Set of supplementary notes.

6.4. TUNNELLING 95
experiments are carried out below 1 K for transition metal compounds, and below 100 mK
for heavy fermion compounds (see Appendix).
6.4 Tunnelling
Tunnelling spectroscopies (injecting or removing electrons) through a barrier have now evolved
to be very important probes of materials. The principle here is that a potential barrier allows
one to maintain a probe (usually a simple metal) at an electrical bias different from the chemical
potential of the material. Thus the current passed through the barrier comes from a nonequilibrium
injection (tunnelling) through the barrier.
A model for a simple metal tunnelling into a more complex material is shown in Fig. 6.8.
With the metal and sample maintained at different electrical potentials separated by a bias eV ,
then the current through the junction can be estimated to be of the form
I ∝
∫ µ
µ+eV
g L (ω)g R (ω)T (ω) (6.7)
where T is the transmission through the barrier for an electron of energy ω and g L and g R
are the densities of states. 2 If the barrier is very high so that T is not a strong function of
energy, and if the density of states in the contact/probe is approximately constant, then the
energy-dependence comes entirely from the density of states inside the material. Notice then
that the differential conductivity is proportional to the density of states (see Fig. 6.8):
dI/dV ∝ g(µ + eV ) . (6.8)
It is difficult to maintain very large biases, so most experiments are limited to probing electronic
structure within a volt or so of the Fermi energy.
Tunnel junctions are sometimes fabricated by deposition of a thin insulating layer followed
by a metal contact.
The technique of scanning tunnelling microscopy (STM) uses a small tip, with vacuum as
the surface barrier. Because the tunnel probability is an exponential function of the barrier
thickness, this scheme provides high (close to atomic, in some cases) spatial resolution, even
though the tip radius will be nm or larger. By hooking this up to a piezoelectric drive in a
feedback loop, it has proved possible to provide not only I − V characteristics at a single point,
but also spatial maps of the surface.
Scanned probe spectroscopies have advanced to become extraordinary tools at the nanoscale.
As well as STM, it is possible to measure forces near a surface (atomic force microscopy, AFM
), which is particularly useful for insulating samples. It has proven possible to manipulate
individual atoms, to measure the magnetism of a single spin, and with small single-electron
transistors to study to motion of single electron charges in the material.
2 Strictly this formula applies when the tunnelling process does not conserve momentum parallel to the
interface, i.e. if the surface is rough or disorded.