New Feature of Quantum Module:
SchrödingerPoisson Solver for Nanowire Application
Introduction
The trend toward ultrashort gate length MOSFET requires a more and more effective control of the channel by the gate leading to new architecture like doublegate, trigate, omegagate, and fourgate (or gateallaround) MOSFETs. Recent advances in nanoscale fabrication techniques have shown that semiconductor nanowires are becoming promising candidates for next generation technologies. In particular, silicon nanowire transistors have been demonstrated by several research groups with crosssectional dimensions in the range of several nanometers.
To address a strong quantum confinement effect in a nanowire crosssection, SILVACO has recently enhanced its Quantum Module [1]. Along with a general 2D Schrödinger solver valid for any crosssectional shape, a new solver was developed to specifically account for the natural cylindrical symmetry of the nanowire. This article presents these new features.
Nanowire: Geometry and Structure
In Figure 1, a scheme of a nanowire channel is shown. It is obvious this type
of device is 3D by nature.
Figure 1. Scheme of the channel of a nanowire (electrodes are not
drawn). 
As the purpose of this article is to demonstrate the capability of ATLASQuantum to compute the wavefunctions, eigen states and carrier concentration in such a device, two structures will be considered, see Figure 2. The first structure (Figure 2a) is a crosssection perpendicular to Zaxis, and the second one (Figure 2b) is the halfplane passing through the axis of the cylinder.
Figure 2. schemes of the 2D structures to be considered
for simulations.

Simulations
To find eigen states and carrier concentration in the crosssection perpendicular to Zaxis (Figure 2a), ATLAS will solve a 2D Schrödinger equation on a general triangular mesh. The anisotropy of effective mass is automatically taken into account in the discretized equation. For example, in silicon nanowire the wave functions and eigen energies will be found for each of the three types of conduction band valleys with effective masses (MX, MY, MZ) along X, Y and Z axes equal to (MT1, ML, MT2), (MT2, MT1, ML) and (ML, MT2, MT1).
In the case of the halfplane structure (Figure 2b) ATLAS will solve a 1D Schrödinger equation in cylindrical coordinates in the radial direction for each slice perpendicular to the Zaxis. In order to employ a cylindrical symmetry, ATLAS has to use an approximate isotropic effective mass in the radial direction. Because of this assumption, now there are only two different types of conduction band valleys as one of the valley becomes doubly degenerate. The effective masses (Mr, Mz) along radial and axial directions for the two types of valleys are equal to (MT1, ML) and (2*MT1*ML/(MT1+ML), MT1). Note also that computational requirements are significantly reduced in this case due to a 1D nature of confinement and smaller number of valleys required.
The choice of the approximate effective mass for the radial direction is motivated by the fact that the wave functions and eigen energies resulting from a cylindrical solver are very close to those of the exact 2D solver.
For the disk plane, one considers a 14nm diameter section of silicon, 2.5e18 cm3 ptype doped, and a gate oxide 2 nm thick. The structure is built in DevEdit and depicted in Figure 3.
Figure 3. Crosssection of the nanowire considered for the simulation
with material and grid shown (Aluminum used as the gate electrode has a
square shape for simplicity reason, a circular shape would lead to the
same result). 
In practical use, the key word to include in the MODELS statement, is 2DXY.SCHRO which makes a call to 2D solver built for general triangular mesh.
The 2D electron concentration
is plotted in Figure 4 for 0.5 V bias applied to the gate. It clearly shows
electrons are repelled from the
interface
Si/Oxide in all radial directions.
Figure 4. (a) 2D contour plot of electron concentration, (b)cutline passing
through the center (both in linear scale). 
In Figure 5, the 6 first wave functions are depicted showing the 2D nature of the problem.
Figure 5. from left to right and top to bottom, the 6 first wavefunctions
displayed in the section of the channel of the nanowire for Vgate=0.5 V. 
The structure of the rectangle
which makes
the nanowire
by revoluting around its left vertical axis is depicted in Figure
6. Only the channel
part is accounted, source and drain are ignored in order to compare
the results with the previous ones.
Figure 6. Structure file built in ATLAS defining the halfplane of revolution,
the mesh and electrodes are displayed. The axis of revolution is on the
left (x=0). 
To obtain the solution, the parameter CYLINDRICAL is added to MESH statement.
Figure 7 shows the electron concentration for both structures along a radius, the agreement is good and the small difference is due to the difference of mesh. This result confirms the effective mass approximation made in the cylindrical case.
Figure 7. Electron concentration in linear scale
in the disk structure (red line) and the rectangle structure (green line).
Gate oxide is on the right 
Conclusion
This article presents new capabilities of ATLASQuantum module. The SchrödingerPoisson solver handles arbitrary geometries and can be used in cylindrical mode. This opens numerous perspectives in the quantum simulation field. As an example, we have presented here the case of a Silicon nanowire.
Acknowledgment
We greatly thank G.Iannaccone from University of Pisa for his
collaboration in the development of the 2D SchrödingerPoisson solver.
References
 Simulation Standard, Vol.16, No.11, November 2006 (available on www.silvaco.com)