Scanning Tunneling Spectroscopy - Tunneling Current

Tunneling Current

Since STS relies on tunneling phenomena and measurement of the tunneling current or its derivative, understanding the expressions for the tunneling current is very important. Using the modified Bardeen transfer Hamiltonian method, which treats tunneling as a perturbation, the tunneling current (I) is found to be

where is the Fermi distribution function, and are the density of states (DOS) in the sample and tip, respectively, and is the tunneling matrix element between the modified wavefunctions of the tip and the sample surface. The tunneling matrix element,

describes the energy lowering due to the interaction between the two states. Here and are the sample wavefunction modified by the tip potential, and the tip wavefunction modified by sample potential, respectively.

For low temperatures and a constant tunneling matrix element, the tunneling current reduces to

which is a convolution of the DOS of the tip and the sample. Generally, STS experiments attempt to probe the sample DOS, but equation (3) shows that the tip DOS must be known for the measurement to have meaning. Equation (3) implies that

under the gross assumption that the tip DOS is constant. For these ideal assumptions, the tunneling conductance is directly proportional to the sample DOS.

For higher bias voltages, the predictions of simple planar tunneling models using the Wentzel-Kramers Brillouin (WKB) approximation are useful. In the WKB theory, the tunneling current is predicted to be

where and are the density of states (DOS) in the sample and tip, respectively. The energy- and bias-dependent electron tunneling transition probability, T, is given by

where and are the respective work functions of the sample and tip and is the distance from the sample to the tip.