**8. High** *Tc* **in cuprates**

High-temperature superconductors (abbreviated high-*Tc* or HTS) are materials that behave as superconductors at unusually high temperatures. The first high-*Tc* superconductor was discovered in 1986 by IBM researchers Georg Bednorz and K. Alex Müller, who were awarded the 1987 Nobel Prize in Physics "for their important break-through in the discovery of superconductivity in ceramic materials".

Whereas "ordinary" or metallic superconductors usually have transition temperatures (temperatures below which they are superconductive) below 30 K (243.2°C) and must be cooled using liquid helium in order to achieve superconductivity, HTS have been observed with transition temperatures as high as 138 K (135°C) and can be cooled to superconductivity using liquid nitrogen. Compounds of copper and oxygen (so-called cuprates) are known to have HTS properties, and the term high-temperature superconductor was used interchangeably with cuprate superconductor. Examples are compounds such as lanthanum strontium copper oxide (LASCO) and neodymium cerium copper oxide (NSCO).

### *SQUID Magnetometers, Josephson Junctions, Confinement and BCS Theory of Superconductivity DOI: http://dx.doi.org/10.5772/intechopen.83714*

Let us take lanthanum copper oxide *La*2*CuO*4, where lanthanum donates three electrons and copper donates two and oxygen accepts two electrons and all valence is satisfied. The *Cu* is in state Cu2+ with electrons in *d*<sup>9</sup> configuration [11]. *d* orbitals are all degenerated, but due to crystal field splitting, this degeneracy is broken, and *dx*<sup>2</sup>*<sup>y</sup>*<sup>2</sup> orbital has the highest energy and gets only one electron (the ninth one) and forms a band of its own. This band is narrow, and hence all *k* states get filled by only one electron each and form Wannier packets that localize electrons on their respective sites. This way electron repulsion is minimized, and in the limit *U*≫*t* (repulsion term in much larger than hopping), we have Mott insulator and antiferromagnetic phase.

When we hole/electron dope, we remove/add electron to *dx*<sup>2</sup>*<sup>y</sup>*<sup>2</sup> band. For example, *La*<sup>2</sup>*xSrxCuO*<sup>4</sup> is hole doped as Sr has valence 2 and its presence further removes electrons from copper. *Nd*<sup>2</sup>*xCexCuO*<sup>4</sup> is electron doped as Ce has valence 4 and its presence adds electrons from copper. These extra holes/electrons form packets. When we discussed superconductivity, we discussed packets of width *ωd*. In d-bands, a packet of this width has many more *k* states as d-bands are narrow and only 1–2 eV thick. This means *N* (k-points in a packet) is very large and we have much larger gap Δ and *Tc*. This is a way to understand high *Tc*, d-wave packets with huge bandwidths. This is as shown in **Figure 26**. As we increase doping and add

#### **Figure 26.**

bring in an extra electron as shown in **Figure 25**? It will go in one of the empty pockets, and then we have only *p* � 1 pockets left to scatter to reducing the binding

*Depiction on how upon tunneling an extra electron, shown in dashed lines, enters the superconducting state.*

*(A) Depiction on how a tiny voltage between two metals separated by an insulating barrier generates current that goes through an insulating barrier through tunneling. (B) If one of the metals is a superconductor, then the*

the energy by Δ, and therefore to offset this increase of energy, we have to apply a

High-temperature superconductors (abbreviated high-*Tc* or HTS) are materials that behave as superconductors at unusually high temperatures. The first high-*Tc* superconductor was discovered in 1986 by IBM researchers Georg Bednorz and K. Alex Müller, who were awarded the 1987 Nobel Prize in Physics "for their important break-through in the discovery of superconductivity in ceramic materials". Whereas "ordinary" or metallic superconductors usually have transition temperatures (temperatures below which they are superconductive) below 30 K

superconductivity, HTS have been observed with transition temperatures as high as

138 K (135°C) and can be cooled to superconductivity using liquid nitrogen. Compounds of copper and oxygen (so-called cuprates) are known to have HTS properties, and the term high-temperature superconductor was used interchangeably with cuprate superconductor. Examples are compounds such as lanthanum strontium copper oxide (LASCO) and neodymium cerium copper oxide (NSCO).

*Np*

*<sup>ω</sup><sup>d</sup>* . Therefore, the new electron raises

*<sup>ω</sup><sup>d</sup>* with a change <sup>Δ</sup> <sup>¼</sup> <sup>4</sup>ℏ*d*<sup>2</sup>

(243.2°C) and must be cooled using liquid helium in order to achieve

energy to � <sup>4</sup>ℏ*d*<sup>2</sup>

**Figure 24.**

**Figure 25.**

**90**

**8. High** *Tc* **in cuprates**

*Np p*ð Þ �1

voltage as big as Δ for tunneling to happen.

*applied voltage has to be at least as big as the superconducting gap.*

*Magnetometers - Fundamentals and Applications of Magnetism*

*(A) Depiction on how in narrow* d *band the electron wave packet comprises all* k *points to minimize repulsion. (B) The wave packet in two dimensions with wave packet formed from superposition of k-points (along the kdirection) inside the Fermi sphere.*

#### **Figure 27.**

*The characteristic phase diagram of a high* Tc *cuprate. Shown are the antiferromagnetic insulator (AF) phase on the left and* dome*-shaped superconducting phase (SC) in the centre.*

more electrons, the packet width further increases till it is ≫*ωd*; then, we have significant offset evolution, and binding is hurt leading to loss of superconductivity. This explains the *dome* characteristic of superconducting phase, whereby superconductivity increases and then decreases with doping. The superconducting *dome* is shown in **Figure 27**.

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*DOI: http://dx.doi.org/10.5772/intechopen.83714*

*SQUID Magnetometers, Josephson Junctions, Confinement and BCS Theory of Superconductivity*

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