Evidence of Andreev blockade in a double quantum dot coupled to a superconductor
Po Zhang, Hao Wu, Jun Chen, Sabbir A. Khan, Peter Krogstrup, David Pekker, Sergey M. Frolov
EEvidence of Andreev blockade in a double quantum dot coupled to a superconductor
Po Zhang, Hao Wu, Jun Chen, Sabbir A. Khan,
3, 4
Peter Krogstrup,
3, 4
David Pekker, and Sergey M. Frolov ∗ Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, 15260, USA Department of Electrical and Computer Engineering,University of Pittsburgh, Pittsburgh, PA, 15260, USA Microsoft Quantum Materials Lab Copenhagen, 2800 Lyngby, Denmark Center for Quantum Devices, Niels Bohr Institute,University of Copenhagen, 2100 Copenhagen, Denmark
We design and investigate an experimental system capable of entering an electron transportblockade regime in which a spin-triplet localized in the path of current is forbidden from entering aspin-singlet superconductor. To stabilize the triplet a double quantum dot is created electrostaticallynear a superconducting lead in an InAs nanowire. The superconducting lead is a molecular beamepitaxy grown Al shell. The shell is etched away over a wire segment to make room for the doubledot and the normal metal gold lead. The quantum dot closest to the normal lead exhibits Coulombdiamonds, the dot closest to the superconducting lead exhibits Andreev bound states and an inducedgap. The experimental observations compare favorably to a theoretical model of Andreev blockade,named so because the triplet double dot configuration suppresses Andreev reflections. Observedleakage currents can be accounted for by finite temperature. We observe the predicted quadruplelevel degeneracy points of high current and a periodic conductance pattern controlled by the occu-pation of the normal dot. Even-odd transport asymmetry is lifted with increased temperature andmagnetic field. This blockade phenomenon can be used to study spin structure of superconductors.It may also find utility in quantum computing devices that utilize Andreev or Majorana states.
Superconductor-semiconductor nanostructures are ofinterest in the context of quantum computing devices.Most prominently, they are a platform investigated asa host of Majorana zero modes that can be used tobuild topological qubits [1–7]. While these qubits havenot been achieved, other types of qubits namely An-dreev, fluxonium and transmon have all been createdout of super-semi structures [8–12]. Spin qubits can alsobe hosted by quantum dots defined in semiconductingnanowires [13–15]. Quantum dots exhibit iconic trans-port blockade phenomena: Coulomb blockade which isused in metrology to set quantum current standard [16]and Pauli spin blockade which is used for readout andinitialization of spin qubits [17].In quantum dots coupled to superconductors Andreevbound states form. They are hybrids of quantum dotenergy levels and many-body Bogoliubov quasiparti-cles [18, 19]. Andreev double quantum dots have alsobeen realized [20–22]. Here, we ask a question: cana blockade phenomenon unique to superconductors bedemonstrated in Andreev quantum dots? Our goal isto observe Andreev blockade which is a suppression ofAndreev reflection by a spin-triplet double dot configu-ration [23].We fabricate a double quantum dot with one super-conducting and one non-superconducting dot in an InAssemiconductor nanowire device (Fig. 1). The right dotexhibits an induced gap and Andreev bound states (seeFigs. 2(a) and (b)). The left dot is characterized byCoulomb diamonds and shows no dramatic supercon-ducting features (see Figs. 2(c) and (d)). Low-bias (sub-gap) transport reveals patterns that theory predicted forthe four-step Andreev charge transport cycle which arises
FIG. 1. (a) Schematic of Andreev blockade. The blockadedconfiguration is indicated with a red cross showing how a spin-triplet in double dot is prevented from forming a spin-singletCooper pair. (b) Scanning electron microscope (SEM) imageof a device similar to the one studied in the main text. Sectionmarked ’Al/InAs’ is an InAs nanowire covered by an Al shell.A section where the shell is etched is marked ’InAs’. Verticalwhite lines mark gate electrodes used in creating the doubledot. when two electrons required to form a Cooper pair aretransported through the double dot (Fig. 3). As evi-dence of Andreev blockade we find asymmetry betweenquadruple charge degeneracy points at even-to-odd andodd-to-even transitions in the normal dot (Figs. 3 and4). The observed asymmetry has the properties predictedby theory [23]: the pattern is flipped at opposite volt-age bias and it disappears at higher temperature andat higher magnetic field (Figs. 5 and S2). Experimen-tally, we find that current is not completely blocked inthe regimes that we label as Andreev blockade. Our nu- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b B i a s ( m V ) B i a s ( m V ) N (mV) 379 354 V N (mV) 379375V S (mV)340 340 V S (mV) 375 0.10.0 d I / d V ( e / h ) d I / d V ( e / h ) Δ
360 V N (mV) 357.4 V N (mV)374.8 373V S (mV) V S (mV)350.7 350.7 360.1 360.1 FIG. 2. Differential conductance spectra for (a-b) QD S and(c-d) QD N . Spectra are taken by fixing one dot at a de-generate state while tuning the other dot with the ( V S , V N )combination. The voltage combinations are indicated in Fig.3(a). The large white arrows and small blue arrows indicateresonance peaks with different amplitudes. merical model, based on Ref. [23], accounts for this byintroducing finite temperature. OUR APPROACH
We experimentally realize conditions required for theobservation of Andreev blockade following a theoreti-cal proposal [23]. We use a double quantum dot totrap a spin triplet state in an semiconductor nanowire(Fig. 1(a)). The right side side of the double dot is con-nected to a superconductor, forming QD S , the left side isconnected to a non-superconductor lead, such that QD N is a normal dot in the multi-electron regime.Fig. 1(b) shows a scanning electron microscope imageof the nanowire device. An InAs nanowire covered with ∼
15 nm epitaxial Al is placed on top of 60 nm pitch gateelectrodes. The double dot is defined electrostatically byvoltages on gate electrodes indicated in the image. V N and V S are the gate voltages primarily used for tuning thedot states. Al on the left section of the wire is selectivelyetched to make the normal lead. The device is measuredin a dilution fridge with a base temperature of about40 mK.We demonstrate that dot QD S exhibits Andreevbound states, while dot QD N exhibits Coulomb dia-monds, a staple of non-superconducting quantum dottransport (Fig. 2). The double dot configuration is setup by tuning all of the electrostatic gates adjacent to thesuperconducting lead. Spectra are then taken by fixingone dot at a degeneracy point while tuning the other dot.Spectra of QD S show induced superconducting hard gapwhich is a stripe of suppressed current at voltage biases below ∆ /e = 0 . N show Coulomb diamonds and no clear inducedgap. To better understand the four panels of Fig. 2, it ishelpful to look at Fig. 3(a) which shows a charge stabil-ity diagram and cuts in V N - V S space that correspond toFig. 2.There are also more subtle conditions that the systemmust meet for Andreev blockade to be observable. Theinduced superconducting gap should be hard in order tosuppress single-particle transport below the gap. Anysingle-particle transport is an Andreev blockade liftingmechanism, thus the softer the induced gap the weakerare the Andreev blockade signatures. The barrier tothe superconducting lead should be low in order to in-duce Andreev bound states. This is in contrast withPauli blockade setups which typically require few elec-tron regimes and hence high barriers to facilitate strongconfinement. The inter-dot charging energy should besmaller than the induced gap because the Andreev trans-port regions shrink rapidly with the increasing of theinter-dot charging energy [23]. To match experimentalresults we set the inter-dot charging energy to 10 µ eV,whic correponds to a weakly coupled double dot regime. PREDICTED SIGNATURES OF ANDREEVBLOCKADE
Following Ref. [23], we are looking for the followingfour experimental signatures (A-D) of Andreev transportand blockade in a N-QD N -QD S -S system. We describethe charge sates in both dots by their parity, which canbe either even or odd. The parity of states in QD N canbe inferred by studying how the degeneracy points shiftin magnetic field, with odd regions expanding and evenregions shrinking at higher fields (see Fig. S7). The parityof states in QD S can be inferred from Andreev spectra,with regions inside loop-like resonances being odd (seeFig. 2(a)-(b)). See supplementary materials and Ref.[23]for theoretical background underlying these signatures.(A) At subgap voltage biases current is confined to tri-angular regions of the charge stability diagram.The triangles do not appear in closely spaced pairsas in non-superconducting double dots where theyform around triple points. Instead, triangles ap-pear at quadruple Andreev degeneracy points, aconsequence of the two-electron charge transfer cy-cle, form a parallelogram grid in V S vs V N space.(B) An alternating pattern of blockade/no-blockade isobserved when quadruple points are tuned by V N . V S does not affect whether blockade is present ornot.(C) The sign of source-drain bias voltage flips Andreevblockade. A quadruple point that is blockaded inpositive bias is not blockaded in negative bias, andvice versa.(D) Andreev blockade is not present when supercon-ductivity is suppressed by magnetic field or tem-perature. Signatures (A)-(C) should no longer bemanifest.Signatures (B) and (C) can be formulated to-gether as follows. Andreev blockade is expected for(odd,odd) → (even,even) charge parity transitions and for(odd,even) → (even,odd) transitions, where the arrows in-dicate the direction of charge transfer, so that the condi-tions are valid for both signs of the applied bias.An ideal blockade corresponds to total suppression ofcurrent below the gap. Blockade can be suppressed bythe presence of sub-gap quasi-particles and thermally ex-cited quasi-particles [23]. If Andreev blockade is onlypartially suppressed, a reduced current, known as leak-age current, indicates blockade.In principle there should be no fine-tuning requiredto observe Andreev blockade. All that is needed is onenormal dot, one superconducting dot and a hard gap su-perconductor lead. Thus we are looking for a region of V S vs V N that includes many charge degeneracy pointsthat exhibit blockade signatures. In practice, mesoscopicfactors such as additional quantum dots in the nanowiresegments covered by the leads can introduce their owncurrent modulations. Thus some gate tuning may stillbe required to clearly observe Andreev blockade. MEASURED ANDREEV BLOCKADESIGNATURES
Signature (A) which is current confined to single, notdouble, triangles in the charge stability diagram is il-lustrated by Figs. 3(a,b). Stability diagrams are takenat two opposite bias voltages. For both bias directionswe observe elongated triangles, rounded due to relativelylow bias voltages required to stay below the induced gapof aluminum. Numerical model results in Figs. 3(c,d)closely reproduce the experiment. Larger gate rangeshown in Fig. 4 confirms the single-triangle character ofthe charge degeneracy points.Signature (B) in experiment presents itself as an al-ternating pattern of high current/low current when theoccupation of QD N is changed. It is illustrated by Figs. 3and 4. We see dim degeneracy points followed by brightones. In Fig. 4 the dim columns are marked by dimarrows, while the bright columns are marked by brightarrows. The region of ( V S , V N ) parameter space depictedcontains 6 × E O(a) (b) 0.1 mV-0.1 mV2a 2b2c2d EOEE375340 V S ( m V )
354 V N (mV) 379 354 V N (mV) 3790.560.0 I experiment (nA)(c) (d) 0 1I simulation (a.u.)simulation simulation FIG. 3. (a-b) Experimental and (c-d) simulated results ofa ”unit” stability diagram with 2 × N and QD S are labeled on top and rightaxes in panel (a), respectively (E - even, O - odd). Dashedlines are traces along which spectra in Fig. 2 are taken. Thesource-drain bias voltage is indicated in white. Parametersfor simulation (in a.u. which match system energies in meVfor convenience): source-drain bias µ S − µ N = -0.1 in (c), 0.1in (d), charging energy U N = 4, U S = 0 .
7, inter-dot chargingenergy U NS = 0 .
01, induced gap ∆ S = 0 .
2, temperature T =0 .
02 . that the blockade is partially lifted. In the simulation ofFigs. 3(c,d) we assume finite temperature to reproducethis behavior. Finite temperature enables single particletunneling into a hard-gap superconducting lead via ther-mally excited quasi-particles, and provides a way aroundthe blocked Andreev processes.Signature (C) is the reversal of the high/low currentpattern in opposite bias. In Fig. 3(a), at − . . × (a) -0.15 mV EEEEOOOO E O EO E350300 V S ( m V ) (b) 0.15 mV350300 V S ( m V )
350 400V N (mV) 20 I ( n A ) FIG. 4. Stability diagrams in a larger ( V S , V N ) parameterspace. The yellow dashed rectangle in panel (a) encloses theregion studied in Figs. 2, 3 and 5. Parities in QD N and QD S are labeled on top and right axes, respectively. Blue (white)arrows indicate columns of conductance triangles with low(high) current. The source-drain bias voltage is indicated inwhite. of QD S is changed AB is not affected, and when the oc-cupation of QD N is changed AB appears at the oppositebias. The full picture is more complicated as certain biasasymmetries are also observed at currents above the in-duced gap. However, Figs. S3 and S4 illustrate that ingeneral bias asymmetry at high bias does not follow thesame pattern as subgap low-bias asymmetry, suggestingthat they are of different origin.Finally, signature (D) which is the disappearance ofother signatures when superconductivity is suppressed ispresented in Fig. 5 and supplementary figures S2, S6, S9,S10. Figs. 5(a-d) reproduce the same regime as in Fig. 3for different bias voltages. When a magnetic field of 0.6 Tis applied, in Figs. 5(e-h), we observe that the alternatingpatterns of bright/dim degeneracy points are no longerpresent, and neither is the bias voltage asymmetry. Thesame holds true at elevated temperature, as illustratedin Fig. S2.Single elongated degeneracy points are replaced by lesselongated points at higher fields where superconductivityof the aluminum shell is suppressed (Figs. 5(e-h)) - in thisregime the charge degeneracy points appear more similarto those of a normal double dot with pairs of triangulartriple points, but with significant rounding and blurringand a weak interdot capacitive coupling. In future ex-periments using a larger gap superconductor such as Snor Pb [27–29] as a shell can provide a larger bias voltagerange for the observation of Andreev blockade and makethis observation more clear by reducing the relative roleof feature broadening. -0.2 mV 0.2 mV-0.5 mV 0.5 mV -0.2 mV 0.2 mV-0.5 mV 0.5 mV(a) (b)(c) (d) (e) (f)(g) (h)375 V S ( m V ) N (mV) I ( n A ) V S ( m V ) I ( n A )
354 3790 T 0.6 T
FIG. 5. Stability diagrams at different bias voltages andmagnetic fields (a-d) B = 0 and (e-h) B = 0.6 T. The biasvoltage is indicated in each panel. The magnetic field is inthe sample plane at a 24 degree angle with the nanowire. ALTERNATIVE EXPLANATIONS
We have considered the possibility that signatures (A-D) were only identified due to fine-tuning in a deliberatesearch for predicted patterns. In this scenario, signaturessuch as alternating bright/dim degeneracy points andbias asymmetries are not due to Andreev blockade, butrather they arise accidentally due to additional states co-existing with the QD N -QD S system in the same nanowire- for example spurious Andreev and normal quantum dotsin the nanowire lead segments. Those other states arefine-tuned to modulate transport in the double dot injust the right way to be consistent with Andreev block-ade.We cannot fully exclude the possibility of the pres-ence of extra mesoscopic states beyond the two quantumdots. We see, e.g. Fig. 4 lower left part, that while thestability points form a dominating double-dot pattern,their intensities vary across a large V S - V N range suggest-ing non-monotonic coupling to states outside the dots ornon-monotonic inter-dot barriers. This is typical for awide variety of quantum dots, including those made inthe Intel cleanroom [30].Additional data in supplementary materials and fullexperimental data on Zenodo [31] also demonstrate thatwithin the same device other regimes do not show pat-terns of Andreev blockade, when all gates used to definea double dot are set differently.Our argument to not favor the above explanation isthat a pattern consistent with Andreev blockade signa-tures (A-C) are observed over a regime covering 6 × FOLLOW-ON WORK
Recently larger hard gaps have been induced in semi-conductor nanowires in experiments with tin (gap of 0.6mV) and lead (gap of 1.2 mV) shells [28, 29]. It wouldbe interesting to repeat Andreev blockade experimentsusing these superconductors. First, larger ratio of gap tomeasurement temperature may result in stronger block-ade. Second, the ability to work at higher bias and largercharging energies would make the observation of variousblockade features such as bias triangles more conclusive,and reduce the role that rounding plays at low biases.Finally, blockade can be studied to higher magnetic fieldallowing for a detailed investigation of spin structure inthe parent superconductor.Several improvements can be done in immediate follow-on work related to materials processing and device fabri-cation. This would impact not only Andreev blockade ex-periments but many works aimed at searching for Majo-rana modes and building superconductor-semiconductorqubits. For instance, the wet etch degrades the qualityof nanowires by introducing defects. The supplementaryinformation in Ref. [27, 32] shows that even using thestandard etchant for InAs-Al wires may result in InAsdamage or Al islands. The use of in-situ shadowing ordry etching are promising avenues to explore.
FUTURE RELEVANCE
At the most basic level, Andreev blockade offers ameans of studying spin-resolved transport in hybrid de-vices at zero magnetic field. We foresee application ofAndreev blockade in experiments that probe spin pairingin superconductors. Much like Pauli blockade was usedto investigate spin mixing mechanisms in semiconduc-tors due to hyperfine, spin-orbit or electron-phonon cou-pling, Andreev blockade can be potentially used to detecttriplet pairing or admixtures thereof, spin-flip scattering,spin polarization or textures such as Larkin-Ovchnnikov-Fulde-Ferrel state in the superconductor. A two-arm An-dreev blockade device with two double dots in parallel canin principle be used as a spin-sensitive probe for crossedAndreev reflection. Quantum dots with superconduct-ing leads are building blocks of Andreev qubits, Kitaevemulators and of topological qubits [7, 10, 33]. Thesedevices may manifest Andreev blockade or utilize it todetect the state of a qubit or an emulator by providing aspin-dependent transport or transition rate element.
COMPARISON TO OTHER WORKS
Several versions of a triplet blockade in quantum dotsclosely related to Andreev blockade have been consid-ered theoretically [34–38], with several works focusing on a parallel combination of quantum dots, which is rele-vant for crossed Andreev reflection [39–41]. Other typesof blockade related to Andreev reflection such as chiralblockade have been proposed [42].A recent experiment in a similar double dot setup withtwo rather than one superconducting lead has studieda triplet blockade that develops at large magnetic field,where spin triplet is the unique ground state of the doubledot [43]. In contrast, Andreev blockade demonstratedin this work occurs at zero magnetic field, due to thestochastic filling of a quantum dot by random spins.
FUNDING SOURCES
S.F. and D.P. are supported by NSF PIRE-1743717.S.F. is supported by NSF DMR-1906325, ONR andARO. P.K. is supported by European Union Horizon2020 research and innovation program under the MarieSk(cid:32)lodowska-Curie Grant No. 722176 (INDEED), Mi-crosoft Quantum and the European Research Council(ERC) under Grant No. 716655 (HEMs-DAM).
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Dayeh, and S. M.Frolov, Phys. Rev. B , 155416 (2017).[46] B. Cord, J. Lutkenhaus, and K. K. Berggren, Jour-nal of Vacuum Science & Technology B: Microelectron-ics and Nanometer Structures Processing, Measurement,and Phenomena , 2013 (2007). Supplementary Materials: Evidence of Andreev blockade in a double quantum dotcoupled to a superconductor
THEORETICAL BACKGROUND: ANDREEV BLOCKADE
Theory of Andreev blockade and the details of our numerical model are contained in a separate paper [23]. Herewe summarize key concepts that are relevant for our interpretation of the experimental data.Andreev blockade is a consequence of a dynamical formation of a stable sub-gap spin-triplet state. The series doubledot setup allows for this to happen: a triplet state with one spin on each dot can have a low chemical potential dueto small exchange interaction between spins.Andreev blockade is the suppression of Andreev reflection due to spin parity mismatch between the double quantumdot and the superconductor. Andreev blockade is most apparent when the superconductor induces a hard gap in thenanowire. The hard gap ensures the suppression of single-particle tunneling and enforces Andreev reflection asdominant means of transport.Andreev reflection transfers a charge of 2e into the superconductor, where ’e’ is the electron charge. In order tomove the two electrons between the normal and the superconducting lead, the double dot transitions through fourcharge configurations per Andreev cycle. Therefore transport is only allowed at quadruple charge degeneracy pointswhere four charge configurations have similar chemical potentials all within a source-drain bias window that does notexceed the superconducting gap. As a point of comparison, in non-superconducting double dots transport involvesmoving just one electron between the leads and takes place at triple, rather than quadruple, degeneracy points, whichresults in the formations of the well-known honeycomb charge stability diagram [44].Andreev blockade is controlled by the occupation of the normal dot QD N , but it is not sensitive to the occupationof the superconducting dot QD S . Quantum dot QD S is strongly coupled to the superconductor, and this couplinghybridizes all states of the same parity. For example, states with 0 and 2 charges are hybridized, and so are states with1 and 3 charges. Coupling to the superconductor imposes an approximate particle-hole symmetry which mandatesthat transport at odd-to-even and even-to-odd degeneracy points is the same. This translates into the insensitivityof Andreev blockade to the occupation of QD S .A complimentary way to see why transport is insensitive to the charging state of QD S is as follows. Over the courseof a transport cycle two electrons must be added to QD S , driving the dot from an even parity state, to an odd paritystate, back to an even parity state. If the normal dot QD N has two charges on it, then they must be of oppositespins and one of them can always escape into QD S . Specifically, if QD S is in the even parity state, either of the spinson QD N can move to QD S thus putting QD S into an odd parity state. If QD S is in an odd parity state, one of theelectrons on QD N has the opposite spin and can move to QD S resulting in a transition to an even parity state. Onthe other hand, if QD N only has one spin, then that spin cannot escape to QD S if QD S is an odd parity state of thesame spin. Hence, the charging state of QD N determines if an Andreev blockade is established. PAULI BLOCKADE VS. ANDREEV BLOCKADE
Both Pauli blockade and Andreev blockade occur in double quantum dots with two charges in spin triplet state.However, they have different origins. Pauli blockade is due to Pauli principle which prevents two electrons of the samespin from occupying the same orbital. Andreev blockade is due to inability of forming Cooper pairs out of spin-tripletpairs. The characteristic energy scale of Andreev blockade is the induced superconducting gap. The characteristicenergy scale of Pauli blockade is the singlet-triplet energy level spacing in the (0,2) charge configuration.Andreev and Pauli blockades appear in different yet overlapping parameter spaces and can be observed in the samedevice. In soft gap Andreev double quantum dots, Pauli blockade has been previously observed in Ref. [20]. Thoughwe did not observe Pauli blockade in the device studied here. This is consistent with what is known of Pauli blockade- it is a relatively rare phenomenon which is observed more frequently in few-electron double dots, due to largersinglet-triplet energies. Whereas, in multi-electron dots, such as those studied here, Pauli blockade appears in one outof every 10 or 100 degeneracy points [45]. In contrast, Andreev blockade is not expected to be as sensitive to quantumdot energy scales and is, in principle, guaranteed by the superconducting gap. Even if singlet-triplet energies weresignificant for all orbitals, Pauli blockade would be expected in 1 out of 4 degeneracy points, while Andreev blockadeis expected in 2 out of 4.
FURTHER READING
Background information on non superconducting single and double quantum dots can be found in Ref. [44]. Anintroduction to Andreev bound states can be found in Refs. [18, 19]. A related subject of Majorana zero modes in1D wires is also discussed in Ref. [19]. Experiments on epitaxial growth of Al on nanowires and the hard gap can befound in Ref. [24, 25].
METHODS
The growth of InAs nanowires with Al shells are performed using molecular beam epitaxy (MBE). First, InAsnanowires are grown from predefined Au catalysts via vapor-liquid-solid mechanism. After the nanowire growth, thegrowth chamber is cooled down and in situ
Al growth is carried out, ensuring the high interface quality of the hybrid.Further discussion about the growth can be found in Ref. [27].Electrostatic 60 nm pitch gates are patterned by 100 kV e-beam lithography (EBL) with PMMA 950 A1 as theresist. The development for gate patterns is performed in 1:3 MIBK/IPA for 1 minute at a low temperature of -15 ◦ Cto enhance the resolution [46]. Then a bilayer of 1.5/6 nm Ti/PdAu is evaporated by electron-beam evaporation. Thegates are covered by 10 nm of HfO as the dielectric layer, patterned by EBL and grown by atomic layer deposition(100 cycles).Nanowires are transferred onto the gate chip using a micro-manipulator under an optical microscope. Windows forAl-etching are defined by 10 kV EBL. The PMMA resist is dried in a vacuum chamber at room temperature to avoidheating the Al layer [26]. AlO x /Al on the InAs wire is selectively etched using MF CD-26 developer/DI water solutionfor 2 minutes, at a volume ratio of 1:20 at room temperature. The etching of the shell over a significant length ofthe nanowire (hundreds of nanometers) is a delicate process which requires optimization. If not optimized, residualgrains of aluminum are left on the nanowire and they are capable of inducing superconductivity in the regions wherethis is not desired, e.g. over the dot QD N .Leads are made with EBL, Ar cleaning, followed by e-beam evaporation of 10/130 nm of Ti/Au.Measurements are performed in a dilution refrigerator with a 40 mK base temperature. AUTHOR CONTRIBUTIONS
S.A.K. and P.K. grew the nanowires. J.C., H.W., and P.Z. fabricated the devices. D.P., P.Z. and S.M.F. developedthe theory and performed numerical simulations. P.Z. and H.W. performed measurements. P.Z. and S.M.F. wrotethe manuscript with inputs from all authors.
VOLUME AND DURATION OF STUDY
The first InAs-Al device for the transport component of the study was measured in March 2019, when we performedcharacterizations of newly arrived wires and tested fabrication recipes. A total of 8 double dot devices were studied.The device discussed in this manuscript was made in March 2020 and measured from June to September 2020,producing 6855 datasets. The interruption between the fabrication and the measurement is due to shutdown causedby COVID-19.
DATA AVAILABILITY
Curated library of data extending beyond what is presented in the paper, as well as simulation and data processingcode are available at [31].
SUPPLEMENTARY DATA FOR THE DEVICE A PRESENTED IN THE MAIN TEXT
100 nmB InAs InAs/AlV N V S DrainSource
FIG. S1. SEM image of device A that corresponds to data in the main text. The image is taken after Al etch but before theevaporation of Ti/Au leads. The double quantum dot is created electrostatically using 6 gate electrodes (yellow lines) and theunderlying doped Si substrate (as a global gate). N (mV)375345 V S ( m V )
355 379V N (mV)375345 V S ( m V ) (b) G ( e / h ) G ( e / h ) lower-lefttop-leftlower-righttop-right FIG. S2. Temperature dependence of the transport region in Fig. 3. (a) Stability diagrams at different temperatures. Thetemperature is labeled in each sub-panel. The source-drain bias voltage is 0.1 mV. (b) Local maximum conductance extractednear four degeneracy points versus the temperature. The asymmetry between the left and right maxima disappears above Al’sbulk critical temperature (1.2 K). B=0. I ( n A ) V S ( m V ) -0.5 mV
320 340 360 380 400 420V N (mV)280300320340360380 V S ( m V ) FIG. S3. Additional data to Fig. 4, at bias voltages above the gap. The source-drain bias voltage is indicated in white. B=0,base temperature. The even/odd pattern of high/low current is not present at high bias.
358 359 360 361 362 363V N (mV)1.00.50.00.51.0 B i a s ( m V ) d I / d V ( e / h ) V N (mV) B i a s ( m V )
300 320 340 360 380 400 4202.50.02.5 0.0000.0250.0500.0750.100 d I / d V ( e / h )
356 357 358 359 360 361V N (mV)1.00.50.00.51.0 B i a s ( m V ) d I / d V ( e / h ) (a)(b)(c)(d) B Horiz =0.6TB
Horiz =0.6TB
Vert =0.6TU N
358 359 360 361 362V N (mV)1.00.50.00.51.0 B i a s ( m V ) d I / d V ( e / h )
325 350 375 400 425V N (mV)300350 V S ( m V ) I ( n A ) bc de FIG. S4. (a) This stability diagram is the same as Fig. S3(b). The dashed lines are traces along which bias-gate spectra inpanels (b-e) are taken. The magnetic fields are noted for each bias-gate panel. B V ert is the magnetic field used in the maintext. B Horiz is used only in this figure. B Horiz is not along the wire and is not in the plane of the chip, but it is perpendicularto B V ert . The Coulomb diamond (blue dashed diamond) in panel (b) yields a charging energy of roughly 4 mV in QD N .Determining the charging energy in QD S is difficult. The blue dashed lines in panel (c) show edges of Column diamonds withnegative slopes. The positive slope edges are hard to distinguish except perhaps near V N = 360 mV. This is because in ourdevice the QD S has a shallow barrier to the superconducting lead causing strong asymmetry. We choose the charging energyin QD S to be 0.3-0.7 meV for numerical simulations. V S ( m V ) (a) -0.05 mV (b) 0.05 mV340375 V S ( m V ) (c) -0.1 mV (d) 0.1 mV340375 V S ( m V ) (e) -0.15 mV (f) 0.15 mV340375 V S ( m V ) (g) -0.2 mV (h) 0.2 mV340375 V S ( m V ) (i) -0.5 mV (j) 0.5 mV354 379340375 V S ( m V ) (k) -1.0 mV 354 379(l) 1.0 mV 0.00.3 I ( n A ) I ( n A ) I ( n A ) I ( n A ) I ( n A ) I ( n A ) V N (mV) V N (mV) FIG. S5. Additional data to Fig. 5, at zero magnetic field. Bias voltages are noted on each panel. At the highest bias voltages(1.0 mV) the large triangular patterns characteristic of normal (non-superconducting) quantum dots are visible in panel (l).The shapes of the triangles are different than at subgap biases, e.g. panels (d) and (f). While in this figure the bias asymmetryin the overall current appears to persist at high biases above the gap, this is not so in the larger range presented in Fig. S3.
354 379V N (mV) 0.00.10.20.3 I ( n A ) V S ( m V ) I ( n A )
354 379V N (mV)340375 V S ( m V )
354 379V N (mV) FIG. S6. Additional data to Fig. 5, at -0.1 mV bias and different fields. The yellow short line in panel (a) indicates V S = 355mV (see Fig. S7).
355 360 365 370 375 380V N (mV)0.00.20.40.6 I ( n A ) (a) (b) V N ( m V ) V N = 1.53 B + 15.26 FIG. S7. (a) Current vs V N extracted from data in Fig. S6, at V S = 355 mV. Curves are shifted vertically for clarity, witha step of 0.1 nA. Black asterisks mark resonance peaks due to levels in QD N . Data at 0.4 T and 0.6 T are multiplied by 10to highlight the peaks. (b) Peak-to-peak distance vs magnetic field shows Zeeman splitting of QD N levels. The dashed line isa linear fit to the points. The charging energy E C of the QD N is about 4 mV (Fig. S4). An effective g-factor of 6.9 can beestimated with equation ∆ V N = α ( gµ B B + E C ), where α is a coefficient be determined, µ B is the Bohr magneton, gµ B B isthe Zeeman energy. These data can be used to assign even and odd occupations in QD N .
320 340 360 380 400 420V N (mV)280300320340360380 V S ( m V )
320 340 360 380 400 420280300320340360380 V S ( m V )
320 340 360 380 400 420280300320340360380 V S ( m V ) I ( n A ) FIG. S8. Zeeman splitting of QD N in the large regime discussed in the main text. (a-b) Stability diagrams at -0.1 mV. Themagnetic field is noted in each panel. (c) Asterisks show peak positions extracted from a series of horizontal linecuts. Dashedlines are linear fitting lines. These data can be used to assign even and odd occupations in QD N . V S ( m V ) I ( n A )
354 379V N (mV)340375 V S ( m V )
354 379V N (mV) 354 379V N (mV) (a) (b) (c)(d) (e) (f) FIG. S9. Additional data for Fig. 5, at -0.2 mV bias and at different magnetic fields. (a) (b)(c) (d)
354 379V N (mV)340375 V S ( m V ) I ( n A )
354 379V N (mV)340375 V S ( m V ) FIG. S10. Additional data for Fig. 5, at -0.5 mV and different applied magnetic fields.
354 379V N (mV)0.50.5 B i a s ( m V )
354 379V N (mV)340375 V S ( m V )
354 379V N (mV)0.50.5 B i a s ( m V ) /h)354 379V N (mV)0.50.5 B i a s ( m V ) bcd (b)(a)(c) (d) FIG. S11. (a) This is a repeat from Fig. 3(a). The white dashed lines are traces along which spectra (b-d) are taken. (b-d)Spectra at different V S values. (b,d) are the same as Fig. 2(c,d). While panel (c) is a new dataset in between (d) and (b). N (mV)0.500.000.50 B i a s ( m V ) /h)369 379V N (mV)340375 V S ( m V ) I ( n A ) N (mV)0.500.000.50 B i a s ( m V ) N (mV)0.500.000.50 B i a s ( m V ) N (mV)0.500.000.50 B i a s ( m V ) N (mV)0.500.000.50 B i a s ( m V ) N (mV)0.500.000.50 B i a s ( m V ) N (mV)0.500.000.50 B i a s ( m V ) (a) (b)(c) (d) (e)(f) (g) (h)b h → FIG. S12. Differential conductance spectra for a series of slightly shifting traces in V S - V N . (a) The same data as Fig. S6(a).Red lines are traces along which spectra (b-h) are taken. The asymmetry between positive-bias and negative-bias Andreevresonance half-loops shows up for cuts taken through the middle of the bias triangles (d-f) but is not apparent for cuts awayfrom that regime. N (mV)200250300350400 V S ( m V )
550 600 650 700V N (mV)400500600700 0.00.20.40.60.81.0 I ( n A ) Configuration 1 Configuration 2(a) (b)
FIG. S13. (a) The rectangle shows the regime in Fig. 4. There is a charge jump near V N = 407 mV due to the instabilitycaused by the large range scan. (b) The same device in another gate voltage configuration obtained by re-tuning all six gates.This panel contains data from seven small-range scans. The rectangles indicate regimes where datasets in Fig. S14 and Fig.S15 are taken. (e) (f) N (mV)0.50.00.5 B i a s ( m V ) N (mV) 0.0000.0050.0100.0150.020 d I / d V ( e / h ) (c) (d) I ( n A ) -0.1 mV V S ( m V ) (b)(a) 0.1 mV V S ( m V ) I ( n A ) -0.2 mV
545 575V N (mV) 0.00.10.20.30.4 I ( n A ) -0.5 mV
545 575V N (mV)475505 V S ( m V ) FIG. S14. A different regime with a different gate-voltage configuration (see Fig. S13). The stability diagrams show biasasymmetry that is similarly consistent with Andreev blockade. However in this regime the pattern does not clearly repeat overmultiple periods as it does in Fig. 4. (a-f) The bias is noted in each panel. Spectra of QD S are taken when QD N is at (g) rightand (h) left degeneracy point.
544 546V N (mV)0.50.00.5 B i a s ( m V )
548 550V N (mV) 544 546V N (mV) 543 544 545V N (mV) 0.000.050.10 d I / d V ( e / h )
540 560V N (mV)620640 V S ( m V )
540 560V N (mV) 540 560V N (mV) 540 560V N (mV) 0.00.20.40.6 I ( n A )
540 560V N (mV)620640 V S ( m V )
540 560V N (mV) 540 560V N (mV) 540 560V N (mV) (a) (b) (c) (d)(e) (f) (g) (h)(i) (j) (k) (l)0 T 0.1 T 0.2 T 0.6 T FIG. S15. A different regime with a different gate-voltage configuration (see Fig. S13). The stability diagrams show biasasymmetry consistent with Andreev blockade at B=0. The asymmetry does not obviously repeat in adjacent degeneracy points.The dot QD S is stronger coupled to the superconducting lead such that the Andreev loop containing the odd-parity region isreduced to a point and we observe a single degeneracy point on the left and a single on on the right at zero magnetic field. Atfinite field, the pattern of four degeneracy points is restored as QD S undergoes a quantum phase transition and the odd-parityregion develops. (a-d) -0.05 mV. (e-h) 0.05 mV. Spectra of QD S are taken along traces near QD N ’s 0-1 transition. DATA FROM OTHER DEVICES (B AND C)
50 100 150V (mV)0.40.20.00.20.4 B i a s ( m V ) d I / d V ( e / h ) V V V V V V S ec ti on w it h A l
200 nm(a) (V)2.252.302.352.402.452.50 V ( V ) (V)1.501.551.601.651.701.75 03 (V)1.501.551.601.651.701.75 0.00.1 I ( n A ) =1.3 (f) (g)(b) FIG. S16. Characterization of Device B. (a) The SEM image. (b) Gate voltage configurations for datasets in panel c (top),d (middle), and e-f (bottom). The device can be tuned from a single dot regime (panel c, spectrum, and panel d, stabilitydiagram) to a double dot regime (panel e-f). The bias voltage is indicated in white in each stability diagram. A backgroundvariation in (f) is subtracted by the current at V = 1 . (V) 0150 I ( p A ) (V)1.661.681.701.721.74 V ( V ) -0.1 mV (a) (b)(d) (e) (f) (g) (c) d efg (V) 0.000.020.04 d I / d V ( e / h ) (V)0.40.20.00.20.4 B i a s ( m V ) (V) 1.52 1.54V (V) FIG. S17. (a-b) Device B in a regime that shows bias asymmetry. The large white and small blue arrows indicate columnsthat have high and low currents. The pattern does not obviously repeat itself along the V direction. We can not concludewhether the bias asymmetry is due to a coincidence or due to Andreev blockade. (c) The rectangle-enclosed regime in panela. Dashed lines are traces along which spectrum (d-g) are taken. Andreev bound states in QD S have weak and squared-shapeloops (d-e), due to relatively large charging energy.
200 nm V6V1 (V)0.20.2 B i a s ( m V ) d I / d V ( e / h ) (V)0.20.2 B i a s ( m V ) d I / d V ( e / h ) V1 V2 V3 V4 V5 V602 V1 V2 V3 V4 V5 V602 (a) (b)(c)
FIG. S18. Characterization of Device C. (a) The SEM picture after etching. The blue polygon is the design pattern forthe normal lead. (b-c) Spectra taken by tuning V Andreev bound states. The gate voltage configurations are shown on therightmost column. (V) 0.00.51.01.52.02.53.0 I ( n A ) -0.05 mV (V)1.681.691.701.711.72 V ( V ) (c)(d) (e)(f)0.05 mV (V)0.500.250.000.250.501.68 1.70 1.72V (V)0.500.250.000.250.50 B i a s ( m V ) (V)0.500.250.000.250.50 0.000.050.100.15 d I / d V ( e / h ) (V)0.500.250.000.250.50 B i a s ( m V ) (a) (b)(c) (d)(e) (f)(a) (b)(c) (d)(e) (f)