Low-loss lumped-element inductors made from granular aluminum

Lumped-element inductors are an integral component in the circuit QED toolbox. However, it is challenging to build inductors that are simultaneously compact, linear, and low-loss with standard approaches that either rely on the geometric inductance of superconducting thin films or on the kinetic inductance of Josephson-junction arrays. In this work, we overcome this challenge by utilizing the high kinetic inductance offered by superconducting granular aluminum (grAl). We demonstrate lumped-element inductors with a few nH of inductance that are up to <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mn>100</a:mn></a:math> times shorter than inductors built from pure aluminum (Al). To characterize the properties of these linear inductors, we first report on the performance of lumped-element resonators built entirely out of grAl with sheet inductances varying from <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mn>30</c:mn></c:math>–<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mn>320</e:mn></e:math> pH/sq and self-Kerr nonlinearities of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mn>0.2</g:mn></g:math>–<i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mn>20</i:mn><i:mspace width="0.2em"/><i:mrow><i:mi>Hz</i:mi><i:mo>/</i:mo><i:mi>photon</i:mi></i:mrow></i:math>. Further, we demonstrate integration of these grAl inductors into hybrid resonators with <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>Al</l:mi></l:math> or tantalum (<n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>Ta</n:mi></n:math>) capacitor electrodes without increasing total internal losses. Interestingly, the measured internal quality factors systematically decrease with increasing room-temperature resistivity of the grAl film for all devices, indicating a trade-off between compactness and internal loss. For our lowest resistivity grAl films, we measure quality factors reaching <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mn>3.5</p:mn><p:mo>×</p:mo><p:msup><p:mn>10</p:mn><p:mn>6</p:mn></p:msup></p:math> for the all-grAl devices and <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mn>4.5</r:mn><r:mo>×</r:mo><r:msup><r:mn>10</r:mn><r:mn>6</r:mn></r:msup></r:math> for the hybrid grAl/<t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mi>Ta</t:mi></t:math> devices, similar to state-of-the-art quantum circuits. Our loss analysis suggests that the surface-loss factor of grAl is similar to that of pure <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mi>Al</v:mi></v:math> for our lowest resistivity films, while the increasing losses with resistivity could be explained by increasing inductor loss in the grAl film.