# Condensation Energy for a Two-Gap Superconducting State in Nanoparticles

Authors: Sergei P. Kruchinin; Hiroyuki Kawabe; Hidemi Nagao; Yasuhiro Nakazawa
Journal: Journal of Nanoparticles
 (2013)

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## Abstract

Recent technology and experiments have fabricated high-quality superconducting MgB2 nanoparticles.
We investigate properties of two-gap superconductivity in nanosized systems by using a
two-sublevel model. In the present work, we analyze the results obtained for superconducting granules
in the case of multiband superconductivity. We discuss the finite size effect in multiband
superconductors. A definition of the critical level spacing of two-gap superconductivity is also presented,
and we discuss the condensation energy and the parity gap of two-gap superconductivity
in relation to the size dependence of those properties with two bulk gaps and the effective pair
scattering process between two sublevels.


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## Body

## 1. Introduction

Recent advances in nanoscience have demonstrated that fundamentally new physical phenomena are found when systems are reduced in size to dimensions that become comparable to the fundamental microscopic length scales of a material under study. Superconductivity is a macroscopic quantum phenomenon, and therefore, it is especially interesting to see how this quantum state is influenced when the samples are reduced to nanometer sizes. In such systems, new states of matter can be engineered that do not occur in bulk materials.The properties of superconducting materials with ultra-small sizes differ from those in bulk [1]. The characteristics of a superconducting granule depend on the discrete energy spectrum, the parity of the number of electrons, and quantum fluctuations [2]. The dependence of superconducting properties on the parity of the number of electrons participating in the Cooper coupling was called the “parity effect” [3]. The analogous phenomenon is known in nuclear physics. The parity effect was experimentally observed in small superconducting aluminum granules [4, 5].After the discovery of multiband superconductors such as MgB2 with Tc=39K [6] and iron pnictides with Tc=45K [7], the physicists were faced with the question about the influence of the number of bands on the properties of superconducting granules. This problem was discussed in recent works [8–11]. Here, we plan to generalize the results of calculations of the thermodynamics of superconducting granules to the case of a two-band model of superconductivity.Recent discovery of superconductivity of MgB2 has much attracted great interest with multigap superconductivity [6]. Although multigap superconductivity had been discussed theoretically [12] in 1958, the multigap superconductivity has been observed experimentally in the 1980s [13].MgB2 is the first material whose effects are so dominant and implications are so thoroughly explored. Magnesium diboride is reported to be an anisotropic superconductor with conventional Bardeen-Cooper-Schrieffer electron-phonon coupling [7, 14]. The band structure of MgB2 calculated in several works since the discovery of the superconductivity is similar to that of graphite and is formed by by the σ and π zones. Magnesium diboride has two superconducting gaps, 4 meV and 7.5 meV, due to the π and σ electron bands. The two-gap structure was established in a number of experiments, and the two-gap superconductivity has been also discussed by many groups.In the recent years, great efforts have been devoted to the fabrication of MgB2  nanostructures that could play a crucial role in the field of applied superconductivity [7]. The ideal candidate is one-dimensional (1D) nanostructures including nanotubes, nanowires, and nanoparticles. As for MgB2, nanoparticles of approximately 20–100 nm in size are available (see Figure 1) [8].Figure 1
The micrograph of 20–100 nm nanoparticles of MgB2.For ultra-small superconducting grains, the experiments [4, 5] by Black et al. have also generated much interest in the size dependence of superconductivity. Properties of ultra-small superconducting grains have been theoretically investigated by many groups [10, 11, 15–20]. Such ultra-small grains were considered by Anderson [1]. The standard BCS theory gives a good description of the phenomenon of superconductivity in large samples. For ultra-small Al grains [4, 5], the bulk gap has been discussed in relation to physical properties of ultra-small grains such as the parity gap [3], condensation energy [18], electron correlation [16], and as well as the size dependence of the level spacing [17] of samples. In this paper, we investigate the properties of two-gap superconductivity in a nanosized system by using a two-sublevel model in the case of strong interaction. We discuss the condensation energy and the parity gap of two-gap superconductivity in relation to the size dependence of those properties in the case of two bulk gaps and the effective pair scattering process between two sublevels.


## 2. The Model and the Solution

In the case of a two-gap superconductor, we can consider a model with two sublevels corresponding to two independent bands. We consider a pairing Hamiltonian with two sublevelsν=1,2 corresponding to two bands 1 and 2 written as
(1)H=H0+Hint,
where
(2)H0=∑j,σ,ν‍[ενj-μ]aνjσ†aνjσ,(3)Hint=-∑ν‍gν∑j,j′∈Iν‍aνj↑†aνj↓†aνj′↓aνj′↑+g12∑j∈I1,k∈I2‍a1j↑†a1j↓†a2k↓a2k↑+H.c.,
where aνjσ†(aνjσ) is the creation (annihilation) operator in sublevels ν with spin σ and the energies ενj. Here, the operators for each sublevel satisfy the anticommutation relations, the operators between sublevels are independent, and μ is the chemical potential. The sums of j and k in (3) are over the set I1 of  N1I1 states corresponding to a half-filled band 1 with fixed width 2ω1D and the set I2 of N2I2 states for band 2, respectively. The second term in (1) is the interaction Hamiltonian, g1 and g2 are the effective electron-electron interaction constants including the e-p interaction for each sublevel, respectively, and g12 is the effective interaction constant that corresponds to the pair scattering process including the phonon scattering between two bands. Note that, in the bulk limit of (1), we can derive a similar coupled gap equation presented in [9].In this study, we assume that the Debye energies for two sublevels are the sameω1D=ω2D=ωD. Within this assumption, N1I1 and N2I2 are relatively estimated by the density of state (DOS) for two bands as N1I1/N2I2=ρ1/ρ2, where ρ1 and ρ2 are DOSs for two bands, respectively. The interaction constants g1 and g2 can be written as d1λ1 and d2λ2, respectively, d1=2ωD/N1I1 and d2=2ωD/N2I2 mean the mean energy level spacing, and λ1 and λ2 are the dimensionless parameters for two sublevels. We consider the intersublevel interaction constant g12=d1d2λ12. In summary, we obtain a relation of ρ1/ρ2=N1I1/N2I2=d2/d1. It is convenient to apply the path integral approach to the treatment of fluctuations of the order parameters. In works [16, 17], the problem of ultra-small superconducting grains was considered within the path integral method
(4)Z(μ,T)=∫‍D2Δ(τ)e-S[Δ],
where T is the temperature and S[Δ] is the action.We now adapt the developed technique of path integration in the case of two-gap superconductivity. In what follows, we will use this method to study the influence of fluctuations on the two-gap superconductivity in ultra-small superconducting grains. This approach gives an exact expression for the canonical partition function of a superconductor [3]. Within the path integral approach, we obtain the canonical partition function for the Hamiltonian of (1) in the following form:
(5)Z(μ,T)=∫‍DΔ1DΔ1*DΔ2DΔ2*e-S[Δ1,Δ2],
where the action S[Δ1,Δ2] is defined as
(6)S[Δ1,Δ2]=-∑j‍[TrlnG1j-1-ξ1jT]-∑k‍[TrlnG2k-1-ξ2kT]+∫01/T‍dτ1g1g2-g122×[g2|Δ1(τ)|2+g1|Δ2(τ)|2222222222222+g12(Δ1(τ)Δ2(τ)*+Δ1(τ)*Δ2(τ))|Δ2(τ)|2].
Here, Δ1 and Δ2 are bulk gaps for sublevels 1 and 2, respectively, ξνj=ενj-μ, and Gνj-1(τ,τ′)=[-d/dτ-ξνjσz-Δν(τ)σ+-Δν*(τ)σ-]δ(τ-τ′), where σ±=σx±iσy and σx,y,z are the Pauli matrices. G1-1 and G2-1 satisfy antiperiodic boundary conditions.In the case of stronger interaction,Δ1≫d1 and Δ2≫d2 can be considered in the mean-field approximation for the order parameters within the path integral approach. Substituting the time-independent order parameters into the action of (6), we have
(7)Ω(μ)=∑j,ν‍(ξνj-ϵνj)+1g1g2-g122×[g2Δ12+g1Δ22+g12(Δ1*Δ2+Δ1Δ2*)],
where ϵ1j=(ξ1j2+Δ12)1/2 and ϵ2k=(ξ2k2+Δ22)1/2. In (7), the values of Δ1 and Δ2 must be chosen in a way that minimizes Ω. From the minimization of Ω, we obtain the coupled gap equation at zero temperature for a two-gap system:
(8)(Δ1Δ2)=(g1∑j‍12ϵ1j-g12∑k‍12ϵ2k-g12∑j‍12ϵ1jg2∑k‍12ϵ2k)(Δ1Δ2).
From this equation, we formally obtain an expression for a bulk gap at zero temperature in the case of two-gap superconductivity:
(9)Δ~1=ωsinh-1(1η1),Δ~2=ωsinh-1(1η2),
where 1/ην=(λν′+α±(-1)ν′[η1,η2]λ12)/(λ1λ2-λ122) (ν≠ν′) and α±[η1,η2]=±sinh(1/η1)/sinh(1/η2). For two-band superconductivity, we can consider two cases for the phases of the gaps: sgn(Δ~1)=sgn(Δ~2) and sgn(Δ~1)=-sgn(Δ~2). For the same phase, α+ is used in (8), and we use α- for the opposite phases. Note that Δ~1=-Δ~2 in the limit of strong intersublevel coupling λ12, that is, for the opposite phases. At λ12=0, we find the same results of two bulk gaps derived from the conventional BCS theory for two independent sublevels.In nanosized single-band superconductivity, the condensation energy can be defined asEN,bC(λ)=EN,bG(0)-EN,bG(λ)-nλd, where EN,bG is the ground state energy of an N-electron system in the interaction band, b is the number of electrons on single occupied levels, and λ and n are the dimensionless coupling parameter and the number of pair occupied levels, respectively. In the case of a nanosized two-band system, the condensation energy can be written in a similar way as follows: EN1,b1;N2,b2C(λ1,λ2,λ12)=EN1,b1;N2,b2G(0,0,0)-EN1,b1;N2,b2G(λ1,λ2,λ12)-∑ν‍nνλνdν, where EN1,b1;N2,b2G(λ1,λ2,λ12) means the ground state energy of the (N1+N2)-electron system. From (5) and (7), the condensation energy of a two-sublevel system can be expressed in terms of the condensation energies of independent single level systems:
(10)EN1,b1;N2,b2C(λ1,λ2,λ12)=EN1,b1C(λ1)+EN2,b2C(λ2)-λ122λ1λ2-λ122(Δ12d1λ1+Δ22d2λ2+2(Δ1*Δ2+Δ1Δ2*)d1d2λ12),
where EN1,b1C(λ1) and EN2,b2C(λ2) correspond to the condensation energies for single band cases and the third term means the cross term between two bands. At the same phases of Δ1 and Δ2, the condensation energy of (10) decreases; that is, there appears the instability by the coupling constant λ12. On the other hand, at the opposite phases, the condensation energy becomes larger, because Δ1*Δ2+Δ1Δ2*<0. We can expect that the condensation energy of two-gap superconductivity becomes more stable than that of two independent systems due to the intersublevel coupling λ12 and the opposite phases.To discuss the critical level spacing for a two-gap system, we start from the coupled gap equation of (8). For the case of the critical level spacing of a two-gap system, we have
(11)1=λ1∑j‍12|ξ~1j|+λ2∑k‍12|ξ~2k|-(λ1λ2-λ122)∑j‍12|ξ~1j|∑k‍12|ξ~2k|,
where ξ~i=ξi/di for sublevels i=1,2. For the odd or even grain, (11) can be approximately solved by using the digamma function: for the odd case, the critical level spacing becomes
(12)d1co=ωDeγexp[-1λ],d2co=d2d1d1co,
and, for the even case,
(13)d1ce=4ωDeγexp[-1λ],d2ce=d2d1d1ce.
Here, we use
(14)1λ=12x[λ1+λ2-ax+(λ1-λ2-ax)2+4λ122]
with x=λ1λ2-λ122, a=log(d1/d2). From these expressions, we find the relations
(15)d1ce=4d1co,d2ce=4d2co,(16)d1/2co≈eγ2exp[1η1/2-1λ]Δ~1/2.
In the case of |λ1-λ2|≫λ12, (16) can be approximately rewritten as
(17)d1/2co≈eγ2exp[λ2-λ1+2αλ12λ1λ2-λ122]Δ~1/2.
On the other hand, in the limit of |λ1-λ2|=λ12, we have
(18)d1/2co≈eγ2exp[(1+α)λ12λ1λ2-λ122]Δ~1/2.
For the case of λ12=0, (16) can be rewritten as d1/2co≈exp[γ]/2exp[1/λ1-1/λ2]Δ~1/2. Therefore, when the coupling constants λ1 and λ2 become the same value, we have a relation similar to that for a single level system: d1/2co≈0.89Δ~1/2. These results suggest that the critical level spacing strongly depends on λ12 and the difference between the effective interaction constants for the sublevels. The relation in (15) is the same relation as in the conventional nanosized BCS theory.In the case of two sublevel spacings, the chemical potential lies halfway between the highest occupied and the lowest unoccupied levels with smaller level spacing in the half-filled case as shown in Figure2(a). We assume that d1<d2 and that the numbers of occupied levels corresponding to each sublevel are n1 and n2, respectively. Then, the total number of electrons becomes N=2n1+2n2. When we consider N=2n1+2n2+1, the chemical potential lies on the level ε1n1+1, as shown in Figure 2(b). Figure 2(c) shows the position of the chemical potential in the case of N=2n1+2n2+2. The parity gap of nanosized two-gap superconductivity is written as
(19)Δp1=E2n1+1+2n2,1G-12(E2n1+2n2,0G+E2(n1+1)+2n2,0G).
From (7) and the ground state energy EN,bG=ΩμN+μNN, we obtain
(20)Δp1=Δ1-d14(ρ1ρ2-1).
In view of Figures 2(c), 2(d), and 2(e), we can obtain another formula for a parity gap:
(21)Δp2=E2(n1+1)+2n2+1,1G-12(E2(n1+1)+2n2,0G+E2(n1+1)+2(n2+1),0G),
which yields
(22)Δp2=Δ2-d24(3ρ2ρ1-1).
The present results suggest two kinds of the dependence of the parity gap on the level spacing. We note that the parity gap does not depend upon the effective interaction λ12, and the structure around the Fermi level plays an important role, by contributing to the size dependence of the parity gap.Positioning of the chemical potential relative to the electronic energy levels in a two-gap superconducting grain. Solid and dotted lines mean two sublevels. (a) Half-filled system with2n1+2n2 electrons, (b) 2n1+1+2n2-electron system, (c) 2(n1+1)+2n2-electron system, (d) 2(n1+1)+2n2+1-electron system, and (e) 2(n1+1)+2(n2+1)-electron system.
(a)
(b)
(c)
(d)
(e)


## 3. Conclusion

Thus, we have investigated the properties of nanosized two-gap superconductivity by using a two-sublevel model in the framework of the mean-field approximation. In view of the discussion for the condensation energy in nanosized two-gap superconductivity, the phases of the gaps are very important to stabilize the superconductivity. At the same phases, the two-gap superconductivity is instable by the coupling constantλ12. On the other hand, at the opposite phases, the superconductivity becomes stable. We can expect that the condensation energy of two-gap superconductivity becomes more stable than that of two independent systems due to the intersublevel coupling λ12 and to the opposite phases.The parity effect can be observed only at sufficiently low temperatures. It becomes especially significant in ultra-small granules about 10 Å in size. The experimental investigation of superconductivity in such granules becomes possible now due to the developed elegant experimental technique [6, 7]. The relations concerning the parity effect and the energy condensation in the case of multiband superconductivity, which are obtained in the present work, can be verified in the same experiments as in [6, 7] but with the use of nanogranules made of MgB2.In summary, a model corresponding to nanosized two-gap superconductivity has been presented, and the expression of the partition function of a nanosized system has been analytically derived by using the path integral approach. A definition of the critical level spacing of the two-gap superconductivity has been also presented, and we discuss the condensation energy and the parity gap of the two-gap superconductivity in relation to the size dependence of those properties with two bulk gaps and the effective pair scattering process between two sublevels. The results of this work can be tested in the tunneling experiments with MgB2 nanoparticles [2, 4]. In work [19], the limits of the application of calculations to one-band nanosuperconductors were considered in detail. In the next article, we will plan to study such limits for two-band nanosuperconductors, as well as the influence of fluctuations on the two-gap superconductivity in ultra-small superconducting grains.



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