Finite-length effects and Coulomb interaction in Josephson junctions based on Ge quantum wells and probed with microwave spectroscopy

Proximitized <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>Ge</a:mi> </a:math> quantum wells have emerged as a platform for studying Andreev bound states (ABSs) due to their expected strong spin-orbit interaction and high mobility. Here, we used microwave spectroscopy techniques to investigate ABSs in Josephson junctions (JJs) realized in proximitized <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>Ge</c:mi> </c:math> quantum wells. Spectroscopic signatures observed in a 350-nm junction indicated the presence of multiple ABSs and were reproduced with a model including finite-length effects. The ABS spectra measured for a <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mn>1.2</e:mn> </e:math> - <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mtext fontfamily="times">μ</g:mtext> <g:mrow> <g:mrow> <g:mi mathvariant="normal">m</g:mi> </g:mrow> </g:mrow> </g:math> junction were explained by a model including three ABSs in two conduction channels and finite Coulomb interaction. Our work highlights the importance of interactions in JJs and serves as a basis for understanding and manipulating ABSs in <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>Ge</k:mi> </k:math> -based hybrid devices.