Abstract
In the effort to develop disruptive quantum technologies, germanium is emerging as a versatile material to realize devices capable of encoding, processing and transmitting quantum information. These devices leverage the special properties of holes in germanium, such as their inherently strong spinâorbit coupling and their ability to host superconducting pairing correlations. In this Review, we start by introducing the physics of holes in low-dimensional germanium structures, providing key insights from a theoretical perspective. We then examine the materials-science progress underpinning germanium-based planar heterostructures and nanowires. We go on to review the most significant experimental results demonstrating key building blocks for quantum technology, such as an electrically driven universal quantum gate set with spin qubits in quantum dots and superconductorâsemiconductor devices for hybrid quantum systems. We conclude by identifying the most promising avenues towards scalable quantum information processing in germanium-based systems.
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References
Pillarisetty, R. Academic and industry research progress in germanium nanodevices. Nature 479, 324â328 (2011).
Kamata, Y. High-k/Ge MOSFETs for future nanoelectronics. Mater. Today 11, 30â38 (2008).
Loss, D. & DiVincenzo, D. P. Quantum computation with quantum dots. Phys. Rev. A 57, 120â126 (1998).
Kitaev, A. Y. Unpaired Majorana fermions in quantum wires. Phys.-Uspekhi 44, 131â136 (2001).
Larsen, T. et al. Semiconductor-nanowire-based superconducting qubit. Phys. Rev. Lett. 115, 127001 (2015).
Dimoulas, A., Tsipas, P., Sotiropoulos, A. & Evangelou, E. K. Fermi-level pinning and charge neutrality level in germanium. Appl. Phys. Lett. 89, 252110 (2006).
Pillarisetty, R. et al. in Proceedings of the 2018 IEEE International Electron Devices Meeting (IEDM) 6.3.1â6.3.4 (IEEE, 2018).
Dobbie, A. et al. Ultra-high hole mobility exceeding one million in a strained germanium quantum well. Appl. Phys. Lett. 101, 172108 (2012).
Sammak, A. et al. Shallow and undoped germanium quantum wells: a playground for spin and hybrid quantum technology. Adv. Funct. Mater. 29, 1807613 (2019).
Gao, F. et al. Site-controlled uniform Ge/Si hut wires with electrically tunable spinâorbit coupling. Adv. Mater. 32, 1906523 (2020).
Hu, Y. et al. A Ge/Si heterostructure nanowire-based double quantum dot with integrated charge sensor. Nat. Nanotechnol. 2, 622â625 (2007).
Hu, Y., Kuemmeth, F., Lieber, C. M. & Marcus, C. M. Hole spin relaxation in GeâSi coreâshell nanowire qubits. Nat. Nanotechnol. 7, 47â50 (2012).
Ares, N. et al. Nature of tunable hole g factors in quantum dots. Phys. Rev. Lett. 110, 046602 (2013).
Watzinger, H. et al. Heavy-hole states in germanium hut wires. Nano Lett. 16, 6879â6885 (2016).
Watzinger, H. et al. A germanium hole spin qubit. Nat. Commun. 9, 3902 (2018).
Xu, G. et al. Dipole coupling of a hole double quantum dot in germanium hut wire to a microwave resonator. New J. Phys. 22, 083068 (2020).
Li, Y. et al. Coupling a germanium hut wire hole quantum dot to a superconducting microwave resonator. Nano Lett. 18, 2091â2097 (2018).
Hendrickx, N. W. et al. Gate-controlled quantum dots and superconductivity in planar germanium. Nat. Commun. 9, 2835 (2018).
Hendrickx, N., Franke, D., Sammak, A., Scappucci, G. & Veldhorst, M. Fast two-qubit logic with holes in germanium. Nature 577, 487â491 (2020).
Hendrickx, N. W. et al. A four-qubit germanium quantum processor. Preprint at arXiv http://arxiv.org/abs/2009.04268 (2020).
Hendrickx, N. W. et al. A single-hole spin qubit. Nat. Commun. 11, 3478 (2020).
Xiang, J., Vidan, A., Tinkham, M., Westervelt, R. M. & Lieber, C. M. Ge/Si nanowire mesoscopic Josephson junctions. Nat. Nanotechnol. 1, 208â213 (2006).
Ridderbos, J. et al. Josephson effect in a few-hole quantum dot. Adv. Mater. 30, 1802257 (2018).
Vigneau, F. et al. Germanium quantum-well Josephson field-effect transistors and interferometers. Nano Lett. 19, 1023â1027 (2019).
Petta, J. R. et al. Coherent manipulation of coupled electron spins in semiconductor quantum dots. Science 309, 2180â2184 (2005).
Koppens, F. H. L. et al. Driven coherent oscillations of a single electron spin in a quantum dot. Nature 442, 766â771 (2006).
Nowack, K. C., Koppens, F. H. L., Nazarov, Y. V. & Vandersypen, L. M. K. Coherent control of a single electron spin with electric fields. Science 318, 1430â1433 (2007).
Kloeffel, C. & Loss, D. Prospects for spin-based quantum computing in quantum dots. Annu. Rev. Condens. Matter Phys. 4, 51â81 (2013).
Bulaev, D. V. & Loss, D. Spin relaxation and decoherence of holes in quantum dots. Phys. Rev. Lett. 95, 076805 (2005).
Bulaev, D. V. & Loss, D. Electric dipole spin resonance for heavy holes in quantum dots. Phys. Rev. Lett. 98, 097202 (2007).
Fischer, J., Coish, W. A., Bulaev, D. V. & Loss, D. Spin decoherence of a heavy hole coupled to nuclear spins in a quantum dot. Phys. Rev. B 78, 155329 (2008).
Trif, M., Simon, P. & Loss, D. Relaxation of hole spins in quantum dots via two-phonon processes. Phys. Rev. Lett. 103, 106601 (2009).
Heiss, D. et al. Observation of extremely slow hole spin relaxation in self-assembled quantum dots. Phys. Rev. B 76, 241306 (2007).
Gerardot, B. D. et al. Optical pumping of a single hole spin in a quantum dot. Nature 451, 441â444 (2008).
Brunner, D. et al. A coherent single-hole spin in a semiconductor. Science 325, 70â72 (2009).
Warburton, R. J. Single spins in self-assembled quantum dots. Nat. Mater. 12, 483â493 (2013).
Nolting, W. & Ramakanth, A. Quantum Theory of Magnetism (Springer, 2009).
Winkler, R. Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (Springer, 2003).
Luttinger, J. M. & Kohn, W. Motion of electrons and holes in perturbed periodic fields. Phys. Rev. 97, 869â883 (1955).
Luttinger, J. M. Quantum theory of cyclotron resonance in semiconductors: general theory. Phys. Rev. 102, 1030â1041 (1956).
Winkler, R., Culcer, D., Papadakis, S. J., Habib, B. & Shayegan, M. Spin orientation of holes in quantum wells. Semicond. Sci. Technol. 23, 114017 (2008).
Lawaetz, P. Valence-band parameters in cubic semiconductors. Phys. Rev. B 4, 3460â3467 (1971).
Terrazos, L. A. et al. Theory of hole-spin qubits in strained germanium quantum dots. Preprint at arXiv http://arxiv.org/abs/1803.10320 (2020).
Bir, G. L. & Pikus, G. E. Symmetry and Strain-Induced Effects in Semiconductors (Wiley, 1974).
van Kesteren, H. W., Cosman, E. C., van der Poel, W. A. J. A. & Foxon, C. T. Fine structure of excitons in type-II GaAs/AlAs quantum wells. Phys. Rev. B 41, 5283â5292 (1990).
Lodari, M. et al. Light effective hole mass in undoped Ge/SiGe quantum wells. Phys. Rev. B 100, 041304 (2019).
Sercel, P. C. & Vahala, K. J. Analytical formalism for determining quantum-wire and quantum-dot band structure in the multiband envelope-function approximation. Phys. Rev. B 42, 3690â3710 (1990).
Harada, Y., Kita, T., Wada, O. & Ando, H. Anisotropic magneto-optical effects in one-dimensional diluted magnetic semiconductors. Phys. Rev. B 74, 245323 (2006).
Csontos, D., Brusheim, P., Zülicke, U. & Xu, H. Q. Spin-\(\tfrac{3}{2}\) physics of semiconductor hole nanowires: Valence-band mixing and tunable interplay between bulk-material and orbital bound-state spin splittings. Phys. Rev. B 79, 155323 (2009).
Kloeffel, C., Trif, M. & Loss, D. Strong spin-orbit interaction and helical hole states in Ge/Si nanowires. Phys. Rev. B 84, 195314 (2011).
Kloeffel, C., RanÄiÄ, M. J. & Loss, D. Direct Rashba spin-orbit interaction in Si and Ge nanowires with different growth directions. Phys. Rev. B 97, 235422 (2018).
Menéndez, J., Singh, R. & Drucker, J. Theory of strain effects on the Raman spectrum of Si-Ge core-shell nanowires. Ann. Phys. 523, 145â156 (2011).
Kloeffel, C., Trif, M. & Loss, D. Acoustic phonons and strain in core/shell nanowires. Phys. Rev. B 90, 115419 (2014).
Kloeffel, C., Trif, M., Stano, P. & Loss, D. Circuit QED with hole-spin qubits in Ge/Si nanowire quantum dots. Phys. Rev. B 88, 241405 (2013).
Nigg, S. E., Fuhrer, A. & Loss, D. Superconducting grid-bus surface code architecture for hole-spin qubits. Phys. Rev. Lett. 118, 147701 (2017).
Maier, F., Meng, T. & Loss, D. Strongly interacting holes in Ge/Si nanowires. Phys. Rev. B 90, 155437 (2014).
Mao, L., Gong, M., Dumitrescu, E., Tewari, S. & Zhang, C. Hole-doped semiconductor nanowire on top of an s-wave superconductor: a new and experimentally accessible system for Majorana fermions. Phys. Rev. Lett. 108, 177001 (2012).
Maier, F., Klinovaja, J. & Loss, D. Majorana fermions in Ge/Si hole nanowires. Phys. Rev. B 90, 195421 (2014).
Ivchenko, E. L., Kaminski, A. Y. & Rössler, U. Heavy-light hole mixing at zinc-blende (001) interfaces under normal incidence. Phys. Rev. B 54, 5852â5859 (1996).
Vervoort, L., Ferreira, R. & Voisin, P. Effects of interface asymmetry on hole subband degeneracies and spin-relaxation rates in quantum wells. Phys. Rev. B 56, R12744âR12747 (1997).
Guettler, T. et al. Optical polarization relaxation in InxGa1âxAs-based quantum wells: Evidence of the interface symmetry-reduction effect. Phys. Rev. B 58, R10179âR10182 (1998).
Vervoort, L., Ferreira, R. & Voisin, P. Spin-splitting of the subbands of InGaAs-InP and other âno common atomâ quantum wells. Semicond. Sci. Technol. 14, 227â230 (1999).
Olesberg, J. T. et al. Interface contributions to spin relaxation in a short-period InAs/GaSb superlattice. Phys. Rev. B 64, 201301 (2001).
Hall, K. C. et al. Spin relaxation in (110) and (001) InAs/GaSb superlattices. Phys. Rev. B 68, 115311 (2003).
Golub, L. E. & Ivchenko, E. L. Spin splitting in symmetrical SiGe quantum wells. Phys. Rev. B 69, 115333 (2004).
Nestoklon, M. O., Ivchenko, E. L., Jancu, J.-M. & Voisin, P. Electric field effect on electron spin splitting in SiGe/Si quantum wells. Phys. Rev. B 77, 155328 (2008).
Prada, M., Klimeck, G. & Joynt, R. Spinâorbit splittings in Si/SiGe quantum wells: from ideal Si membranes to realistic heterostructures. New J. Phys. 13, 013009 (2011).
Furthmeier, S. et al. Enhanced spinâorbit coupling in core/shell nanowires. Nat. Commun. 7, 12413 (2016).
Wojcik, P., Bertoni, A. & Goldoni, G. Enhanced Rashba spin-orbit coupling in core-shell nanowires by the interfacial effect. Appl. Phys. Lett. 114, 073102 (2019).
Hao, X.-J. et al. Strong and tunable spin-orbit coupling of one-dimensional holes in Ge/Si core/shell nanowires. Nano Lett. 10, 2956â2960 (2010).
Higginbotham, A. P. et al. Antilocalization of coulomb blockade in a Ge/Si nanowire. Phys. Rev. Lett. 112, 216806 (2014).
Brauns, M., Ridderbos, J., Li, A., Bakkers, E. P. A. M. & Zwanenburg, F. A. Electric-field dependent g-factor anisotropy in Ge-Si core-shell nanowire quantum dots. Phys. Rev. B 93, 121408 (2016).
Wang, R., Deacon, R. S., Yao, J., Lieber, C. M. & Ishibashi, K. Electrical modulation of weak-antilocalization and spinâorbit interaction in dual gated Ge/Si core/shell nanowires. Semicond. Sci. Technol. 32, 094002 (2017).
Sun, J. et al. Helical hole state in multiple conduction modes in Ge/Si core/shell nanowire. Nano Lett. 18, 6144â6149 (2018).
de Vries, F. K. et al. Spinâorbit interaction and induced superconductivity in a one-dimensional hole gas. Nano Lett. 18, 6483â6488 (2018).
Golovach, V. N., Borhani, M. & Loss, D. Electric-dipole-induced spin resonance in quantum dots. Phys. Rev. B 74, 165319 (2006).
Stano, P. et al. g-factor of electrons in gate-defined quantum dots in a strong in-plane magnetic field. Phys. Rev. B 98, 195314 (2018).
Camenzind, L. C. et al. Hyperfine-phonon spin relaxation in a single-electron GaAs quantum dot. Nat. Commun. 9, 3454 (2018).
Stano, P. et al. Orbital effects of a strong in-plane magnetic field on a gate-defined quantum dot. Phys. Rev. B 99, 085308 (2019).
Winkler, R. Rashba spin splitting in two-dimensional electron and hole systems. Phys. Rev. B 62, 4245â4248 (2000).
Chesi, S., Giuliani, G. F., Rokhinson, L. P., Pfeiffer, L. N. & West, K. W. Anomalous spin-resolved point-contact transmission of holes due to cubic Rashba spin-orbit coupling. Phys. Rev. Lett. 106, 236601 (2011).
Nichele, F. et al. Characterization of spin-orbit interactions of GaAs heavy holes using a quantum point contact. Phys. Rev. Lett. 113, 046801 (2014).
Nichele, F. et al. Spin-orbit splitting and effective masses in p-type GaAs two-dimensional hole gases. Phys. Rev. B 89, 081306 (2014).
Miserev, D. S. & Sushkov, O. P. Dimensional reduction of the Luttinger Hamiltonian and g-factors of holes in symmetric two-dimensional semiconductor heterostructures. Phys. Rev. B 95, 085431 (2017).
Srinivasan, A. et al. Detection and control of spin-orbit interactions in a GaAs hole quantum point contact. Phys. Rev. Lett. 118, 146801 (2017).
Hung, J.-T., Marcellina, E., Wang, B., Hamilton, A. R. & Culcer, D. Spin blockade in hole quantum dots: Tuning exchange electrically and probing Zeeman interactions. Phys. Rev. B 95, 195316 (2017).
Marcellina, E., Hamilton, A. R., Winkler, R. & Culcer, D. Spin-orbit interactions in inversion-asymmetric two-dimensional hole systems: a variational analysis. Phys. Rev. B 95, 075305 (2017).
Liu, H., Marcellina, E., Hamilton, A. R. & Culcer, D. Strong spin-orbit contribution to the hall coefficient of two-dimensional hole systems. Phys. Rev. Lett. 121, 087701 (2018).
Mizokuchi, R. et al. Hole weak anti-localization in a strained-Ge surface quantum well. Appl. Phys. Lett. 111, 063102 (2017).
Moriya, R. et al. Cubic Rashba spin-orbit interaction of a two-dimensional hole gas in a strained-Ge/SiGe quantum well. Phys. Rev. Lett. 113, 086601 (2014).
Chou, C.-T. et al. Weak anti-localization of two-dimensional holes in germanium beyond the diffusive regime. Nanoscale 10, 20559â20564 (2018).
Nenashev, A. V., Dvurechenskii, A. V. & Zinovieva, A. F. Wave functions and g factor of holes in Ge/Si quantum dots. Phys. Rev. B 67, 205301 (2003).
Pryor, C. E. & Flatté, M. E. Landé g factors and orbital momentum quenching in semiconductor quantum dots. Phys. Rev. Lett. 96, 026804 (2006).
van Bree, J. et al. Anisotropy of electron and hole g tensors of quantum dots: an intuitive picture based on spin-correlated orbital currents. Phys. Rev. B 93, 035311 (2016).
Katsaros, G. et al. Hybrid superconductorâsemiconductor devices made from self-assembled SiGe nanocrystals on silicon. Nat. Nanotechnol. 5, 458â464 (2010).
Wimbauer, T., Oettinger, K., Efros, A. L., Meyer, B. K. & Brugger, H. Zeeman splitting of the excitonic recombination in InxGa1âxAs/GaAs single quantum wells. Phys. Rev. B 50, 8889â8892 (1994).
Durnev, M. V. et al. Magnetic field induced valence band mixing in [111] grown semiconductor quantum dots. Phys. Rev. B 87, 085315 (2013).
Drichko, I. L. et al. In-plane magnetic field effect on hole cyclotron mass and gz factor in high-mobility SiGe/Ge/SiGe structures. Phys. Rev. B 90, 125436 (2014).
Simion, G. E. & Lyanda-Geller, Y. B. Magnetic field spectral crossings of Luttinger holes in quantum wells. Phys. Rev. B 90, 195410 (2014).
Pingenot, J., Pryor, C. E. & Flatté, M. E. Electric-field manipulation of the Landé g tensor of a hole in an In0.5Ga0.5As/GaAs self-assembled quantum dot. Phys. Rev. B 84, 195403 (2011).
Maier, F., Kloeffel, C. & Loss, D. Tunable g factor and phonon-mediated hole spin relaxation in Ge/Si nanowire quantum dots. Phys. Rev. B 87, 161305 (2013).
Ares, N. et al. SiGe quantum dots for fast hole spin Rabi oscillations. Appl. Phys. Lett. 103, 263113 (2013).
Marcellina, E. et al. Electrical control of the Zeeman spin splitting in two-dimensional hole systems. Phys. Rev. Lett. 121, 077701 (2018).
Crippa, A. et al. Electrical spin driving by g-matrix modulation in spin-orbit qubits. Phys. Rev. Lett. 120, 137702 (2018).
Venitucci, B., Bourdet, L., Pouzada, D. & Niquet, Y.-M. Electrical manipulation of semiconductor spin qubits within the g-matrix formalism. Phys. Rev. B 98, 155319 (2018).
DiVincenzo, D. P. The physical implementation of quantum computation. Fortschr. Phys. 48, 771â783 (2000).
Itoh, K. et al. High purity isotopically enriched 70Ge and 74Ge single crystals: isotope separation, growth, and properties. J. Mater. Res. 8, 1341â1347 (1993).
Asen-Palmer, M. et al. Thermal conductivity of germanium crystals with different isotopic compositions. Phys. Rev. B 56, 9431â9447 (1997).
Becker, P., Pohl, H.-J., Riemann, H. & Abrosimov, N. Enrichment of silicon for a better kilogram. Phys. Status Solidi A 207, 49â66 (2010).
Tyryshkin, A. M. et al. Electron spin coherence exceeding seconds in high-purity silicon. Nat. Mater. 11, 143â147 (2012).
Veldhorst, M. et al. An addressable quantum dot qubit with fault-tolerant control-fidelity. Nat. Nanotechnol. 9, 981â985 (2014).
Muhonen, J. T. et al. Storing quantum information for 30 seconds in a nanoelectronic device. Nat. Nanotechnol. 9, 986â991 (2014).
Sigillito, A. J. et al. Electron spin coherence of shallow donors in natural and isotopically enriched germanium. Phys. Rev. Lett. 115, 247601 (2015).
Fischer, J. & Loss, D. Hybridization and spin decoherence in heavy-hole quantum dots. Phys. Rev. Lett. 105, 266603 (2010).
Maier, F. & Loss, D. Effect of strain on hyperfine-induced hole-spin decoherence in quantum dots. Phys. Rev. B 85, 195323 (2012).
Zwanenburg, F. A. et al. Silicon quantum electronics. Rev. Mod. Phys. 85, 961â1019 (2013).
Vandersypen, L. M. K. & Eriksson, M. A. Quantum computing with semiconductor spins. Phys. Today 72, 38â45 (2019).
Burkard, G., Loss, D. & DiVincenzo, D. P. Coupled quantum dots as quantum gates. Phys. Rev. B 59, 2070â2078 (1999).
Weiss, K. M., Elzerman, J. M., Delley, Y. L., Miguel-Sanchez, J. & Imamoglu, A. Coherent two-electron spin qubits in an optically active pair of coupled InGaAs quantum dots. Phys. Rev. Lett. 109, 107401 (2012).
Chesi, S. et al. Single-spin manipulation in a double quantum dot in the field of a micromagnet. Phys. Rev. B 90, 235311 (2014).
Wong, C. H., Eriksson, M. A., Coppersmith, S. N. & Friesen, M. High-fidelity singlet-triplet S-T_ qubits in inhomogeneous magnetic fields. Phys. Rev. B 92, 045403 (2015).
Reed, M. D. et al. Reduced sensitivity to charge noise in semiconductor spin qubits via symmetric operation. Phys. Rev. Lett. 116, 110402 (2016).
Martins, F. et al. Noise suppression using symmetric exchange gates in spin qubits. Phys. Rev. Lett. 116, 116801 (2016).
Abadillo-Uriel, J. C., Eriksson, M. A., Coppersmith, S. N. & Friesen, M. Enhancing the dipolar coupling of a S-T0 qubit with a transverse sweet spot. Nat. Commun. 10, 5641 (2019).
Khaetskii, A. V. & Nazarov, Y. V. Spin-flip transitions between Zeeman sublevels in semiconductor quantum dots. Phys. Rev. B 64, 125316 (2001).
Golovach, V. N., Khaetskii, A. & Loss, D. Phonon-induced decay of the electron spin in quantum dots. Phys. Rev. Lett. 93, 016601 (2004).
Stano, P. & Fabian, J. Theory of phonon-induced spin relaxation in laterally coupled quantum dots. Phys. Rev. Lett. 96, 186602 (2006).
Kornich, V., Kloeffel, C. & Loss, D. Phonon-assisted relaxation and decoherence of singlet-triplet qubits in Si/SiGe quantum dots. Quantum 2, 70 (2018).
Li, J., Venitucci, B. & Niquet, Y.-M. Hole-phonon interactions in quantum dots: Effects of phonon confinement and encapsulation materials on spin-orbit qubits. Phys. Rev. B 102, 075415 (2020).
People, R. & Bean, J. C. Band alignments of coherently strained GeSi/Si heterostructures on 001 GeSi substrates. Appl. Phys. Lett. 48, 538â540 (1986).
People, R. Indirect band gap and band alignment for coherently strained bulk alloys on germanium (001) substrates. Phys. Rev. B 34, 2508â2510 (1986).
Virgilio, M. & Grosso, G. Type-I alignment and direct fundamental gap in SiGe based heterostructures. J. Phys. Condens. Matter 18, 1021â1031 (2006).
Giorgioni, A. et al. Strong confinement-induced engineering of the g factor and lifetime of conduction electron spins in Ge quantum wells. Nat. Commun. 7, 13886 (2016).
Paul, D. The progress towards terahertz quantum cascade lasers on silicon substrates. Laser Photonics Rev. 4, 610â632 (2010).
Matthews, J. W. & Blakeslee, A. E. Defects in epitaxial multilayers: III. Preparation of almost perfect multilayers. J. Cryst. Growth 32, 265â273 (1976).
Wagner, G. R. & Janocko, M. A. Observation of a two-dimensional hole gas in boron-doped Si0.5Ge0.5/Ge heterostructures. Appl. Phys. Lett. 54, 66â68 (1989).
Murakami, E., Etoh, H., Nakagawa, K. & Miyao, M. High hole mobility in modulation-doped and strain-controlled p-Si0.5Ge0.5/Ge/Si1âxGex heterostructures fabricated using molecular beam epitaxy. Jpn. J. Appl. Phys. 29, L1059âL1061 (1990).
Murakami, E., Nakagawa, K., Nishida, A. & Miyao, M. Strain-controlled Si-Ge modulation-doped FET with ultrahigh hole mobility. IEEE Electron Device Lett. 12, 71â73 (1991).
Xie, Y. H. et al. Very high mobility two-dimensional hole gas in Si/GexSi1âx/Ge structures grown by molecular beam epitaxy. Appl. Phys. Lett. 63, 2263â2264 (1993).
Schäffler, F. High-mobility Si and Ge structures. Semicond. Sci. Technol. 12, 1515â1549 (1997).
Lee, M. L., Fitzgerald, E. A., Bulsara, M. T., Currie, M. T. & Lochtefeld, A. Strained Si, SiGe, and Ge channels for high-mobility metal-oxide-semiconductor field-effect transistors. J. Appl. Phys. 97, 011101 (2005).
Isella, G. et al. Low-energy plasma-enhanced chemical vapor deposition for strained Si and Ge heterostructures and devices. Solid State Electron. 48, 1317â1323 (2004).
Känel, H. V., Kummer, M., Isella, G., Müller, E. & Hackbarth, T. Very high hole mobilities in modulation-doped Ge quantum wells grown by low-energy plasma enhanced chemical vapor deposition. Appl. Phys. Lett. 80, 2922â2924 (2002).
Rössner, B., Chrastina, D., Isella, G. & von Känel, H. Scattering mechanisms in high-mobility strained Ge channels. Appl. Phys. Lett. 84, 3058â3060 (2004).
Shah, V. A. et al. Reverse graded relaxed buffers for high Ge content SiGe virtual substrates. Appl. Phys. Lett. 93, 192103 (2008).
Shah, V. A., Dobbie, A., Myronov, M. & Leadley, D. R. Reverse graded SiGe/Ge/Si buffers for high-composition virtual substrates. J. Appl. Phys. 107, 064304 (2010).
Colace, L. et al. Metalâsemiconductorâmetal near-infrared light detector based on epitaxial Ge/Si. Appl. Phys. Lett. 72, 3175â3177 (1998).
Gunn, L. C. III, Capellini, G., Rattier, M. J. & Pinguet, T. J. Methods of incorporating germanium within CMOS process. US Patent 6,887,773 (2005).
Hartmann, J. M. et al. Reduced pressureâchemical vapor deposition of Ge thick layers on Si(001) for 1.3â1.55-μm photodetection. J. Appl. Phys. 95, 5905â5913 (2004).
Lu, T. M. et al. Enhancement-mode buried strained silicon channel quantum dot with tunable lateral geometry. Appl. Phys. Lett. 99, 043101 (2011).
Borselli, M. G. et al. Pauli spin blockade in undoped Si/SiGe two-electron double quantum dots. Appl. Phys. Lett. 99, 063109 (2011).
Maune, B. M. et al. Coherent singlet-triplet oscillations in a silicon-based double quantum dot. Nature 481, 344â347 (2012).
Su, Y.-H., Chuang, Y., Liu, C.-Y., Li, J.-Y. & Lu, T.-M. Effects of surface tunneling of two-dimensional hole gases in undoped Ge/GeSi heterostructures. Phys. Rev. Mater. 1, 044601 (2017).
Lodari, M. et al. Low percolation density and charge noise with holes in germanium. Mater. Quantum Technol. https://doi.org/10.1088/2633-4356/abcd82 (in the press).
RöÃner, B., Isella, G. & Känel, H. V. Effective mass in remotely doped Ge quantum wells. Appl. Phys. Lett. 82, 754â756 (2003).
Irisawa, T. et al. Hole density dependence of effective mass, mobility and transport time in strained Ge channel modulation-doped heterostructures. Appl. Phys. Lett. 82, 1425â1427 (2003).
Sawano, K. et al. Magnetotransport properties of Ge channels with extremely high compressive strain. Appl. Phys. Lett. 89, 162103 (2006).
Sawano, K. et al. Strain dependence of hole effective mass and scattering mechanism in strained Ge channel structures. Appl. Phys. Lett. 95, 122109 (2009).
Foronda, J., Morrison, C., Halpin, J. E., Rhead, S. D. & Myronov, M. Weak antilocalization of high mobility holes in a strained Germanium quantum well heterostructure. J. Phys. Condens. Matter 27, 022201 (2015).
Hassan, A. H. A. et al. Anisotropy in the hole mobility measured along the [110] and [1â10] orientations in a strained Ge quantum well. Appl. Phys. Lett. 104, 132108 (2014).
Morrison, C. et al. Observation of Rashba zero-field spin splitting in a strained germanium 2D hole gas. Appl. Phys. Lett. 105, 182401 (2014).
Failla, M., Myronov, M., Morrison, C., Leadley, D. R. & Lloyd-Hughes, J. Narrow heavy-hole cyclotron resonances split by the cubic Rashba spin-orbit interaction in strained germanium quantum wells. Phys. Rev. B 92, 045303 (2015).
Shi, Q., Zudov, M. A., Morrison, C. & Myronov, M. Spinless composite fermions in an ultrahigh-quality strained Ge quantum well. Phys. Rev. B 91, 241303 (2015).
Morrison, C., Casteleiro, C., Leadley, D. R. & Myronov, M. Complex quantum transport in a modulation doped strained Ge quantum well heterostructure with a high mobility 2D hole gas. Appl. Phys. Lett. 109, 102103 (2016).
Holmes, S. N. et al. Spin-splitting in p-type Ge devices. J. Appl. Phys. 120, 085702 (2016).
Mironov, O. et al. Fractional quantum hall states in a Ge quantum well. Phys. Rev. Lett. 116, 176802 (2016).
Morrison, C. & Myronov, M. Electronic transport anisotropy of 2D carriers in biaxial compressive strained germanium. Appl. Phys. Lett. 111, 192103 (2017).
Drichko, I. L. et al. Effective g factor of 2D holes in strained Ge quantum wells. J. Appl. Phys. 123, 165703 (2018).
Berkutov, I. B., Andrievskii, V. V., Kolesnichenko, Y. A. & Mironov, O. A. Quantum effects in a germanium quantum well with ultrahigh mobility of charge carrier. Low Temp. Phys. 45, 1202â1208 (2019).
Laroche, D. et al. Magneto-transport analysis of an ultra-low-density two-dimensional hole gas in an undoped strained Ge/SiGe heterostructure. Appl. Phys. Lett. 108, 233504 (2016).
Lu, T. M. et al. Density-controlled quantum Hall ferromagnetic transition in a two-dimensional hole system. Sci. Rep. 7, 2468 (2017).
Lu, T. M. et al. Effective g factor of low-density two-dimensional holes in a Ge quantum well. Appl. Phys. Lett. 111, 102108 (2017).
Hardy, W. J. et al. Single and double hole quantum dots in strained Ge/SiGe quantum wells. Nanotechnology 30, 215202 (2019).
Failla, M. et al. Terahertz quantum Hall effect for spin-split heavy-hole gases in strained Ge quantum wells. New J. Phys. 18, 113036 (2016).
Gul, Y. et al. Quantum ballistic transport in strained epitaxial germanium. Appl. Phys. Lett. 111, 233512 (2017).
Mizokuchi, R., Maurand, R., Vigneau, F., Myronov, M. & De Franceschi, S. Ballistic one-dimensional holes with strong g-factor anisotropy in germanium. Nano Lett. 18, 4861â4865 (2018).
Gul, Y., Holmes, S. N., Myronov, M., Kumar, S. & Pepper, M. Self-organised fractional quantisation in a hole quantum wire. J. Phys. Condens. Matter 30, 09LT01 (2018).
Lauhon, L. J., Gudiksen, M. S., Wang, D. & Lieber, C. M. Epitaxial coreâshell and coreâmultishell nanowire heterostructures. Nature 420, 57â61 (2002).
Morales, A. M. & Lieber, C. M. A laser ablation method for the synthesis of crystalline semiconductor nanowires. Science 279, 208â211 (1998).
Woodruff, J. H., Ratchford, J. B., Goldthorpe, I. A., McIntyre, P. C. & Chidsey, C. E. D. Vertically oriented germanium nanowires grown from gold colloids on silicon substrates and subsequent gold removal. Nano Lett. 7, 1637â1642 (2007).
Dayeh, S. A. & Picraux, S. T. Direct observation of nanoscale size effects in Ge semiconductor nanowire growth. Nano Lett. 10, 4032â4039 (2010).
Tian, B., Xie, P., Kempa, T. J., Bell, D. C. & Lieber, C. M. Single-crystalline kinked semiconductor nanowire superstructures. Nat. Nanotechnol. 4, 824â829 (2009).
Goldthorpe, I. A., Marshall, A. F. & McIntyre, P. C. Inhibiting strain-induced surface roughening: dislocation-free Ge/Si and Ge/SiGe coreâshell nanowires. Nano Lett. 9, 3715â3719 (2009).
Conesa-Boj, S. et al. Boosting hole mobility in coherently strained [110]-oriented GeâSi coreâshell nanowires. Nano Lett. 17, 2259â2264 (2017).
Dillen, D. C., Kim, K., Liu, E.-S. & Tutuc, E. Radial modulation doping in coreâshell nanowires. Nat. Nanotechnol. 9, 116â120 (2014).
Sistani, M. et al. Highly transparent contacts to the 1D hole gas in ultrascaled Ge/Si core/shell nanowires. ACS Nano 13, 14145â14151 (2019).
Lu, W., Xiang, J., Timko, B. P., Wu, Y. & Lieber, C. M. One-dimensional hole gas in germanium/silicon nanowire heterostructures. Proc. Natl Acad. Sci. USA 102, 10046â10051 (2005).
Kotekar-Patil, D., Nguyen, B.-M., Yoo, J., Dayeh, S. A. & Frolov, S. M. Quasiballistic quantum transport through Ge/Si core/shell nanowires. Nanotechnology 28, 385204 (2017).
Zhang, X., Jevasuwan, W., Sugimoto, Y. & Fukata, N. Controlling catalyst-free formation and hole gas accumulation by fabricating Si/Ge core-shell and Si/Ge/Si core-double shell nanowires. ACS Nano 13, 13403â13412 (2019).
Tersoff, J. & Tromp, R. M. Shape transition in growth of strained islands: spontaneous formation of quantum wires. Phys. Rev. Lett. 70, 2782â2785 (1993).
Mo, Y.-W., Savage, D. E., Swartzentruber, B. S. & Lagally, M. G. Kinetic pathway in Stranski-Krastanov growth of Ge on Si(001). Phys. Rev. Lett. 65, 1020â1023 (1990).
McKay, M. R., Venables, J. A. & Drucker, J. Kinetically suppressed Ostwald ripening of Ge/Si(100) hut clusters. Phys. Rev. Lett. 101, 216104 (2008).
Zhang, J. J. et al. Monolithic growth of ultrathin Ge nanowires on Si(001). Phys. Rev. Lett. 109, 085502 (2012).
Tersoff, J. & LeGoues, F. K. Competing relaxation mechanisms in strained layers. Phys. Rev. Lett. 72, 3570â3573 (1994).
Watzinger, H., Glaser, M., Zhang, J. J., Daruka, I. & Schäffler, F. Influence of composition and substrate miscut on the evolution of {105}-terminated in-plane Si1âxGex quantum wires on Si(001). APL Mater. 2, 076102 (2014).
Brauns, M. et al. Highly tuneable hole quantum dots in Ge-Si core-shell nanowires. Appl. Phys. Lett. 109, 143113 (2016).
Froning, F. N. M. et al. Single, double, and triple quantum dots in Ge/Si nanowires. Appl. Phys. Lett. 113, 073102 (2018).
Roddaro, S. et al. Spin states of holes in Ge/Si nanowire quantum dots. Phys. Rev. Lett. 101, 186802 (2008).
Brauns, M. et al. Anisotropic Pauli spin blockade in hole quantum dots. Phys. Rev. B 94, 041411 (2016).
Zarassi, A. et al. Magnetic field evolution of spin blockade in Ge/Si nanowire double quantum dots. Phys. Rev. B 95, 155416 (2017).
VukuÅ¡ic´, L. et al. Single-shot readout of hole spins in Ge. Nano Lett. 18, 7141â7145 (2018).
VukuÅ¡ic´, L., KukucËka, J., Watzinger, H. & Katsaros, G. Fast hole tunneling times in germanium hut wires probed by single-shot reflectometry. Nano Lett. 17, 5706â5710 (2017).
Hanson, R., Kouwenhoven, L. P., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K. Spins in few-electron quantum dots. Rev. Mod. Phys. 79, 1217â1265 (2007).
Lawrie, W. I. L. et al. Quantum dot arrays in silicon and germanium. Appl. Phys. Lett. 116, 080501 (2020).
Hofmann, A. et al. Assessing the potential of Ge/SiGe quantum dots as hosts for singlet-triplet qubits. Preprint at arXiv https://arxiv.org/abs/1910.05841 (2019).
Higginbotham, A. P. et al. Hole spin coherence in a Ge/Si heterostructure nanowire. Nano Lett. 14, 3582â3586 (2014).
Vandersypen, L. M. K. et al. Interfacing spin qubits in quantum dots and donors â hot, dense, and coherent. NPJ Quantum Inf. 3, 34 (2017).
Lawrie, W. I. L. et al. Spin relaxation benchmarks and individual qubit addressability for holes in quantum dots. Nano Lett. 20, 7237â7242 (2017).
Hutin, L. et al. in Proceedings of the 2018 48th European Solid-State Device Research Conference (ESSDERC) 12â17 (IEEE, 2018).
Froning, F. N. M. et al. Ultrafast hole spin qubit with gate-tunable spin-orbit switch. Preprint at arXiv http://arxiv.org/abs/2006.11175 (2020).
Knill, E. et al. Randomized benchmarking of quantum gates. Phys. Rev. A 77, 012307 (2008).
Bertrand, B. et al. Quantum manipulation of two-electron spin states in isolated double quantum dots. Phys. Rev. Lett. 115, 096801 (2015).
De Franceschi, S., Kouwenhoven, L., Schönenberger, C. & Wernsdorfer, W. Hybrid superconductorâquantum dot devices. Nat. Nanotechnol. 5, 703â711 (2010).
Lutchyn, R. M. et al. Majorana zero modes in superconductorâsemiconductor heterostructures. Nat. Rev. Mater. 3, 52â68 (2018).
Clark, T. D., Prance, R. J. & Grassie, A. D. C. Feasibility of hybrid Josephson field effect transistors. J. Appl. Phys. 51, 2736 (1980).
de Lange, G. et al. Realization of microwave quantum circuits using hybrid superconducting-semiconducting nanowire Josephson elements. Phys. Rev. Lett. 115, 127002 (2015).
Casparis, L. et al. Superconducting gatemon qubit based on a proximitized two-dimensional electron gas. Nat. Nanotechnol. 13, 915â919 (2018).
Sau, J. D. & Sarma, S. D. Realizing a robust practical Majorana chain in a quantum-dot-superconductor linear array. Nat. Commun. 3, 964 (2012).
Ridderbos, J. et al. Multiple Andreev reflections and Shapiro steps in a Ge-Si nanowire Josephson junction. Phys. Rev. Mater. 3, 084803 (2019).
Ridderbos, J. et al. Hard superconducting gap and diffusion-induced superconductors in GeâSi nanowires. Nano Lett. 20, 122â130 (2020).
Hendrickx, N. W. et al. Ballistic supercurrent discretization and micrometer-long Josephson coupling in germanium. Phys. Rev. B 99, 075435 (2019).
De Franceschi, S. et al. Andreev reflection in Si-engineered Al/InGaAs hybrid junctions. Appl. Phys. Lett. 73, 3890â3892 (1998).
Krogstrup, P. et al. Epitaxy of semiconductorâsuperconductor nanowires. Nat. Mater. 14, 400â406 (2015).
Kral, S. et al. Abrupt Schottky junctions in Al/Ge nanowire heterostructures. Nano Lett. 15, 4783â4787 (2015).
El Hajraoui, K. et al. In situ transmission electron microscopy analysis of aluminumâgermanium nanowire solid-state reaction. Nano Lett. 19, 2897â2904 (2019).
Franke, D. P., Clarke, J. S., Vandersypen, L. M. K. & Veldhorst, M. Rentâs rule and extensibility in quantum computing. Microprocess. Microsyst. 67, 1â7 (2019).
Taylor, J. M. et al. Fault-tolerant architecture for quantum computation using electrically controlled semiconductor spins. Nat. Phys. 1, 177â183 (2005).
Veldhorst, M., Eenink, H. G. J., Yang, C. H. & Dzurak, A. S. Silicon CMOS architecture for a spin-based quantum computer. Nat. Commun. 8, 1766 (2017).
Li, R. et al. A crossbar network for silicon quantum dot qubits. Sci. Adv. 4, eaar3960 (2018).
Hill, C. D. et al. A surface code quantum computer in silicon. Sci. Adv. 1, e1500707 (2015).
Nickerson, N. H., Li, Y. & Benjamin, S. C. Topological quantum computing with a very noisy network and local error rates approaching one percent. Nat. Commun. 4, 1756 (2013).
Veldhorst, M. et al. A two-qubit logic gate in silicon. Nature 526, 410â414 (2015).
Watson, T. F. et al. A programmable two-qubit quantum processor in silicon. Nature 555, 633â637 (2018).
Zheng, G. et al. Rapid gate-based spin read-out in silicon using an on-chip resonator. Nat. Nanotechnol. 14, 742â746 (2019).
West, A. et al. Gate-based single-shot readout of spins in silicon. Nat. Nanotechnol. 14, 437â441 (2019).
Urdampilleta, M. et al. Gate-based high fidelity spin readout in a CMOS device. Nat. Nanotechnol. 14, 737â741 (2019).
Crippa, A. et al. Gate-reflectometry dispersive readout and coherent control of a spin qubit in silicon. Nat. Commun. 10, 2776 (2019).
Samkharadze, N. et al. Strong spin-photon coupling in silicon. Science 359, 1123â1127 (2018).
Mi, X. et al. A coherent spinâphoton interface in silicon. Nature 555, 599â603 (2018).
Borjans, F., Croot, X. G., Mi, X., Gullans, M. J. & Petta, J. R. Resonant microwave-mediated interactions between distant electron spins. Nature 577, 195â198 (2020).
Fowler, A. G., Mariantoni, M., Martinis, J. M. & Cleland, A. N. Surface codes: Towards practical large-scale quantum computation. Phys. Rev. A 86, 032324 (2012).
Itoh, K. M. & Watanabe, H. Isotope engineering of silicon and diamond for quantum computing and sensing applications. MRS Commun. 4, 143â157 (2014).
Hu, X., Liu, Y.-x. & Nori, F. Strong coupling of a spin qubit to a superconducting stripline cavity. Phys. Rev. B 86, 035314 (2012).
Sigrist, M. & Ueda, K. Phenomenological theory of unconventional superconductivity. Rev. Mod. Phys. 63, 239â311 (1991).
Sau, J. D., Lutchyn, R. M., Tewari, S. & Das Sarma, S. Generic new platform for topological quantum computation using semiconductor heterostructures. Phys. Rev. Lett. 104, 040502 (2010).
Alicea, J. Majorana fermions in a tunable semiconductor device. Phys. Rev. B 81, 125318 (2010).
Read, N. & Green, D. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Phys. Rev. B 61, 10267â10297 (2000).
Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083â1159 (2008).
Mao, L., Shi, J., Niu, Q. & Zhang, C. Superconducting phase with a chiral f-wave pairing symmetry and Majorana fermions induced in a hole-doped semiconductor. Phys. Rev. Lett. 106, 157003 (2011).
Leijnse, M. & Flensberg, K. Quantum information transfer between topological and spin qubit systems. Phys. Rev. Lett. 107, 210502 (2011).
Leijnse, M. & Flensberg, K. Hybrid topological-spin qubit systems for two-qubit-spin gates. Phys. Rev. B 86, 104511 (2012).
Hoffman, S., Schrade, C., Klinovaja, J. & Loss, D. Universal quantum computation with hybrid spin-Majorana qubits. Phys. Rev. B 94, 045316 (2016).
RancËic´, M. J., Hoffman, S., Schrade, C., Klinovaja, J. & Loss, D. Entangling spins in double quantum dots and Majorana bound states. Phys. Rev. B 99, 165306 (2019).
Choi, M.-S., Bruder, C. & Loss, D. Spin-dependent Josephson current through double quantum dots and measurement of entangled electron states. Phys. Rev. B 62, 13569â13572 (2000).
Leijnse, M. & Flensberg, K. Coupling spin qubits via superconductors. Phys. Rev. Lett. 111, 060501 (2013).
Hassler, F., Catelani, G. & Bluhm, H. Exchange interaction of two spin qubits mediated by a superconductor. Phys. Rev. B 92, 235401 (2015).
Wang, K. et al. Ultrafast operations of a hole spin qubit in Ge quantum dot. Preprint at arXiv http://arxiv.org/abs/2006.12340 (2020).
Acknowledgements
G.S., M.V. and F.A.Z. acknowledge financial support from the Netherlands Organization for Scientific Research (NWO). F.A.Z., D.L. and G.K. acknowledge funding from the European Unionâs Horizon 2020 research and innovation programme under grant agreement no. 862046. G.K. acknowledges funding from FP7 ERC Starting Grant 335497, FWF Y 715-N30 and FWF P-30207. S.D.F. acknowledges support from the European Unionâs Horizon 2020 research and innovation programme under grant agreement no. 81050 and from the Agence Nationale de la Recherche through the TOPONANO and QSPIN projects. J.-J.Z. acknowledges support from the National Key R&D Program of China (grant no. 2016YFA0301701) and Strategic Priority Research Program of the Chinese Academy of Sciences (grant no. XDB30000000). D.L. and C.K. acknowledge the Swiss National Science Foundation and NCCR QSIT.
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Scappucci, G., Kloeffel, C., Zwanenburg, F.A. et al. The germanium quantum information route. Nat Rev Mater 6, 926â943 (2021). https://doi.org/10.1038/s41578-020-00262-z
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DOI: https://doi.org/10.1038/s41578-020-00262-z