PowerPoint プレゼンテーション - V15のESRにおける吸収強度の温度

June 27- July 1, 2005
Trieste, Italy
Numerical study on ESR of V15
IIS, U. Tokyo, Manabu Machida
RIKEN, Toshiaki Iitaka
Dept. of Phys., Seiji Miyashita
Nanoscale molecular magnet V15
K6 V As6O42 H2O8H2O
IV
15
[A. Mueller and J. Doering (1988)]
Vanadiums provide
fifteen 1/2 spins.

(http://lab-neel.grenoble.cnrs.fr/)
Hamiltonian and Intensity
H    J ij Si  S j   Dij  Si  S j  HS  S
i, j
i, j
,T   1 e
IT  
 
 Re 
z
i
i

0
 I,T  d  


0
0
M M t  e
x
H
2
2
R
x
it
dt
,T  d
The parameter set
[H. De Raedt, et al., PRB 70 (2004) 064401]
[M. Machida, et al., JPSJ (2005) suppl.]
J  800K, J1  225K, J2  350K
D  D  D  40K
x
1,2
y
1,2
z
1,2
Difficulty
M M t  
x
x
T re
M M t 
 H
T re
 H
x
x
difficult!
2
O
N
– Direct diagonalization requires memory of  
3
– Its computation time is of ON 
(e.g. S. Miyashita et al. (1999))

Two numerical methods
• The double Chebyshev expansion method
(DCEM)
- speed and memory of O(N)
- all states and all temperatures
• The subspace iteration method (SIM)
- ESR at low temperatures.
DCEM
ESR absorption curves
DCEM
Typical calculation time for one absorption curve
is about half a day.
Background of DCEM
The DCEM =
a slight modification of
the Boltzmann-weighted time-dependent method
(BWTDM).
[T. Iitaka and T. Ebisuzaki, PRL (2003)]
Making use of
the random vector technique and
the Chebyshev polynomial expansion
DCEM (1)

M M t  
x
x
  e
Random phase vector

 Xˆ 

n

 H / 2
x
x
 H / 2

e
M
M
t
e

 


e
 H / 2
N
  n e

av
 H / 2

 T r Xˆ  Xˆ
 T r Xˆ
av
i n
n1
i m  n 
ˆ
n X n  e
 mn
m,n



av
n Xˆ m
av
DCEM (2)
Chebyshev expansions of
the thermal and time-evolution operators.
 H / 2
e
kmax
 
 
 I0  2 T0 H   2  Ik  2 Tk H 


k1
 i Ht
e
kmax
 J0 t T0 H   2 i Jk t Tk H 
k
k1
J
>> HS
small 
Temperature dependence

of intensity Itot     0 I, d
Our calculation
Experiment

[Y.Ajiro et al. (2003)]
SIM
ESR at low temperatures by SIM
I T  


H R2
0
2
,T  
,T  d

e
 E m
 e
 E m
e
 E n

M n
x
m
2
m,n
m
    E n  E m 
We consider
the lowest eight levels.
I1T  
H R2 HS
8
HS 
tanh

 2 

Intensity ratio
RT   I T /I1T 
Temperature dependence of R(T)
With DM
Without DM

Triangle model analysis
J12  J23  J31  J  2.5K
D  D  D  D  0.25K
x
1,2
y
1,2
z
1,2
Energy levels with weak DM
OD
E 8   43 J  32 HS
E 7   43 J  12 HS
E 6   43 J  12 HS
3
 H  J
2
c0
S
E 5   43 J  32 HS
E4  J  HS 
E3  J  HS 
3
2
3
2
D
D
E2  43 J  12 HS 
3
2
D
E1  43 J  12 HS 
3
2
D
3
4
3
4
1
2
1
2

E5
E2
E1
Intensity ratio of triangle model
Rtri T  T

0
3D
1
HS
HS  H
3
H  HS
c0
S
c0
S


At zero temperature


Summary
O(N) algorithms for the Kubo formula
DCEM
■ Random vector and Chebyshev polynomials
ESR of V15
■ High
to low temperatures by DCEM
■ Ultra-cold temperature by SIM
■ Triangle model analysis
M. Machida, T. Iitaka, and S. Miyashita, JPSJ (2005) suppl.
(cond-mat/0501439)

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