Correction to: Nature Communicationshttps://doi.org/10.1038/ncomms16025, published online 06 July 2017 The original version of this Article included the authors Kun Zuo and Vincent Mourik who wish to be removed from authorship. Consequently, the author affiliations for these authors have been removed from the ‘Authors and Affiliations’ section. The original version of the ‘Contributions’ statement, which read “H.Z. and Ö.G. fabricated the devices, performed the measurements and analysed the data. S.C.-B. performed the TEM analysis. M.P.N. and M.W. performed the numerical simulations. K.Z., V.M., F.K.d.V., J.v.V., M.W.A.d.M., J.D.S.B., D.J.v.W., M.Q.-P., M.C.C. and S.G. contributed to the experiments. D.C., S.P. and E.P.A.M.B. grew the InSb nanowires. S.K. prepared the lamellae for the TEM analysis. K.W. and T.T. synthesized the h-BN crystals. L.P.K. supervised the project. All authors contributed to the writing of the manuscript”, has been amended to read “H.Z. and Ö.G. fabricated the devices, performed the measurements and analysed the data. S.C.-B. performed the TEM analysis. M.P.N. and M.W. performed the numerical simulations. F.K.d.V., J.v.V., M.W.A.d.M., J.D.S.B., D.J.v.W., M.Q.-P., M.C.C. and S.G. contributed to the experiments. D.C., S.P. and E.P.A.M.B. grew the InSb nanowires. S.K. prepared the lamellae for the TEM analysis. K.W. and T.T. synthesized the h-BN crystals. L.P.K. supervised the project. All authors contributed to the writing of the manuscript”. This has been corrected in both the PDF and HTML versions of the article.

Nature Communications is publishing an editorial expression of concern on the article “Ballistic superconductivity in semiconductor nanowires”, by H. Zhang et al. On 09 December 2021, the Editorial Staff was alerted by Vincent Mourik and two other researchers to potential problems in the manner in which raw data have been selected, processed and analysed. In response to these concerns, Nature Communications initiated an investigation by contacting the corresponding authors of the article and consulting with two independent experts. The investigation involved technical scrutiny of the additional analyses provided by the corresponding authors, including supplementary data from the repository https://zenodo.org/records/6851435. Based on the evidence presented, the Reviewers endorsed the publication of the correction note appended below. Readers are urged to take this information into consideration when interpreting the data presented in this article. Kun Zuo and Vincent Mourik also informed the editorial staff that they wished to be removed from authorship because in their opinion, the correction does not address the concerns with respect to the data and they do not endorse the validity of the claims and conclusions of the article. The author list in both the PDF and HTML has now been rectified. All authors,with the exception ofKenjiWatanabe and Takashi Taniguchi, disagreewith the publication of this Editorial Expression of Concern.

Correction to: Nature Nanotechnologyhttps://doi.org/10.1038/s41565-017-0032-8, published online 15 January 2018. The Letter reports Majorana signatures in hybrid InSb semiconductor nanowire–NbTiN superconductor devices. The devices exhibit a conductance plateau near the conductance quantum 2e²/h at bias voltages above the superconducting gap (normal conductance), accompanied by an enhanced Andreev conductance at bias voltages below the superconducting gap (subgap conductance). We have attributed these experimental observations to ballistic transport as supported by a theoretical analysis¹, finding mean free paths on the order of or larger than the effective wire segment (the segment covered by the superconducting electrode). Here, we correct errors discovered on reanalysis of the original data², following concerns raised by readers. Due to the age of the paper, it cannot be corrected directly in the original publication, thus the updates are provided via this amendment. We provide additional discussion on the claim of ballistic transport so as to avoid misinterpretations. External peer review of the reanalysis concluded that the claims in the Letter remain. An extended public repository including data obtained from nanowire devices that were not included in the publication can be found in ref. ². We note the lack of a series of flat and precisely quantized conductance plateaus (a staircase), a clear ballistic transport characteristic (see the two newly included Supplementary Figs. 1 and 7 showing larger voltage ranges of Fig. 1 and the original Supplementary Fig. 5). Our earlier studies on ballistic transport in nanowire devices^3,4 indicate that vapour–liquid–solid nanowires do not have the proper geometry for observing a conductance staircase without the application of a magnetic field perpendicular to the wire axis, which requires ideal (Landauer) reservoirs interfacing the ballistic region, absorbing charge carriers with near-unit probability. Similar to our earlier studies, ohmic contacts in the present nanowire devices do not satisfy the conditions of Landauer reservoirs. However, the transport in the effective wire segment can nevertheless be ballistic whose characteristic is a plateau feature near 2e²/h in normal conductance together with an enhanced Andreev conductance. Importantly, precise quantization is not realistic, prevented by the two-terminal device geometry, inevitably decreasing the conductance. In summary, a plateau feature with an enhanced Andreev conductance together with our theoretical analyses indicate that a large fraction of transport is ballistic over distances of the order of our device length. We add a discussion to the main text of the Letter as follows: “… followed by a dip in conductance due to channel mixing²⁰ [ref. ¹ below]. We do not observe higher plateaus (Supplementary Figs. 1 and 7), which we attribute to the contacts not satisfying the conditions of Landauer reservoirs, resulting in residual scattering more effective at larger conductance. This is in line with our earlier studies^25,39 [refs. ^4,5 below] which indicated that vapour–liquid–solid nanowires do not have the proper geometry for observing a conductance staircase without the application of a perpendicular magnetic field. From the absence of quantum dots, the observed induced gap …”. The following text should also have been included in the abstract: “… exhibiting clear ballistic transport properties manifested by a conductance plateau with an Andreev enhancement, albeit lacking a quantized conductance staircase hindered by the device geometry.” The conductance values reported in the publication are ~8% lower (near 2e²/h) than the actual value (corrected Fig. 1). This deviation is due to a drop in the gain of the current-to-voltage amplifier at an ac excitation frequency of 67 Hz⁵. As a result, there is a slight change in the Andreev conductance enhancement factor and the superconducting contact transparency extracted from the enhancement (a comparison between the values quoted in the publication and the corrected ones is given below in B). The general conclusions do not rely on the exact value of the conductance as precise quantization is not expected due to the two-terminal device geometry. The subtracted series resistance of 3 kΩ in the original Fig. 1 was an overestimation (see corrected Fig. 1 in the Supplementary Data file). The subtraction of 3 kΩ was not mentioned in the original publication. A comparison of the original and corrected Fig. 1 is presented in a Supplementary Data file accompanying this correction. For all the figures in the original publication except Fig. 1, we either subtracted a contact resistance value of 0.5 kΩ, which is an underestimation¹, or no resistance at all. We note that in tunneling measurements the overall resistance is significantly higher than the normal metal contact resistance whose contribution can therefore be neglected. Figure 1, however, was used to estimate the superconducting contact transparency and Andreev enhancement in the high conductance regime, requiring a realistic exclusion of the contact resistance. Following our previous paper⁴, which found normal metal contact resistance values between 1.5–3.25 kΩ per contact and was based on fitting the measured conductance using theory (single mode interfacing a superconductor), which provided reasonable agreement after excluding 3 kΩ, we subtracted 3 kΩ to exclude the resistance of the normal metal contact. During our reanalysis, we have discovered that the minimum resistance of this device at the largest applied gate voltages is 2.9 kΩ, a value providing an upper bound on the contact resistance. Here, 2.9 kΩ would be the contact resistance under the assumption that the nanowire itself has zero resistance at largest gate voltages. The contact resistance can be estimated with an alternative method by subtracting a series resistance to match the observed conductance plateau at bias voltages above the superconducting gap to the expected quantized value, a procedure not done in the original publication. By taking the conductance averaged at positive and negative |V| ~ 1.7 mV (around the largest bias voltages available for this analysis) we find that the quantized value is reached for a contact resistance of 0.77 kΩ. (Considering only the positive bias and separately only the negative bias results in a range of 0–2.13 kΩ for the contact resistance.) In our corrected estimate of the contact resistance, we have applied the calibration procedure⁵ that corrects for ac circuit effects, uses calibrated values for the series resistance of the setup where Fig. 1 was measured and directly corrects the error listed in A above. Upon reanalysis we estimate the following contact resistance values, enhancement factors and transparencies: (Table presented.) Contact resistance Enhancement factor Transparency Lower bound 0 kΩ 1.26 0.88 Conservative estimation¹ (used in corrected Fig. 1) 0.5 kΩ 1.32 0.90 Current best estimate 0.77 kΩ 1.36 0.90 Original estimate in paper 3 kΩ >1.5 >0.93 The corrected superconducting contact transparency value of 0.9 does not affect the claim of high transparency. The claim of ballistic transport does not rest on the exact value of the conductance plateau and hence is also unaffected. The original Methods section omits the indication of subtracted series resistances which account for the normal metal contact resistance in each figure. The following is included here for the corrected Methods: The original Methods section omits the indication of subtracted series resistances which account for the normal metal contact resistance in each figure. The following is included here for the corrected Methods: “Contact resistance treatment. A fixed-value series resistance of 0.5 kΩ has been subtracted in Figs. 1 and 4, Supplementary Figs. 1, 2b,c and 4–9 to account for the contact resistance of the normal metal lead. This value is smaller than the lowest contact resistance we have obtained for InSb nanowire devices²⁵ (ref. ⁴ below), which makes the interface transparency estimated from Fig. 1 a lower bound. For the remaining figures, no series resistance has been subtracted to account for the normal metal contact resistance.” In the original Supplementary Fig. 5 (now Supplementary Fig. 6), a charge jump was corrected by removal of 12 line traces (corresponding to +0.15 V to +0.04 V in gate voltage in the measured data) and offset of the gate voltage axis by 0.12 V after the charge jump (–1 V to +0.03 V) to maintain continuity of the axis. This processing was not mentioned in the original publication. The corrected Supplementary Fig. 6 excludes this processing and represents the data as measured. In the original Supplementary Fig. 5 (now Supplementary Fig. 6), a charge jump was corrected by removal of 12 line traces (corresponding to +0.15 V to +0.04 V in gate voltage in the measured data) and offset of the gate voltage axis by 0.12 V after the charge jump (–1 V to +0.03 V) to maintain continuity of the axis. This processing was not mentioned in the original publication. The corrected Supplementary Fig. 6 excludes this processing and represents the data as measured. A comparison of the original and corrected Fig. SI5 (now Fig. SI6) is presented in a Supplementary Data file accompanying this correction. Original Supplementary Fig. 1f (now Supplementary Fig. 2f): The offset mentioned in the caption is erroneously given as 0.006 × 2e²/h but is 0.01 × 2e²/h. Original Supplementary Fig. 4a,b (now Supplementary Fig. 5a,b) were indicated to present data from Fig. 2a (or original Supplementary Fig. 1a). This is incorrect. The data used are from the original Supplementary Fig. 1b (now Supplementary Fig. 2b) which has the same measurement settings as in Fig. 2a except the barrier gate is –1.5 V (the barrier gate is –1.4 V in Fig. 2a or original Supplementary Fig. 1a). In the original panels c–e of Supplementary Fig. 7 (now Supplementary Fig. 9c–e) the bias polarity is mistakenly inverted.

Spin qubits in germanium are a promising contender for scalable quantum computers. Reading out of the spin and charge configuration of quantum dots formed in Ge/Si core/shell nanowires is typically performed by measuring the current through the nanowire. Here, we demonstrate a more versatile approach on investigating the charge configuration of these quantum dots. We employ a high-impedance, magnetic-field resilient superconducting resonator based on NbTiN and couple it to a double quantum dot in a Ge/Si nanowire. This allows us to dispersively detect charging effects, even in the regime where the nanowire is fully pinched off and no direct current is present. Furthermore, by increasing the electro-chemical potential far beyond the nanowire pinch-off, we observe indications for depleting the last hole in the quantum dot by using the second quantum dot as a charge sensor. This work opens the door for dispersive readout and future spin-photon coupling in this system.

The authors of the paper “Epitaxy of advanced nanowire quantum devices”¹ wish to retract this work. When preparing the underlying data for public release², it was discovered that some data had been inappropriately deleted or cropped when preparing the final published figures, and we promptly alerted the editors of Nature. We found unjustified data removal and cropping in Figures 4a and c, and Extended Data Figures 7 and 8, which affect the agreement between the theoretical curves and the experimental data and the claims of ballistic transport. We are accordingly retracting the paper. The authors stand by all the other data, and their contribution to advanced nanowire quantum devices. All authors have agreed to this retraction.

In this Letter, we reported electrical measurements and numerical simulations of hybrid superconducting–semiconducting nanowires in a magnetic field. We reported plateaus in the conductance at 2e2/h, which we interpreted as evidence for the presence of Majorana zero-modes. However, several inconsistencies were pointed out by Sergey Frolov and Vincent Mourik between the raw measurement data that was made available to them and the figures that were published in the paper. We therefore re-analysed all the existing raw data for our original measurements and rebuilt the original experimental set-up for a re-calibration of the conductance values. We established that the data in two of the figures (Fig. 2a and Extended Data Fig. 4b) had been unnecessarily corrected for charge jumps (corrections that were not mentioned explicitly in the paper), and that one of the figure axes was mislabelled (Fig. 4b). The new conductance calibration shifted the plateau values by 8 per cent, above 2e2/h, which affects all the figures1. When the data are replotted over the full parameter range, including ranges that were not made available earlier, points are outside the 2-sigma error bars. We can therefore no longer claim the observation of a quantized Majorana conductance, and wish to retract this Letter. After informing Nature of this decision, Nature issued an Editorial Expression of Concern2 and initiated the retraction process. In ref. 1 we provide all the raw data underlying the published figures as well as the unpublished datasets. Ref. 1 also contains the analysis methods and a side-by-side comparison between the original and the corrected figures. In ref. 3 we provide a new manuscript with corrected and extended datasets, discussed in the context of new insights on zero-energy states in systems with inhomogeneous potentials and disorder. We thank Piet Brouwer, Klaus Ensslin, David Goldhaber-Gordon and Patrick Lee for the expert evaluation report available via ref. 1. We also thank Michael Wimmer and Bernard van Heck for their help with the analyses. We apologize to the community for insufficient scientific rigour in our original manuscript.

The growth of Sn-rich group-IV semiconductors at the nanoscale can enrich the understanding of the fundamental properties of metastable GeSn alloys. Here, we demonstrate the effect of the growth conditions on the morphology and composition of Ge/GeSn core/shell nanowires by correlating the experimental observations with a theoretical interpretation based on a multiscale approach. We show that the cross-sectional morphology of Ge/GeSn core/shell nanowires changes from hexagonal to dodecagonal upon increasing the supply of the Sn precursor. This transformation strongly influences the Sn distribution as a higher Sn content is measured under the {112} growth front. Ab initio DFT calculations provide an atomic-scale explanation by showing that Sn incorporation is favored at the {112} surfaces, where the Ge bonds are tensile-strained. A phase-field continuum model was developed to reproduce the morphological transformation and the Sn distribution within the wire, shedding light on the complex growth mechanism and unveiling the relation between segregation and faceting. The tunability of the photoluminescence emission with the change in composition and morphology of the GeSn shell highlights the potential of the core/shell nanowire system for optoelectronic devices operating at mid-infrared wavelengths.

We show a hard superconducting gap in a Ge-Si nanowire Josephson transistor up to in-plane magnetic fields of 250 mT, an important step toward creating and detecting Majorana zero modes in this system. A hard gap requires a highly homogeneous tunneling heterointerface between the superconducting contacts and the semiconducting nanowire. This is realized by annealing devices at 180 °C during which aluminum interdiffuses and replaces the germanium in a section of the nanowire. Next to Al, we find a superconductor with lower critical temperature (TC = 0.9 K) and a higher critical field (BC = 0.9-1.2 T). We can therefore selectively switch either superconductor to the normal state by tuning the temperature and the magnetic field and observe that the additional superconductor induces a proximity supercurrent in the semiconducting part of the nanowire even when the Al is in the normal state. In another device where the diffusion of Al rendered the nanowire completely metallic, a superconductor with a much higher critical temperature (TC = 2.9 K) and critical field (BC = 3.4 T) is found. The small size of these diffusion-induced superconductors inside nanowires may be of special interest for applications requiring high magnetic fields in arbitrary direction.

Typical measurements of nanowire devices rely on end-to-end measurements to reveal mesoscopic phenomena such as quantized conductance or Coulomb blockades. However, creating nanoscale tunnel junctions allows one to directly measure other properties such as the density of states or electronic energy distribution functions. In this paper, we demonstrate how to realize uniform tunnel junctions on InSb nanowires, where the low invasiveness preserves ballistic transport in the nanowires. The utility of the tunnel junctions is demonstrated via measurements using a superconducting tunneling probe, which reveal nonequilibrium properties in the open quantum dot regime of an InSb nanowire. The method for high-quality tunnel junction fabrication on InSb nanowires is applicable to other III-V nanowires and enables characterization of nanowire local density of states.

Superconducting coplanar-waveguide resonators that can operate in strong magnetic fields are important tools for a variety of high-frequency superconducting devices. Magnetic fields degrade resonator performance by creating Abrikosov vortices that cause resistive losses and frequency fluctuations or suppress the superconductivity entirely. To mitigate these effects, we investigate lithographically defined artificial defects in resonators fabricated from Nb-Ti-N superconducting films. We show that by controlling the vortex dynamics, the quality factor of resonators in perpendicular magnetic fields can be greatly enhanced. Coupled with the restriction of the device geometry to enhance the superconductors critical field, we demonstrate stable resonances that retain quality factors ≃105 at the single-photon power level in perpendicular magnetic fields up to B⊥ ≃20mT and parallel magnetic fields up to B⥠≃6T. We demonstrate the effectiveness of this technique for hybrid systems by integrating an In-Sb nanowire into a field-resilient superconducting resonator and use it to perform fast charge readout of a gate-defined double quantum dot at B=1T.

Low-dimensional high-quality InSb materials are promising candidates for next-generation quantum devices due to the high carrier mobility, low effective mass, and large g-factor of the heavy element compound InSb. Various quantum phenomena are demonstrated in InSb 2D electron gases and nanowires. A combination of the best features of these two systems (pristine nanoscale and flexible design) is desirable to realize, e.g., the multiterminal topological Josephson device. Here, controlled growth of 2D nanostructures, nanoflakes, on an InSb platform is demonstrated. An assembly of nanoflakes with various dimensions and morphologies, thinner than the Bohr radius of InSb, are fabricated. Importantly, the growth of either nanowires or nanoflakes can be enforced experimentally by setting growth and substrate design parameters properly. Hall bar measurements on the nanostructures yield mobilities up to ≈20 000 cm ² V ⁻¹ s ⁻¹ and detect quantum Hall plateaus. This allows to see the system as a viable nanoscale 2D platform for future quantum devices.

We present a Josephson junction based on a Ge-Si core-shell nanowire with transparent superconducting Al contacts, a building block which could be of considerable interest for investigating Majorana bound states, superconducting qubits, and Andreev (spin) qubits. We demonstrate the dc Josephson effect in the form of a finite supercurrent through the junction and establish the ac Josephson effect by showing up to 23 Shapiro steps. We observe multiple Andreev reflections up to the sixth order, indicating that charges can scatter elastically many times inside our junction and that our interfaces between superconductor and semiconductor are transparent and have low disorder.

Strain engineering in Sn-rich group IV semiconductors is a key enabling factor to exploit the direct bandgap at mid-infrared wavelengths. Here, we investigate the effect of strain on the growth of GeSn alloys in a Ge/GeSn core/shell nanowire geometry by controlling the Ge core diameter and correlating the results with theoretical strain calculations. Incorporation of the Sn content in the 10-20 at. % range is achieved with Ge core diameters ranging from 50 nm to 100 nm. While the smaller cores lead to the formation of a regular and homogeneous GeSn shell, larger cores lead to the formation of multifaceted sidewalls and broadened segregation domains, inducing the nucleation of defects. This behavior is rationalized in terms of the different residual strain, as obtained by realistic finite element method simulations. The extended analysis of the strain relaxation as a function of core and shell sizes, in comparison with the conventional planar geometry, provides a deeper understanding of the role of strain in the epitaxy of metastable GeSn semiconductors.

High aspect-ratio InSb nanowires (NWs) of high chemical purity are sought for implementing advanced quantum devices. The growth of InSb NWs is challenging, generally requiring a stem of a foreign material for nucleation. Such a stem tends to limit the length of InSb NWs and its material becomes incorporated in the InSb segment. Here, we report on the growth of chemically pure InSb NWs tens of microns long. Using a selective-area mask in combination with gold as a catalyst allows complete omission of the stem, thus demonstrating that InSb NWs can grow directly from the substrate. The introduction of the selective-area mask gives rise to novel growth kinetics, demonstrating high growth rates and complete suppression of layer deposition on the mask for Sb-rich conditions. The crystal quality and chemical purity of these NWs is reflected in the significant enhancement of low-temperature electron mobility, yielding an average of 4.4 × 10⁴ cm²/(V s), compared to previously studied InSb NWs grown on stems.

Spin-orbit interaction (SOI) plays a key role in creating Majorana zero modes in semiconductor nanowires proximity coupled to a superconductor. We track the evolution of the induced superconducting gap in InSb nanowires coupled to a NbTiN superconductor in a large range of magnetic field strengths and orientations. Based on realistic simulations of our devices, we reveal SOI with a strength of 0.15-0.35 eV Å. Our approach identifies the direction of the spin-orbit field, which is strongly affected by the superconductor geometry and electrostatic gates.

We report on the selective-area chemical beam epitaxial growth of InAs in-plane, one-dimensional (1D) channels using patterned SiO2-coated InP(001), InP(111)B, and InP(011) substrates to establish a scalable platform for topological superconductor networks. Top-view scanning electron micrographs show excellent surface selectivity and dependence of major facet planes on the substrate orientations and ridge directions, and the ratios of the surface energies of the major facet planes were estimated. Detailed structural properties and defects in the InAs nanowires (NWs) were characterized by transmission electron microscopic analysis of cross-sections perpendicular to the NW ridge direction and along the NW ridge direction. Electrical transport properties of the InAs NWs were investigated using Hall bars, a field effect mobility device, a quantum dot, and an Aharonov-Bohm loop device, which reflect the strong spin-orbit interaction and phase-coherent transport characteristic present in the selectively grown InAs systems. This study demonstrates that selective-area chemical beam epitaxy is a scalable approach to realize semiconductor 1D channel networks with the excellent surface selectivity and this material system is suitable for quantum transport studies.

We demonstrate the merits of an unexplored precursor, tetrasilane (Si₄H₁₀), as compared to disilane (Si₂H₆) for the growth of defect-free, epitaxial hexagonal silicon (Si). We investigate the growth kinetics of hexagonal Si shells epitaxially around defect-free wurtzite gallium phosphide (GaP) nanowires. Two temperature regimes are identified, representing two different surface reaction mechanisms for both types of precursors. Growth in the low temperature regime (415 °C-600 °C) is rate limited by interaction between the Si surface and the adsorbates, and in the high temperature regime (600 °C-735 °C) by chemisorption. The activation energy of the Si shell growth is 2.4 ±0.2 eV for Si₂H₆ and 1.5 ±0.1 eV for Si₄H₁₀ in the low temperature regime. We observe inverse tapering of the Si shells and explain this phenomenon by a basic diffusion model where the substrate acts as a particle sink. Most importantly, we show that, by using Si₄H₁₀ as a precursor instead of Si₂H₆, non-tapered Si shells can be grown with at least 50 times higher growth rate below 460 °C. The lower growth temperature may help to reduce the incorporation of impurities resulting from the growth of GaP.

Present day nanowire solar cells have reached an efficiency of 17.8%. Nanophotonic engineering by nanowire tapering allows for high solar light absorption. In combination with sufficiently high carrier selectivity at the contacts, the short-circuit current (J_sc) has presently reached 29.3 mA/cm², reasonably close to the 34.6 mA/cm² theoretical limit for InP. Although further optimization of the current is important, an equally challenging condition to approach the Shockley Queisser (S-Q) limit is to increase the open-circuit voltage (V_oc) towards the radiative limit. The key requirement to reach the radiative limit is to increase the external radiative efficiency at open-circuit conditions towards unity. It is the main purpose of this review to highlight recent progress in nanophotonic engineering to further enhance the open circuit voltage of a nanowire solar cell. In addition to material optimization for increasing the internal photoluminescence efficiency, the light extraction efficiency is a major design criterion for enhancing the external radiative efficiency and thus the V_oc. Since the semiconductor substrate is a sink for internally generated photoluminescence, it is equally important to eliminate the loss of emitted light into the substrate. Even at the S-Q limit, the V_oc is still substantially decreased by a photon entropy loss due to the conversion of a parallel beam of photons from the sun into an isotropic emission pattern, in which each individual photon is emitted into a random direction. The 46.7% ultimate solar cell limit for direct solar irradiation can only be approached, once the cell is capable to focus all emitted photoluminescence back to the sun. We will show that nanophotonic engineering provides a pathway to approach the ultimate limit.