Sending report ...Schred Detailed Description
Dragica Vasileska and Zhibin Ren
Arizona State University, Purdue University
February, 2000
Schred 2.0 calculates the envelope wavefunctions and the corresponding bound-state energies in a typical MOS (Metal-Oxide-Semiconductor) or SOS (Semiconductor-Oxide- Semiconductor) structure and a typical SOI structure by solving self-consistently the one-dimensional (1D) Poisson equation and the 1D Schrodinger equation.
In version 2.0, the program can simulate both p type and n type silicon bodies, the program can also assume both n type and p type polysilicon or metals with specified workfunctions as gate contact. For the quantum simulation, it is assumed that the Si/SiO2 interface is parallel to a [100] plane. The conduction band is represented by the six equivalent valleys, the effective masses are calculated from the valley curvature. Due to the tensor property of the effective masses, in solving the effective mass based Schrodinger equation in the confined direction, two different perpendicular effective masses end up with two set of confined energies and corresponding wavefunctions.
The first set consists of the two equivalent valleys which have the longitudinal mass perpendicular to the interface, whereas the second set consists of four equivalent valleys with light mass perpendicular to the interface (F. Stern, "Iteration Methods for Calculating Self-Consistent Fields in Semiconductor Inversion Layers," J. Comp. Phys., Vol. 6, pp. 56-67, 1970). The exchange and correlation corrections to the ground state energy are included in the solver by using the local density approximation (L. Hedin and B. I. Lundqvist, "Explicit Local Exchange-Correlation Potentials," J. Phys. C: Solid St. Phys., Vol. 4, pp. 2064-2083, 1971). The first set has relatively lower confined energies because of the heavier longitudinal mass, the second set has higher confined energies.
Similarly, the valence band is represented by the heavy hole band and light hole band, the spit-off band is ignored because the spit-off energy is large enough to exclude any hole staying there. In treating holes quantum mechanically, the same effective mass based Schrodinger equation is solved with the masses quoted from references (C. Hu, S. Banerjee, k. Sadra, B.G. Streetman and R. Sivan, "Quantization Effects in Inversion Layers of PMOSFET's on Si (100) Subatrates," IEEE Electron Dev. Lett., Vol. 17, No. 6, pp.276-278, June 1996 and S. Takagi, M. Takayanagi, and A. Toriumi, "Characterization of Inversion-Layer Capacitance of Holes in Si MOSFET's," IEEE Trans. Electron Devices 46, pp. 1446-1450, 1999).Due to their different perpendicular masses, the heavy holes form the first set of energy levels which are relatively low, and the light holes form the second set with higher confined energies.
Schred 2.0 also has the capability of treating the electron/hole density in the inversion layer classically by using either Maxwell-Boltzmann or Fermi-Dirac statistics.
In doing bulk structure quantum mode simulation, the version 2.0 can not only solve the effective mass based Schrodinger equation for inversion layer carriers, but also can solve the equation for accumulation layer carriers, for example, if the bulk is p type silicon, in the inversion range, electrons are treated quantum mechanically, whereas in the accumulation range, holes are treated quantum mechanically. This is a feature that many other simulators do not offer.
In doing SOI quantum mode simulation, both electrons and holes are treated quantum mechanically at the same time. This is because in most cases, the SOI bodies are undoped or lightly doped, and the two dielectric gates confine the carriers in both inversion and accumulation regimes, therefore, the quantum effects can be equally important for both electrons and holes at low biases.
For both simulation modes, (classical or quantum mechanical), if the gate contacts are polysilicon, the charge density on the gates will always be computed classically. The gate dielectric constant can be specified different from SiO2, The new version also allows different dielectrics for the top and bottom gates in a SOI structure.
This eases the simulations of effects of exotic insulator materials on device performance. Typical outputs of the solver are the spatial variations of the conduction-band edge and 3D charge density in the body; 2D surface charge density, average distance of the carriers from the interface; inversion layer capacitance Cinv, depletion layer capacitance Cdepl, total gate capacitance Ctot and in the case of capacitors with poly-silicon gates, it also calculates the poly-gate capacitance Cpoly. When choosing quantum-mechanical description of the electron density in thechannel, it also provides the subband energies the subband population, and wavefunction variations in the body.
Schred is written in Fortran 77. The program is efficient compared to other 1D Schrodinger-Poisson self-consistent simulators. On a SPARC-5 workstation, generally, it takes about 10 seconds per bias point in quantum mode calculation, and about 5 seconds per bias point in classical mode calculation. But for bulk accumulation range simulation, it takes a relatively long time--about 2 to 3 minutes for one bias point. This is because in accumulation range, the band potential energy level bends very little, and the subband energies crowds together, so that a large number of subbands need to be included in the calculation in order to accurately account for the contributions from all lowest subbands. A SOI quantum mode simulation with very thick silicon body (thicker than 0.1 micron) can also involve relatively long computation time.
Examples of the application of Schred can be found in D. Vasileska, D. K. Schroder and D.K. Ferry, "Scaled silicon MOSFET's: Part II-Degradation of the total gate capacitance," IEEE Trans. Electron Devices 44, pp. 584-587 (1997), and D. Vasileska, and D.K. Ferry, "The influence of space quantization effects on the threshold voltage, inversion layer and total gate capacitance in scaled Si-MOSFETs," in the Technical Proceedings of the First International Conference on Modeling and Simulation of Microsystems, Semiconductors, Sensors and Actuators, Santa Clara, California, April 6-8, 1998, pp. 408-413. J. Fossum, Z. Ren, K. Kim and M. Lundstrom "Extraordinarily High Drive Currents in Asymmetrical Double-Gate MOSFETs," submitted for publication at VLSI symposium June, 2000. More details about the solver are also provided in the Schred User's Manual.