Anisotropy and Dzyaloshinsky-Moriya Interaction in V15

Manabu Machida, Seiji Miyashita, and Toshiaki Iitaka

IIS, U. TokyoDept. of Physics, U. TokyoRIKEN

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V15

( )[ ] OH8OHOAsVK 2242615IV

6 ⋅V15

Vanadiums provide fifteen 1/2 spins.

Recently nano-magnets have attracted a lot of attention. Among them, we study the ESR of V15.

[I. Chiorescu et al. (2000)]

Hamiltonian

( ) ∑∑ ∑ −×⋅+⋅=i

xis

ij ijjiijjiij SHSSDSSJH

rrrrr

J=800K

J2

J1

Electron Spin Resonance

( )th

( )ωzIEnergy absorption is calculated by means of the Kubo formula.

• Double Chebyshev Expansion Method

• Subspace Iteration Method

Double Chebyshev Expansion Method(DCEM)

• The DCEM makes it possible to obtain the ESR intensity of V15 at arbitrary temperatures.

• Especially the DCEM has an advantage at high temperatures and strong fields.

Kubo Formula

( ) [ ][ ]H

iHtziHtzH MMtg

β

β

−

−−

=eTr

eeeTr

( ) ( )ωχωω zzRz H

I ′′=2

2€

′ ′ χ ω( ) = 1− e−βω( )Re g t( )

0

∞

∫ e−iωtdt

Intensity (total absorption):

€

I z= dω I z ω( )

0

∞

∫

Energy absorption:

Dynamical susceptibility:

Algorithm

Trace

Hβ−e Chebyshev polynomial expansion

Time evolution

Random vectors

Leap-frog method(Boltzmann-weighted time-dependent method)

Chebyshev expansion(Double Chebysev expansion method)

Chebyshev expansion method also in time domain

Chebyshev vs Leap-frog

J SH

€

˜ H s =gμB

Jmax

⎛

⎝ ⎜

⎞

⎠ ⎟× Hs[T]

[ ][ ] ⎟⎟

⎠

⎞⎜⎜⎝

⎛≈

s

sH

BHA ~ln~

frog-leap of timeComp.

Chebyshev of timeComp.

>>

1000(T) 100(T) 10(T)

Chebyshev

126(min) 187(min) 430(min)

Leap-frog 11(min) 165(min) 1326(min)

Test Parameters for DCEM

• J=800K, J1=54.4K, J2=160K• DM interaction: D=(40K,40K,40K)

Temperature Dependence of Intensity

[Y.Ajiro et al. (2003)]

With and Without DM

32K

Subpeak due to DM

Subspace Iteration Method(SIM)

• Much more powerful than the naïve power method.

• Especially the SIM has an advantage at low temperatures.

Method of Diagonalization

• Combination of (a)Anomalous Quantum Dynamics (Comp.Phy

s.Comm. Mitsutake et al. 1995)amplifies the eigenstates En Δ ｔ＞１

(b)Subspace Iteration Method (F.Chatelin1988)updates the orthogonal basis sets of low energy subspace S of the total Hilbert space.

Subspace and DOS

-100000

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

1e+006

-4000 -3500 -3000 -2500 -2000

DOS (arb. unit)

Energy (K)

'dos' using 1:2

S8

S56

Subspace S152

Energy Levels

-3655

-3650

-3645

-3640

-3635

-3630

-3625

0 0.5 1 1.5 2 2.5 3 3.5 4

Energy (K)

B (T)

'e0000' using 1:2'e0000' using 1:3'e0000' using 1:4'e0000' using 1:5'e0000' using 1:6'e0000' using 1:7'e0000' using 1:8

(DM=0,DD=0)

-3655

-3650

-3645

-3640

-3635

-3630

-3625

0 0.5 1 1.5 2 2.5 3 3.5 4

Energy (K)

B (T)

'e00dm' using 1:2'e00dm' using 1:3'e00dm' using 1:4'e00dm' using 1:5'e00dm' using 1:6'e00dm' using 1:7'e00dm' using 1:8

(DM=40K,DD=0)

-3655

-3650

-3645

-3640

-3635

-3630

-3625

0 0.5 1 1.5 2 2.5 3 3.5 4

Energy (K)

B (T)

'edd00' using 1:2'edd00' using 1:3'edd00' using 1:4'edd00' using 1:5'edd00' using 1:6'edd00' using 1:7'edd00' using 1:8

(DM=0,DD≠0)

-3655

-3650

-3645

-3640

-3635

-3630

-3625

0 0.5 1 1.5 2 2.5 3 3.5 4

Energy (K)

B (T)

'edddm' using 1:2'edddm' using 1:3'edddm' using 1:4'edddm' using 1:5'edddm' using 1:6'edddm' using 1:7'edddm' using 1:8

(DM=40K,DD≠0)

Ene

rgy

(K)

Ene

rgy

(K)

Ene

rgy

(K)

Ene

rgy

(K)

Method of Moments (1)

• Probability function

• Moments

€

χ ' '(ω) = f (ω) =πhω

kTZe−Ea / kT

a,b

∑ aμ x b( )2δ( aE − bE − hω)

∫

∫∞

∞

=

0

0

)(

)(

ωω

ωωω

ωfd

fd n

n

( )

∫

∫∞

∞

− =Δ

0

0

)(

)(

ωω

ωωωω

ωfd

fdn

n

• Total intensity

• Line width

Method of Moments (2)

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I = dωf (ω)0

∞

∫ ω = ω

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W = dωf (ω)(0

∞

∫ ω − ω )2 = Δω2

Test Parameters for SIM

• J1=250K, J2=350K

• DM interaction: D=(40K,40K,40K)

Line Width with DM/DD

0

2e+009

4e+009

6e+009

8e+009

1e+010

0.1 1 10 100 1000 10000

Line width (Hz)

Temperature (K)

'iesrdm00_8.txt' using ($1):5'iesr00dd_8.txt' using ($1):5'iesrdmdd_8.txt' using ($1):5

'iesrdm00_56.txt' using 1:5'iesr00dd_56.txt' using 1:5'iesrdmdd_56.txt' using 1:5

DM interaction ->Line width diverges!

at T=0.5K

0

2e+006

4e+006

6e+006

8e+006

1e+007

1.2e+007

1.4e+007

1.6e+007

1.8e+007

2e+007

0 50 100 150 200

Im chi(omega)

Omega (Hz)

Hext=108GHz

'chidm00_56.txt' using ($1/1e9):2

€

Im χ ω( )

Larmor precession

Peaks due to DM interaction

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ω GHz( )

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ω GHz( )

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ω GHz( )

at T=32K

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

1e+006

0 50 100 150 200

Im chi(omega)

Omega (Hz)

Hext=108GHz

'chidm00_56.txt' using ($1/1e9):8

€

Im χ ω( )

Larmor precession

Peaks due to DM interaction

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ω GHz( )

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ω GHz( )

at T=64K

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

500000

0 50 100 150 200

Im chi(omega)

Omega (Hz)

Hext=108GHz

'chidm00_56.txt' using ($1/1e9):9

€

Im χ ω( )

Larmor precession

Peaks due to DM interaction

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ω GHz( )

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ω GHz( )

at T=128K

0

20000

40000

60000

80000

100000

120000

140000

160000

180000

200000

0 50 100 150 200

Im chi(omega)

Omega (Hz)

Hext=108GHz

'chidm00_56.txt' using ($1/1e9):10

€

Im χ ω( )

Larmor precession

Peaks due to DM interaction

€

ω GHz( )

€

ω GHz( )

at T=256K

€

Im χ ω( )

Larmor precession

Peaks due to DM interaction

0

10000

20000

30000

40000

50000

60000

70000

0 50 100 150 200

Im chi(omega)

Omega (Hz)

Hext=108GHz

'chidm00_56.txt' using ($1/1e9):11

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ω GHz( )

€

ω GHz( )

Summary

• The DCEM reproduces the experimentally obtained temperature dependence of the intensity.

• The DM interaction allows a transition between excited states that is otherwise forbidden.

• Measuring these ESR peaks at higher temperatures may provide a method of estimating the magnitude and direction of the DM interaction.