Quantum Nanomagetism (USA, 2011)

Universitat de Barcelona 1990-2010: quantum

magnetismFall 2011

Javier Tejada, Dept. Física Fonamental

Contenidos

Introduction to magnetismSingle Domain ParticlesQuantum relaxation: 1990-96Resonant spin tunneling: 1996-2010Quantum magnetic deflagrationSuperradiance

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

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

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 mollecular magnets• Identical to single domain particles• Quantum objectsObjetos cuánticos

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 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 magnetic resonance

-Sz

Sz

Sz -Sz


TítuloResonant spin tunneling on mollecular 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


• TB depends on measuring frequency

0

0

/1ln VK

TB

A.C. measurements

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

-4-3-2-1012

34

56

7

8

9

10

B = 2B0

Superradiancie

I

t

τ1

Luminescence

Superradiancie

I

t

τSR

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

L

λ

L ~ λ

Superradiancia

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

cv

1

Shift on frequency due to relative velocity between emitter and observer (non relativistic case):

Frequency seen by the observer Frequency of

the emitter

cv

Relative velocity

Linear 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

Rotational Doppler Effect

EPR Results

In

HFMR 0

In

H n

0

IIHHH

Bnn 2

2

1

B 2

OeH 5.2~ measured particles 1~by produced nmr


HnnI

LLE Bn 1

21

TkE Bn ~

HE

nB

n

~2

2/1

~

HTk

nB

B

KT 2~

mKHB 17.0~100n

Occupied states


• 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


• 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


• 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 interactionsNMR:

ESR and FMR:


Magnetic vortices are bi-dimensional magnetic systems whose magnetic equilibriumconfiguration is essentially non-uniform (the vortex state): the spin field splits into twowell-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 yieldsthe displacement of the vortex core perpendicularlyto 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’), consistingof 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

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

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

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

Conclusions1) 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

Theoretical modelingRigid 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 modulusof 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

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 haveand 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

Quantum Nanomagetism (USA, 2011)

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