Bardeen-Cooper-Schrieffer Theory of Finite-Size Superconducting Metallic Grains

Abstract

We study finite-size effects in superconducting metallic grains and determine the BCS order parameter and the low energy excitation spectrum in terms of the size and shape of the grain. Our approach combines the BCS self-consistency condition, a semiclassical expansion for the spectral density and interaction matrix elements, and corrections to the BCS mean field. In chaotic grains mesoscopic fluctuations of the matrix elements lead to a smooth dependence of the order parameter on the excitation energy. In the integrable case we observe shell effects when, e.g., a small change in the electron number leads to large changes in the energy gap. DOI: 10.1103/PhysRevLett.100.187001 PACS numbers: 74.20.Fg, 05.45.Mt, 74.78.Na Since experiments by Ralph, Black, and Tinkham [1] on Al nanograins in the mid-1990s, there has been considerable interest in the theory of ultrasmall superconductors (see [2,3] for earlier studies). In particular, finite-size corrections to the predictions of the Bardeen-CooperSchrieffer (BCS) theory for bulk superconductors [4] have been studied [5][6][7][8][9][10] within the exactly solvable Richardson model [11]. Pairing in specific potentials, such as a harmonic oscillator potential [12] and a rectangular box, [13,14] and mesoscopic fluctuations of the energy gap [15,16] have been explored as well. Nevertheless, a comprehensive theoretical description of the combined effect of the discrete energy spectrum and fluctuating interaction matrix elements has not yet emerged.In the present Letter we develop a framework based on the BCS theory and semiclassical techniques that permit a systematic analytical evaluation of the low energy spectral properties of superconducting nanograins in terms of their size and shape. Leading finite-size corrections to the BCS mean field can also be taken into account in our approach. Our main results are as follows. For chaotic grains, we show that the order parameter is energy dependent. The energy dependence is universal; i.e., its functional form is the same for all chaotic grains. The matrix elements are responsible for most of the deviation from the bulk limit. In integrable grains, we find that the superconducting gap is strongly sensitive to shell effects, namely, a small modification of the grain size or number of electrons can substantially affect its value.We start with the BCS Hamiltonian H P n n c y n c n ÿ P n;n 0 I n;n 0 c y n" c y n# c n 0 # c n 0 " , where c n (c y n ) annihilates (creates) an electron of spin in state n,are matrix elements of a short-range electron-electron interaction, is the BCS coupling constant, and n and n are eigenstates and eigenvalues of the one-body mean field Hamiltonian of a free particle of mass m in a clean grain of volume V. Eigenvalues n are measured from the Fermi level F and the mean level spacingp is the spectral density at the Fermi level in the Thomas-Fermi approximation. Our general strategy can be summarized as follows: (i) use semiclassical techniques to compute the spectral density P n ÿ n and I ; 0 as ser...

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