Comment on 'High-$T_\mathrm{c}$ superconductivity in H₃S: pressure effects on the superconducting critical temperature and Cooper pair distribution function'

Camargo-Martínez, González-Pedreros and Baquero analysed in detail the effects of pressure on the electron-phonon spectral density, $\alpha^2F(\nu)$, in H₃S and evaluate the superconducting transition temperature, $T_\mathrm{c}$. They use the QUANTUM-ESPRESSO package to implement the density functional theory over the (first) Brillouin zone, with a k-grid of $24\times24\times24$ for the electronic integration, and a q-grid of $8\times8\times8$ to calculate phonon frequencies according to the Monkhorst–Pack (MP) scheme, k being the electron wavenumber and q phonon momentum [1].

Assuming free electrons (spherical Fermi surface) and Debye phonons, it is straightforward to find $q_\mathrm{D}/k_\mathrm{F} = (2/Z)^{1/3}$, or $\theta = 2\sin^{-1}(q_\mathrm{D}/2k_\mathrm{F})$, where $q_\mathrm{D}$ is the Debye momentum, $k_\mathrm{F}$ Fermi wavenumber, Z valency and θ angle of electron-phonon scattering, with $\theta = 78.1^\circ$, 60.0^∘, 51.8^∘, ..., for the maximum angle of normal scattering when Z = 1, 2, 3, etc. However, in the case of electrical resistivity or superconductivity, we must ensure $\theta = 180^\circ$, that is scattered electrons must be able to cover the entire Fermi surface, so that umklapp scattering has to be involved [2]. To solve the problem Carbotte and Dynes recommended to 'extend the q integration from the phonon BZ to a sphere of $2k_\mathrm{F}$ to allow for all possible phonon umklapp processes' [3].

In contrast the MP scheme is designed to serve the purpose of 'integrating periodic functions of a Bloch wave vector over either the entire Brillouin zone (BZ) or over specified portions' [4]. In elementary analysis two plane waves, propagating in opposite directions, can be grouped together to form a sine and/or a cosine function to play the role of base functions for Fourier expansion. In the MP scheme a number of plane waves, propagating over a range of directions, are grouped together for a similar expansion. Different groups are identified by their 'star', which is a set of locations on a spherical shell in real space and characterised by the shell radius. Different groups are orthogonal to each other when integrated over a discrete grid in BZ [4].

The MP scheme can be compared with the scheme of fast Fourier transform (FFT) where the usual functional basis of expansion, in terms of exponential functions of a complex variable, is replaced by a set of vectors orthogonal to each other when integrated over a discrete grid. Indeed, apart from being able to reduce the load of computation significantly, the MP scheme offers further convenience when the 'stars' are made to assume symmetry of the crystal. On the other hand, while there is little difference in applicability between FFT and its counterpart in continuum, phonon expansion in the MP scheme is valid in BZ, where orthogonality of the MP vector set has been proved.

It may not be trivial to prove applicability of the MP scheme beyond the first BZ. In particular the umklapp phonons are not replicas of normal phonons. They arise not from some periodicity but from a complicated phase condition involving both k and q. In the treatment of Carbotte and Dynes the number of umklapp states is not periodical but largely proportional to q³, $0\leq q\leq 2k_\mathrm{F}$ [3]. We also find from computation frequencies of the umklapp phonons in the Carbotte-Dynes sphere are not periodical unless q is in a few special directions [5].

In conclusion, hydride superconductor research is a field of rapid development. Existing computer packages may not be adequate to support front line research activities. For example, in 2016, Wisesa, McGill and Mueller presented a method for rapid generating efficient k-point grids for Brillouin zone integration [6]. In 2018 and 2020, Morgan and collaborators demonstrated that the new grids indeed offer an advantage over the traditional MP grids [7, 8]. It is not clear if the QUANTUM-ESPRESSO package in [1] was updated accordingly in time.

If the MP scheme in the QUANTUM-ESPRESSO package still is in its original form, then the umklapp phonons must all have been excluded from [1]. It has been known for some time the theoretical superconductivity always turns out to be overly strong when both normal and umklapp phonons are included in the first principles evaluations [9]. There is a recent argument calling for exclusions of some of the phonons, with theoretical justifications [5, 9]. Perhaps the practice to use the MP scheme to evaluate superconductivity could somehow be linked to that recent argument.

All data that support the findings of this study are included within the article.