I. Introduction
Tunable inductors occupying a smaller footprint area with better inductance and quality factor operating in a wide range of frequencies are essential to design better radio frequency integrated circuits (RFICs). Tunable inductors can be employed in tunable RF amplifiers, filters, reconfigurable matching networks, and voltage-controlled oscillators. In existing tunable RFICs, varactors (tunable capacitors) are usually used for tuning the frequency. The use of a tunable capacitor and inductor together provides greater variability in frequency ranges. Several tunable inductors with various geometries and tuning techniques have been developed. Among them, microelectromechanical system (MEMS) based inductors are explored with tuning methods, such as MEMS switches [Park 2004, Shirane 2012], coil coupling [Zine-El-Abidine 2005, Kim 2009], etc. Although these tuning methods achieved tunability to a certain extent, they have their own limitations. Tuning obtained using a MEMS switch provides a very high inductance tunability of 230% [Shirane 2012] but in discrete steps, and it depends on the number of switches. Most of the MEMS inductors provide a very high quality factor and tunability. However, they occupy a huge area, and their on-chip integration is difficult [Chen 2020]. Using ferromagnetic material as the core of a solenoidal inductor [Vroubel 2004, Wang 2017, Chen 2020] and integrating ferromagnetic patterns on the inductor [Salvia 2005, Zhang 2009, Rahman 2015, Wang 2016] is another approach that has been studied for tunability. The techniques used for tuning these ferromagnetic inductors are external dc bias and magnetic field biasing. Magnetic field biasing requires external magnets or heavy coils, so this tuning method is not possible for monolithic RFIC. The better method of tuning the inductor on the chip is by controlling the permeability of ferromagnetic material, and it can be done by varying dc current (dc bias method), which is compatible with complimentary metal–oxide–semiconductor (CMOS) RFICs. The most explored ferromagnetic material is Permalloy () because of its high and electrically tunable permeability, low coercivity, and no stress anisotropy [Wang 2016].