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Single Molecule Magnets: Exploring Quantum Magnetization Dynamics at the Nanoscale. Quantum versus classical magnetic clusters Single molecule magnets (SMMs) In particular, Mn 12 and Fe 8 Quantum magnetization dynamics Quantum tunneling, hysteresis, and the role of the environment

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slide1
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
slide2
MESOSCOPIC MAGNETISM

Classical

Quantum

macroscale

nanoscale

nanoparticles

permanent

micron

clusters

molecular

Individual

superparamagnetism

magnets

particles

clusters

spins

100 nm

10 nm

1 nm

23

10

8

6

5

4

3

2

S = 10

10

10

10

10

10

10

10

10 1

Mn12-ac

Ferritin

multi - domain

single - domain

Single molecule

nucleation, propagation and

uniform rotation

quantum tunneling,

annihilation of domain walls

quantum interference

1

1

1

Fe

8

0.7K

S

S

S

0

0

0

M/M

M/M

M/M

0.1K

1K

-1

-1

-1

-40

-20

0

20

40

-100

0

100

-1

0

1

m

m

m

H(mT)

H(T)

H(mT)

0

0

0

size

slide3
down

Up

  • Energy barrier prevents switching - hence bistability....
  • ....unless DE < kBT superparamagnetism [t = toexp(DE/kBT)]

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

  • Anisotropy due to crystal field - in this case, uniaxial.
slide4
down

Up

  • Energy barrier prevents switching - hence bistability....
  • ....unless DE < kBT superparamagnetism [t = toexp(DE/kBT)]

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

  • Anisotropy due to crystal field - in this case, uniaxial.
  • Switching involves rotation of all spins together.
  • Classical continuum of states between "up" and "down".
slide5
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

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)

slide6
Spin projection - ms

Energy

E-4

E4

E-5

E5

Eigenvalues given by:

E-6

E6

E-7

E7

  • Small barrier - DS2
  • Superparamagnet at ordinary temperatures

E-8

E8

DE  DS2

10-100 K

E-9

E9

"up"

"down"

|D | 0.1 - 1 K for a typical single molecule magnet

E-10

E10

Quantum effects at the nanoscale (S = 10)

Simplest case: axial (cylindrical) crystal field

21 discrete ms levels

Thermal activation

slide7
Spin projection - ms

Energy

E-4

E4

E-5

E5

E-6

E6

  • msnot good quantum #
  • Mixing of msstates
  • resonant tunneling (of ms) through barrier
  • Lower effective barrier

E-7

E7

E-8

E8

E-9

E9

"up"

"down"

DEeff < DE

E-10

E10

Quantum effects at the nanoscale (S = 10)

Break axial symmetry:

HT interactions which do not commute with Ŝz

Thermally

assisted

quantum

tunneling

slide8
Spin projection - ms

Energy

E-4

E4

E-5

E5

E-6

E6

  • Ground state degeneracy lifted by transverse interaction:
  • splitting  (HT)n

"down"

"up"

E-7

E7

E-8

E8

  • Ground states a mixture of pure "up" and pure "down".

E-9

E9

"up"

"down"

E-10

E10

Do

{ }

Tunnel splitting

±

Pure quantum tunneling

Quantum effects at the nanoscale (S = 10)

Strong distortion of the axial crystal field:

  • Temperature-independent quantum relaxation as T0
slide9
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/0306303

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 ).
slide10
Another well characterized single molecule magnet

[Fe8O2(OH)12(tacn)6]Br8.9H20

Fe3+

S=5/2

S=10

Wieghardt, 1984

slide11
How do these two S = 10 systems differ?

z-axis is out of

the screen

Mn12-ac, S = 10

Fe8, S = 10

Two-fold axis Rhombic

Four-fold axis Tetragonal

slide12
Spin projection - ms

Spin projection - ms

Thermally

assisted

quantum

tunneling

Below TB 3 K,

t  (H = 0)

  • This is why tunneling in high spin (classical) systems is not observed.

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

  • 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.

Pure quantum tunneling

= 0.1 mK

Mn12-ac, S = 10

Fe8, S = 10

D  0.6 K

D  0.2 K

D/E  5

G0 10-3 Hz

115 GHz

or 5.5 kB

300 GHz

or 14.4 kB

  • To first-order, HT mixes states with Dms= ±k, where k is the power to which Sx or Sy appear in HT.
slide13
X

For now, consider only B//z :

(also neglect transverse interactions)

System off resonance

X

  • Magnetic quantum tunneling is suppressed
  • Metastable magnetization is blocked ("down" spins)

Application of a magnetic field

Spin projection - ms

"down"

"up"

Several important points to note:

  • Applied field represents another source of transverse anisotropy
  • Zeeman interaction contains odd powers of Ŝx and Ŝy
slide14
For now, consider only B//z :

(also neglect transverse interactions)

  • Resonant magnetic quantum tunneling resumes
  • Metastable magnetization can relax from "down" to "up"

Application of a magnetic field

Spin projection - ms

"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

slide15
Tunneling

"on"

Tunneling "off"

Hysteresis and magnetization steps

Mn12-ac

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)

slide16
Both even and oddDmssteps are observed

+10

-9

+10

-8

+10

-7

+10

-6

+10

-5

+10

-4

+10

-3

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, 187203 (2001)]

-500

Tunneling steps are periodic in Hz

slide17
(1)

(3)

(2)

(4)

(5)

M

x

Msat

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.

1) Prepare the sample in a fully magnetized state.

5) Dynamic nuclear fluctuations and electron/nuclear spin cross-relaxation broaden the magnetic energy levels (even at mK temperatures).

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.

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.

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

Mn12

56Fe8

20 mT  0 12 mT

50 mT 10 mT 1.2 mT

  • Relaxation is a complex many-body problem.

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)]

slide18
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)]

slide19
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, 187203 (2001)]

  • Nuclear and dipolar couplings control magnetization dynamics.
  • These couplings also represent unwanted source of decoherence.

What have we learned thus far?

HTprovides the tunneling mechanism (tunnel matrix elements)

slide20
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
slide21
s

z, S4-axis

Hxy

Single-crystal, high-field/frequency EPR

Rotate field in xy-plane and look for symmetry effects

In high-field limit (gmBB > DS), ms represents spin- projection along the applied field-axis

slide22
Subsequent outcomes:

Determination of significant D-strain effect in Mn12 and Fe8.

[Polyhedron 20, 1441 (2001); PRB 65, 014426 (2002)]

Measurement of dipolar couplings in Mn12 and Fe8, and first evidence for significant intermolecular interactions in Fe8.

[PRB 65, 224410 (2002); PRB 66, 144409 (2002)]

Location of low-lying S = 9 state in Fe8.

[cond-mat/030915; PRB in-press (2003)]

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

slide23
q

Susumu Takahashi (2003)

two-axis rotation

capability

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

M. Mola et al., Rev. Sci. Inst. 71, 186 (2000)

slide24
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, 217204 (2003)

four-fold line shape/width modulation due to a quadratic transverse interaction caused by solvent disorder

PRL 90, 217204 (2003)

f

Determination of transverse crystal-field interactions in Mn12-Ac

Hard-plane (xy-plane) rotations

f = 50 GHz

0

30

60

Angle (degrees)

90

Easy

120

150

slide25
Organic surrounding

(not directly bound to core):

4 water molecules

2 acetic acid molecules

Solvent disorder (rogueness)

slide26
+E(Sx2 - Sy2)

-E(Sx2 - Sy2)

+E

-E

Equal population of molecules with opposite signs of E

A. Cornia et al., Phys. Rev. Lett. 89, 257201 (2002)

Disorder lowers the symmetry of the molecules

slide27
b9 does not appear until 2.5 degrees of rotation

a10 vanishes in the first degree of rotation

Rotation away from the hard plane

q

slide28
a10

2.15o spread

Hard

plane

b9

1.25o spread

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

This is the solution to the odd tunneling resonance problem

slide29
The molecular approach is the key
    • Immense control over the magnetic unit and its coupling to the environment

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/0302015]
    • 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"
slide30
Chicken Fat

S = 4 Ni4 system with strong quartic (Ŝ+4 + Ŝ-4) anisotropy

(Ni also predominantly I = 0)

Ni4

Ni4

º

,

E-C. Yang, JACS (submitted, 2003)

slide31
Site-selective reactivity

o

N

c

N

N

N

Mn

Mn

Mn

Mn

Mn

Mn

O

O

O

O

O

O

Mn

R

Mn

O

O

N

Br

O

Mn

Mn

Mn

Mn

Mn

Mn

C

O

O

O

O

O

O

O

O

Mn

Mn

Mn

Mn

Mn

O

O

O

H

H

Mn

H

C

Cl

Mn

Mn

O

Mn

O

O

F

Mn

O

Mn

Mn

Mn

Mn

Mn

O

O

Mn

O

O

O

O

Mn

Mn

Mn4S = 9/2

An untapped gold mine

slide32
No H = 0 tunneling

To lowest order, the exchange generates a bias which each spin experiences due to the other spin within the dimer

Exchange coupling in a dimer of S = 9/2 Mn4 clusters

Wolfgang Wernsdorfer, George Christou, et al., Nature, 2002, 406-409

slide34
Clear evidence for coherent transitions involving both molecules

Experiment

Simulation

(submitted, 2003)

slide35
UF Chemistry

George Christou

Nuria Aliaga-Alcalde

Monica Soler

Nicole Chakov

Sumit Bhaduri

UCSD Chemistry

David Hendrickson

En-Che Yang

Evan Rumberger

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

NYU Physics

Andy Kent

Enrique del Barco

Also: Kyungwha Park (NRL)

Wolfgang Wernsdorfer (Grenoble)

Mark Novotny (MS State U)

Per Arne Rikvold (CSIT - FSU)

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