# Superconducting Incommensurate Host-Guest Phases in Compressed Elemental
Sulfur

Jack Whaley-Baldwin<EMAIL_ADDRESS>TCM Group, Cavendish Laboratory,
University of Cambridge  Michael Hutcheon  TCM Group, Cavendish Laboratory,
University of Cambridge  Chris J. Pickard  Advanced Institute for Materials
Research, Tohoku University, Sendai, Japan

###### Abstract

We use Density Functional Theory (DFT) and structure searching methods to show
that the ground state of elemental sulfur is a Barium-IVa type incommensurate
host-guest (HG) phase between $386$-$679$ GPa, becoming the first group-VI
element predicted to possess such a structure. Within the HG phase, sulfur
undergoes a series of transitions in which adjacent guest chains are not
aligned but rather offset by different amounts, which can be described as a
rearrangement of the stacking order of the chains. We show that these chain
rearrangements are intimately coupled to modulations of the host and guest
atoms which prove crucial to stabilising the HG structure. Unlike the high-
pressure HG phases of other elements, sulfur does not exhibit interstitial
charge localisation, and instead features strongly localised ‘voids’ that are
depleted of electronic charge. Prior to adopting the HG structure, we predict
that sulfur possesses an orthorhombic structure of $Fdd2$ symmetry. We
calculate the superconducting critical temperatures of these newly discovered
phases, and show that $T_{c}$ is expected to peak between $24.8$ and $28.2$ K
at $271$ GPa.

##  I Introduction

Sulfur (S) is an element renowned for its structural diversity, possessing a
considerable array of metastable allotropes at ambient pressure alone Meyer
(1964, 1976). This complexity persists at high pressures, where sulfur adopts
elaborate catenated spiral structures above $1.5$ GPa Degtyareva  _et al._
(2005a) and an incommensurately modulated structure at $83$ GPa Degtyareva
_et al._ (2005b); Luo  _et al._ (1993), which is driven by a charge density
wave (CDW) instability.

Self-hosting incommensurate host-guest (HG) structures, which comprise two
distinct elemental sublattices embedded within one another in such a way that
the ratio of their lattice lengths along a certain crystal axis is an
irrational number, are well-known within the literature Nelmes  _et al._
(1999); Woolman  _et al._ (2018); Schwarz  _et al._ (1999); Chris J. Pickard
(2010); McMahon  _et al._ (2000a); Brown  _et al._ (2018); Häussermann  _et
al._ (2002); McMahon  _et al._ (2000b); Tsuppayakorn-aek  _et al._ (2018a);
Degtyareva  _et al._ (2004); Arapan  _et al._ (2008). The reasoning behind the
stability of such host-guest phases is often explained through so-called Fermi
surface effects, sometimes referred to as the Jones mechanism Jones and
Tyndall (1934), in which lattice distortions introduce Fourier components into
the potential that lower the energy of Fermi level states.

The physical properties of these elemental host-guest structures are of great
interest and vary significantly between elements. Several examples are
superconducting, and often exhibit high critical temperatures owing to
favourable low-frequency quasi-acoustic phonon modes Brown  _et al._ (2018),
with a record $T_{c}$ of $29$ K among all elements (excluding hydrogen) being
calcium in just such a phase (Ca-VII at $210$ GPa Shimizu (2018); Sakata  _et
al._ (2011); Fujihisa  _et al._ (2013)). Conversely, other realisations of
elemental HG structures feature large pseudogaps in their electronic densities
of states at the Fermi level, and form poorly-conducting electrides with
significant amounts of electronic charge localised in interstitial regions
Woolman  _et al._ (2018); Chris J. Pickard (2010).

X-ray diffraction studies on S up to $165$ GPa Degtyareva  _et al._ (2005b);
Luo  _et al._ (1993) and $250$ GPa Gregoryanz  _et al._ (2002); Drozdov  _et
al._ (2015) have determined that S transforms from the incommensurately-
modulated CDW phase into the trigonal $\beta$-Po structure of $R\bar{3}m$
symmetry at ${\sim}150$ GPa. Several first-principles structure searching
investigations Gavryushkin  _et al._ (2017); Whaley-Baldwin and Needs (2020);
Whaley-Baldwin (2020) have predicted that the $R\bar{3}m$ structure remains
stable up to ${\sim}500$ GPa, where S then adopts a body-centered cubic (bcc)
structure.

The correct determination of the crystal structure of S at high pressures is
of interest to experimentalists, in order to identify its presence as a
decomposition product in high pressure hydrogen sulfide experiments on H2S and
H3S and to resolve the resulting diffraction pattern Goncharov  _et al._
(2016); Laniel  _et al._ (2020). Knowledge of the true ground state is also
important from a structure searching standpoint, so that the enthalpy of
candidate H2S / H3S structures can be measured against decomposition into H
and S Cui  _et al._ (2019); Li  _et al._ (2016).

The high-pressure superconducting properties of pure S have long attracted
attention, and at one time it held the highest experimentally verified $T_{c}$
in an element Struzhkin  _et al._ (1997). Further, the fact that the
incommensurately modulated phase of S is superconducting with such a high
$T_{c}$ is unusual, as the Fermi-level gaps opened up by CDW formation often
work to suppress superconductivity Degtyareva  _et al._ (2007). As well as H2S
and H3S, sulfur is a key constituent of several other high-$T_{c}$ compounds,
including a recently claimed room-temperature superconductor Snider  _et al._
(2020).

In this work, we investigate the phase diagram of S up to $700$ GPa and report
that under sufficient compression, S adopts a superconducting incommensurate
HG structure similar to that of Barium in the IVa phase Nelmes  _et al._
(1999), in addition to a previously unreported orthorhombic phase. We show
that the guest structure in S undergoes several changes with increasing
pressure, and link this to modulations of the host and guest atoms. We further
study the electronic structure and charge density distribution of the HG
phase, and compare the latter to other high-pressure HG structures.

##  II Computational Details

We used the ab-initio Random Structure Searching (AIRSS) package Pickard and
Needs (2006, 2011) for our structure searching, with the plane-wave DFT code
`CASTEP` Clark  _et al._ (2005) performing the underlying electronic structure
calculations. We generated a sulfur pseudopotential specifically designed for
high-pressure work using the OTFG code in `CASTEP` (see Supplemental Material
sup ). Due to small enthalpy differences in certain parts of the phase
diagram, we used an $850$ eV plane-wave cutoff and a k-point spacing of
$2\pi\times 0.01$ Å-1, sufficient for absolute enthalpy convergence to $\pm
0.2$ meV. Relative enthalpies between structures were converged to an even
tighter tolerance. Geometry optimisations were carried out using the L-BFGS
algorithm and structures were relaxed until the forces on each atom were
$5\times 10^{-4}$ eV / Å-1 or smaller, and the residual stresses $1\times
10^{-2}$ GPa or smaller.

We performed structure searches at $300$, $400$, $500$ and $750$ GPa, using
randomly generated structures containing between $1$ and $15$ atoms and
possessing between $1$ and $48$ symmetry operations. We then performed
searches at these same pressures with $16$, $21$, $24$ and $32$ atoms.
Finally, we performed much smaller searches in peturbed supercells of our
various host-guest approximants, which involved up to $288$ atoms. In total,
we considered around $7500$ fully relaxed structures.

##  III Results and Discussion

###  III.1 Relative Enthalpies

Figure 1: Top: Static-lattice phase diagram of S calculated with the PBE
functional. ‘HG’ denotes the host-guest phases, and for clarity, they have
been plotted as a single curve. The inset shows the small enthalpy differences
between the $Fdd2$ structure and HG phase. The simple-cubic $Pm\bar{3}m$
structure considered in previous studies Whaley-Baldwin and Needs (2020);
Gavryushkin  _et al._ (2017); Whaley-Baldwin (2020); Rudin and Liu (1999);
Rudin  _et al._ (2001) is shown for comparison. Bottom: Phase diagrams with
vibrational effects included (at the level of the harmonic approximation) in
the vicinity of the $R\bar{3}m\rightarrow Fdd2$ transition (left) and HG
$\rightarrow Im\bar{3}m$ transition (right). The black dashed line on the left
diagram marks the boundary of the second order $Fdd2\rightarrow Fddd$ phase
transition.

(a) Looking down the pseudo-guest chain axes. From top-to-bottom, the pseudo-
guest chains are arranged in an AB stacking.

(b) The same structure, but with the two different types of pseudo-guest
chains artificially coloured red and blue, shown looking down the axis of the
red guests.

(c) The same as the image on the immediate left, but rotated $45\degree$
anticlockwise around an axis running top-to-bottom of the page (i.e.
perpendicular to the chains). The result is an equivalent structure.

Figure 2: The commensurate $Fdd2$ structure, viewed down the ‘pseudo-guest’
chains. The relation of this view to the lattice vectors of the conventional
(64-atom) orthorhombic cell is shown in Fig. 3.

Figure 3: The $Fdd2$ structure, in the same colour scheme as Fig. 2. Two
example chains (like those shown in Fig. 2) have been highlighted with red and
blue lines (top figure only). The conventional (64-atom) orthorhombic unit
cell is indicated in black.

Fig. 1 shows our phase diagram of S calculated using the PBE functional. In
addition to PBE, table 1 shows the (static-lattice) transition pressures
according to the LDA and PBEsol; the order of phase transitions is unaffected
by the choice of functional. Using PBE, we determine that at $266$ GPa, S
transitions from the $R\bar{3}m$ ($\beta$-Po) phase to a structure of $Fdd2$
symmetry with $16$ ($64$) atoms in its primitive (conventional) cell, which
visually resembles a strongly distorted version of the Ba-IVa HG phase.

Functional |  $R\bar{3}m\rightarrow$ $Fdd2$ |  $Fdd2\rightarrow$ HG | HG $\rightarrow Im\bar{3}m$
---|---|---|---
LDA | $249$ | $378$ | $700$
PBE | $266$ | $386$ | $679$
PBEsol | $245$ | $366$ | $644$
Table 1: Static-lattice transition pressures in GPa according to LDA, PBE and
PBEsol.

This $Fdd2$ structure is detailed in Figs. 2 and 3. Whilst it clearly
resembles the Ba-IVa HG structure, it has the property of being self-similar
when rotated $45\degree$ about an axis perpendicular to to the pseudo-guest
chains. We use the term pseudo-guest, as this property implies that there is
no longer a unique chain axis, and therefore it cannot be equivalent to the
‘standard’ Ba-IVa type HG structure (although, as we will show, S does
eventually adopt the Ba-IVa HG structure at higher pressures). The equivalence
through a $45\degree$ rotation means that there are two nonintersecting
pseudo-guest chains, running in different directions in offset planes, as
detailed in Fig. 3. Each pseudo-guest chain forms part of the host structure
when looking down the other chain’s axis (and vice versa), and thus it is no
longer possible to decompose the structure into two distinct host and guest
lattices (see Figs. 2(b) and 2(c)). Whilst the pseudo-guest chains in the same
layer are in alignment, those in the layer below are shifted by half of the
intra-chain atomic separation, thus giving the chains an AB ordering when
viewed from the side.

We tested the stability of this structure for a few other elements that
possess a high-pressure HG phase (such as Al and As), but in each case it was
never the lowest-enthalpy phase. The ‘two-chain’ motif is reminiscent of the
structure of the ternary incommensurate compound Hg3-δAsF6 Axe and Bak (1982);
Buiting  _et al._ (1984); de Groot  _et al._ (1985), in which chains of
mercury atoms, running at $90\degree$ to one another, sit within an AsF6
framework. However, unlike this compound, and the incommensurate Ba-IVa phase,
our calculations suggest that the $Fdd2$ structure is in fact a commensurate
crystal and is not quasi-periodic. To demonstrate that this is the case, we
first show that at higher pressures, S does exhibit a truly incommensurate Ba-
IVa type HG structure, and then use these findings to show that the $Fdd2$
phase cannot be incommensurate.

The Ba-IVa HG structure, which occurs (with minor variations) in multiple
elements across the periodic table Nelmes  _et al._ (1999); Woolman  _et al._
(2018); Schwarz  _et al._ (1999); Chris J. Pickard (2010); McMahon  _et al._
(2000a); Brown  _et al._ (2018); Häussermann  _et al._ (2002); McMahon  _et
al._ (2000b); Tsuppayakorn-aek  _et al._ (2018a); Degtyareva  _et al._ (2004);
Arapan  _et al._ (2008), comprises a tetragonal ‘host’ structure of $I4/mcm$
symmetry with $8$ atoms in the (conventional) unit cell, and one of two
possible coexisting ‘guest’ structures; either a $C$-face centered tetragonal
structure, or an $M$-face centered monoclinic structure, with atoms at
$(0,0,0)$ and $(1/2,1/2,0)$ in both cases. The host and guest structures are
incommensurate along their $c$-axes with an ideal ratio $\gamma_{0}$ that is
remarkably close to $4/3$. In some cases, such as Bi-III and Sb-II McMahon
_et al._ (2007); Kartoon and Makov (2019) or Ca-VII Fujihisa  _et al._ (2013),
the positions of both the host and guest atoms in the HG structure are
distorted/modulated relative to their ‘ideal’ HG positions, giving both the
channels and chains a ‘wiggly’ or worm-like appearance.

We find that above $386$ GPa, a Ba-IVa type HG approximant of $C2/c$ symmetry
($\gamma=4/3$), with $16$ (32) atoms in its primitive (conventional) cell,
becomes lower in enthalpy than the $Fdd2$ structure. Similar to the $Fdd2$
structure, the atoms in the $C2/c$ approximant are distorted from their ideal
positions; however, a unique chain axis for the guest atoms now exists, and
thus the $C2/c$ approximant is of the ‘standard’ Ba-IVa type. Below $386$ GPa,
the $C2/c$ approximant is dynamically unstable, and it develops an imaginary
optical $\Gamma$-point phonon mode that leads to the $Fdd2$ structure. The
structural and enthalpic differences between the $C2/c$ approximant and the
$Fdd2$ structure are small, and over the stability range of the $Fdd2$, they
differ in enthalpy by at most $1.5$ meV per atom (see inset in Fig. 1).

In order to calculate the ideal incommensurate host-guest ratio $\gamma_{0}$
at which the enthalpy is a minimum, we fitted the enthalpy of various
commensurate approximants to a quadratic curve (see Supplemental Fig. 3 sup ),
from which the ideal incommensurate ratio can be read off as the minimum. Fig.
4 shows that $\gamma_{0}$ slowly decreases monotonically with increasing
pressure, and that the choice of exchange-correlation functional does not
affect the results significantly. This slow monotonic decrease in $\gamma_{0}$
with pressure is also seen in the HG phase of sodium Woolman  _et al._ (2018),
but is opposite to the cases of aluminium Chris J. Pickard (2010) and bismuth
Kartoon and Makov (2019). In potassium and rubidium $\gamma_{0}$ initially
decreases with pressure, before this behaviour reverses and $\gamma_{0}$
instead increases with pressure Woolman  _et al._ (2018).

Figure 4: Dependence of the ideal incommensurate ratio with pressure. Data was
taken in $100$ GPa intervals and fitted to a cubic spline.

The procedure used to calculate $\gamma_{0}(p)$ allows us to propose that the
$Fdd2$ structure is not a commensurate approximant to an incommensurate phase,
despite its visual similarity to the Ba-IVa structure; of all the approximants
used to construct our $H(\gamma)$ quadratic fits ($9$ in total), only the
$16$-atom approximant collapsed into the two-chain pattern at low pressures;
approximants with other $\gamma$ values simply remained in the ‘single chain’
Ba-IVa type structure. This suggests that the $Fdd2$ structure does not belong
to a family of HG approximants, but rather is a $16$-atom structure in its own
right. Additionally, we re-emphasise the fact that since one cannot decompose
the $Fdd2$ into a unique host and a unique guest structure, it is not possible
to even define a $\gamma$ value for the $Fdd2$.

Relative to the more densely packed bcc ($Im\bar{3}m$) and trigonal
($R\bar{3}m$) phases, the HG structures are stabilised by a relatively large
reduction in the total electronic energy. At 500 GPa, despite being
${\sim}0.719\%$ larger in volume, the HG phase has a total electronic energy
$0.190$ eV per atom lower than the bcc structure, which is more than enough to
offset the increased $pV$ term and make it a favourable structure. The idea
that S becomes more open-packed at these pressures had previously been
considered by Rudin et. al. Rudin and Liu (1999); Rudin  _et al._ (2001) who
proposed an $R\bar{3}m\rightarrow$ simple-cubic ($Pm\bar{3}m$) transition, but
neither that work, nor two previous high-pressure structure searching studies
Gavryushkin  _et al._ (2017); Whaley-Baldwin and Needs (2020), located the
$Fdd2$ and HG phases. We further note the significance of our findings on the
construction of convex hulls for sulfur compounds (e.g. H2S and H3S) at these
pressures; the HG phase of S is lower in enthalpy than the previously-assumed
ground state ($R\bar{3}m$) by 11 meV per atom at 300 GPa, and up to 57 meV at
500 GPa.

The inclusion of nuclear vibrational effects does not change the phase
transition sequence, and at low temperatures, vibrational Zero-Point Energies
(ZPEs) cause only small deviations from the static-lattice pressure boundaries
(see Fig. 1). For example, at the $R\bar{3}m\rightarrow Fdd2$ transition, we
find the ZPE of the $R\bar{3}m$ phase to be $1.4$ meV per atom higher than
that of the $Fdd2$ structure, which lowers the transition pressure by
${\sim}4$ GPa from the static-lattice value in Table 1. The ZPEs of the $Fdd2$
structure and the $C2/c$ approximant are identical to within our convergence
limits. At the HG $\rightarrow Im\bar{3}m$ transition, we find the HG ZPE to
be higher than that of the $Im\bar{3}m$ phase by $4.8$ meV per atom, and the
transition pressure reduces by ${\sim}22$ GPa compared to the value in Table
1. At higher temperatures ($\gtrsim 300$ K), vibrational effects reduce the
transition pressures of both the $R\bar{3}m\rightarrow Fdd2$ and HG
$\rightarrow Im\bar{3}m$ transitions. The lowering of the transition pressure
for the $R\bar{3}m\rightarrow Fdd2$ transition is particularly interesting for
two reasons. Firstly, it provides an experimental route to access the $Fdd2$
structure at slightly lower pressures, somewhat reducing the difficulty of any
diamond anvil experiments. Secondly, it allows for observation of a new
structure of $Fddd$ symmetry, also with 16 (64) atoms in its primitive
(conventional) unit cell. This $Fddd$ structure is derived from the $Fdd2$
structure, which continuously develops an imaginary $\Gamma$-point phonon mode
below $215$ GPa, pushing it into the $Fddd$ symmetry - the resulting
transition between the phases is thus second order. We note that this $Fddd$
structure is distinct from the well-known ambient-pressure orthorhombic ground
state of sulfur (also of $Fddd$ symmetry) which comprises S8 rings. Our
calculations show that at temperatures above $\sim 1260$ K, the $Fddd$ phase
should become the ground state at $215$ GPa (again, see Fig. 1). We note that
a recent publication Arveson  _et al._ (2019) on the high-pressure melting
curve of S achieved temperatures of $\sim 1700$ K at $51$ GPa using a laser-
heated diamond anvil cell (LHDAC), and extrapolation of their data using a
Simons-Glatzel fit Simon and Glatzel (1929) suggests that the melting
temperature of S should be at least $\sim 3850$ K at $200$ GPa.

###  III.2 Chain Ordering

After establishing the $Fdd2$ structure and $C2/c$ approximant as competitive
phases, we conducted auxiliary searches with peturbed supercells of the $C2/c$
approximant containing $32$, $64$, $128$ and $288$ atoms. These searches
produced two new HG approximants of $P4/ncc$ symmetry ($32$ atoms) and $Pcca$
symmetry ($64$ atoms) with slightly lower enthalpies. These approximants have
the same commensurate ratio as the $C2/c$ approximant ($\gamma=4/3$) and
feature the same host structure, but possess different guest structures. The
bottom of Fig. 5 shows the relative enthalpies of these different host-guest
approximants. It can be seen that the $16$-atom $C2/c$ approximant is actually
favoured only in a narrow pressure range ($384$-$400$ GPa), before the
$64$-atom $Pcca$ approximant becomes favourable between $400$-$590$ GPa. The
$32$-atom $P4/ncc$ approximant is then the ground state between $590$-$679$
GPa, with the bcc $Im\bar{3}m$ phase becoming stable thereafter.

Each of the $C2/c$, $Pcca$ and $P4/ncc$ approximants represent a different
ordering of the guest atom chains, in which adjacent chains are displaced by a
different amount parallel to the direction of the chain axes (i.e. have
different heights in the $z$ direction). We found that this displacement was
always either $1/3$ or $2/3$ of the guest lattice $c$ axis length ($c_{G}$).
Fig. 5 shows the different orderings as viewed along, and perpendicular to,
the axes of the chains. In each case, the chains can clearly be described as
having been stacked in a certain order, with the $C2/c$, $Pcca$ and $P4/ncc$
approximants possessing a stacking order of ABC, ABAC and AB respectively.
Fig. 5 also shows that the aforementioned distortions/modulations of the host
and guest structures are slightly different in each case, being largest for
the $C2/c$ structure and smallest for the $P4/ncc$ structure. This suggests a
coupling between the chain orderings and the distortions; an observation that
we shall return to later.

Figure 5: Top: ABC $(0,\frac{1}{3},\frac{2}{3})$ stacking of the $16$-atom
$C2/c$ approximant. Middle: ABAC $(0,\frac{1}{3},0,\frac{2}{3})$ stacking of
the $64$-atom $Pcca$ approximant. Bottom: AB $(0,\frac{2}{3})$ stacking of the
$32$-atom $P4/ncc$ approximant. The relative stability of the stackings is
shown beneath the structures; the $Fdd2$ and bcc phases become stable
immediately off to the left and right of the plot, respectively.

The lowest-enthalpy stacking ordering at a given pressure is a trade off
between minimisation of the total electronic energy and the $pV$ term. The AB
ordering of the $P4/ncc$ approximant stacks most efficiently to give the
lowest $pV$ term at all pressures, but has the largest total electronic energy
of all approximants. This situation is reversed in the ABC stacking of the
$C2/c$ approximant, which has the lowest electronic energy but the largest
$pV$ term. The $Pcca$ approximant of ABAC stacking is a compromise between
these two motifs and explains why it is favoured over a comparatively large
pressure range. At lower pressures ($<400$ GPa), the $pV$ term is less
important than the electronic term, and the $C2/c$ approximant becomes the
ground state. At much larger pressures ($>590$ GPa), the $pV$ term becomes
dominant, and the $P4/ncc$ approximant is the lowest enthalpy phase.

Chain orderings that differ from the ‘ideal’ all-aligned case have been
discovered in other host-guest phases. The AB-ordered HG phase (our $P4/ncc$
approximant) that S adopts above $590$ GPa is also the ground state of the
group-V elements As, Sb and Bi at much lower pressures McMahon  _et al._
(2000a); Häussermann  _et al._ (2002). The Ca-VII phase of Calcium Fujihisa
_et al._ (2013) is well-described by a $128$-atom approximant in which the
guest chains have different $z$-heights in both the $a$ and $b$ directions (as
opposed to S, where the $z$-heights change only along the $a$ direction). In
the case of Ba-IVc, the structure adopted by Barium at pressures a few GPa
higher than the Ba-IVa phase, the chain displacements are arranged in such a
way as to produce an extremely complicated, mesoscopic-scale, interlocking
S-shape patterning, which is well-represented by a $768$-atom cell Loa  _et
al._ (2012).

Whilst the enthalpy differences between the ABC, ABAC and AB stackings are
mostly ${\sim}1$ meV per atom, we note that the ‘ideal’ stacking (of $I4/mcm$
symmetry), in which all guest chains are perfectly aligned (i.e., AAA…), is
higher in enthalpy at all pressures by at least $20$ meV per atom and up to
$45$ meV, indicating that perfect chain alignment is strongly disfavoured in
S. This raises the question of why any stacking that differs from the all-
aligned case has such a drastically reduced enthalpy. Surprisingly, we find
that explicitly offsetting the chains causes a only a very small enthalpy
reduction in itself. Instead, we show that the answer lies in the
distortions/modulations of the host and guest atoms, which cannot occur unless
the chains are first offset.

Fig. 6 details the situation for the ABC case at $387$ GPa, although the ABAC
and AB cases are analogous. Starting from the all-aligned $I4/mcm$
approximant, we manually offset the chains such as to have an ABC ordering,
keeping all else fixed. This gives rise to an enthalpy reduction of $4$ meV
per atom, which can be attributed to the opening of several directional band
gaps at the Fermi level (see Supplemental Fig. 5 sup ). We subsequently relax
the cell vectors (with the atomic positions fixed), which gives rise to a
further enhalpy reduction of $1$ meV. Finally, we add in the modulation of the
host and guest atoms, which further reduces the enthalpy by $34$ meV; an order
of magnitude larger than the reductions caused by the chain offsets and the
cell vector relaxation. The bottom of Fig. 6 shows that these energetic
reductions arise as a result of shifting eDOS weight to lower energies, whilst
little changes at the Fermi level itself.

Figure 6: Top: Schematic decomposing the $39$ meV enthalpy reduction that
occurs going from the all-aligned approximant to the ABC ordered approximant
at $387$ GPa. The enthalpy axis is logarithmic. Bottom: The corresponding
changes in the eDOS. The Fermi level is set at $0$ eV.

Since the modulations provide the greatest contribution by far to the total
enthalpy reduction, it is natural to ask why offsetting the guest chains - a
change that causes only a very small enthalpy reduction in itself - is
necessary to see these modulations appear. When the guest chains are offset,
the host atoms are exposed to an asymmetric environment which creates forces
that cause the host atoms to distort, which in turn distorts the guest atoms.
If the chains are all-aligned, then the environment on either side of the host
atoms is the same; symmetry implies that these forces do not arise, and there
are therefore no distortions. In the ABC and ABAC stackings, both the host and
guest atoms are distorted in all of the $x$, $y$ and $z$ directions (see Fig.
5). However, in the case of AB stacking, symmetry dictates that the guests are
displaced only in the $z$ direction. These $z$ distortions longitudinally
pinch together pairs of guest atoms along the chain axes and dimerize them
such that the covalent bonds alternate in a short-long-short-long pattern,
with the short bond ${\sim}3.2\%$ smaller than the long bond. This 1D
dimerization is referred to as quasi-pairing in the literature McMahon  _et
al._ (2007), and the same behaviour has recently been predicted to occur in
the CDW phase of sulfur at lower pressures Whaley-Baldwin and Needs (2020).
Detailed experimental McMahon  _et al._ (2007) and theoretical Kartoon and
Makov (2019) analyses of both the host and guest distortions in the Bi-III
structure confirm that the modulations play an indispensable role in the
stability of HG phases.

In addition to the structures found in our structure search, we calculated the
enthalpy of various complex structures that appear in the vicinity of HG
phases in other elements, such as oC52 in rubidium Nelmes  _et al._ (2002),
oC84 in caesium McMahon  _et al._ (2001), oP8 in potassium Lundegaard  _et
al._ (2009), and A7 in the group-V elements Häussermann  _et al._ (2002).
These structures were found to be at least $0.1$ eV per atom above the ground
state at all pressures considered here. We also constructed some $\gamma=4/3$
approximants with longer-period chain stackings such as ABAAC, ABCAB, ABCACB
and permutations thereof, as well as configurations where the chain heights
changed in the $b$\- direction also (see Supplemental Material for a table of
all the stackings that were tested sup ). Most of these structures collapsed
to the 64-atom $Pcca$ ABAC stacking, which seems to have a particularly large
basin of attraction in the configuration space of chain orderings. Those that
did not collapse to the ABAC stacking simply had a higher enthalpy at all
pressures, although we note that an ABCACB stacking was particularly close to
becoming the ground state ($+0.2$ meV above the ABAC stacking at $550$ GPa).
We further considered the effect of spin polarisation in our calculations, and
seeded our structures with varying ferromagnetic and anti-ferromagnetic
orderings, but in all cases, the density relaxed to a spin-unpolarised
configuration.

###  III.3 Charge Density

We compare charge density isosurfaces of HG sulfur in Fig. 7 at moderate
($35\%$ of maximum) and very small ($0.3\%$ of maximum) values, alongside
several 2D slices of the charge density taken perpendicular to the guest
chains. The charge density of S is peaked near atomic sites and along lines
connecting neighbouring atoms, which is representative of a covalently bonded
solid. S exhibits no significant interstitial localisation of charge and
therefore does not constitute a conventional electride, unlike the HG phases
of the alkali metals Woolman  _et al._ (2018) or aluminium Chris J. Pickard
(2010). However, S displays curious behaviour when studied at very small
isosurface values, which correspond to charge-depleted regions. The middle of
Fig. 7 shows that these charge-depleted regions form well localised lobes,
which we refer to as charge ‘voids’, and co-incide with minima in the Electron
Localisation Function (ELF) Becke and Edgecombe (1990). These voids form a
structure equivalent to the host lattice of the alkali metal HG phases Woolman
_et al._ (2018), with one host unit situated at the midpoint of every ‘short’
intra-chain covalent bond. The bottom of Fig. 7 shows the 2D charge density
taken at the midpoints of these ‘short’ bonds for the AB case; when these
slices are superimposed, the full set of voids (as seen in the middle of Fig.
7) is recovered.

Figure 7: Charge density in HG sulfur. Top: Isosurface at $35\%$ of maximum,
displaying clear covalent bonds. Middle: Isosurface at $0.3\%$ of maximum,
showing the charge-depleted ‘voids’. Bottom: 2D slices of the charge density
taken perpendicular to the guest chains for the AB ordering (guest atoms
highlighted). The ABC and ABAC stackings give similar charge distributions,
but the voids have different $z$ coordinates in each case (as they are tied to
the guest structure).

Remarkably, these voids are the exact locations of charge density maxima in
the HG phase of aluminium Chris J. Pickard (2010). The duality between the
charge densities of S and Al can be explained in terms of the intra-chain
bonding between the guest atoms, which is present in S but absent in Al. In S,
the short intra-chain covalent bonds contain significantly more charge than
the weaker long bonds; this excess charge is drawn from the charge voids, and
explains why there is precisely one set of voids per short bond.

We note that in S there is no opening of a pseudogap in the electronic Density
of States (eDOS) at the Fermi level (see Fig. 6 and Supplemental Fig. 6 sup ).
This is in contrast to the behaviour of high-pressure electrides Woolman  _et
al._ (2018); Chris J. Pickard (2010), in which there is often a dramatic
reduction in the eDOS at the Fermi level (in Al, for example, the eDOS reduces
by a factor of ${\sim}4$). Interestingly, we find that there is still
significant $s\rightarrow d$ charge transfer in S; for the HG phase at $500$
GPa, the Fermi-level eDOS comprises $11\%$ $s$ character, $42\%$ $p$ character
and $47\%$ $d$ character.

###  III.4 Superconductivity and Phason Modes

We have calculated the superconducting critical temperatures of our high-
pressure sulfur phases using Migdal-Eliashberg theory within Density
Functional Perturbation Theory (DFPT) using the `QUANTUM ESPRESSO` code et al.
(2009). We used the McMillan-Allen-Dynes formula Allen and Dynes (1975) to
calculate Tc:

$T_{c}=\frac{\omega_{{}_{\mathrm{log}}}}{1.2}\mathrm{exp}\bigg{(}\frac{-1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}\bigg{)}$
(1)

Where $\omega_{{}_{\mathrm{log}}}$ is the logarithmically averaged phonon
frequency, $\lambda$ is the electron-phonon coupling constant, and $\mu^{*}$
is the Coulomb pseudopotential parameter, rescaled from the bare value $\mu$
to account for retardation Morel and Anderson (1962):

$\mu^{*}=\frac{\mu}{1+\mu\
\mathrm{ln}\big{(}\frac{\epsilon_{F}}{\Theta_{D}}\big{)}}$ (2)

Where $\epsilon_{F}$ is the Fermi energy, and $\Theta_{D}$ is the phonon Debye
frequency. We note that a relatively recent publication Monni  _et al._ (2017)
has employed Superconducting DFT (SCDFT) Lüders  _et al._ (2005); Marques  _et
al._ (2005) to calculate the $T_{c}$ of the $R\bar{3}m$ phase at pressures up
to $200$ GPa, and shows that the value of $\mu$ for S in this regime remains
nearly constant at $0.20$. Rescaling this using Eqn. 2, we obtain a $\mu^{*}$
value of $0.09$ that is essentially constant across all the phases, since the
ratio $\epsilon_{F}/\Theta_{D}$ changes very little between the phases and
with increasing pressure. Further, a previous publication Zakharov and Cohen
(1995) derived a $\mu^{*}$ value of $0.08$ for the bcc phase at $500$ GPa
using the static Thomas-Fermi dielectric function. We therefore choose
$\mu^{*}\in[0.08,0.10]$ as a reasonable range of representative $\mu^{*}$
values.

Figure 8: Superconducting critical temperatures of the sulfur phases as a
function of pressure. Dotted vertical lines denote the structural phase
transitions discussed in this work. The shading corresponds to
$\mu^{*}\in[0.08,0.10]$; the grey region connects the lowest-enthalpy
structures at each pressure, and the purple region in the top left is a
possible metastable path if S can be trapped in the $Fdd2$ symmetry below
$270$ GPa. The data has been linearly interpolated.

Fig. 8 shows our calculated $T_{c}$ values as a function of pressure. It can
be seen that $T_{c}$ is expected to peak shortly after the transition to the
$Fdd2$ phase, where we predict $T_{c}$ to be $24.8$ \- $28.2$ K at $271$ GPa.
For a purely elemental solid, this is a high critical temperature and is close
to the record among the elements (excluding hydrogen) of $29$ K in high-
pressure Ca-VII Shimizu (2018); Sakata  _et al._ (2011); Fujihisa  _et al._
(2013). Phonon calculations show that although it is not the lowest enthalpy
phase at such pressures, the $Fdd2$ structure is dynamically stable down to
pressures as low as $215$ GPa (below here, it develops a soft mode that pushes
it into the $Fddd$ structure, as previously mentioned). Since we find the
$T_{c}$ of the $Fdd2$ phase to be nearly constant in this region, it is
possible that upon reducing pressure from $>270$ GPa, hysteresis effects may
allow $T_{c}$ to take a path similar to that indicated in purple in the top
left of Fig. 8. We note that whilst $T_{c}$ falls monotonically with pressure
after the transition to the $Fdd2$ phase, we predict that S will remain
superconducting up to at least $700$ GPa.

Whilst we used isotropic Migdal-Eliashberg theory to calculate $T_{c}$, we
note that the aforementioned SCDFT calculations in Ref. Monni  _et al._ (2017)
account fully for the Fermi surface anisotropy. Our $T_{c}$ results for the
$R\bar{3}m$ phase at $250$ GPa are slightly higher (by ${\sim}3$ K) than the
$T_{c}$ obtained from the use of SCDFT in Ref. Monni  _et al._ (2017),
although our Eliashberg function $\alpha^{2}F(\omega)$ is very similar (see
Fig. 10). Conversely, our $T_{c}$ value for the $R\bar{3}m$ structure is
actually slightly lower (by ${\sim}4$ K) than the experimental value obtained
by Drozdov et. al. Drozdov  _et al._ (2015).

Figure 9: Phonon dispersion relation for the AB-ordered $P4/ncc$ approximant
at $375$ GPa (commensurate ratio $\gamma=4/3$). The $\Gamma\rightarrow$ X
direction runs across the tetragonal face of the HG structure (perpendicular
to the guest chains). Two low-energy phason modes, which correspond to
relative movement of the host and guest lattices, have been highlighted in
green and red. In the limit of a truly incommensurate structure, the
frequencies of these modes vanish.

In the absence of complications such as the formation of finite-sized domains,
the energy of a truly incommensurate HG structure must necessarily be
invariant with respect to relative translation of the host and guest lattices.
Such a relative translation can be achieved with an optical phonon mode
running perpendicular to the guest chains, which displaces the host and guest
structures in opposite directions. The energy of such a phonon mode should be
exactly zero for a true quasi-crystal, although the use of commensurate
approximants in ab-initio studies such as this work means that they appear to
have a small positive frequency in a phonon calculation. Fig. 9 shows the
phonon dispersion relation for the AB-ordered $P4/ncc$ approximant, in which
two low-energy optical modes can be seen. These ‘phason’ modes naturally lead
to large values of the mode-specific electron-phonon coupling constant
$\lambda_{\mathrm{\textbf{q}},\nu}$, which varies with phonon frequency as
$\sim\omega^{-1}_{\mathrm{\textbf{q}},\nu}$, and give rise to strong-coupling
superconductivity in HG phases Brown  _et al._ (2018); however, as mentioned,
our use of a commensurate approximant forces the phason modes in Fig. 9 to
have a nonzero frequency. Consequently in Table 2, which summarises relevant
superconducting properties of interest for several of the structures, the
value of the electron-phonon coupling constant $\lambda$ for the HG phase is
not in the strong-coupling regime where $\lambda>>1$ Allen and Dynes (1975).
It is probable that the use of a larger approximant (with accompanying lower
phason frequencies) would give rise to a larger value of $\lambda$, although
such a calculation is beyond the scope of this work and would be
computationally expensive. We also note that solving the Eliashberg equations
(necessary for the true strong coupling case) gave very similar $T_{c}$ values
to those obtained using the McMillan-Allen-Dynes formula for all of the
structures discussed in this work. Interestingly, we find that $\lambda$ peaks
in the $Fdd2$ phase, which by virtue of being a commensurate structure, cannot
possess a phason mode. Nonetheless, we still predict a relatively high $T_{c}$
of $16.4$ \- $19.8$ K for the HG phase at $425$ GPa, which we partially
attribute to the lack of a significant Fermi-level pseudogap in the eDOS (see
Table 2).

Structure | Pressure (GPa) |  $T_{c}$ (K) | $\lambda$ |  $\langle\omega_{log}\rangle$ (K) | $\mathcal{N}(\epsilon_{F})$
---|---|---|---|---|---
$R\bar{3}m$ | $250$ | $17.6$ | $0.689$ | $469$ | $0.0390$
$Fdd2$ | $271$ | $26.5$ | $0.847$ | $439$ | $0.0414$
HG $(\gamma=\frac{4}{3})$ | $425$ | $18.1$ | $0.738$ | $417$ | $0.0445$
$Im\bar{3}m$ | $700$ | 12.6 | $0.543$ | $660$ | $0.0470$
Table 2: Superconducting critical temperature $T_{c}$, electron-phonon
coupling constant $\lambda$, logarithmic average frequency
$\langle\omega_{log}\rangle$ and Fermi-level electronic density of states
$\mathcal{N}(\epsilon_{F})$ (in units of electrons per eV per Å3) for several
of the structures discussed in this work at various pressures.

Figure 10: Eliashberg function $\alpha^{2}F(\omega)$ for selected structures
at given pressures. The contribution of each phonon branch to
$\alpha^{2}F(\omega)$ has been shaded with a different colour. The
(cumulative) electron-phonon coupling constant $\lambda(\omega)$ has been
plotted with a dashed blue line.

##  IV Conclusions

We have shown that upon increasing pressure beyond the experimentally-known
$R\bar{3}m$ ($\beta$-Po) phase, sulfur adopts a commensurate structure of
$Fdd2$ symmetry in the range $266$-$386$ GPa that resembles a strongly
distorted Ba-IVa phase. Between $386$ and $679$ GPa, S possesses a truly
incommensurate HG structure of the Ba-IVa type. Above $679$ GPa, we find that
S adopts a more closely packed bcc structure. We have accounted for the
effects of finite temperatures on the phase boundaries, and have shown that
all of the transition pressures can be lowered with an increase in
temperature, as well as predicting the existence of an $Fddd$ phase below
$215$ GPa and at temperatures $\gtrsim 1260$ K.

We have studied the properties of the incommensurate HG phase in detail, and
have shown that the ideal incommensurate ratio $\gamma_{0}$ decreases
monotonically with increasing pressure. Over the stability range of the HG
phase, S transforms between several competing chain-ordered variants of the
incommensurate HG structure, with an ABAC chain stacking stable over most of
this pressure range. These different chain orderings are intimately coupled to
modulations of the host and guest atoms, which contribute substantially to the
stability of the HG phases. We find that there is no significant interstitial
localisation of charge in S, instead showing that there are notable localised
absences of charge (‘voids’). We have calculated the superconducting critical
temperature of S for several of the structures, and expect $T_{c}$ to peak
between $24.8$ and $28.2$ K in the $Fdd2$ phase at $271$ GPa. We predict that
S remains a superconductor up to at least $700$ GPa.

##  V Acknowledgements

We wish to thank Malte Grosche for useful discussions. The computational
resources for this project were provided by the Cambridge Service for Data
Driven Discovery (CSD3), and we are further grateful for computational support
from the UK national high performance computing service, ARCHER, for which
access was obtained via the UKCP consortium and funded by EPSRC grant ref
EP/P022561/1.

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SUPPLEMENTAL MATERIAL

##  VI Pseudopotentials

We used a sulfur pseudopotential designed using the OTFG code bundled with
`CASTEP`. Our pseudopotential had a cutoff radius of $0.74$ Å$\ =1.4$ Bohr
(far smaller than half the smallest interatomic separation encountered in this
study) and included the $n=3$ valence shell only. Our tests show that
inclusion of the core $n=2$ states has no effect on resolving enthalpy
differences at these pressures. The local channel was chosen as $l_{loc}=3$.
This pseudopotential was used for all `CASTEP` calculations in the manuscript.

The OTFG code string used the generate the pseudopotential was:

        S 3|1.4|4|10|12|30:31:32(qc=7)

For the `QUANTUM ESPRESSO` (QE) superconducting critical temperature
calculations, the default scalar relativistic, ultrasoft PBE pseudopotential
for S from pslibrary were found to be sufficient in all cases, except for the
$C2/c$ approximant at $425$ GPa, which required a smaller $R_{c}$. For this
calculation, a custom pseudopotential with cutoff radius $R_{c}=1.6$ Bohr was
generated via the `ld1.x` code bundled with QE, using the following input
file:

    &input
       title=’S’,
       zed=16.,
       rel=1,
       config=’[Ne] 3s2 3p4 3d-1’,
       iswitch=3,
       dft=’pbe’
     /
     &inputp
       lpaw=.false.,
       pseudotype=3,
       file_pseudopw=’S.UPF’,
       author=’Jack Whaley-Baldwin’,
       lloc=-1,
       rcloc=1.6,
       which_augfun=’PSQ’,
       rmatch_augfun_nc=.true.,
       nlcc=.true.,
       new_core_ps=.true.,
       rcore=1.4,
       tm=.true.
     /
    6
    3S  1  0  2.00  0.00  1.40  1.60  0.0
    3S  1  0  0.00  6.00  1.40  1.60  0.0
    3P  2  1  4.00  0.00  1.40  1.60  0.0
    3P  2  1  0.00  3.00  1.40  1.60  0.0
    3D  3  2  0.00  0.10  1.40  1.60  0.0
    3D  3  2  0.00  0.30  1.40  1.60  0.0

This custom pseudopotential was tested against its respective default
pslibrary pseudopotential for a few structures, and was found to agree in
volume to within $0.1\%$, and relative enthalpies agreed to $\pm 0.1$ meV.

##  VII Convergence Calculations

Here, we present convergence calculations at $400$ and $800$ GPa, which span
the majority of the pressure range covered. We consider as representative
structures the $R\bar{3}m$ stucture at $400$ GPa and the $Im\bar{3}m$
structure at $800$ GPa. We show only the convergence results for the PBE
pseudopotential here, because the converged cutoffs and k-point spacings for
the the LDA and PBEsol pseudopotentials are nearly identical.

Supplemental Figs. S1 and S2 show that our chosen parameters of a plane-wave
cutoff of $850$ eV, a k-point spacing of $2\pi\times 0.01\ $Å-1, and the
`CASTEP` default electronic smearing temperature ($\approx 1200$ K) were
sufficient to achieve absolute convergence of at least $\pm 0.2$ meV
throughout the pressure range of interest; the relative energies of the phases
are converged to better than this.

Such stringent convergence was necessary as the different chain-ordered host-
guest phases are separated by enthalpy differences of order $\approx 0.5$ meV.

Figure S1: Absolute convergence of the total electronic energy of the
$R\bar{3}m$ structure at $400$ GPa, for the plane-wave cutoff (top), k-point
spacing (middle) and electronic smearing temperature (bottom). $\pm 1$ meV and
$\pm 0.5$ meV lines are shown relative to the most expensive result.

Figure S2: Absolute convergence of the total electronic energy of the
$Im\bar{3}m$ structure at $800$ GPa, for the plane-wave cutoff (top), k-point
spacing (middle) and electronic smearing temperature (bottom). $\pm 1$ meV and
$\pm 0.5$ meV lines are shown relative to the most expensive result.

##  VIII Ideal Host-Guest Ratio Fitting Procedure

Supplemental Fig. S3 details how the ‘ideal’ host-guest $\gamma$-value was
obtained for a given pressure.

In addition to those approximants found in the search, several approximants
with different $\gamma$-values were constructed ‘by hand’, and the resulting
enthalpies were fitted to a quadratic curve. The ideal value $\gamma_{0}$ is
simply read off from the minimum of this curve.

Figure S3: Dependence of the enthalpy of the HG phase on the $c_{H}/c_{G}$
ratio at $400$ GPa using the PBE functional. The black line is a least-squares
fit to a quadratic curve, and the ‘ideal’ value $x_{0}$ is taken from the
minimum of the curve. Each point represents a particular host-guest
approximant, and the label shows the number of atoms in that approximant. By
constructing similar curves at different pressures, we can obtain $\gamma_{0}$
at each pressure and thus plot Fig. 4 in the manuscript.

##  IX Enthalpies of C2/c and Fdd2 Structures

The $C2/c$ HG approximant exhibits an imaginary phonon mode at the
$\Gamma$-point below $380$ GPa, which distorts it into the $Fdd2$ structure.

The $C2/c$ and $Fdd2$ structures are very nearly identical, both structurally
and in enthalpy. The enthalpy difference between them is at most $\approx 1.5$
meV at $275$ GPa. Supplemental Fig. S4 compares the enthalpies of the two
structures, in addition to another enthalpy curve that detects only the $C2$
symmetry group, which is a subgroup of both $C2/c$ and $Fdd2$. This shows that
sulfur ‘selects’ the $Fdd2$ symmetry below $380$ GPa and the $C2/c$ HG
approximant above this pressure.

Figure S4: Relative enthalpies of the $C2$, $C2/c$ and $Fdd2$ structures. $C2$
is a symmetry subgroup of both $C2/c$ and $Fdd2$.

##  X Enthalpies of Other Stacking Orderings

In the manuscript, we state that we tested various other stacking orderings
that turned out to either be not energetically competitive, or simply
collapsed into another chain ordering. Supplemental Table 1 details these
other stacking orderings.

Stacking Sequence | Chain Heights | Smallest Enthalpy (meV/atom) | Pressure (GPa)
---|---|---|---
ABCACB | $(0,1,2,0,2,1)$ | $+0.3$ | 550
ABACBCBC† | $(0,1,0,2,1,2,1,2)$ | $+0.8$ | $500$
AB† | $(0,1)$ | $+1.3$ | $450$
ABB | $(0,1,1)$ | $+16.9$ | $480$
ABACB | $(0,1,0,2,1)$ | collapsed to ABC : $(0,1,2)$ | N/A
ABB | $(0,2,2)$ | collapsed to ABB : $(0,1,1)$ | N/A
ABBB | $(0,1,1,1)$ | collapsed to ABAC : $(0,1,0,2)$ | N/A
ABBC | $(0,1,1,2)$ | collapsed to ABAC : $(0,1,0,2)$ | N/A
ABBC | $(0,2,2,1)$ | collapsed to ABAC : $(0,1,0,2)$ | N/A
Table 1: Explicitly tested stacking sequences that were constructed by hand.
The ‘Chain Heights’ ($c$-axis displacement as one moves along the
$a$-direction) are given in units of a third of the intra-chain atomic
spacing, e.g. ‘2’ corresponds to $\frac{2}{3}$. The ‘Smallest Enthalpy’ is
measured relative to the lowest-enthalpy stacking sequence as given in the
manuscript, which is ABC in the range $384$-$400$ GPa, ABAC in the range
$400$-$590$ GPa, and AB in the range $590$-$679$ GPa. The ‘Pressure’ column in
this table is the pressure at which the enthalpy of that sequence was lowest.
Where the stacking sequence collapsed into another ordering, ‘N/A’ (Not
Applicable) has been placed in the Pressure column. Sequences marked with a
dagger (†) developed (after relaxation) $c$-axis displacements that changed
when moving along the $b$-axis also.

##  XI Band Gaps Opened by Chain Ordering

Supplemental Fig. S5 below shows the bandstructures of the $\gamma=4/3$
approximant with all chains aligned (i.e. AAAAAA…) and then with an ABC
ordering. It can be seen that adding the ABC ordering opens up several band
gaps at the Fermi level, and this contributes to the lowering of electronic
energy as described in Fig. 6 of the main manuscript.

Figure S5: Electronic bandstructures at $387$ GPa along $\Gamma\rightarrow$
X:$(0.5,0,0)$ and $\Gamma\rightarrow$ S:$(0.5,0.5,0)$ high symmetry directions
for a ($96$-atom) $3\times 1\times 1$ tiling of the ideal $I4/mcm$ structure.
Band gap openings are highlighted with cyan boxes. Top: With all chains
aligned. Bottom: With the chains offset ‘by hand’ as to have an ABC ordering
along the $a$-axis.

##  XII Electronic Density of States

The top of Supplemental Fig. S6 compares the electronic Densities of States
(eDOS) of the $Im\bar{3}m$ phase and 64-atom $Pcca$ HG approximant. The HG
phase features a relative reduction in total electronic energy as a result of
transferring its eDOS weight to lower energies. A small reduction in the Fermi
level eDOS is also visible.

For comparison, the bottom of Supplemental Fig. S6 compares the eDOSs of HG
aluminium at $5.20$ TPa and HG sulfur at $500$ GPa (where the 64-atom $Pcca$
approximant was used to model the HG phase). It can clearly be seen that a
drastic drop in the Fermi level eDOS that is present in Al does not occur for
S, where the eDOS at $E_{F}$ is in fact relatively flat.

Figure S6: Electronic Densities of States (eDOS). Top: For the 64-atom $Pcca$
approximant and $Im\bar{3}m$ phase at $500$ GPa. Bottom: For HG Aluminium at
$5.20$ TPa and HG sulfur at $500$ GPa (64-atom $Pcca$ approximant) - These
pressures roughly correspond to the midpoint of the respective HG stability
ranges.

##  XIII Volume Discontinuity

Supplemental Fig. S7 shows that the $R\bar{3}m\rightarrow Fdd2$ and HG
$\rightarrow Im\bar{3}m$ transitions introduce volume discontinuities into the
equation of state of high-pressure sulfur. The fractional volume change
$\frac{|\Delta V|}{V_{0}}$ is $\approx 0.71\%$ upon transition to the $Fdd2$
phase, and $\approx 0.99\%$ upon transition to the $Im\bar{3}m$ phase. The
magnitude of the volume change at the $Fdd2\rightarrow$ HG transition (not
shown here) was less than $0.01\%$.

Figure S7: Volume discontinuities with pressure for the $R\bar{3}m\rightarrow
Fdd2$ transition (top) and the HG $\rightarrow Im\bar{3}m$ transition
(bottom).

##  XIV  Powder Diffraction Patterns

Figure S8: Simulated powder diffraction patterns for the $Fdd2$ phase at $275$
GPa (top), the $C2/c$ ABC-ordered HG approximant at $275$ GPa (middle) and
$R\bar{3}m$ phase at 250 GPa (bottom). The incident radiation wavelength was
set to $0.4$ Å.

Supplemental Fig. S8 shows simulated powder diffraction patterns for the
$Fdd2$, $C2/c$ approximant ($\gamma=4/3$) and $R\bar{3}m$ ($\beta$-Po) phases.
At $275$ GPa, the $Fdd2$ phase should be the ground state structure. However,
we present diffraction patterns for both the $C2/c$ approximant and the $Fdd2$
structure at $275$ GPa, in order to assist experimentalists in identifying any
difference between the diffraction patterns of these two (very) similar
structures.

We note that the most significant difference is the absence of a peak at
$2\theta\sim 14.1^{\circ}$ in the pattern of the $C2/c$ approximant, which is
present in the $Fdd2$ pattern. In fact, there are several peaks present in the
$Fdd2$ pattern that are missing in the $C2/c$. We also note that the relative
heights of a few peaks are changed between the two patterns. We expect the
real host-guest diffraction pattern (to which the $C2/c$ is an approximation)
to possess satellite peaks, which will not be present on the pattern presented
here.

##  XV Aluminium and Arsenic Phase Diagrams

In the manuscript, we mention that the $Fdd2$ structure does not become the
ground state in other elements that possess high-pressure HG phases. In
Supplemental Fig. S9 below, we present, as examples, the phase diagrams of Al
and As with the $Fdd2$ structure added in.

Figure S9: Phase diagrams of aluminium in the terapascal regime (top) and
arsenic between $20$-$100$ GPa (bottom). For Al, the $C2/c$ approximant was
used to represent the HG phase (as per Ref. Chris J. Pickard (2010)), and for
As, the 32-atom $P4/ncc$ approximant was used (as per Ref. Häussermann  _et
al._ (2002)).

Whilst certainly competitive, the $Fdd2$ structure does not become the ground
state in either of these cases.

##  XVI  Superconducting Critical Temperature Calculations

We used the plane-wave DFT code `QUANTUM ESPRESSO` et al. (2009) for electron-
phonon coupling calculations and subsequent calculation of the superconducting
critical temperature $T_{c}$ within Migdal-Eliashberg theory and using the
McMillan-Allen-Dynes formula Allen and Dynes (1975). Density Functional
Perturbation Theory (DFPT) was used for all phonon calculations.

A crucial consideration in accurate $T_{c}$ calculations is the choice of the
double-delta smearing parameter $\sigma$ in the calculation of the Eliashberg
function:

$\displaystyle\alpha^{2}F(\omega)=\frac{1}{g(\epsilon_{F})}\sum_{m,n}\sum_{\vec{q},\nu}\delta(\omega-\omega_{\vec{q},\nu})\sum_{\vec{k}}|g^{\vec{q},\nu,m,n}_{\vec{k}+\vec{q},\vec{k}}|^{2}$
$\displaystyle\times\ \delta(\epsilon_{\vec{k}+\vec{q},m}-\epsilon_{F})\
\delta(\epsilon_{\vec{k},n}-\epsilon_{F})$ (S1)

Where the delta functions are replaced by Gaussians of width $\sigma$:

$\displaystyle\delta(\epsilon_{\vec{q},\nu}-\epsilon_{F})\rightarrow\mathrm{exp}\bigg{(}\frac{(\epsilon_{\vec{q},\nu}-\epsilon_{F})^{2}}{\sigma^{2}}\bigg{)}$
(S2)

This replacement is necessary, as Eqn. S1 can only be used as-is in the
infinite q\- and k\- kpoint sampling limit. The use of finite q\- and k\-
point grids in electronic structure calculations necessitates the replacement
of the delta functions by Gaussians.

Careful selection of the $\sigma$ value is very important in elemental
systems, where $T_{c}$ values are generally of order $10$ K, and thus an error
of a few Kelvin represents a large relative error. The optimal value of
$\sigma$ can be determined by calculating $T_{c}(\sigma)$ for two similarly-
sized k-point grids and finding the point at which the critical temperatures
of the two grids start to diverge. It is a necessary requirement that the
k-point grid be an integer multiple of the q-point grid. As an example, if we
use a 6x6x6 q-point grid, we might calculate $T_{c}(\sigma)$ for a 24x24x24
and a 30x30x30 $k$-point grid, corresponding to $kpq$ (‘k-points per q-point’)
values of 4 and 5, respectively. This procedure is described in detail in ref.
Wierzbowska  _et al._ , and we show an example of such a $T_{c}$ calculation
for the $Fdd2$ structure in Supplemental Fig. S10 below.

Figure S10: $T_{c}$ of the $Fdd2$ structure at $271$ GPa as a function of
smearing width $\sigma$ for two k-point grids - one with $4$ k-points per
q-point, and one with $5$ k-points per q-point. A green arrow indicates the
point where the two curves start to diverge, and the green dotted line shows
that the ideal smearing width is $\sim{0.06}$ Rydberg in this case.

##  XVII  Structural Information

We present (primitive) `.cell` files for various structures below.

$Fddd\ @\ 200$ GPa

        %block LATTICE_CART
        ang
          0.0000000   3.0716500   7.5560978
          2.8136036   0.0000000   7.5560978
          2.8136036   3.0716500   0.0000000
        %endblock LATTICE_CART

        %block POSITIONS_FRAC
          S  0.581656103  0.791591252  0.050624316
          S  0.071197879  0.428802121  0.071197879
          S  0.397001150  0.397001150  0.102998850
          S  0.852998850  0.852998850  0.147001150
          S  0.178802121  0.821197879  0.178802121
          S  0.668343897  0.458408748  0.199375684
          S  0.102998850  0.102998850  0.397001150
          S  0.428802121  0.071197879  0.428802121
          S  0.673871671  0.199375684  0.458408748
          S  0.791591252  0.581656103  0.576128329
          S  0.050624316  0.576128329  0.581656103
          S  0.199375684  0.673871671  0.668343897
          S  0.458408748  0.668343897  0.673871671
          S  0.576128329  0.050624316  0.791591252
          S  0.821197879  0.178802121  0.821197879
          S  0.147001150  0.147001150  0.852998850
        %endblock POSITIONS_FRAC

$R\bar{3}m\ @\ 250$ GPa

        %block LATTICE_CART
        ang
        1.6211882   0.9359935   0.8405054
        -1.6211882   0.9359935   0.8405054
        0.0000000  -1.8719869   0.8405054
        %endblock LATTICE_CART

        %block POSITIONS_FRAC
        S  0.500000000  0.500000000  0.500000000
        %endblock POSITIONS_FRAC

$Fdd2\ @\ 300$ GPa

        %block LATTICE_CART
        ang
          0.0000000   7.2469132   2.6760045
          2.9544756   0.0000000   2.6760045
          2.9544756   7.2469132   0.0000000
        %endblock LATTICE_CART

        %block POSITIONS_FRAC
          S  0.460350598  0.792807744  0.055443387
          S  0.731418288  0.482390943  0.064087521
          S  0.058060264  0.051661061  0.071035871
          S  0.178964129  0.430757196  0.191939736
          S  0.430757196  0.178964129  0.198338939
          S  0.850284770  0.573046329  0.274747620
          S  0.573046329  0.850284770  0.301921281
          S  0.975252380  0.948078719  0.399715230
          S  0.558601729  0.194556613  0.457192256
          S  0.185912479  0.527896752  0.518581712
          S  0.948078719  0.975252380  0.676953671
          S  0.792807744  0.460350598  0.691398271
          S  0.482390943  0.731418288  0.722103248
          S  0.527896752  0.185912479  0.767609057
          S  0.194556613  0.558601729  0.789649402
          S  0.051661061  0.058060264  0.819242804
        %endblock POSITIONS_FRAC

$C2/c\ @\ 400$ GPa

        %block LATTICE_CART
        ang
          2.8740970  -2.5844681   0.0000000
          2.8740970   2.5844681   0.0000000
         -2.5688715   0.0000000   7.0105305
        %endblock LATTICE_CART

        %block POSITIONS_FRAC
          S  0.257215059  0.026326505  0.048178790
          S  0.578217142  0.431995613  0.119203117
          S  0.095583293  0.639743506  0.124195854
          S  0.709544949  0.927654278  0.230208302
          S  0.072345722  0.290455051  0.269791698
          S  0.360256494  0.904416707  0.375804146
          S  0.568004387  0.421782858  0.380796883
          S  0.973673495  0.742784941  0.451821210
          S  0.026326505  0.257215059  0.548178790
          S  0.431995613  0.578217142  0.619203117
          S  0.639743506  0.095583293  0.624195854
          S  0.927654278  0.709544949  0.730208302
          S  0.290455051  0.072345722  0.769791698
          S  0.904416707  0.360256494  0.875804146
          S  0.421782858  0.568004387  0.880796883
          S  0.742784941  0.973673495  0.951821210
        %endblock POSITIONS_FRAC

$Pcca\ @\ 500$ GPa

        %block LATTICE_CART
        ang
         10.1310268   0.0000000   0.0000000
          0.0000000   5.0612578   0.0000000
          0.0000000   0.0000000   7.5490969
        %endblock LATTICE_CART

        %block POSITIONS_FRAC
          S  0.921154366  0.141917384  0.001201932
          S  0.578845634  0.858082616  0.001201932
          S  0.178845563  0.342070917  0.004277600
          S  0.321154437  0.657929083  0.004277600
          S  0.750000000  0.000000000  0.027404472
          S  0.003489518  0.491437926  0.122581360
          S  0.496510482  0.508562074  0.122581360
          S  0.570435274  0.163949693  0.162606534
          S  0.929564726  0.836050307  0.162606534
          S  0.828512529  0.359431377  0.164791163
          S  0.671487471  0.640568623  0.164791163
          S  0.326611856  0.346111455  0.168767685
          S  0.173388144  0.653888545  0.168767685
          S  0.076817651  0.146729300  0.171665485
          S  0.423182349  0.853270700  0.171665485
          S  0.250000000  0.000000000  0.227082257
          S  0.750000000  0.000000000  0.272917743
          S  0.923182349  0.146729300  0.328334515
          S  0.576817651  0.853270700  0.328334515
          S  0.673388144  0.346111455  0.331232315
          S  0.826611856  0.653888545  0.331232315
          S  0.171487471  0.359431377  0.335208837
          S  0.328512529  0.640568623  0.335208837
          S  0.429564726  0.163949693  0.337393466
          S  0.070435274  0.836050307  0.337393466
          S  0.996510482  0.491437926  0.377418640
          S  0.503489518  0.508562074  0.377418640
          S  0.250000000  0.000000000  0.472595528
          S  0.821154437  0.342070917  0.495722400
          S  0.678845563  0.657929083  0.495722400
          S  0.078845634  0.141917384  0.498798068
          S  0.421154366  0.858082616  0.498798068
          S  0.578845634  0.141917384  0.501201932
          S  0.921154366  0.858082616  0.501201932
          S  0.321154437  0.342070917  0.504277600
          S  0.178845563  0.657929083  0.504277600
          S  0.750000000  0.000000000  0.527404472
          S  0.496510482  0.491437926  0.622581360
          S  0.003489518  0.508562074  0.622581360
          S  0.929564726  0.163949693  0.662606534
          S  0.570435274  0.836050307  0.662606534
          S  0.671487471  0.359431377  0.664791163
          S  0.828512529  0.640568623  0.664791163
          S  0.173388144  0.346111455  0.668767685
          S  0.326611856  0.653888545  0.668767685
          S  0.423182349  0.146729300  0.671665485
          S  0.076817651  0.853270700  0.671665485
          S  0.250000000  0.000000000  0.727082257
          S  0.750000000  0.000000000  0.772917743
          S  0.576817651  0.146729300  0.828334515
          S  0.923182349  0.853270700  0.828334515
          S  0.826611856  0.346111455  0.831232315
          S  0.673388144  0.653888545  0.831232315
          S  0.328512529  0.359431377  0.835208837
          S  0.171487471  0.640568623  0.835208837
          S  0.070435274  0.163949693  0.837393466
          S  0.429564726  0.836050307  0.837393466
          S  0.503489518  0.491437926  0.877418640
          S  0.996510482  0.508562074  0.877418640
          S  0.250000000  0.000000000  0.972595528
          S  0.678845563  0.342070917  0.995722400
          S  0.821154437  0.657929083  0.995722400
          S  0.421154366  0.141917384  0.998798068
          S  0.078845634  0.858082616  0.998798068
        %endblock POSITIONS_FRAC

$P4/ncc\ @\ 650$ GPa

        %block LATTICE_CART
        ang
          4.9149353   0.0000000   0.0000000
          0.0000000   4.9149353   0.0000000
          0.0000000   0.0000000   7.3482019
        %endblock LATTICE_CART

        %block POSITIONS_FRAC
          S  0.500000000  0.000000000  0.052805013
          S  0.154488689  0.145587719  0.083832564
          S  0.645587719  0.345511311  0.083832564
          S  0.354412281  0.654488689  0.083832564
          S  0.845511311  0.854412281  0.083832564
          S  0.000000000  0.500000000  0.193048082
          S  0.848729494  0.151270506  0.250000000
          S  0.348729494  0.348729494  0.250000000
          S  0.651270506  0.651270506  0.250000000
          S  0.151270506  0.848729494  0.250000000
          S  0.500000000  0.000000000  0.306951918
          S  0.145587719  0.154488689  0.416167436
          S  0.654488689  0.354412281  0.416167436
          S  0.345511311  0.645587719  0.416167436
          S  0.854412281  0.845511311  0.416167436
          S  0.000000000  0.500000000  0.447194987
          S  0.500000000  0.000000000  0.552805013
          S  0.845511311  0.145587719  0.583832564
          S  0.354412281  0.345511311  0.583832564
          S  0.645587719  0.654488689  0.583832564
          S  0.154488689  0.854412281  0.583832564
          S  0.000000000  0.500000000  0.693048082
          S  0.151270506  0.151270506  0.750000000
          S  0.651270506  0.348729494  0.750000000
          S  0.348729494  0.651270506  0.750000000
          S  0.848729494  0.848729494  0.750000000
          S  0.500000000  0.000000000  0.806951918
          S  0.854412281  0.154488689  0.916167436
          S  0.345511311  0.354412281  0.916167436
          S  0.654488689  0.645587719  0.916167436
          S  0.145587719  0.845511311  0.916167436
          S  0.000000000  0.500000000  0.947194987
        %endblock POSITIONS_FRAC

$Im\bar{3}m\ @\ 700$ GPa

        %block LATTICE_CART
        ang
         -1.1027680   1.1027680   1.1027680
          1.1027680  -1.1027680   1.1027680
          1.1027680   1.1027680  -1.1027680
        %endblock LATTICE_CART

        %block POSITIONS_FRAC
          S  0.000000000  0.000000000  0.000000000
        %endblock POSITIONS_FRAC

@MISCsupp,note=”Supplemental material has been appended to the end of this
document, and contains pseudopotential information, convergence calculations,
ideal host-guest ratio fitting procedure, enthalpy comparisons of other chain
orderings, electronic bandstructure plots, simulated diffraction patterns,
comparisons of S to aluminium and arsenic, details of the superconducting
critical temperature calculations and structural information.”

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