Flux pinning before irradiation

Many studies have shown that both grain boundaries and in some cases, second phase particles are effective in increasing the critical current density in type II superconductors. However Scanlan et al. [112] and Schauer and Schelb [113] found that the grain boundaries were the only defects present in sufficient density to account for flux pinning in multifilamentary Nb₃Sn wires.

They measured the dependence of 𝐽𝐽𝑐𝑐 on grain size combining the four probe technique measurement and TEM analysis; in this way they were able to provide the experimental curve for the pinning force as a function of the inverse of grain size (1/𝐷𝐷), shown in Figure 20. Pande and Suenaga [114], on the basis of interaction of fluxoids with dislocations forming the grain boundaries walls, derived a relationship between the pinning force 𝐹𝐹𝑒𝑒 and grain size 𝐷𝐷 in type II superconductors that approximately confirm the experimentally observed relationship by Scanlan and coworkers [112] and predicted a peak in 𝐹𝐹_𝑒𝑒 at 𝑎𝑎₀/𝐷𝐷 ≈0.2, where 𝑎𝑎₀ is the lattice constant of the flux line lattice.

Three mechanisms have been used to describe the interaction between grain boundaries and flux lattice in a superconductor: the first one is based on the interaction between the grain boundary strain fields and the stress fields of the flux lines lattice; the second one is based on the

anisotropy of the critical field around the grain boundaries; the third one suggests that the electron scattering at the grain boundaries is the responsible for the pinning force [115].

Figure 20 Pinning force maxima vs inverse average grain size for multifilamentary Nb₃Sn. Increase in pinning force is less than linear with 1/d. From W. Schauer and W. Schelb. [113].

In 1978 Das Gupta [116] measured the pinning force of a single grain boundary in Nb bicrystal and he found several indications to reject the first two models and to consider the electron scattering as the main mechanism responsible for grain boundaries pinning.

The first quantitative theory in favor of this grain boundaries pinning mechanism was formulated by Zerweck in 1981 [117]. Starting from the calculation of the mean free path and the coherence length as a function of the distance from the grain boundary he found that the elementary pinning force depends on the material purity: the pinning force is low for very pure materials (𝛼𝛼<< 1), it increases when the purity decreases and reaches a maximum at 𝛼𝛼 = 1, then goes down again to zero for very dirty materials (𝛼𝛼>> 1).

Yetter et al. [118] demonstrated by means of the Electron Scattering Flux Pinning model (ESFP model) that the interaction between flux lines and grain boundaries in Nb3Sn is mainly due to the spatial variations of the electronic mean free path caused by these grain boundaries. The

results obtained by Yetter [118] are in qualitative agreement with those of Zerweck [117] with the only difference that the value of the impurity parameter at 𝐹𝐹𝑒𝑒𝑚𝑚𝑎𝑎𝑒𝑒 is 10 times higher. The last developments of the theory of electron scattering for the grain boundary pinning force mechanism came from Welch in 1985 [119]. As a main improvement with respect to the previous works he took into account the width of the grain boundaries in the calculations of the elementary pinning force: 𝐹𝐹𝑒𝑒𝑚𝑚𝑎𝑎𝑒𝑒 is achieved for grain boundary widths 1.5 times larger than the coherence length ξ, and for an impurity parameter 𝛼𝛼 = 2.

The two features of grain boundaries pinning determining the pinning strength acting on the flux line lattice in A15 superconductors are the grain size and the composition variation of phases near the grain boundaries. The decrease of grain size can be achieved by the variation of the heat treatment conditions or by the introduction of elements to inhibit grain growth, thus causing an improvement of the pinning through the generation of more boundaries to scatter electrons.

Instead, the composition variation of the phases near the grain boundaries would increase the pinning force by the enhancement of the Ginzburg – Landau parameter 𝑏𝑏 near the boundaries.

This 𝑏𝑏 – enhancement is usually thought to be due to the local decrease of the electronic mean free path caused by deviations from stoichiometry close to the grain boundaries [119].

Scaling law for flux pinning

The major goal of the flux pinning theory is the prediction of the functional dependence of 𝐹𝐹_𝑒𝑒 on magnetic field 𝐵𝐵 and temperature 𝑇𝑇 under the influence of the microstructural parameters. Fietz and Webb [120] first demonstrated in a number of Nb – based alloys that, if the temperature is changed, the pinning force obeys a scaling law for which the peak in 𝐹𝐹𝑒𝑒 scales as [𝐻𝐻_𝑐𝑐2(𝑇𝑇)]^2.5: 𝐹𝐹_𝑒𝑒 = [𝐻𝐻_𝑐𝑐2(𝑇𝑇)]^2.5𝑓𝑓(𝑏𝑏), where 𝑓𝑓(𝑏𝑏) is uniquely a function of the reduced magnetic field 𝑏𝑏=_𝐵𝐵^𝐵𝐵

𝑐𝑐2. The importance of this discovery is that knowing 𝐹𝐹𝑒𝑒 at a given temperature the 𝐹𝐹𝑒𝑒 at other temperatures can be simply found by scaling the results by [𝐻𝐻_𝑐𝑐2(𝑇𝑇)]^2.5. The scaling behavior was subsequently identified in several other superconductors including Nb3Sn and many pinning force model expressions were proposed in the past decades. Recently Ekin [121] unified all these

models in that it is known today as the Unified Scaling Law (USL) and the pinning force per unit length is expressed by:

𝐹𝐹𝑒𝑒 = 𝐾𝐾(𝑑𝑑,𝜀𝜀0) 𝑓𝑓 (𝑏𝑏) ³²

where the function 𝐾𝐾(𝑑𝑑,𝜀𝜀₀) is a prefactor that describes the pinning force dependence on reduced temperature 𝑑𝑑 and intrinsic axial strain 𝜀𝜀₀ ≡ 𝜀𝜀 − 𝜀𝜀_𝑚𝑚 defined as zero at the strain 𝜀𝜀_𝑚𝑚 where 𝐼𝐼𝑐𝑐 is maximum , whereas 𝑓𝑓 (𝑏𝑏) is a function of the reduced field b:

𝑏𝑏= 𝐵𝐵

𝐵𝐵_𝑐𝑐2^∗ (𝑑𝑑,𝜀𝜀0)

33

where 𝐵𝐵_𝑐𝑐2^∗ is an effective upper critical field used for scaling (or simply scaling field), which is in general not identical to the upper critical field determined experimentally, for instance by means of resistivity measurements. Similarly, the reduced temperature 𝑑𝑑 is defined by an effective critical temperature 𝑇𝑇_𝑐𝑐^∗(𝜀𝜀₀):

𝑑𝑑 = 𝑇𝑇 𝑇𝑇_𝑐𝑐^∗(𝜀𝜀₀)

34

The field dependence of the pinning force can be expressed as:

𝑓𝑓(𝑏𝑏) =𝑏𝑏^𝑒𝑒(1− 𝑏𝑏)^{𝑞𝑞 35}

where the low- and high-field exponents 𝑒𝑒 and 𝑞𝑞 must be constant for the pinning force to be shape invariant. If these parameters would exhibit a dependence on field, temperature, or strain, scaling would fail. In polycrystalline Nb3Sn the experimentally determined exponents are often close to 𝑒𝑒= 0.5 and 𝑞𝑞 = 2.0, which is consistent with pinning by core interaction between the vortices and normal conducting surfaces, as shown in Table 1, thus implying grain boundary pinning as the dominant mechanism [111]. The reduced field at which 𝑓𝑓(𝑏𝑏) reaches its maximum is given by:

𝑏𝑏_{𝑚𝑚𝑎𝑎𝑒𝑒} = 𝑒𝑒

𝑒𝑒+𝑞𝑞 ³⁶

Thus, in case of pure grain boundary pinning with the scaling exponents 𝑒𝑒= 0.5 and 𝑞𝑞= 2 the pinning force maximum occurs at 𝑏𝑏_{𝑚𝑚𝑎𝑎𝑒𝑒} = 0.2.

Assuming that the temperature and strain dependence of both 𝐾𝐾(𝑑𝑑,𝜀𝜀₀) and 𝐵𝐵_𝑐𝑐2^∗ (𝑑𝑑,𝜀𝜀₀) can be separated, equation 32 can be cast in the convenient separable form [121]:

𝐹𝐹_𝑒𝑒 =𝐶𝐶 𝑏𝑏(𝜀𝜀₀)ℎ(𝑑𝑑)𝑓𝑓(𝑏𝑏)

In the above expression C is a constant, 𝑏𝑏(𝜀𝜀₀) and ℎ(𝑑𝑑) are the dimensionless strain and temperature dependences respectively, and 𝑓𝑓(𝑏𝑏) is the field dependence given by equation ₃₅. In 1972 Kramer [122] assumed that flux motion at low fields is primarily due to depinning, while synchronous shear of the flux line lattice is the dominant mechanism at high fields where a

“peak effect” in the pinning force arises.