Superconducting parity effect across the Anderson limit

How small can superconductors be? For isolated nanoparticles subject to quantum size effects, P.W. Anderson in 1959 conjectured that superconductivity could only exist when the electronic level spacing δ is smaller than the superconducting gap energy Δ. Here we report a scanning tunnelling spectroscopy study of superconducting lead (Pb) nanocrystals grown on the (110) surface of InAs. We find that for nanocrystals of lateral size smaller than the Fermi wavelength of the 2D electron gas at the surface of InAs, the electronic transmission of the interface is weak; this leads to Coulomb blockade and enables the extraction of electron addition energy of the nanocrystals. For large nanocrystals, the addition energy displays superconducting parity effect, a direct consequence of Cooper pairing. Studying this parity effect as a function of nanocrystal volume, we find the suppression of Cooper pairing when the mean electronic level spacing overcomes the superconducting gap energy, thus demonstrating unambiguously the validity of the Anderson criterion.

demonstrating unambiguously the validity of the Anderson criterion.
The addition energy of an electron to a superconducting island, weakly coupled to the environment by the capacitance C Σ , is given by (See Methods): where the first term is the Coulomb energy, the second term depends on the parity of electron occupation number as a consequence of the formation of a Cooper pair 1,2 , the third term is the electronic level spacing in the island.This parity effect has been observed in large ∼ 1 µm micro-fabricated Al islands, through direct measurement of the charge capacitance of the island 2 , through the even-odd modulation of the addition energy in single electron transistors [3][4][5][6] or the parity dependence of the Josephson current in Cooper pair transistors [7][8][9] .
Until now, the parity effect on the addition energy has never been observed in small nanocrystals (NCs) near the Anderson limit 10 , reached at a volume about V Anderson 100 nm 3 , where the mean electronic level spacing < δ > equals the superconducting gap energy ∆.
In single electron transistors fabricated with nanosized superconducting grains of aluminum 11,12 , the 2e modulation of the addition energy could not be observed directly.Also, because only a few devices could be fabricated, testing the Anderson criterion was not possible with this approach.
Indirect indications for the disappearance of superconductivity in small superconducting grains came from magnetization measurements 13,14 ; because these measurements were averaged over macroscopic quantities of NCs, the link to the Anderson limit remained ambiguous.In this work, we present a new system that enables a study of single and isolated NCs across the Anderson limit, where the NCs can be reproducibly obtained in large quantities.The superconducting gap energy and the transition temperature are measured through a study of the superconducting parity effect in the addition energy of the NCs.This constitutes an alternative approach to conventional tunneling measurement of the superconducting gap in the quasi-particle excitation spectrum, which cannot be accessed as a consequence of the Coulomb gap at zero bias.

Sample preparation
The observation of the parity effect is challenging as it requires clean systems, free of impurity states responsible for the so-called quasiparticle poisoning 9,15 .Furthermore, scanning tunneling spectroscopy of isolated NCs requires, in addition to the tip-NC tunnel barrier, a second tunnel barrier between the NC and the conducting substrate 16,17 , as sketched in Supplementary Fig. 1.The data indicate that the Fermi level is in the conduction band of InAs as expected for this ndoped sample.With a sulphur dopant concentration, N D ∼ 6 × 10 16 cm −3 , the Fermi level is 21 meV above the conduction-band minimum.A zoom on these spectra, Fig. 1f, shows multiple peaks that result from the discrete levels of the tip-induced Quantum Dot (QDot), a phenomena that has also been observed in previous works 19 .This demonstrates that Pb deposition on InAs do not produce any significant defects and doping.Indeed, in presence of defects or adsorbate, the surface of III-V semiconductors present interface states that pin the Fermi level at the charge neutrality level 20,21 , Fig. 1g.For InAs, this level is located 150 meV above its conduction band minimum, which leads to the formation of an electron accumulation layer as shown by numerous photoemission experiments 20,22 .In contrast, perfectly clean (110) surfaces do not present any in-terface states and consequently the Fermi level is not pinned.Thus, the electric field from the STM tip can easily shift the conduction band and generates the so-called tip-induced QDot 19 , as sketched Fig. 1h.While the energy of the QDot levels can shift on long distances, see Supplementary Fig. 4, as a consequence of variations in the electrostatic environment due to the random distribution of Pb NCs and sulphur dopants, we see, Fig. 1f, that the QDdot levels are not altered on short distances (< 10 nm) near the NCs.Only a weak broadening of the QDot levels is observed, likely a consequence of their weak tunnel coupling with the Pb NCs.

In this work, the
Coulomb blockade and nature of the tunnel barrier On NCs of three distinct sizes shown As no dielectric insulator has been deposited on the surface and no Schottky barrier exists at metal-InAs interfaces 21,22 , the origin of the tunnel barrier and the meaning of the thickness d appear clearly only after one realizes that the Fermi wavelength of the 2D gas in InAs is larger than the lateral size of the NCs.At the interface between the Pb NC and InAs, the Fermi energy in InAs is at the charge neutrality level, E F = 150 meV, Ref. 21,22 , which gives for the Fermi wavelength λ F = 20 nm.As known from numerous works with quantum point-contacts formed in 2D electron gas 24,25 , the transmission coefficient T decreases for constrictions smaller than the Fermi wavelength.Because a NC covers only a fraction of the area λ 2 F , its transmission coefficient with the 2D gas is significantly smaller than one, which explains the observation of the Coulomb blockade.For a small NC, the weak coupling model Ref. 26 can be used to describe the data, as shown Fig. 2g.This model shows that the contact impedance is of the order of R contact ∼ 10 MΩ, implying that the transmission coefficient T = R contact e 2 /h = 0.0025 is weak as anticipated.In this model, the magnitude of the Coulomb peaks increases with the ratio R tunnel /R contact , as observed on the DC curves measured as function of tip height, Supplementary Fig. 6.Fig. 2h shows the amplitude of the Coulomb peak, normalized to its base value, as function of NC area.The amplitude is constant for small area (< 100 nm 2 ) but decreases quickly for area approaching πλ 2 F /4 300 nm 2 .This behavior cannot be described by the weak-coupling model just discussed, however, it can be understood by considering models of Coulomb blockade in the strong coupling regime 27,28 .These models show that the Coulomb oscillations disappear when T approaches unity, when charge fluctuations between the NC and the substrate become significant.Fig. 3 shows that the Coulomb peaks of the largest NCs have almost completely disappeared.The fact that the amplitude of the Coulomb peaks decreases for NCs area approaching λ 2 F confirms our interpretation that the tunnel barrier is due to a quantum constriction of the electronic wave function at the interface between the NC and the 2D gas.Thus, the dielectric thickness d = 4 nm extracted from C sub above is actually set by the Debye length of the 2D gas and C sub actually corresponds to the quantum capacitance of InAs.
Superconducting parity effect Thanks to this highly clean type of tunnel junction, free from quasi-particle poisoning, the superconducting parity effect in the NCs can be observed through the even-odd modulation of the addition voltage, as shown Fig. 2de, Fig. 3 and Fig. 4. The addition voltages can be precisely extracted thanks to the sharpness of the Coulomb peaks, which voltage positions are obtained through a fit with a Lorentz function, Supplementary Fig. 7.As sketched Fig. 2i and shown by Eq. 1, the addition voltage δV even for injecting an electron in an even parity NC is higher than δV odd for injecting an electron in an odd parity NC, where the energy difference is given by the binding energy of the Cooper pair.For NCs smaller than the Anderson volume, Fig. 4dgj, we observe that δE HT is non-zero, which indicates that the electronic level spacing δ has now a significant contribution to the addition energy, following Eq. 1.The values of δE HT are distinct between successive charge configurations.
Indeed, in metallic systems, the electronic levels are randomly distributed as described by Random Matrix Theory (RMT) 29 .Collecting the values δE HT for all NCs, Fig. 5a shows that, in average, the evolution of δE HT with NC volume can be properly described by the relation: For the smaller NC of volume V /V Anderson = 0.55, Fig. 4g, the level spacing δE HT is larger and has a temperature dependence that dominates the shift of the Coulomb peaks with temperature.
This shift could be the consequence of thermally induced electro-chemical shifts or temperature dependent strain or electric field effects.While the parity effect is barely visible on the raw data, using the procedure employed for the previous NC, the temperature T c 5 K value and the energy gap ∆ ∆ bulk /4 can be extracted, Fig. 4i.
Finally, for the smallest NCs V /V Anderson = 0.43 (0.34), shown respectively Fig. 4j and Supplementary Fig. 8h, they have the largest level spacing δE HT and, even though the addition energies are measured with much higher resolution than the superconducting gap energy, no parity effect can be observed on Fig. 4l and Supplementary Fig. 8j, respectively.
For 13 NCs where the DCs have been acquired as function of temperature, some of which are shown Fig. 4 and Supplementary Fig. 8, the level spacing, the superconducting gap energy and the transition temperature are extracted and plotted Fig. 5a, Fig. 5b and Fig. 5c, respectively.Upon reducing the NC volume, both quantities display a sharp decrease to zero when the level spacing becomes of the order of the superconducting gap energy, 1 meV.See Supplementary Note 2 for a comparison with the results of Bose et al. 17 on a system where the superconducting nanoparticles are strongly coupled to the normal substrate.

Discussion
Fig. 5 suggests that superconductivity disappears when the mean level spacing at the Fermi surface of the electron-type band, Fig. 5e, increases up to the superconducting gap energy.This is consistent with recent theoretical calculations 30 and STM measurements 31 which have shown that electron-phonon coupling is stronger for this electron-type band owing to its p-d character.Regarding the BCS ratio, within the experimental resolution, no significant deviation from the bulk value has been observed.
To summarize, we have found that a 2D electron gas of large Fermi wavelength constitutes an ideal substrate for studying Coulomb blockade in nanosized NCs evaporated in an ultra-high vacuum environment.This discovery leads us to observe, for the first time by STM, the parity effect and quantum confinement in isolated superconducting NCs and enabled the first demonstration of the Anderson criterion for the existence of superconductivity at single NC level.Furthermore, this new insight on the superconductor-InAs interface is of interest for topological superconductivity where Majorana islands are generated by depositing a superconductor on InAs nanowires 32,33 .

Relation between sample bias and energies
The Coulomb gap at zero bias results from Coulomb blockade that prevent charge fluctuations in the NC.As sketched in Supplementary Fig. 1, Coulomb blockade is lifted when the Fermi level of either one of the electrodes is aligned with one of the excited levels of the NC.Thus the amplitude of the Coulomb gap observed in the DC is given by The Coulomb peaks observed at higher voltages result from the shift of the electrochemical potential of the NC upon increasing the voltage bias across the double junction.This shift is given by : with : Charge states with increased number of electrons become accessible when the electrochemical potential changes by 2 × E C .Thus the voltage difference between two charge states is given by : η This formula shows that the addition voltage depends only on the capacitance C tip and not on the capacitance C sub , as shown Supplementary Fig. 1b, where a simulation of the conduction spectrum, using the Hanna and Tinkham model 26 for two distinct values of the capacitance C sub .
Addition energies Following Ref. 1,34 , the total energy of a NC with N electrons is given by: The electrochemical potential of a nanocrystal with an even (odd) N (N+1) number of electrons is given by: From these last equations, one obtains the addition energies for a NC with an even (odd) N (N+1) number of electrons : Thus, the difference of addition energies between two successive charge states is given by : When the electronic spectrum of the NC is discrete, the level spacing δ should be included in the addition energy.
Random level distribution In metallic NCs, the electronic level distribution is described by RMT 29,35 .In a NC with strong spin-orbit coupling, RMT predicts that the level spacing should be described by a Gaussian symplectic ensemble.For this level distribution, shown Supplementary Fig. 9, the width of the distribution, i.e. the standard deviation, is equal to σ < δ >, 29,36 .
Between two successive charge states, the addition energy can fluctuate by an amount of the order of σ, consequently, in average, the difference in addition energies between two successive charge states is given by: At temperatures above the superconducting transition temperature: Thus an estimation of the level spacing can be obtained by a measure of the difference in the addition energies above T C .
Furthermore, the gap amplitude can be obtained from:      Competing Interests The authors declare that they have no competing financial interests.
Correspondence Correspondence and requests for materials should be addressed to H.A. (email: herve.aubin@espci.fr) Pb NCs are obtained by thermal evaporation of a nominal 0.3 monolayer of Pb on the (110) surface of InAs heated at T = 150 • C. The (110) surface is obtained by cleaving a n-type InAs substrate in ultra high vacuum at a base pressure P ∼ 10 −10 mbar.Two distinct samples (A and B) have been prepared with slightly different NC concentrations and sizes.The volume of the NCs ranges from 20 nm 3 0.2 V Anderson to 800 nm 3 8 V Anderson while the heightranges from 1 unit cell (0.495 nm) to 5.2 nm, see Supplementary Fig.2and Supplementary Note 1 for details on NC volume determination.The scanning tunneling microscopy (STM) topographic images, Fig.1abcand Supplementary Fig.3, for sample A and sample B, respectively, show that Pb grows in the Volmer-Weber, i.e. island mode18 .The 3D Laplacian image ∆ xy z(x, y), Fig.1c,shows that the NCs are well crystallized and expose mostly the (111) planes of the cubic facecentered Pb structure, as indicated by the observation of the characteristic hexagonal shape of the (111) facets.Surrounding these NCs, the surface remains free from adsorbate, as atomic resolution images of the (110) InAs surface prove, Fig.1d.Tip-induced QDot on the InAs surface Fig.1eshows the Differential Conductance (DC) dI/dV measured on the InAs surface at several distances, from 0 to 10 nm, of a Pb NC.The data are measured at T = 1.3 K, unless indicated otherwise, using a standard lock-in procedure (See Methods).

Fig. 2abc ,
Fig. 2abc, representative DC spectra are shown Fig. 2de.They display a Coulomb gap at zero bias of width δV sub = e/(C sub + C tip ) where C sub (C tip ) is the capacitance between the NC and the substrate (tip).The data also display sharp Coulomb peaks where the voltage interval between the peaks provides the addition voltage δV add for an electron, which is related to the addition energy by : δV add = E add /eη where η = Fig.4ashows the DCs for a large NC, V /V Anderson = 1.6, as function of temperature.The corresponding addition voltages, shown Fig.4b, are almost equal above T c = 7.2 K, the superconducting transition temperature of bulk Pb.However, an even-odd modulation is observed at low temperature T = 1.3 K.The difference in the addition energies between two successive charge configurations is obtained from δE = eη(δV even − δV odd ).For this large NC, four Coulomb peaks are observed which provide three distinct addition voltages indicated by the horizontal bars.From these addition voltages, two distinct values of the addition energy difference δE between two charge configurations are obtained and given by δE = η(δV Head − δV Tail ), where the head (tail) refers to the colored arrows in the panel.These two values of δE are shown Fig.4cas function of temperature.Their values is near zero at high temperature, δE HT ∼ 0, and increase below T c = 7.2 K to reach, at low temperature, the theoretically expected value |δE LT | ∼ 4∆ bulk , Ref.1  , where ∆ bulk = 1.29 meV is the superconducting gap of bulk Pb.The value δE LT changes sign as one goes from the difference between two addition energies δE = eη(δV even − δV odd ) to the next difference δE = eη(δV odd − δV even ).
Fig. 4f, which shows that the critical temperature T c 6 K is smaller than the bulk value.The amplitude of the superconducting gap is obtained from ∆ = (δE(T = 1.2K) − δE HT ext (T = 1.2 K))/4.For this NC, the superconducting energy gap is about two times smaller than the bulk value, ∆ = ∆ bulk /2.

)
Measurements detailsThe microscope used is a low temperature, T base = 1.3 K, Joule-Thomson JT-STM from SPECS accommodated with a preparation chamber operating in Ultra High Vacuum at a base pressure P ∼ 10 −10 mbar.The differential conductance curves dI/dV are measured with a standard lock-in procedure.An AC signal of amplitude about 1 meV and frequency about 777 Hz is employed.

Figure 1 :
Figure 1: Pb NCs on InAs (110).(a) 1 µm×1 µm topographic STM image (1 V, 30 pA) of Pb NCs grown on the (110) InAs surface of sample A. Scale bar is 300 nm.(b) Zoom on 30 nm×30 nm area, showing a Pb NC.Scale bar is 10 nm.(c) 3D Laplacian ∆ xy z(x, y) image of a NC.(d) 6.5 nm×6.5 nm atomic resolution image of InAs (110) obtained near the NC.Scale bar is 2 nm.(e) DC measured at several distances from the Pb NC along the red arrow on panel b.(f) Zoom at low bias showing the conductance peaks due the discrete levels of the tip-induced quantum dot.(g) Sketch of the band bending below the Pb NC due to the pinning of the Fermi level at the charge neutrality level.(h) Sketch of the band bending induced by the tip leading to the formation of a quantum dot.16

Figure 2 : 2 F / 4 .
Figure 2: Pb NCs in the regime of Coulomb blockade.(a,b,c) 30 nm × 30 nm Laplacian ∆ xy z(x, y) topographic images (30 pA, 1V) of NCs of decreasing size, labeled I to III, where the hexagonal shape of the (111) facets is visible, as shown by the dash line on panel c.The scale bars correspond to 10 nm.The insets show the corresponding topographic STM images.(d) DC measured at the center of NC I and II, indicated by dots on the panels b,c.The addition voltages δV odd and δV even can be identified for each curve.The black arrows indicate the local maxima in the density of states due to quantum well states.The colored symbols identify the corresponding data points in panels f,h and Fig. 5. (e) DC map as function of sample bias and distance measured on NC III along the red arrow shown panel a.The black arrows indicate the Coulomb peak lines.(f) Capacitance C sub extracted from the Coulomb gap at zero bias.It scales linearly with the NC area.(g) Simulation of the DC for NC II using the weak coupling model 26 .(h) Normalized Coulomb peak amplitude A norm = (A peak − A base )/A base , this value decreases at the approach of the area πλ 2 F /4. (i) Sketch of electron occupation of NC II.

Figure 3 :Figure 4 :Figure 5 :
Figure 3: DCs for increasing NC volume.The colored symbols identify the corresponding data points in Fig. 2fh and Fig. 5.For each spectrum, the corresponding NC and the volume ratio V /V Anderson are shown on the right.Note that for the smallest NC (bottom black curve) no Coulomb peaks are observed, instead a large Coulomb gap and broad quantum well peaks are observed.