In this validation example, we show you how to calculate capacitance of an interconnect, consisting of six conductors embedded in a seven dielectric layers, is considered. As indicated in Figure 1, four of the six conductors cross over two ninty degrees bends.
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Figure 1  Six conductors embedded in a set of seven dielectric layers
Each straight conductor has a length of 13mm. The crosssection of all conductors is 1mm × 1mm. As shown in Figure 1, the piecewise lengths of the bent conductors are a = b = 13mm, and S1 = 3.5mm, S2 = 3mm. The relative permittivity of the dielectric layers are, from the bottom, e_{r1} = 2, e_{r2} = 3, e_{r3} = 3, e_{r4} = 4, e_{r5} = 4, e_{r6} = 5, e_{r7} = 5.
The thickness of each layer is 1mm, except the third layer from the bottom. It has a thickness of 2mm. The total height of the structure is 8mm. The air around the structure must be also included in the model. In the published article, the domaindecomposition method (DDM) is used as a capacitance calculator of the considered interconnect. We wish to validate the results of EMS against the published data [1] for the capacitance calculation.
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Figure 2  Solid model of the seven layers interconnect
The Electrostatic module of EMS is used as capacitance calculator for the 3D multilayeredinterconnect structure at hand. After creating the Electrostatic study, or the design scenario, in EMS, three important steps shall always be followed. That is, apply the proper material for all solid bodies, apply the necessary boundary conditions, or the so called Loads/Restraints in EMS, and mesh the entire model.
In the Electrostatic analysis of EMS, the only required material property is the relative permittivity, which is shown in Table 1 for the seven insulators and the surrounding air.
Component/Body

Material Name

Relative Permittivity

Air

Air

1.0

Dielectric 1

e2

2.0

Dielectric 2

e3

3.0

Dielectric 3

e3

3.0

Dielectric 4

e4

4.0

Dielectric 5

e4

4.0

Dielectric 6

e5

5.0

Dielectric 7

e5

5.0

Table 1  Relative permittivity of the seven insulators and the surrounding air
Loads and restraints are necessary to define the electric and magnetic environment of the model. The results of analysis directly depend on the specified loads and restraints. Loads and restraints are applied to geometric entities as features that are fully associative to geometry and automatically adjusted to geometric changes.
In this study a grounded conductor is applied on the top and bottom faces of the seventh and the first dielectric layers, respectively. The conductors are indexed, i.e. numbered, using the so called floating conductors in EMS.
Figure 3  Mesh of the structure without the air region
After a successful run, the Electrostatic module produces three result folders and a result table. The folders contain the electric field E, the electric displacement D, the potential distribution V, respectively. The results table contains the capacitance matrix. Furthermore, all of the results can be visualized in various formats such as fringe, vector, contour, section, line, and, clipping plots. The results can easily be zoomedin, exported, and dissected.
For this particular benchmark, the capacitance matrix is compared against the reported results in [1]. As Table 2 indicates, the results of EMS match those reported by the authors of [1]. You can think of EMS as capacitance calulator for interconnects and VLSI.
Cij (pF)

conductor1

conductor2

conductor3

conductor4

conductor5

conductor6

conductor1

0.745

0.158

0.123

6.515 e03

2.807 e02

4.566 e03

conductor2

0.158

1.369

0.210

0.145

3.278 e02

2.885 e02

conductor3

0.123

0.210

1.743

0.172

0.256

0.262

conductor4

6.516 e03

0.145

0.172

1.689

0.265

0.267

conductor5

2.807 e02

3.277 e02

0.256

0.265

3.469

5.154 e02

conductor6

4.566 e03

2.885 e02

0.262

0.267

5.154 e02

3.448

Table 2  Capacitance matrix (in pF) obtained by EMS

C_{11}

C_{22}

C_{33}

C_{44}

C_{55}

C_{66}

DDM

0.680

1.29

1.57

1.52

2.54

2.54

Spice Link

0.669

1.29

1.60

1.54

2.53

2.53

Table 3  The self capacitance terms (in pF) as reported in [1]
[1] Zhenhai Zhu, Hao Ji, Wei Hong, "An Efficient Algorithm for the Parameter Extraction of 3D Interconnect Structures in the VLSI Circuits: DomainDecomposition Method," IEEE Transactions on Microwave Theory and Techniques, vol. 45, no. 8, August 1997, pp. 11791184.