One-Dimensional Superconductivity in Nanowires (ALTOMARE:SUPERCONDUCTIV. O-BK) || 1D Superconductivity in a Related System

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91D Superconductivity in a Related System

9.1Introduction

In this chapter, we will provide a brief overview of S-SmNW-S (superconductor–semiconductor nanowire–superconductor) devices (see Chapter 5) and supercon-ducting (nanowire) single-photon detectors (SNSPD). Superconductor–nanotube–superconductor devices (Section 5.3) belong to the first class, the understandingof which requires the use of nonequilibrium Usadel equations (Section 5.2.1); be-sides the intrinsic importance of nanotubes from a technological perspective, thissystem has shown indication of a Majorana mode (Section 5.4), which representsa finding of substantial fundamental importance. SNSPDs are all superconductingdevices constituted, in their simplest form, by a single nanowire wrapped to covera large area. The linewidth of the nanowire is approaching the limit in which theycan be considered 1D (10–30 nm) and as such, relevant to the topics treated in thisbook. In arrays of carbon nanotubes, coupled with each other through an insulat-ing medium, the individual nanotubes can be single-walled and as small as 0.4 nmin diameter.

9.2Carbon Nanotubes

Carbon nanotubes were originally discovered in 1991 by S. Iijima [1] and since then,the literature on the subject has flourished. Nanotubes are created by wrapping asingle sheet of graphene on itself (as can be done with a single sheet of paper). Assuch, their band structure combines the band characteristics of graphene togetherwith the quantization condition imposed on the electron momentum in the direc-tion perpendicular to the nanotube. This is also dependent on the angle betweenthe graphene unit cell and the angle of rotation of the wrapping which determinesthe chirality of the nanotube. All this variability directly translates into extremelyvarying behavior regarding their electrical properties, metallic or semiconducting.Not only can they be used as switches or transistors (see [2] for recent results), butthey can also emit light and be used as optical and electrical interconnects; we de-

One-Dimensional Superconductivity in Nanowires, First Edition. F. Altomare and A.M. Chang© 2013 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2013 by WILEY-VCH Verlag GmbH & Co. KGaA.

276 9 1D Superconductivity in a Related System

fer to a paper by P. Avouris [3] for a recent review of the possibilities offered bynanotubes.

In this section, we will review the main results pertaining to 1D superconductiv-ity in nanotube systems. Either intrinsic or due to the proximity effect, supercon-ductivity has been observed in:

single nanotubes: a single rolled graphene sheet a rope of nanotubes: several concentric nanotubes nanotube bundles: nanotubes embedded in a polymer matrix.

As in the case of superconducting nanowires, a certain fragmentation exists in thefield: different groups are working on different types of nanotubes which makes itdifficult to have a comprehensive overview of the field. Additionally, nanotube fab-rication is nontrivial. Typically, nanotube growth starts from a seed particle whichacts as a catalyst in a flow of gas containing carbon and other elements to whichboth the nanotube growth and quality are extremely sensitive. And, as if these de-tails were not enough, results are highly dependent on the type of nanotube understudy – single wall, rope, embedded in a matrix –, on its type – metallic, semicon-ducting –, if it is suspended or not. In addition, some effects are only seen if thenanotube is gated. We will only present the main results, providing references tothe original work for the interested reader.

9.2.1Proximity Effects in SWNT

The first observation of proximity induced superconductivity in nanotube datesback to 1999 [4, 5]. Kasumov et al. [4] measured metallic single-wall nanotubes(SWNT) and crystalline ropes connected to superconducting contacts: rhenium/gold for the ropes and tantalum/gold for the SWNT. They found out that in a rope(L D 1.7 μm, d D 23 nm, 200 tubes) with normal resistance smaller than 6.5 kΩand with a thermal length greater than the rope length1), the resistance of the ropebecomes negligible below 1 K (Figure 9.1a). For the SWNT (L D 0.3 μm, d D 1 nm),the zero resistance state is attained for temperatures smaller than 0.4 K. Increasingthe magnetic field would lead to a reduction of TC up to a field of 2 T, at which pointthere would be a small upturn in the measured resistance for a temperature smallerthan 0.2 K (Figure 9.1b). While the behavior of Ic as a function of temperaturewas consistent with the Ambegaokar–Baratoff expression [6, 7], for the SWNT, theextracted superconducting gap was 40 times larger than in the contact. Whereas theoverall behavior of the devices was somewhat consistent with the large critical fieldnecessary to suppress superconductivity in the SWNT, the large gap and criticalfield are not consistent with the idea that the superconductivity in the nanotube ismerely due to the proximity effect of the superconducting contacts. One suggestion

1) Superconductivity via the proximity effect penetrates in the nanotube for a distance of the order ofthe thermal length.

9.2 Carbon Nanotubes 277

Figure 9.1 (a) Temperature dependence ofthe resistance of ropes RO1, RO2, and RO3mounted on Au/Re contacts measured forzero magnetic field. The arrow indicates thetransition of the Au/Re bilayer. The sampleRO3 becomes superconducting below 1 K,but the existence of resistance steps abovethe transition indicates that superconductivitymay not be homogeneous. (b) Temperaturedependence of the resistance of the single

tube ST1 mounted on Au/Ta measured for dif-ferent values of the magnetic field perpendic-ular to the tube axis. (The labels on the curvescorrespond to the value of magnetic field inTesla.) The arrow indicates the transition tem-perature of the contact in zero magnetic field.(Inset) Field dependence of the transitiontemperature (defined as the inflexion point ofR(T )). From [4].

proposed by the authors is that intrinsic superconductivity might be stabilized bythe superconducting contacts. Another suggestion is that repulsive interaction ina Luttinger liquid2) can become attractive and thus lead to a superconducting tran-sition (see also Section 2.3). The induced proximity effect can be exploited also for

2) Metallic carbon nanotubes behave as Luttinger liquid rather then Fermi liquid.


the fabrication of a sensitive nanoelectromechanical resonator. Reulet et al. [8] irra-diated a suspended carbon nanotube rope with a radio frequency comparable to themechanical vibration frequency of the nanotube and observed the acoustoelectriceffect when measuring the DC resistance and the nanotube critical current (for thesample that becomes superconducting). The quality factor of the nanotube withsuperconducting contacts is larger than that achieved in nanotubes with normalcontacts.

In the same year, Morpurgo et al. [5] studied gated metallic SWNT (L � 0.3 μm,D < 1.8 nm). The nanotubes are grown on a 0.5 μm silicon oxide layer separatingthe nanotubes from a doped silicon substrate which acts as a gate. Two importantresults are observed in this work. The first is the observation of a dip in the differ-ential resistance which depends both on the contacts and the applied gate. This dipis indicative of a proximity effect mediated by Andreev reflection. The other inter-esting effect is the observation, at even lower temperature, of a peak superimposedto the Andreev features. In Figure 9.2a,b, we display the differential resistance of asample with Nb contacts at two different gate voltages and the resistance of a differ-ent sample with normal contact (Figure 9.2c). The effect of the gate voltage betweenpanel (a) and (b) is dramatic. In Figure 9.3, we display the effect of the tempera-ture for temperatures close to the critical temperature of the contacts. Above Tc ofthe contacts, the shape of the dV/d I is quite similar to the differential resistanceof the sample with normal contacts. The effect of the gate voltage is therefore tochange the transparency of the contacts as the applied voltage would not be suffi-cient to appreciably change the population of the electronic subbands in the carbonnanotube3). High-transparency contact leads to efficient Andreev conversion, whichtherefore produces a dip in dV/d I . Low transparency, leading to a small probabilityof Andreev reflection, produces a peak in the differential resistance.

However, this picture, valid in the absence of electron–electron interaction, failsto explain the emergence of small peaks in the Andreev dip (Figure 9.4). Hence, therole of interaction in the nanotubes or possibly intrinsic superconductivity are to beexamined in detail. In the case of moderate contact transparency, interplay betweenproximity induced superconductivity and the Kondo effect has been observed inSWNT [9].

9.2.2Intrinsic Superconductivity in SWNT

This effect is quite intriguing and although reported in 2006 [10], it has, so far,not been reproduced by other groups. Quite startling, a zero bias anomaly (ZBA),commonly associated with Andreev reflection and, therefore, a signature of super-conductivity [5], has been reported in single-wall nanotubes (SWNT). Let’s look atthe experiment in detail.

3) The resistance of nanotubes fabricated with this process is independent of the tube length up to� 10 μm. The electrons move balistically along the nanotube and the resistance is dominated bythe contacts.


Figure 9.2 (a) and (b) show the differentialresistance dV/d I vs. bias voltage V for a car-bon nanotube sample of diameter D < 1.8 nmwith Nb electrodes at (a) Vg D �40 V and(b) Vg D 0 V. (c) For comparison, the dif-ferential resistance of a sample with normalmetal electrodes (Ti-Au) over the same rangeof bias voltage (T D 4.2 K). Note that (a)and (b), respectively, show a dip and a peak at

bias voltages below 2–3 mV, not seen in (c).The expected range of bias voltages for thesefeatures is � 2.9 mV, twice the gap of Nb(the factor of 2 accounting for two Nb-SWNTinterfaces in series). Over this range of biasvoltages, the relative size of the peak in dV/d Iin the normal sample (c), one-tenth of the to-tal resistance, is about 3% of the relative peaksize for the Nb sample (b). From [5].

An isolated semiconducting SWNT, contacted by electrodes, is lying on a dopedsilicon substrate coated with SiO2. Through the substrate, a gate can be applied tothe SWNT. This is the knob the authors use to induce superconductivity in theirdevices. Zhang et al. fabricated three devices with nanotube diameters between 1and 3.3 nm and lengths between 1 and 2 μm. The contacts are either superconduct-ing (bilayer of 5 nm of Pd and 200 nm of Nb) with Tc D 9.1 K or normal (50 nmof Pd). While for the vast majority of the gate voltages (Vg) the differential resis-


Figure 9.3 Differential resistance dV/d Ivs. bias voltage V for a SWNT sample #2 ofdiameter D < 1.8 nm with Nb electrodesover a range of temperatures around Tc ofNb (9.2 K). The magnitude of the Andreev dip

decreases with increasing T and disappearsabove Tc. From [5]. For a color version of thisfigure, please see the color plates at the begin-ning of the book.

Figure 9.4 (a) Differential resistance dV/d I vs. bias volt-age V for a SWNT sample #3 of diameter D < 1.8 nm withNb electrodes measured at 4.2 K (dashed line) and 2 K (solidline), showing a narrow peak emerging from the center of theAndreev dip at lower temperatures (Vg D �40 V). (b) At atemperature of 40 mK, dV/d I may show either a peak or adip, even at large negative gate voltage. This is different fromwhat is found at 4.2 K, where dV/d I will always have a dip atsufficiently negative gate voltage. The low temperature tracesdiffer by only a few volts (dashed curve, Vg D �38 V; solidcurve, Vg D �40 V), but show opposite curvatures around zerobias. The overall behavior at temperatures well below 4 K is notunderstood. From [5].

tance (dV/d I ) decreases from room temperature down to � 30 K and then slowlyincreases down to the lowest temperatures measured, for some small range of Vg,dV/d I has a sharp downturn for T . 12 K. In correspondence with these sharpdownturns, dV/d I has a dip as a function of the source-drain voltage. Notice thatin absence of the downturn, in the same range of source-drain voltages, dV/d I hasa peak. All three samples exhibit these characteristics (Figure 9.5).

One difference that exists between the sample with superconducting contactsand the other is that in the first case, the ZBA starts to appear around 30 K, while inthe others, it appears below 15 K. The authors exclude several possibilities regard-


Figure 9.5 Differential resistance as a func-tion of bias and temperature for CNFET (car-bon nanotube field-effect-transistor) no. 1.Vg D 47 V for the dip on the left. Vg D 48 Vfor the peak on the right. The temperatures for

the curves shown in (c) are 5.5, 8.5, 9.5, 15,20, and 30 K. The temperatures for the curvesshown in (d) 3, 4, 7, 9, 15, and 20 K. From [10].For a color version of this figure, please seethe color plates at the beginning of the book.

ing the appearance of the ZBA. Obviously, the ZBA cannot be attributed exclusivelyto the presence of superconducting contact. The authors are able to exclude reso-nant tunneling through imperfection in the SWNT (shape of differential resistancevs. T), Kondo effects (would not fit the shape of the ZBA) and Fano resonances(Coulomb blockade is not visible). The explanation suggested by the authors is thatthe SWNT themselves are superconducting within the small ranges of voltageswhere the ZBA is observed. This happens because the net effect of the gate voltageis to shift the Fermi energy of the nanotube. For sufficiently large values of Vg, EF

will be shifted to values of energy corresponding to the Van Hove singularities inthe nanotube density of states (see e.g., [11]). Not only are they able to reasonablyreproduce the dip of the differential resistance as a function of the temperature,for devices with and without superconducting contacts, but this hypothesis is sup-ported by the spacing of ZBAs observed at different gating voltages; this spacingis consistent with the distance in energy between different Van Hove singularities,associated with the 1-dimensional subbands, as calculated for the physical param-eters of the SWNT under study.

9.2.3Superconductivity in Ropes Mediated by the Environment

One of the nanotube systems from which a lot of results have been obtained is the4-Å (0.4 nm) nanotubes. Discovered in 1999, they are the smallest nanotubes andare grown inside the linear channel of AlPO4-5 zeolite crystal (IUPAC code:AFI).


Figure 9.6 The framework structure of the AFI crystal viewed along the c-axis. The structure hasbeen rendered with Qutemol [12].

The channel width is about 7 Å (0.7 nm) and the distance between channels is about14 Å (1.4 nm) (Figure 9.6). The first number determines the radius of the nanotube,while the second determines the distance among nanotubes which turns out to berelevant for explaining observed superconductivity. Given the small diameter of thechannel, only extremely small nanotubes can be grown inside. Still, there are threetypes of nanotubes which have such a small diameter: they have chirality (4,2),(5,0), and (3,3), and ab initio calculations show that the first one is semiconductingwhile the other two are metallic. It must be noted that during the growth process,all three chiralities are grown, but as it turns out, only one is relevant to the super-conductivity. The first observation of superconductivity in this system dates backto 2001 [13] when the magnetic susceptibility of a pure AFI crystal and an AFIcrystal with embedded nanotubes was measured. The main observation regardedthe isotropy of the susceptibility signal: while the AFI pure crystals were isotropic,the SWNT@AFI was not. The results are shown in Figure 9.7 after subtractingthe signal from the pure AFI, together with simulation based on phenomenolog-ical calculation using the Ginzburg–Landau formalism. An interesting aspect ofthis calculation is that it allowed one to identify which of the three chiralities isinvolved in the superconductivity; in fact, only the effective electron mass of the(5,0) is consistent with the value used in the calculation (0.36me with me the bareelectron mass).

This experimental work has also spurred a lot of interest in assessing the possi-bility of superconductivity in these nanotubes. However, all calculations have sug-gested that the critical temperature for a normal to superconductor transition innanotubes would be either quite lower than the 10 K observed in [13] or Peierls in-stability would render superconductivity impossible. However, a theoretical inves-tigation from Gonzales and Perfetto [14] (Section 2.3) uncovered the possibility ofsuperconductivity in the array of nanotubes depending on the dielectric constant of


Figure 9.7 Normalized magnetic susceptibil-ity of the SWNTs plotted as a function of tem-perature for five values of the magnetic field.The curves are displaced vertically for clarity.Values shown are for theory (open symbols)and experiment (filled symbols). �0 denotesthe value of the susceptibility at T D 1.6 Kand magnetic field D 0.2 T. The experimentalvalue of �0, when normalized to the volume ofthe SWNTs, is �0.015 (in units where B D 0

denotes �0 D �1). The scatter in the theorypoints reflects statistical fluctuations inher-ent in the Monte Carlo calculations. (Inset)Temperature dependence of magnetizationdensity for zeolite AFI crystallites (curve 1)and for AFI crystallites with SWNTs in theirchannels (curve 2). Both curves are measuredat 2000 Oe. From [13]. For a color version ofthis figure, please see the color plates at thebeginning of the book.

the environment. While this was not clearly recognized in the original experiment,it turns out that the interaction between nanotubes plays a significant role.

A key improvement to test the experimental results came from the marked im-provement of the fabrication technique [15]; again, this is a case where improve-ments in the fabrication have brought about marked improvements in the experi-mental results.

The original fabrication consisted of two steps: growth of AFI crystals and thengrowth of nanotubes. In the original process, TPA (trypropylamince) moleculeswere embedded in the AFI matrix as precursor; after growing the AFI, a heat treat-ment at 580 ıC decomposed the TPA from which the nanotubes can grow. How-ever, the number of carbon atoms available for the formation of a carbon nanotubewas not be sufficient to occupy the full channel. In addition to this, carbon atomscould also escape from the AFI channels, thus further decreasing the actual num-ber of carbon atoms available to form the nanotube. Moreover to this, the nitrogenand hydrogen atoms from the TPA decomposition could form NH3 which cor-rodes the AFI crystals. Thermal gravimetric analysis of these samples reveals amass change of only 1.47%, corresponding to a filling factor of 4.5%. Again, weshould point out that during the nanotube growth, all three types of nanotubes aregrown and only the nanotubes with (5,0) chirality are superconducting. The new


approach starts from the AFI crystals, but then removes the TPA by heating theAFI crystals in vacuum for 8 h. Successively, ethylene gas continuously flows as asource of carbon atoms for the formation of carbon nanotubes. With this approach,the group was able to produce a thermogravimetric change of 3.35%, correspond-ing to a filling factor of 10.3%. With some additional tweaks, the filling factor wasraised up to 16.6%.

In the following, we will only discuss the electrical measurement performed onthese nanotubes and refer to the review by Wang et al. [16] for additional details.

9.2.3.1 Electrical ExperimentsImmediately after the formation of the AFI crystal, FIB is used to etch the crystalin order to define two electrodes with a gap of � 100 nm. As the carbon nanotubesgrown with this method are typically 100–200 nm long, this length serves to makesure a (array of) single continuous nanotube is connected.

After the nanotube growth, a bilayer of Ti/Au (Ti is used for adhesion) is evap-orated on the AFI, FIB is used to form the electrodes by removing the film in a

Figure 9.8 (a) Cartoon of the sample con-tain an array of SWNT in an AFI zeolite crys-tal. Yellow denotes gold and green denotesthe AFI crystal surface exposed by FIB etch-ing. Nanotubes are delineated schematicallyby open circles. (b) SEM image of one sam-ple. The c-axis is along the N-S direction. Thethin, light, horizontal line in the middle is the100 nm separation between the two surfacevoltage electrodes that are on its two sides.The dark regions are the grooves cut by theFIB and sputtered with Au/Ti to serve as the

end-contact current electrodes. (c) and (d)show schematic drawings of the two-probeand four-probe geometries, respectively. Blue-dashed lines represent the current paths. In(d), the two end-contact current pads are4 mm in depth and 30 mm in width. The dif-ference between the two-probe and the four-probe measurements is the transverse re-sistance, delineated by the red circles in (c).From [17]. For a color version of this figure,please see the color plates at the beginning ofthe book.


Figure 9.9 Resistance plotted as a function oftemperature for the 4-terminal sample, mea-sured at 10 nA. The geometry of the sampleis illustrated in the cartoon inset, with yellowindicating gold coating and green the AFI crys-

tal surface. The slice between the 2 groovesis 5 μm thick. The enlarged section showsthe transition starts at 15 K. Up to 4 T, there isno appreciable change in the R(T) behavior.From [18].

predetermined pattern and finally annealing at 450 ıC is used to reduce the con-tact resistance. By appropriately modifying the electrode pattern, both two- andfour-terminal electrical measurements can be performed (Figure 9.8). These mea-surements reveal a normal to superconducting transition that starts at 15 K; themeasured resistance drops from 11 kΩ to approximately zero in a 10 K span, andadditionally, is insensitive to a 4 T magnetic field (Figure 9.9). Both of these areindicative of the transition of an array of 1D superconductors. Additional informa-tion is provided by measurements of differential resistance: measurements up to11 T performed at T D 0.35 K clearly demonstrate the presence of a supercurrentas a dip in the dV/d I . In addition, measurements in zero field at higher tempera-tures reveal a peak in the transition. The peak first appears at 2.7 K, a temperatureat which, from the R vs. T data, the system should be in the 1D superconductingstate. Recent theoretical work [19] has provided indication that for a thin array of(5,0) nanotubes, Peierls distortion/charge density waves may represent an excitedstate of the array and destabilize the superconductivity.

An additional sample has revealed measurements consistent with an overall crit-ical temperature of 15 K and has, in addition, provided indication of other mech-anisms. As already mentioned, wide transition in the R vs. T curve and indepen-dence on the magnetic field is a hallmark of a N/S transition in a 1D superconduc-tor. In a different sample, while the measured TC remained similar, a sharp transi-tion dependent on the magnetic field was observed around 7.5 K. The observed be-havior is found to be consistent with a Kosterlitz–Thouless–Berezinskii-type (KTB)transition: lowering the temperature, the growth of the superconducting conden-sate favors Josephson coupling in the ab plane (perpendicular to the nanotubegrowth). This is analogous to a 2D spin system in which vortex–antivortex excita-tion can be found; these vortices can become bound KTB pairs at sufficiently low


temperature. The nanotube system of 1D superconductors becomes a system of2D superconductors (in the ab plane) weakly linked in the direction of nanotubegrowth; the magnetic field would decrease the superconducting electron density inthe quasi-1D elements which would shift the TKTB transition at lower temperaturesand diminish the amplitude of resistance drop associated with the transition. Thisinterpretation is supported by both nonlinear I �V characteristics and temperaturedependence of the resistance. Additional details can be found in [13] together withthe possible observation of a FFLO state (Fulde, Ferrel Larkin and Ovchnnikov); inthis state, a type-II superconductor would sacrifice part of its volume to a normalstate in order to increase its upper critical fields.

Superconductivity in this nanotube system is also supported by measurementsof the magnetic Meissner effect and thermal specific heat. However, as both ofthese types of measurements measure the bulk effect, they are extremely difficultin the SWNT@AFI system. The signal of the AFI, in fact, typically dominates theoverall signal, and while resolving the effect of the nanotube is quite challenging,the overall picture agrees with the electrical measurements discussed above.

9.3Majorana Experiments

Another class of 1D devices in which the proximity effect plays an important rolein revealing novel physical phenomena includes semiconducting nanowires withstrong spin–orbit interaction, as discussed in Section 5.4. In fact, recently, it wassuggested ([20–23] and references in Section 5.4) that semiconducting nanowirescould be a convenient place to observe Majorana fermions, a special class of realsolutions of the Dirac equations which have the extremely intriguing characteris-tic of the Dirac particles being identical to their antiparticles; in a semiconductingnanowire with strong spin–orbit coupling, in the presence of a normal and a su-perconducting electrode, it would be possible to observe Majorana fermions (seeSection 5.4 for a theoretical overview). An experimental realization of this sugges-tion has recently been reported by Mourik et al. [24] in L.P. Kouwenhoven’s groupat Delft University, and in other groups [25–27]. Another suggestion for the ob-servation of a Majorana mode comes from Fu and Kane [28], who have predictedits existence at the interface between a conventional superconductor and a super-conducting topological surface state; this can be formed by proximitization of atopological insulator by the conventional superconductor. This experiment has re-cently been reported by Williams et al. in D. Goldhaber-Gordon’s group at StanfordUniversity [29].

9.3.1Majorana Experiment in Semiconducting Nanowires

Before reviewing the Majorana experiment, it is useful to review the original ex-periment on semiconducting nanowires [30]. InAs nanowires are grown via a cat-

9.3 Majorana Experiments 287

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(Inset) Measured voltage spacing ΔV (sym-bol) as a function of microwave angular fre-quency ωRF. The solid line (theory) shows theagreement with the AC Josephson relationΔV D „ωRF/(2e). From [30]. For a color ver-sion of this figure, please see the color platesat the beginning of the book.

alytic process: a thin layer of gold is deposited on the surface of a InAs or InPsubstrate. The substrate is heated to a temperature close to the growth tempera-ture, at which point the gold film agglomerates in nanoparticles. The diameter ofthe nanowire is determined by the size of the gold nanoparticles and the semi-conductor material is provided by pulsed laser ablation of an undoped InAs target.The single-crystalline nanowires thus formed are then transferred onto a heavilydoped Si substrate covered with a SiO2 oxide layer; standard e-beam lithography isused to define the electrode pattern which is then deposited by e-beam evaporationof a Ti/Al (10 nm/120 nm) bilayer. Before the metal deposition, a light etching inbuffered hydrofluoric acid is used to improve the contact. While no thermal an-nealing is performed, the contact transparency is extremely high, with nanowireresistances ranging from 0.4 to 4.5 kΩ. As the nanowire acts as a weak link be-tween the two superconducting electrodes, it is necessary, in fact, that the interfaceresistance is extremely small, and thus Cooper pairs from the electrodes can dif-fuse into the semiconductor. Additionally, the length of the nanowire determinedby the distance between the electrodes must be comparable if not smaller than thephase coherence length for electron propagation in the nanowire. One of the mainresults of the paper is summarized in Figure 9.10.

Here, the black line shows the V(I ) curve at 40 mK for the nanowire. The hys-teresis at low currents can clearly be seen. To demonstrate that the behavior ofthis nanowire is similar to the behavior of a Josephson junction, the authors ap-ply microwave radiation to demonstrate the presence of Shapiro steps [31]. Thephase locking between the applied microwave field and the Josephson frequency


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Figure 9.11 (a) Implemented version of the-oretical proposals, using InSb SmNW andNb superconductor. Scanning electron mi-croscope image of the device with normal(N) and superconducting (S) contacts. TheS contact only covers the right part of theInSb nanowire. The underlying gates, num-bered 1 to 4, are covered with a dielectric.(Note that gate 1 connects two gates, andgate 4 connects four narrow gates; see (b).)(b) schematic of our device. (c) an illustrationof energy states. The black rectangle indicates

the tunnel barrier separating the normal partof the nanowire on the left from the wire sec-tion with induced superconducting gap, Δ. (In(a), the barrier gate is also shown in white.)An external voltage, V, applied between Nand S drops across the tunnel barrier. (Red)stars, again, indicate the idealized locationsof the Majorana pair. Only the left Majorana isprobed in this experiment. Adapted from [24].For a color version of this figure, please seethe color plates at the beginning of the book.

produces voltage plateaus at Vn D n„ωRF/(2e) with n as an integer, e as the electroncharge, and „ as the Planck constant. The inset in Figure 9.10 shows the agreementbetween the measured and predicted ΔV as a function of the microwave frequency.

This demonstration of the proximity effect in a semiconducting nanowire is anecessary condition for the study of Majorana fermions. Based on the theoreticalproposal [20–23], Mourik et al. [24] devised an experiment to observe their existence:an InSb nanowire4) grown with a process similar to the one described above iscontacted by a normal (bilayer of Ti/Au) and a superconducting (NbTiN) electrode.A set of gates is also deposited on a substrate in order to provide additional knobsto the experiment. Device and schematics are reproduced in Figure 9.11.

The main results of the paper are shown in Figure 9.12a,b. By tuning one of thefine gates between the normal and superconducting electrodes, the authors create atunneling barrier in the nanowire, which allows the measurement of the nanowiredifferential conductance. The Majorana fermions, while having zero charge andenergy, are observed in tunneling spectroscopy measurements as states at zero en-ergy while changing magnetic fields and gate voltages over wide ranges. To thisextent, it is important to notice that the superconducting electrode only partiallycovers the nanowire; this is intentional so that the effect of the underlying gates isnot completely screened. The traces in Figure 9.12a show the evolution of the differ-ential conductance as a function of the magnetic field applied along the nanowireaxis. While at small fields, only the peaks corresponding to the quasiparticle den-

4) InSb has larger spin–orbit coupling than InAs.


Figure 9.12 Magnetic field-dependent spec-troscopy. (a) d I/dV vs. V at 70 mK taken atdifferent B fields (from 0 to 490 mT in 10 mTsteps; traces are offset for clarity, except forthe lowest trace at B D 0). Data are fromdevice 1. Arrows indicate the induced gappeaks. (b) Colorscale plot of d I/dV vs. V andB. The ZBP is highlighted by a dashed oval;

green dashed lines indicate the gap edges. At� 0.6 T, a non-Majorana state is crossing zerobias with a slope equal to � 3 meV/T (indi-cated by sloped yellow dotted lines). Tracesin (a) are extracted from (b). From [24]. Fora color version of this figure, please see thecolor plates at the beginning of the book.

sity of states are visible (˙250 μeV), at fields between 100 and 400 mT a zero biaspeak (ZBP) appears; this is more clearly observed in Figure 9.12b. This peak isidentified with the left Majorana states of Figure 9.11c. Other features present inFigure 9.12b are used to verify the experimental conditions; in particular, the or-ange lines follow a pair of resonances with a slope of the order of Ez D gμBB/2,where μB is the Bohr magneton and g is the Landé factor for the nanowire. Thetheoretical expectation is to observe Majorana fermions when Ez > (Δ2 C μ2)

12 ,

where Δ is the superconducting gap in the nanowire and μ is the Fermi energy.For μ D 0, and Δ � 250 μeV, we obtain a magnetic field of about 0.15 T, consistentwith the experimental observation.

Additional evidence that the ZBP is due to the presence of Majorana fermions isprovided by extensive experimental checks in devices fabricated with leads whichwere both normal or both superconducting [24], and exploring the dependenceon the magnetic field orientation. In the absence of any of the main ingredients,that is, one normal and one superconducting contact, and an external field alignedalong the nanowire axis, the magnetic field-dependent spectroscopy displays differ-ent characteristics and, while a ZBP can also appear, its presence is not consistentwith the existence of Majorana fermions [24]. While this work does not addressthe topological properties of Majorana fermions, it provides an unequivocal signa-ture of their existence and is a first step toward more sophisticated experimentsthat can demonstrate the usefulness of Majorana fermions in topological quantumcomputing [29, 32, 33, 35].


Figure 9.13 (main) V vs. I for Al-Bi2Se3-Al S-TI-S devices of dimensions (L, W ) D(45 nm, 1 μm) for B D 0, 2, 3, 5, 8, 10 mT andat a temperature of 12 mK. At B D 0, IC is850 nA, which is reduced upon increasing B.For this device, the product ICRN D 30.6 μV,which is much lower than theoretically ex-pected for conventional JJs. (Upper-left in-

set) I–V curves overlap for all values of B atV � 2Δ/e � 300 μV. (Lower-right inset)Sweeps up and down in I show little hys-teresis, indicating that the junction is in theoverdamped regime. Adapted from [36]. Fora color version of this figure, please see thecolor plates at the beginning of the book.

9.3.2Majorana Experiment in Hybrid Superconductor-Topological Insulator Devices

In a recent experiment, Williams et al. [36] developed the original proposal by Fuand Kane [28] and demonstrated features that are compatible with the existence ofthe Majorana mode in an S-TI-S structure, where TI stands for topological insulator(Bi2Se3), rather than in a S-SmNW-S structure. After preparing flakes of Bi2Se3 bymechanical exfoliation [34] standard e-beam lithography is performed in order todeposit superconducting aluminum electrodes whose adhesion is guaranteed by athin layer of Ti. The measurements performed on these two devices are voltage–current trace and the differential resistance as a function of the magnetic field.

One main observation is that the product IcRN, with Ic and RN the weak linkcritical current and normal resistance respectively, is inconsistent with the Ambe-gaokar–Baratoff formula [6, 7] IcRN D πΔ/(2e) with Δ denoting the gap of thesuperconducting electrodes (Figure 9.13). The second observation is that upon ap-plication of a magnetic field in the plane perpendicular to the Bi2Se3 flake, thedifferential resistance as a function of the magnetic field does not show a Fraun-hofer pattern which would correspond to the current–phase relationship (CPR) fora less exotic weak link.


Figure 9.14 (a) Differential resistancedV/d I as a function of B and I showing ananomalous magnetic diffraction pattern for aW D 1.5 μm Al-Bi2Se3-Al S-TI-S junction. Twofeatures are of note: a smaller-than-expectedvalue of BC at 1.70 mT and a nonuniformspacing between minima at values B D 1.70,6.50, 11.80 mT. (b) (Main) IC(B) (dashed linewith circles) extracted from dV/d I in (a) iscompared to the expected Fraunhofer pattern

for the junction (solid line) where a reductionof the scale of the pattern and the nonuniformspacing are evident. (Inset) a comparison ofthe simulated Fraunhofer pattern for a sinu-soidal and an empirically determined, peakedCPR (current–phase relation). The narrowingof the diffraction pattern and the aperiodicminima observed in (a) are captured by thisCPR. From [36].

Including the presence of a 1D Majorana wire (i.e., two Majorana modes at thetwo S-TI interfaces), and assuming that the supercurrent is determined by thephysics of the junction near the zero energy – dominated by the Majorana modes –which in turn produces a peaked CPR, the author can qualitatively reproduce theexperimental results as shown in the inset of Figure 9.14. It should be stressed thatin a control sample in which the Bi2Se3 is replaced with a graphite flake, both theIcRN product and the Fraunhofer diffraction pattern are close to the expectation fora weak link.


9.4Superconducting Nanowires as Single-Photon Detectors

Superconducting nanowires are showing a lot of promise as single-photon detec-tors. The field is quite vast and specialized, and we will only discuss the aspectswhich are more relevant for nanowires of extremely reduced size. Our discussionwill follow Natarajan et al. ([37], and references therein) to which we refer for amore in-depth review of the topic. The principle of operation of a superconducting(nanowire) single-photon detector (SSPD or SNSPD) is relatively simple and wasdemonstrated by Goltsman et al. in 2001 [38]. A superconducting nanowire held attemperature well below its critical temperature is current-biased at a current justbelow the nanowire’s critical current (Ic), or more accurately its switching current(Is). A photon incident on the nanowire will release enough energy to drive a smallsection of the nanowire normal, thus creating a hotspot [39] (Figure 9.15a-ii). Whilethe size of the hotspot is smaller than the nanowire width, the current flowingthrough the nanowire will tend to flow around the normal region (Figure 9.15a-iii).The local current density around the hotspot will thus increase, locally exceedingIc, which, in turn, will increase the size of the hotspot. This process will continueuntil the hotspot is as large as the nanowire (Figure 9.15a-iv). At this point, Jouleheating will continue increasing the size of the normal region (Figure 9.15a-v) un-til the current is stopped (by the external circuit) and the nanowire can cool down(Figure 9.15a-vi) so that the SNSPD is ready for detecting another incident photon.

The simplified equivalent circuit used to understand the detection is shown inFigure 9.15b and a simulation of its time response, for typical parameters, in Fig-ure 9.15c. The switch is initially closed so that all of the current flows through it;the absorption of a photon is equivalent to opening the switch. The response ofthe circuit is limited by the time constant τ1 D Lk/(Z0 C Rn(t)) of the order of1 ns for the parameters in Figure 9.15c, where Z0 D 50 Ω, the kinetic inductanceLk D μ0λ2L/A with λ as the penetration depth, L as the nanowire length and Adenoting the cross-sectional area of the nanowire. When the current is divertedthrough Rn , the nanowire starts to cool down: at this point, the switch is closedand the time constant will be τ2 D Lk/Z0. The total dead time in which the detec-tor is not sensitive to the incoming photon is τ D τ1 C τ2 � τ2 D Lk/Z0. Fromthe last expression and the circuit diagram, it is obvious that to reduce the deadtime, it is necessary to reduce the kinetic inductance (by changing its dimension ormaterial) or placing additional resistors in series with the device.

A SNSPD is typically characterized by its efficiency. This, in turn, is related to:(1) coupling with the incident radiation, (2) detector material and geometry, and (3)the probability that an incident photon is converted into a signal. Other importantmetrics for a detector are: (1) noise that can produce false counts (usually measuredas the dark count rate (DCR) (2) jitter, that is, the time before the arrival of a photonand the generation of the output signal, and (3) recovery time.

Typical SNSPD nanowires are of the order of 100 nm in width; from this pointof view, they are quite far from the reduced size nanowires which we have touchedupon in this work. However, it has been recognized that narrow (20–30 nm wide)

9.4 Superconducting Nanowires as Single-Photon Detectors 293

Figure 9.15 The basic operation principle ofthe superconducting nanowire single-photondetector (SNSPD) (after Goltsman Refer-ence 7 in [37], Semenov et al. Reference 19in [37] and Yang Reference 20 in [37]): (a) Aschematic illustrating the detection cycle. (i)The superconducting nanowire maintainedwell below the critical temperature is directcurrent (DC)-biased just below the criticalcurrent. (ii) When a photon is absorbed bythe nanowire, thus creating a small resistivehotspot. (iii) The supercurrent is forced toflow along the periphery of the hotspot. Sincethe NbN nanowires are narrow, the local cur-rent density around the hotspot increases,exceeding the superconducting critical cur-rent density. (iv) This, in turn, leads to theformation of a resistive barrier across thewidth of the nanowire Reference 7 in [37].(v) Joule heating (via the DC bias) aids thegrowth of the resistive region along the axisof the nanowire Reference 20 in [37] until thecurrent flow is blocked and the bias currentis shunted by the external circuit. (vi) This al-

lows the resistive region to subside and thewire becomes fully superconducting again.The bias current through the nanowire returnsto the original value (i). (b) A simple electri-cal equivalent circuit of a SNSPD. Lk is thekinetic inductance of the superconductingnanowire and Rn is the hotspot resistance ofthe SNSPD. The SNSPD is current-biased atIbias. Opening and closing the switch simu-lates the absorption of a photon. An outputpulse is measured across the load resistorZ0 (Reference 21, 22 in [37]). (c) A simula-tion of the output voltage pulse of the SNSPD(approximating the pulse shape typically ob-served on an oscilloscope after amplification).Values of Lk D 500 nH and Rn D 500 Ω havebeen used for this simulation (for simplicity,the Rn is assumed fixed, although a more de-tailed treatment Reference 20 in [37] showsRn (t)). The solid blue line is the leading edgeof the SNSPD output pulse, whilst the dot-ted red line is the trailing edge of the outputpulse. The time constants relate to the phasesof the detection cycle in (a). From [37].

nanowires may present substantial advantages in terms of registering probabilityand operation in the midinfrared range.

In the following, we will discuss, in detail, recent experimental results on SNSPDwith ultranarrow nanowires. Typical SNSPDs use NbN wires of about 100 nmwidth and 3–5 nm thickness which are typically arranged in meander in order tocover the large illuminated area. However, this comes with the drawback of an


Figure 9.16 The proposed parallel detector.(a) Electron microscope image of the detec-tor showing the coplanar transmission lineused for DC biasing and signal readout, theseries inductor LS (indicated by the arrow)and the parallel structure (in the box). (b) En-

largement of the structure of the five parallelwires: each wire is 5 μm long and 100 nmwide. (c) Electrical model of the parallel de-tector used to simulate its response. The graybox represents the detector. From [40].

extremely large inductance, mostly due to the kinetic inductance of the wire. Asdiscussed above, this increases the detector recovery time and also reduces themaximum count rate. A clever workaround [40] for reducing the kinetic induc-tance of the detector consisted of designing a circuit with five identical nanowiresin parallel (electrical schematics in Figure 9.16). This circuit, dubbed the supercon-ducting nanowire avalanche photodetector (SNAP), is biased near the switchingcurrent Is of the nanowires: the arrival of a photon will cause one section of thenanowire to turn normal, thus redirecting the current in the other nanowires. Ifthe biasing current is large enough, the increase in current will switch the re-maining nanowires as well. Since the total inductance is Ldet D (1 C 1/N )LW, theoutput signal will be larger and the recovery time will be smaller than a meanderednanowire covering the same surface.

Since detectors based on narrow nanowire are expected to be more efficient indetecting midinfrared photons and are able to operate at a lower bias than detectorsbased on larger nanowires, Marsili et al. [41] implemented an SNSPD with a 30 nmwide NbN nanowire. Based on this result, they also developed a SNAP based on30-nm wide nanowires. In particular, the ability of these detectors to work at lowerbias operation, with respect to a 90 nm wide nanowire detector, is a real advantage.

The fabrication starts with the deposition of a 4–5 nm of NbN. Gold contactsare fabricated using optical lithography and deposited using an e-beam evaporator.Electron beam lithography is used to pattern negative resist hydrogen silsesquiox-ane (HSQ); the transfer of this pattern in the film using reactive ion etching formsthe nanowire, thus completing the detector [41]. Since the circuit density is quitehigh, proximity effect correction during e-beam lithography [42] plays a key role inthe successful fabrication of the device.

The first result by Marsili et al. (Figure 9.17) is the demonstration of an SNSPDbased on a 30 nm nanowire. This is a single meandering nanowire with a lat-eral width of 30 nm and a thickness of 4–5 nm. The first result is that a 90 nm

9.4 Superconducting Nanowires as Single-Photon Detectors 295

wide SNSPD, free of constriction, has a lower detection efficiency than the 30 nmwide SNSPD. Additionally, constrictions in 30 nm-SNSPD devices produce a rel-atively small effect when compared with a 90 nm-SNSPD. Intuitively, this shouldnot be surprising as the detector performance depends on its bias current and ifthe switching current of the constriction is much smaller than the switching cur-rent of the detector, the detector performance would suffer. Any constriction willlower the operating bias of the detector because if the constriction switches to thenormal state, the detector operation will be impossible. Assuming a constriction tobe a reduction of the nanowire superconducting cross-sectional area with respectto the nominal value (σc D σn(1 � ISW/IC) implies that if ISW D IC, the area ofthe constriction is zero), we can see how a constriction in the 90 nm-SNSPD suchthat σc D 27 nm2 implies that the constriction has a switching current close to90% of the critical current. If σc D 90 nm2, the constriction will switch at a currentequal to 0.75I 90 nm

c . If we take a 30 nm-SNSPD with the same fractional constric-tion, that is, σc D 27 nm2 and ISW D 0.75I 30 nm

c , we can see how the impact ofsuch a constriction is almost negligible (Figure 9.17).

This should not be surprising if we recall the working of an SNSPD displayedin Figure 9.15. The arrival of a photon will drive a superconducting region of sizecomparable to the superconducting coherence length, in the normal state. If thenanowire is quite larger than the coherence length, in order to have a high prob-ability of registering the photon, the bias current will have to be very close to thenanowire switching current. It is natural to see how the performance of such adetector is dramatically affected by a constriction.

Note that all the above experiments were performed with a light at 1550 nm wave-length. As it is well-known that longer wavelengths shift the detection efficiencycurve of Figure 9.17 at higher currents, these results bode well for the operation ofthese detectors at longer wavelengths [43–45].

However, since the response of an SNSPD is proportional to its bias current,low bias operation inevitably produces a low signal to noise ratio. To overcome thisproblem, Marsili et al. fabricated a superconducting nanowire avalanche photode-tector (SNAP) to exploit its advantages with respect to SNSPD. The experimentalresults for SNAP with 2, 3 and 4 nanowires in parallel are shown in Figure 9.18.

The detection efficiency curves are shown in Figure 9.18 and show plateaus for3-SNAP and 4-SNAP. While the n-SNAP behaves as expected if biased at a cur-rent at which the detection efficiency is independent of the illumination level (IAV),the presence of the plateau regions points to a different behavior for lower cur-rent (Ib < IAV). Contrary to the simplified explanation provided above, at this bias,the arrival of a photon (or dark count) has the effect of “arming” the detector byaltering the current distribution in the remaining nanowires. The arrival of a sub-sequent photon “triggers” the detector, thus making these detectors suitable fordetection of correlated photons. This explanation is consistent with an electrother-mal model [46], which takes into account the heat flow between the detector andsubstrate. While the reduction in nanowire size seems to be a fruitful path to fol-low, in order to increase the performance of SNSPDs, the overall detector geometryneeds to be properly designed. After working on the polymethylmethacrylate cold-


Figure 9.17 SNSPDs based on 30 nm wideNbN nanowires. (a) Scanning electron micro-scope (SEM) images of an SNSPD hydrogensilsesquioxane (HSQ) mask on NbN. Thenanowires are 30 nm wide and the pitch is100 nm (inset), covering an active area of1.03 μm � 1.14 μm (dashed frame). See Sup-porting Information for the device fabricationprocess. (b) Device detection efficiency atλ D 1550 nm as a function of normalizedbias current for constricted and constriction-free SNSPDs based on 30 nm wide and 90 nmwide nanowires. We assumed the constrictionto be a reduction of the nanowire supercon-ducting cross-sectional area with respect tothe nominal value. We quantified the deviceconstriction state by estimating the area ofthe nonsuperconducting part of the nanowirecross section as σc D σn(1 � ISW/IC), whereσn is the nominal nanowire cross section es-timated from the nanowire width (measuredby SEM) and thickness (estimated from thematerial deposition time and rate), ISW is the

device switching current, defined as the biascurrent at which the device switches from thesuperconducting to the normal state, and ICis the device critical current, experimentallydefined as the highest measured ISW of thedevices fabricated on the same film for theultranarrow nanowire SNSPDs (IC D 7.2 μA)and extracted from kinetic inductance vs.IB measurements for the 90 nm nanowire-width SNSPDs (IC D 18.8�20.1 μA). Thedevice detection efficiency was calculatedas η D H(CR � DCR)/Nph, where CR isthe count rate measured when the SNSPDwas illuminated, DCR is the count rate mea-sured when the SNSPD was not illuminated,H is a normalization factor (see SupportingInformation), and Nph is the number of pho-tons per second incident on the device activearea. The Ico (defined to be at the inflectionpoint of the η vs. IB curves) of the 30 nm widenanowire-width SNSPDs is marked with ablack arrow (see Supporting Information).From [41].

development technique (Section 6.10.3), and encouraged by the suggestion of thehigher fill factor (i.e., the ratio between the total area encompassed by the detectorand the active area), Berggren’s group set out to demonstrate an SNSPD with 90%fill factor [47]: a detector of area 3 μm � 3 μm with a 88 nm wide line separated by12 nm gaps. However, the testing of such a detector revealed unexpected results:the switching current of this detector was smaller than the switching current of thedetector with lower fill factor. The explanation to this somewhat surprising effectlies in the geometry of SNSPDs. The nanowire meander is composed by a nanowireregion of 88 nm width connected with a wider region (� 190 nm) in correspon-dence with a turn. The turns, while they have a larger cross-sectional area andthus a critical current larger than the nanowire, can negatively impact the detectorperformance because the current can “crowd” at the edge of the turn [48–50]; thedensity of current lines increases near the shortest current path (inside of a turn).Optimizing the design has improved the performance of the detector in terms of bi-

References 297

Figure 9.18 Detection efficiency and jitterof NbN SNAPs (superconducting nanowireavalanche detectors) vs. normalized biascurrent. (a) η at 1550 nm wavelength andDCR (inset) vs. normalized bias current (ra-tio of bias current and switching currentof each device, IB/ISW) for an SNSPD, a 2-SNAP, a 3-SNAP, and a 4-SNAP based on30 nm wide nanowires (in black, red, green,and blue, respectively). The devices were de-signed by integrating in parallel the same1.47 μm � 230 nm section N D 1, 2,3, 4 times. The series inductance of the N-

SNAPs was designed to satisfy the conditionLS(N � 1)/L0 D 10, where L0 D 13 nH is thekinetic inductance of one section (estimatedfrom the fall time of the SNSPD responsepulse). The detector switching currents wereISW D 7.2 μA (SNSPD); 13.4 μA (2-SNAP);18.1 μA (3-SNAP); 28.4 μA (4-SNAP). Thephoton fluxes (expressed in photons persecond, photons/s) on the detector activearea were μ D 5.5 Mphotons/s (SNSPD);12 Mphotons/s (2-SNAP); 19 Mphotons/s (3-SNAP); 25 Mphotons/s (4-SNAP). Adaptedfrom [41].

asing current and reduced the dark count [50]. The dark count is an additional causeof low performance in detectors. As the pulses created by dark counts are similarto pulses generated by the absorption of photons [51] it is plausible to imaginethat dark counts are generated by a similar mechanism: a hotspot in the detectorbecomes large enough to drive a section of the nanowire, and successively the en-tire nanowire, normal (Figure 9.15). Mechanisms leading to the creation of normalregion in the superconductor are quantum and thermal phase slips, quantum andthermal generation of vortices and vortex–anti-vortex creation [51–53]. While exper-imental evidence is so far incomplete, the most plausible explanation are related tothermal excitation of vortices [52, 53] or vortex–anti-vortex creation [53]. However itmust be noted that, while the continuous reduction in nanowire width might makephase slips relevant, the temperature at which the detector operate is too low forpurely thermally phase slips and too high for quantum processes.

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