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Fun with Anomalous Hall Effect

Fun with Anomalous Hall Effect. Ruoyu Yin Jan. 16 th , 2019. CONTENTS. Ⅰ. Introduction A. A brief history of AHE B. classification of the AHE 1. Intrinsic contribution to 2. skew-scattering contribution to 3. side-jump contribution to Ⅱ. Experiments on typical materials

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Fun with Anomalous Hall Effect

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  1. Fun with Anomalous Hall Effect Ruoyu Yin Jan. 16th, 2019

  2. CONTENTS • Ⅰ. Introduction • A. A brief history of AHE • B. classification of the AHE • 1. Intrinsic contribution to • 2. skew-scattering contribution to • 3. side-jump contribution to • Ⅱ. Experiments on typical materials • A. transition metals • B. ferromagnetic semiconductors

  3. CONTENTS • Ⅰ. Introduction to AHE theories • A. A brief history of AHE • B. classification of the AHE • 1. Intrinsic contribution to • 2. skew-scattering contribution to • 3. side-jump contribution to • Ⅱ. Experiments on typical materials • A. transition metals • B. ferromagnetic semiconductors

  4. Hall effect In 1879, Hall effect was discovered In 1881, Anomalous Hall effect was discovered in ferromagnetic conductors, 10 times stronger than ordinary one. Edwin Hall

  5. An enigmatic problem • No substantive progress in nearly a whole century • AHE problem involves concepts based on topological and geometry • As a reference, general relativity was proposed in 1915 • In 1984, concept of Berry phase was created • In 1954, a concept call “anomalous velocity”, which is tightly connected with Berry phase arose. Problem raised, 1881

  6. Anomalous Hall effect Phys. Rev. 30, 1. AHE in Ni Rev. Mod. Phys. 25, 151.

  7. Anomalous Hall effect Experiments of Pugh (1930) and Pugh and Lippert (1932) established that an empirical relation between , and, applies to many materials over a broad range of external magnetic field. As is already understood, is related to ordinary Hall effect, and it can be explicitly expressed: depends only on the density of carriers. However, is subtly on a variety of material parameters, in particular, on longitudinal resistivity Phys. Rev. 36, 1503 Phys. Rev. 42, 709.

  8. Anomalous Hall effect How varies with is a very important topic in AHE research. Experimentally, people assumed a power low, i.e., Competing theories suggested that either or . The right figure shows , From Kooi, 1954 Phys. Rev. 95, 843

  9. Concept clarification A simple question: In a Hall measurement, if known , what is the relation between and ?

  10. Concept clarification A simple question: In a Hall measurement, if known , what is the relation between and ?

  11. Concept clarification Under normal conditions, , so approximately, Here, if told , one should be clear it is equivalent to tell:

  12. Classification of the AHE 1. Karplus and Luttinger mechanism • For non-magnetic material, the sum of “anomalous velocity” over all occupied band states is zero, but for ferromagnetic, it is nonzero. • Because this contribution depends only on the band structure and is independent of scattering, it has been referred to as intrinsic AHE. • is independent of , so for KL mechanism, , i.e., .

  13. Classification of the AHE 2. skew scattering • scattering of SOI impurity plays an essential role. • In this scattering dominate process, , i.e.

  14. Classification of the AHE 3. side jump • Side jump process is also caused by SOI impurities. • This could be viewed as KL mechanism under the electric field due to impurities. • Side jump mechanism predicts , identical to that of KL mechanism. • Thus, side jump mechanism yields a contribution to AHE seemingly independent of the density and strength of scatters.

  15. Modern treatment of AHE Boltzmann equation for electron distribution function: Berry curvature Coherent interband mixing disorder Intrinsic mechanism Skew scattering Side jump Full AHE

  16. Dependence on relaxation time Given that , how relates to is equivalent to . Skew scattering Because , we get . The dependence on relaxation time undoubtedly indicates this contribution is linked to scattering. Intrinsic mechanism Because , we get . Side jump Similar to intrinsic one, .

  17. Dependence on relaxation time How to tell the difference between Intrinsic contribution and side-jump contribution? Although over all contribution of side jump mechanism is independent of the density and strength of scatters, the fact that side jump contribution originates from scattering process would not alter. At the limit , side jump would not occur, while intrinsic contribution remains. Therefore, experimentally, intrinsic contribution is defined as the extrapolation of the ac-interband Hall conductivity to zero frequency in the limit , with faster than . The left term,

  18. Dependence on relaxation time Comparison of three kinds of contributions

  19. Detailed calculations of 3 contributions 1. Intrinsic contribution to The intrinsic contribution to the conductivity is dependent only on the band structure of the perfect crystal. It can be calculated directly from the simple Kubo formula for the Hall conductivity for an ideal lattice, given the eigenstates and eigenvalues of a Bloch Hamiltonian , Where

  20. Detailed calculations of 3 contributions 1. Intrinsic contribution to Where By noting that The first expression of reduces to: Where is Berry connection, and is Berry curvature, This result indicates the “anomalous velocity” form:

  21. Where complex formula come from

  22. Detailed calculations of 3 contributions 2. Intrinsic contribution to The skew-scattering contribution to the AHE can be sharply defined; it is simply the contribution which is proportional to the Bloch state transport lifetime. It will therefore tend to dominate in nearly perfect crystals. In semiclassical Boltzmann transport theory, the principle of detailed balance is always true: However, in this microscopic sense, detailed balance is not generic. In the presence of SOI, a transition which is right handed with respect to the magnetization direction has a different transition probability than the corresponding left-handed transition. Proposed by Smit, the potential of the impurity scattering reads: Using this matrix elements of potential and involving Born approximation, we derive: This asymmetric scattering contributes to Hall conductivity.

  23. Detailed calculations of 3 contributions 3. Side jump contribution to Formally, this contribution is defined unambiguously, when considering the scattering of a Gaussian wave packet from a spherical impurity with SOI: Potential term: SOI Hamiltonian: For a wave-packet incident with wave vector , Berger found that the wave packet suffers a transverse displacement . For , is far too small to be observed. In solids, however, the effective SOI is enhanced by band structure effects by the factor With the enhancement, the transverse displacement is , which is remarkable

  24. CONTENTS • Ⅰ. Introduction to AHE theories • A. A brief history of AHE • B. classification of the AHE • 1. Intrinsic contribution to • 2. skew-scattering contribution to • 3. side-jump contribution to • Ⅱ. Experiments on typical materials • A. transition metals • B. ferromagnetic semiconductors

  25. Transition Metals Three distinct regimes roughly delimited by the conductivity

  26. Transition Metals a. High conductivity regime Skew-scattering dominated for pure Fe film doped with Cr, Cu, Mn and Si vs at low temperature () Skewness parameter is dependent on material. Phys. Rev. B 7, 4203

  27. Transition Metals b. Good metal regime Measurement of Hall conductivity and resistivity in single-crystals Fe, and in thin films of Fe, Co, and Ni. • is T-dependent, but at specific temperature interval, is nearly in dependent of temperature. Phys. Rev. Lett. 99, 086602.

  28. Transition Metals c. Bad-metal-hopping regime Phys. Rev. B 78, 092405. Phys. Rev. B 79, 014431. J. Phys. C 8, 1010 Phys. Rev. B 77, 100403.

  29. Ferromagnetic semiconductors (DMSs)

  30. Ferromagnetic semiconductors (DMSs) Zero field Phys. Rev. Lett. 98, 026601.

  31. Ferromagnetic semiconductors (DMSs) Nernst effect, If assume: Thus, Zero-field Nernst coefficient Phys. Rev. Lett. 101, 117208

  32. Conclusion • Anomalous Hall effect: much stronger than OHE. where is dependent on material parameters, in particular, on . • A power low of on , i.e., . And is related to scattering process • Classification of AHE: intrinsic, skew-scattering and side-jump. The classification is based on both mechanism and relaxation time dependence. • The AHE of transition metal is divided into three regimes: high conductivity regime, good metal regime and bad-metal-hopping regime. • The AHE of DMSs seems similar to that of other FM material, but actually different. Experiments exclude the contribution of skew-scattering to metallic (Ga,Mn)As.

  33. Reference Pugh, E. M., and N. Rostoker, 1953, Hall Effect in Ferromagnetic Materials. Rev. Mod. Phys. 25, 151. Pugh, E. M., and T. W. Lippert, 1932, Hall e.m.f. and Intensity of Magnetization. Phys. Rev. 42, 709. Pugh, E. M., Hall Effect and the Magnetic Properties of Some Ferromagnetic Materials. Phys. Rev. 36, 1503. Kooi, C., 1954.Hall Effect in Ferromagnetics, Phys. Rev. 95, 843. Majumdar, A., and L. Berger, Hall Effect and Magnetoresistance in Pure Iron, Lead, Fe-Co, and Fe-Cr Dilute Alloys. 1973, Phys. Rev. B 7, 4203. Miyasato, T., Crossover Behavior of the Anomalous Hall Effect and Anomalous Nernst Effect in Itinerant Ferromagnets. Phys. Rev. Lett. 99, 086602. Sangiao, S., Anomalous Hall effect in Fe (001) epitaxial thin films over a wide range in conductivity. Phys. Rev. B 79, 014431.

  34. Reference Feng, J. S.-Y., R. D. Pahsley, Magnetoelectric properties of magnetite thin films. J. Phys. C 8, 1010. Venkateshvaran, D., Anomalous Hall effect in magnetite: Universal scaling relation between Hall and longitudinal conductivity in low-conductivity ferromagnets. Phys. Rev. B 78, 092405. Fernandez-Pacheco, A., Universal scaling of the anomalous Hall effect in Fe3O4 epitaxial thin films. Phys. Rev. B 77, 100403. Chun, S. H., Interplay between Carrier and Impurity Concentrations in Annealed Ga1−xMnxAs: Intrinsic Anomalous Hall Effect. Phys. Rev. Lett. 98, 026601. Pu, Y., Mott Relation for Anomalous Hall and Nernst Effects in Ga1−xMnxAs Ferromagnetic Semiconductors. Phys. Rev. Lett. 101, 117208. Nato, N., Anomalous Hall effect. Rev. Mod. Phys. 82, 1539

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