Generic size dependences of pairing in ultrasmall systems: electronic nano-devices and atomic nuclei

Abstract

We study the average and global pairing behaviours of electronic devices, like films, wires, and grains using semiclassical methods, such as Weyl and Thomas–Fermi approximations, in the ultrasmall, i.e., quantal regime, which exhibits strong quantum fluctuations and shell effects. We discuss how these results, mostly analytic, are elaborated for average size dependencies, in order to be used in other circumstances when the fully quantal calculation become computationally too expensive. We also compare the results with latest experimental one where possible.

Appendix A: Quantal study of a free particle in a hemi-sphere

The Schrödinger equation for a free particle confined in a half-sphere of radius R read

$$\begin{aligned} -\frac{\hbar ^2}{2m}\nabla ^2\psi _{nlm}(r,\theta ,\phi ) = E\psi _{nlm}(r,\theta ,\phi ). \end{aligned}$$

(A1)

The problem has very specific boundary conditions i.e. $\psi _{nlm}(r,\theta ,\phi )$ is equal to zero at the edges of the half sphere. Before entering in a more detailed discussion, it is interesting to specify the system of reference. r represent the radial distance from the centre of the sphere and it can take all values in the range $r\in [0,R]$. $\theta$ is the angle formed by r with the axis of symmetry of the half sphere it goes from $\theta \in [0,\pi /2]$. The value $\theta =0$ indicates a point on the top of the half-sphere, the value $\theta =\pi /2$ is placed on the disk closing the bottom of the half-sphere. $\phi$ is the angle formed by the projection of r on xy-plane and it spans from 0 to $2\pi$.

The boundary conditions can be then translated as

$$\begin{aligned} \left\{ \begin{array}{cc} \psi _{nlm}(R,\theta ,\phi )=0 &{}\theta \ne \frac{\pi }{2},\\ \psi _{nlm}(r,\frac{\pi }{2},\phi )=0 &{}\theta = \frac{\pi }{2}, \end{array}\right. \end{aligned}$$

these two conditions take into account the fact that the wave-function is zero on the hemisphere and the disk delimiting the Hilbert space. To solve such a problem, we use the usual ansatz of

$$\begin{aligned} \psi _{nlm}(r,\theta ,\phi )\propto u_{nl}(r)Y_{lm}(\theta ,\phi ), \end{aligned}$$

(A2)

here nlm are the quantum number of the system. $Y_{lm}(\theta ,\phi )$ is the spherical harmonic. Notice we neglect spin terms since there is no spin-orbit coupling and the terms simply factorise. We replace Eq. (A2) in Eq. (A1) and we get

$$\begin{aligned} -\frac{\hbar ^2}{2m}\left[ \frac{1}{r^2}\frac{\partial }{\partial r}\left( r^2\frac{\partial }{\partial r}\right) - \frac{l(l+1)}{r^2}\right] u_{nl}(r) =E_{nl}u_{nl}(r).\nonumber \\ \end{aligned}$$

(A3)

By defining $k_{nl}^2=\frac{2m}{\hbar ^2} E_{nl}$ and performing simple manipulations, we obtain the Bessel equation whose solution are the spherical Bessel functions $u_{nl}(r)=j_{l}\left( k_{nl}r \right)$.

By imposing the first set of boundary conditions, i.e. $u_{nl}(R)=0$, we obtain the discretisation of the eigen-spectrum. This condition is angle independent. In the case of a closed half-sphere, we also need to impose that the wave function is zero on the bottom disk closing the space. This condition should be valid for any value of r and fixed $\theta =\pi /2$.

As such we have to impose this on the angular part of the wave function

$$\begin{aligned} Y_{lm}\left( \frac{\pi }{2},\phi \right) =0. \end{aligned}$$

(A4)

This is achieved simply by selecting the associated Legendre polynomial that are odd $P_l^m(\cos \theta )$, or in other words they have a node on the $\theta =\pi /2$ plane. This conditions is respected for all polynomials so that $l+m$ is an odd number ($l=0$ excluded). As a consequence, we can solve the half-sphere problem using the same methodology used to solve the sphere, apart from the extra selection rule on the quantum number used to build the basis.