yoshio kuramoto - 京都大学ykis07.ws/presen_files/27/kuramoto.pdf · electronic higher...
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Electronic Higher Multipoles in Solids
Yoshio KuramotoDepartment of Physics, Tohoku University
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Outline
• Elementary examples of multiple moments• Role of multipole moments in solids• Case studies
– octupole order in Ce1-xLaxB6 (Kramers 4f1)– scalar order in Pr skutterudites (non-Kramers
4f2)• Mysterious order in SmRu4P12 (Kramers 4f5
=>octupole?)• Summary
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Electric dipole (O2) and electric octupole (CH4)
Figure 1: Arrangement of the methane molecules in the b-c-plane of the Cmca space group of phase III. a) and c) show cuts through the planes at x = 0, 1/2 with m molecules, b) and d) represent planes x = 1/4, 1/2 with 2-site molecules. The underlying blue colour represents orientations in phase II (measurements at hrpd, isis).
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Multipole momentsdipole quadrupole octupole
magnetic
electric
Electronic state with ang.mom. J => multipoles up to rank 2J
Large spin-orbit coupling in f-electron systems => JÀ 1 possible
even parity
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Multipole oscillations: cosnθ
popular in nuclear physics => http://walet.phy.umist.ac.uk/P615/
monopole (n=0) dipole (n=1)
quadrupole (n=2) octupole (n=3)
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Hidden orders in solids- dependent on the stage of development
• antiferromagnetism (Neel, 1936)• antiferroelectricity• multipoles 2n =2,4,8,16,32,64,…
– n=2 (quadrupole)– n=3 (octupole) octopus– n=4 (hexadecapole)– n=5 (triakontadipole)– n=6 (hexacontatetrapole)
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Special multipoles => scalar in point group
hexadecapole (O4)hexacontatetrapole (O6
c, O6t)
O4 ∝ x4+y4+z4-3r4/5 O6t ∝ (x2-y2) (y2-z2) (z2-x2)
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• Leading to unusual magnetism, elastic anomaly…
• Strong spin-orbit interaction + discrete symmetry – Mixing of multipoles with different ranks (x => Jx)
e.g., Γ5g: xy, yz, zx with xy(7z2-1), yz(7x2-1), zx(7y2-1)
– Coupling to crystalline lattice
diffraction from superlattice? => resonant X-ray scattering (2005),
neutron scattering (2007) in Ce0.7La0.3B6
Role of higher multipoles of localized electrons
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Clathrate structures
RB6 (R=La, Ce, Pr,…) R skutterudite: RT4X12
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Ce 4f1 under cubic crystal field
J=5/2 => Γ7 (2 fold)+ Γ8 (4 fold)
↑”+” orbital
”-” orbital
↑↑
↑
σα, τα : pseudo spins
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Splitting of Γ8 level• Magnetic field (Hz)
• Quadrupole field (O20)
↑↑
↑↑
+
+--
↑
↑
↑
↑
+
-
”+” orbital
”-” orbital
↑↑
↑↑
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(time reversal: even)
(time reversal: odd)
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τz
η+η-
τx
ζ−
ζ+
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Strange ordered phase(phase Ⅳ) in CexLa1-xB6
• No magnetic order by neutrons
• NMR and μSR detect internal fields
• Gigantic elastic anomaly and slight lattice distortion
Tayama et al. JPSJ (1997)
T (K) octupole order?(Kuramoto, Kusunose,
Kubo: ’00-)
magnetic
quadrupole
paramagnetic
Order parameter?
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Magnetization (Exp.)
Magnetic anisotropy in Ce0.7La0.3B6under uniaxial stress
Theory: K. Kubo and Y. Kuramoto: J. Phys. Soc. Jpn. 73 (2004) 216.
Experiment: T. Morie et al.: J. Phys. Soc. Jpn. 73 (2004) 2381.
Theoretical (mean-field) prediction
Single octupole domain under uniaxialstress along [111]
no stress
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octupolar fieldβββzyx TTT ++
Γ8
βββzyx TTT ++
23
23−
0
A
Broken T-reversal and broken orbital degeneracy in Ce0.7La0.3B6
antiferro-octupoles withA,B sublattices =>ferro-quadrupoles =>(0,0,0) lattice distortion (Goto) +(1/2,1/2,1/2) Bragg peak (Mannix)
B
[111]
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Resonant X-ray scattering (RXS)
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Scattering amplitude in RXS
Detectable up to hexadecapoles by E2 scatteringApproximation: energy levels for intermediate states mare all represented by Δ. ⇒ Irreducible tensor technique is applicable(S.W. Lovesey et al: Physics Reports 411 (2005) 233.)
E1: dipole
E2: quadrupole
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Resonant X-ray scatt. on Ce0.7La0.3B6 (D. Mannix et al. PRL’05)
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Azimuthal scan around [111] of Ce0.7La0.3B6
Experiment: D. Mannix et al.: Phys. Rev. Lett 95 (2005) 117206Theory: H. Kusunose and Y.K: JPSJ, 74, (2005) 3139
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Contribution of four domains
[111]μ=[-1,1,1], [1, -1,1], [1, 1,-1]=> threefold pattern around [111]
is possible.
μ
α = σσ’ or σπ’
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Neutron scattering on Ce0.7La0.3B6
Kuwahara et al.: JPSJ 76 (2007) 093702
for each Ce site.However, octupole gives
RcelldrM(r) = 0
RcelldrM(r)eik·r 6= 0
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Clathrate structures
RB6 (R=La, Ce, Pr,…) R skutterudite: RT4X12
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Resistivity ρ (T) in Pr skutteruditesH. Sato et al.: J. Phys.: Condens. Matter 15 (2003) S2063–S2070
Metal-insulator transition
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PrFe4P12 vs PrRu4P12
Aoki et al., Matsuda et al., Sekine et al.
Isotropic χ below TN
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Non-Kramers CEF levels (4f2)
ΔΓ1
Γ? Γ4 and Γ5 mix in the Th point group
Γ4 : dipole, Γ5 : quadrupole
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Neutron scattering of PrRu4P12 (Iwasa et al.: 2004)
CEF: well-defined, but T-dependent!
Cubic symmetry is preserved.
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T-dependent CEF splittings in PrRu4P12 for T < TM-I=63K (Iwasa et al: ’05)
( ) tt
cc
tc OcOcOcyOOxxOWH 6066
064
04664CEF 1 ++=+−+=
scalar operators in Th symmetry
=>Hexadecapole: O4=O40+5O4
4
as the order parameterY.Kuramoto et al.: PTP suppl.(’05)T.Takimoto: JPSJ (2006)
=>Hexacontatetrapole (O6) can be mixed.
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Y. Kuramoto et al.: Prog. Theor. Phys. Suppl. 160 (2005) 134.
Pr3+ (4f2) CEF levels against effective hybridization strength
4f1 4f3
AF under pressure(Hidaka ‘06)
No magnetic orderDown to 60 mK(Sakakibara ’05)
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Landau-type phenomenology A. Kiss and Y.Kuramoto: JPSJ 75 (2006) 103704.
PrFe4P12
Exp. Points: Matsuda et al
λ ∼ 0
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Anisotropy induced by a scalar order
Field angle dependence of the transition temperature
(001)
(111)
(110)
PrFe4P12H =2.7 T
Universal ratio forscalar orders!Tayama et al (’06)
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Other evidences in favor of the scalar order in PrFe4P12
• NMR spectra in the ordered phase show local cubic symmetry.
• Lattice distortion observed by neutron and X-ray does not show lower symmetry.
• Induced staggered moment is parallel or antiparallel to the magnetic field.
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Scalar form factorsRdrρ4f (r) = 2 for all sites
(a) Γ1 singlet(b) Γ3 doublet(c) Γ5 triplet(d) Staggered scalar order
Degeneracies in (b) and (c) should be lifted at T = 0either by Kondo or distortion!
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σ−π’ E2 superlattice scattering from hexadecapole O4
0+5O44
Around [001] axis: Intensity ∝ 2 sin2 4ψ = 1-cos 8ψ
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Outline
• Elementary examples of multiple moments• Role of multipole moments in solids• Case studies
– octupole order in Ce1-xLaxB6 (Kramers 4f1)– scalar order in Pr skutterudites (non-Kramers
4f2)• Mysterious order in SmRu4P12 (Kramers 4f5
=>octupole?)• Summary
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Enigmatic phase transitions in SmRu4P12
Sekine et al (2001)
Matsuhira et al (2005)
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quartet
Four singlets or orbital Kondo
C. Sekine et al. (2003)
Orbital degeneracy remains in phase II =>elastic anomaly at II-III boundary (Nakashima et al)
Order parameters in SmRu4P12 ?
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D. Kikuchi et al.: (2006)
II
I
III
SmRu4P12: magnetic isotropy at I-II transition
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The simplest octupole order (Txyz)
(= Txyz)
Pseudo-scalar –inconsistent with NQR
(drawing by K. Kubo)
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Possible charge & spin density patterns in SmRu4P12
Y. Aoki et al: J. Phys. Soc. Jpn. 76 (2007) 113703
cf Ce0.7La0.3B6
Nature of the second transition under magnetic field?
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• Metal-insulator transition as in PrRu4P12
• Time reversal broken (NMR, muSR)• Trigonal electric field at Sm (NQR)
– cf (111) lattice distortion in Ce0.7La0.3B6
• Emergence of a second transition under H• No superlattice found so far
Order parameter(s) in SmRu4P12 ?
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Summary
• Magnetic octupole order in CexLa1-xB6– Induced quadrupoles => lattice distortion – Superlattice observation by resonant X-ray
scattering and neutron scattering• Scalar multipole orders in skutterudites
– Phase transition keeping the local symmetry• Unsolved problems
– Ground state of PrFe4P12?– SmRu4P12, NpO2, URu2Si2,…