ESR Intensity and Anisotropy of Nanoscale Molecular Magnet V15 IIS, U. Tokyo, Manabu Machida RIKEN,...
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ESR Intensity and Anisotropy of Nanoscale Molecular Magnet
V15
IIS, U. Tokyo, Manabu MachidaRIKEN, Toshiaki Iitaka
Dept. of Phys., Seiji Miyashita
Fa3-4 (LT1175)
August 12, 2005, Florida, USA
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Nanoscale Molecular Magnet V15
(http://lab-neel.grenoble.cnrs.fr/)€
K6 V15IVAs6O42 H2O( )[ ] • 8H2O
Vanadiums provide fifteen 1/2 spins.
[A. Mueller and J. Doering (1988)]
Dzyaloshinsky-Moriya (DM) interaction?
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Outline of The Talk
■ A new O(N) algorithm for ESR.■ Temperature dependence of ESR intensity.
◆ We reproduce the experimental data.◆ The effect of DM is not clearly seen.
■ ESR intensity at very low temperatures.◆ The intensity is prominently affected by DM.◆ The deviation due to DM is estimated as
€
rD HS
Part I
Part II
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Hamiltonian and Intensity
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H = − Jij
v S i ⋅
v S j
i, j
∑ +v D ij ⋅
v S i ×
v S j( )
i, j
∑ − HS Siz
i
∑
€
I T( ) =ωHR
2
2′ ′ χ ω,T( ) dω
0
∞
∫
€
′ ′ χ ω,T( ) = 1− e−βω( ) Re M x M x t( ) e−iωt dt
0
∞
∫
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Difficulty
– Its computation time is of(e.g. S. Miyashita et al. (1999))
€
M x M x t( ) =Tr e−βH M x M x t( )
Tr e−βH
– Direct diagonalization requires memory of
€
O N 2( )
€
O N 3( )
difficult!
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Our New Method
DCEM(The Double Chebyshev Expansion Method)
(1) Speed and memory of O(N).(2) Random vector and Chebyshev polynomial.(3) No systematic error. (4) The scheme of time evolution is improved
from BWTDM[T. Iitaka and T. Ebisuzaki, PRL (2003)].
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DCEM (1)
€
M x M x t( ) =Φ e−βH / 2
( )M x M x t( ) e−βH / 2 Φ( )[ ]av
Φ e−βH / 2( ) e−βH / 2 Φ( )[ ]
av
Random phase vector
€
Φ ˆ X Φ[ ]av
= n ˆ X nn
∑ + e i θ m −θ n( )−δmn[ ]av
n ˆ X mm,n
∑
= Tr ˆ X + Δ ˆ X ≅ Tr ˆ X €
Φ = n e iθ n
n=1
N
∑
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DCEM (2)Chebyshev polynomial expansions of the thermal and time-evolution operators.
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e−βH / 2 = I0 − β2( )T0 H( ) + 2 Ik − β
2( )Tk H( )k=1
kmax
∑
€
e− i Ht = J0 t( )T0 H( ) + 2 −i( )kJk t( )Tk H( )
k=1
kmax
∑
€
J
€
HS>> small
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Comparison with Experiment- Temperature Dependence of
-
[Y.Ajiro et al. (2003)]
Our calculation Experiment
SIM(8): Intensity by the lowest eight levels.
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I T( )
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With and Without DM
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Effect of DM at Low Temperatures
With DM Without DM
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R T( ) = I T( ) /I1 T( )
Intensity ratio
Calculated by SIM(8) (the lowest eight levels).
€
I1 T( ) : a 1/2 spin
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Triangle Model and Its Energy Levels
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HSc =
3
2J ≅ 2.8 T[ ]
Produces energy levels almost equal to those of V15.
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rD
€
Mz =1/2
€
Mz = 3/2
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€
HS⟨⟨HSc
€
HSc⟨⟨HS
€
Rtri T( ) T →0 ⏐ → ⏐ ⏐€
3
€
1+r D HS
Intensity Ratio of Triangle Model
At zero temperature
up to the first order of D
€
rD = (D,D,D),
r D = 3D
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SummaryTemperature dependence of ESR intensity
Intensity ratio at ultra-cold limit
③Intensity ratio at weak fields (Mz=1/2) deviates from 1 due to DM interaction.
④The deviation is given by
M. M., T. Iitaka, and S. Miyashita, J. Phys. Soc. Jpn. Suppl. 74 (2005) 107 (cond-mat/0501439).M. M., T. Iitaka, and S. Miyashita, in preparation.
①O(N) algorithm both for speed and memory.②We reproduce the experimental intensity.
€
rD HS