The isotope effect in H$\rm_{3}$S superconductor

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by  R. Szczesniak

The isotope effect in H S superconductor
3
R. Szcze¸´sniak(1,2)∗ and A. P. Durajski(1)†
1 Institute of Physics, Cze¸stochowa University of Technology,
Ave. Armii Krajowej 19, 42-200 Cze¸stochowa, Poland and
2 Institute of Physics, Jan D lugosz University in Cze¸stochowa,
Ave. Armii Krajowej 13/15, 42-200 Cze¸stochowa, Poland
(Dated: January 8, 2016)
TheexperimentalvalueofH3Sisotopecoefficientdecreasesfrom2.37to0.31inthepressurerange
6
from 130 GPa to 200 GPa. We have shown that the value of 0.31 is correctly reproduced in the
1
framework of theclassical Eliashberg approach. Ontheotherhand,theanomalously large valueof
0
theisotopecoefficient(2.37) maybeassociated withthestrongrenormalization ofthenormalstate
2
bythe electron density of states.

n
a
J Keywords: H SandD Ssuperconductor,isotopeco-
3 3
7 efficient, Eliashberg approach. 240 LaH3
230 ScH3
n] Themetallichydrogenthe mostprobablycouldbethe YAHlH33
o superconductor with the very high value of the critical 120 GSiaHH43
c temperature (TC) [1], [2]. The expected high TC is asso- K) SnH4
- ciated with the large Debye frequency (the mass of the ( 90 GeH4
r C Si2H6
p proton is very small) and the lack of the electrons on T CaH6
u the inner shells, which should significantly increase the B2H6
.s electron-phonon coupling constant (λ) [3], [4], [5]. Un- 60 SGieHH4(4H(H2)22) 2
t fortunately, the pressure of the hydrogen’s metallization
a 30
m is very large (p > 400 GPa [6], [7]). For this reason, the
experimental confirmation of the theoretical predictions
d- has not been obtained to this day. 0
n In 2004Ashcroft suggestedthe existence of the super- 0 150 300 450
o p (GPa)
conducting state in the hydrogen-rich compounds with
c the critical temperature comparable to T of the pure FIG.1: Thepredictedcriticaltemperaturesforthehydrogen-
[ C
hydrogen, whereas the metallization pressure might be rich compounds. The results for tri-hydrides: LaH3, ScH3,

1 subjectedtothe significantdecreasedue to the existence YH3,AlH3,andGaH3 arefrom works[9],[10],and[11]. The
v resultsobtainedforfour-hydridesarepresentedinthepapers:
of the chemical pre-compression [8]. Ashcroft’s predic-
1 SiH4 [12] (experiment), [13], [14], SnH4 [15], GeH4 [16]. The
tions were confirmed in many later papers. The selected
3 results for six-hydrides and eight-hydrides were obtained in
results are presented in Fig.1.
5 worksSi2H6 [17],[18],Ba2H6 [19],CaH6 [20],SiH4(H2)2 [21],
1 Thesuperconductingstateinthehydrogensulfidewith and GeH4(H2)2 [22], [23].
0 the exceptionally high value of the critical temperature
.
1 (TC ∼ 200 K) was discovered in 2014 [24], [25]. The
0 detailed dependence of the critical temperature on the In the first step, on the basis of the experimental re-
16 Fpriges.s2u.reforthe compounds H3S andD3S is presentedin asunldtsT, Dw3eS(dpe)tewrmhiicnhedsertvheedafoprprtohxeimcaaltciuolnatliionnesofTtCHh3eSi(spo)-
: C
v The experimental results [24], [25] and the theoretical tope coefficient:
i papers [27–32] suggestthat the superconducting state in
X
the hydrogen sulfide is induced by the electron-phonon
r
a interaction. In particular, the strong isotope effect was ln TD3S(p) −ln TH3S(p)
observed. However, the values of the isotope coefficient α (p)=− C C , (1)
(α) significantly differ from the canonical value of 0.5 exp h ln[mDi]−ln[hmH] i
predicted by the BCS theory [33], [34].
where m and m are respectively the deuterium’s and
Inthe presentedpaper,weexplainedthe experimental D H
protium’s atomic mass. The shape of the function
data for α on the basis of the classicaland the extended
α (p) is plotted in Fig.3. It can be clearly seen that
Eliashbergformalismbasingonthephononpairingmech- exp
theisotopecoefficientdecreaseswiththeincreasingpres-
anism.
sure. In particular, the following values were obtained:
α (130GPa)=2.37 and α (200GPa)=0.31.
exp exp
Thevalueoftheisotopecoefficientforp=200GPacan
∗Electronicaddress: [email protected] be reproduced in the framework of the classical Eliash-
†Electronicaddress: [email protected] bergformalism. Tothisend,wesolvednumericallyequa-
2
tion renormalization factor. The fermion Matsubara
frequency is given by the formula: ω = π (2n−1),
250 n β
β = 1/k T (k is the Boltzmann constant). The
B B
electron-phonon pairing kernel has the following form:
200
K) K(z) = 2 0+∞dΩΩ2Ω−z2α2F(Ω). The Eliashberg func-
(C150 tions (α2FR(Ω)) for p =130 GPa and p= 200 GPa were
T calculated by Duan et al. [27].
The depairing electron correlations in the Eliashberg
100
formalism are described with the use of the formula:
µ⋆(ω ) = µ⋆θ(ω −|ω |). The quantity µ⋆ denotes the
n C n
50 Coulomb pseudopotential, θ is the Heaviside function.
ω represents the cut-off frequency: ω =3Ω , where
C C max
0 Ωmax is the Debye frequency. It should be noted that
100 125 150 175 200 225 the Coulomb pseudopotential was defined by Morel and
p (GPa) Anderson [37]:
FIG.2: Theinfluenceofthepressureonthevalueofthecrit- µ
ical temperature - H3S (the red circles) and D3S (the green µ⋆ = . (4)
circles) [26]. The lines were obtained using the approxima- 1+µln ωωlen
tion procedure. The squares represent the results obtained (cid:16) (cid:17)
withthehelpoftheclassical Eliashbergequationsinthehar- The symbol µ is given by the formula: µ = ρ(0)U,
monicapproximation,thetrianglerepresentstheanharmonic whereasρ(0)isthevalueoftheelectrondensityofstates
analysis, and the bluespheres denote expression (5). at the Fermi level, and U is the Coulomb integral. The
quantity ω represents the characteristic electron fre-
e
quency and the logarithmic phonon frequency is given
by: ω =exp 2 ΩmaxdΩα2(Ω)F(Ω)ln(Ω) .
2.5 ln λ 0 Ω
InFig.2,wehmaRrkedthevaluesofthecriiticaltempera-
turecalculatedwiththehelpoftheEliashbergequations.
2.0
We considered µ⋆ ∈ {0.1,0.2,0.3}. Additionally, we also
placed the value of T for p = 200 GPa, determined
1.5 C
beyond the harmonic approximation [38]. It turns out
that the numerical results can be reproduced using the
1.0 formula (see also Fig.1):
−(1+λ)
0.5 k T =ω exp , (5)
B C ln λ−µ⋆(1+0.4369λ)
(cid:20) (cid:21)
0.0 where the electron-phonon coupling constant should be
125 150 175 200 calculated from: λ=2 ΩmaxdΩα2(Ω)F(Ω).
p (GPa) 0 Ω
Onthisbasis,itwasfoundoutthatthevaluesµ⋆corre-
FIG.3: Theblueline-theexperimentalvaluesoftheisotope R
spondingto[T ] wereequalto0.239and0.286,respec-
coefficient on the basis of formula (1). The squares were ob- C exp
tivelyforthepressureat130GPaand200GPa(thehar-
tainedintheframework oftheclassical Eliashbergformalism
in the harmonic approximation. The triangle corresponds to monic approximation), and 0.146 (the anharmonic anal-
theclassical Eliashberg formalism - theanharmonic analysis. ysis).
Theredcirclewasobtainedassumingthestrongrenormaliza- The expression on the isotope coefficient was derived
tion of the normal state by theelectron density of states. using the dependence:
ω dT
ln C
α= . (6)
tions [35], [36]: 2T dω
C ln
π 1100 K(iω −iω )−µ⋆(ω ) Thus:
n m m
ϕ = ϕ , (2)
n β ω2 Z2 +ϕ2 m 1 (1+λ)(1+0.4369λ)(µ⋆)2
m=X−1100 p m m m α= 2"1− (λ−µ⋆(1+0.4369λ))2 #. (7)
1100
1 π λ(iωn−iωm) The theoretical results have the following form:
Z =1+ ω Z , (3)
n m m
ωnβ m=−1100 ωm2 Zm2 +ϕ2m α(130GPa) = 0.432 and α(200GPa) = 0.397 (the har-
X monicapproximation),andα(200GPa)=0.477(thean-
where ϕ = ϕ(iω ) reprepsents the order parameter harmonicapproach). It canbe easily seen that the theo-
n n
function, and Z = Z(iω ) denotes the wave func- reticalvalue ofthe isotopecoefficientforp=200GPain
n n
3
harmonicapproximationqualitativelywellreproducethe normal state in the studied system.
experimental data. In the case of p = 130 GPa the dis-
LetusconsidertherenormalizedGreenfunctionofthe
crepancy between the Eliashbergresult and the resultof
normal state, in which the depreciation of the electron
the measure is extremely high, whichmeans the collapse
density of states was taken into account [42]:
of the classical theoretical description.
Thehighvalueoftheisotopecoefficientinthetermsof
thelowerpressurescanbetriedtoexplainbythepairing iωnτ0+εkτ3 iωnτ0+εkτ3
mechanism other than the electron-phonon mechanism Gk(iωn)=−ω2 +ε2 +B2A− ω2 +ε2 (1−A),
n k n k
[39]. However, the modifying of the classical Eliashberg (8)
formalism should also be considered. From the theo- where τ , τ are the Pauli matrices associated with the
0 3
retical point of view it highlights the big change of the normalstate and εk is the electronenergy. The parame-
electron density of states at and near the Fermi surface tersA∈h0,1iandB determinethe depthandthe width
together with the pressure change. The ab initio calcu- of the decrease in electron density of states with respect
lations performed for p=210 GPa suggest the existence to the baseline at the Fermi level. Deriving the Eliash-
of the sharp peak of ρ(ε) very close to the Fermi surface berg equations for the renormalized Green function and
[40]. The peak moves away from the Fermi surface and using the approximationsdiscussed in paper [42], the al-
vanishes for the lower pressures [27], [28], [41]. Hence, gebraic equation on the critical temperature can be ob-
physically this means the significant modification of the tained:
λ ω λ 1 1 iB 1 igB
ln
1= ln − f Ψ +2f ReΨ + +2f ReΨ + , (9)
1 2 3
1+λ 2πk T 1+λ 2 2 2πk T 2 2πk T
(cid:18) B C(cid:19) (cid:20) (cid:18) (cid:19) (cid:18) B C(cid:19) (cid:18) B C(cid:19)(cid:21)
where: A = 0.904 and B = 29.12 meV. Physically this means
the very sharp drop in the electron density of states at
1/2
(1−A)(1+λ)+A andneartheFermilevelinthenarrowenergyrange. The
g = , (10)
1+λ obtained result in the natural manner can be associated
(cid:20) (cid:21)
with the offset of the ρ(ε) peak from the Fermi surface.
(1−A)2
f = , (11)
1 g2 In conclusion, basing on the experimental data we de-
termined the range of variation of the isotopic coeffi-
cientfor H S superconductorinthe functionof the pres-
3
1 1−A−g2 2 sure. We showed that the isotope coefficient accepts the
f2 = 2g2 g2−(1−A)2+ 1−g2 , (12) anomalouslyhighvaluesintheareaofthelowerpressures
" (cid:0) (cid:1) # (∼ 130 GPa). On the other hand, for the higher pres-
sures (∼200 GPa), the values of α are lower than those
in the BCS theory. The conducted theoretical analysis
1 1−A−g2 2 provedthatthelowvaluesoftheisotopecoefficientcould
f =− . (13)
3 2g2 "(cid:0) 1−g2 (cid:1) # be reproduced in the framework of the classical Eliash-
berg formalism. The anomalouslyhigh values of α could
The symbol Ψ denotes the digamma function. be induced by the strong renormalization of the normal
Wesolvednumericallyequation(9)assumingtheinput state associated with the significant changes of the elec-
parametersforthepressureat130GPa. Itturnsoutthat tron density of states with the change in the pressure.
equation(9)allowstoreproducethe experimentalvalues Note that the proposed model does not require the non-
ofthe criticaltemperatureandthe isotopecoefficientfor phonon pairing mechanism.
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