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

2. A brief history charge spin
persistent current in a metal ring (Hund, Ann. Phys. 1934)
related papers on superconducting ring
Byers and Yang, PRL 1961 (flux quantization)
Bloch, PRL 1968 (AC Josephson effect)
persistent current in a metal ring
Imry, J. Phys. 1982
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 observations (Levy et al, PRL 1990; Chandrasekhar et al, PRL 1991)
electron spin 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)
AFM ring (Schutz, Kollar, and Kopietz, PRB 2003)
this work: ferrimagnetic ring
charge
spin

3. Aharonov-Bohm (AB) effect (1959)
magnetic flux Φ
solenoid
path 1
path 2 r0 Φ
Phase shift =
= 2πΦ/Φ0
flux quantum Φ0 = h/e
(0.4×10-6 Gauss-cm2)

4. AB resistance oscillation in a mesoscopic ring (Buttiker, Imry, and Landauer, Phys. Lett. 1983)
Webb et al, PRL 1985

5. Persistent charge current in a normal metal ring
Similar to a periodic system with a large lattice constant
L=2R
Phase coherence length R … …
Persistent current I
/0
1/2
-1/2
Smoothed by elastic scattering… etc

6. Berry phase (1984) Solid angle Ω = area on the sphere / R2 Ω
Berry phase = spin×Ω

7. Metal ring in a textured B field (Loss et al, PRL 1990, PRB 1992)
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”) B C
Ω(C)
Electron energy:

8. Persistent charge and spin current (Loss et al, PRL 1990, PRB 1992)

9. Ferromagnet (FM), antiferromagnet (AFM), and ferrimagnet (FIM)
Spin wave in ferromagnet
Spin-wave quantum is called “magnon”

10. Ferromagnetic Heisenberg ring in a non-uniform B field
(Schütz, Kollar, and Kopietz, PRL 2003)
Large spin limit, using
Holstein-Primakoff bosons:

11. vector/triad on curved surface: rules of parallel transport
mi+2 mi
mi+1
Δ is the sum of the angles of the triangle
defect angle (≡Δ－π)
= solid angle Ω traced out by the path
= rotation angle of a parallel-transported vector
Twist angle of the triad
= solid angle traced out by m
Ref: Littlejohn, PRB 1988; Balakrishnan et al, PRL 1990, PRB 1993.

12. 3
Local triad and parallel-transported triad +) 2 1
solid angle traced out by m
A gauge-invariant expression

13. Longitudinal part to order S,
Transverse part
Choose the triads such that Then, mi
mi+1

14. Hamiltonian for spin wave (NN only, Ji.i+1 ≡－J)
Persistent spin current
Magnetization current
Schütz, Kollar, and Kopietz, PRL 2003
Im vanishes if T= (no zero-point fluctuation!)
Im vanishes if N>>1
Choose a gauge such that Ω spreads out evenly ka
ε(k)

15. Experimental detection (from Kollar’s poster)
measure voltage difference ΔV at a distance L above and below the ring
magnetic field
temperature
Estimate:
L=100 nm
N=100
J=100 K
T=50 K
B=0.1 T
→ ΔV=0.2 nV

16. Antiferromagnetic Heisenberg ring in a textured B field (Schütz, Kollar, and Kopietz, PRB 2004)
Large spin limit
half-integer-spin AFM ring has infrared divergence (low energy excitation is spinon, not spin wave)
consider only integer-spin AFM ring. need to add staggered field to stabilize the “classical” configuration (modified SW)
for a field not too strong v

17. 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

18. Ferrimagnetic Heisenberg ring in a textured B field (Wu, Chang, and Yang, PRB 2005)
no infrared divergence, therefore no need to introduce the self-consistent staggered field
consider large spin limit, NN coupling only
Using HP bosons, plus Bogolioubov transf., one has
where

19. Persistent spin current
At T=0, the spin current remains non-zero
Effective Haldane gap

20. System size, correlation length, and spin current (T=0)
AFM limit
Magnon current due to zero-point fluctuation
Clear crossover between 2 regions
FM limit
no magnon current

21. Magnetization current assisted by temperature
Assisted by quantum fluctuation (similar to AFM spin ring)
At low T, thermal energy < field-induced energy gap (activation behavior)
At higher T, Imax(T) is proportional to T (similar to FM spin ring)

22. 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),
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).
Beware of background (equilibrium) spin current. There is no real transport of magnetization.
Similar problems in spin-orbital coupled systems (such as the Rashba system).

23. Open issues: spin ring with smaller spins
spin ring with anisotropic coupling
diffusive transport
leads and reservoir
itinerant electrons (Kondo lattice model.. etc)
connection with experiments
methods of measurement
any use for such a ring?