Table Of ContentSuperconductivity of monolayer Mo C: the key role of functional groups
2
Jun-Jie Zhang1 and Shuai Dong1,a)
Department of Physics, Southeast University, Nanjing 211189, China
(Dated: 5 January 2017)
Monolayer Mo C is a new member of two-dimensional materials. Here the electronic structure and lattice
2
dynamics of monolayer Mo C are calculated. According to the electron-phonon interaction, it is predicted
2
that monolayer Mo C could be a quasi-two-dimensional superconductor and the effects of functional-groups
2
are crucially important considering its unsaturated surface. Despite the suppressed superconductivity by
7
chalcogen adsorption, our most interesting prediction is that the electron-phonon interaction of monolayer
1
Mo C can be greatly enhanced by bromine absorbtion, suggesting that Mo CBr as a good candidate for
0 2 2 2
2 nanoscale superconductor.
n
a
I. INTRODUCTION
J
4
Two-dimensional(2D) graphene-likecarbidesandcar-
] bonitrides(MXenes,e.g. Ti CandNb C)haveattracted
2 2
l
l enormous interest for their novel chemical and physical
a
properties,sincetheyweresuccessfullyproducedbyetch-
h
- ing the A layers of MAX phase (M is an early transition
s metal, A is an element from group IIIA or IVA, and X
e
is carbon or nitrogen)1–3. Due to the unsaturated sur-
m
face with unpaired electrons, the surfaces of MXenes al-
.
t ways easily adsorb various functional groups (e.g. F, O,
a
or/andOHgroup)duringetching,thusthechemicaland
m
physicalproperties arevaryingwith variousadsorptions.
-
d For this reason, MXenes and their functionalized ones
n have been widely investigated regarding the magnetism, FIG. 1. Side views of atomic structures of monolayer Mo C
2
o electronic structures, as well as catalytic properties and (a) and threeatom-adsorbed configurations (b)-(d).
c energy storage4–8.
[
Recently, 2D layered Mo2C as a new member of
1 MXenes was formed from Mo2Ga2C thin films. The 4K(thickfilm)tonear0Kwhenitsthicknessislessthan
v metallic nature of Mo-Ga bond is weaker than Mo-C 3.5 nm15,16. Moreover, analogous to other members of
4 bond which has a mixed covalent/metallic/ionic char-
MXenes, the surfaces of monolayerMo C are easily cov-
1 2
9 acter, thus monolayer Mo2C can be produced by selec- ered by functional groups, and Khazaei et. al. indicates
tively etching Ga layer9. Structurally, monolayer Mo C
0 2 the full adsorption of functional groups is more stable
0 is constructed by the Mo-C-Mo sandwich, as shown in than the cases of partial adsorption17. For multilayer
1. Fali.g.s1tu(ad)i,edwhthicehtlhoeorkmsoseilmecitlarircstopr1oTp-eMrtoieSs2.ofKmhaoznaoelaiyeetr. Mo2CMXenes,they(stackinglikeMoS2)seemtobeim-
0 possible due to its unsaturatedsurfaces. In experiments,
Mo C, which was found to be a promising candidate as
7 2 the possible stackings are -Mo CX -Mo CX - and Mo-
1 a high-performance thermoelectric material10. In addi- C-Mo-C-Mo. For the former,2the2effec2ts of2functional
v: tion, the structural, electrical, thermal and mechanical groups are unavoidable. For the latter, the stoichiom-
properties of monolayer Mo C were also studied11.
i 2 etry has changed. These facts motive us to investigate
X Superconductivity in ultrathin films owns promising
thesuperconductivityofpureMo Cmonolayerandthose
2
r future for applications, e.g. superconducting computa- with various functional groups.
a
tional devices, thus great efforts have been made to dis-
Inthiswork,thelatticedynamicsandelectron-phonon
cover 2D superconductors12–14. Considering that mono-
coupling (EPC) of monolayer Mo C and functionalized
layer Mo C is non-magnetic and appears strong metal- 2
2 ones have been studied via first-principles density func-
licity(accordingto first-principlesstudy)10,itbringsthe
tional theory (DFT) and density function perturbation
opportunitytobeanultrathinsuperconductor. However,
theory (DFPT). Our calculations find that the strong
althoughthesuperconductivitywasdiscoveredinbulkα-
EPC in monolayer Mo C may lead to superconductiv-
Mo C (an allotrope of 2D layered Mo C), its supercon- 2
2 2 ity,whiletheoxidationwouldmakeitssuperconductivity
ducting transition temperature (T ) was depressed from
C disappear. Besides, the superconducting T is also ob-
C
viously suppressed by absorbtion of sulfur or selenium.
In contrast, the EPC of monolayer Mo C can be greatly
2
enhanced by bromine absorbtion, leading to a predicted
a)Electronicmail: [email protected] T up to 12 K.
C
2
II. MODEL & METHODS the q space22,23:
∞ α2F(ω)
The electronic structure calculations have been per- λ= λ =2 dω (4)
formed using the Vienna ab initio simulation package Xqν qν Z0 ω
(VASP) with projector augmented wave method18,19.
where EPC constant λ for mode ν at wave vector q is
Generalized gradient approximation of Perdew-Burke- qν
defined by the integration22,23:
Ernzerhof (GGA-PBE) are used with a cutoff energy
600 eV. The spin-orbit coupling (SOC) is also included γ
qν
λ = . (5)
in electronic structure calculations. The phonon disper- qν π~N(ε )ω2
F qν
sion calculations are carried out using ultrasoft pseudo-
potential (including the semicore electrons as valence Toobtainaccurateelectron-phononinteractionmatrices,
electrons in case ofMo) as implemented in PWSCF pro- adense36×36×1gridisadoptedfortheEPCcalculation.
gram of the Quantum-ESPRESSO distribution, which
are calculated within the framework of DFPT20. In the
DFPT calculation, GGA-PBE formulation is also used III. RESULTS & DISCUSSION
with a cutoff energy 35 Ry for the expansion of the elec-
tronic wave function in the plane waves, whereas cutoff A. Unfunctionalized monolayer Mo C
2
energy for charge density and potential is set to be 350
Ry. Structure optimization and electronic structure are
The structure of unfunctionalized monolayer Mo C is
2
repeatedbyusing PWSCFandtheresultsareconsistent
layered hexagonal with a space group of D , and the
3d
with those obtained using VASP.
stacking of the Mo-C-Mo is in the ABC-type along the
The vacuumspaceof∼15˚Ais intercalatedintointer-
hexagonal c axis (see Fig. 1(a)), which is similar to 1T-
laminationtoeliminatetheinteractionbetweenlayers. A MoS . Our optimized lattice constant (2.965 ˚A) is con-
2
12×12 2D grid uniform is applied for both the k-points sistent with the value reported by Khazaei et. al.10.
of the self-consistent and q-points of dynamical matrices
The calculated band structures and density of states
calculations. (DOS) are shown in Fig. 2. As previously reported10,
The EPC calculation is estimated according to the
monolayerMo C indicates strongmetallic behavior,and
Migdal-Eliashberg theory [α2F(ω)], which is given by21: 2
the Fermi surfaces are mainly contributed by Mo’s d-
orbits according to the projected DOS (Fig. 2(a)). The
1 γ
α2F(ω)= δ(ω−ω ) qν (1) maximally localized Wannier functions (MLWFs) can
2πN(ε ) qν ~ω
F Xqν qν partion Mo’s d orbital and C’s p orbital, as shown in
Fig. 2(b). C’s p orbital is mainly occupied on energy
where N(εF) is the electronic DOS at Fermi level, ωqν range from −8 eV to −4 eV and highly hybridize with
denotes phonon frequency of the νth phonon mode with Mo’s d orbitals. The p and p orbitals are degener-
x y
wave vector q, and the phonon linewidth γqν is defined ate, which are higher in energy than the pz orbital, as
by22,23: shown in Fig. 2(a). The Mo’s d-orbital are split due to
2πω 2 C3v crystalline field. Thus the dz2 has the lowest on-
γqν = ΩBZqν Xij Z d3k(cid:12)(cid:12)gkqiν,k+qj(cid:12)(cid:12) δ(εki−εF) sainted denxye)rgayn,dth(edxrzesatnsdadryez)p,atirhweilsaetedreogfenwehraicthe:is(sdixg2h−tyly2
(cid:12)×δ(ε (cid:12)−ε ). (2) higher. The SOC can open the degeneracy at Γ¯ point
k+qj F
(see Fig.2(c)), while it has little effect on the Fermi sur-
where ij denote indices of energy bands, ΩBZ is vol- face. The crossing-Fermi-level bands and corresponding
ume of Brillouin zone, εki and εk+qj are eigenvalues of Fermi surfaces are shown in Fig. 2(d) and (e). The anti-
Kohn-Sham orbitals at given bands and wave vectors. bonding dz2 band (the upper one in Fig. 2(e)) forms the
The gk,qν is the EPC matrix element which can be de- hole pocket Fermi surfaces around M¯ points, while the
termined self-consistently by the linear response theory, electron Fermi surface (the lower one in Fig. 2(e)) is the
which describes the probability amplitude for scattering circular-shape around the Γ¯ point which is mainly con-
of an electron with a transfer of crystalmomentum q, is tributed by dx2−y2 and dxy. Such double Fermi surfaces
determined by22,23: correspond to carriers with multiple effective masses,
charges,and mobilities.
~ dV
gij =( )1/2hψ | SCF ·eˆ |ψ i (3) Then the phononproperties and electron-phononcou-
k,qν 2Mωqν i,k duˆqν qν i,k+q pling are calculated. The SOC is not included in these
calculations, considering its negligible effects on the
whereM istheatomicmass, dVSCF measuresthechange Fermi surfaces. For monolayerMo C, the Ramanmodes
duˆqν 2
of self-consistent potential induced by atomic displace- can be decomposed as A1 ⊕ 2E1 at the zone center
1g g
ment, ψ and ψ are Kohn-Sham orbitals. (seeFig.3(a)),andthecalculatedRamanfrequenciesare
i,k i,k+q
The EPC constant λ is obtained by summation over 150.3 cm−1 and 193.1 cm−1 for the E1 and A1 modes
g 1g
the first Brillouin zone or integration of the α2F(ω) in respectively. Moreover, both Raman modes have strong
3
(a) (b) (c)
A1 E
1g g
LA
TA
ZA
FIG. 3. Phonon properties for single Mo C layer. (a) Cal-
2
culated phonon dispersion. Insert: sketch of Raman modes.
(b-c) Phonon DOS, electron-phonon coupling λ, and Eliash-
berg spectral function.
Todiscussthesuperconductivity,theobtainedα2F(ω)
FIG. 2. Electronic structure of monolayer Mo C. (a) Den- and λ(ω) are also plotted in Fig. 3(c). Their similar
2
sity of states (DOS)andProjected DOS.(b) Projected band shapesindicatethatallF(ω)makecontributionstoEPC.
structure. Red: dz2; Green: dx2−y2 and dxy, Blue: dxz and Due to the factor 1/ω in the definition of λ (see Eq. 4),
dyz. (c)Bandstructurewith(black)andwithoutSOC(red). the contributions from low ω region is more prominent.
(d)Three-dimensionalviewoftwobandscrosstheFermilevel. Exactly,thecalculatedλ(ω =250cm−1)is≈0.56which
(e)Energycouterplotsoftwobands. Blackcurves: theFermi
is beyond 90% of the total EPC (λ(ω = ∞) = 0.63),
surface.
indicating that the phonon modes in the frequency re-
gionbelow250cm−1 havethedominantcontribution. In
particular, three low-lying optical branches have strong
couplingtoelectrons(λ is0.11forE1and0.21forA1 )
qν g 1g coupling to electrons, which make 40% contribution to
according to Eq. 5.
EPC.Therefore,itisnaturaltoexpectthestrongEPCin
The calculated phonon dispersions along major high
monolayer Mo C to induce superconducting state. The
symmetry lines and phonon densities of states (PDOS, 2
T can be estimated using the Allen-Dynes modified
F(ω))for monolayerMo C areshowninFig.3(a). More C
2 McMillan equation22:
dense k-meshes (18× 18× 1 and 24× 24× 1) in self-
consistent calculations are also tested. The maximum ω 1.04(1+λ)
ln
error of obtained phonon frequencies are less than 1%, TC = 1.2 exp[−λ−µ∗(1+0.62λ)], (6)
implying the convergence of phonon calculation. In ad-
∗
dition, the different methodology and pseudo-potential where µ is the Coulomb repulsion parameter and ω is
ln
would lead to a few differences regarding the phonon the logarithmically averaged frequency. When taking a
bandstructures8,11. Infact,ourphononstructureisvery typical value µ∗ =0.1, the estimated T is about 5.9 K.
C
close to that in previous report11, although tiny differ-
ences remain unavoidable.
No imaginary frequency exists in the full phonon B. Functionalized monolayer Mo2C
spectra, indicating the dynamical stability of monolayer
Mo C. Meanwhile, the phonon behavior exhibits several In above study, it has been predict that pure mono-
2
remarkable characteristics. First, near the zone center, layer Mo C maybe a quasi-2D superconductor. How-
2
both the LA and TA branches are near linear while the ever,inrealsituations,thefunctionalgroupatsurfacesof
ZA branch (out-of-plane acoustical mode) is quadratic. monolayer Mo C are unavoidable considering its highly
2
These characters reflect the nature of 2D sheet. In de- unsaturated surfaces. In fact, previous studies reported
tail, the ZA phonon in 2D materials like graphene has that monolayer Mo C transforms from metal to semi-
2
a quadratic dispersion over a wide range of the 2D Bril- conductor with F- and Cl-adsorption10, namely the F-
louinzoneω =a q2,wherea isapositiveconstant and Cl-functionalized monolayer Mo C could not be a
ZA ZA ZA 2
andq is the 2Dphononwavevector. The similarconclu- superconductor. In the following, the changes of super-
sions are also found in monolayer black phosphorene24. conductivity by adsorbing various functional group will
Second, according to the partial PDOS (Fig. 3(b)), the bestudied,whichmayshedlighttotuningthesupercon-
vibrational modes of Mo dominate the low-frequency ductivity of monolayer Mo C in real experiments.
2
regime while those of C dominate the high-frequency Due to the full adsorption is more stable than the
regime, due to their large difference in mass. There is partial case10, the 1 × 1 Mo C unit cell (u.c.) with
2
a large gaparound300cm−1, which partions the optical two functional groups X (one on each surface), i.e.
modes of Mo and C. Mo CX , is adopted in our calculation. Four functional
2 2
4
TABLE I. The calculated total energy for Mo CX of dif- TABLEII.ThecalculatedMo CX ’ssuperconductiveparam-
2 2 2 2
ferent configurations for adsorption is in unit of eV/per u.c.. eters of N(εF) (states/eV), ωln (K), λ,and TC (K).
TheenergyofBB-adsorptionissetasthereference. Theopti-
mizedlatticeconstantaisinunitof˚A.Forthemostfavorable Mo2CO2 Mo2CS2 Mo2CSe2 Mo2CBr2
configuration,thecorrespondingbindingenergy(Eb)(inunit N(εF) 1.3 1.5 1.6 3.3
of eV) is also presented. ω 357.4 326.6 283.7 160.7
ln
AB AC BB a a (Ref. 10) E λ 0.2 0.4 0.4 1.1
b
O 0.72 1.29 0 2.891 2.886 −8.67 TC <0.1 1.0 1.4 12.8
S 0.48 0.69 0 3.087 3.078 −6.08
Se 0.39 0.24 0 3.161 −4.57
shown in Fig. 4(a-d). The three-dimensional view of
Br −0.19 −0.65 0 3.428 3.418 −3.13
three bands as well as the Fermi surface of dx2−y2/dxy
anddz2 areshowninFig.4(e-h). Forchalcogen,itsband
structure looks like that of pristine monolayer Mo C,
2
atoms (X=O, S, Se, and Br) are considered. Consider- namely the upper cone-shape band is surrounded by
ing the symmetry, there are three mostly possible site- lower flower-shape band, although the degree of sur-
configuration for adsorption: AB-adsorption (Fig. 1(b)), rounding is suppressed. Due to the different adsorption
AC-adsorption (Fig. 1(c)), BB-adsorption (Fig. 1(d)). site and valence state of Br, the shapes of Fermi sur-
For the AB-adsorption, one X atom is right above C faces are significantly changed to sun-like patterns for
layer,while another X is right below other side Mo. For Mo CBr .
2 2
the AC-adsorption, each X atom is right above/below The calculated phonon dispersion along major high
other side Mo layer. For the BB-configuration,both two symmetry lines are shown in Fig. 5. No imaginary fre-
X atoms stand above/below the C site. quency exists in the full phonon spectra, indicating the
The crystal structures are fully relaxed upon the ab- dynamicalstabilityoftheMo CX . Therefore,itismore
2 2
sorptions, and the calculated lattice constant as well as likelythatMo CX is ableto be obtainedinrealexperi-
2 2
totalenergy are listed in Table I. Our calculationsare in mentconsideringitsstabilityinthermodynamicsandlat-
good agreement with previous reports10,25, and the BB- tice dynamical. Obviously,the vibrationmodes ofMo C
2
adsorptionisthemostfavorablecaseforchalcogen,while are strongly coupled with the surface X, as revealed by
Mo2CBr2 favors the AC-adsorption. The corresponding the phonon dispersion (Fig. 5). Different X atom con-
binding energy (Eb = EMo2CX2 −EMo2C −EX2) is also tribute to the phonon spectrum in different frequency
calculated and also listed in Table I. All binding ener- rangeduetotheinequivalentmassandbondstrength. In
gies are negative which indicate thermodynamic stabil- particular,forMo CO ,thecontributionsfromOmainly
2 2
ity for all structures. And the value of binding energy locate at the intermediate- and high-frequency regimes,
decreases from Mo2CO2 to Mo2CBr2, implying the ad- indicatingthefactofstrongbondofMo-O.Inthecaseof
sorption of oxygen should be quite possible, as observed Mo CS , the frequencies contributed by S mainly locate
2 2
in real experiment9. According to the Lo¨owdin popula- inthe intermediateregime,while forSe andBrcasesthe
tion analysis,the chargetransfer from Mo to X is about contributions from X is in the low frequency side.
0.48, 0.39, 0.34, and 0.24 electron for O-, S-, Se-, and Our results for α2F(ω), F(ω) and PDOS of Mo CX
2 2
Br-adsorption, respectively. Stronger Coulomb attrac- are shown in Fig. 6. As in the monolayer Mo C case,
2
tion between X and Mo ions can lead to larger binding α2F(ω) and F(ω) of Mo CX have similar peaks, indi-
2 2
energy. cating all vibration modes contribute to EPC. Compar-
The electronic structures for Mo CX are calculated. ing to pristine Mo C, the strength of α2F(ω) have been
2 2 2
Due to the unchanged symmetry of crystalline field, the suppressed by O-, S- and Se-adsorption. And such sup-
splitting of Mo’s d orbitals is similar to pristine mono- pression obviously exists in low frequency regime which
layer Mo C. Quantitatively, the absorption of X further havelargecontributionstoEPC(becauseoftheω−1part
2
raises the energy of doubly-degenerate d and d due inEq.4). TheaverageEPCisalsocalculatedusingEq.4,
xz yz
to the strong hybridization with the p orbitals of X. In aslistedinTableII.CorrespondingT isestimatedfrom
C
∗
details,theon-siteenergydifferencebetweend /d and modifiedMcMillanequation(Eq.6)withµ =0.1(listed
xz yz
dx2−y2/dxy is about 1.72, 0.99, 0.71, 0.21 eV for the O-, in Table II). The results indicate that the superconduc-
S-, Se- and Br-adsorption respectively, according to the tivityisgreatlysuppressedinMo CS andMo CSe and
2 2 2 2
MLWFs calculation. The lower of d /d orbitals can almostdisappearsinMo CO ,asaresultofreducedelec-
xz yz 2 2
be also evidenced in the band structures of Mo CX , as tronic DOS’s at Fermi level and suppressed EPC’s.
2 2
shown in Fig. 4(a-d). All band structures show metal- AsshowninFig.6(d),strengthofα2F(ω)hasbeenim-
lic character after X-absorption, different to previously provedinMo CBr ,andthecorrespondingaverageEPC
2 2
studied F-/Cl-absorption. Similar to pristine monolayer is about 1.09. The obtained λ(ω = 300 cm−1) ≈ 0.97
Mo2C, dx2−y2/dxy and dz2 of Mo still make dominant is approximately beyond 88% of the total EPC (λ(ω =
contributions around the Fermi level, although the con- ∞) = 1.09), and estimated T is up to 12.8 K. To be
C
tributions from dx2−y2/dxy are reduced more or less, as exact, EPC of vibration modes (Eg1 and A11g) at Γ¯ point
5
FIG. 4. Electronic structures of Mo2CX2. (a-d) Band structures. Red: dz2; Green: dx2−y2 and dxy; Blue: dxz and dyz. (e-h)
Three-dimensional view of three bandsaround theFermi level and thecorresponding Fermi surface (Black curves).
(a)800 (c) 600 which have Raman activity have been improvedfor 30%
700 comparing to those of pristine monolayer Mo C. In ad-
500 2
600 dition, the large values of EPC for acoustic modes at M¯
−1)500 −1)400 contribute substantially to the average EPC. Thus, the
m400 m300
ω (c300 ω (c200 supIneracdodnidtuiocnti,vweeThCaids tpruiesdhetdheupfuilnl -MOoH2CcoBvre2r.up Mo C
200 2
1000 X=O 1000 X=Se simurafgaicneas.ryHfroewqeuveenrc,yevoefnphaoftneornthaeppaetaormsiactrtehlaexMa¯tiopno,inatn,
x
(b) (d)600 1.0 implying unstable structure. Therefore,structuralphase
600
500 500 0.8 transition would appear which makes the problem more
−1ω (cm)234000000 −1ω (cm)234000000 00..46 cyfuootmnudprelti.hcaetceudr.reTnthuwso,rkthaenddedcoesreartvioensionfdiOviHdugarlosutpudiisesbien-
100 100 0.2
X=S X=Br
0 0 0 IV. CONCLUSION
FIG.5. CalculatedphonondispersionforMo CX . Thecon- We haveanalyzedthe electronicproperties,the lattice
2 2
tribution from X and monolayer Mo2C is distinguished by dynamical properties, and superconductivity of mono-
color. layer Mo C and its functionalized ones. Our calcula-
2
tionshaveconfirmedthestrongEPCinmonolayerMo C,
2
(a) (c)
X=O X=Se which may lead to superconductivity below 5.9 K. Even
though, since the absorption of functional groups is un-
avoidable in real experiment, its superconductivity can
be modified. Our calculationhavefound that for chalco-
gen functionalized monolayerMo C the superconductiv-
2
ity would be seriously suppressed (or even totally disap-
(b) (d) pear). The most interesting prediction is that electron-
X=S X=Br phonon coupling can be greatly enhanced in monolayer
Mo C by bromine absorbtion, and thus its correspond-
2
ing superconductive T may be pushed up to 12.8 K,
C
suggesting that Mo CBr may be a good candidate as
2 2
nanoscale superconductor.
ACKNOWLEDGMENTS
FIG. 6. Phonon DOS (PDOS, F(ω)), projected PDOS of X
atoms, electron-phonon coupling λ(ω), and Eliashberg spec- Work was supported by the National Natural Science
tral function of Mo CX .
2 2 Foundation of China (Grant No. 11674055), the Fun-
6
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