Transcript PowerPoint - Physics - University of Florida

Quantum coherence in an
exchange-coupled dimer of
single-molecule magnets
Stephen Hill and Rachel Edwards
Department of Physics, University of Florida, Gainesville, FL32611
Nuria Aliaga-Alcalde, Nicole Chakov and George Christou
Department of Chemistry, University of Florida, Gainesville, FL32611
PART I
Introduction to single-molecule magnets
•Emphasis on quantum magnetization dynamics
PART II
Monomeric Mn4 single-molecule magnets
Electron Paramagnetic Resonance technique, with examples
Focus on [Mn4]2 dimer
•Quantum mechanical coupling within a dimer
•Evidence for quantum coherence from EPR
Supported by: NSF, Research Corporation, & University of Florida
PART I
Introduction to single-molecule magnets
Emphasis on their quantum dynamics
MESOSCOPIC MAGNETISM
Classical
macroscale
permanent
magnets
nanoparticles
micron
particles
size
Quantum
nanoscale
clusters
molecular
superparamagnetism clusters
100 nm
23
S = 10
1010
10 8
10 6
10 nm
10 5
10 4
Individual
spins
1 nm
10 3
10 2
10
1
Mn12-ac
multi - domain
single - domain
nucleation, propagation and
annihilation of domain walls
uniform rotation
1
Ferritin
Single molecule
quantum tunneling,
quantum interference
1
1
S
S
S
M/M
M/M
M/M
Fe8
0
0
0.7K
0
1K
-1
-40
-1
-20
0
20
m 0H(mT)
40
0.1K
-1
-100
0
m 0 H(mT)
100
-1
W. Wernsdorfer, Adv. Chem. Phys. 118, 99-190 (2001); also arXiv/cond-mat/0101104.
0
m 0H(T)
1
~1.2 nm
A. J. Tasiopoulos et al.,
Angew. Chem., in-press.
The first single molecule magnet: Mn12-acetate
Lis, 1980
Mn(III)
S=2
Mn(IV)
S = 3/2
Oxygen
Carbon
[Mn12O12(CH3COO)16(H2O)4]·2CH3COOH·4H20
R. Sessoli et al. JACS 115, 1804 (1993)
•Ferrimagnetically
coupled
(Jintra  100
Well defined giant spin
(S = magnetic
10) at low ions
temperatures
(T <K)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
Quantum effects at the nanoscale (S = 10)
Energy
E ms
Hˆ  DSˆz2
( D  0)
E-4
E-5
E4
E5
E-6
E6
E(ms )  - D m
E-7
E7
•Small barrier - DS2
E-8
"up"
Simplest case: axial
(cylindrical) crystal field
Spin projection - ms
DE  DS2
10-100 K
E-9
E-10
2
s
•Superparamagnet at
ordinary temperatures
E9 "down"
|D |  0.1 - 1 K
E8
E10
21
discrete
ms levels
Thermal
activation
Eigenvalues given by:
for a typical
single molecule
magnet
Quantum effects at the nanoscale (S = 10)
Energy
E ms
Hˆ  DSˆz2  HˆT
HT  interactions which
do not commute with Ŝz
E-4
E-5
E4
E5
E-6
E6 • ms not good quantum #
E-7
E-8
"up"
Break axial symmetry:
Spin projection - ms
E-9
E-10
Thermally
assisted
quantum
tunneling
DEeff < DE
E7
E8
•Mixing of ms states
 resonant tunneling
(of ms) through barrier
•Lower effective barrier
E9 "down"
E10
Quantum effects at the nanoscale (S = 10)
Energy
E ms
E-4
E-5
E4
E5
E-6
E6
E-7
E7
E-8
"up"
Strong distortion of the
axial crystal field:
Spin projection - ms
E-9
"up"
1
2
{
"down"
±
}
Do
Tunnel splitting
•Ground state degeneracy
lifted by transverse
interaction:
splitting  (HT)n
E8 •Ground states a mixture of
pure "up" and pure "down".
E9 "down"
Pure quantum tunneling
E-10
•Temperature-independent
quantum relaxation as T0
E10
Application of a magnetic field
Spin projection - ms
"up"
"down"
Hˆ  DSˆz2  HˆT  g m B B  Sˆ
B  Sˆ  Bx Sˆx  By Sˆ y  Bz Sˆz
Several important points to note:
•Applied field represents another
source of transverse anisotropy
X
System off resonance
X
•Zeeman interaction contains odd
powers of Ŝx and Ŝy
For now, consider only B//z :
(also neglect transverse interactions)
2
s
s
B
s
E(m )  - D m  g m Bm
•Magnetic quantum tunneling is suppressed
•Metastable magnetization is blocked ("down" spins)
Application of a magnetic field
Spin projection - ms
"up"
"down"
Hˆ  DSˆz2  HˆT  g m B B  Sˆ
B  Sˆ  Bx Sˆx  By Sˆ y  Bz Sˆz
Several important points to note:
Increasing field
System on resonance
•Applied field represents another
source of transverse anisotropy.
•Zeeman interaction contains odd
powers of Ŝx and Ŝy.
For now, consider only B//z :
(also neglect transverse interactions)
2
s
s
B
s
E(m )  - D m  g m Bm
•Resonant magnetic quantum tunneling resumes
•Metastable magnetization can relax from "down" to "up"
Hysteresis and magnetization steps
Mn12-ac
Tunneling
"on"
Low temperature H=0
Tunneling "off" •Friedman, Sarachik,
step is an artifact
This loop represents an ensemble
average of the response of many
molecules
Tejada, Ziolo, PRL (1996)
•Thomas, Lionti, Ballou,
Gatteschi, Sessoli,
Barbara, Nature (1996)
PART II
Quantum Coherence in Exchange-Coupled
Dimers of Single-Molecule Magnets
Mn4 single molecule magnets (cubane family)
C3v symmetry
[Mn4O3Cl4(O2CEt)3(py)3]
MnIII: 3 × S = 2
MnIV: S = 3/2
Distorted
cubane

S = 9/ 2
•MnIII (S = 2) and MnIV (S = 3/2) ions couple ferrimagnetically to give
an extremely well defined ground state spin of S = 9/2.
•This is the monomer from which the dimers are made.
Fairly typical SMM: exhibits resonant MQT
B//z
“down”
Hˆ o  DSˆz2  m B B.g.Sˆ  Hˆ T  Hˆ '
“up”
-S
(S-1)
S
B<0
1W.
Wernsdorfer et al., PRB 65, 180403 (2002).
2W. Wernsdorfer et al., PRL 89, 197201 (2002).
•Barrier  20D  18 K
•Spin parity effect1
•Importance of transverse internal fields1
•Co-tunneling due to inter-SMM exchange2
Note: resonant MQT
strong at B=0, even
for half integer spin.
Energy level diagram for D < 0 system, B//z
Hˆ o  DSˆz2  m B B.g.Sˆ
400
S = 9/2
300
B // z-axis of molecule
Frequency (GHz)
200
100
0
-100
M = 5
- / to 3
S
2
-/
M = 7
2
M = 9
S
- / to 5
S
2
-/
-/
7
t
2
2 o/2
-200
-300
-400
-500
0
1
2
3
Note frequency range Magnetic field (tesla)
4
5
q

Cavity perturbation
Cylindrical TE01n (Q ~ 104 - 105)
f = 16  300 GHz (now 715 GHz!)
Single crystal 1 × 0.2 × 0.2 mm3
T = 0.5 to 300 K, moH up to 45 tesla
•We use a Millimeter-wave Vector Network
Analyzer (MVNA, ABmm) as a spectrometer
Au SS
B
M. Mola et al., Rev. Sci. Inst. 71, 186 (2000)
Energy level diagram for D < 0 system, B//z
Hˆ o  DSˆz2  m B B.g.Sˆ
400
S = 9/2
300
B // z-axis of molecule
Frequency (GHz)
200
100
0
-100
M = 5
- / to 3
S
2
-/
M = 7
2
M = 9
S
- / to 5
S
2
-/
-/
7
t
2
2 o/2
-200
-300
-400
-500
0
1
2
3
Note frequency range Magnetic field (tesla)
4
5
HFEPR for high symmetry (C3v) Mn4 cubane
Cavity transmission (arb. units - offset)
[Mn4O3(OSiMe3)(O2CEt)3(dbm)3]

Field // z-axis of the molecule (±0.5o)
1
3
1

- /2 to - /2
1
- /2 to /2
5
24 K
18 K
14 K
8K
6K
4K
3
- /2 to - /2
7
5
- /2 to - /2
9
f = 138 GHz
7
- /2 to - /2
0.0
1.0
2.0
3.0
Magnetic field (tesla)
4.0
5.0
Fit to easy axis data - yields diagonal crystal field terms
ˆ , where Oˆ 0   Sˆ 2 Sˆ 2   Sˆ 4
Hˆ o  DSˆz2  B40Oˆ 40  m B g zz BS
4
z
z
 z
100

Frequency (GHz)
150
S = 9/2
-1
D = -0.484 cm
0
-1
B4 = -0.000062 cm
gz = 2.00(1)
50
0
0
1
2
3
Magnetic field (tesla)
4
5
Routes to incredible # of SMMs
Core ligands (X): Cl-, Br-, F-
NO3-, N3-, NCOOH-, MeO-, Me3SiO-
Jahn-Teller
points towards
core ligand
Peripheral ligands: (i) carboxylate ligands: -O2CMe, -O2CEt
(ii) Cl-, py, HIm, dbm-, Me2dbm-, Et2dbm-
Antiferromagnetic exchange in a dimer of Mn4 SMMs
[Mn4O3Cl4(O2CEt)3(py)3]
-10
a
1
0.14 T/s
(-9/2,-9/2)
m1
s
M/M
Energy (K)
D1 = -0.72 K
J = 0.1 K
No H = 0
tunneling
0.5
-20
(9/2,-5/2)
0
-30
-0.5
m2
(-9/2,7/2)
(9/2,-7/2)
(-9/2,9/2)
(1)
-40
-1
-1.2
-1.2
(9/2,9/2)
-0.8
-0.8
-0.4
-0.4
(2)(3)
0.4
00
0.4
z (T)
µµ 00 HH (T)
0.04 K
0.3 K
0.4 K
0.5 K
0.6 K
0.7 K
0.8 K
(4)(5)
1.0 K
0.80.8
1.21.2
Hˆ  Hˆ 1  Hˆ 2  JSˆ1  Sˆ2  Hˆ 1  Hˆ 2  JSˆz1Sˆz 2
E  D  m12  m22   g m B B  m1  m2   Jm1m2
To zeroth order, the exchange generates a bias field Jm'/gmB
which each spin experiences due to the other spin within the dimer
Wolfgang Wernsdorfer, George Christou, et al., Nature, 2002, 406-409
Systematic control of coupling between SMMs - Entanglement
This scheme in the same spirit as
proposals for multi-qubit devices
based on quantum dots
D. Loss and D.P. DiVincenzo, Phys. Rev. A 57, 120 (1998).
Heisenberg:
JŜ1.Ŝ2
•Quantum
mechanical
coupling
by the transverse (off•Zeroth order
bias term,
JŜz1Ŝcaused
z2, is diagonal in the mz1,mz2 basis.
diagonal) parts of the exchange interaction Jxy(Ŝx1Ŝx2 + Ŝy1Ŝy2).
•Therefore, it does not couple the molecules quantum mechanically,
•This
term causes
the involve
entanglement,
i.e. itrotations.
truly mixes mz1,mz2
i.e. tunneling
and EPR
single-spin
basis states, resulting in co-tunneling and EPR transitions involving
two-spin rotations.
•CAN WE OBSERVE THIS?
S1 = S2 = 9/2; multiplicity of levels = (2S1 + 1) (2S2 + 1) = 100
Energy (K)
-20
-30
9
9
9
7
9
5
9
9
(- /2,- /2)
(- /2,- /2)S,A
-40
0.0
(- /2,- /2)S,A
(- /2, /2)
0.2
0.4
0.6
0.8
1.0
1.2
Magnetic field (tesla)
Look for additional splitting (multiplicity) and symmetry
effects (selection rules) in EPR or tunneling experiments.
First clues: comparison between monomer and dimer EPR data
[Mn4O3Cl4(O2CEt)3(py)3]
NA11 frequency dependence
NA11 145 GHz easy axis T-dep
9 /)
,2- 2
/
o(
t
)
9 /
9 /)
2
2
3 / ,1 / ,2
2
((+
to
9 /)
9 /)
2
2
3 / ,,
1 /
2
(+
(- 2
to
9 /)
2
1 / ,2
(+
180
Frequency (GHz)
18 K
15 K
10 K
8K
6K
4K
2K

100
80
60
40
0
0
1
2
3
4
5
6
Cavity transmission (arb. units - offset)

150
1
3
1
Monomer
1
- /2 to /2

- /2 to - /2
3
- /2 to - /2
7
5
- /2 to - /2
9
Monomer
f = 138 GHz
7
- /2 to - /2
1.0
2.0
3.0
24 K
18 K
14 K
8K
6K
4K
4.0
5.0
2
3
4
Magnetic field (tesla)
5
100

5
Magnetic field (tesla)
1
120
Magnetic field (tesla)
0.0
0
140
20
Frequency (GHz)
Cavity transmission (arb. units)
160
1
S = 9/2
-1
D = -0.484 cm
0
-1
B4 = -0.000062 cm
gz = 2.00(1)
50
6
0
0
1
2
3
Magnetic field (tesla)
4
5
Full exchange calculation for the dimer
}
Hˆ D  Hˆ S1  Hˆ S 2  JSˆ1.Sˆ2
Monomer Hamiltonians
Isotropic exchange
•Apply the field along z, and neglect the transverse terms in ĤS1 and ĤS2.
•Then, only off-diagonal terms in ĤD come from the transverse (x and y)
part of the exchange interaction, i.e.


ˆ
Hˆ D   Hˆ S1  Hˆ S 2  JSˆz1Sˆz 2   12 J Sˆ1 Sˆ2-  Sˆ1- Sˆ2  Hˆ 0 D  H'
•The zeroth order Hamiltonian (Ĥ0D) includes the exchange bias.
•The zeroth order wavefunctions may be labeled according to the spin
projections (m1 and m2) of the two monomers within a dimer, i.e.
m1 , m2
•The zeroth order eigenvalues are given by
E0 D  D2  m12  m22   D4  m14  m24   m B g z B  m1  m2   Jm1m2
Full exchange calculation for the dimer
•One can consider the off-diagonal part of the exchange (Ĥ') as a
perturbation, or perform a full Hamiltonian matrix diagonalization

Ĥ'  12 J Sˆ1 Sˆ2-  Sˆ1- Sˆ2

•Ĥ' preserves the total angular momentum of the dimer, M = m1 + m2.
•Thus, it only causes interactions between levels belonging to a particular
value of M. These may be grouped into multiplets, as follows...
M  m1  m2
m1 , m2
M  -9
- 92 , - 92
M  -8
- 92 , - 72 , - 72 , - 92
M  -7
- 92 , - 52 , - 52 , - 92 , - 72 , - 72
M  -6
- 92 , - 32 , - 32 , - 92 , - 72 , - 52 , - 52 , - 72
M  -5
- 92 , - 12 , - 12 , - 92 , - 72 , - 32 , - 32 , - 72 , - 52 , - 52
1st order correction
lifts degeneracies between
states where m1 and m2
differ by ±1
1st order corrections to the wavefunctions
1S
M  -9
- 92 , - 92
 2A
M  -8
1
 - 92 , - 72 - - 72 , - 92
2

 3 S
M  -8
1
 - 92 , - 72  - 72 , - 92
2

 4 S
M  -7
1
1
2 1  2 2
 5 A
M  -7
1
 - 92 , - 52 - - 52 , - 92
2
 6 S
M  -7
1
1  2
2
-
-
7
2
9
2
•The symmetries of the states are
important, both in terms of the energy
corrections due to exchange, and in terms
of the EPR selection rules.
, - 52  - 52 , - 92 - 2  - 72 , - 72


, - 72    - 92 , - 52  - 52 , - 92 

•Ĥ' is symmetric with respect to
exchange and, therefore, will
only cause 2nd order interactions
between states having the same
symmetry, within a multiplet.
7A
M  -6
1
1
2 1    '2
-
9
2
, - 23 - - 23 , - 92 -  '  - 72 , - 52 - - 52 , - 72

 7 S
M  -6
1
1
2 1    '2
-
9
2
, - 23  - 23 , - 92 -  '  - 72 , - 52  - 52 , - 72

8 A
M  -6
1
1
2 1    '2
-
7
2
, - 52 - - 52 , - 72   '  - 92 , - 32 - - 32 , - 92

 9 S
M  -6
1
1
2 1    '2
-
7
2
, - 52  - 52 , - 72   '  - 92 , - 32  - 32 , - 92


 '
3J
 0.472
D  12 J
21
  0.578    0.273
8
1st and 2nd order corrections to the energies
 - 72 , - 52 
-18.5
 - 92 , - 32 
-22.5
 - 72 , - 72 
-24.5
-
-26.5
9
2
,-
5
2

Exchange
bias
Full Exchange
(9) - S
(8) - A
(e) (d)
(7) - A & S
(6) - S
(5) - A
(4) - S
(b)
 - 92 , - 72 
(c)
Matrix element
very small
(3) - S
(2) - A
-32.5
(a)
 - 92 , - 92 
-40.5
(1) - S
Magnetic-dipole interaction is symmetric
Magnetic field dependence
7
-0
Energy (K)
9
(- /2, /2)
9 9
(- /2, /2)
1
9
(+ /2,- /2)
5
5
( - / 2, - / 2)
3
7
( - / 2, - / 2)
-0
1
9
( - / 2, - / 2)
5
7
( - / 2, - / 2)
3
9
( - / 2, - / 2)
-0
7
7
5
9
( - / 2 , - / 2)
-0
( - / 2 , - / 2)
7
9
9
9
(- /2,- /2)
-100
This figure does not show all levels
0
1
2
3
4
Magnetic field (tesla)
5
(- /2,- /2)
6
•The effect of Ĥ' is field-independent, because the field does not change
the relative spacings of levels within a given M multiplet.
Clear evidence for coherent transitions involving both molecules

Hˆ D   Hˆ S1  Hˆ S 2  JSˆz1Sˆz 2   12 J Sˆ1 Sˆ2-  Sˆ1- Sˆ2
Experiment
f = 145 GHz

Simulation
Cavity transmission (arb. units)
NA11 145 GHz easy axis T-dep
24 K
18 K
4.2
4.8
5.4
9 MHz
Magnetic field (tesla)
 tf > 1 ns
S. Hill et al., Science 302, 1015 (2003)
Variation of J, considering
only the exchange bias
Variation of J, considering
the full exchange term
J=0K
J = 0.03 K
J = 0.06 K
J = 0.1 K
J = 0.13 K
Absorption (arb. units)
Absorption (arb. units)
J=0
J = 0.03 K
J = 0.06 K
J = 0.1 K
J = 0.13 K
T = 6 K, f = 160 GHz
T = 6 K, f = 160 GHz
0
1
2
3
4
Magnetic field (tesla)
5
6
0
1
2
3
4
5
Magnetic field (tesla)
• Simulations clearly demonstrate that it is the off diagonal part of the
exchange that gives rise to the EPR fine-structure.
• Thus, EPR reveals the quantum coupling.
• Coupled states remain coherent on EPR time scales.
6
Confirmed by hole-digging (minor loop) experiments
R. Tiron et al., Phys. Rev. Lett. 91, 227203 (2003)
Next session: B25.011
Control of exchange
This scheme is in the same spirit
as proposals for multi-qubit
devices based on quantum dots
D. Loss and D.P. DiVincenzo, Phys. Rev. A 57, 120 (1998).
"off"
Light
"on"
+
-
Decoherence – role of nuclear spins
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)]
Reduce via nuclear labeling – may require something other than Mn
Also have to worry about intermolecular interactions
Chicken Fat
Mn4
Mn4

OH
,
OH
Chicken Fat
Mn4
Mn4

OH
,
OH
The molecular approach is the key
• Immense control over the magnetic unit and its
coupling to the environment
1. Control over magnitude and symmetry of the anisotropy through
the choice of molecule:
• Choice of magnetic ion, modifications to molecular core, etc.
2. Reduce electronic spin-spin interactions by adding organic bulk
to the periphery of the SMM, or by diluting with non-magnetic
molecules.
3. Reduce electron-nuclear cross-relaxation by isotopic labeling.
4. Move the tunneling into frequency window where decoherence
may be less severe:
• Achieved with lower spin and lower symmetry molecules,
• or with a transverse externally applied field,
• or by deliberately engineering-in exchange interactions.
5. Move over to antiferromagnetic systems, e.g. the dimer:
• Quantum dynamics of the Néel vector - harder to observe!
What do we currently understand?
 Quantum tunneling is extremely sensitive to SMM symmetry
•Transverse anisotropies provide tunneling matrix elements
 Magnetization dynamics controlled by nuclear and electron
spin-spin interactions
•Fluctuations drive SMMs into and out of resonance
 Such interactions represent unwanted source of decoherence
What do we not understand?
 What are the dominant sources of quantum decoherence?
 What are typical decoherence times for various quantum
states based on SMMs which could be useful?
 How can we reduce decoherence?
 Can we control the spin dynamics coherently?
 Pulsed/time-domain EPR
Many collaborators/students involved
...illustrates the interdisciplinary nature of this work
UF Physics
FSU Chemistry
UF Chemistry
Rachel Edwards
Alexey Kovalev
Konstantin Petukhov
Susumu Takahashi
Jon Lawrence
Norman Anderson
Tony Wilson
Cem Kirman
Naresh Dalal
George Christou
Micah North
David Zipse
Randy Achey
Chris Ramsey
Nuria Aliaga-Alcalde
Monica Soler
Nicole Chakov
Sumit Bhaduri
Muralee Murugesu
Alina Vinslava
Dolos Foguet-Albiol
Shaela Jones
Sara Maccagnano
Enrique del Barco
NYU Physics
Andy Kent
Also: Kyungwha Park (NRL)
Marco Evangelisti (Leiden)
Hans Gudel (Bern)
Wolfgang Wernsdorfer (Grenoble)
Mark Novotny (MS State U)
Per Arne Rikvold (CSIT - FSU)
UCSD Chemistry
David Hendrickson
En-Che Yang
Evan Rumberger