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