Quantifying the Nonadiabaticity Strength Constant in Recently Discovered Highly-Compressed Superconductors

1. Introduction

The discovery of near-room temperature superconductivity in highly compressed sulphur hydride by Drozdov et al [1] manifested a new era in superconductivity. This research field represents one of the most fascinating scientific and technological exploration in modern condensed matter physics where advanced first principles calculations [2,3,4,5,6,7,8,9,10,11] are essential part of the experimental quest for the discovery of new hydrides phases [12,13,14,15,16,17,18,19,20,21], and both of these directions drive the development of new experimental techniques to study highly-pressurized materials [22,23,24,25,26,27,28,29,30,31].

From 2015 till now, several dozens of high-temperature superconducting polyhydride phases have been discovered and studied [1,12,13,14,15,16,17,18,19,20,21,24,32,33,34,35,36,37,38,39,40,41,42,43]. At the same time, high-pressure studies of the superconductivity in non-hydrides are also progressed recently [44,45,46,47,48,49,50,51,52,53], including observation of $T_c>26K$ in highly-compressed elemental titanium [54,55] and scandium [56,57], and $T_c^{onset}\cong 78K$ in ${La}_3{Ni}_2{O}_7$ [58].

First principles calculations [12,13,14,15,16,17,18,19,20,21,59,60,61,62,63,64,65,66,67,68] are essential tool in the quest for room-temperature superconductivity (which was used [65] to explain experimental result [69] for one of the most difficult to explain hydride case, ${AlH}_3$), and primary calculated parameter in these calculations is the transition temperature, $T_c$. As, the confirmation of the predicted $T_c$, as the determination of other fundamental ground state parameters, for instance, the upper critical field, $B_{c2}\left(0\right)$ [5,24,33,39], the lower critical field, $B_{c1}\left(0\right)$ [12,22], the self-field critical current density, $J_c\left(sf,T\right)$ [24,70,71,72], the London penetration depth, $\lambda \left(0\right)$ [22,23,73,74], the superconducting energy gap amplitude, $\mathsf{\Delta }\left(0\right)$ [75,76,77], and gap symmetry [78,79], etc., are the task for experiment and data analysis

Another complication in understanding of the superconductivity in highly-pressurized materials is the phenomenon of nonadiabaticity, which originates from a fact that Migdal-Eliashberg theory of the electron-phonon mediated superconductivity [80,81] is based on primary assumption/postulate that the superconductor obeys the inequality:

$$T_{\theta }\ll T_F,$$

(1)

where, $T_{\theta }$ is the Debye temperature, and $T_F$ is the Fermi temperature. In other words, Eq. 1 implies that the superconductor exhibits fast electric charge carriers and slow ions. This assumption simplifies theoretical model of the electron-phonon mediated superconductivity, however, Eq. 1 is not satisfied for many unconventional superconductors [82,83,84,85,86,87,88,89,90] (which was first pointed out by Pietronero and co-workers [91,92,93,94]) and many highly-compressed superconductors [79,89,95,96,97].

While theoretical aspects of the non-adiabatic effects can be found elsewhere [11,82,88,91,92,93,94,95], in practice, the strength of the nonadiabatic effects can be quantified by the $\frac{T_{\theta }}{T_F}$ ratio [89,90] for which in Ref. [89] three characteristic ranges were proposed:

$$\left\{\begin{array}{c}\frac{T_{\theta }}{T_F}<0.025\to adiabaticsuperconductor;\\ 0.025\lesssim \frac{T_{\theta }}{T_F}\lesssim 0.4\to moderatelystrongnonadiabaticsuperconductor;\\ 0.4<\frac{T_{\theta }}{T_F}\to nonadiabaticsuperconductor.\end{array}\right.$$

(2)

It was found in Ref. [89], and confirmed in Ref. [79], that superconductors with $T_c>10K$ (from a dataset of 46 superconductors from all major superconductors families) exhibit the $\frac{T_{\theta }}{T_F}$ ratio in the range $0.025\lesssim \frac{T_{\theta }}{T_F}\lesssim 0.4$.

This is interesting and theoretically unexplained empirical observation.

In this study we further extended empirical $\frac{T_{\theta }}{T_F}$ database by deriving several fundamental parameters:

(1) the Debye temperature, $T_{\theta }$;

(2) the electron-phonon coupling constant, ${\lambda }_{e-ph}$;

(3) the ground state coherence length, $\xi \left(0\right)$;

(4) the Fermi temperature $T_F$;

(5) the nonadiabaticity strength constant, $\frac{T_{\theta }}{T_F}$;

(6) and the ratio $\frac{T_c}{T_F}$;

for five recently discovered highly-compressed superconductors for which reported raw experimental data are enough to deduce mentioned above parameters, and which represent materials with high or record high $T_c$ in their families:

(1) elemental titanium, $\delta -Ti$ [54,55];

(2) ${TaH}_3$ [21];

(3) ${LaBeH}_8$ [98];

(4) black phosphorous [99,100,101];

(5) violet phosphorous [53].

In the result, we derived the nonadiabaticity strength constant, $\frac{T_{\theta }}{T_F}$, for these superconductors and confirmed previously reported empirical observation [79,89] that superconductors with $T_c>10K$ obey the condition $0.025\lesssim \frac{T_{\theta }}{T_F}\lesssim 0.4$.

2. Utilized models and data analysis tools

2.1. Debye temperature

Debye temperature, $T_{\theta }$, is one of fundamental parameters which determines the superconducting transition temperature, $T_c$, within electron-phonon phenomenology [81,102,103,104,105,106]. This parameter can be deduced as a free-fitting parameter from a fit of temperature dependent resistance, $R\left(T\right)$, to the saturated resistance model within the Bloch-Grüneisen (BG) equation [107,108,109,110]:

$$R\left(T\right)=\frac{1}{\frac{1}{R_{sat}}+\frac{1}{R_0+A{\left(\frac{T}{T_{\theta }}\right)}^5\underset{0}{\overset{\frac{T_{\theta }}{T}}{\int }}\frac{x^5}{\left(e^x-1\right)\left(1-e^{-x}\right)}dx}},$$

(3)

where $R_{sat}$, $R_0$, $T_{\theta }$ and $A$ are free fitting parameters.

2.2. The electron-phonon coupling constant

From the deduced $T_{\theta }$ and measured $T_c$, which we defined by as strict as possible resistance criterion of $\frac{R\left(T\right)}{R_{norm}}\to 0$, where $R_{norm}$ is the sample resistance at the onset of the superconducting transition, the electron-phonon coupling constant, ${\lambda }_{e-ph}$, can be calculated as the root of advanced McMillan equation [103,104,105,106]:

$$T_c=\left(\frac{1}{1.45}\right)\times T_{\theta }\times e^{-\left(\frac{1.04\left(1+{\lambda }_{e-ph}\right)}{{\lambda }_{e-ph}-{\mu }^{*}\left(1+0.62{\lambda }_{e-ph}\right)}\right)}\times f_1\times f_2^{*},$$

(4)

where

$$f_1={\left(1+{\left(\frac{{\lambda }_{e-ph}}{2.46\left(1+3.8{\mu }^{*}\right)}\right)}^{3/2}\right)}^{1/3},$$

(5)

$$f_2^{*}=1+\left(0.0241-0.0735\times {\mu }^{*}\right)\times {\lambda }_{e-ph}^2,$$

(6)

where ${\mu }^{*}$ is the Coulomb pseudopotential parameter, which we assumed to be ${\mu }^{*}=0.13$ (which is typical value utilized in the first principles calculation for many electron-phonon mediated superconductors [54,111]).

2.3. Ground state coherence length

To deduce the ground state coherence length, $\xi \left(0\right)$, we fitted the upper critical field datatset, $B_{c2}\left(T\right)$, to analytical approximant of the Werthamer-Helfand-Hohenberg model [112,113], which was proposed by Baumgartner et al [114]:

$$B_{c2}\left(T\right)=\frac{1}{0.693}\times \frac{{\varphi }_0}{2\pi {\xi }^2\left(0\right)}\times \left(\left(1-\frac{T}{T_c}\right)-0.153\times {\left(1-\frac{T}{T_c}\right)}^2-0.152\times {\left(1-\frac{T}{T_c}\right)}^4\right),$$

(7)

where ${\varphi }_0=\frac{h}{2e}$ is the superconducting flux quantum, $h=6.626{\times 10}^{-34}J\cdot s$ is Planck constant, $e=1.602{\times 10}^{-19}C$, and $\xi \left(0\right)$ and $T_c\equiv T_c\left(B=0\right)$ are free fitting parameters.

2.4. The Fermi temperature

Simplistic approach to calculate the Fermi temperature, $T_F$, is to use the expression of free-electron model [115,116]:

$$T_F=\frac{{\epsilon }_F}{k_B}=\frac{{\left(3{\pi }^{2}n{\hslash }^3\right)}^{\frac{3}{2}}}{2{m_e\left(1+{\lambda }_{e-ph}\right)k}_B},$$

(8)

where $m_e={9.109\times 10}^{-31}kg$ is bare electron mass, $\hslash =1.055{\times 10}^{-34}J\cdot s$ is reduced Planck constant, $k_B=1.381\times {10}^{-23}{m}^2\cdot kg\cdot s^{-2}\cdot K^{-1}$ is Boltzmann constant, and $n$ is the charge carrier density per volume ($m^{-3}$). Equation 8 can be used, if the Hall resistance measurements were analysed to estimate the charge carrier density, $n$.

If Hall resistance measurements were not performed, then to calculate the Fermi temperature, we utilized the equation [58,59]:

$$T_F=\frac{{\pi }^2{m}_e}{8k_B}\times \left(1+{\lambda }_{e-ph}\right)\times {\xi }^2\left(0\right)\times {\left(\frac{\alpha k_B{T}_c}{\hslash }\right)}^2,$$

(9)

where $\alpha =\frac{2\mathsf{\Delta }\left(0\right)}{k_B·T_c}$ is the gap-to-transition temperature ratio and this is the only unknown parameter in Eq. 6.

2.5. The gap-to-transition temperature ratio

To calculate the Fermi temperature by Eq. 9 there is a need to know $\alpha =\frac{2\mathsf{\Delta }\left(0\right)}{k_B·T_c}$. In this study, to determine $\alpha =\frac{2\mathsf{\Delta }\left(0\right)}{k_B·T_c}$ we utilized the following approach. Carbotte [111] collected various parameters for 32 electron-phonon mediated superconductors, which exhibit $0.43\le {\lambda }_{e-ph}\le 3.0$ and $3.53\le \frac{2\mathsf{\Delta }\left(0\right)}{k_B·T_c}\le 5.19$. In Figure 1 we presented the dataset reported by Carbotte in his Table IV [111]. The dependence $\frac{2\mathsf{\Delta }\left(0\right)}{k_B·T_c}$ vs ${\lambda }_{e-ph}$ can be approximate by linear function (Figure 1) [117]:

$$\frac{2\Delta \left(0\right)}{k_B{T}_c}=C+D\times {\lambda }_{e-ph},$$

(10)

where $C=3.26\pm 0.06$, and $D=0.74\pm 0.04$.

As far as one can determine ${\lambda }_{e-ph}$ by utilized Equations (3)-(6), the $\frac{2\mathsf{\Delta }\left(0\right)}{k_B·T_c}$ ratio can be estimated from the Eq. (10).

3. Results

3.1. Highly-compressed titanium

Zhang et al [54] and Liu et al [55] reported on record high $T_c$ in $\delta -Ti$ phase compressed at megabar pressures. In Figure 2 we showed the fit of the $R\left(T\right)$ dataset measured by Zhang et al [54] for the $\omega -Ti$ phase compressed at $P=18GPa$ to Equation 3.

Deduced Debye temperature (Figure 2,a) for $\omega -Ti$-phase ($P=18GPa$) is $T_{\theta }=361\pm 1K$ which is in ballpark value with $T_{\theta }\left(298K\right)=380K$ for uncompressed pure elemental titanium, which exhibits $\alpha -Ti$ phase [118].

To calculate the electron-phonon coupling strength constant, ${\lambda }_{e-ph}$, by Equations 4-6, we defined the superconducting transition temperature, $T_c=2.1K$, by the use of $\frac{R\left(T\right)}{R_{norm}}=0.25$ criterion, which was chosen based on the lowest temperature, at which experimental $R\left(T\right)$ data measured at $P=18GPa$ was reported by Zhang et al [54]. Deduced ${\lambda }_{e-ph}=0.49$, which is very close to the ${\lambda }_{e-ph}=0.43$ of pure elemental aluminium (Figure 1, and Ref. [111]).

We also confirmed the power-law exponent $n=3.1$ (reported by Zhang et al [54]) for the temperature dependent $R\left(T\right)$, which was extracted by Zhang et al [54] from the simple power-law fit of $R\left(T\right)$ at temperature range of $3K\le T\le 70K$:

$$R\left(T\right)=R_0+X\times T^n,$$

(11)

where $R_0$, $X$ and $n$ are free fitting parameters. As we showed earlier [119], Eq. 10 does not always return correct $n$-values, and $R\left(T\right)$ data fit to Eq. 3, where $p$ is free-fitting parameter, is the reliable approach to derive the power-law exponent. However, for the given case, our fit to Eq. 3 (Figure 2,b) returns the same power-law exponent, $p=3.15\pm 0.03$, to the one reported by Zhang et al [54].

In Figure 3 we showed $R\left(T\right)$ data measured by Zhang et al [54] and Liu et al [55] and data fits to Equations 3-7 for the $\delta -Ti$ phase compressed at $P=154GPa$ (Figure 3,a), $P=180GPa$ (Figure 3,b), $P=183GPa$ (Figure 3,c), and $P=245GPa$ (Figure 3,d).

While Liu et al [55] reported first principles calculation result for ${\lambda }_{e-ph}$ and logarithmic frequency ${\omega }_{log}$ for highly compressed titanium over wide range of applied pressure, in Figure 4 we presented a comparison of the deduced ${\lambda }_{e-ph}$ and $T_{\theta }$ values from experiment and calculated ones [55]. To compare ${\omega }_{log}$ (calculated by first principles calculations) and $T_{\theta }$ deduced from experiment, we used theoretical expression proposed by Semenok [120]:

$${\frac{1}{0.827}\times \frac{\hslash }{k_B}\times \omega }_{log}\cong T_{\theta }.$$

(12)

In Figure 4,c we also show $T_F$ values calculated by Eq. 8, where we used derived ${\lambda }_{e-ph}$ and bulk density of charge carriers in compressed titanium, $n$, measured by Zhang et al [54]. Due to Zhang et al [54] reported the $R\left(T\right)$ and $n$ measured at different pressures, for $T_F$ calculations we assumed the following approximations: $n\left(P=18GPa\right)=n\left(P=31GPa\right)=1.72\times {10}^{28}{m}^{-3}$; $n\left(P=154GPa\right)=2.39\times {10}^{28}{m}^{-3}$; $n\left(P=180GPa\right)=n\left(P=183GPa\right)=n\left(P=177GPa\right)=1.70\times {10}^{28}{m}^{-3}$.

The evolution of the adiabaticity strength constant $\frac{T_{\theta }}{T_F}$ vs pressure is also showed in Figure 4,c.

It can be seen (Figure 4) that there is a very good agreement between calculated by first principles calculations and extracted from experiment ${\lambda }_{e-ph}$ and characteristic phonon temperatures, $T_{\theta }$ and ${\frac{1}{0.827}\times \frac{\hslash }{k_B}\times \omega }_{log}$, at low and high applied pressures. More experimental data is required to perform more detailed comparison between calculated and experimental values.

Derived values for highly-compressed titanium are in Figures 5 to 7, which are widely used representation of main superconducting families (while other global scaling laws are utilized different variables [71,121,122,123,124,125,126,127]).

It is interesting to note, that $\delta -Ti$ is located in close proximity to A15 superconductors in all of these plots (Figures 5 to 7). It is more likely, that this is a reflection that the highest performance of the electron-phonon mediated superconductivity in metals and alloys is achieved for these materials.

Figure 5. Uemura plot, where highly-compressed $Ti$, $Ta{H}_3$, ${LaBeH}_8$, and black and violet phosphorous (BP and VP, respectively) are shown together with several families of superconductors: metals, iron-based superconductors, diborides, cuprates, Laves phases, hydrides, and others. References on original data can be found in Refs. [79,89,128,129].

Figure 5. Uemura plot, where highly-compressed $Ti$, $Ta{H}_3$, ${LaBeH}_8$, and black and violet phosphorous (BP and VP, respectively) are shown together with several families of superconductors: metals, iron-based superconductors, diborides, cuprates, Laves phases, hydrides, and others. References on original data can be found in Refs. [79,89,128,129].

Figure 6. The nonadiabaticity strength constant $\frac{T_{\theta }}{T_F}$ vs ${\lambda }_{e-ph}$ where several families of superconductors and highly-compressed $Ti$, $Ta{H}_3$, ${LaBeH}_8$, black and violet phosphorous are shown. References on original data can be found in Refs. [79,89,128,129].

Figure 6. The nonadiabaticity strength constant $\frac{T_{\theta }}{T_F}$ vs ${\lambda }_{e-ph}$ where several families of superconductors and highly-compressed $Ti$, $Ta{H}_3$, ${LaBeH}_8$, black and violet phosphorous are shown. References on original data can be found in Refs. [79,89,128,129].

Figure 7. The nonadiabaticity strength constant $\frac{T_{\theta }}{T_F}$ vs $T_c$ for several families of superconductors and highly-compressed $Ti$, $Ta{H}_3$, ${LaBeH}_8$, black and violet phosphorous are shown. References on original data can be found in Refs. [79,89,128,129].

Figure 7. The nonadiabaticity strength constant $\frac{T_{\theta }}{T_F}$ vs $T_c$ for several families of superconductors and highly-compressed $Ti$, $Ta{H}_3$, ${LaBeH}_8$, black and violet phosphorous are shown. References on original data can be found in Refs. [79,89,128,129].

3.2. Highly-compressed I-43d-phase of TaH₃

Recently, He et al [21] reported on the observation of high-temperature superconductivity in highly-compressed I-43d-phase of TaH₃. In Figure 8 we showed the fit of the $R\left(T\right)$ dataset measured by He et al [21] for the tantalum hydride compressed at $P=197GPa$.

By utilizing Eqs. 4-6, we deduced ${\lambda }_{e-ph}=1.53$ (Figure 8), which is within ballpark value for other highly compressed hydride superconductors [3,106].

Because He et al [21] did not report result of Hall coefficient measurements, we deduced the Fermi temperature by the use of Eq. 9, and this we deduced $B_{c2}\left(T\right)$ dataset from R(T,B) curves reported by He et al [21] in their Figure 2,a [21], for which we utilized the criterion of $\frac{R\left(T\right)}{R_{norm}}=0.02$. Obtained $B_{c2}\left(T\right)$ data and data fit are shown in Figure 8(b). Deduced $\xi \left(0\right)=5.45\pm 0.10nm$.

To calculate the Fermi temperature in I-43d-phase of TaH₃ at P = 197 GPa, we substituted derived ${\lambda }_{e-ph}=1.53$ and $\xi \left(0\right)=5.45nm$ in Eq. 9, where $\alpha =\frac{2\mathsf{\Delta }\left(0\right)}{k_B·T_c}=4.39$ was obtained by substituting ${\lambda }_{e-ph}=1.53$ in Eq. 10 (Figure 1).

In the result of our analysis the following fundamental parameters of the I-43d-phase of TaH₃ ($P=197GPa$) have been extracted:

(1) the Debye temperature, $T_{\theta }=263K$;

(2) the electron-phonon coupling constant, ${\lambda }_{e-ph}=1.53\pm 0.13$;

(3) the ground state coherence length, $\xi \left(0\right)=1.53\pm 0.13$;

(4) the Fermi temperature, $T_F=1324\pm 74K$;

(5) $\frac{T_c}{T_F}=0.019\pm 0.01$, which implies that the this phase falls in unconventional superconductors band in the Uemura plot;

(6) the nonadiabaticity strength constant, $\frac{T_{\theta }}{T_F}=0.20\pm 0.01$.

In Figure 5, Figure 6 and Figure 7 one can see the position of the I-43d-phase of TaH₃ at P = 197 GPa (within other representative materials from main families of superconductors), from which can be concluded that TaH₃ are typical superhydride exhibited similar strength of nonadiabatic effects to its near room temperature counterparts, i.e. H₃S and LaH₁₀.

3.3. Highly-compressed Fm-3m-phase of LaBeH₈

Recently, Song et al [98] reported on the observation of high-temperature superconductivity with in highly-compressed LaBeH₈. Crystalline structure of this superhydride at $P=120GPa$ was identified as $Fm\overline{3}m$, which was predicted (as one of several possibilities) by Zhang et al [130]. In Figure 9(a) we showed the fit of the $R\left(T\right)$ dataset measured by Song et al [98] in the LaBeH₈ compressed at $P=120GPa$.

$B_{c2}\left(T\right)$ dataset was extracted from R(T,B) curves reported Song et al [98] in their Figure 3,a [98]. For $B_{c2}\left(T\right)$ definition we utilized the criterion of $\frac{R\left(T\right)}{R_{norm}}=0.25$. Obtained $B_{c2}\left(T\right)$ data and data fit are shown in Figure 9(b). Deduced $\xi \left(0\right)=2.8nm$.

Data analysis by the same routine described in previous Section 3.2 showed that $Fm\overline{3}m$-phase of LaBeH₈ at P = 120 GPa exhibits the following parammeters:

(1) the Debye temperature, $T_{\theta }=752\pm 6K$;

(2) the electron-phonon coupling constant, ${\lambda }_{e-ph}=1.46$;

(3) the ground state coherence length, $\xi \left(0\right)=2.80\pm 0.02nm$;

(4) the Fermi temperature, $T_F=2413K$;

(5) $\frac{T_c}{T_F}=0.029$, which implies that this phase falls in unconventional superconductors band in the Uemura plot;

(6) the nonadiabaticity strength constant, $\frac{T_{\theta }}{T_F}=0.31\pm 0.01$.

3.4. Highly-compressed black phosphorous

Impact of high pressure on the superconducting parameters of black phosphorous has studied over several decades [99,100,101]. Recent detailed studies in this field have reported by Guo et al [100] and Li et al [99].

To show the reliability of high-pressure studies of superconductors (which was recently questioned by non-experts in the field [131,132]) in Figure 10 we showed raw $R\left(T\right)$ datasets measured at $P=15GPa$ by two independent groups, by Shirotani et al [101] and Li et al [99], whose reports have been published within a time frame of 24 years.

The agreement between deduced ${\lambda }_{e-ph}$ (Figure 10) from two datasets [99,101] is remarkable. It should be noted that the approach, used for this analysis (Figure 10), has been developed to analyze data measured in highly-compressed near-room temperature superconductors [106], which particularly implies that concerns expressed by non-experts in the field [131,132,133,134] in regard of highly-compressed near-room temperature hydride superconductors do not have any scientific background.

In Figure 11 we showed $B_{c2}\left(T\right)$ datasets extracted from raw $R\left(T,B\right)$ datasets measured at very close pressure, $P=15.9GPa$ [100] and $P=15GPa$ [99], which were also reported by two independent groups. For the $B_{c2}\left(T\right)$ definition we utilized the same strict criterion of $\frac{R\left(T\right)}{R_{norm}}=0.01$ for both $R\left(T,B\right)$ datasets in Figure 11.

Average deduced parameters for black phosphorus $P=15GPa$, which we derived from experimental data analysis reported by three different groups and which were used to position the black phosphorus in Figures 1,5-7 are:

(1) the Debye temperature, $T_{\theta }=587K$;

(2) the electron-phonon coupling constant, ${\lambda }_{e-ph}=0.548$;

(3) the ground state coherence length, $\xi \left(0\right)=77nm$;

(4) the Fermi temperature, $T_F=5200K$;

(5) $\frac{T_c}{T_F}=0.001$, which implies that black phosphorus falls in conventional superconductors band in the Uemura plot;

(6) the nonadiabaticity strength constant, $\frac{T_{\theta }}{T_F}=0.11$.

Deduced parameters show that the black phosphorus compressed at $P=15GPa$ exhibits low strength of nonadiabatic effects.

3.5. Highly-compressed violet phosphorous

Recently, Wu et al [53] reported on the observation of the superconducting state in violet phosphorus (vP) with $T_c>5K$ when the material is subjected to high pressure in the range of $3.6GPa\le P\le 40.2GPa$. In Figure 12(a) we showed the $R\left(T\right)$ dataset, and data fit to Eq. 3, measured by Wu et al [53] in the violet phosphorus compressed at $P=40.2GPa$.

$B_{c2}\left(T\right)$ dataset was extracted from the only R(T,B) dataset reported by Wu et al [53] for material compressed at Wu et al [53]. For $B_{c2}\left(T\right)$ definition we utilized the criterion of $\frac{R\left(T\right)}{R_{norm}}=0.14$. Deduced $B_{c2}\left(T\right)$ dataset is shown in Figure 12(b). The fit to Equation 7 (which is single band model) has low quality, because $B_{c2}\left(T\right)$ has an upturn at $T\lesssim 4K$. We interpreted this upturn as an evidence for the second band opening at $T\lesssim 4K$, and, thus, we fitted data used two-band model [128,135]:

$$B_{c2,total}\left(T\right){=B}_{c2,band1}\left(T\right)+B_{c2,band2}\left(T\right),$$

(13)

where $B_{c2,band1}\left(T\right)$ and $B_{c2,band2}\left(T\right)$ exhibit their independent transition temperature and the coherence length. Deduced values are listed in the Figure Caption to Figure 12. However for further analysis we used $T_c=T_{c,band1}=9.0K$ and ${\xi \left(0\right)}_{total}=36\pm 1nm$.

Processing data by the same approach described in previous Sections, we derived the following parameters for violet phosphorus compressed at $P~40GPa$:

(1) the Debye temperature, $T_{\theta }=665\pm 4K$;

(2) the electron-phonon coupling constant, ${\lambda }_{e-ph}=0.607$;

(3) the ground state coherence length, $\xi \left(0\right)=36\pm 1nm$;

(4) the Fermi temperature, $T_F=3240K$;

(5) $\frac{T_c}{T_F}=0.003$, which implies that this phase falls in close proximity to conventional superconductors band in the Uemura plot;

(6) the nonadiabaticity strength constant, $\frac{T_{\theta }}{T_F}=0.21$.

Derived parameters imply that the violet phosphorus compressed at $P~40GPa$ exhibit moderate level of nonadiabatic effects similar to the ones in highly-compressed hydrogen-rich near-room temperature superconductors $H_{3}S$ and ${LaH}_{10}$.

4. Conclusions

In this work, we analyzed experimental data reported for five recently discovered highly-compressed superconductors: $\delta -Ti$ [54,55], ${TaH}_3$ [21], ${LaBeH}_8$ [98], black phosphorous [99,100,101], and violet phosphorous [53], for which we established several superconducting parameters, including the strength of nonadiabaticity, $\frac{{\hslash \omega }_D}{{k_{B}T}_F}=\frac{T_{\theta }}{T_F}$.