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Possible Benefits from Phonon/Spin-Wave Induced Gaps below or above E_{F} for Superconductivity in High-T_{C} Cuprates

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Results

## 4. Possible Spin-Waves

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The result of ${T}_{C}$ from Equation (1) with $V=0.5$ meV for the gap at ${E}_{F}$ (green heavy line), for the gap 5 meV above (thin blue line) and for 5 meV below (red broken line), ${E}_{F}$. Note that ${T}_{C}$ is the transition temperature at which the (gain) of electronic energy equals the (cost) of vibrational energy for each of the three positions of the gap. This means that ${T}_{C}$ is the superconducting gap for the gap at ${E}_{F}$, while the ${T}_{C}$’s for the other two gap positions are just the temperature at which their respective gaps can appear. The blue and red curves in the figure will be pushed downwards if the gaps are more distant from ${E}_{F}$.

**Figure 2.**An example of how a free-electron DOS (the thin horizontal line) is deformed when a phonon sets up a periodic potential with a larger k-vector than ${k}_{F}$. A gap (here, of 16 meV) is then apparent above ${E}_{F}$ (10 meV in this example shown by the bold blue line). The thin red line is the Fermi–Dirac distribution for the gapped DOS, and the broken bold blue line shows the occupied part of the DOS. The thin broken red line is the Fermi–Dirac distribution for the non-gapped DOS, which is equivalent to the Fermi–Dirac occupation of the constant DOS. As observed by the broken lines, the occupied Fermi–Dirac edge moves down by the presence of the gap above ${E}_{F}$. This leads to a gain in electronic energy, but the gain becomes smaller as T (${k}_{B}T=2.5$ meV in the plot) increases.

**Figure 3.**A model of how a constant DOS might be deformed when two phonons, one with k smaller than and one larger than ${k}_{F}$ are present (thin blue line). Disorder can come from ZPM, thermal vibrations or spin-fluctuations, and has a smearing effect on the DOS with a large increase in $N\left({E}_{F}\right)$ (bold red line).

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**MDPI and ACS Style**

Jarlborg, T.
Possible Benefits from Phonon/Spin-Wave Induced Gaps below or above *E _{F}* for Superconductivity in High-

*T*Cuprates.

_{C}*Condens. Matter*

**2022**,

*7*, 41. https://doi.org/10.3390/condmat7020041

**AMA Style**

Jarlborg T.
Possible Benefits from Phonon/Spin-Wave Induced Gaps below or above *E _{F}* for Superconductivity in High-

*T*Cuprates.

_{C}*Condensed Matter*. 2022; 7(2):41. https://doi.org/10.3390/condmat7020041

**Chicago/Turabian Style**

Jarlborg, Thomas.
2022. "Possible Benefits from Phonon/Spin-Wave Induced Gaps below or above *E _{F}* for Superconductivity in High-

*T*Cuprates"

_{C}*Condensed Matter*7, no. 2: 41. https://doi.org/10.3390/condmat7020041