QUANTUM TUNNELING OF MAGNETIZATION

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QUANTUM TUNNELING OF MAGNETIZATION IN MOLECULAR
MAGNETS
Theory
Valery Pokrovsky Department of Physics Texas AM
University
Tutorial Session
APS Meeting, Los Angeles, March 21-25, 2005
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Outline

• Nanomagnets. Brief description
• Molecular magnetic hysteresis
• Simple theoretical models of nanomagnets
• Tunneling in magnetic field
• Symmetry and selection rules
• Berry phases and oscillations in transverse
  magnetic field
• Radiation and absorption of photons
• Landau-Zener tunneling in noisy environment
• Interaction with nuclear spins
• Conclusions

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Nanomagnets Brief description
Third order axis
Rhombic
Tetragonal
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Molecular magnetic hysteresis
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AFM dimer Mn42 , S9/2 , Wernsdorfer et al.,
Nature-2002 Hill et al., Science-2003
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Magnetization curve of Fe-8 (W. Wernsdorfer)
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What must be explained

• Origin of quantum hysteresis
• Steps on hysteresis curve
• Dependence on temperature
• Dependence on sweeping rate

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Simple theoretical models for nanomagnets
Approximation of large unified spin exchange
interaction dominates, adiabatic motion of total
spin
No symmetry (Fe8)
Tetragonal symmetry (Mn12)
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Energy spectra in the absence of magnetic field
Splitting due to biaxial anisotropy
Fe8
Wernsdorfer and Sessoli 1999
Transitions conserve parity of m
Contradicts to experiment with
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Tunneling in magnetic field
Motion of levels in magnetic field
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Landau-Zener tunneling
Avoided level crossing (Wigner-Neumann theorem)
Schrödinger equations
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Adiabatic levels
Center-of mass energy 0
LZ parameter
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Transition probability
Landau-Zener time
Numerical values for Fe8
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Berrys phases and oscillations in transverse
magnetic field
Berrys phase the spin as a top acquires the
rotation phase
2
1
-- solid angle
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Symmetry and selection rules
Tetragonal symmetry
No symmetry
Zero for half-integer S No splitting of doublets
Zero for half-integer and odd integer S.
Kramers degeneration
Kalatsky and Pokrovsky, 1998
Loss, Di Vincenzo, Grinstein, 1992
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Magnetic field in hard direction
Tunneling amplitude must oscillate with the
transverse field
Theoretical prediction A. Garg, 1995
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W. Wernsdorfer and R. Sessoli, Science 284, 133
(1999)
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cond-mat/0503193
Asymmetric molecules
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Photon-assisted tunneling in Fe8
L. Sorace, W. Wernsdorfer, C. Thirion, A.-L.
Barra, M. Pacchioni, D. Mailly, and B. Barbara,
Phys. Rev. B 68, 220407 (2003)
Circularly polarized radiation f115 GHz
Strong nonlinearity spin interaction, heating,
superradiance?
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Landau-Zener tunneling in noisy environment
V. Pokrovsky and N. Sinitsyn, Phys. Rev. B 67,
144303, 2003
Classical fast noise in 2-level system
Noise is fast if
-- Spectral width of noise
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Transitions produced by noise
tacc
It produces transitions until
Accumulation time
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Transitions are produced by a spectral component
of noise, whose frequency equal to its
instantaneous value in the LZ 2-level system.
Fermi golden rule is exact for gaussian fast
noise!
Regular and random field act together
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Interaction with nuclear spins
N. Prokofiev and F. Stamp, Rep. Prog. Phys. 63,
669 (2000)
Central spin S
Nuclear spins
Nuclear spins do not have enough time to relax
during the LZ process
Smearing of effective magnetic field
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Environment
Model of spin bath
Coupling of nuclear spins to central spin and
environment
Central spin
Nuclear spin diffusion
Nuclear spins follow adiabatically the central
spin
Nuclear spins flip independently orthogonality
blocking
Spin diffusion the effective field and its rate
fluctuate during the transition with the
amplitude much larger than D
Depletion of central spins with small effective
magnetic field and square root relaxation
Why Landau-Zener theory works so well in
experiment?
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Explanation proposed by Prokofiev and Sinitsyn,
2004
Effects of spin diffusion and orthogonality
blocking do not change the term in the transition
probability proportional to D2. They
are compensated by interference effects.
Numerical calculations Dobrovitsky and Sinitsyn,
2004

• average number of nuclear spins
• flipped during the LZ transition

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Conclusions

• Nanomagnets display the quantum process of spin
  reversal

• This process is seen as single-molecule
  hysteresis

• The tunneling proceeds at crossing of two levels
  driven by magnetic field

• Minimal gap at avoided crossing is very small
  D10-7 K

• The energy gap oscillates vs. transverse
  magnetic field displaying the
• interference of two tunneling paths with the
  Berrys phase difference

• Interaction with nuclear spins is strong at low
  temperatures, but
• it does not change the transition probability if
  the latter is small

Open questions

• Selection rules at level crossings

• Collective processes at Landau-Zener tunneling

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S10
Fe8
Mn12
Orthorhombic
Tetragonal