Constraining Forces Stabilizing Superconductivity in Bismuth

As shown in former papers, the nonadiabatic Heisenberg model presents a mechanism of Cooper pair formation generated by the strongly correlated atomic-like motion of the electrons in narrow, roughly half-filled “superconducting bands” of special symmetry. The formation of Cooper pairs is not only the result of an attractive electron–electron interaction but is additionally the outcome of quantum mechanical constraining forces. There is theoretical and experimental evidence that only these constraining forces operating in superconducting bands may produce eigenstates in which the electrons form Cooper pairs. Here, we report evidence that also the experimentally found superconducting state in bismuth at ambient as well as at high pressure is stabilized by constraining forces.


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Bismuth shows sequential structure transition as function of the applied pressure, as summarized in an illustrative form by O. Degtyareva et al.  The band structure of Bi-I is depicted in Fig. 1. The Bloch functions of the band highlighted in red are labeled by the single-valued representations It is clear that this band (or any other band in the band structure) does not contain a closed band 59 (Definition 2 of Ref. [5]) with the symmetry of band 1 or band 2 in Table A4, meaning that we cannot According to Table A3, we may unitarily transform the Bloch functions (2) into Bloch functions labeled by the double-valued representations,   Table A1. E F denotes the Fermi level. The band highlighted in red is the superconducting band.

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The band structure of Bi-V is depicted in Fig. 2. The Bloch functions of the band highlighted in red now are labeled by the single-valued representations Again, this band (or any other band in the band structure) does not contain a closed band (Definition 2 of Ref. [5]) with the symmetry of the bands listed in Table A8. Hence, we cannot unitarily transform the Bloch functions into best localized and symmetry-adapted Wannier functions situated at the Bi atoms. According to Table A7, we may unitarily transform the Bloch functions (4) into Bloch functions labeled by the double-valued representations, The underlined representations belong to band 4 listed in Table A9    Consequently, the NHM predicts that both phases become superconducting below a transition 78 temperature T c .

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The superconducting band of Bi-I ( Fig. 1) even comprises all the electrons at the Fermi level. However, the small Fermi surface and the small density of states at the Fermi level results in the 81 extremely low superconducting transition temperature of T c = 0.53mK [3].

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In terms of superconducting bands, the NHM confirms the experimental observations that      The following abbreviation is used in this manuscript: -1 -1 1 Notes to Table A1 1. i = 1, 2, 3.
2. The symmetry elements are labeled in the Schönflies notation as illustrated, e.g., in Table 1 Table 6.14) of Ref. [11]. In this paper the double-valued representations are not explicitly given but are sufficiently defined by this table. Table A4. Single-valued representations of all the energy bands in the space group R3m of Bi-I with symmetry-adapted and optimally localized usual (i.e., spin-independent) Wannier functions centered at the Bi atoms.
Notes to Table A4 1. z = 0.23 . . . [1]; the exact value of z is meaningless in this table. In the hexagonal unit cell, the Bi atoms lie at the Wyckoff positions 6c(00 ± z) [1]. In the trigonal system, their positions in the unit cell are ρ = ±(zT 1 + zT 2 + zT 3 ), where the vectors T 1 , T 2 , and T 3 denote the basic vectors of the trigonal lattice as given, e.g., in Table 3.1 of Ref.
[11]. 2. The notations of the representations are defined in Table A1. The entry "OK" below the time-inversion operator K indicates that the Wannier functions may even be chosen symmetry-adapted to the magnetic group  Table A2. 6. Each row defines one band consisting of two branches, because there are two Bi atoms in the unit cell. Table A5. Double-valued representations of the superconducting band in the space group R3m of Bi-I.
Notes to Table A5 1. z = 0.23 . . . [1]; the exact value of z is meaningless in this table. In the hexagonal unit cell, the Bi atoms lie at the Wyckoff positions 6c(00 ± z) [1]. In the trigonal system, their positions in the unit cell are ρ = ±(zT 1 + zT 2 + zT 3 ), where the vectors T 1 , T 2 , and T 3 denote the basic vectors of the trigonal lattice as given, e.g., in Table 3 of C 3v where d 1 and d 2 are defined in Table A2 and d 1/2 denotes the two-dimensional double-valued representation of O(3) as given, e.g., in Table 6.1 of Ref. [11]. Note that the two representations d 1 ⊗ d 1/2 and d 2 ⊗ d 1/2 are equivalent.
Notes to Table A6  Table A7. Compatibility relations between the single-valued (upper row) and double-valued (lower row) representations of the space group Im3m.
[11]. In this paper the double-valued representations are not explicitly given but are sufficiently defined by this table.
Notes to Table A8 1. The notations of the representations are defined in Table A6. 2. Assume a closed band of the symmetry in any row of this table to exist in the band structure of Bi-V. Then the Bloch functions of this band can be unitarily transformed into Wannier functions that are • localized as well as possible; • centered at the Bi atoms; and • symmetry-adapted to the space group Im3m (229) [5].
The entry "OK" below the time-inversion operator K indicates that the Wannier functions may even be chosen symmetry-adapted to the magnetic group  Table A6.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 16 November 2017 doi:10.20944/preprints201711.0103.v1 Table A9. Double-valued representations of the space group Im3m of all the energy bands of Bi-V with symmetry-adapted and optimally localized spin-dependent Wannier functions centered at the Bi atoms.