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).