**UFRJ**

# Observation of shell effects in superconducting nanoparticles of Sn

(Parte **1** de 2)

Observation of shell effects in superconducting nanoparticles of Sn

Sangita Bose1*, Antonio M. García-García2*, Miguel M. Ugeda1,3, Juan D. Urbina4, Christian H. Michaelis1, Ivan Brihuega1,3* and Klaus Kern1,5

In a zero-dimensional superconductor, quantum size effects1,2 (QSE) not only set the limit to superconductivity, but are also at the heart of new phenomena such as shell effects, which have been predicted to result in large enhancements of the superconducting energy gap3–6. Here, we experimentally demonstrate these QSE through measurements on single, isolated Pb and Sn nanoparticles. In both systems superconductivity is ultimately quenched at sizes governed by the dominance of the quantum fluctuations of the order parameter. However, before the destruction of superconductivity, in Sn nanoparticles we observe giant oscillations in the superconducting energy gap with particle size leading to enhancements as large as 60%. These oscillations are the first experimental proof of coherent shell effects in nanoscale superconductors. Contrarily, we observe no such oscillations in the gap for Pb nanoparticles, which is ascribed to the suppression of shell effects for shorter coherence lengths. Our study paves the way to exploit QSE in boostingsuperconductivityinlow-dimensionalsystems.

Downscaling a superconductor and enhancing superconductivityhasbeenamajorchallengeinthefieldofnanoscalesuperconductivity. The advent of new tools of nanotechnology for both synthesis and measurement of single, isolated mesoscopic superconducting structures has opened up the possibility to explore new and fascinating phenomena at reduced dimensions7–16. One of them, the parity effects in the superconducting energy gap, was demonstrated almost two decades ago in the only experiments that have been able to access the superconducting properties of an individual nanoparticle7 so far. Another exciting prediction is the occurrence of shell effects in clean, superconducting nanoparticles4–6.

The origin of shell effects is primarily due to the discretization of the energy levels in small particles that leads to substantial deviations of the superconducting energy gap from the bulk limit. For small particles, the number of discrete energy levels within a small energy window (pairing region) around the Fermi

energy (EF) fluctuates with very small changes in the system size. Consequently, this leads to fluctuations in the spectral density

around EF. In weakly coupled superconductors, electronic pairing mainly occurs in a window of size ED (Debye energy) around EF; therefore, an increase (decrease) of the spectral density around

EF will make pairing more (less) favourable, thereby increasing (decreasing) the energy gap (∆). As a consequence, the gap becomes dependent on the size and the shape of the particle (see schematic in Fig. 1). The strength of fluctuations also increases with the symmetry of the particle, because symmetry introduces degeneracies in the energy spectrum. It is easy to see that these

1Nanoscale Science Department, Max Planck Institute for Solid State Research, Heisenbergstrasse 1, Stuttgart, D-70569, Germany, 2CFIF, Instituto Superior Técnico, UTL, Av. Rovisco Pais, 1049-001 Lisboa, Portugal, 3Univ. Autonoma Madrid, Dept. Fis. Mat. Condensada, E-28049 Madrid, Spain, 4Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany, 5Institut de Physique de la Matière Condensée, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland. *e-mail:sangita.bose@fkf.mpg.de; ag3@Princeton.EDU ivan.brihuega@uam.es

degenerate levels will enhance the fluctuations in the spectral density and also in the gap as the number of levels within ±ED of EF, and consequently the number of electrons taking part in pairing,fluctuatesmarkedly.Thesedegeneratelevelswillbereferred to as ‘shells’ in analogy with the electronic and nucleonic levels forming shells in atomic, cluster and nuclear physics (see ref. 3 and references therein). For cubic or spherical particles, this might lead to a large modification of ∆. Theoretically, these shell effects are describedquantitativelybyintroducingfinite-sizecorrectionstothe Bardeen–Cooper–Schrieffer (BCS) model5,6. In this letter, through our scanning tunnelling spectroscopic measurements on individual superconducting nanoparticles of Pb and Sn, we demonstrate for the first time the existence of these shell effects and the influence of the superconducting coherence length on them.

Figure 2a shows a schematic of the experimental measurement where a scanning tunnelling microscope (STM) tip is used to measure the tunnelling density of states (DOS) of superconducting nanoparticles of both Pb and Sn. A typical representative STM topographic image for Sn nanoparticles (for Pb nanoparticle topographic image, see ref. 17) with varying size on a BN/Rh(1) substrate (see the Methods section for details) is shown in Fig. 2b. We take the height of the nanoparticle as our reference because it is measured with a high degree of accuracy with the STM. The quasiparticle excitation spectra (conductance plots of dI/dV versus V normalized at +5mV) for a selection of Pb and Sn nanoparticles at a temperature of 1.2–1.4K are plotted in Fig. 2c–e. We fitted each spectrum with the tunnelling equation18,

=Gnn d dV where Ns(E) is the DOS of the superconductor, f (E) is the Fermi– DiracdistributionfunctionandGnn istheconductanceofthetunnel junction for V ∆/e. Ns(E) is given by:

where ∆(T) is the superconducting energy gap and Γ(T) is a phenomenological broadening parameter that incorporates all broadening arising from any non-thermal sources (conventionally it is associated with the finite lifetime (τ) of the quasiparticles,

Γ ∼ h/τ (ref. 19)). (This equation is phenomenological although it works in many different systems.) There is an excellent agreement between the experimental data and the theoretical fits, giving

550 NATURE MATERIALS | VOL 9 | JULY 2010 | w.nature.com/naturematerials

NATUREMATERIALSDOI:10.1038/NMAT2768 LETTERS hi h ~ 6 levels h ~ 12 levels h ~ 4 levels

Number of levels in the pairing region ∝ superconducting energy gap => gap h > gap h > gap h

Figure 1 | Schematic of shell effects. Schematic explaining the physical origin of shell effects in small particles that leads to an oscillation in the gap value with particle size. The left panel shows the energy band diagram of a small particle with a height h where the discretization of the energy levels is arising from quantum confinement. Also for a particle with definite axes of symmetry, each level has further degeneracies and each degenerate level in a small particle is referred to as the ‘shell’. Now, in superconductivity only the levels within the pairing region (Debye window) about the Fermi level, EF, take part in pairing and consequently superconductivity. We show the expansion of this pairing region for three particles with heights h1, h2 and h3 that are very close to each other (so that the mean level spacing is similar). The number of levels in this pairing window fluctuates depending on the position of the Fermi level in the three particles, which leads to the fluctuation in the gap (shell effects).

unique values of ∆ and Γ (plotted as a function of particle size in Fig. 2f and g respectively). Comparing the raw data for the Pb and Sn, we observe that there is a gradual decrease in the zero-bias conductance dip with particle size for Pb nanoparticles (Fig. 2c), whereas for Sn nanoparticles (Fig. 2d,e) there is a non-monotonic behaviour that strongly depends on the particle size regime. We observe that although the large Sn particles (>20nm) differing by a size of 1nm have similar DOS signifying similar gaps, there is a large difference in the DOS and hence ∆, for the smaller Sn particles (<15nm) even if they differ by less than 1nm in size. The difference in the two systems is shown more clearly in Fig. 2f , where we plot the normalized gap (normalized with respect to their bulk values). For Pb, ∆ decreases monotonically with decrease in particle size, whereas there is a huge variation in the gap values for Sn below a particle size of 20nm. For these small sizes, gap values differ even more than 100% for similar-sized Sn particles and enhancements as large as 60% with respect to the Sn bulk gap are found. In both systems however, superconductivity is destroyed below a critical particle size, which is consistent with the Anderson criterion2, according to which superconductivity should be completely destroyed for particle sizes where the mean level spacing becomes equal to the bulk superconducting energy gap (Supplementary Information). (In recent years, this criterion has been substantially refined. It is now accepted that superconductivity is destroyed at sizes depending on the parity of the grain and can be lower than that predicted by the Anderson criterion. See, ref. 20. We think that our experiments are not sensitive to the parity of a particle.) It is also worth noting that the average gap for the large Sn nanoparticles (20–30nm) shows an increase of 20% from the bulk value (Supplementary Information).

From the two parameters characterizing the superconducting state of our nanoparticles, ∆ and Γ, only Γ evolves in a similar way as a function of particle size both for Pb and Sn (Fig. 2g). In both systems, we observe an increase in Γ with reduction in particle size. Interestingly, it seems that superconductivity is limited to sizes where Γ < ∆bulk.. At smaller sizes, superconductivity is completely suppressed in both systems. This indicates that Γ may have a particular significance in our measurements. To understand the behaviour of Γ with particle size, we invoke the role of quantum fluctuations in small particles. It is known from both theoretical calculations and experiments that there should be an increase in the quantum fluctuations in confined geometries21–23, as observed in experiments on nanowires8. Similarly, because in a zero-dimensional (0D) superconductor the number of electrons taking part in superconductivity decreases, we expect an increase in the uncertainty in the phase of the superconducting order parameter18,19 (within a single particle, there will be a decrease in the long-range phase coherence). The increased fluctuations in the superconducting order parameter are expected to increase Γ (as fluctuations act as a pair-breaking effect). Therefore, we associate Γ with the energy scale related to quantum fluctuations. Our results indicate that in 0D systems the presence of quantum fluctuations of the phase (where Γ > ∆bulk) set the limit to superconductivity and this corresponds to the size consistent with the Anderson criterion23.

We focus now on the main result of this work, reflected in the variation of ∆ with particle size in Sn nanoparticles, and the observed striking difference with Pb. To interpret the experimental results we carry out a theoretical study of finite-size corrections in the BCS formalism in line with refs 4–6. We will primarily focus only on the finite-size corrections to the BCS gap equation because the corrections to the BCS mean field approximation5 lead to a monotonic decrease in the gap and are not responsible for the observed oscillations in Sn nanoparticles. (The leading correction to the BCS prediction is given by ∆=∆BCS−δ/2, where

∆BCS is the bulk gap and δ is the discrete energy level spacing of the nanoparticle. This predicts a decrease in the average gap with reduction in particle size and can qualitatively explain the monotonic dependence of the average gap in both Pb and Sn nanoparticles.) For the correction to the BCS gap equation, two types of correction are identified, smooth and fluctuating4,5. The former depends on the surface and volume of the grain and always

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LETTERS NATUREMATERIALSDOI:10.1038/NMAT2768 a

c ed

BN nano-par

Rh(1 )

BN Rh(1)

g Particle height (nm)Particle height (nm)

STM tip

Pb or Sn nanoparticle

30 nm 16.5 nm 13 nm 8.3 nm 5 nm 4.4 nm BN substrate d I /d

10.5 nm, Δ = 0.50 meV 10.0 nm, Δ = 0.89 meV d I /d V (arb. units) 0.5

29.5 nm, Δ = 0.71 meV 29.0 nm, Δ = 0.70 meV

/ bulk

Sn nanoparticles Pb nanoparticles

(meV)

Pb nanoparticles Sn nanoparticles for Pb = 1.36 meV

(nm) d I /d V (arb. units)

Figure 2 | Experimental configuration and low-temperature superconducting properties of single, isolated Pb and Sn nanoparticles: observation of shell effects. a, 3D representation of the experimental set-up. Superconducting nanoparticles deposited on a BN/Rh(1) substrate vary in height between 1 and 35nm and are probed individually with the help of the STM tip. b, 125×90nm2 3D STM image showing the Sn nanoparticles of varying sizes deposited on the BN/Rh (1) substrate. The image is taken at a sample bias voltage of 1V with a tunnelling current of It =0.1nA. This is representative of topographic images of the superconducting Pb and Sn nanoparticles on the substrate. c–e, Normalized conductance spectra (dI/dV versus V, normalized at a bias voltage of 5mV). The circles are the raw experimental data and the solid lines are the theoretical fits using equations (1) and (2) (see text). c, For Pb nanoparticles of different heights at T =1.2K. d, For two large Sn nanoparticles with heights of 29.5 and 29.0nm at T =1.4K. e, For two small Sn nanoparticles with heights of 10.5 and 10.0nm at T =1.4K. f,g, Comparison of the variation of superconducting energy gap (∆) and broadening parameter (Γ) at low temperature (T =1.2–1.4K) for different Pb and Sn nanoparticles respectively as a function of particle height. The gap is normalized with respect to the bulk gaps. The solid lines in f are guides to the eye.

enhances the gap with respect to the bulk. As this contribution decreases monotonically with the system size, it is not relevant in thedescriptionoftheexperimentalfluctuationsof∆.Toexplainthe observed fluctuations of gap in Sn, we start with the self-consistent equation for the BCS order parameter5,6,

a typical length of the grain, ν(0) is the spectral density at the Fermi where εi are the eigenvalues, with degeneracy gi, and ψε(r) are the eigenfunctions with energy ε of a free particle confined inside the grain. For Sn, a weak-coupling superconductor, a simple BCS formalism is capable of providing a good quantitative description of superconductivity. Equation (3) can be further simplified by noting4–6 that for kFL 1 gap oscillations are controlled only by ν(ε). In our experiment (where L ranges between 2 and 60nm) we are always in this limit as the Fermi wavevector kF = 16.4nm−1 in Sn. As explained in the introduction, the gap oscillations arise from the discreteness of the level spectrum (Fig. 1), which is reflected in the expression of the spectral density ν(ε), and hence equation (3) leads to an oscillatory variation of gap with particle size. It can also beseenfromtheexpressionofν(ε)thatthepresenceofdegeneracies

(Parte **1** de 2)