Chapter 3 High Temperature Superconductors
3.3 Superconductivity in YBCO
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36 3.3.2 Anisotropy
YBCO is anisotropic in nature, meaning that its properties vary with respect to different crystallographic orientation. In general, HTSs are known to be inhomogeneous superconductors with superconductivity mainly associated to CU-O ab planes. The Cu-O planes are the main factors which accounts for the domination of anisotropy of normal and superconducting properties of HTSs. A crystal of YBCO has a larger coherence length in the ab plane than along the c-axis. Hence, the critical current is much bigger in the ab plane than along the c-axis as shown in Table 3.1. The energy gap also varies according to the crystallographic planes or directions. The energy gap is greater along the b directions and it is zero between the two axes.
Table 3.1 shows the superconducting characteristics of YBCO [27] and it can clearly deduce that the material is highly anisotropic from the crystallographic dependence of the superconducting parameters.
Due to the anisotropic nature of YBCO, the penetration depth is eventually not constant in all directions. So, an average penetration depth needs to be considered. It is in fact the average of the penetration depth in the ab plane and the c-axis .
Table 3. 1 Anisotropic Properties of YBCO [27]
Parameter Parallel to c-axis Parallel to ab-plane
ξ(Ȧ) 2 to 3 15 to 20
λ(Ȧ) 1500 7500
c (A/cm2 at 77 K) 104 106
(at 77 K) 150T 30T
37 3.3.3 Magnetic Properties of YBCO and HTS in general
Type-I superconductors are mostly elemental and type-II are compounds such as cuprate compounds. The initial distinction between type-I and type-II superconductors derived from the behavioural study of superconductors subjected to a magnetic field. Superconducting YBCO, which is a type-II superconductor, as the other cuprate superconductors, has a higher critical magnetic field than conventional superconductors (type-I). Two critical limits can be reached Hc1 and Hc2. According to Abrikosov analysis of the G-L theory, YBCO has a surface energy σs< 0 (so do other cuprate superconductors) when it is placed in a magnetic field as discussed in section 2.28 [14]. The G-L parameter k>
√ , is given by:
(3.1)
Where is the Ginzburg-Landau coherence length.
Figure 3.8 shows the magnetisation response for type-I and type-II superconductors with the applied magnetic field. As shown in Figure 3.9, below Hc1, YBCO exhibits perfect diamagnetism of the Meissner effect. Conversely to a perfect superconductor, YBCO is a flux expeller below Hc1. However, as Hc1 is exceeded, the material enters into the mixed state and it splits into normal and superconducting domains since, this is energetically favourable.
H c
H c
H a (Am) H c1 H c H c2
H a (Am) H c
Magnetisation (Am) Magnetisation (Am)
(a) (b)
Figure 3. 8 Reversible magnetisation curves for (a) Type-I (b) Type-II superconductors
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Superconducting
Normal
(a) (b) (c)
Superconducting State (perfect diamagnetism)
Mixed State Normal State
Figure 3. 9 Field profile of a superconductor: (a) Meissner state (b) Mixed state type-II (c) normal state for either type-I or type-II.
3.3.4 Critical Current of HTS (YBCO)
The critical current density is the maximum lossless value of current density above which the superconductor reverts into a normal material (undergoes resistive losses). Two types of current contribute to the critical current density: the transport current and the screening current.
Transport current, is the current which flows on the surface of a superconducting specimen, transporting current in and out of a wire. is the current density due to the transport current whereas the screening current, is the current that occurs due to the subjection of a superconductor to a magnetic field, shielding the interior of the material from the magnetic field. is the current density due to the screening current.
. (3.2)
Where, is the total current density.
If exceeds the critical current density, superconductivity breaks. So, the Cooper pairs are imparted as the energy is greater than the binding energy of the pairs (> .
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Sufficiently big current at the surface of the superconductor can lead to a value exceeding the critical current and the associated critical magnetic field. The critical magnetic field is associated with a critical current density, . So, it can be concluded that the stronger is the applied magnetic field, the smaller will be the value of the critical current. When no magnetic field is applied, the only magnetic field generated is due to the transport current.
Consider a circular superconductor carrying a current. The current seems to flow in a dimensionless line down the middle of the conductor. At a distance m away from a line current, according to Ampere‟s law, there is a magnetic field of strength:
. (3.3)
According to Silsbee‟s hypothesis [28],
, (3.4)
, (3.5)
Where:
r: radius of superconductor (x>> λ) I: current,
Ic: critical current,
: critical current density, λ : penetration depth,
: the depairing current density.
The depairing current for YBCO single crystal is very high. However, the critical current decreases significantly for YBCO samples which are granular in nature. The granular nature of YBCO limits the potential of the material for real life application. Complete depairing causes the material to change to normal when the critical current/critical field is exceeded. Before the critical field is reached, the material enters into a mixed state at . Mixed state means that, the HTS will have normal and superconducting domains. The material contains magnetic vortex lines, threading through the bulk in the form of triangular lattice known as flux line lattice (FLL).
A vortex consists of a normal core, in which the magnetic field is large and is surrounded by a
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superconducting region, in which a continuous supercurrent flows maintaining the magnetic field within the core of the material. Abrikosov predicted a square array of vortices [14] which was proved to be in fact triangular in nature due to the lower free energy of a triangular array of vortices. In the centres of the vortices, the order parameter is zero and the energy gap is inexistent leading to a loss in perfect conductivity. If transport current flows at the surface of the material, and is perpendicular to the magnetic vortices, Lorentz force will act on the vortices.
The vortices eventually move in response to the Lorentz force resulting in power dissipation.
This gives rise to resistance in the sample known as flux-flow resistance, which impedes on the current carrying capacity of HTS material.
In practice, YBCO and other HTSs are usually doped to induce pinning centres which will restrain the freedom of motion of the vortices and hence, improve the current carrying capacity of the material. For instance, in superconducting magnet, pinning centres are deliberately inserted to reduce joule losses due to interaction between flux vortices and high transport current.
Each Vortex carries a magnetic flux quantum of:
(3.6)
Where;
h is the Planck constant ( e is the electronic charge of an electron ( .