Analytical and numerical model of spiral inductors on high resistivity silicon substrates

• Q-factor of spiral inductors on high resistivity silicon simulated.
• Beneficial effect of high-r Si saturates at 10 kΩ cm.
• One port analytical model for spiral inductors validated.
• Power loss penetration depth established as 30 μm.

Passive devices and spiral inductors in particular suffer severely from losses in the silicon substrate underneath. This major component in limiting the quality factor of spiral inductors can be mitigated by using higher substrate resistivity or by locally increasing the resistivity below the inductor. These approaches have been modelled mostly ad hoc with phenomenological ππ models guiding the design. In this paper we use HFSS 3D finite element simulations to simulate the response of spiral inductors above high resistivity substrates of various resistivities achievable with Si. We relate these simulations to a simple one-port model of the Q-factor of spiral inductors to show in physical terms that the effective substrate thickness of around 30 μm of the spiral inductor is equal to half the electric field penetration depth. The value of the penetration depth depends on the lay-out of the spiral inductor, but, critically, not on the substrate resistivity, and hence the one-port model maintains the proportionality between resistance parameter and substrate resistivity.