1. Single Molecule Magnets:
Exploring Quantum Magnetization Dynamics at the Nanoscale
Quantum versus classical magnetic clusters
Single molecule magnets (SMMs)
In particular, Mn12 and Fe8
Quantum magnetization dynamics
Quantum tunneling, hysteresis, and the role of the environment
High frequency Electron Paramagnetic Resonance
Solution to the tunneling mechanism in Mn12-acetate
Quantum coherence in a supramolecular dimer of SMMs

2. MESOSCOPIC MAGNETISM macroscale nanoscale Classical Quantum size
permanent
magnets
micron
particles
nanoparticles
clusters
molecular
Individual
spins
S = 10 23 10 8 6 5 4 3 2
multi - domain
single - domain
Single molecule
nucleation, propagation and
annihilation of domain walls
uniform rotation
quantum tunneling,
quantum interference -1 1
-40
-20 20 40
M/M S m
H(mT)
-100
100
H(T) Fe 1K
0.1K
0.7K
Mn12-ac
Ferritin
1 nm
10 nm
100 nm
superparamagnetism
Classical
Quantum
size

3. Magnetic anisotropy  bistability, hysteresis
The case of a classical, single domain, ferromagnetic particle:
Stoner-Wohlfarth
DE = K × Volume
 anisotropy × # of spins
K = energy density associated with the anisotropy
Energy density DE
down Up
Anisotropy due to crystal field - in this case, uniaxial.
Energy barrier prevents switching - hence bistability....
....unless DE < kBT  superparamagnetism [t = toexp(DE/kBT)]

4. Magnetic anisotropy  bistability, hysteresis
The case of a classical, single domain, ferromagnetic particle:
Stoner-Wohlfarth
DE = K × Volume
 anisotropy × # of spins
K = energy density associated with the anisotropy
Energy density DE
down Up
Anisotropy due to crystal field - in this case, uniaxial.
Energy barrier prevents switching - hence bistability....
....unless DE < kBT  superparamagnetism [t = toexp(DE/kBT)]
Switching involves rotation of all spins together.
Classical continuum of states between "up" and "down".

5. The first single molecule magnet: Mn12-acetate
Lis, 1980
Mn(III)
S = 2
S = 3/2
Mn(IV)
Oxygen
Carbon
[Mn12O12(CH3COO)16(H2O)4]·2CH3COOH·4H20
R. Sessoli et al. JACS 115, 1804 (1993)
Ferrimagnetically coupled magnetic ions (Jintra  100 K)
Well defined giant spin (S = 10) at low temperatures (T < 35 K)
Easy-axis anisotropy due to Jahn-Teller distortion on Mn(III)
Crystallizes into a tetragonal structure with S4 site symmetry
Organic ligands ("chicken fat") isolate the molecules

6. Quantum effects at the nanoscale (S = 10)
"up"
"down"
E-10
E-9
E-8
E-7
E10 E9 E8 E7 E6 E5
Spin projection - ms
E-6
E-5
Energy E4
E-4
Simplest case: axial (cylindrical) crystal field
DE  DS2 K
|D |  K for a typical single molecule magnet
Eigenvalues given by:
Small barrier - DS2
Superparamagnet at ordinary temperatures
21 discrete ms levels
Thermal activation

7. Quantum effects at the nanoscale (S = 10)
"up"
"down"
E-10
E-9
E-8
E-7
E10 E9 E8 E7 E6 E5
Spin projection - ms
E-6
E-5
Energy E4
E-4
Break axial symmetry:
HT  interactions which do not commute with Ŝz
DEeff < DE
ms not good quantum #
Mixing of ms states
 resonant tunneling (of ms) through barrier
Lower effective barrier
Thermally
assisted
quantum
tunneling

8. Quantum effects at the nanoscale (S = 10)
"up"
"down"
E-10
E-9
E-8
E-7
E10 E9 E8 E7 E6 E5
Spin projection - ms
E-6
E-5
Energy E4
E-4
Strong distortion of the axial crystal field:
Temperature-independent quantum relaxation as T0
Ground state degeneracy lifted by transverse interaction:
splitting  (HT)n Do
Tunnel splitting
Ground states a mixture of pure "up" and pure "down".
"down"
"up"
{ }
Pure quantum tunneling

9. How is this evidenced?
Mn30 - the largest single molecule magnet displaying a temperature independent relaxation rate
Prokof'ev and Stamp
Phys. Rev. Lett. 80, 5794 (1998)
W. Wernsdorfer, G. Christou, et al., cond-mat/
The physics behind this short-time relaxation requires some detailed explanation (to follow).
Temperature-independent magnetization relaxation as T  0 K.
Highly axially symmetric systems show the slowest relaxation, e.g. Mn12-ac (G-1  ).

10. Another well characterized single molecule magnet
[Fe8O2(OH)12(tacn)6]Br8.9H20
Fe3+
S=5/2
S=10
Wieghardt, 1984

11. How do these two S = 10 systems differ?
z-axis is out of
the screen
Mn12-ac, S = 10
Fe8, S = 10
Four-fold axis Tetragonal
Two-fold axis Rhombic

12. Mn12-ac, S = 10 Fe8, S = 10 D  0.6 K D  0.2 K D/E  5 G0  10-3 Hz
Spin projection - ms
Thermally
assisted
quantum
tunneling
Spin projection - ms
Pure quantum tunneling
D  0.6 K
D  0.2 K
D/E  5
G0  10-3 Hz
This is why tunneling in high spin (classical) systems is not observed.
Below TB  3 K,
t   (H = 0)
115 GHz
or 5.5 kB
300 GHz
or 14.4 kB
Many potential sources of transverse anisotropy for Mn12-ac:
Internal dipolar, exchange and hyperfine fields;
Higher order crystal field interactions;
Disorder, lattice defects, etc..
However, after 10 years, the mechanism remains poorly understood.
Example:
 for Fe8, HT mixes states for which Dms = ±2 in first-order, Dms = ±4 in second-order, and so on. Thus, the ms = ±10 states interact only in 10th order
= 0.1 mK
To first-order, HT mixes states with Dms = ±k, where k is the power to which Sx or Sy appear in HT.

13. Application of a magnetic field
Spin projection - ms
For now, consider only B//z :
(also neglect transverse interactions)
"down"
"up"
Several important points to note:
Applied field represents another source of transverse anisotropy
Zeeman interaction contains odd powers of Ŝx and Ŝy
System off resonance X
Magnetic quantum tunneling is suppressed
Metastable magnetization is blocked ("down" spins)

14. Application of a magnetic field
Spin projection - ms
For now, consider only B//z :
(also neglect transverse interactions)
"down"
"up"
Several important points to note:
Applied field represents another source of transverse anisotropy.
Zeeman interaction contains odd powers of Ŝx and Ŝy.
Increasing field
System on resonance
Resonant magnetic quantum tunneling resumes
Metastable magnetization can relax from "down" to "up"

15. Hysteresis and magnetization steps
Mn12-ac
Tunneling
"on"
Tunneling "off"
Low temperature H=0
step is an artifact
This loop represents an ensemble average of the response of many molecules
Friedman et al., PRL (1996)
Thomas et al., Nature (1996)

16. Periodicity of the magnetization steps
Therefore, HT must contain odd powers of Ŝx and Ŝy.
This, in turn, suggest that transverse fields play a crucial role in the tunneling. [Chudnovsky and Garanin, PRL 87, (2001)]
+10 -3
+10 -4
+10 -5
+10 -6
-500
+10 -7
+10 -8
+10 -9
Tunneling steps are periodic in Hz
Both even and odd Dms steps are observed

17. So, what about internal fields?
These are too weak to explain tunneling at odd resonances in Mn12.
Nevertheless, they play a crucial role in the tunneling mechanism.
field dipolar exchange hyperfine
20 mT  mT
50 mT 10 mT mT
Mn12
56Fe8
Relaxation is a complex many-body problem.
(5)
5) Dynamic nuclear fluctuations and electron/nuclear spin cross-relaxation broaden the magnetic energy levels (even at mK temperatures). x
(4)
4) The change in magnetization drives those molecules that were initially in resonance, out of resonance. Meanwhile, other molecules are brought into resonance, and the hole becomes broader.
Msat
(1)
1) Prepare the sample in a fully magnetized state.
M<Msat
(3)
3) Some molecules will relax, thus burning a hole in the dipolar field distribution; this relaxation results in a change in the magnetization of the sample.
(2)
2) Put the system in resonance so that the magnetization can begin to relax. Note: due to internal dipolar fields, not all molecules will be in resonance - depends on sample shape.
This combination of "dipolar shuffling" and the coupling to the nuclear bath governs the time-evolution of the magnetization, leading to the short time t-1/2 relaxation.
[Prokof'ev and Stamp, Phys. Rev. Lett. 80, 5794 (1998)]

18. Here, we see the importance of the environment
The role of dipolar and hyperfine fields was first demonstrated via studies of isotopically substituted versions of Fe8.
[Wernsdorfer et al., Phys. Rev. Lett. 82, 3903 (1999)]

19. What have we learned thus far?
HT provides the tunneling mechanism (tunnel matrix elements)
For Fe8: rhombic term, E(Ŝx2 - Ŝy2), is the dominant source of transverse anisotropy.
This has been verified via measurements of tunnel splittings D, using an elegant Landau-Zener method.
[Wernsdorfer, Sessoli, Science 284, 133 (1999)]
For Mn12: dominant source of transverse anisotropy not known; 4th order term, B44(Ŝ+4 + Ŝ-4), believed to play a role.
Landau-Zener studies fail to observe clear tunnel splittings; instead, broad distributions are found.
[del Barco et al., Europhys. Lett. 60, 768 (2002)]
For both: odd tunneling resonances suggest additional source of transverse anisotropy, possibly related to disorder.
[Chudnovsky and Garanin, PRL 87, (2001)]
Nuclear and dipolar couplings control magnetization dynamics.
These couplings also represent unwanted source of decoherence.

20. Single-crystal, high-field/frequency EPR
Reminder: field//z
z, S4-axis Hz
ms represents spin- projection along the molecular 4-fold axis
Magnetic dipole transitions (Dms = ±1) - note frequency scale!
EPR measures level spacings directly, unlike Landau-Zener

21. Single-crystal, high-field/frequency EPR
Rotate field in xy-plane and look for symmetry effects s
z, S4-axis
Hxy
In high-field limit (gmBB > DS), ms represents spin- projection along the applied field-axis

22. The first single-crystal study
PRL 80, 2453 (1998)
The advantages are two-fold:
One can perform angle dependent studies to look for symmetry of HT.
One can also carry out detailed line-shape analyses; this is harder to do for powder averages.
Sample: 1 × 0.2 × 0.2 mm3
Subsequent outcomes:
Determination of significant D-strain effect in Mn12 and Fe8.
[Polyhedron 20, 1441 (2001); PRB 65, (2002)]
Measurement of dipolar couplings in Mn12 and Fe8, and first evidence for significant intermolecular interactions in Fe8.
[PRB 65, (2002); PRB 66, (2002)]
Location of low-lying S = 9 state in Fe8.
[cond-mat/030915; PRB in-press (2003)]

23. Cavity perturbation Cylindrical TE01n (Q ~ 104 - 105) f = 16  300 GHz
Single crystal 1 × 0.2 × 0.2 mm3
T = 0.5 to 300 K, moH up to 45 tesla
(now 715 GHz!)
We use a Millimeter-wave Vector Network Analyzer (MVNA, ABmm) as a spectrometer
Susumu Takahashi (2003)
two-axis rotation
capability
M. Mola et al., Rev. Sci. Inst. 71, 186 (2000)

24. Determination of transverse crystal-field interactions in Mn12-Ac
PRL 90, (2003)
Hard-plane (xy-plane) rotations
Four-fold line shifts due to a quartic transverse interaction in HT
Previously inferred from neutron studies
Mirebeau et al., PRL 83, 628 (1999)
B44 is the only free parameter in our fit
PRL 90, (2003)
f = 50 GHz f
four-fold line shape/width modulation due to a quadratic transverse interaction caused by solvent disorder 30 60
Angle (degrees) 90
Easy
120
150

25. Solvent disorder (rogueness)
Organic surrounding
(not directly bound to core):
4 water molecules
2 acetic acid molecules

26. +E -E Disorder lowers the symmetry of the molecules +E(Sx2 - Sy2)
Equal population of molecules with opposite signs of E
A. Cornia et al., Phys. Rev. Lett. 89, (2002)

27. Rotation away from the hard planeq
b9 does not appear until 2.5 degrees of rotation
a10 vanishes in the first degree of rotation

28. Rotation away from the hard plane
0.18o increments
Implies about 1.5o distribution in the orientations of the easy axes of the Mn12 molecules
2.15o spread
a10 b9
1.25o spread
Hard
plane
This is the solution to the odd tunneling resonance problem

29. Returning to issue of decoherence and the environment
Mn12 (Fe8?) probably not ideal quantum system for future work
1st single molecule magnets - novel, yet contrasting behaviors
Physicists have focused almost exclusively on these two systems
Timescale assoc. with nuclear and dipolar fluctuations is slow
Strong coupling to slow spin dynamics  quantum decoherence
Move tunneling into a "coherence window" - different frequency range
[Stamp and Tupitsyn, cond-mat/ ]
Larger tunnel splittings via transverse applied field, or smaller spin S
Other ways to beat decoherence
Isotopically label with I = 0 nuclei
Focus on antiferromagnetic S = 0 systems (immune to dipolar effects)
Isolate molecules, either by dilution, or with "chicken fat"
The molecular approach is the key
Immense control over the magnetic unit and its coupling to the environment

30. Chicken Fat
S = 4 Ni4 system with strong quartic (Ŝ+4 + Ŝ-4) anisotropy
(Ni also predominantly I = 0)
Ni4
Ni4 ,
E-C. Yang, JACS (submitted, 2003)

31. An untapped gold mine Mn4 S = 9/2 Site-selective reactivity o c N BrH Cl F Br
Site-selective reactivity
Mn4 S = 9/2
An untapped gold mine

32. Exchange coupling in a dimer of S = 9/2 Mn4 clusters
No H = 0 tunneling
To lowest order, the exchange generates a bias which each spin experiences due to the other spin within the dimer
Wolfgang Wernsdorfer, George Christou, et al., Nature, 2002,

33. Including the full exchange into the calculation

34. Clear evidence for coherent transitions involving both molecules
Experiment
Simulation
(submitted, 2003)

35. Many collaborators/students to acknowledge...
...illustrates the interdisciplinary nature of this work
UF Physics
Rachel Edwards
Alexey Kovalev
Susumu Takahashi
Sabina Kahn
Jon Lawrence
Andrew Browne
Shaela Jones
Neil Bushong
Sara Maccagnano
FSU Chemistry
Naresh Dalal
Micah North
David Zipse
Randy Achey
UF Chemistry
George Christou
Nuria Aliaga-Alcalde
Monica Soler
Nicole Chakov
Sumit Bhaduri
UCSD Chemistry
David Hendrickson
En-Che Yang
Evan Rumberger
NYU Physics
Andy Kent
Enrique del Barco
Also: Kyungwha Park (NRL)
Wolfgang Wernsdorfer (Grenoble)
Mark Novotny (MS State U)
Per Arne Rikvold (CSIT - FSU)