We are taught in school that all processes in everyday life are associated with the dissipation of energy. For example, driving a car, more energy is spent than required for mere movement of the car. This is because some of the energy obtained by burning the gasoline is dissipated into the environment through imperfect efficiency of the engine, friction of wheels, and air resistance to the car. Similarly, transport of electrical current through a wire is associated with energy loss. This is due to interaction of the electrons traveling through the wire with the atoms in the wire, whereby they give some of their energy to the atoms, heating up the wire. This phenomenon is called electrical resistance. Wherever we look around us, there is abundance of examples like this. However, superconductors are an exception, they do not exhibit electrical resistance [5-7]. Electrical current flows through superconductors without any loss, until it reaches a critical value. So are they not subject to the same physical laws as all the other phenomena around us? The answer to this can be found in the laws of quantum mechanics.
Quantum mechanics is a branch of physics applicable to atomic and subatomic particles. One of its main postulates is that the energy of these particles can have only certain discrete values [8-11]. The quantum state of a particle, or a system of particles, with the lowest allowed energy is called the ground state. The energy of a particle in the ground state can change only by an amount equal to the difference between the energy of higher lying states and the ground state. Therefore, there is a minimum energy required by the particle in the ground state in order to attain a state with the nearest higher energy level. On the other hand, it cannot transfer its energy to its surroundings, because it is already in the state with the lowest allowed energy level. These principles are vital for understanding the superconductivity and consequently for developing the idea of vortex pinning by nanoparticles in high-temperature superconductors.
The first model explaining superconductivity was the Bardeen-Cooper-Schriefer (BCS) model [12, 13]. It described superconductivity as a quantum state where electrons or holes form Cooper pairs. By pairing, they attain a lower energy level and they are in the ground state. The nearest available state is of a higher energy and there is an energy gap between these two states. When in the ground state, the charge carriers cannot exchange energy with their environment, as long as the energy of the environment is lower than this energy gap. Because of this, they can transfer electrical current through a wire without heating it up (i.e., there is no
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Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 7: Pages (207-218)
electrical resistance in this wire). However, if thermal energy of the environment is high enough to overcome the energy gap, the Cooper pairs will be broken up and the charge carriers will attain quantum state with a higher energy level. The temperature at which this occurs is called the critical temperature (Tc). BCS theory predicts that the energy gap is actually 3.52 times the thermal energy at Tc, which is close to the value for most of the known superconductors.
All this implies that the superconducting state is a state of lower energy than the resistive state. Therefore, this is a stable state of the materials exhibiting superconductivity and transition into the resistive state requires input of external energy.
Another characteristic property of superconductors is that they screen out the external magnetic field, so that the field B = ^0(H + M) is zero inside the superconducting volume [5-7]. Here, B, H, and M are vectors of the net magnetic field (also called magnetic induction), external field, and magnetization, respectively. /x0 = 4w x 10-7 T2 m3/J is the permeability of vacuum. This occurs because the magnetic field would otherwise break up the Cooper pairs and the total energy of the system would be higher if the field penetrated superconducting volume. Superconductors are divided into two groups, in regard to their response to external magnetic field. The type I superconductors screen out the external field completely, for small magnetic fields. This is accomplished by maintaining persistent supercurrent near the surface of the superconductor. These supercurrents produce internal field (or magnetization M), which cancels the field H. For higher fields, the thickness of the layer carrying the supercurrents gradually increases and superconductivity is finally broken down at a field called the critical field Hc.
The type II superconductors behave the same way as type I, up to the first critical field, Hc1. For higher fields, they are in a mixed state. Here, the field penetrates the superconductor in the form of thin filaments, called magnetic vortices  (Fig. 1). Each vortex carries magnetic flux of one flux quantum: i>0 = 2.067 x 10-15 Tm2. The vortex
Figure 1. Schematic drawing of the field entering the type II superconductor through magnetic vortices (shaded).
consists of a core, which is in nonsuperconducting state. The diameter of the core is equal to the coherence length, Magnetic flux inside this core is screened out from the rest of the superconducting volume by persistent supercurrents, circulating the core. The value of the supercurrents decreases exponentially with the distance from the vortex center, with characteristic length scale equal to the London penetration depth, A. This is mirrored in exponential decrease of B with distance from the core (Fig. 2). As external field increases, more magnetic vortices move into the superconductors from its surface. The overall screening around the whole of the superconductor, similar to the type I superconductors, still persists in the mixed state. When the external field reaches a value called the second critical field, Hc2, superconductivity is broken down.
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