Illustrated Physics
physics, drawn
Quantum

The mass a curve decides

An electron in a solid answers a force with a mass that is nothing to do with the mass of an electron. It is set by how sharply the band bends, it is smaller than the free value in a wide band and larger in a narrow one, and near the top of any band it is negative — which is why aluminium's Hall voltage has the sign of a positive carrier and no adjustment to an electron count can repair it.
19 min read 9 figures Where the model stopsThe same equation again

Assumes: What happens when the wells get close · Everything has a wavelength, and almost nothing shows it

A free electron has one mass and it is the same everywhere. Put the same electron in a crystal and ask what it does when a field pushes on it, and the answer contains a number that is not 9.109 × 10⁻³¹ kilograms. In gallium arsenide it is a fifteenth of that, and the electrons that carry the current there are the ones near the top of a filled sea. In some of the transition-metal oxides it is a hundred times larger. Near the top of any band at all it is negative, and the electron then accelerates toward the force that is pushing it away.

A band 3.60 eV wide, and the curvature is the whole story. Energy against wavenumber for one tight-binding band, α + 2β·cos(ka), with α = -4 eV, β = -0.9 eV and a repeat of 300 pm. The band is 3.60 eV from floor to ceiling and periodic in k, so the zone boundary at ka = π is not an edge of anything: it is where the curve turns over. Near the bottom it is a parabola, which is the free-electron dispersion with a different coefficient — a mass of 0.470 m_e rather than one — and near the top it is a parabola the other way up, a mass of -0.470. The inflection between them is at ka = π/2, and it is the point at which a constant force stops producing any acceleration.
Fig. 1 One band, from a chain of wells coupled to their neighbours. The energy is α + 2β·cos(ka) — not fitted, but the exact spectrum of the chain whose levels the rung below this one watched arrive — and its whole width is 4|β|, here 3.60 eV. The curve is a parabola at the bottom, a parabola upside down at the top, and straight in between. Everything this essay is about is in the second derivative of that shape, which is positive on the left, zero at ka = π/2 and negative on the right.

The quantity is not a correction and it is not a fitting parameter. It is a second derivative of a curve, it is measurable to four figures by cyclotron resonance, and it is what decides how fast a transistor switches. It is also, like every other mass in mechanics, defined by what a force does — which is the only definition that survives here.

What a force can and cannot do to a band electron

The rung below this one built the band and stopped there. A chain of N identical wells has N levels spread across a width of 4|β| whatever N is, and whether that band is full or half full is the difference between a wire and a window. What it did not ask is what happens when something pushes.

One well, two wells, and the band they become. The energy levels of a chain of identical wells, for 1, 2, 4, 8, 20, 60 of them, with an on-site energy of -4 eV and a coupling of -0.9 eV between neighbours. One well has one level. Two split it into two, 1.80 eV apart. By 60 the levels have filled a band 3.60 eV wide, which is closing on the limit of four times the coupling, 3.60 eV — and no further widening happens however many more wells are added. The count of levels grows with the number of wells; the width of the band does not.
Fig. 2 Where the curve above comes from. One well has one level; sixty have sixty, spread over 3.59 eV, closing on the 4|β| = 3.60 eV that the infinite chain has exactly. The count of levels tracks the count of wells and the width stops growing after the third one — so the band is a property of the coupling rather than of the size of the crystal, and the dispersion curve is what those levels look like when there are too many to draw.

An electron in a periodic potential is a Bloch wave, which is the same theorem that opens a gap: a plane wave of wavenumber k multiplied by something with the period of the lattice. Its velocity is not ħk/m. It is the group velocity of the wave, which is the slope of the band,

v(k)=1ℏdEdk,v(k) = \frac{1}{\hbar}\frac{\mathrm{d}E}{\mathrm{d}k},

and this is the same statement as a wave packet moving at the slope of its own dispersion relation rather than at the speed of its crests. Nothing about a lattice is needed to derive it.

What an external force does is change k, at a rate that has no lattice in it either:

ℏdkdt=F.\hbar\frac{\mathrm{d}k}{\mathrm{d}t} = F.

Put the two together and the acceleration follows by the chain rule:

dvdt=1ℏd2Edk2dkdt=1ℏ2d2Edk2 F.\frac{\mathrm{d}v}{\mathrm{d}t} = \frac{1}{\hbar}\frac{\mathrm{d}^2E}{\mathrm{d}k^2}\frac{\mathrm{d}k}{\mathrm{d}t} = \frac{1}{\hbar^2}\frac{\mathrm{d}^2E}{\mathrm{d}k^2}\,F.

Which is Newton’s second law with a coefficient in front of F, and the coefficient is a curvature. Defining

m∗=ℏ2(d2Edk2)−1m^* = \hbar^2\left(\frac{\mathrm{d}^2E}{\mathrm{d}k^2}\right)^{-1}

restores F=m∗aF = m^*a exactly. The electron has not changed. What has changed is that the force is competing with a lattice that is also pushing, and the whole of that competition has been folded into one number per point on the curve.

The number, across a band

0.47 m_e at the bottom, infinite in the middle, -0.47 at the top. Effective mass against wavenumber across one band, in units of the free electron mass, for a chain of period 300 pm and band width 3.60 eV. At the bottom the mass is 0.470 m_e — lighter than a free electron, from a band that binds it — and it grows without limit toward the inflection at ka = π/2, where the curve is straight and no force changes the velocity at all. Past that it is negative, reaching -0.470 m_e at the zone boundary. Nothing about the electron has changed: the mass is a property of the band it is in, and the sign is a statement about which way the lattice hands momentum back.
Fig. 3 The mass across the band above, in units of the free electron mass. It is 0.470 m_e at the bottom, grows without limit toward ka = π/2, and is −0.470 m_e at the zone boundary. The divergence is not a singularity in anything physical: at the inflection the band is locally straight, so a force changes k and changes no velocity at all, and an infinite mass is the correct description of something that will not accelerate. The generator computes the value at the band bottom twice — once by central difference on the plotted curve and once from the closed form — and refuses to draw unless they agree to a part in a million.

Three readings of that curve are worth having in hand.

A wide band means a light electron. The curvature at the bottom is 2|β|a², so m∗=ℏ2/2∣β∣a2m^* = \hbar^2/2|\beta|a^2 and a strongly coupled chain — big β, wide band — produces a small mass. That is the opposite of the intuition a heavier atom carries, and it is why the semiconductors with the widest bands are the fast ones.

A band 1.00 eV wide, and the curvature is the whole story. Energy against wavenumber for one tight-binding band, α + 2β·cos(ka), with α = -4 eV, β = -0.25 eV and a repeat of 500 pm. The band is 1.00 eV from floor to ceiling and periodic in k, so the zone boundary at ka = π is not an edge of anything: it is where the curve turns over. Near the bottom it is a parabola, which is the free-electron dispersion with a different coefficient — a mass of 0.610 m_e rather than one — and near the top it is a parabola the other way up, a mass of -0.610. The inflection between them is at ka = π/2, and it is the point at which a constant force stops producing any acceleration.
Fig. 4 The same construction for a chain that is coupled four times less strongly and spaced further apart: a band 1.00 eV wide rather than 3.60. The shape is identical because the shape is a cosine whatever the numbers are; only the vertical scale has changed. The parabola fitted at the bottom is therefore much flatter, and the mass correspondingly larger.
0.61 m_e at the bottom, infinite in the middle, -0.61 at the top. Effective mass against wavenumber across one band, in units of the free electron mass, for a chain of period 500 pm and band width 1.00 eV. At the bottom the mass is 0.610 m_e — lighter than a free electron, from a band that binds it — and it grows without limit toward the inflection at ka = π/2, where the curve is straight and no force changes the velocity at all. Past that it is negative, reaching -0.610 m_e at the zone boundary. Nothing about the electron has changed: the mass is a property of the band it is in, and the sign is a statement about which way the lattice hands momentum back.
Fig. 5 And the mass that goes with it: 0.610 m_e at the band bottom against 0.470 for the wide band, from the same expression with different arguments. A narrow band is a flat band, a flat band has little curvature, and little curvature is a large mass — the same statement as a group velocity read off a slope and a spreading read off a curvature, with the packet made of electrons. Taken to its limit this is a bound electron, which has no band at all and does not move — an infinite mass, which is what a completely flat band gives.

The relation is exact rather than a tendency, and it runs the wrong way round twice: a bigger coupling means a wider band, a wider band means a sharper curve, and a sharper curve means a lighter carrier. So the material in which electrons are most tightly held to their atoms is the one whose carriers are heaviest, and the flattest band of all — no coupling, no band — is an electron that does not move at all.

The mass is a local property, not a property of the material. It depends on where in the band the electron sits, so quoting “the effective mass of silicon” always means the mass near the band edge, which is where the carriers that matter actually are. Away from the edge the parabolic approximation fails and there is no single number.

And past the inflection the sign flips. This is not a mathematical curiosity. The upper half of every band has negative curvature, so an electron there answers a push by accelerating the other way.

The mass is only the first term

A curvature is a second derivative at a point, so calling m∗m^* the mass of a carrier is already an expansion truncated after two terms. For the cosine band the truncation can be written down exactly rather than estimated. With E=α+2βcos⁡kaE = \alpha + 2\beta\cos ka the second derivative is −2βa2cos⁡ka-2\beta a^2\cos ka, so

m∗(k)=m∗(0)cos⁡ka,m^*(k) = \frac{m^*(0)}{\cos ka},

and the mass of a carrier is its band-bottom mass divided by the cosine of how far up the band it has climbed. That is a secant, and a secant leaves its own value fast. A sixth of the way to the zone boundary the mass is 1.155 times the edge value, a quarter of the way it is 1.414 times, a third of the way it is exactly twice. In energy those three points sit 0.241 eV, 0.527 eV and 0.90 eV above the bottom of a 3.60 eV band — so a carrier seven per cent of a band width above the edge already answers a force with a mass fifteen per cent wrong, if the edge value is the one being quoted.

The reason the number is quoted anyway is thermal. Room temperature is 26 meV, which on this band puts cos⁡ka=0.9856\cos ka = 0.9856 and the mass 1.5 per cent above its edge value — inside the error of most things one would do with it. The parabolic band is not an idealisation that holds because bands are nearly parabolic. It holds because the carriers that matter are crowded into the bottom few per cent of one, and it fails precisely where a device drives them out of that region, which is what a high field in a short channel does.

At the inflection the expansion has no second term at all, and the honest description there is a first-order one: E≈Ei+ℏvi(k−ki)E \approx E_i + \hbar v_i (k - k_i), a straight line, a fixed speed and no mass in the expression anywhere. An infinite mass and a massless dispersion are the same statement seen from two sides — a carrier that will not change its velocity in answer to a force. Graphene’s carriers live at a point where two bands meet linearly and are described that way throughout, with a velocity of about 10⁶ m/s standing where a mass would be, and the cyclotron mass measured in one is a derived quantity rather than a coefficient in the dispersion relation.

The orbit a steady force produces

The sign flip has a consequence that is easier to draw than to argue about: under a constant force, a band electron does not run away. It oscillates.

A constant force, and the electron turns round after 0.69 ps. Position of a band electron under a steady field of 10.0 MV/m, against time. The force is constant and never reverses; the motion does, because the force acts on k rather than on v, and k sweeps steadily through a zone that is periodic. The electron accelerates while the mass is positive, coasts through the inflection, and is decelerated by the same force through the upper half of the band. The period is 1.379 ps and the amplitude 180.0 nm, which is the band width divided by twice the force — read off the integrated trajectory here and equal to W/2eE to a part in a thousand. Nobody sees this in an ordinary crystal: a collision every 10 fs interrupts it 138 times over before one orbit is complete, which is why a metal has a resistance rather than an oscillation.
Fig. 6 Position against time for an electron in the 3.60 eV band under a steady 10 MV/m, integrated from ħ dk/dt = −eE and v = (1/ħ)dE/dk. The field is constant and points one way throughout. The motion is a closed orbit of amplitude 180 nm and period 1.379 ps, because the force acts on k rather than on v and the band is periodic in k: sweeping k once across the zone takes the electron through the whole cycle of accelerating, coasting and decelerating. The amplitude is the band width divided by twice the force, W/2eE, and the generator reads it off the trajectory and compares it against that expression before drawing.

That is the honest answer to “what does a field do to an electron in a crystal”, and it is not what a metal does. The dashed line on the figure is why.

The orbit never happens. An electron in copper is scattered — by a lattice vibration, by an impurity, by a boundary — about every ten femtoseconds. The Bloch period at ten megavolts per metre is 1.379 picoseconds, which is a hundred and thirty-eight collisions later. The electron is knocked back to the bottom of the band long before it discovers that the band has a top, and what survives of the motion is a small steady drift: Ohm’s law, and a resistance in place of an oscillation. Conduction in a metal is a failure of this figure, and the failure is total.

A constant force, and the electron turns round after 0.21 ps. Position of a band electron under a steady field of 1.0 MV/m, against time. The force is constant and never reverses; the motion does, because the force acts on k rather than on v, and k sweeps steadily through a zone that is periodic. The electron accelerates while the mass is positive, coasts through the inflection, and is decelerated by the same force through the upper half of the band. The period is 0.414 ps and the amplitude 120.0 nm, which is the band width divided by twice the force — read off the integrated trajectory here and equal to W/2eE to a part in a thousand. Nobody sees this in an ordinary crystal: a collision every 200 fs interrupts it 2 times over before one orbit is complete, which is why a metal has a resistance rather than an oscillation.
Fig. 7 The same calculation for a repeat a hundred times longer and a band sixty times narrower — the arrangement a semiconductor superlattice has, where the period is a stack of grown layers rather than a row of atoms. The Bloch period falls to 0.414 ps because the period is inversely proportional to the lattice spacing, and the mean free time rises because the layers are pure. Now the orbit fits inside the collision time, and the oscillation is a real signal at a few terahertz. Nothing about the physics changed; the two timescales swapped order.

The escape from the trap is to change the ratio rather than the argument. Make the repeat long enough and the Bloch period short enough, and the oscillation stops being hypothetical — which is what a superlattice does, and why terahertz emission from one is the direct evidence that the picture above is not a story.

How the number is actually measured

A second derivative of a curve nobody can see sounds like a quantity fitted to whatever it has to explain. It is not, and the measurement that pins it is a resonance.

Put a carrier of charge ee and mass m∗m^* in a magnetic field and it goes round a circle at ωc=eB/m∗\omega_c = eB/m^*, a frequency that contains the mass and nothing else — no density, no scattering time, no band width. Shine radiation on the sample and sweep either the frequency or the field, and absorption peaks where the two agree. For a free electron the constant is 28.0 GHz per tesla; gallium arsenide’s conduction band, at 0.067 me0.067\,m_e, resonates at 418 GHz in the same one-tesla field, and the factor of fifteen between those numbers is read straight off a spectrum. Four significant figures are routine.

There is a condition attached and it is the same condition the oscillation above failed. The carrier has to complete an orbit before it is scattered, ωcτ>1\omega_c\tau > 1, which at a tesla means a mean free time longer than about a picosecond. In copper at room temperature it is ten femtoseconds, a hundred times short, and no resonance appears at all — cyclotron resonance is a measurement of pure crystals at low temperature, and the reason a metal at the bench does not show one is exactly the reason the Bloch orbit above does not happen. Two experiments, one inequality.

That is worth holding beside the alternative. A mass inferred by fitting a conductivity would have the scattering time in it and could absorb any error into that; a resonance has one parameter and either lands on it or does not. Which is why the sentence “the effective mass of gallium arsenide is 0.067” is a measurement rather than a convention, and why a fudge factor is the wrong description of it.

The measurement that will not fit an electron

The negative-mass region is easy to write down and easy to disbelieve. The evidence that a solid really behaves as though it contained positive carriers arrives from a measurement that has no theory in it at all.

Run a current through a strip, put a magnetic field across it, and the magnetic force pushes the carriers sideways until the charge that piles up at the edge balances it. The transverse voltage that results has a sign, and the sign says whether the things carrying the current are positive or negative. It is about as direct as a measurement gets.

3 of 7 metals have the Hall coefficient of a positive carrier. Measured Hall coefficient beside the free-electron prediction −1/ne, computed from each metal's own number density and valence, in units of 10⁻¹⁰ m³/C. sodium: -2.30 against -2.36; potassium: -4.20 against -4.46; silver: -0.90 against -1.07; copper: -0.53 against -0.74 — the right sign and the right order, which is the free-electron model working. And then aluminium: 1.02; indium: 1.60; beryllium: 2.44, where the sign itself is wrong and no adjustment to a density can fix it, because a density cannot be negative. Aluminium's value is one positive carrier per atom to within 1.5 per cent — 1.04 predicted against 1.02 measured — which is what a band with one electron missing per atom looks like from outside.
Fig. 8 Measured Hall coefficients beside the free-electron prediction −1/ne, computed from each metal’s own number density and valence. Sodium, potassium, silver and copper give the right sign and the right order of magnitude — the free-electron model doing its job. Aluminium, indium and beryllium give the opposite sign. No electron density reproduces a positive coefficient, because a density cannot be negative; and aluminium’s +1.02 × 10⁻¹⁰ m³/C is what exactly one positive carrier per atom would give, 1.04 × 10⁻¹⁰ computed from its own atomic density, agreeing to 1.5 per cent.

The resolution is bookkeeping, and it is exact rather than approximate. A completely full band carries no current: for every electron at +k there is one at −k, the velocities cancel, and nothing a field does can change that, because there is no empty state to move into. So the current of a nearly full band is the current the full band would carry — zero — minus the current of the missing electrons. Subtracting the contribution of a negative charge with a negative mass gives, term for term, the contribution of a positive charge with a positive mass. The hole is not a particle that was hiding there; it is the arithmetic of an absence, and it is right about the Hall voltage because the arithmetic is right.

Worth doing the subtraction slowly, because every sign in it matters. An electron near the top of a band has charge −e-e and, by the curvature above, a negative mass; its contribution to a current under a field is proportional to (−e)2/m∗(-e)^2/m^* with m∗m^* negative, which is negative. Removing it from a full band that carries nothing therefore leaves a positive contribution — the same one a particle of charge +e+e and mass ∣m∗∣|m^*| would make. Two sign flips, charge and curvature, and they compose rather than cancel. Nothing in the crystal is positively charged and nothing in it has acquired a positive mass; what has happened is that a sum over 102310^{23} states minus one state has been rewritten as one term, and the term looks like a particle.

Aluminium has three valence electrons per atom. Two of them fill a band; the third goes into a band that ends up almost full, and what is left is close to one vacancy per atom. The number the Hall probe reads is that count.

What the picture cannot show, and what it costs

The whole construction rests on one approximation stated nowhere in the algebra: that the force from outside is slow and weak compared with the lattice’s own scales. Three things it buys, and their price.

The same theorem holds in a medium with no electrons in it at all. A stack of alternating layers has a forbidden band for exactly the reason a crystal does — the Bloch condition has no real solution there — and a wave packet near a band edge in such a stack has an effective mass in the identical sense. That is the strongest evidence that the quantity is a property of a dispersion relation rather than of anything material.

It costs the acceleration theorem its exactness. ħ dk/dt = F holds for the force from an external field and not for the forces from the lattice, which have been absorbed into the band. Push hard enough and the electron does not stay in its band: it is lifted across the gap, the single-band description fails, and what happens instead is Zener breakdown.

It costs the mass its constancy. m∗m^* is a scalar only in one dimension. In a real crystal the second derivative is a tensor with three principal values that need not be equal or even of the same sign, and the “mass” of an electron in silicon depends on which way it is being pushed.

Silicon is the worked case for both of those. Its conduction band minimum is not at the zone centre but on six equivalent axes, and the constant-energy surface round each is a cigar rather than a sphere: the mass along the axis is 0.98 me0.98\,m_e and across it 0.19 me0.19\,m_e, a factor of five depending only on which way the field points. The single number 0.26 that gets quoted is neither of those — it is the combination that reproduces the density of states, and a different combination again is the one that belongs in a mobility. Three numbers, all called the effective mass of silicon, all correct for the question they answer and none interchangeable with the others. The valence band is worse still, carrying two bands degenerate at the top with different curvatures, so that silicon has a light hole and a heavy hole at the same energy and a field accelerates both.

And it hides where the momentum went. ħk is called crystal momentum and it is not momentum: it is not conserved, the lattice takes and gives it in units of a reciprocal lattice vector, and that is exactly what allows the electron to turn round at the zone boundary without anything visible pushing it. The negative mass is the ledger entry for a transfer the picture does not draw.

Aluminium’s conduction electrons fill states up to a Fermi energy of 11.6 eV, and room temperature smears the edge by about 26 meV — a quarter of a per cent of the depth. That is why only the states near the top take part in anything, and why the effective mass that matters in a metal is the one evaluated at the Fermi surface rather than at the band minimum. The band’s curvature varies across it, and the electrons sample only one place.

Where else the same sentence is true

The argument used nothing about electrons, charges or crystals. It used a periodic medium, a dispersion relation and a slowly applied force, and it concluded that the response is governed by the curvature of the dispersion relation.

The same picture three times, with the gap changed. Valence and conduction bands for copper, germanium, silicon, gallium arsenide, with the gap between them drawn to scale in electronvolts and the fraction of electrons thermally promoted across it at room temperature printed underneath: copper, gap 0 eV, no barrier at all; germanium, gap 0.67 eV, 2.4e-6; silicon, gap 1.12 eV, 3.9e-10; gallium arsenide, gap 1.42 eV, 1.2e-12. Nothing about the three drawings differs except the height of one white band, and that one number is the difference between a wire, a transistor and a window.
Fig. 9 Four materials by their band gaps, which is the quantity the rung below this one ended on. The gap decides whether there are carriers; the curvature decides what those carriers do once they exist. Gallium arsenide has both a wider gap than silicon and a conduction-band mass of 0.067 m_e against silicon’s 0.26 — a factor of four in mobility from the shape of a curve, and the reason a gallium arsenide device switches faster than a silicon one made to the same dimensions.

So the same sentence is true of a phonon in a lattice, of light in a photonic crystal, and of a cold atom in an optical lattice — where the Bloch oscillation that no metal will show has been watched directly, because the “lattice” is made of light, the “electron” is an atom, and nothing scatters. The apparatus that made the prediction observable was not a better crystal; it was a system with no impurities in it at all.

Each allowed band has a bottom where the curvature is positive, an inflection where it vanishes and a top where it is negative — the same three regions in every band of every periodic structure. So the effective mass runs from positive through infinite to negative within a single band, and none of those is a statement about anything getting heavier: it is a curvature changing sign, read as a mass because that is the letter it occupies in the equation of motion.

The rung after this one

What this essay has not touched is what happens when there are many carriers rather than one, and they push on each other. The mass measured by cyclotron resonance in a real metal is not the band mass: it is a few per cent to a few tens of per cent larger, because moving one electron means rearranging its neighbours, and the thing that responds to the force is the electron together with the disturbance it drags. The next rung on this ladder is that dressing — where the extra mass comes from, how large it is, and why a picture that treats a solid as a set of independent one-electron states works as well as it does when nothing in it is independent.

Everything above concerns one electron in one band. A metal has 102310^{23} of them stacked into the available states, and what a cyclotron resonance measures is an average over the Fermi surface rather than a single curvature — which is where the next rung begins, and why the tabulated effective mass of a real metal is a tensor with several components rather than a number.

Part 2 of 4

This essay is one argument about Bands. The others:

One well, two wells, and the band they become. The energy levels of a chain of identical wells, for 1, 2, 3, 6, 12, 40 of them, with an on-site energy of -4 eV and a coupling of -0.9 eV between neighbours. One well has one level. Two split it into two, 1.80 eV apart. By 40 the levels have filled a band 3.59 eV wide, which is closing on the limit of four times the coupling, 3.60 eV — and no further widening happens however many more wells are added. The count of levels grows with the number of wells; the width of the band does not. What happens when the wells get close Part 1 — Two atoms brought together split one level into two. A thousand split it into a thousand, packed into a band whose width stops growing after the third. Whether that band is full or half full is the whole difference between a wire and a window. Two levels that refuse to cross. On the left, the energies of a two-state system as one state is swept past the other, for couplings of 0, 0.05, 0.15, 0.35 electronvolts. With no coupling at all the two levels cross, which is the dotted pair. With any coupling whatever they do not: the eigenvalues are plus and minus half the square root of the squared detuning plus four times the squared coupling, so the closest approach is exactly twice the coupling — measured off the drawn curves as 0.000, 0.100, 0.300, 0.700 eV against 2V, agreeing to 0.0e+0. Far from the crossing the curves rejoin the uncoupled lines, so the repulsion is local: it is largest where the two states are degenerate and dies away as the square of the coupling divided by the detuning. On the right is what makes this more than a picture of two hyperbolas — the character of the upper state, meaning how much of the first basis state is in it. It swaps completely across a region whose width is set by the coupling, so the level that arrives as one thing leaves as the other. That exchange of identity is why the phrase avoided crossing is misleading: nothing is avoided, the labels are. The crossing that never happens Part 3 — Two energy levels swept past one another do not cross. Any coupling between them, however small, opens a gap of exactly twice the coupling — and the two levels exchange their identities across it, so the state that arrives as one thing leaves as the other while the labels are what avoided anything.

Where this goes next

One well, two wells, and the band they become. The energy levels of a chain of identical wells, for 1, 2, 3, 6, 12, 40 of them, with an on-site energy of -4 eV and a coupling of -0.9 eV between neighbours. One well has one level. Two split it into two, 1.80 eV apart. By 40 the levels have filled a band 3.59 eV wide, which is closing on the limit of four times the coupling, 3.60 eV — and no further widening happens however many more wells are added. The count of levels grows with the number of wells; the width of the band does not. What happens when the wells get close Two atoms brought together split one level into two. A thousand split it into a thousand, packed into a band whose width stops growing after the third. Whether that band is full or half full is the whole difference between a wire and a window. How much of copper's electron sea a temperature can reach. The probability that a state of a given energy is occupied, in copper, at 4 temperatures, with energy measured in units of the Fermi energy — 7.04 eV here. At absolute zero the curve is a step: every state below the ceiling is full and every state above it is empty. Raising the temperature rounds the step, and rounds it over a range of about kT, which is the whole point — at room temperature kT is 0.0259 eV against a ceiling of 7.04 eV, so the rounding is 1.6 per cent of the way down the sea and everything deeper is untouched. An electron in the deep is not held there by a force; it simply has nowhere to go, because every state it could be promoted to is occupied. at 0 K the step is spread over 0.00 per cent of E_F, at 300 K the step is spread over 1.61 per cent of E_F, at 3000 K the step is spread over 16.13 per cent of E_F, at 20000 K the step is spread over 107.52 per cent of E_F. The pressure that is not a temperature Copper's conduction electrons are at a temperature of eighty thousand kelvin, in a wire that is at room temperature. That is not a figure of speech, it is what the exclusion principle does to a mole of particles, and it explains the largest unexplained number in the theory of metals. The frequencies a repeat will not carry. The band structure of a medium made of quarter-wave layers of index 1 and 2, repeated for ever: frequency against Bloch phase across one cell, in the reduced zone. Inside a band the phase runs from 0 to π and the wave travels. Between bands there is no real phase at all, and the shaded strips are frequencies at which the medium supports nothing — not a weakly transmitted wave, no wave. Gap 1 runs from 0.784 to 1.216; Gap 2 runs from 2.784 to 3.216, in units of the quarter-wave design frequency. The first, measured off the drawn band edges, is 0.4327 wide against the 0.4327 of (4/π)·arcsin|r| — the same number computed from the Fresnel ratio of the two indices alone, agreeing to 2.6e-14 per cent. Every band edge sits where the phase is 0 or π, which is to say where the wave's own period fits the repeat a whole number of times: the gap is a property of the periodicity, and the materials only decide how wide it is. The gap a repeat opens Stack two transparent materials in alternating layers and there is a band of frequencies the stack will not carry — not weakly, not with loss, but not at all. Nothing has been absorbed and neither material has a resonance there. What forbids those frequencies is the repeat itself, and the width of the band has a closed form containing only the ratio of the two indices.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Band gapBloch waveConductivityCurvatureDispersion relationEffective massGroup velocityHall effectHoleMomentumPeriodic mediaTaylor expansion