Table Of ContentHigh temperature superconductivity in sulfur and selenium hydrides at high pressure
José A. Flores-Livas,1 Antonio Sanna,1 and E. K. U. Gross1
1Max-Planck Institut für Microstrukture Physics, Weinberg 2, 06120 Halle, Germany
(Dated: January 29, 2015)
Duetoitslowatomicmasshydrogenisthemostpromisingelementtosearchforhigh-temperature
phononic superconductors. However, metallic phases of hydrogen are only expected at extreme
pressures (400GPa or higher). The measurement of a record superconducting critical temperature
of 190K in a hydrogen-sulfur compound at 200GPa of pressure [1], shows that metallization of
hydrogencanbereachedatsignificantlylowerpressurebyinsertingitinthematrixofotherelements.
5 In this work we re-investigate the phase diagram and the superconducting properties of the H-
1 S system by means of minima hopping method for structure prediction and Density Functional
0 theory for superconductors. We also show that Se-H has a similar phase diagram as its sulfur
2 counterpartaswellashighsuperconductingcriticaltemperature. WepredictH Setoexceed120K
3
superconductivity at 100GPa. We show that both H Se and H S, due to the critical temperature
n 3 3
a and peculiar electronic structure, present rather unusual superconducting properties.
J
PACSnumbers:
6
2
Under high pressure conditions, insulating and semi- ing. Forpressuresabove180GPaanextremelyhightran-
]
n conducting materials tend to become metallic, because, sitiontemperatureofabout190Kwasmeasured[1]. This
o with increasing electronic density, the kinetic energy T ishigherthaninothersuperconductorsknownsofar,
C
c
growsfasterthanthepotentialenergy. Asmetallicityisa including cuprates and pnictides. The experimental ev-
-
r necessary condition for superconductivity, superconduc- idence is supported theoretically [32, 35, 37], and crys-
p
tivity becomes more likely under pressure [2, 3]. Wigner tal prediction methods suggest that the system becomes
u
s and Huntington [4], already in 1935 suggested the possi- superconducting with a SH3 stoichiometry. In this work
. bility of a metallic modification of hydrogen under very were-investigateextensivelytheS-Hphaseswithstateof
t
a
high pressures. Ashcroft and Richardson predicted [5, 6] the art ab-initio material search minima hopping meth-
m
hydrogentobecomemetallicunderpressureandalsothe ods [38–40] (MHM) and compute the superconducting
- possibilitytobeahightemperaturesuperconductor. The properties with the completely parameter-free Density
d
n highcriticaltemperature(TC)ofhydrogen[7–9]isacon- Functional Theory for Superconductors (SCDFT). We
o sequence of its low atomic mass leading to high energy also extend the analysis to the Se-H system, predicting
c vibrational modes and in turn to a large phase space afairlysimilarphasediagramandcomparablesupercon-
[
available for electron-phonon scattering to induce super- ducting properties.
1 conductivity [10]. However, the estimated pressure of
v metallization [11, 12] is beyond the current experimental
6
capabilities and it has been a challenge to confirm this
3 METHODS
hypothesis [13–16].
3
6
It was only recently that hydrogen-rich compounds
0 Electronic and phononic structure calculations are
started to be explored as a way to decrease the
. performed within density-functional theory as imple-
1 tremendous pressure of metallization in pure hy-
0 mented in the two plane-wave based codes abinit [41],
drogen [17], essentially performing a chemical pre-
5 and espresso [42] within the local density approx-
compression. The first system explored experimentally
1 imation LDA exchange correlation functional. The
v: was silane (SiH4) [18]. Soon after, many other pre- core states were accounted for by norm-conserving
compressed hydrogen rich materials have been explored
i Troullier-Martins pseudopotentials [43]. The pseudopo-
X experimentally[19–22]andtheoretically[23–35]. Theim-
tential accuracy has been checked against all-electron
r portance of a systematic search for a crystalline ground (LAPW+lo) method as implemented in the elk code
a
state has been put in evidence for disilane (SiH ), where
6 (http://elk.sourceforge.net/). In order to predict the
structures enthalpically higher lead to transition tem-
ground state structure of sulfur/selenium hydride com-
peratures of the order of 130K. Interesting structures
pounds we use the minima hopping method [38–40] for
have been proved not to be the global minimum and for
the prediction of low-enthalpy structures. This method
the correct ground state was found a rather moderate
has been successfully used for global geometry optimiza-
electron-phonon coupling and T of 25K [36]. In agree-
C tioninalargevarietyofapplications[44–46]. Givenonly
ment with experimental evidence.
the chemical composition of a system, the MHM aims
Recently it was reported that sulfur hydride (SH ), at finding the global minimum on the enthalpy surface
2
when pressurized, becomes metallic and superconduct- whilegraduallyexploringlow-lyingstructures. Moveson
2
Figure1. (Coloronline)CalculatedenthalpiesforH S(a)andH Se(b)structuresandtheirdecompositions. Valuesaregiven
3 3
with respect to the Cccm structure, stable at low pressure. In H S three structures were already reported in Ref. 35 (*) and
3
two in Ref. 32 (**). The other structures of H S as well as all the structures of H Se are predictions of this work by means of
3 3
the Minima Hopping Method.
the enthalpy surface are performed by using variable cell fur+hydrogen), itsdecompositionintoH S+hydrogen
2
shapemoleculardynamicswithinitialvelocitiesapproxi- and H S - hydrogen [57]. At low pressure we find the
4
mately chosen along soft mode directions. We have used Cccm structure up to 95GPa and the R3m (β-Po-type)
1,2,3 formula units of H S and H Se at selected pres- rhombohedralstructurebetween95and150GPa. Above
3 3
suresof100, 150, 200, 250and300GPa. Therelaxations 150GPa,weconfirm[32,37]thecubicIm¯3m(bcc)asthe
to local minima are performed by the fast inertia relax- most stable lattice.
ationengine[47]andbothatomicandcelldegreesoffree-
In a similar way we have studied the Se-H phase dia-
domaretakingintoaccount. Finalstructuralrelaxations
gram. Chemically, selenium is known to have very sim-
and enthalpy calculations were performed with the vasp
ilar physical properties to sulfur and this system is not
code[48]. Theplane-wavecutoffenergywassetto800eV,
an exception. The enthalpies of the phases found in our
and Monkhorst-Pack k-point meshes with grid spacing
MHM runs are shown in Fig. 1b. Once again we use
denser than 2π×0.01Å−1, resulting in total energy con-
the Cccm structure as reference since, as in the case of
vergence to better than 1meV/atom. Superconducting
sulfur, it is the most stable at low pressures and up to
propertieshavebeencomputedwithindensity-functional
70GPa. Between 70GPa to 100GPa, we find that the
theory for superconductors (SCDFT) [49–51]. This the-
H Se + hydrogen decomposition is more stable than the
2
ory of superconductivity is completely ab-initio, fully
H Se stoichiometry. H Se returns to be the most stable
3 3
parameter-freeandprovedtoberatheraccurateandsuc-
composition above 100GPa and at least up to 250GPa.
cessfulindescribingphononicsuperconductors[52–55]. It
allowstocomputeallsuperconductingpropertiesinclud- Therefore from our analysis both systems in the range
ing the critical temperature and the excitation spectrum 50GPa to 250GPa show, with increasing pressure, two
of the system [56]. phase transitions. The S-H system, always stable in the
H Sstoichiometry,hasafirstorderphasetransitionfrom
3
CccmtoR3mat∼100GPa,thentheR3mrhombohedral
CRYSTAL STRUCTURE PREDICTION AND distortiondecreasescontinuouslyupto150GPawhereit
ENTHALPIES transformsintotheIm¯3mcubicstructure. TheSe-Hsys-
tem at low pressure is also stable in the H Se stoichiom-
3
Experimentally little is known on the high pressure etry but becomes unstable to a phase separation into
stability and composition of the S-H system and, to the H S + hydrogen in the range from 70GPa to 100GPa.
2
best of our knowledge, nothing is known about the Se- Above 100GPa another discontinuous phase transition
H. Therefore we investigate their low temperature phase occurs, directly into the Im¯3m cubic structure. Note
diagrambymeansoftheMHMforthepredictionoflow- that100GPaisalsothepressurebelowwhichtheIm¯3m
enthalpy structures. Computed enthalpies as a function structurewoulddistortintotheβ-PoR3m, thereforede-
of pressure are reported in Fig. 1. We consider the H S pending on experimental conditions this rhombohedral
3
stoichiometryaswellasitselementaldecomposition(sul- phase may occur as a metastable one. The sequence
3
of transformation is highlighted in Fig. 1 by means of
shaded areas.
ELECTRONIC AND PHONONIC PROPERTIES
OF H SE AND H S AT HIGH PRESSURE
3 3
We focus now on the properties of H S and H Se in
3 3
the pressure range of stability of the Im¯3m structure.
The two materials present very similar properties. At
200GPa electronic band dispersions and Fermi surfaces
are barely distinguishable, as seen in Fig. 2. And in the
range of pressure between 100 to 200GPa there are no
significantchangesintheelectronicpropertiesapartfrom
the overall bandwidth that increases with pressure. An
important aspect of the electronic structure is the pres-
enceofseveralFermisurfacesheets,withnomarkednest-
ing features and with Fermi states both at low and high
momentum vector. At small momentum (close to the Γ-
point, center of the Brillouin zone in Fig. 2) there are
threesmallFermisurfaces(onlythegreenlargeronecan
be seen in the figure, smaller ones being inside it). How-
ever, these provide a rather small contribution to the
total density of states (DOS) at the Fermi level which
mostly comes from the two larger Fermi surface sheets.
These are of hybrid character, meaning that their Kohn- Figure 2. (Color online) Fermi Surfaces (top) and electronic
band structures (bottom) of H S and H Se at high pressure
Sham (KS) states overlap both with H and S/Se states 3 3
in the Im¯3m phase. The color-scale in the band lines indi-
(the overlap is expressed in the figure by the color-scale
catestheprojectionoftheKSstatesontheatomicorbitalsof
of the band lines), suggesting that they will be coupled
the sulfur/selenium atom normalized by the maximum total
with both hydrogen and S/Se lattice vibrations (more atomicprojectionofthesevalencestatesthatisofabout70%.
details on this point will be given in the next section).
Overall the DOS shows a square root behavior of the 3D
electrongas,themaindeviationfromthisoccurscloseto
the Fermi energy where a peak with an energy width of energy (see Fig. 3c).
about 2eV is present. This structure will play a relevant In spite of these important changes in the phononic
role in the superconducting properties. energy dispersion, the overall coupling strength[58, 59] λ
Unlike the electronic structure, phonons are strongly doesnotshowlargevariationsoverthepressurerange,as
pressure and material dependent. Clearly a key role is we can see from Tab. I. Naturally the coupling increases
played by the occurrence of the II order R3m to Im¯3m near the phase transition due the optical mode soften-
phase transition. Far away from it (i.e. at very high ing, however, as this is restricted to a relatively small
pressure) we have three sets of well separated phonon region near the Γ point, the effect is not dramatic. On
modes: acoustic (below 60meV), optical modes that are the other hand there is definitively a difference in the
transversewithrespecttotheS/Se-Hbond(between100 coupling strength of the Se (λ∼1.5) with respect to the
to200meV)and,above200meV,stretchingmodesofthe S system (λ ∼ 2.5), indicating that selenium, due to its
Se/S-H bond. These are clearly seen in Fig. 3b for H Se larger ionic size, provides a better electronic screening of
3
at 200GPa. As pressure reduces, the bond structure of the hydrogen vibrations.
the system tends to destabilize because, from a four-fold
coordination in the Im¯3m structure, it goes to a three-
foldcoordinationintheR3mone. Thismeansthatoneof SUPERCONDUCTING PROPERTIES
thehigh-energystretchingmodeslowlysoftensatΓ. This
softeningcanbeclearlyseeninH Sat200GPa(Fig.3a) Wehavecomputed, bymeansofSCDFT,criticaltem-
3
whereaH-Sstretchingmodewentdowntoabout60meV. peraturesofH SandH Seinthepressurerangeofstabil-
3 3
Eventually,aspressurelowersthissoftenstozeroenergy, ity of the Im¯3m structure, these are collected in Tab. I.
markingthe occurrence ofthephase transition, at about ThepredictedT fortheH Ssystemis180Kat200GPa
C 3
150GPa in H S and slightly below 100GPa in H Se. In and 195K at 180GPa, in agreement with the measured
3 3
fact, at 100GPa this mode has, in H Se, almost zero valueof185Kat177GPa. Ontheotherhandourpredic-
3
4
tion for the deuterium substituted system D S is 141K,
3
at200GPa. Thatismuchlargerthanthemeasured[1]T
C
of90K.Thishugeexperimentalisotopeeffectistherefore
not consistent with our calculations. However the good
agreementobtainedwiththeT oftheH Ssystemseem
C 3
to exclude an explanation in terms of anharmonic effects
inthehydrogenvibrations,assuggestedinRef.60. Nev-
ertheless, the theoretical isotope coefficients αS = 0.05
and αH = 0.4 (defined as αA = −MA ∂TC , and com-
Tc ∂MA
puted at 200GPa with a three point numerical differen-
tiation) clearly indicate and confirm [60] the dominant
contributionofhydrogenphononmodestothesupercon-
ducting pairing.
Our prediction for H Se at 200GPa is of 131K, this
3
reduction of T is clearly not an isotopic effect of the
C
substitution S to Se. As mentioned in the previous sec-
tion,itiscaused,instead,byadifferentcouplingstrength
of the hydrogen modes in the Se environment. In spite
of the much lower coupling strength λ the reduction of Figure 3. (Color online) Phonon dispersion and α2F func-
TC isnotverylargewithrespecttothesulfursystem,as tions [58, 59] of H3S and H3Se at high pressure in the cubic
expected from the fact that the critical temperature at Im¯3m structure. The color coding gives the projection of
high coupling increases with the square root of λ (while themodedisplacementontheS/Seatom. Displacementsare
visible dominated by H due to its lighter mass.
is exponential at low coupling) [58, 59].
To compute the critical temperatures we have used
SCDFT, this choice allow us to deal with the unusual
sion contribute to the condensation (in unconventional
superconducting properties of these systems from first
superconductors exactly the same happens but directly
principles without relying on conventional assumptions
at the Fermi level). This mechanism is called Coulomb
coming from low pressure experience. There are two as-
renormalization [59] since it renormalizes the repulsive
pects of these systems that are uncommon, that make
Coulombscatteringthatoccursatlowenergy. Thephase
the use of conventional [61] Eliashberg methods difficult
shiftoccursat|(cid:15) |(cid:38)ω butthescatteringprocessesbe-
to apply to these systems. First, the strong variation of k log
comelessandlessimportantas|(cid:15) |increases(goingdown
the electronic density of states at the Fermi level, that is k
as 1/(cid:15)). Therefore the most important energy region is
pinnedtoarathersharppeakintheDOS,secondtheex-
where the DOS of the H S and H Se systems shows a
tremelylargeel-phcouplingandphononfrequenciesthat 3 3
dip, implying that the phase space available for this pro-
lead to a very broad region around E where the inter-
F
cess is small and its effect weak. Note that in order to
actionisdominatedbyphononsoverCoulombrepulsion.
reproducetheT comingfromSCDFTwithintheAllen-
The effect of the energy dependence of the DOS can C
Dynes(AD)formulaoneshouldassumeaµ∗ of0.16,that
be appreciated by comparing Eliashberg results with
is actually much larger than the value of µ itself ((cid:39)0.1).
SCDFT when neglecting the Coulomb interaction (see
MakingclearthatthetheMorel-Andersontheory[66]can
Tab. I). At 200GPa the two theories [65] disagree by
not even be applied.
54K, SCDFT giving 284K while Eliashberg gives 338K.
The superconducting pairing is distributed over many
This difference comes from the energy dependence of
phonon modes and over the Brillouin zone in q-space,
the DOS, while Eliashberg assumes a flat DOS, in the
despite the presence of several Fermi surface sheets and
SCDFT we can easily check this assumption by assum-
with different orbital character across the Fermi level,
ing a flat DOS, and for this case the SCDFT calculation
we obtained a isotropic (weakly k-dependent) gap at the
would lead to 334K, in agreement with the Eliashberg
Fermi level and the effect of anisotropy [67] on T is
result. C
negligible (< 1K).
PhysicallythereductionofT ,occurringwhenthereal
C
DOSisconsidered,arisesfromthefactthatthephononic
pairingextendsinaratherlargeregionaroundtheFermi
level, over the DOS peak structure of these systems (see CONCLUSIONS
Fig. 2). Beyond the range of the phononic pairing the
coupling is dominated by the Coulomb interaction. As, We have presented a theoretical investigation on the
in the static limit, this is repulsive, a superconducting crystal structure and superconductivity of H S. An ex-
3
systemcompensatesitbyaphaseshiftinthegap(i.e. in tensive structural search confirms the H S stoichiometry
3
the quasiparticle orbitals), therefore making this repul- as the most stable configuration at high pressure. By
5
λ ω TSCDFT ∆(T=0) TSCDFT,ph TAD,µ∗=0.1 TAD,µ∗=0
log c c c c
H S 200GPa 2.41 109meV 180K 43.8meV 284K 255K 338K
3
D S 200GPa 2.41 82meV 141K 32.9meV 216K 188K 247K
3
H S 180GPa 2.57 101meV 195K 44.8meV 297K 250K 331K
3
H Se 200GPa 1.45 120meV 131K 28.4meV 234K 174K 246K
3
H Se 150GPa 1.38 107meV 110K 23.4meV 195K 145K 209K
3
H Se 100GPa 1.76 87meV 123K 27.0meV 198K 156K 214K
3
TableI.CalculatedcriticalTemperaturesandgaps. λistheelectronphononcouplingparameter[58,59];ω isthelogarithmic
log
average of the α2F function [58, 59]; TSCDFT is the critical temperature from SCDFT including RPA screened Coulomb
c
repulsion; TSCDFT,ph is the phonon only SCDFT critical temperature; TAD,µ∗=0 is the critical temperature from the Allen-
c c
Dynes modified McMillan formula [59, 62, 63], at µ∗ =0 (i.e. with no Coulomb pairing); TAD,µ∗=0.1 is the same AD formula
c
butwiththeconventionalvalueofµ∗ =0.1;∆(T=0)isthesuperconductinggapcomputed[64]fromtheSCDFTcalculations
meansofparameter-freeSCDFTwehavepredictedaT [8] L. Zhang, Y. Niu, Q. Li, T. Cui, Y. Wang, Y. Ma,
C
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3
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