1. Universitat de Barcelona
1990-2010: quantum magnetism
Fall 2011
Javier Tejada, Dept. Física Fonamental

2. Contenidos
Content
Introduction to magnetism
Single Domain Particles
Quantum relaxation: 1990-96
Resonant spin tunneling: 1996-2010
Quantum magnetic deflagration
Superradiance

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

4. Introduction to magnetism
The magnetic moment of an atom has e
two contributions:
p μorbital
1. The movement of the electrons
around the nucleus. The electric
charges generate magnetic fields
while moving
2. Electron, like the other fundamental particles, has an intrinsic propierty named spin,
which generates a magnetic moment even outside the atom:
e e μspin
S=1/2 S=-1/2
Hence, the magnetic moment of the atom is the sum of both contributions
e
μtotal = μorbital + μspin
p

5. Título
Introduction to magnetism
S 0 S 1
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:
e The system always tends to be at
e
p the lowest energy state:: J ~ TC
ˆ  
ˆ ˆ
eff
J s1 s 2
The overlapping of the wave Summation over
functions decays exponentially. nearest neighbours

6. Título
Introduction to magnetism
Existence of metastable
states
Magnetic Time dependent
hysteresis phenomena
Slow relaxation towards the free energy
minimum.
Global Non-linear
thermodynamic effects.
equilibrium.

7. Título
Single domain particles
• Permanent magnets divide
themselves in magnetic domains to
minimize their magnetic energy.
• There are domain walls between these domains:
E ex Exchange energy E ex
a (nm )
E an Anisotropy energy E an
a Lattice constant

8. Título
Single domain particles
E ex 3 5 The exchange energy is so high that
Tipically 10 10
E an it is difficult to do any non-uniform
rotation of the magnetization.
If the particle has R then no domain
walls can be formed. This is a SDP:
The probabilty of the flip E ex
exp( ) 0 and E ex Tc
of an individual spin is:
T
Hence, at low T, the magnetic moment is a
T Tc S ct
vector of constant modulus:

9. Single domain particles
The rotation of M as a whole needs certain energy called magnetic
anisotropy.
• Relativistic origin:
p
v
– Order of magnitude , with p even.
c
• Classic description:
– Energetic barrier of height: U
U (H )
U kV e T
Anisotropy Volume
constant

10. Single domain particles
Quantum description: Because the spin is a quantum characteristic, it
can pass the barrier by tunnel effect.
The tunnel effect, that reveals the
Easy axis Hard axis quantum reality of the magnetism, allows
the chance of finding the magnetic
moment of the particle in two different
states simultaneously.
U +
The action of the observer on the
particle will determine its final state!!!

11. Single domain particles
Important aspects of SDPs:
• Volume distribution: f R f V f U
• And orientations:
• Their magnetic moments tend to align with the applied magnetic
field.

12. Single domain particles
• The particles relax toward the equilibrium state:
t
M M0 1 S ln
0
Magnetic viscosity
• Thermal behaviour ( S T )
– At high temperatures it is easier to “jump” the barrier.
• Quantum behaviour (independent of T)
– Relaxation due to tunnel effect.

13. Magnets: memory and relaxation
When removing the applied
Magnetic solids (ferromagnets) show
field, these materials keep
hysteresis when an external magnetic field is
certain magnetization that
applied:
slowly decreases with time.
M HH
MR ~ Memory
MR ~ ln t
Hc
H
Magnetic solids have memory,
and they lose it with time!!!
H
H
t ~ 109 years: Paleomagnets
Hc Magnet ~ 5000 Oe
t ~ 10 years: credit cards
Hc Transformer ~ 1 Oe

14. Título
Quantum relaxation: 1990-96
Magnetic viscosity Magnetic viscosity
dependance on T, for low variation with respect
T, of a TbFe3 thin film to the magnetic field.

15. Título
Quantum relaxation: 1990-96

16. Resonant spin tunneling on
mollecular magnets
• Identical to single domain particles
• Quantum objectsObjetos cuánticos
[M i , M j ] 2i M
B ijk k |M| ~ μB Quantum
[M i , M j ] M iM j
M jM i B
M k
|M| » μB Classic
Empirically, the magnetic moment is considered in a quantum way if
|M| ≤ 1000μB
2 2
H A
DS z
ES x
M(H,T) univocally determined by D and E

17. Resonant
Título spin tunneling on
mollecular magnets
• Application of an external field: Zeeman term H S
- 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.

18. Resonant spin tunneling on
mollecular magnets
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
Sz
magnetic resonance
Sz -Sz
-Sz

19. Resonant
Título spin tunneling on
mollecular magnets

20. Resonant spin tunneling on
mollecular magnets
-2-10 1 2
-3 3
-4 4
-5 5
-6 6
-7 7
-8 8
-9 9
-10 10
Magnetic field
B=0

21. Resonant spin tunneling on
mollecular magnets
-3-2-10 1 2
-4 3
-5 4
5
-6
6
-7
7
-8
8
-9
9
-10
10 Magnetic field
B = 0.5B0

22. Resonant spin tunneling on
mollecular magnets
-3-2 12
-4 3
-5 4
-6 5
-7 6
-8 7
-9 8
-10 9
B = B0 10
Magnetic field

23. Resonant spin tunneling on
mollecular magnets
-3-2-10 1
-4 2
-5 3
-6 4
-7 5
-8 6
-9 7
-10 8
9
B = 2B0 Magnetic field
10

24. Resonant
Título spin tunneling on
mollecular magnets
• After a certain time, the relaxation becomes exponential:
M t M eq
t 1 exp H t
• Peaks on the relaxation rate Γ(H) at the resonances:

25. A.C. measurements
• TB depends on measuring frequency
K V0
TB
ln 1 / 0

26. Quantum magnetic deflagration
Avalanche ignition produced by SAW:
Surface Acustic 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
LiNbO3
IDT Mn12 crystal substrate
c-axis
Conducting
stripes

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

28. Quantum magnetic deflagration

29. Quantum magnetic deflagration

30. Quantum magnetic deflagration

31. Superradiancie
– 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 8
9
B = 2B0
10

32. Superradiancia
This kind of emission (SR) has carachteristical propierties that make it
different from other more common phenomena like luminiscence
I
Luminescence
τ1
t
I
L L~λ
Superradiancie τSR
λ
t

33. Milestones
Título
1896 Zeeman Effect (1)
1922 Stern–Gerlach Experiment (2)
1925 The spinning electron (3)
1928 Dirac equation (4)
1928 Quantum Magnetism (5)
1932 Isospin (6)
1940 Spin–statistics connection (7)

34. Milestones
Título
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. Milestones
Título
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)

36. Milestones
Título
1996 Mesoscopic tunnelling of magnetization (22)
1997 Semiconductor spintronics (23)
© 2008 Nature Publishing Group

37. Linear Doppler
Shift on frequency due to relative velocity between emitter and
observer (non relativistic case):
Relative
v velocity
1
Frequency c
seen by the
observer Frequency of
the emitter
v
c

38. Rotational Doppler
Shift on frequency due to relative rotation between emitter and
observer (circularly polarized light):
Relative rotation
Frequency
seen by the Frequency of
observer the emitter

39. Rotational Doppler Effect

40. EPR Results

41. EPR Results

42. Rotational Doppler Effect
2 B
FMR 0
H 
n
I
n
Hn 0
I
2
 
H Hn 1
Hn 
I 2 B
I
measured H ~ 2 . 5 Oe
produced by r ~ 1nm particles

43. Rotational Doppler Effect
Occupied states
L L 1
En n n 1 B
H
2I
2 En
n ~
B
H
1/ 2
k BT
E n ~ k BT n~
B
H
T ~ 2K
n 100
B
H ~ 0 . 17 mK

44. Rotational Doppler Effect
• Change in frequency observed due to rotation:
• RDE in GPS systems (resonance of an LC circuit)
– Resonant frequency insensitive to magnetic fields
Resonance
• RDE in Magnetic Resonance systems
– Resonant frequency sensitive to magnetic fields
Resonance

45. Rotational Doppler Effect
• 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

46. Rotational Doppler Effect
• Ω ≈ 100 kHz
Ω << ωRes << Δω
• ωRes ≈ GHz BUT
• ESR and FMR:
• Δω ≈ MHz Position of maximum can be
determined with accuracy of 100
kHz ≈ Ω
• ωRes ≈ MHz
• NMR: • Δω ≈ kHz Ω ≈ Δω
ESR and Gyromagnetic
FMR: tensor (shape,...)
anisotropy
• κ ≠ 1 needed Hyperfine
NMR: interactions

47. Magnetic Vortices
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.

48. Magnetic Vortices
We have studied an array of These hysteresis loops correspond
permalloy (Fe81 Ni19) disks with to the single domain
diameter 2R = 1.5 μm and thickness (SD)⇐⇒Vortex transitions. For the
L = 95 nm under the application of range of temperatures explored,
an in-plane magnetic field up to the vortex linear regime in the
1000 Oe in the range of ascending branch should extend
temperatures 2 − 300 K. from 300 Oe to 500 Oe at least.

49. Magnetic Vortices
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)

50. Magnetic Vortices
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.

51. Magnetic Vortices
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).

52. Magnetic Vortices
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.

53. Magnetic Vortices
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