Introduction

Superconductivity is the complete loss of electrical resistivity of a material that occurs only below a certain temperature, called superconducting critical temperature T_c. It is a state of matter with technologically impactful applications but with serious difficulties of use on a large scale due to the extreme conditions in which it occurs: low temperatures or high pressures. However, its application on a small scale is a current fact.

Research on the subject from a theoretical approach seeks to establish its fundamental physical mechanisms, the understanding of which will positively lead to the engineering of superconducting materials with a view to their application on a large scale. The best approach, recognized with the Nobel Prize in physics in 1971, is the Bardeen-Cooper-Schrieffer (BCS) theory, which states that superconductivity is the “physics of Cooper pairs” [1].

Here, the effective attraction between electrons forming the Cooper pair is generated by the interaction between the electrons and the lattice vibrations (phonons), called the electron-phonon interaction. This scheme explains the phenomenon for weak electron-phonon coupling systems, leading to T_c below 70 K, (lower than the temperature of liquid nitrogen). Thus, the next step was to generalize the BCS theory to superconductors, in which the electron-phonon interaction is strong and hence has a higher T_c This was the work of G. M. Eliashberg [2] who in his theoretical description, introduced the electron-phonon interaction and the electronic and phononic band structure more precisely. All that information is gathered in a function, the Eliashberg spectral function, or electron-phonon coupling function α²F(ω) (see Fig 1), which can be obtained both theoretically (DFT calculations) and experimentally (tunneling experiment). The Eliashberg spectral function is obtained from the calculated phonon spectrum and the calculated electron–phonon matrix elements [3, 4]. The Coulombic repulsion between electrons is included through a parameter μ.

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Fig 1.

Schematics of (a) the Eliashberg spectral function α²F(ω) and (b) the functional derivative of the superconducting critical temperature T_c concerning the α²F(ω) function, ΔT_c/δα²F(ω).

https://doi.org/10.1371/journal.pone.0286855.g001

On the other hand, the linearization of the Eliashberg equations makes it possible to determine the functional derivative of the superconducting critical temperature T_c concerning the function α²F(ω), δF_c/δα²F(ω). The first numerical calculations of the δF_c/δα²F(ω) in superconductors were performed by Bergmann and Rainer [5]. Their results showed that this function has a universal form (see Fig 1b): it grows from ω = 0 to a maximum at ω ∼ 7K_BT_c and then slowly decreases to 0 as ω → ∞ [5, 6]. From δT_c/δα²F(ω), it is possible to determine the phonon frequency leading to the highest possible T_c in a superconductor [7] and to describe the change in T_c, ΔT_c, given a slight variation in the α²F(ω) function, Δα²F(ω), generated by the action of physical conditions such as pressure, doping, etc. [8–10], thus (Eq (1); (1)

A previous theoretical study showed that there is a correlation between the frequencies of the maxima of the δT_cδα²F(ω) and α²F(ω) functions [11, 12], where the convergence of these frequencies occurs at the optimal electron-phonon interaction conditions leading to the superconductor reaching the maximum possible [13–17]. This convergence frequency is called the optimum frequency ω_opt, which satisfies the relation , where K_B is the Boltzmann constant. The calculation of the δT_c/δα²F(ω) requires the experimental (or test) T_c value for the prior determination of the parameter μ*, which physically accounts for the Coulombic repulsion between electrons in the system under study. The parameter μ* is determined from the fit to the linearized Eliashberg equations (see Materials and methods section- Eq (3) when the pair breakdown parameter ρ tends to zero (ρ → 0), which is valid for T = T_c [15].

Up until now, the physical interpretation and application of the δT_c/δα²F(ω) have failed to consolidate. In 2015 Nicol and Carbotte used the δT_c/δα²F(ω) to demonstrate that the α²F(ω) spectral function of sulfur trihydride H₃S at 200 GPa is highly optimized for T_c [18]. González-Pedreros and Baquero [10] and Camargo-Martínez et al. [19] used the δT_c/δα²F(ω) to determine the trend of T_c as a function of pressure in Nb-bcc (Cubic Niobium) and H₃S respectively, taking the reported experimental T_c as a starting point. In other work, the δT_c/δα²F(ω) was determined to identify possible frequency regions where phonons would be the most effective in increasing T_c [20, 21]. All these results are descriptive and not predictive in nature.

One of the possible contributions of theoretical physics in superconductivity is to clearly establish the fundamental physical foundations of the superconducting phenomenon in order to suggest with certainty, the line of experimental process to obtain superconductivity at room temperature in viable conditions for its application to large-scale. An example of the predictive effect of the theoretical approach on superconductivity was observed in the idea proposed by Ashcroft [22], who stated that hydrogen-rich systems would be viable candidates to be high critical temperature superconductors. This proposal gave rise to experimentation in this field with the discovery of new high-Tc superconductors, as H₃S (T_c of 203 K at 155 GPa [23]) or LaH₁₀ (T_c of 260 K at 180 GPa [24]), called hydride superconductors. This discovery gave a new impetus to this field of study, which had been stuck with the superconducting cuprates (T_c of 164 K) since 1994 [25]. The current difficulty with hydride superconductors is in their high-pressure conditions of formation. In this sense, evaluating possible new ways to predict T_c values in terms of well-defined physical conditions (such as pressure, doping, etc.) is an interesting line of work. Here, the study of the functional derivative δT_c/δα²F(ω) seeks to establish the possible existence of patterns that lead to the determination of an optimum temperature of the system (superconducting critical temperature), which would also avoid the use of test or experimental T_c in first-principles calculations.

From a purely computational point of view, the temperature value can have pivotal implications in the calculation, result, and interpretation of the functional derivative δT_c/δα²F(ω). For this reason, in this manuscript, we present the analysis of the effects of temperature variation, around experimental T_c value, on the calculation of the functional derivative δT_c/δα²F(ω), in the superconductor H₃S, of which a T_c of 203 K at 155 GPa was measured [23].

Materials and methods

This study was developed based on the Eliashberg α²F(ω) spectral functions of H₃S obtained in previous work [12, 19], whose calculations were performed in the range of pressures (155–225 GPa), where the experimental T_c were reported [23]. Here, the functional derivatives were obtained with the procedure widely used by Carbotte et al. [13–16, 18, 26, 27] which is based on the work of G. Bergmann and D. Rainer [5]. The determination of the functional derivative (see Eq (2)) of the superconducting critical temperature T_c with respect α²F(ω) function, δT_c/δα²F(ω), was performed from the relation: (2)

Where ρ was expressed in terms of K_nm for T = T_c, which is obtained as a (kernel) solution of the linearized Eliashberg equations on the imaginary axis [5, 6]: (3) with ), ω_n = πT(2n − 1) the n-th Matsubara frequency and . For more details on the mathematical formulation, see the reference [17]. To evaluate the effects of the T_c parameter on the δT_c/δα²F(ω) calculations, 10 K variations in temperature around the experimental value (reference temperature) were developed for each of the pressures evaluated (155 GPa,175 GPa, 195 GPa, and 215 GPa).

Results and discussion

The δT_c/δα²F(ω) as a function of frequency calculated for H₃S at different pressures and temperatures are presented in Fig 2.

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Fig 2. δT_c/δα²F(ω) as a function of frequency ω for H₃S calculated at different temperatures at pressures of a) 155GPa, b) 175 GPa, c) 195 GPa, and d) 215 GPa.

Stars show critical temperature related to corresponding pressure, horizontal arrows indicate frequency shift of the maximum value of the δT_c/δα²F(ω), , red arrows present how δT_c/δα²F(ω)-curve increases their separation, and vertical arrows point average frequency of intersection of the δT_c/δα²F(ω).

https://doi.org/10.1371/journal.pone.0286855.g002

Each case evaluated (Fig 2), frequency values in every δT_c/δαT2F(ω)-maximum are displaced, δ_MAX, as temperature is increased in accord with relation ω_opt ∼ k_BT_c [5]. The δT_c/δα²F(ω)-maximum allows identifying of the frequency regions ω_opt where phonons are more effective in increasing T_c [28]; this is why it is so important to evaluate their behavior.

To every pressure (Fig 2a–2d), in a temperature interval ΔT = 40K about T_c, δT_c/δα²F(ω) intersects a narrow frequency range (it looks point-like) is verified. Condition of small variation of the δT_c/δα²F(ω) due to temperature effects, below the intersection point, is observed for the system under the lowest compression (155 GPa) and vice versa. This behavior is opposite if the δT_c/δα²F(ω) are compared at frequencies higher than the intersection point. In both cases, the effect of temperature on the calculation of the δT_c/δα²F(ω) could establish patterns (intersection point, variation, or separation between the δT_c/δα²F(ω) and their maxima) that would lead to the determination of optimal physical conditions of the superconducting state and the possible estimation of the T_c.

Now, the intensities (value on the vertical axis) of the δT_c/δα²F(ω)-maximums show two different behaviors (Fig 2). At 155 GPa, such intensities slightly increase their value as the temperature increases, as a consequence of the little separation induced in the δT_c/δα²F(ω). However, for the other pressures (175, 195, and 215 GPa), the behavior of the δT_c/δα²F(ω) intensities is opposite to the 155 GPa case, starting from a higher intensity and decreasing with increasing temperature, being more evident with increasing pressure. It is important to note that there seems to be no relationship between the variation of the frequency of the maximum of the δT_c/δα²F(ω),, and the intensity of the maximum of the δT_c/δα²F(ω).

The patterns of the δT_c/δα²F(ω) vs T_c in the H₃S reveal that these seem to have a characteristic behavior at a specific pressure (155 GPa).

(Fig 3) shows the linearity of the frequency of the maximum of the as a function of temperature for all pressures. Such lines are collinear with mean slope 0.64 meV/K. This means that the moves uniformly toward higher frequencies as the temperature increases. On the other hand, is almost unaffected by the pressure (p) since a considerable change of Δp = 40 GPa induces a small meV. However, each pressure has a limit of , whose maximum value is reached at 155 GPa, leading to a higher T_c.

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Fig 3. Frequency of the maximum of the δT_c/δα²F(ω),, as a function of temperature, at different pressures.

The dashed boxes (yellow) show the slight change in induced by pressure.

https://doi.org/10.1371/journal.pone.0286855.g003

In the calculation of the δT_c/δα²F(ω), the temperature variation involves the determination of the μ* parameter of Fig 3. The comparison between the parameters μ* adjusted at different temperatures for each of the pressures is presented in Fig 4.

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Fig 4. The parameter μ* as a function of temperature T for H₃S at different pressures.

The horizontal band (yellow) marks the Δμ* within which the μ* fitted to the experimental T_c are set, indicated by the diagonal band (purple).

https://doi.org/10.1371/journal.pone.0286855.g004

It is observed in Fig 4 that μ* vs T has a comparable trend between pressures. This could be assumed to be quasi-linear in a first approximation. However, this quasi-linearity is much more evident at higher pressure. The results show that a ΔT_c = 40K induces a . However, the μ* values fitted to the experimental T_c in the pressure range from 155 to 215 GPa are in the Δμ* range of 0.209–0.285, which implies a K, a small range with respect to the 203 K of the experimental maximum T_c of H₃S. This result is interesting because it would allow establishing an initial criterion of theoretical estimation of the T_c around a small range of temperatures, according to the Δμ*. It is then necessary to determine, with this procedure, the Δμ* other systems to evaluate if this presents a universal range or if it varies significantly from one system to another.

In the calculation of the δT_c/δα²F(ω), it was found that for temperatures distant between -60 K and +30 K with respect to the experimental T_c, μ* values of 0,8 and 0,09 are generated, which are outside the values typically used or calculated (between 0,3 and 0,1), and their δT_c/δα²F(ω) presented computational difficulties in their calculation, with behaviors different in form from those observed in the δT_c/δα²F(ω) calculated at temperatures close to the experimental T_c.

Conclusions

This paper presents the preliminary theoretical analysis of the effects of temperature variation around the experimental superconducting critical temperature T_c on the calculation of the functional derivative δT_c/δα²F(ω) for superconducting H₃S in the pressure range from 155 to 215 GPa. These calculations included the determination of the μ* parameters through fitting T_c in the linearized Eliashberg equations. The calculated δT_c/δα²F(ω) revealed temperature- and pressure-induced displacement, intersection, and separation patterns that could be associated with the physical conditions in the superconducting state and the estimation of T_c. The μ* values obtained allowed the determination of a range of values leading to temperatures that could establish an initial criterion for possible theoretical estimation of T_c. This procedure must be evaluated and confirmed in other similar systems to establish the possible generalization of the results presented here.