Polarisation loops, lone pairs, piezoelectrics and second harmonic generation

A paraelectric follows the field, a ferroelectric remembers it

Intuition

Imagine a field of weather vanes. In a paraelectric, the vanes spin freely and only line up while the wind blows. A ferroelectric has stiff vanes in groups: once a strong wind has turned the groups, many stay turned when the wind stops.

For a paraelectric, polarisation is proportional to the field.

$P = \varepsilon_0 \chi_e E \qquad \chi_e = \varepsilon_r - 1$

$P$ is the polarisation, $E$ is the applied field, and $\chi_e$ is the electric susceptibility. At $E = 0$ the molecules move at random and $P = 0$. At $E \neq 0$ they align. The $P$ against $E$ curve is a straight line through the origin, with slope $\varepsilon_0\chi_e$ at the origin.

Try it: worked example

Silicon has relative permittivity 11.68. What is its electric susceptibility chi_e?

Show answer and steps

Answer10.68

1. Use $\chi_e = \varepsilon_r - 1$.
2. Put in $\varepsilon_r = 11.68$.
3. $\chi_e = 11.68 - 1 = 10.68$.

The ferroelectric loop has three landmarks: saturation, remnant and coercive

The lecture builds the loop frame by frame. In plain words, domains are small regions where all the dipoles point the same way. A field makes domains line up, and the loop records how the total polarisation follows.

Slide	Applied field	What happens
p.22	Zero field, $E = 0$	Domains random, $P = 0$
p.23	Small field	Domains start to align, $P > 0$
p.24	Medium field	More alignment, $P > 0$
p.25	Strong field	Complete alignment, saturation polarisation $P_s$
p.26	Zero field after a strong field	Remnant polarisation $P_r$, $P > 0$
p.27	Medium reverse field	$P = 0$ at the coercive field $E_c$
p.28	Strong reverse field	Complete reverse alignment, $P < 0$

Exam detail

Online Test 1 lists “key parameters in the measurement of dielectric and magnetic materials”. The three loop landmarks are $P_s$ (all aligned), $P_r$ (what is left at $E = 0$) and $E_c$ (the reverse field that brings $P$ to zero). The magnetic loop later uses the same three ideas with $M$ and $H$.

Pitfall

Remnant polarisation is at zero field and coercive field is at zero polarisation. They sit on opposite axes of the loop, and exam options often swap them.

Electrons that do not bond can still bend the structure

Lecture 10 gives two cases of non-bonding electrons distorting a structure.

d electrons. They do not generally take part in bond formation, but they do influence the coordination environment. Crystal-field splitting gives smaller radii. Jahn-Teller distortions lower the symmetry. MnO (Mn²⁺, d⁵) is an ideal NaCl type. CuO (Cu²⁺, d⁹) is a distorted NaCl type. The slide also plots ionic radius against number of 3d electrons for M²⁺ and M³⁺, with high-spin and low-spin branches. Those plots did not extract with values, so only the trend is stated.

The inert pair effect. Heavy cations with a valence 2 less than the group valence often have stereochemically active lone pairs. The lecture lists Sn²⁺, Sb³⁺ and Te⁴⁺ as [Kr]4d¹⁰5s², and Tl⁺, Pb²⁺ and Bi³⁺ as [Xe]4f¹⁴5d¹⁰6s². These lone pairs are not spherically symmetric (due to sp-hybridisation), which leads to asymmetry in the structure. Examples on the slides: α-PbO with Pb at the centre of an O square pyramid (Pb-O 2.3 Å and 4.3 Å), and α-Sb₂O₄ with ordered layers of Sb³⁺ and Sb⁵⁺ sites, giving a localised polar structure.

Piezoelectrics turn stress into voltage, and SHG crystals turn red into green

Piezoelectric effect. Applied stress produces a voltage. Uses: microphones, tilt sensors, strain gauges, ignition devices (such as a BBQ), quartz crystal microbalances and scanning probe microscopes. Reverse piezoelectric effect. An applied voltage changes the dimensions. Uses: crystal oscillator timing devices, speakers, inkjet printing, actuators, vibration damping and scanning probe microscopes (to control the needle position).

Second harmonic generation (SHG). KTiOPO₄ and LiB₃O₅ are important SHG materials. Both have space group symmetry $Pna2_1$, which lacks a centre of inversion. In a green laser, red light passes through KTP, and two photons combine into a new photon with twice the energy, which is green light.

Recall

What do polar BaTiO3 and the SHG crystals KTP and LBO share?

Their space groups (P4mm, Pna2_1) have no inversion centre, so they are non-centrosymmetric. The lecture does not state this for the piezoelectric slide.