Phonon Dynamics and Multipolar Isomorphic Transition in beta-pyrochlore KOs2O6

0.36 MB·English

by  Kazumasa Hattori

Typeset with jpsj2.cls <ver.1.2> Letter
Phonon Dynamics and Multipolar Isomorphic Transition in β-pyrochlore KOs O
2 6
Kazumasa HATTORI and Hirokazu TSUNETSUGU
Institute for Solid State Physics, University of Tokyo, 5-1-5, Kashiwanoha, Kashiwa Chiba 277-8581, Japan
We investigate with a microscopic model anharmonic K-cation oscillation observed by neu-
1 tron experiments in β-pyrochlore superconductor KOs2O6, which also shows a mysterious
1 first-order structural transition at T =7.5 K. We have identified a set of microscopic model
p
0 parameters that successfully reproduce the observed temperature dependence and the super-
2 conductingtransitiontemperature.ConsideringchangesintheparametersatT ,wecanexplain
p
n puzzling experimental results about electron-phonon coupling and neutron data. Our analysis
a demonstrates that the first-order transition is multipolar transition driven by the octupolar
J component of K-cation oscillations. The octupole moment does not change the symmetry and

4 is characteristic to noncentrosymmetric K-cation potential.

] KEYWORDS: beta pyrochlore, anharmonic phonon,superconductivity, isomorphic transition
n
o
c
- Theβ-pyrochloreoxidesAOs2O6(A=K,Rb,orCs)are tronscatteringdata20 showthe oscillationamplitude in-
r one of the material families that have unique cage-like creases below T . Its naive interpretation is an increase
p p
u structure, and A-cation inclusions exhibit large anhar- in the electron-phonon coupling.
s monic local oscillations.1–9 Such anharmonic and large- In our previous work,22 we have developed a general
.
t amplitude oscillations strongly interact with conduction theoryofanharmonicionoscillationsintetrahedralsym-
a
electrons, which leads to strong coupling superconduc- metry. A strong coupling theory of superconductivity
m
tivity,10–12 anomalous temperature dependence of elec- mediated by these ion oscillations has been also devel-
-
d tricresistivity4,13andnuclearrelaxationtime.5,14 Toex- oped.Applyingthemtoβ-pyrochlorecompounds,wees-
n plainsuchinterestingpropertiesisabigchallengeforthe timatedtheirT .Wealsostartedtostudythecontradic-
c
o theory of electron-phonon systems and strong coupling tionbetweenthe changeofthe electronphononcoupling
c theory of superconductivity.15 and the neutron data at T in KOs O but that was in
[ p 2 6
As the A-cation size decreases, amplitude of the A- a qualitative level.

1 cation oscillation increases, and thus potassium oscilla- In this Letter, we will resolve by microscopic calcu-
v
tions are the most anharmonic.16 Elastic neutron scat- lations the contradiction between the data of resistiv-
7
teringdatashowthatthetemperaturedependencecurve ity and specific heat4 and the neutron scattering20 in
4
6 of oscillation amplitude is concave (convex upwards) in KOs2O6 at Tp. Our calculation demonstrates that the
0 KOs O , reflecting strong anharmonicity.17,18 A recent first-order transition is a multipolar isomorphic phase
2 6
1. inelasticneutronscatteringexperimentrevealedthatthe transition and the phonon anharmonicity plays a cru-
0 phononpeaksshift to lowerenergywithdecreasingtem- cialroleto explainthe experimentaldata.We will apply
1 peratureandthepeakpositionsareataround3-7meVat
1 the lowesttemperature.19 Thissofteningis thestrongest
:
v in K-, and the weakest in Cs-compound. Superconduct-
i ing transitiontakes places in allthe three, andthe tran-
X
sition temperature is T =9.6, 6.3 and 3.3 K for A=K,
r c
a Rb and Cs, respectively.4,9 The symmetry of the super-
conducting gap function is confirmed to be s-wave.6,7
KOs O , the most anharmonic one, not only has the
2 6
highest T of superconductivity, but also exhibits an-
c
other singularity, a first-order structural transition at
T = 7.5 K.2 It is interesting that no sign of symmetry
p
breaking has been observed.8,20,21 Below T , the tem-
p
perature dependence ofelectric resistivity changesto T2
fromTγ(γ ∼0.5)athighertemperatures,4 andtheabso-
lute value decreases by about 25 % at T . Both of these
p
results show that electron-phonon scatterings are sup-
pressedbelowT ,3 whichisalsosupportedbythe reduc- Fig. 1. (color online) Potential profile, and probability density
tion in the specpific heat jump at T in magnetic fields of the ground state wavefunction |Ψgs(r)|2 for K (solid), Rb
c (dashed), and Cs (dotted line), respectively, along [111] direc-
H when T (H)<T (H)∼T (0).4 However, recent neu-
c p p tion. Energy eigenvalues are indicated by short bars and the
numbersrepresentthedegeneracy.
1
2 J.Phys.Soc.Jpn. Letter AuthorName
the theorydevelopedinref.22toAOs O andcarryout
2 6
moreelaboratenumericalcalculationsinordertounder-
stand the higher-temperature and higher-energy prop-
erties of the anharmonic A-cation oscillations. We will
showthatwecanreproducethetemperaturedependence
of the softening of phonon energy and the amplitude of
A-cation oscillations in a wide range of temperatures in
nice agreement with the experimental data.
WestartwithashortreviewofourmodelforA-cation
oscillation.Since the inelastic neutronexperimentshows
nearly momentum-independent modes of their dynam-
ics,19 weemployalocalmodel.TheA-site hasthe tetra-
hedral T symmetry and this implies that the ion po-
d
tential generally has, in addition to spherical and cu-
bic fourth order terms, a third-order anharmonic term,
whichbreaksspaceinversionsymmetry.TheionicHamil-
Fig. 2. (coloronline)Temperaturedependenceofthevarianceof
tonian is thus given by ion oscillation hx2i. Three solid lines represent calculated hx2i
H = −∇2r + Mω2|r|2+bxyz+c |r|4+c r˜4(,1) fpoerritmheenttharledeactoamtpakoeunndfrso.mSyrmefbso.l1s6a,n1d7daontdte1d8.lineshowtheex-
ion 1 2
2M 2
where~issettobeunity,andr=(x,y,z)isthe iondis- fore, we use positive ω2 for all the three compounds. As
placementfromtheequilibriumposition,r˜4 =x4+y4+z4 we will show later, our potential parameters reproduce
andM istheionmass.Weignorethehigherorderterms the correct temperature dependence of the variance of
of O(|r|5) which is irrelevant for our discussions because ionoscillationhx2i,andalsotheexcitationenergyagree-
of positive forth order terms. We diagonalize Hamilto- ing with the neutron and x-ray scattering data.16–19
nian(1)numericallyintherestrictedHilbertspaceofdi- We first calculated the variance of ion oscillation hx2i
mension62196,whichisaboutthreetimeslargerHilbert as a function of temperature from the calculated eigen-
space used in the previous study22 and sufficient to dis- functions of (1) and the results are shown in Fig. 2. The
cussthephysicalpropertiesbelowroomtemperature,by experimental values determined by neutron17,18 and x-
extrapolating the obtained data to the infinite limit if ray16,17 scattering are also shown for comparison. As
necessary.Detailsofcalculationsareexplainedinref.22. is clearly seen, our results are quantitatively consistent
An important point is the variation in potential pa- with the experimental results. For KOs2O6, we mainly
rameters among different compounds. The Madelung concentrateonfittingtherecentexperimentaldataofref.
partofthepotentialenergyisessentiallythesameamong 18 and the low-temperature limit is consistent with the
the three compounds, since the band structure calcula- recent data ∼ 0.014 ˚A2 just above Tp.20 As for Rb- and
tions23showessentiallythesameelectricstatesforthem. Cs-compounds,thecalculatedhx2iisslightlylargerthan
Thus, the main difference in the potential parameters the data in ref. 17.However,concerning the discrepancy
originatesfromthe therelaxationoflocalchargedensity between the data in refs. 17 and 18 for K-compound,
as discussed in ref. 23. This point can be well captured our results for Rb- and Cs-compounds well capture the
by setting the smaller ω for the smaller ion. We use the essential aspect of the experimental data. We note that
same parameters as those in ref. 22: ω =26.4 K, 54.6 K thevaluesinthelow-T limitareimportanttodiscussthe
and74.8KforK,Rb,andCs,respectively,withkeeping superconducting transitiontemperature, since these val-
the same values for other parameters, b = 9324 K/˚A3 ues directly influence the dimensionless electron-phonon
and c = 4c = 3332 K/˚A4. The validity of our choice coupling constant λ.
1 2
will be checked later by comparing the phonon energy Figure 3 shows the phonon spectrum, i.e., the imag-
and oscillation amplitudes with experimental data. inary part of the phonon Green’s function D(ν) =
Figure 1 shows the potential profile with these pa- −iR∞dt eiνt−ηth[x(t),x(0)]i at six different tempera-
0
rameters along [111] direction for the three compounds. turesforA=K,RbandCscompounds.ηisaphenomeno-
Eigenenergies of Hamiltonian (1) are indicated by bars logicalrelaxationratesetasη =3.5K.Thereareseveral
andthe probability density ofthe groundstate |Ψgs(r)|2 peaks in the spectrum of KOs2O6 for intermediate tem-
is also shown. The potential of K-cation is much shal- peratures in Fig. 3(a) owing to the anharmonic terms
lower than those of Rb and Cs, and correspondingly, its and the smaller ω compared to other members. The re-
|Ψ (r)|2 hasabroadertailthantheothers.Kuneˇsetal., sults are consistent with the inelastic neutron scattering
gs
proposed a potential with ω2 < 0 for KOs O based on spectra,19 particularly in the following three aspects. (i)
2 6
thefirstprinciplecalculation,23butwiththeirchoice,the The peak positions at the lowest temperature are 38.5,
K-cation oscillates about 1 ˚A, which is too large and in- 52.9 and 73.1 K, which correspond to the energy of the
consistentwiththeneutronexperimentdata.17,18There- first excited states ∆, consistent with the neutron data
at 1.5 K:19 Ωexp = 38, 56, and 66 K for K, Rb and Cs,
E
J.Phys.Soc.Jpn. Letter AuthorName 3
Fig. 3. (coloronline)Phononspectrumvsenergyatsixdifferent
temperatures for (a) K, (b) Rb, and (c) Cs compounds. Spec-
trum isbroaden owingto the manyLorentzians whosewidthis
phgiaovrsemintioobnnyicfηiotry=.KI3n.(s5ceitKr:clTae)en,mdRptbhe(rissaqtiuusraterhede)edapinerdnecdCtensc(codenisaoemfqtouhneendch)eitgaohkfeestnthepfreaoanmk- Figw.it4h. fi(xcionlogr co′1n=li5nce′2)=Cc1o,nt(obu)r δpλlo,t(oc)f (δa∆),δT(cd)inδuω2′/-ub′2,plaannde
ref.19.Linesrepresentthepresentcalculations. (e) δu3/u3. In (a), all the five regions are indicated; A:
(sgn(δλ), sgn(δ∆), sgn(δu2), sgn(δu3)) = (+,−,+,+),
respectively. (ii) The softening of the peak energy with B: (+,−,+,−), C: (+,+,+,−), D: (−,+,+,−) and E:
(−,+,−,−). Thesolidlineswithsymbols represent the bound-
decreasing T is the strongest for K, intermediate for Rb
aries and the dashed lines represent the contours. Region D is
and the weakest for Cs. The temperature dependence is
indicatedbyshade.
alsoqualitativelyreproducedbyourcalculationasshown
in the inset of Fig. 3. Note that the relevantmode splits tial parameters at T , we will identify the transition as
p
intotwobranchesowingtotwocagesintheunitcell,and “multipolar” isomorphic one, because the change in the
our data correspond to their average energy. (iii) The octupole u is much larger than in the isotropic scalar
3
spectrum of K at higher temperatures is much broader u . This point is important to explain the experimental
2
than that of Rb and Cs. results of recent neutron scattering20 and specific heat
As discussed above, our model can explain the neu- jump at T ,4 which will be explained below. Our previ-
c
tronexperimentsandwelldescribespotentialofA-cation ous analysis based on the simplified toy model24 fails to
oscillations. These results confirm our previous work,22 describe this point, since the model cannot distinguish
in which the superconducting transition temperature Tc u2 and u3 owing to the lack of the degrees of freedom.
was calculated with the same parameters used in this This kind of phase transition driven by a scalar order
Letter and the result is Tc = 10.5, 5.7 and 3.4 K, for K, parameter is similar to that proposed for the f-electron
Rb and Cs, respectively. The calculated Tc agrees with system PrRu4P12.25
the experimental data within ±1 K, even without opti- Let us discuss the nature of the transition at T and
p
mizing parameters for each compound. The variation in threeexperimentalresultsareimportant.First,themag-
Tc among the three members is mainly attributed to the netic field dependence of the specific heat jump at Tc4
difference in the anharmonicity of A-cation oscillations. indicates that the electron-phonon coupling λ decreases
Indeed,60%ofTc isowingtotheK-cationoscillationin below Tp. Secondly, a recent neutron experiment shows
KOs2O6. that u2 jumps up at Tp and the value is ∼ 0.02 ˚A2 at
As mentioned in the introduction, KOs2O6 exhibits lowtemperature1.5K.20 Thirdly,theuppercriticalfield
a first-order structural transition at Tp = 7.5 K, which Hc2(T) shows that the first-ordertransitiondoes not af-
doesnotbreakanysymmetry.4,8,20,21Inref.24,wehave fectT (H →0).4,26 Thefirsttwoexperimentaldatacan-
c
proposedascenarioofisomorphicstructuraltransitionto notbeexplainedonthebasisofharmonicoscillations.In
explainitbasedonasimpletoymodel.Thepointisthat the harmonic case, λ ∝ u2 and thus, the increase in u
2 2
in the tetrahedral symmetry, not only hr2i = 3hx2i ≡ leadstoenhancedλ.Thesepuzzlingexperimentalresults
3u2 but also the third moment |hxyzi|≡u3 are nonzero arenaturallyexplainedonthebasisofthemultipolariso-
and their changes at Tp do not break the point group morphic transition.
symmetry. Since the transition is first-order, the potential pa-
Here, we will examine this scenario by means of the rameters change discontinuously, which originates from,
presentrealisticmodel.Analyzing the changesinthe os- for example, the changes in volume, oxygen positions,
cillation profile as a function of variations in the poten- and the inter-site ion interactions. The high-T poten-
4 J.Phys.Soc.Jpn. Letter AuthorName
tial parametersω=26.4K, b=9324K/˚A3, c =4c =3332 phic transition and demonstrated that the third order
1 2
K/˚A4 changetolow-T ones,say,ω′,b′,c′ andc′,respec- fluctuation hxyzi is the primary order parameter. Our
1 2
tively.Changesinbandωareparticularlyimportantand theorynaturallyexplainsboththereductioninelectron-
we investigate the effects of their change, while we set phonon coupling constant and the enhanced oscillation
′ ′ ′ ′
c =5c =c forsimplicity. Thus,with varyingω andb, amplitude at T . We hope detailed neutron scattering
1 2 1 p
we repeat the same procedure as before22 and calculate experiments detect its change at T .
p
thedeviationsofdimensionlesselectron-phononcoupling
′ Acknowledgment
δλ≡λ−λ,andgapδ∆,similarly,δu ,δu ,andδT .Here,
2 3 c
the quantities denoted with prime symbol are the low-T TheauthorsthankT.Dahm,K.Ueda,J.Yamauraand
values calculated for the new potential parametersat T Z. Hiroi for discussions and also acknowledge the inter-
c
foreachoftheparameterset.T islowenoughcompared nationalworkshop“New Developments in Theory of Su-
c
with ∆ and thus these quantities are regarded as those perconductivity”heldinInstituteforSolidStatePhysics,
for the low-temperature limit. The results are shown in University of Tokyo, Japan, June 22-July 10, 2009, for
Fig. 4. giving them an opportunity to discuss many aspects of
First, it is noted that there exist five regions, A-E, this work. This work is supported by KAKENHI (No.
distinguished by the sign of δλ, δ∆, δu and δu . In the 19052003and No. 20740189).
2 3
harmonic case, only the variations characterized by the
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2
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p
δλ < 0 mean the suppression of the electron scatterings 12) J.Chang,I.EreminandP.Thalmeier:New.J.Phys.11(2009)
055068.
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3 2
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In summary, we have proposed that the first-order 898.
transition observed in KOs O is a multipolar isomor- 24) K. Hattori and H. Tsunetsugu: J. Phys. Soc. Jpn 78 (2009)
2 6
013603.
J.Phys.Soc.Jpn. Letter AuthorName 5
25) T.Takimoto:J.Phys.Soc.Jpn.75(2006)034714. ref.4.ThisimpliesthatTcforthelow-T potentialparameters
26) Hc2(T) below Tp(H) for H > 8 T can be extrapolated to isapproximatelythesameasTc forthehigh-T ones.
vanishingatthesameTc asthecaseofH =0,seeFig.10in