The product doping cannot move

Pure silicon at room temperature is a poor conductor, because at any moment only about one atom in five million million has had an electron shaken loose from its bond by heat into the conduction band, leaving behind a hole that can also move. Add phosphorus atoms, each carrying one more electron than the silicon it replaces and giving it up easily, at a concentration of one in five million, and the free electrons multiply a millionfold. The holes, which the phosphorus neither supplies nor takes away, fall by a factor of a million at the same time. Nothing was done to them. They fall because electrons and holes are made and destroyed in pairs, and the equilibrium between making and destroying fixes not either concentration but their product.

A reaction with two products #

In a pure semiconductor, the relevant reaction is the breaking of a bond:

$$\text{bound electron} \;\rightleftharpoons\; e^- + h^+ .$$

Heat supplies the band gap’s worth of energy and creates a free electron and a hole together; an electron meeting a hole falls back into it and the pair disappears. At equilibrium the two rates balance. The chemical-potential statement of that balance is the same one that governs any reaction left to settle: the chemical potentials of the reactants equal those of the products, and here that means the electrons’ chemical potential plus the holes’ equals a constant fixed by the material and the temperature.

Each species’ chemical potential is, for a dilute population, $kT$ times the logarithm of its concentration plus a constant — the same logarithm that turned a surface site’s filling into a function of pressure. A fixed sum of two logarithms is a fixed product:

$$n\,p = n_i^2,$$

where $n_i$ is the concentration each species has in the pure crystal. That is the law of mass action, the rule chemists write as an equilibrium constant, applied to a reaction whose products are an electron and the vacancy it left. Donor atoms add electrons; they cannot change the product; so the holes must fall to keep it. Adding a hundred times more donors gives a hundred times fewer holes.

The law needs one more condition to fix both numbers: charge neutrality. Every donor that gives up its electron leaves a fixed positive ion, so the free electrons must outnumber the holes by exactly the donor concentration, $n - p = N_d$. Together with $np = n_i^2$ that is a quadratic, and its solution,

$$n = \frac{N_d}{2} + \sqrt{\frac{N_d^2}{4} + n_i^2},$$

is what the first figure draws. Below $n_i$ the donors are a trace in a sea of thermally made pairs and change nothing. Above it the electrons equal the donors and the holes are $n_i^2/N_d$. At a typical doping of $10^{16}$ per cubic centimetre, silicon has $10^{16}$ free electrons and $10^4$ holes in each cubic centimetre — a ratio of a million million to one — and it is still in equilibrium, with pairs being made and destroyed at the same rate as in the pure crystal.

The same law in a glass of water #

Water does the same thing, and chemists have been writing it down for longer.

$$\text{H}_2\text{O} \;\rightleftharpoons\; \text{H}^+ + \text{OH}^-$$

A tiny fraction of water molecules dissociate at any moment, making a hydrogen ion and a hydroxide ion together. At equilibrium the product of their concentrations is fixed: $[\text{H}^+][\text{OH}^-] = K_w = 10^{-14}$ in moles per litre, squared, at 25 °C. Pure water has $10^{-7}$ moles per litre of each. Add a strong acid and the hydrogen ions rise; the hydroxide ions fall in exact proportion, and the product stays at $10^{-14}$.

The pH scale is the logarithm of the hydrogen-ion concentration, pOH the logarithm of the hydroxide’s, and the statement that pH plus pOH equals 14 is the product law written in logarithms: a fixed sum of logarithms, which is to say a fixed sum of chemical potentials. The figure’s flat plateau in the middle is the region where the added acid is outnumbered by the water’s own ions — exactly the region below $n_i$ in the silicon figure — and the straight lines on either side are the regime where the added species dominates and the other is suppressed in proportion.

The resemblance is not a resemblance.

Measured in their own natural unit — concentrations in units of the pure material’s value — silicon with phosphorus and water with hydrochloric acid are indistinguishable. Both are the solution of $n - p = N$ and $np = 1$. What a chemist calls a strong acid, a solid-state physicist calls an n-type dopant; what the chemist calls the self-ionisation of water, the physicist calls intrinsic carrier generation; and the neutral point, pH 7, is the intrinsic Fermi level. The same arithmetic also sets the ions on either side of a charged membrane, where the product of the mobile positive and negative ions must match across the membrane and a fixed charge on one side forces an imbalance that holds a voltage with no pump running.

Half the energy in the exponent #

How many pairs heat makes in the pure material depends on the energy of making one, and the dependence has a factor of two in it that is easy to miss.

The pair costs the band gap $E_g$, so the product of the two concentrations carries the Boltzmann factor $e^{-E_g/kT}$. Each concentration is the square root of the product, and carries $e^{-E_g/2kT}$. The effective activation energy of the intrinsic carrier concentration is half the gap, and an Arrhenius plot of $n_i$ — the kind that reads an activation energy off a rate — gives a slope of half the gap, not the gap. Water’s ions do the same thing with half the enthalpy of ionisation, and the dependence is strong enough that pure water at the boiling point has a pH near 6, not 7, while being exactly as neutral as it was at room temperature: neutrality means equal concentrations, not pH 7.

The comparison of the curves carries a surprise. Pure water, a textbook insulator, has more charge carriers per cubic centimetre than pure germanium and thousands of times more than pure silicon. It conducts poorly because its carriers are ions, heavy and slow in a viscous liquid, where a semiconductor’s are electrons and holes moving through a crystal nearly unimpeded. The count of carriers is set by the pair energy; how well they carry current is a separate question entirely.

The Fermi level moves like a pH electrode #

The chemical potential of the electrons — the Fermi level — is where the product law shows itself as an energy.

Because each concentration is the exponential of its chemical potential over $kT$, a factor of ten in concentration is $kT \ln 10$ in chemical potential — 59.5 millielectronvolts at room temperature. Doping silicon moves its Fermi level by that amount per decade of dopant, up for donors and down for acceptors, and the holes’ chemical potential moves by the same amount in the other direction, keeping the sum fixed. A glass pH electrode, which measures the chemical potential of hydrogen ions through a membrane that passes only them, produces a voltage that moves by 59.5 millivolts per unit of pH at the same temperature. It is the same number because it is the same thing: a chemical potential that is $kT$ times the logarithm of a concentration, divided by the particle’s charge.

That the Fermi level can be anywhere in the gap, set by doping, is what makes a junction between two differently doped pieces bend its bands: when the two are joined, their Fermi levels must become one, and the difference between them before joining becomes a built-in voltage, $kT/q \ln(N_aN_d/n_i^2)$ — with the product law’s $n_i^2$ in the denominator.

What a voltage does to the product #

The product law holds at equilibrium. A voltage across a junction is exactly a way of holding it away from equilibrium, and the amount by which it does so is simple.

Under a forward voltage the electrons and holes no longer share one chemical potential. Each population settles to its own — two quasi-Fermi levels — separated by the applied voltage times the electron’s charge. The sum of the two chemical potentials is no longer the equilibrium value, and the product of the concentrations is no longer $n_i^2$ but $n_i^2 e^{qV/kT}$. Electrons and holes meeting and recombining do so at a rate proportional to their product, so the recombination current rises as the same exponential, and that is the diode law: a current that multiplies tenfold for every sixty millivolts. In a light-emitting diode the recombination makes light, and the light carries the chemical-potential difference as its own; in a solar cell the process runs backwards, light makes pairs, and the split between the two chemical potentials is the voltage the cell delivers.

Under a reverse voltage the product falls below $n_i^2$, recombination nearly stops, and the only current is the trickle of pairs generated thermally near the junction — which is why a reverse-biased diode’s leakage current rises steeply with temperature, as $n_i^2$ does, doubling every ten degrees or so in silicon.

Why the few carriers are the ones that count #

It would be natural to treat the minority carriers as a rounding error — $10^4$ holes beside $10^{16}$ electrons — and for carrying current through a uniform piece of doped silicon they are. For almost everything a semiconductor device does, they are the whole story.

The reason is that the majority carriers are everywhere already, and a change in their concentration is a small fractional change. The minority carriers are nearly absent, and injecting even a modest number of them changes their concentration by many orders of magnitude, far from the equilibrium the product law demands. Such an excess cannot stay: it recombines, over the minority-carrier lifetime, and in that time it spreads by diffusion over a distance called the diffusion length — typically tens to hundreds of micrometres in good silicon. The diffusion constant that sets the spreading is tied to the carriers’ mobility by the Einstein relation, the same one that connects any random walk to the drag it feels.

A bipolar transistor is built around exactly this. Electrons are injected from a heavily n-type emitter into a thin p-type base, where they are minority carriers; they diffuse across a base thinner than their diffusion length before most of them can recombine, and are collected on the other side. The current gain is a ratio of how many survive the crossing to how many recombine on the way, and it depends on the minority carriers’ lifetime and on nothing about the majority. A solar cell is the same story run by light: photons make pairs, the minority members of the pairs must diffuse to the junction before they recombine, and the cell’s efficiency is set by how far they get. A photodetector, a charge-coupled image sensor and a light-emitting diode all live or die by the minority carriers the product law leaves behind.

Two further details follow directly from the law. A crystal doped with both donors and acceptors — compensated — behaves according to the difference between them, because charge neutrality counts only the net excess of fixed charge; and the minority concentration is set by that net doping even if the individual dopant concentrations are enormous. And the product law is why the minority concentration is so sensitive to temperature: $n_i^2$ rises by a factor of about two for every eight degrees near room temperature in silicon, so a device whose behaviour depends on minority carriers — every one of them — has its leakage, its noise and its threshold voltages drift with temperature at that rate. The electrons and holes themselves are waves in a crystal’s bands, with an effective mass set by the band’s curvature, but none of that enters the product law except through $n_i$; the law is thermodynamics, and it would hold for any two species made together.

Where the product law is an approximation #

Dilute species only. The step from chemical potential to $kT\ln n$ assumes the carriers or ions are dilute enough not to feel each other or the limits of the states available. Heavily doped silicon, above about $10^{18}$ per cubic centimetre, is degenerate — its electrons fill the bottom of the conduction band like the sea in a metal — and the Boltzmann approximation fails; the gap itself shrinks with heavy doping. Concentrated acids are not dilute either, and chemists replace concentrations by activities for the same reason.

All donors ionised. The first figure assumes every phosphorus atom has given up its electron. At room temperature in silicon that is nearly true; at low temperature the donors hold on to their electrons, which is itself a two-state site filling according to the Fermi function, and the free electrons fall away.

Equilibrium, or steady bias. The product law and its biased version describe steady states. Immediately after a flash of light or a change of voltage the product is anything, and it relaxes back over the recombination lifetime — nanoseconds to milliseconds depending on the material’s defects.

The prefactor is measured. The intrinsic concentrations in the temperature figure are calibrated to measured room-temperature values, with the gap’s own variation with temperature included; the structure of the curves — half the gap in the exponent — is derived, and the absolute level is not.

The traffic behind a steady product #

A plot of concentrations is a plot of averages, and it hides the traffic that maintains them. In pure silicon at room temperature, pairs are created and destroyed at a rate of about $n_i$ divided by the recombination lifetime — around $10^{13}$ per cubic centimetre per second in silicon whose lifetime is a millisecond, and far faster in material with more defects; at equilibrium the two rates match exactly, and doping changes both rates without changing the balance. The equilibrium is dynamic, which the static curves do not show, and it is the dynamics that a device exploits when a voltage tips the balance.

Nor do the curves show where the pairs are made and destroyed. In real material most recombination happens not by an electron meeting a hole directly but through defect levels in the gap, which hold an electron and then a hole in turn. The product law does not care how the reaction proceeds — equilibrium is equilibrium whatever the route — but the rate at which the product returns to its equilibrium value after a disturbance depends entirely on the route.

Still open: how far a material’s own defects set its minority-carrier lifetime #

The product law is exact in equilibrium, and a semiconductor device’s performance depends on how quickly the product returns to $n_i^2$ after it has been pushed away — the minority-carrier lifetime, which in silicon ranges from nanoseconds to milliseconds depending on the purity and perfection of the crystal. For the materials now being developed for solar cells, including metal-halide perovskites, the lifetime is surprisingly long given how many defects the crystals contain, and why is not settled: the candidate explanations include defects whose levels lie outside the gap, where they cannot mediate recombination, and electrons and holes that are kept apart by the lattice’s own distortions. How much of the lifetime is a property of the material and how much of the particular defects in particular samples decides how close such cells can come to the limit the product law sets on their voltage.

The next question about chemical potential is what happens when the species being made in pairs are not in one uniform medium but on two sides of an interface — electrons in a metal and ions in a solution — where the difference between two chemical potentials becomes the voltage of a battery. The habit worth carrying from here is to ask, of any reaction that makes two things together, which quantity it actually fixes. An equilibrium that makes a pair fixes the sum of two chemical potentials, and so the product of two concentrations — and every scheme for controlling one of the pair, from doping a crystal to titrating an acid, works by accepting what that does to the other.