Small Superconductors—Introduction
Abstract and Keywords
This article provides an overview of small superconductors, including some of the basic definitions, prominent characteristics, and important effects manifested by such materials. In particular, it discusses size effects, surface effects, electronmeanfreepath effects, phase slips, unusual vortex states, and proximity effects. The article first considers the two characteristic length scales of superconductors, namely the magnetic penetration depth and coherence length, before proceeding with an analysis of two size effects that account for how superconductivity responds when the bulk sample is made smaller and smaller in the nano range: the small size effects and the quantum size effects. It then examines other phenomena associated with small superconductors such as quantum fluctuations, Anderson limit, parity and shell effects, along with the behaviour of nanowires and ultrathin fims. It also describes some of the experimental techniques commonly used in the synthesis of small superconductors.
Keywords: small superconductors, surface effects, vortex states, proximity effect, superconductivity, small size effects, quantum size effects, Anderson limit, Nanowires, ultrathin fims
By small superconductors we mean superconducting objects whose one, two, or all three dimensions are shorter than some small characteristic length. As depicted in Fig. 1.1, this respectively makes a (a) threedimensional (3D) bulk material (b) quasitwo dimensional (2D), (c) quasione dimensional (1D), or (d) quasizero dimensional (0D) in the form of a thin film, a narrow wire, and a fine particle. Although studies of small superconductors began in the 1960s, the field received a significant impetus only 15–20 years ago when technological progress made various new experimental tools of fabrication and characterization of nanosized samples more abundantly available. Studies of small superconductors are important for many reasons. One of the prime motivations is to assess if there is any miniaturization limit for superconducting nanodevices. How do various superconducting parameters such as the critical temperature T_{c}, critical magnetic field H_{c}, critical current density J_{c}, etc. respond to sample size reduction and is there any limiting size just below which superconductivity completely disappears or is destabilized? Further, above all, there exists a perpetual curiosity to know if some entirely new properties or phenomena emerge when the sample size is significantly reduced down to the nanoscale. Finally, in a world obsessed with miniaturization of technology, mesoscopic superconductors are acquiring even greater relevance and timeliness for revolutionary innovations in novel devices and applications.
1.1 Two characteristic length scales of superconductors
A superconductor possesses two fundamental length scales, namely the magnetic penetration depth λ and coherence length ξ. The former, according to F. and H. London (1935), arises when a superconductor is subjected to an external (p. 4) magnetic field, smaller than the critical field H_{c}. Due to one of its fundamental properties, namely, the Meissner effect, when a bulk material turns superconducting it expels the magnetic field from its interior. This, however, leaves a thin surface layer, called the penetration depth (or the London penetration depth) λ over which the magnetic field has exponentially decayed and there exist supercurrents that flow to keep the external magnetic field excluded from the bulk of the sample.
The second characteristic length ξ, after Pippard (1950) and Ginzburg and Landau (1950), represents the spatial range over which superconductivity gets affected by any local perturbation like thermal fluctuation/heating and magnetic field. For typeI, or positive surface energy superconductors, that mostly include superconducting elements, such as Al, In, Pb, Sn, Zn, etc., ξ > λ (Fig. 1.2a), while for typeII, or negative surface energy superconductors, like alloys and compounds that include Chevrel phases, A15 superconductors, hightemperature superconducting (HTS) cuprates, and Febased superconductors (FBS), ξ < λ (Fig. 1.2b). In terms of microscopic theory, the coherence length measures the separation of the two charge carriers (both negative or positive), which are generally of opposite spins, forming a Cooper pair which makes the pairs function as extended entities.
Table 1.1 lists ξ and λ values along with the critical temperature for some representative superconductors. The materials with one or more dimensions smaller than ξ or λ are considered as small superconductors where the two length scales lie between a few nanometers and several hundred nanometers. Small superconductors thus clearly belong to the nanoscale, but more broadly they are also referred to as mesoscopic. The Greek word meso means “in between” and the smallness lies in between the macro, or the bulk, obeying the classical laws of physics, and the micro, or molecular size, where quantum laws dominate. On practical considerations, the nanometer is taken as the lower limit of the length (p. 5) scale for size reduction which classifies the above lowdimensional systems as taking the form (Fig. 1.1) of nanometric thin films, ultrafine nanowires, and isolated or embedded nanoparticles (or nanodots), respectively. The above length scales, in their lower limit, are comparable to the de Broglie or Fermi wavelength which causes a superconducting nanoparticle to exhibit quantum confinement effects and display unusual superconducting behavior.
In general, the sample dimensions smaller than λ give rise to unusual magnetic and electrical transport properties while those smaller than ξ manifest changes in their order parameter and T_{c}related behaviors. The relevance of the former had in fact been realized as early as the mid1930s from the London phenomenology (F. and H. London, 1935), which had showed that for superconducting thin films and fibers of thickness smaller than λ the Meissner effect is incomplete or only partial. As a result, the energy of magnetic field expulsion in them is significantly reduced in comparison to the bulk samples of the same material. Since the condensation energy density (μ_{0}H_{c}^{2}/2) for both bulk and thin samples is the same, the field expulsion from a smaller volume of thin samples would lead to a higher value of their critical (p. 6) field μ_{0}H_{c}. This contention was shortly afterwards corroborated by Pontius (1937) who found the critical field of a lead wire to be markedly increased when its wire diameter was reduced. Bean et al. (1962) forced soft typeI superconducting metals like Hg, In, Pb, etc. into the interconnected 3–5nm sized pores of a Vycor glass which resulted in networks of very fine filaments (Fig. 1.3). The critical fields of the filamentary structures were found to be around 5 T, i.e. more than 100 times their bulk values. Further, at 4 K, the filaments were found to sustain supercurrent densities of 10^{8}A/m^{2}, about 1000 times larger than the bulk values. A similar large increase of the critical field has more recently been reported for Pb and In nanoparticles (Li et al. 2003, 2005), (p. 7) primarily resulting from their incomplete Meissner effect. In typeII superconductors, the Meissner effect is observed only until the lower critical field H_{c1} < H_{c} and here again the results of Blinov et al. (1993) and Fleischer et al. (1996) exhibit an order of magnitude increase in H_{c1} of YBCO fine powders of diameter < λ.
Table 1.1 Critical temperature, coherence length and penetration depth of different superconducting materials. (Data compiled from Narlikar (2014) and Narlikar and Ekbote (1983).
Superconductor 
Critical Temperature (K) 
Coherence Length ξ(0Κ) (nm) 
Penetration Depth λ(0Κ) (nm) 

Al 
1.2 
1600 
16 
Cd 
0.51 
760 
110 
In 
3.4 
364 
64 
Nb 
9.30 
38 
39 
Pb 
7.19 
87 
39 
Sn 
3.7 
230 
34 
V 
5.38 
40 
100 
MgB_{2} 
39 
6.0 
185 
Nb_{3}Sn 
18.3 
3.1 
100 
NbN 
16.1 
4.5 
200 
PbMo_{6}S_{8} 
13 
2.5 
240 
BKBO 
30 
4.0 
250 
MgCN_{i3} 
7.0 
4.6 
225 
Na_{x}CoO_{2} 
4.5 
5.7 
790 
KOs_{2}O_{6} 
9.6 
3.7 
243 
YBa_{2}Cu_{3}O_{7} 
91 
0.3–2.0 
500–150 
SmFeAsO_{0.8}F_{0.2} 
55 
4.0 
200 
Rb_{3}C_{60} 
29.5 
2.0 
440 
k(ET)_{2}Cu(NCS)_{2} 
9.0 
3.0 
510 
It is worth mentioning that some of the superconducting materials structurewise naturally grow as quasilowdimensional superlattices and interestingly a good many of them like HTS cuprates, Febased compounds, MgB_{2}, and A15 compounds are noted for their high T_{c} values. This has markedly emphasized the significance of low dimensionality in superconductivity. For instance, various HTS cuprates are all layered materials (Fig. 1.4c) in which metallic copper oxide layers one or a few atoms thick, where superconductivity occurs, alternate with insulating or blocking layers that make the structure quasi2D. Similarly, the crystal structure of Chevrel phase compounds (Fig. 1.4a) possesses mutually separated localized molybdenum octahedra, important for their superconductivity, and some of the organic superconductors (Fig. 1.4b) are formed as linear stacks of molecules that superconduct. These structural features make the above two types of materials respectively quasi0D and 1D. Interestingly, nanolevel defect structures present in bulk materials can additionally be of 0D/1D/2D (Table 1.2), which accordingly infuse in the latter the (p. 8) traits of lowdimensional systems. For example, noninteracting ultrafine particles embedded in a bulk matrix or the extrafine grain structure of thin films can serve as a 0D system.
Table 1.2 Defect structures with different dimensionality in materials.
Defect Structure 
Naturally Grown 
Artificially Introduced 

Zerodimensional 

Nanoparticles, small pieces of nanorods, nanoribbons of additive materials like SiC, different forms of carbon, iron oxide, different carbides, and various pure metals such as silver, nickel, cobalt, etc. 
Onedimensional 
Crystal dislocations 

Twodimensional 


Threedimensional 
Precipitates of second phase, voids and inclusions, 3D networks of dislocation tangles and various boundaries 
1.2 Two size effects in superconductors
How does superconductivity respond when the bulk sample is made smaller and smaller in the nano range? Although the question has been extensively pursued for the last two decades it is still a long way away from being fully understood. The issue is commonly discussed in terms of two size effects, namely (a) the small size effects, and (b) the quantum size effects. As with almost all nanomaterials, the pristine properties of bulk superconductors get markedly affected when their dimensions are reduced and made smaller than λ or ξ. Generally, the superconducting characteristics remain little affected down to 100 nm. The changes taking place on the length scale of 100 nm to about 20 nm are described by the small size effects (SSE), while in the ultrasmall regime below 20 nm, until the superconductivity is quenched, the behavior is discussed in terms of the quantum size effects (QSE).
The extended size ξ of the Cooper pairs as supercurrent carriers poses an obvious constraint in their mobility through superconducting samples with dimensions smaller than ξ. Besides, there is a relevant question as to how an inherently longrange phenomenon like superconductivity can really survive on a scale that is small or ultrasmall in comparison to ξ. In such a situation the Cooper pairs may get markedly squeezed and their wave functions significantly (p. 9) perturbed to adversely affect or even completely destabilize superconductivity. A third length scale, much smaller than either of λ or ξ, in the regime of QSE, is found to be necessary, which could set the limit to the superconductivity phenomenon. This third length scale is the Anderson limit.
1.3 QSE, quantum fluctuations, Anderson limit, parity and shell effects, etc.
In the ultrasmall regime of 0D superconductors, with the nanoparticle size getting closer to the Fermi wavelength, quantum confinement effects occur which give rise to discretization of energy levels that forms part of the QSE. The mean spacing of the eigenspectrum, d_{k} (the Kubo gap) ≈ (1/N_{0})E_{F}; near the Fermi level E_{F} increases with decrease of the particle diameter σ (Fig. 1.5) and superconducting properties dramatically change. The situation is akin to the electrons in a box model where a small change in the box dimensions produces large changes in the density of energy states. It is worth recalling that in superconductivity the electronic states located within a small shell of width ±hω_{D} at E_{F} are responsible for the phenomenon and these are markedly affected when the particle size is reduced.
(p. 10) Anderson (1959) pointed out that in the ultrasmall regime (< 20 nm diameter) of a 0D superconductor when the particle or grain diameter σ = σ_{c} is indeed so small that it leads to d_{k} ≥ Δ, where 2Δ is the superconducting energy gap at the Fermi level E_{F}, the particle will no longer be superconducting. This is known as the Anderson limit for the critical particle diameter σ_{c}, below which superconductivity is destabilized and ceases to exist. Broadly, the ratio Δ/d_{k} relates to the number of Cooper pairs present in the system and, clearly, at σ = σ_{c}, when the above ratio becomes < 1 (i.e. Δ < d_{k}), the σ_{c} is too small to hold any Cooper pair to sustain superconductivity. When the Anderson limit is approached from above, quantum fluctuations quickly build up to destabilize superconductivity. The series of experiments by Ralph et al. (1995, 1996, 1996a) strongly corroborated the discretization of the eigenspectrum of ultrasmall superconductors and established the validity of the Anderson limit. These experiments also yielded some novel insights linked with (a) discretization of energy levels, and (b) the finite electron number present in small particles exhibiting superconductivity. Besides the suppression or destabilization of superconductivity just below the Anderson limit, (a) and (b) above give rise to a host of unexpected phenomena like quantum fluctuations, shell effects, and shape resonance effects, by modulating various features like electronic density of states (DOS), the superconducting energy gap, critical temperature T_{c}, electron–phonon interaction, resistivity, Hall conductivity, H_{c2}, etc. Most noteworthy of these are the building up of huge oscillations in Δ and T_{c}, first observed in Sn (Bose et al. 2010), when the particle size (or more correctly, the number of electrons therein) was reduced in the ultrasmall range from 20 nm down to the Anderson limit. The behavior is described in terms of what are known as the shell effects and the shape resonance effects. It is worth pointing out that although both Δ and T_{c} monotonically decrease with the particle size, the amplitude of oscillations keeps on increasing as the Anderson limit gets closer. At the optimum level, in the case of nanoparticles of Sn, such oscillations were 60% higher than their corresponding bulk values (Bose et al. 2010). These oscillations are the manifestation of quantum fluctuations in the superconducting order parameter caused by large variations in the electronic DOS in a shell of width ±hω_{D} at E_{F} resulting from changes in the particle size. An increase in DOS implies a rise in Δ and T_{c}, and vice versa. Here the particle shape is also relevant as the symmetric (spherical or cubic, for example) particles contribute further degeneracies to the discrete energy spectrum and thereby enhance the fluctuations and the ensuing variation in Δ and T_{c}. The degenerate levels are analogous to the shells of atomic nuclei, which has prompted the name shell effects for the above phenomena. The maxima of the oscillating pattern are usually referred to as the shape resonances.
The finite electron number present in ultrasmall superconductors makes the application of conventional BCS (Bardeen–Cooper–Schrieffer) theory unsuitable (p. 11) and further leads to an unusual parity effect. Accordingly, ultrasmall grains with even electron numbers exhibit a larger superconducting gap than those with odd numbers, which is, however, consistent with the characteristic pair correlation of a superconducting state. Ultrasmall grains with an odd number of finite electrons always possess an unpaired electron that blocks the scattering of other pairs and thereby weakens the pairing correlations and reduces Δ. Taking into consideration the parity effect, more involved calculations reveal that when the electron number present is even, superconductivity would disappear only when d_{k} = 3.56Δ, but when the number is odd, the phenomenon will vanish when d_{k} = 0.89Δ (Dukelsky and Sierra 2000).
Table 1.3 gives the measured values of the Anderson limit σ_{c} for some of the elemental superconductors. As can be seen, the values are much smaller than the pertinent λ or ξ values of Table 1.1. Similar studies, however, seem to be lacking for superconducting alloys and compounds. The Anderson limit for MgB_{2} has been found to be 2.5 nm (Li and Dou 2006) and their results for nanograined samples are shown in Fig. 1.7. As may be seen, down to about 12 nm grain size there is essentially no change in the bulk T_{c} of MgB_{2}, but below that it decreases fast and no superconductivity is observed at grain sizes of 2.5 nm and below. On theoretical considerations Ivanov et al. (2003) have pointed out that for superconductors with higher T_{c} and larger gap coefficients the values of σ_{c} will be much smaller than for pure elements of Table 1.3 which, experimentally, may not be easy to realize. For HTS cuprates, for example, which have T_{c} > 100 K and a large gap coefficient in the range 4 to 12, the σ_{c} is predicted to be < unit cell length, and therefore the Anderson limit may, in fact, be unattainable (Ivanov et al. 2003).
Interestingly, many of the mentioned effects in fact were long ago theoretically predicted (Blatt and Thompson 1963, Hwang et al. 2000, Jankó et al. 1994, Kresin and Ovchinnikov 2006, Wei and Chou 2002, Yu and Strongin 1976) and a good many of them have also since been experimentally corroborated (Bao et al. 2005, Bose et al. 2009, 2010, Czoschke et al. 2003, GarcíaGarcía et al. 2008, Orr et al. 1984, Ralph et al. 1995, 1996a, 1996b, RomeroBermudez and GarcíaGarcía 2014, Tinkham 2000, Upton et al. 2004, von Delft and Ralph 2001).
Table 1.3 Anderson limit for nanoparticles of some elemental superconductors. The values mentioned are from Bose and Ayuub (2014) and references therein, and Yang et al. (2011).
Superconductor 
Al 
In 
Nb 
Pb 
Sn 
V 

Anderson limit σ_{c} (nanoparticle diameter) (nm) 
6.2 
5.5 
7.0 
3.5 
4.3 
3 
(p. 12) 1.4 Factors influencing small size effects
The superconducting behavior in the regime of small size effects, i.e. in the 100–20 nm range, is commonly linked to two main factors arising from size reduction. (a) The disruption in the periodicity at the sample surface causes the latter to possess vastly altered properties. At the nanolevel, the surface to volume ratio becomes significantly prominent to cause the surface properties to override the bulk properties (surface effects). (b) A decrease in the electron mean free path due to spatial confinement (electron mean free path effects) which, as we will see, has interesting consequences. Besides (a) and (b) there are fluctuations (thermal and quantum) in the form of phase slips that become important as the sample size diminishes. Their effect (Sec. 1.5) is primarily to broaden the superconducting transition and adversely affect superconductivity.
1.4.1 Surface effects
With samples of reduced size, their surface properties grow more relevant than bulk properties and this influences many of their important parameters like the critical temperature, the critical magnetic field, and the critical current. The atoms located on the sample surface possess a lower number of bonds than those located in the bulk interior, which makes the surface phonon modes much softer than the bulk and this thereby lowers its average phonon frequency.
The dimensionless parameter λ_{e}, which measures the electron–phonon interaction strength in conventional superconductors to determine their T_{c}, is related to the electron DOS N(0) at E_{F}, ionic mass M, the electronic matrix element ⟨I^{2}⟩, and mean square phonon frequency ⟨ω^{2}⟩ through the relation,
(1.1)
Decrease in ⟨ω^{2}⟩ in the above situation favors an increase in λ_{e} and consequently also the critical temperature. In this way, fine pressed powders (quasi0D category) of elemental platinum, which is otherwise nonsuperconducting, exhibit superconductivity at 0.02 K (Schindler et al. 2002). The softened surface phonon modes in the case of small superconductors make a positive contribution to their critical temperature. There is also a negative contribution to T_{c} coming from structural distortion occurring as a result of the sample size reduction. This causes a decrease in the electronic DOS N(0) at E_{F} which suppresses T_{c}. As we have seen earlier, discretization of energy levels occurs in small particles due to quantum confinement effects that subsequently get broadened as the particle size is reduced. This again lowers the DOS which makes a negative contribution to T_{c}. The overall effect of these competing factors is found to be different in different materials. The variation of T_{c} with particle size results in three types of behavior (p. 13) which are schematically shown in Fig. 1.6. A pronounced peaklike behavior such as shown by curve II is manifested by weakly superconducting metal. For instance, in the case of In, which is weakly superconducting, the T_{c} goes through a sharp maximum at about 30 nm and thereafter, in the ultrasmall regime, it shows a fast decrease until the Anderson limit is reached, below which the phenomenon disappears (Li et al. 2005). For intermediate electron–phonon coupling superconductors, the behavior is as shown by (III), i.e. the curve is initially flat, and then there is a decrease in T_{c} with particle size which becomes faster as the Anderson limit is approached. This kind of behavior is manifested by Nb which possesses intermediate electron–phonon coupling and its superconductivity is lost when the particle size is smaller than around 7 nm (Bose et al. 2009). In general, the negative contributions in small superconductors dominate over the aforesaid positive contribution when the sample dimensions are lowered below around 20 nm and the overall effect is therefore to suppress T_{c}. The behavior represented by (I) in Fig. 1.6 is for strongly coupled superconductors such as Pb. From bulk down to below 10 nm there is no change in T_{c} which Bose et al. (2009) attribute to the almost exact compensation of the positive phonon softening effect by the negative quantum size effect until the Anderson limit is approached. Interestingly, the behavior (Fig. 1.7) of MgB_{2} (Li and Dou 2006) seems very similar to that of Pb and may also be explained similarly.
Interestingly, in the case of weakly coupled superconductors like In, Al, and Sn, the gap coefficient 2Δ(0)/k_{B}T_{c} increases on size reduction; it remains invariant for Nb whose gap coefficient is of an intermediate value between weak and strongly coupled superconductors; while for Pb which is strongly coupled, its gap coefficient exhibits an increase (Bose and Ayyub 2014) on size reduction.
Surface effects are important also in influencing the critical magnetic field of small superconductors. SaintJames and de Gennes (1963) had showed that the solution of Ginzburg–Landau equations for a typeII superconductor subjected to a magnetic field was quite different at its surface than for the bulk. Even when the magnetic field had exceeded the upper critical field H_{c2} to quench the bulk superconductivity, a thin surface layer of thickness ξ, parallel to the magnetic field, continued to remain superconducting up to a much (p. 14) higher magnetic field, known as the sheath critical field H_{c3} ≅ 1.695H_{c2}. For thin discs the numerical factor exceeds even 2.5 (Schweigert and Peeters 1999). The surface phonon modes, mentioned earlier, are believed to be responsible for the origin of sheath superconductivity. For typeII superconductors of low Ginzburg–Landau parameter κ (see Narlikar 2014), the existence of a surface sheath promotes magnetic irreversibility and a higher critical current density j_{c}. However, for largeκ materials, which are used for winding highfield superconducting magnets, different types of pinning entities in the form of small normal (nonsuperconducting) particles and structural defects are essential for strong flux pinning and enhanced j_{c}.
Surfaces of small precipitates of a second phase, various structural defects, and of artificially introduced fine particles (Table 1.2) embedded in wires, tapes, or thin films of superconductors serve as effective pinning centers for Abrikosov’s flux vortices. Their presence vastly enhances the critical current density of the superconducting matrix in which they are embedded. In general, for optimum pinning, the number density of the pinning entities should be large, their volume fraction small, and the surface area of each entity or the pinning center should be optimum. Ideally, since the normal core of the flux vortex has a radius equal to the range of coherence ξ of the matrix, the size of the pinning centers should be in the nano range, for optimum pinning. This makes the pinning entities mesoscopic. Since the magnetization of the pinning entity, in general, differs from that of the superconducting matrix, there is always a circulating supercurrent around the pinning center at its interface with the matrix. This surface current interacts with the circulating supercurrent of the moving flux vortex and creates an irreversible surface barrier responsible for flux pinning (Bean and Livingston 1964, Campbell et al. 1968). This approach of introducing nanoparticles of different materials and metallurgical phases, such as Gd211 in HTS cuprates (Muralidhar et al. 2004), or nanodiamond (Cheng et al. 2003, Vajpayee et al. 2007), nanoSiC (Dou et al. 2002), nanoC (Yeoh et al. 2006), nanoSiO_{2} (Rui et al. 2004), carbon nanotubes (Yeoh et al. 2006a), etc. as nanosized artificial pinning centers (APCs), has been effectively used to enhance the j_{c} of superconducting MgB_{2}. In fact, this has emerged as a successful route to increase the current carrying capacity of several highfield superconductors for various applications.
1.4.2 Electron mean free path effects
When a metallic particle or a grain is reduced in size, a wire is made narrower, or a film is made thinner, all lead to a decrease in their electron mean free path. The electron mean free path stands out as an important factor in superconductors which in different ways influences several of their important properties and parameters. These prominently include the two characteristic lengths λ and ξ, the normal stat resistivity ρ_{n}, the Ginzburg–Landau parameter κ, and the three critical fields H_{c1}, H_{c2}, and H_{c3}.
(p. 15) 1.4.2.1 λ, ξ, and κ
When the sample size is reduced, the electron mean free path l_{F} decreases, which has mutually opposite effects on the coherence length ξ and the penetration depth λ. The former decreases while the latter increases, following the relations from the Ginzburg–Landau theory (1950):
(1.2)
(1.3)
where ξ_{ο} is the coherence length for the pure material at 0 K. In the case of thin superconducting discs (thickness t) the effective penetration depth, Pearl’s penetration depth (Pearl 1964), is given by Λ = 2λ^{2}/t. The Ginzburg–Landau parameter κ, which varies as λ/ξ, therefore increases as the sample thickness is reduced. Interestingly, this makes a typeI superconductor like Pb behave like a typeII when it is formed as a nanoparticle of around 15 nm diameter (Bose 2007). As a result very thin typeI superconductors show an interesting vortex structure, normally seen only in typeII, but, as we will see later, they also carry many other surprising features (Deo et al. 1997, Schweigert and Peeters 1998, 1999).
1.4.2.2 Critical fields H_{c1} and H_{c2}
The lower and upper critical fields H_{c1} and H_{c2} are respectively related to λ and ξ,
(1.4a)
(1.4b)
where φ_{o} (= h/2e = 2×10^{−15}Wb) is a quantum of magnetic flux.
Consequently, with size reduction the lower critical field would decrease while the upper critical field would increase. However, if the sample size is smaller than λ, due to an incomplete Meissner effect, as discussed in 1.1, the H_{c1} would increase. Thus, in the case of H_{c1} there is a competition occurring between the electron mean free path effect and the consequence of the incomplete Meissner effect, and the latter effect should dominate as the sample size is reduced. As mentioned earlier, the lower critical field does indeed go up with decreasing particle size (Blinov et al. 1993).
There is a similar competition also with H_{c2}. Decrease in the electron mean free path enhances the normal state resistivity of the small superconductor, which is related to the upper critical field through the relation (see Narlikar 2014),
(1.5)
(p. 16) The coefficient of the electronic specific heat in the normal state, γ, and the critical temperature T_{c} do not normally change significantly by electron mean free path effects and consequently an increase of ρ_{n} as per the above equation would make a positive contribution to the upper critical field. However, there is a competing negative contribution that arises from discretization of energy levels due to the quantum confinement effect which, as discussed earlier, causes a decrease in the DOS at E_{F} and thereby suppresses T_{c}. The latter effect would surely dominate when the sample size is reduced down to the ultrasmall regime and the upper critical field would finally vanish when the Anderson limit is reached. The reported observations and their explanation of findings by Bose et al. (2006) of such a nonmonotonic behavior of the upper critical field of nanosized grains of Nb are in accord with the above reasoning.
Nanostructured thin films of some superconductors are noted for their exceptionally high upper critical fields in comparison with their bulk values, which makes them attractive candidates for high magnetic field applications. For instance, for Chevrel phase superconductors the upper critical field for nanostructured films is reported to increase from 50 T to 100 T while for MgB_{2} from 16 T to about 75 T (Narlikar 2014).
1.5 Behavior of nanowires and ultrathin films
1.5.1 Nanowires and fluctuation effects: thermally activated phase slips and quantum phase slips
Fluctuation effects, in general, become relevant for lowdimensional systems and all the more so in 1D superconducting nanowires. Quantum fluctuations were briefly mentioned in Sec. 1.3. In a strictly 1D system, superconducting longrange order and a zeroresistance state are, in fact, forbidden by the Mermin–Wagner (1966) theorem. Langer and Ambegaokar (1967) and McCumber and Halperin (1970) in their LAMH theory had pointed out that the process responsible for transient breaking down of superconductivity just below T_{c} was the socalled phase slips, which originated from thermal fluctuations. In addition to these thermally activated phase slips (TAPS), the experimental evidence (Giordano 1994) has indicated that for ultrathin wires, at much lower temperatures < T_{c}, when thermal excitation is not relevant, quantum phase slips (QPS) can occur due to quantum fluctuations via quantum tunneling.
Superconducting order is characterized by a wave function ψ = ψexp.(iϕ) where ϕ is its spatially coherent phase that is, so to speak, locked everywhere in the superconducting state. Locally, near T_{c}, the coherence can, however, get disturbed by thermal fluctuations or the occurrence of phase slips which results in the local transient loss of superconductivity (t ≈10^{−12} s) over a region of size ξ where the order parameter ψ locally fluctuates to zero and the phase ϕ becomes (p. 17) indeterminate. The minimum energy cost for this is a function of the condensation energy density ½μ_{0}H_{c}^{2} per unit volume, and for a nanowire of crosssectional area A becoming normal over a segment ξ, the corresponding energy needed, i.e. the barrier height, is (Langer and Ambegaokar 1967) ΔF = K.μ_{0}H_{c}^{2}Aξ, where H_{c} is the thermodynamic critical field and K is a constant. Subsequently, as ψ builds back up to its original value, the phase undergoes a slip and rotates by ±2π from its value just prior to the phase slip event. This change of phase corresponds to a voltage pulse, with more slips giving rise to more resistive voltages. The process is represented in Fig. 1.8a by a spiral of constant radius where with each phase slip event the spiral loses one complete loop corresponding to a phase of 2π. As shown in Fig. 1.8b, the process (TAPS) can take place by thermal activation over the barrier of height ΔF, just below T_{c} or, if the wire is ultrathin, it (QPS) can (p. 18) occur quite far below T_{c}, through the above barrier due to quantum fluctuations via tunneling. In both situations the system moves from one local potential minimum into the neighboring one separated by ±2π in the phase space.
In 2D and 3D samples the supercurrent transport is, however, little affected by the creation of the normal (nonsuperconducting) region that can be readily bypassed by moving Cooper pairs without interruption. But, in nanowires of radius smaller than ξ, the normal region formed blocks the entire crosssection of the nanowire, and therefore it can no longer be bypassed by electron pairs. The wire therefore ceases to be superconducting. The experimental results show that the normal state resistance R_{N} of the nanowire has to be smaller than the quantum resistance R_{Q} = h/(2e)^{2} ≈ 6.5 kOhm to follow the predictions of the LAMH theory. When R_{N} > R_{Q}, the nanowire fails to exhibit superconductivity (Bezryadin 2008).
Because of the two characteristic phase slips, TAPS and QPS, mentioned earlier, the resistance (R)–temperature (T) behavior of nanowires differs from that of the bulk wires (thickness/diameter of around 100 nm or more) in two ways. Instead of the very sharp transition (width < 0.01 K) of bulk wires the TAPS results in the rounding and broadening of the transition point at T_{c}onset, while the QPS results in an elongated tail at T < T_{c}, which frequently does not lead to a R = 0 state even at 0 K. The behavior is schematically shown in Fig. 1.9. This happens when the wires are sufficiently small (≤ 30 nm) in diameter or thickness. The behavior has been studied in a number of elemental superconductors like Sn, Zn, Al, Pb, and In (Zgirski et al. 2005, Sharifi et al. 1993, Giordano 1994, Bezryadin et al. 2000, Lau et al. 2001, Tian et al. 2005, Wang et al. 2005). However, the transition from TAPS to QPS in superconducting nanowires still remains to be systematically studied.
As with 0D superconductors, discussed earlier, 1D ultrathin nanowires do exhibit the quantization of electron motion and discretization of energy levels, but only along the directions transverse to the length. These have been experimentally corroborated by observations of shape resonances with large oscillations in T_{c} of ultrathin nanowires of Al, Sn, and Pb (Guo et al. 2004, Shanenko et al. 2006) and also theoretically explained (Shanenko and Croitoru 2006).
1.5.2 Ultrathin films
The 2D superconducting system holds some similarity with the behavior of 1D nanowires described above. When the normal state resistance R_{N} of a thin film exceeds the quantum resistance R_{Q} = 6.5 kOhm, superconductivity is not observed (Haviland et al. 1989, Hebard and Paalnen 1990, Bollinger et al. 2011). Ultrathin superconducting films with thickness < 20 nm, if they are not of a continuous form, comprise small islands and exhibit granular behavior, characteristic of 0D superconductors discussed earlier. The films are formed on substrates which can influence their intrinsic behavior through the proximity effect. Similar to what we previously discussed for 0D superconductors, fluctuations too play (p. 19) a significant role in 2D systems. The Berezinskii–Kosterlitz–Thouless (BKT) transition originating from phase fluctuations adversely affects superconductivity. Interestingly, despite this, superconductivity does occur in ultrathin films of thicknesses much smaller than ξ and, in fact, even down to the level of a single monolayer of the superconducting elements (Zhang et al. 2010) and compounds (Lee et al. 2014, Xi et al. 2015), albeit with reduced T_{c} and broadened transition as compared to the bulk samples.
Fig. 1.10 depicts the results of T_{c} depression in ultrathin films of NbN reported by Marsili et al. (2008) and Kang et al. (2011) when the film thickness varied from 100 nm to about 2.5 nm. Clearly, the two sets of observations are in close mutual agreement. Down to 100 nm the critical temperature remains practically invariant from the bulk value. From 100 to about 10 nm thickness the films display a small decrease of T_{c}, which, however, in the ultrathin regime below 10 nm depicts a much faster drop. As also discussed previously, a decrease in the film thickness can give rise to different competing factors influencing T_{c}. Firstly, due to the previously discussed small size effects (SSE), the structural disorder in the film increases with decreasing thickness, causing a reduction in the DOS at E_{F} which lowers the T_{c}. On the other hand, as the film gets thinner, surface phonons become more dominant to increase the electron–phonon interaction which promotes T_{c}. As with 0D and 1D superconductors, in the ultrathin regime of 2D films, the electron wave function is quantized along the direction normal to the film surface, which results in discretization of energy levels as part of the quantum size effects (QSE). This contributes to a decrease in the DOS at E_{F} and lowers T_{c}. The observed behavior of T_{c} in Fig. 1.10 is the overall impact of the above factors in which the QSE finally dominate and depress T_{c} through decrease in the DOS (Kang et al. 2011) at E_{F}. Similar to 0D and 1D superconductors, discretization of energy levels in ultrathin films gives rise to large oscillations (Bao et al. 2005, Shanenko et al. 2007) in various superconducting parameters in the form of shape resonances as we discussed earlier. (p. 20)
Fig. 1.11a presents the results of R–T measurements of Gao et al. (1999) on ultrathin films of YBCO while Fig. 1.11b depicts the results on MgB_{2} films (Wang et al. 2009). In both cases the transition gets rounded and broadened and T_{c} goes down. In the case of YBCO, the 2.5 nm film does not seem to indicate any possibility of reaching the zeroresistance state even at 0 K. Interestingly, the MgB_{2} films of thickness ≤ 20 nm exhibit a localization effect by displaying the characteristic negative dR/dT at low temperatures, possibly due to the presence of structural disorder occurring at the ultrasmall thickness. However, from the R–T curves of Fig. 1.11a and Fig. 1.11b one cannot unambiguously distinguish TAPS and QPS regimes.
1.6 Vortex states of small superconductors
We have already seen above how size effects influence T_{c} and other related properties of superconductors. In this section we take a quick look at the influence of (p. 21) confinement on the formation of magnetic field induced vortex states in them. Normally, the vortex structure is a characteristic feature only of typeII superconductors. However, typeI superconductors, as we have seen earlier, if they are made sufficiently small, start behaving akin to typeII ones and thus may have flux vortices under an applied magnetic field. When the field exceeds the lower critical field H_{c1} the macroscopic Meissner state gives way to a mixed state, comprising a triangular latticelike arrangement of Abrikosov’s vortex lines (Fig. 1.12a) threading the bulk of the superconductor (Abrikosov 1957). The triangular vortex lattice, as imaged by Nishio using scanning Hall microscopy, is depicted in Fig. 1.12b. Each vortex line has a normal cylindrical core of radius ξ carrying a single quantum of magnetic flux ϕ_{o} (= h/2e = 2 × 10^{−15}Wb), which is surrounded by vortices of supercurrents spread over a radius of λ (Fig. 1.12c). The smallest subdivision of magnetic flux is a consequence of the negative surface energy existing between the normal core and its superconducting surrounding, the perennial characteristic of typeII superconductivity. A single rotational turn around the vortex changes the phase of the order parameter by 2π. In general, for a bulk superconductor, it is energetically impermissible for a flux line, whose selfenergy varies as ϕ_{o}^{2}, to contain n ≥ 2 flux quanta. Such a vortex line, known as the giant vortex, will normally be unstable and split into n vortices, each with a (p. 22) single flux quantum. However, a giant vortex state (GVS), as we will see, is one of the novel and stable vortex states of small superconductors.
The triangular vortex lattice in bulk superconductors is the outcome of mutual repulsion between the macroscopic number of vortex lines. In small or confined superconductors of size <(λ, ξ), where the sample boundary casts its strong influence on the vortex configuration, the latter markedly deviates from the conventional triangular lattice. The vortex configuration is now determined not just by the repulsive intervortex interaction but also by the sample boundary simultaneously imposing the strong effect of its own symmetry in arranging the vortex distribution (Baelus and Peeters 2002). We have already mentioned earlier surface superconductivity and the presence of the Bean–Livingston surface barrier (1964) for flux vortices. Extensive theoretical (Schweigert and Peeters 1998, Schweigert et al. 1998, Palacios 2000, Baelus et al. 2004) calculations, based on the Ginzburg–Landau theory, the London approach, molecular dynamics, and the Bogoliubov–de Gennes formulation (Bogoliubov 1947, de Gennes 1966) have predicted the existence of two distinct vortex states, namely (a) the giant vortex state (GVS) and (b) multivortex state (MVS), respectively consisting of (one or more) multiply quantized and several singly quantized flux vortices. These have been corroborated by a host of experimental studies (Moshchalkov et al. 1995, Bruyndoncx et al. 1999, Geim et al. 1997, 2000, Kanda et al. 2004, Baelus et al. 2005) on mesoscopic superconducting thin (~5 nm) circular discs of lateral size of a few ξ (< 10 ξ). The total number of flux quanta present in the sample is called its vorticity or winding number denoted by L which is an integral number. Alternatively, this parameter is defined as L = Δϕ/2π, where Δϕ is the total phase change of the sample due to magnetization. The spatial arrangement of vortices in mesoscopic superconductors is controlled by the geometrical shape of the sample boundary and the vorticity L. In the case of a small circular disc, instead of Abrikosov’s triangular lattice, the vortices formed prefer to arrange in circular form or a shell along the periphery of the sample edge. Some form of triangular lattice may however appear in the central part of the disc, away from the boundary, where its geometrical effect is relatively less. With an increasing external magnetic field or by reduction of the lateral size, as more such vortices with a single flux quantum are pushed inside, some of them tend to merge to form a single giant vortex at the center of the disc, with its normal core carrying a magnetic flux of (nϕ_{o}). The geometrical boundary effect in small superconductors facilitates the formation of a GVS which, as discussed above, is otherwise not feasible in bulk samples. If the lateral size of the sample is increased, or the magnetic flux density is reduced the GVS would, however, dissociate into the multivortex state.
Unlike the bulk superconductors, in mesoscopic samples only a finite number of vortex lines are accommodated and the confinement effect increases with decrease in the sample size. For instance, if the lateral diameter of the superconducting disc D ≤ 2ξ, which represents the diameter of the normal core of a flux vortex, the disc clearly would not accept it as this would imply the destruction of superconductivity. Insertion of a single vortex in such a sample implies an immediate transition from the Meissner state to the normal state, without passing (p. 23) through the mixed state. This way, effectively, a typeII superconductor under a small magnetic field behaves like a typeI. Interestingly, the reverse situation was discussed previously in the absence of a magnetic field where a small typeI superconductor behaved like a typeII. The ultimate confinement represents the situation where the sample is large enough to accommodate just one vortex inside without the destruction of superconductivity. The above contentions have been experimentally well corroborated by studies of Nishio et al. (2008) on thin nanocrystals of Pb on Si(111) which showed that samples with lateral size smaller than (2.5–2.8)ξ_{effective} are not able to accommodate even a single vortex.
Interestingly, much theoretical work has been done (Baelus et al. 2002, 2004) towards predicting different vortex configurations formed in nanosized superconductors or nanodots of various geometrical shapes like circular discs, equilateral triangles, squares, rectangles, etc., as a function of increasing L. Many of these have also been successfully corroborated through the Bitter decoration (Grigorieva et al. 2006) and various scanning probe techniques, like scanning tunneling, scanning SQUID, and scanning Hall microscopy techniques (Kokubo et al. 2010, 2014, 2015, Nishio et al. 2004, 2008a). In the case of circular discs, the multivortex state formed grows up in the form of discrete concentric circular shells (Xu et al. 2011). Interestingly, the process of the filling up of shells with added vortices resembles the building up of the wellknown periodic table of chemical elements. Starting from the Meissner state, which corresponds to L = 0, until L = 5 all the vortices occupy the first shell. A new shell is formed at L = 6 with the added vortex occupying the center of the previous shell having 5 vortices and such a state is represented by (1,5). With the increase of vorticity up to L = 8 the state changes to (1,7), whereafter (i.e. at L = 9) with added vortices, first only the inner shell grows to the state (2,7). This pertains until L = 13, then is followed by increase of the vortices only in the outer shell, bringing about the state represented by (2,11). Subsequently, at L > 14, the inner shell grows, leading to the state (5,11) at L = 16. The third shell appears at L = 17, with the added vortex again occupying the shell center, and the state is represented by three digits, i.e. (1,5,11). The next three vortices, leading to L = 20, are all added to the outermost shell, i.e. (1,5,14) and thereafter, up to L = 32, all the shells grow intermittently to (5,11,16). The fourth shell is formed at L = 33, again in the form of a single vortex at the center of the existing three shells, and the vortex configuration is represented by four numbers, i.e. (1,5,11,16). The numbers 6, 17, and 33, at which a new shell is formed, are referred to as magic numbers. These magic numbers and the described pattern of shell formations and filling are both theoretically and experimentally in mutual agreement.
It would be interesting to ask how the vortex configuration might change in small noncircular superconducting samples possessing a discrete symmetry, such as squares and equilateral triangles having pointed edges of the size ≈ 5ξ to 10ξ? In this situation the ensuing vortex configuration grows more complex since the vorticity has to adjust with the corresponding boundary shapes respectively of C_{4} and C_{3} symmetries (Chibotaru et al. 2001, 2005, Misko et al. 2003, Teniers et al. 2003, also see Chapter 3 of this book). L = 1 and L = 4 in a squareshaped (p. 24) sample present no problem as one and four vortices, each with a single flux quantum, can respectively be accommodated at the square center and at its four corners or diagonal points without disturbing the fourfold symmetry. Similarly L = 2 can be readily realized by replacing the single quantum vortex of L = 1 by a giant vortex with L = 2 at the square center. The vorticity can be increased to L = 5 by adding a single quantum vortex at the center of the four corner points of the L = 4 configuration and further raised to L = 6 by replacing the central vortex of L = 5 by a giant vortex with L = 2. Both these configurations represent two concentric shells. Vorticities of L = 3 and L = 7 are, however, not consistent with the square symmetry of the sample. Here an additional vortex–antivortex pair is formed to geometrically adjust the vortex distribution to meet the fourfold square symmetry of the sample; here antivortex represents an oppositely directed vortex with a flux quantum –ϕ_{o.} Thus, for L = 3, we have four vortices at the square corners and one antivortex at its center (i.e. 3 vortices + 1 vortex–antivortex pair). The vortex–antivortex pair is also called a vortex–antivortex molecule. Similarly, the configuration for L = 7 is formed with four vortices on each of its two diagonals with an antivortex at the square center (i.e. 7 vortices + 1 vortex–antivortex pair/molecule) constituting a threeshell arrangement (4ϕ_{o.}+ 4ϕ_{o.}–1ϕ_{o.}). A similar situation exists for a triangleshaped sample, for example, for L = 2 which is not compatible with the triangular shape. This can be realized by three ϕ_{o.} vortices, one each at the three apex points, and a single –ϕ_{o.} (antivortex) at the center of the triangle. Interestingly the filling up of the triangle and square centers with magnetic flux quanta, as the vorticity rises, follows distinct repetitive patterns (Table 1.4). The former is 0:+1:–1:0:+1:–1:0:+1: … while for a square it is 0:+1:+2:–1:0:+1:+2:–1:0 … This holds relevance for superconducting technology applications, which can use the triangular and square nanodots as components of supercomputers in quantum computing.
Table 1.4 Filling up of magnetic flux vortices/flux quantum at the center of mesoscopic superconducting triangles and squares with increasing vorticity.
Vorticity L 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 

Mag.flux (ϕ_{o}) at triangle center 
0 
+1 

0 
+1 

0 
+1 

0 
+ 1 

Mag.flux (ϕ_{o}) at square center 
0 
+1 


0 
+1 


0 
+1 


GV: giant vortex state;
AV: antivortex state. In the case of a triangle the repetitive pattern is 0:+1:–1:0:+1: … …, while for a square: 0:+1:+2:–1:0:+1: … ….
The antivortex formation in square and triangleshaped samples, described above, is however symmetry induced which to date remains to be experimentally corroborated. One of the reasons for this is that vortex–antivortex pair formation is sensitive to the presence of defects in the sample and moreover their mutual separation is too small, i.e. < ξ.
(p. 25) 1.7 Proximity effect behaviors
1.7.1 Proximity effect and Andreev reflection
The proximity effect is a mesoscopic phenomenon that takes place when a superconductor (S) is placed at close proximity and in good electrical contact with a normal metal (N). The Cooper pairs from S now, so to speak, leak into N with the result that the latter inherits superconducting characteristics over a mesoscopic range, extending from the nano to the micrometer scale, close to their NS interface. This interesting phenomenon is called the superconducting proximity effect, and the length scale over which it occurs, the proximity range or Thouless range. The proximity behavior was first observed by Holm and Meissner (1932) which could, however, capture more serious attention only after the discovery of the Josephson effect in 1962, which subsequently led to its significant theoretical understanding (Andreev 1964, de Gennes 1966, Deutscher and de Gennes 1969). The advent of nanoscience in the late 1980s and realization of various challenges posed in its understanding gave a fresh impetus to further studies of the proximity effect for basic research and applications. The effect makes it possible to induce superconductivity in varied materials, resulting in a wide combination of diverse properties which normally would be unattainable. Proximity superconductivity has been reported in numerous novel materials such as graphene (Heersche et al. 2007, Hayashi et al. 2010), DNA molecules (Yu et al. 2009), and in a host of nanowires and nanobelts of Al, Au, Co, Pb, Zn, etc. (Singh et al. 2012). Because of the mesoscopic range of the phenomenon, the experimental studies are performed with the two materials processed in thin film form with, of course, a good electrical bonding in between. It is worth emphasizing that superconductivity induced by the proximity effect essentially represents a metastable state resulting from noninteracting charge carriers and is therefore different from the stable phenomenon originating from the regular pairing interaction of interacting electrons of the BCS theory.
At voltages and temperatures below the superconducting gap, single particle tunneling is suppressed, and for electrons with energy less than 2Δ the dominant process for transfer of electrons across the NS interface is the Andreev reflection. In this, an incoming quasielectron from N is reflected at the NS interface as a quasihole. The missing charge 2e is absorbed by the superconducting condensate S as a Cooper pair. The opposite may also happen and accordingly a quasihole may get reflected in the Andreev way as a quasielectron and thereby remove one Cooper pair from S. This is believed to be how the Cooper pairs in the proximity effect may leak from S to N. It is worth pointing out that the Andreev reflection, which is called a retroreflection, is different from a normal reflection. In this there is a conservation of momentum and not of electrical charge and further it is compatible with time reversal, i.e. the reflected wave vector propagates back along the incoming path.
(p. 26) 1.7.2 Inverse proximity effect and the behavior of S–FM hybrid structures
Interestingly, just as the proximity of a superconductor makes a normal metal superconducting, there exists also a simultaneous reverse effect in which the proximity of a normal metal weakens superconductivity due to the depletion of Cooper pairs in S and diffusion of normal electrons from N to S. The T_{c} of S may therefore get significantly lower than the bulk value or may even get completely quenched. This is known as the inverse proximity effect which becomes particularly interesting when N is magnetic.
The proximity range (or Thouless range) may, in general, vary from less than a few nanometers to over many micrometers from the NS interface, depending on the materials used. For instance, in pure nonmagnetic metallic N the length scale at very low temperatures is in the micrometer range which is, however, markedly lowered to a few nanometers or even smaller when component N is ferromagnetic (FM). Clearly, the singlet superconductivity with Cooper pairs having antiparallel electron spins is expected to decay rapidly when leaked inside a ferromagnetic N exhibiting parallel spins. This is found true when the components involved, such as (Fe/Ni)–(In) (Chiang et al. 2007), are of macroscopic size and the proximity length is ≈1 nm. Surprisingly, however, for ultrathin films and nanowires of ferromagnetic materials forming mesoscopic S–FM hybrid structures the superconducting proximity effect is unexpectedly enhanced to the micrometer range and also it is larger for thinner nanowires (Petrashov et al. 1999, Bergeret et al. 2005, Keizer et al. 2006, Wang et al. 2010). Researchers have argued that the large proximity range is due to a change from the conventional spin singlet superconductivity to the unconventional spin triplet superconductivity where parallel electron spins are permitted. In the case of magnetic Ni and Co nanowires proximityconnected to superconducting W electrodes, Wang et al. (2010) find the proximity length is around 500 nm and, interestingly, R–T measurements depict an unusual peak near the T_{c}onset which is absent with nonmagnetic nanowires. According to the authors, the peak is possibly related to the coexistence of superconductivity and ferromagnetic order.
1.7.3 Giant proximity effect
An anomalously large proximity effect, extending over a length scale of 10–100 micrometers, is observed when the component S is an optimally doped highT_{c} cuprate and N is also of the same family, but in the underdoped normal state (Bozovic et al. 1994, 2004, Kasai et al. 1992, Marchand et al. 2008). It is called the anomalous or giant proximity effect. The unconventional normal state is considered one of the distinguishing marks of highT_{c} cuprates (see Narlikar 2014) and in respect of the proximity effect too, their observed behavior is found to be equally anomalous. The normal state of N in the case described is believed to be the socalled pseudogap state, which is quite unusual in that (p. 27) even above T_{c} it carries preformed electron pairs and flux vortices which normally occur only in the superconducting state. HighT_{c} material with a giant proximity effect has the advantage that a more easily synthesized and thicker normal material can be used for applications. Excitingly, hightemperature proximity superconductivity with T_{c} = 80 K has been realized (Zareapour et al. 2012) in the topological insulators Bi_{2}Se_{3} and Bi_{2}Te_{3} by keeping them in proximity to HTS Bi2212. These results hold considerable promise for future device applications.
1.7.4 Antiproximity effect and the proximity behavior of nanowires; minigap state
Yet another category of proximity effect, called the antiproximity effect (APE), was first reported (Tian et al. 2005a, 2006) with Zn nanowires of 40 nm thickness and 6 μm length sandwiched in between two bulk superconducting (BS) electrodes (Sn or In). Surprisingly, the nanowires did not exhibit superconductivity when the BS electrodes were superconducting and they showed superconductivity only after the superconductivity of the electrodes was quenched by an external magnetic field. The effect was, however, not observed for nanowires of thickness t ≥ 70 nm. The observed behavior, being quite contrary to the expected proximity effect, is termed the antiproximity effect and has since been observed by several researchers (Chen et al. 2009, 2011, Singh et al. 2011) and also for Al single crystalline nanowires (Singh et al. 2010). In the case of the latter, the authors (Singh et al. 2011) find that the critical current of an individual nanowire, contacted by superconducting electrodes, in the absence of magnetic field was significantly smaller than that for normal electrodes, suggesting that the origin of the APE is not linked with the application of the magnetic field.
To date, there seems to be no convincing explanation of the APE, however. But, the above studies carried out (de Horne et al. 2007, Wang et al. 2009, Liu et al. 2012, Stanescu and Sarma 2013) with a host of normal metallic, semiconducting, and superconducting nanowires in contact with superconducting electrodes have yielded some unexpected novel insights into mesoscopic proximity superconductivity. The studies of Wang et al. (2009) on 70 nm thick and 1–2 μm length Au nanowires, with proximity superconductivity induced through contact with a superconducting tungsten (W) electrode, have yielded many new results. While the proximity superconductivity of short wires of length L ≤ 1.0 μm displayed the characteristic zero resistance drop in R–T curves, the longer samples with L = 1.9 μm revealed some residual resistance down to the lowest temperature reached. This shows that small lengths of nanowires prepared from normally nonsuperconducting metals can serve as suitable proximity superconductors for various nanodevice applications. Interesting R–T behavior is observed for the samples of intermediate lengths of 1.2 μm, which depicted a resistive drop in two steps (p. 28) corresponding to two T_{c} values. This was further corroborated through R–H curves manifesting two critical fields. The higher values, closer to the contact point with the W electrode, matched well with the reported critical parameters of the W electrode used for the creation of the proximity superconductivity. The lower values are attributed to the formation of the proximityinduced minigap state close to the center of the Au nanowire which constitutes a novel feature of the proximity superconductivity. The gap in the Au wire at the electrode is the same as the bulk gap of W. Low temperature differential magneto resistance behavior of the proximityconnected gold nanowires described shows (He and Wang 2011) uniform oscillatory behavior with a period proportional to the quantum of magnetic flux ϕ_{o}. The previously discussed phase slip process occurring in such nanowires is considered responsible for the observed resistance jumps in the oscillatory behavior. This holds relevance to the possible application of proximity superconductivity in quantum computing.
1.8 Synthesis of small superconductors
Several physical, chemical, and metallurgical techniques are currently being used to fabricate nanoparticles, nanowires, and ultrathin films of a host of different materials. In this section we take a brief look only at some of the processing routes that are commonly used for synthesizing small superconductors.
Historically, Shalnikov (1938) was the first to study the vapor condensation of superconducting Sn and Pb on liquid helium cooled substrate and report their respectively enhanced T_{c} of 4.7 K and 7.5 K. However, this approach gained significance only in the mid1950s when Buckel and Hilsch (1954, 1956) found that the structure of quenchcondensed films of various superconducting elements like Pb, Bi, In, Sn, etc. could be described either as liquid like amorphous or as lattice like amorphous, both showing a wide deviation from their bulk superconductivity. The latter represents really nothing but the state of nanograined films while the former manifests the situation of the subnano structure, having a short range order, that of a liquid. A modified form of this technique—inert gas condensation—is schematically displayed in Fig. 1.13. The method involves vaporizing the molten metal from heated boats placed in a chamber (evaporation rate of about 0.005 nm/s), under an inert gas pressure. The metal vapor is allowed to condense and quench at the cold finger from where the condensed particles formed can be easily scraped off and studied ex situ without substrate. The particle size, from 100 nm down to a few nm, as determined by XRD (Xray diffraction) or TEM (transmission electron microscopy), is primarily controlled by the inert gas pressure of the deposition chamber. The method has been widely used for preparing ultrasmall particles of low melting point superconducting metals (Brun et al. 2009, Buhrman and Halperin 1973, (p. 29) Bernardi et al. 2006, Li et al. 2003, 2005, 2008) to understand their nanostate behavior. Interestingly, nanoparticles of superconductors can also be easily prepared down to the size of a few nanometers without any substrate by using the popular metallurgical technique of ball milling. Performing the ball milling in tungsten carbide bowls purged with argon, Li and Dou (2006) produced nanoparticles of MgB_{2} in the range of 64.1 to 2.5 nm. The heat accumulation during the milling process was kept under control by following a continuous cycle of 5 mins milling and 3 mins of rest time.
Among the established PVD (pulsed vapor deposition) routes, both high pressure dc magnetron sputtering (Ayyub et al. 2001, Banerjee et al. 2003, Bose et al. 2006) and the pulselaser ablation technique (Yang et al. 2011) have been successfully used to synthesize nanostructured films of Nb and V, with average grain diameters respectively down to 5 and 2.5 nm deposited on Si substrates. In the case of the former, the base pressure in the deposition chamber was 10^{−8} torr and the average grain size was controlled from 60 nm to 5 nm by varying the process parameters, namely the Ar gas pressure from 3 to 100 mtorr and the dc power, from 25 to 200W. The structure essentially comprises weakly connected nanosized grains, akin to independent nanoparticles.
Novel approaches have been followed to realize nanowires of various superconducting metals, alloys, and compounds. The two that have emerged as the most popular are the (a) suspended molecular template technique (Bezryadin et al. 2000, Hopkins et al. 2005, Zhang and Dai 2000) and (b) porous template technique (Martin 1994, Michotte et al. 2002). The former can be used down to (p. 30) 10 nm diameter/width, while the latter is suitable for growing single crystalline wires of diameter in the range 40–100 nm. The molecular template technique involves sputter depositing the superconducting material on the surface of a carbon nanotube (CNT) (or a nanotube string/bundle), serving as a template, mounted over a trenchlike configuration (Fig. 1.14) formed on a trilayered Si/SiO_{2}/SiN substrate. Since the template CNT is very thin, with diameter of < 3 nm, the diameter of the coated nanowire, after deposition of the desired superconductor, can be kept below 10 nm. The crosssection of the nanowire formed is determined by the width of the CNT and the amount of material deposited and this can be conveniently measured by TEM. The morphology of the deposited wire is controlled by sputtering parameters like gas pressure and dc power. Instead of CNT, the DNA molecule can alternatively be used as a molecular template for the synthesis of nanowires (Bezryadin et al. 2004).
The porous template technique uses membranes with long aligned arrays of nanosized pores which are filled with the desired superconductor by electrodeposition to form an array of nanowires. Two types of polymer membranes have primarily been used for this application. These are (a) track etched polycarbonate (PC), and (b) anodic aluminum oxide (AAO) membrane, which are chosen according to the requirements. The pore density of the latter is in between 10^{10} and 10^{11} pores/cm^{2}, while the PC possesses a lesser pore density of about 10^{8}/cm^{2}. Both types possess long (a few tens of micrometers in length) parallel arrays of channels free from interconnections which are aligned almost perpendicular to the membrane surface. The pore diameter tends to vary from 40 to 300 nm. Synthesis of metallic nanowires involves growing them inside the aforementioned channels by electrodeposition, using the conventional threeelectrode cell and a suitable electrolyte. A gold layer is coated on one surface of the membrane to serve as the cathode with a graphite rod as the anode. The deposition is stopped when the pores are filled. The nanowires formed are single crystalline and their morphology and microstructure can be controlled by varying the deposition parameters, e.g. deposition potential, pH value, and temperature. As the nanowires are placed inside the membrane they are protected from oxidation and the transport measurements can be made in situ by attaching leads on either sides.
Lithography provides yet another fabrication route for synthesizing nanowires on a substrate. The method is more versatile, albeit more expensive than those (p. 31) described above, and also complicated and is generally used for synthesizing complex nanostructures. Both photo (using UV) and electron beam lithography (EBL) are commonly used although the latter has a higher resolution, but is more costly and time consuming. The process for fabricating an array of nanowires is summarized stepwise in Fig. 1.15. Basically, the process involves five steps. A Si chip is coated with a resist (Fig. 1.15a) suitable for photolithography or EBL to be used and, in turn, it is covered (Fig. 1.15b) with a mask for the synthesis of nanowires. The structure is exposed to UV radiation or an electron beam (p. 32) (in SEM—scanning electron microscope). The exposed regions are chemically altered to get dissolved in a developer (Fig. 1.15c). Next, the superconducting material whose nanowires are to be made is sputter deposited on the structure (Fig. 1.15d). The substrate structure is then subjected to the liftoff process to remove the leftover resist, which leaves an array of nanowires on the substrate (Fig. 1.15e). Low energy (≈ 1 keV) Ar ion irradiation of the nanowires provides a smooth polishing effect. It is also used for progressively reducing the nanowire crosssection to allow measurements on the same sample with reduced thickness/diameter.
Among the chemical routes the solgel method has been successfully pursued (Nath and Parkinson 2006) to synthesize 20 μm lengths of MgB_{2} nanowires of 50–100 nm diameter. In this, as the first step, a gel was prepared by mixing magnesium bromide and sodium borohydride reagents in the presence of cetyltrimethylammonium (CTAB) and leaving this solution open to the atmosphere for many hours. The pyrolysis of this gel was the next step, carried out for about 5 min at 800^{o}C in an atmosphere of diborane and nitrogen and thereafter it was allowed to cool very slowly to room temperature. The XRD and SEM studies of the black powdery product thus formed confirmed it to be mainly MgB_{2} in the form of fine nanowires. Hundreds of micrometers long MgB_{2} nanowires of 50–400 nm diameter have been reported as synthesized by converting long boron nanowires under Mg vapor at high temperature (Wu et al. 2001).
For synthesizing 2D samples of superconducting elements, alloys, and compounds, various PVD techniques like sputtering, laser ablation, thermal evaporation, etc. are commonly used for depositing films of a few nanometers to several tens of nanometers thickness on suitable substrates; the methods are described elsewhere (Narlikar 2014). For example, epitaxial NbN films of thickness 2.5–100 nm have been prepared on MgO substrates using magnetron sputtering (Kang et al. 2011, Marsili et al. 2008). Similarly, highquality ultrathin MgB_{2} films of thickness down to 7.5 nm have been reported on SiC substrates via the hybrid physical–chemical vapor deposition (HPCVD) technique (Wang et al. 2009). Various thin film deposition techniques used for highT_{c} cuprates, Febased superconductors, etc. are described elsewhere (Narlikar 2014). However, as mentioned in Sec. 1.5.2, in the case of small 2D superconductors, one is particularly interested in the behavior of ultrathin films down to the lowest 2D limit of monolayer thickness. To prepare such films, the technique most popularly used is molecular beam epitaxy (MBE), which is a sophisticated form of thermal evaporation that allows a single layer of atoms to be deposited at a time. The high quality of the performance is assured by the combined MBE system having various analytical tools like STM (scanning tunneling microscopy), ARPES (angle resolved photoemission spectroscopy), and RHEED (reflection high energy electron diffraction), all functioning at the base pressure of 10−^{11} torr, for in situ characterization and monitoring of the deposited films.
(p. 33) 1.9 Summary and outlook
In this introductory chapter we saw that as the size of a superconductor was significantly reduced, it exhibited novel properties, displaying a marked departure from the behavior of bulk superconductors. At this level of smallness, which may vary from a fraction of a micrometer to a few nanometers or even shorter, both quantum confinement effects and sample surface effects become significant controlling factors for their physical properties. The bizarre change in the properties and the new phenomena displayed by small superconductors are manifested by all categories of superconductors that may include typeI, typeII, low T_{c}, high T_{c}, metallic, ceramic, organic, hybrid, etc. In this chapter we briefly explored the general mesoscopic superconducting features and phenomena that include phase slips, quantum fluctuations, different types of proximity effects, Andreev reflection, strange vortex matter, etc. The fabrication techniques commonly used for the synthesis of small samples and structures have been briefly presented. The upcoming chapters of this handbook will be taking stock in depth of the recent advances in the field of small superconductors, in the form of basic research, materialsspecific developments, and the exciting progress in their rapidly evolving device technology. Various new challenges posed and the way they are being effectively met have been assessed and discussed in the reviews and extended articles comprising the present book, with each chapter promising an exciting outlook for the future of small superconductors.
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