*3.3.4. Theory of anti-proximity effect*

There are several theoretical models that could be used for the theoretical explanation of the magnetic field-induced or -enhanced superconductivity in nanowires, while some of which were proposed long before the anti-proximity effect was reported. Thus they are not generally accepted.

Field-Induced Superconductors: NMR Studies of λ–(BETS)2FeCl4 9

When the bulk electrodes are driven normal by the applied magnetic field (*H* ≥ 30 mT), the contact resistances R ≠ 0. Similarly, if the electrodes are normal but the nanowire is superconducting (or vice versa), there will be a resistance due to charge conversion processes [10]. This results in a high circuit frequency *f* = 1/(2π*RC*) ~ 1 GHz, with a behavior like a pure resistor (i.e., impedance *XC =*1/2π*fC* → 0). If this shunting resistance is less the quantum of resistance (*h*/4*e*2 ~ 6*.*4 kΩ), then it will damp the superconducting phase fluctuations, and thus stabilizing the superconductivity. On the other hand, in a dissipation environment [Fig. 4 (c)], the superfluid density (σ) of the zinc nanowire cannot screen the interaction between bulk vortices completely. As a result, the superconducting phase

In order words, when the magnetic field is applied (*H* ≠ 0) to the bulk electrodes, the dissipations between the two ends of the electrodes will be enhanced and meanwhile the superconducting phase fluctuations are damped. This leads to the stabilization of the

The interference model proposes that there is an interference between junctions of two superconducting grains, with random Josephson couplings J and J' associated with disorder, as sketched shown in Fig. 5. It produces a configuration-averaged critical current

2 2

′ Φ

*J J*

*J*

*J'*

=+ − ′ <sup>+</sup> <sup>+</sup> ′ <sup>Φ</sup>

<sup>1</sup> <sup>1</sup> cos (2 ) ... <sup>2</sup> *<sup>C</sup>*

Φ

This is a periodic function of Φ [cos2(2πΦ/Φ0)] with a period of half flux quantum (Φ0/2 = hc/4e), where Φ is the magnetic flux through each hole due to the existence of disorder in the sample (note, the sample has an array of holes through each of which there has a

Thus when Φ is small, *<I*C> increases as Φ increases, and this corresponds to a negative magnetoresistance, i.e., when applied magnetic field *H* increases, the electrical resistivity of

0

(1)

π

( )1/2 2 2 <sup>2</sup>

*JJ I JJ*

Here Φ represents for the magnetic flux through an array of each holes due to existence of disorder. **Figure 5.** Schematic of interference between junctions with Josephson couplings J and J'

the nanowires drops down. Thereby the superconductivity is enhanced.

becomes stable for sufficiently small shunt resistance [17].

**b. Interference model** 

*<I*C> as [18]

flux Φ).

superconductivity of the nanowires between the bulk electrodes.

### **a. Phase fluctuation model**

This model proposes that there is an interplay between the superconducting phase fluctuations and dissipative quasiparticle channels [17].

The schematic diagram of this model regarding the anti-proximity effect experiment (Fig. 3) can be re-illustrated as that shown in Fig. 4.

**Figure 4.** (a) Schematic of the anti-proximity effect experiment (Fig. 3). (b) Simplification of (a). (c) Phase fluctuations in a dissipative environment. R – resistance of the bulk electrodes, C – circuit capacitance (between the electrodes), V – voltage, I – current, σL and σR – superfluid (surface vortex densities) [17].

When the bulk electrodes are superconducting, there is a supercurrent flowing between the nanowire and BS electrodes, and the contact resistances (R) vanish (R = 0). Thus the circuit frequency becomes low, and the quantum wire is shunted by the capacitor (C) if the energy (frequency *f*) is less than the bulk superconducting gap energy of the electrodes. In this case, the quantum fluctuations of the superconducting phase drive the superfluid density (σ) of the zinc nanowire to zero (even when temperature *T* = 0). As soon as the superfluid density vanishes, the Cooper pairs dissociate and the nanowire becomes normal (with a normal resistor with resistance *RN*). Thus this explains the resistive behavior at zero applied magnetic field (*H*= 0).

When the bulk electrodes are driven normal by the applied magnetic field (*H* ≥ 30 mT), the contact resistances R ≠ 0. Similarly, if the electrodes are normal but the nanowire is superconducting (or vice versa), there will be a resistance due to charge conversion processes [10]. This results in a high circuit frequency *f* = 1/(2π*RC*) ~ 1 GHz, with a behavior like a pure resistor (i.e., impedance *XC =*1/2π*fC* → 0). If this shunting resistance is less the quantum of resistance (*h*/4*e*2 ~ 6*.*4 kΩ), then it will damp the superconducting phase fluctuations, and thus stabilizing the superconductivity. On the other hand, in a dissipation environment [Fig. 4 (c)], the superfluid density (σ) of the zinc nanowire cannot screen the interaction between bulk vortices completely. As a result, the superconducting phase becomes stable for sufficiently small shunt resistance [17].

In order words, when the magnetic field is applied (*H* ≠ 0) to the bulk electrodes, the dissipations between the two ends of the electrodes will be enhanced and meanwhile the superconducting phase fluctuations are damped. This leads to the stabilization of the superconductivity of the nanowires between the bulk electrodes.

### **b. Interference model**

8 Superconductors – Materials, Properties and Applications

fluctuations and dissipative quasiparticle channels [17].

can be re-illustrated as that shown in Fig. 4.

There are several theoretical models that could be used for the theoretical explanation of the magnetic field-induced or -enhanced superconductivity in nanowires, while some of which were proposed long before the anti-proximity effect was reported. Thus they are not

This model proposes that there is an interplay between the superconducting phase

The schematic diagram of this model regarding the anti-proximity effect experiment (Fig. 3)

**(***a***) (***b***) (***c***)**

**R/2**

**R/2**

**V**

**RN**

**C**

**Figure 4.** (a) Schematic of the anti-proximity effect experiment (Fig. 3). (b) Simplification of (a). (c) Phase fluctuations in a dissipative environment. R – resistance of the bulk electrodes, C – circuit capacitance (between the electrodes), V – voltage, I – current, σL and σR – superfluid (surface vortex

When the bulk electrodes are superconducting, there is a supercurrent flowing between the nanowire and BS electrodes, and the contact resistances (R) vanish (R = 0). Thus the circuit frequency becomes low, and the quantum wire is shunted by the capacitor (C) if the energy (frequency *f*) is less than the bulk superconducting gap energy of the electrodes. In this case, the quantum fluctuations of the superconducting phase drive the superfluid density (σ) of the zinc nanowire to zero (even when temperature *T* = 0). As soon as the superfluid density vanishes, the Cooper pairs dissociate and the nanowire becomes normal (with a normal resistor with resistance *RN*). Thus this explains the resistive behavior at zero applied

*3.3.4. Theory of anti-proximity effect* 

generally accepted.

densities) [17].

magnetic field (*H*= 0).

**a. Phase fluctuation model** 

The interference model proposes that there is an interference between junctions of two superconducting grains, with random Josephson couplings J and J' associated with disorder, as sketched shown in Fig. 5. It produces a configuration-averaged critical current *<I*C> as [18]

$$\left\{\mathbf{I}\_{C}\right\} = \left(\mathbf{I}^{2} + \mathbf{J}^{\*2}\right)^{1/2} \left[1 - \frac{1}{2} \left\langle \frac{\mathbf{J}^{\prime}}{\mathbf{J}^{2} + \mathbf{J}^{\*2}} \right\rangle \cos^{2}(2\pi \frac{\Phi}{\Phi\_{0}}) + \dots\right] \tag{1}$$

$$\xrightarrow{J} \underbrace{\left\{\begin{array}{c} \mathbf{I} \\ \bullet \\ \bullet \\ \bullet \end{array}\right\} \dots$$

Here Φ represents for the magnetic flux through an array of each holes due to existence of disorder.

**Figure 5.** Schematic of interference between junctions with Josephson couplings J and J'

This is a periodic function of Φ [cos2(2πΦ/Φ0)] with a period of half flux quantum (Φ0/2 = hc/4e), where Φ is the magnetic flux through each hole due to the existence of disorder in the sample (note, the sample has an array of holes through each of which there has a flux Φ).

*J'*

Thus when Φ is small, *<I*C> increases as Φ increases, and this corresponds to a negative magnetoresistance, i.e., when applied magnetic field *H* increases, the electrical resistivity of the nanowires drops down. Thereby the superconductivity is enhanced.

## **c. Charge imbalance length model**

This model proposes that there is a charge-imbalance length (or relaxation time) associated with the normal metal - superconductor boundaries of phase-slip centers [20]. Applying magnetic field reduces the charge-imbalance length (or relaxation time), resulting in a negative magnetoresistance at high currents and near *T*c. Thus the superconductivity in the nanowires is enhanced.

Field-Induced Superconductors: NMR Studies of λ–(BETS)2FeCl4 11

**(c)**

**Figure 6.** (a) Crystal structure of λ–(BETS)2FeCl4 in a unit cell. (b) BETS molecule [20]. (c) Phase diagram

Noticeably, the conducting layers comprised of BETS are sandwiched along the *b* axis by the insulating layers of FeCl4− anions. The least conducting axis is *b*, the conducting plane is *ac*, and the easy axis of the antiferromagnetic spin structure is ~30° away from the *c* axis

At the room temperature (298 K), the lattice constants are: *a* = 16.164(3), *b* = 18.538(3),

distance between Fe3+ ions is 10.1 Ao within a unit cell, which is along the *a*-direction, and

In order to study the mechanism of the superconductivity in λ–(BETS)2FeCl4 and to test the validity of the Jaccarino-Peter effect, as well as to understand the multi-phase properties of the material as show in the unusual phase diagram [Fig. 6 (c)], we successfully conducted a

These include both 77Se-NMR measurements and proton (1H) NMR measurements, as a function of temperature, magnetic field and angle of alignment of the magnetic field [20, 23, 24].

= 96.69(1)o, and

γ

= 112.52(1)o. The shortest

β

the nearest distance of Fe3+ ions between neighboring unit cells is 8.8 Ao [21].

of λ–(BETS)2FeCl4 [8].

(parallel to the needle axis of the crystal) [21, 22].

*c* = 6.592(4) Angstrom (Ao), *α* = 98.40(1)o,

**5. NMR studies of** λ**–(BETS)2FeCl4**

series of nuclear magnetic resonance (NMR) experiments.
