Semiconductors
A semiconductor is nothing more exotic than a band insulator whose gap is small enough for the thermal energy \(k_{B}T\), or a deliberately introduced impurity, to put a controllable number of carriers into an otherwise empty band — the identification is Wilson's [Wilson:1931a] [Wilson:1931b], and everything in this chapter is a consequence of it and of the band theory of Electrons in Solids: Band Theory. What makes the subject worth a chapter of its own is that the controllability is quantitative over some twenty orders of magnitude in carrier density, that the resulting devices are the most exhaustively verified artefacts in physics, and that the chain from a measured band gap to a working transistor is short enough to be followed in full. The Nobel prizes of 1956 (transistor) and 2000 (heterostructures and the integrated circuit) bracket that chain.
The chapter therefore runs from the band structure of silicon, germanium and gallium arsenide and the optical measurement of their gaps [Macfarlane:1958], through carrier statistics, doping [Pearson:1949] and the measurement of mobility and lifetime [Haynes:1951], to rectification at a barrier [Davydov:1938] [Mott:1939] [Schottky:1939], the p–n junction [Shockley:1949], the transistor [Bardeen:1948] [Shockley:1951], and the optoelectronic devices that run the junction backwards — solar cell [Chapin:1954], light-emitting diode [Round:1907] [Holonyak:1962] and injection laser [Hall:1962]. It ends in the low-dimensional structures [Dingle:1974] [Esaki:1970] that supply the two-dimensional electron gas of Experiment: The Quantum Hall Effect. The standard device reference is [Sze:2007]; the founding monograph is Shockley's [Shockley:1950].
Semiconductors: all derivations of this chapter are pending.
What a semiconductor is
Wilson's band picture
[Reserved: Wilson's two papers, which identified a semiconductor as a filled-band insulator with a small gap and showed that its conductivity must rise with temperature rather than fall [Wilson:1931a] [Wilson:1931b]; the resulting exponential \(\sigma\propto\ee^{-E_{g}/2k_{B}T}\) and the resistivity range from \(10^{-5}\,\mathrm{\Omega}\,\mathrm{m}\) to \(10^{5}\,\mathrm{\Omega}\,\mathrm{m}\) that separates semiconductors from metals and insulators; the older empirical facts this explained — Faraday's observation that the resistance of silver sulfide falls on heating, the opposite of every metal [Faraday:1833], and the erratic behaviour of “impure” samples now known to be doping.]
The electrical resistance of silver sulfide falls as it is heated — the opposite of the behaviour of every metal, and the first recorded observation of semiconduction [Faraday:1833]. In a pure semiconductor the effect is not a weak drift but an exponential: over several decades of conductivity, and to within a slowly varying prefactor,
so that a plot of \(\ln\sigma\) against \(1/T\) is a straight line whose slope measures a single energy \(E_{g}\) characteristic of the material [Wilson:1931a] [Wilson:1931b]. The same substances have room-temperature resistivities anywhere between about \(10^{-5}\,\mathrm{\Omega}\,\mathrm{m}\) and \(10^{5}\,\mathrm{\Omega}\,\mathrm{m}\), and the value in a given specimen is exquisitely sensitive to impurities present at parts per million — which is why the early measurements were irreproducible and were long dismissed as dirt.
Derivation. Let the conduction band edge be at \(E_{c}\) and the valence band edge at \(E_{v}\), with \(E_{g}=E_{c}-E_{v}\), and let the chemical potential lie several \(k_{B}T\) inside the gap, so that the Fermi–Dirac occupation of Quantum Statistics may be replaced by its Boltzmann tail at both edges. Integrating the band-edge density of states against that tail gives the standard forms
where \(N_{c}\) and \(N_{v}\) are effective densities of states varying only as \(T^{3/2}\). Multiplying the two expressions in Equation (127.2) eliminates the chemical potential entirely:
the law of mass action, which holds whatever the doping and is the single most useful relation in the subject. In an undoped crystal every conduction electron leaves a hole behind, so \(n=p=n_{i}\), and the conductivity \(\sigma=e(n\mu_{n}+p\mu_{p})\) inherits the square root of Equation (127.3), which is Equation (127.1). The factor of two in the exponent is the whole content of the measurement: it is there because the carriers are created in pairs across the gap, so the slope of the Arrhenius plot returns \(E_{g}/2\) and not \(E_{g}\). A metal has no gap and hence no such factor; its resistivity is governed instead by the scattering of Electrons in Solids: Band Theory, which grows with temperature, and the two temperature dependences therefore run in opposite directions.
∎The band structures of silicon, germanium and gallium arsenide
[Reserved: the diamond and zincblende lattices and their Brillouin zones; the indirect gap of silicon and germanium, with conduction minima away from the zone centre — six valleys along \(\Delta\) in silicon, eight half-valleys at \(L\) in germanium — against the direct gap of gallium arsenide at \(\Gamma\); the anisotropic effective-mass tensors measured by cyclotron resonance [Dresselhaus:1955]; the light-, heavy- and split-off hole bands; \(k\cdot p\) theory as the standard analytic parametrization near a band edge [Kane:1957]; tabulated parameters [Madelung:2004].]
Measuring the gap optically
[Reserved: the absorption edge as the direct measurement of \(E_{g}\); the square-root edge of a direct gap against the phonon-assisted structure of an indirect one, resolved into separate phonon-emission and phonon-absorption thresholds by Macfarlane and co-workers in silicon [Macfarlane:1958] — a measurement of the band structure and the lattice modes of Phonons and Lattice Dynamics at once; the shrinking of the gap with temperature and the empirical Varshni form [Varshni:1967]; photoluminescence and photoconductivity as complementary determinations.]
A semiconductor is transparent to photons below a threshold energy and absorbs strongly above it, and the threshold measures the band gap: about \(1.12\,\mathrm{eV}\) in silicon, \(0.67\,\mathrm{eV}\) in germanium and \(1.42\,\mathrm{eV}\) in gallium arsenide, against about \(5.5\,\mathrm{eV}\) in diamond [Madelung:2004]. In silicon the edge is not one threshold but two, separated by a fixed energy of order tens of \(\mathrm{meV}\), and the lower of the two weakens as the crystal is cooled while the upper does not; the separation matches twice the energy of a lattice mode of Phonons and Lattice Dynamics at a definite wavevector [Macfarlane:1958]. One optical measurement therefore returns the electronic gap, the fact that the gap is indirect, and the energy of the phonon that bridges it.
Derivation. A photon of energy of order \(1\,\mathrm{eV}\) has wavevector \(\omega/c\) of order \(10^{7}\,/\mathrm{m}\), three orders of magnitude smaller than the width \(10^{10}\,/\mathrm{m}\) of a Brillouin zone. On the scale of Electrons in Solids: Band Theory the photon therefore carries energy but no crystal momentum, and an optical transition is vertical: it connects states of the same \(\vect{k}\). If the valence maximum and the conduction minimum lie at the same \(\vect{k}\), as in gallium arsenide, absorption begins as soon as \(\hbar\omega\geq E_{g}\) and there is a single threshold. If they lie at different \(\vect{k}\), as in silicon and germanium, no vertical transition of energy \(E_{g}\) exists, and the transition can proceed only with the simultaneous emission or absorption of a phonon of wavevector equal to the offset. Conserving energy in the two cases,
which are two thresholds separated by \(2\hbar\Omega\) and placed symmetrically about \(E_{g}\). The absorption branch requires a phonon to be waiting in the crystal, so its strength carries the Bose occupation factor of that mode and vanishes exponentially as \(T\to0\); the emission branch requires only that a phonon can be created, and survives. That asymmetry is what identifies which threshold is which, and it converts Equation (127.4) into a measurement of \(E_{g}\) and \(\hbar\Omega\) separately. It also explains why an indirect-gap material is a poor light emitter and a good solar absorber only when thick: the transition needs a third particle, and is correspondingly weak.
∎Carriers and their statistics
Intrinsic conduction
[Reserved: electrons thermally promoted across the gap leaving holes behind, so that \(n=p=n_{i}\); the effective density of states, the law of mass action \(np=n_{i}^{2}\) independent of doping, and the intrinsic densities that follow — of order \(10^{16}\,/\mathrm{m}^{3}\) in silicon at \(300\,\mathrm{K}\) against \(10^{29}\,/\mathrm{m}^{3}\) of conduction electrons in copper; the chemical potential near mid-gap and its temperature drift; the Fermi–Dirac statistics of Quantum Statistics and the Boltzmann limit in which they are usually applied.]
Doping and extrinsic conduction
[Reserved: substitutional donors and acceptors; the hydrogenic impurity level, binding energy scaled by \(m^{*}/m\) and \(1/\epsilon_{r}^{2}\) to a few tens of \(\mathrm{meV}\), and the consequent freeze-out at low temperature; the systematic measurements of boron- and phosphorus-doped silicon by Pearson and Bardeen that established the picture quantitatively [Pearson:1949]; compensation, degenerate doping and the Mott criterion for the impurity-band transition [Mott:1968]; zone refining and the purity — better than one part in \(10^{10}\) — without which none of this is reproducible.]
Substituting a group-V atom for a silicon atom at the level of parts per million raises the electron density by many orders of magnitude and leaves the hole density correspondingly depressed, in accordance with Equation (127.3); substituting a group-III atom does the same for holes. Over a wide intermediate range of temperature the carrier density equals the dopant density and is flat in temperature, the impurities being fully ionized; below that range it falls exponentially as the carriers freeze back onto their donors, with an activation energy of a few tens of \(\mathrm{meV}\), and above it the intrinsic pairs of Equation (127.1) take over. The systematic measurements of boron- and phosphorus-doped silicon that established this picture are those of Pearson and Bardeen [Pearson:1949]. The Hall coefficient of Electrons in Solids: Band Theory [Hall:1879] reverses sign between the two doping types, which is the direct evidence that the positive carriers are real; near the intrinsic regime, where both species conduct, it can reverse sign again with temperature in one sample.
The hydrogenic impurity level: a donor treated as a charge screened by the dielectric constant of the host and binding an electron of the band effective mass, giving a binding energy reduced from the hydrogenic value by the effective-mass ratio and by the square of the relative permittivity, hence tens of \(\mathrm{meV}\); and the charge-neutrality condition combined with the law of mass action, which yields the freeze-out, saturation and intrinsic regimes as limits of one expression
The Hall effect as a carrier measurement
[Reserved: the Hall coefficient [Hall:1879] used in a semiconductor for what it does best — returning the sign of the majority carrier and, with the conductivity, both the density and the mobility; the sign reversal on passing from n-type to p-type material as the direct evidence for holes; two-carrier conduction near intrinsic, where the Hall coefficient can change sign with temperature; van der Pauw's theorem, which extracts resistivity and Hall coefficient from a sample of arbitrary shape with four contacts on its edge [vanderPauw:1958], as the working laboratory method.]
Drift, diffusion and mobility
[Reserved: the drift–diffusion equations, the Einstein relation \(D/\mu=k_{B}T/e\) linking them (the same relation proved for Brownian motion in Experiment: Brownian Motion and Avogadro's Number); measured room-temperature mobilities of about \(0.15\,\mathrm{m}^{2}/\mathrm{V}/\mathrm{s}\) for electrons in silicon and eight times that in gallium arsenide; scattering by ionized impurities, by acoustic and polar optical phonons and their differing temperature laws; velocity saturation and the transport formalism of Nonequilibrium Thermodynamics and Transport.]
Recombination and carrier lifetime
[Reserved: radiative band-to-band recombination, Auger recombination, and the trap-assisted Shockley–Read–Hall mechanism that dominates in indirect-gap material [Shockley:1952] [Hall:1952]; minority-carrier lifetimes from nanoseconds to milliseconds and the diffusion length that follows; surface recombination and passivation; why the lifetime, not the mobility, is what limits solar cells and bipolar transistors alike.]
Junctions and rectification
Metal–semiconductor contacts
[Reserved: asymmetric conduction at a metal contact, known empirically since the cat's-whisker detector; the barrier theories advanced independently by Davydov [Davydov:1938], Mott [Mott:1939] and Schottky [Schottky:1939], and the space-charge layer they predict; thermionic emission over the barrier against diffusion through it; Bardeen's demonstration that surface states pin the Fermi level and largely decouple the measured barrier height from the metal work function [Bardeen:1947] — the result that redirected the effort toward the junction transistor; ohmic contacts.]
The p–n junction
[Reserved: the depletion region, built-in potential and internal field of a junction between p- and n-type material; Shockley's theory and the ideal diode law \(I=I_{0}(\ee^{eV/k_{B}T}-1)\) [Shockley:1949]; capacitance–voltage profiling as the measurement of the doping profile; deviations from ideality — recombination current, high-injection, series resistance; avalanche and Zener breakdown [Zener:1934]; the junction as the single component from which every device in Section 127.4 and Section 127.5 is assembled.]
A single crystal doped p-type on one side and n-type on the other conducts asymmetrically: forward biased it passes a current that rises exponentially with the applied voltage, with a logarithmic slope of \(e/k_{B}T\) per volt — about one decade of current per \(60\,\mathrm{mV}\) at room temperature — while reverse biased it passes a small current that saturates and is almost independent of voltage. Over many decades the characteristic follows
with \(I_{0}\) fixed by the doping and the minority-carrier properties and not adjustable [Shockley:1949]. The asymmetry arises inside a homogeneous single crystal with no interface between different substances, and the exponent contains the temperature rather than any device parameter, which is what marks Equation (127.5) as a statement about carrier statistics.
The p–n junction in equilibrium: the diffusion of carriers across the metallurgical boundary, the depletion region and built-in potential that arrest it, and the equality of drift and diffusion currents that fixes that potential through the law of mass action; then, under bias, the Boltzmann factor by which the barrier lowering multiplies the minority-carrier density at the depletion edge, the diffusion of those excess carriers into the neutral regions, and the resulting saturation current — together with the recombination, high-injection and series-resistance terms by which real diodes depart from the ideal law
The Haynes–Shockley experiment
[Reserved: the experiment that made minority carriers visible — a pulse of holes injected into an n-type germanium bar, drifted by a field and detected downstream, giving drift velocity, diffusion constant and lifetime from the arrival time, the width and the area of one measured pulse [Haynes:1951]; the direct test of the Einstein relation it provides; why it settled the question of whether injected minority carriers exist at all, which underpins Section 127.4.1.]
A pulse of holes injected at one point of an n-type germanium bar, where holes are vastly outnumbered, does not vanish into the electron sea. It travels down the bar at a velocity proportional to the applied field, arrives at a downstream contact after a measurable delay, broadens as it goes by an amount that grows as the square root of the transit time, and loses area exponentially with a lifetime of its own [Haynes:1951]. One oscilloscope trace therefore yields the drift mobility from the delay, the diffusion constant from the width and the lifetime from the area. The two transport coefficients so obtained are not independent: they satisfy
about \(25.7\,\mathrm{mV}\) at \(298\,\mathrm{K}\), the same relation between fluctuation and dissipation that governs Brownian motion in Experiment: Brownian Motion and Avogadro's Number.
Derivation. Consider the carriers in thermal equilibrium in a static potential \(\phi(x)\), with no current flowing anywhere — equilibrium, not a steady state, so that drift and diffusion must cancel locally and not merely on average. For electrons of charge \(-e\) the potential energy is \(U=-e\phi\), so the equilibrium density is the Boltzmann distribution
An electron of mobility \(\mu\) drifts at velocity \(-\mu E\), against the field, so the particle flux is \(\Gamma=-n\mu E-D\,\dd n/\dd x\). Setting it to zero and substituting Equation (127.7),
which must hold for every \(E\) and every \(n\), and therefore holds if and only if \(D/\mu=k_{B}T/e\), which is Equation (127.6). The content of the argument is that a single microscopic process — the same collisions — limits the drift and produces the random walk, so the coefficient describing the response to a force and the coefficient describing the spontaneous spreading cannot be chosen independently. Since \(\mu\) and \(D\) are measured from different features of the same recorded pulse, the Haynes–Shockley experiment tests Equation (127.6) rather than assuming it.
∎Tunnelling and the Esaki diode
[Reserved: a junction doped degenerately on both sides, so thin that carriers tunnel through rather than over the barrier; Esaki's observation of negative differential resistance in narrow germanium junctions, the first clear demonstration of tunnelling in a solid [Esaki:1958]; the peak-to-valley ratio and its interpretation; phonon-assisted structure in the tunnelling current as spectroscopy; the line from here to superconducting tunnelling in Experiment: Superconductivity.]
Devices
The transistor
[Reserved: the point-contact transistor of Bardeen and Brattain, with power gain demonstrated in a germanium slab in December 1947 [Bardeen:1948]; Shockley's junction transistor, in which the minority-carrier injection of Section 127.3.3 replaces the fragile point contacts [Shockley:1951] [Shockley:1950]; current gain, base transit time and the Early effect [Sze:2007]; the 1956 Nobel prize; why the device is a physics result and not only an engineering one — it is a working measurement of minority-carrier transport.]
A germanium slab carrying two closely spaced point contacts, one forward biased at low impedance and one reverse biased at high impedance, delivers to a load at the second contact more signal power than is supplied to the first: a small current modulation at the input controls a larger power at the output, with gain measured at audio frequencies in December 1947 [Bardeen:1948]. Gain, not rectification, is the observation — rectifying contacts had been in use for decades — and it requires that carriers injected at one contact survive the journey to the other, which is the minority-carrier transport of Phenomenon 127.5. Replacing the fragile point contacts by two p–n junctions sharing a thin base gives the same effect in a robust and quantitatively predictable form [Shockley:1951] [Shockley:1950].
Transistor action from the junction physics already in hand: injection of minority carriers across a forward-biased emitter junction, their diffusion across a base thin compared with the diffusion length so that few recombine, and their collection by the reverse-biased collector junction; the resulting current gain as the fraction surviving the base, and the power gain as the product of that fraction with the ratio of the collector to the emitter impedance — the transfer of current from a low-resistance to a high-resistance circuit that gives the device its name
The MOSFET and the inversion layer
[Reserved: the field-effect idea, older than the point-contact transistor and blocked for two decades by the surface states of [Bardeen:1947], unblocked by thermally grown silicon dioxide; the metal–oxide–semiconductor structure, the inversion layer and the threshold voltage [Kahng:1976]; the inversion layer as a genuine two-dimensional electron gas, with quantized subbands [Ando:1982] — the same system in which quantized magnetoresistance [Fowler:1966] and then the quantum Hall effect of Experiment: The Quantum Hall Effect were found.]
The integrated circuit
[Reserved: Kilby's demonstration that every element of a circuit can be made in one block of semiconductor [Kilby:1976], and the planar process that made it manufacturable; scaling, Moore's empirical doubling law and its status as an economic observation rather than a physical one [Moore:1965]; the physical limits that eventually bind — gate oxide tunnelling, dopant number fluctuations, power density; the 2000 Nobel prize shared with the heterostructure work of Section 127.5.3.]
Light and semiconductors
Photoconductivity and the solar cell
[Reserved: photogeneration of electron–hole pairs above the absorption edge and the photocurrent that results; the silicon p–n junction photocell of Chapin, Fuller and Pearson, the first solar cell of usable efficiency at about six percent [Chapin:1954]; the open-circuit voltage, short-circuit current and fill factor actually measured; the Shockley–Queisser detailed-balance limit, about \(33\,\mathrm{\%}\) for a single gap under the solar spectrum [Shockley:1961], derived from the thermodynamics of Classical Thermodynamics; measured cell efficiencies against it.]
Light-emitting diodes
[Reserved: Round's note recording light from a silicon carbide contact, the first observation of electroluminescence in a semiconductor [Round:1907]; the recombination of injected carriers across a direct gap as its explanation, and the resulting relation between emitted photon energy and \(E_{g}\); the first visible practical device, the red gallium arsenide phosphide diode of Holonyak and Bevacqua [Holonyak:1962]; the long-delayed blue diode built on gallium nitride [Nakamura:1994] and the 2014 Nobel prize; efficiency measurements and droop.]
A current passed through certain semiconducting contacts produces light. Round recorded a yellowish glow from a silicon carbide detector under bias, the first observation of the effect in a semiconductor [Round:1907]. In a forward-biased junction of a direct-gap material the emission is not thermal: the spectrum is a narrow band whose photon energy is close to the band gap of the material and shifts with it as the material composition is changed, which is how the first practical visible emitter was tuned to red in gallium arsenide phosphide [Holonyak:1962] and, thirty years later, to blue in gallium nitride [Nakamura:1994]. The device is the junction of Phenomenon 127.4 run in reverse in the energetic sense: instead of light generating carriers, carriers generate light.
Radiative recombination of injected minority carriers across a direct gap: the emission spectrum as the joint density of states weighted by the carrier occupations, giving a peak just above the gap with a width of order the thermal energy; the selection rule that suppresses the process in an indirect-gap material, where a phonon must participate, and the competing non-radiative Auger and trap-assisted channels that fix the internal quantum efficiency
The semiconductor laser and heterostructures
[Reserved: population inversion in a degenerately doped junction and the coherent emission observed from gallium arsenide diodes by Hall and co-workers [Hall:1962]; the threshold current and its temperature dependence; Kroemer's double-heterostructure proposal, confining carriers and photons in the same layer [Kroemer:1963], and the room-temperature continuous-wave operation it eventually delivered [Alferov:2001]; quantum-well active regions [Dingle:1974]; the stimulated-emission physics itself belongs to Quantum Optics and the Photon.]
Excitons
[Reserved: the bound electron–hole pair, tightly bound and localized in Frenkel's molecular limit [Frenkel:1931] and weakly bound over many lattice constants in Wannier's [Wannier:1937]; hydrogenic binding energies of a few \(\mathrm{meV}\) in gallium arsenide, and the sharp absorption lines below the gap that measure them; why excitons are visible at room temperature in wide-gap and two-dimensional material and not in bulk silicon.]
Low-dimensional semiconductors
Quantum wells and superlattices
[Reserved: epitaxial growth with monolayer control; the square well of Elementary Quantum Systems realized physically, with confined subbands observed as steps in absorption and as discrete emission lines [Dingle:1974]; the Esaki–Tsu superlattice, its minibands and predicted negative differential conductivity [Esaki:1970], in which the Bloch oscillations of Electrons in Solids: Band Theory become observable; band-gap engineering as an experimental programme.]
The two-dimensional electron gas
[Reserved: modulation doping, which separates the donors from the carriers and so removes the dominant scattering channel [Dingle:1978]; mobilities improved by more than three orders of magnitude, up to \(10^{3}\,\mathrm{m}^{2}/\mathrm{V}/\mathrm{s}\) in the best gallium arsenide heterostructures; the electronic properties of such two-dimensional systems reviewed comprehensively in [Ando:1982]; this is the sample that Experiment: The Quantum Hall Effect takes as its starting point, and the reason the fractional effect was found at all.]