Table Of ContentCubic H S around 200 GPa: an atomic hydrogen superconductor stabilized by sulfur
3
D. A. Papaconstantopoulos1, B. M. Klein2, M. J. Mehl3, and W. E. Pickett2∗
5 1Computational Materials Science Center, George Mason University, Fairfax, VA 22030, USA
1 2Department of Physics, University of California Davis, Davis, CA 95616, USA
0 3Center for Computational Materials Science, Naval Research Laboratory, Washington DC 20375, USA
2 (Dated: May 28, 2015)
y The multiple scattering-based theory of Gaspari and Gyorffy for the electron-ion matrix element
a in close packed metals is applied to Im¯3m H3S, which has been predicted by Duan et al. and
M
Bernstein et al to be the stable phase at this stoichiometry around 190 GPa, thus is the leading
candidate to be the phase observed to superconduct at 190K by Drozdov, Eremets, and Troyan.
7 The nearly perfect separation of vibrational modes into those of S and of H character provides a
2 simplification that enables identification of contributions of the two atoms separately. The picture
that arises is basically that of superconducting atomic H stabilized by strong covalent mixing with
]
n S p and d character. The reported isotope shift is much larger than thetheoretical one, suggesting
o thereislargeanharmonicityintheHvibrations. Giventherelativeunimportanceofsulfur,hydrides
c of lighter atoms at similarly high pressures may also lead to high temperature superconductivity.
-
r
p
I. BACKGROUND ofω areexpectedfromtheanticipatedincreaseofforce
u log
s constants as volume is decreased. The large values of λ,
t. The report by Drozdov, Eremets, and Troyan1 (DET) afactoroftwoormoreovermostotherverygoodsuper-
a conductors (including MgB ), imply that the electronic
m ofsuperconductivityuptoTc=190KinH2Scompressed 2
matrix elements are substantially larger than in nearly
to the 200 GPa regime has reignited excitement in the
- all known conventional superconductors.
d possibility of achieving room temperature superconduc-
n tivity. This reportbuilds onprevious success of pressure In this report we use Gaspari-Gyorffy (GG) theory17
o enhancementofT inavarietyoftypesofmaterials: from to provide insight into why electron-ion matrix elements
c
c 134Kto 164Kinthe cuprateHg2223,2,3 fromzeroto 20- vary,andevidentlyincreasestrongly,withpressure. Such
[
25 K in the simple metals Li, Ca, andY,[4–10] and from understanding is necessary not only to interpret the re-
2 zeroto14Kintheinsulatorsilicon.11 Ananticipatedma- sults described above, but also to provide essential clues
v jor factoris the increaseinthe phononenergy scalewith how to increase matrix elements, and λ, at lower or pos-
0 compression, since it sets the temperature scale for T , sibly ambient pressure. Interestingly, shortly after the
c
5 as pointed out early on12 and reviewed more recently13 formulationofthistheory,twoofthepresentauthorsap-
9
3 bpyerAcosnhdcurocfttiviintyprinedmicettinagllipcohsysdibrloegreono.mtemperature su- fpelwiedMibtatropprersesduicret.1T8c∼250K in metallic hydrogen at a
0
1. The newly reported high values of Tc appear to con- GG theory17 builds on the earlier observation of
0 firm theoretical predictions that predated the experi- Hopfield19 that electron scattering off (moving) ions has
5 ment. Applying particle swarm crystal structure search strong local character. First, metallic screening means
1 techniquesfoundedondensityfunctionaltheory,Liet al. the Thomas-Fermi screening length is short, of the or-
v: predicted14 candidatestablecrystalstructuresofH2Sup derof1˚A,andveryweaklydependentoncarrierdensity
i to220GPa. Thesepredictionswerefollowedbylinearre- (more correctly, the Fermi level density of states [DOS]
X sponsecalculationsofthephononspectrumωQ,electron- N(EF)). GG employed a multiple scattering Green’s
r phononmatrixelements,andfinallytheEliashbergspec- function formalism that facilitated three simplifications.
a tral function α2F(ω), from which Tc can be calculated, The first is that the potential is spherical(very goodap-
depending only mildly on the presumed value of the re- proximation)andisnegligiblebeyondtheatomicsphere;
∗
tarded Coulomb repulsion µ = 0.10-0.13. For pressures the secondis that the linearchange in potential ofa dis-
of 140-180 GPa, they obtained an electron-phonon cou- placed ion can be approximated by a rigid shift of the
pling strength λ=1.0-1.2, an Allen-Dynes characteristic atomic potential. Thirdly, the direction dependence of
frequency15 ωlog 1000K,andTc of40Kat140GPaand thewavefunctioncoefficientsisaveragedout,therebyne-
∼
peaking at 80 K at 160 GPa. While 80K is well short of glectinganyspecialinfluenceofFermisurfaceshape. The
thereportedTc=190K,theresultisconvincingthatvery H3S Fermi surface20 is large and multisheeted, minimiz-
high Tc is predicted in H2S at high pressure. ing the likelihood of Fermi surface effects. It is possible
The sister stoichiometry H S has been explored in that these approximations may improve with reduction
3
very similar fashion by Duan et al.16 Predicting struc- in volume, in any case these approximations should not
tures to more than 200 GPa, their linear response re- degrade as the system become denser. The bands shown
sultsforIm¯3mH Sledtoverylargecalculatedvaluesof by Duan et al.16 and Bernstein et al.20 for the Im¯3m
3
electron-phonon coupling strength λ=2.0-2.2, frequency structure of H S that we discuss show much free elec-
3
scales ω 1300K,and values of T up to 200K.In the tron,sphericalcharacterinthelower75%oftheoccupied
log c
∼
calculations of Li et al. and Duan et al., the high values bands, though less so around the Fermi energy where S
14
Total
sp−−HS eF
12 d−S
Unit) s−S
mula 10
For
States/Ry/ 8
States ( 6
Density of 24
0
−0.5 0.0 0.5 1.0 1.5 2.0
Energy (Ry)
FIG. 1: (Color online) The Im¯3m crystal structure of H3S,
illustratingthetwointerlacedReO3sublattices. Largesphere FIG. 2: (Color online) Total and orbital-projected densities
is S, small sphereis H.
of states of H3S at 210 GPa (a=5.6 a.u.). Note the position
of EF within the sharp and narrow peak, and that the H 1s
contribution is that from all three H atoms.
3p - H 1s hybridization produces structure in the DOS.
II. THEORY AND RESULTS Both H2S and H3S having been shown14,16 to have
strong electron-phonon coupling at high pressure. Bern-
steinet al.20 haveprovidedconvincingevidencethatH S
2
The coupling strength λ, and the frequency weighting
isunstabletodecompositionintobccH Sandsulfur,and
3
spectrum g(ω) normalized to unity, are given in Eliash-
that competing stoichiometries are unlikely. This result
berg theory21 by
confirmsthesuggestionofDET,whoreportedsulfurfor-
α2F(ω) 2α2F(ω) mationin their samples. Becauseof this evidence on the
λ=2 dω; g(ω)= , (1) most likely superconducting phase, we focus on Im¯3m
Z ω λω
H Swith its simple bcc structurebasedintwointerlaced
3
where α2F is the Eliashberg electron-phonon spectral ReO3 structuresublattices. Fromα2F ofDuanet al., we
function that governs many superconducting properties. simplify with a constant α2F (constant for each atom
j
The calculations by Li et al. for H S and Duan et al. species j) with frequency ranges (in kelvin) of [430,820]
2
for H S demonstrate that the lower range of phonon fre- for S and [1250,2500] for H. Results are insensitive to
3
quencies (the acoustic modes) have negligible H charac- these limits, depending mostly on the mean frequencies
ter, while the optic modes above the gap at 20-25 THz and the separationof λ into S and H contributions. The
have negligible S character, making it an ideal platform resultingfrequencymomentsω ,ω ,andω andassoci-
log 1 2
for applying the GG expression to the atoms separately. ateddata,forinsightintoseparateSandHcontribution,
Thus λ = 3λ +λ , where the latter arise from the foruseintheADequation,15andtocomparewithresults
H S
integraloverthelowfrequencySmodes,theformerfrom below from GG theory, are presented in Table I.
the nine higher frequency H branches. In this case the With µ∗=0.15, T =234 K results; the difference from
c
GG expression, given originally for an elemental solid, the value quoted by Duan et al. might be due to neglect
canbeappliedtothe SandHspheresseparately.22 Each of the strong coupling factor f =1.13 factor in the AD
1
atomic (j) coupling constant λj is given by the integral equation, which amounts to a 26 K increase,or partially
over the appropriate frequency region, leading to to our constant α2F modeling. Neglecting the contribu-
tionfromtheSmodes, λisdecreasedfrom2.2to1.5but
λj = N(EMF)jω<j2Ij2 > ≡ Mηjjωj2. (2) bωylogoninlycr1e9asKestforo2m1515K0.0TKhetosu1lf7u7r0cKon,tarnibduTtiocndetcoreTacseiss
8%; H S is basically an atomic hydrogen high temper-
The averaged matrix elements <I2> obtained from GG 3
ature superconductor. Bernstein et al. also suggested
theory are discussed below.
that S vibrations are not essential for the high T . The
c
The separationof mode characteralso allows a simple
H isotope effect can also be obtained. λ is unaffected by
estimate of the total frequency moments that enter the masses;Mω2isafunctionoftheforceconstantsalone,so
Allen-Dynes (AD) equation for Tc, through the weight frequencies, specifically ωH decrease as the square root
function log
of the mass. The resulting critical temperature is re-
duced to 170 K, slightly more than 234 K/√2 because
g(ω)=[λ g (ω)+3λ g (ω)]/(λ +3λ ), (3)
S S H H S H ∼
the smallScontributionremains unchanged. The exper-
where the partial g functions, defined analogously to imental value of DET1 is 90 K; the most likely cause of
j
that of AD (Eq. 1) are separately normalized to unity. this discrepancy is strong anharmonicity of the H optic
2
modes. 7
25
The second moment frequency at 200 GPa of S is
ωS=615K,while thatofHisωH=1840K,thuswiththe
2 2 6
atomicmassesof32and1a.m.urespectively,thedenom-
20
inatorMω2 is32/9=3.5largerforS.Theconsequenceis
2
(etthffhaeotcutaigvhgeiviinnenipnrccaorcnetatirsciiebnugthtλieortnehtaisonnηtoHhecilsseaa3mr.5em×ceo3t∼nhto1rd0ibtouiftmieoeffnsetcmotioηnrSge 22I (eV/Å) 15 45 (eV/Å)h2
such a tradeoff). I2 (S)
10 h (S)
3
I2 (H)
TABLE I: Electron-phonon coupling data for H3S, obtained h (H)
frommodelingtheresultsofDuanetal. withaconstantα2F
5 2
model. Frequencies are in Kelvin. 0 50 100 150 200 250 300
P (GPa)
S H H3S
FIG. 3: (Color online) Plots demonstrating the very strong
ω 595 1770 1500
ωl1og 605 1800 1530 pressureincreaseof<I2(j)>(leftaxis)andηj (rightaxis)for
ω2 615 1840 1560 j=S and H in Im¯3m H3S. For H, the η plotted is the total
Mω22 (eV/˚A2) 9.3 2.6 – for 3 H atoms.
η (eV/˚A2) 4.7 1.48 –
λ 0.5 1.7/3 2.2
N. In addition, a fifth broad band is roughly half filled.
The DOS is noteworthy: free electron like over 20 eV of
In terms of the phase shifts δ for the j-th atom for
ℓ,j the valence band before strong structure arises in a 7
orbitalchannelℓ, the squareelectron-ionmatrixelement ±
eV range centered at E , which lies very close (slightly
F
averaged over the Fermi surface can be written in the
above)thestrongandsharppeakinthe DOS.Thispeak
simple form17 as
atE persistsforallpressuresfromP=0to300GPaand
F
E 1 2 even above, almost as if EF were pinned at this peak,
<I2>= F 2(ℓ+1)sin2(δ δ )ν ν ,(4) while other features of the DOS evolve.
j π2 N(E )2 ℓ− ℓ+1 ℓ ℓ+1
F X Table II shows the Fermi level values of total and an-
ℓ=0
gular momentum components of the electronic densities
(1)
where νℓ = Nℓ(EF)/Nℓ is the ratio of the ℓ-th par- of states across a wide range of volumes. Even though
tialDOSto N(1), the single scattererDOS,for the given Im¯3m H S may not be stable at lower pressures, we
3
atomicpotentialinahomogeneoussystem. <I2>isinde- provide results for the large range P=0-210 GPa to ob-
pendentofN(EF)sinceitcanequallywellbeexpressed23 serve the effect of interatomic distance on the electronic
in terms of the fractions Nℓ(EF)/N(EF) which usually structure and coupling. While the total N(EF) shows a
do not reflect the van Hove singularities of either one. weak non-monotonic variation, the ℓ-components have a
The calculated DOS at 210 GPa is shown in Fig. 2. The stronger lattice constant(pressure) dependence. As ex-
Fermi level falls at a sharp and narrow peak; calcula- pected the sulfur 3p-likestates are the dominantcompo-
tions at other volumes indicate this is a persistent oc- nentbutastrongparticipationof3dcharacterespecially
currence. <I2> will tend to be maximized in the cases at high pressures is present, hybridizing with the also
j
where “neighboring” channels ℓ,ℓ+1 have a large dif- strong and nearly constant H 1s contribution.
ference in phase shifts but ratios νℓ that are as large Now we discuss the electronic factor ηj. It should be
as possible. From the GG expression, for each atom kept in mind that the relative importance for T of H
c
ηj = N(EF)<Ij2>, and the latter factor involves the versus S modes is not simply λj, but more like λjω2,j
sin2(δ δ )factorandproductsofPDOSratiosν ν . which is a factor of ten greater for 3H than for S). For
l l+1 l l+1
M ω2 −can be expressed in terms of the ionic force con- hydrogen, which dominates the contribution to T only
j j c
stants, independent of M , (which we return to below) the s p channel is relevant at all pressures. The phase
j
so thatany isotope effectdifferent fromM−1/2 will arise shift f−actor sin2(δs δp) decreases with pressure. Fig. 3
−
from factors beyond λ (primarily anharmonicity). illustrates the factor of two increase in ηH from P=100
The calculations have been carried out with two GPa to 300 GPa. This dramatic increase results from
all-electron linearized augmented plane wave (LAPW) an even larger increase in the PDOS product νHνH, re-
s p
codes, one developed at NRL,24 and also ELK.25 The flecting transfer of 1s character to 2p character. In the
sphere radii were 1.8 a.u. and 1.0 a.u. for S and H re- spherical harmonic expansion of the atomic wavefunc-
spectively, except for the smallest volume where the S tions this ‘2p’ character represent the expansion of tails
radiuswasreducedbecause the sum ofthe radiimustbe of the orbitals on neighboring atoms that gives rise to
no more than a/2. the increased H-S hybridization under pressure.
The band structure16,20 consists of four nearly filled AsnotedabovefromouranalysisoftheresultsofDuan
bands, leaving some holes at Γ and electrons around et al., the sulfur contribution is less important. As for
3
modes are representative. The lower two of these modes
TABLE II: Total and angular-momentum decomposed den-
areIR-activeinvolvingH-Sbondstretchandbondbend-
sities of electronic states at the Fermi level (states/Ry-both
ing modes, the hardest frequency is a silent mode with
spins)inIm¯3mH3S.Theℓ-componentsareprojectionswithin
quadrupolar H motion with respect to S.
sphere radii R(S)=1.8a.u. and R(H)=1.0 a.u.
From calculations of these Γ frequencies in the 240-
a (a.u.) N(EF) S-s S-p S-d H-s/atom
270 GPa range using the ELK code we calculate a posi-
5.6 6.93 0.325 1.74 0.751 0.435
tivebutmodestpressureincreasedlogω¯ /dP 1.9 10−3
5.8 6.43 0.286 1.53 0.605 0.405 H
6.0 6.47 0.274 1.46 0.528 0.415 GPa−1. From Fig. 3 we obtain dlogηH/dP=≈3.5×10−3
6.2 6.42 0.253 1.18 0.528 0.462 GPa−1, thus λ decreaseswith pressure approxima×telyas
6.4 6.79 0.259 1.36 0.417 0.455 dlogλ /dP -0.3 10−3/GPa. However, to our precision
H
∼ ×
6.6 7.15 0.255 1.38 0.375 0.482 thisisindistinguishablefromzero,sothepressurederiva-
6.8 7.56 0.254 1.42 0.339 0.513 tive in Eq. 5 reduces to the first term, the frequency
derivative. The resulting prediction is a small increase
dT /dP=0.4 K/GPa. This result disagrees in sign with
c
H, the increase in η occurs in spite of a decrease in the Duan et al., who quoted a smaller (in magnitude) nega-
sin2(δ δ ) factor,by 30%fromP=100to P=300GPa.
p d tive value of dT /dP=-0.12 K/GPa from direct calcula-
− c
Over this pressure range, the PDOS ratio product νpSνdS tion, however both numbers are small compared to the
increases by 75%, giving a net η increase by more than largevalueofT itself,sothereisnosignificantdisagree-
c
50%. Inaddition,theSs pandevend f channelsbegin ment.
to contribute, reaching 2−0% of the tota−l of <I2> at 300
GPa. Thustheincreaseisacompositeeffectofincreaseof
bothdandscharacterofS,whichisreflectedalsointhe
III. CONCLUSIONS
growing phase shifts of these channels. These transfers
of atomic character under pressure are consistent with
generalexpectationsofthe evolutionofatomiccharacter The report by Drozdov, Eremets, and Troyan of Tc
under reduction of volume. Interpolating to 200 GPa to up to 190 K in HnS samples has breathed new life into
obtain ηS=5.84 eV/˚A2, ηH=1.71eV/˚A2 (which must be the 50 year old expectation of high Tc in atomic H sys-
multiplied by three), and using the frequency moments tems. Both Li et al. and Duan et al. had found that
fromTable1,weobtainλ=2.6,thusT 250K depend- Eliashbergtheoryandlinearresponseresultsforelectron-
c
ing somewhaton the chosenvalue of µ∗=∼0.10-0.13. This phonon coupling account for Tc in the 80-200 K range
value is very consistent with the value of Tc (above) ob- for H2S and H3S at high pressure, and the analysis of
tainedfromourmodeling ofthe Duandata. Thoughthe Bernstein et al make Im¯3m H3S the primary candidate
numbersmightvaryforotherlow-Zhydrides(H P,H B, to be this record-high temperature superconductor. In
n n
...) the lack of any special role of S in these results sug- this paper wehaveestablishedthatthe couplingofH vi-
gests there should also be a strong increase in T with brations increases strongly for pressures up to and even
c
pressure in other low-Z element hydrides. beyond 210 GPa, and that 90+% of the coupling arises
AquestionofgreatinterestiswhetherT increasesfur- from H vibrations in this hydride that is stabilized by
c
ther at higher pressure. The data presented in Table III hybridization20 with S. This picture is analogous to the
providesthepressuredependenceoftheimportantquan- finding of the essential contribution of H in the super-
tities entering η and η . This data demonstrates that conductor PdH at ambient pressure,26 and the broader
S H
the strongest contributions arise for sulfur from the pd picture of Ashcroft12 of superconducting elemental H at
channel and for hydrogen from the sp channel. Note high pressure. The theoretical isotope shift of Tc based
that for H one should multiply by three to account for onthe harmonicapproximationis notinagreementwith
the three H atoms in the unit cell. the experimental result, suggesting substantial H anhar-
Since λ dominates the sulfur contribution, we can monicity will be necessary to understand before the pic-
H
focus on the H contribution alone. The total pressure ture is complete. Our picture, which relies on coupling
derivative contains several contributions across the large Fermi surface, is at odds with the hole
superconductivity picture of Hirsch and Marsiglio.27
dlogT dlogω dlogf dlogE(λ)
c log 1
= + +
Acknowledgments.
dP dP dP dP
dlogω dlogf dlogE(λ) dλ The authors acknowledge many insightful conversa-
log 1
= +( + ) , (5) tions on the theory and application of GG theory with
dP dλ dλ dP
the late B. L. Gy¨orffy, to whom we dedicate this paper.
where f is the strong coupling correction and E(λ) is We acknowledgediscussions with A. S. Botana,F. Gygi,
1
the exponential term in the Allen-Dynes equation. The and I. I. Mazin. M.J.M. was supported by the Office of
pressure variation of the Mω2 denominator is challeng- Naval Research through the US Naval Research Labora-
H
ing to approximate without full calculations of the spec- tory’s basic researchprogram. W.E.P.was supportedby
trum and α2F. We have modeled the variation of the NSF award DMR-1207622-0. D.A.P. was supported by
H spectrum by assuming the three 3-fold Γ point optic grant N00173-11-1-G002 from the U.S. Naval Research
4
TABLEIII:PressurevariationoftheHopfieldparameterη (eV/˚A2)andelectron-ionsquaredmatrixelement<I2>(eV2/˚A2),
both decomposed intothe threechannels sp,pd,and df for S, and theH spchannel (given for a single atom).
a (a.u.) P (GPa) S ηsp S I2sp S ηpd S I2pd S ηdf S I2df H ηsp H I2sp
5.6 210 0.68 2.66 5.12 20.1 0.31 1.21 1.80 7.06
5.8 142 0.39 1.65 4.45 18.9 0.21 0.89 1.43 6.06
6.0 92 0.24 1.00 4.14 17.4 0.15 0.63 1.24 5.21
6.2 57 0.19 0.78 4.13 17.5 0.13 0.57 1.02 4.34
6.4 31 0.08 0.33 3.52 14.1 0.08 0.31 0.96 3.87
6.6 13 0.05 0.18 3.37 12.7 0.06 0.21 0.87 3.31
6.8 0 0.02 0.09 3.12 11.2 0.04 0.14 0.79 2.83
Laboratory.
∗ Electronic address: [email protected] 16 D. Duan et al.,Sci. Rep.4, 6968 (2014).
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5
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