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					            Persistent spin current
                         in
                     a spin ring




Ming-Che Chang                 Jing-Nuo Wu (NCTU)
Dept of Physics
                               Min-Fong Yang (Tunghai U.)
Taiwan Normal Univ
A brief history

          • precursor: Hund, Ann. Phys. 1934

          • persistent current in a metal ring
               • diffusive regime (Buttiker, Imry, and Landauer, Phys. Lett. 1983)
               • inelastic scattering (Landauer and Buttiker, PRL 1985)
 charge



               • the effect of lead and reservoir (Buttiker, PRB 1985 … etc)
               • the effect of e-e interaction (Ambegaokar and Eckern, PRL 1990)

          • experimental observation (Levy et al, PRL 1990; Chandrasekhar et al, PRL 1991)

          • electron charge and spin current
               • textured magnetic field (Loss, Goldbart, and Balatsky, PRL 1990)
               • spin-orbit coupling (Meir et al, PRL 1989; Aronov et al, PRL 1993 … etc)

          • FM ring (Schutz, Kollar, and Kopietz, PRL 2003)
 spin




          • AFM ring (Schutz, Kollar, and Kopietz, PRB 2003)

          • this work: ferrimagnetic ring
Aharonov-Bohm (AB) effect (1959)


    magnetic Φ
    flux
                      r0
    solenoid

                                   path 1



                                   path 2
                                                               Φ

                                                         e
                                            AB phase =       ∫ A ⋅ dx
                                                      = 2πΦ/Φ0
       B = 0 ( A ≠ 0) for r > r0            flux quantum Φ0 = h/e
                                            (0.4×10-6 Gauss-cm2)
AB effect and resistance oscillation in a metal ring
(Webb et al, PRL 1985)
                                                                     Φ
Persistent charge current in a metal ring             kL → kL + 2π
                                                                     Φ0
                                 2
                 2
                     ⎛    Φ ⎞            Persistent current
      εn (Φ) =       ⎜ n+    ⎟
               2mR 2 ⎝    Φ0 ⎠                       e        e ∂En   ∂E
                                            In = −     vn = −       =− n
                                                     L        L ∂k    ∂Φ

                                                             I



                                           -1/2                           1/2   Φ/Φ0



                                         Smoothed by elastic scattering… etc

A ring with static disorder

                                             Phase coherence length
                        Similar to
                                                       L=2πR
           R
                                     …                                                 …
Magnetic response of a gold ring
(Chandrasekhar et al, PRL 1991)


                                   • magnitude of current 30 times
                                   larger than theoretical prediction


                                   Update:
                                   Bluhm et al, PRL 2009
                                   (using scanning SQUID)
                                   Bleszynski-Jayich et al, Science
                                   2009 (using micro-cantilever)
Another phase accumulated over a cycle: Berry phase (1984)

   Neutron spin guided by a
   helical B field

                                          Ω




                                 Berry phase = spin×Ω
A metal ring in a textured B field (Loss et al, PRL 1990, PRB 1992)
                                                     2
                                 1 ⎛ pθ       ⎞
                             H=    ⎜    + eAθ ⎟ + μ B B ⋅ σ
                                2m ⎝ R        ⎠
            R
                            After circling once, an electron acquires
                            • an AB phase 2πΦ/Φ0 (from the magnetic flux)
                            • a Berry phase ± (1/2)Ω(C) (from the “texture”)
         Ω(C)

                            Electron energy:
                B                                             2
                                        2
                                           ⎛    Φ       Φ ⎞
                            ε nσ   =       ⎜ n + A + σ z Ω ⎟ + σ z μB B
                                     2mR 2 ⎝    Φ0      Φ0 ⎠

                                                     ΦΩ Ω
                                                        ≡
                                                     Φ 0 4π
Persistent charge and spin current
(Loss et al, PRL 1990, PRB 1992)


         1                    e − βε nσ
     I = ∑ nσ (−e)v nσ
         L nσ                    Z
         1 ∂              ∂F
       =        ln Z = −
         β ∂Φ A          ∂Φ A

       β = 1/ kT , F = −(1/ β ) ln Z

         1                e − βε nσ
    I s = ∑ nσ sz v nσ
         L nσ                Z
               ∂                  ∂F
       =−          ln Z =
          2eβ ∂Φ Ω        2e ∂Φ Ω


                                                 1
                                          ΦΩ =     ( cos χ − 1)
                                                 2
From now on, no mobile electrons



Ferromagnet (FM), antiferromagnet (AFM), and ferrimagnet (FIM)




Spin wave in ferromagnet




Quantum of spin-wave = magnon (boson)
Spin current = transport of magnons
Ferromagnetic Heisenberg ring in a non-uniform B field
(Schütz, Kollar, and Kopietz, PRL 2003)




                                                   Expansion using a local triad

                                                   Si = Si// mi + Si⊥
                                                             ˆ
                                                   Si⊥ = Si1ei1 + Si 2 ei2
                                                            ˆ          ˆ
                                                   (hi ≡ g μ B Bi )


                                           1
  Large spin limit, using
                                ⇒ H=
                                           2
                                             ∑ J ij mi ⋅ m j Si// S // − ∑ hi ⋅mi Si// H // = O(S 2 + S )
                                                    ˆ ˆ             j
                                                                               ˆ
  Holstein-Primakoff bosons:
                                           1
                                          + ∑ J ij Si⊥ ⋅ S ⊥                           H ⊥ = O( S )
   Si// = S − bi+ bi                       2
                                                           j



   Si+ = 2 Sbi ; Si− = 2 Sbi+                 ⎛                       ⎞
                                          + ∑ ⎜ ∑ J ij S /j/ m j − hi ⎟ ⋅ Si⊥
                                                             ˆ                      H ' = O( S )
                                            i ⎝ j                     ⎠

                                                   // mi ( ⊥ Si⊥ ) with an error of order O(1)
                                                      ˆ
Hamiltonian for spin wave (NN only, Ji.i+1 ≣-J)

H SW = H // − H classical + H ⊥
                //



       = 2 JS ∑ bi†bi + h∑ bi†bi − JS ∑ {bi†+1bi exp [iΩ / N ] + h.c.}
               i                i                        i




H SW = ∑ ( ε k + h )bk+ bk ,                                        ε(k)
          k

ε k = 2 JS (1 − cos ka )
                   2π   ⎛   Ω⎞
where ka =              ⎜n+    ⎟                                                            ka
                   N    ⎝   2π ⎠                                                     Ω
                                                                                     N



Persistent magnetization current
                                                             • Im vanishes if T=0 (no magnon)
       g μB             g μB                 vk
Im =          Is = −
                          L
                               ∑ e(ε
                               k
                                       k   + h ) / kT
                                                        −1
                                                             • Im vanishes if N>>1
Antiferromagnetic Heisenberg chain
  •AFM chain with half-integer-spins :
   low-energy excitation is spinon, not magnon
  • so consider AFM chain with integer-spins

AFM spin chain with S=1




       White and Huse, PRB 1993                  From Zheludev’s poster
Antiferromagnetic Heisenberg ring in a textured B field
(Schütz, Kollar, and Kopietz, PRB 2004)

Large spin limit




                                             for a field not
                                             too strong




                       v



(and Δ H = 0)
Ferrimagnetic Heisenberg ring in a textured B field
(Wu, Chang, and Yang, PRB 2005)


  • consider large spin limit, NN coupling only
  • no need to introduce the staggered field


  γ ≡ SB / S A < 1

  Using HP bosons, plus Bogolioubov transformation,
  one has




                                                      Gapless branch +
                                                      gapful branch

   where
Ferrimagnetic Heisenberg chain
two separate branches of spin wave:




                                                  (S. Yamamoto, PRB 2004)

  • Gapless FM excitation well described by linear spin wave analysis
  • Modified spin wave qualitatively good for the gapful excitation
       Persistent magnetization current

               γ = SB / S A < 1
                                                   N = 100, γ = 0.8
                                                   (nearly sinusoidal)
                                                             0.020
                                                                0.015
                                                   T / JS A =
                                                                0.010
                                                                0.005

       At T=0, the spin current remains non-zero




    2 g μ B JS A
−                         effective Haldane gap
             N
                               1− γ 2
                          Δγ =
                                4γ
                   System size, correlation length, and spin current (T=0)



     1− γ 2                                                      AFM limit
Δγ =        ∝ a /ξ                                               γ ≈ 1, Δγ 1
      4γ
ξ : spin correlation length                                      ξ L
                                                                 Magnon current due to
                                                                 zero-point fluctuation

                               Clear crossover
                               between 2 regions
FM limit
γ    1, Δγ     1
ξ    L
no magnon current
           Issues on the spin current
           • Charge is conserved locally, and charge current density operator J
           is defined through the continuity eq.
           The form of J is not changed for Hamiltonians with interactions.
           • Spin current is defined in a similar way (if spin is locally conserved),
                      N
                                     ∂Slz
              H = J ∑ Sl ⋅ Sl +1           + ∇jlz = 0
                     l =1             ∂t
                                     jl z = J ( Slx Sly+1 − Sly Slx+1 ) = JS l⊥× S l⊥+1

However,   • Even in the Heisenberg model, Js is not unique when there is a
           non-uniform B field. (Schütz, Kollar, and Kopietz, E.Phys.J. B 2004).
           • Also, spin current operator can be complicated when there are 3-
           spin interactions (P. Lou, W.C. Wu, and M.C. Chang, Phys. Rev. B 2004).
           • Similar problems in spin-orbital coupled systems (e.g., Rashba).
           • experimental measurement?
Related topics:
• spin ring with smaller spins
• spin ring with anisotropic coupling
• diffusive transport
• leads and reservoir
• itinerant electrons
• any application?




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posted:6/15/2011
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