Supplementary Information
An insight into sodiation of antimony from first-principles crystal
structure prediction
Riccarda Caputo*
ETH Zurich, Department of Chemistry and Applied Biosciences, Laboratory of Inorganic Chemistry,
Vladimir-Prelog-Weg 1, CH-8093 Zurich
Additional information are provided for the following:
(1) the crystal structure of Sb;
(2) the influence of the d-states of Sb on the enthalpies of formation;
(3) the structure and energy data of different allotropes of Sb in Table S1;
(4) a view of the structure for x = 0.333 in Figure S2;
(5) the calculated IR spectra in Figure S3;
(6) the calculated XRD patterns in Figure S4;
(7) a sketch of the low energy paths of the motion of Na in Na3Sb in Figure S5;
(8) the energy-volume curve of the low energy structures of Na xSb, found via sodiation of Sb and desodiation of
Na3Sb in Figure S6.
The crystal structure of Sb
The crystal structure of antimony has been studied since the beginning of the last century [1] and along decades several
structures have been reported, of which four allotropic forms are known, indicated as grey, black, yellow and explosive
antimony [2]. The grey, silvery white, antimony is the only stable form so far reported as semimetal. In fact, elemental
antimony is known to be a metalloid, meaning that although it might resemble a metal, it does not present the intrinsic
*Email address of the corresponding author: riccarda.caputo@inorg.chem.ethz.ch
properties to be identified as a metal. The inorganic crystallographic structure database (ICSD) reports two polytypes
[3] R and H of the trigonal structure symmetry space group R-3m, with the rhombohedral and hexagonal axis
representation; three cubic structures, of which two, the Pm-3m (IT No 221) and the Im-3m (IT No 229), are the high
pressure phases [4-6] and one, the Fm-3m (IT No 225) is metastable; the hexagonal P63/mmc (IT No 194), which is
metastable; the monoclinic P21/m (IT No 11) and the tetragonal I4/mmm (IT No 139) [7] (and reference therein). At
GGA level of calculations and by using the norm-conserving pseudopotential, with only the 5s25p3 states in the valence
shell, the crystal structure of antimony, optimized with lattice parameter and angle of 4.341 Å and 58.056 degrees, in
the rhombohedral representation of the symmetry space group R-3m. In the hexagonal representation, the lattice
parameters resulted: a = 4.212 Å and c = 10.786 Å, very close to the reported ones [8]. The Sb atoms are located in
x = 0.2341 on the (2c) sites in the rhombohedral representation and (6c) sites in the hexagonal representation. A view
along the a-direction shows that the atoms lying on the {1 1 1} planes are connected and form a distorted cube of edge
lengths 3.240 Å and 2.834 Å, which centroid is in the special position (0, 0, 0). A cut through the Sb-Sb bonds up to
3.20 Å shows a layered-like connection of Sb atoms, via corner-sharing tetrahedral (see Figure S1). The distorted
cube formed by the nearest neighbor Sb atoms in the rhombohedral lattice can be made undistorted by releasing the
symmetry constraints on the atomic positions and searching for the symmetry space group matching them within the
tolerance of 0.1 Ȧ . So doing the face-centred cubic structure Fm-3c (IT No 226) of Sb containing 8 formula units was
modelled, which can be further reduced to primitive-centred cubic structure Pm-3m (IT No 221) containing one atom
on (1a) site at (0, 0, 0). The corresponding optimized lattice parameter is 2.966 Å and therein the Sb atoms are hexacoordinated with edge-sharing octahedra of length equal to the cubic lattice parameters. The two structures, the cubic
and the rhombohedral, differ by only 3.17 kJ/mol, being lower in energy the rhombohedral one, though they are clearly
distinguishable via X-ray diffraction and phonon dispersion curves. Clearly, a small amount of energy is required to
transform the hexagonal structure into the cubic, which showed lattice instability, as found via phonon calculations.
Interestingly, the AIRSS calculations of the pure Sb phase provided, among other possible local minima, the hexagonal
structure as the most likely one. For literature data on the antimony phases, the readers are addressed to specific works
(e.g. [1, 2, 4, 5, 7-14]).
2
The d-states of Sb
The electronic configurations of sodium, atomic number 11, and antimony, atomic number 51, are Ne[3s1] and
Kr[4d10 5s2 5p3], respectively. In order to evaluate the role of the d-states of Sb in the Na-Sb bond formation, the
structures of the elemental sodium, antimony, NaSb and Na 3Sb were optimized by using different ways to treat the
core electrons by using DMol3 [15, 16]. The All Electron calculations were compared to the density functional semicore pseudo-potential (DSPP) and the effective core potentials (ECP) calculations.
In Table SII the lattice parameters and the enthalpies of formation are reported, compared to the corresponding values
calculated by using CASTEP. The enthalpies of formation are very similar for NaSb. A larger difference is found for
Na3Sb. In our CASTEP calculations the d-states of Sb are not treated explicitly but included into the core region, thus
in the pseudopotential approximation (norm-conserving and ultra-soft) the valence shell contains only the 5s2 and the
5p3 electrons. The projected density of states (PDOS), in all three different ways to treat the core region in DMol3
calculations, showed that the 4d states are almost spread over the entire range of the orbital energy and overlap with
the 5s and 5p states at the Fermi level. The calculated electronic band structure of antimony, independently on how
the core region is treated, reproduced the main bonding features reported in literature [10, 12] (and references therein).
A detailed study of the electronic band structures of the Na-Sb system is out of the scope of the present work,
nevertheless, and besides the well-known limits of the DFT calculations to describe precisely the band gaps, a
qualitatively trend along the composition range investigated can be drown. As increasing the concentration of the
electron-donor element, namely Na, the states at the Fermi level of the semi-metal host lattice of Sb get accordingly
filled. Accordingly, the NaxSb compounds, as a function of x, exhibit features of semimetals for x = 0 to metals for
non-integer x values to semiconductors for x integer. In fact, at x = 1, 2 and 3 the valence bands are filled due to the
contribution of the electrons of Na atoms and valence-conduction band gap opens, in particular, for x equal to 1 and
3, for which literature data are available to compare with. In addition, being the band gap inversely proportional to
the square of the lattice constants [17] any small deviations in the lattice parameters from the experimental ones is
reflected into the band gap energy values. In the monoclinic structure of NaSb (IT No 14), the d-states of the elemental
Sb split into two regions and overlap with the sp-states of Na. The bonding features are dominated by the Sb-Sb
connection along the (Sbx)- infinite chains. The band gap reported in the literature of the monoclinic structure falls in
the range 0.902, 1.1 eV, and that of the optimized structure 1.15 eV. Thus, the lowest energy structure of the Na-Sb
system is semiconductor. The difference can be rationalized in terms of the structure: the optimization provided shorter
3
parameters than the reported ones, especially along the b-direction by 0.121 Å, which is the direction along which the
(Sbx)- spirals unfold. In the hexagonal Na3Sb structure, the d-states of Sb split even wider and the higher energy band
of d-sp-states, 2.7 eV wide from the Fermi level, overlap with the sp-states of Na atoms. The Na-Sb bonds exhibit a
mixture character, dative and covalent [18]. The calculated band gap of the hexagonal Na3Sb is 0.916 eV and the cubic
structure is 0.943 eV. The reported band gap of the hexagonal Na 3Sb is 1.1 eV, calculated by an empirical
pseudopotential method [19]. For the structures, which exhibit a band gap, it was possible to calculate, via linear
response Fourier interpolation scheme of the dynamical matrix, the infrared spectra, which are reported in Figure S3.
The calculated XRD patterns of the lowest energy structures at different composition are shown in Figure S4. The
most likely paths of the motion of the Na atoms is particularly relevant for understanding the mechanism of
desodiation and the dynamics of Na atoms. Clearly, it requires a detailed and a dedicated study to investigate for
example the dependence of the Na motion on the structure changes and to make a connection between the
thermodynamically accessible states, found in the present study, and the dynamically accessible states. Nevertheless,
the present study suggested that in the hexagonal Na3Sb the low energy path is via the tetrahedral-trigonal sites, the
4f-2b sites, while the 4f-4f path is the high-energy path. In the cubic Na3Sb the path is via the octahedral-tetrahedral
sites, the 8c-4a, as sketched in Figure S5.
References
1.
R. W. James and N. Tunstall, Phil. Mag., 40, 233 (1920).
2.
N. C. Norman, Chemistry of Arsenic, Antimony and Bismuth. Springer (1988).
3.
A. Guinier, G. B. Bokij, K. Boll-Dornberger, J. M. Cowley, S. Durovic, H. Jagodzinski, P. Krishna, P.
M. de Wolff, B. B. Zvyagin, D. E. Cox, P. Goodman, Th. Hahn, K. Kuchitsu, S. C. Abrahams, Acta
Cryst., A40, 399 (1984).
4.
L. F. Vereshchagin and S. S. Kabalkina, Soviet Physics JETP, 20, 274 (1965).
5.
T. N. Kolobyanina, S. S. Kabalkina, L. F. Vereshchagin, L. V. Fedina, Soviet Physics JETP, 28, 88
(1969).
6.
A. B. Shick, J. B. Ketterson, D. L. Novikov, A. J. Freeman, Phys. Rev. B, 60, 15484 (1999).
4
7.
H. Iwasaki and T. Kikegawa, High Pressure Research, 6, 121 (1990).
8.
C. S. Barrett, P. Cucka, K. Haefner, Acta Cryst., 16, 451 (1963).
9.
G. Brauer and E. Zintl, Z. Phys. Chem. B, 37, 323 (1937).
10.
L. M. Falicov and P. J. Lin, Physical Review, 141, 562 (1966).
11.
J. Rose and R. Schuchardt, Phys. Stat. Sol. (b), 139, 499 (1987).
12.
X. Gonze, R. Sporken, J. Vigneron, R. Caudano, J. Ghijsen, R. Johnson, L. Ley, H. Richter, Physical
Review B, 44, 11023 (1991).
13.
Antimony (Sb) band structure, energy gap Non-Tetrahedrally Bonded Elements and Binary
Compounds I, Vol. 41C. 1998: Landolt-Börnstein Group III, Condensed Matter, Springer.
14.
H. J. Breunig, Antimony: Inorganic chemistry, Encyclopedia of Inorganic Chemistry, John Wiley &
Sons Ltd. (2006).
15.
B. Delley, J. Chem. Phys., 113, 7756 (2000).
16.
B. Delley, J. Chem. Phys., 92, 508 (1990).
17.
R. Dalven, Physical Review B, 8, 6033 (1973).
18.
G. W. Ossman, The Journal of Chemical Physics, 49, 783 (1968).
19.
O. Madelung, Semiconductors. Data Handbook, Springer-Verlag (2004).
20.
K. Lejaeghere, V. Van Speybroeck, G. Van Oost, S. Cottenier, Critical Reviews in Solid State and
Materials Sciences, 39, 1 (2014).
21.
D. Akhtar, V. D. Vankar, T. C. Goel, K. L. Chopra, Journal of Materials Science, 14, 2422 (1979).
22.
K. Aoki, S. Fujiwara, M. Kusakabe, Solid State Communications, 45, 161 (1983).
23.
L. F. Vereshchagin and S. S. Kabalkina, Soviet Physics - JETP, 20, 274 (1965).
24.
U. Schwarz, L. Akselrud, H. Rosner, A. Ormeci, Yu Grin, M. Hanfland, Physical Review B, 67, 214101
(2003).
25.
S. S. Kabalkina and T.N. Kolobyanina, Zhurnal Eksperimental, noi i Teoreticheskoi Fiziki, 58, 486
(1970).
5
Table S1. The lattice parameters (Ȧ) and the energy differences (eV) of the allotropes of antimony calculated by using
CASTEP. In the trigonal, tetragonal and hexagonal lattices b = a. The monoclinic structure was optimized by starting
with the reference structure, which has symmetry space group P21/m (IT No 11).
Symmetry space group (IT No)
Lattice parameters a, b, c (β)
R-3m H (166)
R-3m R (166)
Fm-3m (225)
Im-3m (229)
Pm-3m (221)
I4/mcm (140)
P63/mmc (194)
C2/m (12)
Cmmm (65)
4.212, 10.786
E
0
4.597
3.628
2.966
7.773, 3.926
3.161, 5.690
4.165, 4.204, 5.985 (90.823)
4.176, 4.181, 2.975
0.563
0.405
0.015
0.334
0.512
0.031
0.021
6
Remarks
[8]
[20]
[21]
[22]
[23]
[24]
[23]
[25]
Table S2. Structure and thermodynamic data by using DMol3 code. In the table are reported: the lattice parameters
(a, b, c and, in brackets, the β angle, where appropriate) of the elements, Na and Sb, and the two phases, NaSb and
Na3Sb; the corresponding enthalpies of formation, calculated by using the three different ways to treat the core region
of antimony implemented in DMol3. The enthalpies of formation calculated by using CASTEP, with the GGA-PBE
functional and norm-conserving pseudo-potentials are reported in brackets for comparison.
Compound Symmetry fu Potential
(IT No)
Na
Sb
NaSb
IM-3m
(229)
R-3m
(166)
2
AE
11
6
P21/c (14)
8
AE
DSPP
ECP
AE
51
15
5
Sb(51)
Na(11)
Sb(15)
Na(11)
Sb(5)
Na(11)
Sb(51)
Na(11)
Sb(15)
Na(11)
Sb(5)
Na(11)
Sb(51)
Na(11)
Sb(15)
Na(11)
Sb(5)
Na(11)
Sb(51)
Na(11)
Sb(15)
Na(11)
Sb(5)
Na(11)
DSPP
ECP
C2 (5)
6
AE
DSPP
ECP
Na3Sb
Zval
P63/mmc
(194)
2
AE
DSPP
ECP
4
Fm-3m
(225)
AE
DSPP
ECP
Lattice
parameters (Å)
306
90
30
496
168
4.409, 11.325
4.405, 11.334
4.442, 11.354
6.922, 6.409,
12.714 (117.872)
6.833, 6.431,
12.560 (117.326)
6.824, 6.383,
12.554 (116.748)
9.346, 6.799,
6.282 (107.559)
9.312, 6.752,
6.288 (107.756)
9.325, 6.7452,
6.268 (109.052)
5.399, 9.598
-190.06 (-179.07)
96
5.387, 9.566
-184.71
76
5.386, 9.541
-194.50
336
7.535
-181.60 (-166.44)
192
7.532
-182.99
152
7.535
-181.60
208
128
372
156
96
7
Δf H
(kJ/mol)
Total No
of valence
electrons
22
4.191
-72.15 (-75.17)
-70.69
-74.25
-62.31(-63.94)
-61.57
-63.67
Fig.S1. A three-dimensional view of the hexagonal representation of Sb, symmetry space group R-3m. (a) A cutthrough view of the bonds at 3.20 Å shows the pyramid-based layer network. The average bond length of the perfect
corner-sharing pyramids is 2.834 Å. (b) A cut-through view of the bonds at 3.25 Å shows the distorted cube formed
by the nearest neighbor Sb atoms (highlighted). The average bond length in the distorted octahedron is 3.037 Å, in
which three bonds are 3.240 Å and three are 2.834 Å. The distortion index of the octahedron is 0.067 Å (in bond
lengths). The crystallographic axes are shown: a (red), b (green), c (blue).
8
(a)
(b)
(c)
Fig.S2.(a) The monoclinic structure for x = 0.333, with symmetry space group C2/m. (b) The antimony atoms arrange in
almost perfect octahedra. The mean Sb-Sb bond length is 2.885 Å, and the distortion index of the polyhedra is 0.002 Å in
bond lengths. That is reflected into the β angle, which is almost 90 degrees. (c)The projection of the ELF on the planes
passing through the antimony bonds shows that the bond is -type and polarize towards the sites occupied by Na atoms.
9
Fig.S3. The calculated IR spectra of NaSb, Na2Sb and Na3Sb. In brackets the IT No of the corresponding symmetry
space group.
10
(a)
(b)
Fig.S4. The calculated XRD patterns of selected Na xSb structures. (a) The structures are
optimized from the direct reaction of Na in Sb; (b) the structures are optimized from the
reverse reaction, the desodiation from the hexagonal and cubic Na3Sb structures.
11
(a)
(b)
Fig.S5. The low energy paths of the motion of Na atoms in (a) the hexagonal and (b) the cubic structures of Na3Sb. (a) The drawn path
connecting the 4f-2b sites (in red) is the path with the shortest Na-Na distance, 3.454 Ȧ. The path is not isotropic, because the path
perpendicular to that but along the c-direction (in blue) connects the same site at different distance: 4f-2b, 3.454 Ȧ; 4f-2c, 4.390 Ȧ. The
path 4f-4f (in black) even if shorter, 3.174 Ȧ, is higher in energy. (b) The path connecting the 4a-8c sites (in red) is isotropic with the NaNa distance of 3.226 Ȧ. The 8c-8c (in black) path requires higher energy.
12
0.333(12)
((123
0.333(123)
((123
1.5(71)
0.5(12)
2(11)
0.75(2)
0.5(12)
1(2)
2.5(166)
)
1.25(12)
1(5) 1.5(71) 2(150)
1(11)
3(225)
3(194)
1(14)
Fig.S6. The calculated enthalpy of reaction, Hr in kJ/mol of Na, as a function of the molar
volume, of the lowest energy structures of NaxSb, modelled via sodiation and desodiation
processes. The numbers next to each point are the composition, x, and, in brackets, the symmetry
space group number.
13