Ionic conductivity: ions hop through gaps, and doping adds more gaps

Ions only move if there is a gap to move into, and they need a push to get there

Intuition

Think of a sliding tile puzzle with one tile missing. Without the gap, nothing moves. With the gap, a neighbour can slide in, then another can slide into the new gap. Ions in a crystal are the tiles and a vacancy is the gap. Heat is the shaking that gives a tile enough energy to hop.

Lecture 18 puts it in three steps. Atoms are generally trapped on their lattice sites, apart from thermal vibration. They lack the energy to move to another site. At high temperature they can move from one site to another. Defects are what make ionic conduction possible.

The lecture’s slide 21 compares conductivities by material class, in S m$^{-1}$. Ionic crystals run from below 10^-16 to 10^-2. Solid electrolytes and strong (liquid) electrolytes both run 10^-1 to 10^3. Metals run 10^3 to 10^7, semiconductors 10^-3 to 10^4, and insulators sit below 10^-10. The slide is a table drawn as a figure, so check these ranges against the slide before quoting them.

A hop over an energy barrier gives an Arrhenius law

Take Na$^+$ moving in NaCl. A neighbouring Na$^+$ site is empty (a vacancy), and the ion has to squeeze through the crystal to reach it. The crystal resists. The energy it needs for the jump is the activation energy, $E_a$. Picture a ball that needs to be pushed over a hill to reach the next valley. The height of the hill is $E_a$.

Because the hop is thermally activated, ionic conductivity usually obeys the Arrhenius equation.

$\sigma = \sigma_0 e^{-E_a/kT}$

$\sigma_0$ is the pre-exponential factor, $k$ is the Boltzmann constant and $T$ is temperature. A big barrier gives small conductivity. A higher $T$ gives bigger conductivity. If you plot $\ln\sigma$ against $1/T$ you get a straight line of slope $-E_a/k$. (The line is the usual rearrangement of the equation; the lecture states the equation and draws $\log\sigma$ against $1/T$ on slide 24.)

Doping puts in extra vacancies, so conductivity rises at low temperature

Conductivity can be increased by adding defects. The lecture’s example is NaCl doped with MnCl$_2$. Mn goes onto a cation site, and the lecture says one cation vacancy comes with each Mn. (The slides do not say why. The usual reason is charge balance, which is not from the lecture: Mn carries a higher charge than Na, so fewer cations are needed in total.)

$\mathrm{Na}_{1-2x}\mathrm{Mn}_x\mathrm{V}_x\mathrm{Cl}$

Here $x$ is the Mn fraction and $\mathrm{V}$ is a cation vacancy.

On a plot of $\log\sigma$ against $1/T$, there are two regimes.

• Low temperature, extrinsic region. The vacancies from doping dominate. The lines have slope $E_m$ (the cost of the ion moving). More impurity shifts the line up.
• High temperature, intrinsic region. Thermally made defects dominate. The line is steeper, with slope $E_m + E_d$ (moving plus making the defect).

Compare

Extrinsic region

Low temperature. Vacancies come from the dopant, so their number is fixed by how much impurity you added. Slope $E_m$. Flatter.

Intrinsic region

High temperature. Vacancies are made by heat itself. Their number grows with $T$. Slope $E_m + E_d$. Steepest.

The slide ends with a question: how does this compare with the increased electrical conductivity when Si is doped? The lecture does not give an answer on the slide. See module 3 on doping for the electronic case.

Try it: worked example

Why is the extrinsic line flatter than the intrinsic line on the log sigma against 1/T plot?

Show answer and steps

AnswerThe extrinsic line only needs the ion to move, slope E_m. The intrinsic line must also make the defect, slope E_m + E_d, so it climbs faster with temperature.

1. Conductivity depends on how many vacancies there are and how easily ions hop into them.
2. With doping at low $T$, the number of vacancies is fixed by the dopant. Only the hop depends on temperature, with energy $E_m$.
3. At high $T$, the number of vacancies is set by $n_d/N = \exp(-\Delta H_d/kT)$, which also depends on $T$.
4. Both effects now depend on temperature, so the total slope is the sum, $E_m + E_d$ (slide 24).
5. A bigger slope means a steeper line on a plot against $1/T$.

Moving ions are what batteries and fuel cells run on

Slide 25 lists the uses of ionic conductivity: batteries, solid-oxide fuel cells, halide-selective electrodes and membranes in separation processes. The lecture gives the reactions for two.

Lithium-ion battery. Li$^+$ ions shuttle between graphite and cobalt oxide.

$\text{Anode: } \mathrm{LiC_6} \longrightarrow \mathrm{Li^+} + 6\mathrm{C} + e^-$

$\text{Cathode: } \mathrm{Li^+} + \mathrm{CoO_2} + e^- \longrightarrow \mathrm{LiCoO_2}$

Solid-oxide fuel cell. The electrolyte is YSZ (yttria-stabilised zirconia). Yttrium doping creates oxygen vacancies, so $\mathrm{O^{2-}}$ ions can hop through the solid.

$\text{Anode: } 2\mathrm{H_2} + 2\mathrm{O^{2-}} \longrightarrow 2\mathrm{H_2O} + 4e^-$

$\text{Cathode: } \mathrm{O_2} + 4e^- \longrightarrow 2\mathrm{O^{2-}}$

Exam detail

Know the Arrhenius equation, the doped NaCl formula $\mathrm{Na}_{1-2x}\mathrm{Mn}_x\mathrm{V}_x\mathrm{Cl}$, and all four electrode reactions. Lecture 18’s summary says that defects allow ionic conductivity, that ionic conductivity has an activation barrier described by an Arrhenius relationship, and that it underpins solid-state batteries.

Recall

Why does doping NaCl with MnCl2 raise its conductivity at low temperature?

Each Mn brings one cation vacancy. More vacancies mean more places for Na$^+$ to hop, so the extrinsic (impurity-controlled) conductivity rises.