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Macroscopic quantum effects generated by the acoustic wave in molecular magnet. 김 광 희 ( 세종대학교 ). Acknowledgements E. M. Chudnovksy (City Univ. of New York, USA) D. A. Garanin (City Univ. of New York, USA). M acroscopic Q uantum P henomena. M acroscopic Q uantum P henomena. N. H. - PowerPoint PPT Presentation
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Macroscopic quantum effects generated by the acoustic wave in molecular magnet
김 광 희 ( 세종대학교 )
Acknowledgements
E. M. Chudnovksy (City Univ. of New York, USA)D. A. Garanin (City Univ. of New York, USA)
Macroscopic Quantum Phenomena
Macroscopic Quantum Phenomena
H
H
H
N
downbupa
microscopic, seen
deadlive
macroscopic, not seen
• Classical Dynamics • Quantum Mechanics
0dX
dVXbXm
CMCMCM
?XCM
XEX)X(VdX
d
m2 2
22
?X
Why is Quantum Cat not seen?
- Rash answer - maybe quantum mechanics does not hold for macroscopic bodies such as cats
- Careful answer-Quantum mechanics is OK, but
- maybe states are not degenerate
- maybe tunneling rate is too small
- maybe temperature is too high
- maybe the environments know the states of the system
DECOHERENCE!!
What is a good candidate to show macroscopic quantum phenomena?
• Josephoson junction-based system: phase difference of the order parameter– A. O. Caldeira and A. J. Leggett, Ann.
Phys. (NY) 149, 374 (1983)
– J. Clarke et al, Science, 239, 992 (1988)
• Magnetic system: Magnetization
S.C S.C
Outline
• Review of magnetization reversal in magnet
– Giant spin approximation
– Stoner-Wohlfarth model in classical magnet
– Landau-Zener model in quantum magnet
• Rabi spin oscillations generated by ultrasound in solids
• Macroscopic quantum effects generated by the acoustic wave
in molecular magnet
– Macroscopic quantum beats of magnetization
• Spintronics in molecular magnet
• Summary
Magnet
Molecular magnet
Cobalt cluster of 3 nm
Blue:1289-atoms truncated octahedronGrey: added atoms, total of 1388 atoms
Truncated octahedron with 1289 atoms for diameters of 3.1nm
HRTEM [110] direction, Fcc-structure, faceting
Hystersis Loop in a Magnet
Ms H
-Ms
M
h-1 +1
(anisotropy energy
+external field) cosMHcosDSHMDSE z
22zz
2z
fcoshcos2
1
DS2
E 22
2z
DS2
MHh
1h
1h
[Wernsdorfer et al. PRL (2001)]
[Zurek, QP 0306072]
Classical vs Quantum
Quantum Steps in Mn12
At resonance,
or
(uniaxial symmetry)HSgDSE zB2z
nmm EE
H)nm(g)nm(DH)m(gDm B2
B2
H=0
ng
DH
Bn
HmgDmE B2
m
Quantum Steps in Mn12
= 0.44 T
D = 0.60 K
cf) 0.61 K [Sessoli et al. ’93]
Bg
DH
H
= 0.44 T
mHgDmE zBm 2
[Barra et al. EPL (1996)]
Governed by Quantum dynamics !!
Source (coherent laser)
Phase interference
Figure(interference)
Young experiment Aharovnov-Bohm effect
and ……
Is Aharonov-Bohm effect is expected in molecular magnets ?
hard axis
[Wernsdorfer and Sessoli, Science (1999)]
To study quantum spin-rotation effects in solid, we need to estimate the magnetic field due to rotation .
),(, 21 trutr
the phonon displacement field
the local rotation of the crystal lattice
),(, 21 trutr
Gaussfu
B 10~1
~ 0
GHzf 3~ nmu 1~0
Rabi Spin Oscillation (Cont’d)
For displacement field in a surface acoustic wave, one obtains
In the presence of deformation of the crystal lattice, local anisotropy axes defined by the crystal field are rotated by the angle.
ztxztkxeuc
tr yk
t
t ˆ,ˆ)cos(2
, 0
SiA
Si eHeHˆˆ ˆˆ
Silat eˆ)(
transzA HSDH ˆˆˆ 2
Laboratory frame
Lattice frame
Rabi Spin Oscillation (Cont’d)
The lattice-frame Hamiltonian
The Rabi oscillation between the two lowest states of
SHH Alat ˆˆˆ )(
ztkxeuct
yk
t
t ˆ)sin(2 0
2
AH
Sm Sm
SS 2
1
SSSS z ˆ
sound wave
Rabi Spin Oscillation (Cont’d)
Project the Hamiltonian on the
“Rotating wave approximation”
31)( ˆ)sin(ˆ
2ˆ tkxh R
lateff
yk
tR
tSeuc 0
2
2
tCtCtlat )()(
statesS
tR
R
RR
Rt
R
R
i
i
et
it
tC
tetC
2
2
2sin
/
2cos
,2
sin
22/ RR
00,10 CC at 0,0 xt
Rabi Spin Oscillation (Cont’d)
The probability to find the spin in the state
Sm Sm
Rabi Spin Oscillation (Cont’d)
The expectation value of the projection of the spin onto the Z axis
xKtkxt
xKtkxtStSt
RRR
RRR
Rz
sincos2
1
2
1sinsin2ˆ 2
2
kK RR
1~ R1~
/9.0
,1.0
,10,0
R
Sx
Rabi Spin Oscillation (Cont’d)
The Rabi oscillations of
0,ˆ1ˆ0,
dxtxSS zxav
z
have a wave dependence on coordinatezS
!!
xKtkxt
xKtkxtStSt
RRR
RRR
Rz
sincos2
1
2
1sinsin2ˆ 2
2
How can you obtain the global Rabi oscillations averaged over the whole sample ?
0,ˆ1ˆ0,
dxtxSS zxav
z
312)( ˆ)sin(ˆˆ
E
tkxctSgh RBlat
eff
dt
dHc z
Longitudinal Field Sweep.
Field sweep(cont’d)
SS
SSS
ai
d
da
ai
akxpS
qpiS
d
da
2
2sin2
)()()( )(ˆ)( latlateff
lat thtt
i
StaStat
tkxctSgh
SSlat
RBlat
eff
)(
312)(
)(
ˆ)sin(ˆˆ
where RB qp
cgt
,
/,,
2
22,ˆ
SSz aSaStxS dxtxSS zxav
z
0,,ˆ1ˆ
Field sweep(cont’d)
The field is changing at a constant rate anda pulse of sound is introduced shortly before reaching the resonance between
S
R,, [G-H Kim and Chudnovsky, PRB (2009)]
/
p
2
cg B
R
To study the electronic and magnetic properties of a SMM and eventually to develop electronic devices
Molecular spintronics using molecular nanomanet
[G-H Kim and T-S Kim, PRL (2004)]
Idea is simple!
But, dynamics is not simple!!
What do we expect in the electronic devices?
1SMMRL HHHHΗ
k
pkpkpkp ccH R,Lp
k k
kRLkLRkk
kRLkLR1 .c.HSccJ.c.HccTH
Tunneling of electrons scattered by the spin of SMM
Direct tunneling between two electrodes
Electric currentLRI ?
SMMH :Hamiltonian of SMM
[J.A. Appelbaum, PRL, 1966; P.W. Anderson, PRL 1966 ]
Example: Fe8(cont’d)
Mgh
e2G sJT
2
Z
Y
XH
A
Bhard axis
easy axis
Mgh
e2G sJT
2
Molecular spintronics
Summary
- Classical vs. quantum dynamics in molecular magnet
- Rabi oscillation generated by the ultrasound in molecular magnet
- Applying a longitudinal magnetic field, we can generate quantum beats of the magnetization in molecular magnet
- Possibility of molecular nanomagnet for molecular spintronics
[T. W. Hansch , Nobel lecuture 2005]
Field sweep (cont’d)
The final magnetization on crossing the step
Field sweep (cont’d)
Another possible situation corresponds to the system initially saturated in the |-S> state, after which the acoustic wave is applied to the system and maintained during the sweep.
MSS
SSMMS
ai
d
da
ai
aikxpS
qpMSi
d
da
2
22sin2
MSMS
M,1
the level that provides significant probability of the transition
Field sweep (cont’d)
Field sweep (cont’d)
The optimal condition for pronounced beats
S
qp2
What does the above condition mean for experiment?
2
2 0
2
t
tR
c
qSuc
S
qu
0
The validity of the continuous elastic theory
0u 1q
Field sweep(cont’d)
Since experiments on MM require T~O(K), we should be concerned with the power of the sound. It should be sufficiently low to avoid the unwanted heating of the sample.
skGc
MHzf
scmc
cmg
t
/1
15.0
/10
/15
3
2
23
220
/200100
2
2
1
cmWatt
S
qc
cuA
P
t
t
S
qu
0
(ex) Fe8
Field sweep(cont’d)
Disorder produces randomness in the local field.
ctxHHH Mzz )()(
xHg B
The critical strength of disorder at which the beats disappear 005.0~
-The field sweep in MM is accompanied by the self-organization of the dipolar field such that the external field in the crystal maintains a very high degree of uniformity. [Garanin and Chudnovsky, PRL (2009)]
-Regardless of this effect, our prediction that the asymptotic value of exhibits a significant decrease in the presence of the sound, is not affected by disorder.
zS
Field sweep