Phase diagram and domain splitting in thin ferroelectric films with incommensurate phase

A. N. Morozovska, E. A. Eliseev, JianJun Wang, G. S. Svechnikov, Yu. M. Vysochanskii, Venkatraman Gopalan, and Long-Qing Chen

Phys. Rev. B 81, 195437 – Published 27 May 2010

Abstract

We studied the phase diagram of thin ferroelectric films with incommensurate phases and semiconductor properties within the framework of Landau-Ginzburg-Devonshire theory. We performed both analytical calculations and phase-field modeling of the temperature and thickness dependencies of the period of incommensurate $180 °$ -domain structures appeared in thin films covered with perfect electrodes. It is found that the transition temperature from the paraelectric into the incommensurate phase as well as the period of incommensurate domain structure strongly depend on the film thickness, depolarization field contribution, surface and gradient energy. The results may provide insight on the temperature dependence of domain structures in nanosized ferroics with inherent incommensurate phases.

Received 23 December 2009

DOI:https://doi.org/10.1103/PhysRevB.81.195437

Authors & Affiliations

A. N. Morozovska^1,*,†, E. A. Eliseev^1,2, JianJun Wang³, G. S. Svechnikov¹, Yu. M. Vysochanskii⁴, Venkatraman Gopalan³, and Long-Qing Chen^3,*,‡

¹V. Lashkarev Institute of Semiconductor Physics, NAS of Ukraine, prospect Nauki 41, 03028 Kiev, Ukraine
²Institute for Problems of Materials Science, NAS of Ukraine, Krjijanovskogo 3, 03142 Kiev, Ukraine
³Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, USA
⁴Institute for Solid State Physics and Chemistry, Uzhgorod University, 88000 Uzhgorod, Ukraine

^*Corresponding author.
^†morozo@i.com.ua
^‡lqc3@psu.edu

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Vol. 81, Iss. 19 — 15 May 2010

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Images

Figure 1
(Color online). $180 °$ -domain structure of thin film covered with perfect electrodes. Break of double electric layers at the film-surface junction (marked by a circle) and polarization inhomogeneity (arrows of different length) cause depolarization field.Reuse & Permissions
Figure 2
(Color online). Phase diagram in the coordinates temperature-thickness of the film deposed on the matched substrate $(u_{m}^{*} \approx 0.003)$ and placed between perfect conducting electrodes. PE is a paraelectric phase, IC is an incommensurate phase and CF is a commensurate ferroelectric phase. Solid and dashed curves correspond to the surface energy coefficient $α^{S} = 1$ and $10 m^{2} / F$ , respectively. Insets schematically show the polarization ${x, z}$ profiles in the points I and II of the phase diagram. Material parameters of $S_{2} P_{2} {Se}_{6}$ : $α_{T} = 1.6 \cdot 10^{6} J \cdot m / (C^{2} \cdot K)$ , $T_{C} = 193 K$ , $β = - 4.8 \cdot 10^{8} J \cdot m^{5} / C^{4}$ , $γ = 8.5 \cdot 10^{10} J \cdot m^{9} / C^{6}$ , $g_{1} = - 5.7 \cdot 10^{- 10} J \cdot m^{3} / C^{2}$ , $w_{1} = 1.8 \cdot 10^{- 27} J \cdot m^{5} / C^{2}$ , $v_{1} = 1.2 \cdot 10^{- 8} J \cdot m^{7} / C^{4}$ , $g_{3} = 5 \cdot 10^{- 10} J \cdot m^{3} / C^{2}$ , and positive $g_{2} \sim g_{3}$ were taken from Refs. 61, 62, $ε_{11} = ε_{33} = 10$ (reference medium is isotropic dielectric).Reuse & Permissions
Figure 3
(Color online). Thickness dependences of modulation periods $q_{-} = 2 π k_{-}^{- 1}$ (top curves above the dashed horizontal line) and $q_{+} = 2 π k_{+}^{- 1}$ (bottom curves below the dashed horizontal line) for different values of temperature (a) $T = 195 K$ and(b) 215 K. Temperature dependences of $q_{-} = 2 π k_{-}^{- 1}$ (top curves) and $q_{+} = 2 π k_{+}^{- 1}$ (bottom curves) for different values of the film thickness (c) $h = 30 nm$ and (d) 100 nm. Curves 1, 2, 3, and 4 correspond to the surface energy coefficient $α^{S} = 0.03$ , 0.1, 0.3, and $1 m^{2} / F$ , respectively. Solid and dotted curves represent exact numerical calculations from Eqs. (7, 8, 9) and approximate analytical dependences (10), respectively. Material parameters are the same as in Fig. 2 and $R_{d} \sim 500 nm$ .Reuse & Permissions
Figure 4
(Color online). (a) Temperature dependence of the modulation period $q_{-}$ (top curves above the dashed horizontal line) and $q_{+}$ (bottom curves below the dashed horizontal line) calculated analytically for the film thickness $h = 40 nm$ . Curves 1, 2, 3, 4, and 5 correspond to the surface energy coefficient $α^{S} = 0.03$ , 0.1, 0.3, 1, and $10 m^{2} / F$ respectively. Solid and dotted curves represent numerical calculations from Eqs. (7, 8, 9) and approximate analytical dependences (10), respectively. (b)–(j) Temperature evolution of the ${x, y}$ modulated domain structure calculated by the phase-field modeling for the film with sizes $100 \times 100 \times 40 nm$ , $α^{S} = 10 m^{2} / F$ , and temperatures $T = 0$ , 20, 40, 60, 80, 100, 140, and 180 K. Other parameters are the same as in Fig. 2, but $g_{2} = g_{1} = - 5.7 \cdot 10^{- 10} J \cdot m^{3} / C^{2}$ and $R_{d} \sim 500 nm$ .Reuse & Permissions
Figure 5
(Color online). (a) Sketch of the 1D $x$ -modulated domain structure. (b) Phase field 1D simulation of the polarization component $P_{3}$ variation for different z positions at $T = 160 K$ and $x = 160$ . (c)–(h) Polarization component $P_{3}$ morphologies in thin plates with sizes $h_{x} = 250 nm$ , $h_{y} = 2 nm$ , and $h_{z} = 40 nm$ at temperatures $T = 0$ , 40, 80, 120, 160, and 180 K. The pink solid circles, navy rhombus and dark cyan squares represent $P_{3}$ at $z = 20$ , 33, and 1, respectively. Other parameters are the same as in Fig. 4, but $g_{1} = - 5.7 \cdot 10^{- 10} J \cdot m^{3} / C^{2}$ and $g_{2} > 0$ .Reuse & Permissions

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