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

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?

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

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rD HS

Part I

Part II

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

∑

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I T( ) =ωHR

2

2′ ′ χ ω,T( ) dω

0

∞

∫

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′ ′ χ ω,T( ) = 1− e−βω( ) Re M x M x t( ) e−iωt dt

0

∞

∫

Difficulty

– Its computation time is of(e.g. S. Miyashita et al. (1999))

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M x M x t( ) =Tr e−βH M x M x t( )

Tr e−βH

– Direct diagonalization requires memory of

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O N 2( )

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O N 3( )

difficult!

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)].

DCEM (1)

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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

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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

∑

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

∑

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e− i Ht = J0 t( )T0 H( ) + 2 −i( )kJk t( )Tk H( )

k=1

kmax

∑

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J

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HS>> small

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( )

With and Without DM

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).

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I1 T( ) : a 1/2 spin

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

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Mz =1/2

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Mz = 3/2

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HS⟨⟨HSc

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HSc⟨⟨HS

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Rtri T( ) T →0 ⏐ → ⏐ ⏐€

3

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1+r D HS

Intensity Ratio of Triangle Model

At zero temperature

up to the first order of D

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rD = (D,D,D),

r D = 3D

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.

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rD HS