Gradient-stress on polarization in Ba_1−xSr_xTiO₃ films

Abstract

A group of position-thickness-dependent stresses are used to modified Landau-Devonshire theory to investigate the second-order phase transition in Ba_1−xSr_xTiO₃ films. The result shows that the short-range interaction between the unit cells of the film and the substrate induces the phase transition dispersion and the rise of the transition temperature in the films. The dependence of the effective dielectric constant on the temperature and the average spontaneous polarization on the film thickness are computed, which qualitatively agree with the experiments.

Graphical abstract

Keywords

ferroelectricity, thin film, stress, phase transition

1 Introduction

Ba_1−xSr_xTiO₃ (BST) is the most common ferroelectric material used for high-storage-density capacitor structures such as dynamic random access memory (DRAM) [1] because of its high static dielectric constant. However, the dielectric/ferroelectric properties of the films differ markedly from those of bulk ferroelectrics. The investigations have indicated that the first-order transition takes place in bulk BST [2]. In the BST films, the dielectric anomalies are substantially diffuse [3, 4] and the transition temperature is higher than that of the corresponding bulk [5]. The experimental results showed that the remnant polarization increases at room temperature and the dielectric constant decreases at 400 K when the film thickness of the Ba_0.5Sr_0.5TiO₃ films decreases [3]. Pertsev et al. developed a thermodynamic theory to investigate the properties of the ferroelectric thin films with a constant strain [6, 7]. However, Sinnamon et al. [3] pointed out that the out-of-plane strain in the BST film is of an exponential form with the film thickness. In this paper, we will apply a group of position-thickness-dependent stress distribution functions to the Modified Landau-Devonshire phenomenological theory to explain the phase transition characteristics of the BST film mentioned above.

2 Methods

Consider a two-dimensional ferroelectric film at high temperature with thickness 2D grown on a cubic substrate. The interface between the film and the substrate is “strained incommensurate” [8]. The x and z axes are chosen parallel and perpendicular to the interface, respectively, shown in Fig.1. A set of position-thickness-dependent stress, which originates from the interaction of unit cells between the substrate and the film, can be obtained by solving the elastic mechanical equations [9]:

\begin{array}{}\text{(1)}& \begin{array}{ll}{\sigma }_{xx}(x\mathrm{,}z)=\hfill & -{\sigma }_0\cos (2\text{π}x/L_{\text{s}})\hfill \\ \hfill & \cdot \cosh (2\text{π}z/L_{\text{s}})/\sinh (2\text{π}D/L_{\text{s}})={\sigma }_1\hfill \end{array}\end{array}

\begin{array}{}\text{(2)}& {\sigma }_{zz}(x\mathrm{,}z)=-{\sigma }_{xx}={\sigma }_3\end{array}

\begin{array}{}\text{(3)}& \begin{array}{ll}{\sigma }_{xz}(x\mathrm{,}z)=\hfill & -{\sigma }_0\sin (2\text{π}x/L_{\text{s}})\hfill \\ \hfill & \cdot \sinh (2\text{π}z/L_{\text{s}})/\sinh (2\text{π}D/L_{\text{s}})={\sigma }_5\hfill \end{array}\end{array}

and

\begin{array}{}\text{(4)}& L_{\text{s}}=b_{\text{f}}{b}_{\text{s}}/|b_{\text{s}}-b_{\text{f}}|\end{array}

where b_f and b_s are the lattice constants of the film and the substrate, respectively; σ_i (i=1,3,5) is the Voigt matrix notation, and σ₀ is a constant. The elastic Gibbs function can be expanded in powers of the polarizations P_i (i=x, z) and stresses σ_ij(i, j=x,z) [10]:

\begin{array}{}\text{(5)}& \begin{array}{ll}G=\hfill & [{\alpha }_1-(Q_{11}-Q_{12}){\sigma }_1]P_x^2+{\alpha }_{11}{P}_x^4+{\alpha }_{111}{P}_x^6+{\alpha }_{12}{P}_x^2{P}_z^2\hfill \\ \hfill & +{\alpha }_{112}(P_x^4{P}_z^2+P_x^2{P}_z^4)-Q_{44}{\sigma }_5{P}_x{P}_z+[{\alpha }_1+(Q_{11}\hfill \\ \hfill & -Q_{12}){\sigma }_1]P_z^2+{\alpha }_{11}{P}_z^4+{\alpha }_{111}{P}_z^6+G_0\hfill \end{array}\end{array}

where

\begin{array}{}\text{(5)}& G_0=(s_{12}-s_{11}){\sigma }_1^2-s_{44}{\sigma }_5^2/2\end{array}

and α₁ is the dielectric stiffness, α_ij, α_ijk (i, j=1,2) the high-order stiffness coefficients at a constant stress, s_ij (i, j=1, 2,4) the elastic compliances of the film measured at a constant polarization, Q_ij (i, j=1, 2, 4) the electrostrictive coupling between stress and polarization. α₁ equals to (T-T_c)/(2C{\epsilon }_0), in which ε₀ is the dielectric permittivity of free space and T_c is the transition temperature of bulk ferroelectric [11].

We apply the model developed above to BST-0.5 films. The parameters are taken from Ref. [3] as follows (in SI unit, T is ˚C): C =1.24×10⁵; T_c=-50˚C; {\alpha }_{11}=2.8\times {10}^8;Q₁₁− Q₁₂ =0.055. The terms of P_x⁶ and P_z⁶ in Eq. (5) can be neglected since α₁₁ > 0 [3, 12]. Two low-temperature phases are possibly present in the film when the film undergoes a high-temperature cubic phase to a low-temperature tetragonal phase. The following notation is used to describe the two phases: c-phase for P_z ≠0 and P_x = 0; a-phase for P_x≠0 and P_z = 0. Between the two low-temperature phases, the one with the minimum elastic Gibbs function with respect to the polarization and stress corresponds to the equilibrium thermodynamic state of the film. From Eq. (5) we know that the region with a tensile stress is in favor of the a-phase, and the region with a compressive stress is in favor of the c-phase under the condition of (Q₁₁−Q₁₂) > 0. Considering c-phase with P_z≠0 and P_x=0, the expression for the spontaneous polarization can be derived from the elastic Gibbs function in terms of the stability criterion of the first derivative (\partial G/\partial P_z=0). The dielectric susceptibility (χ_ij) can be determined by \frac{1}{{\chi }_{33}}={\partial }^{2}G/\partial P_z^2. The transition temperature (T_{\text{c}}^{*}) in the film can be obtained from the criterion G = G₀. We have

\begin{array}{}\text{(6)}& T_{\text{c}}^{*}(x\mathrm{,}z)=T_{\text{c}}+2C{\epsilon }_0(Q_{11}-Q_{12})|{\sigma }_1|\end{array}

When ,

\begin{array}{}\text{(7)}& P_z^2(x\mathrm{,}z)=-\frac{{\alpha }_1-(Q_{11}-Q_{12})|{\sigma }_1|}{2{\alpha }_{11}}\end{array}

and

\begin{array}{}\text{(8)}& {\chi }_{33}(x\mathrm{,}z)=\frac{C}{2(T_{\text{c}}^{*}-T)}\end{array}

When T>T_c^{*},

\begin{array}{}\text{(9)}& P_z(x\mathrm{,}z)=\mathrm{0,}{\chi }_{33}(x\mathrm{,}z)=\frac{C}{T-T_{\text{c}}^{*}}\end{array}

Total effective dielectric constant of the film is derived from:

\begin{array}{}\text{(10)}& {\epsilon }_{33}(T)=\frac{2D}{L_s}{\displaystyle {\int }_0^{L_s/2}}{\left({\displaystyle {\int }_0^D}\frac{1}{1+{\chi }_{33}(x\mathrm{,}z)}\text{d}z\right)}^{-1}\text{d}x\end{array}

Let q=2{\epsilon }_0{\sigma }_0{L}_{\text{s}}(Q_{11}-Q_{12})/(\text{π}D) and r=2(T_{\text{c}}-T)/C. When T < T_c and {\chi }_{33}(x\mathrm{,}z)>>\mathrm{1,}{\epsilon }_{33}(T) can be given as follows:

\begin{array}{}\text{(11)}& {\epsilon }_{33}(T)=\frac{2}{\text{π}\sqrt{q^2-r^2}}\arctan \left(\frac{\sqrt{q^2-r^2}}{q}\right)\end{array}

under r < q and

\begin{array}{}\text{(12)}& {\epsilon }_{33}(T)=\frac{2}{\text{π}\sqrt{r^2-q^2}}\arctan \left(\frac{\sqrt{r^2-q^2}}{q}\right)\end{array}

under r > q.

The transition temperature in BST film T_{\text{c}}^{*}, is always higher than that in bulk BST (T_c) as shown in Eq. (6), which agrees with the experiment by Yuzyuk et al. [13]. The range of T_{\text{c}}^{*} covers from T_c to T_{\text{cmax}}^{*}=T_{\text{c}}+2C{\epsilon }_0(Q_{11}-Q_{12}){\sigma }_0 cot (2\text{π}D/L_{\text{s}}). It means that the phase transition in the BST film is diffuse. As the film cools from high temperature(such as deposited temperature) to low temperature, the PE-FE phase transition will take place continuously from the center to the interface of the film.

The curves of {\epsilon }_{33}(T) for the c-phase in BST-0.5 films are shown in Fig. 2, for various thickness, where we take GPa {\sigma }_0=3.7 and L_s is about 100nm from the lattice parameters in Ref. [3]. The dielectric anomalies are substantially diffuse. The peak value of {\epsilon }_{33}(T) decreases as the film thickness decreases. The dependence of {\epsilon }_{33}(T) on the thick ness agrees with the experiment results when the temperature is lower than 400 K and higher than 223 K [3]. The theoretical value is lower than the experiment results when T < T_c. The (average) spontaneous polarization on the film thickness at 300 K which is higher than T_c of bulk BST has been computed, as shown in Fig. 3. It decreases with the increase of the film thickness, which is in qualitative agreement with the results observed by Sinnamon et al. [3]. The result can be explained by the change of the ferroelectric region with the normalized film thickness 2D/L_s, as shown in Fig.4. The ferroelectric region of the film with 2D/L_s =5.0 is smaller than that of the film with 2D/L_s=1.0 and larger than that of the film with 2D/L_s = 9.0. According to Eqs. (1) and (6), the thinner a film, the bigger the stress exists in it and the higher the transition temperature the film will be with. Thus, the (average) spontaneous polarization of a thin film is relatively large compared with that of a thick film.

3 Summary

In the application of position-thickness-dependent stress to the Modified Landau-Devonshire phenomenological theory, the dielectric/ferroelectric properties in the BST films are investigated. For c-phase, the phase transition is substantially diffuse in the film, which originates from the short-range interaction between the unit cells of the film and the substrate. The dependence of the dielectric constant on the temperature and the dependence of the average spontaneous polarization on film thickness are computed, which are in qualitative agreement with the experiment results.

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