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Superconductivity: small steps towards the “grand unification”

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Abstract

The existence of various families of super conducting materials and their TC values are qualitatively rationalized within a simple model. Novel families of superconducting materials, particularly those based on fluoride and hydride anions, are predicted.

Figure We predict that existing families of moderate- and high-TC superconductors should hopefully be enriched by novel compounds containing hardly polarizable anions (such as F-). Covalent chlorides and hydrides also merit careful exploration.

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Notes

  1. The value of the energy at Qas=0 has been normalized to zero for all systems. This is why Δ needs to be added to the equation for the energy in comparison with Eq. 1 in Ref. [8]

  2. In the three-parameter model one deals with two electronic states coupled through one normal vibration. This means that values of k, Δ and V determined here for real molecules do not refer to any excited state but rather represent the global effect of coupling of the ground state with all excited states of appropriate symmetry. For example, the ground state of the H3 radical transition state (Σu+) couples with all excited Σ g+ states via a normal vibration of σu symmetry (i.e. along Qas). Overall coupling is so strong in this case that distortion leads to an energy decrease of the ground state

  3. Values of V are very large for interhalogen and H-containing compounds. This is why enormously large pressures are required to metallize halogens, while no metallization of H2 has been achieved so far using static pressures

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  1. Department of Chemistry, University of Warsaw, Pasteur 1, 02093, Warsaw, Poland

    Wojciech Grochala

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Correspondence to Wojciech Grochala.

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This work is dedicated to my British friend, Peter P. Edwards, at his 55th birthday. God save dear Peter Paul.

Appendix

Appendix

Determination of vibronic coupling constants in linear symmetric triatomic radicals

Fig. 3 shows the Potential Energy Surface (PES) along the antisymmetric stretching coordinate, Qas, of the symmetric (at Qas = 0 Å) linear triatomic radical, here represented by Br2H

Fig. 3
figure 3

The computed PES along the antisymmetric stretching coordinate, Qas, of the symmetric linear radical, Br2H

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From this plot, two separate force constants have been calculated: one corresponding to the imaginary antisymmetric stretching mode at Qas = 0 Å, further called k′−, and force constant at the minimum of the PES (here at Qas = Qmin = 0.26 Å), further called k′′. The notation used here is identical as that used in Ref. [8]. Typically, we have used between 4 and 10 points on the PES for the quadratic fit (see Figs. 4, 5 and

Fig. 4
figure 4

The computed PES of Br2H in the vicinity of energy minimum. Force constant at a minimum, k′′ = 14,856 eV Å−2, can be calculated from the quadratic fit

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Fig. 5
figure 5

The computed PES of Br2H in the vicinity of energy maximum (at Qas = 0 Å). Force constant at a maximum, k′− = −8,975 eV Å−2, can be calculated from the quadratic fit

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Using the known values of k′− and k′′, the value of the force constant in the hypothetical absence of vibronic coupling, k, has been determined from the best fit to the equation: k′′ = k− k3/(k− k′′)2. In case of Br2H, the value of k = 66.1 eV Å−2 was obtained.

The preliminary estimates of the values of the vibronic coupling constant, V, and of the electronic coupling constant, Δ, was calculated as follows. First, (V2/Δ) = k− k− = 75.08 eV Å−2. Second, V={[(k− 2) − (V2/Δ)− 2]/(Qmin2)}− 0.5. Thus, V = 36.3 eV Å−1 and Δ = 17.5 eV. These preliminary estimates were used as starting values in the fit of the computed PES to the equation E=1/2kQas2 + (Δ2+V2 Qas2 )0.5+Δ Footnote 1. From the fit, new set of parameters has been obtained: k = 75.5 eV Å−2, Δ = 22.0 eV, and V = 43.1 eV Å−1 Footnote 2.

The final values of k, V and Δ for other chemical species have been determined in an analogous way.

Figure 6 shows the comparison of computed and fitted PES for Br2H. The fitted PES reproduces all essential features of computed PES, including the position of Qmin. The fitted and computed curves are virtually undistinguishable.

Fig. 6
figure 6

Comparison of the computed and fitted PES of Br2H

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In Table 1, we show the value of the optimized E–X bond length, R0, for a variety of molecules, the analytical value of the force constant for the antisymmetric stretching, kanal, position of the minimum (along Qas) of PES, ΔQas, value of force constant in the absence of vibronic coupling, k, electronic coupling element, Δ, and the vibronic coupling constant, V, determined from the fitting procedure using a three-parameter model [8].

Table 1 Values of R0, ΔQas, k, Δ, and V, and errors of k, Δ and V (for details see text)
Full size table

For F2H, H3 but also for Li3 (and for other species that do not exhibit an imaginary frequency along Qas), the three parameters of the fitting procedure are strongly correlated with one another, i.e. equally good fits may be obtained for various sets of these parameters. This implies large relative errors in determining k, Δ, and V. We have omitted the fitting procedure for such molecules, while making an exception for Li3, in order to compare it to interhalogen compounds.

Vibronic coupling constants versus Pearson’s hardness of the bridging atom

In Fig. 7 we show values of V plotted versus Pearson’s hardness, η, of the bridging element X (η/eV: 7.01 F, 6.42 H, 4.70 Cl, 4.24 Br, 3.70 I).

Fig. 7
figure 7

Values of vibronic coupling constant, V, versus Pearson’s hardness, η, of bridging element X, for three families of molecules (H2X, Cl2X and Br2X). Values of V for I2X molecules have not been shown as they are nearly the same as those for corresponding Br2X ones. Values of V for molecules, which are not unstable along Qas are very small, and have been taken as null

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The harder the bridging atom (F > H > Cl > Br > I), the larger the value of V. For the same bridging atom, the harder the end atoms (H > Cl > Br), the larger the value of V Footnote 3. Confirmation of a possible decrease of V for very large values of hardness, requires a more representative statistical probe.

The TC values for selected families of materials versus the Mulliken electronegativity of the most electronegative element

In Fig. 8, we show experimental values of TC multiplied by the m1/2 factor (i.e. TC divided by the factor that is proportional to the pre-exponential expression from the BCS theory), plotted versus Mulliken electronegativity, μ, of the most electronegative atom in the compound considered. Numerical data is collected in Table 2

Fig. 8
figure 8

Value of the (TC m0.5) product (TC in K, m in atomic mass units) versus the Mulliken electronegativity, μ, of the most electronegative element from the given compound. Quadratic fit (solid line) is shown for the data for record—holding oxocuprate, MgB2, classical Nb and Li under high pressure (dark blue points). Experimental points for best-known nitride, hydride, phosphide carbide and fluoride materials are shown in pink. We think these values may be improved a lot. Green points correspond to the maximum expected values of the (TC m0.5) product for N, H, P, C, Cl and F-based materials, and were calculated using fitted function of μ. For details see Table 2

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Table 2 Values of the Mulliken electronegativity of the most electronegative element, μ, critical superconducting temperature, TC, square root of the atomic mass of the element considered, m0.5, and of the (TC m0.5) product, for several representative families of superconductors
Full size table

The expected values of (TC m0.5) for N, H, P, C, Cl and F-based materials can be translated back to the expected values of TC in these materials. The fit indicates the possibility of great improvement of the TC values for phosphides (61 K), carbides (102 K), and nitrides (136 K), while it delivers astonishingly high TC values for fluorides (268 K = −5°C) and hydrides (>490 K, >210 C).

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Grochala, W. Superconductivity: small steps towards the “grand unification”. J Mol Model 11, 323–329 (2005). https://doi.org/10.1007/s00894-005-0250-0

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