Auxiliary Material
Structures and Potential Superconductivity in SiH4 at High Pressure
–
en Route to ‘Metallic Hydrogen’
Ji Feng, Wojciech Grochala, Tomasz Jaroń, Roald Hoffmann, Aitor Bergara and N.W. Ashcroft
Contents
A1. Set of thirteen crystal structures considered in our theoretical study.
A2. Lattice parameters, volumes, selected interatomic distances, electronic band gap at the Fermi level, residual stress
and bulk moduli for the optimized unit cells of crystal structures considered in this work (VASP results).
A3. Internal energy vs. pressure and enthalpy vs. pressure diagrams for 12 crystal structures (VASP results).
A4. Variation of selected interatomic distances with increasing external pressure, for structures T1 and O2 (VASP
results).
A5. Electronic density of states as a function of pressure for structures T1 and O2 (VASP results).
A6. Band structure and electronic density of states for structure T1 at 85 GPa (CASTEP and VASP results).
A7. Band structure and electronic density of states for structure O2 at 56 and 116 GPa (CASTEP results).
A8. Band structure and electronic density of states for structure O1 at 9 and 97 GPa (CASTEP results).
A9. Band structure and DOS of SiH4 in structure O2, at two different pressures close to metallization onset (VASP
results).
A10. Band structure and DOS of SiH4 in structure O1, at two different pressures close to metallization onset (VASP
results).
A11. Band structure and DOS of SiH4 in structure T3 at two different pressures close to metallization onset (VASP
results).
A12. Band structure and DOS of SiH4 in structure O3 (VASP and CASTEP calculations) at different pressures and the
charge density map of states near the Fermi level of O3 at 91 and 202 GPa.
A13. Calculated rS parameter [bohr] vs. pressure for selected structures of SiH4 (VASP results).
A14. Derivation of the rS value for SiH4, based on the Goldhammer–Herzfeld criterion.
A15. Electron density at the Fermi level (EF ±0.02 eV) for SiH4 in structures O1 and O2 in the metallic state (CASTEP
results).
A16. Phonon spectra (at the  point) for SiH4 in structures O2, O1, O3 and T3 at their metallization onsets (VASP
results).
A17. Molecular orbitals of the uniaxially compressed SiH62– unit (extended Hückel theory).
A18. Estimation of the TC values for SiH4 in structure O3 at 116 GPa, structure O1 at 108 GPa, and structure O2 at 202
1
GPa (VASP results): the BCS picture.
A19. Calculating pressure from VASP data.
A20. References and notes.
2
A1. Set of thirteen crystal structures considered in our theoretical study. Three of these, labeled T1, T2, T3, O1, O2,
O3 and M1, are discussed in greater detail in the text.
T1.(4+4)-coordination*
T2.(4+12)-coordination
T3. Tetragonal
-38.10 eV
-37.16 eV
-37.86 eV
Insulator
Insulator
Insulator
T4.(4+2+4)-coordination
T5.(4)-coordination
O1. 1-D Chain structure
-38.06 eV
-38.00 eV
-36.08 eV
Insulator
Insulator
Semiconductor
O2.(6+8)-coordination
O3. 6-coordination
O4.(6)-coordination
-36.03 eV
-36.21 eV
-36.06 eV
Semiconductor
Semiconductor
Semiconductor
M1.(8)-coordination
M2. (4+8)-coordination
M3.(6+8)-coordination
-33.35 eV
-32.30 eV
-31.80 eV
Metal
Metal
Metal
3
M4.(12+8)-coordination
-28.10 eV
Metal
* The coordination is described by two numbers in parentheses, (m+n), where m is the number of nearest
neighbors and n the second nearest.
4
The structures considered in our theoretical study are arranged in order of increasing internal energy for the
optimized unit cells (at zero pressure). They typically have cubic or tetragonal unit cells. Coordination of Si by H atoms
is tetrahedral for structures T1, T2, T3, T4, and T5, octahedral for O1, O2, O3, O4 and M2, while it is essentially cubic
(M1), square planar (M2) or even 12-fold (M4) for higher energy polymorphs. The idealized UF4 structure1A (with an
unusual tetragonal prismatic coordination of Si), is shown to be unstable by our calculation; it distorts spontaneously to
O2. Structure O1 contains edge–sharing SiH6 octahedra, which form a chain propagating in one dimension. Structures
O2, O3 and O4 are composed of corner–sharing SiH6 octahedra which form two–dimensional flat sheet structures,
made up of SiH2 layers sandwiched between two H layers. Structure M1 contains an 8–fold coordinated Si, while in M2
Si adopts a square planar coordination (M2 is reached by transition from T3 at pressures as high as ca. 300 GPa, and it
will not be further described here). Not unexpectedly, the M coordination types are not observed for any compounds of
tetravalent Si under ambient pressure. Structures M3 and M4 contain H atoms (in the corners of the unit cell) which do
not form chemical bonds to Si; it is therefore not surprising that even at ambient pressures these polymorphs show
metallic behavior in our calculations.
Much more complex structures might be derived from structures studied herein via small distortions (typically
angular ones); those should not lead to significant energy improvement, nor qualitatively undermine our estimates of the
pmet and TC values.
A2. Lattice parameters, volumes, selected interatomic distances, electronic band gap at the Fermi level, residual
stress and bulk moduli for the optimized unit cells of crystal structures considered in this work (VASP results).
The table summarizes selected parameters calculated for the crystal structures considered here.
5
Structures
Z
Lattice Constant (Å) (1)
Unit Cell
Volume of SiH4
3
3
Residual Pressure
Shortest Si-H
Band Gap
Volume (Å )
unit (Å )
(GPa)
Distance (Å)
(eV)
T1
2
a = b = c = 5.295
148.42
74.21
0.054
1.488
7.0
T2
1
a = b = c = 3.807
55.16
55.16
0.000
1.489
5.6
T3
4
a = 6.351, b = c = 6.627
249.60
62.40
0.069
1.490
5.8
5.6
(2)
T4
2
a = b = 4.568 , c = 5.899
122.89
61.45
-0.054
1.486
T5
2
a = b = c = 5.695
184.67
92.34
0.011
1.486
O1
O2
O3
1
2
2
(3)
a = b = 4.246 , c = 2.570
a = b = 3.330, c = 8.327
a = 4.80, b = 4.13,
43.75
92.36
43.75
-0.054
46.18
0.069
1.514
(4)
1.497
(4)
95.44
47.72
-0.014
1.501
(4)
6.7
1.675
(5)
2.3
1.665
(5)
1.9
1.674
(5)
2.4
c = 4.81
O4
1
a = b = 3.323, c = 5.229
57.75
57.75
0.039
1.499 (4) 1.662 (5)
2.4
M1
2
a = b = c = 4.032
65.54
32.77
0.004
1.746
0.0
M2
2
a = b = 4.287, c = 4.358
80.11
40.06
-0.048
1.558
M3
1
a = 3.322, b = 3.319, c = 3.310
36.50
36.50
0.074
ca. 1.655
0.0
(5)
0.0
ca. 2.873
M4
(1)
1
a = b = c = 3.004
27.11
27.11
0.027
2.124 (5), 2.602
Lattices are orthogonal unless otherwise specified. (2) γ = 93.6°. (3) γ = 110.1°. (4) Terminal hydrogen. (5) Bridging hydrogen. N.D. not determined.
6
0.0
A3. Internal energy vs. pressure and enthalpy vs. pressure diagrams for ten crystal structures (VASP results).
-4
-6
Energy (eV/SiH4)
-8
-10
T1
T2
T5
T4
O1
O2
O3
O4
M1
M3
M4
T3
-12
-14
-16
-18
-20
0
100
200
300
Pressure (GPa)
In figures below, ‘VASP Pressure’ is computed by program from the stress tensor; ‘Corrected Pressure’ is calculated
from differentiation of the interpolated internal energy vs. volume curves.
7
Enthalpy (eV/formula unit)
30
20
10
0
-10
T1
T2
T5
T4
O1
O2
O3
M3
M1
M3
M4
T3
-20
0
50
100
150
200
250
300
350
VASP Pressure (GPa)
Fitted Enthalpy (eV/formula unit)
30
20
10
0
-10
T1
T2
T5
T4
O1
O2
O3
M3
M1
M3
M4
T3
-20
0
50
100
150
200
Pressure (GPa)
8
250
300
350
A4. Variation of selected interatomic distances with increasing external pressure, for structures T1 and O2 (VASP
results).
4.5
1.70
g Si-Si interplane
4.0
T1
1.65
g Si-H terminal
3.5
g Si-H bridging
1.60
3.0
1.55
2.5
1.50
2.0
1.45
1.5
1.0
1.40
0
20
40
60
80
100
120
5.0
1.50
a Si-Si
a Si-H
O2
4.5
1.48
4.0
1.46
3.5
1.44
3.0
2.5
1.42
2.0
1.40
1.5
1.0
1.38
0
20
40
60
80
100
120
140
160
Note the significant reduction of the van der Waals gap at early stages of squeezing (corresponding to the drop in the
Si–Si intermolecular distance), and the progressive decrease of the intramolecular Si–H distances as pressure
increases. At p>50 GPa, the bonds of terminal and bridging H to Si (for structure O4) exhibit a similar
compressibility coefficient of about 0.067 Å/100 GPa. The Si–H bonds in structure T1 are less compressible in this
pressure range, the respective coefficient being 0.050 Å/100 GPa.
9
A5. The electronic density of states as a function of pressure, for structures T1 and O2 (VASP results).
100% vol
80% vol
60% vol
40% vol
33 vol%
25% vol
DOS (a.u.)
P = 0 GPa
1 GPa
6 GPa
31 GPa
62 GPa
154 GPa
-15
-5
5
15
Electronic Energy (eV)
DOS for structure T1 at various pressures. Significant DOSF > 0.1 e /(eV FU) is seen at 154 GPa.
10
25
100% vol
80% vol
64.5% vol
50% vol
43% vol
35% vol
DOS (a.u.)
P = 0 GPa
4 GPa
25 GPa
55 GPa
92 GPa
177 GPa
-15
-5
5
15
Electronic Energy (eV)
DOS for structure O2 at various pressures. Significant DOSF > 0.1 e /(eV FU) is seen above 55 GPa.
11
25
A6. The band structure and electronic density of states for structure T1 at 85 GPa (CASTEP and VASP results).
Band Structure
0
DOS
-5
Energy (eV)
En$rgy ($V)
-10
-15
-20
-25
-30
G
G
X
X
M
M
R
R
12
G
G 0.0
0
1.0
1.0
2.0
2.0
3.0
3.0
A7. The band structure and electronic density of states for structure O2 at 56 and 116 GPa (CASTEP results).
56 GPa
116 GPa
13
A8. The band structure and electronic density of states for structure O1 at 9 and 97 GPa (CASTEP results).
9 GPa
97 GPa
Note that for SiH4 in structures O2 and O1, the DOS at the Fermi level, NF, is composed predominantly of s states at
the onset of metallization (H(1s) and Si(3s)). As pressure is increased, the contribution from Si(p) orbitals
significantly increases, and the overall value of NF increases.
14
A9. Band structure and DOS of SiH4 in structure O2 at two different pressures close to metallization onset (VASP
results).
Band Structure
5
50%vol
55 GPa
DOS
0
Energy (eV)
En$rgy ($V)
-5
-10
-15
-20
-25
-30
-35
A
A
G
G
115 GPa
M
M
Z
Z
G
G 0.0
0
Band Structure
5
40% vol
X
X
1.0
1.0
2.0
2.0
3.0
3.0
DOS
0
Energy (eV)
En$rgy ($V)
-5
-10
-15
-20
-25
-30
-35
A
A
G
G
X
X
M
M
A: (0.5, 0.5, 0.5); X: (0.0, 0.5, 0.0); M: (0.5, 0.5, 0.0); Z: (0.0, 0.0, 0.5)
15
Z
Z
G
G 0.0
0
1.0
1.0
2.0
2.0
3.0
3.0
A10. Band structure and DOS of SiH4 in structure O1 at two different pressures close to metallization onset
(VASP results).
5
63% vol.
20 GPa
0
Energy (eV)
En$rgy ($V)
-5
-10
-15
-20
-25
-30
-35
E
G
G
Y
Y
A
Z
G 0.0
0
Band Structure
42% vol
1.0
1.0
2.0
2.0
DOS
5
108 GPa
0
Energy (eV)
En$rgy ($V)
-5
-10
-15
-20
-25
-30
-35
E
G
G
Y
Y
A
E: (0.5, 0.5, 0.5); Y: (0.0, 0.5, 0.0); A: (0.5, 0.5, 0.0); Z: (0.0, 0.0, 0.5)
16
Z
G 0.0
0
1.0
1.0
2.0
2.0
A11. Band structure and DOS of SiH4 in structure T3 at two different pressures close to metallization onset (VASP
results).
Band Structure
DOS
0
-5
Energy (eV)
En$rgy ($V)
-10
-15
-20
-25
44% vol
Structure T3 at 31 GPa
-30
A
Z
G
X
R
A
G 0.0
0
Band Structure
0.1
0.2
DOS
0
-5
Energy (eV)
En$rgy ($V)
-10
-15
-20
-25
28% vol
Structure T3 at 144 GPa
-30
A
Z
G
X
R
A
A: (0.5, 0.5, 0.5); X: (0.0, 0.5, 0.0); R: (0.0, 0.5, 0.5); Z: (0.0, 0.0, 0.5)
17
G 0.0
0
0.1
0.2
A12. Band structure and DOS of SiH4 in structure O3 (VASP and CASTEP calculations) at different pressures
and the charge density map of O3 at 91 and 202 GPa.
Band Structure
5
DOS
0
En$rgy ($V)
Energy (eV)
-5
-10
-15
-20
-25
-30
Structure O3 at 91 GPa
-35
T
T
G
G
X
X
R
R
Z
Z
G 0 0.0
Band Structure
5
0.1
0.2
0.1
DOS
DOS
0
En$rgy ($V)
Energy (eV)
-5
-10
-15
-20
-25
-30
Structure O3 at 150 GPa
-35
T
T
G
G
X
X
R
R
Z
Z
T: (0.0, 0.5, 0.5); X: (0.0, 0.5, 0.0); R: (0.5, 0.5, 0.5); Z: (0.0, 0.0, 0.5).
18
G 0 0.0
1.0
2.0
0.1
Density of states (per eV per unit cell)
Density of states (per eV per unit cell)
The electronic band structures and corresponding densities of states (per two formula units) generated by CASTEP
for silane in structure O3 at 42% vol. (91 GPa, top) and at 33% vol. (202 GPa, bottom). Note the resemblance of the
density of states below the Fermi level to that of a free-electron band (the dotted black line).The s- and p- projections
of the density of states are also included. F: (0.0, 0.5, 0.0); Q: (0.0, 0.5, 0.5); Z: (0.0, 0.0, 0.5). CASTEP
automatically sets the energy of Fermi level to zero.
19
A
B
The electronic density (green) near the Fermi level (EF ± 0.024 eV) for the metallic form of silane (structure O3);
density isovalue = 0.11 e Å–3. Si – yellow balls, H – white balls. (a) 42% vol. (91 GPa), (b) 33% vol. (202 GPa).
Quasi–2D electronic conductivity of SiH4 in this structure is anticipated, even at pressures exceeding 200 GPa.
20
A13. The calculated rS parameter [bohr] vs. pressure for selected crystal structures of SiH 4 (VASP results).
3
a
T1
2.5
c
rs(ao)
T3
M3
l
M4
m
2
T1
1.5
1
0
50
100
150
200
250
300
350
Pressure (GPa)
A14. Derivation of the rS value for SiH4, based on the Goldhammer-Hertzfeld criterion.
The Goldhammer-Herzfeld criterion says that a material becomes metallic when the quantity (1  fVm )
1
diverges, where α is the molecular polarizability, Vm the volume per molecule in the solid and f a dimensionless
factor determined only by the packing of the molecules in the crystal. The divergence occurs when fα/Vm = 1. For
cubic systems f is 4π/3, and this gives:
Vm  4 / 3 . Using the definition of rs, 4 rs3 / 3  Vm / N ve (where Nve
is number of valence electrons), we then have the following equation for cubic systems,
= 36.76 bohr3,2A one readily gets rS = 1.663 Bohr.
21
rs   / N ve  . Using α
1/ 3
A15. Electron density at the Fermi level (EF ±0.02 eV) for SiH4 in structures O2 and O1 in the metallic state
(CASTEP results).
Structure O2 50% vol. (56 GPa), density isovalue = 0.11 e Å–3
Structure O2 40% vol. (116 GPa), density isovalue = 0.11 e Å–3
2D electronic conductivity of SiH4 in the SnF4 structure is anticipated even at pressures approaching 120 GPa.
Structure O1 70% vol. (9 GPa), density isovalue = 0.11 e Å–3
22
Structure O1 42% vol. (20 GPa), density isovalue = 0.28 e Å–3 (note that the unit cell has been rotated with respect
to upper picture, to get yet another perspective)
1D electronic conductivity of SiH4 in the 1D chain structure is anticipated at pressure of 20 GPa; interestingly, in this
pressure regime silane seems to be a “1s(H) orbital–based” metal, with no Si orbitals participating in the states at the
Fermi level.
23
A16. Phonon spectra (at  point) for SiH4 in structures O2, O1, O3 and T3 at their metallization onsets (VASP
results).
Structure O2.
SiH4 in the SnF4 structure (O2) belongs to space group I4/mmm, with two formula units (2 Si and 8 H atoms)
per unit cell. The isomorphous point group is D4h. Both Si atoms occupy sites of 4/mmm symmetry (Wyckoff a), four
H atoms (in SiH2 planes) of mmm symmetry (c), and four H atoms (apical) of 4mm symmetry (e). The 30 degrees of
freedom transform as total = 2A1g + 2Eg + 6A2u + 2B2u + 8Eu, thus giving rise to 6 Raman active (A1g and Eg), 22 IR
active (A2u and Eu) and 2 silent (B2u) modes. There are also three translational lattice modes (A2u + Eu).
The calculated phonon modes at  (at 56 GPa, 52% volume) are listed below.*
 /cm–1
Symm.
Description
 /cm–1
Symm.
Description
2430
A2u
as stretch Si–Hterm
884
Eu
SiH4 deformation
out of phase
2353
A2u
in phase
as stretch Si–Hterm
879
A2u
in phase
2271
Eu
out of phase
as stretch ip Si–Hbridg
822
Eg
out of phase
2258
Eu
A1g
as stretch ip Si–Hbridg
797
B2u
A1g
sym stretch Si–Hterm
735
A2u
Eg
oop Si–Hbridg deformation
in phase
sym stretch Si–Hterm
386
A2u
in phase
956
oop Si–Hbridg deformation
in phase
out of phase
2186
Si–H deformation
in phase
in phase
2254
oop Si–Hbridg deformation
Si(Hterm)2 deformation
out of phase
sym Si–Hterm bending
341
Eu
SiH4 deformation
out of phase
951
Eu
Si–H deformation
136
Eu
ip SiH4 deformation
out of phase
922
B2u
oop Si–Hbridg deformation
82
in phase
Eu
ip SiH4 deformation
out of phase
* “in phase” and “out of phase” with respect to the coupling of nuclear motions between two different SiH4 subunits
within the unit cell; oop means “out of plane”, ip stands for “in plane”, and plane is an ab one, where a and b are unit
cell vectors of a tetragonal cell.
The phonon spectrum is composed of three major regions: eight Si–H stretching modes (2A2u + 2A1g + 2Eu) in
the high frequency range (2186–2430 cm–1)3A, eight Si–H deformation modes (2A2u + 2B2u +2Eu + 2Eg) in the
intermediate range (735–956 cm–1), and of seven (A2u + 3Eu) low frequency lattice deformation modes (82–386
cm–1). The first two families of modes are dominated by motions of light H atoms, the latter ones typically show
large amplitudes of heavier Si atoms. The ratio of wavenumbers for the stretching Si–Hterm and Si–Hbridg modes
24
(2430/2271 = 1.07) shows that apical interactions are still stronger at this pressure than the in-plane interactions, and
the structure is still quasi–2D. A mutual exclusion principle applies for Raman and IR active modes.
The computed phonon spectrum at  is presented in the figure below.
Eu
Eu
Eu
A 2u
A 2u
A 2u
Eu
Eu
A 2u
E u A 2u
IR
B 2u
0
500
B 2u
1000
1500
2500 ν (cm-1)
2000
RAMAN
A 1g
Eg Eg
A 1g
Structure O1.
SiH4 in structure (O1) belongs to space group Cmmm; the primitive cell (for which phonon modes were done)
contains one formula unit (1 Si and 4 H atoms) per unit cell. The isomorphous point group is D2h. The 15 degrees of
freedom transform as vib = 2 Ag + 1 B1g + 1 B2g + 2 B3g + 2 B1u + 2 B2u + 2 B3u, along 3 translational modes (B1u +
B2u + B3u), thus giving rise to 6 Raman active (2 Ag + 1 B1g + 1 B2g + 2 B3g), 6 IR active (2 B1u + 2 B2u + 2 B3u) modes;
there are no silent modes. The mutual exclusion principle applies. There are 6 Si–H stretching modes (3 Raman and 3
IR active); the remaining are deformation modes. We have adopted the following notation: chain propagation C2 axis
= 1, C2 axis containing Si atom and two Hterm atoms = 2, the remaining C2 axis, perpendicular to the previous two =
3.
25
The calculated phonon modes at  (at 108 GPa, 42% volume) are listed below.
 /cm–1
E /meV
Description
 /cm–1
E /meV
Description
2499
310
stretch Si–Hterm
1426
177
Si–Hbridg deformation + Si–Hterm
out of phase (B2u)
2414
299
stretch (Ag)
stretch Si–Hterm
981
122
Si–Hterm deformation
in phase (Ag)
(B3g)
2021
251
stretch ip Si–Hbridg (B2g)
916
114
Si–H deformation (B3u)
1967
244
stretch ip Si–Hbridg (B1u)
821
102
Si–H deformation (B3u)
1963
243
stretch ip Si–Hbridg (B1g)
682
85
Si–H deformation (B3g)
1900
236
stretch ip Si–Hbridg + Si–Hterm
662
82
Si–H deformation (B1u)
(B2u)
The phonon spectrum is composed of four major regions: two Si–Hterm stretching modes (B2u + Ag) in the high
frequency range (2414–2499 cm–1)4A, four Si–Hbridg stretching modes (B1g + B2g + B1u + B2u) in the moderate
frequency range (1900–2021 cm–1), one Si–Hbridg deformation deformation mode with some contribution from
Si–Hterm stretching (Ag, 1426 cm–1) and five Si–H and lattice deformation modes (2B3g + 2B3u +B1u) in the low
frequency range (662–981 cm–1). The ratio of wavenumbers for the stretching Si–Hterm and Si–Hbridg modes
(2499/2021 = 1.24) points to the strong electronic one-dimensionality of this structure, despite high pressure.
The phonon spectrum at  is presented in the figure below.
B 1u
B 3u
B 3u
B 2u
B 1u
B 2u
IR
0
500
1000
1500
2000
-1
2500 ν (cm )
RAMAN
B 3g
B 3g
Ag
B 2g
B 1g
26
Ag
Structure O3.
SiH4 in structure (O3) belongs to space group Pman; the unit cell contains two formula units per unit cell. The
isomorphous point group is C2h. The 30 degrees of freedom transform as vib = 7 Ag + 5 Bg + 6 Au + 9 Bu, along 3
translational modes (Au + 2 Bu), thus giving rise to 12 Raman active (7 Ag + 5 Bg), and 15 IR active (6 Au + 9 Bu)
modes. There are no silent modes and the mutual exclusion principle applies. Symmetry is low, and not really very
helpful in detailed analysis of the spectra. The calculated phonon modes at  (at 150 GPa, 42% volume), are shown
below.
 /cm–1
E /meV
Sym.
 /cm–1
E /meV
Sym.
 /cm–1
E /meV
Sym.
2538
315
Ag
1819
226
Bu
1092
135
Ag
2518
312
Bu
1789
222
Bg
944
117
Bu
2432
301
Bu
1642
204
Ag
933
116
Bg
2394
297
Ag
1420
176
Ag
521
65
Bu
2363
293
Bu
1379
171
Au
496
61
Au
2313
287
Bg
1344
167
Au
394
49
Bu
2307
286
Au
1288
160
Bg
347
43
Ag
2296
285
Ag
1285
159
Bu
317
39
Au
1867
232
Au
1127
140
Bg
19
2
Bu
The phonon spectrum is composed of three major regions, as for structure O2: eight Si–H stretching modes in
the high frequency range (2296–2538 cm–1), thirteen Si–H deformation modes in the intermediate range (933–1868
cm–1), and of four low frequency lattice deformation modes (19–521 cm–1). There are also three translational lattice
modes (A2u + Eu). The stretching Si–Hterm and Si–Hbridg modes are strongly mixed with one another, due to tilting of
the SiH6 octahedra, which results in puckering of the [SiH2] planes; the apical bonds are not much stronger at this
pressure than the ‘in-plane’ bonds, and the structure is nearly a 3D one. The phonon spectrum at  is presented in the
figure below.
IR
Au
Bu
Au Bu Bu
0
500
Ag
Bu
1000
Bu
Au
Au
Bu Au
1500
Bg AgBg Bg Ag
Ag Bg
RAMAN
27
2000
Au Bu
Bu Bu
2500
Ag Bg Ag Ag
ν (cm-1)
Structure T3
This structure belongs to space group I 4 2m and the point group is D2d. The 60 degrees of freedom of motion of
the 20 atoms yield: vib = 9 A1 + 4 A2 + 4 B1 + 11 B2 + 16 E. Of these three are translational (1 B2 + 1 E), and we have
total of 57 vibrational modes. In the tables that follow, we list the phonon modes of T3 calculated at 31 (metallization
onset) and 144 GPa. Of these possible modes, A1, B1 are only Raman active, B2 and E modes are both IR- and
Raman-active, and A2 modes are silent.
Phonon modes of T3 at 31 GPa:
 /cm–1
E /meV
Sym.
 /cm–1
E /meV
Sym.
 /cm–1
E /meV
Sym.
2425
301
A1
1073
133
E
695
86
E
2405
298
E
993
123
B1
693
86
E
2403
298
E
992
123
B1
691
86
B2
2373
294
E
965
120
E
676
84
A2
2356
292
A1
952
118
A1
661
83
E
2354
292
A1
938
116
E
511
63
B2
2332
289
E
933
116
B1
488
61
A1
2326
288
B2
924
115
A1
454
56
B2
2318
287
E
856
106
B2
356
44
B2
2271
282
A1
850
105
B2
354
44
A2
2269
281
B2
764
95
E
352
44
A1
2269
281
A1
715
89
A2
350
43
E
2256
280
B2
715
89
A2
290
36
E
1157
143
B1
705
87
E
281
35
E
Phonon modes of T3 at 144 GPa
 /cm–1
E /meV
Sym.
 /cm–1
E /meV
Sym.
 /cm–1
E /meV
Sym.
2778
345
A1
1699
211
A1
825
102
B1
2676
331
B2
1698
211
A2
816
101
A2
2676
331
B2
1610
200
E
779
97
E
2608
323
A1
1579
196
B2
726
90
E
2608
323
A1
1556
193
E
678
84
A2
2607
323
B2
1526
189
E
678
84
A2
2583
320
E
1419
176
E
613
76
E
2543
315
E
1408
175
B1
450
56
E
2452
304
E
1309
162
A1
368
46
E
2405
298
A1
1309
162
B2
295
37
B2
2291
284
E
1127
140
E
295
37
B2
1817
225
B2
1012
125
E
228
28
B1
1771
220
A1
933
116
B2
228
28
B1
1754
217
A1
844
105
A2
228
28
B2
28
S17. Molecular orbitals of the uniaxially compressed SiH62– unit (extended Hückel theory).
A1g *ss
A2u *ps
Eu *ps
-9.2 eV
A1g n
B1g n
-13.6 eV
Eu ps
A2u ps
-17.3 eV
A1g ss
29
x6
A18. Estimation of the TC values for SiH4 in structure O1, O2 and O3 (VASP results): the BCS picture.
Estimates of TC values can be obtained from the BCS approximate formula (see page 744 Ref [20] in the paper):
TC = 1.13 D exp [–1/ (NF V)],
(A18-1)
where D is Debye temperature, NF is the density of states at the Fermi level per volume, and V is the effective
electron phonon coupling parameter. D has been calculated using the highest energy phonon mode at , as:
 D  D / kB , where D is the Debye cut-off frequency (kB is the Boltzmann constant).
From the BCS formula we have
Tc  1.13D exp(1/ N FoV o ) 
N Fo V o / N FV
,
where we denote a reference material with superscript “o”. Upon rearrangement we have an equation relating the Tc’s
to the ratio of NFV values,

Tc  1.13D Tc o /1.13 Do

N Fo V o / N FV
.
(A18-2)
As it is difficult to assess the actual value of the coupling constant V in our silane system, we propose to assume that
V of silane is similar to that of Pb, the reference material of our choice. Then, we can estimate the Tc of silane from
the equation below:

Tc  1.13D Tc o /1.13Do

N Fo / N F
.
(A18-3)
A note on the units is worthwhile here. The NF that enters the BCS formula has the dimension of [energy]-1·L-3,
and the effective coupling constant V has that of [energy]·L3. What the computational software (VASP) delivers is the
density of states per unit cell, denoted as NF(prog). We calculate NF by the following equation:
N F  N F (prog) /  uc ,
where uc is the unit cell volume.
The table below shows calculated values of density of states at the Fermi level (NF and NF ), cut-off
wavenumber of the phonon spectrum (cut–off), and Debye temperature (D) for SiH4 for four different structures (O2,
O1, O3 and T3), for pressures higher than the metallization pressures for every structure listed; these values are
compared with those for Pb in its experimental structure at 1 atm. Equation (S18-3) yields TC of ca. 0–258 K for
compressed silane in any of these important structures.
30
P
(GPa)
NF
–1 –3
(eV Å )
NF
–1
(eV per electron)
cut–off
D
Structure
Vcell
(Å3)
–1
(K)
Tc
(K)
O2
48.02
44.5
7.0x10-3
2.1x10-2
2430
3497
7.9
-2
-2
2769
3985
258.4
O2
36.94
115.4
1.5x10
3.5x10
O1
27.73
20.0
5.8x10-3
2.0x10-2
2499
3596
2.2
108.1
1.1x10-2
2.6x10-2
2736
3937
92.9
-7
-6
2435
3504
0.0
O1
18.24
O3
38.41
90.7
8. 6x10
O3
32.9
149.6
9.2x10-3
1.9x10-2
2538
3652
36.8
202.2
-2
2.5x10
-2
2588
3724
165.5
2.1x10
-2
2778
3997
61.5
O3
T3
Pb
*
/cm
29.77
66.69
121.29
144.2
*
1 atm
1.3x10
-2
1.0x10
-2 †
1.6x10
2.1x10
0.13
†
N.D.
‡
89
*
7.2 *
Experimental value. † Calculated value. ‡ N.D. = Not Determined.
A19. Calculating pressures from VASP data.
We have found that taking the derivative of the internal energy (U) with respect to volume (by interpolating between
discrete (U, V) points using a 16th order polynomial) yielded nearly the same values of pressure as the stress values
computed directly by VASP. The entropy term is absent at 0 K.
S20. References and notes.
1A. Larson, A. C., Cromer, D. T., Roof, R. B. Crystal Structure of UF 4. Acta Cryst. 17, 555–& (1964).
2A. CRC Handbook of Chemistry and Physics, Lide, D. R., Editor, CRC Press, New York (2003), page 10-168
(section ‘Average electric polarizabilities’).
3A. There are eight Si–H bonds in the unit cell (2 x [2 + 4/2]), so there are eight Si–H stretching modes.
4A. There are six Si–H bonds in the unit cell (2 + 4]), so there are six Si–H stretching modes.
31