Classical and quantum nonlinear phenomena in Mn

Román LÓPEZ-RUIZ

ICMM’08 – Florencia. 24th _{September 2008}

nonlinear phenomena

Classical and quantum

nonlinear phenomena

in Mn

single molecule magnets

The degree of “quantumness” can be characterized by the ratio:

Ω₀

A practical example: magnetic moment with anisotropy

J. L. García-Palacios and S. Dattagupta PRL 95, 190401 (2005)

λ >> 1 ⇒ classical

λ << 1 ⇒ quantum

0 0

/

Ω

≡

τ

λ

=

λ

An absorption energy spectrum:

Ω

“Rest of the world”

E

0

/

τ

Nonlinear susceptibility of superparamagnets: classical theory

J. L. García-Palacios and P. Svedlindh, PRL 85, 3724 (2000) J. L. García-Palacios and D. A. Garanin, PRB 70, 064415 (2004)

0

τ

τ

λ

≈

τ

τ

The dynamical

nonlinear susceptibility depends on:

How to Measure de Nonlinear Component? ... 5 5 3 3

1 + + +

= H H H

M χ χ χ

... ) ( 5 ) ( 3 ) ( )

(ω = χ₁ ω + χ₃ ω H2 + χ₅ ω H4 +

χ

Ac susceptibility as a function of external magnetic field

Beyond the linear response limit

I. FITTING RESPECT A MAGNETIC FIELD (H=0)

∝ χ₁(ω) h₀

∝ χ₂(2ω) h₀2

m 3

(

ω

)

t i m e

-1.0 -0.5 0.0 0.5 1.0 a c m a g net ic f iel d time Large ac magnetic field

∝ χ₃(3ω) h₀3

m 2 ( ω ) m 1 ( ω )

10-4 10-3 10-2 10-1 100 101 102 103 104

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

ωτ

redu

ced

a

c

su

sce

pti

b

ilit

y

bc tetragonal crystals Alligned easy axes

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

1/T (K-1)

τ

(s)

τ₀ = 2 10-8 s

Experiments on Mn₁₂ (SLOW specie)

S = 10

Ω₀ ≈ 14 K

U₀ =70 K _τ

0 ≈ 3×10-8 s

≈ 3 mK

0

/

τ

=

-45 -30 -15 0 15 30 45 60 75

10-3 10-2 10-1 100 101 102 103 0.0 0.2 0.4 0.6 0.8 1.0 Classical predictions χ 3 /

χ 3T

(a)

T=5 K

χ 3

/

χ 3T

ωτ

Mn

12

(b)

H parallel to anisotropy axes

(

)

(

)

⊥ 2 2

4 2 4 3 3 sin cos cos 1 2 1 cos ) 0 ( ) ( g g i i i T ωτ ωτ ωτ χ ω χ Transverse contribution (dominant for classical spins)

g_⊥∝ ∂2τ_/∂ _B

⊥2∝ 1/λ

Longitudinal contribution (dominant for Mn₁₂ )

g_⎪⎪∝ ∂2τ_/∂ _B

⎪⎪2

• Large contribution to nonlinear dynamical susceptibility not expected in classical Physics:

“Quantum nonlinearity”

Classical ∂2τ_/∂ _H2 _{< 0}

Quantum tunnelling ∂2τ/∂H2 _{> 0}

The large nonlinear response is directly

linked with the existence of tunnelling

It reflects the fragility of quantum tunnelling

to external bias Explanation: “quantum non-linearity”

F. Luis et al, PRL 92, 107201 (2004)

R. López-Ruiz et al, PRB 72, 224433 (2005)

(

)

⎥

⎥

_⎦

⎤

⎢

⎢

⎣

⎡

∂

∂

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

+

+

=

0 500 1000 1500 2000 -1x10-5

-5x10-6 0

H

z(Oe)

χ 3

(emu/Oe

3 mol

)

• The quantum nonlinearity is rapidly supressed by a external bias

• Resonant tunnelling becomes a “source of harmonics” at every resonant field

0 4 8 12 16

-10000 -5000 0 5000 10000 -3

-2 -1 0 1 χ 1

' (em

u

/O

e mol)

χ 2

(1

0

-3 emu/

Oe

2 mol

)

H_z (Oe) -20

-10 0 10 20

M

(

μ B

/m

olecule)

0 10 20 30

0.1 1 10 100 1000 0 10 20 30 χ (em u /mo l Oe)

χ

''

χ

'

ω/2π (Hz)

0.0 1.0x10-9 2.0x10-9 3.0x10-9 4.0x10-9

0.1 1 10 100 1000 0.0 4.0x10-11 8.0x10-11 1.2x10-10 1.6x10-10 2.0x10-10 χ

' 3

(e mu /g Oe 3 ) χ

' 3

(e

mu

/g

Oe

3 )

ω/2π (Hz)

QT

No QT

Suitable method to detect and quantify quantum behaviour

LINEAR NONLINEAR

Applied field

H 1500 Oe H=0 Oe

≥

Usually 2-5 % of the molecules in a crystal are “fast”

However, under special crystallization conditions, Mn₁₂ bz contains up to 98% of them

“Fast relaxing” Mn₁₂ clusters

The degree of “quantumness” can be characterized by λ

10-3 10-2 10-1 100 101 102 -4 -2 0 2 4 6 8

ωτ

/

10-3 10-2 10-1 100 101 102 -45 -30 -15 0 15 30 45 60 75 χ 3 /

χ 3T

ωτ

10 times

smaller !!!

/

ωτ

≈ 1.5 K

U₀ =38 K

τ₀ ≈ 3×10-11 _s

Ω₀ ≈ 7.2 K U₀ =70 K

Ω₀ ≈ 14 K

≈ 3 mK

τ₀ ≈ 3×10-8 _s

0

/

τ

=

/

τ

=

λ ≈ 0.2 “Slow”

λ ≈ 2×10-4

0 0

/

Ω

≡

τ

λ

=

10 nm

Outer shell: apo-ferritin

Gaussian distribution: 〈D〉=7.8(3) nm ⇒ 4500 Fe atoms per protein molecule

σ = 0.13

0 2 4 6 8 10 12 14

0 5 10 15

20 sample 1

sample 2

distr

ibut

ion (

%

)

D (nm)

Inner core: stores and delivers iron

Ferrihydrite

ferritin

S ≈ 110

Ω₀ ≈ 14 k_B

λ ≈ 2×10-4

U ≈ 220 k_B

Ω₀ ≈ 4 k_B

τ₀ ≈ 10-12 _s λ ≈ 10

0 10 20 30 40 50 -1x10-10

-8x10-11 -6x10-11 -4x10-11 -2x10-11 0

χ 3

(emu/g Oe

3 ) 1 Hz

10 Hz 90 Hz 240 Hz

Equilibrium

Mn₁₂

S = 10

The ferritin: towards a classical world

Equilibrium

Equilibrium

≈ 3 mK

0

/

τ

=

≈ 48 K

0

/

τ

• |χ₃ | is smaller than |χ_3T | at any T and ω

• |χ₃ | decreases with increasing ω

Classical behaviour

0 10 20 30 40 50

0.0 0.2 0.4 0.6 0.8 1.0 1.2

χ 3

/

χ 3T

T(K)

Experiments compatible with overdamped dynamics

λ > 1 The ferritin: towards a classical world

Equilibrium

T Equilibrium ω

•

T

he

nonlinear

response contains

a

quantum contribution: “quantum nonlinearity”

The nonlinear response is sensitive to the degree of “quantumness” of a nanomagnet

10-4 10-3 10-2 10-1 100 101

0.1 1 10 100

χ

(

ωτ

=1)/

χ

Classical Mn₁₂ SR

Mn₁₂ FR

ferritin

• Continuous transition between quantum and classical limits by increasing λ

“Quantumness”

0 0

/

Ω

≡

τ

λ

=

Román López-Ruiz

Instituto de Ciencia de Materiales de Aragón CSIC & Universidad de Zaragoza

[email protected]

T

hank

Y

ou

V

ery

M

uch

M

olte

G

razie

M

uchas

G

racias

The head: Dr. Fernado Luis

Theory: Dr. Jose Luis García-Palacios

Chemistry: Dr. Angel Millán Dr. Kunio Awaga Dr. Keiji Takeda

Biochemistry Dr. M. Martinez-Júlvez Dr. Gomez-Moreno

The financial support of:

Molecular clusters Natural ferritin Co nanoparticles

S = 10 S ≈ 100 S ≈ 1000

Mn(CH₃ COO)₂

II

in CH₃ COOH

+ KMnO₄

VII

Mn₁₂ Acetate

III,IV

Mn₁₂ Benzoate

III,IV

Mn₁₂ Acetate

III,IV

in CH₂ Cl₂

+ C₆ H₅ COOH (excess)

hexane Mn₁₂ Benzoate

III,IV REDOX

(comproportion)

Ligand exchange

Dr. A. Millán

Instituto de Ciencia de Materiales de Aragón CSIC Universidad de Zaragoza

Dr. K. Takeda Hokkaido University Dr. K. Awaga

Nagoya University

FAST

SLOW

R -CH3 -C₆ H₅

R R

R

R

R R

Calculated: classical

•Large nonlinear response

•Opposite signs!!!

Experimental

-2 -1 0 1

10-2 10-1 100 101 102 -1

0 1 2

λ = 0.03

λ = 0.1

λ = 1

χ

'/

χ

(a)

χ

"/

χ

ωτ

0 20 40 60 80

10-2 10-1 100 101 102 -40

-20 0 20 40

χ

'/

χ

(b)

Mn₁₂ acetate powder

ωτ

χ

"/

0 500 1000 1500 2000 -1x10-5

-5x10-6 0

H

z(Oe)

χ 3

(emu/Oe

3 mol

)

• The quantum nonlinearity is rapidly supressed by a external bias

• Resonant tunnelling becomes a “source of harmonics” at every resonant field

0 4 8 12 16

-10000 -5000 0 5000 10000 -3

-2 -1 0 1 χ 1

' (em

u

/O

e mol)

χ 2

(1

0

-3 emu/

Oe

2 mol

)

H_z (Oe) -20

-10 0 10 20

M

(

μ B

/m

c

B

Ψ

g_⊥= 0 ⇔ λ ≥ 1

τ_Φ ≈ 10-3 _τ 0

Decoherence takes place faster

than dissipation

Quantum longitudinal contribution

g

∝ ∂

τ/∂

_B

⎪⎪2

Classical transverse contribution

g

∝ ∂

τ

_/

∂

_B