Electron-phonon interaction in the lamellar cobaltate Na$_x$CoO$_2$

0.27 MB·

by  A. Donkov

Electron-phonon interaction in the lamellar cobaltate Na CoO
x 2
A. Donkov1, M.M. Korshunov1,2, and I. Eremin1,3
1Max-Planck-Institut fu¨r Physik komplexer Systeme, D-01187 Dresden, Germany
2L.V. Kirensky Institute of Physics, Siberian Branch of Russian Academy of Sciences, 660036 Krasnoyarsk, Russia and
3Institute fu¨r Mathematische und Theoretische Physik,
TU Braunschweig, D-38106 Braunschweig, Germany
P. Lemmens1, V. Gnezdilov2, F.C. Chou3, and C.T. Lin4
1Institute for Physics of Condensed Matter, TU Braunschweig, D-38106 Braunschweig, Germany
2B. I. Verkin Inst. for Low Temperature Physics, NASU, 61164 Kharkov, Ukraine
8 3 Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan and
0 4Max Planck Institute for Solid State Research, D-70569 Stuttgart, Germany
0
(Dated: February 3, 2008)
2
n We study theoretically and experimentally the dependence of the electron-phonon interaction in
a NaxCoO2 on the sodium concentration x. For the two oxygen phonon modes found in Raman
J experiments, A1g and E1g, we calculate the matrix elements of the electron-phonon interaction.
0 Analyzingthefeedback effect of theconduction electrons on thephonon frequencywecompare the
3 calculated and experimentally observed doping dependence of the A1g mode. Furthermore, due
to the momentum dependence of the electron-phonon coupling for the E1g symmetry we find no

] renormalization of thecorresponding phonon frequency which agrees with experiment. Our results
l shed light on thepossible importance of theelectron-phonon interaction in this system.
e
-
r PACSnumbers: 74.70.-b,74.25.Kc,78.30.-j,71.38.-k
t
s
.
t
a Introduction. The origin of the unconventional su- filledCo-d(t2g)orbitals. Duetothepresenceofatrigonal
m
perconductivity in low-dimensional perovskite systems crystallineelectricfield,thet2g levelssplitintothehigher
d- attracts much attention and belongs to the most chal- lying a1g singlet and the two lower lying e′g states [12].
lenging questions of condensed matter physics. The Angle-Resolved Photo-Emission Spectroscopy (ARPES)
n
o best known example among these materials are high-Tc [13,14]revealsadopingdependentevolutionoftheFermi
c cuprate superconductors. There, one of the possible sce- surface, which shows no sign of the e hole pockets for
′g
[
nariosforthe Cooper-pairformationistheso-calledspin 0.3 x 0.8. The observed Fermi surface is centered

≤ ≤
1 fluctuation mechanism. At the same time, due to the around the Γ point and has mostly a1g character. It has
v complexity of the transition metal oxides, other energy been argued that such an effect may arise due to strong
0 scales are present, and their role in the formation of su- electroniccorrelations[8,15],orNainduceddisorder[16],
5
perconductivity remains under debate. This, in particu- however,no consensusin the literature has beenreached
6
lar,concernsthe electron-phononinteraction. Forexam- yet (see, for example, [17, 18, 19]).
4
. pleitsrelevanceforsuperconductivityinlayeredcuprates Despite of intensive studies of the electronic and mag-
1
and the anomalous normal state has been discussed in,
0 netic properties little is known about the phonon ex-
8 e.g., Ref. 1. Despite some progress, a complete under- citations and their doping evolution in NaxCoO2. At
0 standing of the physics of electron-phonon coupling in thesametime,duetothe relativelylowsuperconducting
: perovskitesisstilllackingbecauseofthecrystallographic
v transitiontemperaturethe possiblerelevanceofphonons
i complexities of these materials. for superconductivity cannot be neglected. For exam-
X
r The discovery of superconductivity with Tc=4.6K in ple,theroleoftheelectron-phononcouplinginNaxCoO2
a water intercalated sodium cobaltate, NaxCoO2 yH2O has been discussed in the context of its relevance to su-
[2], is of great interest on its own and also beca·use of perconductivity and charge ordering on the basis of a
similarities with layeredcuprates. The sodium cobaltate t V model [20]. In addition, due to some similarity
−
hasaquasi-two-dimensionallayeredstructurewithCoO2 with high-Tc cuprates the understanding of the phonon
layersandrichphasediagramasafunctionoftheNacon- renormalization in the sodium cobaltates is of great im-
centration, which includes superconductivity at x 0.3, portance. Initially, the effect of renormalization of the
an insulating phase at x 0.5, and unusual magn≈etism optical phonons by the conduction electrons in layered
forx 0.6[3]. Thereisal∼soincreasingexperimentaland cuprates has been considered in Ref. 21.
≥
theoreticalevidenceforunexpectedstrongcorrelationef- In this Rapid Communication we investigate the
fects as the cobaltates approachthe band insulting limit electron-phonon interaction in the NaxCoO2 as a func-
atx=1[4,5,6,7,8,9,10,11]. InNaxCoO2 theNaions tionofdopingconcentrationanditssuperconductingrel-
reside between the CoO2 layers, with Co ions forming a ative by means of Raman spectroscopy. We observe two
triangularlattice,anddonatexelectronstothepartially oxygen phonon modes at small wave vectors with A1g
2
and E1g symmetries. Then we derive the diagonal and
off-diagonal electron-phonon matrix elements for these
modes. Calculating the renormalization of the phonon
frequencies by conduction electrons we compare our re-
sults with the doping dependent evolution of the A1g
mode and obtain the electron-phonon coupling constant
gA1g = 3meV. Due to the structure of the electron-
off
phonon matrix element for the E1g mode we obtain
no doping dependence of the corresponding phonon fre-
quency in good agreement with experiment. Our results
shed light on the possible role of the electron-phononin-
teraction in this compound.
Experiment details. Raman scattering experiments
havebeenperformedinquasi-backscatteringgeometryon
freshly cleaved single crystal surfaces. The sample have
been fully characterized using basic thermodynamic as
well as spectroscopic techniques [22, 23, 24, 25, 26, 27].
After cleavage the crystals were rapidly cooled down in FIG. 1: (Color online) Schematic illustration of the A1g (a)
Heexchangegastopreventdegradation. InNaxCoO2in- danisdplEac1egm(ebn)tpihnotnhoenCmooOd6eso.cTtahheedarraro.wOsnintdhiecaletefttshiedeoxoyfg(ean)
plane E1g andout-of-planeA1g oxygenmodeshavebeen weindicatethecrystallographica,b,andcdirections. (c)-(d)
observed in Raman scattering [23, 28, 29] and the corre- Thecalculatedmomentumdependenceoftheelectron-phonon
sponding oxygendisplacements are depicted in Fig. 1(a) structure factors, FqΓ 2, in the first BZ for the A1g and E1g
and (b). The Co site is not Raman-active. Modes of phonon modes, res(cid:2)pect(cid:3)ively.
the Na sites have not been identified unambiguously[28].
Thisisprobablyrelatedtodisorderonthepartiallyfilled
sites. The two-dimensionality with respect to structure
hybridizedbandscrossestheFermilevel. We refertothe
and bonding leads to a decoupling of the Na and the diagonalizedbands as εα′ with the new orbital index α.
k ′
CoO2 layers. The observed doping-dependence of the Electron-phonon interaction. In analogy to previ-
A1g and E1g oxygen phonon frequencies are shown in ous considerations for cuprates [30, 31], we derive the
Fig. 2. The cross-over from one to the other crystallo-
graphic phases (shaded areas) given by a different occu- electron-phonon matrix elements, gkq, for the A1g and
pation of the Na sites leads for the A1g modes to small Eto1gobpthaoinnotnhmeomdaeisndecponicttreidbuintioFnigt.o1(tah)eanddia(gbo)n.aNl (aimnterlay-,
additionalfrequencyshiftsandfortheE1g modestonew band) part of the electron-phonon interaction we ex-
modeswithalargerenergyoff-set. Thelatterareomitted
pand the Coulomb energy between Co and oxygen ,
for clarity. The two phonon modes display a markedly
diffTeirgehntt-bdinodpiinngg dmeopdeenl.denTcoe.describe the electronic sub- tHhCes=maeleǫl∗dPispi,lαa′c,σe,mγecn†iαt′sσocfiαt′hσe(cid:16)o|xRyi−g1erni,γi|o+ns.|RHi−er1rei,,−eγ|i(cid:17)s,thine
system we use a tight-binding t2g-band model with pa- electron charge, e∗ = −2e is the oxygen ion charge, ǫ is
rameters (in-plane hoppings and the single-electron en- the dielectric constant, Ri are the Co ion positions, ri,γ
ergies) derived previously from the ab-initio LDA (Lo- arethe vectorpositionsofthe vibratingoxygens,andin-
calDensityApproximation)calculationsusingprojection dexγ( γ)labelsthethreeoxygenpositionswithinCoO6
−
procedure for x=0.33 [8]. unit cell above (below) the Co layer. Here, c†iα′σ refers
The free-electron Hamiltonian of the t2g-band model tothediagonalformoftheHamiltonian(1). Afterintro-
in a hole representation is given by ducing the creation (annihilation) operator b†q (bq) for
thephononwithmomentumq,wearrivetoth−efollowing
H0 =− (ǫα−µ)nkασ− tαkβd†kασdkβσ, (1) form of the electron-phonon interaction
kX,α,σ Xk,σ Xα,β
where nkασ =d†kασdkασ, dkασ (d†kασ) is the annihilation Hedli−agph =k,qX,α′,σgqα′α′c†kα′σck−qα′σ(bq+b†−q). (2)
(creation) operator for the t2g-hole with spin σ, orbital
index α, and momentum k, tαβ is the hopping matrix
k For the sake of simplicity we assume the diagonal
element, and ǫα is the single-electron energy. To obtain
electron-phononinteractionisindependentontheorbital
the dispersionwe diagonalize the Hamiltonian (1) calcu-
lating the chemical potential µ self-consistently. Due to index α′. Thus, for the A1g and E1g optical Raman-
activephononoxygenmodesonefindsgA1g =gA1gFA1g,
the non-zero inter-orbital hopping matrix elements, a1g q diag q
and e′g bands are hybridized. However, only one of the gqE1g = gdEi1aggFqE1g, where the structure factors of the
3
electron-phonon interaction are 600
(a)
FqA1g = cosq1−q2 +cosq1+q3 +cosq2+q3, (3) 590
3 3 3
FqE1g = cosq1−q2 1 cosq1+q3 +cosq2+q3 (.4) -1m) 580
3 − 2(cid:20) 3 3 (cid:21) (cA1g 570
Here,q1 =(√3/2)qx−12qy,q2 =qy,q3 =(√3/2)qx+12qy, ω
in units of 2π/a with a being the in-plane lattice con- 560
sctoarnrets,pgodΓnidaging=ba−reeeǫ∗ph√od2n2L+oΓln23fqreq2Muh¯eωnΓc,yw(Γher=e ωAΓ1g,isE1tgh)e, 550 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
L =d=a/√6isthedistancebetweentheCoandthe
A1g
oxygenplane,L =l =a/√3istheplanardistancebe-
E1g 510 (b)
tweenCoandoxygen,andM istheoxygenmass. Assum-
ing that in the band insulator, Nax=1CoO2, the renor- 490
malizationof the phonons by the conduction electrons is 1)
-m
aTbhseesnet,vawlueeussaereωcAlo1gse=to5t8h9ocsme−ob1taanindedωEby1gth=e4fi7rs0tcmpr−in1-. ω (cE1g 470
ciplescalculations[32]. Theresultingmomentumdepen-
450
denceofthestructurefactorsforthebothmodesisshown
in Fig. 1(c) and (d). Interestingly, one sees that while
430
the gq for the A1g mode shows a maximum at the BZ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
center, the corresponding gq for the E1g mode vanishes x
there. Therefore,forq=0the electron-phononcoupling
FIG.2: (Coloronline)TheexperimentalRamandataat10K
for the E1g channel is zero. Taking ǫ ∼ 20 we estimate isshownby(blue)circles. Thecalculateddopingdependence
gdAi1agg ≈0.05eV. of the A1g (a) and E1g (b) phonon modes is shown by red
The off-diagonal (interband) contribution to the crosses, thesolid curveisa guidetotheeye. Theslight scat-
electron-phononinteractionarisesmainly from the mod- tering of the experimental points around x = 0.5, x = 0.76
and x = 0.9 (shown by the dashed areas) are the result of
ulation of the inter-orbital Co-Co hopping matrix ele-
thedifferent crystallographic phases around these points[27].
ment via oxygen. Assuming the linear terms in the ex-
Thesestructuralmodificationsonlyweaklyaffecttheelectron-
pansionofthenearestneighborshoppingmatrixelement
phononcouplingasitmainlyinvolvesNaordering. Measure-
tαijβ(uγ) = tαijβ +Vαβuγ over the oxygen displacements mentsofA1g modeat290K(notshown)followasimilartrend
uγ, one obtains: as the10K data.
Heolffph = gkαq′β′c†kα′σck qβ′σ(bq+b†q). (5)
− k,q,Xα′=β′,σ − − given by:
6
where gkαq′β′ = goΓffFqΓ(γ(k) + γ(k+q)) with γ(k) = Π(q,ω)= 2 gα′β′ 2 f(εαk+′ q)−f(εβk′) ,
cosk2+cosk3+cosk1 being the Colattice structurefac- − αX′,β′Xk (cid:16) kq (cid:17) ω−εαk+′ q+εβk′ +iδ
tor. Again one could see that for q = 0 the off-diagonal
(7)
electron-phononcouplingfor theE1g channelis zerodue where f(ǫ) is the Fermi function. To find the renormal-
to the momentum dependence of the electron-phonon
izationofthebarephononfrequencyandtocomparethe
structurefactor,FqE1g. Therefore,inRamanexperiments results to the Raman experiments we set q 0 limit
which probes q=0 response this mode shows no doping and solve Eq. (6) numerically as a function o→f the dop-
dependence due to the coupling to the electronicsubsys- ing concentration. The main contribution to the renor-
tem. Thisisalsoconfirmedbythefactthattheobserved malization of the optical phonon modes comes from the
pvahlouneonobmtaoidneedenbeyrgaybf-oinriatiloldcoaplcinuglalteivoenlss[l3ie2s].cloTsheetootnhlye interbandtransitions,i.e. terms with gkαq′6=β′ while intra-
bandtransitionsrenormalizetheacousticphononmodes.
Raman-active optical phonon mode which will couple to
The results of our numerical calculations are shown in
the conduction electrons at q=0 is the A1g mode. Fig. 2. The doping evolution has been deduced by cal-
Inthefollowingweconsidertherenormalizationofthe
culating Π(q 0,ω) for various x values. We obtain
A1g phonon. The corresponding Dyson equation reads: the value of th→e off-diagonalelectron-phonon interaction
D−1(q,ω)=D0−1(ω)−Π(q,ω), (6) pgoAofi1fngts≈. T3mhiesVvabluyecisomanpaorridsoernotfomtahgeneitxupdeerismmeanltlaerl tdhaatna
where D0(ω)= ω2 2ωω2Γ+iδ is the momentum-independent thediagonalcontributiontotheelectron-phononinterac-
− Γ
bare phonon propagator. The polarization operator is tion. One sees that the phonon renormalization changes
4
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∼ ≈
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c
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(2004).
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perconductivityinthiscompound. Atthesametime,the
Souma, T. Sato, T. Takahashi, T. Takeuchi, J.C. Cam-
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c
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·
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c
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M.M.K. acknowledges support from INTAS (YS Grant [22] P.Lemmens,V.Gnezdilov,N.N.Kovaleva,K.Y.Choi,H.
05-109-4891) and RFBR (Grants 06-02-16100, 06-02- Sakurai,E.Takayama-Muromachi,K.Takada,T.Sasaki,
F.C.Chou,D.P.Chen,C.T.Lin,andB.Keimer,J.Phys.:
90537-BNTS).I.E. acknowledges support from Volkswa-
Condens. Matter 16, S857 (2004).
gen Foundation. The experimental studies have been
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supported by DFG and ESF-HFM.
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Kremer,P.Lemmens,C.T.Lin,C.Niedermayer,andJ.
Strempfer, Phys. Rev.B 69, 100410(R) (2004).
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