Wim Schoenmaker ©magwel2005 Electromagnetic Modeling of Back-End Structures on Semiconductors Wim Schoenmaker

Wim Schoenmaker©magwel2005

Electromagnetic Modeling of Electromagnetic Modeling of Back-End Structures on Back-End Structures on

SemiconductorsSemiconductors

Wim Schoenmaker


Outline

• Introduction and problem definition

• Designer’s needs: the numerical approach

• Putting vector potentials on the grid

• Static results

• High-frequency results

• Conclusions


Outline



• Vector potentials: their ‘physical’ meaning

• Static results


• Conclusions


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)


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


Introduction - problem definition


Outline




• Static results


• Conclusions


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)


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


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


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


Outline




• Static results


• Conclusions


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


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


Discretizing A

SIddSC

SJlBJB Stokes theorem:

Stokes theorem once more: '

'C

dΔ lASBAB


Discretizing A

• Now 4 times

• geometrical factor

13

1

|l

lklk AA


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


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


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


• Old system of equations

• New system of equations

)(

)(

AJA

A

jVj

jV

0

)(

)(

2

A

jVj

jV

AJA

A

2

Ghost field

Core Idea


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


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).(


Outline



• Vector potentials: their physical meaning

• Classical ghosts: a new paradigm in physics

• Static results


• Conclusions


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”


Outline




• Static results



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


Outline




• Classical ghosts: a new paradigm in physics

• Static results


• Conclusions


Results:Cylindrical wire (Al)

2 a = 3 m

100 GHz50 GHz25 GHz15 GHz4 GHz


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


Proximity effect

1 GHz1 GHz 3 3 GHzGHz

Current density


• Problem:• Alternating currents alternating fields

• alternating currents …..

Substrate Current


~ 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


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)

Documents

Wim Schoenmaker ©magwel2005 Electromagnetic Modeling of Back-End Structures on Semiconductors Wim Schoenmaker