Persistent spin current in a spin ring

Transcription

1 Persistent spin current in a spin ring Ming-Che Chang Dept of Physics Taiwan Normal Univ Jing-Nuo Wu (NCTU) Min-Fong Yang (Tunghai U.)

2 A brief history precursor: Hund, Ann. Phys spin charge persistent current in a metal ring diffusive regime (Buttiker, Imry, and Landauer, Phys. Lett. 1983) inelastic scattering (Landauer and Buttiker, PRL 1985) 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 003) AFM ring (Schutz, Kollar, and Kopietz, PRB 003) this work: ferrimagnetic ring

3 Aharonov-Bohm (AB) effect (1959) magnetic flux Φ solenoid r 0 path 1 path Φ AB phase = e A dx B= 0 ( A 0) for r > r 0 = πφ/φ 0 flux quantum Φ 0 = h/e ( Gauss-cm )

4 AB effect and resistance oscillation in a metal ring (Webb et al, PRL 1985)

5 Persistent charge current in a metal ring kl kl + π Φ Φ 0 ε n Φ Φ = n + mr Φ ( ) 0 Persistent current I e e E En L L k Φ n n = vn = = I -1/ 1/ Φ/Φ 0 Smoothed by elastic scattering etc A ring with static disorder R Similar to Phase coherence length L=πR

6 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 009 (using scanning SQUID) Bleszynski-Jayich et al, Science 009 (using micro-cantilever)

7 Another phase accumulated over a cycle: Berry phase (1984) Neutron spin guided by a helical B field Ω Berry phase = spin Ω

8 A metal ring in a textured B field (Loss et al, PRL 1990, PRB 199) R Ω(C) 1 pθ H = + eaθ + μbb σ m R After circling once, an electron acquires an AB phase πφ/φ 0 (from the magnetic flux) a Berry phase ± (1/)Ω(C) (from the texture ) B Electron energy: ΦA Φ Ω nσ = n + + z + z B mr Φ0 Φ0 ε σ σ μ B ΦΩ Ω Φ0 4π

9 Persistent charge and spin current (Loss et al, PRL 1990, PRB 199) 1 e I = nσ ( e) v nσ L Z nσ 1 = ln Z β Φ A F = Φ A βε nσ β = 1/ kt, F = (1/ β )ln Z 1 e Is = nσ szv nσ L Z nσ = ln Z = eβ Φ e Ω βε nσ F Φ Ω Φ 1 cos 1 Ω = ( χ )

10 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

11 Ferromagnetic Heisenberg ring in a non-uniform B field (Schütz, Kollar, and Kopietz, PRL 003) Expansion using a local triad // S ˆ i = Si mi + Si 1 S ˆ ˆ i = Si1ei + Siei ( h gμ B) i B i Large spin limit, using Holstein-Primakoff bosons: S = S b b // + i i i S = Sb; S = Sb + + i i i i 1 H = Jm mss hms H = OS + S 1 + JS ij i Sj H = OS ( ) // // // // ˆ ˆ ˆ ij i j i j i i i ( ) // + JSmˆ ij j j hi Si H' = O( S ) i j // mˆ ( S ) with an error of order O(1) i i

12 Hamiltonian for spin wave (NN only, J i.i+1 -J) H = H H + H SW // // classical { + 1 [ Ω ]..} = JS bb+ h bb JS b b i N + h c i i i i i iexp / i i i H k ( ε ) = + h b + SW k k k k ( ka) ε = JS 1 cos π Ω where ka = n + N π b, ε(k) Ω N ka Persistent magnetization current I gμ gμ = I = B B k m s ( k h)/ kt L k e ε + v 1 I m vanishes if T=0 (no magnon) I m vanishes if N>>1

13 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

14 Antiferromagnetic Heisenberg ring in a textured B field (Schütz, Kollar, and Kopietz, PRB 004) Large spin limit for a field not too strong v (and Δ H = 0)

15 Ferrimagnetic Heisenberg ring in a textured B field (Wu, Chang, and Yang, PRB 005) consider large spin limit, NN coupling only no need to introduce the staggered field γ S / S < 1 B A Using HP bosons, plus Bogolioubov transformation, one has Gapless branch + gapful branch where

16 Ferrimagnetic Heisenberg chain two separate branches of spin wave: (S. Yamamoto, PRB 004) Gapless FM excitation well described by linear spin wave analysis Modified spin wave qualitatively good for the gapful excitation

17 Persistent magnetization current γ = S / S < 1 B A N = 100, γ = 0.8 (nearly sinusoidal) 0.00 T / JS A = At T=0, the spin current remains non-zero gμb A JS N effective Haldane gap Δ = γ 1 γ 4γ

18 System size, correlation length, and spin current (T=0) Δ = γ 1 γ a / ξ 4γ ξ : spin correlation length AFM limit γ 1, Δ 1 γ ξ L Magnon current due to zero-point fluctuation FM limit γ 1, Δ 1 γ ξ L no magnon current Clear crossover between regions

19 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 z l z l l 1 + jl = 0 l= 1 t z x y y x jl = J( Sl Sl+ 1 Sl Sl+ 1) = JSl Sl+ 1 H = J S S + S However, Even in the Heisenberg model, J s is not unique when there is a non-uniform B field. (Schütz, Kollar, and Kopietz, E.Phys.J. B 004). 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 004). Similar problems in spin-orbital coupled systems (e.g., Rashba). experimental measurement?

20 Related topics: spin ring with smaller spins spin ring with anisotropic coupling diffusive transport leads and reservoir itinerant electrons any application? g{tç~ çéâ4