Magnetic-field-induced parity effect in insulating Josephson junction chains
Timothy Duty, Karin Cedergren, Sergey Kafanov, Roger Ackroyd, Jared H. Cole
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Magnetic-field-induced parity effect in insulating Josephson junction chains
Timothy Duty, ∗ Karin Cedergren, Sergey Kafanov, Roger Ackroyd, and Jared H. Cole School of Physics, University of New South Wales, Sydney, NSW 2052 Australia Physics Department, Lancaster University, Lancaster, UK LA1 4YB. Chemical and Quantum Physics, School of Science,RMIT University, Melbourne,VIC 3001 Australia
We report the experimental manifestation of even-odd parity effects in the transport characteris-tics of insulating Josephson junction chains which occur as the superconducting gap is suppressedby applied magnetic fields at millikelvin temperatures. The primary signature is a non-monotonicdependence of the critical voltage, V c , for the onset of charge transport through the chain, with theparity crossover indicated by a maximum of V c at the parity field B ∗ . We also observe a distinc-tive change in the transport characteristics across the parity transition, indicative of Cooper-pairdominated transport below B ∗ , giving way to single-electron dominated transport above B ∗ . Forfields applied in the plane of the superconducting aluminum films, the parity effect is found to oc-cur at the field, B ∗|| , such that the superconducting gap equals the single-electron charging energy,∆( B ∗|| ) = E C . On the contrary, the parity effect for perpendicularly applied fields can occur atrelatively lower fields, B ∗⊥ ≃ /A I , depending only on island area, A I . Our results suggest anovel explanation for the insulating peak observed in disordered superconducting films and one-dimensional strips patterned from such films. The ground state of a mesoscopic BCS superconductorhas been shown to contain an even number of electrons,as long as the single-electron charging energy is less thanthe superconducting gap, E C < ∆, and the tempera-ture is less than a characteristic temperature T ∗ . Aneven-odd parity effect occurs as T exceeds T ∗ , or at verylow temperatures, if ∆ becomes lower than E C . This hasbeen demonstrated experimentally as a change from 2 e to e -periodicity in the gate-voltage modulation of supercon-ducting aluminum single-charge transistors and Cooper-pair box qubits[1–5]. A quantitative analysis of the par-ity effect in hybrid superconductor-semiconductor islandshas become an important experimental tool in identifyingMajorana modes[6]. In this Letter, we show that even-odd parity effects strongly affect magnetotransport in in-sulating Josephson junction chains, and lead to a peak inthe critical voltage with magnetic field. Such an insulat-ing peak is observed in homogeneously disordered super-conducting films and strips[7–11], which are conjecturedto behave as Josephson-coupled grains or droplets[12, 13].Josephson junctions chains are also discrete versions ofsuperconducting nanowiresWe recently reported results on the thermal parity ef-fect observed in Josephson junction chains very deep inthe insulating state, where the characteristic Josephsonenergy is much less than the Cooper-pair charging en-ergy, E J ≪ E CP , ( E CP = 4 E C )[2]. The hallmark ofthe insulating state—a voltage gap to conduction—wasfound to vanish sharply at the characteristic tempera-ture k B T ∗ = ∆ / ln N eff ≃ ∆ /
9, which coincides with thepresence of ∼ N eff ( T ) ≈ V ρ (0) p πk B T ∆( T ) is the effec-tive number of states arising from integration over thequasiparticle density of states, V is the volume of the is-land and ρ (0) is the density of states for the normal metal at the Fermi energy[1].In a more recent Letter, we showed that insulatingJosephson junction chains in zero magnetic field, with E J ∼ E CP , behave as 1D Luttinger liquids, pinned byoffset charge disorder, and therefore can be understoodas a circuit implementation of the one-dimensional Boseglass[1]. The key result was that the voltage gap, V c , forthe onset of conduction arises from collective depinningof Cooper-pair quasicharge, and is inversely related tothe localisation length as calculated by Giamarchi andSchultz[16]. We also found V c to be proportional to thenumber of junctions in the chain, N , and decreasing asa power law in bandwidth, W . The Bloch bandwidth W is prescribed by the single-junction theory of quasichargeenergy bands[6, 18], and can be envisioned as the ampli-tude for coherent tunneling of flux quanta. W decreasesexponentially with p E J /E C .In this work, we examine the dependence of V c on both parallel and perpendicularly-applied magneticfields, finding a non-monotonic dependence of the crit-ical voltage with magnetic field. Below an orientation-dependent cross over field, B ∗ , the critical voltage V c in-creases with field in accordance with increasing W , whichoccurs by suppression of the superconducting gap, ∆,and hence decreasing E J . Above B ∗ , however, the crit-ical voltage decreases with field, until finally becomingindependent of magnetic field above the superconduct-ing critical field B c .We show that the peak behavior in the critical voltage,along with the change in current-voltage characteristics(IVC’s) at B ∗ reveal a parity crossover where collectivedepinning of Cooper pairs gives way to that of single-electron charges. Moreover, when the field is appliedperpendicular to the island films, the parity effect canbe accompanied by a change in vorticity of the supercon- TABLE I. DevicesDevice E C ( µ eV) E J0 ( µ eV) g ~B B ∗ (mT) B (mT) A J ( µ m) AS7, N=250 95.2 87.9 0.23 k
310 455 0.0110BS1, N=250 114 60.2 0.13 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ k
332 455 0.0121CS5, N=250 64.8 230 0.89 k
346 434 0.0145CS6, N=250 60.4 275 1.13 k
355 426 0.0161DS1, N=250 68.0 223 0.82 ⊥ k
362 401 -LS2, N=200 57.5 44.1 0.19 k
364 417 - ducting islands. In other words, the parity effect occursin sync with formation of the single-vortex state. Chargetransport above B ∗ for both geometries is found to bemarkedly different from that below, lending additionalsupport to our interpretation of the insulating peak asan experimental signature of an even-odd parity transi-tion.The interplay of parity and vortex dynamics hasbeen discussed theoretically by Khaymovich et al. [19]for single-island devices in the context of chargepumping, and single-vortex trapping was observed ex-perimentally in hybrid normal-superconducting-normaltransistors[20].Although the parity effect in superconducting single-charge transistors and Cooper-pair boxes has been wellknown for some years, it has received very little attentionin studies of Josephson junction arrays. A theoreticalpaper by Feigel’man et al. [21] pointed out the implica-tions of the parity effect on the experimental search forthe Kosterliz-Thouless charge-unbinding transition. Theparity effect for chains in the sequential tunneling limit, E J ≪ E C , was studied theoretically by [22], inspiredby the experimental results of Bylander et al. [23]. De-tailed treatments of parity effects in junction arrays with E J ∼ E C , two-dimensional disordered superconductingfilms, and one-dimensional superconducting nanowires,however, are conspicuously absent.We have fabricated and measured a large ensembleof Al/AlO x /Al single-junction chains[1]. Several fami-lies of devices were produced, where within each fam-ily, we varied the junction area, A J , in order to geo-metrically tune the ratio g = E J0 / E C , where E J0 isthe Josephson tunneling energy at zero magnetic field(see SM[24]). For each device, we first obtain an ac-curate measure of the average junction charging energy, E C , from the voltage offset, V off , of each device foundfrom extrapolating its linear current-voltage characteris- -12-11-10-9 -12-11-10-9-8 B*=0.35 T B*=0.057 T (d)(c)(b)(a)
FIG. 1. Measured current on a logarithmic scale, (log | I | ),versus bias voltage V and magnetic field for (a) device C5 in aparallel magnetic field B || , and (b) for device B6 in a perpen-dicular field B ⊥ . Critical voltage V c ( B || found for device C5(c), and V c ( B ⊥ for device B6 (d). The parity transition occursat B ∗ , the field where critical voltage, V c , is maximum. Thecritical field B c is identified as the field at which V c becomesindependent of field. tic (IVC) from large voltage biases, V ≫ N ∆ /e , where∆ is the superconducting gap in zero magnetic field, and N is the number of junctions in the chain. As noted in[2, 24, 25], the experimentally determined charging en-ergy is found as, E C = eV off /N , and the junction Joseph-son energy E J0 across the chain is found from the nor-mal state conductance using the Ambegaokar-Baratoffrelation, E J0 = ∆ R Q / R N , where R Q is the supercon-ducting quantum of resistance, and R N is the junctionresistance in the normal state The experimentally deter-mined device parameters are listed in Table I.Next we measured the critical voltage V c , deep inthe subgap region, V ≪ N ∆ /e . In addition to thezero-field measurements, IVC’s for some devices werealso measured with varying parallel, or perpendicularly-applied magnetic fields. Most of the measured deviceshave non-hysteretic IVC’s in this region, for all values ofapplied field, however, a few devices exhibited hystereticIVC’s for some values of magnetic fields. In this work,we are interested in using the voltage gap as a probeof the parity effect. We therefore take V c as the returnvoltage, that is, the voltage magnitude at which the de-vice returns to the zero-current state (current less thanthe noise floor), when stepping from the non-zero currentstate. The return voltage is found in all cases to be char-acterized by an extremely narrow distribution (smallerthan the experimental resolution). We note that unlikethe situation for nanowires and films based on disorderedsuperconducting films[11], the voltage gaps we observeare very hard: we do not find an observable zero-bias re-sistance arising from a postulated parallel quasiparticleconductance channel, and therefore we have no need tosubtract a finite current, V /R N , in order to observe arobust critical voltage. This indicates that our junctionchains are significantly more homogeneous than devicesbased on disordered films.Figure 1, (a) and (b), show logarithmic scale (log | I | )image plots of the IVC’s for stepped magnetic fields fordevices C5 and B6. Device C5 was measured in a parallelmagnetic field, while device B6 was measured in a per-pendicular field. Both devices were fabricated during thesame fabrication run in neighboring squares on the samechip. The critical voltage for each value of magnetic fieldis identified as the voltage magnitude for which the mea-sured current becomes less that the noise floor for themeasurement, which is . B ∗|| (or B ∗⊥ ), by fitting to few points of the experimentallydetermined V c around it’s maximum.From Figure 2, one notes that the transport data forboth parallel and perpendicular fields are remarkably dif-ferent above and below their respective parity fields. Thiscan be seen more directly in the individual IVC’s, asshown for example for Fig. 2, where we plot data fordevice C5 at magnetic fields both above and below B ∗|| ,fields where V c is approximately equal. We note that allmeasured devices show qualitatively different transportcharacteristics above compared to below their B ∗ ’s (seeSupplemental Material[24] for more examples).In Figure 2 (lower plot), it is evident that for B || = 0 . < B ∗ ), the conductance dI/dV is stronglypeaked just above ± V c , and strictly decreasing with | V | outside the voltage gap. This supercurrent-like featureis indicative of Cooper-pair dominated transport. Con-versely, for B || = 0 .
47 T ( > B ∗ ), the conductance in-creases monotonically outside of the voltage gap. Wenote that both sets of data asymptotically approacheach other. This can be understood qualitatively, as be-low B ∗ , increasing charge transport invariably populateshigher bands in quasicharge via Landau-Zener tunneling.The ensuing relaxation from higher bands produces BCSquasiparticles, which eventually suppress the even-odd -1 0 1 200.10.20.30.4-1 0 1 2-505
FIG. 2. Transport data I vs. V (upper plot), and dI/dV vs. V (lower plot) for device C5 at selected values of paral-lel magnetic field: below the parity field, B || = 0 . B ∗|| = 0 .
35 (grey), and above the parity field, B || = 0 .
47 T. Although V c is approximately equal at 0.1 Tand 0.47 T (inset), the transport characteristics are foundto be substantially different. The conductance above V c de-creases with V for fields below B ∗ as expected for Cooper-pairdominated transport, and increases with V for B || above B ∗ ,indicative of single-electron transport. Similar behavior is ob-served for perpendicularly-applied fields (see SM). free energy difference that permits Cooper-pair tunnel-ing to dominate.In Figure 3, we plot the experimentally determinedparity fields for six devices in parallel magnetic fields(squares, left axis), and seven devices in perpendicularmagnetic fields (diamonds, right axis), as a function oftheir experimentally determined single-electron chargingenergy. First we focus on the parallel field data. Thesolid line follows from suppression of the superconduct-ing gap according to ∆ = ∆ (cid:0) − B /B (cid:1) , which wepreviously found adequate to describe suppression of thesuperconducting gap in comparably-sized aluminum is-lands in a parallel magnetic fields[2]. Setting E C = ∆,at B ∗ , one finds B ∗ = B p − E C / ∆ . B is known asthe pair-breaking parameter, which is expected to be ofthe same order as the critical field B c . The solid line rep-resents the best fit to the parallel field data, resulting in
60 80 100 1200.250.30.35 0.050.060.07
13 14 15 16505560 || FIG. 3. Measured parity field B ∗ versus experimentally deter-mined single-electron charging energy, E C , for devices mea-sured in parallel (squares, left axis), and perpendicular (dia-monds, right axis), magnetic fields. The solid black line fol-lows from suppression of the homogeneous superconductinggap, ∆( B ) = ∆ [1 − ( B/B ) ], and setting, ∆( B ∗ ) = E C , tofind B ∗ . The dashed line is a fit to B ∗ = f Φ /A I (using E C asa proxy for 1 /A I ), where A I = wl , w is the island film width,and l , the between-junction island length. A fit to perpendicu-lar field devices having E C < µ eV, yields f = 2 . ± . B ∗⊥ vs. A − I for these devices and the fitdirectly. B = 0 .
42 T, which is comparable to the average criticalfield found for these devices, B c = 0 .
48 T.An alternative method to estimate the depairing pa-rameter, is to fit V c ( B ) at fields below B ∗ to the scalinglaw behaviour of V c ( W ), as detailed in Ref.[1]. Since V c depends on E J through W , the dependence of V c on ∆and hence B can be found assuming E J to be given bythe Ambegaokar-Baratoff formula, resulting a field de-pendence E J ( B ) = E J0 (cid:0) − B /B (cid:1) (see SupplementalMaterial[24]). Averaging over the parallel field devices(see Table I), we find B = 0 . ± .
02 T, which agreesextremely well with B = 0 .
42 T found above.Considering now the devices measured in a perpendic-ular field, it is clear that B ∗⊥ ( E C ) does not follow thedependence, B p − E C / ∆ , in particular, for the fivedevices with E C < µ eV. We can, however, estimatethe depairing parameter, B , as discussed in the preced-ing paragraph. Averaging over the perpendicular field de-vices, we arrive at B = 0 . ± . B c = 0 . B = 0 .
086 T. The experimental de-pendence of B ∗⊥ on E C for devices with E C . µ eVis nearly orthogonal to that predicted by the black line.This suggests a different mechanism driving the even-odd parity effect: in a perpendicular field, the parity effect can occur in sync with trapping of a single fluxquanta. Concomitant with formation of the vortex state,the average superconducting gap across the island be-comes nearly zero, destroying the free energy differencebetween even and odd parity.Extensive studies, both theoretical and experimental,have been reported detailing the vortex states of meso-scopic superconducting islands, typically within the con-text of finding eigenvalues of the linearised Ginzburg-Landau equations[26–31]. For a thin superconductingdisc of radius, R , it is found, [28, 29], that the single vor-tex state becomes energetically favourable for an appliedflux Φ ≃ f Φ , or B = f Φ /πR , with f = 1 . f ≃ . l ≃
890 nm, thickness d = 30 nm, and varying width (wevaried the junction area across each family to modulatethe ratio, g = E J /E C , while using the same oxidationparameters[1]). We have analysed SEM images of our de-vices finding device-averaged junction widths, w , rangingfrom 80 to 120 nm. We find that in accordance with aparallel plate capacitor model of the junctions, the charg-ing energy indeed scales inversely with device-averagedjunction area, A J = w . For the family of devices mea-sured in perpendicular field we find, E C = 0 . /A J , with A J in µ m [24]. This also gives us a relation betweencharging energy and average island area, since the islandwidths equal those of the junctions, A I = w (0 . − w ) µ m .If we neglect the region of the islands making up thejunction, i.e. we take for the island area, A I , the areabetween the junctions, remarkably, we find that we canfit the larger area devices ( E C . µ eV), measured inperpendicular fields, to a single curve, B ∗⊥ = f Φ /A I ,with f = 2 . ± .
05, as show by the dashed lines inFigure 3 (main plot and inset).In the dirty limit, which applies to our films, the co-herence length is given by ξ = p ~ D/ ∆ , where D isthe diffusion coefficient, which can be deduced from theconductivity, σ , using Einstein’s relation, σ = N (0) e D ,where N (0) = 2 . × J − m − is the density of statesat the Fermi level for aluminum, and e is the electroncharge. From the measured resistivity of our 30 nm films,we estimate a coherence length, ξ ∼
60 nm. The criticalfield for a large film, B c = Φ / πξ , for ξ ∼
60 nm givesan estimate B c ≃
90 mT, which roughly agrees with theobserved critical fields of our devices in perpendicularfields.For thin films in parallel applied fields, calculationsbased on Ginzburg-Landau theory find that the vortexstate can only occur when the thickness, d , is greaterthan 1 . ξ [12, 32]. This corresponds well to our paral-lel field results. Since d < ξ , we see only homogenoussuppression of the superconducting gap up to the parityfield where ∆( B ∗ ) = E C . For perpendicular fields, analy-ses of the vortex states of mesoscopic islands based on thelinearized Ginzburg-Landau equations show that the sin-gle vortex state is stable only for film widths greater thana critical value w c ∼ ξ [29, 30]. The measured widthsof our devices appear to cross this borderline width. Forsuch strong confinement, ( w & ξ ), however, an analy-sis based on the nonlinear Ginzburg-Landau equations,or the Bogoliubov-de Gennes equations, may be requiredfor a more quantitative comparison.In conclusion, we find that the non-monotonic depen-dence and peak in the voltage gap of insulating Joseph-son junction chains with magnetic field arises from aneven-odd parity effect. Below the parity field, the groundstate of our insulating junctions chains is that of a one-dimensional Bose glass of localized Cooper-pairs, withthe onset of transport arising from depinning of the com-pressible Cooper-pair quasicharge[1]. Above B ∗ , oddparity quasicharge bands become accessible, so that thenature of the ground state becomes that of a Fermi glass,and depinning involves single-electron charges ratherthan Cooper pairs. Moreover, in a perpendicular fieldthe transition can occur simultaneously with trapping ofa single flux quantum in the thin-film superconductingislands.Our results are relevant for the insulating peak ob-served in disordered superconducting films, and stripspatterned from such films, which are postulated toform a network of Josephson-coupled superconductingdroplets[7–11]. A current explanation for such data isthe formation of random SQUID loops in the network[13].Our results suggest such a peak arises from an even-oddparity effect, and may occur with formation of the vortexstate in the effective superconducting droplets. We sug-gest that even-odd parity effects could also be observedin superconducting nanowires.TD acknowledges useful discussions with AlexanderShnirman. This work was supported by the ARCCentre of Excellence for Engineered Quantum Systems,CE11000101. Devices were fabricated at the UNSWNode of the Australian National Fabrication Facility.JHC is supported by the Australian Government’s NCINational Facility through the National ComputationalMerit Allocation Scheme, and the ARC Centre of Ex-cellence in Future Low-Energy Electronics Technologies(FLEET), CE170100039. ∗ corresponding author, [email protected].[1] D. V. Averin and Yu. V. Nazarov, Phys. Rev. Lett. ,1993, (1992).[2] M. T. Tuominen, J. M. Hergenrother, T. S. Tighe andM. Tinkham, Phys. Rev. Lett. , 1997, (1992).[3] A. Amar, D. Song, C. J. Lobb, and F. C. Wellstood, Phys.Rev. Lett. , 3234, (1994).[4] P. Lafarge, P. Joyez, D. Esteve, C. Urbina, and M. H. De- voret. Phys. Rev. Lett. , 994, (1993).[5] P. Lafarge, P. Joyez, C. Urbina, and M. H. Devoret, Na-ture , 422 (1993).[6] S. M. Albrecht, A. P. Higginbotham, M. Madsen, F.Kuemmeth, T. S. Jesperson, J. Nyg˚ard, P. Krogstrup, andC. M. Marcus, Nature , 207 (2016).[7] T. I. Baturina, A. Yu. Mironov, V. M. Vinokur, M. R. Bak-lanov, and C. Strunk, Phys. Rev. Lett. , 257003 (2007).[8] Valerii M. Vinokur, Tatyana I. Baturina, Mikhail V. Fis-tul, Aleksey Yu. Mironov, Mikhail R. Baklanov, andChristoph Strunk, Nature , 613 (2008).[9] H. Q. Nguyen, S. M. Hollen, M. D. Stewart, Jr., J. Shain-line, Aijun Yin, J. M. Xu, and J. M. Valles, Jr., Phys.Rev. Lett. , 157001 (2009).[10] M. Ovadia, D. Kalok, I. Tamir, S. Mitra, B. Sac´ep´e, andD. Shahar, Sci. Rep. , 13503 (2015).[11] I. Schneider, K. Kronfeldner, T. I. Baturina, and C.Strunk, arXiv:1806.01335, (2018).[12] Michael Tinkham, Introduction to Superconductivity ,Second edition, Dover, New York (2004).[13] M. V. Fistul, V. M. Vinokur, and T. I. Baturina, Phys.Rev. Lett. , 086805[14] K. Cedergren, S. Kafanov, J.-L. Smirr, J. H. Cole, andT. Duty, Phys. Rev. B , 104513 (2015).[15] Karin Cedergren, Roger Ackroyd, Sergey Kafanov, Nico-las Vogt, Alexander Shnirman, and Timothy Duty, Phys.Rev. Lett. , 167701 (2017).[16] T. Giamarchi and H. J. Schulz, Phys. Rev. B , 325(1988).[17] K. K. Likharev and A. B. Zorin, J. Low Temp. Phys. ,347 (1985).[18] Samuel A. Wilkinson a, Nicolas Vogt, Dmitry S. Golubev,and Jared H. Cole, Physica E , 24 (2018).[19] I. M. Khaymovich, V. F. Maisi, J. P. Pekola, and A. S.Mel’nikov, Phys. Rev. B , 020501(R) (2015).[20] M. Taupin, I. M. Khaymovich, M. Meschke, A. S.Mel’nikov, and J. P. Pekola, Nature Comm. , 10977(2016).[21] M. V. Feigel’man, S. E. Korshunov, and A. B. Pugachev,Soviet Physics JETP , 566 (1997).[22] Jared H. Cole, Andreas Heimes, Timothy Duty, andMichael Marthaler, Phys. Rev. B , 184505, (2015).[23] Jonas Bylander, Tim Duty, G¨oran Johansson, and P.Delsing, Phys. Rev. B , 020506(R) (2007).[24] Supplemental Material [URL].[25] T. S. Tighe, M. T. Tuominen, J. M. Hergenrother, andM. Tinkham, Phys. Rev. B , 1145, (1993).[26] O. Buisson, P. Gandit, R. Rammal, Y. Y. Wang, and B.Pannetier, Phys. Lett. A , 36 (1990).[27] A. K. Geim, I. V. Grigorieva, S. V. Dubonos, J. G. S. Lok,J. C. Maan, A. E. Filippov, and F. M. Peeters, Nature , 259 (1997).[28] Robert Benoist and Wilhelm Zwerger, Zeit. Phys. ,377 (1997)[29] V. A. Schweigert and F. M. Peeters, Phys. Rev. B ,13817 (1998).[30] Liviu F. Chibotaru, Arnout Ceulemans, Vital Bruyn-doncx, and Victor V. Moshchalkov, Nature , 833(2000).[31] L. F. Chibotaru, A. Ceulemans, M. Morelle, G. Teniers,C. Carballeira, and V. V. Moshchalkov, Journal of Math-ematical Physics , 095108 (2005).[32] H. J. Fink, Phys. Rev. , 732 (1969). Supplemental material to:“Magnetic-field-induced parity effect ininsulating Josephson junction chains”
DEVICES
Al/AlO x /Al junction chains were fabricated by standarde-beam lithography followed by two angle evaporation ofaluminum with in situ oxidation between the two evapo-ration steps. Each film is 30 nm thick. Devices were fabri-cated on silicon substrates (n-doped) with approximately300 nm thermally grown SiO on top. Six devices weremeasured in magnetic fields parallel to the substrate, and7 devices in magnetic fields perpendicular to the substrate.The experimentally determined and parameters for the de-vices are listed in Table I.The chain average single-electron, single-junction chargingenergy, E C = e / C J , where C J is the average junctioncapacitance, is extracted from the voltage offset, V off , ofeach device, found from extrapolating its linear IVC fromlarge voltage bias. This is a standard procedure which hasbeen used previously by many groups, and is described in[1–5]. The experimentally determined charging energy isfound as, E C = eV off /N , where N is the number of junc-tions in the chain. One must take care that V off is extrap-olated from sufficiently high voltages above the Coulombblockade where the conductance dI/dV is measured to beconstant, in order to get an unbiased estimate of E C .The average junction Josephson energy in zero field, E J0 ,is determined from the linear conductance at large voltagebias, eV ≫ N ∆, using the Ambegaokar-Baratoff relation E J0 = ∆ R Q R J , where R Q is the superconducting resistancequantum, and R J the inverse of the large-bias linear con-ductance G Ω of the array divided by N . Here ∆ is takento be 210 µ eV, as found in [2].The experimentally determined parity field is denoted B ∗ ,and the depairing parameter, B , as determined from afit to the critical voltage V c ( B ) for B < B ∗ , assuming E J = ∆( B ) R Q G Ω /
2, with ∆( B ) = ∆ (1 − B /B ), usingthe scaling of V c with Bloch bandwidth W (which dependson E J and E C ), detailed in Ref. [1], described in moredetail below. CRITICAL VOLTAGE
Following the characterisation of the Coulomb and Joseph-son energy scales for each device, we measure the criticalvoltage V c , at biases deep in the subgap region, eV ≪ N ∆, for varying magnetic fields. Devices with low valuesof g typically have non-hysteretic IVC’s in this region. Atlarger values of g , some devices exhibit hysteretic IVC’sfor some values of magnetic field. In this work, we are in-terested in using the voltage gap as a probe of the parityeffect. We therefore take V c as the magnitude of voltage atwhich the device returns to the zero-current state (currentless than the noise floor), when stepping from the non-zerocurrent state, since the return voltage is found in all casesto be characterized by an extremely narrow distribution(smaller than the experimental resolution). TABLE II. DevicesDevice E C ( µ eV) E J0 ( µ eV) g ~B B ∗ (mT) B (mT) A J ( µ m) AS7, N=250 95.2 87.9 0.23 k
310 455 0.0110BS1, N=250 114 60.2 0.13 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ k
332 455 0.0121CS5, N=250 64.8 230 0.89 k
346 434 0.0145CS6, N=250 60.4 275 1.13 k
355 426 0.0161DS1, N=250 68.0 223 0.82 ⊥ k
362 401 -LS2, N=200 57.5 44.1 0.19 k
364 417 -
JUNCTION AREA DETERMINES CHARGINGENERGY
As described in Ref. [1], we varied the junction area acrossdifferent chains, in order to vary the coupling constant g .The junctions of our device can be considered as parallelplate capacitors, that is, the junction capacitance, C J , isproportional to the junction area, A J = w , where w is thewidth of the junctions (and islands) making up the device.We have analysed scanning electron microscope (SEM) im-ages of most of the devices we have measured to extractdevice-averaged junction widths, and correlated the av-erage junction areas with measurement of V off /N , whichprovides an experimental estimate of E c , or equivalently, C J . In Figure S1, we plot both the experimentally deter-mined C J vs. junction area, A J (top plot), or alternatively, E C vs. the inverse of the area, A − J (bottom plot). We findclear linear relations between these quantities, which con-firms the validity of the parallel plate capacitance model.For the purpose of comparing data and theory, this alsoallows us to express the device-averaged junction (and is-land) width, w , from the measured charging energy, E C ,as w − = 1 . × E C . CURRENT-VOLTAGE CHARACTERISTICS FORVARYING MAGNETIC FIELDS
In Figure 2 of the main text, we presented detailed current-voltage characteristics (IVC’s) for device C5, measuredin parallel magnetic fields, at field values below, at, andabove B ∗ . In Fig. S2 and Fig. S3 we show similar datafor device C3 in parallel magnetic fields, and device B6in perpendicular magnetic fields. From Table I, one seesthat these devices have quite similar values for chargingenergy E C , as well as Josephson energy, E J . Correspond-ingly, the transport data for these devices is remarkablysimilar, although the applied parallel fields for device C3are approximately six times higher than for device B6 inperpendicular fields.Clearly the IVC’s for both parallel and perpendicular fieldsare remarkably different above and below their respective FIG. S1. (Upper plot) Experimentally determined C J asfound from C J = e N/V off , as a function of junction areas, A , extracted from SEM images. (Lower plot) The same data,but shown as E C = eV off /N versus A − J .parity fields, B ∗ . For fields lower than their respective par-ity fields (blue data), the conductance dI/dV is peakedjust above ± V c , and strictly decreasing with | V | outsidethe voltage gap. This can be understood as follows: (coher-ent) Cooper-pair dominated transport occurs more read-ily for energy differences approaching zero, which single-electron tunnelling involves breaking Cooper pairs, whichincreases with voltage bias. ESTIMATE OF B FROM V c ( B )We assume that the homogenous superconducting gap,∆( B ), is suppressed by an applied magnetic field as,∆( B ) = ∆ (1 − B /B ), where B is the depairingparameter. Using the Ambegaokar-Baratoff relation, the -1 0 1 2-0.200.20.40.6 -2 -1 0 1 200.020.040.06 B || = 200 mTB || = 332 mTB || = 400 mT FIG. S2. Transport data I vs. V (upper plot), and dI/dV vs. V (lower plot) for device C3 at selected values of parallelmagnetic field: below the parity field, B || = 200 mT (blue), atthe parity field, B ∗|| = 332 mT (black), and above the parityfield, B || = 400 mT (red). Although V c is approximately equalfor blue and red data (inset), the transport characteristics arefound to be substantially different. The conductance above V c decreases with V for fields below B ∗ , as expected for Cooper-pair dominated transport, and increases with V for B || above B ∗ , indicative of single-electron transport. Similar behavioris observed for perpendicularly-applied fieldsfield-dependent Josephson energy is given by E J ( B ) =∆( B ) R Q / R J . The single-junction Bloch bandwidth, W ,is the width of the lowest energy band in quasicharge fora current-biased junction [6], which depends solely on E C and E J , and can readily be calculated numerically for ar-bitrary E C and E J . For E J ≫ E C , W = 16 √ π ~ ω p (2 g ) / e −√ g , (1)where g = E J / E C , and ~ ω p = √ E J E C . In Ref. [1], itwas shown that the scaled critical voltage, v = V c /N ~ ω p ,was set by the scaled, single-junction Bloch bandwidth, w = W/ ~ ω p , as v = aw α , (2)where a is a prefactor, and the exponent α = 4 / − K ,where K is the Luttinger parameter, K = Λ − p E J / E C ,and Λ is the screening length. -1 0 1 2-0.200.20.40.6 -2 -1 0 1 200.020.040.06 B = 32 mTB = 62 mTB = 57 mT
FIG. S3. Transport data I vs. V (upper plot), and dI/dV vs. V (lower plot) for device B6 at selected values of perpendicu-lar magnetic field: below the parity field, B ⊥ = 32 mT (blue),at the parity field, B ∗⊥ = 57 mT (black), and above the parityfield, B ⊥ = 62 mT (red). Although V c is approximately equalfor blue and red data (inset), the transport characteristics arefound to be substantially different. The conductance above V c decreases with V for fields below B ∗ , as expected for Cooper-pair dominated transport, and increases with V for B || above B ∗ , indicative of single-electron transport. Similar behavioris observed for perpendicularly-applied fields One can use these relations, along with the measured V c ( B ), for magnetic fields B < B ∗ , to estimate the de-pairing parameter, B . This is shown in Figs. S2 and S3,for respectively, parallel and perpendicular applied fields.For the fitting we have used a two parameter fit (prefactor a and depairing parameter B ), with the experimentallydetermined E J and E C for each device as fixed parameters.The extracted depairing parameters are listed in Table Iand II above. ADDITIONAL DETAILS
Additional experimental details can be found in the Sup-plemental Material of Cedergren et al. , Ref. [1]. ∗ corresponding author, [email protected].[1] Karin Cedergren, Roger Ackroyd, Sergey Kafanov,Nicolas Vogt, Alexander Shnirman, and Timothy Duty,Phys. Rev. Lett. , 167701 (2017).[2] K. Cedergren, S. Kafanov, J.-L. Smirr, J. H. Cole, andT. Duty, Parity effect and single-electron injection forJosephson junction chains deep in the insulating state ,Phys. Rev. B , 104513 (2015).[3] T. S. Tighe, M. T. Tuominen, J. M. Hergenrother,and M. Tinkham, Measurements of charge soliton mo-tion in two-dimensional arrays of ultrasmall Josephsonjunctions , Phys. Rev. B , 1145 (1993).[4] L. S. Kuzmin, P. Delsing, T. Claeson and K. K.Likharev, Single-Electron Charging Effects in One-Dimensional Arrays of Ultrasmall Tunnel Junctions ,Phys.Rev. Lett. , 2539 (1989).[5] P. Delsing, in Single Charge Tunneling , H. Grabert andM. H. Devoret, eds, pp. 249-274, New York, Plenum,1992.[6] K. K. Likharev and A. B. Zorin, J. Low Temp. Phys. , 347–382 (1985). A7N=200N=100C3
C5C6
FIG. S4. Estimate of the depairing parameter, B , for devicesmeasured in a parallel magnetic field by fitting to Eqn. (2) asdescribed in the text. We have used two fit parameters, theprefactor a , and B , with E C and E J experimentally deter-mined from large scale IVC’s as fixed parameters. See TableI for values of B determined from the fits. B1B2B3B4B6
B7D1
FIG. S5. Estimate of the depairing parameter, B , for devicesmeasured in a perpendicular magnetic field by fitting to Eqn.(2) as described in the text. We have used two fit parame-ters, the prefactor a , and B , with E C and E J experimentallydetermined from large scale IVC’s as fixed parameters. SeeTable I for values of B0