The quantum description of a particle such as an electron: not a dot at one place but a wave spread through space. Where the wave is large the electron is likely to be found – the chance goes with the square of its height – and like any wave it can spread, bend round obstacles and interfere with itself. The shapes of these waves decide most of what a material does: how its atoms bond, which bands form, how easily electrons tunnel through a barrier or leak into the next layer. A can map them.
Going deeper
An electron is described by a wave; the square of its height gives where it is likely to be found. In a crystal each wave is a plain wave reshaped in every cell by the atoms – a Bloch state. A scanning tunnelling microscope sees the square of the wave: in a ring of 48 iron atoms on copper, the standing waves of the electrons trapped inside.
A wave for a particle
Quantum mechanics describes an electron by a wavefunction, a wave whose shape is set by the Schrödinger equation. Where the electron is held – by an atom, a box, a crystal – only certain shapes fit, each with its own energy: the of an atom, the levels of a , the bands of a solid. The wavefunction itself cannot be seen. What can be measured is the square of its height at each point, which gives the probability of finding the electron there.
It is still a wave, with a phase as well as a height. Two waves that meet add where crest meets crest and cancel where crest meets trough, so an electron with two routes open can interfere with itself. Its tail also reaches into places a classical particle could never go, which is how electrons tunnel through barriers.
Waves in a crystal
In a crystal the atoms repeat, and Bloch’s theorem says each wavefunction is a plane wave running through the whole crystal, multiplied by a pattern that repeats from cell to cell – a Bloch state, labelled by its band and its . Built from atomic orbitals, these waves spread from atom to atom, and how strongly the orbitals of neighbours overlap sets how wide the bands are.
Much of modern 2D physics lives in the details of the repeating pattern. In graphene it has two parts, one on each of the two sets of carbon atoms, and the balance between them behaves like a – the ; carried once around a it gives the wavefunction a phase of π, which shows up in graphene’s (Castro Neto and colleagues, 2009). How the pattern twists across the Brillouin zone sets the and the invariants that tell a from an ordinary one. At the surface of a material, wavefunctions do not stop dead but decay over a few ångström – enough to couple neighbouring layers, to leak metal states into a at a contact, and to let one layer borrow properties from another.
Seeing one
A scanning tunnelling microscope measures how many states sit at each energy right under its tip, and so maps the square of the wavefunctions at that energy. The classic picture is the quantum corral: 48 iron atoms placed one by one in a ring of radius 71.3 Å on a copper surface trapped the surface electrons inside, and the microscope showed the concentric standing waves of an electron in a round two-dimensional box, along with the discrete energies that go with them (Crommie and colleagues, 1993). The same measurement shows how electrons spread across a cell or gather at defects and edges in .
can do it from momentum space. Measuring how electrons leave a film at a fixed energy over all angles gives a map of an orbital in momentum; for molecules lying on a surface, a Fourier transform turns that map into the orbital’s shape in real space, as done for pentacene films and a of p-sexiphenyl (Puschnig and colleagues, 2009).
For specialists
The complex amplitude ψ(r) – a spinor once spin is included – that solves the Schrödinger equation, with |ψ|2 the probability density; for many electrons it depends on all their coordinates and changes sign when two are exchanged. In a crystal Bloch’s theorem makes each eigenstate a plane wave times a function with the period of the lattice, labelled by band and crystal momentum; how that periodic part changes across the Brillouin zone defines the Berry curvature, the and topological invariants. In graphene it has two components, one per sublattice – the pseudospin – and acquires a Berry phase of π around a Dirac point. Scanning tunnelling spectroscopy images |ψ|2 through the local , as for the standing waves in a quantum corral of radius 71.3 Å, and photoemission orbital tomography reconstructs orbitals by Fourier-transforming momentum maps.
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