SEMITIP, VERSION 5, DOCUMENTATION

A program for computing the electric potential and tunnel current due to a probe tip in proximity to a semiconductor, with circular symmetry. Prolate spheroidal coordinates are used in the vacuum, and a carefully chosen updating scheme is used to ensure stability of the iterative solution. Includes capability for a user specified distribution of surface states. Self-consistency, important for problems of accumulation or inversion, can be achieved.

Version 5.0 - written by R. M. Feenstra, Carnegie Mellon University, Sept 2009
Version 5.1 - posted Dec 13, 2010
Version 5.2 - posted Jan 13, 2011

All routines are written in standard FORTRAN.

A complete description of the background theory of this program is contained in Refs. 1-5. Some examples of running the program are provided in the SEMITIP V5 Introduction. Also, a user should carefully study the documentation for VERSION 1, VERSION 2, VERSION 3, and VERSION 4 of the program. Changes to VERSION 5 relative to VERSION 4 include:

1. A self-consistency loop has been added, whereby the charge densities are computed in the same routine INTCURR.F used to computed the tunnel currents, these charge densities are passed to the finite-element routine SEMISTIP2.F for solving the electrostatics, and this procedure is looped through until convergence is achieved. New parameters to control this self-consistency loop are on lines 47-49 of FORT.9. Line 47 gives the maximum number of times through the loop. Line 48 is a convergence parameter: if the potential at the surface point opposite the tip apex changes in one iteration to the next by less than this amount, then the loop is terminated. This value should be relatively small, like 1.e-5, in order to achieve reasonable convergence. It should be noted in this regard that the self-consistency loop is computed only for the maximal grid size of the electrostatic solution (i.e. scaling of the grid is not performed during each self-consistency iteration), and hence the convergence of the self-consistency is somewhat slow. Line 49 provides the capability of performing a exponentially weighted running average of the charge densities in the loop over this value (e.g. a value of 2.5 would mean a 1/e decay in the averaging over 2.5 iterations). A value of 0 produces no averaging, which works fine in the cases tested.
2. Rather high accuracy is required in the computation of the charge densities (especially for extended states) in order for this self-consistency procedure to suitably converge. Thus, line 42 for FORT.9, the number of energies at which the integrals for charge density or tunnel current are performed, must be quite high. A value of 5000 is found to produce good convergence (although it requires quite a lot of computation time).
3. A couple of small points to note about the computations of charge density:
   • The same routine used to compute the tunnel current (intcurr.f) is also used to compute the wavefunctions and charge densities needed for the self-consistency loop. However, when just computing charge densities (without the current), the computation is only performed at a single value of parallel wavevector.
   • The wavefunctions and charge densities are computed on a finer grid than the electrostatic potential (i.e. according to the expansion parameter in line 43 of FORT.9). As a consequence, it may occur that the grid for the electrostatic solution is rather coarse compared to the spatial extent of the charge densities (see example 3 below for an illustration of this). In that case, when converting the computed charge densities into discrete values on the grid used in the electrostatic solution, some discretization error may result. In particular, a lot of charge density might exist between the surface and the first electrostatic grid point within the semiconductor. Therefore, evaluation of the charge densities at the electrostatic grid points is done by integrating the density for half a grid spacing on either side of the grid point. (In addition, for VERSIONS 5.0 and 5.1, the charge density contained in the half-grid-spacing right beside the surface is summed together and treated as surface charge. But, this procedure is not necessary for VERSIONS 5.2 and higher since the z grid spacing in the semiconductor is changed).
4. In addition to the parameter on line 44 defining the depth into the semiconductor to perform the integration of Schrödinger's equation (as described in the VERSION 4 documentation) there's new, related parameters on lines 45 and 46. The parameter on line 45 is the fractional number of grid points into the semiconductor that the quantum charge density is computed over, with semi-classical charge densities used for larger depths. This value must be less than the entry on line 44, although the difference need not be large between them (sometimes it's good to extend the distance slightly over which the wavefunctions are computed, but still use the semi-classical charge densities at those large distances), e.g. 0.9 and 0.8, respectively, for an accumulation layer case (as in example 1), or smaller values for inversion (example 3). Line 46 gives a limit on the fractional number of radial coordinates that the integration of Schrodinger's equation and the computation of quantum charge density are computed over. This parameter is purely a time saving device, for inversion situations in which the wavefunctions and charge densities are radially localized around r=0. The user must be sure that whatever value is entered here is large enough so that it does not restrict the wavefunctions (this can be checked by explicitly plotting the wavefunctions, or by examining some other result of the computations as a function of this line 46 parameter). For accumulation, a value of 1.0 should be used on line 46.
5. Additional output options have been implemented. As listed at the bottom of the FORT.9 files, these are:

   output parameter:
   in general, values<5 produce most output only at end of computation,
      whereas values>=5 produce output after each iteration of the
      self-consistency and/or finite element loops
   0=minimal output
   1=current and conductance values, potential profiles, localized state energies
   2=also equi-potential curves, localized state wavefunctions, charge density images
   3=also full potential and full charge densities
   4=also all extended wavefunctions (use with caution!)
   5=minimal output after each iteration
   6=also localized state energies at each iteration
   7=also localized state wavefunctions, charge densities at each iteration
   8=also potential profiles at each iteration
   9=also all extended wavefunctions at each iteration (use with caution!)
   

Changes to VERSION 5.1 compared to 5.0 are the same as described for VERSION 4.1 compared to 4.0, as described in the detailed documentation for VERSION 4 (the relevant lines numbers for VERSION 5 are 58-59 of the main program and 734, 742, 842, 850, 954, 969, 1063, and 1078 in intcurr.f). Similarly, changes to VERSION 5.2 compared to 5.1 are the same as described for VERSION 4.2 compared to 4.1.

References:
1. R. M. Feenstra, Electrostatic Potential for a Hyperbolic Probe Tip near a Semiconductor, published in J. Vac. Sci. Technol. B 21, 2080 (2003). For preprint, see http://www.cmu.edu/physics/stm/publ/52/.
2. R. M. Feenstra, S. Gaan, G. Meyer, and K. H. Rieder, Low-temperature tunneling spectroscopy of Ge(111)c(2x8) surfaces, Phys. Rev. B 71, 125316 (2005). For preprint, see http://www.cmu.edu/physics/stm/publ/65/.
3. R. M. Feenstra, Y. Dong, M. P. Semtsiv, and W. T. Masselink, Influence of Tip-induced Band Bending on Tunneling Spectra of Semiconductor Surfaces, Nanotechnology 18, 044015 (2007). For preprint, see http://www.cmu.edu/physics/stm/publ/74/.
4. Y. Dong, R. M. Feenstra, M. P. Semtsiv and W. T. Masselink, Band Offsets of InGaP/GaAs Heterojunctions by Scanning Tunneling Spectroscopy, J. Appl. Phys. 103, 073704 (2008). For preprint, see http://www.cmu.edu/physics/stm/publ/79/.
5. N. Ishida, K. Sueoka, and R. M. Feenstra, Influence of surface states on tunneling spectra of n-type GaAs(110) surfaces, Phys. Rev. B 80 075320 (2009). For reprint, see http://www.cmu.edu/physics/stm/publ/85/.