Global Modeling of High-Frequency Devices

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Global Modeling of High-Frequency Devices

  1. 1. Nanostructures
Research
Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  2. 2. Full-Band

 Full-Wave
 Simulator
 Simulator
 6 4 20-2-4-6 Γ X U,K L L Γ Nanostructures
Research
Group
 CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  3. 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. 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. 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. 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 4energ 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. 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 xPML
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. 8. Sheen, et. al. , IEEE- MTT, 38(7), 1990.Nanostructures
Research
Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  9. 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. 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. 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 
 StopNanostructures
Research
Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  12. 12. • 

Transverse
E-fields
computed
via
2D
Poisson
solver
and
applied
to
source
plane
at
each
timestep.
 − (t −t0 )2Vgs (t ) = Δυ gs e T2 z
 y
 x
 Nanostructures
Research
Group
 CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  13. 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 10.1µm gate MESFET (125µm width) [80 x 25 x 30 uniform mesh n 0 iGaussian pulse excitation (0.1V peak AC amplitude) a100,000 particles G -1 170 GHzΔtPoisson= 5x10-15 s -2ΔtFDTD = 4x10-17 s Current
gain
 -310-layer PML ABC -4Simul. 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. 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
  15. 15. Nanostructures
Research
Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH

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