Emergence of $h/e$-period oscillations in the critical temperature of small superconducting rings threaded by magnetic flux

As a function of the magnetic flux threading the object, the Little-Parks oscillation in the critical temperature of a large-radius, thin-walled superconducting ring or hollow cylinder has a period given by $h/2e$, due to the binding of electrons into Cooper pairs. On the other hand, the single-electron Aharonov-Bohm oscillation in the resistance or persistent current for a clean (i.e., ballistic) normal-state system, having the same topological structure, has a period given by $h/e$. A basic question is whether the Little-Parks oscillation changes its character, as the radius of the superconducting structure becomes smaller, and if it is even comparable to the zero-temperature coherence length. We supplement a physical argument that the $h/e$ oscillations should also be exhibited with a microscopic analysis of this regime, formulated in terms of the Gor’kov approach to BCS theory. We see that, as the radius of the ring is made smaller, an oscillation in the critical temperature of period $h/e$ emerges in addition to the usual Little-Parks $h/2e$-period oscillation. We argue that, in the clean limit, there is a superconductor-normal transition at nonzero flux as the ring radius becomes sufficiently small and that the transition can be either continuous or discontinuous, depending on the radius and the external flux. In the dirty limit, we argue that the transition is rendered continuous, which results in continuous quantum phase transitions tuned by flux and radius.

Figure 1

(Color online) A ring with a magnetic flux threading through it. The radius of the ring is $R$ and the thickness in the cross section is $d$.

Figure 2

Pairing of states (Ref. 3). (a) Upper panel: the external flux is $\Phi =0$, the pairing is between $n_1+n_2=0$. (b) Lower panel: $\Phi =h/2e$, the pairing is between $n_1+n_2=-1$. The pairing configuration changes from type (a) to type (b) at external flux $\Phi =h/4e$ (see Ref. 3).

Figure 3

(Color online) Phase diagram: reduced critical temperature $t\left(\varphi ,R\right)=T_c\left(\varphi ,R\right)/T_c\left(0,R\right)$ vs flux and radius $v=\pi \left|\varphi -m/2\right|\left({\xi }_0/R\right)$ in the clean limit, and in the limit that the finiteness corrections are ignored (i.e., bulk limit). In the case of the exchange-field effect, discussed in Refs. 19, 20, the horizontal axis becomes $v={\mu }_{B}H/{\Delta }_0$, where ${\mu }_B$ is Bohr magneton and $H$ is the exchange field. The curve ABC is the solution to Eq. (14) while ABD is the curve representing the true equilibrium transition line. From A to B, the transition is continuous, whereas from B to D the transition is discontinuous, construction first made in Refs. 19, 20 in the context of the exchange-field effect. The curve BE represents the metastability limit for “superheating,” whereas the curve BC represents the metastability limit for supercooling (Ref. 20).

Figure 4

Critical temperature (normalized to its zero-flux value) $t$ vs flux $\varphi =\Phi /\left(h/e\right)$ for $R=5{\xi }_0$ (upper) and $R=1.5{\xi }_0$ (lower) in the clean limit. The plots are made with $\text{cos}\left(2\pi \overline{n}\right)=1$ and with only the $k=1$ term in Eq. (19) retained.

Figure 5

(Color online) Reduced critical temperature $t\left(\varphi \right)\equiv T_c\left(\varphi \right)/T_c\left(0\right)$ (normalized to its zero-flux value) vs flux $\varphi \equiv \Phi /\left(h/e\right)$ for $R=1.25{\xi }_0$ with $\text{cos}\left(2k\pi \overline{n}\right)=1$ in the clean limit. The higher $T_c$ branches (blue) are calculated by retaining only the first two terms in Eq. (19), whereas the lower $T_c$ branches (red) are calculated by retaining as many as ten such terms. The plot shows multiple solutions to Eq. (19) and signifies the emergence of discontinuous transitions. The upper and lower branches of the solutions are expected to merge at certain values of $\varphi$; the appearance of a gap between them is an artifact of our considering only a finite number of values of $\varphi$.

Figure 6

(Color online) Schematic depiction of the reduced critical temperature $t\left(\varphi \right)\equiv T_c\left(\varphi \right)/T_c\left(0\right)$ (normalized to its zero-flux value) vs flux $\varphi =\Phi /\left(h/e\right)$ for small-radius rings in the clean limit. As illustrated in Fig. 5 for sufficiently small radii, there are multiple solutions for $T_c$, as exemplified there by the higher (blue) branches and the lower (red) branches. Near $\varphi =0$ and $\pm 1/2$, the upper branches are the equilibrium phase boundary and the transition to normal state is continuous. Away from these regions, the globally stable equilibrium phases must be sought by free-energy consideration, and the corresponding phase boundaries are indicated schematically by the solid lines and represent discontinuous transitions. Upper panel: For all flux values, there exists a superconducting state. Lower panel: For smaller radius, it can happen that there are flux values for which no superconducting state exists.

Figure 7

(Color online) Critical temperature (normalized to its zero-flux value) for the disordered regime vs flux $\varphi =\Phi /\left(h/e\right)$ for $R=0.2{\xi }_0$ and the mean-free path $l_e=0.2{\xi }_0$ (solid blue line), and for the same radius but $l_e⪡R$ (dashed red line). The plots are made with $\text{cos}\left(2\pi k\overline{n}\right)=1$ and ${\left(-1\right)}^k$ for the upper and lower panels, respectively. Moreover, only the $k=0$, 1, and $2$ terms in Eq. (33) have been retained as the convergence is rather good.

Figure 8

(Color online) Phase boundaries between normal state (to the left of the curves) and the superconducting state in the temperature [$t\left(\varphi ,R\right)=T_c\left(\varphi ,R\right)/T_c\left(\varphi =0,R\right)$, normalized to its zero-flux value] vs radius ($R$, in unit of ${\xi }_0$) plane. Upper panel: $\varphi =\Phi /\left(h/e\right)=0,0.1,0.2,0.25$ (from the top down). The solid blue boundaries take into account the finite-radius corrections, whereas the dashed red boundaries are solutions to the bulk equation [Eq. (35)]. Lower panel: $\varphi =\Phi /\left(h/e\right)=0.5,0.40,0.30,0.25$ (from the top down). In all plots, we have chosen $l_e=0.2{\xi }_0$ and $\text{cos}\left(2\pi k\overline{n}\right)=1$, and have kept terms up to $k=2$ in Eq. (33). Note the significant deviations associated with the finite-radius corrections near $\varphi =0.5$.