# Pseudogap Isotope Effect as a Probe of Bipolaron Mechanism in High Temperature Superconductors

## Abstract

**:**

## 1. Introduction

## 2. The Critical Temperature of a Pseudogap Phase

## 3. Isotope Coefficient for the Pseudogap Phase

## 4. Isotope Coefficient for Pseudogap Phase in Magnetic Field

## 5. Discussion

_{2}, Bi-2223, Bi-2212, YBaCuO, etc. at temperatures substantially above critical temperature ${T}_{c}$ electron pairs in the form of TI-bipolarons represent the noninteracting strongly bound bosons, demonstrating with a decreasing temperature a transition to the Bose–Einstein condensation. This concept can explain a lot of experiments such as excess conductivity [22], ultrafast carrier localization into the polaronic state [23], Nernst effect as one of the most convincing demonstrations for the existence of the preformed pairs [3], tunneling, and spectroscopic properties [16,24].

_{2}HTSC seem to be of importance [38]. As distinct from oxide ceramics, MgB

_{2}does not have a magnetic order and, as proponents of an external nature of the pseudogap state suggest, should not have a pseudogap. To exclude any other possibilities concerned with disorder, low-dimensionality effects, etc., in experiment [38] use was made of highly perfect crystals. Experiments made in [38] convincingly demonstrated the availability of the pseudogap state in MgB

_{2}and the responsibility of EPI for this state. The results obtained provide good evidence for the TI bipolaron mechanism of the formation of the pseudogap state.

_{2-x}Sr

_{x}CuO

_{4}in the vicinity of the temperature of charged stripe ordering when replacing

^{16}O by

^{18}O [39]), clusters, other types of interactions, etc. The suggested TI bipolaron mechanism of the pseudogap phase formation and explanation of isotopic effects in HTSC materials on its basis are also important in view of the universality of this mechanism. Most HTSC is spatially inhomogeneous. Therefore, one can think that the translation invariant bipolarons are not applicable there. However, this can be not correct as long as inhomogeneity is not very large. As shown in [40], the crystal defects can capture and destroy TI- bipolaron only if their potential well is sufficiently large. Otherwise, TI-bipolaron remains stable.

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Derivation of Formula (13)

## References

- Norman, M.R.; Pines, D.; Kallin, C. The pseudogap: Friend or foe of highTc? Adv. Phys.
**2005**, 54, 715–733. [Google Scholar] [CrossRef][Green Version] - Vishik, I.M.; Lee, W.S.; He, R.-H.; Hashimoto, M.; Hussain, Z.; Devereaux, T.P.; Shen, Z.-X. ARPES studies of cuprate Fermiology: Superconductivity, pseudogap and quasiparticle dynamics. New J. Phys.
**2010**, 12, 105008. [Google Scholar] [CrossRef] - Timusk, T.; Statt, B. The pseudogap in high-temperature superconductors: An experimental survey. Rep. Prog. Phys.
**1999**, 62, 61–122. [Google Scholar] [CrossRef][Green Version] - Hüfner, S.; Hossain, M.A.; Damascelli, A.; A Sawatzky, G. Two gaps make a high-temperature superconductor? Rep. Prog. Phys.
**2008**, 71, 062501. [Google Scholar] [CrossRef][Green Version] - Lee, P.A.; Nagaosa, N.; Wen, X.-G. Doping a Mott insulator: Physics of high-temperature superconductivity. Rev. Mod. Phys.
**2006**, 78, 17–85. [Google Scholar] [CrossRef] - Randeria, M.; Trivedi, N. Pairing correlations above Tc and pseudogaps in underdoped cuprates. J. Phys. Chem. Solids
**1998**, 59, 1754–1758. [Google Scholar] [CrossRef] - Franz, M. Importance of fluctuations. Nat. Phys.
**2007**, 3, 686–687. [Google Scholar] [CrossRef] - Emery, V.J.; A Kivelson, S. Importance of phase fluctuations in superconductors with small superfluid density. Nature
**1995**, 374, 434–437. [Google Scholar] [CrossRef] - Curty, P.; Beck, H. Thermodynamics and Phase Diagram of High Temperature Superconductors. Phys. Rev. Lett.
**2003**, 91, 257002. [Google Scholar] [CrossRef] [PubMed][Green Version] - Moon, E.G.; Sachdev, S. Competition between spin density wave order and superconductivity in the underdoped cuprates. Phys. Rev. B
**2009**, 80, 035117. [Google Scholar] [CrossRef][Green Version] - Sadovskii, M.V. Pseudogap in high-temperature superconductors. Physics-Uspekhi
**2001**, 44, 515–539. [Google Scholar] [CrossRef][Green Version] - Bardeen, J.; Cooper, L.N.; Schrieffer, J.R. Theory of Superconductivity. Phys. Rev.
**1957**, 108, 1175–1204. [Google Scholar] [CrossRef][Green Version] - Lakhno, V.D. Superconducting Properties of 3D Low-Density Translation-Invariant Bipolaron Gas. Adv. Condens. Matter Phys.
**2018**, 2018, 1–12. [Google Scholar] [CrossRef][Green Version] - Lakhno, V. Superconducting properties of a nonideal bipolaron gas. Phys. C Supercond.
**2019**, 561, 1–8. [Google Scholar] [CrossRef][Green Version] - Lakhno, V.D. Superconducting Properties of 3D Low-Density TI-Bipolaron Gas in Magnetic Field. Condens. Matter
**2019**, 4, 43. [Google Scholar] [CrossRef][Green Version] - Lakhno, V.D. Translational-Invariant Bipolarons and Superconductivity. Condens. Matter
**2020**, 5, 30. [Google Scholar] [CrossRef] - Kagan, M.Y. Modern Trends in Superconductivity and Superfluidity; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar] [CrossRef][Green Version]
- Temprano, D.R.; Mesot, J.; Janssen, S.; Conder, K.; Furrer, A.; Mutka, H.; Müller, K.A. Large Isotope Effect on the Pseudogap in the High-Temperature SuperconductorHoBa
_{2}Cu_{4}O_{8}. Phys. Rev. Lett.**2000**, 84, 1990–1993. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bendele, M.; Von Rohr, F.; Guguchia, Z.; Pomjakushina, E.; Conder, K.; Bianconi, A.; Simon, A.; Bussmann-Holder, A.; Keller, H. Evidence for strong lattice effects as revealed from huge unconventional oxygen isotope effects on the pseudogap temperature in La
_{2−x}SrxCuO_{4}. Phys. Rev. B**2017**, 95, 014514. [Google Scholar] [CrossRef][Green Version] - Furrer, A. Neutron Scattering Investigations of Charge Inhomogeneities and the Pseudogap State in High-Temperature Superconductors. In Superconductivity in Complex Systems; Structure and Bonding book series (STRUCTURE, v. 114); Müller, A., Bussmann-Holder, A., Eds.; Springer: Berlin/Heidelberg, Germany, 2005. [Google Scholar] [CrossRef]
- Bill, A.; Kresin, V.Z.; Wolf, S.A. The isotope Effect in Superconductors. In Pair Correlations in Many-Fermion Systems; Plenum Press: New York, NY, USA, 1998; pp. 25–56. [Google Scholar] [CrossRef]
- Solov’Ev, A.L.; Dmitriev, V.M. Fluctuation conductivity and pseudogap in YBCO high-temperature superconductors (Review). Low Temp. Phys.
**2009**, 35, 169–197. [Google Scholar] [CrossRef] - Madan, I.; Kurosawa, T.; Toda, Y.; Oda, M.; Mertelj, T.; Mihailović, D. Evidence for carrier localization in the pseudogap state of cuprate superconductors from coherent quench experiments. Nat. Commun.
**2015**, 6, 6958. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kordyuk, A. Pseudogap from ARPES experiment: Three gaps in cuprates and topological superconductivity (Review Article). Low Temp. Phys.
**2015**, 41, 319–341. [Google Scholar] [CrossRef][Green Version] - Labbé, J.; Bok, J. Superconductivity in Alcaline-Earth-Substituted La
_{2}CuO_{4}: A Theoretical Model. EPL Europhys. Lett.**1987**, 3, 1225–1230. [Google Scholar] [CrossRef] - Radtke, R.J.; Norman, M.R. Relation of extended Van Hove singularities to high-temperature superconductivity within strong-coupling theory. Phys. Rev. B
**1994**, 50, 9554–9560. [Google Scholar] [CrossRef] [PubMed][Green Version] - Schüttler, H.-B.; Pao, C.-H. Isotope Effect ind-Wave Superconductors. Phys. Rev. Lett.
**1995**, 75, 4504–4507. [Google Scholar] [CrossRef] [PubMed][Green Version] - Nazarenko, A.; Dagotto, E. Possible phononic mechanism for dx2−y2 superconductivity in the presence of short-range antiferromagnetic correlations. Phys. Rev. B
**1996**, 53, R2987–R2990. [Google Scholar] [CrossRef] [PubMed][Green Version] - Greco, A.; Zeyher, R. Electronic correlations, electron-phonon interaction, and isotope effect in high-Tccuprates. Phys. Rev. B
**1999**, 60, 1296–1302. [Google Scholar] [CrossRef][Green Version] - Perali, A.; Innocenti, D.; Valletta, A.; Bianconi, A. Anomalous isotope effect near a 2.5 Lifshitz transition in a multi-band multi-condensate superconductor made of a superlattice of stripes. Supercond. Sci. Technol.
**2012**, 25. [Google Scholar] [CrossRef][Green Version] - Valletta, A.; Bianconi, A.; Perali, A.; Saini, N. Electronic and superconducting properties of a superlattice of quantum stripes at the atomic limit. Eur. Phys. J. B
**1997**, 104, 707–713. [Google Scholar] [CrossRef] - Medicherla, V.R.R.; Patil, S.; Singh, R.S.; Maiti, K. Origin of ground state anomaly in LaB
_{6}at low temperatures. Appl. Phys. Lett.**2007**, 90, 62507. [Google Scholar] [CrossRef] - Chainani, A.; Yokoya, T.; Kiss, T.; Shin, S.; Nishio, T.; Uwe, H. Electron-phonon coupling induced pseudogap and the superconducting transition in Ba
_{0.67}K_{0.33}BiO_{3}. Phys. Rev. B**2001**, 64, 180509. [Google Scholar] [CrossRef][Green Version] - Yokoya, T.; Chainani, A.; Kiss, T.; Shin, S.; Hirata, K.; Kameda, N.; Tamegai, T.; Nishio, T.; Uwe, H. High-resolution photoemission study of low-Tc superconductors: Phonon-induced electronic structures in low-Tc superconductors and comparison with the results of high-Tc cuprates. Phys. C: Supercond.
**2002**, 378-381, 97–101. [Google Scholar] [CrossRef] - Sacépé, B.; Chapelier, C.; Baturina, T.I.; Vinokur, V.M.; Baklanov, M.R.; Sanquer, M. Pseudogap in a thin film of a conventional superconductor. Nat. Commun.
**2010**, 1, 140. [Google Scholar] [CrossRef][Green Version] - Mondal, M.; Kamlapure, A.; Chand, M.; Saraswat, G.; Kumar, S.; Jesudasan, J.; Benfatto, L.; Tripathi, V.; Raychaudhuri, P. Phase Fluctuations in a Strongly Disordereds-Wave NbN Superconductor Close to the Metal-Insulator Transition. Phys. Rev. Lett.
**2011**, 106, 047001. [Google Scholar] [CrossRef][Green Version] - Thakur, S.; Biswas, D.; Sahadev, N.; Biswas, P.K.; Balakrishnan, G.; Maiti, K. Complex spectral evolution in a BCS superconductor, ZrB12. Sci. Rep.
**2013**, 3, 3342. [Google Scholar] [CrossRef] [PubMed] - Patil, S.; Medicherla, V.R.R.; Ali, K.; Singh, R.S.; Manfrinetti, P.; Wrubl, F.; Dhar, S.K.; Maiti, K. Observation of pseudogap in MgB
_{2}. J. Phys. Condens. Matter**2017**, 29, 465504. [Google Scholar] [CrossRef][Green Version] - Lanzara, A.; Zhao, G.-M.; Saini, N.L.; Bianconi, A.; Conder, K.; Keller, H.; A Müller, K. Oxygen-isotope shift of the charge-stripe ordering temperature in La
_{2−x}Sr_{x}CuO_{4}from x-ray absorption spectroscopy. J. Phys. Condens. Matter**1999**, 11, L541–L546. [Google Scholar] [CrossRef][Green Version] - Lakhno, V.D. Translation invariant theory of polaron (bipolaron) and the problem of quantizing near the classical solution. J. Exp. Theor. Phys.
**2013**, 116, 892–896. [Google Scholar] [CrossRef][Green Version] - Lakhno, V.D. Isotope Effect in the Translation-Invariant Bipolaron Theory of High-Temperature Superconductivity. Condens. Matter
**2020**, 5, 80. [Google Scholar] [CrossRef] - Alexandrov, A.S. Superconducting Polarons and Bipolarons. In Polarons in Advanced Materials; Alexandrov, A.S., Ed.; v.103; Springer: Berlin/Heidelberg, Germany, 2007; pp. 257–310. [Google Scholar]

**Figure 2.**Dependence of the isotope coefficient for the pseudogap temperature ${T}^{*}$ on the phonon frequency ${\omega}_{0}\left({\omega}^{*}\approx {T}_{c}\right)$.

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Lakhno, V.D. Pseudogap Isotope Effect as a Probe of Bipolaron Mechanism in High Temperature Superconductors. *Materials* **2021**, *14*, 4973.
https://doi.org/10.3390/ma14174973

**AMA Style**

Lakhno VD. Pseudogap Isotope Effect as a Probe of Bipolaron Mechanism in High Temperature Superconductors. *Materials*. 2021; 14(17):4973.
https://doi.org/10.3390/ma14174973

**Chicago/Turabian Style**

Lakhno, Victor D. 2021. "Pseudogap Isotope Effect as a Probe of Bipolaron Mechanism in High Temperature Superconductors" *Materials* 14, no. 17: 4973.
https://doi.org/10.3390/ma14174973