## STM: The Agonizing Details

## Tunneling Current

Based on the Bardeen's tunneling current formalism

where

f (E) is the Fermi function, V is the applied voltage, M_{TS } the tunneling matrix element between the tip wavefunction,
and the sample surface wavefunction, ,
the delta function, E_{i} the energy of a state in the absence of tunneling ( i = T, S), and m the chemical potential, k_{B} is Boltzman constant, T is temperature.

The matrix element M_{TS} is otained from

where m is the free electron mass, the integral is taken over a surface lying entirely within the vacuum barrier region between the tip and the sample,

* denote the complex conjugate, and

For small tunneling voltages and low temperature, f(E) can be expanded to obtain

Carrying out the integral in one dimension (plane wave) by taking
and
as the wavefunctions of the tip and the sample, respectively, gives the explicit exponential dependence of the tunneling current as a function of tip-sample separation. In particular, if the tip and sample are identical materials, the wavefunctions in the cacuum gap can be written as

where K is the inverse decay length

with
, the local barrier height or average work function. Therefore, the tunneling current exhibits an exponential dependence on the separation d given by

With the spherical tip configuration shown in the picture, M_{TS} may be evaluated to assess the potential spatial resolution of STM. The surface wavefunctions are expanded in plane waves parallel to the surface with decaying amplitude into the vacuum

where
is the sample volume, K is inverse decay lenght, k is the surface wavevector and G is a two-dimensional reciprocal lattice vector of the surface. The first few factors a_{G} are typically of the order unity

The spherical tip (R >> k ^{-1} ) wavefunction are expanded in similar form

whereis the probe volume and R is the radius of curvature of the tip. The resulting matrix element is

where r_{o} is the position of the center of curvature of the tip. The tunneling current is

where D_{T}(E_{F}) is the tip density of states per unit volume.

Divide the current by bais voltage, the tunneling conductance,
can be obtained as

where

Differentiate the logarithmic of the tunneling gives the local barrier height,
becomes

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**Author: Tit-Wah Hui** email: <thui@uoguelph.ca>

**Curator: ***Dan Thomas* email: <thomas@chembio.uoguelph.ca>

**Last Updated: ***Mon, Apr 14, 1997 15:04 EST*