## New Research In

### Physical Sciences

### Social Sciences

#### Featured Portals

#### Articles by Topic

### Biological Sciences

#### Featured Portals

#### Articles by Topic

- Agricultural Sciences
- Anthropology
- Applied Biological Sciences
- Biochemistry
- Biophysics and Computational Biology
- Cell Biology
- Developmental Biology
- Ecology
- Environmental Sciences
- Evolution
- Genetics
- Immunology and Inflammation
- Medical Sciences
- Microbiology
- Neuroscience
- Pharmacology
- Physiology
- Plant Biology
- Population Biology
- Psychological and Cognitive Sciences
- Sustainability Science
- Systems Biology

# High-temperature superconductivity using a model of hydrogen bonds

Contributed by Yoseph Imry, April 16, 2018 (sent for review March 6, 2018; reviewed by Eugene Chudnovsky and Michel Devoret)

## Significance

In this paper, we suggest a simple model for high-temperature superconductivity in hydrogen-containing compounds. We show several features: starting with the nonmonotonic behavior of

## Abstract

Recently, there has been much interest in high-temperature superconductors and more recently in hydrogen-based superconductors. This work offers a simple model that explains the behavior of the superconducting gap based on naive BCS (Bardeen–Cooper–Schrieffer) theory and reproduces most effects seen in experiments, including the isotope effect and

Recently, there has been a surge of interest concerning the discovery of high-temperature superconductivity in hydrogen compounds (1⇓–3). These studies showed that metallization and superconductivity in such materials can be obtained by changing the pressure of the system. Furthermore, the discovery of the inverse isotope effect in hydrides (notably, PdH) (4) helped fuel interest in these materials. Historically, arguments have been put forward that the maximal value of

## The Harmonic Oscillator Case

We first consider the emergence of a superconducting gap by assuming that electrons interact with a harmonic oscillator,

### Electron–Proton Coupling.

We explore two forms of this term. For simplicity, we show this in 1D but the approach is easily generalized to 3D. First, we approximate the electron–proton interaction by a delta function, with the form **3** becomes_{3}S, in the metallic phase,

### The BCS Superconducting Gap.

We focus on a particular longitudinal phonon mode, which, by work in ref. 2, has been shown to be particularly dominant in these materials. Consider the matrix element of Eq. **2**. Substituting **4** we find*M* is the proton mass. The relevant dimensionless parameter is the BCS interaction parameter, which we take to be

The BCS gap has the form (9, 13)

### The Isotope Effect.

Having found Δ it is possible to investigate the dependence of the gap on the mass of the proton. We calculate

### Gap Equation at Finite Temperature.

For completeness, we include the gap equation at finite temperature, which permits an accurate determination of *f* is the Fermi–Dirac distribution. At the transition temperature

## A Simple Model of the Superconducting Transition in H 3 S

Numerical as well as experimental work on the structure of

We model the potential seen by the proton, using a sum of two Lennard-Jones (L-J) potentials

The parameter governing the form of the potential is then

Typical values for **3**, Fig. 5 shows the first two levels splitting, as a function of

Fig. 5 is obtained by finding the ground state and first excited state of the potentials given in Fig. 4 as a function of **2** and **8**, we can now obtain the superconducting gap. We replace above, in Eq. **8**, **13**. The red line depicts the energy difference **3**. The combination of these, detailed above, gives the gap, which is plotted in blue.

Fig. 6 clearly illustrates that Δ possesses a maximum. This is obtained at

## Conclusions

This work demonstrates qualitatively that *H*−*S* potential, can, we believe, allow us to fully account for Δ and its dependence on physical parameters. At present, very high pressures (of the order of 100 GPa) are needed to bring about the superconductivity in these compounds. However, judicious chemistry with larger and smaller ions might create internal pressures in the material to simulate such conditions.

## Acknowledgments

Y.I. acknowledges the late Prof. I. Pelah for conversations on hydrogen bonds and the late Prof. W. Kohn on superconductivity and density functional theory. D.K. thanks Or Ben Zvi for his immeasurable support and attention. Financial support from the Weizmann Institute of Science is gratefully acknowledged.

## Footnotes

↵

^{1}D.K. and Y.I. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: yoseph.imry{at}weizmann.ac.il or daniel.kaplan{at}weizmann.ac.il.

Author contributions: Y.I. designed research; D.K. performed research; D.K. analyzed data; and D.K. wrote the paper.

Reviewers: E.C., The City University of New York Lehman College and Graduate School; and M.D., Yale University.

The authors declare no conflict of interest.

Published under the PNAS license.

## References

- ↵.
- Errea I, et al.

- ↵
- ↵.
- Flores-Livas JA,
- Sanna A,
- Gross EK

- ↵
- ↵
- ↵.
- Anderson PW

- ↵
- ↵.
- Bednorz JG,
- Müller KA

- ↵.
- De Gennes PG

- ↵.
- Havriliak S,
- Swenson RW,
- Cole RH

- ↵.
- Imry Y

- ↵.
- Kudryashov NA,
- Kutukov AA,
- Mazur EA

- ↵
- ↵.
- Ge Y,
- Zhang F,
- Yao Y

- ↵.
- Imry Y,
- Pelah I,
- Wiener E

- ↵.
- Galliero G,
- Boned C

- ↵.
- Goncharov AF,
- Lobanov SS,
- Prakapenka VB,
- Greenberg E

- ↵.
- Eremets MI,
- Drozdov AP

## Citation Manager Formats

## Sign up for Article Alerts

## Jump to section

## You May Also be Interested in

### More Articles of This Classification

### Physical Sciences

### Related Content

- No related articles found.

### Cited by...

- No citing articles found.