An electron can pass through a barrier it does not have the energy to climb over. Because it behaves partly as a wave, a little of it reaches into the barrier and, if the barrier is thin enough, out the other side. The chance falls off steeply with thickness, so it matters only across a nanometre or two – the scale of . It is how a sees single atoms, and it is the leak that limits how thin the in a can be.
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Continuum models need parameters – interlayer tunnelling, the AA/AB tunnelling ratio, dielectric screening – that are themselves uncertain.
What Raman spectroscopy, photoemission, tunnelling and photocurrent show for Bi2O2Se, from films tens of nanometres thick to the cleaved surface of its crystal.
A quantum particle does not stop dead at a barrier: its wave fades inside and, if the barrier is thin, part of it emerges beyond. The current through hBN falls exponentially with every added layer, which makes it a barrier of exactly known thickness – and, between two graphene sheets with a gate below, the heart of a tunnelling transistor.
Through, not over
In classical physics a ball without the energy to roll over a hill rolls back. A quantum particle is described by a wave, and at a barrier the wave does not stop dead: inside, it decays exponentially, and if the barrier ends before the wave has died away, part of it carries on beyond. The size of that part is the probability of tunnelling.
The decay is steep. For an electron and a barrier a few electronvolts high, the probability drops roughly tenfold for every extra tenth of a nanometre – which is why a scanning tunnelling microscope, whose current falls this way as the tip moves away, draws almost all its current through the one atom closest to the sample. Thickness matters more than height: the exponent grows only with the square root of the barrier’s height.
Barriers one atom thick
Layered insulators make tunnel barriers of exactly known thickness. Between electrodes of graphite, graphene or gold, hexagonal boron nitride passes a tunnel current that falls exponentially with the number of its layers, down to a single one, and found the current uniform across terraces of equal thickness – the mark of a barrier without pinholes (Britnell and colleagues, 2012).
The same physics sets limits. A a layer or two thick leaks by tunnelling, which is why hBN is questioned as a and why silicon technology gave up thinning its silicon dioxide in favour of thicker oxides with a higher . It can also be put to work. Graphene holds few electrons near its , so a gate shifts its far enough to change the barrier the electrons see: a graphene–hBN–graphene stack works as a tunnelling transistor, switching about fiftyfold at room temperature, and about ten-thousandfold with molybdenum disulfide as the barrier.
Between layers
In a stack of two layers, electrons tunnel from one to the other, and that interlayer tunnelling is what couples them. In graphene it is the term that turns two sets of Dirac cones into moiré bands: treating it as hopping between the π of the two sheets, Bistritzer and MacDonald found that the velocity of the Dirac electrons falls to zero and the lowest band flattens at a set of .
Tunnelling also probes. Tunnelling spectroscopy, with a microscope tip or across a barrier of a few hBN layers, measures how many electronic states lie at each energy. Across a magnetic barrier such as few-layer CrI3, the current depends on how the in its layers are aligned – .
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Transmission through a classically forbidden region, with a probability that falls exponentially with the barrier’s width times the square root of its height above the particle’s energy. Across vacuum it gives the scanning tunnelling microscope its atomic resolution; through hBN the current falls exponentially with the number of layers down to a monolayer, which makes thin hBN a uniform tunnel barrier and sets the leakage floor when it serves as a gate dielectric. In twisted bilayers, tunnelling between the π orbitals of the two layers is the coupling that produces moiré bands and the magic angle.
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