Novel high pressure structures and superconductivity of niobium disulfide

2.3 MB·

by  Zhong-Li Liu

Novel high pressure structures and superconductivity of niobium disulfide
Zhong-Li Liu,1,2,∗ Ling-Cang Cai,2 and Xiu-Lu Zhang3
1College of Physics and Electric Information, Luoyang Normal University, Luoyang 471022, China
2Laboratory for Shock Wave and Detonation Physics Research,
Institute of Fluid Physics, P.O. Box 919-102, 621900 Mianyang, Sichuan, China
3Laboratory for Extreme Conditions Matter Properties,
Southwest University of Science and Technology, 621010 Mianyang, Sichuan, China
(Dated: January 16, 2014)
4
1 We have investigated the pressure-induced phase transition and superconducting properties of
0 niobium disulfide (NbS2) based on the density functional theory. The structures of NbS2 at pres-
2 sures from 0 to 200 GPa were predicted using the multi-algorithm collaborative (MAC) structure
n prediction technique. The previously known 1T-, 2H-, and 3R-NbS2 were successfully reproduced.
Inaddition,manymetastablestructureswhicharepotentialtobesynthesizedwerealsodiscovered.
a
Based on the enthalpy calculations, we found that at 26 GPa NbS transits from the double-
J 2
hexagonal (2H) structure to the tetragonal I4/mmm structure with a 10.6% volume reduction.
5
The calculated elastic constants and phonon dispersion curves of I4/mmm-NbS confirm its me-
1 2
chanical and dynamical stability at high pressure. More interestingly, the coordination number of

NbinI4/mmmstructureiseightwhichislargerthanthatinthetraditionalmetaldichalcogenides,
]
n indicating a new type of bondings of Nb and S atoms. In the new Nb-S bondings, one Nb atom
o and neighboring eight S atoms form a [NbS8] hexahedron unit. Furthermore, I4/mmm-NbS2 ex-
c hibitsahighersuperconductingcriticaltemperaturethan2H-NbS ,asisresultedfromthestronger
2
- electron-phonon coupling coefficients.
r
p
u
s I. INTRODUCTION NbS2 is a two-gap superconductor, similar to NbSe2.
t. Theheatcapacityofa2H-NbS2hasbeenmeasureddown
a
to0.6Kandinmagneticfieldsupto14TbyKaˇcmarˇcik
m Transition metal dichalcogenides (TMDs) MX (M =
2 etal.9 Thetemperaturedependenceoftheelectronicspe-
Nb, Ta, Mo, W, X = S, Se, Te) have intriguing proper-
- cific heat can be attributed to either the existence of a
d ties, ranging from insulator to metal and superconduc-
strongly anisotropic single-energy gap or a two-gap sce-
n tor, and thus always attract extensive interests of ex-
o perimentalists and theorists. Thanks to their in-plane nariowiththelargegapabouttwicebiggerthanthesmall
c one. The field dependence of the Sommerfeld coefficient
covalent bondings and weak interlayer van der Waals
[ inducesamagneticfielddependenceofthesuperconduct-
interactions, they could be easily exfoliated down to a
ing anisotropy.9 The two-gap scenario conclusions are
1 monolayerwhichshowsveryexoticproperties. Forexam-
v ple,bulkMoS isanindirect-band-gapsemiconductor,1,2 supported by the absence of in-plane gap anisotropy in
8 2 recent STM imaging of the vortex lattice in NbS .8
while the monolayer MoS is a direct-band-gap semicon- 2
9 2
3 ductor.3 Consequently, the TMDs have shown exciting 2H-NbS2 has a layered structure and therefore has
3 prospects for a variety of applications, such as catalysts large anisotropic electrical, optical, and magnetic prop-
. andlubricantsinthepetroleumindustry4,promisingap- erties. It has been applied as catalyst for the purifi-
1 plications in nanoelectronics and optoelectronics,5 and cation of petroleum,14 cathode materials in secondary
0
energy storage applications.6 batteries,15 humidity sensors,16,17 and so on. In exper-
4
1 The typical representative of TMDs is 2H-NbSe , iment, presently the one-layer trigonal 1T-NbS2,18 two-
: showing a large charge density wave (CDW) (at 33 K2) layer hexagonal 2H-NbS2,19 and three-layer rhombohe-
v
i that coexists with superconductivity (Tc = 7.2 K).7 Nio- dral 3R-NbS220,21 heve been synthesized. Large-scale
X synthesis of 3R-NbS nanosheets has also been recently
biumdisulfide(NbS )alsobelongstothefamilyofTMD 2
r compounds. But the2CDW order appeared in 2H-NbSe realized.17 Different low dimensional structures of NbS2
a is absent in 2H-NbS ,8–10 and its occurrence is sup2- have different physical and chemical properties. Low
2
pressed by the large anharmonic effects.10 However, it dimensional materials depend on and can be exfoliated
from bulk materials. It is necessary to uncover as many
alsoshowssuperconductivityatasimilartransitiontem-
perature of T = 6 K.11–13 More interestingly, the T of crystal structures of NbS2 as possible. From some new
c c
crystals, it is expected to exfoliate some new-functional
2H-NbS increasessmoothlyfrom6Katzeropressureto
2
∼8.9Kat20GPa,13 alsosimilartothebehaviorofT in low dimensional materials.
c
2H-NbSe which increases to ∼ 8.5 K at 10 GPa.13 The It is known that pressure is able to modulate the
2
upper critical field of 2H-NbS has an initial decrease properties of materials through changing their crystal
2
as pressure increases, contrary to the increase of T , but structures. Furthermore, the structures of NbS un-
c 2
above 8.7 GPa it increases again with pressure.13 der high pressure are fundamental to understand its su-
2
perconductive properties. The mechanism of pressure- TABLEI:Thecomparisonofthecalculatedlatticeconstants
a,c,andc/aofdifferentNbS structureswiththecorrespond-
inducedsuperconductivityandthesuperconductingtem- 2
ing experimental values.
perature in NbS above 20 GPa still remain unknown to
2
us. This motivates us to investigate the superconductiv- Structure Method a(˚A) c(˚A) c/a P (GPa) Reference
ity of NbS2 at higher pressures. In this work, we first 1T-NbS2 VASP-LDA 3.253 5.341 1.642 0 This work
predicted the high-pressure structures of NbS and de- Experiment 3.420 5.938 1.736 0 18
2
2H-NbS VASP-LDA 3.287 11.421 3.475 0 This work
termined its phase transition sequence using the multi- 2
Experiment 3.418 11.860 3.470 0 18
algorithm collaborative (MAC) crystal structure predic-
Experiment 3.310 11.890 3.592 0 19
tiontechniquecombinedwiththedensityfunctionalthe-
Experiment 3.330 11.950 3.589 0 10
ory(DFT).Thenwecalculatedthesuperconductingcrit-
3R-NbS VASP-LDA 3.286 17.577 5.349 0 This work
2
ical temperature through electron-phonon coupling cal-
Experiment 3.335 17.834 5.336 0 17
culations.
The paper is organized as follows. Section II contains
thecomputationaldetails. Theresultsanddiscussionare
presented in Sec. III. Conclusions follow in Sec. IV.
III. RESULTS AND DISCUSSION
A. Structure prediction for NbS at high pressures
2
In our MAC structure searches,22 the structures were
II. COMPUTATIONAL DETAILS
generated with symmetry constraints in the first gener-
ation and optimized with vasp package at fixed pres-
In order to determine the high-pressure structures of sures. The pressures applied to crystal structures in
NbS , we searched its low-energy structures from 0 to optimizations go from 0 to 200 GPa with the interval
2
200GPausingourdevelopedMACcrystalstructurepre- of 20 GPa. At each fixed pressure, the enthalpies of
diction technique.22 The multi algorithms including the these structures were calculated and compared to find
evolutionary,thesimulatedannealing,andthebasinhop- the proper path towards the lowest-enthalpy structure.
ping algorithms are combined to collaboratively search Results show that the previously known 2H-NbS2 has
theglobalenergyminimaofmaterialswiththefixedsto- the lowest enthalpy at lower pressures (below 20 GPa)
ichiometry. The MAC algorithm and all the relevant and the I4/mmm structure has the lowest enthalpy at
techniquesareincorporatedintheMusecode.22 There- higher pressures (above 20 GPa). The previously known
sults were also carefully cross checked and confirmed by 1T-and3R-NbS2 structureswerealsoeasilyreproduced.
the Calypso code,23–25 which is based on the particle Thelarge-scale3R-NbS2 nanosheetsaresynthesizedvery
swarm optimization algorithm. recently.17 More interestingly, we found a new two-layer
hexagonal structure whose energy is very close to 3R-
The ab initio optimizations for every structure gen- NbS at ambient pressure. We refer to it as 2H(cid:48)-NbS .
2 2
erated by the Muse code were performed with vasp According to energy criterion, it is potential to be syn-
package.26,27 We tested the local density approxima- thesized in experiment. Meanwhile, many other struc-
tion (LDA) and the generalized gradient approximation tures were found to be energetically competitive, includ-
(GGA) parametrized by Perdew, Burke and Ernzerhof ing P3m1, P6 22, P6 22 structures, and so on. Among
4 2
(PBE)28 for exchange correlation energy. The two ap- these structures, the trigonal P3m1 structure has lower
proximations give the similar structures order in struc- energy with respective to the known 3R-NbS in the
2
ture prediction. While the LDA calculated lattice con- whole pressure range of interest. So, it is also expected
stantsarebetterthantheGGAforNbS2. Sointhestatic to be synthesized in experiment.
calculations, we adopted the LDA exchange-correlation The calculated lattice constants of 1T-, 2H-, and 3R-
functional. Theelectron-ioninteractionsaredescribedby NbS are listed in Table. I, in comparison with experi-
2
the projector augmented wave (PAW) scheme.29,30 The mental values.10,17–19 We note that the lattice constants
pseudopotentialsforNbandShavethevalenceelectrons’ a and c of the three structures are all slightly under-
configurations of 4p65s14d4 and 3s23d4, respectively. To estimated in our LDA calculations. But the calculated
achieve good convergences the kinetic energy cutoff and c/a values are all in good agreement with experiments.
the k-point grids spacing were chosen to be 500 eV and To further examine the three structures, we also simu-
0.02˚A−1,respectively. Theaccuraciesofthetargetpres- lated their X-ray diffraction (XRD) patterns and com-
sureandtheenergyconvergenceforalloptimizationsare pare them with experimental data. The calculated XRD
better than 0.1 GPa and 10−5 eV, respectively. The patternsofthethreestructuresareallinexcellentagree-
searched systems contained 6, 9, 12, 15 and 18 atoms mentwithcorrespondingexperiments(Fig.1),indicating
in the unit cell. that each structure is identical to the known one.
3
(a)
Expt.: 1T-NbS
2
P3m1
unit) (b)
arb. Expt.: 2H-NbS2 (b) top view (c) front view
sity ( P63/mmc
n
e
nt
I (c)
Expt.: 3R-NbS
2
R3m
10 20 30 40 50 60 70
2θ (○) (a) 2H'-NbS (d) left view
2
FIG.1: (coloronline). SimulatedXRDpatternsof2H-NbS ,
2
3R-NbS , and 1T-NbS , in comparison with the correspond-
2 2
ing experimental results (1T: 18, 2H: 18, 3R: 17).
Thenew2H(cid:48)-NbS2 crystalhasthe2H-MoS2 structure (e) 2H-NbS2 (f) shift of 2H-NbS2
and can be formed by shifting one layer of atoms in 2H-
FIG. 2: (color online). The 2H-NbS and 2H(cid:48)-NbS crystal
NbS . The shifting distance is 0.577 lattice constant a 2 2
2 structures. (a) The structure of 2H(cid:48)-NbS . (b) Top view of
alongtypicaldirection. 2H(cid:48)-NbS hassixatomsinprim- 2
2 2H(cid:48)-NbS . (c)Frontviewof2H(cid:48)-NbS . (d)Leftviewof2H(cid:48)-
itivecellwiththelatticeconstantsof3.28,3.28and11.65 2 2
NbS . (e) The structure of 2H-NbS . (f) The shift direction
˚Aatambientpressure. TheNbandSatomsareatWyck- of on2e layer of 2H-NbS to form 2H2(cid:48)-NbS (top view).
2 2
off’s 2c positions (1/3, 2/3, 1/4) and 4f positions (1/3,
2/3, 0.62), respectively. We show the 2H(cid:48)-NbS struc-
2
ture and the shifting direction in Fig. 2. The shifting
direction is parallel to the layer plane (Fig. 2 f). That B. Phase transition and structural stability of
is to say that the structures of the two layers are the NbS
2
same. Theuniquedifferencebetween2H(cid:48)-and2H-NbS
2
is the relative positions of the two layers. The coordi-
In order to obtain the phase-transition sequence of
nation numbers of Nb atoms in both 2H(cid:48)- and 2H-NbS
2 NbS under compression, we calculated the energies for
2
are six. One Nb atom and the neighboring six S atoms
its different phases at 0 K and pressures from 0 to 200
form a [NbS ] trigonal prismoid. Accordingly, the coor-
6 GPa. The enthalpies vs pressure data of different struc-
dination numbers of S atoms in both 2H(cid:48)- and 2H-NbS
2 tures with respective to 2H-NbS are plotted in Fig. 4,
2
are three.
from which we note at 0 K the previously known hexag-
ThenewI4/mmmstructureisplottedinFig.3. Ithas onal 2H-NbS is stable up to 26 GPa. Above 26 GPa,
2
sixatomsinconventionalunitcell(threeinprimitivecell) NbS transitstothetetragonalI4/mmmstructurewhich
2
with the lattice constants of 3.15, 3.15 and 7.91 ˚A at 26 remains stable up to a very high pressure, 200 GPa, the
GPa. The Nb and S atoms are at Wyckoff’s 2a positions upper limit of our interest. Upon compression, NbS ex-
2
(0.0, 0.0, 0.0) and 4e positions (0.0, 0.0, 0.34), respec- hibits a volume reduction of 10.6% at 26 GPa (Fig. 5).
tively. Moreinterestingly,thecoordinationnumberofNb This volume reduction directly results in the decrease of
in I4/mmm is eight. In this new type of bondings, one interlayerdistanceandtheaggregationofSatomsaround
Nb atom and neighboring eight S atoms form a [NbS8] Nbatoms. Althoughatambientconditions,1T-and3R-
hexahedron. The coordination number of S is four. To NbS haverelativelyhigherenergiesthan2H-NbS ,they
2 2
our knowledge, this type of covalent bondings has not have been synthesized successfully. The energies of 2H(cid:48)-
beenreportedinTMDcrystals. Ingeneral,inTMDsthe and P3m1-NbS are close to that of 3R-NbS , so we be-
2 2
metal atom has traditional six nearest neighbors.6 This lieve they are both potential to be synthesized in exper-
new type of eight nearest neighbors in I4/mmm-NbS2 iment. After all, the trigonal P3m1 structure has lower
implies new potential chemical and physical properties, energy than the known 3R-NbS in the whole pressure
2
especially in two-dimensional crystals. range. The energies of P6 22 and P6 22 structures are
2 4
4
20
2H
18 I4/mmm
u.) 16
3/f.Å
me ( 14 -10.6%
u
(b) top view (c) front view ol
V
12
10
0 50 100 150 200
Pressure (GPa)
FIG. 5: (color online). The equation of states of NbS . The
2
(a) I4/mmm (d) left view verticaldashcurveindicatesthevolumereductionofNbS2 at
the phase transition point, 26 GPa.
FIG.3: (coloronline). PredictedI4/mmmcrystalstructure.
(a)ThestructureofI4/mmm-NbS . (b)Topview. (c)Front
2
view. (d) Left view.
are confirmed by their elastic constants (shown in Ta-
ble II and Fig.6) according to the elastic criteria of the
hexagonal systems,31
C >|C |, (C +2C )C >2C2 , C >0, (1)
both much higher than that of I4/mmm structure. So 11 12 11 12 33 13 44
theGibbsfreeenergybarrieristohighforNbS toover-
2 and tetragonal systems,
come.
C >0, C >0, C >0, C >0,
11 33 44 66
C >C , C +C >2C , (2)
11 12 11 33 13
1 2H' 2(C +C )+C +4C >0,
11 12 33 13
3R
respectively. The increasing of the elastic constants of
nce (ev/f.u.) 0.50 12PTH3m1 Iiits4ya/almssompmmrees-cNshubarnSei2cinawcllirytehassptearsbesl(esFuairgte.6ar)me.flbTeicehtnetitnsaenewdnh2haHing(cid:48)chesdptrrusetcsastbuuirrlee-
e according to the elastic criteria of the hexagonal crys-
differ 26 GPa tals.31 While, the P3m1 structure is only stable at low
y −0.5 pressures. It becomes mechanically unstable as pressure
alp P6422 increases because of the appearance of negative shear
h
Ent −1 P6222 modulus C14. It is also worthy to note that the shear
modulus C of 2H-NbS increases with pressure, but
I4/mmm 44 2
the C values of 2H(cid:48)-NbS remain small as pressure in-
44 2
creases. This implies 2H-NbS is more stable than 2H(cid:48)-
−1.5 2
0 50 100 150 200 NbS . So, it is easier to synthesize 2H-NbS in experi-
2 2
Pressure (GPa) ment other than 2H(cid:48)-NbS .
2
To further check the dynamical stability of the new
FIG. 4: (color online). Enthalpy differences of predicted structures, 2H(cid:48)- and I4/mmm-NbS , we determined
2
structuresrelativeto2H-NbS2structureunderhighpressure. theirvibrationalfrequenciesusingdensityfunctionalper-
turbation theory (DFPT),32,33 as implemented in the
QUANTUM-ESPRESSO package.34 For the exchange-
The mechanical stability of 2H- and I4/mmm-NbS correlation functional we used the Perdew Zunger local-
2
5
TABLEII:TheelasticconstantsofdifferentNbS2 structures (a) 5
under high pressure. The pressure (P) and elastic constants
are all in GPa. 2.5
Structure P C C C C C C
2H 0.00 1741.116 77.1324 9.7130 58.3830 65.4246 -14 V) 0
e
11.90 247.87 94.07 48.92 232.36 99.38 - y (
g
21.69 314.77 127.03 70.09 324.01 139.63 - er
n
2H(cid:48) 0.00 163.26 83.75 15.49 52.47 2.35 - E −5
19.93 267.04 121.63 82.27 314.33 10.43 -
47.17 336.47 186.91 134.25 593.40 15.98 - −7.5
P3m1 0.00 186.36 68.43 20.35 103.46 15.88 1.07
−10
13.73 262.74 81.66 57.44 272.51 64.46 -10.50 Γ XY Σ Γ Z Σ N P Y Z
1 1
23.10 324.97 104.56 72.18 369.49 94.47 -20.06
500
(b)
400
1200 -1m)
C11 c300
C12 y (
1000 C13 nc
C14 ue200
Pa) 800 CC3434 Freq
G 100
nts (
consta 600 0Γ XY Σ Γ Z Σ1N P Y1 Z
c
asti 400
El FIG. 7: (color online). The band structure and phonon dis-
persion curve of I4/mmm-NbS at 60 GPa. (a) the band
2
200 structure, (b) the phonon dispersion curve (left) and phonon
density of states (right).
0
20 40 60 80 100 120
Pressure (GPa)
are 2×4×4, also giving 8 wave vectors.
FIG. 6: The high-pressure elastic constants of I4/mmm-
Phonon dispersion curves (Figs. 7(b) and 8(a)) do not
NbS .
2 show any imaginary frequencies, indicating dynamical
stability of I4/mmm- and 2H(cid:48)-NbS . So we believe
2
I4/mmm- and 2H(cid:48)-NbS are both mechanically and dy-
2
namicalstable. Thephonondispersioncurveof2H-NbS
2
are also presented in Fig. 8(b), compared to the experi-
mental data.10 The agreement of our calculated phonon
frequencies and the 300 K experimental data is quite
density approximation (LDA)35 and ultrasoft pseudopo-
good. By comparing the phonon dispersion curves of
tential.36 We applied the Vanderbilt ultrasoft pseudopo-
2H- and 2H(cid:48)-NbS , we note the phonon frequencies of
2
tentials for Nb and S with the valence electrons con-
2H(cid:48)-NbS exhibit softening near to A point (along the
figurations 4s24p64d25s2 and 3s23p4, respectively. The 2
Γ-A,A-LandH-Adirections),implyingitsmetastability
ultrasoft pseudopotentials were generated with a scalar-
compared to 2H structure. This is consistent with the
relativistic calculation.
conclusions from the elastic constants calculations.
We careful tested on k and q grids, the kinetic energy
cutoff, and other technical parameters to ensure good
convergence of phonon frequencies. The kinetic energy
cutoff, the energy cutoff for the electron density, and C. Electron-phonon coupling and
superconductivity
the k grids were chosen to be 40 Ryd., 400 Ryd., and
16×16×16 Monkhorst-Pack (MP)38 meshes in both to-
tal energy and phonon dispersion calculations, respec- We calculated the superconducting transition temper-
tively. We applied the Gaussian smearing method with ature T of NbS using the Allen-Dynes39 form of the
c 2
the smearing width of 0.05 Ryd. For the dynamical ma- McMillan40 equation,
tricesoftheI4/mmmstructure,weuseda2×2×2qgrid,
(cid:20) (cid:21)
giving 8 wave vectors q in the irreducible wedge of the ω 1.04(1+λ)
T = lnexp − , (3)
first BZ. For the 2H- and 2H(cid:48)-NbS2, the q grid meshes c 1.2 λ−µ∗(1+0.62λ)
6
(a) The Eliashberg spectral function, α2F(ω), which mea-
400 sures the contribution of the phonons with frequency ω
-1m) 300 to the scattering of electrons,41 can be written as,40
c
y (
Frequenc 120000 α2F(ω)= 2πN1((cid:15)F)(cid:88)qν ωγqqννδ(ω−ωqν), (5)
0Γ M K ΓA L H A where N((cid:15)F) is the EDOS at the Fermi level. The
(b) linewidth of the phonon mode was calculated from,40
400
(cid:88)
-1m) 300 γqν =2πωqν |gkqν+qj(cid:48),kj|2δ((cid:15)kj−(cid:15)F)δ((cid:15)k+qj(cid:48)−(cid:15)F), (6)
c
y ( kjj(cid:48)
nc 200
ue where gqν is the electron-phonon coupling matrix
q k+qj(cid:48),kj
Fre 100 element. The Coulomb pseudopotential µ∗ was taken
the typical value 0.10 in all the superconducting critical
0
Γ M K ΓA L H A temperatures (T ) calculations.
c
ThecalculatedT of2H-andI4/mmm-NbS areplot-
c 2
tedinFig.9, comparedwithrecentexperimentaldata.37
FIG. 8: (color online). The phonon dispersion curve of 2H(cid:48)-
TheresultingT sof2H-NbS areinverygoodagreement
(a) and 2H-NbS (b) at 0 GPa (lines). The experimental c 2
2 with experiment and increase with pressure. It is inter-
data [Ref. 10] of 2H-NbS at 2 K (open diamonds) and 300
2
esting that the T of I4/mmm structure is higher than
K (filled circles) are also plotted for comparison. c
that of 2H structure and decreases with pressure. This
is resulted from the stronger electron-phonon coupling
coefficients λ in I4/mmm-NbS (Fig. 9). The phonon
2
calculations indicate that I4/mmm is unstable below 10
20 GPa. ThehighestT ofI4/mmm(at10GPa)is17.83K.
c
From the electronic energy band structure of I4/mmm-
18
1.5 NbS2 (Figs. 7(a)), we note it is metallic. It is previously
16 known that 2H-NbS2 is also metallic, so pressure does
notchangethemetallicpropertiesofNbS ,butenhances
2
14
the electron-phonon coupling effects and thus increases
K) the the superconducting critical temperature.
(C 12 λ
T
1
10
IV. CONCLUSIONS
8
6 In conclusion, we predicted three new 2H(cid:48)-, P3m1-,
and I4/mmm-NbS structures using the MAC crystal
2
4 0.5 structure prediction technique. The new 2H(cid:48)-NbS can
0 20 40 60 80 100 2
P (GPa) be formed by shifting the layer of atoms along typical
direction parallel to the layer plane. Based on enthalpy
calculations, we found 2H-NbS transits to the tetrago-
FIG. 9: (color online). The comparison of calculated super- 2
nal I4/mmm structure at 26 GPa. The new bondings in
conducting critical temperatures with experimental results
I4/mmmforma[NbS ]hexahedron,whichhasnotbeen
37. The calculated electron-phonon coupling coefficients λ 8
reportedinTMDcrystals. Moreinterestingly,thesuper-
are also plotted.
conductingtemperatureofI4/mmm-NbS ishigherthan
2
that of 2H-NbS and decreases as pressure increases, re-
2
sulted from the stronger electron-phonon coupling coef-
ficients λ in I4/mmm-NbS . In the stability region of
2
I4/mmm structure, the highest T is 17.83 K.
c
where λ (= 2(cid:82)∞α2F(ω)/ωdω) is the electron-phonon
0
couplingconstant,ω thelogarithmicaveragefrequency,
ln
and µ∗ the Coulomb pseudopotential. The logarithmic V. ACKNOWLEDGMENTS
average frequency is calculated by
The research was supported by the National Natural
2 (cid:90) ∞ Science Foundation of China (11104127, 11104227), the
ω =exp{ dωα2F(ω)lnω/ω}. (4)
ln λ NSAF of China under grant No. U1230201/A06, the
0
7
Grand No. 2011A140019.
Project2010A0101001fundedbyCAEP,andtheScience
ResearchSchemeofHenanEducationDepartmentunder
∗ Electronic address: [email protected] Commun. 31, 577 (1979).
1 L. Wei, C. Jun-fang, H. Qinyu, and W. Teng, Physica B 22 Z. L. Liu, arXiv:1303.2802 .
405, 2498 (2010). 23 Y. Wang, J. Lv, L. Zhu, and Y. Ma, Phys. Rev. B 82,
2 Y. Ding, Y. Wang, J. Ni, L. Shi, S. Shi, and W. Tang, 094116 (2010).
Physica B 406, 2254 (2012). 24 Y.Wang,J.Lv,L.Zhu, andY.Ma,Comput.Phys.Com-
3 S. Leb´egue and O. Eriksson, Phys. Rev. B 79, 115409 mun. 183, 2063 (2012).
(2009). 25 J. Lv, Y. Wang, L. Zhu, and M. Y., J. Chem. Phys. 137,
4 P. Raybaud, J. Hafner, G. Kresse, and H. Toulhoat, J. 084104 (2012).
Phys.: Condens. Matter 9, 11107 (1997). 26 G. Kresse and J. Hafner, Phys. Rev. B 49, 14251 (1994).
5 Q. Wang, K. Kalantar-Zadeh, A. Kis, J. Coleman, and 27 G. Kresse and J. Furthmu¨ller, Comput. Mater. Sci. 6, 15
M. Strano, Nature Nanotech. 7, 699 (2012). (1996).
6 M. Chhowalla, H. S. Shin, G. Eda, L.-J. Li, K. P. Loh,
and H. Zhang, Nature Chem. 5, 263 (2012). 28 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.
7 D. Moncton, D. JAxe, and F. DiSalvo, Phys. Rev. B 16, Lett. 77, 3865 (1996).
801 (1977). 29 P. E. Bl¨ochl, Phys. Rev. B 50, 17953 (1994).
8 I. Guillamo´n, H. Suderow, S. Vieira, L. Cario, P. Diener, 30 G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
and P. Rodi`ere, Phys. Rev. Lett. 101, 166407 (2008). 31 Z.-j.Wu,E.-j.Zhao,H.-p.Xiang,X.-f.Hao,X.-j.Liu, and
9 J. Ka˘cmar˘cik, Z. Pribulova´, C. Marcenat, T. Klein, J. Meng, Phys. Rev. B 76, 054115 (2007).
P. Rodi`ere, L. Cario, and P. Samuely, Phys. Rev. B 82, 32 S. Baroni, P. Giannozzi, and A. Testa, Phys. Rev. Lett.
014518 (2010). 58, 1861 (1987).
10 M. Leroux, M. Le Tacon, M. Calandra, L. Cario, M.- 33 S.Baroni,S.D.Gironcoli,A.D.Corso, andP.Giannozzi,
A. M´easson, P. Diener, E. Borrissenko, A. Bosak, and Rev. Mod. Phys. 73, 515 (2001).
P. Rodi`ere, Phys. Rev. B 86, 155125 (2012). 34 P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,
11 J. Wilson, F. DiSalvo, and S. Mahajan, Adv. Phys. 24, C.Cavazzoni,D.Ceresoli,G.L.Chiarotti,M.Cococcioni,
117 (1975). I.Dabo,A.D.Corso,S.D.Gironcoli,S.Fabris,G.Fratesi,
12 H. Y. and R. Aoki, J. Phys. Soc. Jpn. 55, 1327 (1986). R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj,
13 V. G. Tissen, M. R. Osorio, J. P. Brison, N. M. Nemes, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri,
M.Garc´ıa-Herna´ndez,L.Cario,P.Rodi`ere,S.Vieira, and R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto,
H. Suderow, Phys. Rev. B 87, 134502 (2013). C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen,
14 C. Geantet, J. Afonso, M. Breysse, N. Allali, and A.Smogunov,P.Umari, andR.M.Wentzcovitch,J.Phys.
M. Danot, Catal. Today 28, 23 (1996). Condens. Matter 21, 395502 (2009).
15 N. Kumagai and K. Tanno, Electrochim. Acta 36, 935 35 J.P.PerdewandA.Zunger,Phys.Rev.B23,5048(1981).
(1991). 36 D. Vanderbilt, Phys. Rev. B 41, 7892 (1990).
16 W.M.R.Divigalpitiya,R.F.Frindt, andS.R.Morrison, 37 V. G. Tissen, M. R. Osorio, J. P. Brison, N. M. Nemes,
J. Phys. D: Appl. Phys. 23, 966 (1990). M.Garc´ıa-Herna´ndez,L.Cario,P.Rodi`ere,S.Vieira, and
17 W. Ge, K. Kawahara, M. Tsuji, and H. Ago, Nanoscale H. Suderow, Phys. Rev. B 87, 134502 (2013).
5, 5773 (2013). 38 H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188
18 C. J. Carmalt, T. D. Manning, I. P. Parkin, E. S. Peters, (1976).
and A. L. Hector, J. Mater. Chem. 14, 290 (2004). 39 B. Allen and R. C. Dynes, Phys. Rev. B 12, 905 (1975).
19 F. Jellinek, G. Brauer, and H. Mu¨ller, Nature 185, 376 40 W. L. McMillan, Phys. Rev. 167, 331 (1968).
(1960). 41 G. M. Eliashberg, Sov. Phys.-JETP 16, 780 (1962).
20 W.B.Clark,J.Phys.C:SolidStatePhys.9,L693(1976).
21 S.Onari,T.Arai,R.Aoki, andS.Nakamura,SolidState