Journal of the Korean Physical Society, Vol. 58, No. 3, March 2011, pp. 599∼603
Scaling Behavior of Amplitude-dependent Ferroelectric Hysteresis Loops in an Epitaxial PbZr0.2 Ti0.8 O3 Thin Film Sang Mo Yang, Seung Yup Jang, Tae Heon Kim and Hun-Ho Kim ReCFI, Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
Ho Nyung Lee Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
Jong-Gul Yoon∗ Department of Physics, University of Suwon, Hwaseong 445-743, Korea (Received 15 March 2010, in final form 20 May 2010) We investigated the scaling behavior of ferroelectric (FE) hysteresis loops as a function of the applied field amplitude (E0 ) in a high-quality epitaxial PbZr0.2 Ti0.8 O3 (PZT) thin film. We observed that the areas of the polarization-electric field hysteresis loops (A) followed the scaling law A ∝ E0α , with the exponent α = 0.45 ± 0.01. This result is in excellent agreement with the theoretical prediction of α by the two-dimensional Ising model. In addition, we found that the coercive field (EC ) showed EC ∝ E0γ with the exponent γ = 0.28 ± 0.01. We attribute this relationship to the difference in the sweep rate of the field amplitude E0 . From the obtained γ value, the growth dimension of FE domains is found to be about 1.68 in our epitaxial PZT thin film. PACS numbers: 77.80.Dj, 77.80.Bj, 77.55.Fg Keywords: Ferroelectric, Hysteresis, Scaling, PZT, Epitaxial film, Coercive field, Dimension DOI: 10.3938/jkps.58.599
dynamics simulation [4]. Ferroelectrics (FEs) are good model systems for validating the scaling of A for polarization-electric field (P E) hysteresis loops. FE thin films have been extensively studied over the past decades due not only to scientific interest but also to their potential application in, e.g., FE random access memories [5,6]. For this reason, there have been many studies in the FE thin films on the microscopic domain evolution imaging [7–10], fatigue property [11], domain switching dynamics [12, 13], and so on. As for a dynamical scaling law of hysteresis loops in FE films, there have been extensive experimental reports [14–19], and a scaling behavior of A ∝ E0α f β has been observed. However, several obtained exponents were slightly different from those of theoretical expectations. In the highly (111)-oriented SrBi2 Ta2 O9 (SBT) films, Park et al. reported the exponent α = 0.40 [15]. Pan et al. studied 400-nm-thick polycrystalline SBT films with a preferred (115)-orientation, and obtained α = β = 0.66 [16]. Yang et al. observed the exponent α = 0.66 for 400nm-thick polycrystalline Mn-doped Pb0.5 Sr0.5 TiO3 films and 0.7Pb(Mg1/3 Nb2/3 )O3 –0.3PbTiO3 films [18]. We attribute these slight differences in exponents to complicated structures, such as polycrystalline structure, or to the relaxor property of FE films. In those films, the hys-
I. INTRODUCTION Hysteretic behavior is one of the most commonly observed nonequilibrium phenomena and has become a great concern of theoretical works in statistical mechanics and in condensed matter physics. Theoretical studies have been carried out to understand the dynamical response of hysteresis loops to external applied fields. In particular, the area of dynamical hysteresis loops (A), corresponding to the dissipation energy during a period of domain reversal, has been intensively studied. Moreover, theoretical works have predicted a scaling behavior of A ∝ E0α f β , where E0 is the amplitude and f is the frequency of the applied ac field . The exponents α and β were found to have strong dependences on the dimension (D), the symmetry, and the simulation size of the system. Rao et al. showed α = 0.66 and β = 0.33 for the 3D continuous N -vector model in the limit N → ∞ . Dhar and Thomas also studied the same model system, but obtained α = β = 0.5 for all D > 2 . For a 2D Ising model, a Monte Carlo simulation study of the hysteresis curve resulted in α = 0.46 and β = 0.36 , and similar results were obtained, α = 0.47 and β = 0.40, from a cell ∗ E-mail:
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Journal of the Korean Physical Society, Vol. 58, No. 3, March 2011
teresis loops should be affected by various domain configurations, grain boundaries, polar nanoregions, and so on, which were not considered in the theoretical predictions. Therefore, if the scaling of A in real FE thin films is to be validated, more careful studies with structurally well-defined films are required. In this paper, we report the scaling behavior of FE hysteresis loops in a high-quality epitaxial PbZr0.2 Ti0.8 O3 (PZT) thin film. Quite recently, we already investigated the f -dependence of P -E hysteresis loops for the PZT film in terms of ac field-induced nonequilibrium domain wall dynamics [20]. Here, we focus on the scaling law for E0 -dependent hysteresis loops of an epitaxial PZT film. We observed that A followed a scaling behavior of A ∝ E0α quite well with the exponent α = 0.45 ± 0.01. This result is in excellent agreement with theoretical predictions for the 2D Ising model. In addition, we investigated the E0 -dependence of coercive field (EC ) in epitaxial PZT film and found that EC ∝ E0γ with the exponent γ = 0.28 ± 0.01. The obtained γ value could provide us with a FE domain growth dimension of about 1.68 in our PZT film.
II. EXPERIMENT
We fabricated an epitaxial PZT (100-nm thickness)/SrRuO3 heterostructure on a SrTiO3 (001) substrate by using pulsed laser deposition. Sputterdeposited Pt top electrodes were patterned using a photolithography lift-off process to have a diameter of 100 µm. A more detailed description of the growth procedures can be found elsewhere [21]. From high-resolution X-ray diffraction (HRXRD) studies, we confirmed that our PZT film was epitaxially grown with its c-axis normal to the substrate planes. PbZr1−x Tix O3 (x > 0.48) has a tetragonal structure at room temperature. PZT films are well known to have ferroelectric-ferroelastic 90◦ domains (i.e., a-axisoriented domains) in addition to FE 180◦ domains (i.e., c-axis-oriented domains). However, as shown in Fig. 1(a), only 001, 002, and 003 reflections of the tetragonal phase are revealed in the HRXRD θ - 2θ scans. This, together with results of reciprocal space mapping reported in Ref. 21, indicates that our PZT film is composed of purely c-axis oriented domains with uniaxial symmetry. Figure 1(b) shows the HRXRD φ scans around the diffraction peak from the (103) planes of PZT and SrTiO3 . The presence of four distinct PZT peaks with 90◦ separations along with the SrTiO3 peaks indicates the cube on cube type epitaxial growth of the PZT film. We measured the P -E hysteresis loops by using a TF analyzer 2000 (aixACCT) while varying the E0 values from 0.1 to 1.2 MV/cm at a fixed frequency of 2 kHz at room temperature.
Fig. 1. (Color online) (a) HRXRD θ - 2θ scan of a Pt/PZT/SrRuO3 heterostructure on a SrTiO3 (001) substrate. STO and SRO stand for SrTiO3 and SrRuO3 , respectively. (b) HRXRD φ scans of the 103 peaks for PZT and SrTiO3 .
III. RESULTS AND DISCUSSION Figures 2(a) and 2(b) show the E0 -dependent hysteresis loops and the switching current (I) of the PZT film, respectively. The high FE quality of our PZT film is manifested by the fully-saturated and well-defined square-like hysteresis loop. A negligibly small imprint (i.e., less than 0.04 MV/cm) was observed, probably due to the different top and bottom electrodes used. As shown in Fig. 2(b), the thin film is highly insulating with nearly zero leakage current. Therefore, our PZT capacitors are very suitable for investigating the dynamical hysteresis loops of FE thin films and their scaling behaviors. The detailed shape of the P -E hysteresis loop depended on E0 significantly, as shown in Fig. 2(a). At E0 > 0.4 MV/cm, the P -E hysteresis loop saturated fully. We could also confirm the saturation of hysteresis loops from the two clear peaks in the I(E) data, as shown in Fig. 2(b). As E0 increased, EC increased slightly from 0.25 to 0.35 MV/cm for the fully-saturated hysteresis loops. Due to the small imprint, the values of EC were determined by (|EC+ | + |EC− |)/2, where EC+ and EC− are the positive and the negative EC in the P -E hysteresis loops, respectively. Switchable polarization (2Pr )
Scaling Behavior of Amplitude-dependent Ferroelectric Hysteresis Loops · · · – Sang Mo Yang et al.
Fig. 2. (Color online) (a) Amplitude E0 -dependent hysteresis loops and (b) E0 -dependent switching current I of a PZT film. The inset shows the I data at E0 = 0.1 and 0.2 MV/cm.
also increased slightly with increasing E0 from 130 to 140 µC/cm2 . We observed that A followed a scaling behavior of A ∝ E0α quite well. Figure 3(a) shows a plot of log A as a function of log E0 . We can see clearly that A increased with increasing E0 , indicating that the dissipation energy during one period of P reversal becomes larger with increasing E0 . For the fully-saturated hysteresis loops (E0 > 0.4 MV/cm), A shows a power law behavior with α = 0.45 ± 0.01, as represented by the solid (blue) linear fitting line. Surprisingly, the obtained α value is in excellent agreement with those of theoretical predictions for the 2D Ising model [3,4]. We need to check the validity of the consideration that our PZT film can be regarded as an example of a 2D Ising model system. When applying an ac field to FE thin films, the P reversal process starts with the creation of domain nuclei at E > EN , where EN is the E value required for nucleation [20,22]. The nucleated domains grow rapidly via forward growth along the c-axis. After the rapid forward growth, the penetrated domains propagate sideways and merge with one another, consequently P reversal is completed, as shown in the recent studies using piezoresponse force microscopy [7, 8, 10]. The time for the forward growth is typically 1 ns in a film
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Fig. 3. (Color online) (a) Areas of hysteresis loops A as a function of amplitude E0 . The solid (blue) line indicates the linear fitting result with the slope α = 0.45 ± 0.01. (b) The plot of the coercive field E0 versus E0 . The solid (red) line shows the linear fitting result with the slope γ = 0.28 ± 0.01.
with a 100-nm thickness and that of sideways growth is relatively slow, from about a few ns to several ms depending on the magnitude of E [5,9,12]. Since our total measurement time (1/f ) is 500 µs, the time for the forward growth along the c-direction is negligible. In other words, the PZT thin film under our measurement condition can be regarded as an effective 2D system in view of P reversal. In addition, we already confirmed that our PZT film had only 180◦ domains, corresponding to up or down spin in the Ising model. Therefore, it is valid to regard our PZT film as an example of a 2D Ising model system. The energy dissipation during P reversal should be closely related to the domain wall motion in a FE thin film with random pinning sites. The motion of FE domain walls undergoes pinning-depinning transitions at numerous pinning sites, so the system energy dissipates. Such pinning effects are known to increase with increasing f [5]. Since we applied a triangular wave with a fixed f to the PZT film, the variation of E0 changed the field sweep rate (dE/dt). Therefore, we can understand that the increase of A with increasing E0 results from an increase of pinning effects with increasing dE/dt. To obtain further insight into the domain wall dynam-
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We argue that the power law behavior of EC ∝ E0γ can also be explained by the difference in dE/dt. Ishibashi et al. developed a phenomenological model for the volume fraction of the reversed domains (q(E)) as a function of dE /dt (or f at a fixed E0 ) based on the Avrami model [23,24]. According to the model of Ishibashi et al., q(E) is expressed as [23]
ics and the scaling behavior in FE thin films, we investigated the E0 -dependence of EC in P -E hysteresis loops, because the change in EC could provide information on the domain wall dynamics in FE thin films [20]. Figure 3(b) shows the EC values as a function of E0 . Interestingly, for the fully-saturated hysteresis loops, EC showed a simple power law relation, EC ∝ E0γ with γ = 0.28 ± 0.01, as represented by the solid (red) linear fitting line.
Z q(E) = 1 − exp −
E
Z
E
Cd 0
E0
00 dt v(E ) dE
!d dE 00
nE (E 0 )dE 0 ,
(1)
E=E 00
where v is the domain wall velocity and nE (E)dE is the number of nuclei per unit volume. d is the growth dimension of FE domains, and Cd is a constant. For example, d = 1 and Cd = 2 for domains with planar walls parallel to each other, d = 2 and Cd = π for cylindrical domains, and so on. We derived the relationship for the E0 -dependent EC value, EC (E0 ). For a triangular wave, E(t) = 4E0 ft during the first 1/4 period (0 ≤ t < 1/4f ). Note that it is enough to consider only the positive part due to the symmetric shape of the P -E hysteresis loops. Thus, dE/dt = 4E0 f . From Eq. (1), we get q(E) = 1−exp −(4E0 f )−d Ψ(E) = 1−exp[−E0−d Ψ0 (E)], (2)
field amplitude E0 -dependent polarization-electric field hysteresis loops at room temperature and obtained the power-law relation A ∝ E0α with α = 0.45 ± 0.01. This result is in excellent agreement with the predicted α value for the 2D Ising model. In addition, we found that the coercive field EC showed EC ∝ E0γ with γ = 0.28 ± 0.01, originating from the difference in the field sweep rate. From the obtained γ value, we determined the growth dimension of the ferroelectric domains in our epitaxial PZT film to be 1.68.
where Ψ(E) and Ψ0 (E) are functions of E, including integrals. It is known that Ψ0 (E) can be approximated by Ψ0 (E) = C 0 E k , where k is approximately equal to 6 and C’ is a constant [5,24]. Using the fact that EC is the E value at q(E) = 1/2, we obtain EC (E0 ):
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2009-0080567 and No. 2010-0020416). J.-G. Yoon acknowledges support from Korea Research Foundation Grant funded by the Korean Government (Grant No. KRF-2008-313-C00234). The work at Oak Ridge National Laboratory (HNL) was sponsored by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U. S. Department of Energy.
k 1 − exp[−E0−d · C 0 EC ] = 1/2,
EC =
d/k CE0 ,
(3) (4)
where C is a constant. Therefore, we obtained a simple power law relation with the exponent γ = d/k. From the obtained γ = 0.28 ± 0.01 and the fact that k ∼ 6, the growth dimension of the FE domains, d, is estimated to be about 1.68. The obtained d value is in reasonable agreement with the result of a study on the domain growth in an epitaxial PZT thin film by using switching current measurements, which showed 1 < d < 2 [13]. This fact also supports that our PZT thin film can be considered to be a 2D system.
IV. CONCLUSION We investigated the scaling behavior for the areas of ferroelectric hysteresis loops A in a high-quality epitaxial PbZr0.2 Ti0.8 O3 thin film. We measured the applied
ACKNOWLEDGMENTS
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