Quantum dynamics in Single-Molecule Magnetsschnack/... · Landau-Zener tunneling Mn 12 with D =...

Quantum dynamics inSingle-Molecule Magnets

Wolfgang WernsdorferLaboratoire de

Magnétisme Louis NéelC.N.R.S. - Grenoble

S = 102 to 106 S = 1/2 to ≈ 30

Magnetic structures

S = 1020

1010

108 10

6 10

5 10

4 10

3 10

2 10 1

clusters atomsmolecularclustersnanoparticlesmicron

particlespermanentmagnets

macroscopic atomic

multi - domains

1 mm

single - domains

20 nm 3 nm

spins

1 nm

Magnetization reversal in magnetic structures

S = 1020

1010

108 10

6 10

5 10

4 10

3 10

2 10 1

clusters atomsmolecularclustersnanoparticlesmicron

particlespermanentmagnets

single - domains spins

macroscopic atomic

-1

0

1

-40 -20 0 20 40

M/M

S

µ0H(mT)

nucleation, propagation andannihilation of domain walls

-1

0

1

-1 0 1

M/M

S

µ0H(T)

Fe8

1K0.1K

0.7K

quantum tunneling,interference, coherence

-1

0

1

-100 0 100

M/M

S

µ0H(mT)

uniform rotation,curling, etc.

multi - domains

Magnetization reversal in magnetic structures

S = 1020

1010

108 10

6 10

5 10

4 10

3 10

2 10 1

clusters atomsmolecularclustersnanoparticlesmicron

particlespermanentmagnets

single - domains spinsmulti - domains

macroscopic atomic

nucleation, propagation andannihilation of domain walls

quantum tunneling,interference, coherence

uniform rotation,curling, etc.

“Classical” magnetismMicromagnetics

Landau Lifshitz Gilbert equation

Quantum magnetismSchrödinger equationOperator formalism

Path intergralsab-initio calulations

etc.

Photons Phonons

Spin bath- nuclear spins- paramagnetic spins- intermolecular interactions

Interactions in magnetic quantum systems (decoherence)

Conductionelectrons

etc.

Magnetic quantum system- spin- molecule- nanoparticle- spin chain

Single-moleculemagnets (SMM)

Mn12 S = 10

Fe8 S = 10

V15 S = 1/2

Ni12 S = 12

Giant spins

Mn84 S ≈ 6

Winpenny, 1999

Lis, 1980

Christou, 2004

Wiegart, 1984

Müller, 1993

Micro-SQUID magnetometry

• sensitivity : 10-4 Fo≈ 102 - 103 µB i.e. (2 nm)3 of Co

≈ 10-18 - 10-17 emu

• fabricated by electron beam lithography(D. Mailly, LPN, Marcoussis - Paris)

particle

Josephson junctions

stray field

≈ 1 µmB

A. Benoit, CRTBT, 1989

Roadmap of the micro-SQUID technique

Crystal of SMMs

Micro-SQUID array

B

crystal

50 µm

• crystal size > few µm• 10-12 to 10-17 emu• temperature 0.03 - 7 K• field < 1.4 T and < 20 T/s• rotation of field• transverse field• several SQUIDs at different positions

• µ-Hall Effect

– Based on Lorentz Force– Measures magnetic field

– Large applied in-planemagnetic fields (>20 T)

– Broad temperature range– Single magnetic particles– Ultimate sensitivity ~102 µB

Micro-magnetometry• µ-SQUID

– Based on flux quantization– Measures magnetic flux– Applied fields below the

upper critical field (~1 T)– Low temperature (below Tc)– Single magnetic particles– Ultimate sensitivity ~1 µB

Hall bars

sampleB

1 to 10 µm

VH =!I

neM

sampleB

1 µm

Jospheson Junctions

OutlineI. A simple tunnel picture (Mn12, Fe8)

• Giant spin model• Landau Zener tunneling • Berry phase

II. Coupling with environment

Mn12-tBuAc

MnIII: s = 2MnIV: s = 3/2

[Mn12O12(O2CCH2But)16(CH3OH)4].CH3OH

not aligned • well aligned !!• larger unit cell• less disorder

Mn12-Ac Mn12-tBuAcComparison between

[Mn12O12(O2CCH3)16(H2O)4].2CH3CO2H.4H2O [Mn12O12(O2CCH2But)16(CH3OH)4].CH3OH

Giant spin approximation (Mn12)MnIII: s = 2MnIV: s = 3/2S = 10

Giant spin approximationExample: Mn12-ac

quantum number M-10 -8 -6 -4 -2 0 2 4 6 8 10

0

40

80

120

160

Energ

y(K

)S = 10

D = 0.63 K

S = 9

D' = 0.57 K

I. Tupitsyn (2000)

J1 ≈ -215 KJ2 ≈ J3 ≈ -86 KJ4 < 43 K

J 4

J 1 J 2

J 3

Hilbert space: (2x2 + 1)8(2x3/2 + 1)4 ≈ 108

Mn3+

Mn4+

≈ 30 K

Giant spin modelSpin Hamiltonian:

!

" = #D Sz2 #B Sz

4 +" tr # gµB

r S

r H

(2S + 1) energy states: M = -S, -S+1, …, S

-10 -5 0 5 10

Ener

gy

quantum number M

H = 0

with S = 10, D = 0.56 K, B = 1.5 mK

Energy levels: Zeeman diagram

energ

y

magnetic field

!

-10

8 -10

8

Tunneling probability at an avoided level crossingLandau-Zener model (1932)

!

c ="

2h gµBm #m' µ0

L. Landau, Phys. Z. Sowjetunion 2, 46 (1932); C. Zener, Proc. R. Soc. London, Ser. A 137, 696, (1932); E.C.G.Stückelberg, Helv. Phys. Acta 5, 369 (1932); S. Miyashita, J. Phys. Soc. Jpn. 64, 3207 (1995); V.V. Dobrovitskiand A.K. Zvezdin, Euro. Phys. Lett. 38, 377 (1997); L. Gunther, Euro. Phys. Lett. 39, 1 (1997); G.Rose andP.C.E. Stamp, Low Temp. Phys. 113, 1153 (1999); M. Leuenberger and D. Loss, Phys. Rev. B 61, 12200 (2000);M. Thorwart, M. Grifoni, and P. Hänggi, Phys. Rev. Lett. 85, 860 (2000); …

!

P =1" exp "c#2

dH /dt

$

% & &

'

( ) )

energ

y

magnetic field

!

| S, m >

| S, m' >

1 P

1 - P

| S, m >

| S, m' >

-10

-9

-8

-7

10 9 87

Application ofLandau-Zener

tunneling

Mn12

with D = 0.56 K, B = 1.5 mK

S = 10

6

5

4

3

21

!

" = #D Sz2 #B Sz

4 +" tr # gµB

r S

r H

0.1 K

Photons Phonons

Spin bath- nuclear spins- paramagnetic spins- intermolecular interactions

Conductionelectrons

etc.

Magnetic quantum system- spin- molecule- nanoparticle- chain

Interactions in magnetic quantum systems (decoherence)

-10

-9

-8

-7

10 9 87

Landau-Zenertunneling &temperature

Mn12

with D = 0.56 K, B = 1.5 mK

S = 10

6

5

4

3

21

!

" = #D Sz2 #B Sz

4 +" tr # gµB

r S

r H

Field derivative of M(H) at different temperaturesMn12-tBuAc

W. Wernsdorfer, M. Murugesu, G. Christou, cond-mat/0508437

Temperature dependence of the peak positions of dM/dHMn12-tBuAc

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5 3 3.5 4

µ0H

z (

T)

T (K)

(9:0)

(10:1)(10:2)

(10:0)

(9:1)(9:2)

(9:3)(8:0) (8:1)

(8:2)(8:3)

(7:2) (7:3) (7:4)

(6:2) (6:3)(6:4)

(5:3) (5:4)(5:5)

(4:3) (4:4)(4:5)

(3:4) (3:5)(3:6)

(2:4)(2:5) (2:6)

(1:6) (1:7)

2 mT/s

(1:5)

(10:3)

(7:0)(7:1)

(6:0)(6:1)

W. Wernsdorfer, M. Murugesu, G. Christou, cond-mat/0508437

Mn12-Ac Mn12-tBuAcComparison between

W. Wernsdorfer, M. Murugesu, G. Christou, cond-mat/0508437

I. Chiorescu, B. Barbara, et al., Phys. Rev. Lett. 85, 4807 (2000)

Mn12-Ac Mn12-tBuAcComparison between

W. Wernsdorfer, M. Murugesu, G. Christou, cond-mat/0508437

A. D. Kent, et al., EPL 49, 521 (2000) and J. Appl. Phys. 87, 5493 (2000)

K. M. Mertes, M. P. Sarachik, et al. Phys. Rev. B 65, 212401 (2002)

Mn12-Ac Mn12-tBuAcComparison between

W. Wernsdorfer, M. Murugesu, G. Christou, cond-mat/0508437

I. Chiorescu,B. Barbara, et al., PRL 85, 4807 (2000)

Mn12-Ac Mn12-tBuAcComparison between

L. Bokacheva, A. D. Kent, et al., PRL 85, 4803 (2000)

W. Wernsdorfer, M. Murugesu, G. Christou, cond-mat/0508437

“sharp” crossover

“smooth” crossover

“sharp” crossoverin agreement with theory

E. M. Chudnovsky and D. A. Garanin, Phys. Rev. Lett. 79, 004469 (1997)

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

v=140 mT/sv=70 mT/sv=14 mT/sv=2.8 mT/s

M/M

S

µ 0H(T)

40 mK

-1 -0.5 0 0.5 1-40

-30

-20

-10

0

Energ

y (K

)

µ0Hz (T)

-10

-9

-8

-7

10

9

8

7

Application ofLandau-Zener

tunneling

Fe8

! = "D Sz

2 + E Sx2"Sy

2( ) + gµB

r S

r H

with S = 10, D = 0.27 K, E = 0.046KA.-L. Barra et al. EPL (1996)

S = 10

Giant spin Hamiltonian of Fe8

! = "D Sz

2 + E Sx2"Sy

2( ) + gµB

r S

r H

Z

Y

X

easy axis

hard axis

easy plan YZ

Htransj

SxHx + SyHy + SzHz

Hysteresis loops at different transverse fields

-1

-0.5

0

0.5

1

-0.2 0 0.2 0.4 0.6 0.8

M/M

S

µ0Hz(T)

Htrans =

0.000 TdHz/dt = 14 mT/s

0.056 T

0.112 T

0.196 T

-1

-0.98

-0.96

-0.94

-0.92

-0.9

-0.04 -0.02 0 0.02 0.04

0 T

dHz/dt =

1.4 mT/s

Htrans =

0.056 T

0.112 T

0.196 T

Quantum phase interference (Berry phase)in single-molecule magnets

Z

Y

X

easy axis

hard axis

easy plan YZ0 0.2 0.4 0.6 0.8 1 1.2 1.4

0.1

1

10

Tu

nn

el

sp

litt

ing

!(1

0-7

K)

Transverse field (T)

M = -10 10

! ! 90°

! ! 0°

! ! 7°

! ! 20°! ! 50°

Htransj

W. Wernsdorfer and R. Sessoli, Science 284, 133 (1999)

Transverse field dependence of tunnel splitting(operator formalism)

0 0.2 0.4 0.6 0.8 1 1.2

0.1

1

10

Tu

nn

el

sp

itti

ng

!(1

0-

7 K

)

Magnetic tranverse field (T)

0°

! = 90°

50°

30°

20°

10°

5°

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0.1

1

10

Tu

nn

el

sp

litt

ing

!(1

0-

7 K

)

Magnetic transverse field (T)

M = -10 -> 10

! " 0°

! " 7°

! " 20°! " 50°! " 90°

W. Wernsdorfer and R. Sessoli, Science 284, 133 (1999)

Path integrals (Feynman)Path-integral partition function:

• extremal trajectories that minimize the Euclidian action, at T = 0

where LE is the Euclidean magnetic Lagrangian related to the real-time Lagrangian L through LE = - L (t -it )

Z

Y

X

H

!

A

B

destructive interference occures wheneverthe shaded area is kp/S, for odd k.

A. Garg, Europhys. Lett. 22, 205 (1993)

Transverse field dependence of tunnel splitting (path integrals formalism)

Z

Y

X

H

!

A

B

0 0.5 1 1.5 2 2.5 30.001

0.1

10

1000

105

107

! (1

0-

8 K

)

µ 0Ht r a n s( T )

0°

! = 90° 50° 30°

20°

10°

5°

! = "D Sz

2 + E Sx2"Sy

2( ) + gµB

r S

r H

A. Garg, Europhys. Lett. 22, 205 (1993)

Quantum phase interference and spin parity in Mn12

10-5

10-4

10-3

10-2

-1.2 -0.8 -0.4 0 0.4 0.8 1.2!M

/M

sµ0H (T)

1.2 K

1.1 K

1.0 K

10-4

10-3

10-2

10-1

-1.2 -0.8 -0.4 0 0.4 0.8 1.2

!M

/M

s

µ0H (T)

1.9 K

1.8 K

1.7 K

1.6 K

1.5 K

[Mn12]-e

S = 19/2[Mn12]-2e

S = 10

W. Wernsdorfer, N. E. Chakov, G. Christou, PRL 95, 037203 (2005)

Spin ground states of Mn based SMMs

0

5

10

15

20

25

1 10 100

S

number of Mn-ions

2 3 4 5 7 20 30 50

Mn12Mn4

Mn25

Mn84

Mn70Mn30

Mn18

Mn2

Mn9

2004

2004

Anisotropy barriers of Mn based SMMs

0

10

20

30

40

50

60

70

1 10 100

!E

(K)

number of Mn-ions

2 3 4 5 7 20 30 50

Mn12

Mn4

Mn25

Mn84

Mn70

Mn30

Mn18

Mn2

Mn9

2004

2004

1.5 nm

0.5 nm

Quantum Tunneling of Magnetization in Lanthanide Single-Molecule MagnetsBis(phthalocyaninato)terbium

E (K

)

H (T)N.Ishikawa, et al.,

J.Phys.Chem. A 106, 9543 (2002) J. Am.Chem.Soc. 125, 8694 (2003)

Inorg.Chem. 42, 2440 (2003)J.Phys.Chem. A 107, 9543 (2003)J. Phys.Chem.B 108, 11265 (2004)

≈ 600 K

Naoto Ishikawa, Department of Applied Chemistry, Chuo University, Tokyo

Quantum Tunneling of Magnetization in Lanthanide Single-Molecule MagnetsBis(phthalocyaninato)terbium

E (K

)

H (T)

≈ 600 K

N. Ishikawa, M. Sugita, W. Wernsdorfer, Angew. Chem. Int. Ed. 44 ,2 (2005)N. Ishikawa, M. Sugita, W. Wernsdorfer, J. Am. Chem. Soc. 127, 3650 (2005)

-1

-0.5

0

0.5

1

-1.2 -0.8 -0.4 0 0.4 0.8 1.2

0.04 K2.0 K7 K

M/M

s

µ0H (T)

0.14 T/s

2% Tb

Naoto Ishikawa, Department of Applied Chemistry, Chuo University, Tokyo

-1

-0.5

0

0.5

1

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

0.280 T/s0.140 T/s0.070 T/s0.035 T/s0.017 T/s0.008 T/s0.004 T/s0.002 T/s0.001 T/s

M/M

s

µ0H (T)

0.04 K2% Tb

- 644.6

- 644.4

- 644.2

- 644.0

- 643.8

- 60 - 40 - 20 0 20 40 60

Ahf=0.0173cm-1

Pquad=0.010cm-1

:avoided crossing occurs byoff-diagonal LF term

:avoided crossing occurs bytransversefield

Others :avoided crossing doesnot occur by either LF termnor transverse field

H = Zeeman + LF term + AhfJ·I + Pquad{Iz2 + (1/3)I(I+1)}

N. Ishikawa, M. Sugita, W. Wernsdorfer, Angew. Chem. Int. Ed. 44 ,2 (2005)N. Ishikawa, M. Sugita, W. Wernsdorfer, J. Am. Chem. Soc. 127, 3650 (2005)

Photons Phonons

Spin bath- nuclear spins- paramagnetic spins- intermolecular interactions

Conductionelectrons

etc.

Magnetic quantum system- spin- molecule- nanoparticle- chain

Interactions in magnetic quantum systems (decoherence)

Exchange-biased quantum tunnelling in asupramolecular dimer of single-molecule magnets

W. Wernsdorfer, N. Aliaga-Alcalde, D. N. Hendrickson & G. ChristouNature 416, 406 (28 March 2002)

9/2

-9/2

-9/2

9/2J J

Single-chain magnets (SCM)

See talk of L. BoganiMonday, 16:15

Mn2Ni - chainof S = 3 units

R. Clérac, H. Miyasaka, M. Yamashita, and C. Coulon, J. Am. Chem. Soc.124, 12837 (2002).

Quantum nucleation in a single-chain magnetWW et al.

cond-mat/0509309

Photons Phonons

Spin bath- nuclear spins- paramagnetic spins- intermolecular interactions

Conductionelectrons

etc.

Magnetic quantum system- spin- molecule- nanoparticle- chain

Interactions in magnetic quantum systems (decoherence)

Quantum computing in molecular magnetsMichael N. Leuenberger & Daniel Loss

NATURE, 410, 791 (2001)• implementation of Grover's algorithm• storage unit of a dynamic random access memory

device.• fast electron spin resonance pulses can be used to

decode and read out stored numbers of up to 105

with access times as short as 0.1 nanoseconds.

Photon assisted tunnelingAbsorption of circular polarized microwaves

-1 -0.5 0 0.5 1-40

-30

-20

-10

0

Energ

y (K

)µ0Hz (T)

!M = ±1h!

-10

-9

-8

-7

10

9

8

7

tunneling∆M = +1hw

-10 -5 0 5 10

Ener

gy

quantum number m

H = 0

hw∆M = -1

PRB 68, 220407(R) (2003)

Photon assisted tunnelingAbsorption of circular polarized microwaves

115 GHz

PRB 68, 220407(R) (2003)

Absorption of circular polarized microwaves(115 GHz)

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

0

0.119

0.151

0.190

0.237

0.256

0.320

0.458

M/M

s

µ0Hz (T)

60 mK

115 GHz

0.007 T/s

P/P0 =

PRB 68, 220407(R) (2003)

Photons Phonons

Spin bath- nuclear spins- paramagnetic spins- intermolecular interactions

Conductionelectrons

etc.

Magnetic quantum system- spin- molecule- nanoparticle- chain

Interactions in magnetic quantum systems (decoherence)

T. Lis, Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem. 36, 2042 (1980)R. Sessoli, D. Gatteschi, M. Novak, D. Hendrikson, G. Christou, et al. (1989-93)

M. Novak, C. Paulsen, B. Barbara, et al. (1994)J. Friedman, M. Sarachik, et al., PRL (1996)L. Thomas, B. Barbara et al., Nature (1996)

L. Thomas, B. Barbara, et al., Nature (1996)

ConclusionBeginning: Mn12-ac

Followed by:Mn, Mn2, Mn3, Mn4, [Mn4]2, Mn5, Mn6, Mn7, Mn8, Mn9, Mn10,Mn11, Mn12, Mn13, Mn16, Mn18, Mn21, Mn22, Mn24, Mn26, Mn30 ,Mn70 , Mn84

Fe2, Fe3, Fe4, Fe5, Fe6, Fe7, Fe8, Fe10, Fe11, Fe13, Fe17/19 , Fe19 , Fe30

Co4, Co5, Co6, Co7, Co10

Ni4, Ni5, Ni6, Ni8, Ni12, Ni21, Ni24

Co2Gd2, Co2Dy2, Cr12, CrNi6, CrNi2, CrCo3, Fe10Na2, Fe2Ni3,Mn2Dy2, Mn2Nd2, V15, Ho, Fe2Ho2, Mn11Ln4, ...

• • • • • •

S = 0, 1/2, 1, 3/2, 2, 5/2, 4, 9/2, 5, …………. 51/2

>250 systems (in our group)

Only few ofthese molecules

are SMMs !!

Development of molecular Spin-Electronics

APS March Meeting 2004

1 10 100 1000N

Quantum world Classical world

Mn4Mn12 Mn84

Mn30

Mesoscopic Physics

A. J.Tasiopoulos, A. Vinslava, W. Wernsdorfer, K. A.Abboud, and G. Christou,Angew. Chem. Int. Ed., 43, 2117 (2004)

4 nm

Collaborations (Physics)

V. Bouchiat, C. Paulsen, A. Benoit, CRTBT, CNRS, Grenoble

L. Sorace, post-doc 2003: GHzA.-L. Barra, LCMI - CNRS, Grenoble

J. Villain, CEA, Grenoble

D. Mailly, LPN, CNRS, Marcoussis

L. Thomas PhD 1996: Mn12-acF. Lionti PhD 1997: Mn12-ac, Fe17/19I. Chiorescu PhD 2000: Mn12-ac, V15R. Giraud PhD 2002: Ho3+

C. Thirion PhD 2003: nanoparticles, GHzR. Tiron PhD 2004: [Mn4]2S. Bahr PhD started 2005: GHzK. Petukhov post-doc 2004-5: GHz

T. Ohm PhD 1998: Fe8V. Villar PhD 2001: Fe8 , chainesE. Lhotel PhD 2004: chaines

F. Balestro, E. Bonet, W. Wernsdorfer, B. Barbara, LLN, CNRS, Grenoble

Collaborations (Chemistry)Group of G. Christou, Dept. of Chemistry, FloridaGroup of R. Sessoli, D. Gatteschi, Univ. de Firenze, ItalieGroup of A. Cornia, Univ. de Modena, ItalieGroup of R.E.P. Winpenny, Univ. de Manchester, UKGroup of E. Brechin, Univ. de Manchester, UKGroup of T. Mallah, OrsayGroup of V. Marvaud, Univ. P. et M. Curie, ParisGroup of A. Müller, Univ. de Bielefeld, GermanyGroup of A. Powell, Univ. de Kahlsruhe, GermanyGroup of D. Hendrickson, Dept. of Chemistry, San DiegoGroup of E. Coronado, Univ. de Valence, SpainGroup of D. Luneau, Univ. of Lyon, FranceGroup of G. Royal, Univ. J. Fourier, Grenoble

Group of R. Clerac & C. Coulon, Univ. Bordeaux, PessacGroup of H. Miyasaka, Tokyo Metropolitan Uni.Group of M. Verdaguer, Univ. P. et M. Curie, ParisGroup of M. Julve, Univ. de Valence, Spain

• • •

Mn84

Christou, 2004

Winpenny, 2003

SMMs

SCMs