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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