2. Contenidos
Introduction to magnetism
Single Domain Particles
Quantum relaxation: 1990-96
Resonant spin tunneling: 1996-2010
Quantum magnetic deflagration
Superradiance
Content
3. • Electrostatic interaction + Quantum Mechanics
Overlapping of wave
functions
12
2
r
e
12
2
r
e 0
S
Is different for 1
S
and
Term j
i s
s In the Hamiltonian
Heisenberg
hamiltonian
Introduction to magnetism
4. e
The magnetic moment of an atom has
two contributions:
1. The movement of the electrons
around the nucleus. The electric
charges generate magnetic fields
while moving
p
2. Electron, like the other fundamental particles, has an intrinsic propierty named spin,
which generates a magnetic moment even outside the atom:
e
S=1/2 S=-1/2
μspin
Hence, the magnetic moment of the atom is the sum of both contributions
e
p
μtotal = μorbital + μspin
μorbital
e
Introduction to magnetism
5. Título
Introduction to magnetism
0
S 1
S
2
1
ˆ
ˆ
ˆ s
s
J
eff
e
e
p
Atoms can be found with two or
more interacting electrons.
Considering two of them in an
atom, the energy of the spin
interaction can be calculed:
The system always tends to be at
the lowest energy state::
The overlapping of the wave
functions decays exponentially.
Summation over
nearest neighbours
C
T
J ~
6. Título
Introduction to magnetism
Existence of metastable
states
Time dependent
phenomena
Magnetic
hysteresis
Slow relaxation towards the free energy
minimum.
Global
thermodynamic
equilibrium.
Non-linear
effects.
7. • Permanent magnets divide
themselves in magnetic domains to
minimize their magnetic energy.
Título
Single domain particles
)
(nm
a
E
E
an
ex
• There are domain walls between these domains:
Exchange energy
Anisotropy energy
Lattice constant
ex
E
an
E
a
8. Título
Single domain particles
5
3
10
10
an
ex
E
E
0
)
exp(
T
Eex
ct
S
T
T c
Tipically
The exchange energy is so high that
it is difficult to do any non-uniform
rotation of the magnetization.
R
If the particle has then no domain
walls can be formed. This is a SDP:
The probabilty of the flip
of an individual spin is:
Hence, at low T, the magnetic moment is a
vector of constant modulus:
c
ex T
E
and
9. Single domain particles
• Relativistic origin:
– Order of magnitude , with p even.
• Classic description:
– Energetic barrier of height:
p
c
v
U
V
k
U
Anisotropy
constant
Volume
T
H
U
e
)
(
The rotation of M as a whole needs certain energy called magnetic
anisotropy.
10. U
Because the spin is a quantum characteristic, it
can pass the barrier by tunnel effect.
The tunnel effect, that reveals the
quantum reality of the magnetism, allows
the chance of finding the magnetic
moment of the particle in two different
states simultaneously.
+
The action of the observer on the
particle will determine its final state!!!
Quantum description:
Easy axis
Single domain particles
Hard axis
11. Important aspects of SDPs:
• Volume distribution:
• And orientations:
• Their magnetic moments tend to align with the applied magnetic
field.
U
f
V
f
R
f
Single domain particles
12. • The particles relax toward the equilibrium state:
• Thermal behaviour ( )
– At high temperatures it is easier to “jump” the barrier.
• Quantum behaviour (independent of T)
– Relaxation due to tunnel effect.
0
0 ln
1
t
S
M
M
Magnetic viscosity
T
S
Single domain particles
13. Magnetic solids (ferromagnets) show
hysteresis when an external magnetic field is
applied:
MR ~ Memory
H
M
H
H
H
H
Magnetic solids have memory,
and they lose it with time!!!
MR ~ ln t
When removing the applied
field, these materials keep
certain magnetization that
slowly decreases with time.
t ~ 109 years: Paleomagnets
t ~ 10 years: credit cards
Hc Magnet ~ 5000 Oe
Hc Transformer ~ 1 Oe
Hc
Magnets: memory and relaxation
16. |M| ~ μB
|M| » μB
Quantum
Classic
Empirically, the magnetic moment is considered in a quantum way if
|M| ≤ 1000μB
Resonant spin tunneling on
mollecular magnets
• Identical to single domain particles
• Quantum objectsObjetos cuánticos
k
ijk
B
j
i M
i
M
M
2
]
,
[
k
B
i
j
j
i
j
i M
M
M
M
M
M
M
]
,
[
M(H,T) univocally determined by D and E
2
2
x
z
A ES
DS
H
17. Título
• Application of an external field: Zeeman term
- Longitudinal component of the field (H easy axis)
Moves the levels.
- Transverse component of the field (H easy axis)
Allows tunnel effect.
• The tunnel effect is possible for certain values of the field;
resonant fields.
S
H
||
Resonant spin tunneling on
mollecular magnets
18. The spin energy levels are moved by an applied magnetic field
For multples of the resonant field (HR, 2HR, 3HR, …) the
energy of two levels is the same, producing quantum
superposition, allowing the tunneling. This is known as
magnetic resonance
-Sz
Sz
Sz -Sz
Resonant spin tunneling on
mollecular magnets
24. Título
• After a certain time, the relaxation becomes exponential:
• Peaks on the relaxation rate Γ(H) at the resonances:
t
H
t
M
t
M eq
exp
1
Resonant spin tunneling on
mollecular magnets
25. • TB depends on measuring frequency
0
0
/
1
ln
V
K
TB
A.C. measurements
26. Avalanche ignition produced by SAW:
IDT
LiNbO3
substrate
Conducting
stripes
Coaxial cable
Mn12 crystal
c-axis
Hz
Coaxial cable connected to an Agilent microwave signal generator
Change in magnetic moment registered in a rf-SQUID magnetometer
Surface Acustic Waves (SAW) are low frequency acoustic phonons
(below 1 GHz)
Quantum magnetic deflagration
27. • The speed of the avalanche
increases with the applied
magnetic field
• At resonant fields the
velocity of the flame front
presents peaks.
• The ignition time shows peaks at
the magnetic fields at which spin
levels become resonant.
f
B
0 T
2k
U(H)
exp
τ
κ
v
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
Quantum magnetic deflagration
31. – All spins decay to the fundamental level coherently, with the
emission of photons.
-10
-9
-8
-7
-6
-5
-4
-3-2
-1012
3
4
5
6
7
8
9
10
B = 2B0
Superradiancie
34. Título
Milestones
1946 Nuclear Magnetic Resonance (8)
1950s Development of Magnetic devices (9)
1950–1951 NMR for chemical analysis (10)
1951 Einstein–Podolsky–Rosen argument in spin variables (11)
1964 Kondo effect (12)
1971 Supersimmetry (13)
1972 Superfluid helium-3 (14)
35. Título
Milestones
1973 Magnetic resonance imaging (15)
1976 NMR for protein structure determination (16)
1978 Dilute magnetic semiconductors (17)
1988 Giant magnetoresistance (18)
1990 Functional MRI (19)
1990 Proposal for spin field-effect transistor (20)
1991 Magnetic resonance force microscopy (21)
37.
c
v
1
Shift on frequency due to relative velocity between emitter and
observer (non relativistic case):
Frequency
seen by the
observer Frequency of
the emitter
c
v
Relative
velocity
Linear Doppler
38.
Shift on frequency due to relative rotation between emitter and
observer (circularly polarized light):
Frequency
seen by the
observer
Frequency of
the emitter
Relative rotation
Rotational Doppler
42. I
n
H
FMR
0
I
n
Hn
0
I
I
H
H
H
B
n
n
2
2
1
B
2
Oe
H 5
.
2
~
measured
particles
1
~
by
produced nm
r
Rotational Doppler Effect
43. H
n
n
I
L
L
E B
n
1
2
1
T
k
E B
n ~
H
E
n
B
n
~
2
2
/
1
~
H
T
k
n
B
B
K
T 2
~
mK
H
B 17
.
0
~
100
n
Occupied states
Rotational Doppler Effect
44. • Change in frequency observed due to rotation:
• RDE in GPS systems (resonance of an LC circuit)
– Resonant frequency insensitive to magnetic fields
• RDE in Magnetic Resonance systems
– Resonant frequency sensitive to magnetic fields
Resonance
Resonance
Rotational Doppler Effect
45. • Article:
S. Lendínez, E. M. Chudnovsy, and J. Tejada Phys. Rev. B 82, 174418
(2010)
• Expression for ω’Res are found for ESR, NMR and FMR.
Resonance
• Exact expression depends on type of resonance (ESR, NMR or FMR)
• Depends on anisotropy
Rotational Doppler Effect
46. • Ω ≈ 100 kHz
• ESR and FMR:
• NMR:
• κ ≠ 1 needed
Ω << ωRes << Δω
BUT
Position of maximum can be
determined with accuracy of 100
kHz ≈ Ω
Ω ≈ Δω
• ωRes ≈ GHz
• Δω ≈ MHz
• ωRes ≈ MHz
• Δω ≈ kHz
anisotropy
Gyromagnetic
tensor (shape,...)
Hyperfine
interactions
NMR:
ESR and
FMR:
Rotational Doppler Effect
47. Magnetic vortices are bi-dimensional magnetic systems whose magnetic equilibrium
configuration is essentially non-uniform (the vortex state): the spin field splits into two
well-differentiated structures, 1) the vortex core consisting of a uniform out-of-plane
spin component whose spatial extension is ∼ 10nm and 2) the curling magnetization
field (in-plane spin component), characterized by a non-zero vorticity value.
We study disk-shaped magnetic vortices.
The application of an in-plane magnetic field yields
the displacement of the vortex core perpendicularly
to the field direction.
The vortex core entirely governs the low
frequency spin dynamics: applying a superposition
of a static magnetic field (∼ 100Oe) and an AC
magnetic field (∼ 10Oe), the vortex shows a
special vibrational mode (called ’slow
translational/gyrotropic mode’), consisting
of the displacement of the vortex core as a whole, following a precessional/
gyrotropic movement around the vortex centre. Its characteristic frequency belongs
to the subGHz range.
Magnetic Vortices
48. We have studied an array of
permalloy (Fe81 Ni19) disks with
diameter 2R = 1.5 μm and thickness
L = 95 nm under the application of
an in-plane magnetic field up to
1000 Oe in the range of
temperatures 2 − 300 K.
These hysteresis loops correspond
to the single domain
(SD)⇐⇒Vortex transitions. For the
range of temperatures explored,
the vortex linear regime in the
ascending branch should extend
from 300 Oe to 500 Oe at least.
Magnetic Vortices
49. a) Temperature dependence of
both MZFC(H) and MFC(H).
b) Isothermal magnetic
measurements along the
descending branch of the
hysteresis cycle, Mdes(H), from
the SD state (H = 1KOe)
Magnetic Vortices
50. The FC curve is the magnetic
equilibria of the system.
a) Normalized magnetization (M(t)
− Meq)/ (M(0) − Meq) vs. ln t
curves measured for two
different applied fields (H = 0
and 300 Oe) at T = 2 K.
b) Thermal dependence of the
magnetic viscosity S(T) for H =
0 and 300 Oe.
Magnetic Vortices
51. Conclusions
1) The existence of structural defects in the disks could be a
feasable origin of the energy barriers responsible for the magnetic
dynamics of the system. We consider that these defects are
capable of pinning the vortex core,when the applied magnetic is
swept, in an non-equilibrium position.
2) Thermal activation of energy barriers dies out in the limit T →
0. Our observation that magnetic viscosity S(T) tends to a finite
value different from zero as T → 0 indicates that relaxations are
non-thermal in this regime (underbarrier quantum tunneling).
Magnetic Vortices
52. Theoretical modeling
Rigid model of the shifted vortex ⇒ The vortex core is described as a
zero-dimensional object whose dynamics is ruled by Thiele’s equation.
The Langrangian is given by L = Gy·x − W(r), where r = (x, y) are the
coordinates of the vortex core in the XY plane, G is the modulus of its
gyrovector and W(r) is the total magnetic energy of the system.
We consider the vortex core as a flexible line that goes predominantly
along the z direction, so that r = r(z, t) is a field depending on the
vertical coordinate of the vortex core, z. The whole magnetic energy
(including the elastic and the pinning potential) is described via a
biparametric quartic potential given by
where μ and h are the magnetic moment of the dot, respectively the
modulus
of external magnetic field (applied in the y direction), λ is the elastic
coefficient and κ and β are the parameters of the potential energy.
Magnetic Vortices
53. In absence of applied magnetic field (h = 0), the obtained expressions for the
crossover temperature Tc and the depinning exponent Seff are
,
respectively, where c is a numerical factor of order unity. Experimentally we
have
and for a measurable tunneling rate Seff cannot exceed 25−30. From all these
we deduce the estimates and
Finally, from these values of the parameters of the pinning potential we can
estimate the width of the energy barrier, which is given by the expression
and the order of magnitude of the heigth of the barrier, which is
Magnetic Vortices
