Topological superconductivity in metallic nanowires fabricated with a scanning tunneling microscope

We report on several low-temperature experiments supporting the presence of Majorana fermions in superconducting lead nanowires fabricated with a scanning tunneling microscope (STM). These nanowires are the connecting bridges between the STM tip and the sample resulting from indentation–retraction processes. We show here that by a controlled tuning of the nanowire region, in which superconductivity is confined by applied magnetic fields, the conductance curves obtained in these situations are indicative of topological superconductivity and Majorana fermions. The most prominent feature of this behavior is the emergence of a zero bias peak in the conductance curves, superimposed on a background characteristic of the conductance between a normal metal and a superconductor in the Andreev regime. The zero bias peak emerges in some nanowires when a magnetic field larger than the lead bulk critical field is applied. This field drives one of the electrodes into the normal state while the other, the tip, remains superconducting on its apex. Meanwhile a topological superconducting state appears in the connecting nanowire of nanometric size.


Introduction
Since Kitaev's proposal [1] in 2001 that Majorana fermions could be found in condensed matter systems, several efforts, both theoretical and experimental, have been addressed to establish the conditions and requirements to detect such elusive particles. It was realized that topological insulators were a platform for Majorana states [2]. Most of these activities involve the use of the superconducting state, because of the close similarity between a Majorana fermion (a particle that is its own antiparticle) and the quasiparticles of a p-wave spinless superconducting condensate that could be found close to (or at) its ground energy state (i.e. Fermi level) under some specific conditions and requirements. The different possibilities, requisites and conditions to be fulfilled by a superconducting system in order to present Majorana fermions have been extensively discussed in recent years [3][4][5][6][7][8][9]. One of the 'easiest' ways to obtain such p-wave spinless superconducting condensate consists of using a standard s-wave superconductor, whose electronic bands are split depending on the spin polarization by means of a spin-orbit coupling, and an external magnetic field which opens a Zeeman gap and a region without spin degeneracy. The effective spinless regime, where the standard superconducting states become a topological one, will be strongly dependent on the values of the superconducting energy gap, the Zeeman gap, the spin-orbit coupling and the filling of the bands (i.e. Fermi level). The existence of well-defined energy bands and levels will favor the fulfilment of the required conditions. Therefore, one-dimensional (1D) or quasi-1D systems with a small number of quantum modes will be more suited to the emergence of Majorana fermions.
Experimental efforts toward the realization of these requirements have focused mainly on the use of quasi-1D semiconducting nanowires [10][11][12][13], which present a strong spin-orbit coupling, and where superconductivity is induced by proximity to an s-wave superconductor. Mourik et al [11] have recently reported experiments on such a system, showing evidence of the detection of Majorana fermions in a semiconducting nanowire by means of tunneling spectroscopy measurements.
In a recent publication [14] we have shown a different approach to the experimental realization of the above-mentioned 'recipe'. Instead of using a semiconductor as the 'source' of the spin-orbit coupling, we use lead, a metallic s-wave superconductor below 7 K, which presents a moderate spin-orbit coupling.
Indeed, there are several differences between the superconducting state developed in a semiconductor and the one present in a metal, mainly related to the characteristic energy and length scales of the superconducting condensate which are closely related to the Fermi wavelength in each type of material.
Another important aspect is related to the requirement of a small number of quantum electronic modes involved in the experimental object. In the case of semiconductors, due to the low electronic density of states, this is quite easily achieved by using quasi-1D nanowires, with diameter in the range of 100 nm and 1 µm length. Such nanowires may be gated in order to move the Fermi level so that the above-mentioned requirements are achieved. However, if we use a metal, the condition of having a small number (of the order of one) of quantum modes, or channels, is only achieved if the diameter of the nanowire where topological superconductivity will be induced is of the order of a few atoms [15]. This small (atomic) dimension of the nanowire implies that the effective superconducting coherence length will also be strongly reduced. Random scattering at the boundaries implies that the elastic mean free path, , will be of the order of the sample width, which, for a small number of channels within the nanowire, cannot be larger than a few nanometers. Then, the superconducting coherence length ξ ≈ √ ξ 0 will be significantly reduced. For ξ 0 ≈ 80 nm and ≈ 1 nm, we find ξ ≈ 9 nm, which is comparable with typical nanowire lengths. We have addressed the realization of such atomic scale nanowires by using a scanning tunneling microscope (STM), which allows one to establish atomic scale contacts between metallic electrodes (usually called 'tip' and 'sample') [16,17]. By using the different capabilities and features of the STM system, it is possible to control and modify this atomic scale contact, detecting the addition of individual atoms to the contact [18]. As an example, it has been possible to create gold nanowires consisting of a chain of single atoms between the tip and sample gold electrodes, where the electric conductance involves a single quantum channel [19].
The low value of the spin-orbit coupling in lead, compared with the value in some semiconducting materials, is an added difficulty in order to accomplish the requirements to create a topological superconducting phase in the nanowire. Lead is a type-I superconductor, with a superconducting gap of 1.35 meV at zero field and zero temperature, and a rather low bulk critical field (75 mT at 300 mK). However, the nanoscopic apex of the tip, and sharp elongated nanostructures, may remain superconducting well above the bulk critical field [17]. The estimated values of the Landé g factor for bulk lead are in the range g ≈ 4-6 [20,21], which can be enhanced by interaction effects in nanoscopic samples [22]. A magnetic field, B, in the range of 100 mT leads to a Zeeman energy, E Z = gµ B B/2, in the range 0.04-0.06 meV and, as the gap is expected to go smoothly to zero as the magnetic field increases, it is possible that a regime where the Zeeman coupling is larger than the superconducting gap is present in some of these lead nanowires. The position of the Fermi energy at the nanowire will depend on details of the electrostatic potential which, in turn, is determined by the geometry of the contact and the voltage distribution within it. Therefore, it is possible that some of such Pb nanowires present a Fermi energy level located within the Zeeman gap, due to random fluctuations in the electrostatic potential. It might be possible to tune the parameters of the device by changing the bias voltage, although a detailed analysis of this topic lies outside the scope of this work.
Our experimental configuration, shown schematically in figure 1, presents another peculiarity that poses a relevant difference compared with the above-mentioned experiments involving semiconductors. The lead nanowire, which will become a topological superconductor under an external magnetic field, is created by indenting the STM Pb tip (which in the end is of nanoscopic dimensions) into a Pb surface (which can be considered 'flat' compared to the tip).
We have shown in previous work [17,[23][24][25] that sharp and elongated nanotips or nanoprotrusions on the sample surface resulting from the tip-sample indentations remain superconducting for magnetic fields much larger than the bulk critical field of lead, depending on their sharpness and dimensions compared with the bulk coherence length and penetration depth, while the larger bulk, flat or blunt parts of the Pb tip and sample have become normal.
Thus, we have a system where the topological superconducting Pb nanowire, which may present Majorana fermions, is in direct contact with (has emerged from) a bulk of the same material in normal state, and it is probed by a superconducting electrode (the nanotip) using and involving a very small number of quantum channels (typically between three and ten).
The probe, the nanotip, is in direct contact with the nanowire. The spectroscopic measurements will take place in a transmission regime where Andreev reflections play a key role [26,27]. The shape and features of the spectroscopic curves (IV curves) in this Andreev regime are strongly dependent on the coupling and transmission probability of each individual quantum channel [25,[28][29][30], thus providing a detailed quantification of the number of quantum channels involved in the conduction process (and at the nanowire), and their individual transmission.
In the following sections we present our recent experiments on these topological nanowires. We discuss the experimental evidence of the emergence of a topological superconducting phase at the nanowire/nanocontact region; the comparison of this type of result with those that could be considered 'expected' or 'normal' for standard S-S or S-N situations; the observability of such 'unexpected' results, which we assign to the presence of Majorana fermions, depending on the number of quantum channels involved and their transmissions; and the analysis of the zero bias current detected in the different superconducting regimes of the nanowire. The evolution of this zero bias current may be of key importance in the study and detection of Majorana fermions, as it would be a direct indication of the splitting of Cooper pairs injected from the superconducting nanotip into pairs of Majorana bound states in a topological superconducting nanowire with an extremely small superconducting gap. STM topographic image of a Pb surface where a nanometer size protusion, created after a tip-sample indentation process, is clearly visible. The nanostructure is 35 nm wide at its base, and 25 nm high. The actual lateral dimensions of the nanostructure will be smaller, as it is scanned by an equivalent protusion at the tip apex. The image was taken at 0.3 K, for a tunneling resistance of 10 M and a voltage bias of 10 mV.

Sample fabrication
The experiments were performed at 0.3 K, with the STM installed in a 3 He cryostat equipped with a superconducting solenoid. The STM was used to produce indentations, in the range of a few tens of nanometers, of a Pb tip on a Pb sample, in order to fabricate sharp elongated nanotips and nano-protrusions on the sample surface. Bulk Pb tip and sample pieces were cut from a high-purity lead bar (Goodfellow, purity 99.999%) just prior to being installed in the vacuum chamber of the STM cryostat system. Along the indentation processes we recorded the variation of the current across the contact as a function of the relative displacement between tip and sample, for a fixed bias voltage (typically in the range of 10 mV). The analysis of the current versus displacement curves allows one to extract information about the sharpness and dimensions of the nanostructures resulting from the indentation process [16]. When the contact is broken (by receding the tip), it is possible to scan and visualize the part of the nanostructure remaining on the sample surface ( figure 2). An equivalent nanostructure remains at the tip apex. This kind of process allows us to create sharp elongated nanotips, with a cone-like apex with typical dimensions ranging from 20 to 100 nm in length and 20 to 100 nm in diameter [17].
Once a sharp Pb nanotip has been created, we move to a 'flat' region of the sample (far from the nanoprotusion) and we proceed with the spectroscopic characterization of the nanotip. Making use of the STM feedback control loop we go from tunneling conditions toward atomic scale contact between tip and sample. This process is followed in detail by recording the current versus displacement variations and the acquisition of current versus voltage curves along the process. The IV (and conductance, dI/dV versus V) curves evolve from the usual gapped structure in tunneling to a much richer structure (the subharmonic gap structure) as the contact is established, due to the increasing contribution of Andreev reflection processes. show the corresponding quantum channels and transmissions. The curves were taken after small atomic rearrangements of a nanocontact, at 350 mK and zero magnetic field.
As mentioned above, the subharmonic gap structure in the spectroscopic curves allows us to determine precisely the number of quantum channels and their individual coupling or transmission (figure 3). It was shown that in atomic scale contacts the individual channel transmissions, {τ n }, can be obtained from the experimental current-voltage, IV, curve by fitting to the sum of N one-channel IV curves, I (V ) = N n=1 i(τ n , V ) [30,31], which have been calculated elsewhere for arbitrary transmission τ [32,33].
A zero bias current is easily observed in the curves, signature of transmission of Cooper pairs between the electrodes (dc Josephson effect). Typically, voltage bias is ramped between ±10 mV, and the conductance is obtained by the numerical derivative of the IV data.
Once the atomic scale contact is created, by indenting the sharp elongated nanotip onto the Pb sample, we can perform small indentation-elongation cycles, with nominal displacements smaller than 10 nm. This leads, in each cycle, to atomic scale rearrangements in the nanocontact region ( figure 4). Eventually, these mechanical processes result in a nanometric length wire, a nanowire, whose cross section is made up of a small number of atoms, as extracted from the quantum channel analysis of the conductance curves. These nanowires are expected to present a topological superconducting state, and eventually Majorana fermions, when a magnetic field is applied [14].
In these experiments, different nanostructures, with conductance values at the constriction ranging from 2G 0 to 50G 0 (G 0 = 2e 2 / h is the conductance quantum), were created, and we have analyzed the evolution of their spectroscopic characteristics versus magnetic field in order to investigate the presence of Majorana fermions. The high quality of the fittings of the experimental curves to the theoretical modelizations ( figure 3(a)), and the high barrier value (∼3 eV) extracted from the exponential tunneling regime in figure 4(b), indicate that the experiments actually involve clean lead nanostructures.

Sample characterization
As shown in figure 5, different types of conductance curves (i.e. dI/dV versus V) can be obtained depending on the geometry of the electrodes at the nanoscale close to the contact, and the values of the external magnetic field [24,34]. Figure 5(a) shows the typical result obtained at zero field, and for H < H c bulk. The full subharmonic gap structure due to multiple Andreev reflections, and a sharp zero bias peak due to Josephson current are obtained, the exact features depending on the transmissions of the different quantum channels involved in the process. The other panels show the more representative types of conductance curves that can be obtained for H > H c bulk. We also show a sketch of the geometry of the nanostructure leading to each observed result. If the contact is established between similar sharp protusions in the tip and sample, both remain superconducting at H > H c , showing results similar to the ones at zero field. If the sample side of the nanostructure is more rounded or blunt than the nanotip, S-S features are still present, but very rounded due to pair-breaking effects at the sample, now more sensitive to the effect of the applied magnetic field ( figure 5(b)). When the nanotip contacts a flat region of the sample, we obtain results like the one in figure 5(c), corresponding to a situation involving NS Andreev reflections, where only the nanotip remains superconducting, and all the samples are in normal state. Finally, in figure 5(d) we show a type of conductance curve that can be obtained after modifying situations like the ones shown in figure 5(c) by means of indentation-retraction cycles. The new feature is a zero bias peak in the conductance, corresponding to a finite zero bias current, which is superimposed on the NS Andreev-like conductance for finite bias. We consider that, in this case, a nanowire is created between the tip and sample.
In the following, we will describe the evolution with magnetic field of the conductance curves in the different nanostructures presented above. , the tip was located on a bump on the surface, resulting in rounded spectra for H > H c , due to the bump being still in superconducting state, but under strong pair breaking effects due to the field. Panel (c) shows the conductance curves that we consider due to the presence of Majorana states at the constriction: for H > H c a zero bias peak appears, superimposed on a conductance background corresponding to an apparent NS situation, similar to those in (a). Curves are normalized at their high-voltage value, and shifted vertically for clarity.

Experimental results and discussion
In figure 6(a) we show the evolution of the conductance curves versus magnetic field in a situation where no 'anomalous' or topological superconducting state is expected to happen. The conductance of this nanostructure is 6G 0 . The atomically sharp nanotip is contacting a 'flat' region of the sample and no indentation-elongation cycles were performed. In this situation, as we increase the external magnetic field, once the bulk critical field of lead is crossed, we observe a sharp transition from SS spectroscopic curves (showing a very rich subharmonic gap structure and Josephson current) below H c , to NS like curves, with no indication of zero bias current. Obviously, for H > H c only the nanotip remains superconducting, and the bulk parts of both electrodes have become normal. If the contact is established with the nanotip located onto a 'bumpy' spot of the sample, this region may remain superconducting above H c , but showing strong pair breaking effects (rounding and shift of the peaks corresponding to the subharmonic gap structure, accompanied by a progressive decrease of the signal at zero bias), and eventually becomes normal. An example of this case is shown in figure 6(b), corresponding to a nanostructure with conductance 4G 0 , and it is also considered as an expected one [24].
As we showed in recent work [14], some nanostructures present peculiar behavior at magnetic fields above H c . The corresponding spectroscopic curves show, for V = 0, clear and well-defined NS features, which can be fitted in terms of quantum channels and NS Andreev reflections (indicating that only one electrode remains superconducting, the nanotip), while a finite current at zero bias, like the one corresponding to the dc Josephson effect in the S-S case, can still be clearly detected (figure 6(c)). We have observed that, under H > H c , the zero bias feature may disappear, and eventually reappear, after performing small indentation-elongation cycles.
We consider this behavior, the emergence of a zero peak, superimposed onto a standard NS Andreev-like quasiparticle curve, as a signature of Majorana fermions in the nanowire that has been created between tip and sample. As discussed earlier, we describe the wire as a dirty superconductor, with the mean free path, , limited by the width of the wire. The effective coherence length within the wire is reduced to ξ ≈ 9 nm. Hence, the two ends of a sufficiently long wire are decoupled, and the Andreev states pinned at them can be considered independently, see figure 7, left, where we also sketch the process needed to obtain zero bias current between a superconductor and a normal electrode, mediated by the presence of a topological superconductor with Majorana fermions. In the standard N-S case, when the topological superconducting nanowire is not present, no current can be obtained at zero bias, because of the presence of the energy gap in the superconductor, and the requirement of finite difference between the Fermi levels in N and S for Andreev reflection processes to be allowed.
Next, we investigate the experimental conditions that allow us to observe the phenomena that we ascribe to the presence of a topological superconducting nanowire with Majorana fermions. As was mentioned in the introduction, it is possible to extract the number of quantum channels and their transmissions involved in the transmission processes leading to the different spectroscopic curves. This can be done both in the low-field regime (H < H c bulk, SS In figure 8 we show calculated one-channel conductance curves corresponding to different values of the transmission probability. The NS conductance curves (bottom panels in figure 8) are calculated following the BTK formalism [27] while the SS curves (top panels in figure 8) are calculated following [32,33]. The actual conductance curve corresponding to a given experimental situation will be the result of the addition of the conductance from different channels with different transmissions, thus resulting in a unique type and shape of the conductance curve in each case.
The analysis of tens of nanowires showed that, for similar values of the total conductance, typically in the range of 3-10G 0 , only those that presented high transparency channels (with transmission probability very close to one) showed a well-defined zero bias current together with an NS quasiparticle Andreev regime, independently of the channel distribution in the SS low-field regime. In figure 9 we show two examples illustrating the different situations that we have found regarding the conductance characteristics of the nanowires under a magnetic field. The value of the zero bias current in the topological regime (magenta curves, in figures 9(d)-(f)) was in these cases always in the range of 1/4-1/5 of the value obtained at zero field (or for H < H c bulk). This observation can be taken as a proof to rule out the possibility that this zero bias current in the topological regime is just Josephson current between the superconducting tip (with a gap value equal to the one at zero field, 0 ) and a superconducting region of the sample with a very reduced gap. In the conductance curve no signature of a second gap is detected. We can estimate that the second gap, 2 , would be less than 2% of 0 , while the gap at the tip, 1 , equals 0 . Josephson current in such an S 1 -S 2 situation would be of the order of 1 2 /( 1 + 2 ), leading to a value of the order of 0 /50, a quantity 25 times smaller than 0 /2, the value corresponding to identical gaps. Note that in our experiments, the zero bias current is reduced by just a factor of 4 or 5, not 25, with respect to the case when both electrodes are in the superconducting state.
Let us note that the topological superconductor, with a reduced gap value and Majorana zero modes, is probed by a superconductor (the nanotip). In this situation, the zero bias conductance should not present the typical limiting value of 2e 2 / h per channel due to Majoranainduced resonant Andreev reflection, as obtained when the topological superconductor is probed by a normal metal [35]. In our experiment, due to the superconducting probe, multiple Andreev reflections should be present, leading to a larger zero bias conductance arising from these Majorana-induced multiple resonant Andreev reflections. This is a situation equivalent to the one present in the 'standard' NS versus SS cases. For a high transparency (τ = 1) barrier, the zero bias conductance is 2G 0 (4e 2 / h) per channel in the NS case, while it diverges in the SS case (see the black curves in the left panels of figure 8).
The relevance of the presence of high transmission channels in order to observe the zero bias feature (i.e. Majorana fermions) has been addressed by studying nanowires whose total conductance is slightly modified in a controlled way. With the nanowire in the topological regime (H > H c bulk), we vary the conductance by a value equivalent to 1G 0 or less. The example in figure 10(a) shows that the curve with larger total conductance presents a slightly smaller zero bias peak. This behavior is very similar to the one observed in S-S nanocontacts [36], where it was seen that the value of the zero bias current depends mainly on the individual transmissions of the quantum channels, and not on the total conductance of the nanocontact. These observations are compatible with the analysis in [37], where the robustness of the Majorana signatures in high transparency situations was studied.
Finally, we show an example of the evolution of the zero bias current with temperature. We wish to note that different nanowires could present slightly different behaviors, but the overall behavior is the one depicted in figure 11. We prepare a nanostructure with a conductance value  of 5G 0 , and apply a magnetic field of 200 mT. As temperature increases, the Andreev-gaprelated features in the conductance curves are rounded and move toward zero bias (i.e. the gap at the nanotip decreases as temperature increases), disappearing at about 6 K in this case, while the signature corresponding to zero bias current was observed to disappear at a much lower T, between 1.5 and 2 K. This is an indication of a critical temperature for the topological superconducting nanowire lower, between 25 and 33%, than that of the 'parent' superconducting material.
In recent work, Potter and Lee [38] proposed a system where surface states in a metallic gold nanowire (in contact with an s-wave superconductor) may fulfill the requirements to present topological superconductivity and Majorana fermions. They suggest that Majorana fermions, arising from surface states in the metallic nanowire, could be detected by an STM tip. The nanowire would have dimensions similar to the ones used in semiconducting wires (in the range of 100 nm width and 1000 nm length), and should be grown in the (111) orientation. In our experimental realization, a metallic conducting nanowire is also considered to develop topological superconductivity and Majorana fermions. However, the nanowire itself is made up of a superconducting material, lead, and brought to atomic scale in order to carry a few quantum channels. Moreover, in our case, the superconducting gap at the nanowire is reduced (or almost destroyed) by the external magnetic field, and it is not proximity induced by a 'back-up' s-wave superconductor. We consider that these differences, shown schematically in figure 12, should be considered in further theoretical work dealing with the presence and detection of Majorana fermions in topological superconducting metallic nanostructures.