anisotropy and dzyaloshinsky- moriya interaction in v15 manabu machida, seiji miyashita, and...
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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)
€
I = dωf (ω)0
∞
∫ ω = ω
€
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
€
ω GHz( )
€
ω GHz( )
€
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
€
ω GHz( )
€
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
€
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
€
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.