Concept

Band bending — where it appears

The curving of a semiconductor's energy bands near a junction, surface or contact, where charge has moved to line up the Fermi levels. The bent region is stripped of free carriers, and the height of the bend is the barrier a diode's current must climb.

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

Also named here as depletion region, fermi level — the same set of essays touches all of them, so they are one junction rather than several.

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

A barrier the metal was supposed to choose. The barrier height at a metal contact on n-type silicon against the metal's work function, from Bardeen's model with interface-state densities of none, 10¹² per cm² per eV, 10¹³ per cm² per eV, 10¹⁴ per cm² per eV, a 0.5 nm gap and a neutrality level a third of the gap above the valence band. With no interface states the barrier is Schottky and Mott's φₘ − χ, rising one for one with the work function: from aluminium to platinum, 0.23 to 1.60 eV. With 10¹² per cm² per eV the slope is 0.92 and the same metals give 0.27 to 1.53 eV; with 10¹³ per cm² per eV the slope is 0.53 and the same metals give 0.48 to 1.20 eV; with 10¹⁴ per cm² per eV the slope is 0.10 and the same metals give 0.71 to 0.84 eV. Dense enough interface states pin every barrier near 0.76 eV, and the metal hardly matters.

The barrier the metal cannot choose

Put a metal on a semiconductor and a barrier forms at the interface. The obvious theory says its height is the difference between the metal's work function and the semiconductor's electron affinity. Platinum and aluminium differ by nearly an electronvolt and a half in work function, so they should make wildly different contacts. On silicon and gallium arsenide they do not: nearly every metal gives nearly the same barrier. Bardeen explained why in 1947. A thin layer of electronic states at the interface screens the metal, and it takes only a few such states per hundred surface atoms to pin the barrier wherever they put it.

quantum · Bands

Named alongside it

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

Depletion region Fermi level The Boltzmann factor Built in potential Doping Drift diffusion Fermi level pinning Interface states Pn junction Poisson equation Rectification Schottky barrier

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