Quantum Magnetism
Javier Tejada Palacios
What is magnetism?
• Electrostatic Interaction + Quantum Mechanics
Overlapping of wave funcions12
2
r
e
12
2
r
e0SIs different for 1Sand
Term jiss in the Hamiltonian
Heisenberg hamiltonian
Exchange interaction
• Interaction as function of spin operators:
Overlap of wave functions decays exponentally
Sumation over nearest neighbors
CTJ ~
Magnetic Anisotropy
• Relativistic origin– order of magnitude , where p is even.
• Classic description:– Energy barrier of height:
• Quantum description:
p
c
v
VkU
Anisotropy constant
volume
Easy axis Hard axis
Macroscopic solid to Single Domain Particle
• Domains and domain walls
• If a particle has size :
– No domain walls
– Single Domain Particle!
• Probability of individual spin flip:
R
aE
E
A
ex
531010
A
ex
E
E
Tipically
nm
0expT
Eex
At low T (<< TC): S = constant
Important aspects of SDP
• Distribution of sizes:
• And orientations:
• Their magnetic moments tend to align the external magnetic field
UfVfRf
Important aspects of SDP
• Particles relax towards the equilibrium state:
• Thermal behaviour ( )– At higher temperatures, it is easier to “jump” the
barrier
• Quantum behaviour (independent of T)– Relaxation due to tunneling
tSMM ln10
Magnetic viscosity
TS
Blocking temperature
• Flipping between ↓ and ↑ states occurs in a certain characteristic time that depends on temperature:
• Below the blocking temperature TB the magnetic moment cannot flip above the barrier
• Above the blocking temperature the magnetic moment can flip, following a Curie law:
Tk
Utt
B
fexp
0
SUPERPARAMAGNETIC state
T > TBT < TB
Molecular clusters
• Identical to SDP
• Quantum objects
• M(H,T) univocally determined by D and E
Magnetization curve
• Application of an external field: Zeeman Term
– Longitudinal field (H || easy axis)
• Moves levels
– Transverse field (H easy axis)
• Allows tunneling
• Tunneling is possible at resonant fields
SH
Spin resonant tunnel effect
-10
-9
-8
-7
-6
-5-4
-3-2-10 1 2
34
5
6
7
8
9
10
B=0Magnetic field
-10
-9
-8
-7
-6-5
-4-3
-2-10 1 23
45
6
7
8
9
10
B = 0.5B0
Magnetic field
Spin resonant tunnel effect
B = B0
-10
-9
-8
-7
-6-5
-4-3-2 1 2
34
5
6
7
8
9
10 Magnetic field
Spin resonant tunnel effect
-10
-9
-8
-7
-6-5
-4-3-2-10 1
23
4
5
6
7
8
9
10
B = 2B0
Magnetic field
Spin resonant tunnel effect
Relaxation in Molecular Magnets
• After a certain time, relaxation goes exponential
• Peaks of the relaxation rate Γ(H) at resonances
tHtMtMeq
exp1
A.C. measurements
• TB depends on measuring frequency
0
0
/1ln
VKT
B
New trends in magnetism
• Magnetic deflagration
• Superradiance
• Rotational Doppler Shift
Magnetic Deflagration
• There are two characteristic timescales which are important here. The first is the thermal diffusion timescale is approximately equal to
• The second is the burning timescale that strongly decreases with temperature, typically as
Energy released
Thermal diffusion
Ignition
(barrier overcoming)
AF
FM
Energy Barrier
Deflagration is a technical term describing subsonic combustion that usually propagates through thermal conductivity
From Magnetisation jumps to magnetic deflagration.
-30 -20 -10 0 10 20 30
-1.0
-0.5
0.0
0.5
1.0
M/M
s
H (kOe)
T = 1.8 K
Molecule magnets
Field jumps 1999
Deflagration-like description 2005
0 10 20 30
0.0
0.5
1.0
M/M
S
H (kOe)
T = 3 K
ManganitesField jumps 1999Deflagration-like description 2007
Intermetallic compoundsField jumps 2002Deflagration-like description 2009
0 5 10 15 20 25 30 35 40 45 50
0.0
0.2
0.4
0.6
0.8
1.0M
/MS
H (kOe)
Molecule Magnets
Magnetic deflagration:
Propagation of a front of reversing spins
at constant velocity along the crystal
Problem: Sweeping H we cannot control the magnetic field at which it occurs.
Y. Suzuki et. al. PRL 95, 147201 (2005)
A. Hernández-Mínguez et. al. PRL 95 17205 (2005)
H
ΔE
Igniting avalanches with SAW
IDT
LiNbO3
substrate
conducting stripes
coaxial cable
Mn12 crystalc-axis
Hz
The coaxial cable is connected to an Agilent microwave signal generator.
The change of the magnetic moment is registered by a rf-SQUID magnetometer.
Surface acoustic waves (SAWs) are low frequency acoustic phonons (below 1 GHz)
Quantum magnetic deflagration in nanomagnets
• 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.
fB0 T2k
U(H)exp
τ
κv
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
Associated to the deflagration...
• Superradiance emission (?)
– All spins decay to the fundamental level coherently, with the emission of photons.
-10
-9
-8
-7-6
-5-4
-3-2-10 12
34
5
6
7
8
9
10
B = 2B0
Linear Doppler
c
v1
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
Rotational Doppler
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 Effect
EPR results
EPR results
I
n
HFMR 0
I
nH
n
0
IIHHH
B
nn2
2
1
B2
OeH 5.2~ measured
particles 1~by produced nmr
Rotational Doppler Effect
Rotational Doppler Effect
HnnI
LLE
Bn1
2
1
TkEBn
~
H
En
B
n~2
2/1
~H
Tkn
B
B
KT 2~
mKHB
17.0~100n
Occupied states
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
• RDE in Magnetic Resonance systems
– Resonant frequency sensitive to magnetic fields
Resonance
Resonance
Rotational Doppler Effect
• Article by S. Lendínez, E. M. Chudnovsy, and J. Tejada:
arXiv:1008.2142v1 [cond-mat.other]
• 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
• Ω ≈ 100 kHz
• ESR and FMR:
• NMR:
• κ ≠ 1 needed
Ω << ωRes << Δω
BUTPosition 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:
With free rotors: Ω≈ 100 MHz