1. Introduction
1.1. Preamble
The discovery of H_{2}S having a critical temperature (T_{c}) ≈ 200 K when subjected to a pressure of about 150 GPa [1] marks a significant advance towards the goal of room temperature superconductivity. This feature is now attributed to H_{3}S which is generally believed to be realized via the reaction 2H_{2}S → (H_{3}S)^{+} + (SH)^{−} [2] . As a consequence, several properties of H_{3}S have been under intensive investigation, properties such as the values of its gaps (Δs), coherence length (ξ), lower and upper critical magnetic fields (H_{c}_{1}, H_{c}_{2}), critical current density (J_{c}) and the London penetration depth (λ_{L}). Knowledge of these properties enables one to calculate the Ginsberg-Landau parameter (κ) and the critical current density (J_{c}_{1}) due to H_{c}_{1}. Based on the value of J_{c}_{1}, which is one of the criteria among several, Hirsch and Marsiglio (H & M) [3] , and references therein, have expressed the opinion that H_{3}S cannot be classified as a conventional superconductor (SC) because it does not meet the mandatory requirement of displaying the Meissner effect.
The publication of a paper in Nature in 2020 entitled “Room-temperature superconductivity in a carbonaceous sulphur hydride” (C-S-H) by Snider, et al. [4] was an exciting development which led Dogan and Cohen (D&C) to carry out a detailed theoretical study of this SC in [5] , where they presented some valuable results pertaining to H_{3}S. While the paper by Snider, et al. was subsequently retracted in 2022, the approach followed by D&C for carrying out a comprehensive study of the properties of any SC and their results concerning H_{3}S remain valid. The template given by D&C, a variant of which we employ here, enables one to obtain at T = 0 the values of ξ, Fermi velocity v_{F}, number density of the charge carriers N_{s}, λ_{L} and κ by sequentially employing the following well-known equations:
${\xi}_{0}=\sqrt{\frac{{\varphi}_{0}}{2\pi {H}_{c2}\left(0\right)}}$ (1)
${v}_{F}\left(0\right)=\frac{\pi {\Delta}_{0}{\xi}_{0}}{\hslash}$ (2)
${N}_{s}\left(0\right)=\frac{1}{3{\pi}^{2}}{\left(\frac{{m}^{*}{v}_{F}^{2}}{{\hslash}^{2}}\right)}^{3/2}$ (3)
${\lambda}_{L}\left(0\right)=\sqrt{\frac{{m}^{*}}{{\mu}_{0}{N}_{s}{e}^{2}}}$ (4)
$\kappa \left(0\right)=\frac{{\lambda}_{L}\left(0\right)}{{\xi}_{0}},$ (5)
where
${\varphi}_{0}=2.0678\text{weber}$ , Δ_{0} is the value of the gap at T = 0,
${m}^{*}=\eta {m}_{e}$ is the effective mass of the electron, m_{e} being the free electron mass, μ_{0} = 4π × 10^{−7} h/m_{e} and e is the electronic charge.
We note that an alternative equation for v_{F} is
${v}_{F}=\sqrt{\frac{2\mu}{\eta {m}_{e}}}c$ (μ and m_{e} in units of electron-Volts) (6)
D&C’s findings about H_{3}S are:
$\kappa \left(0\right)=\text{77}$ ,
$\Delta {T}_{c}/{T}_{c}=0.15$
which signify that it is a strongly type II SC.
The line of argument followed by H&M is based on the calculation of the critical current density J_{c}_{1} that comes into play due to the lower critical field H_{c}_{1} and opposes or expels, in accord with Lenz’s law, the entry of the magnetic flux into the interior of an SC when it is subjected to an external magnetic field. The equation for J_{c}_{1} employed by H&M is the London equation
${J}_{c1}\left(0\right)=\frac{c}{4\pi {\lambda}_{L}\left(0\right)}{H}_{c1}\left(0\right),$ (7)
where c is the velocity of light. We note that the value of H_{c}_{1} employed by H&M is based on the demagnetization factor. An alternative expression is
${H}_{c}{}_{1}=\frac{{\varphi}_{0}}{4\pi {\lambda}_{L}^{2}}\mathrm{ln}\left(\kappa \right)$ (8)
Other features of H_{3}S that provide a backdrop to the present work are the data for the variation of H_{c}_{2} with temperature given in Mozaffari, et al. [6] . These authors also obtained fits to their data by first employing the conventional Ginsberg- Landau theory and subsequently the Werthamer-Helfand-Hohenberg [7] formalism where the temperature dependence of H_{c}_{2}(T) is defined by orbital and spin- paramagnetic effects in the dirty limit. Results of the latter fit that concern us are
T_{c} of the sample H_{c}_{2}(0) ξ_{0}
197 K 88 T 1.84 nm
Following a phenomenological approach, excellent fits to the aforesaid data were also obtained by Talantsev [8] via four equations each of which employed two or more parameters from the following set of properties of H_{3}S
Σ1 = {T_{c}, Δ, ξ, λ_{L}, jump in sp.ht.} (9)
The values of and H_{c}_{2}(0), μ(0) and ξ_{0} arrived at in [8] are
T_{c} of the sample H_{c}_{2}(0) μ_{0} ξ_{0}
191K 100 T 0.47 eV 1.86 nm
1.2. The Scope and the Plan of the Present Paper
We study in this paper the properties of H_{3}S that were the concern of Talantsev [8] , H&M, D&C and Minkov, et al. [9] [10] by employing the framework of the chemical potential (μ)-incorporated Bethe-Salpeter equation (BSE). With a superpropagator as its kernel, the BSE leads to a generalization of the BCS equations (GBCSEs) [11] which have been shown to explain several properties of a wide variety of SCs that includes elemental SCs, MgB_{2}, YBCO, Tl-2212, Bi-2212, SrTiO_{3}, NbN, heavy-fermion and Fe-based SCs. For references to papers where these SCs have been dealt with, see [12] .
Insofar as H_{3}S is concerned, the GBCSEs have been shown to shed light on 1) most of its empirical and inferred features [13] , 2) the empirical values of its J_{c}(T) [14] and 3) the value of its ξ_{0} [15] . Further features of H_{3}S studied here are based on the data for the variation of H_{c}_{2} with temperature given in Mozaffari, et al.
The basic premise of our approach, which is an alternative to the approaches followed in [8] and [6] , is that the chemical potential μ(T) of an SC encapsulates most of its physical features such as size, shape, the manner of preparation and the nature of dopants and that it is predominantly responsible for the T-depen- dent properties of an SC. This idea is of course not new. It was employed earlier by Eagles [16] in the context of the superconductivity of SrTiO_{3} by simultaneously considering the μ-dependent gap equation and the equation for μ. Eagle’s paper is generally considered to mark the beginning of what is now called the BCS-BEC crossover, which was subsequently dealt with in a similar manner in the work of Leggett [17] by additionally employing scattering length theory (SLT). For a study of cross-over physics without appeal to SLT, we draw attention to [18] . It is notable that while employment of the number equation in [16] [17] [18] in cross-over physics is a matter of course, Schrieffer [19] had made the general remark as early as 1964 that “one must simultaneously solve the gap equation and the constraint condition (i.e., the number equation) to determine Δ_{k} and μ.” By and large, such practice until recently has not percolated into the main stream of the high-T_{c} SCs for which, in general, the inequality
${E}_{F}\gg k\theta $ is not satisfied. To put in perspective the work reported here, we are dealing for the first time with H_{c}_{2} of a system at T ≠ 0 in accord with Schrieffer’s remark.
The μ-, T- and H-dependent core equations via which the H_{c}_{2}(T) and J_{c}(T) of any SC can be dealt with are given in Section 2. The procedural details of our approach are given in Section 3 where we follow up on [20] , in which we gave fits using our approach to two data-sets reported in [6] , for the purpose of shedding light on several features associated with
a) H&M’s result
${J}_{c1}\left(0\right)=1.87\times {10}^{9}\text{A}/{\text{cm}}^{\text{2}}$ ^{1}, (10)
which is obtained by employing (7) with the estimates of demagnetizing factor N = 0.8, H_{c}_{1} = 0.48 T and λ_{L} = 20.4 nm, which are based on the experimental set-up of Minkov, et al. [9] . The above value of J_{c}_{1} is much greater than that of any other known SC, whether type I or type II, hard or soft;
b) the following more recent results of Minkov, et al. [10] for a sample with T_{c} ≈195 K
$\xi \left(10\text{\hspace{0.17em}}\text{K}\right)=1.85\text{\hspace{0.17em}}\text{nm},\text{}{\lambda}_{L}\left(10\text{\hspace{0.17em}}\text{K}\right)=37\text{\hspace{0.17em}}\text{nm},\text{}\kappa \left(10\text{\hspace{0.17em}}\text{K}\right)=20,$ (11)
and
${J}_{c}\left(30\text{\hspace{0.17em}}\text{K}\right)=7.1\times {10}^{6}\text{\hspace{0.17em}}\text{A}\cdot {\text{cm}}^{-2}.$ (12)
A salient feature of our approach [21] is that, rather than employing any model to calculate J_{c}, it is based directly on the definition of J_{c} = N_{s} e v_{c}, where e is the electronic charge and v_{c} the critical velocity of the electrons. As is well known, BCS theory sets the momentum (P) of the centre-of mass of the pairs as zero at the outset, which necessitates the employment of a model for the calculation of J_{c} based indirectly on its dependence on the applied magnetic field. A virtue of the Bethe-Salpeter formalism is that it enables one to extend BCS to the situation where P ≠ 0. The equation catering to this situation is one of our core equations.
In the final section we deal afresh with the issue of the Meissner effect in the context of H_{3}S raised in [3] because the approach followed in this paper differs significantly from the earlier approaches.
The paper includes two appendices for convenience of the reader: The first of these is a glossary of the symbols we have employed and the other comprises two flow charts that give an overview of our procedure.
2. The Core Equations
The generic μ-, T- and H-dependent pairing equation that we employ to calculate H_{c}(T) or j_{c}(T) is
$Eq1\left(t,h\right)\equiv 1-{\lambda}_{m1}X{\displaystyle \underset{{z}_{-}\left(\mathrm{..}\right)}{\overset{{z}_{+}\left(\mathrm{..}\right)}{\int}}{\displaystyle \underset{n=0}{\overset{{N}_{L}\left(\mathrm{..}\right)}{\sum}}\frac{\mathrm{tanh}\left[{A}_{+}\left(\mathrm{..}\right)\right]+\mathrm{tanh}\left[{A}_{-}\left(\mathrm{..}\right)\right]}{{z}^{2}-1+\left(n+1/2\right)\frac{\hslash {\Omega}_{1}\left(h\right)}{{\mu}_{1}Y}}}}=0,$ (13)
where
$t=T/{T}_{c}$ ,
$h\left(t\right)={H}_{c2}\left(T\right)/{H}_{c20}$ _{ }
${z}_{\pm}\left(\mathrm{..}\right)=\sqrt{\frac{Y\rho \pm 1\mp 1/y}{3Y\rho}},\text{}{N}_{L}\left(\mathrm{..}\right)=floor\left[\frac{2k\theta}{3\hslash {\Omega}_{1}\left(h\right)}\left(1+Y\rho -1/y\right)-\frac{1}{2}\right]$ _{ }
${A}_{\pm}\left(\mathrm{..}\right)=\frac{\rho \theta Y}{2t{T}_{c}}\left[{z}^{2}-1+\left(n+1/2\right)\frac{\hslash {\Omega}_{1}\left(h\right)}{{\mu}_{1}Y}\pm \frac{1}{{\mu}_{1}Yy}\right]$
${\mu}_{1}=\rho k\theta ,\text{}y=k\theta /\alpha ,\text{}\alpha =\left|P\right|\left|p\right|\mathrm{cos}\left(P,p\right)/2{m}^{*},\text{}{m}^{*}=\eta {m}_{e}$
$\Omega \left(h\right)={\Omega}_{0}h{H}_{c}{}_{2}\left(0\right)/\eta ,\text{}{\Omega}_{0}=e/{m}_{e}c$
θ is the Debye temperature (DT) of the ions that cause pairing and μ_{1} the chemical potential at t = 1; P is the momentum of the pairs in the lab frame, p the relative momentum of their constituents in the center-of-mass frame and m* is the effective mass of an electron, m_{e} being the free electron mass; the unit for the applied field is Gauss.
For (X, Y) in (13), we shall employ the following values
$X=\text{1},\text{\hspace{0.17em}}\text{\hspace{0.05em}}Y=\text{1}$ (14)
$X=Y=q\left(t\right)$ (15)
$X=\frac{1}{\sqrt{q}}\frac{h}{{h}_{1}},\text{}Y=q$ (16)
Substitution of (14) into (13) yields the value of the interaction parameter λ_{m}_{1} with the input of θ, ρ, η, t = t_{1} and h_{1} = H_{1}/H_{c}_{20}, H_{1} being the value of the self-field at t = t_{1} in the absence of any applied field and P = 0 (
$y=\infty $ ).
Substitution of (15) into (13) determines how λ_{m}_{1}, μ_{1} and h_{1} change with t. It has been shown [20] that each of the following models for q(t) provides an excellent fit to the data of Mozaffari, et al.:
$\begin{array}{c}q\left(t\right)=1+{a}_{0}\left(1-t\right)\text{(i)}\\ =\text{1}+{a}_{0}\left(1-{t}^{{a}_{1}}\right)\text{(ii)}\\ =\text{1}+{a}_{0}{\left(1-{t}^{2}\right)}^{{a}_{2}}.\text{(iii)}\end{array}$ (17)
However, these fits came with the caveat that μ(0) = q(0)ρkθ > 250 eV for all these models, in stark contrast with its value < 1 eV noted in (9). Because any value of ρ < 1 causes the limit
${z}_{-}$ in (13) to become pure imaginary and leads to complex roots, the fits in [20] were obtained by employing ρ = 1. In order to seek fits that also lead to μ(0) of the desired order, we stipulated that
${z}_{-}=\mathrm{Re}\left(\sqrt{\left(\rho -1\right)/3\rho}\right)$ in (13) order to employ values of ρ < 1. Remarkably, we thus found that innumerable such values also lead to fits as good as those obtained earlier. Because model (17iii) had led to the least value of μ(0), this is the model we adopt for values of T<T_{c} which enable us to shed light on the values of J_{c} noted in (10) and (12) by employing the number equation given below. Because of the employment of (17iii), we call this the (a_{0}, a_{2}) approach.
Interestingly, the J_{c} values of both H&M and Minkov, et al. [10] can also be addressed by appealing simultaneously to the number equation and (13) with the choice of X, Y as in (16). This approach leads to the values of (q, y) corresponding to each pair of (h, j_{c}) at any T; we therefore call it the (q, y) approach which enables one to calculate several T-dependent properties of the SC such as N_{s}, ξ, etc. The derivation of (13) with X and Y as in (16) has been given in [21] .
Our other core equation is the number equation derived in [22]
$\begin{array}{l}{N}_{s}\left(t,h\right)=C{\left(h{H}_{c2}\right)}^{3/2}{\displaystyle {\int}_{0}^{\sqrt{L\left(t,h\right)}}\left[{\displaystyle \underset{n=0}{\overset{{N}_{m}\left(t,h\right)}{\sum}}\left\{\mathrm{....}\right\}}\right]\text{d}z},\\ \left\{\mathrm{....}\right\}=\left\{1-\mathrm{tanh}\left(\frac{\hslash \Omega \left(h\right)}{2kt{T}_{c}}\left[n+1/2+{z}^{2}-\frac{q\left(t\right){\mu}_{1}}{\hslash \Omega \left(h\right)}\right]\right)\right\}\end{array}$ (18)
where
$\begin{array}{l}C=2.1213\times {10}^{9},\text{}L\left(t,h\right)=\frac{1}{3}\frac{k\theta \left[\rho q\left(t\right)+1\right]}{\hslash \Omega \left(h\right)}\text{}\\ {N}_{m}\left(t,h\right)=floor\left[\frac{2}{3}\frac{k\theta \left[\rho q\left(t\right)+1\right]}{\hslash \Omega \left(h\right)}-\frac{1}{2}\right].\end{array}$
Since we employ (18) in lieu of (3), it is prudent to compare the results one obtains when the inputs employed for both the equations are nearly the same. Insofar as the latter equation is concerned, for C-S-H it yields N_{s} = 5 × 10^{27} m^{−3} [5] for η = 1, E_{F} = 1.06 eV (which follows from v_{F} = 6.1 × 10^{5} m/s), T = 0 and H = 0. For the same values of η and E_{F}, T = 10^{−6} K and H = 300 G (which is the sf that exists in the absence of an applied field), (18) yields N_{s} = 1.75 × 10^{27} m^{−3}.
Remark: With the employment of (18) in the following in mind, we note that the result N_{s} = 1.75 × 10^{27} m^{−3} remains unchanged whether H_{c}_{2} = 0.03 or 120 T for any value of the temperature. It is notable in this context that J_{c} = 0 at H = H_{c} because v_{c}= 0 and not because N_{s} given by (18) vanishes.
The equation for the critical velocity v_{c} in terms of (q, y) is [21]
${v}_{c}\left(\mathrm{...}\right)=\frac{c}{2y}\sqrt{\frac{6k\theta}{q\rho \eta {m}_{e}}},$ (19)
whence the number equation may be written as
$1-\frac{{J}_{c}\left(\mathrm{..}\right)}{e{N}_{s}\left(\mathrm{..}\right){v}_{c}\left(\mathrm{..}\right)}=0.$ (20)
3. Fits to an Empirical Data-Set of Mozaffari, et al. and a Study Pertaining to the Results in (10)-(12)
We undertake in this section the task of shedding light on the results given in the above-noted equations. For this purpose, we employ the H_{c}_{2}(T) data of Mozaffari, et al. which covers the range T_{c} = 191 K to 105.1 K. Because we need the values of μ(t) [= q(t)μ_{1}] and H_{c}_{2} at 0, 10 and 30 K, vide (10 - 12), we now need a model for q(t) that provides a fit to the data and thence to the values of μ and H_{c}_{2} at any specific temperature.
3.1. Procedure for Finding a Fit to Any Set of H_{c}_{2}(T) Values
a) Resolve the Debye temperature (DT) of H_{3}S given by Talantsev [23] as θ(H_{3}S) = 1531 K into the DTs of its constituents by employing the double- pendulum model [24] , whence
${\theta}_{H}=1983.2\text{\hspace{0.17em}}\text{K},\text{}{\theta}_{S}=174.5\text{\hspace{0.17em}}\text{K}\text{.}$ (21)
b) Both of these DTs are needed [13] for an explanation of the empirical value of the T_{c} of H_{3}S and its inferred gap-values of about 40 meV and 28 meV because the 1-phonon exchange mechanism (1PEM) per se due either to the H or the S ions violates the Bogoliubov constraint [11] . However, since a magnetic field considerably weakens the strength of the interaction, it turns out that the data being addressed here can be dealt with via the consideration of the H ions alone.
c) With θ = θ_{H}, solve Eq1(t, h), vide (13), to obtain λ_{m}_{1} for any chosen value of ρ with the input of η = 2.76 (following [8] where it was shown to lead to reasonable values of several parameters), t = 1, X = Y =1,
$y=\infty $ , h = 300/H_{c}_{20} = 3 × 10^{−4}, where 300 G is the self-field sf and H_{c}_{20} = 10^{6} G (chosen for convenience because it enables one to look for numerical solutions near h ≈ 1).
d) Solve Eq1(t_{1}, h_{1}) and Eq1(t_{2}, h_{2}), vide (13), with X and Y as in (15) and
$y=\infty $ to fix the constants a_{0} and a_{2} in the employed model (17iii) by choosing two (t, h) points from the empirical data of 23 such points. For all values of ρ that we employed, the points chosen were
$\left({t}_{1}=145/{T}_{c},{h}_{1}=30.27/{H}_{c}\right)$ and
$\left({t}_{2}=105/{T}_{c},{h}_{2}=61.49/{H}_{c}\right)$ . (22)
3.2. On H&M’s Result Noted in (10)
3.2.1. The (a_{0}, a_{2}) Approach
H&M have given the value of J_{c}_{1}(0) = 1.87 × 10^{9} A/cm^{2} as corresponding to H_{c}_{1} = 0.48 T and λ_{L} = 20.4 nm. The issues we address here are a) Can these results be validated via the (a_{0}, a_{2})-approach? b) If so, what are the values of several other parameters such as v_{F}, ξ and κ corresponding to this triplet?
A salient feature of our approach is that all the results it leads to via a chain of calculations depend only on three parameters, viz., ρ (which determines the chemical potential at t = 1), η and sf. Among these, by far the most important parameter is ρ [20] . For this reason, while seeking to obtain H&M’s value of J_{c}_{1}(0), we kept the values of η and the sf fixed at 2.76 and 0.03 T, respectively, and varied only ρ. We thus found that H&M’s triplet follows by employing, at T = T_{c}, the value of ρ as 0.164, which we arrived at after carrying out calculations for many values of ρ. The detailed results are as follows.
$\rho =0.164,\text{}{\mu}_{1}=2.80\times {10}^{-2}\text{eV,}\eta =\text{2}\text{.76,}h=sf=\text{300G,}t=\text{1}$
$\text{At}\text{\hspace{0.17em}}\text{T}={\text{10}}^{-3}\text{\hspace{0.17em}}\text{K:}\mu =\text{9}\text{.88eV,}{H}_{c2}=\text{332}\text{.2T,}{\lambda}_{m}=2.206\times {10}^{-3},\text{}{N}_{L}=479$ (23)
$\text{At}\text{\hspace{0.17em}}\text{T=10K:}\mu =\text{9}\text{.85eV,}{H}_{c2}=\text{145}\text{.1T,}{\lambda}_{m}=2.199\times {10}^{-3},\text{}{N}_{L}=1095$ (24)
Obtained for either of the above values of μ via (6):
${v}_{F}=1.12\times {10}^{6}\text{m}/\text{s}$
Obtained via (2), with Δ = 38 meV, (18), (4) and (5), respectively,
$\xi =\text{6}\text{.18nm,}{N}_{s}=1.91\times {10}^{29}\text{\hspace{0.17em}}{\text{m}}^{-3}\text{,}{\lambda}_{L}=20.2\text{\hspace{0.17em}}nm\text{,}\kappa =3.2\text{7}$
Obtained via (8): H_{c}_{1} = 0.48 T; obtained via (7): J_{c}_{1} = 1.88 × 10^{9} A/cm^{2}
3.2.2. The (q, y) Approach
In this approach, we seek to find the value of ρ which leads with the input of J_{c}_{1} and H_{c}_{1} to the value of the third member of the H&M’s triplet, i.e., λ_{L}, as ≈ 20.4 nm at 10 K. Since the H&M’s triplet is given at T = 0 and Minkov, et al. have given the values of ξ, λ_{L}, and κ at T = 10 K and the values of J_{c}_{1} at 30 K, a meaningful comparison of various other parameters associated with these necessitates that we should first obtain the value of the H&M’s triplet at T = 10 K from its value at T = 0. We have done so by appealing to the 2-fluid theory and found that, to an accuracy of second place after the decimal, we need to change only J_{c}_{1} from 1.87 × 10^{9} to 1.86 × 10^{9} A/cm^{2}. After repeating the chain of calculations for several value of ρ, we were led to ρ = 37.8 as the value which leads to the desired value of λ_{L}. The details of this exercise which also gives the values of several other parameters associated with H&M’s triplet at 10 K are given below.
$\rho =37.8,\text{}{\mu}_{1}=6.46\text{eV,}\eta =\text{2}\text{.76,}h=sf=\text{300G,}t=1$
Obtained via (13) with X = Y =1:
${\lambda}_{m1}=8.273\times {10}^{-7}$ ,
${N}_{L}=3513054$
Obtained by simultaneously solving (20) and (13) for T = 10, t = 10/T_{c} with X, Y as in (16) and the input of
H_{c}_{1} = 0.48 T, J_{c}_{1} = 1.86 × 10^{9} A/cm^{2}
$q=1.5249,\text{}y=27.646,\text{}\mu =9.851\text{eV,}{\lambda}_{m}=1.072\times {10}^{-5}$
Obtained via the equations noted above for the (a_{0}, a_{2}) approach
${v}_{F}=1.12\times {10}^{6}\text{m}/\text{s}\text{,}\xi =\text{6}\text{.17nm,}{N}_{s}=1.90\times {10}^{29}\text{\hspace{0.17em}}{\text{m}}^{-3}$
v_{c} = 6 × 10^{4} m/s, λ_{L} = 20.22 nm, κ = 3.27
Our findings related with H&M’s values of J_{c}_{1}, H_{c}_{1} and λ_{L} will be discussed below.
3.3. On Minkov, et al.’s [9] Result Noted in (11)-(12)
3.3.1. The (a_{0}, a_{2}) Approach
As above, we begin with an assumed value of ρ < 1, employ η = 2.76 and sf = 300 G and solve (13) to obtain the value of λ_{m}_{1} at t = 1. We then employ model (17iii) and find the values of a_{0} and a_{2}. This is followed up by the chain of calculations as before leading finally to a value of κ. Repeating this procedure by varying ρ, we were able to find the values of this parameter that lead to κ ≈ 20 as noted in (11). Given in Table 1 are five values of ρ that lead to a value of κ in the desired ballpark and the values of various other corresponding parameters.
Given in Figure 1 is the fit to the empirical data provided by model (17iii) for ρ = 0.065 which leads to κ = 19.
3.3.2. The (q, y) Approach
Let us first note that by adopting the values of the diameter (d), thickness (th) and the trapped magnetic flux [m_{trap}(30 K)] of their sample as 85 μm, 2.8 μm and 1.60 × 10^{−8} A m^{2}, respectively, Minkov et al. obtained the result noted in (12) as follows:
${J}_{c}\left(30\text{K}\right)=\frac{3}{\pi}\frac{{m}_{trap}\left(30\text{K}\right)}{d}{\left(\frac{2}{th}\right)}^{3}=7.1\times {10}^{6}\text{A}/{\text{cm}}^{2}.$ (25)
Next, we note that if we employ the values of λ_{L} and κ at T = 10 K as given in (11), then we are led to the corresponding value of H_{c} via (8) as
${H}_{c}{}_{1}=0.36\text{T}\text{.}$ (26)
We now follow the procedure given above by employing (25) and (26) in lieu of H&M’s values of J_{c} and H_{c}_{1}. Given in Table 2 are the results of this exercise for a few select values of ρ.
4. Summing Up
1) Based on the premise that the variation in H_{c}_{2}(T) is caused predominantly by μ(t), we have obtained in this paper a fit to an empirical H_{c}_{2}(T) data-set of Mozaffari, et al. by employing (13). While excellent fits to the same data were obtained earlier by Talantsev [8] via four phenomenological equations each of which employed two or more parameters from the set Σ_{1} in (9), our approach has been shown to lead to a similar fit, vide Figure 1 which relates H_{c}_{2}(T) with a set of variables altogether different from Σ_{1}, viz.,
${\Sigma}_{2}=\left\{\theta ,{\lambda}_{m}\left(T\right),\mu \left(T\right),{L}_{N}\left(T\right)\right\}.$ (27)
For the variation of μ(t) = μ_{1}q(t) (μ_{1} = ρkθ), the model we employed for q(t) was specified in (17iii). The work in [20] and this paper shows that while almost equally good fits to the same data can be obtained by numerous values of ρ, the values of v_{F}, ξ and λ_{L}, etc., that they lead to are different.
2) The issue of the Meissner effect in the context of H_{3}S was addressed by dealing with H&M’s values of the triplet {J_{c}_{1}, H_{c}_{1}, λ_{L}} and Minkov, et al.’s values of ξ, λ_{L} and κ at 10 K and j_{c} at 30 K by employing both the (a_{0}, a_{2}) and the (q, y)
Table 1. Values of various parameters of H_{3}S at T = 10 K obtained by following the procedure of the (a_{0}, a_{2}) approach given in Section 3.2.1.
Figure 1. For the sample with T_{c} = 191 K, the variation of the upper critical field of H_{3}S with temperature obtained via (13) by employing ρ = 0.065 and q(t) as in (17iii). The filled circles are the plot of the empirical values.
approaches. We note that in the former approach, the equation which we solve to obtain the value of H_{c}_{2} for any value of temperature has multiple roots. Since these roots are extremely close to each other, one may employ any one of them for the calculation of any parameter dependent on them. An example: for ρ = 0.164, there are three roots at h =1.4583, 1.4593 and 1.4603.
3) Re: H&M’s values of the triplet {J_{c}_{1}, H_{c}_{1} and λ_{L}}
a) The (a_{0}, a_{2}) approach. Beginning with ρ = 1, η =2.76 and sf = 300 G and employing model (17iii) for q(t), we determined via (13) the values of a_{0} and a_{2} that provide a fit to the H_{c}_{2}(T) data of Mozaffari, et al. With H_{c}_{2}(t) and μ(t) =
Table 2. Sample with T_{c} = 191 K. Values of λ_{m}_{1} obtained by solving (13) with the inputs η = 2.76, sf = 300 G. Inputs for (13) to obtain λ_{m}_{1}: X =1, Y = 1, t = 1, h = 300/H_{c}_{20} (H_{c}_{20} = 10^{6} G). Values of (q, y) obtained by solving (13) and (20) simultaneously, with X, Y as in in (16), and the inputs t = 10/T_{c}, h = 0.36 × 10^{4}/H_{c}_{0}, J_{c}(10 K) = 7.27 × 10^{6} A/cm^{2} (obtained from J_{c}(30 K) = 7.1 × 10^{6} A/cm^{2}). Equations for the calculation of v_{F}, ξ, N_{s}, λ_{L} and κ as given in the text.
μ_{1}q(t) known,we were enabled to sequentially obtain, at any value of t, the values of v_{F}, ξ, N_{s}, λ_{L},H_{c}_{1} and finally J_{c}_{1} via equations that have been specified in Section 3.2.1. Because the value of J_{c}_{1} we obtained for ρ = 1 disagreed with H&M’s value, we repeated the chain of calculations by varying ρ, till we found that ρ = 0.164 leads to J_{c}_{1} = 1.88 × 10^{9} A/cm^{2}, which is very close to H&M’s value of 1.87 × 10^{9} A/cm^{2}. Corresponding to this value of J_{c}_{1}, the values of H_{c}_{1} = 0.48 T and λ_{L} = 20.2 nm were found to be in remarkable agreement with the H&M’s values, viz., 0.48 T and 20.4 nm, respectively. These results provide a validation of the H&M’s triplet because they have been obtained via an altogether different approach without employing the value of any member of it as an input. However, the values of v_{F} = 1.12 × 10^{6} m/s, ξ = 6.18 nm corresponding to the triplet are not in accord with current wisdom because it is believed that the upper limit of the universal Fermi velocity for hydrides is 3.8 × 10^{5} m/s and ξ lies in the range 1.2 - 3 nm. Besides, the value of and κ = 3.27 is much lower than several estimates of it.
b) The (q, y) approach. In this approach, by simultaneously solving (13) and (20) for q any ywith the input of the J_{c}_{1} and H_{c}_{1} values of the H&M’s triplet, we sought to obtain the value of ρ which leads to a value of λ_{L} at T = 10 K close to its value in the triplet. By repeating this process for several values of ρ, we found its desired value to be ≈ 37.8, i.e., μ (T = T_{c}) = 6.46 eV. The values of v_{F}, ξ, λ_{L} and κ at T = 10 K corresponding to it were found to be, respectively, 1.12 × 10^{6} m/s, 6.18 nm, 20.22 nm and 3.27—which are not too different from their values in the (a_{0}, a_{2}) approach.
4) Re: Minkov, et al.’s values of ξ, λ_{L} and κ at 10 K
a) The (a_{0}, a_{2}) approach. Our objective now was to find the value of ρ for which the chain of calculations culminates in the value of κ rather than J_{c}. Beginning with ρ = 1 and repeating the procedure with progressively lower values, we found that ρ = 0.065 (μ_{1} = 0.011 eV) leads to κ = 19, which is very close to Minkov, et al.’s value of 20. However, our values of ξ = 2.98 nm and λ_{L} = 58.0 nm were greater than Minkov, et al.’s values, viz., 1.85 nm for the former and 37 nm for the latter. Besides, the corresponding value of v_{F} = 5.41 × 10^{5} m/s was also found to exceed its universal upper limit. We were hence led to explore if we could simultaneously obtain a lower value for each of these parameters by a different choice of ρ. As is seen from Table 1, we could not succeed because we found that while increasing the value of ρ decreases λ_{L} and ξ, it increases the values of v_{F} and ξ. Decreasing the value of ρ had the opposite effect.
b) The (q, y) approach. From Minkov, et al.’s value of J_{c} (30 K) = 3.1 × 10^{6} A/cm^{2}, we estimated via the 2-fluid theory the value of J_{c} (10 K) to be 3.27 × 10^{6} A/cm^{2}. Employing it together with H_{c} (10 K) = 0.36 T, vide (26), we could obtain the values of q and y corresponding to any value of ρ by simultaneously solving (13) and (20). The values of v_{F}, ξ, λ_{L}, κ, etc., for six triplets of {ρ, q, y} are given in Table 2. Again, while our values of κ for two of these triplets matched Minkov, et al.’s value, the values of λ_{L} and ξ did not. Also, our value of v_{F} exceeded its universal upper limit.
5) Concluding remarks
a) Some of the values of κ that we obtained via two different approaches are not as alarming as 77 given by D&C.
b) Re: the difference between our values of ξ and λ_{L} and those of Minkov, et al. corresponding to the same value of κ ≈ 20, we note that in both of our approaches the values of all the parameters that we calculated depend predominantly on ρ which determines the chemical potential at T = T_{c}; the other two parameters that need to be specified at the beginning, viz., η and sf were not varied. It is plausible that by fine tuning the values of η and sf, the said disagreement is reduced. We draw attention in this context to the last row in Table 1 where changing η from 2.76 to 3.20 is seen to have the effect of reducing v_{F}, ξ and λ_{L} and marginally increasing κ. Similarly, changing sf from 300 to 350 G was found to decrease v_{F} and ξ and λ_{L} and to increase κ.
c) Our finding that there are two values of μ- one obtained via the (a_{0}, a_{2}) approach and the other via the (q, y) approach, corresponding to the H&M’s or Minkov, et al.’s triplet of (J_{c}_{1}, H_{c}_{1}, λ_{L}) is reminiscent of SrTiO_{3}, the plot of the T_{c} of which against concentration has a dome-like structure [16] [11] .
d) Drawing attention to (23) and (24), we note that as the temperature is lowered from 10 K to 10^{−3} K, the increase in the value of μ from 9.85 eV to 9.88 eV is accompanied by a disproportionate increase in the value of H_{c}_{2} from 145 T to 332 T. This super-linear upturn of the H_{c}_{2} curve at low values of temperature is also seen in Figure 1 and is an unfamiliar feature. However, recent plots of the trapped magnetic moments in hydrides given by Minkov, et al. [10] show a similar behavior, vide upper panels c and d in their Figure 1 and panels a and b in Figure 4. Although the authors have attributed this feature to sulphur, the fact nonetheless remains that it has been observed in an experiment concerned with H_{3}S.
e) We would like to emphasize that the above feature of H_{c}(T) close to 0 K plays no role in our work other than providing the values of μ at T ≈ 0 and 10 K. This is so because H_{c}_{2}(T) is employed only while using the number equation where its value is immaterial, vide the Remark below (18).
f) Salient features of our findings in this paper are
1) we have shown that the Hirsch and Marsiglio’s triplet of {J_{c}_{1}, H_{c}_{1}, λ_{L}} leads to values of v_{F} and ξ that are considerably greater than their currently believed upper limits, and to a value of κ which is much lower than several available estimates of it as in [5] and [8] .
2) while we were led via two different approaches to the values of κ in the same ballpark as its value reported by Minkov, et al. [10] , our values of λ_{L} and ξ corresponding to it differ from their values. We also believe to have shown that this is an issue which can be resolved by monitoring μ (equivalently, N_{s}) at T = T_{c} which, via the chain of calculations carried out in this paper, predominantly determines a host of the properties of the SC, including H_{c}_{1} and J_{c}_{1} which are associated with the Meissner effect.
g) We conclude by noting that the repository of knowledge gained from the study of the conventional SCs may not always be a good guide in dealing with SCs like compressed H_{3}S because none of the former category of SCs has been studied at such high pressures as the latter.
Appendix A: Notation
(a_{0}, a_{2}): Constants in
$q\left(t\right)=1+{a}_{0}{\left(1-{t}^{2}\right)}^{{a}_{2}}$ _{ }which determines how the chemical potential varies via μ(t) = μ_{1}q(t), where μ_{1} is the chemical potential at t = 1, i.e., T = T_{c}
c: velocity of light
e: electronic charge
E_{F}: Fermi energy
h = H_{c}/H_{c}_{2}_{0}, the reduced critical field
H_{c}: The applied critical magnetic field
H_{c}_{1}: The lower critical magnetic field
H_{c}_{2}: The upper critical magnetic field
H_{c}_{20}: Assumed value of the upper critical field at T = 0
J_{c}: Critical current density
J_{c}_{1}: Critical current density due to H_{c}_{1}
J_{c}_{2}: Critical current density due to H_{c}_{2}
k: Boltzmann constant
m*: Effective mass of an electron
m_{e}: Mass of a free electron
N: Demagnetizing factor
N_{s}: Number density of charge carriers
p: Relative momentum of the constituents of a Cooper pair in the center-of -mass frame
P: momentum of a Cooper pair in the lab frame
q = μ(t)/μ_{1}
sf: Self-field
t = T/T_{c}, the reduced temperature
T: Temperature (also employed for Tesla, which is clear from the context).
T_{c}: Critical temperature
v_{c}: Critical velocity
v_{F}: Fermi velocity
y = kθ/α
α = |P||p|cos(P, p)/2m*
Δ: Energy gap
η = m*/m_{e}, Effective mass parameter
θ = θ_{H}, Debye temperature of H ions in H_{3}S
θ(H_{3}S): Debye temperature of H_{3}S
θ_{S}: Debye temperature of S ions in H_{3}S
κ = λ_{L}/ξ, Ginsberg-Landau parameter
λ_{L}: London penetration depth
λ_{m}: Magnetic interaction parameter
λ_{m}_{1}: Magnetic interaction parameter at T = T_{c}, i.e., t = 1
μ: Chemical potential
μ_{1}: Chemical potential at T = T_{c}, i.e., t = 1
ξ: Coherence length
ρ = μ_{1}/kθ
Ω(h) = Ω_{0}hH_{c20}/η
Ω_{0} = e/m_{e}c
Appendix B: Flow Charts
1) The (a_{0}, a_{2}) approach:
Step 1: Set values of T_{c}, η and sf. Choose a value of ρ.
Step 2: For T=T_{c} (t = 1), use (13) to solve for λ_{m}_{1}.
Step 3: Select two pairs of data points (T, H_{c}_{2}), use (13) to solve for (a_{0}, a_{2}).
Step 4: With (a_{0}, a_{2}) known, solve (13) to obtain the values of H_{c}_{2} and μ at T = 10 K. Employ these to sequentially obtain the values of v_{F}, ξ, N_{s}, λ_{L}, κ, H_{c}_{1} and, finally, J_{c}_{1} via equations given in the text.
Step 5: If the above value of J_{c}_{1} does not agree with the sought value, e.g., 1.87 × 10^{9} A/cm^{2} in the case of H&M, repeat the entire chain of calculations starting with a different value of ρ till the desired value of J_{c}_{1} is obtained. Following this procedure, remarkably, we were led via ρ = 0.164 to not only the value of J_{c}_{1}, but also of H_{c}_{1} and λ_{L} which are in agreement with the H&M’s values.
2) The (q, y) approach:
Step 1: Set the values of T_{c}, η and sf. Choose a value of ρ.
Step 2: For T = T_{c} (t = 1), use (13) to solve for λ_{m}_{1}.
Step 3: Employing as input the obtained value of λ_{m}_{1} and the H&M’s or the Minkov, et al.’s values of J_{c}_{1} and H_{c}_{1}, solve (13), with X, Y as in (16), and (20) simultaneously for q and y.
Step 4: Employing the equations given in the text, sequentially calculate the values of v_{F}, ξ and N_{s}, culminating with the value of λ_{L}.
Step 5: If the above value of λ_{L} does not agree with the sought value, repeat the entire procedure starting with a different value of ρ.
Remark: The above flow-charts bring out the role of μ = ρkθ as the predominant governor of several properties of the SC under study.
NOTES
^{1}Hirsch, J.E. (private communication) agrees that this is the value that should have been given in [3] and not 1.23 x 109 A/cm2.