Main

Pb has the peculiar property that the wavelength of the highest occupied electron level matches the atom-layer spacing along the 〈111〉 crystal direction almost perfectly, so that two atomic layers of Pb accommodate one-and-a-half ‘Fermi wavelengths’. This electronic property can have profound consequences for the hetero-epitaxial growth of very thin Pb(111) films10, provided that certain kinetic growth conditions are met11,12,13,14,15,16,17,18. Between 200 and 250 K, atomically flat crystalline Pb films on Si(111) substrates evolve in a quasi bilayer-by-bilayer fashion, beginning at five monolayers (ML)11,12,13,14,15,16,17,18. The bilayer growth is periodically interrupted by the growth of a single-atom layer or even a trilayer11. This remarkable growth mode can be attributed to strong Friedel oscillations in the electron density, in conjunction with the nearly perfect matching of lattice spacing and Fermi wavelength11. In this ‘quantum growth’ regime, lattice strain is of minor importance because the metal is soft and quantum size effects are strong. The atomically flat Pb terraces are limited in size by the width of the underlying Si terraces6,11. This level of smoothness, which is astonishing for metal growth on semiconductors, provides the opportunity to explore the superconductive properties of highly crystalline films with atomically controlled thickness.

Using the contactless magnetic techniques described in the Methods section, we obtained the thermodynamic critical temperature Tc and the upper critical magnetic field Hc2(T), shown in Fig. 1, and the non-equilibrium critical current density Jc(T,H) for films with thickness d=5–18 ML, where 1 ML=0.286 nm. This and all layer counts exclude the interfacial wetting layer, which is one atomic layer thick15,18.

Figure 1: Equilibrium superconductive properties of ultrathin, atomically flat Pb films.
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The main panel shows upper critical fields as a function of temperature. Notice the systematic thickness dependence of the slope −dHc2/dT, and the rounding and flattening of Hc2 in the vicinity of Tc, particularly for the thinner films. The rounding suggests the existence of a Tc*, as indicated for the 9 ML film. The inset shows superconductive Tc and Tc* plotted as function of the inverse of the film thickness d (excluding the 1 ML wetting layer). The critical temperatures Tc are obtained from the onset of diamagnetic screening, and the values Tc* are obtained from the extrapolated Hc2(T ) data, using a 10 mG a.c. probing amplitude. For , Tc* extrapolates to (7.03±0.13 K), about standard deviations from the bulk Tc0 (7.2 K). In the blue-shaded region, ξ is limited by boundary scattering and, accordingly, Hc2(T ) is linear in temperature. The yellow region, where ξGL(T ) becomes very large, indicates a strongly superconductive regime where the pair function averages over short-range scattering inhomogeneities22 and becomes increasingly insensitive to them.

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The inset of Fig. 1 shows Tc plotted as a function of 1/d. The Tc(d) data are perfectly linear in 1/d, and extrapolate to the bulk . Accordingly, Tc(d)=Tc0(1−dc/d). The extrapolated threshold dc for the emergence of superconductivity is roughly 1.5 ML=4.3 Å. Note that if the wetting layer is included in the layer count, Tc(d) no longer extrapolates to the bulk Tc0. This suggests that the interfacial wetting layer does not strongly participate in the superconductivity. Our Tc data do not exhibit quantum-size-effect oscillations, which are only observable in the classical layer-by-layer growth regime6. Theoretically, a 1/d dependence (solid line in Fig. 1, inset) arises naturally from the inclusion of a surface-energy term in the Ginzburg–Landau (GL) free-energy of a superconductor19.

The fundamental length scales in the films are significantly affected by the two-dimensional (2D) geometry. For instance, the GL coherence length ξGL(T) of the films is reduced significantly below the Bardeen–Cooper–Schrieffer (BCS) value of bulk Pb, ξ0bulk=905 Å. The in-plane ξGL(T) was obtained from Hc2(T) measurements through the relation Hc2(T)=Φ0/2πξGL2(T), where Φ0 is the flux quantum20. Figure 1 shows Hc2(T) of a few thin films. Hc2(T) varies nearly linearly with temperature, conforming to the standard GL dependence20 ξGL(T)∝(1−T/Tc)−1/2. The linear Hc2(T) was extrapolated to 0 K using Bulaevskii’s expression Hc2(0)≈0.65(−dHc2/dT)Tc* for 2D superconductors21, giving ξGL(0)≈230 Å for a 9 ML Pb film. This significant reduction in coherence length, as compared with the BCS value ξ0bulk of bulk Pb, is attributed primarily to the reduced electronic mean free path l(d) in the films. Using the ‘full value’ expression20

one estimates that . This corresponds to sheet resistances R=200–20 Ω per square for d=5–18 ML, which lie far below the critical value ∼h/4e2=6,454 Ω per square for the suppression of superconductivity2, where h is Planck’s constant and e is the electron charge. Here we renormalized the BCS coherence length, ξ0(d)=ξ0bulk×Tc0/Tc(d), to account for the slightly lower Tc in thin films. Evidently, l(d) and, consequently, ξGL are limited by boundary scattering.

Thus far, the anisotropic GL formalism works surprisingly well in this quantum-confined regime. However, Hc2(T) deviates significantly from linearity as T approaches Tc from below. Although a slight rounding of Hc2(T) near Tc may arise from structural inhomogeneities in conjunction with the boundary conditions for the pair wavefunction22,23, the rounding observed here changes systematically with the fundamental nanoscale dimension d. Extrapolation of the linear Hc2(T) segments to a zero d.c. field (Fig. 1) produces a set of extrapolated temperatures Tc*(d), which also scale linearly with 1/d (inset). Accordingly, in the GL regime below Tc*(d), the slope −dHc2/dT should be proportional to (d−dc*)−1 as is indeed observed experimentally ( is the thickness where Tc*(d) extrapolates to 0 K). One might interpret Tc*(d) as an extrapolated GL or ‘mean field’ transition temperature. This, however, would imply that the region between Tc*(d) and Tc(d) is dominated by 2D fluctuations with only a fluctuation conductivity. Instead, as demonstrated below, the Hc2(T) phase boundary coincides with the onset of irreversible magnetization, indicating the presence of a robust critical current density above Tc*(d).

Figure 2a–d shows scanning tunnelling microscopy images of 7- and 9-ML-thick Pb films on Si(111), with the corresponding d.c. magnetization loops in Fig. 2e–h. Because we intentionally underdosed or overdosed the amount of Pb, we observed either nanoscale voids, shown in Fig. 2a and c, or nanoscale mesas, shown in Fig. 2b and d. The voids and mesas are exactly two atom layers deep or two atoms layers tall, respectively11. They are stabilized by strong quantum-size effects and constitute quantum-growth ‘defects’. The contrast in magnetic properties of the underdosed and overdosed films is striking. In particular, the highly irreversible magnetization of the underdosed films with voids (Fig. 2e and g) implies a near-perfect Bean-like critical state (that is, a very hard superconductor) with exceptionally strong vortex pinning20,24. Using the critical state relation Jc=30m/V r, where m is the d.c. magnetic moment, V is the film’s volume, and r the macroscopic radius of the sample (≈1.5 mm), we obtain Jc=2.0 MA cm−2 for the underdosed 9 ML film at 2 K in 5 Oe d.c. field. In contrast, small d.c. fields quickly depress the magnetization of the overdosed films (Fig. 2f and h), indicating that vortex pinning with mesas is weak.

Figure 2: Scanning tunnelling microscopy images and the corresponding d.c. magnetization loops.
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Quantum growth defects in 7-ML Pb film consist of eithera, two-atom layer deep voids or b, two-atom layer tall mesas, in films with a small shortage or excess of Pb, respectively. Similar quantum growth defects are evident in 9-ML films, with c, voids or d, mesas. The 700×700 nm2 images are obtained with STM. The area density of defects is comparable to the density of vortices in a ‘matching field’ of ∼5 kG. The corresponding d.c. magnetic response of these films, nominally 3×3 mm2, is shown in e–h. Quantum voids produce ‘hard’ hysteresis loops as shown in e and g, whereas quantum mesas produce ‘soft’ hysteresis loops shown in f and h. Hard hysteresis indicates strong vortex pinning by the voids.

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A comprehensive set of Jc(H,T) data was obtained by measuring χ′′ as a function of d.c. magnetic field with various superimposed a.c. probing fields. Within the critical-state model, the peak position of the imaginary a.c. susceptibility χ′′=(m′′/ha.c.) corresponds to the condition Jc=1.03ha.c.d−1, where ha.c. is the a.c. modulation amplitude25,26. For the Pb films, the onset of χ′′ always coincides with the onset of diamagnetic screening (or χ′), indicating that the thermodynamic Hc2(T) phase boundary closely coincides with the onset of irreversible magnetization and establishment of a Bean-like critical state. Notice that the Jc obtained in this way does not depend on the sample radius r. Hence, the close agreement between Jc values from the a.c. susceptibility (2.8 MA cm−2) and d.c. magnetization (2.0 MA cm−2) implies that the critical current paths closely follow the circumference of the sample. The ∼2 MA cm−2 critical currents are truly macroscopic despite the extremely thin geometry and despite the presence of surface steps. In fact, the Jc of the underdosed films is as high as ∼10% of the depairing current density, Jd≈Hc/λeff, which is about 20 MA cm−2 for a 9 ML thin film, an amazing result indeed. Here, we estimated the scattering-increased London penetration depth λeff and thermodynamic critical field Hc assuming that, in accordance with the Anderson theorem20, the product λ ξ=(λLξ0′)bulk≈(λeffξGL)film is independent of scattering and thus independent of the film thickness. Accordingly, λeff(0)≈1,500 Å for a 9 ML film.

Figure 3 shows the temperature-dependent critical currents for 7 and 9 ML films in 5 G d.c. fields. Again, the contrasting behaviour between the voids and mesas is clearly evident. Presumably, the nanoscale voids greatly enhance Jc, which must be attributed to strong vortex pinning. Here, we show that the magnitude and temperature dependence of the critical current can be calculated quite precisely from the known geometry of the voids. The scale of vortex line-energy per unit length is given by20 ɛ0=Φ02/(4πλeff)2. Consequently, 2-ML-deep voids are effective trapping centres for cores of the Pearl vortices as they significantly reduce their line energy inside the voids. In other words, nano-voids attract and pin vortices, whereas nano-mesas repel them. The pinning centres can be viewed as a uniform-depth segment of a columnar defect. A slight modification of Nelson and Vinokur’s expression for columnar defects27 provides the theoretical estimate,

where Δd=2 ML is the depth of the pinning centres (c is the speed of light). This gives Jc(T=0) of about 4 MA cm−2 and a vortex pinning energy U0=ɛ0Δd of ∼500 K. The estimated Jc value is remarkably close to the experimental 2.8 MA cm−2. Furthermore, in the GL regime below Tc*, the critical current Jc∝(T−Tc*)−3/2 (Fig. 3 inset), consistent with the (1−T/Tc)−1/2 variation of λeff and ξ in GL theory, and the observed Hc2(T).

Figure 3: Critical current density of 7 and 9 ML thick Pb films.
figure 3

Open symbols refer to films with nanoscale voids; filled symbols refer to films with nanoscale mesas. The inset shows that Jc∝(Tc*−T )−3/2 for the 7 and 9 ML films below their respective Tc*, as expected from the modified Nelson–Vinokur expression for Jc(T ), in conjunction with the (1−T/Tc)−1/2 variation of λeff and ξ in GL theory.

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The robustness of the critical state can be quantified in terms of the creep exponent n in the current–voltage relation E=Ec(J/Jc)n (E is the electric field)25,26. Large values of n imply a sharp demarcation between loss-free current flow and dissipative conduction. The Bean critical state corresponds to , whereas n=1 corresponds to ohmic transport. The creep exponent can be deduced by plotting χ′ versus χ′′ in a Cole–Cole diagram25,26. Figure 4a shows that the magnetic data of the 9 ML film collapse beautifully, lying very near the theoretical Cole–Cole curve for the Bean critical state. Direct comparison between the data and theoretical Cole–Cole plots for finite creep exponents25,26 indicates that n>100.

Figure 4: Cole–Cole plot of real and imaginary diamagnetic susceptibility, and a plot showing current decay with time.
figure 4

a, Cole–Cole plot of the real and imaginary diamagnetic susceptibility of an underdosed 9 ML thin film. The (χ′,χ′′) data span a wide range of temperatures (1.8 K to Tc) and a.c. modulation amplitudes (200–1,100 mG). The d.c. magnetic field is 5 G. The data collapse perfectly, adjoining the theoretical Cole–Cole curve of the Bean critical state () of a thin disk, which is indicated by the solid line. Similar results were obtained in d.c. fields up to 5 kOe. b, Current decay with time. A solid line is fitted to the theoretical time dependence (given in the text)29. The current decay was analysed within the ‘flux creep’ framework of Anderson–Kim26. The Cole–Cole plot shown in a, and the current decay rate shown in b, both indicate a very sharp demarcation between loss-free current flow and dissipative conduction at the upper critical field, and consistently indicate a very large value of the current exponent (n≥100), meaning very hard superconductivity.

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This large n-value was verified independently by real-time measurements of the current decay rate (Fig. 4b). This experiment was done with the sample stationary28, and the data were analysed according to J(t)=J0−J1ln(1+t/τ), where τ is an initial transient time29. The normalized creep rate S then follows from the relation in a Kim–Anderson formulation20. This procedure yields a creep exponent n=1+S−1 of ∼100, consistent with conclusions from the Cole–Cole analysis; it gives a pinning energy , in good agreement with our previous estimate U0=ɛ0Δd≈500 K. At the temperatures accessible, there is no evidence for quantum tunnelling of vortices.

The experiments reveal, with considerable precision, the thermodynamics and non-equilibrium response of a hard nanoscale superconductor. The surprising robustness follows from the minimal disorder in these high-quality films, coupled with efficient quantum-growth-based trapping of magnetic flux lines. In nanostructures, thickness variations are atomically discrete and comparable, in magnitude, to the overall thickness or size. Therefore, strong vortex pinning may be realized in many superconducting nanostructures whose size can be controlled with atomic precision, as demonstrated here.

Finally, the applicability of 3D anisotropic GL theory for this extreme 2D geometry may be rationalized on the basis of the time-energy uncertainty principle ΔE≥ħ/τ, where τ(d)=l(d)/vF;vF is the Fermi velocity and l(d) is the mean free path. In this extremely thin limit, ΔE is at least a few tenths of an electron volt, which is comparable to the inter-subband spacing of quantum-confined Pb15,17. Hence, carriers may easily be scattered between 2D subbands, so that superconductivity is no longer strictly 2D. Realizing strict 2D superconductivity (Cooper pairing) thus requires long mean free paths, meaning specularly reflecting interfaces. It should be possible to push superconductivity closer to this ‘clean limit’ by judicious choices of the substrate and capping layer. Hopefully this work will inspire new efforts towards creating atomically abrupt nanoscale superconductors with nearly perfect interfaces. Such structures not only present an ideal testing ground for theories of low-dimensional superconductivity with quantifiable parameters, but may also unveil novel and unexpectedly robust critical-state properties that could be useful for superconductive nanodevices.

Methods

Pb films were grown11 in ultrahigh vacuum and protected by an amorphous Ge cap layer. Their superconductive properties were measured inductively as a function of temperature and perpendicular magnetic field, using a SQUID magnetometer. For a.c. studies, a small 100 Hz a.c. probing field was superimposed on the d.c. field. These external fields generate circulating screening currents and an associated magnetic moment m. The a.c. moment contains a diamagnetic in-phase term and a lossy, out-of-phase component, so that m=m′−i m′′. The Tc and Hc2(T) were obtained from the onset of m′(T) measured in zero and finite d.c. fields, respectively, while sweeping temperature T.