Illustrated Physics
physics, drawn
Concept

Doping — where it appears

The deliberate addition of impurity atoms to a semiconductor to supply carriers, at concentrations from parts per billion to parts per thousand. It sets the Fermi level, and with it almost every electrical property the material has, without changing the crystal.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The bands bend by the amount the Fermi levels differed. A junction between 1e+17 cm⁻³ p-type and 1e+16 cm⁻³ n-type silicon at equilibrium, with the Fermi level flat by construction — that is what equilibrium means — and the two band edges carrying the whole of the 0.774 V drop. The bending happens over 331.8 nm, and it is not symmetric: the depletion reaches 301.6 nm into the lightly doped side and only 30.2 nm into the heavily doped one, because the same exposed charge is reached sooner where there is more of it. An electron in the n-side conduction band therefore faces an uphill barrier of 0.774 V to reach the p side, while an electron already on the p side rolls downhill without any barrier at all — which is the asymmetry the whole device is, and it is drawn here before any current has been mentioned.

One level, and the field that bends the bands

Two pieces of the same crystal doped differently have their Fermi levels at different heights. Joining them cannot leave both, because a difference in electrochemical potential is precisely what makes charge move — and everything a diode does is the accounting of what had to happen for that one difference to reach zero.

quantum · Bands One dimensionless group between an engine and Carnot. The efficiency of a thermoelectric couple against the temperature of its hot side, with the cold side at 300 kelvin, for 4 values of the figure of merit, and the Carnot ceiling drawn above them. The expression has exactly one material quantity in it — the dimensionless group formed from the Seebeck coefficient squared, the electrical conductivity, the temperature and the thermal conductivity — and everything else is the two temperatures. At 600 kelvin, a figure of merit of one gives 10.8 per cent against a Carnot ceiling of 50.0, and a figure of merit of four gives 22.6. The approach to the ceiling is slow: every doubling of the group buys less than the last, so the difference between a good material and a perfect one is smaller than the difference between a poor material and a good one.

An engine with one number in it

A thermoelectric couple has no moving part and no working fluid, and its efficiency is the Carnot value multiplied by a factor containing exactly one dimensionless group of material properties. Sixty years of effort have moved that group from about one to about two, and the reason it is hard is that its three ingredients are not independent: raising the conductivity ruins the coefficient it is squared against, and the only lever that is really free is the heat the lattice carries.

thermodynamics · Heat engines Electrons and holes in silicon against the donors added. The concentrations of free electrons and holes in silicon at 300 K against the concentration of donor atoms added, on logarithmic axes, solved from charge neutrality and the product law n·p = nᵢ². Below the intrinsic concentration of 10¹⁰ per cubic centimetre the donors change almost nothing. Above it the electrons follow the donors one for one and the holes fall in exact proportion: a hundred times more donors, a hundred times fewer holes. At 10¹⁶ donors the electrons outnumber the holes by 10¹² to one. The dashed line is √(n·p), which does not move at all across ten decades of doping — checked to a part in a billion.

The product doping cannot move

Add one phosphorus atom for every five million silicon atoms and the crystal's free electrons multiply a millionfold — and its holes fall a millionfold, although nothing was done to them. Add a drop of acid to water and the hydroxide ions vanish in exactly the proportion the hydrogen ions grow. Both are one law: when a reaction makes two species together, equilibrium fixes the sum of their chemical potentials, and so the product of their concentrations. Every semiconductor device is built on the minority carriers that law leaves behind.

thermodynamics · Chemical potential

Named alongside it

The objects these essays reach for when they reach for this one.

SemiconductorThe Boltzmann factorBand bendingBand gapBuilt in potentialCarnot efficiencyChemical equilibriumChemical potentialConductivityDepletion regionDetailed balanceDrift diffusion

All concepts