Explaining the thickness-dependent dielectric permittivity of thin films

The dielectric properties of thin films are of paramount importance in a variety of technological applications, and of fundamental importance for solid-state research. In spite of this, there is currently no theoretical understanding of the dependence of the dielectric permittivity on the thickness of thin films. We develop a confinement model within the Lorentz-field framework for the microscopic Langevin-equation description of dielectric response in terms of the atomic-scale vibrational modes of the solid. Based on this, we derive analytical expressions for the dielectric permittivity as a function of thin-film thickness, in excellent agreement with the experimental data of barium-strontium-titanate thin films of different stoichiometry. The theory shows that the decrease of dielectric permittivity with decreasing thickness is directly caused by the restriction in $k$ space of the available eigenmodes for the field-induced alignment of ions and charged groups.

Figure 1

(a) The thin-film geometry in real space (confined along the $z$ axis but unconfined along the $x$ and $y$ axis), with the maximum wavelength that corresponds to a certain polar angle $\theta$. (b) The corresponding geometry of $k$ space, where the outer Debye sphere (of radius $k_D$) contains two symmetric spheres of forbidden states, i.e., states in $k$ space that remain unoccupied due to confinement along the $z$ axis. See Ref. [21] for a detailed mathematical derivation of this result. (c) An illustrative calculation of the dielectric permittivity using the confined model, with typical values of speed of sound (of the order of kHz and of microscopic friction $\nu$ and Debye frequency both of the order of ${10}^{13}$ Hz. (a) and (b) have been adapted from Ref. [21] with permission.

Figure 2

Comparison between the theoretical prediction given by Eq. (15), continuous line, and experimental data (circles). The latter are in arbitrary physical units, as customary for dielectric spectroscopy measurements. The upper curve refers to experimental data of the dielectric permittivity of (${\mathrm{Ba}}_{0.7}$, ${\mathrm{Sr}}_{0.3}){\mathrm{TiO}}_3$ (BST) thin films measured at $\omega =4$ kHz from Ref. [3], fitted by Eq. (15) with $K_1=1.5$ and $K_2=0.001$, and ${\epsilon }_{\infty }=100$. The lower curve refers to experimental data of the dielectric permittivity of (${\mathrm{Ba}}_{0.5}$, ${\mathrm{Sr}}_{0.5}){\mathrm{TiO}}_3$ thin films averaged between $\omega =400$ Hz and $\omega =10$ kHz from Ref. [24], fitted by Eq. (15) with $K_1=1.2$ and $K_2=0.0015$, and ${\epsilon }_{\infty }=10$. All experimental measures were made at room temperature.