S. Bose, M.M. Ugeda, C.H. Michaelis, I. Brihuega, M. Ternes, K. Kern, A.M. Garc´ıa-Garc´ıa1 and J.D. Urbina2
Downscaling a superconductor and enhanc-ing superconductivity has been a major chal-lenge in the field of nanoscale superconductiv-ity. The advent of new tools of nanotechnol-ogy for both synthesis and measurement of sin-gle, isolated mesoscopic superconducting struc-tures has opened up the possibility to explore new and fascinating phenomena at reduced di-mensions. One exciting prediction is the occur-rence of shell effects in clean, superconducting nanoparticles [1].
Figure 7: Schematic description of shell effects that lead to an oscillation in the gap value with particle size. Left panel: Energy band diagram of a small particle where the discretization of the energy lev-els is arising from quantum confinement and where each level has further degeneracies due to geomet-ric symmetries. In superconductivity only the lev-els within the pairing region (Debye window), take part in pairing and consequently superconductivity.
Right panels: Expansion of this pairing region for three particles with slightly different heights so that the mean level spacing is similar. The number of lev-els 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 width.
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
dis-crete energy levels within the energy window around the Fermi energyEFin which the super-conducting pairing occurs fluctuates with very small changes in the system size. Consequently, this leads to fluctuations in the spectral density aroundEF. In weakly coupled superconductors, electronic pairing mainly occurs in a window around EFwhich has the size of the Debye en-ergy ED. Therefore, an increase (decrease) of the spectral density around EF will make pair-ing more (less) favorable, and will thus increase (decrease) the superconducting energy gap Δ. As a consequence, the gap becomes dependent on the size and the shape of the particle as schematically illustrated in Fig. 7.
The strength of fluctuations also increases with the symmetry of the particle, because symme-try introduces degeneracies in the spectrum. It is easy to see that these 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 num-ber of electrons taking part in pairing, fluctuates markedly. These degenerate levels are referred to as ‘shells’ in analogy with the electronic and nucleonic levels forming shells in atomic, clus-ter and nuclear physics (see Ref. [1] and refer-ences therein). For cubic or spherical particles, this might lead to a large modification of Δ.
Theoretically, these shell effects are described quantitatively by introducing finite-size correc-tions to the Bardeen-Cooper-Schrieffer (BCS) model.
In a recent letter, we have demonstrated for the first time through low-temperature scanning tunneling spectroscopic measurements on in-dividual superconducting nanoparticles of Pb and Sn, the existence of these shell effects and the influence of the superconducting coherence length on them [2].
Figure 8: (a) 3D representation of the experimen-tal setup. Superconducting nanoparticles deposited on a BN/Rh(111) substrate vary in height between 1 nm and 35 nm and are probed individually with the tip of the STM. (b) Representative 3D im-age (125×90 nm2) showing the Sn nanoparticles of varying sizes. (c), (d), Normalized conductance spectra of Pb (c) and Sn (d) nanoparticles of differ-ent height. Symbols are raw experimdiffer-ental data, solid lines are fits using equations (1) and (2).
Figure 8(a) shows a schematic of the exper-imental measurement where a scanning tun-neling microscope (STM) tip is used to mea-sure the tunneling density of states (DOS) of superconducting nanoparticles of both Pb and Sn. The nanoparticles of 1–35 nm height were grownin situon top of a BN/Rh(111) surface by means of buffer layer-assisted growth where the
BN having a bandgap of≈6 eV acts as a decou-pling layer from the underlying metal surface.
A typical representative STM topographic im-age for Sn nanoparticles with varying size on a BN/Rh(111) substrate is shown in Fig. 8(b). We take the height of the nanoparticle as our refer-ence because it is measured with a high degree of accuracy with the STM. The differential con-ductance dI/dV versus V for a selection of Pb and Sn nanoparticles at a temperature of 1.2 – 1.4 K are plotted in Figs. 8(c) and (d). We fitted each spectrum with the tunneling equation G(V) =GN d
dV
∫∞
−∞
NS(E)(f(E)−f(E−eV))dE, (5) where NS(E) is the quasiparticle excitation spectrum of the superconductor, f(E) is the Fermi-Dirac distribution function, and GN is the normal-state conductance of the tunnel junction.The quasiparticle excitation spectrum NS(E)is given by
NS(E,Γ,Δ) =ℜ
⎡
⎣√ ∣E∣+iΓ(T) [∣E∣+iΓ(T)]2−Δ(T)2
⎤
⎦, (6)
where Δ(T) is the superconducting energy gap andΓ(T)is a phenomenological broadening pa-rameter that incorporates all broadening aris-ing from any non-thermal sources and that is conventionally associated with the finite life-time (Γ∼ℏ/τ) of the quasiparticles. There is an excellent agreement between the experimental data and the theoretical fits, giving unique val-ues ofΔandΓ(Fig. 9(a) and (b)).
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. 8(c)), whereas for Sn nanoparticles (Fig. 8(d)) there is a non-monotonic behavior that strongly depends on the particle size regime. We observe that al-though the large Sn particles (>20 nm) differ-ing by a size of ≈1 nm have similar DOS sig-nifying similar gaps, there is a large difference in the DOS and henceΔ, for the smaller Sn par-ticles (<15 nm) even if they differ by less than 1 nm in size.
Figure 9: (a)–(b) Comparison of the variation of the superconducting energy gap (Δ) normalized to the bulk gap and the broadening parameter (Γ) at low temperature (T= 1.2 – 1.4 K) for Pb and Sn nanoparticles with different height. The solid lines in (a) are guides to the eye. (c) Variation of normalized Sn gap with particle height. (d) Variation in the average oscillations in the gap for Pb and Sn with particle height. Filled symbols are obtained from the experimental data and the solid lines are obtained from theoretical calculations.
The difference in the two systems is shown more clearly in Fig. 9(a), where we plot the nor-malized gap width. For Pb,Δdecreases mono-tonically with decrease in particle size, whereas there is a huge variation in the gap values for Sn below a particle size of 20 nm. 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, supercon-ductivity is destroyed below a critical particle size, which is consistent with the Anderson cri-terion [3], according to which superconductiv-ity should be completely destroyed for parti-cle sizes where the mean level spacing becomes equal to the bulk superconducting energy gap.
From the two parameters characterizing the su-perconducting state of our nanoparticles,Δand Γ, onlyΓevolves in a similar way as a function of particle size both for Pb and Sn (Fig. 9(b)).
In both systems, we observe an increase with reduction in particle size. Interestingly, it seems that superconductivity is limited to sizes where Γ<ΔBulk. At smaller sizes superconductivity is completely suppressed in both systems. This in-dicates thatΓmay have a particular significance in our measurements. To understand the behav-ior 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 geome-tries. Similarly, because in a zero-dimensional
superconductor the number of electrons taking part in superconductivity decreases, we expect an increase in the uncertainty in the phase of the superconducting order parameter. The in-creased fluctuations in the superconducting or-der parameter are expected to increaseΓas fluc-tuations act as a pair-breaking effect. Therefore, we associateΓwith the energy scale related to quantum fluctuations. Our results indicate that in 0D systems the presence of quantum fluctua-tions of the phase whereΓ>ΔBulk set the limit to superconductivity and this corresponds to the size consistent with the Anderson criterion [3].
We focus now on the main result, the variation ofΔwith particle size in Sn nanoparticles, and the observed striking difference with Pb. To in-terpret the experimental results we carried out a theoretical study of finite-size corrections in the BCS formalism [2]. There, we primarily focus only on the finite-size corrections to the BCS gap equation because the corrections to the BCS mean field approximation lead to a monotonic decrease in the gap and are not responsible for the observed oscillations in Sn nanoparticles.
For the correction to the BCS gap equation, two types of correction are identified, smooth and fluctuating. The former depends on the surface and volume of the grain and always enhances the gap with respect to the bulk. As this contri-bution decreases monotonically with the system size, it is not relevant in the description of the experimental fluctuations.
For Sn, a weak-coupling superconductor, a sim-ple BCS formalism is capable of providing a good quantitative description of superconduc-tivity. As explained in the introduction, the gap oscillations arise from the discreteness of the level spectrum (Fig. 7). Furthermore, solving the BCS equations self-consistently shows that the presence of degeneracies will enhance the gap fluctuations [2]. Large degeneracies are typ-ical for grains with symmetry axes in which the energy levels are degenerate in a quantum num-ber. A typical example is the sphere with three
axes of symmetry. In this case, each level in the energy spectrum with an angular momen-tum quanmomen-tum numberlis 2l+ 1 times degener-ate.
An important parameter in the BCS formalism is the effective coupling constant λwhich acts between pairing electrons and is given by the attractive electron-phonon coupling minus the Coulomb repulsion. A natural choice for Sn is λ= 0.25 as this leads to the bulk gap and the co-herence length consistent with the experimen-tal values of these observables. The magnitude of the fluctuations will strongly depend on the shape of the grain as expected from the theory of shell effects [1]. From the experimental topo-graphic images of the nanoparticles, we can in-fer that the shape is very close to being a hemi-sphere. However, because of the convolution between the radius of the probing tip with the radius of the nanoparticles in STM images this cannot be said with certainty. A statistical anal-ysis of the nanoparticle images revealed that the deviations from an ideal hemispherical shape should not be larger than 15%. Hence, for cal-culations, we modeled the shape of the nanopar-ticles as being a spherical cap withh/R>0.85.
In Fig. 9(c) we plot the calculated normalized gap as obtained from BCS theory as a function of h and superimpose the experimental results of Sn nanoparticles as filled symbols. Here, the data are normalized with respect to the average gap value obtained experimentally for which we divided the particle size in small bins of 2 nm width and determined the average of the super-conducting energy gap in each bin. For a ratio ofh/Rranging between 0.9 and 0.95, we obtain a reasonably good quantitative matching with the theoretical results, indicating that finite-size corrections can satisfactorily explain the results of Sn nanoparticles [2].
The natural question that follows is why such oscillations are not observed for Pb nanopar-ticles (filled triangles in Fig. 9(a)). We recall that fluctuations in 0D systems have their ori-gin in the discreteness of the spectrum and any
mechanism that induces level broadening will suppress these oscillations. The superconduct-ing coherence length ξ of Pb (ξ= 80 nm) is much shorter than that of Sn (ξ= 240 nm) and will introduce a much stronger level broaden-ing which is proportional to νF/ξ with νF as the spectral density at the Fermi energy. More-over, because interactions are much stronger in Pb, the lifetime of the quasiparticles is shorter and a further level broadening is expected. In Fig. 9(d) we plot the average oscillations ob-tained from both experiments and theory as a function of particle height for Pb and Sn nanoparticles. These average oscillations are the standard deviation of the gap from the av-erage value. We observe a good matching be-tween theory and experiments. We would like to point out that for Pb, the BCS description is an oversimplified model which is not able to sat-isfyingly obtain the correct average gap values.
However, to compute the oscillations in the gap and to check the suppression of the shell effects,
BCS gives a reasonably good description for the strong-coupling Pb.
Our results indicate that for any classical BCS superconductor with large quantum coherence lengths it is possible to enhance the supercon-ducting energy gap by large factors (here up to 60%) by tuning only the particle size. This may prove to be very useful in the case of fullerides or hexaborides that are known to show a rela-tively highTc in the bulk.
References:
[1] Kresin, V.Z. and Y.N. Ovchinnikov.Physical Review B74, 024514 (2006).
[2] Bose, S., A.M. Garc´ıa-Garc´ıa, M.M. Ugeda, J.D. Urbina, C.H. Michaelis, I. Brihuega and K. Kern.Nature Materials9, 550–554 (2010).
[3] Anderson, P.W.Journal of Physics and Chemistry of Solids11, 26–30 (1959).
1UTL, Lisboa
2Universit¨at Regensburg