Coupling Maxwell\'s Equations to Particle-Based Simulators

Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH

•  Introduction / Motivation

•  Full Band Simulator

•  Finite-Difference Time Domain Method
(FDTD) / Maxwell Solver

•  Coupling of Maxwell/Monte Carlo methods

•  Simulation results / Conclusions


Traditional methods of simulating semiconductor devices involve a
solution of Poisson’s equation on a discrete mesh. However, the
static field distribution that results is unable to fully account for the
time-varying nature of the total electromagnetic environment
that exist within and surrounding the device.

•  As operating frequencies increase, must treat signals as
electromagnetic waves propagating along transmission
lines in devices.

•  Must take into account absorption /emission of EM
energy throughout system.


Development of devices operating in this new high-frequency
regime is occuring on two separate ends of a gap.

Terahertz Gap

Current research motivated by a desire to accurately simulate and
capture radiated EM field patterns emanating from ultrafast, high
-frequency devices

Experimental measurements of high-field transport in GaAs and InP
under extreme non-equilibrium conditions have been reported by
Leitenstorfer et. al.1

Recent numerical experiments of transient responses in GaAs and
InP by Wigger et. al.2 have provided further motivation for this work

1A. Leitenstorfer, S. Hunsche, J. Shah, M.C. Nuss, and
W.H. Knox: Phys. Rev. Lett. 82 5140 (1999).

2S. Wigger, M. Saraniti, S. Goodnick, A. Leitenstorfer:
J. Comp. Elec. 1:475-480 (2002)


•  Particle-Based Simulator


•  Coupling of Maxwell/Monte Carlo Methods



Use a semiclassical description of carrier transport via stochastic solution of
the Boltzmann Transport Equation (BTE),

Boltzmann Transport Equation:

drift diffusion

where,


initialization
Flowchart of Simulator:
calculate charge

Poisson solver

free flight

particle dynamics

NO
end simulation time ?

YES
calculate averages

end

initialization

parabolic dispersion: fullband dispersion:

6
4

energy [eV]
2
0
non-parabolic dispersion: -2
-4
-6
L X U,K
L L

wave vector


Initialization: initialization

Empirical Pseudopotential Method
density of states [10 cm eV ] 5

density of states [10 cm eV ]
-1

-1
7 InP GaAs
-3

-3
6 4
22

5

22
3
4
3 2
2
1
1

L L
L L
wave vector

wave vector
X X

U,K U,K

L L
-10 -5 0 5 10 -10 -5 0 5 10
energy [eV] energy [eV]

Initialization: initialization
Phonon Dispersion
Valence Shell Method

0 .0 5 0 .0 4

LO
InP LO GaAs
0 .0 4
TO 0 .0 3 TO
energy [eV]

energy [eV]
0 .0 3

0 .0 2
0 .0 2

LA LA
0 .0 1
0 .0 1 LA LA

TA TA TA
TA
0 L L 0
L X U,K
L L
L X U,K
wave vector wave vector


Free Flight: Drift
free flight

Newton’s Equations of motion:


Ensemble Monte Carlo vs.“Cellular Monte Carlo”
The Ensemble Monte Carlo The Cellular Monte Carlo method
method tabulates the scattering computes and tabulates the
rate integrated over the entire scattering rates from an initial
momentum space. The final state momentum state to all possible
is then obtained by inverting the final states, which satisfy the
energy-momentum dispersion appropriate conservation laws.
relation, which is also tabulated
for full band.
choose scattering
choose new k

new energy
  computationally fast
find new k with   high memory requirements
dispersion relation
Scattering Mechanisms:
  computationally slow   polar scattering
  low memory requirements   deformation potential scattering
  impact ionization

Hybrid EMC/CMC
Idea:
use MC scattering in regions of band structure where scattering is low.

  Nearly as fast as CMC.
  Reduces memory usage.

Hybrid/MC performance ratio

time per iter. [sec/5000 e ]
-
6
4
energ y [eV]

2
0
-2
-4 EMC
-6 CMC
X U,K
L L
L
wave vector
field [V/m]





•  Coupling of Maxwell/Monte Carlo Methods



“Finite-Difference Time-Domain”

• First introduced by K. S. Yee in 1966.

• Method remained relatively unused for ~10 yrs.

  Inadequate processing power.
  Method lacked proper boundary conditions.


Curl form of necessary Maxwell
equations are:

Constitutive Relationships

By applying the curl operator and equating the vector components
of the previous (2) equations, we arrive at the following set of (6)
scalar equations:


Now, using a centered-difference scheme, each expression can
be rewritten in appropriate finite-difference form shown here:
Note that E & H Fields are offset
Magnetic Field Update Equation from each other.

Electric Field Update Equation


•  Scheme is often referred to as a “Leapfrog” Method

i-2 i-1 i i+1 i+2
Ex
t – Δt/2

i-1 1/2 i-1/2 i+1/2 i+1 1/2
Hy
t

i-2 i-1 i i+1 i+2
Ex
t +Δt/2


•  In 3D, the E and H fields can be visualized as existing on separate
but interlaced grids over a cubic cell,

z
Ey
Ex Ex
Hz

Ey
Ex Ez

Ez
Hy
Hx

Ex
Hy “Yee cell”
Hx
Ex Ey Ex

Hz

Ey

y

x


•  An upper bound is imposed on the simulation timestep due to
Courant-Freidrich-Levy1 (CFL) condition for finite-difference
solutions of the wave equation,

where c is the wave velocity, Δt is the timestep, and Δx, Δy, and Δz are
the spatial dimensions of each grid cell.

1R.
Courant, K. Friedrichs, and H. Lewy. “On the
Partial Difference Equations of Mathematical
Physics.” IBM Journal, pp 215-234, Mar. 1967

•  EM wave problems are defined, in general, with OPEN or
UNBOUNDED domains that extend out to infinity.

•  However, computationally impossible to store unlimited amount
of data required

Domain must be truncated so that it:
•  Fully contains structure of interest.
•  Resolves any region of interest within/around device.
•  Allows for wave propagation while minimizing
reflections of outward traveling waves at boundaries.


We have chosen to implement the Perfectly Matched Layer (PML)
absorbing boundary condition recently developed by Berenger1.

•  Formulation involves a “field-splitting” approach
creating boundary layer that can:
  absorb any kind of traveling wave.
  regardless of direction of travel.
  without reflection back into domain.

1J. P. Bérenger, “Perfectly matched layer for
the FDTD solution of wave-structure
interaction problems,” IEEE Trans. Antennas
Propagat., vol 44, pp. 110-117, Jan. 1996.

•  Berenger introduced (1) complex permittivity/permeability
(2) split the field components into 2 parts.

•  This resulted in the following set of (12) equations,





•  Coupling of Maxwell/Monte Carlo



Coupling FDTD solver to EMC

Initialize Device Initialize Device

Calculate Charge Calculate Current Density

Poisson Solver Maxwell Solver

Free Flight Free Flight

Particle Dynamics Particle Dynamics

No No
End of End of
Simulation? Simulation?
Yes Yes
Calculate Averages Calculate Averages

End End


Curl form of necessary Maxwell equations are:

The current density, J can be
calculated directly via temporal
and spatial evolution of charge
from Ensemble Monte Carlo.


Current density is computed at every timestep using weighted summation
of particle velocities in each grid cell,

charge velocity


Determine carrier
distribution

EMC/CMC
Solver
Calculate current density
J(i,j,k)

Calculate E and H fields
using J(i,j,k)
Maxwell
Solver

Determine Lorentz Force


In experimental setup, a mode-locked Ti: Sapphire laser with a
pulse duration of 12fs, central photon frequency of 1.49eV, and bandwidth
of 120meV used to optically excite electron-hole pairs in GaAs and InP pin
Diodes with intrinsic region 500 nm long

pin diode:

h
p+ i n+

VA
momentum space

EC
real space
-
EC h
EG

- +
EV
EFp
EV h
qVA
EFn h = EG+
+


EX=-VA/LX
empirical generation rate:
+ -
+ -
+ -
+ -
change in carrier density: + -

LX

ninj=5x1014 cm-3
tp = 10fs
t0 = 20 fs
t = 0.0167 fs


simulated region

p+ i n+
h
Ly EX=-VA/LX

+ -
+ -
LX +
+
-
- 10 grid cells
+ -
h = 1.49 eV
50 grid cells
Lx = 500 nm
Ly = 100 nm
Lz = 100 nm

Ne = Nh = 25,000


500 nm 500 nm

PML

Region

300 nm Filament
GaAs

Air

Snapshot #2 at 480nm from left Snapshot #1 at 50nm from left y
GaAs
contact surface. contact surface.

x
z

Magnitude of Longitudinal Field (Ex) at 50 nm from contact surface.


Transverse field (Ey) at 50 nm from Displacement of carriers vs. time
contact surface.


Longitudinal field (Ex) at 50 nm from contact surface.


Transverse field (Ey) at 480 nm from contact surface.


Simulation results for 100 kV/cm Simulation results due to Wigger et. al.


•  Presented direct simulation and capture of THz
transient field patterns from simple device structure
using a fullband simulator coupled with a Maxwell solver.

•  Demonstrated usefulness of Global Solver to model
EM characteristics of a simple device.

• Future work will involve implementation of FDTD
Method not constrained by timestep criterion.


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