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@ISSP August 20 th, 2003 Spin-chirality induced anomalous Hall effect in pyrochlore ferromagnets Y. @ISSP August 20 th, 2003 Spin-chirality induced anomalous Hall effect in pyrochlore ferromagnets Y. Taguchi 1, Y. Oohara 2, H. Yoshizawa 2, N. Nagaosa 3, 4, T. Sasaki 1, S. Awaji 1, Y. Iwasa 1, T. Tayama 2, T. Sakakibara 2, T. Ito 4, S. Iguchi 3, K. Ohgushi 3 and Y. Tokura 3, 4, 5 1 Institute for Materials Research, Tohoku University, Japan 2 Institute for Solid State Physics, University of Tokyo, Japan 3 Department of Applied Physics, University of Tokyo, Japan 4 Correlated Electron Research Center, AIST, Japan 5 Spin Super-Structure Project, ERATO, JST, Japan

Contents 1. Introduction of R 2 Mo 2 O 7 2. Anomalous temperature dependence Contents 1. Introduction of R 2 Mo 2 O 7 2. Anomalous temperature dependence of sxy in Nd 2 Mo 2 O 7 3. Sign reversal of r. H in Nd 2 Mo 2 O 7 4. Sensitive dependence of sxy on the band-filling in (Sm 1 -x. Cax) 2 Mo 2 O 7 5. Summary

1. Introduction Pyrochlore-type structure : R 2 Mo 2 O 7 Mo-sublattice, composed of 1. Introduction Pyrochlore-type structure : R 2 Mo 2 O 7 Mo-sublattice, composed of corner-sharing teterahedra. (111)-plane is Kagome lattice. Mo R

Geometrical frustration in pyrochlore lattice AF-coupled spin system ? F-coupled spin system with strong Geometrical frustration in pyrochlore lattice AF-coupled spin system ? F-coupled spin system with strong single-ion anisotropy ? “Spin glass” “Spin Ice” M. J. Harris et al. , Phys. Rev. Lett. 79, 2554 (1997) A. P. Ramirez et al. , Nature 399, 333 (1999)

Electron configuration Mo 4+ : 4 d 2 (111) Mo ion is basically coordinated Electron configuration Mo 4+ : 4 d 2 (111) Mo ion is basically coordinated octahedrally by O ions, but the site-symmetry is D 3 d. eg Mo 4 d eg eg t 2 g O 2 - a 1 g Oh D 3 d

Phase diagram small Mo 4+ : 4 d 2 (constant filling) W/U large Phase diagram small Mo 4+ : 4 d 2 (constant filling) W/U large

Electronic structure ・small Drude weight in Sm 2 Mo 2 O 7 even in Electronic structure ・small Drude weight in Sm 2 Mo 2 O 7 even in the ground state. Sm 2 Mo 2 O 7 Y 2 Mo 2 O 7 Eg~ 0. 1 e. V Mo 4 d O 2 p Mott-Hubbard type (as opposed to CT type) EF

2. Anomalous temperature dependence of sxy in Nd 2 Mo 2 O 7 Strong 2. Anomalous temperature dependence of sxy in Nd 2 Mo 2 O 7 Strong single-ion anisotropy in Nd - moment Anisotropy of Nd moments is transmitted to Mo spins via the f-d interaction. “two-in, two-out” Nd 4 f Nd Jfd Mo

Resistivity, magnetization, and neutron diffraction TC Below TC ~ 90 K Nd 2 Mo Resistivity, magnetization, and neutron diffraction TC Below TC ~ 90 K Nd 2 Mo 2 O 7 30 Nd 4 f (localized moment) Mo 4 d (conduction electron) I(200) I(111) 20 10 1 1 0. 5 0 0 m B/Mo) 1. 5 0 M(H= 0. 5 T) ( Resistivity (m. Wcm) I ( m B 2/2 Nd 2 Mo 2 O 7) T* 0. 5 0 100 150 Temperature (K) * T 50 Below T* ~ 40 K T* is a crossover temperature where Nd moments begin to grow rapidly, resulting in a decrease of total M.

Magnetic Structure determined by the neutron diffraction experiment “umbrella structure” at 8 K q. Magnetic Structure determined by the neutron diffraction experiment “umbrella structure” at 8 K q. N ~ 70~ 80° m. N ~ 2. 2 m. B q. M< 10° m. M ~ 1. 4 m. B A magnetic unit cell contains four Nd-moments and four Mo-spins. Nd 4 f at 40 K( = T*) m. N~ 0. 2 m. B m. M~ 1. 3 m. B Mo 4 d

anomalous Hall effect in magnetic metals r. H = Ro. H + 4 p anomalous Hall effect in magnetic metals r. H = Ro. H + 4 p Rs. M Ey jx r. H H 4 p Rs. M M r. H = Ey/jx = Ro. H + 4 p. Rs. M Ro. H : ordinary term (proportional to H) 4 p. Rs. M : anomalous term (proportional to M) Ro. H Magnetic Field

r. H = Ro. H + ordinary term 6 4 p. Rs. M anomalous r. H = Ro. H + ordinary term 6 4 p. Rs. M anomalous term 6 2 K 5 Nd 2 Mo 2 O 7 H = 0. 5 T H || (100) r H (10 -6 W cm) 10 K 4 TC = 89 K 20 K 30 K 40 K 3 50 K 60 K 2 r H (10 -6 W cm) H || (100) 4 TC 2 70 K 1 0 80 K 90 K 100 K 0 2 4 6 8 10 Magnetic Field (T) 0 0 50 100 Temperature (K)

low temperature data down to 0. 5 K 6 Saturation of r. H is low temperature data down to 0. 5 K 6 Saturation of r. H is observed only below 2 or 1. 5 K. Nd 2 Mo 2 O 7 H = 0. 5 T r H (10 -6 W cm) 5 H || (100) 4 H || (110) 3 H || (111) 2 0 2 4 6 8 Temperature (K) 10

T* 20 15 10 s Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa, and T* 20 15 10 s Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa, and Y. Tokura, Science 291, 2573 (2001) 1 5 0 0 0 50 100 150 Temperature (K) 3 H || (100) H || (110) 2 H || (111) -1 -1 0 s xx(H=0. 5 T) (10 W cm ) 1 -1 -1 ・sxy for every direction continues to increase down to 2 K. 2 xy(H=0. 5 T) ( W cm ) MMo ( m B/Mo) Anomalous T-dependence of transverse conductivity sxy =r. H / (rxx 2+r. H 2) TC

Existing theories for anomalous Hall effect Karplus and Luttinger, Phys. Rev. 95 (1954) 1154 Existing theories for anomalous Hall effect Karplus and Luttinger, Phys. Rev. 95 (1954) 1154 L-S coupling of itinerant electron and imbalance of upand down-spin electron R ∝ r 2 s J. Kondo, Prog. Theoret. Phys. 27 (1962) 772 interaction between conduction electron and localized moment r. H ∝〈 -〈 )3〉 (m m〉 Both theory predict r. H 0 when T 0.

Experimental results for other ferromagnetic metals Fe La 1 -x. Cax. Mn. O 3 Experimental results for other ferromagnetic metals Fe La 1 -x. Cax. Mn. O 3 P. Matl et al. , Phys. Rev. B 57, 10248 (1998) C. H. Chun, M. B. Salamon, Y. Tomioka, and Y. Tokura, Phys. Rev. B 61, R 9225 (2000)

Berry phase theory of anomalous Hall effect J. Ye, Y. B. Kim, A. J. Berry phase theory of anomalous Hall effect J. Ye, Y. B. Kim, A. J. Millis, B. I. Shiraiman, P. Majumdar, and Z. Tesanovic, Phys. Rev. Lett. 83, (1999) 3737 K. Ohgushi, S. Murakami, and N. Nagaosa, Phys. Rev. B 62, (2000) R 6065 JH Si Sj Carrier moving in a spin background with strong Hund's rule coupling JH ≫ t tij = t 0 cos(qij /2) exp(i aij) A carrier moving in a topologically nontrivial spin background acquires a “Berry phase” and feels fictitious magnetic field b. anomalous Hall effect

Theoretical calculation based upon the Berry phase scenario ・Experimental result is reproduced with Mo-spin Theoretical calculation based upon the Berry phase scenario ・Experimental result is reproduced with Mo-spin tilting angle of 4 - 5 degree.

3. Sign reversal of rxy Prediction by the Berry phase theory: changes its sign 3. Sign reversal of rxy Prediction by the Berry phase theory: changes its sign for H || [111] approaches zero without changing sign for H || [100] and [110] High-field measurements Hall effect: up to 27 T at 1. 6 K Magnetization: up to 23 T at 1. 7 K, Vibrating-Sample Magnetometer High magnetic-field was provided by a hybrid magnet @ IMR, Tohoku University. Low-T measurements Magnetization: down to 50 m. K for H < 12 T @ISSP, Univ. of Tokyo

Field-dependence of r. H and M for H || [100] and [110] Magnetization M Field-dependence of r. H and M for H || [100] and [110] Magnetization M (m. B/Nd. Mo. O 3. 5) 3 [100] [110] H || [100] 2 H || [110] 1 T = 1. 7 K T-independent below 1. 7 K 70 m. K for H || [100] 50 m. K for H || [110] gradual magnetization process Mo r. H (m. W cm) 0 H || [100] 5 Nd 2 Mo 2 O 7 T = 1. 6 K Hall effect r. H monotonously approaches zero. H || [110] 0 0 10 20 Magnetic Field (T) 30 c. f. T. Kageyama et al. JPSJ 70, 3006 (2001)

Field-dependence of r. H and M for H || [111] Magnetization Mo T-independent below Field-dependence of r. H and M for H || [111] Magnetization Mo T-independent below 1. 7 K gradual magnetization process c. f. Dy 2 Ti 2 O 7 Hall effect r. H changes its sign at 7. 5 T in accord with the prediction.

Field dependence of Spin Chirality H || [100] fictitious field that penetrates the Mo-tetrahedron Field dependence of Spin Chirality H || [100] fictitious field that penetrates the Mo-tetrahedron Low Field High Field

Field dependence of Spin Chirality H|| [111] Low Field High Field Field dependence of Spin Chirality H|| [111] Low Field High Field

4. Sensitive dependence of transverse conductivity on the band-filling in (Sm 1 -x. Cax)2 4. Sensitive dependence of transverse conductivity on the band-filling in (Sm 1 -x. Cax)2 Mo 2 O 7 anomalous Hall effect in (Sm 0. 9 A 0. 1)2 Mo 2 O 7 : A = Ca 2+, Y 3+ In both cases, rxx and M show little variation. In case of A = Ca, sxy shows large variation. In case of A= Y, sxy shows little variation. role of Ca-doping ・ to introduce scattering center ・ to partially remove f-d interaction (Ca 2+ is non-magnetic ) ・ to change the band-filling Important !

rxx and magnetization Resistivity (m W cm) 1. 5 R 2 Mo 2 O rxx and magnetization Resistivity (m W cm) 1. 5 R 2 Mo 2 O 7 1 0. 5 R = Sm Sm 0. 9 Ca 0. 1 Sm 0. 9 Y 0. 1 0 Magnetization ( m B/Mo) In general, The variation of rxx and M is within 30 %. 1 We would expect little change of sxy. 0. 5 0 However…. . . Sm Sm 0. 9 Ca 0. 1 Sm 0. 9 Y 0. 1 0 20 40 60 80 Temperature (K) 100

sxy is enhanced by as large as 800% ! Sm 0. 9 Ca 0. sxy is enhanced by as large as 800% ! Sm 0. 9 Ca 0. 1 20 s -1 -1 xy(H=0. 5 T) (W cm ) 30 10 Sm Sm 0. 9 Y 0. 1 0 0 20 40 60 80 Temperature (K) 100

Explanation based upon the Berry phase mechanism M. Onoda and N. Nagaosa, J. Phys. Explanation based upon the Berry phase mechanism M. Onoda and N. Nagaosa, J. Phys. Soc. Jpn. 71, 19 (2002). : Fermi distribution func. sxy : gauge flux density Gauge flux density exhibits sharp peaks at band crossing points. As a result, sxy sensitively depends on the position of chemical potential. ky kx EF

Summary The sxy in Nd 2 Mo 2 O 7 continuously increases down to Summary The sxy in Nd 2 Mo 2 O 7 continuously increases down to 2 K. The r. H in Nd 2 Mo 2 O 7 changed its sign when the field was applied along [111] direction while it monotonously approached zero when applied along [100] and [110] direction. The sxy in (Sm 1 -x. Cax)2 Mo 2 O 7 shows large variation with the change of band-filling in spite of little change of sxy and M. These results suggest that the transverse conductivity in these compounds are induced by the spin chirality.