Fig 1.
Structural analysis of simulated a-Bi.
(A) Computer-generated amorphous supercell of bismuth, a-Bi216. (B) Computational g(r) for a-Bi216 compared with the experimental results obtained by Fujime [2]. Positions of the peaks for the rhombohedral x-Bi are included for contrast with the broader, amorphous ones.
Fig 2.
PAD for the a-Bi216 supercell.
The nearest neighbor values of the plane angle in x-Bi are 89° and 91°, whereas the maximum for the amorphous cell occurs at 82.5°. Notice the shoulder located at 56.5° which is reminiscent of angles in an equilateral triangular structure. The bump at about 155° may be due to non-planar surfaces bounding deformed cubes in this amorphous structure.
Fig 3.
Some structures within a-Bi216.
(A) Semi-planar structure associated with angles in the neighbourhood of 155°. (B) Triangular structures associated with the bump at 56.5°, reminiscent of quasi-equilateral triangles. (C) Deformed cubic-like arrangements reminiscent of the crystalline structure of bismuth, corresponding to 82°. Linked atoms are nearest neighbors.
Fig 4.
Electronic crystalline and amorphous behavior.
(A) Comparison of our normalized result for x-Bi216 with the theoretical eDOS for x-Bi from Gonze et al. [32] and the experimental data taken from Jezequel et al. [30], Ley et al. [22] and Kakizaki et al. [31]. Normalization indicates that the area under each curve, up to the Fermi level, EF − E = 0, is set equal to 1. (B) The calculated eDOS for the amorphous structure a-Bi216 (full line) and for the crystalline x-Bi216 (dotted line) are shown; the relevant feature for the amorphous is that the pseudo-gap has essentially disappeared and that therefore there are more electrons at N (EF) in a-Bi than in x-Bi.
Fig 5.
Vibrational behavior for the crystalline and the amorphous.
(A) Comparison of the experimental vDOS (extracted and processed from Kakizaki [31] and taken from the results of Dijk and Dingenen [20]) and our computational calculations for x-Bi. The curves have been normalized so that the area under each curve is 1. (B) Comparison of the calculated vDOS for crystalline and amorphous supercells x-Bi216 and a-Bi216. The area under each curve has been set equal to 3NA.
Fig 6.
Dependence of Tc(λ) on λ from McMillan’s Eq 4 for θD = 129K.
The behavior indicates that the Tc dependence on λ is not very strong for high values of λ. For low values, Tc changes rapidly with λ. The inset is an extended plot. The dotted lines show the value of λ (0.236) for the estimated Tc.
Fig 7.
The modified undermelt-quench process.
This is the amorphization process used to generate amorphous bismuth supercells. The heating ramp stops just below the melting point whereas the cooling ramp has the same (negative) slope.
Fig 8.
The figure shows the amorphous g(r) (grey filled figure) and its 2-point smoothed curve (continuous black line) for a-Bi216.
Fig 9.
The vibrational distributions vDOS (or F(ω)).
The figures show the vDOS (grey filled figures) and their 3-point smooth curves (continuous black lines) for (A) the experimental x-Bi (10) (B) the theoretical x-Bi216 calculated in this work and (C) the theoretical a-Bi216 calculated in this work.
Fig 10.
The electron-phonon coupling function for Bi.
(A) α2(ω) for a-Bi obtained from the ratio between the experimental data of Chen et al. [35] and our calculated Fa(ω) for amorphous bismuth. That is, the result of dividing Chen’s experimental data by our simulational calculations for Fa(ω) point by point. (B) α2(ω)F(ω) calculated from the α2(ω) obtained in (A) and the crystalline Fc(ω) from this work, compared with Chen’s experimental results for a-Bi.