The push that makes an electron swing

What happens when the wells get close builds a band out of atoms. Bring identical atoms together into a lattice and each atomic level spreads into a band of energies, of width set by how easily an electron hops from one atom to the next. In the simplest model, where the electron hops only between neighbours with an energy $J$, the band is $E(k) = -2J\cos ka$, a cosine of the crystal momentum $k$ with the lattice spacing $a$, running from $-2J$ at the bottom to $+2J$ at the top. The mass a curve decides then shows that an electron in such a band responds to a force with an effective mass set by the band’s curvature: light near the bottom, infinite at the inflection, negative near the top.

That last fact has a consequence that is easy to state and was hard to see. A constant force pushes the electron’s crystal momentum steadily up. Its velocity, which is the slope of the band, rises, peaks at the inflection and falls back to zero at the zone edge, where the crystal momentum folds round to the opposite edge. The velocity then goes negative. An electron pushed steadily in one direction moves forward, stops and comes back, and the cycle repeats. It oscillates rather than accelerating.

Felix Bloch saw this in his 1928 thesis, which founded the theory of electrons in crystals, and Clarence Zener discussed it in 1934. Then it remained a theoretical curiosity for sixty years, and often an embarrassment in textbooks, which had to explain why a property implied by the theory of every metal was never seen in any of them.

A force, a momentum and a velocity #

The equation of motion for crystal momentum is the same as Newton’s for ordinary momentum: $\hbar\,dk/dt = F$. Applied to an electron in a band, it says that a constant force makes $k$ grow linearly with time. What it does not say is that the velocity grows too.

The crystal momentum increases at a steady rate, and each time it reaches the edge of the Brillouin zone, $+\pi/a$, it reappears at $-\pi/a$, because the two labels describe the same state. The velocity is $(2Ja/\hbar)\sin ka$, which is positive for half of each cycle and negative for the other half. Integrated, it gives a position that rises from zero to $4J/F$ and returns, with period

$$T_B = \frac{h}{Fa}.$$

For a force of a quarter of the hopping energy per lattice spacing, the swing spans sixteen spacings. A free particle with the effective mass of the band’s bottom, pushed by the same force, would follow a parabola, drawn dashed, and the two agree only for the first part of a period, while the electron is still near the bottom of the band where the effective-mass picture holds.

The momentum the force supplies does not vanish. The lattice takes it. Each time the electron’s wave reaches the zone edge, its wavelength matches twice the lattice spacing and it is Bragg-reflected by the planes of atoms, which absorb a quantum of momentum $2\pi\hbar/a$ and reverse the electron’s direction. A swinging electron is continually transferring the force’s impulse to the crystal as a whole. Nothing stops the electron; something turns it round.

Why a full band carries nothing #

The same equation explains the oldest fact in solid-state physics: why some crystals conduct and others do not. A force moves every electron’s crystal momentum by the same amount, so in a band it slides the whole occupied set of states round the Brillouin zone together. If the band is completely full, sliding it changes nothing: every state is still occupied, the electrons at the zone edge reappear at the opposite edge, and for every electron moving forward there is one moving back at the same speed. A full band carries no current in any field. Every one of its electrons is executing a Bloch oscillation, and the oscillations cancel.

That is why an insulator insulates. Its electrons are not held in place; they are in bands that are entirely full, with an energy gap to the next one, and no force short of one strong enough to drive them across the gap can make their momentum distribution lopsided. A metal conducts because its highest band is only partly filled, so a force can shift the occupied states slightly, leaving more electrons moving with it than against it, and scattering then holds that small shift steady against the force. A semiconductor sits between: nearly full and nearly empty bands, into which a dopant that is a hydrogen atom inside a crystal puts a few electrons or removes a few, and whose conductance is set by how many. Even the empty states near the top of a nearly full band behave as positive charges, holes, because an electron there has the negative effective mass of the band’s top and accelerates backwards.

The rule that a full set of states carries no net current reappears in the resistance counted in whole numbers, where electrons fill Landau levels in a magnetic field rather than Bloch bands, and the filled levels carry a current only along the sample’s edges, quantised in units of $e^2/h$. And it is the band-theory version of an argument the collisions the exclusion principle forbids makes about scattering: in a full set of states nothing can change, because there is nowhere for anything to go. The swing in the figure above is the motion of one electron alone in an empty band. Put it among others that fill the band and the swings are still there, but they add up to nothing.

Why no metal shows it #

The Bloch period depends on the force and the lattice spacing and on nothing else. For an electron in an electric field $E$ the force is $eE$, and the period is $h/eEa$.

In an ordinary crystal with a lattice spacing of three-tenths of a nanometre, a field of a hundred thousand volts a metre — already strong, a hundred volts across a millimetre — gives a Bloch period of 138 picoseconds. The electron is scattered by phonons, impurities and defects about every ten femtoseconds at room temperature, ten thousand times sooner. Each scattering event randomises its crystal momentum, usually back near the bottom of the band, and the electron never gets more than a small fraction of the way to the zone edge. What survives, averaged over many short partial swings, is the steady drift of Ohm’s law. Raising the field to shorten the period runs into the breakdown of the material long before the period approaches the scattering time.

The way out was to change the lattice. A semiconductor superlattice is a stack of alternating thin layers of two semiconductors, gallium arsenide and aluminium gallium arsenide, a few nanometres each, grown one atomic layer at a time. Electrons moving perpendicular to the layers see a periodic potential with a period of ten nanometres or so, thirty times the natural lattice spacing, and the superlattice’s own narrow bands, called minibands, have that period. The Bloch period is thirty times shorter for the same field; at a million volts a metre it is four-tenths of a picosecond, comparable with or shorter than the scattering time in clean samples at low temperature. Esaki and Tsu proposed exactly this in 1970. In 1992 Jochen Feldmann, Karl Leo and colleagues saw the oscillation in the coherent beating of optically excited electrons, and in 1993 Christian Waschke and colleagues detected the terahertz radiation that electrons swinging in unison emit, at the frequency the theory predicted.

A current that falls as the field rises #

Even when the scattering time is shorter than the Bloch period, the oscillation leaves a mark on the steady current, and it was this that Esaki and Tsu were after.

Suppose each scattering event resets an electron to the bottom of the band, and the events happen randomly at rate $1/\tau$. Between events the electron follows its Bloch orbit. Its average velocity is then the velocity along the orbit averaged over the distribution of times since the last reset, which gives

$$v = v_{max}\,\frac{\Omega\tau}{1 + (\Omega\tau)^2}, \qquad \Omega = \frac{eEa}{\hbar}.$$

For weak fields, where $\Omega\tau$ is small, the electron is always reset long before it reaches the zone edge, its velocity is proportional to the field, and the result is Ohm’s law with the band-bottom effective mass. As the field rises, a growing fraction of electrons get past the inflection of the band before they are reset, and some get round to the return half of their swing. The drift peaks at $\Omega\tau = 1$, at half the band’s maximum velocity, and then falls. Beyond the peak, more field means less current.

That is a negative differential resistance, and it matters in practice because a circuit element with negative differential resistance can amplify and oscillate: the Gunn diode, which uses a different mechanism with the same shape, is the microwave oscillator of radar speed guns. Superlattice devices built on Esaki and Tsu’s mechanism have produced oscillation at hundreds of gigahertz. The shape of the curve is also a measurement: where it peaks gives the scattering time.

The evenly spaced levels a tilted band becomes #

The oscillation has an equivalent description in terms of energy levels, which is how the optical experiments first saw it.

A constant force adds a potential energy that rises steadily across the crystal, so the band, drawn against position, tilts. An electron with a given energy can then exist only where that energy lies inside the tilted band: a strip of the crystal of width $4J/F$, the same as the swing. The stationary states are confined to such strips. Each one is centred on some lattice site $m$, its shape across the neighbouring sites is the square of a Bessel function, and its energy is exactly $mFa$. The continuous band of travelling states has become an evenly spaced set of localised levels, named after Gregory Wannier, who predicted it in 1960, and Johannes Stark, after the analogous splitting of atomic levels in a field.

The levels and the oscillation are the same thing. Energy levels evenly spaced by $Fa$ mean that any superposition of them repeats itself after a time $h/Fa$, which is the Bloch period. In a superlattice, light absorbed across the band gap sees those levels as a series of absorption lines whose spacing grows in proportion to the field — a direct, spectroscopic picture, which was observed in 1988, four years before the oscillation in time. A single quantum fact appears as a swing in time and an evenly spaced row of levels in energy, exactly as a pendulum’s period appears as the spacing of its quantum levels.

Swinging or breathing #

The textbook oscillation is that of a wave packet with a definite crystal momentum, spread over many lattice sites. A packet prepared differently does something different.

An electron placed on a single site has no definite crystal momentum: it is a superposition of every state in the band, half moving forward and half moving back. Under a constant force all of them move round the Brillouin zone together, and the packet’s halves move out in both directions, stop, and return. The centre of the packet never moves at all; instead the packet breathes, its spread opening to $(4J/F)/\sqrt 2$ lattice spacings at half a period and closing again on the starting site. Any intermediate preparation does something intermediate. What every preparation shares is the period, because the period belongs to the spacing of the energy levels, not to any particular state. The same breathing appears in light, where the light that spreads in two beams shows a row of coupled waveguides with a steadily increasing propagation constant across it: light launched into one guide spreads outward and refocuses back into it, the optical Bloch oscillation, seen in a photograph of the light from above.

Lattices made of light #

The cleanest observations came from atoms. In 1996 Maxime Ben Dahan, Christophe Salomon and colleagues in Paris loaded laser-cooled caesium atoms into an optical lattice — the standing wave of two counter-propagating laser beams, whose intensity pattern traps atoms in a periodic potential with a period of half the light’s wavelength — and accelerated the lattice by sweeping the frequency of one beam. In the lattice’s frame the atoms felt a constant inertial force, and their momentum distribution, measured by switching off the lattice and photographing the atoms’ flight, showed the crystal momentum sweeping across the zone and being Bragg-reflected at its edge, period after period. Nothing scatters an atom in an optical lattice; the oscillations lasted as long as the atoms could be held.

The force can just as well be gravity. In a vertical optical lattice the force on each atom is $mg$, and the Bloch frequency $mga/h$ is a measurement of $g$. For strontium-88 atoms in a lattice of 532-nanometre light, with a spacing of 266 nanometres, it is 574 hertz, and since the lattice spacing is known from the laser’s wavelength, measuring that frequency to a part in a million measures the local acceleration of gravity to the same precision, over a height of a few micrometres. Such atom gravimeters have been used to test gravity at short distances and to compare the free fall of different isotopes, a quantum version of the comparisons in the fall that does not depend on what is falling.

When the force is strong enough to jump the gap #

The tight-binding picture keeps the electron in one band. A real crystal has many bands separated by gaps, and a strong enough force can drive an electron from one band to the next when it reaches the zone edge, instead of reflecting it. That is Zener tunnelling, and it is a case of the crossing that never happens: the two bands at the zone edge are two levels with a gap between them, and the force sweeps the electron through the near-crossing at a rate proportional to the field. Slowly, it follows its own band and is Bragg-reflected; quickly, it jumps across, with a probability that falls exponentially with the square of the gap divided by the force. In a strong enough field the Bloch oscillation leaks away into higher bands. In Zener diodes, which regulate voltages in power supplies with a junction whose bent bands put an intense field across a few nanometres, that leakage across a semiconductor’s gap is the whole point, and it arises from the same Bragg physics that makes the oscillation possible.

One cosine band and a perfectly even force #

The figures use the simplest band there is, a single cosine from nearest-neighbour hopping, and a force that is perfectly uniform; real bands have more structure, and real fields in a semiconductor are screened by the electrons themselves and vary across a device. The scattering in the drift calculation is the Esaki–Tsu model, in which every collision resets an electron to the bottom of the band at a fixed average rate; real scattering depends on energy and does not always reset the momentum fully. The periods and fields quoted are representative; scattering times in superlattices depend strongly on temperature and on the quality of the interfaces. The Wannier–Stark states are those of an infinite lattice; a finite one adds edge states and modifies the levels near its ends. Interband tunnelling is described only qualitatively. Interactions between electrons, which in a superlattice can drive the electric field into domains rather than letting it stay uniform, are ignored.

Still open: a Bloch oscillator worth building #

A device whose output frequency is set by the applied voltage and the superlattice period, tunable across the terahertz range, has been the promise of Bloch oscillations since 1970, because the terahertz band between microwaves and the infrared has long lacked compact, bright, tunable sources. The obstacle is the same negative differential resistance that makes oscillation possible: it also makes the electric field inside the superlattice unstable, so that it breaks into domains of high and low field that travel through the device, and the clean oscillation is replaced by a domain instability at a lower frequency. Whether that instability can be suppressed well enough for a superlattice to become a practical terahertz source, or whether quantum cascade lasers, which use the same layered materials differently, have already made the question moot, is not settled.

The habit worth carrying away is to ask which momentum a force changes. A steady force on an electron in a crystal increases its crystal momentum steadily, but its velocity is the slope of a band that turns over — so the electron swings over 4J/F and returns after h/Fa, the lattice taking the impulse at every Bragg reflection, and in a metal a scattering every ten femtoseconds hides a swing that would take a hundred and forty picoseconds. Bloch’s oscillation was always there; it took a crystal thirty times coarser, or one made of light, to give it time.