A Fully Physical Model for Leakage Distribution under Process Variations in Nanoscale Double-Gate CMOS Liu Cao Lin Li

Outline Introduction Nanoscale Double-Gate Background Double-Gate Leakage Model Leakage Distribution Multiple-Device Stacks

Introduction Multi-gate technologies are the forerunners to replace bulk CMOS.

Why multi-gate is better? Thicker gate insulator Superior short channel characteristics Improved mobility in the undoped body Elimination of random dopant effects

Double-Gate vs Bulk

Process Variation in Circuit Design

Which model should we use? Compact SPICE-like models [3, 2]? Analytical Model: Semi-physical Vth [11, 4, 5]? Physical and Analytical Model: This paper.

Contribution of this paper Derive a fully physical model for double-gate leakage Develop a model for single device leakage distribution due to gate length and body thickness variations Extend the model to multiple-device stacks

Outline Introduction Nanoscale Double-Gate Background Double-Gate Leakage Model Leakage Distribution Multiple-Device Stacks

Ideal Double-Gate MOSFET Structure

Physical Model of this paper In this paper, only symmetric tox , VGS and ∆ΦMS model is considered

Short Channel Effect (SCE) The 2-D potential distribution is derived as a solution of Poisson’s equation [9].

Quantum Confinement Effect (QCE) QCE leads to quantization of energy levels, and finite probability of occupation of these levels Both the wave functions and quantized energy levels are solutions of the 1-D Schrodinger’s equation.

Process Variations tox and ∆ΦMS are maintained the nominal values. L and t have variations.

Outline Introduction Nanoscale Double-Gate Background Double-Gate Leakage Model Leakage Distribution Multiple-Device Stacks

Subthreshold Diffusion Current In which nc (x, y) represents the effective carrier concentration:

SCE – Electrostatic potential a [9] Poisson’s Equation: With the boundary conditions

Derivation for a Using superposition, the electrostatic potential in DG can be written as: Where vxx is the 1-D solution to equation Sssssssssssss are the solutions to satisfy the top and bottom boundary conditions

Dielectric Boundary Conditions Take ratios of these equations and get the Eigen Value Equation

Fully potential expression in Subthreshld region

Derivation for Subthreshold current The current density can be written as: Integrating in x and z directions gives: Where Is the inversion charge per gate area.

Final Subthreshold current [9] Current continuity requires Ids be independent of y.

QCE – intrinsic carrier concentration [13] It is calculated from the density of states, and the probability of occupation based on the 1-D wavefunction solution.

Compact Solution for Leakage Current KEY: Isub can be captured by accounting for the potential at the center of the ultra-thin body (x=0), and at the top of the source-drain barrier (y=ytop).

Verification for the Analytical Model Worst case (WC) means highest leakage Best case (BC) means lowest leakage

Outline Introduction Nanoscale Double-Gate Background Double-Gate Leakage Model Leakage Distribution Multiple-Device Stacks

Leakage distribution Expand as 2-D Taylor series

Leakage distribution(cont.) is the i-th partial derivative of with respect to L

Leakage distribution(cont.) Thus, we have: where and The expectation of this random variable can be calculated as:

Leakage distribution(cont.)  kth central moment of a real-valued random variable X is the quantity μk := E[(X − E[X])k], where E is the expectation operator. For a continuous univariate probability distribution with probability density function f(x) the moment about the mean μ is

Leakage distribution(cont.) Assuming independent distribution, if L and tsi are distributed normally, then their central moments can be evaluated as: the variance of

Leakage distribution(cont.)

1.4

Outline Introduction Nanoscale Double-Gate Background Double-Gate Leakage Model Leakage Distribution Multiple-Device Stacks

Multiple-Device Stacks 1

Multiple-Device Stacks(cont.) Expand as 3-D Taylor Series

Multiple-Device Stacks(cont.)

Multiple-Device Stacks(cont.)

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