SEMITIP, VERSION 5, Electrostatics and Tunnel Current Computations for a Probe Tip near a Semiconductor

R. M. Feenstra
Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213

This program computes the electrostatic potential and resulting tunnel current produced by a metallic probe tip near a semiconductor. This program is an updated version of the prior VERSION 4, VERSION 3, VERSION 2, and VERSION 1. Basic usage is described below, and additional details are available in the SEMITIP V5 Technical Documentation describing changes relative to VERSION 4. The major change in VERSION 5 relative to VERSION 4 is the added feature of self-consistency, important for situations of accumulation or inversion.

Some background information on the methods used in the program is contained in:

• Surface Charge arising from a distribution of Surface States
• Contact Potential
• Coordinate System with accompanying Diagram
• Integration of Schrödinger's equation
• Computation of Tunnel Current

Self-consistency is achieved by computing both the charge densities and the tunnel current fully quantum mechanically, in the z-direction (normal to the surface), as described in Phys. Rev. B 80 075320 (2009). (In the lateral x- and y-directions, both the charge densities and the wavefunctions are semi-classical). For situations of semiconductor depletion, as commonly occurring in experiment, self-consistency occurs automatically in the computations (since the only charge densities are due to the background dopant atoms). However, for accumulation or inversion, self-consistency will in general produce a noticable effect on the tunnel current. See example 1 below for additional details.

A compiled version of the code, which should run on any Windows PC, is available in the file semitip_v5.exe. Input for the executable code comes from the file FORT.9. Download these two files, into filenames "semitip_v5.exe" and "fort.9". Then, run the code just by double clicking on it. Using a text editor, the input parameters in FORT.9 can be changed to whatever values are desired. In addition to the parameter values, this file also contains comments as to the meaning of each parameter.

Output from the program is contained in the following files (output depends on the value of the output parameter IWRIT as specified in the input file FORT.9):

• FORT.10 - gives the numerical results for the following quantities:
  • tip radius of curvature (nm)
  • tip-sample separation (nm)
  • sample-tip bias voltage (V)
  • contact potential (eV)
  • PHI0 - the surface potential at a point directly opposite the tip apex (eV)
• FORT.11 - provides the potential (eV) along the central axis, as a function of z-distance (output for IWRIT>=1) (the third column gives auxiliary output which are the potential values for all grid points at the maximal radial value in the problem, although the z-values for these points are not output and in particular do not correspond to the values in column 1).
• FORT.12 - provides the potential (eV) along the surface, as a function of the radial distance from the central axis (output for IWRIT>=1)
• FORT.13 - gives the entire array of potential values (eV) (output for IWRIT>=3); see VERSION 3 documentation for more details.
• FORT.14 - provides the current (A/nm^2) (column 2) as a function of sample voltage (V) (column 1). Also, columns 3 and 4 provide the contributions to the current of extended states and localized states, respectively. Separate contributions from the valence band and conduction band go in to FORT.91 and FORT.92, respectively.
• FORT.15 - provides the conductance dI/dV (A/(V nm^2)) (column 2) as a function of sample voltage (V) (column 1). Also, columns 3 and 4 provide the contributions to the conductance of extended states and localized states, respectively. Separate contributions from the valence band and conduction band go in to FORT.93 and FORT.94, respectively.
• FORT.16 - gives an exact copy of the output to the console
• FORT.17 - provides the charge densities on the central axis (column 2) as a function of z-distance along the central axis (column 1). Also, columns 3 and 4 provide the contributions to the charge densities of extended states and localized states, respectively.
• FORT.18 - provides the surface charge densities (column 2) as a function of radial distance away from the central axis (column 1). Also, columns 3 and 4 provide the contributions to the charge densities of extended states and localized states, respectively.
• FORT.19 - provides the surface charge density (column 2) at the point opposite the tip apex (i.e. on the central axis), as a function of bias voltage. Also, columns 3 and 4 provide the contributions to the charge densities of extended states and localized states, respectively.
• FORT.20, FORT.21, ... - contour lines (nm) of the potential (output for IWRIT>=2)
• FORT.30,FORT.40,FORT.50,FORT.60 - listing of voltage (column 1), principal quantum number (column 2), energy of localized state (column 3), and the difference in electron occupation between sample and tip at that energy (column 4), for the light-hole, heavy-hole, split-off and conduction bands, respectively
• FORT.31,FORT.41,FORT.51,FORT.61 - wavefunctions of localized states, as a function of distance along central axis, for individual bands as above. Only listed at final bias voltage in the problem (output for IWRIT>=7).
• FORT.32,FORT.42,FORT.52,FORT.62 - charge densities of localized states, as a function of distance along central axis, for individual bands as above. Only listed at final bias voltage in the problem (output for IWRIT>=7).
• FORT.33,FORT.43,FORT.53,FORT.63 - wavefunctions of extended states for zero parallel wavevector, as a function of distance along central axis, for individual bands as above. Only listed at final bias voltage in the problem (output for IWRIT>=9). CAUTION: these files could be large!
• img1.PGM, img2.PGM, and img3.PGM provide images (PGM format) of the charge densities due to the localized, extended and total charge densities, respectively. The extent of these images is limited by the parameters that limit the region over which the quantum charge densities are computed, lines 47 and 48 of FORT.9. By construction, the number of pixels in the images is expanded (using linear interpolation) by 3x in each dimension from the number of grid points, and the images are placed on a 512x512 background. Parameters for the gray-scale plotting of the images can be modified in the program code. Interpretation of these images is discussed in example 2 below.

All of the parameters in the program can be varied using the input file FORT.9, with the exception of the array sizes (NRDIM,NVDIM,NSDIM, and NEDIM), the specification of an auxiliary function other than a hemisphere which defines a protrusion on the end of the probe tip, and the specification of a surface state density other than a uniform or Gaussian shaped one. Modification of those parameters can be accomplished by changing the source code of the program. The source code is available, in the files:

• SEMITIP_V5.F - main routine.
• SEMISTIP2.F - performs the detailed finite element solution of Poisson's equation.
• POTCUT1.F - takes a cut of the electrostatic potential along the central axis of the problem.
• POTCUT2.F - expands the profile obtained from POTCUT1.F, to produce a sufficient number of points for integration of the Schrödinger equation.
• INTCURR.F - computes the tunneling current, by integrating the Schrödinger equation along the central axis.
• SEMIRHO.F - auxiliary routines for computing semiconductor charge densities.
• SURFRHO.F - auxiliary routines for handling surface charge densities.
• CONTR.F - plots contour lines of the computed potential.

Illustrative Examples of Running the Code

1. n-type GaAs(110), with no surface states.
2. n-type GaAs(110), viewing charge density images.
3. n-type Si surface, plotting energies of inversion states.