Magnesium compounds as future high temperature superconductors

Previously proposed method for calculating the superconducting transition temperature (Tc) of the different substances was used to predict Tc of magnesium compounds. Theoretical calculations show the good agreement with experimental results for MgB2, the only magnesium compound with experimentally proved Tc = 39 K (theoretical value is 40.5 K). The performed computations for binary compounds Mg2X (X: Ge, Sn, Pb) reveal the high potential of magnesium compounds as high-temperature conductors at normal pressure, with predicted superconducting transition temperature up to 70 K.


Introduction
In 1911, the Dutch physicist Heike Kamerling-Onnes discovered the phenomenon of superconductivity [1,2]. More than 100 years later, superconductivity at helium and nitrogen temperatures was found in many metals and alloys, in intermetallic compounds, in ceramics [3], in fullerene and graphene, as well as in hydrides (H2S, LaH10) at ultrahigh pressure [4].
Inaugurating a new era of high-temperature superconductivity, lanthanum and yttrium ceramics, display critical superconducting temperature Tc > 60 K [3]. In [5] we highlighted the common to all above-mentioned substances properties that determine the attainable Tc level: the heats of formation of both elements and compounds; the energy of formation of defects in the substances; the melting temperature; the Debye frequency; the harmonic oscillator energy, considering the lattice atoms as harmonic oscillators. The paper considers the potential of magnesium compounds as high-temperature superconductors on the basis of physicochemical analysis of phase diagrams.

Theoretical aspects
In [5] we proposed the equation for calculating Tc: where h -is the Planck constant, D  -is the Debye frequency (referred to 0 K), k -is the Boltzmann constant, n1,2,3 -is for one-, two-, and three-dimensional oscillator, НV,1 -is the energy of the formation of defects in them. One can calculate the formation energy of mono-vacancies as TmSm for the solid phase, as well as using the formula: , where G0V0 is the elastic shear energy at 0

K, and rel
 -is the average relaxed energy of the oscillator, that is equal to the heat of formation of the substance from the liquid phase. Moreover, the value of HV,1 (determined by quenching a solid from T, close to Tm -the melting temperature) includes practically all the co-occurring defects (change in bond lengths, angle between them, i.e. their energy).

Calculation results
To start with consider magnesium diboride MgB2, nowadays commercially used superconductor with Tc = 39 K [6]. Pure Mg has a low Tc ≈ 0.05 K (at normal pressure), Hv = 50.000 kJ/mol,  = 8.954 kJ/mol; 0 D  , the characteristic temperatures, determined from the elastic constants and the calorific value are 386 and 406 K, respectively, so we have Tc1 = 1.45 and Tc2 = 1.53 K. The calculated heat of formation of MgB2 is  = 22.931 kJ/mol. Assuming that Mg precipitations during the low-temperature decomposition of this compound will determine superconductivity, we calculate Tc taking into account the fact that Hv,Mg = 50.000 kJ/mol, 0 D  = 386 K (for boron, Нv = 95.000 kJ/mol). Thus, for MgB2 we have Tc = 43.6 K, but if we use the value Hv = 54.056 kJ/g-at, Tc would be 36.5 K. The average value is 40.05 K, which is close to the experimental (39 K). Note that the Mg-B diagram at normal pressure has not yet constructed. A certain temperature of the equilibrium is 1050°C (1323 K), and the powders of MgB2 samples are sintered at 973 K.
One should pay attention to magnesium compounds of the Mg2X type, where X-Ge, Sn, Pb, La, as well as others, for example, Mg3Sb and Mg3Bi2. Preliminary calculations show the possibility of reaching Tc > 50 K for these compounds. These compounds can be prepared both by the powder and by the melting method. The major liability of the above magnesium compounds is that they are hygroscopic when aged in the air under normal conditions. The Mg2Ge compound (e/a = 2.7, the same as for MgB2) has Tm = 1388 K (see Figure 1), and the high heat of phase formation (heat of fusion) or the maximum relaxed energy of a one-dimensional oscillator l Re  (for the solid phase). Taking into account that Mg and Ge have the similar defect formation energies (Нv respectively, 50.000 and 55.000 kJ/mol, l Re  = 28.570 kJ/mol, at = 386 K) we have Tc = 67 K, i.e. no worse than that of pnictides. Of course, in the presence of covalent bonds, the result can be lower (~ 12 K), however, when choosing the appropriate "doping" additives, good results can be obtained (for example, Ba or oxides can break covalent bonds).
Mg2Sn phase holds much promise having Тm = 1051 K (Figure 2),  calc. = 19.494 kJ/mol. Taking into account the excess of Mg in this compound, calculations give Tc = 30.9 K (in the case of covalent bonds, the "dopants" are necessary, since Tc = 2.3 K would be). An excess of tin ( 0 D  = 201 K) with HvSn = 50.000 kJ/mol gives Tc = 15.3 K, but if we apply the concept of a two-dimensional oscillator to Sn (that is typical for free tin), we obtain Tc = 55.5 K without "doping additives" that can enhance this value. This system, with affordable and inexpensive metals, also deserves development.
The Mg-Pb system is distinguished by one Mg2Pb compound with a low melting temperature, Tm = 823 K (Figure 3), HvPb = 48.000 kJ/mol; HvMg = 50.000 kJ/mol; then Tc, Mg = 6.15 K and Tc,Pb = 2 K, however, in the case of excess Pb (data for free Pb calculated from a three-dimensional oscillator), we have Tc = 29 K, i.e. just below Tc = 30 K for high-temperature superconductors (HTSC). A calculation with the experimental value  = 13.395 kJ/mol for this phase and 0 D  = 111 K gives Tc = 33.6 K.     (Figure 4), we calculate Tc for antimony. Sb has Нv = 40.000 kJ/mol, its l Re  = 19.842 kJ/mol, 0 ,Sb D  = 200 K, then, given the fact that antimony melts as Sb2, we have Tc = 3.55 K. The metallic properties of antimony are more pronounced (i.e. electrical conductivity, thermal conductivity), and under normal pressure conditions, it is a superconductor at Tc = 2.6 -2.7 K; and at P > 85 kbar, antimony has Tc = 3.6 K [1]. For the Mg3Sb2 compound, Tm = 1501 K.
Low-temperature decay from the magnesium side (902 K) and from antimony side (852 K),  exp = 30.558 kJ/mol. The calculations of the temperature of the superconducting transition give for Mg excess: Tc = 75.2 K, and from the antimony side Tc = 54 K; this is the high result (for  exp = 30.558 kJ/mol), however, it can be less if the compound (and this is the case) will have covalent bonds, then for the two cases we have Tc,Mg = 14.6 K and Tc,Sb = 14.6 K, i.e. the same result, but high enough to investigate this system and to make use of doping. Similar calculations one can make for the Mg3Bi2 and other phases of the Mg-Bi system, where two eutectics are observed, widely different in temperature. It is also necessary to study the Mg3La phase, the other phase based on the elements lying behind the Zintl line, as is true for the MgB2 compound (Tc = 39 K). Hexagonal lattices can be unstable to vibrations in the basal plane (see c66 constant). If the structures listed above are stable to low temperatures, then it is difficult to expect that relaxation vibrations will arise in such the structures at low temperatures (the promising Mg2Si phase). Mg2Si structure is similar to that of fluorspar (4 AB2-molecules in a cube cell), is not hexagonal (like MgВ2, Р6mmm). In any case, these structures could be surface-unstable, and doping can make them volume-unstable. And low Hv / l Re  ratio indicates the high concentration of defects, but the destruction of long-range order in structures with covalent metal bonds hindered. Mg3Sb2 intermetallic compound is of particular interest, for it undergoes the polymorphic β → α transformation, while in the other intermetallic compounds order-disorder transitions are second-order transformations.

Conclusions
On the basis of physicochemical analysis of binary Mg-X (X: B, Ge, Sn, Pb, Sb) phase diagrams and the thermodynamic properties of the constituents the superconducting transition temperatures were calculated for MgB2 (Tc = 40.5, the experimental Tc = 39 K), Mg2Ge (Tc = 67 K), Mg2Sn (Tc = 55.5 K), Mg2Pb (Tc = 29 K), and Mg3Sb2 (54 < Tc < 75). Thus, magnesium compounds of the Mg2X type (X -Ge, Sn, Sb) can serve as the basis for the development of a new-generation HTSC with Tc ~ 60 -70 K, i.e. no lower than that of compounds based on FeAs, and without ultrahigh pressures application.