^{1}, Eric Polizzi

^{2}and Mark Lundstrom

^{3}

### Abstract

The siliconnanowiretransistor (SNWT) is a promising device structure for future integrated circuits, and simulations will be important for understanding its device physics and assessing its ultimate performance limits. In this work, we present a three-dimensional (3D) quantum mechanical simulation approach to treat various SNWTs within the effective-mass approximation. We begin by assuming ballistic transport, which gives the upper performance limit of the devices. The use of a mode space approach (either coupled or uncoupled) produces high computational efficiency that makes our 3D quantum simulator practical for extensive device simulation and design. Scattering in SNWTs is then treated by a simple model that uses so-called Büttiker probes, which was previously used in metal-oxide-semiconductor field effect transistor simulations. Using this simple approach, the effects of scattering on both internal device characteristics and terminal currents can be examined, which enables our simulator to be used for the exploration of realistic performance limits of SNWTs.

This work was supported by the Semiconductor Research Corporation (SRC) and the National Science Foundation (NSF) Network for Computational Nanotechnology (NCN). The authors would like to thank Professor Supriyo Datta, Anisur Rahman, Dr. Avik Ghosh, Dr. Ramesh Venugopal (currently at Texas Instruments), Jing Guo, and other group members for their sincere help.

I. INTRODUCTION

II. THEORY FOR BALLISTICSILICONNANOWIRETRANSISTORS

A. The coupled mode space approach

B. The uncoupled mode space approach

C. A fast uncoupled mode space approach

III. RESULTS FOR BALLISTICSILICONNANOWIRETRANSISTORS

A. Benchmarking of the FUMS approach

B. Device physics and characteristics

IV. TREATMENT OF SCATTERING WITH BÜTTIKER PROBES

A. Theory

B. Results

V. SUMMARY

### Key Topics

- MOSFETs
- 21.0
- Nanowires
- 20.0
- Ballistics
- 13.0
- Fermi levels
- 10.0
- Trajectory models
- 10.0

## Figures

The simulated SNWT structures in this work. (a) A schematic graph of an intrinsic SNWT with arbitrary cross sections (for clarity, the substrate is not shown here). (b) The grid used in the simulation of SNWTs. (c) The cross sections of the simulated triangular wire (TW), rectangular wire (RW), and cylindrical wire (CW) FETs. is the silicon body thickness, is the silicon body width, and is the wire width. For the TW, the direction normal to each gate is ⟨111⟩, so the channel is ⟨101⟩ oriented. In contrast, for the channel of the RW, both ⟨101⟩ and ⟨100⟩ orientations are possible. For the CW, we assume the channel to be ⟨100⟩ oriented.

The simulated SNWT structures in this work. (a) A schematic graph of an intrinsic SNWT with arbitrary cross sections (for clarity, the substrate is not shown here). (b) The grid used in the simulation of SNWTs. (c) The cross sections of the simulated triangular wire (TW), rectangular wire (RW), and cylindrical wire (CW) FETs. is the silicon body thickness, is the silicon body width, and is the wire width. For the TW, the direction normal to each gate is ⟨111⟩, so the channel is ⟨101⟩ oriented. In contrast, for the channel of the RW, both ⟨101⟩ and ⟨100⟩ orientations are possible. For the CW, we assume the channel to be ⟨100⟩ oriented.

The 2D modes [the square of the modulus of the electron wave functions in the (010) valleys] in a slice of (a) triangular wire (TW), (b) rectangular wire (RW), and (c) cylindrical wire (CW) transistors. For clarity, the substrates for TW and RW FETs are not shown here.

The 2D modes [the square of the modulus of the electron wave functions in the (010) valleys] in a slice of (a) triangular wire (TW), (b) rectangular wire (RW), and (c) cylindrical wire (CW) transistors. For clarity, the substrates for TW and RW FETs are not shown here.

The electron subband profile in a cylindrical SNWT with gate length ( and ). The numbers of nodes in the direction is equal to 128. The silicon body thickness [as shown in Fig. 1(c)] is , and the oxide thickness is . The source∕drain doping concentration is and the channel is undoped (the channel region is located from ). The solid lines are for the approximation method (solving a 2D Schrödinger equation only once) used in the FUMS approach, while the circles are for the rigorous calculation (solving 2D Schrödinger equations times) adopted in the UMS and CMS approaches.

The electron subband profile in a cylindrical SNWT with gate length ( and ). The numbers of nodes in the direction is equal to 128. The silicon body thickness [as shown in Fig. 1(c)] is , and the oxide thickness is . The source∕drain doping concentration is and the channel is undoped (the channel region is located from ). The solid lines are for the approximation method (solving a 2D Schrödinger equation only once) used in the FUMS approach, while the circles are for the rigorous calculation (solving 2D Schrödinger equations times) adopted in the UMS and CMS approaches.

The vs curves for a cylindrical SNWT in logarithm (left) and linear (right) scales . The device structure is the same as that in Fig. 3. The crosses are for the CMS approach, the circles are for the UMS approach, and the dashed lines are for the FUMS approach.

The vs curves for a cylindrical SNWT in logarithm (left) and linear (right) scales . The device structure is the same as that in Fig. 3. The crosses are for the CMS approach, the circles are for the UMS approach, and the dashed lines are for the FUMS approach.

The computed LDOS [in ] and electron subbands (dashed lines) of a ballistic cylindrical SNWT with gate length and Si body thickness (the channel region is located from to and the details of the device geometry are described in Fig. 3 caption) ( and ).

The computed LDOS [in ] and electron subbands (dashed lines) of a ballistic cylindrical SNWT with gate length and Si body thickness (the channel region is located from to and the details of the device geometry are described in Fig. 3 caption) ( and ).

The 1D electron density profile along the channel of the simulated cylindrical SNWT (the channel region is located from to and the details of the device geometry are described in Fig. 3 caption). The solid line is for while the dashed line is for ( and ).

The 1D electron density profile along the channel of the simulated cylindrical SNWT (the channel region is located from to and the details of the device geometry are described in Fig. 3 caption). The solid line is for while the dashed line is for ( and ).

The transmission coefficient and electron subbands in the simulated cylindrical SNWT (the channel region is located from to and the details of the device geometry are described in Fig. 3 caption) ( and ).

The transmission coefficient and electron subbands in the simulated cylindrical SNWT (the channel region is located from to and the details of the device geometry are described in Fig. 3 caption) ( and ).

The vs curves for the triangular wire (TW) FET with ⟨101⟩ oriented channels, rectangular wire (RW) FET with ⟨101⟩ oriented channels and cylindrical wire (CW) FET with ⟨100⟩ oriented channels. . All the SNWTs have the same silicon body thickness , oxide thickness , gate length , and gate work function . The Si body width of the RW is . In the calculation of the TW and RW FETs, whose channels are ⟨101⟩ oriented, the effective masses of electrons in the (100) and (001) valleys are obtained from Ref. 22 as , , and .

The vs curves for the triangular wire (TW) FET with ⟨101⟩ oriented channels, rectangular wire (RW) FET with ⟨101⟩ oriented channels and cylindrical wire (CW) FET with ⟨100⟩ oriented channels. . All the SNWTs have the same silicon body thickness , oxide thickness , gate length , and gate work function . The Si body width of the RW is . In the calculation of the TW and RW FETs, whose channels are ⟨101⟩ oriented, the effective masses of electrons in the (100) and (001) valleys are obtained from Ref. 22 as , , and .

A generic plot of the 1D device lattice (solid line with dots, along the direction) for a SNWT with the Büttiker probes attached. Each probe is treated as a virtual 1D lattice (dashed line with dots, along the direction) that is coupled to a node in the device lattice. The coupling energy between this virtual lattice and the node with which it is attached to is , and that between two adjacent device lattice nodes is . The probe Fermi levels are labeled as .

A generic plot of the 1D device lattice (solid line with dots, along the direction) for a SNWT with the Büttiker probes attached. Each probe is treated as a virtual 1D lattice (dashed line with dots, along the direction) that is coupled to a node in the device lattice. The coupling energy between this virtual lattice and the node with which it is attached to is , and that between two adjacent device lattice nodes is . The probe Fermi levels are labeled as .

The computed LDOS [in ] and electron subbands (dashed lines) of a dissipative cylindrical SNWT with gate length and Si body thickness (the channel region is located from to and the details of the device geometry are described in Fig. 3 caption). ( and ). The mobility is and the channel mobility is .

The computed LDOS [in ] and electron subbands (dashed lines) of a dissipative cylindrical SNWT with gate length and Si body thickness (the channel region is located from to and the details of the device geometry are described in Fig. 3 caption). ( and ). The mobility is and the channel mobility is .

The vs curves for a cylindrical SNWT with gate length and Si body thickness (the details of the device geometry are described in Fig. 3 caption) in logarithm (left) and linear (right) scales . The dashed lines are for the ballistic limit while the solid lines are for the case with scattering [i.e., the mobility is and the channel mobility is ].

The vs curves for a cylindrical SNWT with gate length and Si body thickness (the details of the device geometry are described in Fig. 3 caption) in logarithm (left) and linear (right) scales . The dashed lines are for the ballistic limit while the solid lines are for the case with scattering [i.e., the mobility is and the channel mobility is ].

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