Electronic Higher Multipoles in Solids

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1 Electronic Higher Multipoles in Solids Yoshio Kuramoto Department of Physics, Tohoku University

2 Outline Elementary examples of multiple moments Role of multipole moments in solids Case studies octupole order in Ce 1-x La x B 6 (Kramers 4f 1 ) scalar order in Pr skutterudites (non-kramers 4f 2 ) Mysterious order in SmRu 4 P 12 (Kramers 4f 5 =>octupole?) Summary

3 Electric dipole (O 2 ) and electric octupole (CH 4 ) 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).

4 Multipole moments dipole quadrupole octupole electric magnetic even parity Electronic state with ang.mom. J => multipoles up to rank 2J Large spin-orbit coupling in f-electron systems => JÀ 1 possible

5 Multipole oscillations: cos n θ monopole (n=0) dipole (n=1) quadrupole (n=2) octupole (n=3) popular in nuclear physics =>

6 Hidden orders in solids - dependent on the stage of development antiferromagnetism (Neel, 1936) antiferroelectricity multipoles 2 n =2,4,8,16,32,64, n=2 (quadrupole) n=3 (octupole) octopus n=4 (hexadecapole) n=5 (triakontadipole) n=6 (hexacontatetrapole)

7 Special multipoles => scalar in point group hexadecapole (O 4 ) hexacontatetrapole (O 6c, O 6t ) O 4 x 4 +y 4 +z 4-3r 4 /5 O 6t (x 2 -y 2 ) (y 2 -z 2 ) (z 2 -x 2 )

8 Role of higher multipoles of localized electrons Leading to unusual magnetism, elastic anomaly Strong spin-orbit interaction + discrete symmetry Mixing of multipoles with different ranks (x => J x ) e.g., Γ 5g : xy, yz, zx with xy(7z 2-1), yz(7x 2-1), zx(7y 2-1) Coupling to crystalline lattice diffraction from superlattice? => resonant X-ray scattering (2005), neutron scattering (2007) in Ce 0.7 La 0.3 B 6

9 Clathrate structures RB 6 (R=La, Ce, Pr, ) R skutterudite: RT 4 X 12

10 Ce 4f 1 under cubic crystal field J=5/2 => Γ 7 (2 fold)+ Γ 8 (4 fold) + orbital - orbital σ α, τ α : pseudo spins

11 Splitting of Γ 8 level Magnetic field (H z ) Quadrupole field (O 20 ) orbital - orbital

12 (time reversal: even) (time reversal: odd)

13 τ z η - η + ζ τ x ζ +

14 Strange ordered phase (phase Ⅳ) in Ce x La 1-x B 6 quadrupole No magnetic order by neutrons NMR and μsr detect internal fields Gigantic elastic anomaly and slight lattice distortion magnetic paramagnetic T (K) Tayama et al. JPSJ (1997) Order parameter? octupole order? (Kuramoto, Kusunose, Kubo: 00-)

15 Magnetic anisotropy in Ce 0.7 La 0.3 B 6 under uniaxial stress Magnetization (Exp.) Theoretical (mean-field) prediction no stress Single octupole domain under uniaxial stress along [111] 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.

16 Broken T-reversal and broken orbital degeneracy in Ce 0.7 La 0.3 B 6 [111] T + T + T Γ 8 β x β y β z octupolar field B T + T + T β x β y β z antiferro-octupoles with A,B sublattices => ferro-quadrupoles => (0,0,0) lattice distortion (Goto) + (1/2,1/2,1/2) Bragg peak (Mannix) A 3 2

17 Resonant X-ray scattering (RXS)

18 Scattering amplitude in RXS E1: dipole E2: quadrupole Detectable up to hexadecapoles by E2 scattering Approximation: energy levels for intermediate states m are all represented by Δ. Irreducible tensor technique is applicable (S.W. Lovesey et al: Physics Reports 411 (2005) 233.)

19 Resonant X-ray scatt. on Ce0.7La0.3B6 (D. Mannix et al. PRL 05)

20 Azimuthal scan around [111] of Ce 0.7 La 0.3 B 6 Experiment: D. Mannix et al.: Phys. Rev. Lett 95 (2005) Theory: H. Kusunose and Y.K: JPSJ, 74, (2005) 3139

21 Contribution of four domains α = σσ or σπ μ=[-1,1,1], [1, -1,1], [1, 1,-1] => threefold pattern around [111] is possible. [111] μ

22 Neutron scattering on Ce 0.7 La 0.3 B 6 Kuwahara et al.: JPSJ 76 (2007) R cell drm(r) =0 for each Ce site. However, octupole gives R cell drm(r)eik r 6=0

23 Clathrate structures RB 6 (R=La, Ce, Pr, ) R skutterudite: RT 4 X 12

24 Resistivity ρ (T) in Pr skutterudites H. Sato et al.: J. Phys.: Condens. Matter 15 (2003) S2063 S2070 Metal-insulator transition

25 PrFe 4 P 12 vs PrRu 4 P 12 Isotropic χ below T N Aoki et al., Matsuda et al., Sekine et al.

26 Non-Kramers CEF levels (4f 2 ) Δ Γ? Γ 1 Γ 4 : dipole, Γ 5 : quadrupole Γ 4 and Γ 5 mix in the T h point group

27 Neutron scattering of PrRu 4 P 12 (Iwasa et al.: 2004) CEF: well-defined, but T-dependent! Cubic symmetry is preserved.

28 T-dependent CEF splittings in PrRu 4 P 12 for T < T M-I =63K (Iwasa et al: 05) =>Hexadecapole: O 4 =O 40 +5O 4 4 as the order parameter Y.Kuramoto et al.: PTP suppl.( 05) T.Takimoto: JPSJ (2006) =>Hexacontatetrapole (O 6 ) can be mixed. ( c t ) 0 0 c 0 t xo + x O + yo = c O + c O c O H CEF = W c 6 + 6t 6 scalar operators in T h symmetry

29 Pr 3+ (4f 2 ) CEF levels against effective hybridization strength AF under pressure (Hidaka 06) No magnetic order Down to 60 mk (Sakakibara 05) 4f 1 4f3 Y. Kuramoto et al.: Prog. Theor. Phys. Suppl. 160 (2005) 134.

30 Landau-type phenomenology A. Kiss and Y.Kuramoto: JPSJ 75 (2006) PrFe 4 P 12 Exp. Points: Matsuda et al λ 0

31 Anisotropy induced by a scalar order Field angle dependence of the transition temperature (111) PrFe 4 P 12 H =2.7 T (110) (001) Tayama et al ( 06) Universal ratio for scalar orders!

32 Other evidences in favor of the scalar order in PrFe 4 P 12 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.

33 Scalar form factors R drρ4f (r) = 2 for all sites (a) Γ 1 singlet (b) Γ 3 doublet (c) Γ 5 triplet (d) Staggered scalar order Degeneracies in (b) and (c) should be lifted at T = 0 either by Kondo or distortion!

34 σ π E2superlattice scattering from hexadecapole O 40 +5O 4 4 Around [001] axis: Intensity 2 sin 2 4ψ = 1-cos 8ψ

35 Outline Elementary examples of multiple moments Role of multipole moments in solids Case studies octupole order in Ce 1-x La x B 6 (Kramers 4f 1 ) scalar order in Pr skutterudites (non-kramers 4f 2 ) Mysterious order in SmRu 4 P 12 (Kramers 4f 5 =>octupole?) Summary

36 Enigmatic phase transitions in SmRu 4 P 12 Matsuhira et al (2005) Sekine et al (2001)

37 Order parameters in SmRu 4 P 12? C. Sekine et al. (2003) quartet Four singlets or orbital Kondo Orbital degeneracy remains in phase II => elastic anomaly at II-III boundary (Nakashima et al)

38 SmRu 4 P 12 : magnetic isotropy at I-II transition III II I D. Kikuchi et al.: (2006)

39 The simplest octupole order (T xyz ) Pseudo-scalar inconsistent with NQR (drawing by K. Kubo) (= T xyz )

40 Possible charge & spin density patterns in SmRu 4 P 12 cf Ce 0.7 La 0.3 B 6 Y. Aoki et al: J. Phys. Soc. Jpn. 76 (2007) Nature of the second transition under magnetic field?

41 Order parameter(s) in SmRu 4 P 12? Metal-insulator transition as in PrRu 4 P 12 Time reversal broken (NMR, musr) Trigonal electric field at Sm (NQR) cf (111) lattice distortion in Ce 0.7 La 0.3 B 6 Emergence of a second transition under H No superlattice found so far

42 Summary Magnetic octupole order in Ce x La 1-x B 6 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 PrFe 4 P 12? SmRu 4 P 12, NpO 2, URu 2 Si 2,