Advanced Quantum Effects Simulation in ATLAS
Introduction
The trend toward smaller MOSFET devices with thinner gate oxide and greater doping is resulting in the increased importance of quantum mechanical effects, which are observed as shifts in threshold voltage and gate capacitance. Predicting these quantum effects requires solving the Schrodinger equation. This article presents the PoissonSchrodinger solver and recent enhancements implemented in ATLAS from Silvaco.
SchrodingerPoisson
To model the effects of quantum confinement, Quantum allows the selfconsistent solution of the Schrodinger equation with Poisson’s equation. Poisson’s equation is solved in two dimensions over the entire device while Schrodinger’s equation is solved in one dimensional slices across the device.
These solutions provide calculations of the bound sate energies (Eigen energies), the carrier wave functions (Eigen functions), and carrier concentrations in the presence of quantum mechanical confining potential variations.
Considering ml, mt1 and mt2 the electron longitudinal effective mass and the electron transverse effective masses respectively, the electron density is written as:
where x is the position along a vertical slice (normal to the gate oxide), _{li}, E_{li} (resp. _{i}, E_{ti}) are the ith longitudinal (resp. transverse) eigenvector and eigenvalue, k_{B} is the Boltzmann constant, T is the temperature, h is the Planck constant and E_{F} is the Fermi level. For the holes, a similar expression is obtained with the light and heavy holes effective masses.
Operational Modes
In Quantum, solutions to the SchrodingerPoisson system are used in various modes to accommodate various applications. The table above summarizes these modes.
Table 1 shows that Quantum offers three basic operational modes of solutions to the SchrodingerPoisson system of equations: onedimensional Schrodinger “slices” embedded in a twodimensional Poisson solution mesh, twodimensional Schrodinger solutions on the same twodimensional Poisson solution mesh and twodimensional Schrodinger “plane slices” embedded in a threedimensional Poisson solution mesh.
2D Poisson 1D Schrodinger 
2D Poisson 2D Schrodinger 
3D Poisson 2D Schrodinger 

Regular Grid 
X 
X 
X 
Unstructured Grid  X 

Post Processing 
X 

NonEquilibrius 
X 
X 
X 
Strained Silicon 
X 
X 
X 
Radiative Models 
X 
X 
Table 1. Schrodinger Poisson Operational Modes
For the rectangular grid approach, the grid points used for the Schrodinger solution exactly coincide with the grid points used in the Poisson solution. These solutions have the best selfconsistency since they involve no interpolation between the Schrodinger and Poisson solution grids. Conversely, the unstructured grid approach requires a separate grid definition for the Schrodinger solution from the Poisson grid. This requires interpolation between the grids, but offers the advantage of allowing selfconsistent SP solutions for unstructured meshes such as are generated by the process simulator ATHENA or the interactive general purpose structure creation tool DevEdit.
The postprocessing approach does not solve Schrodinger’s and Poisson’s equation selfconsistently. Instead Poisson’s equation is solved selfconsistently with the electron and hole continuity equations in the standard driftdiffusion approach. The Schrodinger solutions are then obtained using the classical Poisson solutions. This has the advantages of providing fast solutions and solutions can be taken far from zero bias (i.e. with currents flowing).
In the nonequilibrium mode, Quantum first calculates the selfconsistent solution of Poisson’s equation with the electron and hole continuity equations as in the standard driftdiffusion model. The SchrodingerPoisson solutions are then calculated selfconsistently using the quasiFermi levels from the classical solutions to estimate the nonclassical carrier concentrations. This has advantages similar to the postprocessing approach but gives selfconsistent SP solutions.
The strained silicon model provides for strain induced splitting in the conduction and valence bands in the solution of Schrodinger’s equation.
For radiative modeling, Quantum provides SchrodingerPoisson solutions to obtain bound state eigen energies which are used in the calculation of the momentum matrix elements. The SchrodingerPoisson solutions also give the wave functions which can be used to calculate overlap integrals. The momentum matrix elements and overlap integrals are used in the calculation gain spectra (for lasers) and spontaneous emission spectra (for all light emission devices).
TwoDimensional SchrodingerPoisson Example
To demonstrate the twodimensional SchrodingerPoisson solver we constructed a simple 3 gate capacitor to demonstrate charge confinement in two dimensions. The device, shown in Figure 1, is composed of a heavily doped ptype silicon region completely embedded in silicon dioxide. Gates are placed at the top and either end of the silicon region. A substrate contact is at the bottom.
Figure 1. 3 gate capacitor.
Figure 2 shows a contour plot of the electron concentration when the gates are biased to 0.5 V. In the figure and the zoomed inset the effects of quantum confinement can be clearly seen.
Figure 2 show electrons concentration in linear scale at Vgate=0.5V
This can be contrasted with the classical solution shown in Figure 3.
Figure 3. Classical electron concentration.
Conclusion
The ATLAS Quantum model offers a variety of operational
modes to accommodate a range of applications to address the effect of quantum
confinement.