Alvaro-AHE

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					Spin Chirality, Berry Phase,
and Anomalous Hall Effect
     Summary:
     •Anomalous Hall Effect
     •Anderson-Hasegawa Transformation
     •Spin chirality in Manganites
     •AHE in frustrated magnets

                   Lorentz Force on Electron

            ρ xy = R0 B




                 Alvaro S. Núñez, Group Meeting
Anomalous Hall Effect

  ρ xy = R0 B + 4π Rs M

Hall resistance deviates from its standard
behavior in ferromagnets.

Karplus and Luttinger proposed that the AHE current arises from a
general property of how electrons move in a periodic lattice.
 r            r     Position operator x in a periodic lattice
 v ≠ ∇ kr ε ( k )   does not commute with itself.
                                                     [ X , X ] = iΩ
                                                        i   j         k   ε ijk
             r
  v = i[ H , X ]
  r                               r r
                        H = Ho + eE ⋅ X
                    r            r      r r
                    v = ∇ kr ε ( k ) + eE × Ω

        http://www.princeton.edu/~npo/AHE-Berry.html
Berry Curvature Term
G. Sundaram and Q. Niu, Phys. Rev. B 59, 14915 (1999).
 r        r     r r
                &
 & = ∇ rε(k ) + k × Ω
 r                      Magnetic Field in Momentum Space
       k
                            r r              r           r
 r&    r er r               Ω ( k ) = ∇ × u( k ) ∂ kr u( k )
             &
 k = eE + r × B
          c
Anomalous Hall Effect: extrinsic mechanisms
                            Skew Scattering:
                            Spin-orbit interaction +
                            Ferromagnetism=
                            One channel of scattering
                            Is favored over the other




                                   Side Jump:
                                   Lateral displacement
                                   different for spin up vs.
                                   spin down




             ρ H = a SKEW ρxx + a S . Jump ρxx
                                            2
Anderson-Hasegawa Physics




r                    r
Ω1                   Ω2             LaMn03
     Mn         Mn

                           +                      r r        r r
                     H = t c c2σ
                           1σ      + h. c.+ J H ( s1 ⋅ Ω 1 + s2 ⋅ Ω 2 )

     The limit JH >> t represent a physical
     situation where the spin directions are
     quenched but the hopping term is modulated
     by a factor involving the spin states
http://research.yale.edu/boulder/Boulder-2003/reading/arovas_lectures.
      Assa Auerbach, Interacting Electrons and Quantum Magnetism




                                                           iaij       θij 
                                         t    eff
                                                    =t e          cos 
                                             ij
                                                                      2


    The phase (Spin-Berry phase) can be
    regarded as an effective magnetic field
                                                        ia12
The total phase is:                                te
                      r r        r
  a12 + a 23 + a 31 ∝ S1 ⋅ S 2 × S 3

                  Spin Chirality
                                            ia31
                                       te
                                                           ia23
                                                    te



Total Phase:

               Spin Aharonov-Bohm Effect
       Φ magnetic flux       b magnetic chirality
Colossal MagnetoResistance Manganites




                               Carriers: Mn eg symmetry
                La1-xCaxMnO3   Moments: Mn t2g symmetry
 Anderson Hasegawa Representation:




                                       Effective Magnetic Field
       1 r
       4
                (   r       r
   bk = εijk n ⋅ ∂i n × ∂ j n   )   Spin Chirality


Continuum limit:




             1       Φ0 
     ρH =         H+   b
          neff ec    2π 
                                    Anomalous Hall Effect
Integrating out the fermions


    1 r
    4
              (  r       r
bk = εijk n ⋅ ∂i n × ∂ j n   )   Spin Chirality




  Low T: “Diluted gas” of
        skyrmion-antiskyrmion pairs
Numerical simulations:




Well agreement with
experiments:
Spin chirality in a frustrated magnet
Can we have spin chirality at 0K?
Can we manipulate the spin chirality?


Kagome Lattice and pyrochlores




  Kagome Lattice                 Pyrochlore Structure
    A
B       C

        a AB + a BC + a CA = φ




                                 r
                                 a1 = AB
                                 r
                                 a 2 = BC
                                 r
                                 a 3 = CA
Hall Effect:




          Clower = − sgn(sin φ )
          Cmiddle = 0
          Cupper = + sgn(sin φ )

If density is 1/3 per atom we have a QHE

Hubbard Model on a Kagome Lattice:
same physics
A2B2O7
                Molybdate family:
                Transition from spin glass
                insulator to ferromagnetic metal
T(K)




              Nd atoms: J=5/2 localized spins
                        of 4f electrons.
              Mo atoms: itinerant 4d-electrons
                         (t2g) per site.




       Jfd: antiferromagnetic coupling




       Mo 4d-electrons become ferromagnetic
       due to a double exchange mechanism
Temperature evolution of magnetic
and transport properties


                                    Curie T=89K
                                    Local Moment Alingment Tc~40K:
                                       •Decrease of total Magnetization
                                       •Increase of neutron scattering
                                                  o -1
                                        (ki=2.559 A )




                                    Different behavior of AH
                                    conductivity it raises with
                                    lowering T
Rare earth 4f moments strong spin anisotropy
easy axes along the center of tetrahedron: Net
Chirality
                                                 Net chirality

        6-fold degeneracy




                                 Chirality is spreaded into the
                                 4d-electrons by the AF exchange


                                  Umbrella-like structure
Test of the theory: Chirality mediated AHE

                      Absence of the effect in Gd2Mo2O7

                      Gd tetrahedrons have Heisenberg
                      interaction




                                         Chirality
                                         Manipulation
                                         AHE manipulation
Quantitative Theory




        •Double exchange between 4d-Mo electrons
        •AF Jfd coupling
        •Ising anisotropy for the localized moments

                                            Effective Exchange
                   Hopping Mo-Mo            Field (MF)




 r                                              AF Jfd coupling
 nj   Along the center of the tetrahedron
Mean Field + Fluctuations




RKKY interaction
Conclusions                          cond-mat/0405634




  Importance of Spin Chirality in for the AHE

   More general the importance of:
   Geometry and Berry Phases

				
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posted:10/16/2012
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