Persistent spin current

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

• 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
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Basics of a superconducting ring (Byers and Yang,
PRL 1961)

• the energy levels (and hence the free energy F)
  are an even periodic function with period ?0h/2e
• SC equilibrium state is given by the min of F
• away from a min, body current
• the flux inside the SC ring has to be quantized
• this in turn implies the Meissner effect

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Phase coherence in a mesoscopic ring (Buttiker,
Imry, and Landauer, Phys. Lett. 1983)
Webb et al, PRL 1985
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Persistent charge current in a normal metal ring
Similar to a periodic system with a large lattice
constant
R


Persistent current
Smoothed by elastic scattering etc
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Diamagnetic response of an isolated gold ring
(Chandrasekhar et al, PRL 1991)
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Persistent current, Drude weight, and the
Meissner fraction
Refs W. Kohn, PRB 1964, M. Himmerichs thesis,
Mainz 2004

• for insulators, D0
• for a clean metal (bulk), D?0 (even at finite T)
  
• D?0 does not imply superconducting behavior
  (Meissner effect)

Drude weight (or charge stiffness)
Meissner fraction

• Same as the Drude weight when T0
• for a clean metal, ?s0 at finite T
  (no Meissner effect)
• for mesoscopic normal metal, ?s?0 even at finite
  T (can show Meissner effect)

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Metal ring in a textured B field (Loss et al, PRL
1990, PRB 1992)

• After circling once, an electron acquires
• an AB phase 2pF/F0 (from the magnetic flux)
• a Berry phase (1/2)O(C) (from the texture)

Electron energy
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Persistent charge and spin current (Loss et al,
PRL 1990, PRB 1992)
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Ferromagnetic Heisenberg ring in a non-uniform B
field
(Schütz, Kollar, and Kopietz, PRL 2003)
Large spin limit, using Holstein-Primakoff bosons
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Longitudinal part to order S,
Transverse part
Choose the triads such that Then,
(rule of connection)
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Local triad and parallel-transported triad
Anholonomy angle of parallel-transported e1
solid angle traced out by m
Gauge-invariant expression
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Hamiltonian for spin wave (NN only, Ji.i1 -J)
Choose a gauge such that O spreads out evenly
e(k)
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Experimental detection (from Kollars poster)

• measure voltage difference ?V at a distance L
  above and below the ring
• magnetic field
• temperature

Estimate L100 nm N100 J100 K T50 K B0.1 T ?
?V0.2 nV
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Antiferromagnetic Heisenberg chain (S1/2)
Free fermion excitation Particle-hole
excitation
Jordan-Wigner transformation
Free fermions for XY-model
From Broholms Cargese lecture
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Bethe ansatz calculation (Karbach et al,
cond-mat/9809162 0008018)
Generation of 2 spinons by a spin flip
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AFM spin chain with S1 (Haldane spin chain)
Low-lying excitation
White and Huse, PRB 1993
From Zheludevs poster
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Gapped phases in isotropic spin systems (from
Broholms talk)
LSM theorem

• n number of spins per primitive unit cell
• S the spin quantum number
• m the magnetization per spin
• n(S-m)
• Oshikawa, Yamanaka, and Affleck (1997)
  and Oshikawa (2000)
• gaps in non-magnetized spin chains?
• Uniform spin ½ chain 1.½ ½ no gap
• Alternating spin ½ chain 2.½ 1
  perhaps
• (2n1) leg spin ½ ladder (2n1).½ n½ no
  gap
• 2n leg spin ½ ladder 2n.½ n perhaps
• Uniform spin 1 chain 1.1 1 perhaps

Integer gap possible
Non-Integer gap impossible
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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
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Antiferromagnetic Heisenberg chain (S1/2 case)
With twisted boundary magnetic field
Zhuo et al, cond-mat/0501693
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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

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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
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Persistent spin current
At T0, the spin current remains non-zero
Effective Haldane gap
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System size, correlation length, and spin current
(T0)
AFM limit
Magnon current due to zero-point fluctuation
Clear crossover between 2 regions
FM limit
no magnon current
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Magnetization current assisted by temperature
Assisted by quantum fluctuation (similar to AFM
spin ring)

• At low T, thermal energy lt field-induced energy
  gap (activation behavior)
• At higher T, Imax(T) is proportional to T
  (similar to FM spin ring)

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Issues on the spin current

• Charge is conserved, 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 conserved),

• 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.
• Spin is not always conserved. Will have more
  serious problems in spin-orbital coupled systems
  (such as Rashba system).

However,
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• Other 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?

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Thank You !