Parity Effect in a mesoscopic superconducting ring

– We study a mesoscopic superconducting ring threaded by a magnetic flux when the single particle level spacing is not negligible. It is shown that, for a superconducting ring with even parity, the behavior of persistent current is equivalent to what is expected in a bulk superconducting ring. On the other hand, we find that a ring with odd parity shows anomalous behavior such as fluxoid quantization at half-integral multiples of the flux quantum and paramagnetic response at low temperature. We also discuss how the parity effect in the persistent current disappears as the temperature is raised or as the size of the ring increases. What happens to superconductivity when the sample is made very small? Anderson [1] already addressed this fundamental question in 1959 and argued that as the size of a superconductor decreases and, accordingly, the average level spacing δ becomes larger than the BCS gap ∆, superconductivity is no longer possible. Recent experiments on ultrasmall “superconducting” nanoparticles [2] have led to reconsider this old, but fundamental question. In a series of experiments the authors of [2] studied transport through nanometer-scale Al grains and succeeded to get the discrete eigenspectrum of a single superconducting grain. The results were found to depend on the parity, i.e. on the electron number in the grain being even or odd. These experiments initiated several theoretical investigations. von Delft et al. [3] used a model of uniform level spacing in a parity-projected mean field theory [4] and found that the breakdown of superconductivity occurs at a value of δ/∆ ∼ O(1) which is parity-dependent. This parity effect has been shown to increase when including the effects of level statistics [5]. Effects of quantum fluctuations [6], canonical description of BCS superconductivity [7], as well as transport theory for a nanoparticle coupled to superconducting leads [8] have also been subjects of the study along this line. Another interesting example for studying the size effect on superconductivity is a mesoscopic superconducting ring threaded by a magnetic flux Φ. It is well known that a conventional BCS superconducting ring exhibits fluxoid quantization at integer multiples of the flux quantum (∗) Electronic address: [email protected] Typeset using EURO-TEX 2 EUROPHYSICS LETTERS Φ0 = hc/2e and a diamagnetic response at Φ = nΦ0 with n being an integer [9]. In this Letter, we address the following question which is essentially the same as in a simply connected grain: What happens in a superconducting ring when the size of the ring becomes very small? For this purpose, we adopt the parity-projected mean field theory [4] for an ideal mesoscopic ring. It is shown that the order parameter strongly depends on the parity when the size of the ring is small enough, as in the case of a grain. The most dramatic feature we show in our study is the behavior of the supercurrent (or persistent current) which strongly depends on the parity. For a ring with even parity, the behavior of the supercurrent is identical to that of a bulk superconducting ring. It exhibits fluxoid quantization at integer multiples of the flux quantum, and a diamagnetic response for small deviations of the flux from the integer multiples of the flux quantum. On the other hand, the characteristics are found to be very different for a small, odd parity ring. In a superconductor with odd parity, there is one unpaired quasiparticle. We find that at low temperature the existence of this quasiparticle drives the superconducting ring to a half-integral fluxoid quantization and a paramagnetic response at small values of the flux. We further show that the anomalous behavior in an odd-parity superconductor disappears at temperatures higher than the level spacing of single electron spectra, where one recovers the behavior of a conventional superconducting ring. Finally, this parity-dependent behavior of supercurrent is shown to disappear in the thermodynamic limit. The ideal superconducting ring can be described by the Hamiltonian: H = ∑ jσ εjc † jσcjσ − λδ ∑ i,j c†i↑c † ī↓ cj̄↓cj↑. (1) The single particle energy is given by εj = h̄ 2mR (j − f/2) , where R is the radius of the ring, f = Φ/Φ0 is the external flux divided by the flux quantum Φ0 = hc/2e, and j is an integer which corresponds to an angular momentum quantum number. This is obtained by solving the Schrödinger equation in a 1D noninteracting ring. Note that j̄ = −j. λ is the dimensionless BCS coupling constant. δ = h̄N/8mR2 is the level spacing at the Fermi energy, where N is the number of electrons in the ring. We don’t take into account the Zeeman splitting, namely h, because it is negligible unless the radius of the ring is very small. For Φ ∼ Φ0 with a uniform magnetic field, the ratio of the level spacing δ to the Zeeman splitting is proportional to R which is estimated as δ/h ∼ 10rsR/m , where rs is the average distance between electrons. For example, in a typical superconductor such as Al with R ∼ 1μm, δ/h ∼ 10. A simple way of describing a mesoscopic superconductor with fixed number parity P (denoted by e for even, and o for odd parity) is to adopt the parity-projected grand canonical partition function [4] ZP (μ) = Tr 1 2 [1± (−1) ]e. (2) We evaluate ZP using the BCS-type mean field approximations, which consists in neglecting quadratic terms of the fluctuations : c†i↑c † ī↓ cj̄↓cj↑ ≃ 〈c†i↑c † ī↓ 〉cj̄↓cj↑ + c † i↑c † ī↓ 〈cj̄↓cj↑〉 − 〈c † i↑c † ī↓ 〉〈cj̄↓cj↑〉 (3) + δij ( c†i↑ci↑〈c † ī↓ cī↓〉+ 〈c † i↑ci↑〉c † ī↓ cī↓ − 〈c † i↑ci↑〉〈c † ī↓ cī↓〉 ) . The ensemble average 〈· · ·〉 should be evaluated in a given parity P = e or P = o. The first three terms on the r.h.s of Eq.(3) correspond to the mean-field approximation for the superconducting pairing. The last three terms are usually not considered since they give K. Kang PARITY EFFECT IN A MESOSCOPIC SUPERCONDUCTING RING 3 no contribution in the thermodynamic limit. However, those terms cannot be ignored in mesoscopic systems, though they were neglected in the previous mean-field description for ultrasmall superconducting grains [3]. Note that the validity of our mean-field treatment is limited to the δ < ∆ limit where the usual BCS approximation can be applied. As a result we get the following expression for the mean-field Hamiltonian H = CP + ∑ jσ Ẽjγ † jσγjσ + μN , (4) where γjσ (γ † jσ) destroys (creates) a quasiparticle; γjσ = ujcjσ − σvjc † j̄σ , and the constant CP and the quasiparticle energy Ẽj are given by CP = ∑