Novel superconductivity: from bulk to nano systems^*

In 1958 Gor'kov reformulated the BCS theory of superconductivity in the language of quantum field theory [11]. Soon after, Gor'kov made an important contribution to relate phenomenological GL theory with the microscopic BCS within some limitations. We present here briefly the equivalence.

1.1.1. GL theory

In GL theory the free energy density, f of a superconductor near the normal to superconductive transition is expressed in terms of a complex order parameter (field) $\Psi$, which is nonzero below ${{T}_{c}}$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{f}_{s}} & = & {{f}_{n0}}+\alpha {{\left| \Psi \right|}^{2}}+\frac{\beta }{2}{{\left| \Psi \right|}^{4}}+\frac{1}{2m*} \\ {} & {} & \times {{\left| \left( -{\rm i}\hbar \nabla -\frac{2e}{c}{\boldsymbol{A}} \right)\Psi \right|}^{2}}+\frac{{{{\boldsymbol{h}} }^{2}}}{8\pi }. \\ \end{array}\end{eqnarray} \tag{ 1 }$$

Here $\alpha$ and $\beta$ are two phenomenological parameters. $\alpha$ is $T$-dependent and chosen in the form $\alpha =a(T-{{T}_{c}})$. ${{f}_{n0}}$ is the normal state free energy density, $m*$ is the effective mass and $e$ is the charge of an electron. ${\boldsymbol{A}}$ is the magnetic vector potential given by ${\boldsymbol{h}} =\nabla \times {\boldsymbol{A}}$, ${\boldsymbol{h}}$ being the magnetic field. By varying the free energy with respect to $\Psi$ and ${\boldsymbol{A}}$, one obtains two GL equations

$$\begin{eqnarray}&&\alpha \Psi +\beta {{\left| \Psi \right|}^{2}}\Psi +\frac{1}{2m*}{{\left( -{\rm i}\hbar \nabla -\frac{2e}{c}{\boldsymbol{A}} \right)}^{2}}\Psi =0,\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {\boldsymbol{j}} & = & \frac{c}{4\pi }\nabla \times {\boldsymbol{h}} \\ {} & = & \frac{e\hbar }{m*i}\left( \Psi *\nabla \Psi -\Psi \nabla \Psi * \right)-\frac{4{{e}^{2}}}{m*c}{{\left| \Psi \right|}^{2}}{\bf A}. \\ \end{array}\end{eqnarray} \tag{ 3 }$$

Here ${\boldsymbol{j}}$ is the current density. Solution of these two equations (with appropriate boundary conditions) determines the order parameter and the current density.

For a homogeneous superconductor the first GL equation gives ${{\left| \Psi \right|}^{2}}=-a(T-{{T}_{c}})/\beta$ for $T\lt {{T}_{c}}$. It is a nontrivial solution typical for a second-order transition.

For non-zero magnetic field and non-zero gradient of $\Psi$, one can calculate the characteristic lengths $\xi$ (coherence length) and $\lambda$ (penetration depth), as mentioned earlier. Here it is noted that the basis of the theory rests on two approximations correct around the critical temperature ${{T}_{c}}$. The order parameter $\mid \Psi \mid$ is small near ${{T}_{c}}$ and $\left| \Psi \right|\tilde{\ }{{\tau }^{1/2}}$ where $\tau =\left( 1-T/{{T}_{c}} \right)$. Despite these limitations there is a myth that the GL theory applies not only around ${{T}_{c}}$, rather it is useful for very low temperature. More in the next section.

1.1.2. BCS theory

It is developed with an assumption that there is an attractive interaction between a pair of electrons by the mediation of a phonon in conventional metallic systems. The theory can be generalised such that any boson can mediate the interaction to produce a bound pair. The many body reduced Hamiltonian is written as

$$\begin{eqnarray}&&H=\mathop \sum \limits_{{\bf k}\sigma } {{\xi }_{{\bf k}}}c_{{\bf k}\sigma }^{\dagger }{{c}_{{\bf k}\sigma }}+{{V}_{int}},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{{V}_{int}}=\frac{1}{N}\mathop \sum \limits_{{\bf kk}^{\prime} } {{g}_{{\bf kk}^{\prime} }}c_{{\bf k}\uparrow }^{\dagger }c_{-{\bf k}\downarrow }^{\dagger }{{c}_{-{\bf k}^{\prime} \downarrow }}{{c}_{{\bf k}^{\prime} \uparrow }},\end{eqnarray} \tag{ 5 }$$

where ${{\xi }_{{\bf k}}}$ is the electron excitation energy, ${{g}_{{\boldsymbol{kk}} ^{\prime} }}$ is the pairing interaction involving ${\bf k}$ and $-{\bf k}$ and ${\bf k}^{\prime}$ and $-{\bf k}^{\prime}$ with up and down spins, respectively. Using any one of the techniques (BCS variational method, Bogolyubov–Valatin canonical transformation, Bogolyubov–de Gennes approach or Gor'kov's thermodynamic Green functions) we obtain the BCS gap equation using the definition

$$\begin{eqnarray}&&{{\Delta }_{{\bf k}}}=-\frac{1}{N}\mathop \sum \limits_{{\bf k}^{\prime} } {{g}_{{\bf kk}^{\prime} }}\left\langle {{c}_{-{\bf k}^{\prime} \downarrow }}{{c}_{{\bf k}^{\prime} \uparrow }} \right\rangle .\end{eqnarray} \tag{ 6 }$$

After going through some algebra [12–14], at $T=0$ one obtains the gap equation

$$\begin{eqnarray}&&{{\Delta }_{{\bf k}}}=-\frac{1}{N}\mathop \sum \limits_{{\bf k}^{\prime} } {{g}_{{\bf kk}^{\prime} }}\frac{{{{\Delta }}_{{\bf k}^{\prime} }}}{2\sqrt{\xi _{{\bf k}^{\prime} }^{2}+{{\left| {{{\Delta }}_{{\bf k}^{\prime} }} \right|}^{2}}}}\;.\end{eqnarray} \tag{ 7 }$$

Assuming ${{g}_{{\bf kk}^{\prime} }}=-{{g}_{0}}$ for $\left| {{\xi }_{{\bf k}}} \right|,\left| {{\xi }_{{\bf k}^{\prime} }} \right|\lt {{\omega }_{D}}$ (Debye cut-off), the k-independent gap equation is

$$\begin{eqnarray}&&{{\Delta }_{0}}=\frac{{{g}_{0}}}{N}\mathop \sum \limits_{\begin{array}{ccccccccccccccc} {\bf k}^{\prime} \\ \left| {{{\boldsymbol{ \xi }} }_{{\rm k}^{\prime} }} \right|\lt {{{\boldsymbol{ \omega }} }_{{\boldsymbol{D}} }} \\ \end{array}} \frac{{{\Delta }_{0}}}{2\sqrt{\xi _{{\bf k}^{\prime} }^{2}+{{\left| {{\Delta }_{0}} \right|}^{2}}}}\;.\end{eqnarray} \tag{ 8 }$$

If the density of states near ${{E}_{{\rm F}}}$ is nearly constant within $\pm {{\omega }_{D}}$, the gap is given by

$$\begin{eqnarray}&&{{\Delta }_{0}}=\frac{{{\omega }_{D}}}{{\rm sinh} \left( 1/{{g}_{0}}N(0) \right)}\;,\end{eqnarray} \tag{ 9 }$$

$N\left( 0 \right)$ is the density of states at the Fermi energy. In the weak-coupling case ${{g}_{0}}N\left( 0 \right)$ being small, the gap is now

$$\begin{eqnarray}&&{{\Delta }_{0}}=2{{\omega }_{D}}{\rm exp} \left( 1/{{g}_{0}}N(0) \right).\end{eqnarray} \tag{ 10 }$$

At finite temperature, this equation is

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \Delta (T) & = & {{g}_{0}}N\left( 0 \right)\int \limits_{0}^{{{\omega }_{D}}} {\rm d}\xi \frac{\Delta (T)}{\sqrt{{{\xi }^{2}}+{{\left| \Delta (T) \right|}^{2}}}} \\ {} & {} & \times \;{\rm tanh} \left( \frac{\sqrt{{{\xi }^{2}}+{{\left| \Delta (T) \right|}^{2}}}}{2T} \right). \\ \end{array}\end{eqnarray} \tag{ 11 }$$

1.1.3. Equivalence of GL with BCS near T_c

Following Gor'kov [13] one can obtain after some algebra the following equation

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \left[ \frac{1}{{{g}_{0}}N\left( 0 \right)}-{\rm ln} \left( \frac{1.14{{\omega }_{D}}}{T} \right) \right]\Delta \left( T \right) \\ \quad +\frac{7\zeta (3)}{8}\frac{1}{{{\pi }^{2}}T_{c}^{2}}{{\Delta }^{3}}\left( T \right)=0. \\ \end{array}\end{eqnarray} \tag{ 12 }$$

Near T_c the coefficient of $\Delta (T)$ may be expanded as ${{g}_{0}}N(0)(1-T/{{T}_{c}})$, then in the uniform limit (${\boldsymbol{A}} =0$), the order parameter $\Psi$ is proportional to $\Delta$. The final results are

$$\begin{eqnarray}&&\psi =\Delta \sqrt{\frac{7\zeta (3)}{8}\frac{n}{{{\pi }^{2}}T_{c}^{2}}},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&\alpha =-\frac{6{{\pi }^{2}}}{7\zeta (3)}\frac{T_{c}^{2}}{{{E}_{F}}}\left( 1-\frac{T}{{{T}_{c}}} \right),\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\beta =-\frac{12{{\pi }^{2}}}{7\zeta (3)}\frac{{{\left( {{k}_{B}}{{T}_{c}} \right)}^{2}}}{{{E}_{F}}n},\end{eqnarray} \tag{ 15 }$$

$n$ is the electron density. Here we presented the order parameter $\Psi$ proportional to the energy gap $\Delta$ of the BCS theory and the coefficients $\alpha$ and $\beta$ of GL theory in terms of physical parameters ${{T}_{c}}$, ${{E}_{{\rm F}}}$ and $n$. In view of these results the GL theory is almost complete in a limited sense, which is correct near ${{T}_{c}}$. We are reminded that $\left| \Psi \right|\tilde{\ }{{\tau }^{1/2}}$, where $\tau =\left( 1-T/{{T}_{c}} \right)$. Here we have a clear understanding that the GL theory is just not phenomenological, rather it has a microscopic basis as discussed above.

Before we proceed further we give an outline of the rest of the paper. Novel superconductors, particularly the multiband (gap) ones are covered in section 2. We present GL theory, BCS theory and GL theory derivable from the BCS. In section 3 we discuss further expansion as mentioned above to include higher power of ${{\tau }^{(2n+1)/2}}$ ($n\geqslant 1$) and their interesting consequences. We present some selected results on the gaps (both single and multiband materials) from our published calculations. In section 4 we discuss some current activity in nano-superconductors based on both BCS and GL theories. There have been many interesting reports on the novelty of nano-superconductors. This is a new area that is coming up vigorously. We present a balanced review of what confinement does to the smallness of nano-superconductors. Section 5 gives a brief summary and conclusions.

Multiband GL theory can be seen as a straight forward generalisation of the standard GL theory. The free energy density can be expanded as powers of multi-order parameters [34, 35]. Here one has total superconducting free energy composed of individual parts and mutual interaction terms arising out of various order parameters and gradient terms. Here we consider the case of two order parameters. Generalization to multi-order parameters is a straight forward case. The simplest inter-order parameter coupling is the internal Josephson coupling term that we consider below. The free energy density is written as

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{f}_{s}} & = & {{f}_{n0}}+\mathop \sum \limits_{j=1,2} {{\alpha }_{j}}{{\left| {{\Psi }_{j}} \right|}^{2}}+\frac{{{\beta }_{j}}}{2}{{\left| {{\Psi }_{j}} \right|}^{4}} \\ {} & {} & +\frac{1}{2m_{j}^{*}}{{\left| \left( -{\rm i}\hbar \nabla -\frac{2e}{c}{\boldsymbol{A}} \right){{\Psi }_{j}} \right|}^{2}}\; \\ {} & {} & -\Gamma \left( \Psi _{1}^{*}{{\Psi }_{2}}+\Psi _{2}^{*}{{\Psi }_{1}} \right)+\frac{{{{\boldsymbol{h}} }^{2}}}{8\pi }. \\ \end{array}\end{eqnarray} \tag{ 16 }$$

Here the last but one term is the Josephson coupling term, with interaction strength $\Gamma$. Let us consider the applied field is zero and we are in the bulk at the temperature $T$ close to ${{T}_{c}}$. By neglecting the ${{\left| {{{\Psi }}_{j}} \right|}^{4}}\;$ term, the free energy density (with reference to normal state) is given by

$$\begin{eqnarray}&&f={{\alpha }_{1}}{{\left| {{\Psi }_{1}} \right|}^{2}}+{{\alpha }_{2}}{{\left| {{\Psi }_{2}} \right|}^{2}}-2\Gamma \left| {{\Psi }_{1}} \right|\left| {{\Psi }_{2}} \right|{\rm cos} \left( \Delta \phi \right),\end{eqnarray} \tag{ 17 }$$

where $\Delta \phi$ is the phase difference between two condensates.

When $\Gamma =0$ two bands are fully decoupled and two order parameters have two critical temperatures. For $\Gamma \gt 0$ the free energy is minimized with $\Delta \phi =0$. For $\Gamma \lt 0$ the free energy is stable with $\Delta \phi =\pi$. This implies the condensation energy for two-band superconductivity is more stable than the two independent states. More on two-band nanosystems in section 4.f.i.

The GL equations for ${{\Psi }_{i}}$ are obtained from the minimization of the free energy and similarly the equation for the current. The self-consistent solutions of these ($i+1$) equations describe the properties of multiband superconductors within the traditional GL theory (see [27–35]).

In the recent past it has been argued [28, 36–40] that the original GL theory is incomplete beyond the order parameter ${{\Psi }_{{\rm i}}}\sim {{\tau }^{1/2}}$. Therefore, one has to incorporate the higher order terms (${{\tau }^{(2n+1)/2}}$ for $n\geqslant 1$) as a systematic expansion, see Fetter and Walecka [41]. For the derivation of this result one has to expand the order parameter equation in powers of $\Psi$ and its spatial gradients. Details of this theory are in our recent paper [40].

One band theory is generalised to a multiband [22, 23] situation. For simplicity we consider a two-band case. Multiband generalization is straightforward

$$\begin{eqnarray}&&H=\mathop \sum \limits_{i{\bf k}\sigma } {{\xi }_{i{\bf k}}}c_{i{\bf k}\sigma }^{\dagger }{{c}_{i{\bf k}\sigma }}+\frac{1}{\Omega }\mathop \sum \limits_{ij{\bf kk}^{\prime} } {{g}_{ij}}c_{i{\bf k}\uparrow }^{\dagger }c_{i-{\bf k}\downarrow }^{\dagger }{{c}_{j-{\bf k}^{\prime} \downarrow }}{{c}_{j{\bf k}^{\prime} \uparrow }},\end{eqnarray} \tag{ 18 }$$

here $i,j$ stand for the band indices 1, 2. ${{g}_{ij}}$ is the intra- and interband couplings, and $\Omega$ is the volume.

Using the same variational principle as in the one-band case, the gap equation is

$$\begin{eqnarray}&&{{\Delta }_{i}}=\frac{1}{\Omega }\mathop \sum \limits_{j{\bf k}} {{g}_{ij}}\frac{{{\Delta }_{j}}}{2{{E}_{j{\bf k}}}}{\rm tanh} \left( \frac{{{E}_{j{\bf k}}}}{2{{k}_{B}}T} \right),\end{eqnarray} \tag{ 19 }$$

where ${{E}_{j{\bf k}}}=\sqrt{\xi _{j{\bf k}}^{2}+{{\Delta }_{{\bf k}}}^{2}}$.

Explicitly, two gaps are given by

$$\begin{eqnarray}&&{{\Delta }_{1}}={{{\rm g}}_{11}}{{A}_{1}}{{\Delta }_{1}}+{{g}_{12}}{{A}_{2}}{{\Delta }_{2,}}\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{{\Delta }_{2}}={{{\rm g}}_{21}}{{A}_{1}}{{\Delta }_{1}}+{{g}_{22}}{{A}_{2}}{{\Delta }_{2}},\end{eqnarray} \tag{ 21 }$$

where

$$\begin{eqnarray}&&{{{\rm A}}_{{\rm i}}}=\frac{1}{\Omega }\mathop \sum \limits_{{\bf k}} \frac{1}{2{{E}_{i{\bf k}}}}{\rm tanh} \left( \frac{{{E}_{i{\bf k}}}}{2T} \right),\;\end{eqnarray} \tag{ 22 }$$

g_ij are interaction matrix involving diagonal ${{g}_{ii}}$ and off-diagonal ${{g}_{ij}}$ parameters related to bands $i$ and $j$. The equation for ${{T}_{c}}$ as largest $\Delta \to 0$, ${{T}_{c}}=1.13{{\omega }_{D}}{\rm exp} \left( 1/\Lambda \right)$, where $\Lambda$ is the largest eigenvalue of ${{\lambda }_{ij}}={{g}_{ij}}{{N}_{j}}\left( 0 \right)$

$$\begin{eqnarray}&&\Lambda =\frac{1}{2}\left[ {{\lambda }_{11}}+{{\lambda }_{22}}+\left. \sqrt{{{\left( {{\lambda }_{11}}-{{\lambda }_{22}} \right)}^{2}}+4{{\lambda }_{12}}{{\lambda }_{21}}} \right| \right].\end{eqnarray} \tag{ 23 }$$

${{T}_{c}}$ of the two band system is enhanced by the second term of the equation below

$$\begin{eqnarray}&&{{{\rm T}}_{{\rm c}}}\approx {{T}_{1c}}\left[ 1+\frac{{{\lambda }_{12}}{{\lambda }_{21}}}{\lambda _{11}^{2}\left( {{\lambda }_{11}}-{{\lambda }_{22}} \right)}+{\rm O}\left( \lambda _{12}^{4} \right) \right].\end{eqnarray} \tag{ 24 }$$

${{T}_{1c}}$ is the critical temperature for band 1.

As we have done in the one-band case (uniform without gradient and ${\boldsymbol{A}} =0$), we follow the similar procedure in the two-band case. Self-consistency condition through third order in ${{\Delta }_{i}}$ gives the same equation as in the one-band case. Near ${{T}_{c}}$ the coefficient of ${{\Delta }_{i}}(T)$ may be expanded to lowest order in $\tau$. Now by comparing the first GL equation, the values of ${{\alpha }_{i}}$ and ${{\beta }_{i}}$ can be obtained. ${{\Psi }_{i}}$ is proportional to ${{\Delta }_{i}}$. Results are [40]

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{\alpha }_{i}} & = & \frac{{{N}_{i}}\left( 0 \right)}{{{N}_{T}}\left( 0 \right)}\frac{6{{\pi }^{2}}{{({{k}_{B}}{{T}_{c}})}^{2}}}{7\zeta \left( 3 \right){{\varepsilon }_{F}}}\left( 1-\frac{T}{{{T}_{c}}}+\mathcal{A} \right. \\ {} & {} & \left. -\frac{{{g}_{jj}}}{\left( {{g}_{11}}{{g}_{22}}-{{g}_{12}}{{g}_{21}} \right){{N}_{i}}\left( 0 \right)} \right), \\ \end{array}\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{{\beta }_{i}}=\frac{{{N}_{i}}\left( 0 \right)}{{{N}_{T}}\left( 0 \right)}\frac{6{{\pi }^{2}}{{\left( {{k}_{B}}{{T}_{c}} \right)}^{2}}}{7\zeta \left( 3 \right)n{{\epsilon }_{F}}}\;,\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&\mathcal{A}={\rm log} \left( \frac{2\hbar {{\omega }_{D}}{{e}^{\gamma }}}{\pi {{k}_{B}}{{T}_{c}}} \right).\end{eqnarray} \tag{ 27 }$$

In 1962, Kubo [50] proposed a theory for metallic particles, which are small in size unlike the bulk. For such systems it is only quantum mechanics that accounts for observed physical properties. In the jargon of the field, 'quantum confinement' means that the de Broglie wavelength of the particles is comparable to the size of the system that contains them. Small size implies strong quantum confinement effects. Bulk extended matter, when sufficiently curtailed in one or more of its dimensions, will behave as a quasi-two dimensional (two-DEG), a quasi-one dimensional (quantum wire) or a quasi-zero dimensional (quantum dot) system. The price to be paid for this is to lose the long-range extendedness of the wave functions. The energy of motion becomes quantized as the wave function is confined. Subject to confinement, the continuum of the density of states becomes discrete. However, the energy gaps between the states are tiny, in the meV range, if the particle size is nanometric.

Kubo recognised the inherent statistical nature of the problem. He argued the energy spectrum depends on the precise geometry of the system, even though the average energy gaps may be same. Energy spectrum in many respects leads to anomalies in the physico-chemical properties of small metallic particles at low temperatures. In brief, the Kubo gap depends on the spectrum. This gap ${{d}_{K}}\tilde{\ }1/D\left( {{E}_{{\rm F}}} \right)$, inversely proportional to the electronic density of states at the Fermi level. Kubo gap is essentially the energy required to remove an electron from the small metal particle. See earlier reviews by Halperin [51] and von Delft and Ralph [52]. In comparison to the thermal energy, if ${{d}_{K}}\leqslant T$, the material is a metal and otherwise it is an insulator. Our further discussion is about only metallic nanoparticles.

Following the BCS theory (meant for clean superconductors), Anderson [53] presented a theory for dirty superconductors, where the elastic scattering strength is large compared to the energy gap. Independently similar results are obtained by Abrikosov and Gor'kov [54], see also [55] for more detailed calculations. The scattering mean free path does not change the Gor'kov's anomalous Green function, which is the same as the energy gap.

Anderson's idea emphasises that often in the presence of some disorder, a superconducting state can be more stable and less likely to be suppressed [56]. This loss of superconductivity has been explored not only in nanoparticles and in nanowires, but also in more complex systems exhibiting many exciting and novel effects. For a small number of particles, as mentioned before, the allowed energy levels are quantized, discrete and the mean energy level spacing ${{d}_{K}}$ (the Kubo gap) becomes bigger for the smallness of the sample. When the superconducting energy gap $\Delta$ approaches ${{d}_{K}}$, superconductivity is suppressed at this point by the Anderson's criterion [57]. In this limit $\Delta /{{d}_{K}}\lt 1$, there are no available energy levels correlated by the pairing interaction. More precisely, it has been argued [58] that superconductivity will vanish for ${{d}_{K}}$ greater than a critical value ${{d}_{Kc}}=3.56\Delta$ for even $N$ and ${{d}_{Kc}}=\Delta /4$ for odd $N$. Presently Kubo-Anderson criteria are utilised prolifically to predict superconductivity in nanosystems [59].

Although Anderson's criterion remained unexplored for some time, from the mid-nineties in view of many experimental investigations on nanoparticles, theoretical activities began vigorously based on the BCS reduced Hamiltonian. While the BCS theory is used for macoscopic systems, we are reminded of the Richardson solution [60], originally developed for finite nucleus. The latter approach is based on the canonical ensemble, where the number of particles is fixed. This is in contrast with the BCS theory, where grand canonical ensemble is used to deal with the macroscopic limit. While the BCS theory provides a mean-field solution, Richardson's solution is exact [61–63]. More on this in the next sub-section.

Richardson solution depends on ${{d}_{K}}$ near the Fermi level ${{E}_{{\rm F}}}$. The property of Cooper pairing depends on the 'parity' of number of electrons $N$. For odd $N$, one electron is unpaired and carries an additional energy ${{\Delta }_{P}}=\Delta$. This feature is experimentally detected [57, 64, 65] exploring the Coulomb blocked phenomenon and theoretically analysed by Matveev and Larkin [66].

During the past two decades there have been a lot of activities in this area of finite size nanosuperconductors. We give here a brief review. Most of the theoretical work reported here is carried out with the BCS theory applied to a finite number of particles appropriate for nanosystems. For thin films Blatt and Thompson [66] showed the energy gap is an oscillatory as a function of film thickness passing through resonances, where the period of oscillation is the Fermi wavelength. The gap increases with a decrease of the thickness. The enhancement of superconductivity by the size effect was also discussed by Parmenter [67]. With the decrease of size the gap increases, where the weak coupling limit goes to the strong coupling case ($\Delta \left( 0 \right)/{{T}_{c}}$ varies from 3.528 to 4). Thermodynamical properties, particularly specific heat, spin susceptibility of small grains are investigated in [68] by using both GL and BCS theories. Since GL is unable to deal with quasi-particles, a static approximation within the BCS was used, where contributions due to quasi particles are incorporated explicitly.

In recent years, the traditional GL theory has been applied to meso/nanosuperconductors. As mentioned earlier, it is important to apply the correct boundary conditions if the superconductivity is occurring in a small volume. On the boundary the order parameter has to vanish. On application of a magnetic field above a certain critical field a vortex can be created. The authors of this work [69] have applied traditional 3-dimensional GL theory in a parallelepiped cell containing a small mesoscopic disk at the centre. Vortex state solutions are obtained by minimising the free energy numerically by the method of simulated annealing. See reference [69] for details of calculations. The result has been shown in figure 5 as a simple example. Clearly we notice the vanishing of order parameter near the edges of the disk and at the centre where the magnetic field has penetrated.

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By using the same traditional GL theory Peeters and co-workers have studied mesoscopic superconductivity for vortex phase diagram [70]. Free energies of different vortex configurations are calculated by increasing both the disk thickness and magnetic field. The authors showed a smooth transition from a multi-vortex state to a giant vortex state (figure 6). With increment of magnetic field with steps of 0.01 ${{H}_{c2}}$ the superconductor goes from a Meissner state to vortex states as shown in the figure 6. This picture is reminiscent of formation of vortices in rotating ⁴He superfluid and atomic Bose–Einstein condensate [71]. In this sub-field of mesoscopic superconductors there has been plenty of theoretical and experimental activities. Some of the references are given here [72]. We also point out more recently available extended GL theory for nano/meso superconductors in the later section 4.f.ii.

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We mentioned earlier that the BCS model for bulk superconductor deals with grand canonical ensemble. For small grains it is inappropriate for use due to the fluctuations in smallness of $N$. This requires a treatment by canonical ensemble (see details in [58]). In metallic condensed systems with large N pairing occurs due to virtual exchange of phonons between the conduction electrons. In atomic nucleus with a fixed number of fermions (small N) pairing appears due to short-range nature of nuclear interaction and it includes contributions from the singlet-S and triplet-P channels. In both cases the pairs are correlated in the time-reversed states. A canonical treatment of BCS states are applied to nuclear physics soon after the former's appearance. In 1963 Richardson [60] provided an exact solution in a simplified form of the BCS, for application to nuclear physics with some more papers followed soon. Despite this, the condensed matter community did not pay much attention until the superconductivity in small metallic grains was reported in the mid-nineties [57]. Currently Richardson model is an attractive picture for nanosuperconductors.

Richardson's reduced Hamiltonian is given by

$$\begin{eqnarray}&&H=\sum \limits_{j=0,\sigma }^{N-1} {{\varepsilon }_{j}}c_{j\sigma }^{\dagger }{{c}_{j\sigma }}-\lambda d\sum \limits_{j,j^{\prime} =0}^{N-1} c_{j\uparrow }^{\dagger }c_{j\downarrow }^{\dagger }{{c}_{j^{\prime} \downarrow }}{{c}_{j^{\prime} \uparrow }}.\end{eqnarray} \tag{ 28 }$$

Here ${{c}_{j\sigma }}\;\left( c_{j\sigma }^{\dagger } \right)$ destroys (creates) electrons in free time-reversed states $\left. \mid j,\sigma \right\rangle$ with discrete uniformly spaced degenerate eigenenergies ${{\varepsilon }_{j}}=jd+{{\varepsilon }_{0}}$. $d$ is the (Kubo)gap of the discrete spectrum, $\lambda$ is dimensionless pairing strength. In BCS model $g=\lambda d$. The interaction $g$ scatters only time-reversed electron pairs within the Debye cutoff frequency ${{\omega }_{D}}$ of the energy shell around the Fermi energy ${{\varepsilon }_{F}}$. $\lambda$ is related to the parameters bulk gap $\Delta$ and the cutoff frequency ${{\omega }_{D}}$ via the bulk gap equation $1/\lambda ={{\omega }_{D}}/\Delta$. In this model $N=2{{\omega }_{D}}/d$.

We notice here an interesting point that the nanostructure boundary condition is entering in a subtle way through the discrete spectrum. Reference [62] considers finite size corrections to the Richardson model, where the low energy spectrum of the problem is included as an expansion ($1/N$) in the inverse of the number of electrons in the grain.

Going back to the original BCS the ground state is written as

$$\begin{eqnarray}&&\left| {\rm BCS} \right.\propto {\rm exp} \left[ \mathop \sum \limits_{j} \frac{{{v}_{j}}}{{{u}_{j}}}c_{j\uparrow }^{\dagger }c_{j\downarrow }^{\dagger } \right]\left. \mid 0 \right\rangle ,\end{eqnarray} \tag{ 29 }$$

where ${{u}_{j}}$ and ${{v}_{j}}$ are variational parameters of the original BCS theory. These two parameters are obtained as BCS solutions from a variational minimization of the reduced Hamiltonian.

$$\begin{eqnarray}&&u_{j}^{2}=\frac{1}{2}\left( 1+\frac{{{\varepsilon }_{j}}-\mu }{{{E}_{j}}} \right),\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&v_{j}^{2}=\frac{1}{2}\left( 1-\frac{{{\varepsilon }_{j}}-\mu }{{{E}_{j}}} \right),\end{eqnarray} \tag{ 31 }$$

with the quasi-particle excitation energy given by ${{E}_{j}}=\sqrt{\left. \varepsilon _{j}^{2}+{{\left| \Delta \right|}^{2}} \right]}$. $\Delta$ is the energy gap that we have discussed earlier. In the grand canonical ensemble this BCS ground state is asymptotically exact for $N\to \infty$. For finite $N$ it requires a canonical ensemble, for which a projected BCS (PBCS) ansatz is used, which is given by

$$\begin{eqnarray}&&\left| {\rm PBCS} \right.\propto {{\left( \mathop \sum \limits_{j} \frac{{{v}_{j}}}{{{u}_{j}}}c_{j\uparrow }^{\dagger }c_{j\downarrow }^{\dagger } \right)}^{N}}\left| 0 \right..\end{eqnarray} \tag{ 32 }$$

It is an 'N conserving theory' [61]. Apart from BCS and PBCS ansatz, there are other numerical approaches, namely Lanzcos method [74], perturbative renormalization group method [65] and density matrix renormalization group technique [58]. These methods have been extensively used for nanosuperconductivity. See reviews [52, 73].

For the novel physics for finite N, we have size, shape and shell effects, effects of fluctuations and odd–even parity [75–79]. Randomness of eigenvalue distributions raises new question of statistics of ensembles in the Dyson–Wigner sense. These features are in the presence of pairing correlations unlike in single-particle physics of disordered systems [80]. Thermodynamic properties of small superconducting grains are studied in detail in [81] by analytical and numerical methods. Here the authors have studied in the limit of d/Δ > 1 and the crossover of superconducting to normal grains.

An interesting point of all this important work is to emphasise how a crossover of bulk to a nanosystem is possible. Although Anderson's criterion has been in the literature for over five decades, the ground breaking experiments of Tinkham and co-workers [57] brought new light on ultra-thin nanosuperconductivity. Without going through details, a list (incomplete) gives an idea of experimental activities [59].

On the other hand, by using the BCS theory appropriate for nanosystems, namely Richardson model, it opened a new panorama in this novel area of superconductivity.

A illustrations we present here some results of calculations. In figure 7 (upper panel) calculations of condensation energy given by the difference of ground state energy of the pairing Hamiltonian and the energy of the filled Fermi state, $E_{b}^{c}=E_{b}^{GS}-FS\left| {{H}_{BCS}} \right|FS$. The condensation energy is scaled by the bulk superconducting gap. They are shown for even (b = 0) and odd (b = 1) parity state in the PBCS and exact calculations. The parity effect in terms of the gap parameter is given by the difference of ground state energy of an odd grain and the mean energy of the neighbouring even grains by adding or removing an electron. This gives the Matveev–Larkin (ML) gap by the expression [65] ${{\Delta }_{ML}}={{E}_{1}}\left( N \right)-1/2\left[ {{E}_{0}}\left( N+1 \right)+{{E}_{0}}\left( N-1 \right) \right]$. The lower panel of figure 7 gives the ML gap scaled to the bulk gap as a function of grain size by the PBCS (dashed line) and exact (continuous line) methods. One can see the asymptotic limits of both PBCS and exact results. It is also interesting to note that the Anderson's criterion $d/\Delta \tilde{\ }1$ is satisfied. For details see [73].

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Fluctuation effects are calculated in [62]. Figure 8 shows the condensation energies for even and odd N both for uniformly spaced and randomly spaced levels (discussions in [62, 80 and 73]).

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Earlier, in section 2, we have discussed multiband theories in the BCS and GL models. Particularly, two-band systems were analysed in some detail. From the material perspective Nb, Ta and V are the elemental metals which have two-band superconductivity [15]. Superconducting studies have been made in Nb-doped SrTiO₃ in bulk [16]. After the discovery of two-band superconductivity in MgB₂ [82] and Fe pnictides [83], important work continued to be done in multiband materials (see these two reviews [84, 85]).

In the area of nanosuperconductivity a number of studies have been reported. Quantum size effect (QSE) modulated superconductivity has been observed in nanograins of Pb on Si [86]. QSE has been studied in Pb, In and V [87–89], where nanosuperconductivity seems to obey the Anderson criterion. Authors of [90] have investigated size dependent T_c, coherence length, upper critical field ${{H}_{c2}}$, and irreversible field ${{H}_{irr}}$, in Nb. Reduction of the gap in ultra-thin films of Pb on Si by variation of thickness is reported in [91]. Magnetization in Pb nanoparticles has been studied in the range of 4–1000 nm [92]. The Meissner effect is clearly seen for particle sizes >30 nm, consistent with the Anderson criterion. Multilayer ultra-thin Pb films (5–15 monolayers) are the subject of a study of scanning tunnelling spectroscopy [93], where layer dependent superconducting gap and ${{T}_{{\rm c}}}$ are reported. Superlattices of artificially engineered pnictide superconductors are fabricated for new interfacial phenomena and many device applications [94].

While experimental progress on multiband superconductivity is heading forward, there are some developments on the theoretical front. The possibility of superconductivity in doped SrTiO₃ in films and interfaces is reported [95] by using a two-dimensional two-band model. Shape resonances and shell effects are investigated for multiband superconductors in thin film form [96]. In this work the one-band model of Blatt and Thompson [66] has been extended for two band superconductors relevant to MgB₂ systems. From this work we show in figure 9 the calculated ${{T}_{{\rm c}}}/{{T}_{{\rm c}}}\left( {\rm Bulk} \right)$ value as the film thickness varies. The implication of this calculation is that the pattern of the shape resonances is more regular as the offset ${{e}_{0\pi }}$ decreases. (Note, this offset is between the two σ and π bands.) In the one- band limit the shape resonances (blue colour) have larger amplitudes compared to the two-band case (black colour). Therefore, the size effect is suppressed by the multiband structure. For details see [96].

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In a long paper, Kruchinin and Nagao (KN) [97] considered a two-band superconductivity model (for MgB₂) by extending Richardson's exact model. We describe this approach in some detail. They considered the two-band Hamiltonian as in the bulk for the nanograin.

$$\begin{eqnarray}&&H=\mathop \sum \limits_{i{\bf k}\sigma } {{\xi }_{i{\bf k}}}c_{i{\bf k}\sigma }^{\dagger }{{c}_{i{\bf k}\sigma }}+\frac{1}{\Omega }\mathop \sum \limits_{ij{\bf kk}^{\prime} } {{g}_{ij}}c_{i{\bf k}\uparrow }^{\dagger }c_{i-{\bf k}\downarrow }^{\dagger }{{c}_{j-{\bf k}^{\prime} \downarrow }}{{c}_{j{\bf k}^{\prime} \uparrow }}.\end{eqnarray} \tag{ 33 }$$

Unlike in the bulk, ks are now discrete state indices, js—band indices. $g$ is the interaction matrix with attraction for the intra-band (${{g}_{ii}}\lt 0$) and repulsion (${{g}_{ij}}\gt 0$) for the inter-band couplings. Since the states are discrete, the sums in the Hamiltonian are over the set $I$ of ${{N}_{1I}}$ states corresponding to band 1 with fixed width $2{{\omega }_{1D}}$ and the set $J$ of ${{N}_{2J}}$ states for band 2, respectively. For simplicity one can assume Debye energies ${{\omega }_{1D}}$ and ${{\omega }_{2D}}$ are the same and equal to ${{\omega }_{D}}$. Then ${{N}_{1I}}/{{N}_{2I}}={{\rho }_{1}}/{{\rho }_{2}}$, where ${{\rho }_{1}}$ and ${{\rho }_{2}}$ are density of states for respective bands. As in the Richardson's model the interaction energies are ${{g}_{11}}={{d}_{1}}{{\lambda }_{11}}$ and ${{g}_{22}}={{d}_{2}}{{\lambda }_{22}}$, ${{d}_{1}}=2{{\omega }_{D}}/{{N}_{1I}}$ and ${{d}_{2}}=2{{\omega }_{D}}{{N}_{2I}}$. The last two are the mean energy level spacings. ${{\lambda }_{1}}$ and ${{\lambda }_{2}}$ are dimensionless parameters for two sub-bands. The inter-band coupling is written as ${{g}_{12}}={{\lambda }_{12}}\sqrt{{{d}_{1}}{{d}_{2}}}$ The ratio ${{\rho }_{1}}/{{\rho }_{2}}={{d}_{2}}/{{d}_{1}}$.

KN [98] employed the functional integral approach, commonly used in the problems of pairing correlations of a finite number of fermions [99, 100] With this technique the coupled gap equations at $T=0$ are obtained (for bulk) as ${{\Delta }_{i}}={{\omega }_{D}}/\;{\rm sinh} \;\left( 1/{{\eta }_{i}} \right)$ with $i=1,2$.

$$\begin{eqnarray}&&\frac{1}{{{{\eta }}_{1}}}=\frac{{{\lambda }_{22}}+{{\alpha }_{\pm }}\left[ {{\eta }_{1}},{{\eta }_{2}} \right]{{\lambda }_{12}}}{{{\lambda }_{11}}{{\lambda }_{22}}-\lambda _{12}^{2}}\;,\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&\frac{1}{{{{\eta }}_{2}}}=\frac{{{\lambda }_{11}}+\alpha _{\pm }^{-1}\left[ {{\eta }_{1}},{{\eta }_{2}} \right]{{\lambda }_{12}}}{{{\lambda }_{11}}{{\lambda }_{22}}-\lambda _{12}^{2}}.\end{eqnarray} \tag{ 35 }$$

The phase of the gap is ${{\alpha }_{\pm }}\left[ {{\eta }_{1}},{{\eta }_{2}} \right]=\pm {\rm sinh} \left( 1/{{\eta }_{2}} \right)/{\rm sinh} \left( 1/{{\eta }_{1}} \right)$. Here there are two cases, (i) the phases can be same and (ii) the phases are opposite. When the inter-band coupling ${{\lambda }_{12}}$ is large, ${{\Delta }_{1}}=-{{\Delta }_{2}}$ and for ${{\lambda }_{12}}=0$, bulk gaps are independent two gaps. These results we have obtained earlier by the GL method (section 2.2) for the bulk two-band superconductivity. KN calculates the condensate energy ${{E}_{{\rm c}}}$ for the two-band nanosuperconductor. As conventionally defined it is the difference of ground state energies of normal and superconducting states including the interaction term. The final expression for the total condensation energy ${{E}_{{\rm c}}}$ is given by

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{E}_{{\rm c}}} & = & {{E}_{{\rm c}}}\left( {{N}_{1}},{{\lambda }_{11}} \right)+{{E}_{{\rm c}}}\left( {{N}_{2}},{{\lambda }_{22}} \right)-\Delta {{E}_{{\rm c}}} \\ {} & {} & \times \left( {{\Delta }_{1}},{{\Delta }_{2}},{{d}_{1}},{{d}_{2}},{{\lambda }_{11}},{{\lambda }_{22}},{{\lambda }_{12}} \right), \\ \end{array}\end{eqnarray} \tag{ 36 }$$

where the first two terms on the right hand side are condensation energy of two independent sub-systems and the last term is the contribution of inter-band contribution given by

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {\Delta }{{E}_{{\rm c}}} & = & \frac{\lambda _{12}^{2}}{{{\lambda }_{11}}{{\lambda }_{22}}-\lambda _{12}^{2}}\left[ \frac{\Delta _{1}^{2}}{{{d}_{1}}{{\lambda }_{11}}}+\frac{\Delta _{2}^{2}}{{{d}_{2}}{{\lambda }_{22}}} \right. \\ {} & {} & +\left. \frac{2\left( \Delta _{1}^{*}{{\Delta }_{2}}+{{\Delta }_{1}}\Delta _{2}^{*} \right)}{{{\lambda }_{12}}\sqrt{{{d}_{1}}{{d}_{2}}}} \right]. \\ \end{array}\end{eqnarray} \tag{ 37 }$$

The quantity $\Delta {{E}_{{\rm c}}}$ increases or decreases depending on the value of the phase difference. $\left( \Delta _{1}^{*}{{\Delta }_{2}}+{{\Delta }_{1}}\Delta _{2}^{*} \right)=2\left( \left| {{\Delta }_{1}} \right|\left| {{\Delta }_{2}} \right|{\rm cos} \left( \Delta \phi \right) \right)$. If its value is $\gt 0$, it leads to an instability, whereas its value $\lt 0$, makes ${{E}_{{\rm c}}}$ more stable. This we have discussed in section 2.1 under the multiband GL theory.

Among other things KN discusses critical level spacing, the parity gap, Kondo effect and application to quantum computing for this model.

This is a new paradigm of nanofilms involving multiband superconductors. Quantum confinement and fluctuations are important in quasi-2D systems. Therefore, it will be alluring to implement the extended GL method as worked out in detail for multiband bulk system in [39, 40]. Shanenko et al [100] have made a preliminary attempt in this realm. They have included terms of order ${{\tau }^{3/2}}$ in the extended GL equation. Yet the validity of GL equations remains $\tau \ll 1$.