- Open Access
Atypical Josephson effect in hybrid superconductor-hole systems
Phys. Rev. Research 8, 013289 – Published 17 March, 2026
DOI: https://doi.org/10.1103/dqgc-8crs
Abstract
We consider hybrid systems consisting of a hole-doped semiconductor coupled to electronic states of finite-size superconductors, where the opposite sign of the masses in the two subsystems gives rise to insulating gaps at subband anticrossings. Consequently, increasing the coupling strength to the superconductor can paradoxically suppress the proximity-induced superconductivity in the semiconductor by enhancing these insulating gaps. We demonstrate that the presence of such induced insulating gaps leads to a characteristic atypical behavior of the critical supercurrent in Josephson junctions based on these hybrid structures. Our findings provide important insights for the design of robust quantum computing platforms utilizing hybrid superconductor-hole systems.
Physics Subject Headings (PhySH)
Article Text
References (76)
- G. Scappucci, C. Kloeffel, F. A. Zwanenburg, D. Loss, M. Myronov, J.-J. Zhang, S. De Franceschi, G. Katsaros, and M. Veldhorst, The germanium quantum information route, Nat. Rev. Mater. 6, 926 (2020).
- R. Pillarisetty, Academic and industry research progress in germanium nanodevices, Nature (London) 479, 324 (2011).
- G. Burkard, T. D. Ladd, A. Pan, J. M. Nichol, and J. R. Petta, Semiconductor spin qubits, Rev. Mod. Phys. 95, 025003 (2023).
- C. Kloeffel, M. Trif, and D. Loss, Strong spin-orbit interaction and helical hole states in Ge/Si nanowire, Phys. Rev. B 84, 195314 (2011).
- X.-J. Hao, T. Tu, G. Cao, C. Zhou, H.-O. Li, G.-C. Guo, W. Y. Fung, Z. Ji, G.-P. Guo, and W. Lu, Strong and tunable spin-orbit coupling of one-dimensional holes in Ge/Si core/shell nanowires, Nano Lett. 10, 2956 (2010).
- Y. Hu, F. Kuemmeth, C. M. Lieber, and C. M. Marcus, Hole spin relaxation in Ge/Si core-shell nanowire qubits, Nat. Nanotechnol. 7, 47 (2012).
- L. A. Terrazos, E. Marcellina, Z. Wang, S. N. Coppersmith, M. Friesen, A. R. Hamilton, X. Hu, B. Koiller, A. L. Saraiva, D. Culcer, and R. B. Capaz, Theory of hole-spin qubits in strained germanium quantum dots, Phys. Rev. B 103, 125201 (2021).
- F. N. M. Froning, C. L. C., O. A. H. van der Molen, A. Li, E. P. A. M. Bakkers, D. M. Zumbühl, and F. R. Braakman, Ultrafast hole spin qubit with gate-tunable spin-orbit switch functionality, Nat. Nanotechnol. 16, 308 (2021).
- H. Liu, T. Zhang, K. Wang, F. Gao, G. Xu, X. Zhang, S.-X. Li, G. Cao, T. Wang, J. Zhang, X. Hu, H.-O. Li, and G.-P. Guo, Gate-tunable spin-orbit coupling in a germanium hole double quantum dot, Phys. Rev. Appl. 17, 044052 (2022).
- C. Adelsberger, M. Benito, S. Bosco, J. Klinovaja, and D. Loss, Hole-spin qubits in Ge nanowire quantum dots: Interplay of orbital magnetic field, strain, and growth direction, Phys. Rev. B 105, 075308 (2022).
- D. V. Bulaev and D. Loss, Spin relaxation and decoherence of holes in quantum dots, Phys. Rev. Lett. 95, 076805 (2005).
- S. Bosco, M. Benito, C. Adelsberger, and D. Loss, Squeezed hole spin qubits in Ge quantum dots with ultrafast gates at low power, Phys. Rev. B 104, 115425 (2021).
- S. Bosco, B. Hetényi, and D. Loss, Hole spin qubits in FinFETs with fully tunable spin-orbit coupling and sweet spots for charge noise, PRX Quantum 2, 010348 (2021).
- S. Bosco, P. Scarlino, J. Klinovaja, and D. Loss, Fully tunable longitudinal spin-photon interactions in Si and Ge quantum dots, Phys. Rev. Lett. 129, 066801 (2022).
- M. J. Carballido, S. Svab, R. S. Eggli, T. Patlatiuk, P. Chevalier Kwon, J. Schuff, R. M. Kaiser, L. C. Camenzind, A. Li, N. Ares, E. P. A. M. Bakkers, S. Bosco, J. C. Egues, D. Loss, and D. M. Zumbühl, Compromise-free scaling of qubit speed and coherence, Nat. Commun. 16, 7616 (2025).
- J. C. Abadillo-Uriel, E. A. Rodríguez-Mena, B. Martinez, and Y.-M. Niquet, Hole-spin driving by strain-induced spin-orbit interactions, Phys. Rev. Lett. 131, 097002 (2023).
- N. W. Hendrickx, L. Massai, M. Mergenthaler, F. J. Schupp, S. Paredes, S. W. Bedell, G. Salis, and A. Fuhrer, Sweet-spot operation of a germanium hole spin qubit with highly anisotropic noise sensitivity, Nat. Mater. 23, 920 (2024).
- C.-A. Wang, H. E. Ercan, M. F. Gyure, G. Scappucci, M. Veldhorst, and M. Rimbach-Russ, Modeling of planar germanium hole qubits in electric and magnetic fields, npj Quantum Inf. 10, 102 (2024).
- S. Bosco and D. Loss, Fully tunable hyperfine interactions of hole spin qubits in Si and Ge quantum dots, Phys. Rev. Lett. 127, 190501 (2021).
- Y. Hu, H. O. H. Churchill, D. J. Reilly, J. Xiang, C. M. Lieber, and C. M. Marcus, A Ge/Si heterostructure nanowire-based double quantum dot with integrated charge sensor, Nat. Nanotechnol. 2, 622 (2007).
- J. Fischer, W. A. Coish, D. V. Bulaev, and D. Loss, Spin decoherence of a heavy hole coupled to nuclear spins in a quantum dot, Phys. Rev. B 78, 155329 (2008).
- L. C. Camenzind, S. Geyer, A. Fuhrer, R. J. Warburton, D. M. Zumbühl, and A. V. Kuhlmann, A hole spin qubit in a fin field-effect transistor above 4 kelvin, Nat. Electron. 5, 178 (2022).
- S. Geyer, B. Hetényi, S. Bosco, L. C. Camenzind, R. S. Eggli, A. Fuhrer, D. Loss, R. J. Warburton, D. M. Zumbühl, and A. V. Kuhlmann, Anisotropic exchange interaction of two hole-spin qubits, Nat. Phys. 20, 1152 (2024).
- S. D. Liles, D. J. Halverson, Z. Wang, A. Shamim, R. S. Eggli, I. K. Jin, J. Hillier, K. Kumar, I. Vorreiter, M. J. Rendell, J. Y. Huang, C. C. Escott, F. E. Hudson, W. H. Lim, D. Culcer, A. S. Dzurak, and A. R. Hamilton, A singlet-triplet hole-spin qubit in MOS silicon, Nat. Commun. 15, 7690 (2024).
- C.-A. Wang, V. John, H. Tidjani, C. X. Yu, A. S. Ivlev, C. Déprez, F. van Riggelen-Doelman, B. D. Woods, N. W. Hendrickx, W. I. L. Lawrie, L. E. A. Stehouwer, S. D. Oosterhout, A. Sammak, M. Friesen, G. Scappucci, S. L. de Snoo, M. Rimbach-Russ, F. Borsoi, and M. Veldhorst, Operating semiconductor quantum processors with hopping spins, Science 385, 447 (2024).
- F. Borsoi, N. W. Hendrickx, V. John, M. Meyer, S. Motz, F. van Riggelen, A. Sammak, S. L. de Snoo, G. Scappucci, and M. Veldhorst, Shared control of a 16 semiconductor quantum dot crossbar array, Nat. Nanotechnol. 19, 21 (2024).
- X. Zhang, E. Morozova, M. Rimbach-Russ, D. Jirovec, T.-K. Hsiao, P. C. Fariña, C.-A. Wang, S. D. Oosterhout, A. Sammak, G. Scappucci, M. Veldhorst, and L. M. K. Vandersypen, Universal control of four singlet–triplet qubits, Nat. Nanotechnol. 20, 209 (2024).
- D. Jirovec, A. Hofmann, A. Ballabio, P. M. Mutter, G. Tavani, M. Botifoll, A. Crippa, J. Kukucka, O. Sagi, F. Martins, J. Saez-Mollejo, I. Prieto, M. Borovkov, J. Arbiol, D. Chrastina, G. Isella, and G. Katsaros, A singlet-triplet hole spin qubit in planar Ge, Nat. Mater. 20, 1106 (2021).
- J. Saez-Mollejo, D. Jirovec, Y. Schell, J. Kukucka, S. Calcaterra, D. Chrastina, G. Isella, M. Rimbach-Russ, S. Bosco, and G. Katsaros, Exchange anisotropies in microwave-driven singlet-triplet qubits, Nat. Commun. 16, 3862 (2025).
- J. Xiang, A. Vidan, M. Tinkham, R. M. Westervelt, and C. M. Lieber, Ge/Si nanowire mesoscopic Josephson junctions, Nat. Nanotechnol. 1, 208 (2006).
- M. Valentini, O. Sagi, L. Baghumyan, T. de Gijsel, J. Jung, S. Calcaterra, A. Ballabio, J. Aguilera Servin, K. Aggarwal, M. Janik, T. Adletzberger, R. Seoane Souto, M. Leijnse, J. Danon, C. Schrade, E. Bakkers, D. Chrastina, G. Isella, and G. Katsaros, Parity-conserving Cooper-pair transport and ideal superconducting diode in planar germanium, Nat. Commun. 15, 169 (2024).
- M. Valentini, Mesoscopic phenomena in hybrid semiconductor-superconductor nanodevices: From full-shell nanowires to two-dimensional hole gas in germanium, Ph.D. dissertation, Institute of Science and Technology Austria, 2023.
- W. Chang, S. Albrecht, T. Jespersen, F. Kuemmeth, P. Krogstrup, J. Nygård, and C. M. Marcus, Hard gap in epitaxial semiconductor–superconductor nanowires, Nat. Nanotechnol. 10, 232 (2015).
- E. J. H. Lee, X. Jiang, M. Houzet, R. Aguado, C. M. Lieber, and S. De Franceschi, Spin-resolved Andreev levels and parity crossings in hybrid superconductor–semiconductor nanostructures, Nat. Nanotechnol. 9, 79 (2014).
- S. Park and A. L. Yeyati, Andreev spin qubits in multichannel Rashba nanowires, Phys. Rev. B 96, 125416 (2017).
- M. Hays, V. Fatemi, D. Bouman, J. Cerrillo, S. Diamond, K. Serniak, T. Connolly, P. Krogstrup, J. Nygård, A. L. Yeyati, A. Geresdi, and M. H. Devoret, Coherent manipulation of an Andreev spin qubit, Science 373, 430 (2021).
- M. Spethmann, X.-P. Zhang, J. Klinovaja, and D. Loss, Coupled superconducting spin qubits with spin-orbit interaction, Phys. Rev. B 106, 115411 (2022).
- A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys.-Usp. 44, 131 (2001).
- M. Luethi, K. Laubscher, S. Bosco, D. Loss, and J. Klinovaja, Planar Josephson junctions in germanium: Effect of cubic spin-orbit interaction, Phys. Rev. B 107, 035435 (2023).
- M. Luethi, H. F. Legg, K. Laubscher, D. Loss, and J. Klinovaja, Majorana bound states in germanium Josephson junctions via phase control, Phys. Rev. B 108, 195406 (2023).
- C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008).
- F. Maier, J. Klinovaja, and D. Loss, Majorana fermions in Ge/Si hole nanowires, Phys. Rev. B 90, 195421 (2014).
- K. Laubscher, J. D. Sau, and S. Das Sarma, Majorana zero modes in gate-defined germanium hole nanowires, Phys. Rev. B 109, 035433 (2024).
- C. Reeg, D. Loss, and J. Klinovaja, Finite-size effects in a nanowire strongly coupled to a thin superconducting shell, Phys. Rev. B 96, 125426 (2017).
- C. Reeg, D. Loss, and J. Klinovaja, Proximity effect in a two-dimensional electron gas coupled to a thin superconducting layer, Beilstein J. Nanotechnol. 9, 1263 (2018).
- C. Reeg, D. Loss, and J. Klinovaja, Metallization of a Rashba wire by a superconducting layer in the strong-proximity regime, Phys. Rev. B 97, 165425 (2018).
- N. B. Kopnin and A. S. Melnikov, Proximity-induced superconductivity in two-dimensional electronic systems, Phys. Rev. B 84, 064524 (2011).
- N. van Loo, G. P. Mazur, T. Dvir, G. Wang, R. C. Dekker, J.-Y. Wang, M. Lemang, C. Sfiligoj, A. Bordin, D. van Driel, G. Badawy, S. Gazibegovic, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Electrostatic control of the proximity effect in the bulk of semiconductor-superconductor hybrids, Nat. Commun. 14, 3325 (2023).
- M. Sütő, T. A. Prok, P. Makk, M. Kirti, G. Biasiol, S. Csonka, and E. Tóvári, Near-surface InAs two-dimensional electron gas on a GaAs substrate: Characterization and superconducting proximity effect, Phys. Rev. B 106, 235404 (2022).
- W. Mayer, J. Yuan, K. S. Wickramasinghe, T. Nguyen, M. C. Dartiailh, and J. Shabani, Superconducting proximity effect in epitaxial Al-InAs heterostructures, Appl. Phys. Lett. 114, 103104 (2019).
- H. Takayanagi and T. Kawakami, Superconducting proximity effect in the native inversion layer on InAs, Phys. Rev. Lett. 54, 2449 (1985).
- C. Adelsberger, H. F. Legg, D. Loss, and J. Klinovaja, Microscopic analysis of proximity-induced superconductivity and metallization effects in superconductor-germanium hole nanowires, Phys. Rev. B 108, 155433 (2023).
- T. D. Stanescu and S. Das Sarma, Proximity-induced superconductivity generated by thin films: Effects of Fermi surface mismatch and disorder in the superconductor, Phys. Rev. B 106, 085429 (2022).
- N. W. Hendrickx, D. P. Franke, A. Sammak, M. Kouwenhoven, D. Sabbagh, L. Yeoh, R. Li, M. L. V. Tagliaferri, M. Virgilio, G. Capellini, G. Scappucci, and M. Veldhorst, Gate-controlled quantum dots and superconductivity in planar germanium, Nat. Commun. 9, 2835 (2018).
- J. Ridderbos, M. Brauns, J. Shen, F. K. de Vries, A. Li, E. P. A. M. Bakkers, A. Brinkman, and F. A. Zwanenburg, Josephson effect in a few-hole quantum dot, Adv. Mater. 30, 1802257 (2018).
- N. W. Hendrickx, M. L. V. Tagliaferri, M. Kouwenhoven, R. Li, D. P. Franke, A. Sammak, A. Brinkman, G. Scappucci, and M. Veldhorst, Ballistic supercurrent discretization and micrometer-long Josephson coupling in germanium, Phys. Rev. B 99, 075435 (2019).
- F. Vigneau, R. Mizokuchi, D. C. Zanuz, X. Huang, S. Tan, R. Maurand, S. Frolov, A. Sammak, G. Scappucci, F. Lefloch, and S. De Franceschi, Germanium quantum-well Josephson field-effect transistors and interferometers, Nano Lett. 19, 1023 (2019).
- J. Ridderbos, M. Brauns, F. K. de Vries, J. Shen, A. Li, S. Kölling, M. A. Verheijen, A. Brinkman, W. G. van der Wiel, E. P. A. M. Bakkers, and F. A. Zwanenburg, Hard superconducting gap and diffusion-induced superconductors in Ge-Si nanowires, Nano Lett. 20, 122 (2020).
- K. Aggarwal, A. Hofmann, D. Jirovec, I. Prieto, A. Sammak, M. Botifoll, S. Martí-Sánchez, M. Veldhorst, J. Arbiol, G. Scappucci, J. Danon, and G. Katsaros, Enhancement of proximity-induced superconductivity in a planar Ge hole gas, Phys. Rev. Res. 3, L022005 (2021).
- A. Tosato, et al., Hard superconducting gap in germanium, Commun. Mater. 4, 23 (2023).
- E. Zhuo, Z. Lyu, X. Sun, A. Li, B. Li, Z. Ji, J. Fan, E. Bakkers, X. Han, X. Song, F. Qu, G. Liu, J. Shen, and L. Lu, Hole-type superconducting gatemon qubit based on Ge/Si core/shell nanowires, npj Quantum Inf. 9, 51 (2023).
- A. G. Moghaddam, T. Kernreiter, M. Governale, and U. Zülicke, Exporting superconductivity across the gap: Proximity effect for semiconductor valence-band states due to contact with a simple-metal superconductor, Phys. Rev. B 89, 184507 (2014).
- S. S. Babkin, B. Joecker, K. Flensberg, M. Serbyn, and J. Danon, Superconducting proximity effect in two-dimensional hole gases, Phys. Rev. B 111, 214518 (2025).
- D. M. Pino, R. S. Souto, M. J. Calderón, R. Aguado, and J. C. Abadillo-Uriel, Theory of superconducting proximity effect in hole-based hybrid semiconductor-superconductor devices, Phys. Rev. B 111, 235443 (2025).
- J. A. Steele, P. J. Strohbeen, C. Verdi, A. Baktash, A. Danilenko, Y.-H. Chen, J. van Dijk, L. Wang, E. Demler, S. Salmani-Rezaie, P. Jacobson, and J. Shabani, Superconductivity in substitutional Ga-hyperdoped Ge epitaxial thin films, Nat. Nanotechnol. 20, 1757 (2025).
- L. Lakic, W. I. L. Lawrie, D. van Driel, L. E. A. Stehouwer, Y. Su, M. Veldhorst, G. Scappucci, F. Kuemmeth, and A. Chatterjee, A quantum dot in germanium proximitized by a superconductor, Nat. Mater. 24, 552 (2025).
- E. Marcellina, A. R. Hamilton, R. Winkler, and D. Culcer, Spin-orbit interactions in inversion-asymmetric two-dimensional hole systems: A variational analysis, Phys. Rev. B 95, 075305 (2017).
- R. Hess, H. F. Legg, D. Loss, and J. Klinovaja, Josephson transistor from the superconducting diode effect in domain wall and skyrmion magnetic racetracks, Phys. Rev. B 108, 174516 (2023).
- A. Levy Yeyati, A. Martín-Rodero, and F. J. García-Vidal, Self-consistent theory of superconducting mesoscopic weak links, Phys. Rev. B 51, 3743 (1995).
- F. K. de Vries, T. Timmerman, V. P. Ostroukh, J. van Veen, A. J. A. Beukman, F. Qu, M. Wimmer, B.-M. Nguyen, A. A. Kiselev, W. Yi, M. Sokolich, M. J. Manfra, C. M. Marcus, and L. P. Kouwenhoven, superconducting quantum interference through trivial edge states in InAs, Phys. Rev. Lett. 120, 047702 (2018).
- W. Himmler, R. Fischer, M. Barth, J. Fuchs, D. A. Kozlov, N. N. Mikhailov, S. A. Dvoretsky, C. Strunk, C. Gorini, K. Richter, and D. Weiss, Supercurrent interference in HgTe-wire Josephson junctions, Phys. Rev. Res. 5, 043021 (2023).
- C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, kwant: A software package for quantum transport, New J. Phys. 16, 063065 (2014).
- C. Ciaccia, R. Haller, A. Drachmann, T. Lindemann, M. Manfra, C. Schrade, and C. Schoenenberger, Charge-4e supercurrent in a two-dimensional InAs-Al superconductor-semiconductor heterostructure, Commun. Phys. 7, 41 (2024).
- D. Willsch, et al., Observation of Josephson harmonics in tunnel junctions, Nat. Phys. 20, 815 (2024).
- H. F. Legg, K. Laubscher, D. Loss, and J. Klinovaja, Parity-protected superconducting diode effect in topological Josephson junctions, Phys. Rev. B 108, 214520 (2023).
- P. D. Johannsen, H. Legg, S. Bosco, D. Loss, and J. Klinovaja, Data and code for “Atypical Josephson effect in hybrid superconductor-hole systems”, Zenodo (2026), https://doi.org/10.5281/zenodo.18409848.