STM: The Agonizing Details

Tunneling Current

Based on the Bardeen's tunneling current formalism

Bardeen's tunneling current formalism


fermi function

f (E) is the Fermi function, V is the applied voltage, MTS the tunneling matrix element between the tip wavefunction, Psi sub T sign and the sample surface wavefunction, Psi sub s sign , Delta (x) the delta function, Ei the energy of a state in the absence of tunneling ( i = T, S), and m the chemical potential, kB is Boltzman constant, T is temperature.

The matrix element MTS is otained from

matrix element

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

complex congugate

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

Tunneling current

Carrying out the integral in one dimension (plane wave) by taking Psi sub To sign and Psi sub s o 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

wavefunction of sample

wavefunction of tip

where K is the inverse decay length

Inverse Decay Length

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

spherical configuration

With the spherical tip configuration shown in the picture, MTS 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

Surface wavefunction

where Ohm sub s sign 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 aG are typically of the order unity

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

tip wavefunction

whereOhm sub t signis the probe volume and R is the radius of curvature of the tip. The resulting matrix element is

matrix element

where ro is the position of the center of curvature of the tip. The tunneling current is

tunneling current

where DT(EF) is the tip density of states per unit volume.

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

Local Density of States


density of states

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

Local Barrier Height


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Author: Tit-Wah Hui email: <>
Curator: Dan Thomas email: <>
Last Updated: Mon, Apr 14, 1997 15:04 EST