Fractal Loop Inductors

We report a significant fractal-scaled increase in inductance for conductive pathways oriented in a fractal loop-based layout for a suite of planar inductors, which effectively occupy the same layout space and require only a single fabrication layer. These are closely predicted using a fractal scaling model for generalized loop-based inductors elucidated in this paper. Single-layer planar fractal structures were investigated with the loop inductor serving as the base construct. Thin sub-200 nm Cr/Au metal films compatible with flexible and stretchable substrates were used. It was found that while higher fractal orders did improve the inductive performance (over nine times from zeroth to third order), it is met with increased resistance. However, when compared with its equivalent series orientation, the effective sheet resistance of the simple fractal strategy demonstrated a clear advantage of up to four times. When the inductor is normalized for thickness, a quality factor ${Q}$ greater than 40 is observed for all structures. Finally, the inductance quality gain figure of merit showing the optimal geometrical approach is introduced to quantify the quality of the structure’s inductance over its resistance.