Abstract
Superconductors and semiconductors are crucial platforms in the field of quantum computing. They can be combined to hybrids, bringing together physical properties that enable the discovery of new emergent phenomena and provide novel strategies for quantum control. The involved semiconductor materials, however, suffer from disorder, hyperfine interactions or lack of planar technology. Here we realise an approach that overcomes these issues altogether and integrate gatedefined quantum dots and superconductivity into germanium heterostructures. In our system, heavy holes with mobilities exceeding 500,000 cm^{2} (Vs)^{−1} are confined in shallow quantum wells that are directly contacted by annealed aluminium leads. We observe proximityinduced superconductivity in the quantum well and demonstrate electric gatecontrol of the supercurrent. Germanium therefore has great promise for fast and coherent quantum hardware and, being compatible with standard manufacturing, could become a leading material for quantum information processing.
Introduction
Solid state quantum computing is actively pursued using superconducting and semiconducting materials^{1,2,3}. The groupIV semiconductors Si and Ge come with central advantages for the realisation of spin quantum bits (qubits). Not only has their purity and technology been refined to a formidable level, they also possess an abundant isotope with zero nuclear spin^{4,5}, enabling spin qubits to reach extremely long coherence times^{6,7} and high fidelity^{8}. These powerful properties have led to demonstrations of twoqubit logic gates^{9,10} and quantum algorithms^{11}. The exchange interaction that is central in these demonstrations is local and cannot directly be used to couple qubits at a distance. Instead, longrange coupling of spin qubits is being explored by incorporating superconductivity and in a first step strong spin–photon coupling has been achieved^{12,13}.
Hole quantum dots in Ge are particularly promising in this context. Ge has the highest hole mobility of all known semiconductors^{14}, reaching values up to 1,500,000 cm^{2} (Vs)^{−1} in doped heterostructures^{15} and is expected to host strong spin–orbit coupling^{16,17}, which facilitates electrical driving for fast qubit operations^{18}. Furthermore, the valence band in Ge has no valley degeneracy, so, compared to electrons^{19}, hole qubits do not have the complication of these close quantum levels^{16}. Experiments have shown readout of holes in Ge/Si nanowires^{20,21}, selfassembled quantum dots^{22} and hut wires^{23}, and promising spin lifetimes have been found^{23,24}. In addition, the strong Fermilevel pinning at the valence band edge leads to ohmic behaviour for all metal–(ptype) Ge contacts^{25}. The resulting strong coupling between metal and semiconductor enables the fabrication of hybrid devices of quantum dot and superconducting structures^{26,27}.
Now, the crucial next step is the development of a planar platform that can bring together low disorder, potential for fast driving and avenue for scaling. Here we address this challenge and present the formation of a quantum dot in a planar Ge quantum well. Furthermore, we implement direct Albased ohmic contacts that eliminate the need for dopant implantation. In addition, the Al leads can proximityinduce superconductivity in the quantum well and we can control the associated supercurrent by tuning electrical gates.
Results
Ge/SiGe heterostructure
Si and Ge are completely miscible and the lattice constant of their alloy, SiGe, varies continuously between its constituents. This is exploited by using strainrelaxed compositionally graded SiGe layers as virtual substrates to define Ge/SiGe heterostructures as shown schematically in Fig. 1a. We use a highthroughput reducedpressure chemical vapour deposition (RPCVD) reactor to grow the complete heterostructure in one deposition cycle on a 100 mm Si(001) substrate. The Ge quantum well is deposited pseudomorphically on a strainrelaxed Si_{0.2}Ge_{0.8}/Ge/Si virtual substrate obtained by reverse grading^{28} (see Methods). The resulting inplane compressive strain in the quantum well splits the valence band states, which further increases the hole mobility^{29}. The quantum well layer is separated from the surface by a Si_{0.2}Ge_{0.8} spacer and a Si cap.
Because of the typeI band alignment in Gerich Ge/SiGe heterostructures, the valence band maximum is energetically higher in Ge than in SiGe, such that holes accumulate in the Ge quantum well^{30}. Heavy and light hole (HH and LH, respectively) electronic states and the related band profiles have been calculated by solving the Schrödinger–Poisson equation for low temperatures as a function of vertical electric field. In our simulations, both a multivalley effective mass approach^{31} and an atomistic tightbinding model^{32} have been used, obtaining consistent data. The calculated valence band edge profile is shown in Fig. 1b, where the wave function of the fundamental state HH_{0}, which has HH symmetry and is well confined in the Ge layer, is also sketched. The straininduced splitting of the HH and LH band edges in the Ge region increases the energy of the fundamental LH state LH_{0} (not shown), which is found at about 46 meV above HH_{0}. Furthermore, we estimate a separation of 14 meV for the first excited HH level HH_{1} (not shown). Notice that these energy splittings are significantly larger than those resulting from the valley interaction in Si/SiO_{2} and Si/SiGe devices, and thus excellent conditions for the operation of spin qubits are expected in this material.
The hole mobility is measured in magnetotransport experiments at 50 mK using a heterostructure field effect transistor as shown in the inset of Fig. 1c. Here the yellow boxes indicate metallic ohmic contacts to the quantum well created by diffusion into the top SiGe layer and the green structure represents an isolated Hallbarshaped top gate, which is used to control the hole density in the quantum well. Clear Shubnikov–de Haas oscillations with zeroresistivity minima and Zeeman splitting of the Landau levels at higher fields are observed in the longitudinal resistance ρ_{xx} as a function of the magnetic field B (Fig. 1c). Furthermore, we observe flat quantum Hall effect (QHE) plateaus at values 1/ν for integer ν in the transverse resistance ρ_{xy} in units of h/e^{2}, where h is Planck’s constant and e the elementary charge. The investigated heterostructures support a density of up to 6 × 10^{11} cm^{−2} and have a maximum mobility of more than 500,000 cm^{2} (Vs)^{−1}, corresponding to a mean free path of L_{m} = 6.4 μm and providing new benchmarks for holes in undoped structures. Figure 1d shows the Hall conductivity σ_{xy} in units of e^{2}/h as a function of the hole density p, controlled by the top gate potential. Again, zeroresistivity minima in ρ_{xx} and clear linear quantisation steps in σ_{xy} are observed. This demonstrates the control of the hole density over a large range using the top gate, which is a central prerequisite for the formation of electrostatically defined quantum dots.
Quantum dot
A scanning electron microscope (SEM) image of the quantum dot nanostructure is shown in Fig. 2a. Here the ohmic Al contacts are coloured in yellow and the isolated Ti/Pd gates are shown in green. In a first step, the Al contacts to the Ge quantum well are defined by electron beam lithography, local etching of the Si capping layer and thermal evaporation of Al. Subsequently, an Al_{2}O_{3} gate dielectric is grown by atomic layer deposition at 300 °C, which also serves as an annealing step to enable the diffusion of Al into the SiGe spacer. In the Ti/Pd gate layer, we design a circular top gate between the two Al leads under which a single quantum dot will be formed. In addition, a central plunger gate P is included to control the dot occupation, as well as barrier gates (BS and BD) and additional finger gates (FS and FD) in the corners of the device. These allow for additional control of the dot size and the tunnelling rates between the quantum dot and the source and drain leads.
A conceptual drawing of the device crosssection is shown in Fig. 2b, where the diffused Al leads that contact the Ge quantum well are indicated by the stripepatterned regions. Because of the ohmic nature of the contact, the transport through the quantum dot can be measured without the need for additional reservoir gates and dopant implantation. As a result, no annealing step at temperatures higher than the quantum well deposition temperature (500 °C) is needed during the fabrication, avoiding harmful Ge/Si intermixing at the interface^{29}.
When measuring the source drain conductance dI/dV as a function of the top gate voltage, conductance peaks are expected when the dot potential aligns between the source and drain potentials, which are the socalled Coulomb oscillations. This is shown in Fig. 2c, where dI/dV was measured as a function of the top gate voltage V_{TG}. The expected oscillations are observed, which is a clear sign of the formation of a quantum dot. The spacing between the peaks is quite regular, indicating that the quantum dot is operated in the manyhole regime. From the period of the conductance peaks, we can extract a top gate capacitance of ~56 aF, which is in very good agreement with the expected capacitance of 52 aF of a parallel plate capacitor using the lithographic dimensions of the top gate. When the top gate voltage is increased and the quantum dot is depleted, the amplitude of the observed peaks is reduced and eventually vanishes, as the tunnelling rates to source and drain reservoirs drop as an effect of the reduced size of the dot. As shown in the supplementary material, we are able to circumvent this effect using an additional accumulation gate in a different device, which allows us to reach the fewhole regime (Supplementary Fig. 1). When TG is tuned to the quantum dot regime, similar oscillations are observed as a function of the plunger gate voltage V_{P}. Here a larger spacing of the Coulomb peaks is observed, corresponding to a gate capacitance of ~6.4 aF, in agreement with the expected weaker coupling of P to the quantum dot. Note that because of the structure of the source and drain leads no additional tuning of the device is necessary. Equivalent to a classical transistor, the quantum dot can be defined using a single gate (TG), which bodes well for the scaling up of qubits in this system.
As shown in Fig. 2d, e, the dot occupancy can also be controlled using the barrier gates BS and BD. The observed conductance lines are diagonal and very equivalent for each of the two barriers and the plunger, indicating that the coupling to the quantum dot is nearly identical. This confirms that the quantum dot is formed in a central position under TG and that, as an effect of low disorder in the heterostructure, a very high level of control is achieved. Furthermore, by implementing a second layer of Ti/Pd gates we can also independently control the tunnelling rate between the quantum dot and the reservoir (Supplementary Fig. 2).
To further characterise the quantum dot, we measure dI/dV as a function of V_{P} and the DC bias voltage V_{SD}. As shown in Fig. 3a, Coulomb diamonds are observed^{33}. From the height and width of these diamonds, the charging energy E_{C} and the lever arm of the corresponding gate α can be extracted. In the regime shown here (V_{BS} = V_{BD} = 550 mV and V_{FD} = 600 mV), we find α_{P} = 0.037(3) and E_{C} = 1.3(1) meV.
Similar diamonds are observed as a function of V_{TG} (Fig. 3b). Here the lever arm is found to be significantly larger (α_{TG} = 0.41(3)), as is expected because the dot is formed directly under TG. A substructure is clearly visible in the conducting areas. These additional lines could correspond to either charge transport via excited states of the dot or to a modulation of the density of states within the source and drain reservoirs^{34}.
When an external magnetic field B is applied, the energy levels of spin degenerate states are expected to split as a result of the Zeeman effect^{33} (cf. Fig. 3c). This becomes apparent as a splitting of the conductance lines related to odd hole occupations 2N + 1 of the quantum dot, as shown in Fig. 3d in an inplane magnetic field B = 2.9 T. Both the ground state and the excited state are subject to this splitting, which we extract by fitting the observed conductance for V_{SD} = −0.26 mV using Gaussian profiles and multiplying the splitting in voltage with the measured lever arm α_{P} (Fig. 3e).
A linear trend is observed as a function of the applied magnetic field as shown in Fig. 3f, g. Note that small splittings ΔE_{Z} < 20 μeV could not be resolved because of the finite width of the conductance peaks. We find the effective gfactors g_{GS} = 0.29(2) and g_{ES} = 0.27(5) for the ground state and excited state, respectively, from the linear fits in Fig. 3h. For the excited state, our data point to either a nonlinearity at lower fields or a significant zerofield spin splitting ΔE_{Z,0} ≈ −11 μeV. The gfactor of the pure HH state is expected to vanish completely for an inplane field. However, the additional confinement of the holes in the x,yplane leads to a significant admixture of LH states and a nonzero inplane Zeeman splitting^{35}.
The observed spin splitting of the first line parallel to the diamond edge ground state identifies it as belonging to the first excited state rather than being connected to the reservoir. The measured energy splitting with respect to the ground state ΔE ≈ 100 μeV remains unchanged as a function of magnetic field strength. It can be compared to the expected level splitting for a twodimensional (2D) quantum dot with area A and effective hole mass m^{*}, which is given by ΔE = πℏ^{2}/m^{*}A. From the temperature dependence of the Shubnikov–de Haas oscillations measured for the Hall bar structures, we find m^{*} = 0.08 m_{e}, with the electron mass m_{e}, and with our device geometry (A = 0.019 μm^{2}) we obtain ΔE ≈ 150 μeV, in good agreement with the measured value.
The creation of ohmic leads by the diffusion of Al in the direct vicinity of the quantum dot could be suspected of creating additional charge traps that have a negative influence on the coherence of a potential qubit. To quantify this effect, we measure the charge noise acting on our quantum dot by recording the transport current I_{dot} in a sensitive region, i.e. on the slope of a Coulomb peak. A 100slong time trace of I_{dot} is acquired at a sampling rate of 30.5 kHz and is decomposed into 15 traces of equal lengths. The discrete Fourier spectra obtained from these traces are averaged, yielding the noise spectral density S_{E} presented in Fig. 4 in comparison to the corresponding spectrum measured in a low sensitivity region (Coulomb blockade). The difference between the two traces confirms that the measured lowfrequency noise spectrum is indeed dominated by charge noise acting on the quantum dot. The noise spectrum follows a typical 1/f trend for low frequencies^{36,37} (solid line in Fig. 4). We find the equivalent detuning noise at 1 Hz to be 1.4 μeV/\(\sqrt {{\mathrm{Hz}}}\). This compares well to noise figures at 1 Hz in other materials, such as 7.5, 0.5, or 2.0 μeV/\(\sqrt {{\mathrm{Hz}}}\) in GaAs^{36}, Si/SiO_{2}^{37} and Si/SiGe^{37}, respectively, showing the suitability of our approach for the creation of lownoise qubits.
Josephson field effect transistor (JoFET)
The direct ohmic contact of the Al leads to the Ge quantum well can proximityinduce superconductivity in the semiconductor system. To demonstrate this effect, we fabricate a JoFET device with two thicker (d = 30 nm) and broader (w = 1 μm) aluminium leads. These are positioned with a separation of 100 nm and are overlaid with a single top gate; a SEM image of the device is shown together with a schematic drawing of the layer structure in Fig. 5a, b. When the sourcedrain voltage is measured as a function of the sourced current I_{SD} at negative top gate potential, a zeroresistance plateau is observed as a clear sign of a supercurrent. This is shown in Fig. 5c, where the blue and orange traces represent measurements in different sweep directions, respectively. Sweeping towards higher absolute values of I_{SD}, the critical current I_{c} is reached at ~12 nA. Beyond these values, linear (ohmic) behaviour is observed. A hysteresis can be observed when sweeping back, which can be expected for proximity induced SNSjunctions and is usually caused by selfheating of the junction when it exits the superconducting state^{38}. To demonstrate that this supercurrent is induced in the quantum well, we measure the observed bias current dependence of the sourcedrain resistance dV/dI as a function of the top gate potential V_{TG}, shown in Fig. 5d. The critical current, which defines the borders of the zero resistance range observed as a black region in the colour plot, is reduced towards increasing V_{TG}. This is an effect of the decrease in the carrier density and the resulting increase of the normal state resistance. For voltages above V_{TG} ≈ −1.27 V, I_{c} cannot be resolved at the limited instrumental resolution in our measurements. For decreasing gate potential, we find that the increase in I_{c} is more significant than the reduction in the normal resistance R_{n}. The characteristic voltage I_{c}R_{n} hence increases and reaches values higher than ~10 μV. Additionally, a modulation of the critical current can be observed as a function of magnetic field, which is a clear hallmark of the Josephson effect (Supplementary Fig. 4). These proofofprinciple experiments should be followed by a detailed characterisation of the proximityinduced superconductivity in this platform. Interesting future paths of research include the study of ballistic transport and the investigation and optimisation of the sharpness of the Al–Ge interface, which could significantly increase the I_{c}R_{n} product.
Discussion
In conclusion, we have shown the operation of a hole quantum dot in a planar, undoped and buried Ge quantum well with a record hole mobility. The Al ohmic leads to the quantum dot significantly simplify the fabrication and tuning processes without an increase of the measured charge noise in comparison to other systems. The strong capacitive coupling between the superconductor and the quantum dot makes this system ideal for reaching strong spin–photon coupling, while the strong spin–orbit coupling present in HHs could be exploited for fast electrical qubit driving. Furthermore, the demonstration of gatetunable superconductivity opens up new research directions, including Majorana modes^{39} and gatemons^{27}. Hole quantum dots in planar Ge constitute thereby a versatile platform that can leverage semiconductor manufacturing to advance and broaden the field of quantum computing.
Methods
Heterostructure growth
The Ge/SiGe heterostructures were grown in one deposition cycle in an Epsilon 2000 (ASMI) RPCVD reactor on a 100 mm ntype Si(001) substrate (resistivity 5 Ω cm). The growth starts with a 1μmthick layer of Ge, using a dualstep process with initial lowtemperature (400 °C) growth of a Ge seed layer followed by a highertemperature (625 °C) overgrowth of a thick relaxed Ge buffer layer. A cycle anneal at 800 °C is performed to promote full relaxation of the Ge. Subsequently, a 700 nm reversegraded Si_{1−x}Ge_{x} layer is grown at 800 °C with x changing linearly from 1 to 0.8. A relaxed 300 nm buffer Si_{0.2}Ge_{0.8} is then grown in two steps at an initial high temperature of 800 °C followed by a lowtemperature growth at 500 °C. The growth continues with a 18nmthick strained Ge quantum well, a 22nmthick Si_{0.2}Ge_{0.8} barrier and a 1nmthick Si cap. The final layers are all deposited at a temperature of 500 °C.
Device fabrication
The photolithography process for fabricating the Hall bar field effect transistors includes the following steps: mesa etching, deposition of 60nmthick Pt layer ohmic contacts, and atomic layer deposition (ALD) at 300 °C of a 30nmthick Al_{2}O_{3} dielectric to isolate a Ti/Au top gate (thicknesses 10/150 nm).
For the quantum dot, the contact and gate structures are created by electron beam lithography, electron beam evaporation of Al and Ti/Pd and liftoff. Following the Al contact layer (20 nm), ALD is used to grow 17 nm of Al_{2}O_{3} at 300 °C as a gate dielectric, followed by the Ti/Pd (5/35 nm) gate structures. For the JoFET device, the same process is done with layer thicknesses 30 nm (Al), 25 nm (Al_{2}O_{3}) and 5/35 nm (Ti/Pd).
Measurement system
Magnetotransport data has been obtained in a ^{3}He dilution refrigerator with a base temperature of 50 mK, equipped with a 9 T magnet. The carrier density \(p = \left( {\left e \right\frac{{\mathrm{d}\rho _{yx}}}{{\mathrm{d}B}}} \right)^{  1}\) in the quantum well is derived from the measured Hall resistivity for fields B < 0.3 T, where no QHE steps are observed. This density can also be derived from the Fourier transform of the longitudinal resistance data as a function of 1/B. From the observed peak width of 1.5 T, we estimate an upper bound for the zerofield spin splitting in our twodimensional hole gas of 1.5 meV. All quantum dot and JoFET measurements were performed in a ^{3}He dilution refrigerator with a base temperature of <10 mK, equipped with a 3 T magnet. All quantum dot measurements of dI/dV are performed using lockin amplification with typical modulation amplitude and frequency of δV_{SD} = 10–100 μV and f_{mod} = 73.5 Hz, respectively, and a small offset in bias voltage due to the measurement electronics is subtracted. The JoFET device was measured in a fourpoint configuration, sourcing a current and measuring the potential across the superconducting junctions. The plotted voltage is corrected by a small offset of the measurement electronics. The differential resistance is measured using lockin amplification with typical a modulation amplitude of δI_{SD} = 0.3 nA.
Data availability
All data underlying this study are available from the 4TU ResearchData repository at https://doi.org/10.4121/uuid:6a4c90c91fd7442fbc218fbe6876e1ea.
References
 1.
Nakamura, Y., Pashkin, Y. A. & Tsai, J. S. Coherent control of macroscopic quantum states in a singleCooperpair box. Nature 398, 786–788 (1999).
 2.
Petta, J. R. et al. Coherent manipulation of coupled electron spins in semiconductor quantum dots. Science 309, 2180–2184 (2005).
 3.
Basov, D. N., Averitt, R. D. & Hsieh, D. Towards properties on demand in quantum materials. Nat. Mater. 16, 1077–1088 (2017).
 4.
Itoh, K. et al. High purity isotopically enriched 70Ge and 74Ge single crystals: Isotope separation, growth, and properties. J. Mater. Res. 8, 1341–1347 (1993).
 5.
Itoh, K. M. & Watanabe, H. Isotope engineering of silicon and diamond for quantum computing and sensing applications. MRS Commun. 4, 143–157 (2014).
 6.
Veldhorst, M. et al. An addressable quantum dot qubit with faulttolerant controlfidelity. Nat. Nanotechnol. 9, 981–985 (2014).
 7.
Sigillito, A. et al. Electron spin coherence of shallow donors in natural and isotopically enriched germanium. Phys. Rev. Lett. 115, 247601 (2015).
 8.
Yoneda, J. et al. A quantumdot spin qubit with coherence limited by charge noise and fidelity higher than 99.9. Nat. Nanotechnol 13, 102–106 (2017).
 9.
Veldhorst, M. et al. A twoqubit logic gate in silicon. Nature 526, 410–414 (2015).
 10.
Zajac, D. M. et al. Resonantly driven CNOT gate for electron spins. Science 359, 439–442 (2018).
 11.
Watson, T. F. et al. A programmable twoqubit quantum processor in silicon. Nature 555, 633–637 (2018).
 12.
Mi, X. et al. A coherent spin–photon interface in silicon. Nature 555, 599–603 (2018).
 13.
Samkharadze, N. et al. Strong spinphoton coupling in silicon. Science 359, 1123–1127 (2018).
 14.
Pillarisetty, R. Academic and industry research progress in germanium nanodevices. Nature 479, 324–328 (2011).
 15.
Failla, M. et al. Terahertz quantum Hall effect for spinsplit heavyhole gases in strained Ge quantum wells. New J. Phys. 18, 113036 (2016).
 16.
Terrazos, L. A. et al. Lightmass holespin qubits formed in a Ge quantum well. Preprint at http://arxiv.org/abs/1803.10320 (2018).
 17.
Moriya, R. et al. Cubic Rashba spinorbit interaction of a twodimensional hole gas in a strainedGeSiGe quantum well. Phys. Rev. Lett. 113, 086601 (2014).
 18.
Watzinger, H. et al. Ge hole spin qubit. Preprint at http://arxiv.org/abs/1802.00395 (2018).
 19.
Yang, C. H. et al. Orbital and valley state spectra of a fewelectron silicon quantum dot. Phys. Rev. B 86, 115319 (2012).
 20.
Lu, W., Xiang, J., Timko, B. P., Wu, Y. & Lieber, C. M. Onedimensional hole gas in germanium/silicon nanowire heterostructures. PNAS 102, 10046–10051 (2005).
 21.
Hu, Y. et al. A Ge/Si heterostructure nanowirebased double quantum dot with integrated charge sensor. Nat. Nanotechnol. 2, 622–625 (2007).
 22.
Ares, N. et al. Nature of tunable hole g factors in quantum dots. Phys. Rev. Lett. 110, 046602 (2013).
 23.
Vukušić, L. et al. Singleshot readout of hole spins in Ge. Preprint at http://arxiv.org/abs/1803.01775 (2018).
 24.
Hu, Y., Kuemmeth, F., Lieber, C. M. & Marcus, C. M. Hole spin relaxation in Ge–Si core–shell nanowire qubits. Nat. Nanotechnol. 7, 47–50 (2012).
 25.
Dimoulas, A., Tsipas, P., Sotiropoulos, A. & Evangelou, E. K. Fermilevel pinning and charge neutrality level in germanium. Appl. Phys. Lett. 89, 252110 (2006).
 26.
Katsaros, G. et al. Hybrid superconductor–semiconductor devices made from selfassembled SiGe nanocrystals on silicon. Nat. Nanotech. 5, 458–464 (2010).
 27.
Larsen, T. et al. Semiconductornanowirebased superconducting qubit. Phys. Rev. Lett. 115, 127001 (2015).
 28.
Shah, V. A. et al. Reverse graded relaxed buffers for high Ge content SiGe virtual substrates. Appl. Phys. Lett. 93, 192103 (2008).
 29.
Dobbie, A. et al. Ultrahigh hole mobility exceeding one million in a strained germanium quantum well. Appl. Phys. Lett. 101, 172108 (2012).
 30.
Virgilio, M. & Grosso, G. TypeI alignment and direct fundamental gap in SiGe based heterostructures. J. Phys. Condens. Matter 18, 1021 (2006).
 31.
Virgilio, M. et al. Physical mechanisms of intersubbandabsorption linewidth broadening in sGe/SiGe quantum wells. Phys. Rev. B 90, 155420 (2014).
 32.
Busby, Y. et al. Near and farinfrared absorption and electronic structure of GeSiGe multiple quantum wells. Phys. Rev. B 82, 205317 (2010).
 33.
Hanson, R., Kouwenhoven, L. P., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K. Spins in fewelectron quantum dots. Rev. Mod. Phys. 79, 1217–1265 (2007).
 34.
Escott, C. C., Zwanenburg, F. A. & Morello, A. Resonant tunnelling features in quantum dots. Nanotech 21, 274018 (2010).
 35.
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).
 36.
Basset, J. et al. Evaluating charge noise acting on semiconductor quantum dots in the circuit quantum electrodynamics architecture. Appl. Phys. Lett. 105, 063105 (2014).
 37.
Freeman, B. M., Schoenfield, J. S. & Jiang, H. Comparison of low frequency charge noise in identically patterned Si/SiO2 and Si/SiGe quantum dots. Appl. Phys. Lett. 108, 253108 (2016).
 38.
Courtois, H., Meschke, M., Peltonen, J. T. & Pekola, J. P. Origin of hysteresis in a proximity Josephson junction. Phys. Rev. Lett. 101, 067002 (2008).
 39.
Mourik, V. et al. Signatures of Majorana fermions in hybrid superconductorsemiconductor nanowire devices. Science 336, 1003–1007 (2012).
Acknowledgements
The authors acknowledge support through a FOM Projectruimte of the Foundation for Fundamental Research on Matter (FOM), associated with the Netherlands Organisation for Scientific Research (NWO).
Author information
Affiliations
Contributions
N.W.H. fabricated the quantum dot devices, M.K. fabricated the JoFET devices and N.W.H. and D.P.F. performed the experiments, supported by M.L.V.T. and R.L. A.S. prepared the heterostructures, D.S. fabricated the Hall bar transistors measured at low temperature by D.S., L.Y. and G.S. G.S. supervised the material development. N.W.H., D.P.F., G.S. and M.V. analysed the data; M.Vi. and G.C. carried out additional analysis. N.W.H. and D.P.F. wrote the manuscript with input from all authors. M.V. conceived and supervised the project.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Hendrickx, N.W., Franke, D.P., Sammak, A. et al. Gatecontrolled quantum dots and superconductivity in planar germanium. Nat Commun 9, 2835 (2018). https://doi.org/10.1038/s4146701805299x
Received:
Accepted:
Published:
Further reading

An array of four germanium qubits
Nature (2021)

A fourqubit germanium quantum processor
Nature (2021)

New signatures of the spin gap in quantum point contacts
Nature Communications (2021)

Electron–hole superfluidity in strained Si/Ge type II heterojunctions
npj Quantum Materials (2021)

The germanium quantum information route
Nature Reviews Materials (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.