Modeling the dependence of properties of ferroelectric thin film on thickness

We present a segregated strain model that can describe the thickness-dependent dielectric properties of epitaxial ferroelectric films. Using a phenomenological Landau approach, we present results for two specific materials, making comparison with experiment and with first-principles calculations whenever possible. We also suggest a “smoking gun” benchtop probe to test our model.

Figure 1

Relative permittivity (dielectric constant) as a function of temperature $\left[ϵ\left(T\right)\right]$ for $(\mathrm{Ba},\mathrm{Sr})\mathrm{Ti}{\mathrm{O}}_3$ (BST) films of different thicknesses grown on $\mathrm{Sr}\mathrm{Ru}{\mathrm{O}}_3∕\mathrm{Mg}\mathrm{O}$ (SRO) substrates (Ref. 23), where the ${ϵ}_{\mathit{max}}\left(T\right)$ at temperature $T_{\mathit{max}}$ is indicated with an arrow.

Figure 2

A schematic of the planar ferroelectric capacitor under consideration, with the key length scales and regions clearly demarcated. Note that the mismatch defects are segregated in a buffer layer of thickness $l_B$ and that the polarization and strain are homogeneous in the majority of the film. Incomplete charge compensation at the ferroelectric-electrode interfaces results in a residual depolarization field, as shown.

Figure 3

Schematic of the segregated strain model. Here, the elastic defects reside within $l_B$ $(⪡l)$ of the film-substrate interface so that the majority of the film is homogeneously strained.

Figure 4

Thickness dependence of the three distinct temperature scales $T_{\mathit{ferro}}$ (solid line), $T_{\mathit{max}}$ (dashed line), and $T_c^{*}$ (dotted line), with $E_{\mathit{ext}}=0$ in the segregated defect model described in the text. Here, Landau coefficients for BST on SRO (see Tables in Sec. 4) have been used and $T_{\mathit{max}}\left(l\right)$ is noted to display a peak at $l^{*}=60\mathrm{nm}$.

Figure 5

Calculated relative permittivity as a function of temperature for BST films on SRO substrates of thicknesses $l=10\mu \mathrm{m}$, $950\mathrm{nm}$, $660\mathrm{nm}$, $340\mathrm{nm}$, $280\mathrm{nm}$, and $175\mathrm{nm}$, with $E_{\mathit{ext}}=0$. The highest permittivity corresponds to the thickest film; the divergence for $10\mu \mathrm{m}$ film at the bulk Curie temperature of $235\mathrm{K}$ is indicated. Reduction in the permittivity for thin films is observed; the peak of the permittivity shifts toward higher temperatures, in agreement with Fig. 1.

Figure 6

Calculated relative permittivity as a function of temperature for BST films epitaxially grown on PSS substrates of thicknesses $l=40$, 68, 150, 300, and $580\mathrm{nm}$, with $E_{\mathit{ext}}=0$. The highest permittivity corresponds to the thickest film. Reduction in the permittivity for thin films is observed, and the peak of permittivity shifts toward lower temperatures.

Figure 7

Calculated nonswitchable polarization $P(\mathrm{C}∕{\mathrm{m}}^2)$ (see Fig. 4) for BST films on SRO substrates of various thicknesses $\left(l\right)$ at temperature $300\mathrm{K}$ and with $E_{\mathit{ext}}=0$; polarization as a function of temperature for films of thickness $175\mathrm{nm}$ (solid line) and $620\mathrm{nm}$ (dashed line) is shown in the inset.

Figure 8

Transition temperature $T_{\mathit{ferro}}$ as a function of applied external electric field $E_{\mathit{ext}}$ for BST films on SRO substrates of thicknesses $100\mathrm{nm}$ (solid line), $175\mathrm{nm}$ (dashed line), and $620\mathrm{nm}$ (dotted line). The corresponding line divides the region into switchable and nonswitchable polarization phases for each film.

Figure 9

Calculated hysteresis polarization loops for BST films on SRO substrates of thicknesses 100 and $175\mathrm{nm}$ at temperature $290\mathrm{K}$.

Figure 10

Spontaneous polarization $P(\mathrm{C}∕{\mathrm{m}}^2)$ as a function of film strain $\left(u_l\right)$ for $100\mathrm{nm}$ STO film at temperatures $T=200\mathrm{K}$ (dotted line), $T=250\mathrm{K}$ (dashed line), and $T=300\mathrm{K}$ (solid line) and zero total field, $E_l^T=0$. Dots correspond to ab initio values of the polarization (Ref. 35).

Figure 11

Calculated paraelectric relative permittivity as a function of external electric field $E_{\mathit{ext}}$ and film strain $u_l=-0.5\%$, $-0.4\%$, $-0.3\%$, $-0.2\%$, $-0.1\%$, and 0.0% for $100\mathrm{nm}$ STO film at temperature $T=200\mathrm{K}$. The permittivity data are shown in the limit of zero bias field $W_l$ in order to make comparison with the ab initio data (Ref. 35).

Figure 12

Calculated relative permittivity as a function of temperature for STO films of thicknesses $l=50$, 100, 200, 300, 400, 500, and $600\mathrm{nm}$ at misfit compressive strain $u_m=-0.9\%$ with $E_{\mathit{ext}}=0$. The highest permittivity corresponds to the thickest film. Reduction in the permittivity for thin films is observed.

Figure 13

Calculated relative permittivity as a function of temperature for STO films of thicknesses $l=50$, 300, and $600\mathrm{nm}$ at misfit strain $u_m=-0.9\%$ (solid line) and $u_m=-3.0\%$ (dotted line) with $E_{\mathit{ext}}=0$. Higher values of misfit compressive strain shifts the permittivity curve toward higher temperatures, where the larger shifts of $T_{\mathit{max}}$ are observed for the thinner films.

Figure 14

Calculated nonswitchable polarization $P(\mathrm{C}∕{\mathrm{m}}^2)$ of STO films as a function of film thickness $\left(l\right)$ for misfit strains $u_m=-3.0\%$, $-2.5\%$, $-2.0\%$, $-1.5\%$, and $-0.9\%$ at temperature $300\mathrm{K}$ and with $E_{\mathit{ext}}=0$. In the inset, the polarization as a function of temperature $\left(T\right)$ is plotted for $100\mathrm{nm}$ (solid line) and $500\mathrm{nm}$ (dashed line) films at misfit strain $u_m=-0.9\%$ and zero external field.

Figure 15

Transition temperature $T_{\mathit{ferro}}$ separating switchable (below) and nonswitchable (above the temperature curve) polarization phases as a function of STO film thickness $l$ and misfit strain $u_m=-3.0\%$, $-2.5\%$, and $-2.0\%$ with $E_{\mathit{ext}}=0$. $T_{\mathit{ferro}}$ becomes negative for compressive misfit strain values smaller than 1.5% for all film thicknesses, resulting in a nonswitchable polarization regime.

Figure 16

Schematic of the inhomogeneous strain model (Ref. 9), where the effective bias field $W\left(z\right)$ is spatially dependent due to flexoelectric coupling. Here, the characteristic length scales associated with the strain relaxation $\left(l_c\right)$ and the buffer layer $\left(l_B\right)$, where the elastic defects reside, are assumed comparable with the overall film thickness $\left(l\right)$.

Figure 17

Schematic of a benchtop probe to test for the segregated strain model: the field offset $\left(W_l\right)$ in the observed hysteresis loop can be tuned to zero by the application of an electric field $E_{\mathit{ext}}^{*}\left(l\right)=-W_l$; in this case, the relative permittivity sharpens since the net (thickness-dependent) field $E_l^T=E_{\mathit{ext}}^{*}\left(l\right)+W_l=0$.