Semiconductor Device Noise Models Based on Semiclassical Transport 51153

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Semiconductor Device Noise Models Based on
Semiclassical Transport 5115-3

• Can E. Korman
• Department of ECE
• The George Washington University
• Washington, DC 20052
• Collaborators
• A. Piazza (Applied Wave Research, Inc.)
• I. D. Mayergoyz (University of Maryland)

Noise Information in Nanoelectronics, Sensors
Standards SPIE's First International Symposium on
Fluctuations and Noise - Santa Fe, NM - June 1-4,
2003
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Outline of the Talk

• Introduction
• Semiclassical Transport Theory
• Stochastic Differential Equations Theory
• Terminal Current Spectral Density Computation
• Numerical Results
• Numerical Techniques
• Conclusion

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1- Introduction

• Motivation
• Noise characterization of devices
• Goal Connecting Electron Kinetics to Terminal
  Noise Characteristics
• Related noise calculation techniques

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Motivation

• Low power low voltages and currents
• Low voltage or current noise becomes critical
  problem
• Accurate noise modeling is an essential element
  in modern device simulators

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Noise Characterization of DevicesTerminal Noise
Spectral Density

• Thermal noise
• GR noise
• Flicker or 1/f noise
• Shot noise

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Connecting Electron Kinetics to Terminal Noise
Characteristics

• Comprehensive approach for device noise
  characterization Electron kinetics is described
  by the Semiclassical Transport Theory

• Develop noise models in the framework of
  Semiclassical Transport Theory suitable for
  efficient numerical simulation and CAD

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Connecting Electron Kinetics to Terminal Noise
Characteristics
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Related Noise Calculation Techniques

• Probabilistic approach reasonable assumption on
  the underlying microscopic process
• Langevin Method add random source to
  deterministic differential equation
• Monte Carlo Method Direct simulation of random
  electron kinetics - computationally intensive
• SDE theory based model Connect electron
  scattering characteristics to terminal noise
  spectral density

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2- Semiclassical Transport Theory

• The adiabatic principle allows to separate
  crystal vibration from electron motion
• The independent electron approximation treats
  electron-electron interactions by one electron
  effective potential
• Each electron moves independently in a perfectly
  periodic potential due to lattice and all other
  electrons
• Scattering mechanisms describe interaction of
  electrons with the lattice

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2- Semiclassical Transport Theory (cont.)

• Semiclassical Transport Theory predicts how, in
  the absence of collisions, the position and wave
  vector of each electron evolve in the presence of
  macroscopic electric and magnetic fields
• Maxwells Equations describe the macroscopic
  fields
• Electron position and wave vector correspond to a
  wave packet of Bloch electrons
• Scattering causes random sudden jumps in the
  electron wave vector

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2- Semiclassical Transport Theory (cont.)

• Electron scattering from one wave vector to
  another is characterized by the transition
  probability function
• Paulis exclusion principle limits the scattering
  rates
• Such a process can be described in terms of
  occupation probability that satisfies Boltzmann
  transport equation (BTE)

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Motion of wave packet of Bloch electrons

• All distinct electron states are confined to a
  unit cell in reciprocal space first Brillouin
  zone
• Velocity of wave packet

• Force due to External Fields - Wave packet
  behaves classically

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Macroscopic External Fields Poissons Equation

• Electrostatic potential ?
• Electric field E
• Conduction valance band electron distributions
  f C and fV
• Ionized impurity densities of donors acceptors
  ND and NA

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Random Force Electron Scattering

• Almost instantaneous changes in the value of the
  electron wave vector
• Phonons deformation potential
• Thermal generation-recombination Shockley Read
  Hall generation-recombination model
• Ionized impurities screened Coulomb potential
• Phonon scattering elastic, inelastic, isotropic,
  unisotropic, intervalley, intravalley, etc

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Scattering Transition in k-space

• Scattering probability per unit time

• Transition rate Born-Von Karman boundary
  conditions

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Paulis Exclusion Principle Scattering
transition from k to k

• Transition into an occupied level is forbidden

• Transition rate into partially filled group of
  level

• Steady-state transition rate

• f0 steady-state electron disribution function

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Electron motion in reciprocal-space
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Boltzmann Transport Equation (BTE) in terms of
electron occupation probability f0
LHS denotes flux in phase space

• RHS denotes flux due to scattering events

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Summary of Semiclassical Transport Model

• Electrons drift due to macroscopic electrostatic
  field
• Electrons scatter due to deviations from perfect
  periodicity of potential
• Paulis exclusion principle limits scattering
  rates
• Semiclassical transport embedded in Poissons and
  Boltzmann transport equations

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Electron Kinetics Stochastic Differential
Equation (SDE) Electron motion can be
interpreted as a differential equation driven by
inhomogeneous randomly weighted Poisson process
SDE

• Scattering rate (Poisson)
• Transition rate

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3 - Stochastic Differential Equations (SDE) Theory

• Differential equations describing processes
  driven by random excitation
• Wiener processes (continuous in time) - diffusion
• Poisson processes (discontinuous in time) -
  scattering
• These are continuous-state Markov processes
• Process is completely characterized by its
  transition probability density function
• Focker-Planck Equation
• Kolmogoroff-Feller equations

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Stochastic Differential Equation General Case

• Differential equation driven by stochastic
  process j(y,t)

• Stochastic process j(y,t) is characterized by

• Intensity function (scattering rate)
• Transition rate

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Forward Kolmogoroff-Feller Equation (KFE)
General Case

• The stochastic process y is characterized by the
  transition probability density function

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Kolmogoroff-Feller Equation for Electron Motion
Described by Semiclassical Transport Theory
The electron transition probability density
function ? satisfies
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3- Summary of SDE

• Electron motion corresponds to Markov process
• Process completely characterized by transition
  probability r
• Transition probability function satisfies KFE
• KFE identical to linear BTE
• All stochastic information on electron kinetics
  is embedded in KFE or equivalently, linear BTE

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4 - Terminal Current Spectral Density Computation
Goal compute spectral density of the terminal
current

• Calculate the transition probability function r
• Relate electron motion in phase-space state to
  the induced current at the device terminals
  Ramo-Schockley theorem
• Combine the results of Steps (1) and (2) to
  calculate the autocovariance function or
  spectral density of induced terminal noise.

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Ramo-Shockley Theorem

• The extension of Ramos theorem by Cavalleri et
  al establishes a connection between induced
  terminal currents and motion of carriers inside a
  device
• The model admits arbitrary mobile charge
  distribution and conduction current density

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Ramo-Shockley Theorem (cont.)

• The model shows that for terminal potentials Vj
  and terminal currrents Ij

E(0) is the electrostatic field due exclusively
to the applied potentials at the terminals j1,
, n
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Ramo-Shockley Theorem (cont.)

• At contact j the induced terminal current due to
  each electron with position x and wave vector k
  is

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Autocovariance function computation of terminal
current
By definition, the autocovariance function of the
observed terminal current is
In terms of the Ramo-Shockley function ij and an
auxiliary function g it can be shown that the
autocovariance function of terminal current j is
as follows
where ? is the transition probability density
function and f is the steady-state electron
distribution function
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Autocovariance function computation(cont.)

• Terminal current autocovariance function
• Auxiliary function g satisfies the transient BTE
• Subject to the following special initial
  conditions

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Current Spectral Density Computation

• The spectral density is defined as the Fourier
  transform of the autocovariance function
• Employing this definition it can be shown that
  the spectral density can be calculated in terms
  of an auxiliary function G

• G satisfies the Fourier transform of the
  transient BTE

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4 Summary of Terminal Current Spectral Density
Computation

• Key computations for the terminal current
  spectral density function are reduced to solution
  of Kolmogorov-Feller Equation or Boltzmann
  Transport Equation
• Ramo-Schockley theorem directly connects electron
  motion to current induced at the device terminals
• A computationally efficient approach in terms of
  the effective distribution function G(x,k,w) was
  presented

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5 Sample Numerical Results

• Calculations for n-doped bulk silicon device
• Thermal noise
• Generation-Recombination noise
• Employ Spherical harmonics technique for
  numerical calculations

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Spectral Density Thermal Noise
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Autocovariance Thermal Noise
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Autocovariance Thermal Noise
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Spectral Density GR Noise
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Spectral Density GR Thermal Noise
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5 - Summary

• Simulation results are presented for bulk n-type
  silicon as a function of
• Electric field
• Temperature
• Electron capture cross section
• Results for thermal and Generation-Recombination
  noise are presented
• Numerical results are in agreement with measured
  data
• Results are obtained using computationally
  efficient Spherical Harmonic technique

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6 - Numerical Implementation

• Problem Steady state solution of Poisson and
  non-linear Boltzmann transport equations
• Problem Transient solution of linear BTE with
  special initial condition in frequency domain
• The BTE/KFE is a non-linear 7D equation (3D
  space, 3D momentum and 1D time)
• Spherical harmonics technique is employed to
  reduce the number of dimensions in momentum space

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Mathematical techniques

• Poissons and non-linear Boltzmann transport
  equations are consistently solved employing
  Gummel block iteration method
• BTE is iteratively solved in real-space
• At a given point in real space, 3D non-linear
  equation in reciprocal-space is solved using
  spherical harmonics
• No time iteration is required equation is solved
  in frequency domain

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Spherical Harmonics Polynomials

• Scattering involves transitions between spheres
• Spherical harmonics give a more efficient
  representation of the distribution function

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Gummel Iteration
start
Poissons equation
BTE
stop
n
y
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Real-space iteration
start
BTE at xi
i0
i i1
n
stop
Last i
y
n
y
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6 Summary of Numerical Techniques

• The approach requires the numerical solution of
• the steady-state Poisson equation
• the steady-state non-linear Boltzmann transport
  equation
• the transient linear Boltzmann transport equation
  
• Gummel iteration used to decouple Poisson and
  Boltzmann transport equations
• BTE solved in space iteratively
• BTE solved in reciprocal-space using spherical
  harmonics
• Transient solution performed in frequency domain

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7 - Conclusion

• Unified approach to model noise in the framework
  of semiclassical transport theory and stochastic
  differential equations
• Noise is exclusively due to electron scattering
• Model accounts for Paulis exclusion principle
• Random electron motion in phase-space directly
  connected to induced terminal current Ramos
  theorem
• Adapted spherical harmonics technique to solve
  transient BTE equation in frequency domain
• Numerical solution approach based on well
  established techniques practical for CAD

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Thank you for your attention

• Questions?

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Reference Publications and Presentations

• C. E. Korman and I. D. Mayergoyz, Semiconductor
  Noise in the Framework of Semiclassical
  Transport,'' Phys. Rev. B, Vol. 54, No. 24, pp.
  17620-17627, 15 December, 1996.
• A. J. Piazza, C. E. Korman and Amro M. Jaradeh,
  A Physics Based Semiconductor Noise Model
  Suitable for Efficient Numerical Implementation,
  IEEE Tran. On CAD of Integrated Circuits and
  Systems, Vol. 18, No. 12, pp. 1730-1740, December
  1999.
• A. J. Piazza and C. E. Korman, Semiconductor
  Noise Computation Based on the Deterministic
  Solution of Poisson and Boltzmann Transport
  Equations, VLSI Design, 1998, vol. 8, Nos. (1-4),
  pp. 381-385.

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Reference Publications and Presentations (cont.)

• C. E. Korman and A. J. Piazza, Computation of
  Semiconductor Noise for Semiclassical Transport,
  proceedings of the International Semiconductor
  Device Research Symposium, Charlottesville,
  Virginia, 1995.
• A. J. Piazza and C. E. Korman, Computation of the
  Spectral Density of Noise in Bulk Silicon Based
  on the Solution of the Boltzmann Transport
  Equation, VLSI Design, 1998, Vol. 6, Nos. (1-4),
  pp. 185-189.