**2.1. Normal state**

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

#### 6 Superconductors 348 Superconductors – Materials, Properties and Applications Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in Superconducting Loops <sup>7</sup>

electron with spin *s* on the lattice site *i*:

$$\mathcal{H}\_0 = -t \sum\_{\langle i,j \rangle,s} e^{i\rho\_{ij}} c\_{is}^\dagger c\_{js} - \mu \sum\_{i,s} c\_{is}^\dagger c\_{is}. \tag{2}$$

(φ) 0

(a)

(b)

(c)

maximum value of the Doppler shift.

D

D

D

D

(φ) 0

(φ) 0

0 1/2 1

for both integer and half-integer values of *ϕ*; *E*(*ϕ*) is therefore Φ0/2 periodic.

**2.2. Superconducting state: Emergence of a new periodicity**


φ

**Figure 4.** Energy spectrum of a discrete one-dimensional ring with *N* lattice sites, *μ* = 0 and (a) *N*/4 is an integer, (b) *N*/4 a half integer, and (c) *N* an odd number. In (a) and (b), levels cross *E*<sup>F</sup> = 0 at integer (a) or half-integer (b) values of *ϕ*. For odd *N*, two different spectra are possible [*N* = 4*n* + 1 (left) and *N* = 4*n* − 1 (right)], and for both, two levels cross *E*<sup>F</sup> within one flux period (red points). *�*<sup>D</sup> denotes the

The energy *E*(*ϕ*) is maximal for those values of *ϕ* where an energy level reaches *E*<sup>F</sup> (red points) and has minima in between. If *N*/4 is an integer, then the minima of *E*(*ϕ*) are at half-integer values of *ϕ* (figure 5 (a), light blue curve), whereas if *N*/4 is a half-integer, the minima are at integer values of *ϕ* (figure 5 (a), dark blue curve). If *N* is odd, two different (but physically equivalent) spectra for *N* = 4*n* ± 1 are possible, and for both, two levels cross *E*<sup>F</sup> in one flux period. This results in a superposition of the two previous cases and there are minima of *E*(*ϕ*)

The normal persistent current *J*(*ϕ*) = −(*e*/*hc*) *∂E*(*ϕ*)/*∂ϕ* (see equation 12 below) jumps whenever an energy level crosses *E*F, because the population of left and right circulating states changes abruptly [figure 5 (b)]. The occupied state closest to *E*<sup>F</sup> contributes dominantly to the current, because all other contributions tend to almost cancel in pairs. The Doppler shift decreases with the ring radius like 1/*R* [c.f. equation (4)] and so does the persistent current.

The theory of flux threaded superconducting loops was first derived by Byers and Yang [13], Brenig [10], and Onsager [38] on the basis of the BCS theory. They showed the thermodynamic equivalence of the two superconducting states discussed above in the thermodynamic limit. However, in a strict thermodynamic limit the persistent (super-) current vanishes, and therefore a more precise statement is necessary with respect to the Φ0/2 periodicity of the supercurrent. Here we analyze the crossover from Φ<sup>0</sup> periodicity in the normal metal loop

0 1/2 1

Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in Superconducting Loops 349

Here �*i*, *j*� denotes all nearest-neighbor pairs *i* and *j*, *s* =↑, ↓. The magnetic field **B** = ∇ × **A** enters into the Hamiltonian (2) through the Peierls phase factor *ϕij* = (*e*/*hc*) *j <sup>i</sup> d***l** · **A**. The chemical potential *μ* controls the number of electrons in the ring. The flux periodicity is easiest to discuss for a particle-hole symmetric situation with *μ* = 0, for which the Fermi energy is *E*<sup>F</sup> = 0. We will later address the changes introduced through an arbitrary *μ*.

We assume that the *N* lattice sites are equally spaced along a ring with circumference 2*πR* = *Na* (figure 3). It follows that the Peierls phase factor for a magnetic field focused through the center of the ring simplifies to *ϕij* = 2*πϕ*/*N*, where *ϕ* = Φ/Φ<sup>0</sup> is the dimensionless magnetic flux. The Hamiltonian (2) is then written in momentum space as:

$$\mathcal{H}\_0 = \sum\_{k,s} \varepsilon\_k(\varphi) c\_{ks}^\dagger c\_{ks} \tag{3}$$

where *c*† *ks* creates an electron with angular momentum *k*. The energy dispersion is

$$\epsilon\_k(\varphi) = -2t \cos \left( \frac{k - \varphi}{R/a} \right) - \mu. \tag{4}$$

The eigenenergies depend on the flux only in the combination *k* − *ϕ*, as is shown in figure 4 for three different cases: (a) *N*/4 is an integer, (b) *N*/4 is a half integer, and (c) *N* is an odd number. The *ϕ* dependent shift in *�k*(*ϕ*) is known as the Doppler shift since it is proportional to the velocity of the corresponding electron. In all three cases, the spectrum has obviously the periodicity 1 with respect to *ϕ*. However, the flux values, for which an energy level crosses *E*F, are different. This number dependence, sometimes referred to as the "parity effect" , is characteristic for discrete systems and not restricted to one dimension. It was discussed in detail in the context of the persistent current in metallic loops [15, 27, 47] and also in metallic nano clusters [36]; it is also essential for the discussion of superconducting rings.

Physical quantities of the normal metal ring can be expressed through the thermal average *ns*(*k*) of the number of electrons with angular momentum *<sup>k</sup>* and spin *<sup>s</sup>*: *ns*(*k*) = �*c*† *kscks*� = *f*(*�k*(*ϕ*)), with the Fermi distribution function *f*(*�*) = 1/(1 + *e�*/*k*B*T*) for the temperature *T*. The groundstate is given by the minimum of the total energy *E* of the system

$$E(\varphi) = \langle \mathcal{H}\_0 \rangle = \sum\_{k,s} \epsilon\_k(\varphi) n\_s(k) \,\prime \tag{5}$$

which is a piecewise quadratic function of the magnetic flux. The momentum distribution function *ns*(*k*) also depends on the magnetic flux only in the combination *k* − *ϕ*. The sum over *k* in equation (5) directly renders the Φ<sup>0</sup> flux periodicity of *E*(*ϕ*). However, the position of the minima of *E*(*ϕ*) depends on the highest occupied energy level and therefore also shows a parity effect (see figure 5).

6 Superconductors

Here �*i*, *j*� denotes all nearest-neighbor pairs *i* and *j*, *s* =↑, ↓. The magnetic field **B** = ∇ × **A**

chemical potential *μ* controls the number of electrons in the ring. The flux periodicity is easiest to discuss for a particle-hole symmetric situation with *μ* = 0, for which the Fermi energy is

We assume that the *N* lattice sites are equally spaced along a ring with circumference 2*πR* = *Na* (figure 3). It follows that the Peierls phase factor for a magnetic field focused through the center of the ring simplifies to *ϕij* = 2*πϕ*/*N*, where *ϕ* = Φ/Φ<sup>0</sup> is the dimensionless magnetic

*ks* creates an electron with angular momentum *k*. The energy dispersion is

The eigenenergies depend on the flux only in the combination *k* − *ϕ*, as is shown in figure 4 for three different cases: (a) *N*/4 is an integer, (b) *N*/4 is a half integer, and (c) *N* is an odd number. The *ϕ* dependent shift in *�k*(*ϕ*) is known as the Doppler shift since it is proportional to the velocity of the corresponding electron. In all three cases, the spectrum has obviously the periodicity 1 with respect to *ϕ*. However, the flux values, for which an energy level crosses *E*F, are different. This number dependence, sometimes referred to as the "parity effect" , is characteristic for discrete systems and not restricted to one dimension. It was discussed in detail in the context of the persistent current in metallic loops [15, 27, 47] and also in metallic

Physical quantities of the normal metal ring can be expressed through the thermal average

*f*(*�k*(*ϕ*)), with the Fermi distribution function *f*(*�*) = 1/(1 + *e�*/*k*B*T*) for the temperature *T*.

*k*,*s*

which is a piecewise quadratic function of the magnetic flux. The momentum distribution function *ns*(*k*) also depends on the magnetic flux only in the combination *k* − *ϕ*. The sum over *k* in equation (5) directly renders the Φ<sup>0</sup> flux periodicity of *E*(*ϕ*). However, the position of the minima of *E*(*ϕ*) depends on the highest occupied energy level and therefore also shows

*�k*(*ϕ*)*c*†

 *<sup>k</sup>* <sup>−</sup> *<sup>ϕ</sup> R*/*a*

*iscjs* − *<sup>μ</sup>*∑

*i*,*s c*†

*iscis*. (2)

*kscks* (3)

− *μ*. (4)

*�k*(*ϕ*)*ns*(*k*), (5)

*j*

*<sup>i</sup> d***l** · **A**. The

*kscks*� =

H<sup>0</sup> = −*<sup>t</sup>* ∑

�*i*,*j*�,*s e <sup>i</sup>ϕij c*†

enters into the Hamiltonian (2) through the Peierls phase factor *ϕij* = (*e*/*hc*)

*E*<sup>F</sup> = 0. We will later address the changes introduced through an arbitrary *μ*.

H<sup>0</sup> = ∑ *k*,*s*

*�k*(*ϕ*) = −2*t* cos

nano clusters [36]; it is also essential for the discussion of superconducting rings.

The groundstate is given by the minimum of the total energy *E* of the system

*<sup>E</sup>*(*ϕ*) = �H0� = ∑

*ns*(*k*) of the number of electrons with angular momentum *<sup>k</sup>* and spin *<sup>s</sup>*: *ns*(*k*) = �*c*†

flux. The Hamiltonian (2) is then written in momentum space as:

electron with spin *s* on the lattice site *i*:

where *c*†

a parity effect (see figure 5).

**Figure 4.** Energy spectrum of a discrete one-dimensional ring with *N* lattice sites, *μ* = 0 and (a) *N*/4 is an integer, (b) *N*/4 a half integer, and (c) *N* an odd number. In (a) and (b), levels cross *E*<sup>F</sup> = 0 at integer (a) or half-integer (b) values of *ϕ*. For odd *N*, two different spectra are possible [*N* = 4*n* + 1 (left) and *N* = 4*n* − 1 (right)], and for both, two levels cross *E*<sup>F</sup> within one flux period (red points). *�*<sup>D</sup> denotes the maximum value of the Doppler shift.

The energy *E*(*ϕ*) is maximal for those values of *ϕ* where an energy level reaches *E*<sup>F</sup> (red points) and has minima in between. If *N*/4 is an integer, then the minima of *E*(*ϕ*) are at half-integer values of *ϕ* (figure 5 (a), light blue curve), whereas if *N*/4 is a half-integer, the minima are at integer values of *ϕ* (figure 5 (a), dark blue curve). If *N* is odd, two different (but physically equivalent) spectra for *N* = 4*n* ± 1 are possible, and for both, two levels cross *E*<sup>F</sup> in one flux period. This results in a superposition of the two previous cases and there are minima of *E*(*ϕ*) for both integer and half-integer values of *ϕ*; *E*(*ϕ*) is therefore Φ0/2 periodic.

The normal persistent current *J*(*ϕ*) = −(*e*/*hc*) *∂E*(*ϕ*)/*∂ϕ* (see equation 12 below) jumps whenever an energy level crosses *E*F, because the population of left and right circulating states changes abruptly [figure 5 (b)]. The occupied state closest to *E*<sup>F</sup> contributes dominantly to the current, because all other contributions tend to almost cancel in pairs. The Doppler shift decreases with the ring radius like 1/*R* [c.f. equation (4)] and so does the persistent current.

### **2.2. Superconducting state: Emergence of a new periodicity**

The theory of flux threaded superconducting loops was first derived by Byers and Yang [13], Brenig [10], and Onsager [38] on the basis of the BCS theory. They showed the thermodynamic equivalence of the two superconducting states discussed above in the thermodynamic limit. However, in a strict thermodynamic limit the persistent (super-) current vanishes, and therefore a more precise statement is necessary with respect to the Φ0/2 periodicity of the supercurrent. Here we analyze the crossover from Φ<sup>0</sup> periodicity in the normal metal loop

#### 8 Superconductors 350 Superconductors – Materials, Properties and Applications Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in Superconducting Loops <sup>9</sup>

assumption of condensation into a state with one selected angular momentum *q* for all pairs,

*<sup>q</sup>*(*ϕ*)*c*−*k*+*q*↓*ck*<sup>↑</sup> <sup>+</sup> <sup>Δ</sup>*q*(*ϕ*)*c*†

where the order parameter is defined as <sup>Δ</sup>*q*(*ϕ*)=(*V*/2) <sup>∑</sup>*k*�*ck*↑*ck*↓�. The mean-field

*�k*(*ϕ*) − *�*−*k*+*q*(*ϕ*)

2 ±

where the Doppler shift term arises from the different energies of the two paired states with momenta *k* and −*k* + *q*. The order parameter Δ*q*(*ϕ*) is determined self-consistently from

*f*(*E*−(*k*, *ϕ*)) − *f*(*E*+(*k*, *ϕ*))

For the discussion of the periodicity of this system, we first disregard the self-consistency condition for the order parameter and set Δ*q*(*ϕ*) ≡ Δ to be constant. Importantly, while H<sup>0</sup> is strictly Φ<sup>0</sup> periodic, H is *not* periodic in *ϕ* if Δ > 0. The question of periodicity is therefore: *which* periodicity is restored by minimizing *E*(*ϕ*) = �H� with respect to *q* and how is this

Figure 6 shows *E*(*ϕ*) for two different values of Δ for *q* = 0 and *q* = 1. For small Δ, *E*(*ϕ*) is still a series of parabolae with minima at integer values of *ϕ*, but the degeneracy of the minima is lifted [figure 6 (a) and (b)]. For even *q*, the energy minimum at *ϕ* = *q*/2 is lowered relative to the other minima, whereas for odd *q*, one new minimum emerges at *ϕ* = *q*/2, which is absent in the normal state. If Δ exceeds a certain threshold Δc, this new odd *q* minimum becomes deeper than the neighboring ones [figure 6 (d)]. We have thus identified the second class of states with minima in *E*(*ϕ*) at half-integer flux values anticipated above and we find that the even and odd *q* minima become equal if Δ becomes large compared to Δc, a ring size dependent value which we will determine below. It is to be understood that the energies *E*(*ϕ*) in figure 6 are not periodic in *ϕ* because the *q*-values are fixed, either to *q* = 0 in (a, c) or to *q* = 1 in (b, d). In loops thicker than the penetration depth, screening currents drive the system always into an energy minimum. In this case, the flux is then quantized in units of

Δ*q*(*ϕ*)<sup>2</sup> + *�*2(*k*, *ϕ*)

*<sup>k</sup>*−, *<sup>c</sup>*−*k*+*q*<sup>↓</sup> <sup>=</sup> *<sup>u</sup>*(*k*)*a*†

and *<sup>v</sup>*2(*k*) =

Δ2

 Δ2

<sup>=</sup> <sup>1</sup>

Hamiltonian (7) is diagonalized with the standard Bogoliubov transformation

*k*↑*c*† −*k*+*q*↓ + Δ2 *<sup>q</sup>*(*ϕ*)

Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in Superconducting Loops 351

*<sup>V</sup>* , (7)

*<sup>k</sup>*<sup>−</sup> <sup>−</sup> *<sup>v</sup>*(*k*)*ak*<sup>+</sup> (8)

, (9)

<sup>1</sup> <sup>−</sup> *�*(*k*, *<sup>ϕ</sup>*) *E*(*k*, *ϕ*)

*<sup>q</sup>* + *�*2(*k*, *ϕ*) and *�*(*k*, *ϕ*)=[*�k*(*ϕ*) +

*<sup>q</sup>* + *�*2(*k*, *ϕ*), (10)

*<sup>V</sup>* . (11)

which allows us to write H in the decoupled form

*k* Δ∗

*�*(*k*, *ϕ*) *E*(*k*, *ϕ*)

which depend on *<sup>ϕ</sup>* and *<sup>q</sup>* through *<sup>E</sup>*(*k*, *<sup>ϕ</sup>*) =

*E*±(*k*, *ϕ*) =

1 *<sup>N</sup>* ∑ *k*

2 

*�*−*k*+*q*(*ϕ*)]/2. The energy spectrum splits into the two branches

H = H<sup>0</sup> + ∑

*ck*<sup>↑</sup> <sup>=</sup> *<sup>u</sup>*(*k*)*ak*<sup>+</sup> <sup>+</sup> *<sup>v</sup>*(*k*)*a*†

1 +

with the coherence factors

achieved?

Φ0/2.

*<sup>u</sup>*2(*k*) =

**Figure 5.** (a) Total energy *E*(*ϕ*) of a ring with *N* sites as a function of the magnetic flux *ϕ*. If *N*/4 is an integer, then the minima are at half-integer values of *ϕ* (dark blue). If *N*/4 is a half-integer, the parabolae are shifted by 1/2 (light blue). The gray lines above the crossing points of the parabolae correspond to possible excited states. (b) Persistent current *J*(*ϕ*) corresponding to the systems described in (a). The purple curve shows the current obtained for odd *N*.

to the Φ0/2 periodicity in the superconducting loop upon turning on the pairing interaction. The discussion of this crossover enables precise statements about the periodicity.

For a one-dimensional superconducting loop (or any loop thinner than the penetration depth *λ*), finite currents flow throughout the superconductor. The magnetic flux is consequently not quantized, only the fluxoid Φ� = Φ + (Λ/*c*) <sup>d</sup>**<sup>r</sup>** · **<sup>J</sup>**(**r**) is, which was introduced by F. London [35]. The flux Φ is the total flux threading the loop, including the current induced flux, and Λ = 4*πλ*2/*c*2. In the absence of flux quantization, *ϕ* is a continuous variable also in a superconducting system with a characteristic periodicity in *ϕ*.

In this section we focus on the emergence of a new periodicity when a superconducting order parameter arises. We therefore include an attractive on-site interaction of the general form <sup>1</sup>

$$\mathcal{H} = \mathcal{H}\_0 - \frac{V}{2N^2} \sum\_{k,k'} \sum\_q c\_{k\uparrow}^\dagger c\_{-k+q\downarrow}^\dagger c\_{-k'+q\downarrow} c\_{k'\uparrow} \tag{6}$$

where *V* > 0 is the interaction strength. In BCS theory it is *assumed* that electron pairs have zero center-of-mass (angular) momentum, i.e., the pairs are condensed in a macroscopic quantum state with *q* = 0, similar to a Bose-Einstein condensate of bosonic particles. In the case of a flux threaded ring, Byers and Yang [13], Brenig [10], and Onsager [38] showed that *q* is generally finite and has to be chosen to minimize the kinetic energy of the Cooper pairs in the presence of a magnetic flux. Nevertheless it is still assumed that pairing occurs only for one specific angular momentum *q*. For conventional superconductors, this assumption is generally true, although for superconductors with gap nodes, the situation may be different, as was shown for *d*-wave pairing symmetry in reference [33]. In this section, we use the

<sup>1</sup> In the literature the symmetric Hamiltonian <sup>H</sup>˜ <sup>=</sup> <sup>H</sup><sup>0</sup> + (*V*/2*N*<sup>2</sup>) <sup>∑</sup>*k*,*k*� <sup>∑</sup>*<sup>q</sup> <sup>c</sup>*† *<sup>k</sup>*+*q*/2↑*c*† <sup>−</sup>*k*+*q*/2↓*c*−*k*�+*q*/2↓*ck*�+*q*/2<sup>↑</sup> is often used [37]. <sup>H</sup>˜ is naturally *hc*/*<sup>e</sup>* periodic in *<sup>ϕ</sup>*, but it is not well defined, although it yields the same physical quantities as <sup>H</sup>. The introduction of half-integer angular momenta in <sup>H</sup>˜ leads to two different limits <sup>Δ</sup> <sup>→</sup> 0 for even or odd *<sup>q</sup>*, corresponding to the two spectra for *<sup>N</sup>*/2 even or odd. Therefore the symmetric <sup>H</sup>˜ is unsuitable for the discussion of the flux periodicity.

assumption of condensation into a state with one selected angular momentum *q* for all pairs, which allows us to write H in the decoupled form

$$\mathcal{H} = \mathcal{H}\_0 + \sum\_k \left[ \Delta\_q^\*(\boldsymbol{\varphi}) \boldsymbol{c}\_{-k+q\downarrow} \boldsymbol{c}\_{k\uparrow} + \Delta\_q(\boldsymbol{\varphi}) \boldsymbol{c}\_{k\uparrow}^\dagger \boldsymbol{c}\_{-k+q\downarrow}^\dagger \right] + \frac{\Delta\_q^2(\boldsymbol{\varphi})}{V},\tag{7}$$

where the order parameter is defined as <sup>Δ</sup>*q*(*ϕ*)=(*V*/2) <sup>∑</sup>*k*�*ck*↑*ck*↓�. The mean-field Hamiltonian (7) is diagonalized with the standard Bogoliubov transformation

$$\boldsymbol{c}\_{k\uparrow} = \boldsymbol{\mu}(\boldsymbol{k})\boldsymbol{a}\_{k+} + \boldsymbol{\upsilon}(\boldsymbol{k})\boldsymbol{a}\_{k-\prime}^{\dagger} \qquad \qquad \qquad \boldsymbol{c}\_{-k+q\downarrow} = \boldsymbol{\mu}(\boldsymbol{k})\boldsymbol{a}\_{k-}^{\dagger} - \boldsymbol{\upsilon}(\boldsymbol{k})\boldsymbol{a}\_{k+} \tag{8}$$

with the coherence factors

8 Superconductors

**Figure 5.** (a) Total energy *E*(*ϕ*) of a ring with *N* sites as a function of the magnetic flux *ϕ*. If *N*/4 is an integer, then the minima are at half-integer values of *ϕ* (dark blue). If *N*/4 is a half-integer, the parabolae are shifted by 1/2 (light blue). The gray lines above the crossing points of the parabolae correspond to possible excited states. (b) Persistent current *J*(*ϕ*) corresponding to the systems described in (a). The

to the Φ0/2 periodicity in the superconducting loop upon turning on the pairing interaction.

For a one-dimensional superconducting loop (or any loop thinner than the penetration depth *λ*), finite currents flow throughout the superconductor. The magnetic flux is consequently

London [35]. The flux Φ is the total flux threading the loop, including the current induced flux, and Λ = 4*πλ*2/*c*2. In the absence of flux quantization, *ϕ* is a continuous variable also in

In this section we focus on the emergence of a new periodicity when a superconducting order parameter arises. We therefore include an attractive on-site interaction of the general form <sup>1</sup>

where *V* > 0 is the interaction strength. In BCS theory it is *assumed* that electron pairs have zero center-of-mass (angular) momentum, i.e., the pairs are condensed in a macroscopic quantum state with *q* = 0, similar to a Bose-Einstein condensate of bosonic particles. In the case of a flux threaded ring, Byers and Yang [13], Brenig [10], and Onsager [38] showed that *q* is generally finite and has to be chosen to minimize the kinetic energy of the Cooper pairs in the presence of a magnetic flux. Nevertheless it is still assumed that pairing occurs only for one specific angular momentum *q*. For conventional superconductors, this assumption is generally true, although for superconductors with gap nodes, the situation may be different, as was shown for *d*-wave pairing symmetry in reference [33]. In this section, we use the

used [37]. <sup>H</sup>˜ is naturally *hc*/*<sup>e</sup>* periodic in *<sup>ϕ</sup>*, but it is not well defined, although it yields the same physical quantities as <sup>H</sup>. The introduction of half-integer angular momenta in <sup>H</sup>˜ leads to two different limits <sup>Δ</sup> <sup>→</sup> 0 for even or odd *<sup>q</sup>*, corresponding to the two spectra for *<sup>N</sup>*/2 even or odd. Therefore the symmetric <sup>H</sup>˜ is unsuitable for the discussion of

The discussion of this crossover enables precise statements about the periodicity.

<sup>2</sup>*N*<sup>2</sup> ∑ *k*,*k*� ∑ *q c*† *k*↑*c*†

J(φ)

0

0.5




(b)

<sup>d</sup>**<sup>r</sup>** · **<sup>J</sup>**(**r**) is, which was introduced by F.

<sup>−</sup>*k*+*q*↓*c*−*k*�+*q*↓*ck*�↑, (6)

*<sup>k</sup>*+*q*/2↑*c*†

<sup>−</sup>*k*+*q*/2↓*c*−*k*�+*q*/2↓*ck*�+*q*/2<sup>↑</sup> is often

1

E(φ)

2

0


(a)

purple curve shows the current obtained for odd *N*.

not quantized, only the fluxoid Φ� = Φ + (Λ/*c*)

a superconducting system with a characteristic periodicity in *ϕ*.

<sup>H</sup> <sup>=</sup> <sup>H</sup><sup>0</sup> <sup>−</sup> *<sup>V</sup>*

<sup>1</sup> In the literature the symmetric Hamiltonian <sup>H</sup>˜ <sup>=</sup> <sup>H</sup><sup>0</sup> + (*V*/2*N*<sup>2</sup>) <sup>∑</sup>*k*,*k*� <sup>∑</sup>*<sup>q</sup> <sup>c</sup>*†

the flux periodicity.

1

3

$$u^2(k) = \left(1 + \frac{\epsilon(k,\varphi)}{E(k,\varphi)}\right) \qquad\qquad\text{and}\qquad\qquad v^2(k) = \left(1 - \frac{\epsilon(k,\varphi)}{E(k,\varphi)}\right),\tag{9}$$

which depend on *<sup>ϕ</sup>* and *<sup>q</sup>* through *<sup>E</sup>*(*k*, *<sup>ϕ</sup>*) = Δ2 *<sup>q</sup>* + *�*2(*k*, *ϕ*) and *�*(*k*, *ϕ*)=[*�k*(*ϕ*) + *�*−*k*+*q*(*ϕ*)]/2. The energy spectrum splits into the two branches

$$E\_{\pm}(k,\varphi) = \frac{\varepsilon\_k(\varphi) - \varepsilon\_{-k+\eta}(\varphi)}{2} \pm \sqrt{\Delta\_{\eta}^2 + \varepsilon^2(k,\varphi)},\tag{10}$$

where the Doppler shift term arises from the different energies of the two paired states with momenta *k* and −*k* + *q*. The order parameter Δ*q*(*ϕ*) is determined self-consistently from

$$\frac{1}{N} \sum\_{k} \frac{f(E\_{-}(k,\rho)) - f(E\_{+}(k,\rho))}{2\sqrt{\Delta\_{\eta}(\varphi)^{2} + \varepsilon^{2}(k,\rho)}} = \frac{1}{V}.\tag{11}$$

For the discussion of the periodicity of this system, we first disregard the self-consistency condition for the order parameter and set Δ*q*(*ϕ*) ≡ Δ to be constant. Importantly, while H<sup>0</sup> is strictly Φ<sup>0</sup> periodic, H is *not* periodic in *ϕ* if Δ > 0. The question of periodicity is therefore: *which* periodicity is restored by minimizing *E*(*ϕ*) = �H� with respect to *q* and how is this achieved?

Figure 6 shows *E*(*ϕ*) for two different values of Δ for *q* = 0 and *q* = 1. For small Δ, *E*(*ϕ*) is still a series of parabolae with minima at integer values of *ϕ*, but the degeneracy of the minima is lifted [figure 6 (a) and (b)]. For even *q*, the energy minimum at *ϕ* = *q*/2 is lowered relative to the other minima, whereas for odd *q*, one new minimum emerges at *ϕ* = *q*/2, which is absent in the normal state. If Δ exceeds a certain threshold Δc, this new odd *q* minimum becomes deeper than the neighboring ones [figure 6 (d)]. We have thus identified the second class of states with minima in *E*(*ϕ*) at half-integer flux values anticipated above and we find that the even and odd *q* minima become equal if Δ becomes large compared to Δc, a ring size dependent value which we will determine below. It is to be understood that the energies *E*(*ϕ*) in figure 6 are not periodic in *ϕ* because the *q*-values are fixed, either to *q* = 0 in (a, c) or to *q* = 1 in (b, d). In loops thicker than the penetration depth, screening currents drive the system always into an energy minimum. In this case, the flux is then quantized in units of Φ0/2.

E±(φ)

0



parts [12, 14].


areas in (a), the energy gap has closed due to the Doppler shift.

E±(φ)

**Figure 7.** Eigenenergies *E*±(*k*, *ϕ*) (10) as a function of flux *ϕ* for *N* = 50 and a fixed order parameter: (a) "large gap" regime with Δ = 0.2 *t*, (b) "small gap" regime with Δ = 0.05 *t*. Blue lines: occupied states, grey lines: unoccupied states. The bold line marks the highest occupied state for all *ϕ*. In the blue shaded

both in the normal and in the superconducting state. The Φ<sup>0</sup> periodic part in the normal state is contained exclusively in *J*+(*ϕ*). A flux window where *E*+(*k*, *ϕ*) is partially occupied exists in each *q* sector when the energy gap has closed [shaded blue areas in figure 7 (a)]. These windows decrease for increasing Δ until *J*+(*ϕ*) vanishes for Δ = Δc. In the "large gap" regime, the supercurrent is carried entirely by *J*−(*ϕ*) and is therefore Φ0/2 periodic and essentially independent of Δ. The discontinuities in *J*−(*ϕ*) are not caused by energy levels crossing *E*F, but by the reconstruction of the condensate when the pair momentum *q* changes to the next integer at the flux values *ϕ* = (2*n* − 1)/4. Figure 9 shows the periodicity crossover of the persistent current in four different steps from the Φ<sup>0</sup> periodic normal current to the Φ0/2 periodic supercurrent. A very similar type of crossover was discussed earlier for loops consisting of a normal and a superconducting part. The flux periodicity of such a system changes from Φ<sup>0</sup> to Φ0/2 with an increasing ratio of superconducting to normal conducting

Further insight into the current periodicity is obtained by analyzing Δc. Close to *E*F, the maximum energy shift is *�*<sup>D</sup> = *at*/2*R*, and the condition for a direct energy gap (or *E*+(*k*, *ϕ*) > 0 for all *k*, *ϕ*) and a Φ0/2-periodic current pattern is therefore Δ > Δ<sup>c</sup> = *�*D. The corresponding critical ring radius is *R*<sup>c</sup> = *at*/2Δ. It is instructive to compare *R*<sup>c</sup> with the BCS coherence length *ξ*<sup>0</sup> = *v*F/*π*Δ, where *v*<sup>F</sup> is the Fermi velocity and Δ the BCS order parameter at *T* = 0. On the lattice we identify *v*<sup>F</sup> = *k*F/*m* with *k*<sup>F</sup> = *π*/2*a* and *m* = 2/2*a*2*t*, and obtain *ξ*<sup>0</sup> = *at*/Δ and thus 2*R*<sup>c</sup> = *ξ*0. This signifies that the current response of a superconducting ring with a diameter smaller than the coherence length, is generally Φ<sup>0</sup> periodic [34]. In these

We have hereby identified the basic mechanism underlying the crossover from Φ<sup>0</sup> periodicity in the normal state to Φ0/2 periodicity in the superconducting state. It is the crossing of *E*<sup>F</sup> of energy levels as a function of the magnetic flux that leads to kinks in the energy and to discontinuities in the supercurrent (or the persistent current in the normal state). If the superconducting gap is large enough to prevent all energy levels from crossing the Fermi energy, the kinks and jumps occur only where the pair momentum *q* of the groundstate

rings the Cooper-pair wavefunction is delocalized around the ring.

0




(b)

0.5

Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in Superconducting Loops 353

(a)

0.5

**Figure 6.** Energy *E*(*ϕ*) in the superconducting state with *q* = 0 (a, c) and *q* = 1 (b, d) for *N* = 50. The upper panels (a, b) show the "small gap" case with Δ = 0.05 *t* and the lower panels (c, d) the "large gap" case with Δ = 0.2 *t*.

Let us for the moment assume that the flux value, at which the energy minimizing *q* changes from one integer to the next, is well approximated by the half way between two minima: *q* = floor(2*ϕ* + 1/2) (floor(*x*) is the largest integer smaller than *x*, e.g., *ϕ* = 0 → *q* = 0; *ϕ* = 1/4 → *q* = 1; *ϕ* = 3/4 → *q* = 2). Small deviations from these values will be discussed for the self-consistent solution in section 2.3. The energy spectrum is then Φ<sup>0</sup> periodic, but discontinuous at the flux values where *q* changes, as shown in figure 7. Clearly distinguishable are now the "small gap" (a) and the "large gap" (b) regime: Δc represents the maximum of the flux-induced shift of the energy levels close to *E*F, before *q* changes. If Δ < Δc, the energy gap closes at certain values of *ϕ*, whereas if Δ > Δc, an energy gap persists for all *ϕ*.

Although the spectra are Φ<sup>0</sup> periodic both in figure 7 (a) and (b), the closing of the gap in the "small gap" regime has significant effects on the periodicity of physical quantities like *E*(*ϕ*). Even more prominent is the periodicity crossover for the persistent current in the ring. The supercurrent is given by *J*(*ϕ*) = *J*+(*ϕ*) + *J*−(*ϕ*)=(*e*/*h*)*∂E*(*ϕ*)/*∂ϕ*, where

$$J\_{\pm}(\boldsymbol{\varrho}) = \frac{e}{\hbar c} \sum\_{k} \frac{\partial \epsilon\_{k}(\boldsymbol{\varrho})}{\partial k} n\_{\pm}(k) \tag{12}$$

with *<sup>n</sup>*+(*k*) = *<sup>u</sup>*2(*k*)*f*(*E*+(*k*, *<sup>ϕ</sup>*)) and *<sup>n</sup>*−(*k*) = *<sup>v</sup>*2(*k*)*f*(*E*−(*k*, *<sup>ϕ</sup>*)). *<sup>J</sup>*+(*ϕ*) and *<sup>J</sup>*−(*ϕ*), as well as *J*(*ϕ*), are plotted in figure 8. The contribution *J*−(*ϕ*) forms a Φ0/2 periodic saw-tooth pattern,

10 Superconductors

E(φ) -

E(φ) -

**Figure 6.** Energy *E*(*ϕ*) in the superconducting state with *q* = 0 (a, c) and *q* = 1 (b, d) for *N* = 50. The upper panels (a, b) show the "small gap" case with Δ = 0.05 *t* and the lower panels (c, d) the "large gap"

Let us for the moment assume that the flux value, at which the energy minimizing *q* changes from one integer to the next, is well approximated by the half way between two minima: *q* = floor(2*ϕ* + 1/2) (floor(*x*) is the largest integer smaller than *x*, e.g., *ϕ* = 0 → *q* = 0; *ϕ* = 1/4 → *q* = 1; *ϕ* = 3/4 → *q* = 2). Small deviations from these values will be discussed for the self-consistent solution in section 2.3. The energy spectrum is then Φ<sup>0</sup> periodic, but discontinuous at the flux values where *q* changes, as shown in figure 7. Clearly distinguishable are now the "small gap" (a) and the "large gap" (b) regime: Δc represents the maximum of the flux-induced shift of the energy levels close to *E*F, before *q* changes. If Δ < Δc, the energy gap

Although the spectra are Φ<sup>0</sup> periodic both in figure 7 (a) and (b), the closing of the gap in the "small gap" regime has significant effects on the periodicity of physical quantities like *E*(*ϕ*). Even more prominent is the periodicity crossover for the persistent current in the ring. The

> *hc* ∑ *k*

with *<sup>n</sup>*+(*k*) = *<sup>u</sup>*2(*k*)*f*(*E*+(*k*, *<sup>ϕ</sup>*)) and *<sup>n</sup>*−(*k*) = *<sup>v</sup>*2(*k*)*f*(*E*−(*k*, *<sup>ϕ</sup>*)). *<sup>J</sup>*+(*ϕ*) and *<sup>J</sup>*−(*ϕ*), as well as *J*(*ϕ*), are plotted in figure 8. The contribution *J*−(*ϕ*) forms a Φ0/2 periodic saw-tooth pattern,

*∂�k*(*ϕ*)

closes at certain values of *ϕ*, whereas if Δ > Δc, an energy gap persists for all *ϕ*.

supercurrent is given by *J*(*ϕ*) = *J*+(*ϕ*) + *J*−(*ϕ*)=(*e*/*h*)*∂E*(*ϕ*)/*∂ϕ*, where

*<sup>J</sup>*±(*ϕ*) = *<sup>e</sup>*

0

0.05

0.10

E(0)0.15

0

0.05

0.10



(d)

*<sup>∂</sup><sup>k</sup> <sup>n</sup>*±(*k*) (12)


(b)


E(0)0.15

E(φ) -

E(φ) -

0

case with Δ = 0.2 *t*.

0.05

0.10

E(0)0.15

0

0.05

0.10



(c)


(a)


E(0)0.15

**Figure 7.** Eigenenergies *E*±(*k*, *ϕ*) (10) as a function of flux *ϕ* for *N* = 50 and a fixed order parameter: (a) "large gap" regime with Δ = 0.2 *t*, (b) "small gap" regime with Δ = 0.05 *t*. Blue lines: occupied states, grey lines: unoccupied states. The bold line marks the highest occupied state for all *ϕ*. In the blue shaded areas in (a), the energy gap has closed due to the Doppler shift.

both in the normal and in the superconducting state. The Φ<sup>0</sup> periodic part in the normal state is contained exclusively in *J*+(*ϕ*). A flux window where *E*+(*k*, *ϕ*) is partially occupied exists in each *q* sector when the energy gap has closed [shaded blue areas in figure 7 (a)]. These windows decrease for increasing Δ until *J*+(*ϕ*) vanishes for Δ = Δc. In the "large gap" regime, the supercurrent is carried entirely by *J*−(*ϕ*) and is therefore Φ0/2 periodic and essentially independent of Δ. The discontinuities in *J*−(*ϕ*) are not caused by energy levels crossing *E*F, but by the reconstruction of the condensate when the pair momentum *q* changes to the next integer at the flux values *ϕ* = (2*n* − 1)/4. Figure 9 shows the periodicity crossover of the persistent current in four different steps from the Φ<sup>0</sup> periodic normal current to the Φ0/2 periodic supercurrent. A very similar type of crossover was discussed earlier for loops consisting of a normal and a superconducting part. The flux periodicity of such a system changes from Φ<sup>0</sup> to Φ0/2 with an increasing ratio of superconducting to normal conducting parts [12, 14].

Further insight into the current periodicity is obtained by analyzing Δc. Close to *E*F, the maximum energy shift is *�*<sup>D</sup> = *at*/2*R*, and the condition for a direct energy gap (or *E*+(*k*, *ϕ*) > 0 for all *k*, *ϕ*) and a Φ0/2-periodic current pattern is therefore Δ > Δ<sup>c</sup> = *�*D. The corresponding critical ring radius is *R*<sup>c</sup> = *at*/2Δ. It is instructive to compare *R*<sup>c</sup> with the BCS coherence length *ξ*<sup>0</sup> = *v*F/*π*Δ, where *v*<sup>F</sup> is the Fermi velocity and Δ the BCS order parameter at *T* = 0. On the lattice we identify *v*<sup>F</sup> = *k*F/*m* with *k*<sup>F</sup> = *π*/2*a* and *m* = 2/2*a*2*t*, and obtain *ξ*<sup>0</sup> = *at*/Δ and thus 2*R*<sup>c</sup> = *ξ*0. This signifies that the current response of a superconducting ring with a diameter smaller than the coherence length, is generally Φ<sup>0</sup> periodic [34]. In these rings the Cooper-pair wavefunction is delocalized around the ring.

We have hereby identified the basic mechanism underlying the crossover from Φ<sup>0</sup> periodicity in the normal state to Φ0/2 periodicity in the superconducting state. It is the crossing of *E*<sup>F</sup> of energy levels as a function of the magnetic flux that leads to kinks in the energy and to discontinuities in the supercurrent (or the persistent current in the normal state). If the superconducting gap is large enough to prevent all energy levels from crossing the Fermi energy, the kinks and jumps occur only where the pair momentum *q* of the groundstate

z

numerically with a discretized radial coordinate.

**2.3. Multi channels and self consistency**

effect. For spin singlet pairing the BdG equations are [16]

2*m i*∇ +

 1 2*m* 

*<sup>E</sup>***n***u***n**(**r**) = <sup>1</sup>

*E***n***v***n**(**r**) = −

θ

fact that it is Φ0/2 periodic illustrates the remark by Parks and Little.

**Figure 10.** Flux threaded annulus with inner radius *R*<sup>1</sup> and outer radius *R*2. For a magnetic flux threading the interior of the annulus, the radial part of the Bogoliubov - de Gennes equations is solved

of electrons leads to a reduction of the fundamental flux period Φ<sup>0</sup> to Φ0/2, the pairing of pairs to quartets would lead to the quarter-period Φ0/4. Then the saw-tooth pattern of the supercurrent becomes Φ0/4 periodic and the maximum current is only half the value for unpaired Cooper pairs. If the oscillation in *E*(*ϕ*) were due to the kinetic energy of the pairs, then the formation of quartets and the Φ0/4 periodicity would be energetically favorable. The

Although the one-dimensional ring discussed above comprises all the qualitative features of the periodicity crossover at *T* = 0 upon entering the superconducting state, some additional issues need to be considered. First, the spectrum of a one dimensional ring is special insofar as only two energy levels exist that cross the Fermi energy in one flux period. This situation is ideal to investigate persistent currents, because there are maximally two jumps in one period. In an extended loop all radial channels contribute at the Fermi energy and have to be taken into account. Second, the self-consistency condition of the superconducting order parameter leads to corrections of the results obtained above, and third, the periodicity crossover upon entering the superconducting state by cooling through the transition temperature *T*c is somewhat different from the *T* = 0 crossover. These points are addressed in this section.

Here we extend the ring to an annulus with an inner radius *R*<sup>1</sup> and an outer radius *R*2, as shown in figure 10. For such an annulus, we choose a continuum approach on the basis of the Bogoliubov - de Gennes (BdG) equations, for which no complications arise from the parity

*u***n**(**r**)+ Δ(**r**) *v***n**(**r**),

*v***n**(**r**)+ Δ∗(**r**)*u***n**(**r**),

(13)

*e c* **A**(**r**) 2 − *μ* 

*<sup>i</sup>*<sup>∇</sup> <sup>−</sup> *<sup>e</sup> c* **A**(**r**) 2 − *μ*  R1

Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in Superconducting Loops 355

**Φ**

R2

**Figure 8.** (a) The Φ<sup>0</sup> periodic persistent current *J*(*ϕ*) (in units of *t*/Φ0) in the normal state. (b) The contribution *J*−(*ϕ*) (dark blue) is Φ0/2 periodic and identical in the normal and the superconducting state. The Φ<sup>0</sup> periodicity in the "small gap" regime is entirely due to *J*−(*ϕ*) (light blue), which vanishes in the "large gap" regime.

**Figure 9.** Crossover from the Φ0-periodic normal persistent current to the Φ0/2-periodic supercurrent in a ring with *N* = 26 at *T* = 0. *J*(*ϕ*) is in units of *t*/Φ0. For this ring size Δ<sup>c</sup> ≈ 0.24 *t*. The discontinuities occur where the *ϕ*-derivative of the highest occupied state energy changes sign. From left to right: Δ = 0, 0.08 *t*, 0.16 *t*, 0.24 *t*.

changes. The latter is true, if the ring diameter is larger than the coherence length *ξ*<sup>0</sup> of the superconductor.

To conclude the discussion of the supercurrent we mention an issue raised by Little and Parks [39]. A simple theoretical model to predict the amplitude of the oscillations of *T*c is the following: For all non-integer or non-half-integer values of *ϕ*, there is a persistent current *J*(*ϕ*) circulating in the cylinder. The kinetic energy *E*kin(*ϕ*) associated with this current is proportional to *J*2(*ϕ*), as is the energy *E*(*ϕ*) in figure 5 (a). It is therefore suggestive to subtract *E*kin(*ϕ*) from the condensation energy of the superconducting state and deduce the oscillations of *T*<sup>c</sup> from those of *E*(*ϕ*). This was done in a first approach by Little and Parks [30], by Tinkham [44], and by Douglass [19] within a Ginzburg-Landau ansatz, yet it was later shown to be incorrect by Parks and Little in a subsequent article [39]. They wrote that "the microscopic theory [i.e., the BCS theory] shows that it is *not the kinetic energy of the pairs* which raises the free energy of the superconducting phase ..., but rather it is due to *the difference in the energy* of the two members of the pairs", i.e., *�k*(*ϕ*) − *�*−*k*+*q*(*ϕ*). It is remarkable that the results of Tinkham and Douglass are nevertheless identical to the microscopic result [40]. The notion whether it is the kinetic energy of the screening current that causes the oscillations, or rather an internal cost in condensation energy in the presence of a finite flux, is important insofar as it provides an explanation for an intriguing problem: In the same way as the pairing

**Figure 10.** Flux threaded annulus with inner radius *R*<sup>1</sup> and outer radius *R*2. For a magnetic flux threading the interior of the annulus, the radial part of the Bogoliubov - de Gennes equations is solved numerically with a discretized radial coordinate.

of electrons leads to a reduction of the fundamental flux period Φ<sup>0</sup> to Φ0/2, the pairing of pairs to quartets would lead to the quarter-period Φ0/4. Then the saw-tooth pattern of the supercurrent becomes Φ0/4 periodic and the maximum current is only half the value for unpaired Cooper pairs. If the oscillation in *E*(*ϕ*) were due to the kinetic energy of the pairs, then the formation of quartets and the Φ0/4 periodicity would be energetically favorable. The fact that it is Φ0/2 periodic illustrates the remark by Parks and Little.

### **2.3. Multi channels and self consistency**

12 Superconductors

**Figure 8.** (a) The Φ<sup>0</sup> periodic persistent current *J*(*ϕ*) (in units of *t*/Φ0) in the normal state. (b) The contribution *J*−(*ϕ*) (dark blue) is Φ0/2 periodic and identical in the normal and the superconducting state. The Φ<sup>0</sup> periodicity in the "small gap" regime is entirely due to *J*−(*ϕ*) (light blue), which vanishes

J±(φ)

0


<sup>φ</sup> -1 -1/2 0 1/2 <sup>1</sup>

**Figure 9.** Crossover from the Φ0-periodic normal persistent current to the Φ0/2-periodic supercurrent in a ring with *N* = 26 at *T* = 0. *J*(*ϕ*) is in units of *t*/Φ0. For this ring size Δ<sup>c</sup> ≈ 0.24 *t*. The discontinuities occur where the *ϕ*-derivative of the highest occupied state energy changes sign. From left to right:

changes. The latter is true, if the ring diameter is larger than the coherence length *ξ*<sup>0</sup> of the

To conclude the discussion of the supercurrent we mention an issue raised by Little and Parks [39]. A simple theoretical model to predict the amplitude of the oscillations of *T*c is the following: For all non-integer or non-half-integer values of *ϕ*, there is a persistent current *J*(*ϕ*) circulating in the cylinder. The kinetic energy *E*kin(*ϕ*) associated with this current is proportional to *J*2(*ϕ*), as is the energy *E*(*ϕ*) in figure 5 (a). It is therefore suggestive to subtract *E*kin(*ϕ*) from the condensation energy of the superconducting state and deduce the oscillations of *T*<sup>c</sup> from those of *E*(*ϕ*). This was done in a first approach by Little and Parks [30], by Tinkham [44], and by Douglass [19] within a Ginzburg-Landau ansatz, yet it was later shown to be incorrect by Parks and Little in a subsequent article [39]. They wrote that "the microscopic theory [i.e., the BCS theory] shows that it is *not the kinetic energy of the pairs* which raises the free energy of the superconducting phase ..., but rather it is due to *the difference in the energy* of the two members of the pairs", i.e., *�k*(*ϕ*) − *�*−*k*+*q*(*ϕ*). It is remarkable that the results of Tinkham and Douglass are nevertheless identical to the microscopic result [40]. The notion whether it is the kinetic energy of the screening current that causes the oscillations, or rather an internal cost in condensation energy in the presence of a finite flux, is important insofar as it provides an explanation for an intriguing problem: In the same way as the pairing


<sup>φ</sup> -1 -1/2 0 1/2 <sup>1</sup>

φ

J\_

(b)

J+

1

J-(φ) + J+(φ)

J(φ)0.2

> -0.2 -0.4

0

0.4

0


in the "large gap" regime.


Δ = 0, 0.08 *t*, 0.16 *t*, 0.24 *t*.

superconductor.


<sup>φ</sup> -1 -1/2 0 1/2 <sup>1</sup>

(a)

1

Although the one-dimensional ring discussed above comprises all the qualitative features of the periodicity crossover at *T* = 0 upon entering the superconducting state, some additional issues need to be considered. First, the spectrum of a one dimensional ring is special insofar as only two energy levels exist that cross the Fermi energy in one flux period. This situation is ideal to investigate persistent currents, because there are maximally two jumps in one period. In an extended loop all radial channels contribute at the Fermi energy and have to be taken into account. Second, the self-consistency condition of the superconducting order parameter leads to corrections of the results obtained above, and third, the periodicity crossover upon entering the superconducting state by cooling through the transition temperature *T*c is somewhat different from the *T* = 0 crossover. These points are addressed in this section.

Here we extend the ring to an annulus with an inner radius *R*<sup>1</sup> and an outer radius *R*2, as shown in figure 10. For such an annulus, we choose a continuum approach on the basis of the Bogoliubov - de Gennes (BdG) equations, for which no complications arise from the parity effect. For spin singlet pairing the BdG equations are [16]

$$\begin{aligned} E\_{\mathbf{n}}u\_{\mathbf{n}}(\mathbf{r}) &= \left[ \frac{1}{2m} \left( i\hbar \nabla + \frac{e}{c} \mathbf{A}(\mathbf{r}) \right)^{2} - \mu \right] u\_{\mathbf{n}}(\mathbf{r}) + \Delta(\mathbf{r}) u\_{\mathbf{n}}(\mathbf{r}), \\ E\_{\mathbf{n}}v\_{\mathbf{n}}(\mathbf{r}) &= - \left[ \frac{1}{2m} \left( i\hbar \nabla - \frac{e}{c} \mathbf{A}(\mathbf{r}) \right)^{2} - \mu \right] v\_{\mathbf{n}}(\mathbf{r}) + \Delta^{\*}(\mathbf{r}) u\_{\mathbf{n}}(\mathbf{r}), \end{aligned} \tag{13}$$

#### 14 Superconductors 356 Superconductors – Materials, Properties and Applications Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in Superconducting Loops <sup>15</sup>

with the self-consistency condition (gap equation) for the order parameter Δ(**r**):

$$\Delta(\mathbf{r}) = V \sum\_{\mathbf{n}} u\_{\mathbf{n}}(\mathbf{r}) v\_{\mathbf{n}}^{\*}(\mathbf{r}) \tanh\left(\frac{E\_{\mathbf{n}}}{2k\_{\rm B}T}\right) \tag{14}$$

where *V* is the local pairing potential. For an annulus of finite width we separate the angular part of the quasi-particle wavefunctions *u***n**(**r**), *v***n**(**r**) using polar coordinates **r** = (*r*, *θ*) with the ansatz

$$\begin{aligned} \mu\_{\mathbf{n}}(r,\theta) &= \mu\_{\mathbf{n}}(r)e^{\frac{i}{2}(k+q)\theta}, \\ \upsilon\_{\mathbf{n}}(r,\theta) &= \upsilon\_{\mathbf{n}}(r)e^{\frac{i}{2}(k-q)\theta}, \end{aligned} \tag{15}$$



φ

(b)

Δ(

**Figure 11.** Self-consistent calculations for a discretized annulus with an inner radius *<sup>R</sup>*<sup>1</sup> = <sup>100</sup>*a*<sup>⊥</sup> and an outer radius *R*<sup>2</sup> = 150*a*⊥. The (super-) current (a) *J*(*ϕ*) jumps whenever an energy level crosses *E*F. The energy levels *�*(*ϕ*) of the normal state are indicated by the grey lines. (b) displays the self-consistent order parameter Δ as a function of *ϕ*. The lines correspond to the pairing interaction *V* = 0 (orange), *V* = 0.28 *t* (green), *V* = 0.32 *t* (light blue), and *V* = 0.38 *t* (dark blue). The black arrows mark the

(*R*<sup>2</sup> − *<sup>R</sup>*1)/*a*<sup>⊥</sup> for *<sup>n</sup>* = 1. For each crossing, a jump appears in the current as a function of *<sup>ϕ</sup>*, as shown in figure 11 (a). The persistent current is therefore proportional to the level spacing at *E*<sup>F</sup> for each flux value, i.e., it is small for a large density of states and large for a small density of states and vanishes in the limit of a continuous density of states (c.f. reference [32]). The amplitude of the normal persistent current is thus a measure for the difference of the energy spectrum at integer and half integer flux values. It is maximal for the one dimensional case, but might be very small in real metal loops (c.f. measurements of the Aharonov-Bohm effect

Upon entering the superconducting state, an energy gap develops around *E*<sup>F</sup> preventing energy levels from crossing *E*F. Thus the persistent supercurrent arises as in the one dimensional ring independent of the density of states. The large jumps in the supercurrent appear at the value of *ϕ* where the energies of the even-*q* and odd-*q* states become degenerate and *q* switches to the next integer. In the flux regimes with no crossings of *E*<sup>F</sup> the supercurrent is linear and the total energy quadratic in *ϕ*. For the largest value shown (Δ = 0.006 *t*), there is a direct gap for all values of *ϕ*. Even for this large Δ, the current and the energy are not precisely Φ0/2-periodic because of the energy difference of the even and odd *q* states in finite systems [34, 45]. The offset of the *q*-jump is only relevant for values of the pairing interaction *V* for which Δ is finite for all *ϕ*. In figure 11 (b), the offset is clearly visible for the largest two values of *V* (marked with black arrows). Its sign depends on the shape of the annulus and the

The introduction of self-consistency for the order parameter does not fundamentally change these basic observations [figure 11 (b)]. One finds that Δ(*R*1) < Δ(*R*2), but if (*R*<sup>2</sup> − *R*1)/*R*<sup>1</sup> 1, the difference is small. In the following, we denote the average of Δ(*r*) by Δ. The crossover

positions of the *<sup>q</sup>*-jump for *<sup>V</sup>* <sup>=</sup> 0.38 *<sup>t</sup>*. Here the energy units are *<sup>t</sup>* <sup>=</sup> 2/2*mea*⊥.

value of *V*— the offset changes sign for increasing *V* (cf. reference [45]).

0

0.002

0.004

0.01

Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in Superconducting Loops 357

φ)

(a)

(φ)

J(φ)

0

8

4



0

in metal rings [4]).

where *k* and *q* are either both even or both odd integers. Thus *k* is the angular momentum as for the one dimensional ring and **n** = (*k*, *ρ*) with a radial quantum number *ρ*. The order parameter factorizes into Δ(*r*, *θ*) = Δ(*r*)*eiq<sup>θ</sup>* where the radial component

$$
\Delta(r) = V\_0 \sum\_{\mathbf{n}} \mu\_{\mathbf{n}}(r) v\_{\mathbf{n}}^\*(r) \tanh\left(\frac{E\_{\mathbf{n}}}{2k\_{\mathbf{B}}T}\right) \tag{16}
$$

is real. For a magnetic flux *ϕ* threading the interior of the annulus we choose the vector potential **A**(*r*, *θ*) = **e***<sup>θ</sup> ϕ*/(2*πr*), where **e***<sup>θ</sup>* is the azimuthal unit vector. With

$$\left(-i\nabla \pm \frac{\varrho}{r} \mathbf{e}\_{\theta}\right)^{2} = -\frac{1}{r} \partial\_{r} (r \partial\_{r}) + \frac{1}{r^{2}} (-i\partial\_{\theta} \pm \varrho)^{2} \tag{17}$$

the BdG equations therefore reduce to radial differential equations for *u***n**(*r*) and *v***n**(*r*):

$$\begin{split} E\_{\mathbf{n}}u\_{\mathbf{n}}(r) &= -\left[\frac{\hbar^2}{2m}\frac{\partial\_r}{r}(r\partial\_r) - \frac{\hbar^2 l\_u^2}{2mr^2} + \mu\right]u\_{\mathbf{n}}(r) + \Delta(r)v\_{\mathbf{n}}(r), \\ E\_{\mathbf{n}}v\_{\mathbf{n}}(r) &= \left[\frac{\hbar^2}{2m}\frac{\partial\_r}{r}(r\partial\_r) - \frac{\hbar^2 l\_v^2}{2mr^2} + \mu\right]v\_{\mathbf{n}}(r) + \Delta(r)u\_{\mathbf{n}}(r), \end{split} \tag{18}$$

with the canonical angular momenta

$$
\hbar \hbar l\_{\mu} = \frac{\hbar}{2} (k + q - 2\varrho),
\tag{19}
$$

$$
\hbar \hbar l\_{\upsilon} = \frac{\hbar}{2} (k - \eta + 2\wp). \tag{20}
$$

For integer and half-integer flux values, equations (18) can be solved analytically whereas for an arbitrary magnetic flux a numerical solution is required (for details, see reference [31]). Within this procedure, the radial coordinate is discretized into *<sup>N</sup>*<sup>⊥</sup> values *rn* separated by the distance *<sup>a</sup>*<sup>⊥</sup> = (*R*<sup>2</sup> − *<sup>R</sup>*1)/*N*⊥. The number *<sup>q</sup>* plays the same role as in the previous section. Here we choose *q* for each value of the flux to minimize the total energy of the system. The flux for which *q* changes to the next integer can therefore deviate from the values (2*n* − 1)/4, at which we fixed the jump to the next *q* for the one-dimensional model.

In the normal state (Δ = 0), the number of eigenstates sufficiently close to *E*<sup>F</sup> which may cross *E*<sup>F</sup> as a function of *ϕ* is controlled by the average charge density *n* and is approximately

14 Superconductors

where *V* is the local pairing potential. For an annulus of finite width we separate the angular part of the quasi-particle wavefunctions *u***n**(**r**), *v***n**(**r**) using polar coordinates **r** = (*r*, *θ*) with

where *k* and *q* are either both even or both odd integers. Thus *k* is the angular momentum as for the one dimensional ring and **n** = (*k*, *ρ*) with a radial quantum number *ρ*. The order

*u***n**(*r*)*v*∗

<sup>=</sup> <sup>−</sup><sup>1</sup> *r*

the BdG equations therefore reduce to radial differential equations for *u***n**(*r*) and *v***n**(*r*):

*<sup>r</sup>* (*r∂r*)<sup>−</sup> <sup>2</sup>*<sup>l</sup>*

*<sup>r</sup>* (*r∂r*)<sup>−</sup> <sup>2</sup>*<sup>l</sup>*

*lu* <sup>=</sup> 2

*lv* <sup>=</sup> 2

at which we fixed the jump to the next *q* for the one-dimensional model.

is real. For a magnetic flux *ϕ* threading the interior of the annulus we choose the vector

**<sup>n</sup>**(**r**)tanh

*i* <sup>2</sup> (*k*+*q*)*<sup>θ</sup>* ,

*i* <sup>2</sup> (*k*−*q*)*<sup>θ</sup>* ,

**<sup>n</sup>**(*r*)tanh

*<sup>∂</sup>r*(*r∂r*) + <sup>1</sup>

2 *u* <sup>2</sup>*mr*<sup>2</sup> <sup>+</sup>*<sup>μ</sup>*

2 *v* <sup>2</sup>*mr*<sup>2</sup> <sup>+</sup>*<sup>μ</sup>*

For integer and half-integer flux values, equations (18) can be solved analytically whereas for an arbitrary magnetic flux a numerical solution is required (for details, see reference [31]). Within this procedure, the radial coordinate is discretized into *<sup>N</sup>*<sup>⊥</sup> values *rn* separated by the distance *<sup>a</sup>*<sup>⊥</sup> = (*R*<sup>2</sup> − *<sup>R</sup>*1)/*N*⊥. The number *<sup>q</sup>* plays the same role as in the previous section. Here we choose *q* for each value of the flux to minimize the total energy of the system. The flux for which *q* changes to the next integer can therefore deviate from the values (2*n* − 1)/4,

In the normal state (Δ = 0), the number of eigenstates sufficiently close to *E*<sup>F</sup> which may cross *E*<sup>F</sup> as a function of *ϕ* is controlled by the average charge density *n* and is approximately

 *E***<sup>n</sup>** 2*k*B*T* *u***n**(*r*)+ Δ(*r*)*v***n**(*r*),

(*k* + *q* − 2*ϕ*), (19)

(*k* − *q* + 2*ϕ*). (20)

*v***n**(*r*)+ Δ(*r*)*u***n**(*r*),

*<sup>r</sup>*<sup>2</sup> (−*i∂θ* <sup>±</sup> *<sup>ϕ</sup>*)<sup>2</sup> (17)

 *E***<sup>n</sup>** 2*k*B*T* , (14)

(15)

(16)

(18)

with the self-consistency condition (gap equation) for the order parameter Δ(**r**):

*u***n**(**r**)*v*∗

*u***n**(*r*, *θ*) = *u***n**(*r*)*e*

*v***n**(*r*, *θ*) = *v***n**(*r*)*e*

<sup>Δ</sup>(**r**) = *<sup>V</sup>* <sup>∑</sup>**<sup>n</sup>**

parameter factorizes into Δ(*r*, *θ*) = Δ(*r*)*eiq<sup>θ</sup>* where the radial component

potential **A**(*r*, *θ*) = **e***<sup>θ</sup> ϕ*/(2*πr*), where **e***<sup>θ</sup>* is the azimuthal unit vector. With

<sup>Δ</sup>(*r*) = *<sup>V</sup>*<sup>0</sup> <sup>∑</sup>**<sup>n</sup>**

 <sup>2</sup> 2*m ∂r*

 <sup>2</sup> 2*m ∂r*

*E***<sup>n</sup>** *u***n**(*r*) =−

*E***<sup>n</sup>** *v***n**(*r*) =

with the canonical angular momenta

<sup>−</sup>*i*<sup>∇</sup> <sup>±</sup> *<sup>ϕ</sup> r* **e***θ* 2

the ansatz

**Figure 11.** Self-consistent calculations for a discretized annulus with an inner radius *<sup>R</sup>*<sup>1</sup> = <sup>100</sup>*a*<sup>⊥</sup> and an outer radius *R*<sup>2</sup> = 150*a*⊥. The (super-) current (a) *J*(*ϕ*) jumps whenever an energy level crosses *E*F. The energy levels *�*(*ϕ*) of the normal state are indicated by the grey lines. (b) displays the self-consistent order parameter Δ as a function of *ϕ*. The lines correspond to the pairing interaction *V* = 0 (orange), *V* = 0.28 *t* (green), *V* = 0.32 *t* (light blue), and *V* = 0.38 *t* (dark blue). The black arrows mark the positions of the *<sup>q</sup>*-jump for *<sup>V</sup>* <sup>=</sup> 0.38 *<sup>t</sup>*. Here the energy units are *<sup>t</sup>* <sup>=</sup> 2/2*mea*⊥.

(*R*<sup>2</sup> − *<sup>R</sup>*1)/*a*<sup>⊥</sup> for *<sup>n</sup>* = 1. For each crossing, a jump appears in the current as a function of *<sup>ϕ</sup>*, as shown in figure 11 (a). The persistent current is therefore proportional to the level spacing at *E*<sup>F</sup> for each flux value, i.e., it is small for a large density of states and large for a small density of states and vanishes in the limit of a continuous density of states (c.f. reference [32]). The amplitude of the normal persistent current is thus a measure for the difference of the energy spectrum at integer and half integer flux values. It is maximal for the one dimensional case, but might be very small in real metal loops (c.f. measurements of the Aharonov-Bohm effect in metal rings [4]).

Upon entering the superconducting state, an energy gap develops around *E*<sup>F</sup> preventing energy levels from crossing *E*F. Thus the persistent supercurrent arises as in the one dimensional ring independent of the density of states. The large jumps in the supercurrent appear at the value of *ϕ* where the energies of the even-*q* and odd-*q* states become degenerate and *q* switches to the next integer. In the flux regimes with no crossings of *E*<sup>F</sup> the supercurrent is linear and the total energy quadratic in *ϕ*. For the largest value shown (Δ = 0.006 *t*), there is a direct gap for all values of *ϕ*. Even for this large Δ, the current and the energy are not precisely Φ0/2-periodic because of the energy difference of the even and odd *q* states in finite systems [34, 45]. The offset of the *q*-jump is only relevant for values of the pairing interaction *V* for which Δ is finite for all *ϕ*. In figure 11 (b), the offset is clearly visible for the largest two values of *V* (marked with black arrows). Its sign depends on the shape of the annulus and the value of *V*— the offset changes sign for increasing *V* (cf. reference [45]).

The introduction of self-consistency for the order parameter does not fundamentally change these basic observations [figure 11 (b)]. One finds that Δ(*R*1) < Δ(*R*2), but if (*R*<sup>2</sup> − *R*1)/*R*<sup>1</sup> 1, the difference is small. In the following, we denote the average of Δ(*r*) by Δ. The crossover

where we used the relations Δ<sup>c</sup> = *at*/2*R* and *E*<sup>F</sup> = 2*t* for a discretized one-dimensional ring at half filling, and the BCS relation <sup>Δ</sup>2(0) <sup>≈</sup> 3.1(*k*B*T*c)2. For a ring with a radius of 2500 lattice constants (≈ 10 μm) and Δ(0) = 0.01 *t* (≈ 3 meV) one finds the ratio (*T*<sup>c</sup> − *T*∗)/*T*<sup>c</sup> ≈ 1.3 <sup>×</sup> <sup>10</sup><sup>−</sup>4. This is in reasonable quantitative agreement with the experimental results of Little and Parks [30, 39], discussed also by Tinkham in reference [44]. Their prediction was similar to equation (22), up to a factor in which they include a finite mean free path. But they did not included the difference introduced through even and odd *q* states. This difference was considered in calculations of *T*c by Bogachek *et al.* [9] in the single-channel limit and found to be exponentially small. A detailed study of the normal- to superconducting phase boundary was also done by Wei and Goldbart in reference [49] in which they considered the Φ<sup>0</sup> periodic contributions. In equation (22) the value of Δ(0) is in fact different for even and odd *q*. Although quantitative predictions of *T*<sup>c</sup> − *T*<sup>∗</sup> of the theory presented here might be too large as compared to the experiment, it serves as an upper limit because it describes the maximum possible persistent current. Inhomogeneities and scattering processes in real systems further reduce the difference of the energy spectra in the even- and odd-*q* flux sectors

Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in Superconducting Loops 359

For temperatures close to *T*c, the difference of the eigenenergies of even and odd *q* states is less important than at *T* = 0. Thus the deviation from the Φ0/2 periodicity of the current and of the order parameter is smaller. Furthermore, persistent currents in the normal state are exponentially small compared to the supercurrents below *T*c. Their respective Φ<sup>0</sup> periodic behavior is therefore essentially invisible for the flux values where Δ = 0. In figure 12, the difference between Δ(*ϕ* = 0) and Δ(*ϕ* = 1/2) is still visible, but the corresponding differences in the current are too small. Only for a superconductor with very small *T*c, we expect the

Although we found within the framework of the BCS theory, that the crossover to a Φ0/2 periodic supercurrent takes place slightly below *T*c, detailed studies by Ambegaokar and Eckern [5] and by von Oppen and Riedel [48] including superconducting fluctuations reveals that the crossover might actually take place above *T*c. This fluctuation driven crossover is broader than the BCS crossover with a similarly, exponentially suppressed Φ<sup>0</sup> periodic normal current contribution. For a superconductor with a *T*c small enough to observe a normal persistent current above *T*c, Eckern and Schwab suggested that the crossover regime, where both Φ<sup>0</sup> and Φ0/2 periodic current contributions are present, should be observable at

The discussion of the periodicity crossover in a multi-channel loop also gives insight into the flux periodicity of loops of unconventional superconductors with gap nodes like a *d*-wave superconductor. In nodal superconductors the density of states is finite arbitrarily close to *E*F. Therefore some energy levels cross *E*<sup>F</sup> as a function of the flux, regardless of the size of the order parameter and consequently, the "small gap" situation extends to arbitrarily large loops [6, 25, 32, 34]. Of course, the number of energy levels crossing *E*<sup>F</sup> decreases with increasing ring size and thus also the Φ<sup>0</sup> periodic contribution to the supercurrent. The dependence of this Φ<sup>0</sup> periodic contribution on the ring size depends on the order parameter symmetry. The careful study in reference [32] revealed that for *d*-wave superconductors, the relation between the Φ<sup>0</sup> and Φ0/2 periodic current contributions is proportional to 1/*R*1. It

and thereby reduce *T*<sup>c</sup> − *T*∗.

periodicity crossover to be visible.

a temperature *T* ≈ 2*T*<sup>c</sup> [20, 21].

**Figure 12.** The order parameter Δ(*ϕ*) and the persistent current *J*(*ϕ*) for the temperature driven transition from the normal to the superconducting state in an annulus with inner radius *R*<sup>1</sup> = 30*a*<sup>⊥</sup> and outer radius *R*<sup>2</sup> = 36*a*⊥. The pairing interaction is *V*<sup>0</sup> = 0.7 *t*, with a critical temperature of *k*B*T*<sup>c</sup> ≈ 0.0523 *t* for zero flux. For these parameters Δ(*T* = 0) ≈ 0.1 *t*. The lines (from top to bottom) correspond to the temperatures *k*B*T* = 0.0513 *t* (dark blue), *k*B*T* = 0.0520 *t* (light blue), and *k*B*T* = 0.0522 *t* (green). Notice that Δ is slightly different for the flux values *ϕ* = 0 and *ϕ* = ±1/2.

is then controlled by the pairing interaction strength *V*, for which we chose such values as to reproduce the crossover from the normal state to a state with direct energy gap for all flux values. The order parameter Δ is now also a function of *ϕ*. If Δ(*ϕ* = 0) 0.006 *t* [c.f. figure 11 (b)], the gap closes with *ϕ*, and Δ decreases whenever a state crosses *E*F. Unlike in a one dimensional ring, Δ does not drop to zero at the closing of the energy gap, but decreases stepwise. This is because in two or three dimensions, Δ is stabilized beyond the depairing velocity by contributions to the condensation energy from pairs with relative momenta perpendicular to the direction of the current flow; the closing of the indirect energy gap does not destroy superconductivity [7, 50].

Experimentally more relevant is to control the crossover through temperature. With the pairing interaction *V* sufficiently strong to produce a *T* = 0 energy gap much larger than the maximum Doppler shift, the crossover regime is reached for temperatures slightly below *T*c. For the annulus of figure 10, the crossover proceeds within approximately one percent of *T*c. The crossover regime becomes narrower for larger rings, proportional to the decrease of the Doppler shift. In the limit of a quasi one-dimensional ring of radius *R* we can be more precise: If we define the crossover temperature *T*<sup>∗</sup> by Δ(*T*∗) = Δ<sup>c</sup> and assuming Δ<sup>c</sup> � Δ, we can use the Ginzburg-Landau form of the order parameter

$$\frac{\Delta(T)}{\Delta(0)} \approx 1.75 \sqrt{1 - \frac{T}{T\_{\odot}}} \tag{21}$$

and obtain

$$\frac{T\_{\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\text}}}}}}}}}}\mathbfright)}}\,}\text{\text{\tiny\text{\text{\tiny\text{\tiny\text{\tiny\text{\textdegree}}}}}\text{)}}\text{)}}{3.1\,\text{\text{\textdegree C}}\,\text{(\text{\textdegree C})}\,\text{(\text{\tiny\text{\textdegree C})}\text{)}}\,\text{(\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\text}}}}}}}}}\mathbfright)}}\mathbf}}\mathbf}}\,\text{\}}}\,\text{)}}\,\text{)} + \frac{\text{\text{\tiny\text{\tiny\text{\textell}}}\text{)}\text{)}}{12.1\,\text{\text{\textdegree C}}\,\text{(\text{\tiny\text{\textgreater}}\text{)}}\text{)}\text{)}}\,\text{}$$

where we used the relations Δ<sup>c</sup> = *at*/2*R* and *E*<sup>F</sup> = 2*t* for a discretized one-dimensional ring at half filling, and the BCS relation <sup>Δ</sup>2(0) <sup>≈</sup> 3.1(*k*B*T*c)2. For a ring with a radius of 2500 lattice constants (≈ 10 μm) and Δ(0) = 0.01 *t* (≈ 3 meV) one finds the ratio (*T*<sup>c</sup> − *T*∗)/*T*<sup>c</sup> ≈ 1.3 <sup>×</sup> <sup>10</sup><sup>−</sup>4. This is in reasonable quantitative agreement with the experimental results of Little and Parks [30, 39], discussed also by Tinkham in reference [44]. Their prediction was similar to equation (22), up to a factor in which they include a finite mean free path. But they did not included the difference introduced through even and odd *q* states. This difference was considered in calculations of *T*c by Bogachek *et al.* [9] in the single-channel limit and found to be exponentially small. A detailed study of the normal- to superconducting phase boundary was also done by Wei and Goldbart in reference [49] in which they considered the Φ<sup>0</sup> periodic contributions. In equation (22) the value of Δ(0) is in fact different for even and odd *q*. Although quantitative predictions of *T*<sup>c</sup> − *T*<sup>∗</sup> of the theory presented here might be too large as compared to the experiment, it serves as an upper limit because it describes the maximum possible persistent current. Inhomogeneities and scattering processes in real systems further reduce the difference of the energy spectra in the even- and odd-*q* flux sectors and thereby reduce *T*<sup>c</sup> − *T*∗.

16 Superconductors

Δ(

**Figure 12.** The order parameter Δ(*ϕ*) and the persistent current *J*(*ϕ*) for the temperature driven transition from the normal to the superconducting state in an annulus with inner radius *R*<sup>1</sup> = 30*a*<sup>⊥</sup> and

is then controlled by the pairing interaction strength *V*, for which we chose such values as to reproduce the crossover from the normal state to a state with direct energy gap for all flux values. The order parameter Δ is now also a function of *ϕ*. If Δ(*ϕ* = 0) 0.006 *t* [c.f. figure 11 (b)], the gap closes with *ϕ*, and Δ decreases whenever a state crosses *E*F. Unlike in a one dimensional ring, Δ does not drop to zero at the closing of the energy gap, but decreases stepwise. This is because in two or three dimensions, Δ is stabilized beyond the depairing velocity by contributions to the condensation energy from pairs with relative momenta perpendicular to the direction of the current flow; the closing of the indirect energy

Experimentally more relevant is to control the crossover through temperature. With the pairing interaction *V* sufficiently strong to produce a *T* = 0 energy gap much larger than the maximum Doppler shift, the crossover regime is reached for temperatures slightly below *T*c. For the annulus of figure 10, the crossover proceeds within approximately one percent of *T*c. The crossover regime becomes narrower for larger rings, proportional to the decrease of the Doppler shift. In the limit of a quasi one-dimensional ring of radius *R* we can be more precise: If we define the crossover temperature *T*<sup>∗</sup> by Δ(*T*∗) = Δ<sup>c</sup> and assuming Δ<sup>c</sup> � Δ, we

2

<sup>1</sup> <sup>−</sup> *<sup>T</sup> T*c

12.4Δ2(0)(*R*/*a*)<sup>2</sup> <sup>=</sup> *<sup>E</sup>*<sup>2</sup>

F

3.1(*k*B*T*c)2(*R*/*a*)<sup>2</sup> , (22)

outer radius *R*<sup>2</sup> = 36*a*⊥. The pairing interaction is *V*<sup>0</sup> = 0.7 *t*, with a critical temperature of *k*B*T*<sup>c</sup> ≈ 0.0523 *t* for zero flux. For these parameters Δ(*T* = 0) ≈ 0.1 *t*. The lines (from top to bottom) correspond to the temperatures *k*B*T* = 0.0513 *t* (dark blue), *k*B*T* = 0.0520 *t* (light blue), and *k*B*T* = 0.0522 *t* (green). Notice that Δ is slightly different for the flux values *ϕ* = 0 and *ϕ* = ±1/2.

0.008

0.016

0


(21)

(b)

φ)

J(φ)

0

0.010



and obtain


(a)

gap does not destroy superconductivity [7, 50].

*T*<sup>c</sup> − *T*<sup>∗</sup> *T*c

can use the Ginzburg-Landau form of the order parameter

<sup>≈</sup> <sup>Δ</sup><sup>2</sup> c

Δ(*T*) <sup>Δ</sup>(0) <sup>≈</sup> 1.75

3.1Δ2(0) <sup>=</sup> *<sup>t</sup>*


For temperatures close to *T*c, the difference of the eigenenergies of even and odd *q* states is less important than at *T* = 0. Thus the deviation from the Φ0/2 periodicity of the current and of the order parameter is smaller. Furthermore, persistent currents in the normal state are exponentially small compared to the supercurrents below *T*c. Their respective Φ<sup>0</sup> periodic behavior is therefore essentially invisible for the flux values where Δ = 0. In figure 12, the difference between Δ(*ϕ* = 0) and Δ(*ϕ* = 1/2) is still visible, but the corresponding differences in the current are too small. Only for a superconductor with very small *T*c, we expect the periodicity crossover to be visible.

Although we found within the framework of the BCS theory, that the crossover to a Φ0/2 periodic supercurrent takes place slightly below *T*c, detailed studies by Ambegaokar and Eckern [5] and by von Oppen and Riedel [48] including superconducting fluctuations reveals that the crossover might actually take place above *T*c. This fluctuation driven crossover is broader than the BCS crossover with a similarly, exponentially suppressed Φ<sup>0</sup> periodic normal current contribution. For a superconductor with a *T*c small enough to observe a normal persistent current above *T*c, Eckern and Schwab suggested that the crossover regime, where both Φ<sup>0</sup> and Φ0/2 periodic current contributions are present, should be observable at a temperature *T* ≈ 2*T*<sup>c</sup> [20, 21].

The discussion of the periodicity crossover in a multi-channel loop also gives insight into the flux periodicity of loops of unconventional superconductors with gap nodes like a *d*-wave superconductor. In nodal superconductors the density of states is finite arbitrarily close to *E*F. Therefore some energy levels cross *E*<sup>F</sup> as a function of the flux, regardless of the size of the order parameter and consequently, the "small gap" situation extends to arbitrarily large loops [6, 25, 32, 34]. Of course, the number of energy levels crossing *E*<sup>F</sup> decreases with increasing ring size and thus also the Φ<sup>0</sup> periodic contribution to the supercurrent. The dependence of this Φ<sup>0</sup> periodic contribution on the ring size depends on the order parameter symmetry. The careful study in reference [32] revealed that for *d*-wave superconductors, the relation between the Φ<sup>0</sup> and Φ0/2 periodic current contributions is proportional to 1/*R*1. It was estimated that for a ring of a cuprate superconductor with a circumference of ∼ 1 μm, this ratio is about 1% and should be observable experimentally.

as an angular momentum. Agterberg and Tsunetsugu showed within a Ginzburg-Landau approach, that a PDW superconductor indeed allows for vortices carrying a Φ0/4 flux

Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in Superconducting Loops 361

Based on a microscopic model, it was shown in reference [33] that a PDW state cannot result from an on-site pairing interaction. However, the PDW state can be a stable alternative for unconventional superconductors with gap nodes, specifically for *d*-wave superconductivity as realized in the high-*T*c cuprate superconductors. More work is, however, needed to analyze under which conditions the PDW state indeed develops an energy minimum at multiples Φ0/4, and these minima become degenerate in the limit of large loops. The notion of the absence of coexistence of Cooper pairs with different center-of-mass momenta for conventional superconductors (as, e.g., in the PDW state) justifies the reduction of the sum over *q* in the Hamiltonian (6) in section 2.2 to one specific *q* in order to derive the periodicity

Here we mention a further system where a similar mechanism as described above may lead to Φ0/4 flux periodicity: Sr2RuO4. Experimental evidence exists that Sr2RuO4 is a spin triplet *p*-wave superconductor. A triplet superconductor can be represented by the two-component

and the *sz* = −1 condensates. This realizes a similar situation as for the PDW state where Φ0/4 periodicity is possible [46]. Indeed Jang *et al*. observed recently that the flux through a

The authors acknowledge discussions with Yuri Barash, Ulrich Eckern, Jochen Mannhart, and Christof Schneider. This work was supported by the Deutsche Forschungsgemeinschaft

*Experimental Physics VI & Theoretical Physics III, Center for Electronic Correlations and Magnetism,*

*Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics,*

*Experimental Physics VI, Center for Electronic Correlations and Magnetism, Institute of Physics,*

[1] Abrikosov, A. A. [1957]. On the magnetic properties of superconductors of the second

[2] Agterberg, D. F., Sigrist, M. & Tsunetsugu, H. [2009]. Order parameter and vortices in

the superconducting *q* phase of CeCoIn5, *Phys. Rev. Lett.* 102: 207004.

*<sup>q</sup>*<sup>2</sup> (*ϕ*)}, with the center-of-mass momenta *q*<sup>1</sup> and *q*<sup>2</sup> for the *sz* = 1

quantum [2, 3].

crossover in superconducting rings.

*<sup>q</sup>*<sup>1</sup> (*ϕ*), Δ↓↓

*University of Augsburg, 86135 Augsburg, Germany*

*University of Augsburg, 86135 Augsburg, Germany*

group, *Soviet Physics – JETP* 5: 1174.

microscopic Sr2RuO4 ring is quantized in units of Φ0/4 [24].

*Institute of Physics, University of Augsburg, 86135 Augsburg, Germany*

order parameter {Δ↑↑

**Acknowledgements**

through TRR 80.

**Author details**

Loder Florian

Kampf Arno P.

Kopp Thilo

**4. References**
