Table Of ContentQuest for high T in layered structures: the case of LiB
c
Matteo Calandra
Institut de Min´eralogie et de Physique des Milieux condens´es,
case 115, 4 place Jussieu, 75252, Paris cedex 05, France
7
Aleksey N. Kolmogorov and Stefano Curtarolo
0
Department of Mechanical Engineering and Material Science,
0
2 Duke University, Durham, North Carolina 27708, USA
(Dated: February 3, 2008)
n
a Usingelectronicstructurecalculationwestudythesuperconductingpropertiesofthetheoretically-
J devised superconductor MS1-LiB (LiB). Wecalculate the electron-phonon coupling (λ=0.62) and
0 the phonon frequency logarithmic average (hωilog = 54.6 meV ) and show that the LiB critical
1 temperature is in the range of 10-15 K, despite the frozen-phonon deformation potential being of
thesameorderofMgB2. Asaconsequence,LiBcapturessomeoftheessentialphysicsofMgB2 but
] (i) theelectron-phonon coupling dueto σ states is smaller and (ii) theprecious contribution of the
n
o π carriers to the critical temperature is lacking. We investigate the possible change in Tc that can
c beinduced bydoping and pressure and find that these conditions cannot easily increase Tc in LiB.
-
r PACSnumbers: 63.20.Kr,63.20.Dj,78.30.Er,74.70.Ad
p
u
s
INTRODUCTION ture and the existence of σ and π bands originated from
.
t
a the carbon 2p orbitals, the electronic structure close to
m theFermileveliscompletelydifferentfromthatofMgB .
The quest for superconductivity in layered structures 2
Theπbands,reminiscentofthegraphiteones,andanin-
- has become the focus of intense research since the dis-
d tercalant free-electron-like band [12, 13] cross the Fermi
n covery of superconductivity in MgB2 (Tc = 39 K) [1]. energy. The intercalant band forms a spherical Fermi
o The layeredstructure of MgB generates one of its most
2 surface [14, 15]. The electron-phonon coupling of CaC
c prominent features, namely the B 2p orbitals form σ- 6
[ xy (λ = 0.83) is mainly due to coupling of the interlayer
bands [2, 3, 4] which are weakly dispersing along the
band with C vibrations perpendicular to the graphite
1 k direction and have a marked two dimensional charac-
z layers and with Ca vibrations. So, even though miss-
v ter. In MgB the σ bands are hole doped, but the top
9 of these ban2ds is on−ly 0.5 eV higher than the Fermi ing the σ-bands, CaC6 reaches an interesting 11.5 K Tc.
9 ≈ This temperature is substantially enhanced by pressure
level. Theσ bandsFermisurfacesheets[4],twoslightly
1 − (T =15.1Kat 8GPa[16]),contrarytowhathappens
1 warped cylinders with axis perpendicular to the boron c ≈
in MgB .
0 layers, generate a huge electron-phonon coupling along 2
7 thek direction. Thecarriersintheπ bands,formedby Ascanbeseenfromtheaboveexamples,evenifonere-
z
0 the B 2p orbitals,further enhance th−e averageelectron- strictstosandwichstructuresformedbyboronorcarbon
/ z
t phonon coupling [5]. layers, the details of the electronic and phonon spectra
a
and, subsequently, the critical temperature can change
m The formationof σ andπ states is typicalof graphite-
dramatically when the intercalantis included. As a con-
- like structures composed by boron or carbon atoms.
d GiventhesuccessofMgB itisnaturaltolookforhighT sequence, a theoretical approach is absolutely necessary
2 c
n toidentifythemostprobablesuperconductorsoratleast
superconductivity in structures having similar features.
o to exclude the less probable ones.
The problem is that, given the boron layers, small vari-
c
: ations in the valence or mass of the intercalant or in the An attempt in this direction has been recently made
v structural parameters are sufficient to considerably alter in Ref. [17], where by using ab initio methods the au-
i
X the σ or π bands positions or the shape of their Fermi thors studied the possible hole-doping of LiBC, a 1
≈
r surfacesandconsequentlydestroysuperconductivity. For eV gap semiconductor. The authors suggested that a Tc
a
one or some of these reasons AlB2, ZrB2, NbB2, MoB2, of the order of MgB2 could be reached if the insulating
YB2, TaB2, TiB2, HfB2, VB2 and CrB2 are not super- LiBCissubstantiallydopedwithholestoobtainLi0.5BC.
conducting [6, 7, 8]. Successive experimental studies have indicated that the
structural response to the introduction of holes unfavor-
The hope of finding new superconducting materials in
ably modifies the electronic structure of Li BC, and so
layeredstructureswasrecentlyincreasedbythediscovery x
far no high T superconductivity has been found in this
of superconductivity in the graphite intercalated com- c
system [18].
pounds, YbC and CaC [9, 10]. This is particularly
6 6
promising since a huge number of intercalants are avail- Ideally designing new superconductors ab initio re-
ableforgraphite[11]. InCaC ,despitethelayeredstruc- quires three steps. The first is the determination of the
6
2
moststablestructuresgivena setofatomic species. The ture basing our choice on the following considerations.
second is the calculation of the electronic structure to On the one hand, it has the smallest unit cell of all MS
verify that the given structure is at least metallic or can structures,whichofferscomputationalefficiency. Onthe
be made metallic easily. The third is the determination other hand, even though other stacking sequences are
of the phonon dispersion and of the electron-phononpa- possible (e.g. MS2, Refs. [23, 24]), MS1 is a good rep-
rameters. resentative model of the layered lithium monoboride be-
The first point is a daunting task even if one restricts cause the long-period shifts are expected to have little
one’s search to a specific set of likely candidates[19]. A effect on its superconducting properties [25]. MS1-LiB
systematic approach to tackle this problem has been re- has a rhombohedral unit cell with R¯3m space group.
cently offered in the way of data mining of ab initio cal- There are four atoms in the primitive unit cell with
culations [20, 21, 22]. In this method one uses the in- Wyckoff positions Li(2c)(1/2 z ,1/2 z ,1/2 z ),
Li Li Li
formations obtained from ab initio calculations of many Li(2c)(1/6+z ,1/6+z ,1/−6+z ), B−(2c)( δ, −δ, δ)
Li Li Li
− − −
different structures to build a database that can be then and B(2c)(2/3 δ,2/3 δ,2/3 δ). The fully relaxed
− − −
usedtojudgethestabilityofnewstructures. Application parameters are a=b=c=5.92 ˚A, α=β =γ =29.8o.
of this method to intermetallics has led to the identifi- Density functional theory (DFT) calculations are per-
cation of new layered lithium monoboride phases which formed using the Quantum Espresso code [26] within
have a good chance to form under proper synthesis con- the generalized gradient approximation (GGA) [27]. We
ditions [23]. use norm-conserving pseudopotentials [28] with configu-
Once a stable metallic structure is given, a calcula- ration 2s12p0 and non-linear core correction [29] for Li,
tion of the phonon spectra and of the electron-phonon and configuration 2s22p1 for B. The wavefunctions are
coupling needs to be performed to obtain T . Indeed, expanded using a 50 Ry cutoff. The dynamical matrices
c
whilesomequalitativeinformationcanbeextractedfrom and the electron-phonon coupling are calculated using
electronic-structure [2], for a quantitative analysis step density functional perturbation theory in the linear re-
three is absolutely necessary. sponse [26]. For the electronic integrationin the phonon
In this work we investigate the superconducting prop- calculation we use an N = 12 12 12 uniform k-
k
× ×
ertiesofthepreviouslydeterminedmetalsandwich(MS) point mesh and Hermite-Gaussian smearing from 0.05
lithium monoboride [23] by calculating its phonon spec- Ry. For the evaluation of the electron-phonon coupling
trum and electron-phonon parameters. This system is we use anN =40 40 40 Monkhorst-Packmesh. For
k
× ×
metallic and, from qualitative arguments, one can infer the λ average over the phonon momentum q we use an
that T is of the same order of that of MgB [23]. In- N =4 4 4 q pointmesh. The phonondispersionis
c 2 q
× × −
deed this system has an electronic structure which is a obtained by Fourier interpolation of the dynamical ma-
hybrid between those of MgB and CaC , since there trices computed on the N mesh.
2 6 q
are hole-doped σ bands forming cylindrical Fermi sur- The pressure-and doping-inducedchangesin the elec-
−
faces andthere is anintercalantbandcrossingthe Fermi tronicpropertiesofLiBarestudiedwithViennaAbinitio
level. Moreover the deformation potential is compara- Simulation Package VASP [30, 31] within the GGA [27].
ble to that of MgB [23]. From this point of view, LiB Weuseprojectoraugmentedwaves(PAW)[32]pseudopo-
2
is a much more promising material than Li BC, be- tentials, in which Li semi-core states are treated as va-
0.5
cause even without doping it has a significant density of lence; the energy cutoff is set at 30 Ryd. The projected
σ-states at the Fermi level. electronicdensityofstates(EDOS)isfoundbydecompo-
Unfortunately,thefullelectron-phononcouplingcalcu- sition of the wavefunction within a sphere of the default
lations performed in this paper indicate that LiB should PAWradiusof1.7a.u. FortheMS1unitcell,the2 2 3-
× ×
haveaT ina10-15Krange. WeshowthatLiBcaptures MS1 and 2 2 1-MS2 supercells we use 31 31 31,
c
× × × ×
some of the important physics of MgB , namely the role 18 18 6, and 18 18 10 Monkhorst-Packk-meshes,
2
× × × ×
of the σ bands, but it lacks the contribution of the π respectively.
−
states to the electron-phonon coupling and it is only a
far relative of CaC because the interlayer band is very
6
weakly coupled with the phonons. In an attempt to im- BAND-STRUCTURE, DOS AND FERMI
provethesituationweexaminewhatrolethehydrostatic SURFACE
pressure and doping can play in determining the critical
temperature. The band structure of Li B is presented in Fig. 1
2 2
(see footnote [33] for high-symmetry points notation).
Similarly to what happens in MgB [2], there are two
2
TECHNICAL DETAILS boron σ-bands crossing the Fermi energy ǫ . Compared
F
tothe σ-bandsinMgB , these bandsareevenmoretwo-
2
In all our calculations of the layered lithium mono- dimensional (due to the larger interlayer distance) and
boride we have used the MS1 theoretical crystal struc- shifted by more than 0.6 eV to higher energies at the Γ
3
4
2
) 0
V
e
(
y
g -2
r
e
n
E
-4
B states
-6 Li states
FIG. 2: (Color online) Fermi surface of MS1-LiB. For conve-
nience, theBrillouin zone is stretched along the z-direction.
-8
Γ T UXU’L Γ S
) total
)
FemIGp.ty1:(f(uCllo)lodrotosnrlienper)esBeanntdthsetraumctouurnetooffLLiBi(.BT)hcehsairzaecotefrthaet f.u.0.6 B-px,y
n B-pz
a given k-point. See footnote [33] for high-symmetry points i
p Li
notation. s
V
e0.4
(
/
point. As in MgB2, they generate two cylindrical Fermi ates
surfaces (in our case with axes along the ΓT direction, st
(
Fig. 2). The boron π states in LiB resemble more the S 0.2
O
π states of graphite, as they cross exactly at ǫF, so that D
LiBislackingπFermisurfacesaltogether. InMgB these E
2
states cross at about 2 eV above ǫ , which leads to the
F
appearance of an extended π Fermi surface [4]. Another 0.0
-12 -10 -8 -6 -4 -2 0 2 4
importantdifference betweenthe electronicstructuresof
E (eV)
thetwoboridesisthepresenceofalithiumbandatǫ in
F
LiB. The position of this band resembles the intercalant
band in CaC6 [13], although in LiB it has substantial FIG. 3: (Color online) Electronic density of states projected
hybridization to boron states close to the T-point. The overselected atomic orbitals.
correspondingFermisurface(acompressedsphere)isde-
picted in Fig. 2.
Thetotaldensityofstates(EDOS)andtheEDOSpro- samesymmetrybutdifferenteigenvectorsweuse the fol-
−
jmecatiendcoomveproanteonmtiactoǫFrbiistaglsiveisnibllyusbtorarotendpσinstFaitges3..ATshine lAo∗1wgin[Lgizn]o,tAa−2tuion[(:LiA−1gB)[zB],z]E,g∗E[gLi[xByx],yA],+2Euu[(L[(iL+iB−)zB],)xEyu+],
graphite the boron pπ EDOS at ǫF is zero and increases [(Li+B)xy],whereinbracketswegivethecorresponding
slowly and linearly immediately after ǫ . atoms and vibrations. For convenience, we label phonon
F
brancheseverywhereintheBrillouinzoneusingthename
of their representation at Γ.
PHONON SPECTRUM AND Exceptforacousticmodes,aclearseparationexistsbe-
SUPERCONDUCTING PROPERTIES tweenopticalLiandBvibrations. Limodesareconfined
inthe 40-55meVregionandare notdispersive,meaning
The phonon dispersion and the phonon density of that Li-vibrations behave essentially as Einstein modes.
states (PHDOS) are illustrated in Fig. 4. The phonon Boron in-plane vibrations are softened along the ΓT di-
modes at the Γ-point are decomposed as 2A +2A + rectiondue to coupling to the σ bands. The softening at
1g 2u
2E +2E [34]. To distinguish between modes with the Γ of the E phonon branches is approximately 20 meV,
g u g
4
0,5
0,4 Total
B xy
0,3 Li xy (a)
0,2
0,1
0
0 20 40 60 80 100
0,5
0,4 Total
B z
0,3
Li z (b)
0,2
0,1
0
0 20 40 60 80 100
3 α2F 4 x
4 x λ
2
(c)
1
0
0 20 40 60 80 100
ω
(meV)
FIG. 4: (Color online) Phonon dispersion of MS1-LiB with
decomposition in in plane Li and B vibrations (labeled Lixy
and Bxy respectively) and out-of-plane Li and B vibrations
(labeled Liz and Bz respectively). Notation for the phonon FIG. 5: (Color online) Phonon density of states (PHDOS),
modes at the Γ-point, given next to the vertical axis, is ex- partialphonondensityofstatesprojectedoverselectedvibra-
plained in the text. tions,EliashbergfunctionandintegratedEliashbergfunction
for MS1-LiB. For clarity, the integrated Eliashberg function
has been multiplied by a factor of 4
tobe comparedwiththe almost40meVinMgB forthe
2
E modes[5,35]. Thissuggestsastrongcouplingofthe
2g
σ bandstothe in-planevibrationsinLiB[23],butnotas tively).
strong as in the case of MgB . The Eliashberg function
2
The three acoustic modes along the ΓT direction are
substantially softer with respect to the other directions. α2F(ω)= 1 λ ω δ(ω ω ) (2)
qν qν qν
At the zone border,T,these modes areformedby (i) in- 2N −
q Xqν
plane(Li+B) vibrationsatenergies 8.5meVand(ii)
xy
≈
out-of-plane(Li+B)z vibrationsat≈8.0meV.They can and the integral λ(ω) = 2 ωdω′α2F(ω′)/ω′ are shown
be related to the soft modes at the Γ point in MS2-LiB, 0
inFig. 5. AscanbeseenmRostofthecontributioncomes
discussed in Ref. [23].
fromphononstatesinthe60-90meVregionandasmaller
Thesuperconductingpropertiescanbeunderstoodcal-
contribution comes from low energy states.
culating the electron-phonon coupling λ for a phonon
qν
An estimate of the different contributions of the in-
mode ν with momentum q:
plane (Li and B ) and out-of-plane (Li and B ) vi-
xy xy z z
4 brations to λ can be obtained from the relation
λ = gν 2δ(ǫ )δ(ǫ )
qν ω N(0)N | kn,k+qm| kn k+qm
qν k kX,n,m (1) λ= Λiα,jβ = 1 [Gq]iα,jβ[Cq−1]jβ,iα (3)
N
where the sum is over the Brillouin Zone. The matrix iXαjβ iXαjβ q Xq
element is gν = knδV/δu k+qm / 2ω ,
where uqν iksn,tkh+eqmamplihtud|e of tqhνe| displaciempentqoνf where i,α indices indicate the displacement of the
the phonon, V is the Kohn-Sham potential and N(0) ith atom in the Cartesian direction α, [Gq]iα,jβ =
∗
is the electronic density of states at the Fermi level. k,n,m4g˜iαg˜jβδ(ǫkn)δ(ǫk+qm)/[N(0)Nk], and g˜iα =
The calculated average electron-phonon coupling is λ = PknδV/δx k+qm /√2. The C matrix is the
qiα q
h | | i
λ /N 0.62 (N and N are the previously Fourier transform of the force constant matrix (the
qν qν q ≈ k q
Pdefined k-space and q-space mesh dimensions, respec- derivative of the forces with respect to the atomic dis-
5
placements). The decomposition leads to: in Ref. [13]). In LiB, on the contrary, the Li modes are
much higher in energies ( 50 meV) meV and disper-
Bxy Bz Lixy Liz sionless. The main reason∼for this difference comes from
B .46 .00 .02 .00
xy − the mass of Li which is 5.77 times smaller than that of
Λ = Bz .00 .13 .02 −.05 (4) Ca leading to frequencies which are onaverage2.4 times
Li .02 .02 .08 .01
xy − − larger.
Liz .00 .05 .01 .07
− − Thecriticalsuperconductingtemperatureisestimated
The off-diagonal terms are small (but not negligible) using the McMillan formula [38]:
compared to the total λ. Most of the coupling is to the
in-plane B vibration; contributions from the Li and the ω 1.04(1+λ)
log
T = h i exp (5)
out-of-plane B vibrations are smaller. Since the σ-bands c 1.2 (cid:20)−λ µ∗(1+0.62λ)(cid:21)
−
donotcoupletothe B vibrationsandsincethereareno
z
∗
π Fermi surfaces,the coupling to B vibrations is due to where µ is the screened Coulomb pseudopotential and
z
the intercalant band. Note that the decomposed values
of λ contain contributions from different modes and are ω log =eλ2R0+∞α2F(ω)log(ω)/ωdω (6)
h i
summed over all the q-points in the Brillouin zone. For
example, Λ = 0.46 includes the coupling to the the phonon frequencies logarithmic average. We obtain
in-plane EgB,xEy,u−Bx,yand Eu+ branches. By examining the hωilo∗g =54 meV leading to Tc of approximately 10-15K
integrated Eliashberg function λ(ω) in Fig. 5(c) and the forµ =0.14 0.1. Thisvaluecouldbefurtherenhanced
−
phonon characters in Fig. 4 one can infer that the E by multiband effects.
g
branch is the most important of the three: among them
it has the highest PHDOS in the 70-100 meV range, in
which λ gains most of its total value. The soft in-plane DOPING AND PRESSURE EFFECTS
E+ branch is far less important, as the net contribution
u
from all the soft modes having energy under 20 meV is Even though a theoretically-devised from scratch su-
only 0.08 (Fig. 5(c)). perconductor with T = 10 15 K could be considered
It i≈s instructive to compare our result with other lay- a success of the matcerials p−rediction methodology, the
ered superconductors. In MgB2 the coupling of the σ- stoichiometric LiB compound falls short of expectations
bands to the phonon modes is λMσ,σgB2 = 0.62 ± 0.05 to compete with the record-holding binary MgB2. In
[5], while in LiB the corresponding value is less than thissectionweinvestigatewhetheritispossibletofavor-
0.46, as discussed above. This difference can be clari- ably modify the electronic properties of LiB and achieve
fied by noting that the E2g phonon linewidth γq,E2g = higher Tc by doping or applying pressure. We pay spe-
2πN(0)ωq2E2gλqν along ΓA in MgB2 happens to be com- cialattentionto the evolutionof the π states, since their
parableinmagnitudewiththatoftheEg modealongΓT reintroduction at ǫF may soften the Eg mode and lead
in LiB. Therefore, the reduced electron-phononcoupling to a larger coupling.
in LiB is mainly due to the Eg phonon frequency being As has been pointed out previously [23], the bonding
harder than the E2g one in MgB2 (ωELgiB/ωEM2ggB2 ≈ 1.3 π states are completely filled under ambient conditions.
at Γ). This unfortunate result can be linked to the ab- Because the band crossing in LiB at the Fermi level is
sence of the π carriers, which play an important role in accidental,itmay be possible to movethe crossingpoint
softening of the E mode in MgB [36, 37]. withpressureandincreasetheπ-bandsEDOSatǫ . Fig-
2g 2 F
WefindthatLiBandgraphiteintercalatedcompounds ure6(a)revealsthatthereisindeedarapidchangeinthe
have few similarities in terms of superconducting fea- σ and π EDOS, followed by a plateau after 5 GPa. This
tures. In particular, in CaC the intercalant modes behaviorisareflectionoftwodistinctlydifferentregimes
6
are responsible of 50% of the total electron-phonon ofthe LiB structuralchanges: i)inthe 0-5-GPapressure
∼
coupling, and the rest comes from vibrations of carbon rangetheLi-Liinterlayerspacingquicklyshrinksandthe
modesinthedirectionperpendiculartothegraphitelay- B-B bond length slightly expands so that at 5 GPa they
ers. InCaC onehasλ +λ =0.33andλ =0.33 become about 0.5 and 1.02 of their zero-pressure values,
6 Caxy Caz Cz
[13] . In LiB the overallcontributionof B , Li andLi respectively; ii) for pressures above 5 GPa the Li atoms
z xy z
vibrationsis less thanhalfofthatofCaC , whichmeans in the bilayer start experiencing the hard-core repulsion
6
that while LiB captures some of the physics of MgB , and the compound compresses more isotropically. The
2
it does not capture the physics of graphite intercalated inset in Fig. 6(a) illustrates that by reducing the Li-Li
compounds to full extent. This is also clear from the interlayer distance one forces the charge from the inter-
phonon spectrum of CaC where the intercalant modes calant band (completely emptied at about 6 GPa) into
6
are at energies lower than 20 meV and one of the Ca theboronπ andσ states(loweredby0.7eVatthatpres-
modes undergoes a marked softening with a correspond- sure). Oncethe chargeredistributioniscomplete,noap-
ing large electron-phononcoupling (at point X of Fig. 2 preciable changes in the EDOS are seen for the boron
6
is observed in MgB . Another way to tweak the elec-
2
tronic structure could be to hole-dope LiB as it is done
0.16 a) 1 b) 0.16 for Li BC [18]. The known limitations of this approach
n f.u.))0.14 E (eV)0 Mg 0.14 adreesttahbxeilbizuactkiolinngofotfhtehecohmexpaoguonndalulpaoynerhseaanvdytLhiedeevpelnettiuoanl
pi0.12 -1 0.12
ates/(eV s00..0180 B-px,y -20 2 P4 (G6Pa)8 10 B-px,y eAl 00..0180 [c1h8aW]r.geefidrcsetllsiwmiuthlaatentehuetrealleicztinrognp-o(shitoilvee-)(ndeogpaitnigveu)sbinagcka-
S (st0.06 0.06 ground. Normally, in this approach one can safely relax
O the unit cell parameters and obtain valuable informa-
D0.04 e 0.04
E tion about the bond length variation under small dop-
0.02 B-pz B-pz Al 0.02 ing. However, in the case of the electron-doped LiB the
Mg
0.00 0.00 repulsion between the negatively charged boron layers
0 2 4 6 8 10 0.0 0.1 0.2
overcomes the weak binding between the lithium layers,
P (GPa) doping (e/boron)
causing the c-axis to undergo unphysical expansion even
at small levels of doping. Therefore, we fix the c-axis at
FIG. 6: (Color online) a) EDOS at theFermi level projected thezero-dopingvalueandrelaxonlytheremainingthree
overσandπstatesasafunctionofpressure;b)thesameasa parameters. The set of data, shown as hollow points in
function of doping level: thehollow points represent charged Fig. 6(b), supports our earlier conclusion that the π-
LiB unit cell in a neutralizing background; the solid points
band EDOS cannot be easily increased. Note that the
correspond tosubstitutional Mg and Aldopingof LiB in the
approximationsusedinthistest,i.e. the fixedc-axisand
MS1andMS2superstructuresdescribed inthetext[41]. For
the use of a neutralizing background, may influence the
comparison, theσ andπ EDOSinMgB2 are0.098 and0.064
states/(eV·spin·f.u.), respectively. The inset shows pressure- results to some extent. For example, the positive elec-
induced changes in the position of different states near the trostatic potential from ionized dopants could bring the
Fermilevel: σ-boronandintercalant (labeledζ)statesat the delocalized π-states down (in addition to the rigid band
∗
Γpointandcrossingoftheπ-π statesalongtheΓSdirection. downshift) and could potentially be an important factor
in increasing the π EDOS.
To address these limitations we use a more realistic
states up to at least 30 GPa. Therefore,the peculiar be- model of the electron-doped LiB by substituting Li with
havior of the nearly free electron intercalant states (also MgorAl. Smalldopinglevelsareobtainedonlyforlarge
observedinothersystems[12,39])istheonlymeaningful unitcells;weusethehexagonal2 2 3-MS1and2 2 1-
factorallowingmodificationoftheLiBboronstateswith MS2 supercells with 48 and 32 a×tom×s, respectivel×y. ×Re-
pressure. placement of one or two Li atoms in these structures re-
Thesesimulationsdemonstratethatthecompressionof sults in the 1/24, 1/16, 1/12, and 1/8 concentrations of
LiB does not lead to the desired π-bands EDOS values dopantsperboron,andthelevelofdopingisfoundinthe
comparabletothoseinMgB2. Moreover,thehydrostatic assumptionthattheygiveupalltheir valencecharge. In
pressure causes such a quick drop in the σ-bands EDOS allthecasesthec-axisexpansioninthefullyrelaxedunit
that this will likely negate any possible enhancement in cells does not exceed 6%. The resulting averaged boron
theelectron-phononcouplingduetothereintroductionof EDOS for the Mg and Al sets are shown in Fig. 6(b)
the π-states at ǫF. The phononmodes are also expected as solid points [41]. The scattered presence of dopants
to harden under pressure, further reducing the electron- in the lattice should cause some dispersion of the local
phonon coupling in LiB [40]. boron properties. A general trend observed in our su-
LiB has plenty of available bonding σ states, there- percell calculations is that a downshift of the π and σ
fore the compound should be easy to electron-dope. A states happens only for B layers in direct contact with
quick examinationofthe boronEDOSstates aroundthe thedopant. Typicalvaluesofthe downshiftthatasingle
Fermi level (Fig. 3 or Fig. 4 in Ref. [23]) gives an Mg(Al)atominducesinalleightatomsinaneighboring
idea onwhatpossible changesinthe Fermisurfaces and, Blayerareabout0.2(0.5)eV.Itisnoteasytoisolatethe
eventually, in the electron-phonon coupling the doping importanceofdifferentfactorsdefiningthelevelofBdop-
could lead to. At small doping levels, when the rigid ing,i.e. thesimplechargetransfer,theelectrostaticeffect
bandapproximationnormallyholds,the EDOSfromthe discussedaboveandthestructuralchanges(expansionof
two dimensional σ bands should only slightly fluctuate the c-axis and contraction of the B-B bond). However,
until the states are completely filled, which happens at Fig. 6(b) demonstrates that the net effect of the substi-
∆q eN(0)∆E 0.70 (e/eV) (1 eV) = 0.70 e/f.u. = tutional doping is described reasonably well within the
≈ ≈
0.35e/boron. The EDOSfromtheπ bands growsslowly rigid band model.
and even at the relatively high electron-doping of 0.35 Insummary,ourtestsindicatethatitisratherdifficult
e/boron it would amount only to about a half of what toreintroduceasignificantamountofπ statesatǫ with
F
7
hydrostatic pressure or small doping, because the band would be to find a suitable LiB-based ternary alloy; this
crossing in LiB happens to be exactly at ǫ , about 2 eV question is currently under investigation.
F
lower than in MgB2. To have a chance of substantially Theoretical development of potentially important su-
increasing T , one should search for more radical ways perconducting materials is a difficult task because T
c c
of modifying the electronic structure of the MS metal critically depends on their band structure features and
borides. vibrational properties. The challenge is even greater if
one attempts to design a superconductor from scratch,
since one first needs to ensure its thermodynamic sta-
CONCLUSIONS AND PERSPECTIVES bility. The case of LiB shows that it is possible to the-
oretically predict a compound that both i) has a good
Inthisworkwehaveinvestigatedelectronandphonon chance to form and ii) possesses interesting supercon-
propertiesoftherecentlytheoretically-devisedsupercon- ducting properties. Study of such promising candidates
ductorLiB[23]. Bystudyingindetailsthephononprop- givesimportantinsightsinto howto performamoretar-
erties of this hypothetical material we have found that geted search for novel superconducting materials.
its criticaltemperatureisoftheorderof10-15K.Super- While we were finishing writing this paper, a preprint
conductivity is mainly of the MgB kind with planar pσ on the related structure MS2-LiB appeared on-line [37].
2
bands strongly coupled with phonons. Differently from The results of the paper are similar to ours except for
MgB , LiB has no Fermi surface generated by π states some numerical details that can probably be related to
2
but an additional intercalant one. Thus, its electronic the different unit cells and k,q-point samplings used[43].
structure can be seen as a hybrid between MgB and However, our conclusions concerning the possibility of
2
CaC . However, the intercalant electronic states of LiB increasing the T in LiB by doping are rather different,
6 c
areweaklycoupledwiththeLiandB vibrationssothat as explained in the previous Section.
z
the overallelectron-phonon coupling is only λ=0.62.
Since the discovery of MgB , no other diborides have
2
been found with high Tc (for a full list, see Ref. [42]). ACKNOWLEDGMENTS
If we compare LiB with the known diborides, our calcu-
lated10-15Kcriticaltemperatureisnotsolow,although
We acknowledge many fruitful discussions with Igor
it is far from the 39 K of MgB . Nevertheless, the study
2 Mazin, Francesco Mauri, Michele Lazzeri, and Roxana
of LiB gives an important understanding. The common
Margine. Calculations were performed at the San Diego
belief is that the main effect for the singular and unique
supercomputing center and at IDRIS supercomputing
behaviorofMgB isgivenbythepresenceofalmosttwo-
2 center (project 061202).
dimensionalσ bands. LiB hasevenmoreplanarσ bands
and even higher σ EDOS at ǫ relative to that in MgB
F 2
[23,24];however,theircontributiontothetotalelectron-
phonon coupling turns out to be at least 25% smaller.
This reduction can be attributed to the differences be- [1] J. Nagamatsu et al.,Nature(London) 410, 63 (2001).
tween the in-plane boron vibrations in the two borides, [2] J. M. An and W. E. Pickett, Phys. Rev. Lett. 86, 4366
(2001).
caused mainly by the lack of the π carriers at ǫ in LiB.
F
[3] K. D. Belashchenko, M. van Schilfgaarde and V. P.
Namely, the softening of the E mode in LiB is substan-
g Antropov , Phys. Rev.b 64, 092503 (2001)
tiallysmaller(byaboutafactoroftwo)thanthatofE
2g [4] J.Kortus,I.I.Mazin,K.D.Belashchenko,V.P.Antropov,
in MgB2, which makes the former mode be noticeably L.L. Boyer, Phys.Rev. Lett., 86, 4656 (2001).
harder (ωLiB/ωMgB2 1.3 at Γ). Therefore,in addition [5] Y. Kong, O.V. Dolgov, O. Jepsen,O.K. Andersen, Phys.
Eg E2g ≈
to the direct loss of the π states contribution to λ, their Rev. B, 64, 020501(R)
absence at ǫ in LiB also has a strong indirect negative [6] A. S. Cooper, E. Corenzwit, L. D. Longinotti, B.T.
F
Matthias and W.H. Zachariasen, Proc. Natl. Acad. Sci.
effect on the overall electron-phonon coupling.
USA 67, 313
We have investigated whether this somewhat unex-
[7] V.A. Gasparov, N. S. Sidorov, I. I. Zverkova and M. P.
pected obstacle, preventing LiB to be a truly high T
c Kulakov, cond-mat/0102421
superconductor, could be overcomewith moderate mod- [8] L. Leyarovska and E. Leyarovski, J. Less-common Met.
ificationsofthecompound’sproperties. Behaviorofelec- 67, 249
tronic features important for the LiB superconductivity [9] T. E. Weller et al., NaturePhys.1, 39 (2005)
hasbeenexaminedasafunctionofsmalldopingandpres- [10] N. Emery, C. Herold,M. dAstuto,V. Garcia, C. Bellin,J.
F. Mareche, P. Lagrange,G. Loupias, Phys. Rev. Lett.
sure. OurresultsindicatethattheπEDOSincreasesvery
95, 087003 (2005)
slowly and cannot reach the desired values, at least not
[11] M. S.Dresselhaus andG. Dresselhaus, Adv.in Phys. 51
beforethe σ EDOSissubstantiallyreduced. Thus,small
1, 2002
dopingandpressurearenotexpectedtosignificantlyim- [12] G. Cs´anyi, P. B. Littlewood, A. H. Nevidomskyy, C. J.
proveTc inLiB.Apromisingdirectiontofixtheproblem Pickard,andB.D.Simons,NaturePhysics1,42(2005).
8
[13] M. Calandra and F. Mauri Phys. Rev. Lett. 95, 237002 [30] G. Kresse and J. Hafner, Phys.Rev. B 47, 558 (1993).
(2005) [31] G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169
[14] I.I. Mazin Phys. Rev.Lett. 95, 227001 (2005) (1996).
[15] MatteoCalandraandFrancescoMauriPhys.Rev.B74, [32] P.E. Blochl, Phys. Rev.B 50, 17953 (1994).
094507 (2006) [33] The special points are: Γ = (0,0,0),T= (1/2,1/2,1/2),
[16] A. Gauzzi, S. Takashima, N. Takeshita, C. Terakura, H. U=(1/2+η,τ,τ),X=(1/2,0,0),U′ =(1/2−η,−τ,−τ),
Takagi, N. Emery, C. Herold, P. Lagrange, G. Loupias, L= (1/4,−1/4,−1/4),S= (1/2,−1/2,0). Given the
cond-mat/0603443 rhombohedral angle α, the parameters η and τ are ex-
[17] H. Rosner, A. Kitaigorodsky and W. E. Pickett, Phys. pressed as η = ˆ1−2/(3tan2(α/2)−1)˜/12 and τ =
Rev.Lett. 88, 127001 (2002) ˆ1+1/(3tan2(α/2)−1)˜/12.
[18] A.M. Fogg, J.B. Claridge, G.R. Darling and M.J. Ros-
[34] D.L. Rousseau, R.P. Bauman, and S.P.S. Porto, J. Ra-
seinsky, Chemical Communications 12, 1348 (2003); man Spectrosc. 10, 253 (1981).
A.M.Fogg,J.Meldrum,G.R.Darling,J.B.Claridge,and
[35] A. Shukla, M. Calandra, M. d’Astuto, M. Lazzeri, F.
M.J. Rosseinsky,J. Am.Chem. Soc. 128, 10043 (2006).
Mauri, Ch. Bellin, M. Krisch, J. Karpinski, S. M. Kaza-
[19] D. de Fontaine, in Solid State Physics, edited by H.
kov, J. Jun, D. Daghero, and K. Parlinski, Phys. Rev.
EhrenreichandD.Turnbull(AcademicPress,NewYork, Lett. 90, 095506 (2003)
1994), Vol. 47, pp. 33-176; G. Ceder, Comput. Mater. [36] I.I.MazinandV.P.Antropov,PhysicaC385,49(2003).
Sci. 1, 144 (1993).
[37] Amy Y. Liu and I. I. Mazin, cond-mat/0610057
[20] S. Curtarolo, D. Morgan,K. Persson,J. Rodgers, and G. [38] McMillan, Phys. Rev. 167, 331 (1968).
Ceder Phys.Rev.Lett. 91, 135503 (2003) [39] E.R. Margine and V.H. Crespi, Phys. Rev. Lett. 96,
[21] D.Morgan,G.Ceder,andS.Curtarolo,Meas.Sci.Tech-
196803(2006);J.S.Kim,L.Boeri,R.K.Kremer,andF.S.
nol. 16, 296-301 (2005)
Razavi, cond-mat/0603530, to appear on Phys.Rev. B.
[22] S. Curtarolo, D. Morgan, and G. Ceder, Calphad 29,
[40] We do not observe any significant softening of the in-
163-211 (2005).
planeboronphononmoderesultingfromthe 2%expan-
[23] A. N. Kolmogorov and S. Curtarolo, Phys Rev. B 73,
sionoftheboron-boronbondundersmall(below5GPa)
180501(R) (2006)
pressures.Thefrequenciesofthesoft slidingmodeshave
[24] A.N.Kolmogorov andS.Curtarolo, PhysRev.BTBA,
been shown to nearly double at 2 GPa [23].
TBA (2006)
[41] Weplacethedopantsinthesupercellasuniformlyaspos-
[25] For example, the long-period shifts have little effect on
sible. Duetothehigh sensitivity of theprojected EDOS
the MS-LiB energy: MS1 and MS2 have been demon-
to k-sampling we use a 0.2 eV smearing in the MS1 and
strated to be nearly degenerate under ambient condi-
MS2 supercell calculations.
tions, so that the synthesized lithium monoboride could
[42] C.BuzeaandT.Yamashita,SuperconductorScienceand
bea mixtureof thetwo crystal structure phases [23].
Technology 14, R115 (2001)
[26] http://www.pwscf.org,S.Baroni,etal.,Rev.Mod.Phys.
73, 515-562 (2001) [43] IncompoundssuchasMgB2,λqν variessubstantiallyall
over the Brillouin zone. Thus a better q-point sampling
[27] J.P.Perdew,K.Burke,M.Ernzerhof,Phys.Rev.Lett.77,
thanthatpossibleinstate-of-the-artab-initiosimulations
3865 (1996)
is needed to obtain estimates of λ comparable with that
[28] N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993
achievedforsimplesuperconductorssuchasNb.Forthis
(1991).
[29] S. G. Louie, S. Froyen and M. L. Cohen, Phys. Rev. B reason, in systems like MgB2, the errors in parameters
such as λ or ω is probably larger than expected.
26, 1738 (1982) log
