Heterogeneous Integration Project
Quarterly Report
Neil Goldsman, Akin Akturk, and Zeynep Dilli, Bruce Jacob
In the previous quarter we focused on modeling electrical coupling between interconnects
and we continued our investigation of the effects of contact pads on circuits performance.
Modeling and Extraction of Interconnect Capacitive Parasitics:
The goal of this work is to find the cross talk between the metal wires quantitatively. The
cross talk implies the formation of a capacitive network between different nodes that
represent the metals. This network is shown in fig. 1. The web of these self and mutual
capacitances induce voltages on the floating metals, which characterize wires or bus lines
with both sides in high impedance state. From a circuit point of view, this could result in
spurious voltage high-levels, which would cause unwanted switching of circuit elements.
We have written software to extract the capacitance matrix of an arbitrary rectangular
geometry in which metals of different sizes are scattered in an oxide region. Once we
have the capacitance matrix, it can be used with any voltage bias distribution to calculate
the coupling of electric fields and induced voltages between conductors.
The extraction of the capacitances requires the knowledge of the potential distribution
over the region of interest. In our case, we have perfect conductors in a perfectly
insulated environment, so the solution of the Poisson equation with appropriate boundary
conditions would suffice to find the potential distribution.
The simple Q=CV relation states that the capacitance between any two nodes can be
calculated if the voltage difference and the induced charge due to this difference is known
between two nodes. In order to find the self and the mutual capacitance of each node in
the given problem, the node under investigation is set to 1V while all the other nodes are
kept at the ground level. This avoids the formation of induced charges on these
conductors due to wires other than the node of observation. The field distribution in this
scheme can be calculated by solving the Poisson equation,  2  0 since the net charge
is zero, in the region that fills the space between the metals. The boundary conditions can
be specified as follows: The metals are all tied to a supply, so their potential is either 1V
or 0V; and at the edges of the solution domain, the normal electric field is zero. This
problem is solved by conjugate gradient method for each node. After each successive
solution of this scheme for a specific node, the self and mutual capacitances for this node
is calculated as follows: Since the metals are assumed to be perfect conductors, there can
not be any induced charge within the conductor. However, the induced charges form on
the surface due to the discontinuity of the displacement field on the metal-insulator
interface. The amounts of these surface charges are proportional to the magnitude
differences of the normal displacement vectors on the boundary point of interest. The
sign reference can be chosen either inside field minus outside being the positive or vice
versa. Since the electric field is zero within the metal, the associated displacement vector
is also zero and the surface charge is directly proportional to the normal displacement
vector in the insulator adjacent to the boundary. The addition of all the surface charges on
a conductor is the total induced charge on it. So the mutual capacitances between the
metal that is tied to 1V and the conductors (that are grounded) can be calculated by taking
a simple ratio; surface charge on the grounded object divided by 1V. The determination
of the induced charge due to the interaction of the two bodies should be measured on the
grounded one because charges induce on the reference metal due to all grounded metals.
Finally, the self-capacitance of the metal box can be calculated by adding all surface
charges on itself and dividing the sum by 1V.
The specific components of the capacitance matrix can be obtained by using the
algorithm described above. For any voltage distribution on the metals, the induced
potentials on the floating ones can be determined by the utilization of this matrix. Since
the net charge on the floating metals is zero, unless intentionally charged and left to float,
Furthermore, the elements of the matrix provide the SPICE values for the parasitic
capacitors. Thus, these values can be used directly for circuit simulation. In essence,
using differential equation-based modeling, we have extracted lumped circuits elements
for efficient circuit analysis and design. Figure 2 below shows the calculated potential
distribution obtained from this method. Figure 3 summarizes the solution method.
Figure 1: The capacitive network.
Figure 2: Potential Distribution due to the biases shown above
Q1  C11V1  C12V2  ...  C1NVN
Q2  C21V1  C22V2  ...  C2 NVN
:
QN  C N 1V1  C N 2V2  ...  C NN VN
C ji 
Qi
Vi
V j 0, j i
 C11 C12 ... C1N 
C
C22 ... C1N 
21

C 
 :
: 


C N 1 C N 2 ... C NN 
C F1F1VF1  C F1F2VF2  ...  C F1FL VFL  CM 1M1VM1  CM1M 2VM 2  ...  CM1VM M
C F2 F1VF2  C F2 F2VF2  ...  C F2 FL VFL  CM 2 M1VM1  CM 2 M 2VM 2  ...  CM 2VM M
:
C FL F1VF1  C FL F2VF2  ...  C FL FL VFL  CM M M1VM 1  CM M M 2VM 2  ...  CM M VM M
Figure 3: Summary of the algorithm. The subscript M and F refer to fixed and floating
metals respectively.