Study of the Isotope Effects of Novel Superconducting LaH10-LaD10 and H3S-D3S Systems

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1. Introduction

Superconductivity was first discovered in 1911 [1], in mercury cooled below 4 K. The temperature below which a material becomes superconducting is called the critical temperature. It was quickly appreciated that a state exhibiting zero electrical resistance could be tremendously useful, if materials that have critical temperatures much higher than 4 K could be found. Over the past century, as more superconductors have been discovered, the record for the highest critical temperature achieved has progressed towards the ultimate goal of room temperature.

Since the discovery of Bednorz and Müller in 1986 [2] on oxide superconductors with critical temperature *T _{c}* approximately equal to 35 K, there are a great number of laboratories all over the world involved in research of superconductors with high

2. Isotope Effect Coefficient

Isotope Effect in BCS Theory

According to the weak coupling BCS theory, the relation between transition temperature *T _{c}*, typical phonon frequency

${k}_{B}{T}_{c}=1.14\hslash \omega \mathrm{exp}\left(-\frac{1}{VN\left({E}_{f}\right)}\right)$ (1)

*V* is the pairing potential arising from electron-phonon interaction, *N*(*E _{f}*) is the electron density of states at Fermi surface and

${T}_{c}=A{M}^{-\alpha}$ (2)

where *A* is a constant, *M* is the mass of the element substituted by its isotope and *α* is the isotope effect coefficient, which is defined as:

$\alpha =-\frac{\partial \mathrm{ln}{T}_{c}}{\partial \mathrm{ln}M}\approx -\frac{M}{{T}_{c}}\frac{\Delta {T}_{c}}{\Delta M}$ (3)

where ∆*T _{c}* is the shift of the critical temperature substitution of isotopic mass and ∆

${\alpha}_{i}=\frac{\mathrm{ln}{T}_{c}\left(i+1\right)-\mathrm{ln}{T}_{c}\left(i\right)}{\mathrm{ln}{M}_{i+1}-\mathrm{ln}{M}_{i}}\approx \frac{{M}_{i}}{{T}_{c}\left(i\right)}\frac{{T}_{c}\left(i+1\right)-{T}_{c}\left(i\right)}{{M}_{i+1}-{M}_{i}}$ (4)

In the formula (4), two set of adjacent data (*T _{c}*(

Vora [17] has deduced from the best fit to the data of about twenty-five materials, the following equation for *T _{c}*:

${T}_{c}=\left(\frac{\langle \omega \rangle}{10.71}\right)\left(\lambda -0.3362\right)$ (5)

where
$\langle \omega \rangle $ is the average phonon frequency and *λ* is the electron-phonon coupling strength. As the electron-phonon coupling strength is unaffected by the isotope substitution for harmonic phonon dispersion, and by using Equation (5), the isotope-effect coefficient can be written in terms of the phonon frequency for the LaH_{10}-LaD_{10} system as example:

$\alpha =-\frac{M}{\Delta M}\frac{{\langle \omega \rangle}_{{\text{LaD}}_{\text{10}}}-{\langle \omega \rangle}_{{\text{LaH}}_{\text{10}}}}{{\langle \omega \rangle}_{{\text{LaH}}_{\text{10}}}}$ (6)

The isotope effect evaluation using Equation (6) requires only the knowledge of the phonon frequencies which can be measured by the infrared or Raman spectra or predicted by the first principal density functional theory DFT. This equation indicates that the isotope effect causes a phonons frequency shift (energy shift) which differs than the original BCS theory which is given in terms of the superconducting temperature shift. Both of these shifts are due to the internal heavy atom effects.

It is noticeable that the D-derived optical phonon modes shifts towards lower frequencies, relative to the corresponding H-derived modes. For instances, at 250, 300 and 350 GPa, the lowest optical modes Γ point shift from 109.44, 118.36 and 123.12 meV in LaH_{10} to 77.52, 83.44 and 86.93 in LaD_{10}, respectively [18]. According to these phonon frequencies, Equation (6) gives *α*-values: 0.293, 0.297 and 0.295 for the pressures of 250, 300 and 350 GPa respectively. The average value of *α* is 0.295, while the experimental value from the critical temperature shift is 0.35 [3]. The error percent between the two methods is 15.7%. This discrepancy is due to the neglect of the acoustic phonon frequencies contribution, which are much less than the optical phonon frequencies. Equation (6) only calls the optical phonon frequencies as a proxy for critical temperatures.

Isotope effect in strong coupling constant

The BCS theory did not completely succeed in explaining isotope effect in superconductors, but it paved the way for a deeper understanding of electron-phonon coupling. Eliashberg model [19] assumed strong coupling between electrons and phonons and calculated the spectrum and the damping excitations. All superconductors are characterized as having weak ( ${\lambda}_{opt}\ll 1$ ), intermediate ( ${\lambda}_{opt}\approx 1$ ), and strong coupling ( ${\lambda}_{opt}\gg 1$ ) [20]. McMillan-Dynes [21] [22] performed advanced analysis of the problem by utilizing the Eliashberg theory and proposed the critical temperature equation:

${T}_{c}^{\xb0}=\frac{{\varpi}_{opt}}{1.2}\mathrm{exp}\left[-\frac{1.04\left(1+{\lambda}_{opt}\right)}{{\lambda}_{opt}-{\mu}^{*}\left(1+0.62{\lambda}_{opt}\right)}\right]$ (7)

Here *μ*^{*} is the effective Coulomb repulsion, which is assumed to be within a range of *μ*^{*} = 0.1 – 0.2. Equation (7) is highly accurate for a wide range of coupling strength *λ _{opt}* ≤ 1.5, and it is widely used to evaluate the

$\alpha =\frac{1}{2}\left[1-{\left({\mu}^{*}\mathrm{ln}\frac{\hslash \omega}{1.45{k}_{B}{T}_{c}}\right)}^{2}\left(\frac{1+0.62{\lambda}_{opt}}{1+{\lambda}_{opt}}\right)\right]$ (8)

For *λ _{opt}* ˃1.5, Allen and Dynes [23] proposed a correction factors should be included in the Equation (7), so that it becomes:

${T}_{c}={T}_{c}^{\xb0}{f}_{1}{f}_{2}$ (9)

where *f*_{1} is the “strong-coupling correction”, and *f*_{2} is the “shape correction”. *f*_{1} must scale as
${\lambda}_{opt}^{1/2}$. For the *f*_{2} calculation, the empirical relation deduced from Table 1 in ref. 23 is used [20]:

${f}_{2}=1+\left(0.0241-0.0735{\mu}^{*}\right){\lambda}_{opt}^{2}$ (10)

This parabolic function is deduced by the fit of tabulated *f*_{2} values for all materials reported by Allen and Dynes [23].

A modified form of the isotope effect coefficient α is developed by Gor’kov and Kresin (GK) [24] and shown to provide the relative contributions of optical and acoustic branches of infrared or Raman spectrum. The GK model is based on a hypothesis that the isotope effect originates from high frequency phonons and differs in the two phases. The value of the isotope coefficient is written as [24]:

$\alpha \approx \frac{1}{2}\left[1-\frac{{\lambda}_{ac}}{{\lambda}_{opt}}\frac{{\rho}^{2}}{{\left({\rho}^{2}+1\right)}^{2}}\right]$ (11)

Here $\rho =\frac{{\varpi}_{ac}}{\pi {T}_{c}^{\xb0}}$.

*ϖ _{ac}* is the average frequency of the acoustic phonons and

3. Results and Discussion

Isotope Effects in the fcc (LaH_{10}-LaD_{10}) System

First-principles calculations based on density functional theory suggested that a new family of superconducting hydrides that possess clathrate-like structure in which the host atom (lanthanum) is at the center of a cage formed by hydrogen atoms. This nearly spherical structure can be considered as standard for the study of the electrons-electrons and the electrons-phonons interactions and then the isotope effects. Table 1 shows the data used for calculating the isotope effects in superconducting LaH_{10}-LaD_{10} system under high compression. The Coulomb pseudopotential *μ*^{*} = 0.2 was assumed. The isotope coefficient *α* was determined by using the EM-model, Equation (8) and the GK-model, Equation (11). Both models mainly depend on the values of the critical temperatures, but new variables, acoustic phonon frequency and the acoustic coupling coefficient were added in the GK model. The predicted critical temperatures *T _{c}* were between 150 and 266 K. The reported superconductivity critical temperature of around 250 K at about 170 GPa [3]. When calculating the isotope effect from the Equation (11), the phonon contributions were taken into accounts using the optical and the acoustic branches which they have different frequencies and coupling constants. On this basis, it was introduced the average frequencies

Table 1. Calculated values of electron phonon coupling *λ*, average phonon frequency *ϖ*, *T _{c}* and the isotope effect coefficient

relative contributions of the optical and the acoustic phonons. Depending on [24], *λ _{opt} *≈ 3

Figure 1_{ }demonstrates the variations of the superconducting parameters *T _{c}*,

Figure 1. The calculated superconducting parameters at different pressures for the LaH_{10}-LaD_{10} system (a) calculated *T _{c}* of LaH

than 1.5. The predicted *T _{c}* values are around 266 K at the pressure of 250 GPa and 151 K at the pressure of 350 GPa. The recent reported value for LaH

Table 2. Calculated values of electron-phonon coupling coefficient*λ*, average phonon frequency *ϖ*, *T _{c}* and the isotope effect coefficient

Table 3. Calculated values of superconducting parameters at different pressure. Electrons-phonons coupling coefficient *λ*, average phonons frEquationuency *ϖ*, *T _{c}* and the isotope effect coefficient α for Im-3m structure of sulfur hydride. The Coulomb pseudopotential

0.34, in excellent agreement with measured value of *α* = 0.35 [29] and with the theoretical prediction [24].

For Im-3m structure, the *α*-values are constant with a pressure change and are 0.49 when using Equation (8) (Table 3), while it was found that the average value of *α* when using Equation (11) is 0.41. The error percent for the experimental value is 17%. In H_{3}S-D_{3}S system, it is noticed that the GK model achieves *α*-values better than the EM model.

4. Conclusion

The isotope effect on the superconductivity transition temperature *T _{c}* is one of the hallmarks of phonon-induced superconductivity in conventional superconductors. The dependence of the superconductivity transition temperature on the isotope mass provides an important probe of the pairing mechanism. Precise values of

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