9. Scanning Tunneling Microscopy (Bardeen’s formalism)

In this section, we present the steps to obtain the current flowing from a sample to a tip using Bardeen’s formalism. We refer the readers to [GW06, L14] for a basic review of Bardeen’s theory. The main assumptions underlying the theory:

  1. Tunneling is weak enough to employ first-order time-dependent perturbation theory;

  2. Tip and sample states are nearly orthogonal.;

  3. The electron-electron interaction can be ignored;

  4. The occupation of the tip and sample are independent of each other;

  5. The tip and sample are each in electrochemical equilibrium.

Using the above hypotheses, the following current equation is derived

(9.1)\[I = \frac{2\pi e}{\hbar}\sum\limits_{\mu\nu} \left[ f_\text{FD}(E^S_\mu-E^S_F) - f_\text{FD}(E^T_\mu-E^T_F)\right] |M_{\mu\nu}|^2 \delta(E^T_\mu - E^S_\nu - eV)\]

where \(e\) is the charge of the electron, \(\hbar\) is the reduced Planck constant, \(\mu\), \(\nu\) are band indices, \(f_\text{FD}\) is the Fermi-Dirac function, \(E^{S,T}_\mu\) is an eigenvalue of the sample or tip, \(E^{S,T}_F\) is Fermi level of the sample or tip and \(V\) is the bias applied between the sample and tip. Finally, \(M_{\mu\nu}\) are the tunneling matrix elements defined as

(9.2)\[M_{\mu\nu} = \frac{\hbar^2}{2m}\int\limits_{z=z_0} dx dy \left[ \psi^S_\mu \frac{\partial \psi^T_\nu}{\partial z} - \frac{\partial \psi^S_\mu}{\partial z} \psi^T_\nu \right]\]

In (9.2), the domain of the surface integral is limited to a single unit cell.

RESCU evaluates the continuum version of (9.1), (9.3).

(9.3)\[I = \frac{2\pi e}{\hbar} \int d\epsilon \left[ f_\text{FD}(\epsilon + eV - E^S_F) - f_\text{FD}(\epsilon-E^T_F) \right] n^S(\epsilon + eV) n^T(\epsilon) M(\epsilon+eV,\epsilon)\]

This is done in an stm-current calculation, during which RESCU computes (9.3) as follows:

  1. Solve the KS equation of the sample plus tip system (as a whole system). This provides the eigenvalues \(E^{S,T}_\mu\), which are the same for sample and tip since the whole system is calculated, and eigenstates \(\psi_\mu\).

  2. Compute the PDOS of the sample \(n^S(\epsilon)\) and tip \(n^T(\epsilon)\).

  3. Split the whole system wavefunctions into sample and tip wavefunctions \(\psi = a^S \psi^S + a^T \psi^T\) by computing \(\psi^{S,T} = \sum a^{S,T}_\eta \phi^{S,T}_\eta\) where \(\phi^{S,T}_\eta\) are atomic orbital centred at atoms in the sample and tip respectively, and \(a^{S,T}_\eta = \langle \phi_\eta | \psi \rangle\).

  4. Compute the tunneling matrix elements \(M_{\mu\nu}\) using (9.2).

  5. Evaluate (9.3), interpolating \(M_{\mu\nu}\) to obtain \(M(\epsilon+eV,\epsilon)\).

This new feature of RESCU has limited functionality, here are the main constraints:

  1. It is implemented for atomic orbital calculations (LCAO.status = 1).

  2. The tip is above the sample in the +z direction.

  3. Their is a plane \(z = z_0\) which separates the sample and tip.

  4. The k-sampling is limited to \(\Gamma\) for now, because the feature was develop to study large realistic samples and tips.

[GW06]

Alex Gottlieb and Lisa Wesoloski. Bardeen’s tunnelling theory as applied to scanning tunnelling microscopy: A technical guide to the traditional interpretation. <10.1088/0957-4484/17/8/R01> Nanotechnology - NANOTECHNOL 17 (Apr. 2006).

[L14]

Samir Lounis. Theory of Scanning Tunneling Microscopy. 2014. <10.1016/S0076-695X(08)60006-X> Methods in Experimental Physics, Academic Press, Volume 27, 1993, Pages 1-29.