Inductive Coupling Using Permeable Layer

In single-flux-quantum (SFQ) circuits, the ground plane can severely limit the coupling factor between adjacent inductors. If higher coupling factors are required, ground plane holes can be place directly below the adjacent microstrip lines [99]. However, when using multiple ground planes, punching holes through all the layers is not always possible. In this section, it is shown

that permeable materials can be used to increase the mutual coupling (k) between two inductors, without creating a hole in the ground plane.

Figure 4.6.13 shows the GDS layout of two superconducting microstrip lines, with a 44 µm × 38 µm permeable rectangle sandwiched between the lines

and the ground plane. Using InductEx and the Hypres 4.5 kA/cm2 _{Nb fabri-}

cation process, a 3D model is constructed from the GDS layers, as shown in Fig. 4.6.15. Layer M 2 is used for the microstrip line, layer M 0 for the ground plane, and layer M 1 was changed to a permeable (magnetic) layer.

The inductance and mutual inductance between the two mictrostrip lines

were calculated for a range of µr values, as shown in Table 4.6.2, with an

excitation voltage of 1 V at 10 GHz. The coupling factor increases several order as the relative permeability increases, as can be seen in Fig. 4.6.14. Figure 4.6.15a and 4.6.15b show respectively the direction of current flow of both the electric, J, and magnetic, M, current densities. Once again, the current in the ground plane is displaced by the magnetic layer, which increases the total inductance and mutual inductance, similar to a hole being placed in the ground plane [99]. These results still have to be verified experimentally for superconducting structures, but this falls outside the scope of this dissertation.

64 μm 30 μm 6 μm 7 μm 38 μm 44 μm

Figure 4.6.13: GDS layout of a two microstrip lines with a 44 µm × 38 µm permeable rectangle sandwich between layers M 2 and M 0 of the Hypres

Table 4.6.2: Inductance and mutual inductance between microstrip lines in Fig. 4.6.13.

µr Inductance Mutual Coupling

[pH] inductance [pH] factor (k) 1 12.01 0.094 0.008 10 15.99 0.435 0.027 102 _{33.95 6.631 0.195} 103 _{58.69 23.901 0.407} 104 _{65.01 29.343 0.451} 100 101 102 103 104 Relative permeability 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Coupling factor (k)

Figure 4.6.14: Coupling factor between the two microstrip lines for a range of relative permeability values.

(a) (b)

Figure 4.6.15: Microstrip lines above substrate with relative permeability of

µr = 1000. (a) Vector field of electric current density, J. (b) Vector field of magnetic current density, M, within permeable material.

4.7

Conclusion

Modification were made to TetraHenry (TTH) to support electric and mag- netic currents inside inhomogeneous dielectric and magnetic materials. The integral equations, required for EMQS and Full-wave analysis, with support for superconductivity, are presented. VL basis functions, consisting of Half-SWG functions, are used to account for charge accumulation at material interfaces of both dielectric and magnetic materials. The FMM is used to accelerate the computation of vector and scalar potential fields. It is shown that Diagonal-L and Pattern-R preconditioners are still effective at accelerating the convergence rage of the GMRES iterative solver.

Numerical results confirm the accuracy of TTH for non-superconducting structures, when extracting impedance over a wide range of frequencies. EMQS analysis can be used to accurately calculate the impedance of structures on the order of a wavelength, given that the separation between the structures are small relative to the wavelength. For larger structures, Full-Wave analy- sis should be used. The calculation time of TTH is also significantly faster, compared to FastImp, when using EMQS analysis.

It is shown that the frequency behaviour of a superconducting transmission line is affected, if the transmission line has vias punching through the ground plane. The frequency behaviour also depends on the size of the hole in the ground plane.

The effect of magnetic materials on non- and superconducting materials are analysed. It is shown that magnetic materials can significantly increase the self- and mutual inductance of non- and superconducting structures. The inductance extracted with TTH for non-superconducting structures, in the presence of permeable materials, correspond with existing results and with

CST Studio. The results obtained with TTH for superconducting structures,

in the presence of permeable materials, still has to be verified experimentally, but this falls outside the scope of this dissertation.

Chapter 5

External Magnetic Field

5.1

Introduction

When superconducting integrated circuits are exposed external magnetic fields, currents are induced inside the superconductors to expel the magnetic field. This phenomenon is known as the Meissner effect [26]. Due to the high order of sensitivity of these circuits, any small current can lead to a catastrophic failure. Understanding the limitations and operating margins of the circuits, in the presence of magnetic fields, can help designers optimize layouts and prevent circuit failure.

Obtaining the operating margins of a superconducting integrated circuits, in the presence of magnetic fields, requires simulating the structure for each possible angle and amplitude of the field. This can be time consuming when using VIE-based solvers, such as TTH. A better approach would be to use the VIE-based solver to derive an equivalent circuit model for each x-, y- and z- component of the field. This circuit model can then be reused to rapidly simu- late magnetic fields at different amplitudes and obtain the operating margins.