![Page 1: Wim Schoenmaker ©magwel2005 Electromagnetic Modeling of Back-End Structures on Semiconductors Wim Schoenmaker](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e7e5503460f94b8182c/html5/thumbnails/1.jpg)
Wim Schoenmaker©magwel2005
Electromagnetic Modeling of Electromagnetic Modeling of Back-End Structures on Back-End Structures on
SemiconductorsSemiconductors
Wim Schoenmaker
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Wim Schoenmaker©magwel2005
Outline
• Introduction and problem definition
• Designer’s needs: the numerical approach
• Putting vector potentials on the grid
• Static results
• High-frequency results
• Conclusions
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Wim Schoenmaker©magwel2005
Outline
• Introduction and problem definition
• Designer’s needs: the numerical approach
• Vector potentials: their ‘physical’ meaning
• Static results
• High-frequency results
• Conclusions
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Wim Schoenmaker©magwel2005
Introduction - interconnects
On-chip connections between transistors
Dimensions/pitches still decreasingIncreasing clock frequencies
Frequency dependent factors become more and more important:e.g. Cross-talk, skin effect, substrate currents
Transistors (gates, sources and drains)
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Wim Schoenmaker©magwel2005
Introduction - integrated passives
Passive structures in RF systems – e.g. antennas, switches, …
Cost reduction by integration in IC’s
Simulation of RF componentsIncreases reliability Increases production yieldDecreases developing cycle
RF section WLAN receiver - 5.2 GHz
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Wim Schoenmaker©magwel2005
Introduction - problem definition
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Wim Schoenmaker©magwel2005
Outline
• Introduction and problem definition
• Designer’s needs: the numerical approach
• Putting vector potentials on the grid
• Static results
• High-frequency results
• Conclusions
![Page 8: Wim Schoenmaker ©magwel2005 Electromagnetic Modeling of Back-End Structures on Semiconductors Wim Schoenmaker](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e7e5503460f94b8182c/html5/thumbnails/8.jpg)
Wim Schoenmaker©magwel2005
Designer’s needs - Problem definition
• constraints:– Use the ‘language’ of designers
not electric and magnetic fields (E,B) but
Poisson field V and at high frequency also vector potential A
– Full 3D approach• exploit Manhattan structure, i.e. 3D grid
– Include high frequency consideration• work in frequency space
• provide designers with R(), C(), L(),G() parameters
(resistance R, capacitance C, inductance L, conductance G)
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Wim Schoenmaker©magwel2005
Designer’s needs - numerical approach
AB
AE
t
V
t
t
DJH
BE
B
D
0
0
HB
ED
)(
)(
AJA
A
jVj
jV
WARNING!!Engineers write:
j 1
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Wim Schoenmaker©magwel2005
Constitutive laws
Conductors
j j
Ej
),(
),(
pnUpjq
pnUnjq
p
n
j
j
pn
ppn
nnn
AD
pkTμpqμ
nkTμnqμ
)NNnq(pρ
jjj
Ej
Ej
Semi-conductors
AB
AE
t
Vwith ... gives ...
V, A, n and p as independent variables
Jn is the electron current densityJp is the hole current densityU is recombination – generation termp local hole concentrationn electron concentrationT is lattice temperature k is Boltzmann’s constantQ is elementary charge
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Wim Schoenmaker©magwel2005
Designer’s needs - numerical approach
• Question: How to put V and A on a discrete grid ?– Answer (1): V is a scalar and could be assigned to
nodes
– Answer (2): A= (Ax,Ay,Az) and with
we obtain a 3-fold scalar equation
(Ax,Ay,Az) on nodes too!
BUT
VECTOR = 3-FOLD SCALAR
0 A
...2 JA
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Wim Schoenmaker©magwel2005
Outline
• Introduction and problem definition
• Designer’s needs: the numerical approach
• Putting vector potentials on the grid
• Static results
• High-frequency results
• Conclusions
![Page 13: Wim Schoenmaker ©magwel2005 Electromagnetic Modeling of Back-End Structures on Semiconductors Wim Schoenmaker](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e7e5503460f94b8182c/html5/thumbnails/13.jpg)
Wim Schoenmaker©magwel2005
Continuum vs discrete
• So far A.dx infinitesimal 1-form:• On the computer dx : • x are the distances between grid nodes i.e. connections
on the grid
• A is a connection
• A exists between the grid nodes
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Wim Schoenmaker©magwel2005
Implementation for numerical simulation
• Equations to be solved
• Discretization of V is standard: V is put in nodes
• A is put on the links
)(
)(
AJA
A
jVj
jV
Vi=V(xi ) Vj=V(xj )
Aij=A(xi ,xj )=A.em
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Wim Schoenmaker©magwel2005
Discretizing A
SIddSC
SJlBJB Stokes theorem:
Stokes theorem once more: '
'C
dΔ lASBAB
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Wim Schoenmaker©magwel2005
Discretizing A
• Now 4 times
• geometrical factor
13
1
|l
lklk AA
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Wim Schoenmaker©magwel2005
Counting nodes & links & equations
• Grid with N3 nodes N3 unknowns (Vi)
3N3(1-1/N) links 3N3(1-1/N) unknowns (Al)
• N3 equations for V
there are 3N3(1-1/N) equations for A
BUT
not all A are independent
PROBLEM!
# equations = # unknows
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Wim Schoenmaker©magwel2005
Gauge condition
• Solution: Select a gauge condition– Coulomb gauge– Lorentz gauge– ….
• Coulomb gauge
0 A 06
1
ll
l hA
Each node induces a constraint between A-variables total =N3
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Wim Schoenmaker©magwel2005
Implementation of gauge condition
• Old proposal: build a ‘gauge tree’ in the grid– highly non-local procedure– difficult to program
• New proposal: force # equations to match # unknowns– introduce extra field such that
solutionis0and02 A
)(r‘ghost’ field
Local procedure sparse matriceseasy to program N3 variables i
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Wim Schoenmaker©magwel2005
• Old system of equations
• New system of equations
)(
)(
AJA
A
jVj
jV
0
)(
)(
2
A
jVj
jV
AJA
A
2
Ghost field
Core Idea
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Wim Schoenmaker©magwel2005
Easy to program
creates a
Regular matrix
Local procedure SparseDiagonal dominant
A
A
μ
1
χ
χγμ
1
2
A
A
So .. we do not solve But ...
Gauge implementation
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Wim Schoenmaker©magwel2005
Gauge implementation
• Exploit the fact that• We can make a Laplace operator for one-forms
on the grid by
• This is an alternative for the ghost field method
0
AAA 2).(
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Wim Schoenmaker©magwel2005
Outline
• Introduction and problem definition
• Designer’s needs: the numerical approach
• Vector potentials: their physical meaning
• Classical ghosts: a new paradigm in physics
• Static results
• High-frequency results
• Conclusions
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Wim Schoenmaker©magwel2005
Why a paradigm ?
• What is a paradigm ?– paradigm shift = change of the perception of the world
(Thomas Kuhn)• Examples
– Copernicus view on planetary orbits– Einstein’s view on gravity ~curvature of space-time
• Scientific revolutions with periodicity of – 1 year ? Management (pep) talk –software vs 3.4 --> vs 3.5– 300 years ? Ok, see examples above– 25 years ? Acceptable and operational use of the word
“paradigm”
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Wim Schoenmaker©magwel2005
Outline
• Introduction and problem definition
• Designer’s needs: the numerical approach
• Vector potentials: their physical meaning
• Static results
• High-frequency results
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Wim Schoenmaker©magwel2005
Spiral inductor
Emagn1=1.16E-12 J Emagn2=1.20E-12 J
L=2.041E-11 H
Emagn1=1.82E-19 J Emagn2=1.87E-19 J
L=3.69E-13 H
dvdv magnmagn B.HJ.A2
1E||
2
1E 21
Static B-field of spiral
Static B-field of ring
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Wim Schoenmaker©magwel2005
Outline
• Introduction and problem definition
• Designer’s needs: the numerical approach
• Vector potentials: their physical meaning
• Classical ghosts: a new paradigm in physics
• Static results
• High-frequency results
• Conclusions
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Wim Schoenmaker©magwel2005
Results:Cylindrical wire (Al)
2 a = 3 m
100 GHz50 GHz25 GHz15 GHz4 GHz
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Wim Schoenmaker©magwel2005
Results:Cylindrical wire
Resistance (analytical)Resistance (solver)Reactance (analytical)
Reactance (solver)
85DC 55304 14 GHz
]/)1[(
]/)1[(
2
1
1
0int
ajI
ajI
a
jZ
Current density in cilindrical conductor (100 GHz)
0.00E+00
2.00E+09
4.00E+09
6.00E+09
8.00E+09
1.00E+10
1.20E+10
1.40E+10
3 3.5 4 4.5 5 5.5 6
Analytical resultSolver
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Wim Schoenmaker©magwel2005
Proximity effect
1 GHz1 GHz 3 3 GHzGHz
Current density
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Wim Schoenmaker©magwel2005
• Problem:• Alternating currents alternating fields
• alternating currents …..
Substrate Current
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Wim Schoenmaker©magwel2005
~ V
Results: ring
Boundary conditions:
• A-field on boundary vanishes (DC = no perp B-field)
• dV/dn perp to edge of simulation domain vanishes(DC = no perp E-field)
• On the contacts a harmonic signal for V
• Consider only first harmonic variables
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Wim Schoenmaker©magwel2005
Results: current densities and the ring
Top view showing skin effect(100 MHz)
Top view of substrate showingeddy currents (100 MHz)
Side view of the ports showing theproximity effect (100 MHz)
Top view showing skin effect(500 MHz)
Top view of the substrate showingeddy currents (500 MHz)
Side view of the port showing theproximity effect (500 MHz)