2. Full-Band Full-Wave
Simulator Simulator
6
4
2
0
-2
-4
-6
Γ X U,K
L
L Γ
Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
3. When devices are operated at high frequencies:
• Coupling between fluctuation in charge distribution and
propagating EM fields must be included into simulation model.
• As operating frequencies increase, period of EM waves
approaches relaxation time of carriers in semiconductor material.
• Finite amount of time for carrier to react to changes in applied
fields (i.e. changes in particle velocities)
Transport directly affected
by EM wave propagation
Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
4. Poisson solvers are unable to:
• directly capture inherent “carrier-wave” interaction.
• account for existing magnetic fields in real device.
Full-wave solver can:
• directly solve full set of EM field equations.
• account for externally applied sources and changes in the
field due to charge fluctuations.
• directly simulate absorption/emission of EM energy in/out of
system (i.e. optical excitation, radiative processes, THz
devices.)
Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
5. M. Saraniti and S.M. Goodnick, IEEE TED, 47, 1909 (2000)
K. Kometer, G. Zandler, and P. Vogl, Phys. Rev., B46(3), 1382 (1992)
particle dynamics
choose scattering
Ensemble Monte Carlo (EMC)
new energy
computationally slow
low memory requirements
find new k with
dispersion relation
VS.
Cellular Monte Carlo (CMC)
computationally fast choose new k
high memory requirements
Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
6. Idea:
use MC scattering in regions of band structure where scattering is low.
Nearly as fast as CMC.
Reduces memory usage.
H ybrid/ MC perf ormance 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
field [V/m]
wave vector
Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
7. z K.S. Yee, IEEE Trans. Antennas Propagat., 14(302) 1966
“Yee cell”
Maxwell’s equations • Most direct explicit solution of
Ey
Maxwell’s equations available (i.e.
Ex Ex
no matrix inversion required).
Hz
∂H Ez
∇ × E = −µ
Ex
Ey
• A complete “full-wave” method
∂t Hx without approximation (i.e. no pre
Hy Hy
-selection of output modes or
∂E
Ez Ex
solution form necessary.)
∇× H = ε +J Ex
Hx
Ey Ex
y
∂t Hz
Ey
x
PML Absorbing Boundary Conditions
• Introduces “artificial” anisotropic electric
/magnetic* conductivities within domain
boundaries allowing for absorption
/attenuation waves.
• Employs a numerical “split-field” approach
allowing perfect transmission into absorbing
layer (regardless of frequency, polarization, or
angle of incidence).
J. P. Bérenger, IEEE Trans. Antennas Propagat., 44(110) 1996.
Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
8. Sheen, et. al. , IEEE- MTT, 38(7), 1990.
Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
9. • Stability limit, called the CFL criterion severely limits maximum timestep for
solution of PDEs on a finite grid.
1
Δt FDTD ≤
2 2 2
⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞
υ max ⎜ ⎟ +⎜
⎜ Δy ⎟ + ⎜ Δz ⎟
⎟
⎝ Δx ⎠ ⎝ ⎠ ⎝ ⎠
• CFL criterion can be relaxed using newly reported ADI-FDTD
method.
Requires both implicit and explicit field updates thus more
time spent per FDTD timestep.
Allows for timesteps several orders of magnitude larger than
conventional limit.
Tradeoff b/w accuracy and chosen timestep.
T. Namiki, IEEE MTT 47(10), 2003 (1999).
F. Zheng, et. al, Microwave Guided Wave Lett., 9(11), 441 (1999).
Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
10. Steps full-wave simulation:
FDTD:
Initialization
∂H
∇ × E = −µ 1. Obtain steady-state solution for specific dc
∂t bias point (CMC/Poisson) and store E fields
∂E and J.
∇× H = ε +J
∂t 2. Initialize H field in FDTD solver using:
CMC:
∇× E = 0
dc dc
∇× H = J
1 ⎛ N (i , j ,k ) ⎞
J (i, j , k ) = ⎜ ∑ S n vn ⎟
ΔxΔyΔz ⎜ n =1 ⎟ 3. Apply excitation source and begin
⎝ ⎠ updating fields:
J tot ∂E 1
∂t ε
[ ac tot dc
= ∇× H − J − J( )]
CMC FDTD ∂H
1
= − ∇× E
(Etot , H tot ) ∂t µ
Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
11. E AC + E DC
Start (
H AC + H DC )
Run CMC for DC
bias point. Update particles using
newly computed fields.
DC
E x,y,z (x, y, z)
DC
J x,y,z (x, y, z) Total
.
J x,y,z (x, y, z;t)
Apply small-signal
excitation source Update E, H Fields
AC
E x,y,z (x, y, z;t)
FDTD Solver AC
H x,y,z (x, y, z;t)
t = t MAX ?
No
Yes
Stop
Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
13. ⎡ ℑ( out (ω , zi )⎤
V
Voltage gain: Gain = 20 log ⎢ ⎥
⎣ ℑ( in (ω , z0 )⎦
V
− S 21
Current gain: h21 =
(1 − S11 )(1 + S 22 )+ S12 S 21
Voltage Gain
S11 : input reflection coefficient
S 22 : input reflection coefficient
S12 : reverse transmission coefficient
S 21 : forward transmission coefficient 5
4
3
] 2
B
d 1
0.1µm gate MESFET (125µm width) [
80 x 25 x 30 uniform mesh n 0
i
Gaussian pulse excitation (0.1V peak AC amplitude) a
100,000 particles G -1
170 GHz
ΔtPoisson= 5x10-15 s -2
ΔtFDTD = 4x10-17 s Current gain
-3
10-layer PML ABC
-4
Simul. time = 6.5 days (3GHz 64-bit Xeon, 8GB RAM)
-5
0 50 100 150 200 250 300
Frequency [GHz]
Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
14. Start Time-Stepping (t =0 )
n+1 2
Update E x implicitly along y direction for all x, y, z
• Coupling ADI-FDTD with CMC simulator. Update E y
n+1 2
implicitly along z direction for all x, y, z
Sub-Iteration #1
n+1 2
Update Ez implicitly along z direction for all x, y, z
• Timestep is split into (2) sub-iterations. t = (n + 1 2)Δt
n+1 2
Update H x explicitly for all x, y, z
• E-fields are updated implicitly along Update H y
n+1 2
explicitly for all x, y, z
specific directions. Update H z
n+1 2
explicitly for all x, y, z
• H-fields are updated explicitly
throughout. Update E x
n+1
implicitly along z direction for all x, y, z
n+1
Sub-Iteration #2
Update E y implicitly along x direction for all x, y, z
n+1
Update Ez implicitly along y direction for all x, y, z
t = (n + 1)Δt
n+1
Update H x explicitly for all x, y, z
n+1
Update H y explicitly for all x, y, z
Larger ΔtFDTD possible
n+1
Update H z explicitly for all x, y, z
Shorter simulation times NO (t < t max )
Time-Stepping
Complete?
YES (t = t max )
End Time-Stepping
Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
