Detecting Superconductivity in the High Pressure Hydrides and Metallic Hydrogen from Optical Properties

J. P. Carbotte, E. J. Nicol, and T. Timusk

Phys. Rev. Lett. 121, 047002 – Published 26 July 2018

Abstract

We present a new technique for measuring the critical temperature $T_{c}$ in the high pressure, high $T_{c}$ electron-phonon-driven superconducting hydrides. This technique does not require connecting leads to the sample. In the region of the absorption spectrum above the sum of the optical gap and maximum phonon energy, the reflectance mirrors the temperature variation of the superconducting order parameter. For an appropriately chosen value of fixed photon energy, the temperature dependence of the reflectance varies much more rapidly below $T = T_{c}$ than above. It increases with increasing temperature in the superconducting state while it decreases in the normal state. Examining the temperature dependence of the reflectance at a fixed photon energy, there is a cusp at $T = T_{c}$ which provides a measurement of the critical temperature. We discuss these issues within the context of the recently reported atomic metallic phase of hydrogen, but our proposed technique should prove useful for other hydrides with large coupling to high energy phonons.

Received 15 February 2018
Revised 23 April 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.047002

Physics Subject Headings (PhySH)

Optical conductivity Pressure effects Superconducting gap Superconducting phase transition

High-temperature superconductors

Eliashberg theory Methods in superconductivity

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

J. P. Carbotte^1,2,*, E. J. Nicol^3,†, and T. Timusk^1,2

¹Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada
²The Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada
³Department of Physics, University of Guelph, Guelph, Ontario N1G 2W1, Canada

^*carbotte@mcmaster.ca
^†enicol@uoguelph.ca

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Vol. 121, Iss. 4 — 27 July 2018

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Images

Figure 1
The real part of the superconducting state optical conductivity $4 π σ (T, Ω) / ω_{p}^{2}$ of metallic hydrogen as a function of photon energy $Ω$ for various values of temperature relative to $T_{c}$ , with $t = T / T_{c}$ . The conductivity is multiplied by $1 0^{4}$ for this plot, and $ω_{p}$ is the plasma frequency. The red distribution is the electron-phonon spectral density $α^{2} F (ω)$ displaced by twice the energy gap at zero temperature $2 Δ_{0}$ and scaled up by a factor of 1.5 on this plot. The spectral density is from Ref. [18] for metallic hydrogen at 500 GPa with $T_{c} = 240 K$ . The inset shows the value of $T_{c} (ω)$ (solid black dots) obtained from our Eliashberg calculations when only the phonon modes below $ω$ are included. The red curve is the partial area under $α^{2} F (ω)$ up to $ω$ divided by the total area under $α^{2} F (ω)$ integrated up to $ω_{\max} \sim 360 meV$ .
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Figure 2
The reflectance $R_{S} (T, Ω)$ of superconducting metallic hydrogen as a function of photon energy $Ω$ for several temperatures, with $t = T / T_{c}$ . The arrows at $Ω_{c 1} \sim 400 meV$ and $Ω_{c 2} \sim 900 meV$ define the range of photon energies for which the reflectance in the superconducting state increases as temperature increases. This behavior is opposite to that expected in a normal metal and is taken to be the signature of superconductivity.
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Figure 3
The normalized reflectance $R_{S} (T, Ω) / R_{S} (T_{c}, Ω)$ as a function of photon energy in the interval $Ω_{c 1} = 400 meV$ to $Ω_{c 2} = 900 meV$ indicated by arrows in Fig. 2. In this energy range, the reflectance increases with increasing temperature, where $t = T / T_{c}$ . The increase is small at low temperature but accelerates as $T$ approaches $T_{c}$ . This variation reflects directly the temperature dependence of the order parameter (energy gap) associated with the second order phase transition of BCS theory.
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Figure 4
The reflectance of metallic hydrogen at fixed photon energy $Ω_{c} = 600 meV$ as a function of temperature. The solid red curve is the normal state, while the solid blue is for the superconducting state. The temperature at which these two curves meet defines the value of the critical temperature of the sample, which here is 240 K. The dashed black curve is for comparison and is a scaling of the mean field temperature dependence of BCS theory, i.e., $[a (1 - b Δ (T) / Δ_{0})]$ , where $a$ and $b$ are scaling constants.
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Figure 5
The normalized reflectance of metallic hydrogen $R (T, Ω_{c}) / R (T_{c}, Ω_{c})$ at fixed photon energy $Ω_{c}$ as a function of temperature. The various curves illustrate how the cusp at $T = T_{c}$ in this quantity changes for different choices of parameters in the reflectance calculation. Here, $1 / τ_{imp} = 100 meV$ . Increasing $1 / τ_{imp}$ to 500 meV produces no noticeable change in this normalized quantity. The solid curves refer to different frequencies, with $ω_{p} = 33 eV$ and the diamond refractive index $n_{0} = 2.417$ as has been used throughout this paper. The dashed curves are for fixed frequency $Ω_{c} = 600 meV$ , but showing the effect of halving the plasma frequency and of removing the correction of the diamond $n_{0}$ , restoring it to the free space value of $n_{0} = 1$ .
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