Techniques for electromigration stress determination in interconnects of an integrated circuit
View Patent ↗A technique for determining stress in interconnects of an integrated circuit design includes generating a maximal spanning tree of a directed graph. The directed graph represents an interconnect of an integrated circuit design. The technique also includes locating a first point on the spanning tree that has a lowest stress and a second point on the spanning tree that has a highest stress. The technique further includes determining whether a maximum first stress between the first and second points is less than a critical stress.
1. A method of determining stress in interconnects of an integrated circuit design, comprising:
generating, using a data processing system, a maximal spanning tree of a directed graph, wherein the directed graph represents an interconnect network of an integrated circuit design;
locating, using the data processing system, a first point on the spanning tree having a lowest stress and a second point on the spanning tree having a highest stress; and
determining, using the data processing system, whether a maximum first stress between the first and second points is less than a critical stress.
2. The method of claim 1 , further comprising:
indicating, using the data processing system, a pass condition in response to the maximum first stress being less than the critical stress, wherein the maximum first stress is given by:
Δσ
ij
=
c
R
sh
×
Δ
V
ij
where ‘c’ is a material-dependent constant, R sh is the sheet resistance, and ΔV ij is the potential from node ‘i ‘to node ‘j’.
3. The method of claim 1 , further comprising:
determining, using the data processing system, whether a second stress at a node along a path between the first and second points is less than the critical stress in response to the maximum first stress being greater than or equal to the critical stress; and
indicating, using the data processing system, in response to the second stress at the node along the path between the first and second points being less than the critical stress, a passing condition on an edge connected to the node.
4. The method of claim 1 , further comprising:
determining, using the data processing system, whether a second stress at a node along a path between the first and second points is less than the critical stress in response to the maximum first stress being greater than or equal to the critical stress; and
indicating, using the data processing system, in response to the second stress at the node along the path between the first and second points being greater than or equal to the critical stress, a failing condition on an edge connected to the node.
5. The method of claim 1 , wherein the lowest stress occurs at a highest potential.
6. The method of claim 1 , wherein the highest stress occurs at a lowest potential.
7. The method of claim 3 , wherein the second stress at node ‘j’ is given by:
σ
j
=
-
c
2
R
sh
×
∑
i
=
1
i
≠
j
N
(
a
T
i
×
Δ
V
ij
)
A
where ‘A’ is the total area of the interconnect, R sh is the sheet resistance, ‘c’ a material-dependent constant, α Ti a partial interconnect area at node ‘i’, ΔV ij is the potential difference between node ‘i’ and node ‘j’; and ‘N’ is the total number of nodes in the directed graph.
8. A data processing system for determining stress in interconnects of an integrated circuit design, the data processing system comprising:
a storage subsystem; and
a processor coupled to the storage subsystem, wherein the processor is configured to generate a maximal spanning tree of a directed graph that represents an interconnect of an integrated circuit design, locate a first point on the spanning tree having a lowest stress and a second point on the spanning tree having a highest stress, and determine whether a maximum first stress between the first and second points is less than a critical stress.
9. The data processing system of claim 8 , wherein the processor is further configured to indicate a pass condition in response to the maximum first stress being less than the critical stress, and wherein the maximum first stress is given by:
Δσ
ij
=
c
R
sh
×
Δ
V
ij
where ‘c’ is a material-dependent constant, R sh is the sheet resistance, and ΔV ij is the potential from node ‘i ‘to node ‘j’.
10. The data processing system of claim 8 , wherein the processor is further configured to determine whether a second stress at a node along a path between the first and second points is less than the critical stress, in response to the maximum first stress being greater than or equal to the critical stress, and indicate, in response to the second stress at the node along the path between the first and second points being less than the critical stress, a passing condition on an edge connected to the node.
11. The data processing system of claim 8 , wherein the processor is further configured to determine whether a second stress at a node along a path between the first and second points is less than the critical stress, in response to the maximum first stress being greater than or equal to the critical stress, and indicate, in response to the second stress at the node along the path between the first and second points being greater than or equal to the critical stress, a failing condition on an edge connected to the node.
12. The data processing system of claim 8 , wherein the lowest stress occurs at a highest potential.
13. The data processing system of claim 8 , wherein the highest stress occurs at a lowest potential.
14. The data processing system of claim 10 , wherein the second stress at node ‘j’ is given by:
σ
j
=
-
c
2
R
sh
×
∑
i
=
1
i
≠
j
N
(
a
T
i
×
Δ
V
ij
)
A
where ‘A’ is the total area of the interconnect, R sh is the sheet resistance, ‘c’ a material dependent constant, α Ti a partial interconnect area at node ‘i’, ΔV ij is the potential difference between node ‘i’ and node ‘j’; and ‘N’ is the total number of nodes in the directed graph.
15. A method of determining stress in interconnects of an integrated circuit design, comprising:
generating, using a data processing system, a maximal spanning tree of a directed graph, wherein the directed graph represents an interconnect network of an integrated circuit design;
locating, using the data processing system, a first point on the spanning tree having a lowest stress and a second point on the spanning tree having a highest stress;
determining, using the data processing system, whether a maximum first stress between the first and second points is less than a critical stress;
indicating, using the data processing system, a pass condition in response to the maximum first stress being less than the critical stress, wherein the maximum first stress is given by:
Δσ
ij
=
c
R
sh
×
Δ
V
ij
where ‘c’ is a material-dependent constant, R sh is the sheet resistance, and ΔV ij is the potential from node ‘i ‘to node ‘j’, and wherein the lowest stress occurs at a highest potential and the highest stress occurs at a lowest potential.
16. The method of claim 15 , further comprising:
determining, using the data processing system, whether a second stress at a node along a path between the first and second points is less than the critical stress in response to the maximum first stress being greater than or equal to the critical stress; and
indicating, using the data processing system, in response to the second stress at the node along the path between the first and second points being less than the critical stress, a passing condition on an edge connected to the node.
17. The method of claim 16 , wherein the second stress at node ‘j’ is given by:
σ
j
=
-
c
2
R
sh
×
∑
i
=
1
i
≠
j
N
(
a
T
i
×
Δ
V
ij
)
A
where ‘A’ is the total area of the interconnect, R sh is the sheet resistance, ‘c’ a material-dependent constant, α Ti a partial interconnect area at node ‘i’, ΔV ij is the potential difference between node ‘i’ and node ‘j’; and ‘N’ is the total number of nodes in the directed graph.
18. The method of claim 15 , further comprising:
determining, using the data processing system, whether a second stress at a node along a path between the first and second points is less than the critical stress in response to the maximum first stress being greater than or equal to the critical stress; and
indicating, using the data processing system, in response to the second stress at the node along the path between the first and second points being greater than or equal to the critical stress, a failing condition on an edge connected to the node.
19. The method of claim 18 , wherein the second stress at node ‘j’ is given by:
σ
j
=
-
c
2
R
sh
×
∑
i
=
1
i
≠
j
N
(
a
T
i
×
Δ
V
ij
)
A
where ‘A’ is the total area of the interconnect, R sh is the sheet resistance, ‘c’ a material-dependent constant, α Ti , a partial interconnect area at node ‘i’, ΔV ij is the potential difference between node ‘i’ and node ‘j’; and ‘N’ is the total number of nodes in the directed graph.
20. The method of claim 15 , further comprising:
determining, using the data processing system, whether a second stress at a node along a path between the first and second points is less than the critical stress in response to the maximum first stress being greater than or equal to the critical stress;
indicating, using the data processing system, in response to the second stress at the node along the path between the first and second points being less than the critical stress, a passing condition on an edge connected to the node; and
indicating, using the data processing system, in response to the second stress at the node along the path between the first and second points being greater than or equal to the critical stress, a failing condition on an edge connected to the node.