NYU Spins Dynamics in Nanomagnets Outline Andrew …physics.nyu.edu/kentlab/Lectures/Kent_Lecture3.pdf · Magnetic Bistability in a Molecular Magnet Nature 1993, and Sessoli et al.,

NYU

Department of Physics, New York University

Andrew D. Kent

2009 Boulder School, Lecture 3

Spins Dynamics in Nanomagnets

1

Lecture 1: Magnetic Interactions and Classical Magnetization DynamicsLecture 2: Spin Current Induced Magnetization DynamicsLecture 3: Quantum Spin Dynamics in Molecular Nanomagnets

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Outline

I. Introduction

Quantum Tunneling of Magnetization

Initial Discoveries

Single molecule magnets

II. Magnetic Interactions in SMMs

Energy Scales, Spin Hamiltonians

Mn12-acetate

III. Resonant Quantum Tunneling of Magnetization

Thermally activated, thermally assisted and pure

Crossover between regimes

Quantum Phase Interference

IV. Experimental Techniques

Micromagnetometry

SQUIDs and Nano-SQUIDs


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Magnetic Nanostructures

3

2 INTRODUCTION

Fig. I.1 Scale of size which goes from macroscopic down to nanoscopic sizes. The unit of

this scale is the number of magnetic moments in a magnetic system (roughly corresponding to

the number of atoms). The hysteresis loops are typical examples of magnetization reversal via

nucleation, propagation and annihilation of domain walls (left), via uniform rotation (middle),

and quantum tunneling (right).

Image from, W. Wernsdorfer, Advances in Chemical Physics 2001 and ArXiv:0101104


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+m -m

U

Ma

gnetic F

ield

Quantum Tunneling

Thermal relaxation (over the barrier)

Quantum

Temperature

Re

laxatio

n r

ate


also, Enz and Schilling, van Hemmen and Suto (1986)

Tc

Tc = U/kBB(0)

Thermal

4 2009 Boulder School, Lecture 3

Magnetic Bistability in a Molecular Magnet

Nature 1993, and Sessoli et al., JACS 1993

Magnetic hysteresis at 2.8 K and below (2.2 K)S=10 ground state spin


Quantum Tunneling in Single Molecule Magnets

Single crystal studies of Mn12: L. Thomas, et al. Nature 383, 145 (1996)Susceptibility studies: J.M. Hernandez, et al. EPL 35, 301 (1996)


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Single Molecule Magnets

7

=

Mn84

S=6

Physics-Individual molecule can be magnetized and exhibit

magnetic hysteresis 1993

-Quantum Tunneling of Magnetization 1995

-Quantum Phase Interference 1999

-Quantum Coherence 2008

SMM Characteristics

•Molecules

•High spin ground state

•Uniaxial anisotropy

•Single crystals

•Synthesized in solution

•Modified chemically

•Peripheral ligands

•Oxidized/reduced

•Soluble

•Bonded to surfaces

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[Mn12O12(O2CCH3)16(H2O)4].2CH3COOH.4H2O

First SMM: Mn12-acetate

Micro-Hall magnetometer

• S4 site symmetry

• Tetragonal lattice a=1.7 nm, b=1.2 nm

• Strong uniaxial magnetic anisotropy (~60 K)

• Weak intermolecular dipole interactions (~0.1 K)

2 acetic acid molecules

4 water molecules

Magnetic CoreCompeting AFM

Interactions

S=10

Ground state8 Mn3+ S=2

4 Mn4+! S=3/2

Organic Environment

Single

Crystal

8

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Intra-molecular Exchange Interactions

J1 ~ 215 K

J2,J3 ~ 85 K

J4 ~ 45 K

(2S1+1)8(2S2+1)4=108 ! S=10 and 2S+1=21

S=3/2J1

J4

J2

J3

J1

J2J3

J4

!

H = Jij

! S i "! S j +

<ij>

#! S i "Di "

! S i + ...

i

#

"S=2

9

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Spin Hamiltonian

!

H = "DSz

2" gµB

! S #! H

Magnetic Anisotropy and Spin Hamiltonian

(Ising-like) Uniaxial anisotropy

2S+1 spin levels

up

down

x

y

z

Sz|m >= m|m >

Em = !Dm2

U=DS2

10

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Relaxation processes in SMMs

up

x

y

z

U=U0(1-H/Ho)2

Thermal activation (over the barrier)

Magnetic relaxation at high temperature

M/M

s

!oHz (T)

Resonant Quantum Tunneling of Magnetization

down

Magnetic field

11

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up

down

x

y

z

Magnetic field

U=U0(1-H/Ho)2

with Zeeman term

“Resonance” fields where antiparallel spin projections are coincident, Hk =kD/gµB, levels m and m’; k=m+m’

!

HA =2DS

gµB

Anisotropy Field:

!

H = "DSz2" gµBSzHz

Em = !Dm2! gµBHzm

Sz|m >= m|m >


12

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QT on (fast relaxation)

on

-re

son

an

ce

9

-10

-9

-87

8

on-resonance

HL = kHR

10

QT off (slow relaxation)

off-r

eso

na

nce

10

-9

-8

9

-10

7

8

off-resonance

kHR < HL< (k+1)HR

0 0.5 1.0 1.5 2.0-0.5

HL(T )


on

-re

son

an

ce

-9

-10

-8 7

8

9

10

on-resonance

HL = (k+1)HR

13

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up

down

x

y

z

U=U0(1-H/Ho)2

Thermally assisted tunneling

!

"TAT = f (T)

Magnetic relaxation at intermediate temperature

U

Hz

k=6

k=7k=8



Magnetic field

14

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up

down

x

y

z

U=U0(1-H/Ho)2

Pure quantum tunneling

Magnetic relaxation at low temperature

3 4 5

0.4 T/min

0.2 T/min

0.1 T/min

0.05 T/min

0.02 T/min

-1

-0.5

0

0.5

1

M/M

s

Applied Field (Tesla)

Magnetic hysteresis

Field sweep rate

k=6

k=7

k=8


Magnetic field


15

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!

"m,m '

#DH

x

HA

$

% &

'

( )

m*m '

Upper levels Large splitting

Low Boltzmann population

Lowest levels Small splitting

High Boltzman population

Tunnel splitting on resonance

!

H = "DSz2" gµBSzHz " gµBSxHx

#m,m’

Relaxation pathway = f(T)

Crossover from Thermally Assisted to Pure QTM

Theory of thermally assisted tunneling: Villian (1997), Leuenberger & Loss (2000) and

Chudnovsky & Garanin (1999)

Crossover: Chudnovsky & Garanin, PRL 1997

16

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Thermal to Quantum Crossover in Magnetization Relaxation

17

ln $

TT0

QuantumTunneling

Thermalactivation

crossover

T0

ln $

TT0

T

E

T0

Top

E

TT0

U(x) = – x2 + x4

– x2 ± x3U(x) = – x2 – x4 ± x5

First order crossoverChudnovsky '92

Second order crossoverLarkin and Ovchinnikov '83

ln $

T

Chudnovsky and Garanin '97Uniaxial nanomagnets for smalltransverse magnetic fields!!!

1/S1/S

Bottom

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0 2 4 6 8 10

0.0

0.5

1.0

Energy/(DS2)

Hz/H

o

|m|=10

|m|=9

|m|=8

|m|=7

|m|=6

n = 0 1 2 3 4 5 6 7 8 9

Energy relative to the lowest level in the metastable well

D=0.548(3) K gz=1.94(1) !

B=1.17(2) ! 10-3 K (EPR: Barra et al., PRB 97)

H0= D/gzµB = 0.42 T

K. Mertes et al. PRB 200118

NYU

2009 Boulder School, Lecture

Experiments on the Crossover to Pure QTM in Mn12-acetate

0 2 4 6 8 10

0.0

0.5

1.0

Energy/(DS2)

Hz/H

o

|m|=10

|m|=9

|m|=8

|m|=7

|m|=6

n = 0 1 2 3 4 5 6 7 8 9

~1 K

~0.1 K

Thermal

Quantum

dM/dH versus Hz

Schematic: dominant levels as a function of temperature

ADK et al. EPL 2000

L. Bokacheva, PRL 2001

K. Mertes et al. PRB 2001

W. Wernsdorfer et al. PRL 200619

VOLUME 85, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 27 NOVEMBER 2000

0

0.2

0.4

0.6

0.8

1

m(t

)

1.40 K

0.99 K

1.10 K

1.19 K

1.29 K

0.56 K ! T ! 0.74 K "=0o

n=60.83 K

a)

0.1

0.2

0.3

0.4

0 1000 2000 3000

m(t

)

Time (seconds)

"=35o

n=4

1.04 K

0.60 Kc

b)

FIG. 4. Relaxation of the magnetization vs time at differenttemperatures for (a) n ! 6, u ! 0±, showing a crossover to aquantum regime at approximately 1 K, and (b) n ! 4, u ! 35±,showing no temperature independent regime. m!t" is a reducedmagnetization: m!t" ! #Ms 2 M!t"$%2Ms. In (a) the fivecurves below 0.74 K overlap (0.56, 0.58, 0.63, 0.68, and0.74 K). These curves can be fit with m!t" ! m0 exp#!2t%t"b$,where m0 ! 0.94 6 0.01, t ! !5.45 6 0.15" 3 104 s, b !0.48 6 0.02. The fit overlaps the data. In (b) the unmarkedcurves from top to bottom correspond to T ! 0.68, 0.70, 0.75,0.83, 0.91, and 0.95 K.

temperature it changes significantly. This temperature cor-responds to the crossover temperature seen in Fig. 3—consistent with pure QT. In contrast, for the peak n ! 4,u ! 35±, the magnetization relaxation rate changes signifi-cantly as the temperature decreases in the entire studiedrange. Relaxation curves can be fit by a stretched ex-ponential function m!t" ! m0 exp#2!t%t"b$, where b &0.4 0.6. This form of relaxation has been observed pre-viously in Fe8 [3] and in Mn12 [21], although it is notcompletely understood [22].

In summary, we have presented new low temperaturemagnetic studies of thermally assisted and pure quantumtunneling in Mn12. The crossover between these tworegimes was found to be either abrupt or gradual, depend-ing on the magnitude and orientation of applied magneticfield. Higher longitudinal and transverse fields broaden thecrossover, consistent with a recent model [23]. We havealso shown that below the crossover temperature the mag-netization relaxation becomes temperature independent.

We note that the measured crossover temperature ('1.1 K)is significantly higher than predicted (0.6 K) [24]. Thismay be due to an intrinsic mechanism promoting tunnelingin Mn12 such as a transverse anisotropy. Further studies ofthis crossover should lead to a better understanding of themechanisms of tunneling in Mn12.

This work was supported by NSF-INT (GrantNo. 9513143) and NYU.

[1] See E. M. Chudnovsky and J. Tejada, Macroscopic Tunnel-ing of the Magnetic Moment (Cambridge University Press,Cambridge, UK, 1997), Chap. 7.

[2] J. R. Friedman, M. P. Sarachik, J. Tejada, and R. Ziolo,Phys. Rev. Lett. 76, 3830 (1996); L. Thomas, F. Lionti, R.Ballou, D. Gatteschi, R. Sessoli, and B. Barbara, Nature(London) 383, 145 (1996).

[3] C. Sangregorio, T. Ohm, C. Paulsen, R. Sessoli, and D.Gatteschi, Phys. Rev. Lett. 78, 4645 (1997).

[4] R. Sessoli, D. Gatteschi, A. Caneschi, and M. A. Novak,Nature (London) 365, 141 (1993).

[5] M. Novak and R. Sessoli, in Quantum Tunneling of Mag-netization-QTM’94, edited by L. Gunther and B. Barbara(Kluwer, Dordrecht, 1995), p. 171; B. Barbara et al., J.Magn. Magn. Mater. 140–144, 1825 (1995).

[6] J. M. Hernandez et al., Europhys. Lett. 35, 301 (1996).[7] W. Wernsdorfer and R. Sessoli, Science 284, 133 (1999).[8] S. Hill et al., Phys. Rev. Lett. 80, 2453 (1998).[9] M. Hennion et al., Phys. Rev. B 56, 8819 (1997).

[10] A. L. Barra, D. Gatteschi, and R. Sessoli, Phys. Rev. B 56,8192 (1997).

[11] I. Mirebeau et al., Phys. Rev. Lett. 83, 628 (1999).[12] Y. Zhong et al., J. Appl. Phys. 85, 5636 (1999).[13] E. M. Chudnovsky and D. A. Garanin, Phys. Rev. Lett. 79,

4469 (1997).[14] G.-H. Kim, Phys. Rev. B 59, 11 847 (1999); G.-H. Kim, Eu-

rophys. Lett. 51, 216 (2000); H. J. W. Müller-Kirsten, D. K.Park, and J. M. S. Rana, Phys. Rev. B 60, 6662 (1999), andreferences therein.

[15] The analogy to phase transitions is a purely formal one asdiscussed in [13]. The first-order, second-order terminol-ogy for the escape rate crossover is originally due to Larkinand Ovchinnikov [16].

[16] A. I. Larkin and Y. N. Ovchinnikov, Sov. Phys. JETP 59,420 (1984).

[17] A. D. Kent et al., Europhys. Lett. 49, 512 (2000).[18] W. Wernsdorfer et al., Europhys. Lett. 50, 552 (2000).[19] A. D. Kent, S. von Molnar, S. Gider, and D. D. Awschalom,

J. Appl. Phys. 76, 6656 (1994).[20] T. Lis, Acta Crystallogr. Sect. B 36, 2042 (1980).[21] L. Thomas, A. Caneschi, and B. Barbara, Phys. Rev. Lett.

83, 2398 (1999).[22] N. V. Prokof’ev and P. C. E. Stamp, Phys. Rev. Lett. 80,

5794 (1998); E. M. Chudnovsky, Phys. Rev. Lett. 84, 5676(2000); N. V. Prokof’ev and P. C. E. Stamp, ibid., 84, 5677(2000); W. Wernsdorfer, C. Paulsen, and R. Sessoli, ibid.,84, 5678 (2000).

[23] D. A. Garanin, X. Martinez Hidalgo, and E. M. Chud-novsky, Phys. Rev. B 57, 13 639 (1998).

[24] L. Bokacheva, A. D. Kent, and M. A. Walters, Polyhedron(to be published).

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VOLUME 85, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 27 NOVEMBER 2000

0

0.2

0.4

0.6

0.8

1

m(t

)

1.40 K

0.99 K

1.10 K

1.19 K

1.29 K

0.56 K ! T ! 0.74 K "=0o

n=60.83 K

a)

0.1

0.2

0.3

0.4

0 1000 2000 3000

m(t

)

Time (seconds)

"=35o

n=4

1.04 K

0.60 Kc

b)

FIG. 4. Relaxation of the magnetization vs time at differenttemperatures for (a) n ! 6, u ! 0±, showing a crossover to aquantum regime at approximately 1 K, and (b) n ! 4, u ! 35±,showing no temperature independent regime. m!t" is a reducedmagnetization: m!t" ! #Ms 2 M!t"$%2Ms. In (a) the fivecurves below 0.74 K overlap (0.56, 0.58, 0.63, 0.68, and0.74 K). These curves can be fit with m!t" ! m0 exp#!2t%t"b$,where m0 ! 0.94 6 0.01, t ! !5.45 6 0.15" 3 104 s, b !0.48 6 0.02. The fit overlaps the data. In (b) the unmarkedcurves from top to bottom correspond to T ! 0.68, 0.70, 0.75,0.83, 0.91, and 0.95 K.

temperature it changes significantly. This temperature cor-responds to the crossover temperature seen in Fig. 3—consistent with pure QT. In contrast, for the peak n ! 4,u ! 35±, the magnetization relaxation rate changes signifi-cantly as the temperature decreases in the entire studiedrange. Relaxation curves can be fit by a stretched ex-ponential function m!t" ! m0 exp#2!t%t"b$, where b &0.4 0.6. This form of relaxation has been observed pre-viously in Fe8 [3] and in Mn12 [21], although it is notcompletely understood [22].

In summary, we have presented new low temperaturemagnetic studies of thermally assisted and pure quantumtunneling in Mn12. The crossover between these tworegimes was found to be either abrupt or gradual, depend-ing on the magnitude and orientation of applied magneticfield. Higher longitudinal and transverse fields broaden thecrossover, consistent with a recent model [23]. We havealso shown that below the crossover temperature the mag-netization relaxation becomes temperature independent.

We note that the measured crossover temperature ('1.1 K)is significantly higher than predicted (0.6 K) [24]. Thismay be due to an intrinsic mechanism promoting tunnelingin Mn12 such as a transverse anisotropy. Further studies ofthis crossover should lead to a better understanding of themechanisms of tunneling in Mn12.

This work was supported by NSF-INT (GrantNo. 9513143) and NYU.

[1] See E. M. Chudnovsky and J. Tejada, Macroscopic Tunnel-ing of the Magnetic Moment (Cambridge University Press,Cambridge, UK, 1997), Chap. 7.

[2] J. R. Friedman, M. P. Sarachik, J. Tejada, and R. Ziolo,Phys. Rev. Lett. 76, 3830 (1996); L. Thomas, F. Lionti, R.Ballou, D. Gatteschi, R. Sessoli, and B. Barbara, Nature(London) 383, 145 (1996).

[3] C. Sangregorio, T. Ohm, C. Paulsen, R. Sessoli, and D.Gatteschi, Phys. Rev. Lett. 78, 4645 (1997).

[4] R. Sessoli, D. Gatteschi, A. Caneschi, and M. A. Novak,Nature (London) 365, 141 (1993).

[5] M. Novak and R. Sessoli, in Quantum Tunneling of Mag-netization-QTM’94, edited by L. Gunther and B. Barbara(Kluwer, Dordrecht, 1995), p. 171; B. Barbara et al., J.Magn. Magn. Mater. 140–144, 1825 (1995).

[6] J. M. Hernandez et al., Europhys. Lett. 35, 301 (1996).[7] W. Wernsdorfer and R. Sessoli, Science 284, 133 (1999).[8] S. Hill et al., Phys. Rev. Lett. 80, 2453 (1998).[9] M. Hennion et al., Phys. Rev. B 56, 8819 (1997).

[10] A. L. Barra, D. Gatteschi, and R. Sessoli, Phys. Rev. B 56,8192 (1997).

[11] I. Mirebeau et al., Phys. Rev. Lett. 83, 628 (1999).[12] Y. Zhong et al., J. Appl. Phys. 85, 5636 (1999).[13] E. M. Chudnovsky and D. A. Garanin, Phys. Rev. Lett. 79,

4469 (1997).[14] G.-H. Kim, Phys. Rev. B 59, 11 847 (1999); G.-H. Kim, Eu-

rophys. Lett. 51, 216 (2000); H. J. W. Müller-Kirsten, D. K.Park, and J. M. S. Rana, Phys. Rev. B 60, 6662 (1999), andreferences therein.

[15] The analogy to phase transitions is a purely formal one asdiscussed in [13]. The first-order, second-order terminol-ogy for the escape rate crossover is originally due to Larkinand Ovchinnikov [16].

[16] A. I. Larkin and Y. N. Ovchinnikov, Sov. Phys. JETP 59,420 (1984).

[17] A. D. Kent et al., Europhys. Lett. 49, 512 (2000).[18] W. Wernsdorfer et al., Europhys. Lett. 50, 552 (2000).[19] A. D. Kent, S. von Molnar, S. Gider, and D. D. Awschalom,

J. Appl. Phys. 76, 6656 (1994).[20] T. Lis, Acta Crystallogr. Sect. B 36, 2042 (1980).[21] L. Thomas, A. Caneschi, and B. Barbara, Phys. Rev. Lett.

83, 2398 (1999).[22] N. V. Prokof’ev and P. C. E. Stamp, Phys. Rev. Lett. 80,

5794 (1998); E. M. Chudnovsky, Phys. Rev. Lett. 84, 5676(2000); N. V. Prokof’ev and P. C. E. Stamp, ibid., 84, 5677(2000); W. Wernsdorfer, C. Paulsen, and R. Sessoli, ibid.,84, 5678 (2000).

[23] D. A. Garanin, X. Martinez Hidalgo, and E. M. Chud-novsky, Phys. Rev. B 57, 13 639 (1998).

[24] L. Bokacheva, A. D. Kent, and M. A. Walters, Polyhedron(to be published).

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M(t)

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Tunneling Selection Rules

Form of HA determined by the site symmetry of the molecule

Mn12-acetate

S4-site symmetry (tetragonal)

Fe8

C2-site symmetry (rhombic)

180o

90o

H = !DS2

z ! gµBSzHz + HA

HA = E(S2

x ! S2

y) HA = C(S4+ + S

4!

)

20

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Hysteresis Loops as a Function of Transverse Magnetic Field

P = e!!

! = "!2/(2!v)v = 2SgµBdBz/dt

Landau-Zener Transitions

21

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Oscillations and Parity Effect in the Tunnel Splitting

22

Predicted for a biaxial system by Garg 1992!

Spin-parity effects in QTM: Loss et al., 1992 von Delft & Henley, 1992

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Quantum Phase Interference in Fe8

HT

HT

23

H = !DS2z + E(S2

x ! S2y)! gµBSxHx

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Micromagnetometry

•" 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

sample

B

1 to 10 µm

sample

B1 µm

Josephson Junctions

see, A. D. Kent et al., Journal of Applied Physics 1994 W. Wernsdorfer, JMMM 1995

24

• µ-Hall Effect

– Based on Lorentz Force

– Measures magnetic field

– Large applied in-plane magnetic fields (>20 T)

– Broad temperature range

– Single magnetic particles

– Ultimate sensitivity ~102 µB

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Nano-SQUID

25

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Summary

26

I. Introduction


Initial Discoveries

Single molecule magnets

II. Magnetic Interactions in SMMs

Energy Scales, Spin Hamiltonians

Mn12-acetate

III. Resonant Quantum Tunneling of Magnetization

Thermally activated, thermally assisted and pure

Crossover between regimes

Quantum Phase Interference

IV. Experimental Techniques

Micromagnetometry

SQUIDs and Nano-SQUIDs

Documents

NYU Spins Dynamics in Nanomagnets Outline Andrew …physics.nyu.edu/kentlab/Lectures/Kent_Lecture3.pdf · Magnetic Bistability in a Molecular Magnet Nature 1993, and Sessoli et al.,