Introduction

When superconductivity was discovered by Kamerlingh Onnes in mercury in 1911 (Tc ~ 4.2 K)1 after he produced liquid helium for the first time in 1908 (boiling point of 4.2 K at atmospheric pressure) this phenomenon was assumed only to consist of a vanishing electrical resistance for some metallic materials. It was not until the Meissner effect (the expulsion of magnetic fields) was observed that this discovery became a puzzle. Now superconductivity is displayed by many and sundry materials and is characterized by exhibiting zero electrical resistance and the above-mentioned Meissner effect below a characteristic temperature Tc, the superconducting transition temperature.

In 1957 Bardeen, Cooper and Schrieffer (BCS)2 developed the first successful theory of superconductivity based on two simple but revolutionary and decisive concepts. They proposed that electrons pair through the atomic vibrations in the material due to the now known Cooper pairing potential, and that this pairing gives rise to the transition to the superconducting state; also, that the coherent motion of the paired electrons gives them the inertia to sustain electrical currents without dissipation2. Alternative ideas have appeared since then and even different concepts to substitute the initial ones but BCS has withstood the passing of time. Since phonons are invoked to be responsible for the electron pairing, a manifestation of this interaction should appear in superconductivity, and it does: the so-called isotope effect which is the dependence of Tc on the isotopic mass, Tc√M = const. The Meissner effect is also borne out; hence the two main aspects of superconductivity are duly accounted for by the BCS theory.

Due to the variety and abundance of materials that superconduct, it has also been ventured that in principle all materials may become superconductors if cooled down to low enough temperatures. In what follows we demonstrate that invoking the corresponding electronic densities of states, N(E), and the vibrational densities of states, F(ω), for the various solid phases of bismuth under pressure, the superconducting transition temperatures can be calculated if the Cooper attraction sets in with a strength comparable for all phases. This facile approach, if proven correct, can be generalized to study phases of other materials similarly related.

BCS was a breakthrough that fostered a remarkable growth of the field propitiating progress and new questionings and generating alternative theories to explain specific phenomena. However, in 2016 based on the original BCS we predicted that at ambient pressure bismuth becomes a superconductor at a temperature below 1.3 mK3. A year later an experimental group corroborated that in fact Bi is a superconductor with a transition temperature of 0.5 mK, a result that eluded previous work4. Analogously we have studied low dimensional structures of bismuth, bilayers or bismuthene, and have predicted that they may become superconductive, with a transition temperature of TcL = 2.61 K, provided the Cooper pairing potential manifests itself in a similar manner as in the bulk5,6. Finally, since it is very suggestive that all phases of bismuth under pressure should superconduct we calculated N(E) and F(ω) for Bi-IV and predicted a transition temperature of 4.25 K7, to be corroborated.

Why does Bi display several crystalline phases at moderate pressures? Five solid phases, from Bi-I to Bi-V, have been identified for pressures below 7 GPa and for temperatures between 400 and 600 K7,8. It seems that all crystalline structures are energetically accessible in this interval of the P-T plane, and that differences among them respond more to subtle changes in the binding energy per atom; mass densities are also relatively similar, from 9.8 g/cm3 for Bi-I to 12.64 g/cm3 for Bi-V. Figure 1 displays an accepted present-day classification of the bismuth phases where an increasing tendency for the experimental Tc exists as the pressure is increased, at least up to 7 GPa7,9.

Figure 1
figure 1

Phases of bismuth in the pressure-temperature plane taken from ref. 7. The broken lines were obtained by linear and quadratic fits to experiment8. The experimental superconducting transition temperatures are included9. For the Wyckoff structure (Bi-I)10 we report the Tc we predicted3 and the one subsequently measured4.

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The origin of these series of papers has to do with the question: why was amorphous Bi a superconductor and the crystalline Wyckoff phase10 was not? No clue existed and to find an answer we undertook the computer generation of an amorphous sample of bismuth, whose pair distribution function agreed with experiment. We then calculated its N(E) and its F(ω) and observed that the amorphous phase had a metallic-like character, since N(EF), the electronic density of states at the Fermi level, was reminiscent of a metallic system3. The amorphous phase had been found to be a superconductor with a Tc ~ 6 K so this led us to infer that the Wyckoff phase should become superconductive also, although at very low temperatures, prediction experimentally corroborated a year later4.

If life can be simple, why complicate it? In this work we report a study of the electronic and vibrational properties of the solid phases of bismuth under pressure, calculated from first principles. We obtain Tc s very close to experiment, except for Bi-III, where the argued incommensurate nature of the crystal structure does not allow an unambiguous evaluation. For this phase we consider two different structures, one due to Chen et al. and the other constructed by us based on the results of McMahon et al. Superconductivity appears in most phases of bismuth because the Cooper pairing potential may not be drastically affected in going from one structure to the next as the pressure changes, and because the electronic properties, through the electron density of states N(E), and the vibrational properties, through the vibrational density of states F(ω), become the relevant factors. As Cooper points out in his decisive publication concerning the electron pairing as the phenomenon giving rise to superconductivity, “because of the similarity of the superconducting transition in a wide variety of complicated and differing metals, it is plausible to assume that the details of metal structure do not affect the qualitative features of the superconducting state”11.

Procedure: The BCS Approach

Calculating Tc from basics has proven to be a very difficult task. The electron–phonon interactions are very complex processes and approximations have to be invoked that may mask the essential features of these interactions. However, it is possible to obtain results like the ones reported in this work that give an indication as to what materials may become superconductors if the N(E) and the F(ω) are known, and if the Cooper attraction sets in. In what follows we shall recur to some of our previous publications3,5,6,7 since the procedure is the same for these series of calculations.

We base our discussion on the BCS expression for the transition temperature:

$${T}_{c}=1.13\,{\theta }_{D}\exp \,[\frac{-1}{N({E}_{F}){V}_{0}}],$$
(1)

where θD is the Debye temperature and represents the role played by the vibrations, typified by F(ω), the vibrational density of states (vDoS). N(EF) is the electron density of states (eDoS) at the Fermi level EF, and V0 is the Cooper pairing potential that binds pairs of electrons2,11. Figure 2 shows Tc as a function of the parameters that appear in Eqn. 1 for the specific value of the pairing potential calculated in this work, where the strong dependence of the transition temperature on the factor N(EF) can be observed.

Figure 2
figure 2

The superconducting transition temperature Tc as a function of the electronic N(EF) and vibrational θD (the Debye temperature) parameters for the value of the Cooper pair potential V0 calculated in this work. To be consistent the units for N(EF) are: states eV-1 atom-1. The units for Tc and θD are kelvins.

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This indicates that under certain circumstances N(EF) can play a more important role than θD, especially when different phases of the same material are compared since in this situation it can be assumed that the strength of the pairing potential would not be altered much by the phase changes. Also, the Debye temperatures may not change drastically whereas the electronic properties could change radically, going from being a semimetal to becoming a conductor as is the case for bismuth7.

From Eqn. 1, if the N(EF) and the F(ω) are known, together with the pairing potential V0, the superconducting transition temperature can be obtained, as long as we can calculate θD from the function F(ω). The computational developments that have taken place in the recent decades allow the precise calculation of N(EF) and F(ω) for a variety of atomic structures of various materials; crystalline, amorphous, porous, liquids, etc. However, the precise calculation of the pairing potential V0 has eluded the efforts of the simulational community.

Suppose that we know the transition temperature, the eDoS and the vDoS for a given phase of bismuth that we identify as Δ and want to obtain the transition temperature of a Γ phase for which we also know the eDoS and the vDoS, but do not know the pairing potential for either one. Let us assume that the two phases of a given element have similar V0 s; we may consider for simplicity that the pairing potential is the same and this can be used to obtain the unknown Tc of the phase under scrutiny. The facile procedure is as follows. All quantities related to the Γ phase will be identified with a Γ superscript in the BCS equation,

$${T}_{c}^{{\rm{\Gamma }}}=1.13\,{\theta }_{D}^{{\rm{\Gamma }}}\exp [\frac{-1}{{N({E}_{F})}^{{\rm{\Gamma }}}{V}_{0}}].$$

Analogously, for the Δ phase we get

$${T}_{c}^{{\rm{\Delta }}}=1.13\,{\theta }_{D}^{{\rm{\Delta }}}\exp [\frac{-1}{{N({E}_{F})}^{{\rm{\Delta }}}{V}_{0}}].$$

If we take the ratio \({T}_{c}^{\Gamma }/{T}_{c}^{{\rm{\Delta }}}\) and find an expression for \({T}_{c}^{\Gamma }\) we obtain:

$${T}_{c}^{\Gamma }={\{{T}_{c}^{{\rm{\Delta }}}\}}^{1/\eta }\delta {\{1.13{\theta }_{D}^{{\rm{\Delta }}}\}}^{(\eta -1)/\eta },$$
(2)

where

$$N{({E}_{F})}^{\Gamma }=\eta N{({E}_{F})}^{{\rm{\Delta }}}\,{\rm{and}}\,{\theta }_{D}^{\Gamma }=\delta \,{\theta }_{D}^{{\rm{\Delta }}},$$
(3)

and the pairing potential V0 cancelled out since we assumed it to be the same for all phases. How strong this supposition is will become apparent when we calculate the superconducting transition temperatures of all solid phases in what follows.

The Debye temperatures θD, as required by Eqn. (1), is calculated from the vibrational spectra of each structure. For this we use an expression due to Grimvall, for the Debye frequency ωD12

$${\omega }_{D}=\exp [\frac{1}{3}+\frac{{\int }_{0}^{{\omega }_{max}}{\rm{l}}{\rm{n}}(\omega )F(\omega ){\rm{d}}\omega }{{\int }_{0}^{{\omega }_{max}}F(\omega ){\rm{d}}\omega }],$$
(4)

that we utilized in our previous work with good results5,6,7. F(ω) is the vDoS of the supercell under consideration, and ωmax is the maximum frequency of the corresponding vibrational spectrum. Using Eqn. 4, the calculations for θD = ћωD/kB give the values for each phase structure and with these results and those for N(EF) we are in a position to proceed. To obtain these results we removed the translational modes around 0 THz (ω ≈ 0) for all structures; this are more relevant the smaller the number of atoms in a supercell.

Now we may calculate the Tc of the Γ phase by using the values of the reference Δ phase. Since we have calculations of the eDoS and the vDoS for the amorphous bismuth structure, for the Wyckoff phase, for the bilayer structure and for the Bi-IV phase we can choose any of them as reference. To be consistent, we have been referring all previous calculations to the results obtained for the Wyckoff structure for which we obtained the N(EF), the F(ω), the θD, and the Tc by using the results for the amorphous structure that we studied originally. So from now on the Wyckoff phase shall be used as the Δ phase, i.e., the reference phase, the phase we originally predicted to be superconductive; the Γ phase shall be the corresponding solid phase under pressure whose Tc we want to calculate.

The Phases: Calculations

The method

In what follows we shall report calculations obtained for the five phases of bismuth shown in Fig. 1. For the sake of completeness, we include results previously published for the Bi-I and the Bi-IV phases to present an integrated work. Since the crystalline structures are experimentally known we shall use them to construct the corresponding crystalline supercells, with periodic boundary conditions, to proceed to the calculation of the quantities N(E) and F(ω), from which N(EF) and θD shall be obtained. Once these quantities are known we use Eqn. (2) to calculate all the superconducting transition temperatures for the solid phases of bismuth under pressure, as a function of the reference structure Δ.

As described elsewhere7, the quantities N(E) and F(ω) were obtained using the DMol3 code which is part of the Dassault Systèmes BIOVIA Materials Studio suite13. For the various supercells used, based on the experimental structures, a single-point energy calculation was performed first, using a double-numeric basis set and a fine mesh within the LDA-VWN approximation14; also, an unrestricted spin-polarized calculation for the energy was done. Since bismuth is a heavy element with many (83) electrons, the density-functional semi-core pseudo-potential (DSPP) approximation was used15. This pseudo-potential has been investigated by Delley where an all electron calculation is compared to the DSPP; the rms errors are essentially the same for both methods, 7.7 vs 7.515. Scalar relativistic corrections are incorporated in these pseudopotentials, essential for heavy atoms like Bi. Since DSPPs have been designed to generate accurate DMol3 calculations, it is expected that their use represents a good approximation; considerations of symmetry were left out for all structures. The calculation parameters were the same for the respective N(E) and F(ω) for all structures, so meaningful comparisons can be made. For example, an energy convergence of 10−6 eV was used throughout, the real space cutoff was set to 6.0 Å, and the integration grid was set to fine; the calculations were carried out using a Monkhorst-Pack mesh adequate to each structure as shall be seen later for each phase. For the vibrational calculations, the finite-displacement approach was employed within DMol3, with a step size of 0.005 Å to calculate the Hessian using a finite-difference evaluation. Also, since the number of atoms in the supercells are different we have normalized our results to make them comparable and give N(E) as the number of states per electron-volt per atom and F(ω) as the number of modes per frequency unit per atom everywhere.

For the eDoS the DMol3 analysis tools included in the Materials Studio suite were used, set to eV, and an integration method with a smearing width of 0.2 eV. The number of points per eV was 100. For the vDoS the results were analyzed with the OriginPro software, the calculated normal modes were imported in THz. To obtain the vDoS a frequency count with a 0.11 THz bin width was used, and the resulting bins were smoothed with a two-point FFT filter. The integral under the curve is normalized to three. The results for both densities of states are given per atom everywhere. Finally, the equations we shall use for the Γ phase are

$${T}_{c}^{\Gamma }={\{{T}_{c}^{W}\}}^{1/\eta }\,\delta {\{1.13{\theta }_{D}^{W}\}}^{(\eta -1)/\eta },$$
(5)

where we have explicitly included the Wyckoff phase as the reference phase. Also,

$$N{({E}_{F})}^{\Gamma }=\eta \,N{({E}_{F})}^{W},$$
(6)

and

$${\theta }_{D}^{\Gamma }=\delta \,{\theta }_{D}^{W}.$$
(7)

Antecedents

Solid Bi phases under pressure have been studied for decades7. Bismuth has been an appealing material, perhaps due to its low melting point and to its unusual physical properties. Since 1935 the phases of solid bismuth under pressure have been studied systematically. Bridgman, in his seminal work16,17, established the existence of several phases18,19 that have evolved into 5 presently accepted structures somewhat different to the original classification20. All 5 structures have been determined experimentally, inclusive the so-called incommensurate one of Bi-III. It is now taken for granted that these 5 phases extend to low temperatures, except for the ill-studied Bi-IV that exists in a very well-defined area in the P-T plane, 2.5 GPa ≤ P(Bi-IV) ≤ 5.0 GPa, and in the neighborhood of 500 K7.

Bismuth is a puzzling material; a versatile substance; it is the heaviest element of group 15, the highest atomic-number semimetal. At ambient pressure and temperature Bi is a crystalline solid, frustrated since it would like to be cubic but ends up being a rhombohedral (layered-like) structure, a semimetal for which the conducting properties are limited. At very low temperatures amorphous Bi (a-Bi) is stable and becomes a conductor, but more surprisingly it becomes a superconductor, and this was an incomprehensible result until we studied it and determined that the superconductivity is mainly due to changes in the electronic structure at the Fermi level3. It would seem that making it amorphous by lifting all the symmetry restrictions of the Wyckoff crystal frees the electrons and lets them Cooper-pair to superconduct.

In what follows we shall refer the reader to ref.20, Appendix A1, for the parameters that characterize the crystalline structures of the solid phases of bismuth under pressure used in this work, except for the proposed structures of Bi-III due to Chen et al.21 and to McMahon et al.22 since the debate exists as to what structure may be more representative of the phase. These parameters were determined at room temperature and we shall specify in each case what the pressure is for the parameters used so a precise structure is kept in mind. All pressure values have an uncertainty of 0.05 GPa and for a, b and c the uncertainties are less than 0.001 Å, and for β it is less than 0.01°20. Tables 1 and 2 list some crystallographic parameters taken from ref.20 for the phases, and some physical data calculated in this work.

Table 1 Crystallographic data for the 5 solid phases of bismuth. Bi-III is the so-called incommensurate structure and the debate still exists as to which commensurate one may be more representative.
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Table 2 Physical data and supercell information for the 5 solid phases of bismuth.
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For the eDoS and the vDoS calculations we tried to construct supercells with the same number of atoms as closely as possible. The figures for the phases in this section have been drawn with a default bond length of 3.45 Å where the covalent radius is taken as 1.46 Å. For the Wyckoff phase and for Bi-IV we include our previously published results3,7 for the sake of completeness.

Bismuth I. The Wyckoff Phase

Both the amorphous bismuth structure and the crystalline one have been fully analyzed in ref.3 and the crystalline one was revisited in ref.7. The results are very similar in both references except for the fact that in the present work we have removed the translational modes. For the Wyckoff structure10, Bi-I, we used a 6-atom crystalline cell that we multiply by 5 × 4 × 2 to obtain a supercell with 240 atoms for the calculations of N(E) and of F(ω). The crystalline structure of this phase has two near-neighbor peaks located in 3.11 Å and 3.49 Å; these two peaks coalesce to give rise to a single wide peak for the amorphous sample3. The integration grid was set to fine and the calculations were carried out using a Monkhorst-Pack mesh of 2 × 3 × 3 in k-space. Since this phase is the reference phase, in this section we shall not reproduce N(E) and F(ω) since they appear in the figures where comparisons are made (see also refs3,7).

As mentioned before, based on these ideas in ref.3 we carried out some estimations that led us to propose an upper bound for the superconducting transition temperature of the Wyckoff phase of Bi. We obtained a Tc for crystalline Bi at atmospheric pressure of 1.3 mK, or lower, by comparing values of N(E) and F(ω) between the Wyckoff phase and the amorphous phase, calculated using ab initio computational simulations. Several months later our predictions were corroborated experimentally (Tc = 0.53 mK)4.

For Bismuth I the pressure considered was 0 GPa, with 6 atoms in the hexagonal representation of the rhombohedral primitive cell. The following lattice parameters were used: a = b = 4.55 Å and c = 11.86 Å and with α = β = 90° and γ = 120°, and the space group is \(R\bar{3}m\), as reported in Table 1. Figure 3 is a representation where the bilayers are clearly indicated.

Figure 3
figure 3

Representation of the rhombohedral structure of the Wyckoff phase displaying the bilayers. The bonds correspond to a default of 3.45 Å where the covalent radius is taken as 1.46 Å, as mentioned at the end of Section 3.2. This is the case for all structures.

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Bismuth II. The non-low temperature Phase

Bismuth II is a phase that narrows as the temperature is decreased, Fig. 1, and the fact that an experimental superconductive transition temperature has been reported, Tc = 3.9 K, indicates that it remains a well-defined structure at very low temperatures9. The supercell used for the ab initio calculation of N(E) and F(ω) has 240 atoms, result of a 4 × 3 × 5 multiplication of the 4-atom unit cell and the parameters considered are a = 6.65 Å, b = 6.09 Å and c = 3.29 Å, with α = 90°, β = 110.37° and γ = 90° with a C2/m space group registered in Table 1, all at a pressure of 2.7 GPa20. A representation of the corresponding cell is shown in Fig. 4 where the monoclinic structure can be observed, together with two (dark and light) distinct monoatomic layers displayed. The integration grid was set to fine and the calculations were carried out using a Monkhorst-Pack mesh of 2 × 3 × 3 in k-space.

Figure 4
figure 4

Representation of monoclinic C2/m structure of Bi-II where two distinct monoatomic layers (black and white spheres) can be observed.

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The results obtained for the electronic density of states are given in Fig. 5a where a comparison is made with the results obtained for the Wyckoff phase. The density of electron states at the Fermi level for the Wyckoff phase is 0.15 electrons per atom and for the Bi-II phase is 0.50 electrons per atom. The ratio η is 3.33. In Fig. 5b a comparison of the results obtained for the vibrational density of states is presented. The calculations for the Debye temperatures give 134.2 K for the Wyckoff phase and 115.5 K for Bi-II. The ratio δ is 0.86. The calculated transition temperature for the Bi-II phase is 3.9 K which is to be compared with the experimental result of 3.9 K.

Figure 5
figure 5

(a)N(E) and (b) F(ω) for Bi-II (black solid line) compared to the results for the Wyckoff structure (grey). Since the supercells may have a different number of atoms we plot the results per atom everywhere to compare them meaningfully.

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Bismuth III. The incommensurate Phase

Paraphrasing Hamlet: To commensurate, or not to commensurate, that is the question.

Bismuth III is a low temperature phase that extends to pressures higher than 7 GPa, Fig. 1. The experimental superconductive transition temperature reported, Tc = 7 K, is the highest second to that of Bi-V. This phase has been somewhat controversial since it was described as a regular periodic cell when first identified21, and nowadays it is considered as an incommensurate structure20,22. Since an incommensurate structure is very difficult to represent and even more difficult to calculate, there have been attempts to propose representative commensurate periodic crystalline structures that supposedly would reflect with some precision the real structure23. This gave rise to two structures, one due to Chen and coworkers21, the other due to McMahon and coworkers22; Häussermann et al. proposed a commensurate approximation24. In this work we consider Chen’s representation and, based on the 3:4 Häussermann approximation, we construct a commensurate one for McMahon’s data and calculate their properties and their superconducting transition temperature. As we shall see, the structure proposed by Chen et al. leads to a Tc closer to experiment than ours.

The Chen supercell used for the ab initio calculation of N(E) and F(ω) has 120 atoms, result of a 2 × 2 × 3 multiplication of the 10-atom representative cell. Figure 6 is a representation where the tetragonal structure is illustrated. The parameters, registered in Table 1, at a pressure of 3.8 GPa, are: a = b = 8.66 Å, c = 4.24 Å; α = β = γ = 90° and the space group is P4/n21. The integration grid was set to fine and the calculations were carried out using a Monkhorst-Pack mesh of 2 × 3 × 3 in k-space.

Figure 6
figure 6

Tetragonal structure proposed by Chen et al.21 to describe the experimentally studied incommensurate Bi-III phase at 3.8 GPa. (a) The x-y plane of the structure. (b) The horizontal in-the-plane axis is the z-axis of the structure.

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On the other hand, we propose a slightly different commensurate host-guest structure, after McMahon et al. concluded that Bi III is, in fact, formed by tetragonal host-guest structures incommensurate with one another along the z-axis22. The supercell we used for the ab initio calculation of N(E) and F(ω) has 128 atoms, result of a 2 × 2 × 1 multiplication of our 32-atom representative cell and the parameters are: a = b = 8.52 Å, cH = 4.2 Å and cG = 3.1 Å for the host and guest structures, respectively; α = β = γ = 90° and the space group is P1, at a pressure of 4.2 GPa (compare with Table 1)22. The integration grid was set to fine and the calculations were carried out using a Monkhorst-Pack mesh of 2 × 3 × 3 in k-space. Our basic 32-atom cell is represented in Fig. 7. The host structure is tetragonal with 8 atoms in the unit cell (white spheres), whereas the guest structure is body centered tetragonal with two atoms in the unit cell (black spheres).

Figure 7
figure 7

Our commensurate Bi-III 3:4 host-guest structure based on McMahon incommensurate structure22, at 4.2 GPa. (a) The x-y plane of the structure. (b) The horizontal in-the-plane axis is the z-axis of the structure.

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For the Chen structure the eDoS is given in Fig. 8(a) compared to the Wyckoff phase. The eDoS for the Wyckoff phase is 0.15 electrons per atom and the one for the Bi-III phase is 0.62 electrons per atom. The ratio η is 4.13. In Fig. 8(b) the vDoS are given. The Debye temperatures are 134.2 K for the Wyckoff phase as before, and 96.4 K for Bi-III. The ratio δ is 0.72. The calculated transition temperature for this structure is 6.5 K which is to be compared with the experimental result of 7 K.

Figure 8
figure 8

(a) N(E) and (b) F(ω) for Bi-III (black solid line) according to Chen et al., compared to the results for the Wyckoff structure (grey). Negative energy vibrational modes were removed from Bi-III.

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Figure 9 are the corresponding results for our structure, a la Häussermann, based on the data reported by McMahon et al., compared to Wyckoff’s. The density of electron states for this structure at the Fermi level is 0.45 electrons per atom and the ratio η is 3.00. The Debye temperature 144.4 K and the ratio δ is 1.08. The calculated transition temperature is then 3.5 K.

Figure 9
figure 9

(a) N(E) and (b) F(ω) for Bi-III (black solid line) according to our commensurate representation of McMahon’s results, compared to the Wyckoff structure (grey).

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So, what then? to commensurate or not to commensurate?

Doing the incommensurate cell would be more representative but given the difficulty23 we decided to calculate Bi-III a la Chen that leads to Tc of 6.5 K compared to the experimental result of 7 K, as opposed to a Tc of 3.5 K for our commensurate cell a la Häussermann. Chen’s seems more adequate for our purposes but it is necessary to find a better commensurate representation of McMahon’s results

Bismuth IV. The “non-superconducting” phase

Here we reproduce some results reported in ref.7, for completeness. Experimentally, Bismuth IV has not been found to be a superconductor. However, since all other phases are superconductive, it seems highly probable that Bi-IV would also be, provided the proper conditions are obtained. That is why we decided to investigate this phase using the same facile approach that we describe in this work. We found that, in fact, this phase could become a superconductor if its crystalline structure could be maintained while cooling it to a transition temperature of 4.25 K. Corroboration is pending.

Experimental results by Chaimayo et al.25 for bismuth IV have confirmed that its crystalline structure is orthorhombic at 465 K and 3.2 GPa with the space group given as Cmca, with a = 11.19 Å, b = 6.62 Å and c = 6.61 Å and with the crystalline angles given by α = β = γ = 90°, see Table 1. A 2 × 2 × 3 Monkhorst-Pack scheme in k-space was used and a fine integration grid utilized. The 16-atom crystalline structure of this phase is represented in Fig. 10 with the bilayers depicted with white spheres and intercalated between gray and black layers. By multiplying this cell 4 × 2 × 2 times a supercell with 256 atoms is obtained. It is this supercell that was used to calculate N(E). To obtain F(ω) a supercell with 128 atoms was used, result of a 2 × 2 × 2 multiplication of the depicted cell7. In 1958 Bundy identified this phase and it was then assumed to be cubic19.

Figure 10
figure 10

Crystalline cell of Bi-IV with 16 atoms, taken from ref.7. The bilayers (white spheres) and the layers with grey and black spheres are shown.

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The results obtained for the eDoS and the vDoS of the Chaimayo structure are given in Fig. 11. Figure 11(a) shows that N(E) at the Fermi level is, as always, 0.15 electrons per atom for the Wyckoff phase and 0.53 electrons per atom for Bi-IV; the ratio η being 3.53. Figure 11(b) presents a comparison of the two vibrational densities of states. The calculations for the Debye temperatures lead to 134.2 K for the Wyckoff phase and 102.1 K for Bi-IV; the ratio δ is 0.76. The predicted superconducting transition temperature for this crystalline phase is 4.25 K, result to be corroborated.

Figure 11
figure 11

(a) N(E) and (b) F(ω) for Bi-IV (black solid line) compared to the Wyckoff structure (grey) as previously reported in ref.7.

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The θD now obtained for the Wyckoff phase, 134.2 K, is somewhat larger than that reported in ref.3, 129 K, since as mentioned before we have ignored the translational “static” modes (ω ≈ 0), proportionally more relevant as the number of atoms in the supercell diminishes. As mentioned before, the experimental values for θD reported by DeSorbo26 for crystalline bismuth at ambient pressure goes from 140 K at high temperatures to 120 K at low temperatures. Our calculations indicate that θD = 134.2 K, a value that lies between the experimental ones. It should be noticed that the vibrational spectrum for Bi-IV is more localized than that for the Wyckoff phase, although one would expect it to be more extended since the application of pressure would bring the atoms together (the mass density does increase, 9.80 vs 11.33 g/cm3, Table 2) augmenting the force constant; evidently however, the change in the crystalline structure supersedes this factor.

Bismuth V. The high pressure phase

The structure of Bi-V at a pressure of 8.5 GPa is given by a = b = c = 3.80 Å, and the angles are α = β = γ = 90° and, as indicated in Fig. 12, the structure is a body-centered cubic one (white spheres) with the space group Im\(\bar{3}\)m. The supercells used for the calculation of N(E) and F(ω) have 250 atoms, result of a 5 × 5 × 5 multiplication of the 2-atom unit cell. A Monkhorst-Pack mesh of 3 × 3 × 3 in k-space was used.

Figure 12
figure 12

The body centered cubic structure of the Bi V phase can be seen in white20.

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The results for N(E) and F(ω) for both the Wyckoff and Bi-V phases are presented in Fig. 13 (a) and (b), respectively. The calculations give a result of 0.56 electrons per atom at the Fermi level, which when compared to the value for the Wyckoff phase (0.15), Fig. 13(a), leads to a ratio η of 3.73. In Fig. 13(b) a comparison of the results obtained for the vibrational density of states is given. As before the Debye temperature is 134.2 K for the Wyckoff phase; the Debye temperature for Bi-V is 137.8 K and the ratio δ is 1.03. The calculated superconducting transition temperature for this phase is 6.8 K compared with the experimental result of 8 K. The vibrational spectrum for Bi-V has practically lost the gaps or pseudogaps observed in F(ω) for other phases, since the bcc crystal lacks the layered arrangements of other structures.

Figure 13
figure 13

(a) N(E) and (b) F(ω) for Bi-V (black solid line) compared to the results for the Wyckoff structure (grey).

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Results and Discussion

Figure 14 shows the diagram of the bismuth phases under pressure together with the experimentally determined transition temperatures and our predictions and calculations. The agreement is satisfactory and indicates several features. The fact that the calculations were conducted under the BCS approximation indicates that this approach is adequate for these materials. It has been argued that bismuth is a strong coupling substance and that, in fact, McMillan equation27 should be used when dealing with this material. Tunneling experiments conducted more than 50 years ago by Smith and coworkers28 showed that for amorphous bismuth, 2∆0/kBTc = 4.60 indicating a strong coupling regime, compared to the limit of weak coupling 2∆0/kBTc = 3.5; however, the 5 bismuth phases under pressure, including Bi-I may not be strong coupling since our calculations agree well with experiment and we hope that our predictions will be corroborated by experiment, as was the case for the Wyckoff structure. Table 3 is a compendium of parameters for these solid phase structures.

Figure 14
figure 14

Bismuth phases under pressure with predicted \({T}_{c}^{p}\), calculated \({T}_{c}^{c}\), and experimentally determined \({T}_{c}^{m}\), superconducting transition temperatures. The Bi-III structure used was the one reported by Chen et al.21.

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Table 3 Values for the parameters η and δ defined in Eqs (6) and (7) for the various phases (See Table 2 also).
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Also, the comparison of the results for the higher-pressure phases was made against the Wyckoff phase. Clearly this comparison could have been made against other structure, based on experiment, or against the amorphous, etc. However, being systematic gives consistency.

All our calculations are based on a reasonable assumption, the fact that the Cooper pairing potential is assumed to be the same for all phases. Since there are no estimates for the value of this potential for bismuth we decided to obtain it from the BCS equation and from the calculated results for N(EF) and F(ω) (or θD) and the superconducting transition temperatures reported in Table 3, for each phase. As expected Vo should be a constant and the same for all phases (as was assumed) except for the Bi III which are pathological. In Fig. 15(a) we plot this value for the five phases, acknowledging that for Bi-III two structures were considered. The calculation is carried out with both, the experimental transition temperatures and the superconducting transition temperatures obtained by us. It can be observed that the average of Vo excluding both representations of Bi-III, gives 2.32 meV, a value that we used in the graph of Fig. 2. The representations of the Bi-III commensurate structures give results away from experiment so these structures should be analyzed with more care. To include the effect of the density of states N(EF) in Fig. 15(b) we plot the product N(EF)Vo where it can be observed that all structures comply with the weak coupling criterion established in the literature, that N(EF)Vo ≪ 1. More work should be done to investigate discrepancies and the implications of our results.

Figure 15
figure 15

Cooper pairing potentials, Vo, for the solid phases of Bi under pressure, obtained from Equation 1. The units of Vo are meV and the Bi III structures are excluded from the average.

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Conclusions

Two important assumptions have been made when predicting the superconducting transition temperatures of the solid phases of Bismuth under pressure. These are:

Assumption 1.- All the phases considered can be described by the Bardeen, Cooper, Schrieffer equation for the superconducting transition temperature. This implies that we consider their superconductivity as phononic, or the conventional type;

Assumption 2.- Moreover, we assume that the Cooper pairing potential V0 is the same for all phases, since the material is the same albeit in different crystalline structures and for different densities.

However, the fact that we have been able to predict the superconducting transition temperature of Bi-I, already corroborated by experiment, and the fact that we have adventured predictions for the superconductivity in bismuth bilayers and for the phase Bi-IV, not experimentally studied yet, gives us confidence that, at least for bismuth, our approach is sound. The present work ascertains the validity of our approach, since the calculated superconducting transition temperatures are all very close to experiment. On the other hand, the assumption that the pairing potential is the same for all phases a priori, is validated by the calculation of its value a posteriori. The average of V0 is 2.32 meV and when considering the factor (N(EF) V0) this is much smaller than 1 for all phases, a condition indicating that we are in the weak coupling limit. The horizontal line in Fig. 15(b) represents the value that most classic superconductors display29. The pairing mechanism considered is due to phonons as indicated by the factor θD that appears in the BCS expression, Eqn. 1 and in Fig. 2.

Calculating the pairing potential is a challenge. Computational simulations are helping to accomplish this but meanwhile, using the present approach, we can obtain reasonable approximations to the superconducting transition temperatures if one deals with the same material in different phases, case at point. Also, computational simulations have evolved favorably to allow one to obtain reliable results for the electronic and vibrational densities of states to be able to compare them meaningfully and infer conclusions. As far as the vibrational properties are concerned, one can observe interesting changes in F(ω) for all phases associated to the presence of low dimensional structures and, in some of them, associated to the presence of low frequency modes, often invoked as very important in the manifestation of superconducting properties.

Certainly, the electronic and vibrational properties are conducive to propitiating superconductivity if the pairing potential manifests itself and for that the temperature has to be low enough until superconductivity appears; for most of them the liquid helium temperature is adequate, as shown in Fig. 14. The results obtained and displayed in this figure, gives us confidence in our procedure. Other materials should be considered to see the breadth of our surmises.

Data Availability

The datasets generated and analysed during the current study are available from the corresponding on reasonable request.