Quantum Tunneling
Through Walls Too High to Climb
A Wall It Cannot Climb
Roll a ball at a hill too high for it, and it rolls back. Every time: to reach the top it would need more energy than it has, and energy does not come from nowhere. A particle as light as an electron does something else. Now and then it turns up on the far side of a hill it could not climb, with the same energy it came with. This is tunneling, one of the first strange things quantum physics explained, and it lights the Sun, sets how long uranium lasts and writes the photos into your phone’s memory.
The reason is that a particle travels as a wave, and the square of the wave’s size at each place gives the chance of finding the particle there. A wave does not stop dead at the edge of a wall: that would take a corner of infinite sharpness, and only a wall of infinite height bends a particle’s wave that hard. A real wall bends it less, so the wave reaches into it and dies away. Where the wall’s height stays the same, it loses the same share of its size at every step deeper. If the wall is thin, some of the wave is still left at its far side, and that part goes on.
It goes on smaller, but not weaker. A particle’s energy of motion is set by how closely the crests of its wave are packed, and beyond the wall they are packed just as before. The smaller wave only makes the particle less likely to be found there.
Looked for, the particle is always found whole, on one side or the other, never in pieces. The chance of each side is the share of the wave that went there, counted by its size squared. In the scene the wave that gets through carries about a sixth of the chance, so about one try in six ends beyond the wall.
Any wave does this. Light striking the inside face of a glass block at a slant is turned back entirely, yet if a second block is held less than a wavelength away, some of it leaks across the gap of air into it, more the thinner the gap. What quantum physics adds is that matter travels as a wave too, and that whatever gets through, light included, is caught whole, one particle at a time.
The particle does not borrow energy to get over and pay it back later: its wave keeps the same energy before the wall, inside it and after. Could it be caught inside the wall? A detector placed there can find it. But to catch a particle in so thin a place, the detector has to squeeze its wave into that place, and a wave squeezed that tight carries about as much extra energy as the particle lacked.
Every Atom’s Width Counts
Inside a wall of even height the wave shrinks by the same share at every step. So the chance of getting through shrinks by about the same factor for every step of thickness, and the factors multiply. A wall twice as thick does not halve the chance; it roughly squares it. If one wall lets through one particle in a hundred, a wall twice as thick lets through only one in some tens of thousands.
The scanning tunneling microscope is built on that steepness. A metal needle, its point ending in a single atom, is held over a metal surface a few atoms’ widths away, with a small voltage between the two. Electrons in a metal lack the energy to leave it, so the empty gap is a wall to them. Yet a current flows: electrons tunnel across. Widen the gap by a tenth of a nanometer, about a third of an atom’s width, and the current falls about tenfold.
Because of that steepness, nearly all the current passes through the one atom at the very end of the needle, the one nearest the surface. Moved across the surface, the needle maps the cloud of the surface’s electrons near the top of their energies, the only ones that take part in the crossing. That cloud is usually thickest at each atom, so the map shows the atoms one by one. Many of the pictures of single atoms you may have seen, rows of bumps like eggs in a tray, were made this way.
Mass counts too. Short of the top by the same energy, a heavier particle’s wave dies away faster. A proton is 1,836 times heavier than an electron, and its wave dies away 43 times faster, so a stretch of wall that cuts an electron’s chance tenfold cuts a proton’s about 10⁴³-fold.
Yet hydrogen nuclei, the lightest, still tunnel in chemistry. An ammonia molecule is a squat pyramid, a nitrogen atom above a triangle of three hydrogen atoms, and it can turn inside out like an umbrella in the wind, though it lacks the energy to climb the wall between the two shapes. Started on one side, it tunnels over and back about 24 billion times a second. Made with heavy hydrogen, twice the mass, it flips not twice but about 15 times more slowly. A reaction between hydrogen molecules and heavy hydrogen ions has been measured that goes only by tunneling. It is the slowest reaction involving a charged particle ever seen, and its rate matches what tunneling predicts.
A person is about 10³² times heavier than an electron, and a wall is a hundred million times thicker than the gap under the microscope’s needle. Both sit in the multiplication. Written as a decimal, the chance of walking through a wall would begin with more zeros than anyone could write down in the age of universe.
How the Sun Gets Through
The Sun shines by fusing protons, the nuclei of hydrogen. Two protons both carry positive charge and push each other away, harder the closer they come. To touch, they would have to climb that push: a wall of energy hundreds of times higher than what a typical proton has in the Sun’s core, at about 15 million degrees. By the old physics, the share of protons with enough energy is less than one in 10¹⁷⁰. Not one of the Sun’s 10⁵⁷ protons would ever make it. The Sun would still glow as it slowly shrank, but for some tens of millions of years, not the billions it has shone.
Tunneling lets them meet below the top. Two protons with the core’s typical energy, meeting head on, get through about once in two hundred million tries. The few with four or five times the typical energy meet a thinner part of the wall and get through about once in ten thousand. Though rare, these faster few do most of the fusing. Every star that shines by fusion does it this way.
Getting through is only half of it. In the instant two protons touch, one of them must turn into a neutron, by the weak force, or they fly apart again, and that is rarer still. So a proton in the core waits billions of years for its turn. The Sun burns slowly because both chances are small, and brightly only because its core is vast. It burns steadily for another reason: a core that heats up swells and cools again.
The wall has been measured at the Sun’s own energies. Under 1,400 meters of rock in a mountain in Italy, out of reach of the cosmic rays that would swamp the count, a small accelerator fires helium-3 nuclei at helium-3, a later step in the Sun’s chain, at the very energies they meet with in the Sun. The lower the energy, the steeper the count falls, just as tunneling says; at the lowest, the reaction comes about twice a month.
Out of the Nucleus
Tunneling works outward too. A heavy nucleus such as uranium’s can throw out an alpha particle, a tight bundle of two protons and two neutrons. Inside, the bundle is held by the short grip that the strong force gives neighboring protons and neutrons, and that grip ends at the nucleus’s edge. Outside, the electric push of all the other protons drives it away. Between the two stands a wall: the bundle has the energy to fly off once it is out, but not to climb over. It tunnels out.
That explains a striking rule: the more energy an alpha particle leaves with, the shorter its nucleus lives, and by far more than seems reasonable. A bundle with more energy meets the wall nearer its top, where it is thinner, and every step of thinness multiplies its chance. Polonium-212’s alpha particles leave with about twice the energy of uranium-238’s. Half of a sample of polonium-212 is gone in three tenths of a millionth of a second; half of uranium-238 lasts 4.5 billion years, 24 powers of ten longer.
At the far end of the rule sits bismuth-209, long counted as stable. Physicists cooled a crystal holding bismuth to two hundredths of a degree above absolute zero and caught its alpha particles, 128 in five days. They leave with about a third of the energy of polonium-212’s, and half of the bismuth lasts 2 × 10¹⁹ years, more than a billion times the age of universe.
Built Into Chips
The memory in your phone keeps your photos as electrons trapped in billions of tiny cells. Each cell has a pocket for electrons, walled in by an insulator a few nanometers thick. To write, a strong voltage tilts the wall, so that the electrons meet only its thin top corner, and they tunnel in. With the voltage off, the full wall stands again, and at that thickness they would take many years to tunnel out. The cell keeps its bit with no power at all. To erase, the voltage is turned around, and tunneling through the same wall clears the pocket.
In most of today’s chips the pocket is a thin layer of silicon nitride that traps electrons. Earlier chips, and some still, hold them on an island of silicon, called a floating gate. Either way, the same exponential makes the wall both a door, under the voltage, and a lock, without it.
Elsewhere on a chip, tunneling is the enemy. As transistors shrank, the insulator under each one’s gate thinned to silicon oxide 1.2 nanometers thick, about five atoms, and electrons tunneled straight through it, wasting power. Chip makers turned to hafnium oxide, which does the same job at a thickness that stops most of the leak.
Two walls can do what one cannot. Put two thin walls a few atoms apart, and an electron should get through both even more rarely than through one, the two small chances multiplied. But its wave bounces back and forth between them, and at certain energies the bounces add up so that nothing is turned back: if the two walls are alike, all of the wave gets through both. Only a wave can do that.
A resonant tunneling diode is built this way. A tunneling electron keeps its energy, so current flows mostly while the voltage lines up the incoming electrons with one of those energies; past it, more voltage gives less current. That twist lets the diode make electrical waves that swing nearly two trillion times a second.
A Whole Circuit Tunnels
Everything so far was one particle, or a few moving as one. Yet something large can tunnel as a whole: an electric circuit on a chip you could hold in your hand. It is built round a junction, two superconductors with an insulator a few atoms thick between them. In a superconductor, billions of electrons, bound in pairs, move in one shared quantum state, so the circuit’s electrical state is one quantity for the whole circuit, not a crowd of separate ones.
Cooled almost to absolute zero, the circuit carries a steady current with no voltage across it, held there by a hill of energy it lacks the energy to climb. This hill is not a place: it stands between two states of that one quantity, one with no voltage across the junction and one with a voltage. Now and then a voltage appears all the same: the whole circuit has tunneled out at once. Shone on with microwaves, it takes in only certain frequencies, so its energy comes in steps, like an atom’s.
Steps like these, in circuits built round the same kind of junction, are the working parts of superconducting qubits. The lowest two serve as 0 and 1 in the quantum computers that Google, IBM and others build.
Why can a circuit do it and a person not? Mass and width are part of the answer. The rest is quiet. The circuit, cold and shut away, keeps its one quantity a single wave. Anything warm and large is struck by air and light all the time, and each blow records where it is; such a thing is never a single wave on both sides of anything.
How Long Inside?
How long does a tunneling particle spend inside the wall? The question turns out to have more than one answer. Work out when the peak of a tunneled wave comes out, and past a certain thickness the delay stops growing. A thicker wall then seems crossed faster, and at last faster than light.
Single photons show it. Timed through a mirror coating about a thousandth of a millimeter thick, the photons that got through arrived 1.47 millionths of a billionth of a second sooner than twins that crossed the same distance in air, as if they had gone at 1.7 times the speed of light. Nothing can be signaled that way. The peak that comes out is rebuilt from the front of the wave that went in, which was already on its way, because the wall lets through more of the front than of the rest. One reading is that this delay is not a crossing time at all, but how long the wall holds the wave before letting it go.
A clock can also be built that runs only inside the wall. Physicists cooled rubidium atoms to a billionth of a degree above absolute zero and sent them at a wall of laser light about a thousandth of a millimeter thick. A second pair of lasers turned each atom’s spin, as a magnet would, but only inside the wall. The atoms that came through had turned by an angle that said 0.61 thousandths of a second, a long time only because atoms that cold crawl, at about 4 millimeters a second.
Timed another way, the delay comes out as none. Electrons pulled out of hydrogen atoms by a strong laser field leave with no delay, to within 1.8 billionths of a billionth of a second. The clocks need not disagree: each times a different thing. Whether one of them is the time a particle spends inside, or the question has no single answer, is not settled. A recent experiment sent light along a guide into a step it could not climb, and timed it there by how much of it leaked meanwhile into a second guide alongside. It has started a new argument over the question between rival readings of quantum physics.
