Particle-Based Device Simulation - Advanced Device Simulation - Lecture Slides, Slides of Computer Science

Download Particle-Based Device Simulation - Advanced Device Simulation - Lecture Slides and more Slides Computer Science in PDF only on Docsity!

Monte Carlo Device Simulation

(Particle-Based Device Simulation)

Outline

• Introductory Concepts
• Review of Bulk MC Code for Si
• MC Device Simulation
  • Motion in real space
  • Charge assignment and force interpolation
  • Boundary conditions
  • Current calculation
  • Example of Simulation of a MESFET
  • Example of Simulation of a 50 nm Gate Length MOSFET
  • Example of Simulation of 250 nm FibMOS Device
  • Example of inclusion of tunneling via simulation of Si

MESFET devices

Goals of Device Modeling

• Analysis - Simulating the behavior of a device (or

circuits) with a physical model to understand the

dependencies and limiting physical mechanisms in the

device/circuit performance (e.g. effects of noise, limits

on frequency/gain, trap effects, effects of geometry).

• Design - Systematic use of a device/circuit model to

achieve a desired functionality. For device design, and

low level circuit design, the process is mainly an

iterative, trial and error approach prior to actual

physical implementation of a device or a circuit.

Physical Device Simulation

• There are 2 main components in physical device

simulation:

– Charge motion due to driving forces and diffusion

(transport) (1)

– Fields due to charge distribution and motion (i.e

current) (2)

• Analytical solutions are only possible in 1D.

Numerical solutions require discretization of (1)

and (2) above onto a mesh, and solution of

simultaneous algebraic equations

• (1) and (2) must be solved simultaneously (self-

consistently)

Hierarchy of Semiconductor Simulation Models

Quantum Approaches

Boltzmann Equation Monte Carlo Particle Approaches

Moments of Boltzmann Equation (Hydrodynamic and Energy Transport Approaches)

Drift-Diffusion Approaches

Compact Approaches

Normalization of Variables

Standard way of scaling due to de Mari:

Space: Intrinsic L D (N=n i )

Extrinsic L D (N=Nmax)

Potential: Thermal voltage

Carrier density: Intrinsic density N=n i

Extrinsic density N=Nmax

Diffusion Coeff: D 0

Mobility μ

Recomb./Gen. R

Time: T

L (^) D = ε kBT /( e^2 N )

VT = kBT / e

1 cm^2 / s

μ = D 0 / V T

R = D 0 N / LD

0 T L^2 / D = D

Scattering Rates for Silicon

1011

1012

1013

1014

0 0.2 0.4 0.6 0.8 1

Acoustic rate [1/s]

Energy [eV]

1010

1011

1012

1013

1014

0 0.2 0.4 0.6 0.8 1

intervalley zero_g (absorption) intervalley zero_g (emission) intervalley zero_f (absorption) Intervalley rates [1/s]^ intervalley zero_f (emission)

Energy [eV]

Acoustic Deformation Potential Scattering Rate

Zeroth Order Intervalley Scattering Rate

First Order Intervalley Scattering Rate

1010

1011

1012

1013

0 0.2 0.4 0.6 0.8 1

intervalley first_g (absorption) intervalley first_g (emission) intervalley first_f (absorption) Intervalley rates [1/s]^ intervalley first_f (emission)

Energy [eV]

• Initial energy of the carriers
• Scattering process (isotropic)

Ensemble Monte Carlo Technique

t< tmax

start

stop

Initial condition

Scattering table

Drift

Scattering

YES (^) NO

0

100

200

300

400

500

0 0.05 0.1 0.15 0.2 0.25 0.3 0.

Count

Energy [eV]

ln( 1 ) 2

energy^3 k T ran =− B

E '= E ± ω ϕ = 2 π⋅ ran 2 , cos θ= 1 − 2 ⋅ ran 3



 



= θ

= θ ϕ

= θ ϕ

' ' cos

' 'sin sin

' 'sin cos

k k

k k

k k

z

y

x

Monte Carlo Simulation Results for Bulk Silicon

-5x10^6

0

5x10^6

1x10^7

1.5x10^7

2x10^7

2.5x10^7

0 0.5 1 1.5 2 2.5 3 3.5 4

1 kV/cm 5 kV/cm 10 kV/cm 50 kV/cm

time [ps]

Drift velocity [cm/s]

0

0 0.5 1 1.5 2 2.5 3 3.5 4

1 kV/cm 5 kV/cm 10 kV/cm 50 kV/cm Energy [eV]

time [ps]

106

107

1 10 100

Current simulations Yamada simulations Canali exp. data Drift velocity [cm/s]

Electric field [kV/cm]

Time Evolution of Average Kinetic Energy

Time Evolution of Mean Drift Velocity

Mean Drift Velocity vs. Electric Fields

• The extension of the k -space Monte Carlo to simulate semicon-

ductor devices requires that the real space position of each carrier be calculated, and the resulting charge used to solve Poisson’s equation simultaneously with the particle dynamics.*

• The semiconductor is discretized

using either the finite difference or the finite elements approach for the solution of Poisson’s eq.

• The charge of the particles (super

particles) is then assigned to the grid points.

*R.W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, McGraw-Hill, 1981

Monte Carlo Device Simulation

Initial potential, fields positions and velocity of carriers

t = 0

t = t + ∆t

Free-flights acceleration displacement

Scattering events final states

All electrons?

t = N∆t?

Assign charge to mesh points

Calculate potentials and fields at each mesh point

End of simulation ?

stop

N Y N Y N Y

Requirements on MESH size and

TIME step

• The time step and the mesh size may correlate

to each other in connection with the

numerical stability.

• The time step ∆ t must be related to the

plasma frequency

• The Mesh size is related to the Debye length.

• However, the ∆ t chosen must be checked

again by calculating the distance l , defined

2

m

e n

s

p

l max = v max ×∆ t

Docsity.com

NGP C IC

2. Charge Assignment and Force Interpolation

• There are two methods most commonly used for the charge

assignment: Nearest Grid Point (NGP) and Cloud in Cell (CIC) scheme (see figures below):

Nearest grid point method

Nearest element cell

Cloud-in-cell method