**2. The periodicity crossover**

4 Superconductors

60

30

Jc (μA)

0


(b)

hc/2e

μ0H (μT)

hc/4e

<sup>1</sup> 2 3 10-8

Γ [hc/2e]

visible is the abrupt change of periodicity at *μ*0*H* ≈ ±5 μT [42].

temperature, and the magnetic-field range of the SQUID.

(a)

4 5 6

**Figure 2.** (a) Fourier transform *J*c(Γ) of the critical current *J*c(*H*) measured by Schneider *et al*. on a 24◦ grain boundary SQUID at *T* = 77 K as a function of the applied magnetic field where Φ0/2 = 6.7 μT [41].

has the advantage that flux oscillations can be observed at any temperature *T* < *T*c, and they are most clearly visible in the critical current *J*c. SQUIDs fabricated from conventional superconductors have been used in experiments and applications for five decades, and they proved to oscillate perfectly with the expected flux period Φ0/2. It was therefore a surprise that flux oscillations with different periodicities were found in 2003 by Lindström *et al*. [28] and Schneider *et al*. [41, 42] in SQUIDs fabricated from films of the high-*T*c superconductor YBa2Cu3O*<sup>y</sup>* (YBCO) where the Josephson junctions arise from grain boundaries. Flux trapping experiments in loops showed that flux quantization in the cuprate class of high-*T*c superconductors occurs in units of Φ0/2 [23], identically to what has been observed with conventional superconductors. In addition, Schneider *et al*. observed a variety of oscillation periods, depending on the geometry of the SQUID loop, the grain-boundary angle, the

Two distinct patterns of unconventional oscillations in YBCO SQUIDs have to be discerned. The first kind consists of oscillations which have a basic period of Φ0/2, overlaid by other periodicities, such that the Fourier transform *J*c(Γ) of *J*c(Φ) contains peaks appear which do not correspond to the period Φ0/2 [41]. An example for such a measurement are shown in figure 2 (a). The peaks at integer values of Γ correspond to higher harmonics of Φ0/2, and their appearance is natural. However, there are clear peaks at Γ = 1/2 (red arrow) and Γ = 5/2, which correspond to Φ<sup>0</sup> periodicity and higher harmonics thereof. The origin of the Φ<sup>0</sup> periodicity in those experiments is so far not conclusively explained. There was, however, extensive research on the flux periodicity of unconventional (mostly *d*-wave) superconductors, which revealed that the periodicity of the normal state persists in the superconducting state if the energy gap symmetry allows for nodal states [6, 25, 32, 34, 51]. This effect derives directly from the analysis in this book chapter and is discussed in detail in

The red arrow points out the Fourier peak corresponding to a Φ<sup>0</sup> periodic current contribution. (b) Critical current *J*c(*H*) over a 24◦ grain boundary SQUID at *T* = 4.2 K, where Φ0/2 = 2.7 μT. Clearly

Jc (Γ)

10-2

10-4

10-6

reference [32].

In this section we introduce the periodicity crossover and consider first the simplest model containing the relevant physics: a one dimensional ring consisting of *N* lattice sites and a lattice constant *a* (figure 3). The ring is threaded by a magnetic flux Φ focused through the center and not touching the ring itself. We use a tight-binding description with nearest-neighbor hopping parameter *t*, which sets the energy scale of the system. We start from the flux periodicity of the normal metal state of the ring, which varies for different numbers of electrons in the ring. On this basis we introduce a superconducting pairing interaction and investigate the flux periodicity of the groundstate upon increasing the interaction strength. For a ring with a finite width (an annulus) we investigate the flux dependence of the self-consistently calculated superconducting order parameter and study the temperature driven periodicity crossover when cooling the ring through the transition temperature *T*c.
