Conductors dielectrics and semiconductors

Solids differ first of all in how many of their electrons are free to move. In a conductor (a metal) the valence band overlaps the conduction band, so free carriers — electrons — are abundant, $n\sim10^{28}\ \text{m}^{-3}$, and the resistivity is tiny: $\rho\sim10^{-8}\ \Omega\cdot\text{m}$. In a dielectric (insulator) the band gap is wide, $E_g > 3\ \text{eV}$, thermal motion hardly ever lifts electrons into the conduction band, $\rho > 10^{8}\ \Omega\cdot\text{m}$, and a charge placed on an insulator stays where it is. A semiconductor has a narrow gap, $E_g \approx 1\ \text{eV}$ (silicon: $1.12\ \text{eV}$), so a few carriers appear at room temperature, the conductivity lies between the two extremes, and it grows rapidly with temperature. For all three classes the local relation between field and current is Ohm's law in differential form, $\vec{j} = \sigma \vec{E}$, with $\sigma = 1/\rho$; microscopically $\sigma = n e \mu$, where $n$ is the carrier density and $\mu$ the mobility. The classification is therefore quantitative: it is the carrier density $n$ (equivalently the band gap $E_g$) that decides whether a body conducts charge, stores it, or does something in between.

A bar of material has resistivity $\rho = 5.0\times10^{-8}\ \Omega\cdot\text{m}$. An electric field $E = 0.10\ \text{V/m}$ is applied along it. (a) Classify the material. (b) Find the current density.

A semiconductor sample has electron density $n = 1.0\times10^{22}\ \text{m}^{-3}$ and mobility $\mu = 0.050\ \text{m}^{2}/(\text{V}\cdot\text{s})$.

(a) Find $\sigma$ and $\rho$.

(b) What is the electron drift velocity in a field $E = 100\ \text{V/m}$?

(c) For copper $n = 8.5\times10^{28}\ \text{m}^{-3}$, $\mu = 4.5\times10^{-3}\ \text{m}^{2}/(\text{V}\cdot\text{s})$. By what factor is copper's conductivity larger?

Band gaps: silicon $E_g = 1.12\ \text{eV}$, gallium arsenide $E_g = 1.42\ \text{eV}$, diamond $E_g = 5.5\ \text{eV}$. Using $hc = 1240\ \text{eV}\cdot\text{nm}$, find the threshold (longest) wavelength of light able to create free carriers in each material. Which material suits a visible-light photodetector ($400$–$700\ \text{nm}$), and how should diamond be classified at room temperature?

Three experiments with a multimeter — battery-powered only (ohmmeter and diode-test modes), never connected to mains.

1. Conductor. With a soft pencil draw a graphite strip about $6\ \text{cm}$ long on paper and measure its resistance $R$. Repeat for $L = 2, 4, 6\ \text{cm}$ and plot $R(L)$: it is linear. Estimate $\sigma = L/(R A)$ and compare with the tabulated value for graphite.
2. Dielectric. Measure $R$ of a dry glass rod, a plastic ruler and dry paper — the meter reads '> $20\ \text{M}\Omega$' (off-scale). Then rub the plastic ruler with wool and check that a minute later it still attracts small paper bits: the charge does not leak away.
3. Semiconductor. In diode-test mode measure the forward voltage drop of a silicon diode (about $0.6$–$0.7\ \text{V}$) and the reverse connection (off-scale). Warm the diode in your hand for $1$–$2$ minutes and measure again: the forward drop decreases, because at higher temperature the carrier density rises and less voltage is needed for the same current.

If warming the device is inconvenient or unsafe, replace step 3 with any band-structure simulation (for example PhET 'Band Structure'): vary temperature and doping while recording the carrier density $n$ and the conductivity.

Results table: material — measured quantity — class (conductor / dielectric / semiconductor). Discuss why a conductor's resistance rises with temperature while a semiconductor's falls.

Show how the band gap controls the number of free carriers and therefore fixes the class of a solid.

Sketch: three band diagrams side by side.

• Conductor: the conduction and valence bands overlap, and the Fermi level lies inside the partly filled conduction band: almost all occupied electron states sit at or below the Fermi level, and above it there is only a narrow thermal layer of width of order $kT$ containing a few excited electrons.
• Semiconductor: gap $E_g \approx 1\ \text{eV}$, only a few dots at the top, labelled 'thermal excitation'.
• Dielectric: gap $E_g > 3\ \text{eV}$, conduction band empty.

Beneath the diagrams runs a shared logarithmic axis of resistivity $\rho$ in $\Omega\cdot\text{m}$, marked $10^{-8}$ (metals), $10^{-5}$–$10^{4}$ (semiconductors), $>10^{8}$ (dielectrics). An arrow $T\uparrow$ at the semiconductor shows $n$ growing with temperature.

A phone screen is a sheet of glass about $0.5\ \mathrm{mm}$ thick (relative permittivity $\varepsilon_r \approx 6$), coated on the inside with a transparent conductor, indium tin oxide. When a fingertip (a conductor) comes within about $1\ \mathrm{mm}$ of the glass, the controller measures a change of capacitance of a few picofarads — for example from $20\ \mathrm{pF}$ to $19\ \mathrm{pF}$ — and registers a touch.

The glass is a dielectric: its charges are bound inside atoms and shift only by a fraction of an atomic size, so they polarize and support the field $E = U/d$ across the $0.5\ \mathrm{mm}$ thickness without carrying any steady current. The finger is a conductor: its free electrons rearrange in about $10^{-12}\ \mathrm{s}$ so that the finger surface becomes an equipotential. The capacitance change comes from both kinds of behaviour at once.

A lightning stroke carries a peak current of about $30\ \mathrm{kA}$ for a few tens of microseconds and transfers a charge of roughly $15\ \mathrm{C}$. The steel body of a parked car is about $0.8\ \mathrm{mm}$ thick. Passengers stay unhurt although the outer surface briefly reaches potentials of millions of volts relative to the ground.

Steel is a conductor with about one free electron per atom, number density $n \approx 8.5 \times 10^{28}\ \mathrm{m^{-3}}$; these electrons move in about $10^{-6}\ \mathrm{s}$ and place the excess charge on the outer surface. In electrostatic equilibrium the field inside the metal is zero, $E_\text{in} = 0$, so the interior stays at one potential — a Faraday cage. The air around the car is a dielectric that normally carries no current and breaks down only when the field reaches its dielectric strength, about $3 \times 10^{6}\ \mathrm{V/m}$.

A household panel of area $1.6\ \mathrm{m^2}$ delivers about $300\ \mathrm{W}$ in sunlight of intensity $1000\ \mathrm{W/m^2}$, with each cell giving about $0.6\ \mathrm{V}$. Its active material is crystalline silicon with band gap $E_g = 1.12\ \mathrm{eV}$; at $300\ \mathrm{K}$ the number density of its own free carriers is only about $1.5 \times 10^{16}\ \mathrm{m^{-3}}$.

Silicon is a semiconductor: at room temperature it releases only about $1.5 \times 10^{16}$ carriers per cubic metre, roughly twelve orders of magnitude fewer than the $8.5 \times 10^{28}\ \mathrm{m^{-3}}$ of copper, and a carrier appears only when thermal motion or an absorbed photon supplies at least $E_g = 1.12\ \mathrm{eV}$. Adding one dopant atom per $10^{6}$ silicon atoms raises the carrier number by many orders of magnitude without turning the material into a metal, which is what keeps it between the conductor and the dielectric.