The discovery by Drozdov et al. [1] of H₂S as a superconductor (SC) having a critical temperature (T_c) of ≈ 200 K at the ultra-high pressure of ≈ 150 GPa in 2015 is a significant landmark in the more than a century long quest for room temperature superconductivity. Understandably therefore a huge effort—experi- mental as well as theoretical—has been expended in the intervening years to unravel various features of this SC, which has led to some valuable insights and concrete results; for a partial summary of these, see [2] . In this paper, we concern ourselves with the temperature-dependent upper critical field (H_c2(T)) of compressed H₂S not only because it sheds light on some conceptual aspects of its superconductivity, but also because it is an important property from the point of view of its practical usage.

In a recent paper, Talantsev [3] obtained excellent fits to the temperature (T)- dependent upper critical field B_c2 data of highly compressed H₃S reported by Mozaffari et al. [4] by employing the following four models:

1) The Baumgartner-Werthamer-Helfand-Hohenberg (B-WHH) model

This model is based on the following relation given by Werthamer, Helfand and Hohenberg [5]

ln ( T T c ( B = 0 ) ) = ψ ( 1 2 ) − ψ ( 1 2 + ℏ D B c 2 ( T ) 2 ϕ 0 k T ) , (1)

where Ψ is the Euler psi-function or the digamma function, T_c is the critical temperature, D is the diffusion constant for the normal electrons/holes in the conduction band, ϕ 0 = 2.07 × 1 0 − 15 Wb is the flux quantum and k the Boltzmann constant. This relation has two free fitting parameters viz., T_c (B = 0) and D. In a modification of (1) where the free parameter D is replaced by ξ(0)—the coherence length at T = 0, Baumgartner et al. [6] proposed a relation which has been found to be quite accurate. Designated above as the B-WHH model, this relation is

B c 2 ( T ) = ϕ 0 1.386 π ξ 2 ( 0 ) [ ( 1 − t ) − 0.153 ( 1 − t ) 2 − 0.152 ( 1 − t ) 4 ] , (2)

where t = T/T_c is the reduced temperature.

2) The Jones-Hulm-Chandrasekhar model [7]

B c 2 ( T ) = B c ( T ) 2 λ ( 0 ) 1.77 ξ 2 ( 0 ) ( 1 − t 2 1 + t ) , (3)

where B_c(T) is the thermodynamic critical field and λ(0) is the London penetration depth at T = 0.

3) The Gor’kov model [8] as recast in [3]

B c 2 ( T ) = ϕ 0 3.54 π ξ 2 ( 0 ) ( 1.77 − 0.43 t 2 + 0.07 t 4 ) ( 1 − t 2 ) . (4)

4) The tribrid model

We have named the fourth model in [3] as the tribrid model because it is based on the Gorter-Casimir two-fluid theory, the Ginsberg-Landau theory and the BCS theory. This model employs the following equation

B c 2 ( t ) = ϕ 0 2 π ξ 2 ( 0 ) [ F 1 ( t ) F 2 ( t ) ] , (5)

where

F 1 ( t ) = ( 1.77 − 0.43 t 2 + 0.07 t 4 1.77 ) 2

and

F 2 ( t ) = [ 1 − 1 2 k t T c ∫ 0 ∞ d ε / cosh 2 ( ε 2 + Δ 2 ( t T c ) 2 k t T c ) ] − 1 .

Talantsev’s work [3] relates the empirical B_c2(T) data of H₃S with the set of four of its fundamental parameters, viz., T_c, ξ(0), Δ(0) and ΔC/C, where Δ(0) is the gap of the SC at T = 0 and ΔC/C is the jump in its electronic specific heat at T = T_c.

It seems apt to label the four models described above as phenomenological because they are not directly based on the dynamical variables of the problem as, for example, λ_m, the magnetic interaction parameter. In this paper, we meet the need to explain the B_c2(T) data reported in [4] as dealt with in [3] via a dynamical equation for pairing that incorporates the chemical potential μ, T and the applied magnetic field H—it being our basic premise that the Fermi energy (E_F) or μ plays a fundamental role in determining the properties of a superconductor, regardless of its size, shape and the manner of preparation all of which determine its E_F at T = 0 or μ when T ≠ 0. This equation and the procedure for solving it are given in the next section. A notable feature of our approach is that it sheds light on a set of parameters corresponding to any value of H_c2(T), which is altogether different from the set of parameters on which the study reported in [3] was based.

1) In order to deal with the above data, we first need to fix the value of θ to be used in (6). Given that H₃S is believed to result when H₂S is subjected to ultra-high pressure via the reaction 2 H 2 S → ( H 3 S ) + + ( H S ) − [10] and that θ(H₃S) = 1531 K [11] , we need to resolve θ(H₃S) into the DTs of the H and the S ions because the two gaps of H₃S have been explained by invoking the 2-phonon exchange mechanism due to these ions [2] . Employing the double-pendulum model [12] , we have

θ H = 1983.2 K, θ S = 174.5 K . (7)

We note that both of these DTs are needed [2] for an explanation of the empirical value of the T_c of H₃S 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 [13] . Since a magnetic field considerably weakens the strength of the interaction, it turns out—as will be seen—that the data being addressed here are attributable to the interactions caused by the H ions alone.

2) With θ fixed as θ_H as in (7), we obtain via (6) the value of λ_m at T = T_c, i.e., t = 1. For this purpose, we need to specify the values of H_c2 at t = 1, ρ (since μ = ρ k θ) and η. This is done as follows. We employ H_c2 (1) = 300 G, which is the self-field that remains even when there is no applied field; ρ = 1, i.e., μ = k θ_H = 170.9 meV, which is in accord with the estimate of the E_F of H₃S given by Gor’kov and Kresin as 500 ≥ E_F ≥ 100 (meV) [14] , and η = 2.76 because this is the value employed in [3] . The effect of the values of ρ and η other than these will be addressed below.

3) As per our premise in Section 1.2, we now need to specify how μ varies with t between 1 and 0. We note in this context that the t-dependent values of the critical current density (j_c) of H₃S have recently been explained by making in an equation similar to (6) and the number equation the following replacements:

μ → q ( T ) μ , λ m → λ m / q ( t ) [15] .

Observing that the form of h_c2(t) is predominantly determined by the form of λ_m(t) and that a good phenomenological description for the variation of the former is provided by h c 2 ( t ) = ( 1 − t 2 ) , we assume that

μ ( t ) = μ ( 1 ) q ( t ) , λ m ( t ) = λ m ( 1 ) q ( t ) , (8)

and

q ( t ) = 1 + a 0 ( 1 − t a 1 ) a 2 . (9)

Remarks:

a) It seems pertinent to note that a model is needed to extend or supplement the experimental data up to 0 K because such data are available only up to T ≈ 100 K, which depends on the size of the sample. This is so, because the size of the sample is necessarily small—diameter ≈ 50 μm and thickness ≈ a few μm—in order to subject them to the desired high pressure in a diamond anvil cell. It has so far not been possible to cool such samples up to 0 K.

b) Note that regardless of the values of a₀, a₁ and a₂, q(t) = 1 for t = 1 whence μ(t) and λ_m(t) in (8) reduce to their respective values at t = 1. In the following we deal with the empirical data by employing three models of q(t) distinguished by the values of the constants in (9).

We now illustrate the above procedure by considering the data-sets of two samples in [4] .

Sample 1

The raw data for this sample comprise 23 values of H_c2(T) for 105.1 ≤ T ≤ 191 K. With

θ = 1983.2 , ρ = 1 ( μ ( 1 ) = 170.9 meV ) , η = 2.76 , T c = 191 , H c 0 = 100 × 10 4 G t = 191 / T c = 1 , h = 300 / H c 0 ,

the solution of (6) yields

λ m ( 1 ) = 1.1281 × 10 − 6 . (10)

The value of H_c0 ≡ H_c2(0) (an unknown) noted above has been chosen as a matter of convenience because it enables us to look for the solutions by varying h between 0 and ≈ 1.

We note that our results of course do not depend on the choice of the value of H_c0 and that this step which yields (10) is common to all the three forms of q(t) dealt with below.

1) The linear form of q(t)

If in the generic form of q(t) in (9) we let a₁ = a₂ = 1, then we obtain the linear form of q(t), viz.,

q 1 ( a 0 , t ) = 1 + a 0 ( 1 − t ) .

As noted in (8), this is the factor via which we cause μ to vary with t. Since μ = ρ k θ, we need to replace ρ in (6) by ρ times q(a₀, t), whence

E q 1 ( t , h ) → E q 1 ( a o , t , h ) ,

L 1 ( ρ ) → L 3 ( ρ , a 0 , t ) = q 1 ( a 0 , t ) ρ − 1 3 q 1 ( a 0 , t ) ρ , L 2 ( ρ ) → L 4 ( ρ , a 0 , t ) = q 1 ( a 0 , t ) ρ + 1 3 q 1 ( a 0 , t ) ρ

N L ( h , ρ ) → N L ( h , ρ , a 0 , t ) = f l o o r [ 2 k θ [ q 1 ( a 0 , t ) ρ + 1 ] 3 ℏ Ω ( h ) − 1 2 ] . (11)

To fix a₀, we need the input of one (t, h) point from the data under consideration. Choosing a value of t in the mid-range of the data, i.e.,

t = 150 / T c , h = 26.47 × 10 4 / H c 0 . (12)

We obtain

a 0 = 2165.4. (13)

We can now employ (11) to calculate the value of h corresponding to any t. We do so for 16 points between t = 1 (T = 191 K) and 0 and obtain the limiting values of the parameters of interest as:

H c 0 = 601.1 × 10 4  G; for 1 ≥ t ≥ 0:  0.171 ≤ μ ( t ) ≤ 370.2   ( eV ) ;  1.128 × 10 − 6 ≤ λ m ( t ) ≤ 2.444 × 10 − 3 ;   1.811 × 10 5 ≥ N L ( t ) ≥ 9.793 × 10 3 . (14)

The plot obtained for the variation of H_c2(T) for all the 16 points in the range 191 ≥ T ≥ 0 ( K ) is given below (Figure 1).

2) The “parabolic” form of q(t)

Let a₁ = 2 in (9), then

q 2 ( t ) = 1 + a 0 ( 1 − t 2 ) a 2 , (15)

whence

E q 1 ( t , h ) → E q 1 ( a 0 , a 2 , t , h ) ,

L 1 ( ρ ) → L 3 ( ρ , a 0 , a 2 , t ) = q 2 ( a 0 , a 2 , t ) ρ − 1 3 q 2 ( a 0 , a 2 , t ) ρ , L 2 ( ρ ) → L 4 ( ρ , a 0 , a 2 , t )

N L ( h , ρ ) → N L ( h , ρ , a 0 , a 2 , t ) ,

where L₄ and N_L are given by the expressions for them as in (11) with q₁(a₀, t) replaced by q₂(a₀, a₂, t).

We now need the input of two (t, h) points from the empirical data to fix a₀ and a₂ by simultaneously solving Eq1(a₀, a₂, t₁, h₁) and Eq1(a₀, a₂, t₂, h₂). Choosing (t₁, h₁) and (t₂, h₂) as

( t 1 = 145 / T c , h 1 = 30.27 × 10 4 / H c 0 ) , ( t 2 = 105.1 / T c , h 2 = 61.49 × 10 4 / H c 0 ) , (16)

we obtain

a 0 = 1475.0 , a 2 = 1.208. (17)

These lead to the limiting values of the parameters of interest as:

H c 0 = 242.9 × 10 4 G;for1 ≥ t ≥ 0 : 0 .171 ≤ μ ( t ) ≤ 252 .2 ( eV ) , 1.128 × 10 − 6 ≤ λ m ≤ 1.665 × 10 − 3 , 1.811 × 10 5 ≥ N L ≥ 1.652 × 10 4 . (18)

The plot now obtained for the variation of H_c2(T) for 15 points in the range 191 ≥ T ≥ 0 ( K ) is given below (Figure 2).

3) The concave-upward form of q(t)

For a₂ = 1 in (9), we have

q ( t ) = 1 + a 0 ( 1 − t a 1 ) , (19)

whence we again need to solve two simultaneous equations, viz., Eq1(a₀, a₁, t₁, h₁) and Eq1(a₀, a₁, t₂, h₂), in order to fix a₀ and a₁ with the input of two (t, h) points from the empirical data. Choosing the values of (t₁, h₁) and (t₂, h₂) as

( t 1 = 125 / T c ,   h 1 = 44.77 × 10 4 / H c 0 ) ,   ( t 2 = 105.1 / T c ,   h 2 = 61.49 × 10 4 / H c 0 ) , (20)

we obtain

a 0 = 2195.1 , a 1 = 0.9549. (21)

The limiting values of the parameters of interest now are:

H c 0 = 579.0 × 10 4 G;for 1 ≥ t ≥ 0 : 0.171 ≤ μ ( t ) ≤ 252.2 ( eV ) , 1.128 × 10 − 6 ≤ λ m ( t ) ≤ 2.477 × 10 − 3 , 1.811 × 10 5 ≥ N L ( t ) ≥ 1.031 × 10 4 (22)

and the plot obtained for the variation of H_c2(T) for 17 points in the range 191 ≥ T ≥ 0 ( K ) is as given below (Figure 3).

Sample 2

The raw data for this sample comprise 24 values of H_c2(T) for 55.9 ≤ T ≤ 173.7 K. With the same values of θ, ρ and η, the plots of the variation of H_c2(T) for this sample are similar to those for Sample 1 and are hence not given.

For both the samples dealt with above, the values of ρ and η employed for each of the three forms of q(t) were 1 and 2.76, respectively. Repeating the above exercise for any form of q(t) with different values of ρ and η, e.g., ρ = 1, η = 2.25, 3.25, 4.0 and ρ = 2, η = 2.76, we found that while the values of a₀, a₁, etc., were now different, the plots of H_c2(T) that we were led to were almost indistinguishable from the corresponding plots that we had obtained earlier.