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