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High Temperature Superconductivity

Liz Prettner February 17, 2010 History of High Temperature Superconductors (HTS)

A superconductor can be defined as a material that exhibits zero resitivity, but infinite conductivity below the critical temperature.[7] The search for high temperature superconductors (HTS) has been ongoing since the discovery of the first superconductor. A graph of the high temperature superconductors and the year of their discovery can be found in figure 1. In 1986, the high temperature superconductor LaBaCuO which has a Tc of 35K was discovered by Karl Muller and Johannes Bednorz. [1] This discovery led to the a new interest in a class of superconductors known as cuprates that consist of copper-oxygen planes.[1] The critical temperature was increased to 90K, above the liquid temperature of nitrogen, with the discovery of YBaCuO in 1987.[5] Cuprates Fig 1. Graph of the highest temperature superconductors and the year remained (and still do) a high priority of discovery.[16] due to its high Tc until the discovery of superdconductivity in iron-pnictides by a collaborative group at the Tokyo Institute for Technology led by Yoichi Kamihara who discovered that the CuO plane is not a requirement for superconductivity. [3] While the mechanism for superconductivity in both cuprates and pnictides is still to a great extent unknown, there have been many attempts made to explain the phenomenon. HTS are type-II superconductors.[7] The meaning and repercussions of this will be explored as well as the mechanism for superconductivity in the relatively new discovered pnictides.

Type-II Superconductors

HTS are type-II superconductors. While type-I superconductors are typically metallic, type-II superconductors are alloys. Type-II superconductors occur when the coherence length, is shorter than the penetration depth, . This also means that the electronic mean free path is short. [7] The penetration depth can be derived from the London equations. It results from the characterization of the superconductor.[7] ,[1]

L =

mc 2

nq 2

1

The coherence length is a measurement of the distance between a normal state and a superconducting state. It is governed by the Landau-Ginzburg equations. The coherence length is determined by the energy gap. [7],[1]

o =

2h f

E g

Fig 2 (a) Type-I superconductor where the penetration depth is shorter than the coherence length. (b) Type-II superconductor where the penetration depth is longer than the coherence length.[16]

Instead of only having one critical field, as in a type-I superconductor, type-II superconductors have two critical temperatures, which results in three regions. Both type-I and type-II superconductors display the Meissner-Ochsenfeld phenomenon below the Hc (Hc1 in the case of type-II superconductors) preventing the field from penetrating into the bulk sample until a critical field where the field then is able to completely penetrate the sample resulting in a normal state. However, a type-II superconductor above Hc1 but below Hc2 allows the field to penetrate in regions, known as vortices, which are discussed in greater detail below. The vortices, also known as fluxoids, only form when it results in a lower energy state. The region below Hc1 is a superconducting state while the region above Hc2 is a normal state. The region between Hc1 and Hc2 is the vortex state, which allows the magnetic field to penetrate into the bulk sample in small regions. 7],[10]

Fig 3 (a) Typical H vs T phase diagram for a Type-I superconductor. Note there is only one critical field. (b) Typical phase diagram for a type-II superconductor displaying the existence of two critical fields. The type-I critical field dashed line is drawn only for comparison.[10]

The next question is how are Hc, in the case of the type-I superconductor, and Hc1 and Hc2, in the case of the type-II superconductor calculated. For the answer, we return to the Landau-Ginzburg equations. 2

The formula for Hc1, where o is the quantum of flux and is the penetration depth, is as follows. [7]

H c1

o 2

Hc2 on the other hand is dependent on the coherence length, and is where the packets of flux are packed as closely as possible. [7],[1]

H c2

o 2

This means that there is one vortex within a 2 region and that the vortices are within a coherence length of one another. [7],[1] It is known that Hc2 is higher than Hc by the equation given in Kittel, which was calculated by adding the stabilization energy for the core and the surrounding material. The stabilization energy must be less than zero, so it is set to zero for determining the maximum amount. By going through these calculations, the following can be determined.[7],[1]

H c 2 ~ H c

is defined as the Landau-Ginzburg parameter and is equal to the penetration depth divided by the coherence length. [7],[1]

=

Vortex

Vortices exist when the magnetic field is not completely expelled from the sample itself and may penetrate into the sample in order to lower the magnetic energy. A superconducting current then forms, returning the region within the vortex to a normal state. This is a second order transition since there is no latent heat involved. Vortices typically form when the external field is parallel to the c-axis of the crystal, as is depicted in figure 4 and solely exist in type-II superconductors between the Hc1 and Hc2 state. As the field increases, so do the number of vortices and their size. At the second critical field, Hc2, the packets are within a coherence length, , away from one another. [7],[10],[1]

Fig 4 (a) Type-II superconductor with a vortex that has formed. Note the single vortex, the region it encompasses is in a normal state while it is surrounded by a superconducting state. (b) The green lines depict the external magnetic field, while the pink lines depict the superconducting current. The region within the superconducting current is a normal state, while the region outside of the current is still a superconducting state.[10]

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Vortex Pinning

Alexei Abrikosov predicted that when a type-II superconductor contains multiple vortices, they may form a triangular lattice in the a-b plane. This is known as an Abrikosov lattice.

Fig 5 (a) A triangular lattice to better depict the idea of an Abrikosov lattice. [10] (b) Vortex pinning evident in a niobium superconductor. This image was produced by an electron microscope. Not the lattice formation. This is known as an Abrikosov lattice.[16]

The vortices will prefer to remain in the lattice formation, but may move as a result of the Lorentz force if there are temporary currents in the a-b plane. This is known as flux creep. The vortices then move around the crystal structure. This may be prevented if there is a defect or contaminant in the crystal structure, which will cause the vortex to become pinned. Vortex pinning in high temperature superconductors is crucial because if vortices are allowed to creep, they can result in net currents within the crystal, which can cause portions of the sample to return to the normal state. As the field is increased, the number of vortices increases and the probability that the vortices will cause net currents increases.[10]

Vortex Quantization

Fig 6 Click to view video of flux creep in NbSe2[11]

By implementing basic electrodynamics equations and Bohr-Sommerfeld quantization , London discovered that flux is quantized in the vortex.

=

BCS Theory

nhc = n o q

BCS theory was developed in 1957 by Bardeen, Cooper, and Schrieffer to explain superconductivity. It was suggested that lattice distortions creates a cooper pair. The Cooper pairs results in a lower Fermi energy.

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Fig 7 (a) The video depicts how the cooper pairs form when the lattice is distorted. (b) Depicts how the lowered energy causes the electrons to move sans resistance.[16]

These pairs act like bosons, and thus are able to move through the lattice with relative ease. The simulations in figure 7 best demonstrate this idea. The electrons pair through s-wave pairing. This means that in the momentum space, the Fermi surface forms an s-wave and the electrons must have a momentum of k and -k as seen in figure 8.[7],[10]

Fig 8 (a) s-wave Fermi surface typical in the BCS theory. (b) d-wave Fermi surface typically found in cuprates. Note the four nodes. These are the normal states.10]

Cooper pairs cause a decrease in the Fermi energy, which thus results in an energy gap. This is because cooper pairs act like bosons and thus can occupy the same energy levels. In order to break the cooper pair, there needs to be enough energy to overcome the energy gap.[10] BCS theory explains the phenomenon seen in conventional superconductors, but does not thoroughly explain high temperature superconductors. BCS theory Fig 9 Cooper pair formation creating an predicts phonons are responsible for the formation of [16] Cooper pairs, but this only works with low temperature energy gap. superconductors. Cooper pairs are still thought to exist in HTS, but the mechanism for the pairing is thought to be different due to theoretical predictions. 5

Instead of phonons being responsible for the pairs, magnetic spin fluctuations are thought to be the cause.[17]

Cuprates

Cuprates are mentioned only for a comparison to pnictides. Although cuprates are still not fully understood, extensive experimentation has been performed on them as opposed to pnictides which are still considered fairly new.

Interest

The main reason for interest in cuprates is the high critical temperature. Superconductivity in cuprates is attributed to the copper-oxygen planes, which are depicted in figure 10. It has been debated as to whether cuprates have s-wave pairing and/or follow BCS theory, but it has been determined that cuprates have d-wave pairing. It has also been determined that Cooper pairings still take place in cuprates, but it is the spin fluctuations that induce Cooper pair formation and thus a superconducting state exhibiting a dx2-y2 symmetry.[6]

Structure

Cuprates have a copper-oxygen plane that is responsible for the superconducting properties. Pairing of electrons occurs within the conducting plane.

Fig 10 (a) Cu-O planes found in cuprates. The black circles represent oxygen atoms while the white circles represent the copper.[17] (b) k-space depiction of YBCO. Note the d-wave symmetry.[10]

Phase Diagram

The phase diagram consists of a parent antiferromagnetic state. There is the typical dome of superconductivity with a pseudogap.

Fig 11 Typical phase diagram for cuprates.[18]

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Pnictides Interest

Pnictides are a relatively new class of superconductors discovered by group at the Tokyo Institute for Technology led by Yoichi Kamihara. They are named for the pnictogen group, which is any member of the nitrogen group including Phosphorous, Arsenic, etc. as depicted in figure 12. Pnictides are considered a type-II superconductor and thus produce vortices between Hc1 and Hc2. Before pnictides, it was thought to need the copper oxygen plane to produce layers sufficient for controlling the charge carrier density.[3]

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Structure

Pnictides are layered, similar to cuprates. The iron-arsenic plane is considered of pnictogens. to produce the same effect as the copper-oxygen plane in cuprates and to be responsible for the superconductivity found in pnictides. For example, the critical temperature of LaOFeAs was found to increase with a doping of fluorine. The doping was found to increase the number of carriers in the conductive layer. Figure 13, from an article on LaOFFeAs shows the layered structure with the formation of carriers in the conductive layer.[12]

Fig 12 Group consisting

Fig 13 (a) LaOFFeAs. Doping of the F in the oxygen spaces causes an increase in the number of electron carriers.(b)Resistivity data for LaO1-xFxFeAs for different pressures. Note as the pressure increases, the critical temperature appears to increase.[12]

The iron, Fe2+, forms tetrahedrons within the layers. According to the article by Yoichi Kamihara, this means that the pnictide's Fermi level is formed by 3dxy, 3dyz, or 3dzx orbitals. This is markedly different from cuprates, which form a square, planar structure and where there is 3dx2-y2 symmetry.[3]

Critical Temperature

One of the first discovered pnictide superconductors, LaOFeP, possessed a critical temperature of ~3K. By doping some of the oxygen sites with fluorine, the critical temperature could be increased to ~26K. Shortly thereafter, it was discovered that one could increase the critical temperature by increasing the 7

pressure. In the article by Hiroki Takahashi, it was found that for LaO1-xFxFeAs 3GPa of pressure could increase the critical temperature from 26K to 43K. Figure 13b depicts the resistivity data for various pressures for LaO1-xFxFeAs.[12] It should be mentioned that not all pnictides need to have iron. It has recently been discovered that LiCu2P2 demonstrates iron pnictide superconductors as can be seen best in figure 14.[13] This may be due to the similar structure shared by both the iron pnictides and LiCuP.[13]

Phase Diagram

Fig 14 LiCu2P2 is an iron free pnictide The phase diagram for a pnictide is similar to that for that has the same structure as the iron cuprates. Both have regions of Fermi-liquids for high doping pnictides.[13] and both have dome-like superconducting regions. The main difference lies within the parent compound, which for pnictides is can be spin density waves or an antiferromagnet, depending upon the compound.

Fig 15 (a) Phase diagram for CeFeAsOF. (b) Phase diagram for LaOFFeAs.[19]

Superconductivity: Spin Fluctuations or phonons?

Pnictides do not completely obey BCS theory since there is no Hebel-Slichter peak in NMR relaxation data below the critical temperature. The question then is what dominates the superconductivity if not BCS? This has been an ongoing debate between two possible solutions. There has been an ongoing battle within the world of pnictides whether the superconductivity is a result of electron-phonon interactions or if it caused by magnetic fluctuations. There have been many papers on both sides of this argument, but the most recent papers seem to hint at magnetic fluctuations. One theoretical paper in particular by Boeri, Dolgov, and Golubov calculates the maximum Tc in LaFeAsO1-xFx if electron-phonon coupling were solely responsible for the superconductivity. They calculate using the Migdal-Eliashberg theory that the Tc for such a system would be 0.8K as opposed to the actual Tc~26K. In order to obtain a reasonable Tc close to the experimental value, they would need five times higher electron-phonon coupling constant. So the electron-photon interaction cannot be the only explanation for the presence of superconductivity. [9] Another idea proposed was that the magnetic spin fluctuations could possibly be responsible for the 8

superconductivity. A team of researchers, led by Drew at the University of Fribourg has shown by using muon spin rotation that for SmFeAsO1-xFx the magnetic fluctuations seem to increase rapidly around the critical temperature.[15]

Conclusion

In conclusion, it has been demonstrated that pnictides are a new class of high temperature superconductors, specifically type-II superconductors. This poses interesting experimentation problems, especially in the vortex region. In NMR it is important to have enough spins in the superconducting or normal state in order to produce a reasonable signal. The combination of both produces unique issues and often a different line shape. This needs to be kept in mind especially when working with high magnetic fields since the number of vortices increases as the magnetic field increases. Pnictides are similar to cuprates in the sense that they are both unconventional superconductors. The layers themselves vary slightly from one another, where cuprates form a square plane while pnictides are more tetragronal. The electron-phonon interactions do not dominate their Cooper pairing. They have different symmetries and pairing mechanism (d-wave/s-wave). Pnictides and cuprates are both also layered superconductors, which results in a conductive band where the carriers are able to move freely. Further studies will hopefully reveal a more conclusive reasoning for the mechanism of superconductivity. With this knowledge, the hope is to better understand superconductivity at a basic level to one day develop a room temperature superconductor.

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References

[1] Tinkham, M. Introduction to Superconductivity, 2nd ed.; Dover Publications, Inc, 1996. [2] Bednorz, J.G.; Muller, K.A.; Zeitschrift fur Physik B: Condensed Matter 1986, 64, 189-193. [3] Kamihara, Y.; Hiramatsu, H.; Hirano, M.; Kawamura, R.; Yanagi, H.; Kamiya, T.; Hosono, H. Journal of the American Chemical Society 2006, 128, 10012-10013. [4] Yang, W.L.; Sorini, A.P.; et al. Physical Review B 2009, 80, 014508.1-10. [5] Wu, K.M; Ashburn, J.R.; Torng, C.J. Physical Review. Letters 1987, 58 (9): 908. [6] Monthoux, P.; Balatsky, A.V.; Pines, D. Physical Review B 1992, 42, 22. [7] Kittel, C. Introduction to Solid State Physics, 7th ed.; John Wiley & Sons, Inc. 1996. [8] Cao, C. et al. Physical Review B 2008, 77, 220506(R) [9] Boeri, L. et al. Physical Review Letters 2008, 101, 026403 [10] Shinn, M.A.; Wirth, F.; Amarasiriwardena, D.; Kossler, W.J. Vortex Pinning in Type-II Superconductors and the d-wave model. Hampshire College. 2008. [11] University of Oslo. Real-Time Magneto-Optical Imaging of Vortex Lattice. http://www.fys.uio.no [12] Takahashi, H.;Igawa, K.; Arii, K.; Kamihara, Y.; Hirano, M.; Hosono, H. Nature, 2008, 453, 376-378. [13] Han, J.T.; Zhou, J.S.; Cheng, J.G.; Goodenough, J.B. Journal of the American Chemical Society, 2010, 3, 132. [14] Pascher,N.; Deisenhofer, J; Krug von Nidda, H.A.; Hemmida, M.; Jeevan, H.S.; Gegenwart, P; Loidl, A. 2010. Arxiv 1001.1302v1 [15] Drew, A.J; Pratt, F.L; Lancaster, T.; et. Al. Physical Review Letters 2008, 101, 097010. [16] University of Cambridge. Teaching and Learning Packages. 12 September 2008. [17] Tsuei, C.C.; Kirtley, J.R. Review of Modern Physics, 2000, 4, 72. [18] Broun, D.M. Nature Physics 2008, 4, 170-172. [19] Normal, M. Physics 2008, 1, 21.

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