Parity lifetime of bound states in a proximitized semiconductor nanowire
A. P. Higginbotham, S. M. Albrecht, G. Kirsanskas, W. Chang, F. Kuemmeth, P. Krogstrup, T. S. Jespersen, J. Nygard, K. Flensberg, C. M. Marcus
PParity lifetime of bound states in a proximitized semiconductor nanowire
A. P. Higginbotham,
1, 2, ∗ S. M. Albrecht, ∗ G. Kirˇsanskas, W. Chang,
1, 2
F. Kuemmeth, P. Krogstrup, T. S. Jespersen, J. Nyg˚ard, K. Flensberg, and C. M. Marcus Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen,Universitetsparken 5, 2100 Copenhagen Ø, Denmark Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Dated: January 22, 2015)
Quasiparticle excitations can compromise theperformance of superconducting devices, caus-ing high frequency dissipation, decoherence inJosephson qubits [1–6], and braiding errors inproposed Majorana-based topological quantumcomputers [7–9]. Quasiparticle dynamics havebeen studied in detail in metallic superconduc-tors [10–14] but remain relatively unexplored insemiconductor-superconductor structures, whichare now being intensely pursued in the contextof topological superconductivity. To this end,we introduce a new physical system comprised ofa gate-confined semiconductor nanowire with anepitaxially grown superconductor layer, yieldingan isolated, proximitized nanowire segment. Weidentify Andreev-like bound states in the semi-conductor via bias spectroscopy, determine thecharacteristic temperatures and magnetic fieldsfor quasiparticle excitations, and extract a par-ity lifetime (poisoning time) of the bound statein the semiconductor exceeding 10 ms.
Semiconductor-superconductor hybrids have been in-vestigated for many years [15–19], but recently have re-ceived renewed interest in the context of topological su-perconductivity, motivated by the realization that com-bining spin-orbit interaction, Zeeman splitting and prox-imity coupling to a conventional s-wave superconductorprovides the necessary ingredients to create Majoranamodes at the ends of a one-dimensional (1D) wire. Suchmodes are expected to show nonabelian statistics, allow-ing, in principle, topological encoding of quantum infor-mation [20–22] among other interesting effects [23, 24].Transport experiments on semiconductor nanowiresproximitized by a grounded superconductor have recentlyrevealed characteristic features of Majorana modes [25–28]. Semiconductor quantum dots with superconduct-ing leads have also been explored experimentally [29–32],and have been proposed as a basis for Majorana chains[33–35]. Here, we expand the geometries investigatedin this context by creating an isolated semiconductor-supercondutor hybrid quantum dot (HQD) connected tonormal leads. The device forms the basis of an isolated ∗ These authors contributed equally to this work.
Majorana system with protected total parity, where boththe semiconductor nanowire and the metallic supercon-ductor are mesoscopic [36, 37].The measured device consists of an InAs nanowire withepitaxial superconducting Al on two facets of the hexag-onal wire, with Au ohmic contacts (Figs. 1a,b). Four de-vices showing similar behavior have been measured. TheInAs nanowire was grown without stacking faults usingmolecular beam epitaxy with Al deposited in situ to en-sure high-quality proximity effect [38, 39]. Differentialconductance, g , was measured in a dilution refrigeratorwith base electron temperature T ∼
50 mK using stan-dard ac lock-in techniques. Local side gates, patternedwith electron beam lithography, and a global back gatewere adjusted to form an Al-InAs HQD in the Coulombblockade regime, with gate-controlled weak tunneling tothe leads. The lower right gate, V R , was used to tune theoccupation of the dot, with a linear compensation fromthe lower left gate, V L , to keep tunneling to the leadssymmetric. We parameterize this with a single effective a g (e /h) V S D ( m V ) V G (V) c o e o b Au AlInAs B V SD V R V L V BG I B
FIG. 1:
Nanowire-based hybrid quantum dot. a , Scanningelectron micrograph of the reported device, consisting of an InAsnanowire (gray) with segment of epitaxial Al on two facets (blue)and Ti/Au contacts and side gates (yellow) on a doped silicon sub-strate. b , Device schematic and measurement setup, showing ori-entation of magnetic field, B . c , Differential conductance, g , asa function of effective gate voltage, V G , and source-drain voltage, V SD , at B = 0. Even (e) and odd (o) occupied Coulomb valleyslabeled. a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n gate voltage, V G (see Supplement).Differential conductance as a function of V G andsource-drain bias, V SD , reveals a series of Coulomb dia-monds, corresponding to incremental single-charge statesof the HQD (Fig. 1c). While conductance features athigh bias are essentially identical in each diamond, atlow bias, V SD < E C = 1 . µ eV, satisfy the condition(∆ < E C ) for single electron charging [42, 43]. Differ-ential conductance at low bias occurs in a series of nar-row features symmetric about zero bias, suggesting trans-port through an Andreev-like bound state, with negativedifferential conductance (NDC) observed at the borderof odd diamonds. NDC arises from slow quasiparticleescape, as discussed below, similar to current-blockingseen in metallic superconducting islands in the oppositeregime, ∆ > E C [44, 45].To gain quantitative understanding of these features,we model transport through a single Andreev bound statein the InAs plus a Bardeen-Cooper-Schriffer (BCS) con-tinuum in the Al. The model assumes symmetric cou-pling of both the bound state and continuum to theleads, motivated by the observed symmetry in V SD ofthe Coulomb diamonds. Transition rates were calculatedfrom Fermi’s golden rule and a steady-state Pauli mas-ter equation was solved for state occupancies. Conduc-tance was then calculated from occupancies and transi-tion rates (see Supplement).Measured and model conductances are compared inFigs. 2a,b. The coupling of the bound state to eachlead, noting the near-symmetry of the diamonds, was es-timated to be Γ = 0 . E =58 µ eV at zero magnetic field, was measured using finitebias spectroscopy (Fig. 2e). The normal-state conduc-tance from each lead to the continuum, g Al = 0 . e /h ,was estimated by comparing Coulomb blockaded trans-port features in the high bias regime ( V SD = 0 . µ eV, was foundfrom the onset of NDC, which is expected to occur at eV SD = ∆ − E (Fig. 2f). While the rate model showsgood agreement with experimental data, some featuresare not captured, including broadening at high bias, withgreater broadening correlated with weaker NDC, andpeak-to-peak fluctuations in the slope of the NDC fea-ture. These features may be related to heating or cotun- V S D ( m V ) V G (mV) a b -0.2 0 0.2 0.4 g (e /h) EXPERIMENT THEORY-0.2 0 0.2 0.4 g (e /h) v S D c d e f v = SD v = G SD v = G -E e v = SD e ( Δ / e ) v G ( Δ / e ) FIG. 2:
Subgap bias spectroscopy, experiment and model.a , Experimental differential conductance, g , as a function of gatevoltage V G and source-drain V SD , shows characteristic patternincluding negative differential conductivity (NDC). b , Transportmodel of a . v G = αV G up to an offset, where α is the gate leverarm. Axis units are ∆ /e = 180 µ V, where ∆ is the supercon-ducting gap. See text for model parameters. c , Source and drain(gold) chemical potentials align with the middle of the gap in theHQD density of states. No transport occurs due to the presenceof superconductivity. d , Discrete state in resonance with the leadsat zero bias. Transport occurs through single quasiparticle states. e , Discrete state in resonance with the leads at high bias. Trans-port occurs through single and double (particle-hole) quasiparticlestates. f , Discrete state and BCS continuum in the bias window.Transport is blocked when a quasiparticle is in the continuum, re-sulting in NDC. neling, not accounted for by the model.The observation of negative differential conductanceplaces a bound on the relaxation rate of a single quasi-particle in the HQD from the continuum (in the Al) tothe bound state (in the InAs nanowire). Negative differ-ential conductance arises when an electron tunnels intothe weakly coupled BCS continuum, blockading trans-port until it exits via the lead. The blocking condition isshown for a hole-like excitation in Fig. 2f. Unblocking oc-curs when the quasiparticle relaxes into the bound state,followed by a fast escape to the leads. NDC thus indi-cates a long quasiparticle relaxation time, τ qp , from thecontinuum to the bound state. Using independently de-termined parameters, the observed NDC is only compat-ible with the model when τ qp > . µ s (see Supplement).This bound on τ qp is used below to similarly constrainthe characteristic poisoning time for the bound state.Turning our attention to the even-odd structure ofzero-bias Coulomb peaks (Figs. 3a,b), we observed con-sistent large-small peak spacings (Fig. 3), associating the g ( e / h ) V G (V) T = 50 mK B = 08575 S ( m V ) V G (V) T = 350mKB = 150mT E F N ab d eo eo ≈≈ g ( e / h ) g ( e / h ) c V G (mV) oe e FIG. 3:
Even-odd Coulomb peak spacings a , Measured zero-bias conductance, g , versus gate voltage, V G , at temperature T ∼
50 mK, and magnetic field B = 0. b , Peak spacing, S , versus gatevoltage. Black points show spacings from a calculated using thepeak centroid (first moment), red points T = 350 mK and B = 0,purple points B = 150 mT and T ∼
50 mK. c , Right-most peaks in a . Peak maxima ( (cid:52) ) and centroids ( (cid:4) ) are marked. d , Free energy, F , at T = 0 versus gate-induced charge, N , for different HQDoccupations, where N = CV G /e up to an offset and C is the gatecapacitance. Parabola intersection points are indicated by circles,corresponding to Coulomb peaks. BCS continuum (shaded), shownfor odd occupancy. Odd Coulomb diamonds carry an energy offset E for quasiparticle occupation of the sub gap state, resulting in adifference in spacing for even and odd diamonds. larger spacings with even occupation, as expected the-oretically [42, 43] and already evident in Fig. 1. Oc-casional even-odd parity reversals on the timescale ofhours were observed in some devices, similar to whatis seen in metallic devices [14]. Peak spacing alterna-tion disappears at higher magnetic fields, B , consistentwith the superconducting-to-normal transition, and alsodisappears at elevated temperature, T > . T c ∼ even valleys, the opposite of the Kondo effect. The asym-metric shape is most pronounced at low temperature, T < .
15 K, and decreases with increasing magnetic field.The degree of asymmetry is not predicted by the ratemodel, even taking into account the known small asym-metry due to spin degeneracy [46]. In the analysis below,we consider peak positions defined both by peak maximaand centroids.A model of even-odd Coulomb peak spacing that in- cludes thermal quasiparticle excitations follows earliertreatments [40, 41, 43], including a discrete subgap stateas well as the BCS continuum [41] (Fig. 3d). Even-oddpeak spacing difference, S e − S o , depends on the differ-ence of free energies, S e − S o = 4 αe ( F o − F e ) , (1)where α is the (dimensionless) gate lever arm. The freeenergy difference, written in terms of the ratio of parti-tion functions, F o − F e = − k B T ln (cid:18) Z o Z e (cid:19) , (2)depends on D ( E ), the density of states of the HQD, Z o Z e = (cid:90) ∞ d E D ( E ) ln coth[ E/ (2 k B T )] , (3)where D ( E ) consists of one subgap state and the contin-uum. For ∆ (cid:29) k B T , this can be written F o − F e ≈ − k B T ln( N eff e − ∆ /k B T + 2 e − E /k B T ) , (4)where N eff = ρ Al V √ πk B T ∆ is the effective number ofcontinuum states for Al volume, V , and normal densityof states ρ Al [40, 41] (see Supplement).Within this model, one can identify a characteristictemperature, T ∗ ∼ ∆ / [ k B ln( N eff )], less than the gap,above which even-odd peak spacing alternation is ex-pected to disappear. Note in this expression N eff it-self depends on T , and also that T ∗ does not dependon the bound state energy, E . A second (lower) char-acteristic temperature, T ∗∗ ∼ (∆ − E ) / [ k B ln( N eff / E , is where the even-odd alter-nation is affected by the bound state, leading to satu-ration at low temperature [40, 41]. For a spin-resolvedzero-energy ( E = 0) bound state—the case for unsplitMajorana zero modes—these characteristic temperaturescoincide and even-odd structure vanishes, as pointed outin Ref. [36]. In the opposite case, where the bound statereaches the continuum ( E = ∆), the saturation temper-ature vanishes, T ∗∗ = 0, and the metallic result with nobound state is recovered [40, 41].Experimentally, the average even-odd peak spacing dif-ference, (cid:104) S e − S o (cid:105) , was determined by averaging over a setof 24 consecutive Coulomb peak spacings, including thoseshown in Fig. 3, at each temperature. Figure 4 showseven-odd peak spacing difference appearing abruptly at T onset ∼ . T sat ∼ . V = 4 E / ( αe ). Fig-ure 4 shows good agreement between experiment and themodel, Eq. (1), using a density of states determined inde-pendently from data in Fig. 2, with V = 7 . × nm asa fit parameter, consistent with the micrograph (Fig. 1a),and ρ Al = 23 eV − nm − [14].
100 0.50.20.10.05 T ** T *(B=0) T (K) 4 V ( B =0) peak max peak centroid model ( = 0) model ( = 50 neV) V (150 mT)4 V (100 mT)4 V (80 mT)4 V (40 mT) T *(40 mT) T *(100 mT) T (K)100 200 g ( - e / h ) V G (V) ‹ S e - S o ( m V ) ‹ S e - S o ( m V ) ‹ ‹ S o S e FIG. 4:
Temperature and magnetic field dependence of the even-odd peak spacings.
Average even-odd spacing difference, (cid:104) S e − S o (cid:105) , versus temperature, T . Spacing between peak maxima (triangle) and centroids (square) are shown. Spacing expected fromlower Zeeman-split bound state, 4 V ( B ) = 4 E ( B ) / ( αe ), indicated on right axis. Quasiparticle activation temperature, T ∗ , and crossovertemperature, T ∗∗ , indicated on top axis. Solid curve is Eq. (1) with a HQD density of states measured from Fig. 2 (∆ = 180 µ eV, E = 58 µ eV, α = 0 . V = 7 . × nm . Dotted curve includes a discrete state broadening, γ = 50 neV, fit to the centroid data. Left inset:
Same as main, but at B = 40 , , ,
150 mT, from top to bottom. Curves are fit to twoshared parameters: g-factor, g = 6, and superconducting critical field, B c = 120 mT, with other parameters fixed from main figure. Rightinset:
Representative Coulomb peaks showing even ( S e ) and odd ( S o ) spacings. The asymmetric peak shape complicates measurementof even-odd spacings, as one can either use the centroidsor maxima to measure spacings, the two methods givingdifferent results. Larger peak tails on the even valley sidecause the centroids to be more regularly spaced than themaxima. This is evident in Fig. 4, where the centroidmethod shows a decreasing peak spacing difference atlow temperature, while with the maximum method thespacing remains flat. The thermal model of S e − S o canalso show a decrease at low temperature if broadening ofthe bound state is included (See Methods). We do notunderstand at present if the low temperature decrease inthe centroid data is related to the decrease seen in themodel when broadening is included. It is worth noting,however, that the fit to the centroid data gives a broaden-ing γ = 50 neV, reasonably close to the value estimatedfrom the lead couplings, ( h Γ ) / ∆ = 20 neV.Applied magnetic field (direction shown in Fig. 1b)reduces the characteristic temperatures T onset , T sat , andsaturation amplitudes. Field dependence is modeled byincluding Zeeman splitting of the bound state and orbitalreduction of the gap and bound state energy, taking the g -factor and critical magnetic field as two fit parametersapplied to all data sets. The fit value g = 6 lies within the typical range for InAs nanowires [47, 48], supporting ourinterpretation of the bound state residing in the InAs.The fit value of critical field, B c = 120 mT, is typical forthis geometry.Good agreement between the peak spacing data andthe thermodynamic model (Fig. 4) suggests that thenumber of thermally activated quasiparticles obeys equi-librium statistics, N eq ( T ) = N e − /k B T (see Supple-ment for derivation). Saturation caused by the boundstate means that even-odd amplitude loses sensitivity asa quasiparticle detector below T sat . We therefore take N eq ( T sat ) ∼ − (for T sat ∼ . T sat . The corresponding upper bound of the quasiparticlefraction, x qp = N eq ( T sat ) / ( ρ Al V ∆) ∼ − , is compara-ble to values in the recent literature, 10 − − − , formetallic superconducting junctions and qubits [3–6, 13].We now discuss the implications of our measurementsfor determining the poisoning time, τ p , of the boundstate. For the present geometry, the dominant sourceof poisoning of the bound state is not tunneling of elec-trons from the leads, which is negligible in the stronglyblockaded regime, but is rather the continuum in thestrongly-coupled Al, within the isolated structure itself.Theoretical estimates [8] suggest an inverse relationshipbetween τ p and the number of available quasiparticles,with a proportionality that depends on system details.Taking the bound on single quasiparticle relaxation timefrom the continuum into the bound state, τ qp > . µ s,from above, as the poisoning time when a single quasipar-ticle is present, we estimate τ p by scaling for the actualnumber of quasiparticles in equilibrium, N eq , giving apoisoning time τ p = τ qp /N eq (cid:38)
10 ms.We expect τ p to depend weakly on the bound stateenergy for low-energy bound states [11, 49, 50], includ-ing for Majorana zero modes at E = 0. Device geom-etry may somewhat alter the number of quasiparticlesavailable to relax into the bound state, i.e. by chang-ing N eff , but any increase can be compensated by ex-ponentially small decreases in the quasiparticle temper-ature. The long poisoning time obtained here suggeststhat a large number of braiding operations in Majoranasystems should be readily achievable within the relevanttime scale. Methods
Sample preparation:
InAs nanowires were grown inthe [001] direction with wurzite crystal structure with Alepitaxially matched to [111] on two of the six { } side-facets. They were then deposited randomly onto a dopedsilicon substrate with 100 nm of thermal oxide. Electron-beam lithographically patterned wet etch of the epitaxialAl shell (Transene Al Etchant D, 55 C, 10 s) resulted in asubmicron Al segment (310 nm, Fig. 1a). Ti/Au (5/100nm) ohmic contacts were deposited on the ends following in situ Ar milling (1 mTorr, 300 V, 75 s), with side gatesdeposited in the same step. For the present device, theend of the upper left gate broke off during processing.However, the device could be tuned well without it.
Master equations:
The master equations (used forFig. 1b) consider states with fixed total parity, composedof the combined parity of quasiparticles in the thermal-ized continuum and the 0, 1, or 2 quasiparticles in thebound state (see Supplement).
Free energy model:
Even and odd partition functionsin Eq. 2, F o − F e = − k B T ln( Z o /Z e ), can be written assums of Boltzmann factors over respectively odd and evenoccupancies of the isolated island. For even-occupancy, Z e = 1 + (cid:88) i (cid:54) = j e − E i /k B T e − E j /k B T + ..., (5)where the first term stands for zero quasiparticles, thesecond for two (at energies E i and E j ), and additionalterms for four, six, etc. Z o similarly runs over odd oc-cupied states. Rewriting these sums as integrals overpositive energies yields F o − F e = − k B T ln tanh (cid:90) ∞ d E D ( E ) ln coth( E/ k B T ) , (6) where D ( E ) is the density of states of the HQD, D ( E ) = ρ BCS ( E ) + 12 ρ +0 ( E ) + 12 ρ − ( E ) . (7)We take ρ BCS ( E ) to be a standard BCS density of states, ρ BCS ( E ) = ρ Al V E (cid:112) E − ∆( B ) θ ( E − ∆) (8)( θ is the step function), and ρ to be a pair of Lorentzian-broadened spinful levels symmetric about zero, ρ ± ( E ) = γ/ π ( E − E ± ) + ( γ/ + γ/ π ( E + E ± ) + ( γ/ . (9)Zeeman splitting of the bound state and pair-breaking bythe external magnetic field are modeled with the equa-tions E ± ( B ) = ∆( B )∆ E ± gµ B B, (10)∆( B ) = ∆ (cid:115) − (cid:18) BB c (cid:19) , (11)where E is the zero-field state energy and ∆ is the zerofield superconducting gap. In the event that a boundstate goes above the continuum, E + s > ∆( B ), we nolonger include the state in the free energy. Equation (6)was integrated numerically to obtain theory curves inFig. (4).Equations (10) and (11) are reasonable provided thelower spin-split state remains at positive energy, E − > B c , the bound state will reach zeroenergy, resulting in topological superconductivity andMajorana modes, the subject of future work.We thank Leonid Glazman, Bert Halperin, RomanLutchyn and Jukka Pekola for valuable discussions,and Giulio Ungaretti, Shivendra Upadhyay and ClausSørensen for contributions to growth and fabrication. Re-search support by Microsoft Project Q, the Danish Na-tional Research Foundation, the Lundbeck Foundation,the Carlsberg Foundation, and the European Commis-sion. APH acknowledges support from the US Depart-ment of Energy, CMM acknowledges support from theVillum Foundation. [1] Lang, K. M., Nam, S., Aumentado, J., Urbina, C. &Martinis, J. M. Banishing quasiparticles from Josephson-junction qubits: Why and how to do it. Applied Super-conductivity, IEEE Transactions on , 989–993 (2003).[2] Aumentado, J., Keller, M. W., Martinis, J. M. & Devoret,M. H. Nonequilibrium quasiparticles and 2 e periodicityin single-Cooper-pair transistors. Physical Review Letters , 066802 (2004). [3] Martinis, J. M., Ansmann, M. & Aumentado, J. Energydecay in superconducting Josephson-junction qubits fromnonequilibrium quasiparticle excitations. Physical ReviewLetters , 097002 (2009).[4] De Visser, P. J. et al.
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1. Effective gate voltage definition2. Bound on the single quasiparticle relaxation time3. Detailed interpretation of Coulomb diamonds4. Derivation of transport and thermal model5. Comparison of free energy approximations6. Effect of the bound state on the free energy
1. Effective gate voltage definition
We define an effective gate voltage in software to tune the HQD. The physical gate voltages, V R and V L , are relatedto the effective gate voltage, V G , by V R = V R, + κ V G ,V L = V L, + (cid:112) − κ V G , with κ = 0 . V R, = − .
41 V, V L, = − .
96 V.These transformation rules ensure that V G = ( V R − V R, ) + ( V L − V L, ) , so that V G can be interpreted as thedistance from ( V R, , V L, ) in the V R − V L plane.All measurements are performed at backgate voltage V BG = 2 .
39 V.
2. Bound on the single quasiparticle relaxation time
The effect of quasiparticle relaxation is shown in Figs. S1a-e. Quasiparticle relaxation results in a disappearanceof the negative differential conductance, in combination with the appearance of an extra conductance threshold. Wequantify this observation by introducing the relative conductance ratio R = g (cid:48) + g NDC g (cid:48) − g NDC (S1)where g NDC is the minimum of the negative differential conductance, and g (cid:48) is the maximum of the extra conductancethreshold that appears when τ qp → R -value is a metric for the relative strength of the negativedifferential conductance.Figs. S1f-j show example conductance traces at constant bias and their associated R -values. The traces show that R ≈ − R ≈ R -value calculated as a function of single quasiparticle relaxation time, τ qp . Also shown is ameasured R -value averaged over all negative differential conductance features in Fig. 1 of the main text. The measured R -value is consistent with τ qp > . µ s, giving the experimental bound on the single quasiparticle relaxation time.
3. Detailed interpretation of Coulomb diamonds
Each conductance threshold in the Coulomb diamond plots can be interpreted with the help of the transport model,as shown in Fig. S3. For example, the highest bias at which NDC is observed occurs at the intersection of black andgreen lines, when v SD = (∆ + E ) /e . -0.200.2 V s d ( m V ) -101 v S D -0.200.20.4 g ( e / h ) EXPERIMENT τ qp = ∞ τ qp = 0.1 μs τ qp = 5 ns τ qp = 0 a b c d e -1 0 10.10-0.1 g ( e / h ) G (mV) 0.10-0.1 g ( e / h ) -1 0 1 -1 0 1-1 0 1R = -0.95 R = -0.8 R = 0.1 R = 1 f g h i j g NDC g’ ( Δ / e ) v G ( Δ / e ) v G ( Δ / e ) v G ( Δ / e ) v G ( Δ / e ) Fig. S1:
Effect of quasiparticle relaxation. a
Measured conductance g versus source-drain bias V SD and gate V G . b ,Transport model of a , with τ qp = ∞ . v G ≡ αV G up to an offset, where α is the gate lever arm. Axis units are ∆ /e = 180 µ V. c-e , Model with τ qp = 0 . µ s, τ qp = 5 ns, and τ qp = 0 respectively. f-j , Conductance versus gate at constant bias indicated in a . Relative conductance ratio, R = ( g (cid:48) + g NDC ) / ( g (cid:48) − g NDC ), for theory curves is labeled (see text). -101 R qp ( μ s ) Fig. S2:
Single quasiparticle relaxation bound.
Relative conductance ratio, R = ( g (cid:48) + g NDC ) / ( g (cid:48) − g NDC ), versus singlequasiparticle relaxation time τ qp . Dashed curve is theory derived as shown in Fig. S1. Data is the average over all chargetransitions in Fig. 1, with vertical error the standard deviation of the mean, and horizontal error propagated from vertical.
4. Derivation of transport and thermal model
This section gives a detailed derivation of the transport and thermal model used in the main text.To describe the electron transport through a metallic superconducting quantum dot we consider the followingmodel: H = H LR + H D + H T , (S2) -202 -1 0 1 -202 -1 0 1-202 -1 0 1 v G a b c g ( e / h ) E = 0.1Δ E = 0.3Δ E = 0.5Δ ( Δ / e ) v G ( Δ / e ) v G ( Δ / e ) v S D ( Δ / e ) v S D ( Δ / e ) v S D ( Δ / e ) Fig. S3:
Interpreting conductance thresholds. a , Calculated conductance g versus v SD and v G . E = 0 . v SD / ± ( v G + ∆ /e ) [black], v SD / ± ( v G + E /e ) [blue], v SD / ± ( v G − E /e ) [green], v SD / ± ( v G − ∆ /e ) [red]. b , E = 0 . c , E = 0 . where the Hamiltonian H LR = (cid:88) ανs ( ε ανs − µ α ) c † ανs c ανs (S3)describes the normal metallic leads with c † ανs being an electron creation operator in the lead α ∈ { L , R } , with anorbital quantum number ν and spin s ∈ {↑ , ↓} . The leads have chemical potentials given by µ α = ± V /
2, where V denotes symmetrically applied bias. For the semiconductor-superconductor hybrid quantum dot, we use a simplifiedmodel consisting of a Bardeen-Cooper-Schrieffer (BCS)[S1] continuum and an Andreev bound state for fixed numberof particles [S2]: H D = (cid:88) s,p = e,h (cid:104) E γ † s,p γ s,p + (cid:88) n E n γ † ns,p γ ns,p (cid:105) + E cN , (S4) E cN = U ( N − N g ) , N = (cid:104) (cid:88) s d † s d s + (cid:88) n d † ns d ns (cid:105) , (S5)where d † ns creates an electron in the continuum with quantum number n (e.g. momentum of electrons on the dot),and d † s denotes electron creation in a localized level, which gives rise to a subgap state in the BCS spectrum. Thecharging effects on the quantum dot are described by constant interaction model given by the term E cN , where thecharging energy is given by E C = 2 U and the number of electrons on the dot is controlled by a gate voltage V g , whichis parameterized by dimensionless number N g = V g /E C . The operator N gives the total number of electrons on thedot. With superconducting pairing, the dot Hamiltonian is diagonal in the basis of the quasiparticle operators γ † ,which are given by [S3–S5] d ns = u n γ ns,e + sv n γ †− n ¯ s,h , (S6a) γ ns,e = u n d ns − sv n d †− n ¯ s S, (S6b) γ ns,h = u n S † d ns − sv n d †− n ¯ s , (S6c) γ † ns,e = S † γ † ns,h , γ ns,h = S † γ ns,e , (S6d)with S † denoting the Cooper pair creation operator and γ † ns,e/h denoting the quasiparticle creation operator, whichadds an electron/hole to the system. Here a state with quantum numbers − ns is the time-reversed partner of a statewith quantum numbers ns . The quasiparticle excitation energy E n and the BCS coherence factors u n , v n are givenin terms of superconducting gap ∆ and electron dispersion on the dot ε n as E n = (cid:112) ε n + ∆ , u n = (cid:115) (cid:18) ε n E n (cid:19) , v n = (cid:115) (cid:18) − ε n E n (cid:19) . (S7)Similarly, the subgap state operator is expressed as d s = u γ s,e + sv γ † s,h , (S8)with u , v being model dependent coherence factors, which we set to u = v = 1 / √ H T = (cid:88) ανs (cid:104) t S ,α c † ανs d s + t ∗ S ,α d † s c ανs + (cid:88) n (cid:0) t C ,α c † ανs d ns + t ∗ C ,α d † ns c ανs (cid:1) (cid:105) = (cid:88) ανs (cid:104) t S ,α c † ανs ( u γ s,e + sv γ † s,h ) + t ∗ S ,α ( u ∗ γ † s,e + sv ∗ γ s,h ) c ανs + (cid:88) n (cid:8) t C ,α c † ανs ( u n γ ns,e + sv n γ †− n ¯ s,h ) + t ∗ C ,α ( u ∗ n γ † ns,e + sv ∗ n γ − n ¯ s,h ) c ανs (cid:9)(cid:105) , (S9)where t C ,α gives the tunneling amplitude to the continuum and t S ,α gives the tunneling amplitude to the subgapstate. Thermodynamics of the even/odd effect
We now present the free energy difference between the superconducting metallic island having even or odd numberof electrons. The parity of the number of quasiparticles has to be equal to the parity of the number of electrons N on the island. The free energy difference δF between the odd and even occupation is expressed as [S6, S7] δF = F o − F e = − β ln (cid:18) Z o Z e (cid:19) , (S10)in terms of the partition functions for different parities2 Z o / e = (cid:89) n,s (1 + e − βE n ) ∓ (cid:89) n,s (1 − e − βE n ) . (S11)where β = 1 /k B T denotes the inverse temperature of the island. For a sufficiently large island the single particlespectrum can be described by the spectrum of a grounded superconductor where the single particle spectrum E n isgiven by Eq. (S7). Without a subgap state the free energy difference Eq. (S10) is expressed as δF BCS = − k B T ln tanh (cid:34) (cid:88) n,s ln coth (cid:18) βE n (cid:19)(cid:35) = − β ln tanh (cid:90) + ∞ ∆ d E ρ
BCS ( E ) ln coth (cid:18) βE (cid:19) , (S12)where ρ BCS ( E ) is the BCS density of states for quasiparticles on the island given by ρ BCS ( E ) = ρ D E √ E − ∆ , (S13)with ρ D = ρ Al V denoting the normal state density of states, including spin, and ρ Al is aluminum density of states pervolume, and V is the volume of the island. For small temperatures β ∆ (cid:29)
1, the free energy difference (S12) can beapproximated as δF BCS ≈ − k B T ln tanh (cid:20) (cid:90) + ∞ ∆ d E ρ
BCS ( E ) e − βE (cid:21) = − k B T ln tanh (cid:2) N eff e − β ∆ (cid:3) ≈ ∆ − k B T ln( N eff ) , (S14)where an effective number of quasiparticle states N eff is given by N eff = 2 (cid:90) + ∞ ∆ d E ρ
BCS ( E ) e − β ( E − ∆) = 2 ρ D ∆ e β ∆ K ( β ∆) ≈ ρ D (cid:112) πk B T ∆ (S15)and K ν ( x ) denotes the modified Bessel function of the second kind. With a subgap state the free energy difference Eq. (S10) acquires an additional term and one gets δF ABS = − k B T ln tanh (cid:20)(cid:90) + ∞ ∆ d E ρ
BCS ( E ) ln coth (cid:18) βE (cid:19) + ln coth( βE / (cid:21) . (S16)See also Eq. (S42) where the approximate expression for the first term in used.In the main text we discuss how the low-temperature data deviates from the above Andreev-bound-state model interms of a life-time broadening of the subgap state. This is done by including a phenomenological broadening withwidth γ into the subgap density of states, which then gives the free energy difference δF ABS − LB = − k B T ln tanh (cid:34) (cid:90) + ∞ ∆ d E ρ
BCS ( E ) ln coth (cid:18) βE (cid:19) + 12 (cid:88) τ = ± s = ↑ , ↓ (cid:90) + ∞ d ω π γ ln coth (cid:16) βω (cid:17) ( ω − τ E ) + ( γ/ (cid:35) . (S17)In the kinetic equation calculation presented below, the equilibrium distributions of quasiparticle in the continuum withan even or odd number of quasiparticles are needed. Since we will assume that the particles occupying the continuumare effectively equilibrated, we find the distribution functions by modifying the usual Fermi-Dirac distribution functionas f P ( E ) = 1 e βE ( Z P /Z ¯ P ) + 1 → f e ( E ) = 1 e β ( E + δF BCS ) + 1 ,f o ( E ) = 1 e β ( E − δF BCS ) + 1 , (S18)where P ∈ { e , o } , and ¯ P represents the opposite of P . Number of quasiparticles
Using the above results, we derive a simple expression for the number of quasiparticles in the absence of a boundstate. At low temperature, when δF BCS = ∆ − k B T ln( N eff ), the distribution functions take the form f e = N eff e − β ( E +∆) , (S19) f o = 1 N eff e − β ( E − ∆) , (S20)where N eff is given by Eq. (S15).The number of quasiparticles in each parity state can then be calculated using N P = 2 (cid:90) + ∞ ∆ d E ρ
BCS ( E ) f P ( E ) . (S21)Substituting the above expression for f o gives N o = 1, as expected. Substituting f e gives the quasiparticle number N e = 2 (cid:90) + ∞ ∆ d E ρ
BCS ( E ) N eff e − β ( E +∆) = N eff e − β ∆ (cid:90) + ∞ ∆ d E ρ BCS ( E ) e − β ( E − ∆) = (cid:0) N eff e − β ∆ (cid:1) . (S22)Because of the large charging energy N e is the square of the bulk value N eff e − β ∆ , indicating that quasiparticles mustbe created in pairs. Incoherent transport
We now calculate the current through the quantum dot by a set of master equations, with transition rates calculatedby Fermi’s golden rule, which is valid in the weak tunneling regime or the so-called sequential tunneling regime.According to the Fermi’s golden rule the transition rate from the initial state i to the final state j caused by aperturbation V is given by [S8]: Γ fi = 2 π |(cid:104) f | V | i (cid:105)| δ ( E f − E i ) . (S23)In our case we are interested in the transitions caused by the tunneling Hamiltonian (S9) between the states |D(cid:105)| LR (cid:105) and |D (cid:48) (cid:105)| LR (cid:48) (cid:105) : Γ D (cid:48) D , LR (cid:48) LR = 2 π |(cid:104) LR (cid:48) |(cid:104)D (cid:48) | H T |D(cid:105)| LR (cid:105)| δ ( E D (cid:48) − E D + E LR (cid:48) − E LR ) , (S24)where the many-body eigenstates of the lead Hamiltonian H LR are denoted as | LR (cid:105) and of the dot Hamiltonian H D are denoted as |D(cid:105) . Also E D gives the energy of the state |D(cid:105) and E LR gives the energy of the state | LR (cid:105) . So we seethat we need the following matrix element |(cid:104) LR (cid:48) |(cid:104)D (cid:48) | H T |D(cid:105)| LR (cid:105)| = (cid:88) ανsn | t α | (cid:110) | u n | |(cid:104) LR (cid:48) | c † ανs | LR (cid:105)| |(cid:104)D (cid:48) | γ ns,e |D(cid:105)| + | v n | |(cid:104) LR (cid:48) | c † ανs | LR (cid:105)| |(cid:104)D (cid:48) | γ †− n ¯ s,h |D(cid:105)| + | u n | |(cid:104) LR (cid:48) | c ανs | LR (cid:105)| |(cid:104)D (cid:48) | γ † ns,e |D(cid:105)| + | v n | |(cid:104) LR (cid:48) | c ανs | LR (cid:105)| |(cid:104)D (cid:48) | γ − n ¯ s,h |D(cid:105)| (cid:111) . (S25)Note that we have not written out the terms with the subgap state, but they have an analogous structure.In order to form the tunneling rates we need to specify what kind of states |D(cid:105) on the metallic superconductingquantum dot we consider and what kind of (thermal) averaging procedure for these states we employ. We classify thestates |D(cid:105) according to the number of electrons N , the number of excitations N S in the continuum, quantum number l labeling the configuration of the excitations in the state, and the occupancy of the subgap state x ∈ { , ↑ , ↓ , } , i.e., |D(cid:105) ≡ | N, N S , l, x (cid:105) . (S26)The above states have the energy E N,N S ,l,x = E cN + E x + N S (cid:88) n ∈ l E n . (S27)If the number of electrons N is even/odd then the total number of excitations N tot ,S also has to be even/ood. Thismeans that the number of excitations in the continuum N S depends on the subgap state occupancy x and the chargeon the dot N , i.e., N S → even for N even and x ∈ { , } , odd for N even and x ∈ {↑ , ↓} , odd for N odd and x ∈ { , } , even for N odd and x ∈ {↑ , ↓} . (S28)Now we want to thermally average over all quasiparticle states N S of the continuum and their configurations l .The thermal averaging assumption is valid if the relaxation rate of quasiparticles is much faster than the rate of thetunneling events between the leads and the island. From the expression (S25) we see that we need to consider thethermal averages of the following expressions for the dot f S ( E n , N, x ) = (cid:88) l,N S W N,N S ,l,x |(cid:104) N, N S , l, x | γ † ns,p γ ns,p | N, N S , l, x (cid:105)| , (S29a)¯ f S ( E n , N, x ) = (cid:88) l,N S W N,N S ,l,x |(cid:104) N, N S , l, x | γ ns,p γ † ns,p | N, N S , l, x (cid:105)| = 1 − f S ( E n , N, x ) , (S29b)where p = e, h . Here W N,N S ,l denotes a thermal distribution for which we have W N,N S ,l,x = 1 Z x e − βE N,NS,l,x , Z x = (cid:88) N S ,l W N,N S ,l,x . (S30)By using Eq. (S18) and following the prescription (S28), we get the distributions f S ( E, N, x ) = f e ( E ) , for N even and x = 0 , ,f o ( E ) , for N even and x = ↑ , ↓ ,f o ( E ) , for N odd and x = 0 , ,f e ( E ) , for N odd and x = ↑ , ↓ . (S31)After thermally averaging over the source-drain lead states | LR (cid:105) , using grand-canonical ensemble, and the continuumstates of the dot, we obtain the following tunneling rates from and to the continuum of the dotΓ αN + χ ← NN S + χ (cid:48) ← N S ≈ γ α (cid:90) + ∞| ∆ | E d E (cid:112) E − | ∆ | f N ( E cN + χ − E cN + χ (cid:48) E − χµ α ) f S , − χ (cid:48) ( E, N, x ) , with f S , + ( E, N, x ) = f S ( E, N, x ) , f S , − ( E, N, x ) = ¯ f S ( E, N, x ) . (S32)Here γ α = 2 × π × ρ D ρ α | t C ,α | with ρ α denoting density of states of the normal leads. The continuum coupling γ α = 2 × π × ρ D ρ α | t C ,α | is related to the normal-state conductance by g Al = ( π/ e /h ) γ α . Note that we have setthe chemical potential of the metallic superconducting dot at zero µ D = 0 in order not to complicate the calculations,and also used that ε n = ε − n .Additionally, there are tunneling rates from and to the subgap state. When the starting state has no quasiparticlesin the subgap state, i.e., | (cid:105) , we get the following ratesΓ αN − ← Ns ← ≈ Γ α v [1 − f N ( E cN − E s − E cN − − µ α )] , (S33a)Γ αN +1 ← Ns ← ≈ Γ α u f N ( E cN +1 + E s − E cN − µ α ) . (S33b)For the state with single quasiparticle | s (cid:105) we getΓ αN − ← N ← s ≈ Γ α u [1 − f N ( E cN + E s − E cN − − µ α )] , (S34a)Γ αN − ← N ← s ≈ Γ α v [1 − f N ( E cN − E ¯ s − E cN − − µ α )] , (S34b)Γ αN +1 ← N ← s ≈ Γ α u f N ( E cN +1 + E ¯ s − E cN − µ α ) , (S34c)Γ αN +1 ← N ← s ≈ Γ α v f N ( E cN +1 − E s − E cN − µ α ) , (S34d)and for the state with two quasiparticles | (cid:105) we getΓ αN − ← Ns ← ≈ Γ α u [1 − f N ( E cN + E ¯ s − E cN − − µ α )] , (S35a)Γ αN +1 ← Ns ← ≈ Γ α v f N ( E cN +1 − E ¯ s − E cN − µ α ) . (S35b)Here Γ α = 2 πρ α | t S ,α | , and s ∈ {↑ , ↓} with ¯ s denoting the opposite of s . We include the relaxation from the continuumto the subgap state by introducing the following rates within the same charge stateΓ N o ← N o ← s = Γ N e ← N e ← s = Γ relax . (S36)Now we want to find the current through the superconducting metallic quantum dot. To do this we need to obtainthe occupation probabilities P N,x of the states described by a number of electrons N on the dot and the occupancyof the subgap state x . We write the following steady state Pauli master equation for the probabilities P N,x dd t P N,x = − (cid:88) N (cid:48) x (cid:48) Γ N (cid:48) ← Nx (cid:48) ← x P N,x + (cid:88) N (cid:48) x (cid:48) Γ N ← N (cid:48) x ← x (cid:48) P N (cid:48) ,x (cid:48) = 0 , (S37)with the condition (cid:88) N,x P N,x = 1 . (S38)The rates entering in (S37) are given by Γ N (cid:48) ← Nx (cid:48) ← x = Γ LN (cid:48) ← Nx (cid:48) ← x + Γ RN (cid:48) ← Nx (cid:48) ← x , (S39)and for x = x (cid:48) we have Γ αN (cid:48) ← Nx ← x = Γ α,xN (cid:48) ← NN S − ← N S + Γ α,xN (cid:48) ← NN S +1 ← N S . (S40)When the occupation probabilities P N,x are obtained, the current through the quantum dot can be written as I seq = ( − e ) (cid:88) N,xx (cid:48) (cid:18) Γ LN +1 ← Nx (cid:48) ← x − Γ LN − ← Nx (cid:48) ← x (cid:19) P N,x . (S41)These formulae are then solved numerically to produce the plots in Fig. S1 and Fig. 2b in the main text.
5. Comparison of free energy approximations
This section gives examples of the free energy difference, F o − F e , calculated under different approximations,considering the case without broadening γ = 0 and without an applied field B = 0. Under these conditions the freeenergy difference is given by Eq. (S16). When β ∆ > βE/ ≈ e − βE can be used for thefirst term. Applying the identity (cid:82) + ∞ ∆ d E ρ
BCS ( E ) e − βE = ρ Al V ∆ K ( β ∆) then gives F o − F e ≈ − k B T ln tanh (cid:34) ρ Al V ∆ K ( β ∆) + ln coth (cid:18) βE (cid:19) (cid:35) , (S42)where K ( x ) is a Bessel function of the second kind. In the very low temperature limit β ∆ (cid:29) βE > K ( β ∆) ≈ (cid:112) π/ (2 β ∆) e − β ∆ , ln coth( βE / ≈ e − βE , and tanh( x ) ≈ x can be used, giving F o − F e ≈ k B T ln (cid:34) N eff e − β ∆ + 2 e − βE (cid:35) , (S43)where N eff = ρ Al V √ πk B T ∆.Equations (S42) and (S43) constitute two levels of accuracy at which Eq. (S16) can be evaluated. Figure S4compares the methods. Equation (S42) is an excellent approximation to Eq. (S16) over the experimentally relevanttemperature range. Equation (S43) is poor approximation at intermediate temperatures.
6. Effect of the bound state on the free energy
Figure S5 shows a comparison of the free energy difference, F o − F , with and without the subgap bound state.As the lowest energy unoccupied state, the bound state causes the free energy to saturate at F o − F e = E at lowtemperature. It should be noted that the free energy difference with a subgap bound state was also shown in Fig. 5of Lafarge et al. [S7]. [S1] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. , 1175 (1957).[S2] M. Tinkham, Phys. Rev. B , 1747 (1972).[S3] B. D. Josephson, Phys. Lett. , 251 (1962).[S4] J. Bardeen, Phys. Rev. Lett. , 147 (1962).[S5] M. Tinkham, Introduction to Superconductivity (Dover Publications, 2004).[S6] M. Tuominen, J. Hergenrother, T. Tighe, and M. Tinkham, Phys. Rev. Lett. , 1997 (1992).[S7] P. Lafarge, P. Joyez, D. Esteve, C. Urbina, and M. Devoret, Phys. Rev. Lett. , 994 (1993).[S8] H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics (Oxford University Press, 2004). E F o - F e ( μ e V ) T (K) Eq. (S16) Eq. (S42) Eq. (S43)
Fig. S4:
Comparison of free energy approximations.
Free energy difference F o − F e versus temperature T for threedifferent expressions for the free energy. All parameters same as main text (∆ = 180 µ eV, E = 58 µ eV, γ = 0, B = 0). Blackcrosses are numerically exact values from Eq. (S16), red line is Eq. (S42), blue line is Eq. (S43). E F o - F e ( μ e V ) T (K) No bound state Bound state
Fig. S5:
Effect of bound state on free energy.