Iwpsd Poster V1

Channel Quantization Models for NanoScale MOSFETs

A. Appaswamy, M. Bajaj*, R. Pandey, S. S. Fukay1 and K. V. R. M. Murali IBM Semiconductor Research and Development Center, Bangalore-54, India IBM Semiconductor Research and Development Center, Essex Junction, VT, USA *Email: [email protected]

Motivation: Predictive Quantization Channel Quantization

Modeling

Modeling Channel Quantization: Implementation of MLDA Model Implementation of MLDA Model in TCAD Simulator

of

Channel

• Charge Confinement in 2D Channel Under Strong Inversion • Drastic Modification of Band Structure and Electron Distribution Channel

Quantum corrected density is formulated in terms of an equivalent quantum potential

gate source n+

SiO2 channel

Fermi Integral of order 1/2

drain

IBM proprietary device simulator FIELDAY is used for TCAD modeling.

n+

FIELDAY

Drift-Diffusion Equations:

p-type substrate

• Electro-thermal device simulator solves the 6 coupled PDEs using finite volume

SiO2 Metal

Electron Density Profile Electron Density

• Inversion Capacitance • Threshold Voltage (Vt) • Sub-Threshold Slope

Classical

Current Density Equation

Quantum Corrected

Distance from the interface

TCAD Simulation: MLDA Model Calibration for MOSCap PolySilicon

Triangular Wall Approximation (Analytical Solution)

Schrodinger Equation SiO2

Gate

tox = 2 nm

Substrate

E1 E0

Poisson Equation

Recombination Rate

Fast but typically not predictive – Need repeated calibration

Modeling Channel Quantization: SchrodingerPoisson Solution chrodinger-Poisson Solution Accurate solution using:

Generation Rate

method. • Solves for Carrier (hole and electron) mass and energy transport • Solves 1D, 2D and 3D semiconductor structures

Ef Depth Under Interface Electrical Potential

Numerical Self-Consistent Solution for Non-linear Potential Distribution

n-type MOS Capacitor

Electron Density Hole Density

Electron concentration as a function of depth under interface

Donor Concentration Acceptor Concentration

• Accurate and Predictive • Convergence Issues • Computationally Intensive – ‘Slow’

Charge Profile – Good Fit between Schrodinger-Poisson and MLDA Predictions

Fixed Charge Density Material Permittivity

TCAD Simulation: MLDA Model Predictions

Wave Function Effective Electron Potential Electron Energies

Source

• Electrons treated homogeneous locally. • Constant potential replaced by spatially varying potential. • Applicability for slowly varying potential

Potential

Modeling Channel Quantization: Local Density Approximation (LDA) ocal Density Approximation (LDA)

Lg = 40 nm

Drain

Tox = 0.7 nm

Insulator

Schrodinger Equation

Distance

40 nm Floating Body nFET on SOI Triangular Wall Approximation (Analytical Solution)

V(r) = Slowly varying potential

SiO2

Solving Schrodinger Equation for

TCAD Simulation: MLDA Model Predictions

E1 E0

Carrier density near the Si/SiO2 interface:

Ef Depth Under Interface

Effective density of states in conduction band Fermi Level

Abrupt Change in Potential at semiconductor-insulator Interfaces: LDA not Applicable

Modeling Channel Quantization: Modified Local Density Approximation dified Local Density Approximation (MLDA) SiO2

Transfer characteristics of 40 nm gate length nFET with substrate doping scaled by 4 X.

Transfer characteristics of 40 nm floating gate pFET. The MLDA model matches the SP solution within 10%

• Tox – 0.7 nm (40nm nFET), Model Calibrated for 2 nm Tox (MOSCap) • ~75% Less Computational Time for MLDA Simulations

Triangular Wall Approximation

Schrodinger Equation

The MLDA solution is within 10% of the SP solution across all the gate and drain biases.

Conclusions and References Conclusions

E1 E0

Solving Schrodinger Equation for

Ef Depth Under Interface

• Excellent match of MLDA model predictions to Schrodinger-Poisson solution for both the nFET and pFET structures • Verified predictive nature of MLDA model for varying oxide thicknesses and substrate doping

References Carrier density near the Si/SiO2 interface:

Effective density of states in conduction band Fermi Level Bessel Function of zeroth order Distance from Si/SiO2 interface

TEMPLATE DESIGN © 2007

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1. G. Paasch, and H. Ubensee,” A modified local density approximation,” Physica Status Solidii (b), vol. 113, pp. 165-178, 1982. 2. W. Hansch, T. H. Vogelsang, R. Kircher, and M. Orlowski,” Carrier transport near the Si/SiO2 interface of a MOSFET,” Solid State Electronics, vol. 32, no. 10, pp. 839-849, 1989.