**1. Introduction**

342 Superconductors – Materials, Properties and Applications

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One of the most important properties of superconductors is their perfectly diamagnetic response to an external magnetic field, the Meissner-Ochsenfeld effect. It is a pure quantum effect and therefore reveals the existence of a macroscopic quantum state with a pair condensate. A special manifestation of the diamagnetic response is observed for superconducting rings threaded by a magnetic flux: flux quantization and a periodic current response.

Persistent currents and periodic flux dependence are also known in normal metal rings and best known in form of the Aharonov-Bohm effect predicted theoretically in 1959 [4]. Since the wavefunction of an electron moving on a ring must be single valued, the phase of the wave function acquired upon moving once around the ring is a integer multiple of 2*π*. A magnetic flux threading the ring generates an additional phase difference 2*πϕ* = (*e*/*c*) *<sup>C</sup>* d**r** · **A**(**r**) = (2*πe*/*hc*) Φ, where *C* is a closed path around the ring and **A**(**r**) the vector potential generating the magnetic flux Φ threading the ring. Here, *e* is the electron charge, *c* the velocity of light, and *h* is Planck's constant. Thus, the electron wave function is identical whenever *ϕ* has an integer value and therefore the system is periodic in the magnetic flux Φ with a periodicity of

$$
\Phi\_0 = \frac{hc}{e},
\tag{1}
$$

the flux quantum in a normal metal ring. In particular, the persistent current *J*(*ϕ*) induced by the magnetic flux is zero whenever *ϕ* = Φ/Φ<sup>0</sup> is an integer.

The periodic response of a superconducting ring to a magnetic flux is of similar origin as in a normal metal ring, though the phase winding of the condensate wavefunction has to be reconsidered. On account of the macroscopic phase coherence of the condensate, flux oscillations must be more stable in superconductors, and London predicted their existence in superconducting loops already ten years before the work of Aharonov and

without magnetic flux, as schematically shown in figure 1 (a). The Cooper pairs in this state have a center-of-mass angular momentum (pair momentum) *q* = 0. The pairing wavefunctions of the superconducting state for all flux values Φ, which are integer multiples of Φ<sup>0</sup> and correspond to even pair momenta *q* = 2Φ/Φ0, are related to the wavefunction for Φ = 0 by a gauge transformation. For a flux value Φ0/2, pairing occurs between the electron states with angular momenta *k* and (−*k* + 1), which have equal energies in this case [figure 1 (b)]. This leads to pairs with momentum *q* = . The corresponding pairing wavefunction is again related by a gauge transformation to those for flux values Φ which are

Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in Superconducting Loops 345

half-integer multiples of Φ<sup>0</sup> and correspond to the odd pair momenta *q* = 2Φ/Φ0.

effects of many channels and finite temperatures in section 2.3.

unconventional superconductors were studied.

For the system to be Φ0/2 periodic, it is required that the free energies of the two types of pairing states are equal. Byers and Yang, Onsager as well as Brenig showed that this is in fact the case in the thermodynamic limit. The free energy consists then of a series of parabolae with minima at integer multiples of Φ<sup>0</sup> (corresponding to even pair momenta) and half integer multiples of Φ<sup>0</sup> (corresponding to odd pair momenta). If the arm of the ring is wider than the penetration depth *λ*, the flux is quantized and the groundstate is given by the minimum closest to the value of the external flux. However, in microscopic finite systems this degeneracy of the even and odd *q* minima is lifted, although their position is fixed by gauge invariance to multiples of Φ0/2. The restoration of the Φ0/2 periodicity in the limit of large rings was studied only much later [26, 31, 45, 52]. We study the revival of the Φ0/2 periodicity in sections 2.1 and 2.2 for a one-dimensional ring at zero temperature and investigate the

From the flux periodicity of the free energy, the same flux periodicity can be derived for all other thermodynamic quantities [44]. A clear and unambiguous observation of flux oscillations is possible in the flux dependence of the critical temperature *T*c of small superconducting cylinders. Such experiments have been performed first by Little and Parks in 1962 [29, 30, 39]. They measured the resistance *R* of the cylinder at a fixed temperature *T* within the finite width of the superconducting transition and deduced the oscillation period of *T*<sup>c</sup> from the variation of *R*. These experiments confirmed the Φ0/2 periodicity in conventional superconductors very accurately. At this stage the question of the flux periodicity in superconductors seemed to be settled and understood. The interest then shifted to the amplitude of the supercurrent and also the normal persistent current and their dependence on the ring size, the temperature, and disorder [11, 15, 21, 27, 47]. However, the influence of finite system sizes on the flux periodicity remained unaddressed. Earlier, certain experiments had already indicated some unexpected complications. E.g., Little and Parks pointed out in reference [29] that in tantalum cylinders they could not detect any flux oscillations in *R* at all. Even more peculiar were the oscillations observed in an indium cylinder where signs of an additional Φ0/8 periodicity were clearly visible [29]. This was surprising because indium is a perfectly conventional superconductor otherwise. These results remained unexplained and drew attention only years later, when flux oscillations of

In the meantime a new type of flux sensitive systems was advanced: superconducting quantum-interference devices (SQUIDs). The measurement of flux oscillations in SQUIDs is similar to the Little-Parks experiment. Here the flux dependence of the critical current *J*c through a superconducting loop including one or two Josephson junctions is measured. This

**Figure 1.** Scheme of the pairing of angular-momentum eigenstates in a one dimensional metal loop for (a) Φ = 0 and (b) Φ = Φ0/2, as used by Schrieffer in [43] to illustrate the origin of the Φ0/2 periodicity in superconductors. Paired are always electrons with equal energies, leading to center-of-mass angular momenta *q* = 0 in (a) and *q* = 1 in (b).

Bohm [35]. London expected that the magnetic flux threading a loop is quantized in multiples of Φ<sup>0</sup> because the interior of an ideal superconductor was known to be current free. Although London the pairing theory of superconductivity was not known yet, he anticipated the existence of electron pairs carrying the supercurrent and speculated that the flux quantum in a superconductor might be Φ0/2. This point of view became generally accepted with the publication of the 'Theory of Superconductivity' by Bardeen, Cooper, and Schrieffer (BCS) in 1957 [8]. Direct measurements of magnetic flux quanta Φ0/2 trapped in superconducting rings followed in 1961 by Doll and Näbauer [18] and by Deaver and Fairbank [17], corroborated later by the detection of flux lines of Φ0/2 in the mixed state of type II superconductors [1, 22].

It is tempting to explain the Φ0/2 flux periodicity of superconducting loops simply by the charge 2*e* of Cooper pairs carrying the supercurrent, but pairing of electrons alone is not sufficient for the Φ0/2 periodicity. The Cooper-pair wavefunction extends over the whole loop, as does the single-electron wavefunction, and it is not obvious whether the electrons forming the Cooper pair are tightly bound or circulate around the ring separately. A microscopic model on the basis of the BCS theory is therefore indispensable for the description of the flux periodicity of a superconducting ring. In this chapter, we analyze this problem in detail and focus on a previously neglected aspect: how do the Φ<sup>0</sup> periodic flux oscillations in a normal metal ring transform into the Φ0/2 periodic oscillations in a superconducting ring?

A theoretical description of the origin of the half-integer flux quanta was first found independently in 1961 by Byers and Yang [13], by Onsager [38], and by Brenig [10] on the basis of BCS theory. They realized that there are two distinct classes of superconducting wavefunctions that are not related by a gauge transformation. An intuitive picture illustrating these two types can be found in Schrieffer's book on superconductivity [43], using the energy spectrum of a one-dimensional metal ring. The first class of superconducting wavefunctions, which London had in mind in his considerations about flux quantization, is related to pairing of electrons with angular momenta *k* and −*k*, which have equal energies in a metal loop without magnetic flux, as schematically shown in figure 1 (a). The Cooper pairs in this state have a center-of-mass angular momentum (pair momentum) *q* = 0. The pairing wavefunctions of the superconducting state for all flux values Φ, which are integer multiples of Φ<sup>0</sup> and correspond to even pair momenta *q* = 2Φ/Φ0, are related to the wavefunction for Φ = 0 by a gauge transformation. For a flux value Φ0/2, pairing occurs between the electron states with angular momenta *k* and (−*k* + 1), which have equal energies in this case [figure 1 (b)]. This leads to pairs with momentum *q* = . The corresponding pairing wavefunction is again related by a gauge transformation to those for flux values Φ which are half-integer multiples of Φ<sup>0</sup> and correspond to the odd pair momenta *q* = 2Φ/Φ0.

2 Superconductors

k

**Figure 1.** Scheme of the pairing of angular-momentum eigenstates in a one dimensional metal loop for (a) Φ = 0 and (b) Φ = Φ0/2, as used by Schrieffer in [43] to illustrate the origin of the Φ0/2 periodicity in superconductors. Paired are always electrons with equal energies, leading to center-of-mass angular

Bohm [35]. London expected that the magnetic flux threading a loop is quantized in multiples of Φ<sup>0</sup> because the interior of an ideal superconductor was known to be current free. Although London the pairing theory of superconductivity was not known yet, he anticipated the existence of electron pairs carrying the supercurrent and speculated that the flux quantum in a superconductor might be Φ0/2. This point of view became generally accepted with the publication of the 'Theory of Superconductivity' by Bardeen, Cooper, and Schrieffer (BCS) in 1957 [8]. Direct measurements of magnetic flux quanta Φ0/2 trapped in superconducting rings followed in 1961 by Doll and Näbauer [18] and by Deaver and Fairbank [17], corroborated later by the detection of flux lines of Φ0/2 in the mixed state

It is tempting to explain the Φ0/2 flux periodicity of superconducting loops simply by the charge 2*e* of Cooper pairs carrying the supercurrent, but pairing of electrons alone is not sufficient for the Φ0/2 periodicity. The Cooper-pair wavefunction extends over the whole loop, as does the single-electron wavefunction, and it is not obvious whether the electrons forming the Cooper pair are tightly bound or circulate around the ring separately. A microscopic model on the basis of the BCS theory is therefore indispensable for the description of the flux periodicity of a superconducting ring. In this chapter, we analyze this problem in detail and focus on a previously neglected aspect: how do the Φ<sup>0</sup> periodic flux oscillations in a normal metal ring transform into the Φ0/2 periodic oscillations in a superconducting ring? A theoretical description of the origin of the half-integer flux quanta was first found independently in 1961 by Byers and Yang [13], by Onsager [38], and by Brenig [10] on the basis of BCS theory. They realized that there are two distinct classes of superconducting wavefunctions that are not related by a gauge transformation. An intuitive picture illustrating these two types can be found in Schrieffer's book on superconductivity [43], using the energy spectrum of a one-dimensional metal ring. The first class of superconducting wavefunctions, which London had in mind in his considerations about flux quantization, is related to pairing of electrons with angular momenta *k* and −*k*, which have equal energies in a metal loop


(b)

k

k


(a)

momenta *q* = 0 in (a) and *q* = 1 in (b).

of type II superconductors [1, 22].

k

For the system to be Φ0/2 periodic, it is required that the free energies of the two types of pairing states are equal. Byers and Yang, Onsager as well as Brenig showed that this is in fact the case in the thermodynamic limit. The free energy consists then of a series of parabolae with minima at integer multiples of Φ<sup>0</sup> (corresponding to even pair momenta) and half integer multiples of Φ<sup>0</sup> (corresponding to odd pair momenta). If the arm of the ring is wider than the penetration depth *λ*, the flux is quantized and the groundstate is given by the minimum closest to the value of the external flux. However, in microscopic finite systems this degeneracy of the even and odd *q* minima is lifted, although their position is fixed by gauge invariance to multiples of Φ0/2. The restoration of the Φ0/2 periodicity in the limit of large rings was studied only much later [26, 31, 45, 52]. We study the revival of the Φ0/2 periodicity in sections 2.1 and 2.2 for a one-dimensional ring at zero temperature and investigate the effects of many channels and finite temperatures in section 2.3.

From the flux periodicity of the free energy, the same flux periodicity can be derived for all other thermodynamic quantities [44]. A clear and unambiguous observation of flux oscillations is possible in the flux dependence of the critical temperature *T*c of small superconducting cylinders. Such experiments have been performed first by Little and Parks in 1962 [29, 30, 39]. They measured the resistance *R* of the cylinder at a fixed temperature *T* within the finite width of the superconducting transition and deduced the oscillation period of *T*<sup>c</sup> from the variation of *R*. These experiments confirmed the Φ0/2 periodicity in conventional superconductors very accurately. At this stage the question of the flux periodicity in superconductors seemed to be settled and understood. The interest then shifted to the amplitude of the supercurrent and also the normal persistent current and their dependence on the ring size, the temperature, and disorder [11, 15, 21, 27, 47]. However, the influence of finite system sizes on the flux periodicity remained unaddressed. Earlier, certain experiments had already indicated some unexpected complications. E.g., Little and Parks pointed out in reference [29] that in tantalum cylinders they could not detect any flux oscillations in *R* at all. Even more peculiar were the oscillations observed in an indium cylinder where signs of an additional Φ0/8 periodicity were clearly visible [29]. This was surprising because indium is a perfectly conventional superconductor otherwise. These results remained unexplained and drew attention only years later, when flux oscillations of unconventional superconductors were studied.

In the meantime a new type of flux sensitive systems was advanced: superconducting quantum-interference devices (SQUIDs). The measurement of flux oscillations in SQUIDs is similar to the Little-Parks experiment. Here the flux dependence of the critical current *J*c through a superconducting loop including one or two Josephson junctions is measured. This

**Φ**

**Figure 3.** The simplest description of the many-particle state in a flux threaded loop, we use a

for future research on unconventional superconductors.

**2. The periodicity crossover**

temperature *T*c.

**2.1. Normal state**

tight-binding model on a discrete, one-dimensional ring with *N* lattice sites, lattice constant *a* and radius *R* = *Na*/2*π*. The magnetic flux Φ is confined to the interior of the ring and does not touch the ring itself.

The second kind of unconventional oscillations is more intriguing. In several different YBCO SQUIDs, the periodicity of sinusoidal oscillations changes abruptly with increasing magnetic flux. In the measurement shown in figure 2 (b), the period is Φ0/4 for small flux, and changes to Φ0/2 at a critical flux. As a possible explanation for the appearance of Φ0/4 periodicity, an unusually pronounced second harmonic in the critical current *J*c of transparent Josephson junctions was proposed or, more fundamentally, an effect of interactions between Cooper pairs, leading to the formation of electron quartets [42]. The observation of similar abrupt changes to other fractional periodicities like Φ0/6 and Φ0/8 render this finding even more striking since it could indicate a transition into a new, non-BCS type of superconductivity. This concept, which we sketch briefly in the Conclusions, is a complex and promising topic

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

The tight-binding Hamiltonian for an electronic system including a magnetic field is straightforwardly formulated using the annihilation and creation operators *cis* and *c*†

*is* for an

R

a

Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in Superconducting Loops 347

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

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

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 reference [32].

**Figure 3.** The simplest description of the many-particle state in a flux threaded loop, we use a tight-binding model on a discrete, one-dimensional ring with *N* lattice sites, lattice constant *a* and radius *R* = *Na*/2*π*. The magnetic flux Φ is confined to the interior of the ring and does not touch the ring itself.

The second kind of unconventional oscillations is more intriguing. In several different YBCO SQUIDs, the periodicity of sinusoidal oscillations changes abruptly with increasing magnetic flux. In the measurement shown in figure 2 (b), the period is Φ0/4 for small flux, and changes to Φ0/2 at a critical flux. As a possible explanation for the appearance of Φ0/4 periodicity, an unusually pronounced second harmonic in the critical current *J*c of transparent Josephson junctions was proposed or, more fundamentally, an effect of interactions between Cooper pairs, leading to the formation of electron quartets [42]. The observation of similar abrupt changes to other fractional periodicities like Φ0/6 and Φ0/8 render this finding even more striking since it could indicate a transition into a new, non-BCS type of superconductivity. This concept, which we sketch briefly in the Conclusions, is a complex and promising topic for future research on unconventional superconductors.
