Abstract
Superconductivity is inevitably suppressed in reduced dimensionality^{1,2,3,4,5,6,7,8,9}. Questions of how thin superconducting wires or films can be before they lose their superconducting properties have important technological ramifications and go to the heart of understanding coherence and robustness of the superconducting state in quantumconfined geometries^{1,2,3,4,5,6,7,8,9}. Here, we exploit quantum confinement of itinerant electrons in a soft metal, Pb, to stabilize superconductors with lateral dimensions of the order of a few millimetres and vertical dimensions of only a few atomic layers^{10}. These extremely thin superconductors show no indication of defect or fluctuationdriven suppression of superconductivity, and sustain supercurrents of up to 10% of the depairing current density. Their magnetic hardness implies a Beanlike critical state with strong vortex pinning that is attributed to quantum trapping of vortices. This study paints a conceptually appealing, elegant picture of a model nanoscale superconductor with calculable criticalstate properties and surprisingly strong phase coherence. It indicates the intriguing possibility of exploiting robust superconductivity at the nanoscale.
Main
Pb has the peculiar property that the wavelength of the highest occupied electron level matches the atomlayer spacing along the 〈111〉 crystal direction almost perfectly, so that two atomic layers of Pb accommodate oneandahalf ‘Fermi wavelengths’. This electronic property can have profound consequences for the heteroepitaxial growth of very thin Pb(111) films^{10}, provided that certain kinetic growth conditions are met^{11,12,13,14,15,16,17,18}. Between 200 and 250 K, atomically flat crystalline Pb films on Si(111) substrates evolve in a quasi bilayerbybilayer 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 singleatom layer or even a trilayer^{11}. 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 wavelength^{11}. 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 terraces^{6,11}. 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 T_{c} and the upper critical magnetic field H_{c2}(T), shown in Fig. 1, and the nonequilibrium critical current density J_{c}(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 thick^{15,18}.
The inset of Fig. 1 shows T_{c} plotted as a function of 1/d. The T_{c}(d) data are perfectly linear in 1/d, and extrapolate to the bulk . Accordingly, T_{c}(d)=T_{c0}(1−d_{c}/d). The extrapolated threshold d_{c} 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, T_{c}(d) no longer extrapolates to the bulk T_{c0}. This suggests that the interfacial wetting layer does not strongly participate in the superconductivity. Our T_{c} data do not exhibit quantumsizeeffect oscillations, which are only observable in the classical layerbylayer growth regime^{6}. Theoretically, a 1/d dependence (solid line in Fig. 1, inset) arises naturally from the inclusion of a surfaceenergy term in the Ginzburg–Landau (GL) freeenergy of a superconductor^{19}.
The fundamental length scales in the films are significantly affected by the twodimensional (2D) geometry. For instance, the GL coherence length ξ_{GL}(T) of the films is reduced significantly below the Bardeen–Cooper–Schrieffer (BCS) value of bulk Pb, ξ_{0}^{bulk}=905 Å. The inplane ξ_{GL}(T) was obtained from H_{c2}(T) measurements through the relation H_{c2}(T)=Φ_{0}/2πξ_{GL}^{2}(T), where Φ_{0} is the flux quantum^{20}. Figure 1 shows H_{c2}(T) of a few thin films. H_{c2}(T) varies nearly linearly with temperature, conforming to the standard GL dependence^{20} ξ_{GL}(T)∝(1−T/T_{c})^{−1/2}. The linear H_{c2}(T) was extrapolated to 0 K using Bulaevskii’s expression H_{c2}(0)≈0.65(−dH_{c2}/dT)T_{c}^{*} for 2D superconductors^{21}, giving ξ_{GL}(0)≈230 Å for a 9 ML Pb film. This significant reduction in coherence length, as compared with the BCS value ξ_{0}^{bulk} of bulk Pb, is attributed primarily to the reduced electronic mean free path l(d) in the films. Using the ‘full value’ expression^{20}
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/4e^{2}=6,454 Ω per square for the suppression of superconductivity^{2}, where h is Planck’s constant and e is the electron charge. Here we renormalized the BCS coherence length, ξ_{0}(d)=ξ_{0}^{bulk}×T_{c0}/T_{c}(d), to account for the slightly lower T_{c} in thin films. Evidently, l(d) and, consequently, ξ_{GL} are limited by boundary scattering.
Thus far, the anisotropic GL formalism works surprisingly well in this quantumconfined regime. However, H_{c2}(T) deviates significantly from linearity as T approaches T_{c} from below. Although a slight rounding of H_{c2}(T) near T_{c} may arise from structural inhomogeneities in conjunction with the boundary conditions for the pair wavefunction^{22,23}, the rounding observed here changes systematically with the fundamental nanoscale dimension d. Extrapolation of the linear H_{c2}(T) segments to a zero d.c. field (Fig. 1) produces a set of extrapolated temperatures T_{c}^{*}(d), which also scale linearly with 1/d (inset). Accordingly, in the GL regime below T_{c}^{*}(d), the slope −dH_{c2}/dT should be proportional to (d−d_{c}^{*})^{−1} as is indeed observed experimentally ( is the thickness where T_{c}^{*}(d) extrapolates to 0 K). One might interpret T_{c}^{*}(d) as an extrapolated GL or ‘mean field’ transition temperature. This, however, would imply that the region between T_{c}^{*}(d) and T_{c}(d) is dominated by 2D fluctuations with only a fluctuation conductivity. Instead, as demonstrated below, the H_{c2}(T) phase boundary coincides with the onset of irreversible magnetization, indicating the presence of a robust critical current density above T_{c}^{*}(d).
Figure 2a–d shows scanning tunnelling microscopy images of 7 and 9MLthick 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, respectively^{11}. They are stabilized by strong quantumsize effects and constitute quantumgrowth ‘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 nearperfect Beanlike critical state (that is, a very hard superconductor) with exceptionally strong vortex pinning^{20,24}. Using the critical state relation J_{c}=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 J_{c}=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.
A comprehensive set of J_{c}(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 criticalstate model, the peak position of the imaginary a.c. susceptibility χ′′=(m′′/h_{a.c.}) corresponds to the condition J_{c}=1.03h_{a.c.}d^{−1}, where h_{a.c.} is the a.c. modulation amplitude^{25,26}. For the Pb films, the onset of χ′′ always coincides with the onset of diamagnetic screening (or χ′), indicating that the thermodynamic H_{c2}(T) phase boundary closely coincides with the onset of irreversible magnetization and establishment of a Beanlike critical state. Notice that the J_{c} obtained in this way does not depend on the sample radius r. Hence, the close agreement between J_{c} 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 J_{c} of the underdosed films is as high as ∼10% of the depairing current density, J_{d}≈H_{c}/λ_{eff}, which is about 20 MA cm^{−2} for a 9 ML thin film, an amazing result indeed. Here, we estimated the scatteringincreased London penetration depth λ_{eff} and thermodynamic critical field H_{c} assuming that, in accordance with the Anderson theorem^{20}, the product λ ξ=(λ_{L}ξ_{0}′)_{bulk}≈(λ_{eff}ξ_{GL})_{film} is independent of scattering and thus independent of the film thickness. Accordingly, λ_{eff}(0)≈1,500 Å for a 9 ML film.
Figure 3 shows the temperaturedependent 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 J_{c}, 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 lineenergy per unit length is given by^{20} ɛ_{0}=Φ_{0}^{2}/(4πλ_{eff})^{2}. Consequently, 2MLdeep voids are effective trapping centres for cores of the Pearl vortices as they significantly reduce their line energy inside the voids. In other words, nanovoids attract and pin vortices, whereas nanomesas repel them. The pinning centres can be viewed as a uniformdepth segment of a columnar defect. A slight modification of Nelson and Vinokur’s expression for columnar defects^{27} provides the theoretical estimate,
where Δd=2 ML is the depth of the pinning centres (c is the speed of light). This gives J_{c}(T=0) of about 4 MA cm^{−2} and a vortex pinning energy U_{0}=ɛ_{0}Δd of ∼500 K. The estimated J_{c} value is remarkably close to the experimental 2.8 MA cm^{−2}. Furthermore, in the GL regime below T_{c}^{*}, the critical current J_{c}∝(T−T_{c}^{*})^{−3/2} (Fig. 3 inset), consistent with the (1−T/T_{c})^{−1/2} variation of λ_{eff} and ξ in GL theory, and the observed H_{c2}(T).
The robustness of the critical state can be quantified in terms of the creep exponent n in the current–voltage relation E=E_{c}(J/J_{c})^{n} (E is the electric field)^{25,26}. Large values of n imply a sharp demarcation between lossfree 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 diagram^{25,26}. 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 exponents^{25,26} indicates that n>100.
This large nvalue was verified independently by realtime measurements of the current decay rate (Fig. 4b). This experiment was done with the sample stationary^{28}, and the data were analysed according to J(t)=J_{0}−J_{1}ln(1+t/τ), where τ is an initial transient time^{29}. The normalized creep rate S then follows from the relation in a Kim–Anderson formulation^{20}. 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 U_{0}=ɛ_{0}Δd≈500 K. At the temperatures accessible, there is no evidence for quantum tunnelling of vortices.
The experiments reveal, with considerable precision, the thermodynamics and nonequilibrium response of a hard nanoscale superconductor. The surprising robustness follows from the minimal disorder in these highquality films, coupled with efficient quantumgrowthbased 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 timeenergy uncertainty principle ΔE≥ħ/τ, where τ(d)=l(d)/v_{F};v_{F} 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 intersubband spacing of quantumconfined Pb^{15,17}. 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 lowdimensional superconductivity with quantifiable parameters, but may also unveil novel and unexpectedly robust criticalstate properties that could be useful for superconductive nanodevices.
Methods
Pb films were grown^{11} 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 inphase term and a lossy, outofphase component, so that m=m′−i m′′. The T_{c} and H_{c2}(T) were obtained from the onset of m′(T) measured in zero and finite d.c. fields, respectively, while sweeping temperature T.
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Acknowledgements
We acknowledge discussions with L. Bulaevskii, A. Gurevich, V. G. Kogan, Q. Niu, and Z. Zhang. This work was funded primarily by the National Science Foundation under Contract No. DMR0244570, and sponsored in part by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, US Department of Energy, under contract DEAC0500OR22725 with Oak Ridge National Laboratory, managed and operated by UTBattelle, LLC.
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Özer, M., Thompson, J. & Weitering, H. Hard superconductivity of a soft metal in the quantum regime. Nature Phys 2, 173–176 (2006). https://doi.org/10.1038/nphys244
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