains um1 and um2 . So in-plane misfit strains can be attained by um1 = (as1 − a0 )/as1 and um2 = (as2 − a0 )/as2 [17,23], where a0 is film lattice parameter, as1 and as2 are lattice parameters of substrate along [100] direction and [010] direction, respectively. We can derive ˜ in an external electric field as a functhe modified potential G tion of the polarization components Pi (i = 1, 2, 3) [16,17]: 4 ∗ ∗ ˜ = α1∗ P12 + α2∗ P22 + α3∗ P32 + α11 G P1 + P24 + α33 P34 2 ∗ ∗ P1 + P22 P32 + α12 + α13 P12 P22 + α111 P16 + P26 + P36 + α112 P14 P22 + P32 + P24 P12 + P32 + P34 P12 + P22 + α123 P12 P22 P32 − E1 P1 − E2 P2 − E3 P3 +
[s11 (u2m1 + u2m2 ) − 2s12 um1 um2 ] 2 − s2 ) 2(s11 12
,
(1)
where Ek (k = 1, 2, 3) are components of the applied external electric field, um1 (Q12 s12 − Q11 s11 ) + um2 (Q11 s12 − Q12 s11 ) , 2 − s2 s11 12 (2) um2 (Q12 s12 − Q11 s11 ) + um1 (Q11 s12 − Q12 s11 ) ∗ α2 = α 1 + , 2 − s2 s11 12 (3) (u + u ) Q 12 m1 m2 α3∗ = α1 − (4) , s11 + s12
α1∗ = α1 +
∗ α11 = α11 +
1 [(Q211 + Q212 )s11 − 2Q11 Q12 s12 ] , 2 − s2 2 s11 12
(5)
∗ α33 = α11 +
Q212 , s11 + s12
(6)
∗ α12 = α12 − ∗ α13 = α12 +
[(Q211 + Q212 )s12 − 2Q11 Q12 s11 ] 2 + s2 s11 12
Q12 (Q11 + Q12 ) . s11 + s12
+
Q244 , 2s44
(7) (8)
Here α1 , αij (i, j = 1, 2, 3), and αij k (i, j, k = 1, 2, 3) are the dielectric stiffness coefficients, sij (i, j = 1, 2) are the elastic compliances at constant polarizations, and Qij (i, j = 1, 2, 3) are the electrostrictive constants. The temperature dependence of ferroelectricity is mainly governed by the dielectric stiffness coefficient α1 = (T − T0 )/2ε0 C since the higher-order stiffness coefficients αij and αij k may be taken as temperatureindependent parameters, where T0 and C are the Curie–Weiss temperature and Curie constant, ε0 is the permittivity of free space. In Eq. (1), only α1∗ , α2∗ and α3∗ show the effects of anisotropic strains in films. In other words, anisotropy of strains changes phases and properties of films by second-order polarization terms. ˜ and considering boundary Using the derived potential G conditions, Zembilgotov et al. [16] drew the misfit strain– temperature phase diagram of epitaxial PT film deposited on orthorhombic substrate at a given misfit strains um1 = −um2 when external electric field is equal to zero. Compared with the diagram of PT films grown on cubic substrates [10], the phase diagram of PT films deposited on orthorhombic substrates has new ferroelectric phases. All possible six stable phases are paraelectric phase, three tetragonal phases: a phase (P1 = 0, P2 = P3 = 0), a phase (P2 = 0, P1 = P3 = 0) and c phase (P1 = P2 = 0, P3 = 0), two orthorhombic phases: a c phase (P1 = P3 = 0, P2 = 0) and a c phase (P2 = P3 = 0, P1 = 0). The corresponding phases may be calculated by the ˜ with respect to relevant polarization components. minima of G Restricting our analysis to the small-signal dielectric responses (E → 0), the relative permittivities as functions of polarization components Pi can be derived from the thermodynamic po˜ Using the condition for thermodynamic equilibrium tential G. ˜ ∂ G/∂P i = 0 (E = 0), we can get the spontaneous polarization components Pi (i = 1, 2, 3). Because of reciprocal dielectric ˜ 2 /∂Pi ∂Pj (i, j = 1, 2, 3), the matrix susceptibilities χij = ∂ G inversion then enables us to find the dielectric susceptibilities
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ηij = 1/χij and relative permittivities εij = 1 + ηij /ε0 [24]. It should be noticed that the matrix of reciprocal dielectric susceptibilities χij are diagonal only in tetragonal phases. The important dielectric characteristics of a ferroelectric thin film are the in-plane permittivities ε11 , ε22 and out-plane permittivity ε33 , which can be measured in experiment. Basing on polarization components Pi in different phases, we calculate and summarize the results of relative permittivities. The a phase: 1 1 , =1+ ∗ ∗ ε0 χ11 ε0 (2α1 + 2α12 P22 + 2α112 P24 ) 1 1 =1+ , ε22 = 1 + ∗ P 2 + 30α 4 ε0 χ22 ε0 (2α2∗ + 12α11 111 P2 ) 2 1 1 =1+ . ε33 = 1 + (9) ∗ ∗ χ33 ε0 (2α3 + 2α13 P22 + 2α112 P24 )
ε11 = 1 +
The a c phase: ∗ P22 + 2α112 P24 + 2α123 P22 P32 χ11 = 2α1∗ + 2α12 ∗ + 2α13 P32 + 2α112 P34 , ∗ P22 + 30α111 P24 + 12α112 P22 P32 χ22 = 2α2∗ + 12α11 ∗ + 2α13 P32 + 2α112 P34 , ∗ P22 + 2α112 P24 + 12α112 P22 P32 χ33 = 2α3∗ + 2α13 ∗ + 12α33 P32 + 30α111 P34 , ∗ P2 P3 + 8α112 P23 P3 + 8α112 P2 P33 , χ23 = 4α13 1 ε11 = 1 + , ε0 χ11 χ33 , ε22 = 1 + 2 ) ε0 (χ22 χ33 − χ23 χ22 ε33 = 1 + . 2 ) ε0 (χ22 χ33 − χ23
(10)
ε22 = 1 +
1 , ε0 χ22
ε33 = 1 +
χ11 . 2 ) ε0 (χ11 χ33 − χ13
487
(12)
The a phase: 1 1 , =1+ ∗ P 2 + 30α 4 ε0 χ11 ε0 (2α1∗ + 12α11 111 P1 ) 1 1 1 , =1+ ε22 = 1 + ∗ ∗ ε0 χ22 ε0 (2α2 + 2α12 P12 + 2α112 P14 ) 1 1 =1+ . (13) ε33 = 1 + ∗ ∗ ε0 χ33 ε0 (2α3 + 2α13 P12 + 2α112 P14 ) ε11 = 1 +
In order to analyze the dielectric responses in external electric fields, we apply external electric fields. Because only εii is important in devices, T uii (i = 1, 2, 3) is investigated in our quantitative calculation. We assume that the external electric field is uniform and along axis. When we apply an electric field along spontaneous polarization component Pi , the polarization PiE is composed of two parts: spontaneous polarization and induced polarization. Using the stability criterion of first ˜ partial derivative ∂ G/∂P iE = Ei , the influence of the external electric field on film polarization PiE was taken into account. Substituting PiE for Pi in Eqs. (9)–(13), we can work out the permittivities in the presence of the external field. The tunability is defined as the variation of the permittivity in applied field with respect to the permittivity of zero electric field T uii = εii (0) − εii (E) /εii (0). (14) Thus, if in-plane misfit strains of film and electric field are given, the dielectric properties and tunabilities of thin films grown on dissimilar orthorhombic substrates can be calculated. 3. Results and discussion
The c phase: 1 1 , =1+ ∗ ∗ ε0 χ11 ε0 (2α1 + 2α13 P32 + 2α112 P34 ) 1 1 =1+ , ε22 = 1 + ∗ ∗ ε0 χ22 ε0 (2α2 + 2α13 P32 + 2α112 P34 ) 1 1 . (11) =1+ ε33 = 1 + ∗ P 2 + 30α 4 ε0 χ33 ε0 (2α3∗ + 12α33 111 P3 ) 3 ε11 = 1 +
The a c phase: ∗ P12 + 30α111 P14 + 12α112 P12 P32 χ11 = 2α1∗ + 12α11 ∗ + 2α13 P32 + 2α112 P34 , ∗ P12 + 2α112 P14 + 2α123 P12 P32 χ22 = 2α2∗ + 2α12 ∗ + 2α13 P32 + 2α112 P34 , ∗ P12 + 2α112 P14 + 12α112 P12 P32 χ33 = 2α3∗ + 2α13 ∗ + 12α33 P32 + 30α111 P34 , ∗ P1 P3 + 8α112 P13 P3 + 8α112 P1 P33 , χ13 = 4α13
ε11 = 1 +
ε0 χ33 , 2 ) ε0 (χ11 χ33 − χ13
In the following discussion, we will analyze the effects of misfit strains on dielectric properties and tunabilities of epitaxial PT film deposited on orthorhombic substrates at RT . In our calculation, the parameter values of coefficients for PT thin film are listed in Table 1 [10,16]. The value of misfit strain um1 is from −0.015 to 0.015. Table 1 The parameter values of PT used to calculate renormalized coefficients (in SI units) α1 α11 α12 α111 α112 α123 Q11 Q12 Q44 s11 s12 s44
−1.725 × 108 m/F −7.3 × 107 m5 /C2 F 7.5 × 108 m5 /C2 F 2.6 × 108 m9 /C4 F 6.1 × 108 m9 /C4 F −3.7 × 109 m9 /C4 F −0.089 m4 /C2 −0.026 m4 /C2 0.0675 m4 /C2 8.0 × 10−12 m2 /N −2.5 × 10−12 m2 /N 9.0 × 10−12 m2 /N
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Fig. 1. The spontaneous of PT film grown on orthorhombic substrates.
Fig. 2. Relative permittivities of PT film when um1 = −um2 .
According to the misfit strain–temperature phase diagram [16], a phase, a c phase, c phase, a c phase and a phase are the stable phases at RT . Because in-plane strains are anisotropic, the spontaneous polarization components Pi of films are remarkably different from those of films grown on cubic substrates. The misfit strain dependence of polarization components is represented in Fig. 1. As can be seen from Fig. 1, P1 only exists in a phase and a c phase, P2 is in a phase and a c phase, and P3 is in a c phase, c phase and a c phase. With the increase of tensile strain um1 , P1 is faintly discontinuous at the boundary between c phase and a c phase, then remarkably increases from a small non-zero value of 1.7 × 10−3 C/m2 . At the boundary between a c phase and a phase, P1 is obviously discontinuous. However, the effect of misfit strains on P1 in a c phase is greater than that in a phase. The same profile of P2 can be found while compressive strain um1 increase in a c phase and a phase. Fig. 1 shows that P3 is especial and varies non-monotonously with the change of misfit strains. Firstly, P3 increases with the decrease of compressive strain um1 in a c phase. Then, it becomes stable as film goes into c phase where misfit strain um1 varies from compressive strain to tensile strain. At last, when film transits to a c phase, P3 decreases corresponding to change of misfit strains. This can be ∗ ascribed to misfit strains which we set um1 = −um2 . α3∗ and α33 are independent of misfit strains. Therefore in the tetragonal c phase, P3 is independent of misfit strains and remains constant. Misfit strains have indirect effects on P3 by polarization components of P1 or P2 in the orthorhombic a c phase and a c phase. Based on assumption um1 = −um2 , in-plane polarization component P1 turns into P2 when um1 changes from tensile strain into compressive strain. Fig. 1 also shows that P1 and P3 are discontinuous at boundary between a c phase and a phase, and P2 and P3 are discontinuous at boundary between a c phase and a phase. Furthermore, P1 and P2 are discontinuous at boundaries between a c phase and c phase, and a c phase and c phase, respectively. As a result, all structure phase transitions a -phase/a c-phase, a c-phase/c-phase, c-phase/a c-phase, a c-phase/a -phase are first-order phase transitions. Permittivities are the most important property of ferroelectric films. The relative permittivities ε11 , ε22 and ε33 of PT epitaxial thin films grown on orthorhombic substrates are plot-
ted in Fig. 2 as functions of misfit strains. Fig. 2 reveals the misfit strains dependence of permittivities. It can be seen that the film permittivities vary non-monotonously with the misfit strains existing in the epitaxial system. In a and a c phase, P1 does not exist but ε11 has a small and anomalous jump accompanied by the first-order a -phase/a c-phase transition, as shown in Fig. 2. It can be explained that in-plane permittivity ε11 is related to the polarization components P2 and P3 which are discontinuous at the boundary between a and a c phases. At the boundary of c-phase/a c-phase transition, polarization component P1 increases from a small non-zero value (seeing Fig. 1), however structural transition results in anomalous divergence of in-plane permittivity ε11 [25]. The obvious discontinuity of P1 across the boundary between a c and a phases leads to a big step-like drop of in-plane permittivity ε11 . It is attractive that in-plane permittivities ε11 and ε22 symmetrically distribute about the line on which in-plane misfit strains are zero. Same symmetry of out-plane permittivity ε33 can be found in Fig. 2. These are consistent with the distributions of polarization components. In c phase, value of ε33 is a constant because in the present case we set um1 = −um2 . Although P3 is continuous at the phase boundary between c and a c phases, ε33 has a abnormal jump. This small jump, which appears at misfit strains slightly exceeding the phase boundary, may be explained by the relation ε33 in Eq. (12). This relation shows that ε33 increases due to the presence of a non-zero reciprocal susceptibility χ13 in a c phase. Accompanied by the firstorder a c-phase/a -phase transition, ε33 has a drastic jump and anomalous divergence, which may be related to structural transition [23]. When misfit strain um1 is compressive strain, the same phenomenon can be observed at the boundaries of phase transitions in Fig. 2. The high tunability has important action on tunable microwave devices, it is necessary to be investigated. Taking account of symmetry of dielectric coefficients, we only study the effects of misfit strains on tunabilities along [100] and [001] directions with applied external field of 50 kV/cm along the same directions respectively using the definition of the tunability Eq. (14) and the permittivities Eqs. (9)–(13). In thus electric field, field-induced polarization components have reached to saturation. Fig. 3(a) and (b) show that tunabilities are strongly
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dissimilar substrates, the misfit strain along a crystalline axis will differ from the misfit strain along the other crystalline axis, which changes polarization of film compared with films on cubic substrates, and new phases will appear. Pb0.35 Sr0.65 TiO3 films grown on orthorhombic NdGaO3 substrates, which exists in-plane strain um1 = −5.4 × 10−4 and um2 = −8 × 10−5 , have new phases considering the polarization state [16]. The result is in good agreement with the experimental data. The similar consequence can be found when films are grown on tetragonal substrates [17]. 4. Conclusion (a)
(b) Fig. 3. Tunabilities T u11 (a) and T u33 (b) of PT film as functions of in-plane misfit strains in external electric field of E = 50 kV/cm.
dependent on the misfit strains for different stable phases and theoretically maximum tunabilities can be attained at critical misfit strains corresponding to phase transitions. For applied electric field E1 [100], the tunability T u11 is almost 100% at the critical tensile strain um1 = 1.060%, which corresponds to the c-phase/a c-phase transition. For tunability T u33 , it is interesting that there are two maximums 100% in T u33 for applied filed E3 along [001] direction, which can be attributed to symmetry of permittivity ε33 about um1 = 0. Fig. 3 clearly reveals that T u33 reaches to maximum for um1 = −1.114% and um1 = 1.114%. Referring back to Fig. 1, these critical misfit strains correspond to the first-order a -phase/a c-phase transition and the first-order a -phase/a c-phase transition respectively. Connecting Fig. 2 to Fig. 3, the tunabilities as well as permittivities display similar characteristics as functions of the misfit strains indicating that epitaxy-induced internal stresses can optimize the dielectric properties of thin films. In the condition of um1 = −um2 , the in-plane strain anisotropy results in the appearance of new phases which do not form in films grown on cubic substrates [16]. Furthermore, we find that permittivities and tunabilities of thin film have anomalies at the boundaries of phase transitions, such as two jumps in ε11 and ε22 . Though we only take into account the strain state of um1 = −um2 , the similar consequence can be attained when the strain condition was altered. If ferroelectric film grown on
In summary, a theoretical formalism has been used to investigate the dielectric behaviors of single-domain PT films grown on orthorhombic substrates. Ulteriorly, the combined effects of misfit strains and external electric fields on dielectric properties are investigated. Quantitative calculation shows that the anisotropic in-plane strains um1 = −um2 lead to the symmetry of in-plane permittivities ε11 and ε22 about um1 = 0. Likewise ε33 symmetrically distributes to the each side of um1 = 0. Moreover, permittivities and tunabilities of thin film have anomalies at the boundaries of phase transitions. For in-plane tunability T u11 , it attain maximum at the boundary between c and a c phases. However, out-plane tunability T u33 have two maximums accompanied by first-order a -phase/a c-phase transition and a -phase/a c-phase transition. Our calculation reveals that dielectric tunability reaches to maximums at boundaries of phase transitions where the same direction polarization component appears or disappears. These are important properties to microwave devices. So the permittivities and tunabilities can be chosen to satisfied different demands by controlling applied fields and selecting substrates. Acknowledgements This work was supported by the National Natural Science Foundation of China under the Grant No. 10374069, Jiangsu Provincial Natural Science Foundation under the Grant No. BK2003032, and Jiangsu Key Laboratory of film Materials of China. References [1] J. Im, O. Auciello, P.K. Baumann, S.K. Streiffer, D.Y. Kaufman, A.R. Krauss, Appl. Phys. Lett. 76 (2000) 625. [2] W. Chang, C.M. Gilmore, W.-J. Kim, J.M. Pond, S.W. Kirchoefer, S.B. Qadri, D.B. Chirsey, J.S. Horwitz, J. Appl. Phys. 87 (2000) 3044. [3] J.F. Scott, C.A. Araujo, Science 246 (1989) 1400. [4] O. Auciello, J.F. Scott, R. Ramesh, Phys. Today 51 (7) (2000) 22. [5] P.K. Larsen, G.L.M. Kampshoer, M.J.E. Ulenaers, G.A.C.M. Spierings, R. Cuppens, Appl. Phys. Lett. 59 (1991) 611. [6] R. Takayama, Y. Tomita, K. Iijima, Ueda, J. Appl. Phys. 61 (1987) 411. [7] A.B. Kaufman, IEEE Trans. Electron DEvices ED-16 (1969) 511. [8] A. Mukherjee, S.R.J. Brueck, A.Y. Wu, Opt. Lett. 15 (1998) 151. [9] J.W. Li, F. Duewer, C. Gao, X.-D. Xiang, Y.L. Lu, Appl. Phys. Lett. 76 (2000) 769. [10] N.A. Pertsev, A.G. Zembilgotov, A.K. Taganatsev, Phys. Rev. Lett. 90 (1998) 1988.
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[19] W.K. Simon, E.K. Akdogan, A. Safari, J.A. Bellotti, Appl. Phys. Lett. 88 (2006) 132902. [20] J.A. Bellotti, W. Chang, S.B. Qadri, J.M. Pond, Appl. Phys. Lett. 88 (2006) 012902. [21] W. Chang, S.W. Kirchoefer, J.A. Bellotti, S.B. Qadri, J.M. Pond, J.H. Haeni, D.G. Schlom, J. Appl. Phys. 98 (2005) 024107. [22] M.J. Haun, E. Furman, S.J. Jang, H.A. McKinstry, L.E. Cross, J. Appl. Phys. 62 (1987) 3331. [23] S.P. Alpay, I.B. Misirlioglu, A. Sharma, Z.-G. Ban, J. Appl. Phys. 95 (2003) 8118. [24] S.H. Oh, H.M. Jang, Phys. Rev. B 62 (2000) 14757. [25] E.V. Balashova, A.K. Tagantsev, Phys. Rev. B 48 (1993) 9979.
Dielectric behavior of ferroelectric thin films grown on orthorhombic substrates P.F. Liu, J.H. Qiu, Q. Jiang ∗ Department of Physics, Suzhou University, Suzhou 215006, China Received 21 January 2006; received in revised form 8 April 2006; accepted 25 April 2006 Available online 3 May 2006 Communicated by R. Wu
Abstract Using Landau–Devonshire (LD)-type phenomenological model, we investigate the dielectric behaviors of single-domain PT thin films grown on orthorhombic substrates. Effects of in-plane misfit strains and external electric fields are combined on dielectric properties. In our calculation of permittivities, we take account of the susceptibility matrix. Basing on the assumption that in-plane misfit strains um1 = −um2 , we find that in-plane permittivities ε11 and ε22 are symmetrical about the line where in-plane misfit strains are zero. Same is out-plane permittivity ε33 . Anisotropic in-plane misfit strains result in anomalies of permittivities at boundaries of phase transitions. Further calculation reveals that dielectric tunability reach to maximums at boundaries of phase transitions where the same direction polarization component appears or disappears. So the permittivities and tunabilities can be chosen to satisfy different demands by controlling applied fields and changing in-plane misfit strains. © 2006 Elsevier B.V. All rights reserved. PACS: 77.55.+f; 77.22.Ej; 77.84.-s
1. Introduction During the recent years, considerable experimental and theoretical efforts have been devoted to the study of ferroelectric thin film materials that have remarkably ferroelectric, dielectric, piezoelectric, and electro-optic properties. The perovskite-type ferroelectric thin films have dependence of permittivity in an electric field, which make them attractive for application in tunable microwave devices [1,2], non-volatile memory devices [3–5], infrared sensors [6], and electro-optic devices [7–9]. Previous investigations show that properties of ferroelectric films are greatly affected by dimension of grain, stress, composition, etc. As has been known, internal stress originates from misfit of lattice parameters between film and substrate. When substrate lattice parameter is larger than that of film, the tensile stress occurs and strain um is a positive quantity. In contrast, com* Corresponding author.
E-mail address: [email protected] (Q. Jiang). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.04.084
pressive stress appears and the strain um is negative. Basing on LD-type phenomenological model, Pertsev et al. [10] developed a new form of the thermodynamic potential which corresponds to the actual mechanical boundary conditions in order to construct misfit strain–temperature phase diagrams of BaTiO3 (BT) and PbTiO3 (PT) thin films epitaxially grown on cubic substrates. Ban et al. [11–13] took into account the formation of misfit dislocations at the growth temperature and analyzed theoretically dielectric coefficients, tunability, and pyroelectric coefficients of epitaxial Bax Sr1–x O3 (BST) and PbZrx Ti1–x O3 (PZT) thin films on different substrates as a function of film thickness. Moreover, Lang Chen et al. [14] introduced spontaneous strain of film which roots in spontaneous polarization to study non-linear electric field dependence of piezoelectric response in epitaxial ferroelectric PZT thin films. As a conclusion, strain mechanism has an important effect on phase transitions and properties of epitaxial ferroelectric thin films. Although many groups are interested in effects of stress, just those films grown on cubic substrates have been investigated widely, in which in-plane stresses are isotropic. Recently, experiments and theories pay more attention to the
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films of anisotropic in-plane strains which are not grown on cubic substrates [15–21]. In theory, Zembilgotov et al. [16] firstly considered phase diagrams of a single-domain PT film and a Pb0.35 Sr0.65 TiO3 film grown on orthorhombic substrates and predicted the equilibrium polarization states of thin films. Wang and Zhang [17] presented misfit strain–misfit strain and temperature–misfit strain phase diagrams for single-domain BT and PT thin films grown on tetragonal substrates. In experiment, Simon et al. [18,19] investigated the permittivity of BST 60/40 film grown on NdGaO3 by pulsed laser deposition. Bellotti et al. [20,21] studied in-plane anisotropy in BST films and SrTiO3 films. In this Letter, we investigate the permittivities and tunabilities of single-domain PT film deposited on orthorhombic substrates by using the non-linear thermodynamic theory brought forward by Ref. [16]. Compared with usual ways, permittivities are gained by working out the susceptibility matrix. Further, we investigate the effects of external electric fields on permittivities by calculating tunabilities. Taking into consideration that films in devices are usually applied at room temperature (RT = 25 ◦ C) and there are new ferroelectric phases [16] which do not exist on cubic substrates at RT , we take T = 25 ◦ C in our calculation. According to phase diagram of Ref. [16], we assume that value of in-plane misfit strain um1 along [100] direction is equal to value of in-plane misfit strain um2 along [010] direction. And if one in-plane strains is compressive strain, the other must be tensile strain. 2. Theoretical development For single-domain ferroelectric films grown on thick dissimilar substrates, the elastic Gibbs energy function is introduced in Refs. [10,22]. It is well known that in-plane misfit strains are governed by substrate lattice parameters and lattice parameters of film. Because the thickness of substrate is much larger than the film thickness, the substrate is stress free and the internal stresses are concentrated in the film. In this Letter, we consider a single-domain ferroelectric film in the cubic paraelectric phase grown on a dissimilar orthorhombic substrate which induces two different in-plane strains um1 and um2 . So in-plane misfit strains can be attained by um1 = (as1 − a0 )/as1 and um2 = (as2 − a0 )/as2 [17,23], where a0 is film lattice parameter, as1 and as2 are lattice parameters of substrate along [100] direction and [010] direction, respectively. We can derive ˜ in an external electric field as a functhe modified potential G tion of the polarization components Pi (i = 1, 2, 3) [16,17]: 4 ∗ ∗ ˜ = α1∗ P12 + α2∗ P22 + α3∗ P32 + α11 G P1 + P24 + α33 P34 2 ∗ ∗ P1 + P22 P32 + α12 + α13 P12 P22 + α111 P16 + P26 + P36 + α112 P14 P22 + P32 + P24 P12 + P32 + P34 P12 + P22 + α123 P12 P22 P32 − E1 P1 − E2 P2 − E3 P3 +
[s11 (u2m1 + u2m2 ) − 2s12 um1 um2 ] 2 − s2 ) 2(s11 12
,
(1)
where Ek (k = 1, 2, 3) are components of the applied external electric field, um1 (Q12 s12 − Q11 s11 ) + um2 (Q11 s12 − Q12 s11 ) , 2 − s2 s11 12 (2) um2 (Q12 s12 − Q11 s11 ) + um1 (Q11 s12 − Q12 s11 ) ∗ α2 = α 1 + , 2 − s2 s11 12 (3) (u + u ) Q 12 m1 m2 α3∗ = α1 − (4) , s11 + s12
α1∗ = α1 +
∗ α11 = α11 +
1 [(Q211 + Q212 )s11 − 2Q11 Q12 s12 ] , 2 − s2 2 s11 12
(5)
∗ α33 = α11 +
Q212 , s11 + s12
(6)
∗ α12 = α12 − ∗ α13 = α12 +
[(Q211 + Q212 )s12 − 2Q11 Q12 s11 ] 2 + s2 s11 12
Q12 (Q11 + Q12 ) . s11 + s12
+
Q244 , 2s44
(7) (8)
Here α1 , αij (i, j = 1, 2, 3), and αij k (i, j, k = 1, 2, 3) are the dielectric stiffness coefficients, sij (i, j = 1, 2) are the elastic compliances at constant polarizations, and Qij (i, j = 1, 2, 3) are the electrostrictive constants. The temperature dependence of ferroelectricity is mainly governed by the dielectric stiffness coefficient α1 = (T − T0 )/2ε0 C since the higher-order stiffness coefficients αij and αij k may be taken as temperatureindependent parameters, where T0 and C are the Curie–Weiss temperature and Curie constant, ε0 is the permittivity of free space. In Eq. (1), only α1∗ , α2∗ and α3∗ show the effects of anisotropic strains in films. In other words, anisotropy of strains changes phases and properties of films by second-order polarization terms. ˜ and considering boundary Using the derived potential G conditions, Zembilgotov et al. [16] drew the misfit strain– temperature phase diagram of epitaxial PT film deposited on orthorhombic substrate at a given misfit strains um1 = −um2 when external electric field is equal to zero. Compared with the diagram of PT films grown on cubic substrates [10], the phase diagram of PT films deposited on orthorhombic substrates has new ferroelectric phases. All possible six stable phases are paraelectric phase, three tetragonal phases: a phase (P1 = 0, P2 = P3 = 0), a phase (P2 = 0, P1 = P3 = 0) and c phase (P1 = P2 = 0, P3 = 0), two orthorhombic phases: a c phase (P1 = P3 = 0, P2 = 0) and a c phase (P2 = P3 = 0, P1 = 0). The corresponding phases may be calculated by the ˜ with respect to relevant polarization components. minima of G Restricting our analysis to the small-signal dielectric responses (E → 0), the relative permittivities as functions of polarization components Pi can be derived from the thermodynamic po˜ Using the condition for thermodynamic equilibrium tential G. ˜ ∂ G/∂P i = 0 (E = 0), we can get the spontaneous polarization components Pi (i = 1, 2, 3). Because of reciprocal dielectric ˜ 2 /∂Pi ∂Pj (i, j = 1, 2, 3), the matrix susceptibilities χij = ∂ G inversion then enables us to find the dielectric susceptibilities
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ηij = 1/χij and relative permittivities εij = 1 + ηij /ε0 [24]. It should be noticed that the matrix of reciprocal dielectric susceptibilities χij are diagonal only in tetragonal phases. The important dielectric characteristics of a ferroelectric thin film are the in-plane permittivities ε11 , ε22 and out-plane permittivity ε33 , which can be measured in experiment. Basing on polarization components Pi in different phases, we calculate and summarize the results of relative permittivities. The a phase: 1 1 , =1+ ∗ ∗ ε0 χ11 ε0 (2α1 + 2α12 P22 + 2α112 P24 ) 1 1 =1+ , ε22 = 1 + ∗ P 2 + 30α 4 ε0 χ22 ε0 (2α2∗ + 12α11 111 P2 ) 2 1 1 =1+ . ε33 = 1 + (9) ∗ ∗ χ33 ε0 (2α3 + 2α13 P22 + 2α112 P24 )
ε11 = 1 +
The a c phase: ∗ P22 + 2α112 P24 + 2α123 P22 P32 χ11 = 2α1∗ + 2α12 ∗ + 2α13 P32 + 2α112 P34 , ∗ P22 + 30α111 P24 + 12α112 P22 P32 χ22 = 2α2∗ + 12α11 ∗ + 2α13 P32 + 2α112 P34 , ∗ P22 + 2α112 P24 + 12α112 P22 P32 χ33 = 2α3∗ + 2α13 ∗ + 12α33 P32 + 30α111 P34 , ∗ P2 P3 + 8α112 P23 P3 + 8α112 P2 P33 , χ23 = 4α13 1 ε11 = 1 + , ε0 χ11 χ33 , ε22 = 1 + 2 ) ε0 (χ22 χ33 − χ23 χ22 ε33 = 1 + . 2 ) ε0 (χ22 χ33 − χ23
(10)
ε22 = 1 +
1 , ε0 χ22
ε33 = 1 +
χ11 . 2 ) ε0 (χ11 χ33 − χ13
487
(12)
The a phase: 1 1 , =1+ ∗ P 2 + 30α 4 ε0 χ11 ε0 (2α1∗ + 12α11 111 P1 ) 1 1 1 , =1+ ε22 = 1 + ∗ ∗ ε0 χ22 ε0 (2α2 + 2α12 P12 + 2α112 P14 ) 1 1 =1+ . (13) ε33 = 1 + ∗ ∗ ε0 χ33 ε0 (2α3 + 2α13 P12 + 2α112 P14 ) ε11 = 1 +
In order to analyze the dielectric responses in external electric fields, we apply external electric fields. Because only εii is important in devices, T uii (i = 1, 2, 3) is investigated in our quantitative calculation. We assume that the external electric field is uniform and along axis. When we apply an electric field along spontaneous polarization component Pi , the polarization PiE is composed of two parts: spontaneous polarization and induced polarization. Using the stability criterion of first ˜ partial derivative ∂ G/∂P iE = Ei , the influence of the external electric field on film polarization PiE was taken into account. Substituting PiE for Pi in Eqs. (9)–(13), we can work out the permittivities in the presence of the external field. The tunability is defined as the variation of the permittivity in applied field with respect to the permittivity of zero electric field T uii = εii (0) − εii (E) /εii (0). (14) Thus, if in-plane misfit strains of film and electric field are given, the dielectric properties and tunabilities of thin films grown on dissimilar orthorhombic substrates can be calculated. 3. Results and discussion
The c phase: 1 1 , =1+ ∗ ∗ ε0 χ11 ε0 (2α1 + 2α13 P32 + 2α112 P34 ) 1 1 =1+ , ε22 = 1 + ∗ ∗ ε0 χ22 ε0 (2α2 + 2α13 P32 + 2α112 P34 ) 1 1 . (11) =1+ ε33 = 1 + ∗ P 2 + 30α 4 ε0 χ33 ε0 (2α3∗ + 12α33 111 P3 ) 3 ε11 = 1 +
The a c phase: ∗ P12 + 30α111 P14 + 12α112 P12 P32 χ11 = 2α1∗ + 12α11 ∗ + 2α13 P32 + 2α112 P34 , ∗ P12 + 2α112 P14 + 2α123 P12 P32 χ22 = 2α2∗ + 2α12 ∗ + 2α13 P32 + 2α112 P34 , ∗ P12 + 2α112 P14 + 12α112 P12 P32 χ33 = 2α3∗ + 2α13 ∗ + 12α33 P32 + 30α111 P34 , ∗ P1 P3 + 8α112 P13 P3 + 8α112 P1 P33 , χ13 = 4α13
ε11 = 1 +
ε0 χ33 , 2 ) ε0 (χ11 χ33 − χ13
In the following discussion, we will analyze the effects of misfit strains on dielectric properties and tunabilities of epitaxial PT film deposited on orthorhombic substrates at RT . In our calculation, the parameter values of coefficients for PT thin film are listed in Table 1 [10,16]. The value of misfit strain um1 is from −0.015 to 0.015. Table 1 The parameter values of PT used to calculate renormalized coefficients (in SI units) α1 α11 α12 α111 α112 α123 Q11 Q12 Q44 s11 s12 s44
−1.725 × 108 m/F −7.3 × 107 m5 /C2 F 7.5 × 108 m5 /C2 F 2.6 × 108 m9 /C4 F 6.1 × 108 m9 /C4 F −3.7 × 109 m9 /C4 F −0.089 m4 /C2 −0.026 m4 /C2 0.0675 m4 /C2 8.0 × 10−12 m2 /N −2.5 × 10−12 m2 /N 9.0 × 10−12 m2 /N
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Fig. 1. The spontaneous of PT film grown on orthorhombic substrates.
Fig. 2. Relative permittivities of PT film when um1 = −um2 .
According to the misfit strain–temperature phase diagram [16], a phase, a c phase, c phase, a c phase and a phase are the stable phases at RT . Because in-plane strains are anisotropic, the spontaneous polarization components Pi of films are remarkably different from those of films grown on cubic substrates. The misfit strain dependence of polarization components is represented in Fig. 1. As can be seen from Fig. 1, P1 only exists in a phase and a c phase, P2 is in a phase and a c phase, and P3 is in a c phase, c phase and a c phase. With the increase of tensile strain um1 , P1 is faintly discontinuous at the boundary between c phase and a c phase, then remarkably increases from a small non-zero value of 1.7 × 10−3 C/m2 . At the boundary between a c phase and a phase, P1 is obviously discontinuous. However, the effect of misfit strains on P1 in a c phase is greater than that in a phase. The same profile of P2 can be found while compressive strain um1 increase in a c phase and a phase. Fig. 1 shows that P3 is especial and varies non-monotonously with the change of misfit strains. Firstly, P3 increases with the decrease of compressive strain um1 in a c phase. Then, it becomes stable as film goes into c phase where misfit strain um1 varies from compressive strain to tensile strain. At last, when film transits to a c phase, P3 decreases corresponding to change of misfit strains. This can be ∗ ascribed to misfit strains which we set um1 = −um2 . α3∗ and α33 are independent of misfit strains. Therefore in the tetragonal c phase, P3 is independent of misfit strains and remains constant. Misfit strains have indirect effects on P3 by polarization components of P1 or P2 in the orthorhombic a c phase and a c phase. Based on assumption um1 = −um2 , in-plane polarization component P1 turns into P2 when um1 changes from tensile strain into compressive strain. Fig. 1 also shows that P1 and P3 are discontinuous at boundary between a c phase and a phase, and P2 and P3 are discontinuous at boundary between a c phase and a phase. Furthermore, P1 and P2 are discontinuous at boundaries between a c phase and c phase, and a c phase and c phase, respectively. As a result, all structure phase transitions a -phase/a c-phase, a c-phase/c-phase, c-phase/a c-phase, a c-phase/a -phase are first-order phase transitions. Permittivities are the most important property of ferroelectric films. The relative permittivities ε11 , ε22 and ε33 of PT epitaxial thin films grown on orthorhombic substrates are plot-
ted in Fig. 2 as functions of misfit strains. Fig. 2 reveals the misfit strains dependence of permittivities. It can be seen that the film permittivities vary non-monotonously with the misfit strains existing in the epitaxial system. In a and a c phase, P1 does not exist but ε11 has a small and anomalous jump accompanied by the first-order a -phase/a c-phase transition, as shown in Fig. 2. It can be explained that in-plane permittivity ε11 is related to the polarization components P2 and P3 which are discontinuous at the boundary between a and a c phases. At the boundary of c-phase/a c-phase transition, polarization component P1 increases from a small non-zero value (seeing Fig. 1), however structural transition results in anomalous divergence of in-plane permittivity ε11 [25]. The obvious discontinuity of P1 across the boundary between a c and a phases leads to a big step-like drop of in-plane permittivity ε11 . It is attractive that in-plane permittivities ε11 and ε22 symmetrically distribute about the line on which in-plane misfit strains are zero. Same symmetry of out-plane permittivity ε33 can be found in Fig. 2. These are consistent with the distributions of polarization components. In c phase, value of ε33 is a constant because in the present case we set um1 = −um2 . Although P3 is continuous at the phase boundary between c and a c phases, ε33 has a abnormal jump. This small jump, which appears at misfit strains slightly exceeding the phase boundary, may be explained by the relation ε33 in Eq. (12). This relation shows that ε33 increases due to the presence of a non-zero reciprocal susceptibility χ13 in a c phase. Accompanied by the firstorder a c-phase/a -phase transition, ε33 has a drastic jump and anomalous divergence, which may be related to structural transition [23]. When misfit strain um1 is compressive strain, the same phenomenon can be observed at the boundaries of phase transitions in Fig. 2. The high tunability has important action on tunable microwave devices, it is necessary to be investigated. Taking account of symmetry of dielectric coefficients, we only study the effects of misfit strains on tunabilities along [100] and [001] directions with applied external field of 50 kV/cm along the same directions respectively using the definition of the tunability Eq. (14) and the permittivities Eqs. (9)–(13). In thus electric field, field-induced polarization components have reached to saturation. Fig. 3(a) and (b) show that tunabilities are strongly
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dissimilar substrates, the misfit strain along a crystalline axis will differ from the misfit strain along the other crystalline axis, which changes polarization of film compared with films on cubic substrates, and new phases will appear. Pb0.35 Sr0.65 TiO3 films grown on orthorhombic NdGaO3 substrates, which exists in-plane strain um1 = −5.4 × 10−4 and um2 = −8 × 10−5 , have new phases considering the polarization state [16]. The result is in good agreement with the experimental data. The similar consequence can be found when films are grown on tetragonal substrates [17]. 4. Conclusion (a)
(b) Fig. 3. Tunabilities T u11 (a) and T u33 (b) of PT film as functions of in-plane misfit strains in external electric field of E = 50 kV/cm.
dependent on the misfit strains for different stable phases and theoretically maximum tunabilities can be attained at critical misfit strains corresponding to phase transitions. For applied electric field E1 [100], the tunability T u11 is almost 100% at the critical tensile strain um1 = 1.060%, which corresponds to the c-phase/a c-phase transition. For tunability T u33 , it is interesting that there are two maximums 100% in T u33 for applied filed E3 along [001] direction, which can be attributed to symmetry of permittivity ε33 about um1 = 0. Fig. 3 clearly reveals that T u33 reaches to maximum for um1 = −1.114% and um1 = 1.114%. Referring back to Fig. 1, these critical misfit strains correspond to the first-order a -phase/a c-phase transition and the first-order a -phase/a c-phase transition respectively. Connecting Fig. 2 to Fig. 3, the tunabilities as well as permittivities display similar characteristics as functions of the misfit strains indicating that epitaxy-induced internal stresses can optimize the dielectric properties of thin films. In the condition of um1 = −um2 , the in-plane strain anisotropy results in the appearance of new phases which do not form in films grown on cubic substrates [16]. Furthermore, we find that permittivities and tunabilities of thin film have anomalies at the boundaries of phase transitions, such as two jumps in ε11 and ε22 . Though we only take into account the strain state of um1 = −um2 , the similar consequence can be attained when the strain condition was altered. If ferroelectric film grown on
In summary, a theoretical formalism has been used to investigate the dielectric behaviors of single-domain PT films grown on orthorhombic substrates. Ulteriorly, the combined effects of misfit strains and external electric fields on dielectric properties are investigated. Quantitative calculation shows that the anisotropic in-plane strains um1 = −um2 lead to the symmetry of in-plane permittivities ε11 and ε22 about um1 = 0. Likewise ε33 symmetrically distributes to the each side of um1 = 0. Moreover, permittivities and tunabilities of thin film have anomalies at the boundaries of phase transitions. For in-plane tunability T u11 , it attain maximum at the boundary between c and a c phases. However, out-plane tunability T u33 have two maximums accompanied by first-order a -phase/a c-phase transition and a -phase/a c-phase transition. Our calculation reveals that dielectric tunability reaches to maximums at boundaries of phase transitions where the same direction polarization component appears or disappears. These are important properties to microwave devices. So the permittivities and tunabilities can be chosen to satisfied different demands by controlling applied fields and selecting substrates. Acknowledgements This work was supported by the National Natural Science Foundation of China under the Grant No. 10374069, Jiangsu Provincial Natural Science Foundation under the Grant No. BK2003032, and Jiangsu Key Laboratory of film Materials of China. References [1] J. Im, O. Auciello, P.K. Baumann, S.K. Streiffer, D.Y. Kaufman, A.R. Krauss, Appl. Phys. Lett. 76 (2000) 625. [2] W. Chang, C.M. Gilmore, W.-J. Kim, J.M. Pond, S.W. Kirchoefer, S.B. Qadri, D.B. Chirsey, J.S. Horwitz, J. Appl. Phys. 87 (2000) 3044. [3] J.F. Scott, C.A. Araujo, Science 246 (1989) 1400. [4] O. Auciello, J.F. Scott, R. Ramesh, Phys. Today 51 (7) (2000) 22. [5] P.K. Larsen, G.L.M. Kampshoer, M.J.E. Ulenaers, G.A.C.M. Spierings, R. Cuppens, Appl. Phys. Lett. 59 (1991) 611. [6] R. Takayama, Y. Tomita, K. Iijima, Ueda, J. Appl. Phys. 61 (1987) 411. [7] A.B. Kaufman, IEEE Trans. Electron DEvices ED-16 (1969) 511. [8] A. Mukherjee, S.R.J. Brueck, A.Y. Wu, Opt. Lett. 15 (1998) 151. [9] J.W. Li, F. Duewer, C. Gao, X.-D. Xiang, Y.L. Lu, Appl. Phys. Lett. 76 (2000) 769. [10] N.A. Pertsev, A.G. Zembilgotov, A.K. Taganatsev, Phys. Rev. Lett. 90 (1998) 1988.
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