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