## 3D Simulation Models For Quantum Mechanical Effect

**Quantum 3D**^{™} provides a set of models for simulation of the various effects of quantum confinement and quantum transport of carriers in semiconductor devices. A self consistent Schrodinger – Poisson solver allows calculation of bound state energies and associated carrier wave function self consistently with electrostatic potential. Schrodinger solvers can be combined with Non-equilibrium Green’s Function (NEGF) Approach in order to model ballistic quantum transport in 3D devices with strong transverse confinement.
An alternative approach to modeling subband transport in nanoscale devices is given by Mode-Space Drift-Diffusion Model, which combines transverse Schrodinger with 1D drift-diffusion equations. A quantum moment transport model allows simulation of confinement effects on carrier transport and yet keeps the simplicity of a conventional drift diffusion approach. It also allows quantum confinement effects to be included in the energy balance/hydrodynamic transport model. Quantum 3D also models the effects of oxide tunneling.

## Schrodinger-Poisson

To model the effects of quantum confinement, Quantum 3D allows the self-consistent solution of the 1D or 2D Schrodinger and 3D Poisson equations. The eigen energies and wave functions obtained are used to find the quantum electron density, which is plugged into a 3D Poisson equation. Fast convergence is achieved by utilizing a predictor-corrector scheme. In addition to a real space 2D Schrodinger solver, a fast product-space solver is available, which finds 2D wave functions as a linear combination of products of 1D solutions in two directions. This method is an order of magnitude faster and allows Atlas to simulate larger devices with simple rectangular geometries.

## Ballistic Quantum Transport with Non Equilibrium Green’s Function Approach

Silicon nanowires have recently become feasible candidates for the channel of nanoscale MOSFETs. The cylindrical geometry of a nanowire allows perfect electrostatic control by the surrounding gate and therefore allows further downsizing of device dimensions into the nanometre scale. In order to model electron transport in nanowire transistors and other structures with strong transverse confinement, Quantum 3D provides a new quantum mechanical model based on the Non Equilibrium Green’s Function (NEGF) approach. This is a fully quantum mechanical approach, which treats such effects as source-to-drain tunneling, ballistic transport and quantum confinement on an equal footing. The new NEGF solver is able to model ballistic quantum transport in 3D devices such as Surround Gate MOSFET. The modeling starts with a solution of a 2D Schrodinger equation in the transverse slices of the device in order to find eigenfunctions and eigen energies. The NEGF quantum transport equations are then solved for electron densities and currents in various subbands (modes), propagating from source to drain. In general, Coupled Mode Space (CMS) method is used to account for mixing of electron modes. In the simpler case of uniform cross-section, a much faster Uncoupled Mode Space (UMS) method can be used.

Schematics (left) and I-V characteristics (above) of a Si nanowire transistor with uniform channel cross-section, computed with uncoupled mode space NEGF approach. |

## Drift-Diffusion Mode-Space Model

The Drift-Diffusion Mode-Space model (DDMS) is a semiclassical approach to transport in devices with strong transverse confinement and is a simpler alternative to mode-space NEGF approach. Similarly to the mode-space NEGF, the solution is decoupled into 2D Schrodinger equation in transverse direction and 1D transport equations in each subband. In this model, however, a classical drift-diffusion equation is solved instead of a quantum transport equation. Thus, the model captures quantum effects in transverse direction and yet inherits all familiar Atlas models for mobility, recombination, impact ionization and band-to band tunneling. The model allows additional output of 1D subband-resolved quantities such as eigen energies, carrier densities, currents, quasi-Fermi levels and generation-recombination rates.

The figure shows eigen energies of several electron subbands (black), quasi-Fermi level of the lowest electron subband (blue) and quasi-Fermi level of the highest hole subband (red) in a FinFET under drain and gate bias of 0.5 V. |
Id-Vd characteristics of FinFET, computed by DDMS model. A concentration and field dependent mobility model was assumed. |

## Bohm Quantum Potential

The Bohm Quantum Potential (BQP) model includes quantum effects by modifying classical drift-diffusion or hydrodynamic calculations with a position dependent quantum potential. The model is derived from Bohm interpretation of quantum mechanics, but retains two fitting parameters. This flexibility allows the model to be calibrated with Schrodinger-Poisson calculations to approximate the quantum nature of carriers in all three dimensions. The effects of quantum confinement will then naturally be contained in device I-V characteristics.

Typical FinFet device structure used for a simulation using Bohm quantum potential. | A surface of constant electron Bohm quantum potential, with a value of -0.10 V. This results in significant reduction in electron density, and it is localized near the perimeter of the channel. The cutplane corresponding to the results is also shown. |

Electron concentration in the middle of the FinFET channel The effect of the EBQP is to reduce the electron density around the edges of the channel, where EBQP is negative and increase electron density in the center, where EBQP is positive. |
Drain characteristics for a gate bias of 0.9 V for the FINFET. The effect of using the BQP model is to reduce the drain current, compared to the current obtained when quantum confinement is not included. |

Rev. 110613_04