Ferroelectric dielectric metamaterials

In [ ]:
import sys
sys.path.append("../ferromtm/ferromtm")
from ferromtm.visualization.postpro2D import *
from aotomat.tools.plottools import *
%matplotlib inline
sns.set_context("notebook", font_scale=1.5, 
                rc={"lines.markersize": 12, "lines.linewidth": 2.5, "axes.labelsize":20})
import matplotlib as mpl
colorlist = ["#A64C41", "#8d8d8d", "#416cad", "#46995d", "#c27442", "#d9cf4a", "#875da2"]
mpl.rcParams['axes.prop_cycle'] = mpl.cycler(color=colorlist) 
from research.material.example_bst import *

$ \newcommand{\B}[1]{\boldsymbol{#1}} \newcommand{\tens}[1]{\B{#1}} \newcommand{\re}{\mathrm{Re}} \newcommand{\im}{\mathrm{Im}} \newcommand{\grad}{\B{\mathrm{\nabla}}} \renewcommand{\div}{\B{\mathrm{\nabla\cdotp}}} \newcommand{\ddroit}{\mathrm{d}} \newcommand{\epsf}{\varepsilon^{\rm f}} \newcommand{\epsftens}{\tens{\varepsilon}^{\rm f}} \newcommand{\epstens}{\tens{\varepsilon}} \newcommand{\epsd}{\varepsilon^{\rm d}} \newcommand{\epsvac}{\varepsilon_{0}} \newcommand{\epshom}{\tilde{\epstens}} \newcommand{\epshom}{\tilde{\epstens}} $

Citations

References are nice [Tagantsev et al. 2018] and [Bouchitté and Felbacq 2004]

Parameters for ferroelectrics

  • tunability: $$n = \frac{\varepsilon(0)}{\varepsilon(E)}$$
  • loss tangent: $$\tan\delta(E)$$
  • commutation quality factor: $$K = \frac{(n -1)^2}{n\, \tan\delta(0)\,\tan\delta(E)}$$

Permittivity model

Landau potential

$$F(P,E) = F_0 + a P^2/2 + b P^4/4 + cP^6/6 - EP$$

where $E$ is the applied electric field and $P$ is the polarization

Equation of state

$$\frac{\partial F (P, E)}{\partial P} = a P_0 + b P_0^3 + c P_0^5 - E = 0$$

$P_0$ is the equilibrium polarization

[Zhou et al. 2008]

Permittivity model

\begin{equation} \epsf(E) = \left[\epsvac \frac{\partial^2 F (P, E)}{\partial P^2} \right]^{-1} = \frac{\epsf(0)}{1 + \beta P_0^2 + \gamma P_0^4}, \label{eq_epsf} \end{equation}

where $\beta = 3 b \epsvac \epsf(0) /a$ and $\gamma = 5 c \epsvac \epsf(0) /a$.

  • permittivity without bias $\epsf(0)$ was measured to be 120
  • fitting parameters: $a = 0.992/\epsvac$, $b=0.086/(\epsvac^3\,E_{\rm ref}^2)$, $c=0.014/(\epsvac^5\,E_{\rm ref}^4)$, $E_{\rm ref}=1$ kV/mm.

Dieletric tensor

\begin{equation} \epsftens (\B E) = \begin{pmatrix} \epsf_{xx}(E_x) & 0 & 0 \\ 0 & \epsf_{yy}(E_y) & 0 \\ 0 & 0 & \epsf_{zz}(E_z) \end{pmatrix} \label{eq_epsftens} \end{equation}

Measurements and fit

In [2]:
sample = 4
E_exp, eps_norm_exp = get_sample(sample)
f_eps_fit = retrieve_params(sample=sample)
E_fit = np.linspace(-11, 11, 101)
eps_fit = f_eps_fit(E_fit)


fig, ax = plt.subplots(figsize=(13,8))
ax.plot(E_exp, eps_norm_exp.real, "o",  label="measured", color = colorlist[1])
ax.plot(E_fit, eps_fit.real, "--", color=colorlist[0], label="fit")
ax.set_ylabel(r"normalized permittivity $\varepsilon^{\rm f}/\varepsilon^{\rm f}(E=0)$")
ax.set_xlabel("electric field (kV/mm)")
_ = ax.legend()

Effective parameters

In [3]:
fig, ax = plt.subplots(ncols=2, nrows=2, figsize=(12,8))
plot_eff_par(fig, ax)
plt.tight_layout()

References

[Tagantsev et al. 2018]

Tagantsev, A. K., V. O. Sherman, K. F. Astafiev, J. Venkatesh, and N. Setter “Ferroelectric Materials for Microwave Tunable Applications,” J. Electroceram., 11/1-2 (2018), 5–66.

[Bouchitté and Felbacq 2004]

Bouchitté, G. and D. Felbacq “Homogenization near resonances and artificial magnetism from dielectrics,” Comptes Rendus Mathematique, 339/5 (2004), 377–382.

[Zhou et al. 2008]

Zhou, K., S. A. Boggs, R. Ramprasad, M. Aindow, C. Erkey, and S. P. Alpay “Dielectric response and tunability of a dielectric-paraelectric composite,” Applied Physics Letters, 93/10 (2008), 102908.

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