Current phase relations of few-mode InAs nanowire Josephson junctions

Transcription

1 In the format provided by the authors and unedited. Current phase relations of few-mode InAs nanowire Josephson junctions Eric M. Spanton 1,2, Mingtang Deng 3,4, Saulius Vaitiekėnas 3,5, Peter Krogstrup 3, Jesper Nygård 3, Charles M. Marcus 3 and Kathryn A. Moler 1,2,6,* 1 Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA, USA 2 Department of Physics, Stanford University, Stanford, CA, USA 3 Center for Quantum Devices and Station Q Copenhagen, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark 4 State Key Laboratory of High Performance Computing, NUDT, Changsha, 4173, China 5 Department of Physics, Freie Universität Berlin, Arnimallee 14, Berlin, Germany 6 Department of Applied Physics, Stanford University, Stanford, CA, USA * Corresponding Author: kmoler@stanford.edu June 16, 217 NATURE PHYSICS 1

2 Supplementary Figures.3 5 a A 1 (na) Figure S1: Shape parameter (a 2 ) vs. CPR amplitude (A 1 ) Fitted a 2 vs. A 1 for many rings at V BG = V and T<5 mk. The CPRs fitted here are the same CPRs presented in Fig. 5. NATURE PHYSICS 2

3 Figure S2: Scanning electron micrograph of the gated nanowire measured in Figs. 2-4 The length and diameter of the nanowire junction are L= 15 nm and D = 5 nm, respectively. Scale bars are 2 µm in the main figure and 2 nm in the inset. NATURE PHYSICS 3

4 Supplementary Table Table S1: Nanowire Josephson junction length (L) and diameter (D) for CPRs presented in Fig. 5a ordered from lowest to highest A 1 (bottom to top in Fig. 5a. Unknown dimensions are due to the rings being damaged between the scanning SQUID measurements and scanning electron micrographs were taken. L (nm) D (nm) unk. unk unk. unk. unk unk. unk unk. unk unk. unk unk. unk. unk. unk NATURE PHYSICS 4

5 Supplementary Section 1: Temperature dependence of the CPR We measured the CPR of the gated nanowire junction at fixed V BG (5.55V ) as a function of temperature (Fig. S3). The CPR s amplitude goes to zero between T= 1.2 K and 1.4 K (Fig. S3 b). The diamagnetic response of the Al ring also goes to zero at T=1.4 K (not shown). At temperatures close to T c, the CPR was sinusoidal, and higher harmonics only contributed to the CPR s shape at lower temperatures (Fig. S3 a,c). The temperature dependencies of CPRs for many nanowires in all three samples show similar behavior: A reduction of both A n and a n at high temperatures. The fitted shape parameter a 2 of the CPR vs. V BG showed reduced fluctuations at higher temperatures (Fig. S4). The average value of a 2 at high gate voltages was also reduced with elevated temperature. NATURE PHYSICS 5

6 A n (na) a A 1 A 2 A 3 A n (na) T (K) b T (K) 1. a n c a 2 a T (K) Figure S3: Temperature dependence of CPR at fixed V BG (a,b) A n and (c) a n vs. temperature at fixed V BG =5.55V. Error bars are 9% confidence intervals obtained by bootstrapping 1 times. NATURE PHYSICS 6

7 a 2 a 2 a T =.3 K T = 5 K T =.5 K V BG - V Onset (V) Figure S4: Gate voltage dependencies of the forward-skewness at elevated temperatures. The shape of the CPR (as captured by a 2 ) vs. V BG - V Onset. V Onset is the gate voltage at which the first non-zero CPR was observed. V Onset = V, -1.8 V, and -1.9 V for 3 mk, 25 mk, and 5 mk respectively. NATURE PHYSICS 7

8 Supplementary Section 2: Fitting the most forward-skewed mode We fit the CPR shown in Fig. 3c to the expression: I(φ) =(1+ɛ) e (T ) 2 h τsin(φ φ ) [1 τsin 2 ((φ φ )/2)] 1/2 tanh( (T ) 2k B T [1 τsin2 ((φ φ )/2)] 1/2 ), (S1) We allowed T to vary due to possible electron heating effects which would not be detected by our mixing chamber thermometer. In previous measurements in our scanning SQUID system in a dilution refrigerator, the electron temperature of devices in a similar geometry was estimated to be 1 mk [S1]. A fit with fixed T =.3 K and ɛ = resulted in large residual structure (Fig. S5 a,b), while a fit with T =.3 K and ɛ free to vary gave better results (Fig. S5 c,d), although this fit is still qualitatively worse than the fit presented in the main text. We plotted χ 2 N k=1 (I k I fit )2 /N vs. τ and T fit to determine the behavior of errors k of our fit (Fig. S5 e). We found that the global minimum is at T =.13 K and τ =1., as stated in the main text. Doubling of χ 2 takes place on the intervals τ = [.97, 1.] and T fit = [.3,.13], confirming that we have observed a CPR which is consistent with a single, perfectly transmitting Andreev bound state. To confirm the claim of a single Andreev bound state, we fit to a version of Eqn. S1 with two modes with transmissions τ 1,2 (Fig. S5 f). We find that the χ 2 minimum is achieved by setting one mode to τ = 1. and the other to τ =., confirming that the CPR is of a single, close to perfectly transmitting mode. NATURE PHYSICS 8

9 I RING (na) residual (na) I RING (na) residual (na) a b c d 2π π π 2π (rad) 2π π π 2π (rad) T =.3 K T c = 1.4 K τ = 1. ε =. T =.3 K T c = 1.4 K τ fit =.98 ε =.12 T fit (K) τ e f...4 τ χ 2 (na 2 ) χ 2 (na 2 ) τ 1 Figure S5: Results of fits to the most forward-skewed CPR We fitted the CPR presented in Fig. 3c to a single-mode short junction expression (equation (S1)). (a,b) Fit and resulting residual with fixed T =.3 K and no amplitude scaling factor (ɛ = ). This fit, which has obvious residual structure, results in a fitted τ =1.. (c,d) Fit and resulting residual with fixed T =.3 K and a free amplitude scaling factor (ɛ). This fit has has much smaller residuals than (b) and results in a fitted τ =.98. A fit with free T and ɛ (Fig. 3c,d) gives no obvious residual structure (e) A false color plot of χ 2 N k=1 (I k I fit k )2 /N vs. τ and T fit. We allowed ɛ and φ to vary for each pixel. (f) χ 2 plot of a two-mode fit to the same data. τ 1 and τ 2 were fixed for each pixel, while all other parameters were left free. The global minimum is at τ 1,2 =., 1.. The colorscales in e,f were saturated at two times the observed minimum in χ 2. NATURE PHYSICS 9

10 Supplementary Section 3: Phase shifts in CPRs in Fig. 5 We encountered phase shifts in our measurements of the CPR, all of which we believe to be instrumental or trivial in nature. For measurements on sample B (Fig. 5), phase shifts between higher harmonics on the same sweep were observed. To account for this, for Fig. 5 we fit to the expression: N n=1 I(φ) = A nsin(n(φ + φ FW n )) if dφ/dt >, (S2) N n=1 A nsin(n(φ + φ BW n )) if dφ/dt < which allows for phase shifts between harmonics. Fits to equation (S2) yielded better fits for the CPRs in Fig. 5 (Fig. S6 c,d). The shape of the CPR in the most forward-skewed CPRs measured on sample B was different between forward and backward sweeps, owing to this effect (Fig. S6 a,b). The changes in phase for higher harmonics were opposite signs for forward and backward sweeps, indicating that filtering of the signal was to blame. Fits without phases (equation (3) in main text) were used for Fig Fits to equation (S2) did not change the measured harmonics A n by more than couple percent even for the highest harmonics, no out of phase component was observed in the FFT (see Fig. 1e), and forward and backward sweeps did not show different shapes. This was in part due to careful tuning of the the PI controller in measurements of sample C, which were not performed for the data taken in Fig. 5. NATURE PHYSICS 1

11 I RING (na) a c I RING (na) b residuals (na) d -1 2π π π 2π 2π π π 2π (rad) φ φ 1 (rad) -1 - e φ bw φ n (rad) n φ fw Figure S6: Different phases between harmonics are required to correctly fit some CPRs (a,b) Forward and backwards sweep of the same CPR from an L=5 nm ring on sample B. The positive and negative peaks have different shapes, and the shape depends on sweep direction, indicating phase shifts which are different depending on the sweep direction. (c,d) Example of a free phase fit (Eq. S2, blue line) to the forward sweep and the residuals. (e) The fitted phases of the harmonics vs. harmonic number n. NATURE PHYSICS 11

12 Supplementary Section 4: Hysteretic behavior Many rings were in the hysteretic regime at low temperatures (e.g., Fig. S7). For the majority of the rings where hysteresis was observed, obvious underetching of the nanowire resulted in Al bridging the would-be junction (Fig. S7 inset). The open shape of the measured response of the ring is due to the presence of multiple local minima in the Josephson energy, which arise only in the when β 2πL self I c /Φ > 1 [S2]. The height of the individual jumps is given by I jump =Φ /2L self, while total maximum current is just given by the critical current I C. The height of the jumps was similar for many rings and consistent with the calculated self inductance of 13.9 ph. The overall height of the hysteretic pattern varied from ring to ring, indicating that the critical current of the junctions varied. NATURE PHYSICS 12

13 5 Φ SQ (mφ ) I FC (ma) Figure S7: Example of a hysteretic ring A typical measurement of the measured flux through the SQUID (Φ SQ ) as function of current through the field coil (I FC ) on a hysteretic ring on sample C. Arrows indicate the sweep direction of the field coil current. (Inset) SEM image of the junction measured in the main figure, showing under-etching of the Al which resulted in a very high critical current and hysteretic behavior. Scale bar is 2 nm Supplementary Section 5: Simulations We performed two types of simulations to elucidate the behavior the gate voltage dependence of the InAs nanowire Josephson junctions. The first, a simulation based on Ref. [S3], involves simulating an InAs 2DEG with similar length and width to the wire we studied. This simulation is useful because it does not rely on the short junction approximation to calculate the CPR, and it allows us to introduce interfacial barriers which depend on the Fermi velocity mismatch between the superconductor and the normal region. Because the geometry is only loosely related to the experimental geometry, the subband structure, relative spacing of resonances, and exact dependence of the mismatch on gate voltage are not well simulated, however, it does give us some useful intuition. We made minor modifications to the calculation performed in section V of Ref. [S3] to calculate the CPR vs. electron density in the nanowire. In particular, we calculated Z i, the effective interface transparency, for each subband i, which is given by Z i = (1 r i ) 2 /(4r i ), where r i vf S /vn F,i, the ratio of the Fermi velocity in the superconductor and the ith subband of the normal material. We fitted the calculated CPR to extract A n and a n in a similar way to real data. The low density, single subband behavior exhibits peaks that match what we observed NATURE PHYSICS 13

14 3 25 a.5.4 b a 2 A n (na) A n (na) A n (na) A 2 a c e n 1D (nm -1 ) n 1D (nm -1 ) A 1 a n a n a n d f n 1D (nm -1 ) n 1D (nm -1 ) n 1D (nm -1 ) n 1D (nm -1 ) Figure S8: Simulations following Ref. S3 Simulations of the CPR for an InAs 2DEG with parameters: w InAs = 5 nm, w SC = 11 nm, m =.23m e, L = 15 nm, vf S = m/s. An interfacial barrier which scales with the mismatch in Fermi velocities between the superconductor and semiconductor resulted in Fabry-Perot like oscillations in the Fourier amplitudes and shape parameters (See Text). The fitted CPR parameters A n and a n for a single subband (a-d) and many subbands (e,f). NATURE PHYSICS 14

15 a z (nm) y (nm) x (nm) 12 b A n (na) 8 4 A 1 A 2.4 c a n a 2 a Electron density (a.u.) Figure S9: Effective tight-binding simulations of the CPR of an InAs nanowire Effective tight-binding simulation of the CPR for an InAs nanowire with parameters: w InAs = 5 nm, w SC = 11 nm, m =.23m e, L = 15 nm, r = 25 nm to 55 nm. (a) A plot of the effective tight binding geometry, the chemical potential in the dark region was modulated while the light region was fixed at µ = 4 mev. (b,c) Fourier amplitude A n (b) and shape parameter, a n (c) vs. total density of states in the nanowire. 15 NATURE PHYSICS 15

16 experimentally close to depletion of the nanowire (Fig. S8 a,b). In particular, the observed shape of the peaks in A n and a n are asymmetric and cusp-like. Here, the asymmetry arises due to the density-dependent barrier, which decreases at higher densities due to a smaller Fermi velocity mismatch. The height of a 2 for all peaks except the first peak are.4, consistent with our the maximum allowed forward-skew in short junction theory and the observed peak in Fig. 3. At much higher densities, the resonant peaks begin to look more like oscillations, due to a decrease in the Fermi velocity mismatch (Fig. S8 c,d). The behavior of other subbands are exactly the same as in (Fig. S8 c,d), but offset by the energy onset of each subband. The result is the behavior observed in (Fig. S8 e,f), which shows peak-like behavior at low densities, but more fluctuation-like behavior when multiple subbands are occupied. We performed effective tight-binding simulations using the Kwant Python package [S4], using a more realistic nanowire geometry (Fig. S9 a). In the real nanowire, the thickness of the nanowire is increased due to epitaxial Al, however in our simulation everything is InAs. Wave function mismatch should occur in both, due to different spatial distribution of the wave functions and differences in chemical potential. In the physical nanowire, Al should lead to strong band bending which causes the wave function probability to exist mostly near the Al/InAs interface. In the tight binding model, the wave function mismatch is simulated by a change in the diameter of the nanowire and an applied offset to the chemical potential in the thicker regions of the nanowire. We solved the following Hamiltonian on a square lattice with a lattice spacing a =5nm: H = (6t µ i,j ) i, j i, j t( i +1,j i, j + i, j i +1,j + i, j +1 i, j + i, j i, j +1 ), i,j (S3) where t = h 2 /2m a 2 and µ was varied as a function of space. In the thick part of the nanowire, the chemical potential was held fixed at µ = 8 mev. In the gated region µ was varied to tune the density of the nanowire. We applied a gradient to µ along the z direction to mimic the effect of a close bottom gate, which acts to split up normally degenerate subbands. Defining NATURE PHYSICS 16

17 z= at the center of the nanowire, we set µ inside the nanowire to µ etched + µ etched z/4r, where r is the radius of the nanowire. Using Kwant s solver, we obtained the normal state S-matrix and calculated the transmission eigenvalues at each µ core. We then used the short junction equation (equation (S1)) to calculate the CPR and fitted it to extract A n and a n (Fig. S9 b-e). The fitted CPR parameters display qualitative behavior very similar to the measured CPR (Fig. 2), specifically resonant peaks at low density, a very forward-skewed CPR at low densities, and fluctuationlike behavior at higher densities. Without added onsite disorder, the simulated forwardskewness observed at high densities are very close to the experimental result (a 2,exp 1 and a 2,sim ). Random onsite disorder leads to an increase in the frequency of fluctuations, presumably due to extra induced resonant behavior, as well as a reduction of the forward-skewness at high densities, as is expected for a ballistic to diffusive transition. The observed gate voltage dependence of the CPR and the presence of a highly forwardskewed mode are reproduced in both WKB and tight-binding simulations in the presence of a Fermi velocity or wave function mismatch. The addition of disorder to our model is not required to reproduce the qualitative behavior of fluctuations we observed. Supplementary Section 6: Short-junction limit The short junction limit is often written as L ξ, where L is the length of the junction and ξ is the superconducting coherence length. An equivalent definition of the short junction limit [S5] (and one more useful for our purposes) is in terms of the superconducting energy gap ( ) and the Thouless energy (E C ) of the normal junction: and v F E C h τ φ = hv F l φ, where τ φ and l φ are the phase coherence time and length, respectively, is the Fermi velocity of the junction material. The short junction limit is satisfied when E C >> [S5]. In a gate-tuned junction, the short junction limit can be valid for some subbands and not for others if their v F are different. In particular, v F approaches zero close to each subband onset. Based on simulations in Kwant for a 5 nm radius nanowire, we estimate the subband NATURE PHYSICS 17

18 spacing to be of order tens of mev. Assuming parabolic dispersion, we can rewrite the short junction limit in terms of Fermi energy from the band onset E F > 2 L 2 m /(2 h 2 ). This is satisfied when E F > 9 µev for the L = 15 nm junction, a much smaller energy scale than the approximate subband spacing. Therefore, the regions of gate voltage where the short junction limit is violated are very small compared to the overall swept gate voltage range and are not relevant to our analysis. For the longest studied nanowires (L 6 nm, Fig. 5), the short junction limit in terms of Fermi energy is E F > 1.4 mev, still an order of magnitude smaller than the subband spacing. It is possible that the last occupied subband in these junctions is contributing a mode which is not in the short junction limit, however this does not directly affect our conclusions. Supplementary Section 7: Multi-mode fitting The CPR provides information that can be used to extract the transmissions of multiple modes, because higher-transmission modes have CPRs that are not only more skewed but that also have a specific functional form. In particular, the amplitudes of the higher harmonics (A n ) of the CPR of a single mode depend non-linearly on transmission in the short junction limit (equation (1) and Fig. S1). For example, two τ =.5 modes will give a very different CPR than one with τ = 1.. This non-linear dependence on transmission can allow the identification of multiple transmission coefficients from fits to a single CPR, especially for high transmissions which have greater skewness and more unique shapes. Determining multiple transmissions depends on the ability to resolve higher harmonics at low amplitudes. Our high sensitivity magnetic measurements allow us to resolve harmonics with amplitudes as low as 1 na in the presented data (see Fig. 1e). Our high sensitivity to changes in the amplitude of higher harmonics, or in other words our ability to detect small changes in the shape of the CPR, allows us to explore the possibility of extracting the transmission of multiple modes from experimentally-obtained CPRs. A full study of the limits of many-mode fits would include robust analysis of the statistical and systematic errors, and would ideally also include comparison to other ways of characteriz- NATURE PHYSICS 18

19 4 a τ = 1. I RING (na) 2-2 τ =.9 τ =.5-4 2π π 2π π (rad) b A c A n (na) 1 A 4-1 A τ A 3 a n τ a 2 a 3 Figure S1: Harmonic analysis of the CPR of a single short-junction mode (a) The predicted short-junction CPR for a selection of transmissions (τ). (b,c) Harmonic amplitudes (A n ) and shape parameters (a n ) vs. transmission, showing their non-linear dependence. ing Andreev bound states, but we comment here on our initial work on studying fits involving many modes. CPR data at high gate voltages, where multiple conducting modes contributed to the CPR, show multiple local minima in fits with many free transmission coefficients (τ p ), indicating that they contain information about multiple modes with different transmissions. Fits to a simulated CPR show similar behavior to those of the real data and correctly extracted all of the inputted τ, indicating that the fits are able to robustly find the true global minima. Some fits show well developed minima at high τ, while others only show diffuse minima at intermediate τ and strongly exclude high transmission modes. A full error analysis would include both statistical errors and systematic errors such as errors from the imperfectly NATURE PHYSICS 19

20 known position of the SQUID pickup loop, as discussed in section 3 for a two-mode fit. More work needs to be done to robustly determine the errors in the many-mode regime. Another known method for extracting independent transmissions is by fitting multiple Andreev reflections (MAR) in the I-V characteristics of Josephson junctions. Higher-order multiple Andreev reflections only occur for high transmission modes, and the pattern of higher-order reflections can be fit to extract multiple τ. According to Ref. [S6], MAR fits yield independent transmissions for all modes when N 3, and is able to pick out dominant high transmission modes for higher N [S6]. Supplementary Section 8: Possible origins of backward skew CPR A backward-skewed CPR (Fig. 4), defined here as a CPR with a 2 <, cannot arise due to the short junction theory presented in the main text (equations (1,2)). However, the inclusion of interactions in the form of a charging energy (U) can lead to qualitatively similar behavior of the CPR. Ref. [S7] outlines an extension to the Breit-Wigner formula (equation (2)) which includes the effects of charging energy in a quantum dot Josephson junction, in the limit of strong interaction between the superconducting leads and the dot (Γ ). The primary effect captured in the model is that the spin-degeneracy of Andreev levels is split by the charging energy by an amount E ex, a local exchange field. The resulting Andreev levels are given by the formula: ( ω ± ) 2 = cos2 (φ/2)+2e 2 + Z 2 (Z 2 + sin 2 (φ/2)) ± 2XS(φ) Z 4 + 2(X 2 + E 2 )+1 (S4) where ω ± are the energies of the Andreev levels, X E ex /2Γ, E ɛ /2Γ, Z 2 X 2 E 2, Γ=Γ L =Γ R are the tunnel rates into the junction, ɛ is the energy of the resonance without interactions with respect to the Fermi level, and S(φ) Z 2 cos 2 (φ/2) + E 2 + sin 2 (φ)/4. In the limit of no interactions (E ex = ) this formula exactly reproduces the Breit-Wigner formula (Eqn. 2). As E ex increases, equation (S4) captures the to π-junction transition as NATURE PHYSICS 2

21 the Andreev levels shift due to interactions. Increasing E ex switches the energy ordering of two of the Andreev levels(fig. S11 a-d) and the corresponding CPR (proportional to dω/dφ at zero temperature) changes dramatically (Fig. S11 e-h) [S7]. We calculated the CPR when the position of the Fermi level is resonant with the level in the junction before interactions are taken into account (ɛ =, Fig. S12). Performing Fourier analysis on the CPR, we found that in addition to the -π transition when A 1 changes sign, higher harmonics exhibit sign changes at smaller E ex (Fig. S12 a), leading to negative a 2 (backward skew) for a finite range of charging energies (Fig. S12 b). We fixed E ex at two values prior to the π transition to study the CPR s dependence on Fermi energy (Fig. S13). At E ex =.53/2Γ, the shape parameter a 2 on resonance agrees with the observed minimum a 2 =.4 in the backward-skew region (Fig. S13 b), and coincides with a local minimum in critical current (Fig. S13 a). However the higher harmonics in the simulation are much higher compared to their measured values (Fig. S13 b and Fig. 4). At higher E ex =.7/2Γ, we find the qualitative behavior of the shape parameters more closely match what is observed (minimum in a 2 while higher harmonics are close to zero, Fig. S13 e) and the overall dip in A 1 to 5 na at resonance is close to what was observed (Fig. S13 d, Fig. 4). However, the theoretical value of a 2 = 2 at higher charging energy is much larger than our observation. The CPRs calculated using equation (S4) show that backward-skew (and indeed more exotic CPR shapes) can arise in a junction with finite charging energy, even when the superconducting leads are strongly coupled to the junction. We found qualitative, but not quantitative, agreement between the shape of the CPR we observed and the calculated CPR near a -π transition, following Ref. [S7]. Equation (S4) does not include the effects of spin-orbit coupling, which may change the shape of the CPR near the transition region due to hybridization of the crossed Andreev levels. Spin-orbit induced hybridization between spin branches, asymmetric tunnel barriers, or strong tunnel barriers could all lead to a CPR which more quantitatively matches what we observe. NATURE PHYSICS 21

22 The role of spin-orbit coupling is of great interest due to the realization of Majorana quasiparticles in an appropriately directed magnetic field. At low, but finite, magnetic fields a φ -junction can lead to finite supercurrent at zero phase [S8]. In a single-level quantum dot junction with no charging energy, spin-orbit coupling does not change the CPR [S9]. Spin-orbit coupling can rearrange the resonant structure of a dot with multiple modes, but does not result in a deviation from short-junction behavior in the strong tunneling limit [S9]. In the presence of charging energy and spin-orbit coupling, φ -junction behavior can be recovered for vanishing Zeeman energy, but requires asymmetric tunnel barriers [S1]. The qualitative features of our data can be explained without spin-orbit coupling, although the detailed resonant structure most likely relies on its presence. The presence of a resonance which gives a perfect transmission mode (Fig. 3) requires the absence of a charging energy, while the backward-skewed mode (Fig. 4) may require a charging energy which is comparable to the tunneling rate into the junction (U Γ). The observation of both behaviors in the same junction is not necessarily surprising, as the tunnel rates and charging energy may both be strongly modulated with gate voltage if the barriers arise due to wave function mismatch (See Supplementary Section 5). NATURE PHYSICS 22

23 a E ex / 2Γ =. 5 e E E E b c E ex / 2Γ =.3 E ex / 2Γ =.7 I (na) I (na) I (na) f g d E ex / 2Γ = h E I (na) 2π φ (rad) 2π -1 2π φ (rad) 2π Figure S11: Dependence of the CPR on charging energy Calculations of the Andreev states and CPR following Ref. [S7] and Eqn. S4 using parameters relevant to our junction: T c = 1.4 K, T =.13 K, and therefore = µev, with ɛ held fixed at. As the charging energy is increased (E ex ) the normally spin degenerate Andreev levels (a) split due to interactions (b-d). The CPR starts forward skewed (e), as seen in Fig. 3a, and the CPR starts to deviate from the short junction formula (Eq.1) as the charging energy increases (f-g). At large charging energy, pi-junction behavior is observed (h). NATURE PHYSICS 23

24 4 A 1 a A n (na) 2 A 3 A 5 A 4-2 A E ex / a 2 a 4 b a n a a E ex / 2 Figure S12: Dependence of the CPR on charging energy Calculations of the CPR following Ref. [S7] and Eqn. S4 using parameters relevant to our junction: T c = 1.4 K, T =.13 K, and therefore = µev, with ɛ held fixed at. (a,b) Dependence of harmonic amplitudes (A n ) and shape parameters (a n ) on exchange energy E ex = U/2, showing that a finite charging energy can lead to non-standard CPR shapes, specifically large negative a n values, which are not allowed in the short junction non-interacting case. NATURE PHYSICS 24

25 2 a E ex / 2Γ = d E ex / 2Γ =.7 A n (na) 1 A 1 A 4 A 2 A n (na) 1 5 A n (na) b A 3 1 E F - ε / 2 Γ a a 5 2 a 4 a 3 A n (na) e E F - ε / 2 Γ c E F - ε / 2 Γ f E F - ε / 2 Γ I (na) I (na) -2 2π φ (rad) 2π -2 2π φ (rad) 2π Figure S13: Dependence of the CPR on Fermi Energy at fixed charging energy Calculated CPR parameters (from equation (S4)) near resonance for intermediate fixed charging energies E ex /2Γ =.53 (a-c) and.7 (d-f). The harmonic amplitudes (a,d) and shape parameters (b,e) show behavior similar to the observed backward-skew CPR (Fig. 4), including a local minimum in A 1 at resonance and a large negative (a 2 ). The model presented in equation (S4) gives a qualitatively similar behavior to Fig. 4, but fails to quantitatively reproduce the observed a 2 and a 3 values at resonance. NATURE PHYSICS 25

26 Supplementary References [S1] H. Bluhm, N. C. Koshnick, J. A. Bert, M. E. Huber, and K. A. Moler. Persistent currents in normal metal rings. Phys. Rev. Lett., 12(13):13682, 29. [S2] L. D. Jackel, W. W. Webb, J. E. Lukens, and S. S. Pei. Measurement of the probability distribution of thermally excited fluxoid quantum transitions in a superconducting ring closed by a Josephson junction. Phys. Rev. B, 9: , Jan doi: 1.113/Phys- RevB URL [S3] A. Furusaki, H. Takayanagi, and M. Tsukada. Josephson effect of the superconducting quantum point contact. Phys. Rev. B, 45(18):1563, [S4] C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal. Kwant: a software package for quantum transport. New J. Phys., 16(6):6365, 214. [S5] C. W. J. Beenakker. Three universal mesoscopic Josephson effects. In Transport Phenomena in Mesoscopic Systems, pages Springer, [S6] E. Scheer, P. Joyez, D. Esteve, C. Urbina, and M. H. Devoret. Conduction channel transmissions of atomic-size aluminum contacts. Phys. Rev. Lett., 78: , May doi: 1.113/PhysRevLett URL [S7] E. Vecino, A. Martín-Rodero, and A. Levy Yeyati. Josephson current through a correlated quantum level: Andreev states and π junction behavior. Phys. Rev. B, 68:3515, Jul 23. doi: 1.113/PhysRevB URL [S8] D. B. Szombati, S. Nadj-Perge, D. Car, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven. Josephson φ-junction in nanowire quantum dots. Nat. Phys., 216. [S9] L. Dell Anna, A. Zazunov, R. Egger, and T. Martin. Josephson current through a quan- NATURE PHYSICS 26

27 tumdot withspin-orbit coupling. Phys. Rev. B, 75:8535, Feb 27. doi: 1.113/Phys- RevB URL [S1] Aldo Brunetti, Alex Zazunov, Arijit Kundu, and Reinhold Egger. Anomalous josephson current, incipient time-reversal symmetry breaking, and majorana bound states in interacting multilevel dots. Phys. Rev. B, 88:144515, Oct 213. doi: 1.113/Phys- RevB URL NATURE PHYSICS 27