Charge of a quasiparticle in a superconductor

Non-linear charge transport in SIS Josephson junctions has a unique signature in the shuttled charge quantum between the two superconductors. In the zero-bias limit Cooper pairs, each with twice the electron charge, carry the Josephson current. An applied bias $V_{SD}$ leads to multiple Andreev reflections (MAR), which in the limit of weak tunneling probability should lead to integer multiples of the electron charge $ne$ traversing the junction, with $n$ integer larger than $2{\Delta}/eV_{SD}$ and ${\Delta}$ the superconducting order parameter. Exceptionally, just above the gap, $eV_{SD}>2{\Delta}$, with Andreev reflections suppressed, one would expect the current to be carried by partitioned quasiparticles; each with energy dependent charge, being a superposition of an electron and a hole. Employing shot noise measurements in an SIS junction induced in an InAs nanowire (with noise proportional to the partitioned charge), we first observed quantization of the partitioned charge $q=e^*/e=n$, with $n=1-4$; thus reaffirming the validity of our charge interpretation. Concentrating next on the bias region $eV_{SD}{\approx}2{\Delta}$, we found a reproducible and clear dip in the extracted charge to $q{\approx}0.6$, which, after excluding other possibilities, we attribute to the partitioned quasiparticle charge. Such dip is supported by numerical simulations of our SIS structure.


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Excitations in superconductors (Bogoliubov quasiparticles) can be described according to the BCS theory (Bardeen-Cooper-Schrieffer) [1], as an energy dependent superposition of an electron with amplitude u(), and a hole with amplitude v(); where the energy  is measured relative to the Fermi energy [2]. Evidently, the expectation value of the charge operator (applied to the quasiparticle wave-function), which we address as the quasiparticle charge e * =q()e, is smaller than the charge of an electron, 22 ( ) ( ) ( ) qu      [3]. Solving the Bogoliubov-de Gennes equations one finds that 22  -vanishing altogether at the superconductor gap edges [3]. Note, however, that the quasiparticle wave-function is not an eigen-function of the charge operator [3,4]. Properties of quasiparticles, such as the excitation spectra [5], lifetime [6][7][8][9][10], trapping [11], and capturing by Andreev bound states [12,13], had already been studied extensively; however, studies of their charge is lagging. In the following we present sensitive Shot noise measurements in a Josephson junction, resulting in a clear observation of the quasiparticle charge being smaller than e, q(eVSD~2)<1, and evolving with energy, as expected from the BCS theory.
In order to observe the BCS quasiparticles in transport we study a Superconductor-Insulator-Superconductor (SIS) Josephson junction in the non-linear regime. The overlap between the wave functions of the quasiparticles in the source and in the drain is expected to result in a tunneling current of their effective charge. This is in contrast with systems which are incoherent [14,15] or with an isolated superconducting island, where charge conservation leads to traversal of multiples of e -Coulomb charge [16]. As current transport in the non-linear regime results from 'multiple Andreev reflections' (MAR), it is prudent to make our measurements credible by first measuring the charge in this familiar regime. 3 In short, the MAR process, described schematically in Fig. 1, carries a signature of the shuttled charge between the two SCs, being a consequence of n traversals through the junction (as electron-like and hole-like quasiparticles), with n an integer larger than 2/eVSD. A low transmission probability t (via tunneling through a barrier) in the bias range 2 2 1 SD / n eV / ( n )      assures dominance of the lowest order MAR process (higher orders are suppressed as t n ); with the charge evolving in nearly integer multiples of the electron charge. While there is already a substantial body of theoretical [3,[17][18][19][20][21][22][23] and experimental [24][25][26][27][28][29] studies of the MAR process, charge determination without adjustable parameters is still missing. An important work by Cron et al. [27] indeed showed a staircase-like behavior of the charge using 'metallic break-junctions'; however, limited sensitivity and the presence of numerous conductance channels, some of which with relatively high transmission probabilities, Our SIS Josephson junction was induced in a back-gate controlled, single channel nanowire (NW). The structure, shown in Fig. 2 coherence length is expected to be much larger than 200nm -assuring coherence of electron-hole quasiparticles along the junction [30].
Since the probability of each single-path MAR process is t, and the probability for n paths scales as n tt   , a sufficiently small t is necessary to single out the most probable (lowest n) MAR process. This evidently leads to a minute shot noise signal, requiring sensitive electronics and weak background noise. A 'cold' (~1K), low noise, homemade preamplifier was employed, with a sensitivity limit better than ~10 - where kBT is the thermal energy and r is the total resistance of the SNS junction and the load resistance, namely, Rsample+5Ω in parallel with RL (see Fig. 2). Hence, the 'zero frequency excess noise' for a stochastically partitioned single quantum channel at sufficiently low temperature (our kBT~2eV while eVSD=50-300eV) [33][34][35][36] is: Two comments regarding Eq.3 are due here: (i) Using the differential conductance G for calculating the transmission probability at energies near eVSD is justified since most of the current is carried by quasiparticles emitted in a narrow energy window; much narrower than  due to the van Hove singularity of the density of states in the 1D NW (see more in the discussion part); and (ii) When the transmission is small so 5 that G/gq~0, one resorts to the familiar Schottky (Poissonian) expression of a classical shot noise [37].
While details of the measurement setup and the algorithm used in determining the true excess noise and the extracted charge are provided in the S3 and S4; a short description is due here. As seen in Fig. 2, conductance and noise were measured in the same configuration at an electron temperature of ~25mK. The differential conductance was measured by applying 1µV at 600kHz in addition to a variable DC source' for VSD (rather than a 'current source') allowed access to quiescent regions of negative differential conductance, which otherwise would have been inaccessible (being within hysteretic loops in the I-VSD characteristics).
We start with RL=1kOhm and junction conductance tuned by the back-gate to a partly transmitted single channel in the bare part of the NW. Four MAR conductance peaks were observed at VSD=2Δ/en=300V/n (note that the induced gap in the InAs NW is nearly that of the Al superconductor). The static I-VSD characteristic, required for the determination of the energy dependent charge, was obtained by integrating the differential conductance (Fig. 3b). After a careful subtraction of the background noise, (S4), we extracted the charge as shown in Fig. 3c. Clear steps are seen at values of q=n, with 1 n 4 . Higher charge values (for n> 4) are averaged out mostly due to the successively narrowing MAR region as ~1/n 2 and possibly some inelastic scattering events. It should be stressed out here that while the conductance (and thus the deduced t * ) and the total noise fluctuate violently, the extracted charge evolves smoothly between each of the quantized charge valuesreassuring the process of charge extraction. 6 We performed numerical simulations of the conductance and the excess noise at various junction transmission coefficients and energies, with a Fano factor defined as in which the dip appears, can once again be attributed to the sharper profile of the 7 density of states. Another difference from the theoretical calculation that should be noted is the decrease in the apparent charge from 2e at eVSD<2. We relate this decrease to unavoidable processes of charge e transport which are of order t (not t 2 ); such as quasiparticles excited by noise or temperature, and sub-gap current originating due to the soft induced gap.
In order to further test the validity of the dip in the extracted charge, we fabricated and tested a Normal-I-S (NIS) junction. Here too, a conductance peak develops at the gap's edge (this time at eVSD=); however, the charge evolves monotonically from e to 2e, without any sign of a dip (Fig. 5a). This result is backed by our numerical simulations ( Fig. 5b), while in S9 we give a more intuitive physical picture that reflects why charge partition should not be observed in NIS junction.
Our assertion of observing the quasiparticle charge near the gap's edge requires a discussion. One may consider three possible models of single quasiparticles transport       The I-V characteristics as obtained by integrating the differential conductance. Inset:

Methods and Supplementary Information
In this Supplementary Section we add details that could not find room in the main text. We placed a brief review of the theoretical background as well as the simulation.
In addition, a few details of the NWs growth process followed by the fabrication process are provided, as well as more details on the conductance and noise measurements.

Scattering theory of multiple Andreev reflections
Following [1,2], we here outline the calculation of the current and current noise through a SNS-Josephson junction in the formalism of multiple Andreev reflections.
Where electrons in the normal part are Andreev reflected from the superconducting leads. The normal region contains a barrier whose transmission amplitude squared is t.
It is assumed that the length of the normal region is much smaller than the superconducting coherence length, and that the Fermi energy in the normal region is much larger than the superconducting gap . As a simplified setup we consider a short one-dimensional normal metal piece connected to one-dimensional semi-infinite superconducting leads. This process repeats until the particle has gained enough energy to overcome the superconducting gap and can be absorbed into the continuum of quasi-particle excitations in one of the leads.
We are using a joint index i,    to indicate the origin and energy of the incident electron ( i l,r  for the left and right lead). The wave-function     u t / t   is the respective electron/hole amplitudes obtained from solving the recursive relations denoted above.
Our main interest in this work is the low-frequency current-noise, which is given by the zero Fourier component of the time averaged, symmetrized current correlation: , and the upper bar stands for time averaging (see [1,2] for the explicit expressions). To comply with the experimentally measured quantity

Details on the numerical calculations
The solution to the recurrence relations described above is found using the method of continued fractions, following [3,4]. First, the amplitudes n n

Additional simulations
Noise in a NIS junction: We here calculate the current and current noise in a junction of a normal metal (N) and a superconductor (S) with a tunneling barrier in the middle to model the insulating region (see Fig. S1). Transport through this kind of systems has been studied abundantly in the literature and we here adapt the formalism of Refs. [5,6] The reflection amplitudes   ee r  and   he r  are found from an infinite series expansion considering all possible paths through which an incident electron is reflected as an electron or as a hole respectively. Taking the distance between the normal barrier and the S-N interface to zero, one obtains  25 We calculated the ratio 2 S / eI which does not show a dip around eV  even though the current shows a peak from enhanced tunneling into the superconductor due to a singularity in the density of states. thick) was subsequently evaporated in the same chamber. Following transfer to the growth chamber, the substrate temperature was ramped up to ~550 O C for ripening the Au layer into droplets with a rather uniform size and density distribution [7].

S2 -MBE Growth and sample fabrication
Lowering the growth temperature (to ~400 O C), InAs growth was initiated with an characteristic. Two electrical relays were employed, one at the input (at 300K) and one at the output (at 25mK); allowing switching from low frequency measurement (mode 1) -using the lock in technic, to a higher frequency (mode 2) -using a function generator and a spectrum analyzer. The actual measurements were done in mode 2, while measurements in mode 1 were performed in order to calibrate the higher frequency measurement.

Mode 1low frequency measurement
A calibration line allows calibrating the 5Ohms resistor after cooling. Applying DC voltage plus an AC signal and measuring the two-terminal AC current, allows calculating the static and dynamic conductance. The current was amplified by an external current amplifier [8], with 10 7 V/A conversion factor, followed by a DMM or a lock-in amplifier. The measured differential conductance was used to calibrate the higher frequency measurements.

Mode 2higher frequency measurement
While at DC the Drain is shorted through the coil L, the 600 kHz signal is divided between the junction resistance and RL. The external voltage amplifier, SA-220F5, has a gain of 200, while the home-made 'cold' voltage preamplifier has a gain ~5. Noise measurements were performed by replacing the function generator (needed for the conductance measurements) with a DC source, and increasing the bandwidth of the spectrum analyzer. In our setup we also have two kinds of low pass filters, LPF1 and LPF2 which differ by their cut-off frequency. LPF1 is placed both in RT and in basetemperature has a cutoff frequency of 80MHz (mini-circuit BHP-100+). LPF2 is placed between them, also in base temperature and has a cut-off frequency of 2MHz.

S4 -Estimating the background noise
The total voltage noise per unit frequency at the input of the 'cold' preamplifier: where Sexc is the excess current noise per unit frequency, iamp and amp are the amplifier's current and voltage noises, respectively, T is the temperature and r is the resistance that the amplifier 'sees' at the resonance frequency (with a small frequency window): here Rsample is the differential resistance of the sample and RL is the frequency independent load resistance. Note, that the 1/ f contribution to the noise at f0=600kHz is negligible. This is justified both from our measurements at high magnetic field as explained in S6 as well as from previous noise measurements done in our system to accurately extract integer and fractional charges of excitations in various 2DEG systems.
The background noise, subtracted from the total noise, is: Since the differential resistance is strongly dependent on biasing voltage VSD, we first describe the procedure of determining the background noise. Since this noise (measured at zero bias) is laden with an emerging large Josephson current, it is quenched by applying a magnetic field stronger than Bc (B~200mT), where the superconductivity is quenched. The differential conductance and the background noise were then measured as a function of the back-gate voltage, and thus as function of r, in the relevant range (Fig. S4). The values of the amplifier's noises obtained by fitting are in good agreements with the values we measure using other calibration methods. The electron temperature agrees well with that measured by other shot noise measurements.

S5 -The critical magnetic field
In order to find the critical magnetic field, MAR conductance peaks are measured as a function of the magnetic field, with the spacing between the peaks directly proportional to the diminishing superconducting gap with magnetic field (Fig. S5).

S6 -Number of conducting channels in the bare NW
Under high enough magnetic field the quantum charge passing the junction is that of the electron. The expression for shot noise provided in the text is that of a singly occupied spin-degenerate conducting channel, where G is the conductance and The differential conductance and the I-VSD characteristic were measured after quenching superconductivity (but not lifting spin degeneracy) at the working voltage of the back-gate corresponding to the actual experiment (Figs. S6a and S6b). The noise is then measured as a function of VSD (blue curve in Fig. S6c), and the background noise is subtracted (red curve in Fig. S6c), and the excess noise is plotted in Fig. S6d. The theoretical curve, calculated using Eq. S4, plotted in a black dashed line, seems to agree nicely with the data. In order to test this further, we also plot the expected excess noise assuming two spin-degenerate channels, namely, With I1 and I2 the current carried by each of the two channels, while t1 and t2 are the transmission of each of the two channels. If the total current, I, splits between the two channels in the following way then, Therefore, since we know I and G, we can plot Sexc for a given α. In Fig. S6d

S7 -Nature of tunneling quasiparticles
Three possible models are suggested to account for the single quasiparticle tunneling taking place in the junction. We calculated the Fano-factor (F) for the models in order to see which one of them can account for the measured charge at the superconducting gap's edgebeing smaller than e at a low transmission. In each table, we express the probability of an event to take place P(x) and its charge (X): Model 1: Quasiparticles of charge e tunneling with probability t.
x P(x)  It is important to mention that the models above are considering only single quasiparticle tunneling across the junction neglecting higher order MAR contributions. When we lower the transmission we suppress the higher order MAR contributions and reveal a dip in the charge. This is clearly seen in our data as well as in the results of the theoretical model of S1. Once we suppressed these high order MAR contributions we observe a Fano factor which is smaller than e, which is only consistent with Model 3 above. In other words, the only way to observe a Fano factor that is lower than e is both to suppress enough the high order MAR (going to low transmissions) as well as having a tunneling of quasiparticles carrying a fractional charge.

S8 -Induced superconductivity on a single band
In this section we aim to support our claim in the manuscript for having a non-BCS density of state and specifically a sharper one in our 1D system.
Observing figure 3a in the main text it is possible to see negative differential conductance, this effect which is more apparent as the transmission is decreased originates as we will show from the change in the usual BCS density of states. In figure S7 (a&b) we plot a measurement of the differential conductance as a function of the applied bias and the I-V curve. In figure S7 (c&d) we plot the theoretical predicted I-V curve and differential conductance based on the BCS density of state assuming a uniform transmission. The negative differential resistance which is clearly seen in the measurement and manifested in the experimental I-V curve as a peak in the current is not visible in the theoretical I-V, this suggests a different theoretical model should be given.
The origin of this discrepancy is the assumption of a linear dispersion which usually one considers in calculating the DOS. In a 1D wire, which has a parabolic dispersion, as the Fermi level is lowered to the bottom of the conduction band this assumption 31 fails. Hence, the change in the DOS is more apparent as the fermi energy gets closer to the 'Van-Hove singularity'.
To show this we calculated the DOS and I-V curves as a function the fermi energy position. Figure 8(a, d and g) are the density of state for EF=5Δ, 2Δ and Δ respectively. In figure 8(b, e and h) we plot each DOS when the fermi energy is defined as zero energy. It is already clear that as the Fermi level is pushed towards the bottom of the band the DOS is modified. In Fig 8(c, f and i) we calculate the I-V curves and show that the modified DOS gives rise to a peak in the I-V curve, similar to the one we showed in fig S7b. In conclusion, the negative differential resistance which is seen in experiment as the device is pinched suggests a modification from the usual BCS density of state.

S9 -Charge partition in SIN junction
In the SIS junction, the overlap between filled states of quasiparticles' wave-function and empty states (above the gap) in the two superconductors allows tunneling of quasiparticles with fractional charge. However, in the case of SIN, in the N side there are quasiparticles with charge e while in the S side there are quasiparticles with a smaller charge. Our physical picture suggests that tunneling of electrons, being of the higher charge is always dominant. In one polarity, the electrons that tunnel from N to S breaks to multiple quasiparticles; while in the opposite polarity, quasiparticles bunching to an electron (in N) takes place. This is similar to the known bunching in the 3 / 1 =  fractional quantum hall states where 3 quasiparticles, each with charge e/3, tunnel together to form an electron.
Moreover, and in general, tunneling between two different materials, with different quasiparticles in each side, the current fluctuations will correspond to the larger charge transfer. For example, when the bias is smaller than Δ, electrons from the N region "bunch" to form Cooper pairs and the measured charge (via shot noise) is 2e (via Andreev reflection).