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Quantum Nanomagnetism
and related phenomena
Columbia-Rice Frontier CMP Lecture
October 31st, 2013
Professor Javier Tejada.Dept. Física Fonamental, Universitat de
Barcelona.
Contenidos
Introduction to magnetism: exchange and anisotropy energiesSingle Domain ParticlesMolecular MagnetsResonant spin tunneling on Molecular MagnetsQuantum magnetic deflagrationSuperradianceConclusions
Content
• Electrostatic interaction + Quantum Mechanics
Overlapping of wave functions
12
2
re
12
2
re
0SIs different for and 1S
Term in the Hamiltonianji ss Heisenberg hamiltonian
Introduction to magnetism
TítuloExchange interaction
eep
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 calculated:
The system always tends to be at the
lowest energy state:
The overlapping of the wave functions decays exponentially. Summation over nearest neighbours
TítuloMagnetic anisotropy
• Orbital motion of electrons makes them feel , where is the local electric field.
• Action of 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
where and are tensors of second rank and
fourth rank respectively
EvB )(rE
B
2
2
c
vOb
4
4
c
vOc
),,,,,( zyx
n
nnnnn
nnA SSSScSSbH 4
1
2
1
TítuloMacroscopic solid to single domain particles
53 1010 an
ex
E
E
• Domains and domain walls:
• Tipically
• If the particle has then no domain
wall can be formed. This is a SDP:
• The probabilty of an individual spin flip is:
with
The exchange energy is so high that it is
difficult to do any non-uniform rotation of
the magnetization ( ).
Hence, at low T, the magnetic moment is
a vector of constant modulus:
Single domain particles (SDP)
• Classical description: energy barrier of height
• Blocking temperature is defined via the condition
which leads to:
Analogously, we can also define the Blocking volume:
Anisotropy constant
Volume
TVUe /)(
U
mt/1
mB tKVT ln/
mB tK
TV ln
Microscopic attempt frequency
Important aspects of SDPs
• The particles relax toward the equilibrium state:
SDP: magnetic relaxation
Initial remanent magnetization Viscosity
• The dependence of S on T shows two different regimes:1) Thermal regime: at high temperatures it
is easier to “jump” the barrier. In this regime,
2) Quantum regime: at low temperatures, magnetic relaxation is due to tunnel effect. In this regime S is independent of T.
tSMtM RR ln10
Quantum
ClassicalEmpirically, the magnetic moment is considered to behave quantumly if |M| ≤ 100μB holds.
Quantum magnetic entities
• Their magnetic moment M is a quantum operator: it verifies the commutation relation
which yields
0,
,
10Spin 2
z
z
SH
HDSΗ
S
Molecular Magnets (MM): example of Mn12 acetate
Quantum counterpart of a SDP.
Discrete projection of the spin onto the easy axis.
-3/2
+2
-3/2-3/2
-3/2
+2
+2+2
+2
+2
+2
+2
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.
Título• 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.
Resonant spin tunneling on MM
-10
-9
-8
-7
-6-5
-4-3-2-10 1 2 3
45
6
7
8
9
10
H=0Magnetic field
Mag
netiz
atio
n
Resonant spin tunneling on MM
-10
-9
-8
-7-6
-5-4
-3-2-10 1 23
45
6
7
8
9
10
H = 0.5HR
Magnetic field
Mag
netiz
atio
n
Resonant spin tunneling on MM
H = HR
-10
-9
-8
-7-6
-5-4
-3-2 1 23
45
6
7
8
9
10 Magnetic field
Mag
netiz
atio
n
Resonant spin tunneling on MM
-10
-9
-8
-7-6
-5-4-3
-2-10 1 23
45
6
7
8
9
10
H = 2HR
Magnetic field
Mag
netiz
atio
n
Resonant spin tunneling on MM
• As we only have a single barrier height, relaxation goes exponential.
Relaxation MM
Mn12 Ac relaxation measurements from the remanent state at different temperatures.
tHeq eTMtM 1
• Peaks of the relaxation rate Γ(H) at the resonant fields
Relaxation MM
Relaxation rates of Mn12 acetate at different fields
Landau-Zener effect
m 'm
tW
EEW mm
'
m m
m'm 'm
'm 'm
m
E
E22 WEE
Transition probability
Size of magnetization step
2/2eP
P1
TítuloResonant spin tunneling on MM
22
Energy released ∆E Ignition (barrier overcoming) ∆U
Thermal diffusion kCharacteristic length of propagation δ
Two important characteristic timescales:
• Thermal diffusion
• Burning timescale
MetastableState
∆ U
Deflagration is a technical term describing subsonic combustion that usually propagates through thermal conductivity
∆ EStableState
22
τb= τd Deflagration Flame width
What is a deflagration?
23
ManganitesField jumps 1999Deflagration-like description 2007
Intermetallic compoundsField jumps 2002Deflagration-like description 2010
Molecule magnetsField jumps 1999Deflagration-like description 2005
23
From magnetization jumps to magnetic Deflagration (MD)
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
First evidences of MD
Avalanche ignition produced by SAW:
IDTLiNbO3
substrate
Conducting stripes
Coaxial cable
Mn12 crystalc-axis
Hz
Coaxial cable connected to an Agilent microwave signal generator
Change in magnetic moment registered in a rf-SQUID magnetometer
Surface Acoustic Waves (SAW) are low frequency acoustic phonons
(below 1 GHz)
Quantum magnetic deflagration
• 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
Quantum magnetic deflagration
Quantum Magnetic Deflagration
Quantum Magnetic Deflagration
29
Associated to magnetic avalanches: magnetoresistive avalanches in manganites
I
t
τ1
Luminescence
Superradiance
I
t
τSR
This kind of emission (SR) has characteristic properties that make it different from other more common phenomena like luminescence
L
λL ~ λ
Superradiance
Proposed by Robert H. Dicke in 1954.
NI
2NI
N is the number of dipoles
All spins decay to the fundamental level coherently, with the emission of photons.
-10
-9
-8-7
-6-5
-4-3-2-1012
34
56
7
8
9
10
B = 2B0
Superradiance
Superradiance??
Sharp peak shows a signal which isequivalent to the sample being at 20 K (the expected self heating is about 3 K).
Magnetic deflagration in pulsed fields
Δt
dB/dt (kT/s)
1.61.92.53.23.74.87.0
200 400
0
2
4
6
dM
/dt
t (s)
coil 1 coil 2
1000 2000 3000 4000 5000 6000 7000-2
0
2
4
6
8
10
12
14
16
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
)
dHz/dt (T/s)
Quantum magnetic detonation
t
-0.010 -0.005 0.000 0.005 0.010 0.015 0.020 0.025
0.00
0.02
0.04
0.06
0.08
0.10 Scaling with fixed HR
Mn12
-Ac powder in araldite
1104 T/s 1321 T/s 1725 T/s 2132 T/s 3113 T/s
G:s
cale
dFix
edH
R in
G:\M
olec
ular
Mag
nets
\Mn1
2\P
ulse
d fie
lds
(KU
Leuv
en)\
Mn1
2_B
+C(p
aper
) M
N12
BD
N /
M
N12
BG
N /
M
N12
BIN
/
MN
12B
JN /
M
N12
BM
N
r1/2 d
M/d
H
(H - HR) / r1/2
1.2 1.4 1.6 1.8 2.00.00
0.05
0.10
0.15
0.20
0.25
0.30 1066 T/s 1270 T/s 1660 T/s 2050 T/s 2970 T/s
dM/d
B (a
.u)
B(T)
Superradiance: indirect evidence
r
Hr
dH
dM~
Scaling
• 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!!
Future
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).
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