Quantum Nanomagnetism

Proceedings of the International Conference Nanomaterials: Applications and Properties

QUANTUM NANOMAGNETISM

September 2012

Javier Tejada, Dept. Física FonamentalUniversitat de Barcelona

Contenidos

Introduction to magnetismSingle Domain ParticlesQuantum relaxation: 1990-96Resonant spin tunneling: 1996-2010Quantum magnetic deflagrationSuperradianceFree rotorsMagnetic vorticesQuantum effects in type-I superconductors

Content

• Electrostatic interaction + Quantum Mechanics

Overlapping of wave functions12

2

re

12

2

re

0SIs different for 1Sand

Term ji ss In the HamiltonianHeisenberg hamiltonian

Introduction to magnetism

eThe 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 contributionse

pμtotal = μorbital + μspin

μorbital

e

Introduction to magnetism

TítuloIntroduction to magnetism

0S 1S

21ˆˆˆ ssJeff

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 calculed:The system always tends to be at

the lowest energy state::

The overlapping of the wave functions decays exponentially.

Summation over nearest neighbours

CTJ ~

TítuloIntroduction to magnetism

Existence of metastable states

Time dependent phenomena

Magnetic hysteresis

Slow relaxation towards the free energy minimum.

Global thermodynamic

equilibrium.

Non-linear effects.

• Permanent magnets divide themselves in magnetic domains to minimize their magnetic energy.

TítuloSingle domain particles

)(nmaE

E

an

ex

• There are domain walls between these domains:

Exchange energy

Anisotropy energy

Lattice constant

exE

anE

a

TítuloSingle domain particles

53 1010 an

ex

E

E

0)exp( T

Eex

ctSTT c

TipicallyThe exchange energy is so high that

it is difficult to do any non-uniform

rotation of the magnetization.

RIf 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:

cex TE and

Single domain particles

• Relativistic origin:

– Order of magnitude , with p even.

• Classic description:– Energetic barrier of height:

p

cv

U

VkU Anisotropy constant

Volume

T

HU

e)(

The rotation of M as a whole needs certain energy called magnetic anisotropy.

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


Hard axis

Important aspects of SDPs:

• Volume distribution:

• And orientations:

• Their magnetic moments tend to align with the applied magnetic field.

UfVfRf


• 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.

00 ln1

t

SMM

Magnetic viscosityTS


Magnetic solids (ferromagnets) show hysteresis when an external magnetic field is

applied:

MR ~ Memory

H

M

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

TítuloQuantum relaxation: 1990-96

Magnetic viscosity

dependance on T, for low

T, of a TbFe3 thin film

Magnetic viscosity

variation with respect

to the magnetic field.

TítuloQuantum relaxation: 1990-96

|M| ~ μB

|M| » μB

Quantum

Classic

Empirically, the magnetic moment is considered in a quantum way if|M| ≤ 1000μB

Resonant spin tunneling on molecular magnets• Identical to single domain particles• Quantum objects

kijkBji MiMM 2],[

kBijjiji MMMMMMM ],[

M(H,T) univocally determined by D and E22xzA ESDSH

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.

SH

||

Resonant spin tunneling on molecular 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 magnetic resonance


TítuloResonant spin tunneling on molecular magnets

-10

-9

-8

-7

-6-5

-4-3-2-10 1 2 3

45

6

7

8

9

10

B=0Magnetic field

Mag

netiz

atio

n


-10

-9

-8

-7-6

-5-4

-3-2-10 1 23

45

6

7

8

9

10

B = 0.5B0

Magnetic field

Mag

netiz

atio

n


B = B0

-10

-9

-8

-7-6

-5-4

-3-2 1 23

45

6

7

8

9

10 Magnetic field

Mag

netiz

atio

n


-10

-9

-8

-7-6

-5-4-3

-2-10 1 23

45

6

7

8

9

10

B = 2B0

Magnetic field

Mag

netiz

atio

n


Título

• After a certain time, the relaxation becomes exponential:

• Peaks on the relaxation rate Γ(H) at the resonances:

tHtMtM eq exp1


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 Acustic 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



– All spins decay to the fundamental level coherently, with the emission of photons.

-10

-9

-8

-7

-6-5

-4-3-2-10 1

23

4

5

6

7

8

9

10

B = 2B0

Superradiance

I

t

τ1

Luminiscence

Superradiance

I

t

τSR

This kind of emission (SR) has characteristic propierties that make it different from other more common phenomena like luminiscence

L

λ

L ~ λ

Superradiance

TítuloMilestones

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)

TítuloMilestones

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)

TítuloMilestones

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)

TítuloMilestones

1996 Mesoscopic tunnelling of magnetization (22)

1997 Semiconductor spintronics (23)

© 2008 Nature Publishing Group

Magnetic Vortices (MV): IntroductionVortex State• The spin field splits into two well-differentiated

structures: 1) the vortex core consisting of a uniform out-of-plane spin component (spatial extension about the exchange length) and 2) the curling magnetization field (in-plane spin component), characterized by a non-zero vorticity value.

• The application of an in-plane magnetic field yields the displacement of the vortex core perpendicularly to the field direction.

• The vortex shows a special vibrational mode (called gyrotropic mode) consisting of the displacement of the vortex core as a whole, following a precessional movement around the vortex centre. Its characteristic frequency belongs to the subGHz range.

Experimental set-upWe have studied several arrays of permalloy (Fe19Ni81) disks with diameter 2R = 1.5 μm and different thickness (L = 60, 95 nm) under the application of an in-plane magnetic field up to 0.1 T in the range of temperatures 2-300 K. Samples were prepared stacking four 5x5 mm2 arrays with parallel sides.

MV: Magnetic Characterization

Hysteresis loop of a sample with thickness L = 95 nm.

The vortex linear regime in the ascending branch should extend from H=-300 Oe to 500 Oe (at least) in the L = 95 nm sample. Analogous behaviour has been observed in the other sample.

a) ZFC-FC curves for an applied magnetic field H = 300 Oe (L = 95 nm). b) Isothermal magnetic measurements along the descending branch, Mdes(H), from the SD state (H = 0.1 T) to H = 300 Oe (L = 95 nm).

MV: Relaxation ExperimentsRelaxation mesurements of magnetic vortices from the metastable states of the descending branch at H = 0 to the equilibrium state (which corresponds to M=0) for both samples.

Normalized relaxation measurements for samples L = 60 nm (left) and L = 95 nm (right). M0 corresponds to the initial point of the relaxation.

MV: Viscosity and conclusions

Magnetic viscosity S versus temperature T for both samples.

Conclusions:

• Logarithmic time dependence of the magnetization broad distribution of energy barriers in our system.

• Thermal activation of energy barriers dies out in the limit T 0. 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.

• The presence of structural defects in the disks could be a feasible origin of the energy barriers. We consider them to be capable of pinning the vortex core, when the applied magnetic is swept, in a non-equilibrium position.

Below T = 6 K, magnetic viscosity reaches a plateau with non-zero value.

Type I SC: Topological hysteresis

Flu

x pe

ne

tratio

n: b

ub

ble

s Flu

x e

xpu

lsio

n:

lam

ella

e

There is a GEOMETRICAL BARRIER controlling both the penetration and the expulsion of magnetic flux in the intermediate state. It is the responsible for both the intrinsic irreversibility of a pure defect-free samples and the formation of different flux patterns.

Different flux structures appear in the Intermediate state depending on the magnetic history: Tubes/bubbles are formed during magnetic field penetration and laberinthic patterns appear upon expulsion

Type I SC: Magnetic relaxation

Relaxation mesurements of a Pb sample from the metastable states of the descending branch for different reduced fields h in steps of 0.027 at T=2 K (a) and at different T for the fixed reduced field h=0.10 (b).

We observe a perfect logarithmic time dependence of m(t) for several T and h:

This law holds for any system having a broad distribution of energy barriers as the source of its metastability

Quantum tunneling of N-SC interfaces

Left figure: Temperature dependence of the magnetic viscosity for different reduced fields (a) and reduced magnetic field dependence of S at different T (b). Right figure: Reduced magnetic field vs. T phase diagram showing the different dynamical regimes.

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Quantum Nanomagnetism