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## Nanowire

Simulate 3D nanowire transport in the effective mass approximation with phonon scattering and 3D Poisson self-consistent solution

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#### Abstract

Silicon nanowire transistors are promising device structures for future integrated circuits. Short channel effects are becoming more and more important in the nanoscale regime, and therefore effective gate control will be necessary to achieve good device performance. Devices based on silicon nanowires can be manufactured with multigate and gate-all-around transistors and you can explore them with this tool.

In contrast to planar MOSFETs which have uniform charge and potential profiles in the transverse direction (i.e., normal to both the gate and the source-drain direction), a silicon nanowire transistor has a genuinely 3D distribution of electron density and electrostatic
potential. Therefore self-consistent 3D simulations are mandatory, and you run them with this tool. One of the transport models assumes ballistic transport, which gives the upper performance limit of the devices. The effective-mass mode space approach (either coupled or uncoupled) produces high computational efficiency that makes this
simulator practical for extensive device simulation and
design.

In the previous version of nanowire tool, it can only dealt with ballistic transport. However, the currently fabricated nanowire transistors usually have length of several hundreds of nanowires, which is obviously in the region of dissipative transport. Even in the region of sub-100 nm, the current is far below its ballistic limit, only 50% of ballistic limit. All of these evidences indicate the important role of phonon scattering in the carrier transport of nanowire transistor. This is the main driven force prompting us to release this new version.

In this new version of nanowire tool, we assume that the phonon system is in thermal equilibrium state, and we use the self-consistent Born approximation to calculate phonon-electron interaction.

For intra-valley phonon scattering self-energy, although there exists the anisotropy of the deformation-potential interaction between electrons and acoustic phonons, we assume the usual scalar deformation potential for the intra-valley phonon scattering. If we use the scalar deformation potential, the matrix element vanishes for the transverse acoustic modes, and the matrix element for the longitudinal acoustic (LA) mode remains. Since the energy of LA phonon is small, we treat it as elastic scattering in our program.

Electron transitions between states in two different equivalent valleys can be induced by both acoustic and optical phonon scatterings. In our program, we also take this kind of inter-valley phonon scattering into account.

Since in this new version, we introduce phonon scattering into it, the computation complexity increases thus lead to a much longer simulation time. In order to reduce the simulation time, the C++ program is parallelized using MPI and OpenMP.

The mode space approach treats quantum confinement and transport separately. The simulations you can perform consist of the following steps:

Solve the 3D Poisson equation for the electrostatic potential.
Solve the 2D Schrodinger equation with closed boundary conditions for each cross section (or slice) of the nanowire transistor to obtain the electron subbands (along the nanowire) and eigenfunctions.
Solve the coupled or uncoupled nonequilibrium Green function (NEGF) transport equations for the electron charge density.
If phonon scattering is switched on in the simulation, tthe self-consistent Born approximation to calculate phonon-electron interaction.
Go to step (1) to calculate the electrostatic potential. If the self-consistent loop has converged, calculate the electron current using the NEGF approach and show the results.

In summary you can use four transport models:

Uncoupled mode space with ballistic transport. This is the fastest option.Default setting would take 30 min per bias point
Uncoupled mode space with phonon scattering. This option takes much more time. Default setting would take about 45 min per bias point
Coupled mode space with ballistic transport. Due to the coupling of the modes, this option is also much slower than the first one. But no worries; you can start the simulation and login back later to check the results.
Coupled mode space with phonon scattering. This is the most time consuming job but will give you the most accurate results.

Improvements / modifications in subsequent releases:

- 0 – It is a totally new version. The engine was replaced by simsn which can treat carrier transport with phonon scattering. The simulations now are MPI and OpenMP hybrid parallelized to save time. Also this new version can simulate four kinds of semiconductor materials: Si, Ge, GaAs and InAs. Meanwhile, the GUI of the new version keeps almost unchanged.

- 1 – Enabled quick runs for Uncoupled mode.

- 0.2 – Added E Vs T(E) curve

- 0.1 – Fixed for error in units 1D electron density

- 0 – Improvements in Rappture Input – Added mesh fineness factor, Added different orientation < 100>,< 110> & < 111>, Added material properties, Added different transport orientation, Added pre-run examples- Channel formation, Uncoupled Mode Space with averaging, Coupled Mode Space, Uncoupled Mode Space with scattering. Improvements in Rappture Output-3D Eigenfunctions,3D potential,3D electron density,3D Density of states,1D electron density,1D Potential profile in sequence for each bias point.

Tool Limitations Based on Effective Mass, not using exactly band structure. Phonon dispersion is treated as bulk case. Using mode space approach to save time, but lose some accuracy.

#### Powered by

SIMSN

#### Credits

This tool is based on the work of Hong-Hyun Park.He has developed this effective mass NEGF simulator SIMSN in his Ph.D period. Lang Zeng has contribution to the GUI development.

Hong-Hyun Park | ... Core C++ simulator, beginning with Version 3.0 |

Lang Zeng | ... GUI development, Code Matainance and improvements |

Siqi Wang, Matthew Buresh | ... summer intern on the GUI development |

Other authors | ... developed the previous version of this tool and some parts of the old scripts is still in use. |

#### Sponsored by

NCN@Purdue, MSD FCRP, SRC

One of the authors (Lang Zeng) is sponsored by Chinese Scholarship Council.

#### Publications

*PHYSICAL REVIEW B*

**82**(4), pages 125328, 2010.

*Journal of Applied Physics*

**105**(4), pages 023712, 2009.

#### Cite this work

Researchers should cite this work as follows:

- Hong-Hyun Park and Gerhard Klimeck, "Quantum approach to electronic noise calculations in the presence of electron-phonon interactions"
*PHYSICAL REVIEW B***82**(4), pages 125328, 2010. - Hong-Hyun Park, Seonghoon Jin, Young June Park, and Hong Shick Min, "Quantum simulation of noise in silicon nanowire transistors with electron-phonon interactions"
*Journal of Applied Physics***105**(4), pages 023712, 2009.