1. Quantum Nanomagnetism
and related phenomena
Professor Javier Tejada.
Dept. Física Fonamental, Universitat de Barcelona.
Columbia-Rice Frontier CMP Lecture
October 31st, 2013

2. Contenidos
Content
Introduction to magnetism: exchange and anisotropy energies
Single Domain Particles
Molecular Magnets
Resonant spin tunneling on Molecular Magnets
Quantum magnetic deflagration
Superradiance
Conclusions

3. Introduction to magnetism
• Electrostatic interaction + Quantum Mechanics
2
e
r12
e2
r12
Term
si s j
Overlapping of wave
functions
Is different for S
in the Hamiltonian
0 and S
1
Heisenberg
hamiltonian

4. Título
Exchange
interaction
Atoms can be found with two or
S =0
S =1
more interacting electrons.
Considering two of them in an
atom, the energy of the spin
interaction can be calculated:
e
e
p
The system always tends to be at the
lowest energy state:
J ~ TC
 
ˆ ˆ
ˆ = -J s × s
Þ Heff
1
2
The overlapping of the wave
functions decays exponentially.
Summation over nearest neighbours

5. Título
Magnetic
anisotropy
• Orbital motion of electrons makes them feel B
is the local electric field.
v E , where E(r )
• Action of B on the electron spins
correlation between the
direction of the spin and the orientation of the crystallographic axes.
•Quantum description: crystal-field hamiltonian is given by
HA
where b
1
b
2
v2
O 2
c
Sn Sn
n
and c
fourth rank respectively ( , , ,
1
c
4
v4
O 4
c
Sn Sn Sn Sn

n
are tensors of second rank and
x, y, z )

6. Título
Macroscopic solid to single domain particles
•
Domains and domain walls:
•
Eex
Tipically
Ean
10
3
10
The exchange energy is so high that it is
5
difficult to do any non-uniform rotation of
the magnetization (
•
If the particle has
then no domain
wall can be formed. This is a SDP:
•
The probabilty of an individual spin flip is:
with
Hence, at low T, the magnetic moment is
a vector of constant modulus:
).

7. Single domain particles (SDP)
•
Classical description: energy barrier of height
U=kV
Anisotropy
constant
•
Þ
e
Volume
U (V ) / T
Microscopic attempt frequency
Blocking temperature is defined via the condition
1 / tm
which leads to:
TB
KV / ln
tm
Analogously, we can also define the Blocking
volume:
VB
T
ln
K
tm
U

8. Important aspects of SDPs

9. SDP: magnetic relaxation
• The particles relax toward the equilibrium state:
MR t
M R 0 1 S ln
Initial remanent magnetization
• The dependence of S on T shows two
different regimes:
1)
2)
Thermal regime: at high temperatures it
is easier to “jump” the barrier. In this
regime, S µ T
Quantum regime: at low temperatures,
magnetic relaxation is due to tunnel
effect. In this regime S is
of
T.
t

10. Quantum magnetic entities
• Their magnetic moment M is a quantum operator: it verifies the
commutation relation
which yields
Quantum
Classical
Empirically, the magnetic moment is considered to behave quantumly if
|M| ≤ 100μB holds.

11. Molecular Magnets (MM): example
of Mn12 acetate
+2
+2
+2
-3/2
-3/2
-3/2
+2
+2
-3/2
+2
+2
+2
Spin S
Η
10
DS z2
H , Sz
0
Quantum counterpart of a SDP.
H ,
Discrete projection of the spin
onto the easy axis.

12. Magnetic bistability of Mn12 acetate
Degenerate ground states for the
Mn12 acetate molecule.
There exists an anisotropy energy
barrier between these two spin
orientations.
The effect of an external magnetic field applied along the easy axis.

13. Resonant
Título
spin tunneling on MM
• Application of an external field: adds a Zeeman term
Longitudinal component of the field (H // easy axis)
Shifts the levels.
Transverse component of the field (H easy axis)
Allows tunnel effect.
• The tunnel effect is possible for certain values of the field: the resonant
fields.

14. Resonant spin tunneling on MM
-2-10 1 2
-3
3
-4
4
-5
5
-6
6
-7
7
-8
-9
-10
H=0
8
9
10
Magnetic field

15. Resonant spin tunneling on MM
-2-10 1 2
-3
3
-4
4
-5
5
-6
6
-7
-8
-9
7
8
9
-10
10
H = 0.5HR
Magnetic field

16. Resonant spin tunneling on MM
-3-2
-4
-5
-6
-7
-8
-9
-10
H = HR
12
3
4
5
6
7
8
9
10
Magnetic field

17. Resonant spin tunneling on MM
-3-2-10 1
-4
2
-5
3
-6
4
-7
5
-8
6
-9
-10
H = 2HR
7
8
9
10
Magnetic field

18. Relaxation MM
• As we only have a single barrier height, relaxation goes
exponential.
Mn12 Ac relaxation measurements
from the remanent state at different
temperatures.
Mt
M eq T 1 e
H t

19. Relaxation MM
• Peaks of the relaxation rate Γ(H) at the resonant fields
Relaxation rates of
Mn12 acetate at
different fields

20. Landau-Zener effect
E
m
m'
m'
m'
W
Em
Em '
t
Transition probability
Size of magnetization step
E
E
W
P
1 P
m
m'
m
m
W
m
2
2
e
m'
E
2
/ 2

21. Resonant
Título
spin tunneling on MM

22. What is a deflagration?
Deflagration is a technical term describing subsonic combustion that usually
propagates through thermal conductivity
Metastable
State
∆U
Two important characteristic timescales:
∆E
τb= τd
Energy released  ∆E
Ignition (barrier overcoming)  ∆U
Thermal diffusion  k
Characteristic length of propagation  δ
•
Stable
State
Thermal diffusion
•
Burning timescale
Deflagration
Flame width
22

23. From magnetization jumps to magnetic
Deflagration (MD)
Molecule magnets
Field jumps 1999
Deflagration-like description 2005
Intermetallic compounds
Field jumps 2002
Deflagration-like
description 2010
Manganites
Field jumps 1999
Deflagration-like description 2007
23

24. First evidences of MD
H
ΔE
Magnetic deflagration:
Propagation of a front of reversing spins
at constant velocity along the crystal
A. Hernández-Mínguez et. al. PRL 95 17205 (2005)
Problem: Sweeping H we cannot
control the magnetic field at
which it occurs.
Y. Suzuki et. al. PRL 95, 147201 (2005)

25. Quantum magnetic deflagration
Avalanche ignition produced by SAW:
Surface Acoustic Waves (SAW) are low frequency acoustic phonons
(below 1 GHz)
Coaxial cable connected to an Agilent microwave signal generator
Change in magnetic moment registered in a rf-SQUID magnetometer
Hz
Coaxial cable
IDT
Mn12 crystal
c-axis
Conducting
stripes
LiNbO3
substrate

26. Quantum magnetic deflagration
v
κ
τ0
exp
U(H)
2kB Tf
This velocity is well fitted:
κ = 0.8·10-5 m2/s
Tf (H = 4600 Oe) = 6.8 K
Tf (H = 9200 Oe) = 10.9 K
• The speed of the avalanche
increases with the applied
magnetic field
• At resonant fields the • The ignition time shows peaks at
the magnetic fields at which spin
velocity of the flame front
levels become resonant.
presents peaks.

27. Quantum Magnetic Deflagration

28. Quantum Magnetic Deflagration

29. Associated to magnetic avalanches:
magnetoresistive avalanches in manganites
29

30. Superradiance
Proposed by Robert H. Dicke in 1954.
This kind of emission (SR) has characteristic properties that make it
different from other more common phenomena like luminescence
I
I
N
Luminescence
τ1
t
I
L
τSR
Superradiance
L~λ
I
N2
λ
N is the number of dipoles
t

31. Superradiance
All spins decay to the fundamental level coherently, with the
emission of photons.
-1
-3-2 0 1 2
-4
-5
3
-6
4
-7
5
-8
6
-9
7
-10
B = 2B0
8
9
10

32. Superradiance??
Sharp peak shows a signal which is
equivalent to the sample being at 20 K
(the expected self heating is about 3 K).

33. Magnetic deflagration in pulsed fields
dB/dt (kT/s)
7.0
6
4.8 3.7 3.2
2.5
1.9
1.6
coil 1
coil 2
Δt
dM/dt
4
2
0
200
400
t (s)

34. Quantum magnetic detonation
t
16
14
12
10
Time-difference between the observation of a
magnetisation-reversal in a coil on the left
and on the right of a Mn12Ac-sample in function
of (high) magnetic field-sweeprates.
All observations were done in pulsed fields
at a temperature of about 500mK (in liquid 3He).
t ( s)
8
6
4
2
0
-2
1000
2000
3000
4000
5000
dHz/dt (T/s)
6000
7000

35. Superradiance: indirect evidence
0.30
1066 T/s
1270 T/s
1660 T/s
2050 T/s
2970 T/s
0.20
Mn12-Ac powder in araldite
Scaling with fixed HR
1104 T/s
1321 T/s
1725 T/s
2132 T/s
3113 T/s
0.15
0.10
0.10
Scaling
0.05
0.00
1.2
1.4
1.6
0.08
1.8
2.0
r
1/2
dM/dH
B(T)
dM
dH
r ~
H
r
0.06
0.04
0.02
0.00
-0.010
-0.005
0.000
0.005
0.010
(H - HR) / r
1/2
0.015
0.020
G:scaledFixedHR in G:MolecularMagnetsMn12Pulsed fields (KULeuven)Mn12_B+C(paper)
MN12BDN /
MN12BGN /
MN12BIN /
MN12BJN /
MN12BMN
dM/dB (a.u)
0.25
0.025

36. Future
• Energy barrier between opposite orientations of the magnetic
moment is formed by weak relativistic interactions
whether
stable molecular magnets can ever break liquid nitrogen
temperature of 77K.
• Making identical molecules comparable to mesoscopic magnetic
particles will be a challenging task for chemists.
• Another challenging question would be whether magnetic molecules
can ever become ultimate memory units of conventional computers
or even elements of quantum computers.
• I hope to see answers to these questions in the near future!!

37. References
[1] E. M. Chudnovsky, J. Tejada, Macroscopic Quantum Tunneling of the Magnetic Moment
(Cambridge Univ. Press, 1998).
[2] J.R. Friedman, M.P. Sarachik, J. Tejada and R. Ziolo. Phys. Rev. Lett. 76, 3830–3833 (1996).
[3] A. Hernández-Mínguez et al. Phys. Rev. Lett. 95, 217205 (2005).
[4] Macià et al. Phys. Rev. B 76, 174424 (2007).
[5] Macià et al. Phys. Rev. B 77, 012403 (2008).
[6] F. Macià et al. Phys. Rev. B 79, 092403 (2009).
[7] S. Vélez et al. Phys. Rev. B 81, 064437 (2010).
[8] W. Decelle et al. Phys. Rev. Lett. 102, 027203 (2009).
[12] P. Subedi et al. Phys. Rev. Lett. 110, 207203 (2013). Physics 6, 55 (2013).