Atomic Structure and Energy Bands

1 The Bohr Atom and the One Idea That Survived

Rutherford's scattering experiment of 1911 established that an atom is a tiny dense positive nucleus surrounded by electrons occupying a volume some ten thousand times larger. It also created a crisis. An electron in a circular orbit is accelerating, an accelerating charge radiates electromagnetic energy according to classical electrodynamics, and an orbiting electron that radiates must spiral into the nucleus. The calculation gives a lifetime of the order of \(10^{-11}\) seconds. Matter exists, so classical physics was wrong about something.

Niels Bohr's response in 1913 was not to repair the physics but to forbid the offending behaviour by decree. He kept the classical orbit and added two postulates. First, only those orbits are allowed whose angular momentum is an integer multiple of \(\hbar = h/2\pi\):

The integer \(n\) is the principal quantum number. An electron in an allowed orbit does not radiate, however hard classical theory insists it should.

Second, radiation is emitted or absorbed only when an electron moves between two allowed orbits, and the photon carries the whole energy difference:

Setting the Coulomb attraction equal to the centripetal force, \(\dfrac{e^2}{4\pi\varepsilon_0 r^2} = \dfrac{mv^2}{r}\), and eliminating \(v\) with the quantisation condition gives the orbit radius and the total energy for hydrogen:

The energy is negative because the electron is bound; \(E = 0\) is the free electron at rest, infinitely far away. The 13.6 eV needed to remove the ground-state electron is the ionisation energy of hydrogen, and it is measured to that accuracy.

Three things about this result deserve attention, because all three carry through to the solid. The levels are discrete: an electron in hydrogen may have \(-13.6\), \(-3.40\) or \(-1.51\) eV, and nothing in between. The levels crowd together as \(n\) grows, since \(E_n \propto 1/n^2\); the step from \(n = 1\) to \(n = 2\) is 10.20 eV, while the step from \(n = 3\) to \(n = 4\) is only 0.66 eV. And the electron is bound most tightly nearest the nucleus, so the outermost electrons of a many-electron atom are the loosely held ones that chemistry and electronics actually use. Those outermost electrons in the last unfilled shell are the valence electrons, and from here on they are the only electrons we care about. Silicon has fourteen electrons; ten of them sit in filled inner shells and take no part in anything, and the four valence electrons in the third shell do all the work.

The Bohr model also explains the hydrogen emission spectrum quantitatively. The \(n = 2 \to 1\) transition releases \(13.6(1 - 1/4) = 10.20\) eV, and a photon of that energy has a wavelength of 121.5 nm, which is exactly where the Lyman-alpha line is found. For a model built on a postulate with no justification, that is a remarkable success, and it is why the picture is still taught.

2 Where the Bohr Picture Fails, and What Replaced It

The trouble with the Bohr model is that its successes stop at hydrogen. Applied to helium, which has two electrons, it gives an ionisation energy in error by about 5 per cent, and for heavier atoms it fails outright. It has nothing to say about why some spectral lines are bright and others faint, and nothing about the fine structure — the splitting of single lines into closely spaced multiplets — that better spectrometers revealed. Nor does it explain why the periodic table has the shape it has, with periods of 2, 8, 8, 18 elements. Most damagingly, the quantisation condition is simply asserted. Nothing in the model says why angular momentum should come in units of \(\hbar\).

Louis de Broglie supplied the missing reason in 1924 by proposing that a particle of momentum \(p\) has an associated wavelength \(\lambda = h/p\). If the electron is a wave running round the orbit, it must close on itself after a whole number of wavelengths or interfere itself out of existence:

Bohr's arbitrary postulate is the standing-wave condition for an electron wave on a closed loop. The quantisation was never arbitrary; it was a boundary-value problem in disguise.

Schrödinger turned this into a proper wave equation in 1926, and the consequences are more radical than a tidier derivation. The solutions, called orbitals, are not trajectories. The quantity \(|\psi|^2\) gives the probability per unit volume of finding the electron at a point, and Heisenberg's uncertainty principle, \(\Delta x\,\Delta p \gtrsim \hbar/2\), says that a definite orbit — a definite position and a definite momentum at once — is not merely unknown but meaningless. The electron in the hydrogen ground state is a spherical cloud of probability whose density peaks at 0.529 Å, the Bohr radius. Bohr's orbit is the most likely radius, not a path.

Solving the equation for a central Coulomb potential produces the allowed states labelled by four quantum numbers rather than one: \(n\) for the principal shell, \(l = 0, 1, \dots, n-1\) for the orbital shape (the states written s, p, d, f), \(m_l = -l, \dots, +l\) for its orientation, and \(m_s = \pm\tfrac{1}{2}\) for spin. Pauli's exclusion principle then states that no two electrons in one atom may share all four, which limits an s subshell to 2 electrons, a p subshell to 6, and shell \(n\) to \(2n^2\). That single rule fixes the periods of the periodic table and, more to the present purpose, fixes how many electrons a band in a solid can hold.

One further idea from wave mechanics is needed later. An electron moving through the periodic potential of a crystal does not respond to a force as a free electron would, because the lattice is pushing on it too. The whole of that interaction can be absorbed into a single parameter, the effective mass \(m^*\), after which the carrier may be treated as a free particle obeying \(F = m^*a\). For silicon the density-of-states effective masses at 300 K are about \(1.08\,m_0\) for electrons and \(0.56\,m_0\) for holes; in gallium arsenide the electron effective mass is only \(0.067\,m_0\), which is precisely why GaAs electrons are so fast. Chapter 3 uses these numbers, and Chapter 4 shows the mobility that follows from them.

3 From Isolated Levels to Bands in a Solid

A single silicon atom has sharp levels. A crystal of silicon contains about \(5.0 \times 10^{22}\) atoms per cubic centimetre — a figure worth deriving, since it is used constantly: the diamond cubic cell has a side of 5.431 Å and contains 8 atoms, so the density is \(8/(5.431\times10^{-8}\,\text{cm})^3 = 4.99\times10^{22}\ \text{cm}^{-3}\). What happens to the sharp levels when that many atoms are packed a few ångströms apart?

Pauli's principle answers it. The exclusion principle applies to the whole system, not to one atom, so once the atoms are close enough that their outer orbitals overlap, no two of the \(N\) atoms may keep an electron in an identical state. The level cannot stay single. It splits into \(N\) distinct levels, one for each atom, and because \(N\) is of order \(10^{22}\) in any real crystal the levels lie so close together — a few electronvolts spread over \(10^{22}\) states, giving spacings of order \(10^{-22}\) eV — that they behave as a continuum. That near-continuum of allowed states is an energy band.

The splitting is not the same for every level. It depends on how far the orbital reaches out from its nucleus, so the tightly bound inner shells barely notice their neighbours and stay almost atomic, while the valence orbitals overlap strongly and spread into wide bands. Figure 1.1 plots the effect as a function of interatomic spacing: on the right the atoms are far apart and the 3s and 3p levels of silicon are sharp; moving left, they broaden, then merge, and then — this is peculiar to the group IV elements — separate again into two bands.

The state count is worth following through, because it explains why silicon behaves as it does. In \(N\) isolated silicon atoms the 3s subshell offers \(2N\) states, all occupied, and the 3p subshell offers \(6N\) states, of which only \(2N\) are occupied — silicon's configuration is \(3s^2 3p^2\), four valence electrons in eight available valence states. As the atoms approach, the 3s and 3p bands first overlap and then, through \(sp^3\) hybridisation, reorganise into two bands of \(4N\) states each separated by a gap. The \(4N\) valence electrons (four per atom, \(N\) atoms) exactly fill the lower band and leave the upper one empty.

"Exactly fill" is the phrase that matters. A band with every state occupied cannot carry current. To carry current the electron distribution must become asymmetric in momentum — more electrons moving one way than the other — and that requires some electrons to change state. If every state in the band is already taken, there is no state to change into, and the applied field produces nothing. This is why the argument from "plenty of electrons" fails: quartz has abundant electrons, all of them in full bands.

Two escape routes exist. Either the band is only partly filled, in which case empty states sit immediately above the occupied ones and an arbitrarily small field produces a current; or the band is full but some electrons can be lifted across the gap into the empty band above, leaving vacancies behind them. The first case is a metal. The second is a semiconductor or an insulator, and which of the two depends entirely on the size of the gap.

4 Valence Band, Conduction Band and the Forbidden Gap

Only the top two bands need names. The highest band that is full at 0 K is the valence band, its upper edge written \(E_V\); the band above it, empty at 0 K, is the conduction band, its lower edge written \(E_C\). Between them lies a range of energies for which the Schrödinger equation has no propagating solution in the periodic lattice, so no electron can have an energy in that range. It is the forbidden gap or band gap, of width

Note the word forbidden. The gap is not a region where states are unlikely; it is a region where states do not exist. An electron crossing it is never found part-way across — it absorbs or emits the whole \(E_g\) at once.

An electron promoted from the valence band to the conduction band becomes a mobile negative carrier, free to accelerate because the conduction band is almost entirely empty and offers unlimited nearby states. Its departure leaves an empty state in the valence band, which is no longer full and can therefore also carry current. Chapter 2 develops the bookkeeping by which that vacancy is treated as a positive particle — the hole — and Chapter 2's whole argument depends on the picture established here.

The energy for the promotion can be delivered in several ways, and the equivalence of these is a point students often miss. Thermal energy from lattice vibrations is the usual source: at 300 K the characteristic thermal energy \(kT\) is \(0.0259\) eV, and although that is far smaller than a 1.12 eV gap, the Boltzmann distribution has a tail, and a small fraction of electrons at any instant carry many times the average. A photon does it if \(h\nu \ge E_g\), which is why the gap sets the longest wavelength a photodetector can respond to. A sufficiently strong electric field does it, which is the avalanche mechanism of Chapter 12. And a fast particle does it, which is why radiation upsets memories.

The gap is not a constant of the material; it shrinks as the crystal is heated, because thermal expansion increases the interatomic spacing and the electron-phonon interaction strengthens. The Varshni relation fits the measurements well:

For silicon \(E_g(0) = 1.170\) eV, \(\alpha = 4.73\times10^{-4}\) eV/K and \(\beta = 636\) K, giving \(E_g(300\ \text{K}) = 1.124\) eV and \(E_g(400\ \text{K}) = 1.097\) eV — a slope of about \(-0.25\) meV/K near room temperature. Small as that sounds, Chapter 2 shows it moving the carrier concentration by orders of magnitude.

5 Classifying Solids by the Width of the Gap

With the gap defined, the classification is a matter of arithmetic. The fraction of valence electrons with enough thermal energy to reach the conduction band goes roughly as the Boltzmann factor \(\exp(-E_g/kT)\), and because \(E_g/kT\) sits in the exponent, a change of a factor of five in the gap changes the answer by seventy orders of magnitude.

In a conductor the two bands overlap, so there is no gap at all and the conduction band is partly occupied at every temperature including absolute zero. The carrier concentration is fixed at roughly one per atom, about \(10^{22}\ \text{cm}^{-3}\), and cannot be altered by anything an engineer can do. Heating a metal raises its resistance, because more lattice vibration means more scattering while the carrier count stays put — the positive temperature coefficient familiar from a lamp filament.

In an insulator the gap exceeds roughly 3 eV, the Boltzmann factor is so small that the number of thermally generated carriers in a laboratory-sized crystal is a handful, and the material conducts only when the field is large enough to break it down. The value of diamond's 5.5 eV and silicon dioxide's 9 eV is not that the material is a slightly better insulator than plastic; it is that the insulation is essentially perfect, which is what makes a gate oxide 2 nm thick usable.

A semiconductor occupies the awkward and useful middle, conventionally a gap between about 0.5 and 3 eV. Two properties follow, and both are indispensable. First, the carrier concentration rises steeply with temperature, since \(\exp(-E_g/kT)\) is a rapidly increasing function of \(T\); that gives a semiconductor its negative temperature coefficient of resistance, the opposite of a metal's, and it is the basis of the thermistor. Second, and far more importantly, the carrier concentration is so small in the pure material — about \(10^{10}\ \text{cm}^{-3}\) against \(5\times10^{22}\) atoms — that adding impurities at the level of parts per million swamps it completely. Chapter 2 shows one dopant atom in five million changing the conductivity by a factor of six hundred thousand. That controllability, and not any intrinsic virtue, is what makes semiconductors the material of electronics.

6 Covalent Bonding in the Diamond Lattice

The band picture is complete but abstract, and for reasoning about doping and about broken bonds it helps to have a picture in real space alongside it. Silicon, germanium and carbon all crystallise in the diamond cubic structure: two interpenetrating face-centred cubic lattices offset by one quarter of the body diagonal, so that every atom sits at the centre of a regular tetrahedron formed by its four nearest neighbours, with bond angles of 109.47°.

The tetrahedral geometry follows from the bonding. An isolated silicon atom has configuration \(3s^2 3p^2\); in the crystal the one 3s and three 3p orbitals mix into four equivalent \(sp^3\) hybrid orbitals pointing to the corners of a tetrahedron, each holding one electron. Each hybrid overlaps with one on a neighbouring atom, and the two electrons pair up, spins opposed, in a shared bonding orbital. This is a covalent bond: the electrons belong to the pair of atoms jointly, not to either one, and there is no transfer of charge and hence no ionic component. Every silicon atom therefore has eight electrons around it — its own four, plus one shared from each of four neighbours — and a closed octet is a stable, chemically inert arrangement.

Figure 1.3 flattens the tetrahedron onto the page, drawing four bonds at right angles instead of at 109.47°. The geometry is wrong and the bookkeeping is right, which is the only thing the diagram is for. The correspondence with the band diagram is exact and worth stating explicitly: an electron in a covalent bond is an electron in the valence band, and an electron that has escaped its bond is an electron in the conduction band. The energy needed to break the bond is \(E_g\).

At 0 K every bond is intact, every valence state is filled, and the crystal is a perfect insulator despite containing \(2\times10^{23}\) valence electrons per cubic centimetre. Raise the temperature and the lattice vibrates; occasionally a vibration is violent enough to break a bond, and the freed electron wanders off through the crystal leaving an incomplete bond behind. That pair of events is the generation of an electron-hole pair, and everything in Chapter 2 is built on counting how many exist at a given temperature.

Two features of the diamond lattice have direct engineering consequences. It is an open structure — only 34 per cent of the volume is occupied by hard spheres, against 74 per cent for a close-packed metal — and that openness is why dopant atoms can be driven into interstitial sites and made to diffuse through the crystal at manageable temperatures, which is what every diffusion furnace in a fabrication line relies on. And the bonds are strong and highly directional: silicon melts at 1414 °C against germanium's 938 °C, so silicon tolerates the high-temperature oxidation and diffusion steps that germanium cannot survive.

7 Why Silicon Displaced Germanium

The first transistor, at Bell Labs in 1947, was germanium, and germanium remained the material of choice for a decade. Its band gap is smaller, so its carriers ionise more easily; its mobilities are higher, \(3900\ \text{cm}^2/\text{V}\!\cdot\!\text{s}\) for electrons against silicon's 1350; and it was easier to purify with the techniques then available. Silicon nevertheless displaced it almost completely by the mid-1960s, for three reasons that are worth separating because students routinely give only the first.

1. The oxide. This is the decisive one. Silicon oxidises in steam or dry oxygen to silicon dioxide, which is an excellent insulator with a 9 eV gap and a breakdown field near \(10^7\) V/cm, adheres perfectly to the silicon beneath it, is chemically stable, is easily patterned by photolithography and hydrofluoric acid, and blocks the diffusion of the common dopants. That combination makes the planar process possible: grow an oxide, open windows in it, diffuse dopant only through the windows, regrow, and repeat. Germanium's oxide, GeO\(_2\), is water-soluble, thermally unstable above about 800 °C and useless as a mask. No oxide means no planar process, no self-aligned gate and no integrated circuit. Germanium was not out-performed; it was made obsolete by a manufacturing process it could not support.

2. Leakage. A reverse-biased junction passes a current set by the minority-carrier concentration, which is proportional to \(n_i^2\). At 300 K germanium's intrinsic concentration is \(2.4\times10^{13}\ \text{cm}^{-3}\) against silicon's \(1.0\times10^{10}\ \text{cm}^{-3}\), a ratio of 2400, so \(n_i^2\) differs by a factor of \(5.8\times10^{6}\). A germanium diode leaks microamperes where an otherwise identical silicon diode leaks nanoamperes, and in a circuit with any appreciable resistance in it that difference decides whether an amplifier's bias point stays where you put it.

3. Temperature. The two effects compound. Because \(n_i\) rises with temperature roughly as \(T^{3/2}\exp(-E_g/2kT)\), the smaller gap makes germanium lose control sooner. In germanium doped to \(10^{16}\ \text{cm}^{-3}\), \(n_i\) reaches the doping level at about 191 °C; in silicon at the same doping it does not do so until about 375 °C. Well before that point the doping no longer determines the carrier concentration and the device stops behaving as a device at all. In practice germanium junctions are limited to roughly 75–100 °C and silicon to 150–200 °C, which is the difference between a part that survives an engine bay and one that does not.

Gallium arsenide sits alongside both as a specialist. Its 1.42 eV gap is direct, meaning the conduction-band minimum and valence-band maximum occur at the same crystal momentum, so an electron can recombine with a hole by emitting a photon without needing a lattice vibration to conserve momentum. Silicon's gap is indirect, which is why silicon cannot be made into an efficient light emitter and why every LED and laser diode in Chapter 14 is a compound semiconductor. GaAs also has an electron mobility around \(8500\ \text{cm}^2/\text{V}\!\cdot\!\text{s}\), six times silicon's, which suits it to microwave work. Against this it is brittle, expensive, has no usable native oxide and conducts heat poorly, so it stays in the niches where its two advantages are decisive.

8 Summary and Key Results

9 Common Mistakes

10 Chapter Review