On-chip inductance and
coupling
Zeynep Dilli, Neil Goldsman
Thanks to Todd Firestone and John Rodgers for providing the
laboratory equipment and expertise for measurements.
EM-sensitive components on
semiconductor chips

Modern RF circuits often feature on-chip
inductors required by circuit design


Operating frequencies are high enough to
make this feasible
Increasing circuit complexity also creates
other inductive components

Long transmission (bus) lines; signal/clock
distribution networks…
Motivation for modeling

Investigating parasitic effects
Vulnerability to external EM interference
 Potential to create on-chip interference

Radiation
 Substrate current


System-on-a-chip RF circuits require onchip inductors with high L, small area and
high Q

Automated design and speedy evaluation of
geometrical tradeoffs.
Issues in modeling

Semiconductor substrates are conductive
unable to treat system as
metal/dielectric/ground plane


New processes feature higher doping, higher
conductivity
Device circuits underneath metal structures
display variable doping

Non-uniform substrate: n+ and p+ active
regions, n-wells, p-wells, lightly doped chip
substrate…
Inductor modeling---theory
Modeling Approach: Divide a spiral inductor into segments and treat each
current segment separately.
 V1   L11 Lm,12
V   L
L22
 2    m,21
  
  
VN   Lm, N 1 Lm, N 2
Lm,1N   I 
Lm,1N   I 
s 
  
  
LNN   I 
Lkk=self-inductance (external+internal) of segment k
Sources: Frequency-dependent current distribution within the segment and the magnetic flux
linkage to the loop formed by the segment and its return current.
Lkl=mutual inductance between segments k and l
Sources: Magnetic flux linkage of the current in the first segment to the loop formed by the
second segment and its return current.
Lossy substrate effect: The return current has an effective distance into the substrate; this is
frequency-dependent and can be modeled as a complex distance to account for the losses.
Other frequency dependency: Skin effect in the metal; current crowding in the metal
Internal self-inductance
Frequency-dependent current distribution creates an internal self-impedance
Solve Helmholtz Eqn. for current
distribution:


2 J  j   2  J
Obtain resistance and inductance from J:
Z 
E0
 J  ds
cond .
 R ( )  Lint,self ( )
Internal self-inductance
Physical current in cylindrical conductor, f=5GHz
center
J
r
edge
External self-inductance
Flux of magnetic field linked to surface between
interconnect and its return current
Lext ,self
;


I
 
   B  ds
Need to define the return current path to
determine the flux linkage area. If the
frequency is low enough or substrate has low
conductivity, the physical ground plane below
the substrate is used for this purpose.
But silicon substrates are not dielectric: With higher doping levels in modern
process technologies to optimize the active devices on chips, the substrate
conductivities rise.
And our operating frequencies of interest are high.
Skin depth of semiconductor
substrate
Within our frequency range
the skin depth will fall
below our substrate thickness
(around 5 GHz for p-type sub.,
around 2 GHz for n-well,
lower for active regions)
External self-inductance
Weisshaar et.al. showed in 2002 that an image current with a complex distance can be
defined for the metal-oxide-lossy substrate system.
Signal Current
;
hox
hsub
Insulator
D

Substrate
 1  j  hsub 
heff  hox  1  j     tanh 

    
Metal Plate
Effective virtual ground plane distance
from the signal current
Image Current
Return current depth
Mutual inductance
Mutual inductance: The magnetic flux created
by the current on one loop linking to the area
of other loop
Lij 
 1
Calculate  from the magnetic
vector potential and I from the
4 ai
current distribution; the mutual Lm ,ij 
inductance between two
current segments is then
 ij
Ij
ci
   
ai
bi
aj
J j d li  d l j
dai da j
Rij
cj
bj
J
j
da j
aj
p
Frequency dependency: The signal
current of a current segment and its
image current both induce voltages
on the “target” current segment; the
distribution of the image current
varies with frequency on a
semiconductor substrate.
ẑ
ŷ
q
J xq 
y p2
Virtual Ground Plane
yq 2
h pq
y p1
hqq '
yq1

Wp x p Wp
2
2

Wq xq Wq
2
2
x̂
 J xq 
q' (image)
Inductor modeling---Design issues

Variations in layout:
Metal layer
 Length
 Number of turns
 Metal trace width
 Metal trace spacing
 Substrate doping
 Shape
 …

Some Results
Length Variation
Increasing the length of the inductor increases inductance, but leads to a decline in Q
due to increasing serial resistance as well, this effect worsening at higher frequencies as
skin effect increases the resistance faster than linearly with length.
Some Results
Number of Turns Variation
An inductor with the same length but with more turns has higher inductance, but the
resistance does not rise quite so high so the detrimental effect to Q is less: Increasing the
number of turns is a better way to increase inductance.
Some Results
Trace Spacing Variation
Narrower spacing yields a higher inductance, but will probably increase capacitive
coupling between turns.
Some Results
Substrate Doping Variation
Overall, higher doping reduces inductance (closer return current, smaller loops) and
makes it more freq-dependent (low enough doping pushes all current to bottom).
Relationship between resistance and doping is not straightforward, since conductivity
of substrate affects return current distribution, composition, and its frequency
dependence all at the same time and these effects interact.
Inductors--- Test Chip
Designed for
RF-probe station
measurements
Manufactured
through MOSIS
AMIS 0.5 μm
3 Metal layers
Inductors--- Test Chip
Planar inductor
on substrate
Planar inductor
on grounded poly
Stacked inductor
on substrate
Planar inductor
on n-well
Transformer
RF probe tip
Inductors--- Test Chip, measurements
Cascade probes used with Hewlett-Packard Network Analysis
Measured: S22
S 22
Z L  Z0
1  S 22

 Z L  Z0
Z L  Z0
1  S 22
Inductors--- Test Chip, measurements
Inductors--- Test Chip, measurements
EM Coupling Test Chip 2
For RF-probe
station
measurements
Manufactured
through MOSIS
AMIS 0.5 μm
3 Metal layers
EM Coupling Test Chip 2
NMOS 3
inverter
NMOS 1
Planar inductor
N-wells
NMOS 2
On-Chip EM Coupling
• Coupling between On-chip Inductors
Left: Results from
literature and circuit
model for coupled onchip inductors
Right: Our test structure
for measuring coupling
between inductors on
different metal layers
On-Chip EM Coupling: Inductor to inductor
On-Chip EM Coupling: N-well to n-well
Metal Contact
Port 1
Port 2
Oxide
N Well
N Well
P-Type Silicon Substrate
28.8m
102.45m
28.8m
Need substrate current
analysis….