Dielectrics, capacitors and the structure of barium titanate

A dielectric is an insulator whose charges lean when a field is applied

Intuition

Put a magnet near a pile of iron filings and each filing turns to line up. A dielectric does something similar with electric charge. The material does not conduct, but its charges lean a little towards the field. That leaning is polarisation.

A dielectric is an electrical insulator that can be polarised by an applied electric field. Electric charges shift slightly from their average equilibrium positions, causing dielectric polarisation. Lecture 10 lists four mechanisms.

• Electronic: electron density shifts relative to the nucleus it surrounds.
• Ionic: cations and anions move relative to each other.
• Orientation: molecular dipoles rotate.
• Space charge: ions diffuse over several interatomic distances.

Dielectrics are sorted by their behaviour in a field. Paraelectric materials polarise when a field is applied and lose the polarisation when it is removed. Ferroelectric materials have spontaneous polarisation that is kept when the field is removed.

Exam detail

Lecture 10 slide 2 says Online Test 1 covers “understanding of key parameters in the measurement of dielectric and magnetic materials”. The slide does not say which parameters it means. The quantities this lecture measures or defines for dielectrics are capacitance, relative permittivity, polarisation, susceptibility, the Curie temperature and the points on the hysteresis loop, so know all of them. The same slide also lists the other tested items: point groups, symmetry operations, Miller indices, the Bragg equation, diffraction patterns, ionic lattices and Born-Mayer lattice enthalpy trends. Online Test 1 is worth 5% of the overall grade, has 15 multiple choice questions, one hour and one attempt, and covers lectures 2 to 11.

A dielectric between capacitor plates multiplies the capacitance

A capacitor is two conductors (plates) of area $A$ separated by a distance $d$, carrying equal and opposite charges. Capacitance is the charge on one plate over the voltage between them.

$C = \frac{Q}{\Delta V} \qquad C = \frac{\varepsilon_0 A}{d}$

The stored energy is

$U = \tfrac{1}{2} C (\Delta V)^2$

Insert a dielectric and the capacitance rises by the factor $\varepsilon_r$, the relative permittivity.

$C = \varepsilon_r \varepsilon_0 \frac{A}{d}$

$\varepsilon_r$ says how readily a material becomes polarised. It has no units and is relative to vacuum, which has $\varepsilon_r = 1$ by definition. The lecture table gives examples: air 1.00058986, Teflon 2.1, silicon dioxide 3.9, silicon 11.68, water 80, strontium titanate 310, barium titanate 1200 to 10,000 (20 to 120 °C), lead zirconate titanate 500 to 6000 and calcium copper titanate more than 250,000. The lecture notes that $\varepsilon_0$ also appears in the Born-Mayer equation (Lecture 7) and in Maxwell’s equations (Lecture 1).

Try it: worked example

A vacuum capacitor has plate area A and separation d. A dielectric with epsilon_r = 80 (water) fills the gap. How does the capacitance change?

Show answer and steps

AnswerIt increases by a factor of 80.

1. Vacuum: $C_0 = \varepsilon_0 A / d$.
2. With the dielectric: $C = \varepsilon_r \varepsilon_0 A / d$.
3. Divide: $C / C_0 = \varepsilon_r = 80$.

Barium titanate has an enormous permittivity because titanium sits off-centre

The relative permittivity of BaTiO₃ depends on temperature. It reaches very high values, compared with 5 to 10 for normal dielectrics, with a maximum at 120 °C. The slide plot shows $\varepsilon'$ of about 2500 to 3000 at 40 to 80 °C, rising sharply to a peak near 8500 at $T_c \approx 120$ °C, then falling. These readings are from a figure, so use them as a shape.

BaTiO₃ is a perovskite, general formula ABX₃, where the BX₃ atoms form the ReO₃ lattice. A is a cation in 12-fold coordination by X. B is a cation in octahedral coordination by X. X is an anion such as O²⁻ or F⁻.

The polarisation comes from the titanium ions moving off the centre of the TiO₆ octahedron. This is ionic polarisation, and it is non-centrosymmetric (it has no inversion centre). In principle there are six equivalent directions, so the structure splits into domains with different polarity directions.

Compare

Polar BaTiO3

Space group P4mm. No inversion symmetry. Ti off-centre. Ferroelectric.

Non-polar BaTiO3

Space group Pm-3m. Has inversion symmetry. Ti centred. Paraelectric.

On heating, a solid-solid phase transition generally leads to a higher symmetry structure. The change from ferroelectric to paraelectric happens at the Curie temperature, $T_C$. The lecture table gives: BaTiO₃ 120 °C, LiNbO₃ 1210, PbTiO₃ 490, Bi₄Ti₃O₁₂ 159, potassium dihydrogen phosphate −150, Rochelle salt −18 to +24, and Pb(Zr₁₋ₓTiₓ)O₃ depends on $x$. The Curie temperature varies with structure and composition and can be tailored for an application.

Lecture 10 revises inversion on the way. A centre of symmetry ($i$) is a point through which every atom can move in a straight line to an identical atom on the other side, at the same distance. It can sit on an atom or on a bond. Non-centrosymmetric symmetries have no inversion centre. Why symmetry matters: the lecture says materials properties depend on symmetry.

Pitfall

Paraelectric and ferroelectric are not two different materials. In BaTiO₃ they are the same compound below and above $T_C$. Do not call the high-temperature form ferroelectric just because the permittivity is highest near $T_C$.

Recall

Plain words: why is a ferroelectric called ferroelectric?

It keeps its own built-in polarisation with no field applied, just as a ferromagnet keeps its own magnetisation. The built-in polarisation comes from off-centre ions.