Condensation Energy for a Two-Gap Superconducting State in Nanoparticles

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.

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 𝑇_𝑐=39K [6] and iron pnictides with 𝑇_𝑐=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 sublevels 𝜈=1,2 corresponding to two bands 1 and 2 written as 𝐻=𝐻₀+𝐻_int, (1) where 𝐻₀=_{𝑗,𝜎,𝜈}𝜀_𝜈𝑗−𝜇𝑎^†_𝜈𝑗𝜎𝑎_𝜈𝑗𝜎,𝐻_int=−_𝜈𝑔_{𝜈𝑗,𝑗′∈𝐼𝜈}𝑎^†_𝜈𝑗↑𝑎^†_𝜈𝑗↓𝑎_{𝜈𝑗′↓}𝑎_{𝜈𝑗′↑}+𝑔_{12𝑗∈𝐼1,𝑘∈𝐼2}𝑎^†_1𝑗↑𝑎^†_1𝑗↓𝑎_2𝑘↓𝑎_2𝑘↑+H.c., (2)(3) where 𝑎^†_𝜈𝑗𝜎(𝑎_𝜈𝑗𝜎) is the creation (annihilation) operator in sublevels 𝜈 with spin 𝜎 and the energies 𝜀_𝜈𝑗. 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  𝑁_1𝐼1 states corresponding to a half-filled band 1 with fixed width 2𝜔_1𝐷 and the set 𝐼₂ of 𝑁_2𝐼2 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 same 𝜔_1𝐷=𝜔_2𝐷=𝜔_𝐷. Within this assumption, 𝑁_1𝐼1 and 𝑁_2𝐼2 are relatively estimated by the density of state (DOS) for two bands as 𝑁_1𝐼1/𝑁_2𝐼2=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=2𝜔_𝐷/𝑁_1𝐼1 and 𝑑₂=2𝜔_𝐷/𝑁_2𝐼2 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=𝑁_1𝐼1/𝑁_2𝐼2=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 𝑍𝜇,𝑇=𝐷²Δ(𝜏)𝑒^−𝑆[Δ], (4) 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: 𝑍𝜇,𝑇=𝐷Δ₁𝐷Δ^∗₁𝐷Δ₂𝐷Δ^∗₂𝑒^{−𝑆[Δ1,Δ2]}, (5) where the action 𝑆[Δ₁,Δ₂] is defined as 𝑆Δ₁,Δ₂=−_𝑗Trln𝐺⁻¹_1𝑗−𝜉_1𝑗𝑇−_𝑘Trln𝐺⁻¹_2𝑘−𝜉_2𝑘𝑇+₀^1/𝑇1𝑑𝜏𝑔₁𝑔₂−𝑔²₁₂×𝑔₂Δ₁(𝜏)²+𝑔₁Δ₂(𝜏)²+𝑔₁₂Δ₁(𝜏)Δ₂(𝜏)^∗+Δ₁(𝜏)^∗Δ₂(𝜏). (6) Here, Δ₁ and Δ₂ are bulk gaps for sublevels 1 and 2, respectively, 𝜉_𝜈𝑗=𝜀_𝜈𝑗−𝜇, and 𝐺⁻¹_𝜈𝑗(𝜏,𝜏^′)=[−𝑑/𝑑𝜏−𝜉_𝜈𝑗𝜎^𝑧−Δ_𝜈(𝜏)𝜎⁺−Δ^∗_𝜈(𝜏)𝜎⁻]𝛿(𝜏−𝜏^′), where 𝜎^±=𝜎^𝑥±𝑖𝜎^𝑦 and 𝜎^{𝑥,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 Ω𝜇=_𝑗,𝜈𝜉_𝜈𝑗−𝜖_𝜈𝑗+1𝑔₁𝑔₂−𝑔²₁₂×𝑔₂Δ²₁+𝑔₁Δ²₂+𝑔₁₂Δ^∗₁Δ₂+Δ₁Δ^∗₂, (7) where 𝜖_1𝑗=(𝜉²_1𝑗+Δ²₁)^1/2 and 𝜖_2𝑘=(𝜉²_2𝑘+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: Δ₁Δ₂=𝑔_1𝑗12𝜖_1𝑗−𝑔_12𝑘12𝜖_2𝑘−𝑔_12𝑗12𝜖_1𝑗𝑔_2𝑘12𝜖_2𝑘Δ₁Δ₂. (8) From this equation, we formally obtain an expression for a bulk gap at zero temperature in the case of two-gap superconductivity: Δ₁=𝜔sinh⁻¹1𝜂₁,Δ₂=𝜔sinh⁻¹1𝜂₂, (9) where 1/𝜂_𝜈=(𝜆_𝜈′+𝛼^{(−1)𝜈′}_±[𝜂₁,𝜂₂]𝜆₁₂)/(𝜆₁𝜆₂−𝜆²₁₂) (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(Δ₁)=sgn(Δ₂) and sgn(Δ₁)=−sgn(Δ₂). For the same phase, 𝛼₊ is used in (8), and we use 𝛼₋ for the opposite phases. Note that Δ₁=−Δ₂ 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 as 𝐸^𝐶_𝑁,𝑏(𝜆)=𝐸^𝐺_𝑁,𝑏(0)−𝐸^𝐺_𝑁,𝑏(𝜆)−𝑛𝜆𝑑, where 𝐸^𝐺_𝑁,𝑏 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: 𝐸^𝐶_{𝑁1,𝑏1;𝑁2,𝑏2}(𝜆₁,𝜆₂,𝜆₁₂)=𝐸^𝐺_{𝑁1,𝑏1;𝑁2,𝑏2}(0,0,0)−𝐸^𝐺_{𝑁1,𝑏1;𝑁2,𝑏2}(𝜆₁,𝜆₂,𝜆₁₂)−∑_𝜈𝑛_𝜈𝜆_𝜈𝑑_𝜈, where 𝐸^𝐺_{𝑁1,𝑏1;𝑁2,𝑏2}(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: 𝐸^𝐶_{𝑁1,𝑏1;𝑁2,𝑏2}𝜆₁,𝜆₂,𝜆₁₂=𝐸^𝐶_𝑁1,𝑏1𝜆₁+𝐸^𝐶_𝑁2,𝑏2𝜆₂−𝜆²₁₂𝜆₁𝜆₂−𝜆²₁₂Δ²₁𝑑₁𝜆₁+Δ²₂𝑑₂𝜆₂+2Δ^∗₁Δ₂+Δ₁Δ^∗₂𝑑₁𝑑₂𝜆₁₂, (10) where 𝐸^𝐶_𝑁1,𝑏1(𝜆₁) and 𝐸^𝐶_𝑁2,𝑏2(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 Δ^∗₁Δ₂+Δ₁Δ^∗₂<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 1=𝜆_1𝑗12𝜉̃_1𝑗+𝜆_2𝑘12𝜉̃_2𝑘−𝜆₁𝜆₂−𝜆²_12𝑗12𝜉̃_1𝑗𝑘12𝜉̃_2𝑘, (11) where 𝜉̃_𝑖=𝜉_𝑖/𝑑_𝑖 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 𝑑^𝑜_1𝑐=𝜔_𝐷𝑒^𝛾−1exp𝜆,𝑑^𝑜_2𝑐=𝑑₂𝑑₁𝑑^𝑜_1𝑐, (12) and, for the even case, 𝑑^𝑒_1𝑐=4𝜔_𝐷𝑒^𝛾−1exp𝜆,𝑑^𝑒_2𝑐=𝑑₂𝑑₁𝑑^𝑒_1𝑐. (13) Here, we use 1𝜆=1𝜆2𝑥¹+𝜆₂−𝑎𝑥+𝜆₁−𝜆₂−𝑎𝑥²+4𝜆²₁₂ (14) with 𝑥=𝜆₁𝜆₂−𝜆²₁₂, 𝑎=log(𝑑₁/𝑑₂). From these expressions, we find the relations 𝑑^𝑒_1𝑐=4𝑑^𝑜_1𝑐,𝑑^𝑒_2𝑐=4𝑑^𝑜_2𝑐,𝑑^𝑜_1/2𝑐≈𝑒^𝛾21exp𝜂_1/2−1𝜆Δ_1/2. (15)(16) In the case of |𝜆₁−𝜆₂|≫𝜆₁₂, (16) can be approximately rewritten as 𝑑^𝑜_1/2𝑐≈𝑒^𝛾2𝜆exp²−𝜆₁+2𝛼𝜆₁₂𝜆₁𝜆₂−𝜆²₁₂Δ_1/2. (17) On the other hand, in the limit of |𝜆₁−𝜆₂|=𝜆₁₂, we have 𝑑^𝑜_1/2𝑐≈𝑒^𝛾2()𝜆exp1+𝛼₁₂𝜆₁𝜆₂−𝜆²₁₂Δ_1/2. (18) For the case of 𝜆₁₂=0, (16) can be rewritten as 𝑑^𝑜_1/2𝑐≈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: 𝑑^𝑜_1/2𝑐≈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 Figure 2(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 𝜀_1𝑛1+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 Δ¹_𝑝=𝐸^𝐺_{2𝑛1+1+2𝑛2,1}−12𝐸^𝐺_{2𝑛1+2𝑛2,0}+𝐸^𝐺_{2𝑛1+1+2𝑛2,0}. (19) From (7) and the ground state energy 𝐸^𝐺_𝑁,𝑏=Ω_𝜇𝑁+𝜇_𝑁𝑁, we obtain Δ¹_𝑝=Δ₁−𝑑₁4𝜌₁𝜌₂−1. (20) In view of Figures 2(c), 2(d), and 2(e), we can obtain another formula for a parity gap: Δ²_𝑝=𝐸^𝐺_{2𝑛1+1+2𝑛2+1,1}−12𝐸^𝐺_{2𝑛1+1+2𝑛2,0}+𝐸^𝐺_{2𝑛1𝑛+1+22+1,0}, (21) which yields Δ²_𝑝=Δ₂−𝑑₂43𝜌₂𝜌₁−1. (22) 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.

(a)

(b)

(c)

(d)

(e)

Figure 2 

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 with 2𝑛1+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.

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 constant 𝜆₁₂. 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.

Acknowledgments

The authors thank Professors Jun Akimitsu and Karl Bennemann for their continued encouragement and helpful discussions. S. P. Kruchinin thanks the Research Center for Structural Thermodynamics, Osaka University, for hosting the author as a guest professor to accomplish this work.