2.7: Anisotropic dielectric permittivity
- Page ID
- 8311
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)When an electric field, E, is applied to a dielectric solid, positive and negative charges are displaced in opposite directions within the solid, creating polarisation, P. This is defined as the net dipole moment per unit volume. (An electrical dipole is created by a small separation of equal and opposite charges.)
In an isotropic material, these vectors are related by:
\[\boldsymbol{P}=(e-1) e_{0} \boldsymbol{E}\]
eo is the permittivity of free space, and e is the relative dielectric permittivity (a scalar constant in this case). As with the other examples, in anisotropic materials this scalar has to be replaced by a tensor. Often the occurrence of highly anisotropic dielectric permittivity is associated with ferroelecticity (spontaneous polarisation reversible by an electric field) and pyroelecticity (temperature dependent generation of polarisation).
Example: barium titanate
The high temperature form of BaTiO3 has the cubic perovskite structure with a primitive cubic lattice. At 150°C, a = 0.401 nm. In the temperature range 0ºC to 120ºC, BaTiO3 is tetragonal. At 100°C it has a = b = 0.400 nm and c = 0.404 nm.
https://www.doitpoms.ac.uk/tlplib/an...vskite-vid.mp4
(Video of cubic perovskite structure)
https://www.doitpoms.ac.uk/tlplib/an...ovskite-vr.mp4
(Rotatable model of cubic perovskite structure)
https://www.doitpoms.ac.uk/tlplib/an...vskite-vid.mp4
(Video of tetragonal perovskite structure)
https://www.doitpoms.ac.uk/tlplib/an...ovskite-vr.mp4
(Rotatable model of tetragonal perovskite structure)
The tetragonal-cubic phase transition is highlighted in the following video. It shows a thin section of barium titanate viewed between crossed-polars, which is heated through the transition temperature, and then allowed to cool naturally. Initially, the sample is below the transition temperature, and since the domains of the anisotropic tetragonal phase exhibit birefringence, it is brightly coloured when viewed between crossed-polars. When the sample reaches the transition temperature, the isotropic cubic phase forms, which appears black. The heat source is then removed, so the sample cools down and again undergoes a phase transition to return to the anisotropic tetragonal phase.
https://www.doitpoms.ac.uk/tlplib/an...deos/BaTiO.mp4
(Video of barium titanate phase transition)
In the tetragonal form, the Ti ion is displaced by a small distance, from the centre of the surrounding octahedron of nearest neighbour oxygen ions, along the z-direction. A spontaneous polarisation along the z-axis is generated, but by symmetry there is no polarisation in the x-y plane. Note that the polarsation can be orientated forwards or backwards along the tetragonal axis. This polarisation is easily changed by applying an electric field parallel to the z-axis, but a field applied to the x-y plane has little effect on the polarisation. Consequently the dielectric permittivity is anisotropic.
The refractive index, n, is given by the square root of the relative dielectric permittivity, i.e. \(\sqrt{\mathcal{E}}\). The resulting optical effects are considered in the next section.