Scaling Relations And Magnetic Properties Of Nanoparticles

Universidade Federal do Rio de Janeiro Instituto de Física

Scaling relations and magnetic properties of nanoparticles José d’Albuquerque e Castro

PAN AMERICAN ADVANCED STUDIES INSTITUTE Ultrafast and Ultrasmall; New Frontiers and AMO Physics March 30 - April 11, 2008

Magnetic storage media: thin films ⇒  polycrystalline alloys of Co, Cr, and Pt with B or Ta ⇒  segregation of the non-magnetic elements: reduction of the coupling between grains ⇒  grain size ≈ 10 nm ⇒  areal density: 50 Gbit in-2 ( year 2000)

net magnetization

Problem: superparamagnetism limit 100 Gbit in-2

Superparamagnetism m

magnetic particle

energy barrier EB

Superparamagnetism m

magnetic particle

energy barrier EB

Superparamagnetism m

⇒  energy barrier for reversal: EB = KV volume: V net anisotropy: K ⇒  rate of switching: 1/τ = f0 exp(- EB/kT) f0 ≅ 109 s-1 storage time ≈ 10 years ⇒ EB ~ 40 kT

Patterned thin films Nanoimprint lithography ⇒  arrays of circular nanomagnets ⇒  cylinders with diameter D and height H 5 nm ≤ H ≤ 150 nm 50 nm ≤ D ≤ 500 nm

Lebib et al., JAP 89, 3892 (2001)

Patterned thin films

⇒  grains within particles are strongly coupled ⇒  particles cannot accommodate a domain wall ⇒  patterned thin films as recording media 200-260 Gbit in-2 (periodicity 50 nm) 1 Tbit in-2  periodicity 25 nm

R. P. Cowburn et al., Phys. Rev. Lett 83, 1042 (99)

A. Lebib et al., J. Appl. Phys. 89, 3892 (01) MFM vortex

single-domain

R. P. Cowburn et al., Phys. Rev. Lett. 83, 1042 (99) experimental phase diagram

Configurations I

in-plane

II

out-of-plane

III

vortex

core

Theoretical approach

Eij = classical dipolar interaction µi = magnetic moment on site i Jij = exchange coupling K = anisotropy constant

µi

{α1,α2,α3} = angles between µi and the principal (easy) crystalline axes

configuration of lowest energy ?

For smaller systems ( e.g. N ~ 106) ⇒ single domain configurations only Reason: strength of the exchange interaction J

Scaling approach J. d’A. e C., D. Altbir, J. C. Retamal, and P. Vargas, Phys. Rev. Lett. (2002)

J’ = x J

with x < 1

x = 0.04

x = 0.04 x = 0.06

x = 0.04 x = 0.06 x = 0.08

x = 0.04 x = 0.06 x = 0.08 x = 0.10

triple point

Ht (x) = Ht (1) xη

with η = 0.55

Scaling relations J’ = x J From the relation

Ht (x) = Ht (1) xη

⇒

Ht (1) = Ht (x)/ xη

we expect to find

H(1) = H(x) / xη D(1) = D(x) / xη

Truncated conical particles

C. A. Ross et al., J. Appl. Phys. 89, 1310 (2001) C. A. Ross et al., Phys. Rev. B 62, 14252 (2000)

Calculation for Ni particles ζ = 0.3 x = 0.010 x = 0.015 x = 0.020

J. Escrig, P. Landeros, J.C. Retamal, D. Altbir, and J. d’A. C., Appl. Phys. Lett. (2003)

Calculation for Ni particles ζ = 0.3 η = 0.55

J. Escrig, P. Landeros, J.C. Retamal, D. Altbir, and J. d’A. C., Appl. Phys. Lett. (2003)

Important point: value of η “Scaling relations for magnetized nanoparticles” P. Landeros, J. Escrig, D. Altbir, D. Laroze, J. d’A.C., and P. Vargas Phys. Rev. B 71, 094435 (2005)

  value of η may depend on the model for the magnetic configurations!

model for the core

 vortex core

Fast Monte-Carlo method Could we combine the scaling technique and the Monte-Carlo method? “Fast Monte-Carlo method for magnetic nanoparticles” P. Vargas, D. Altbir and J. d'Albuquerque e Castro Phys. Rev. B, 73, 092417 (2006)

Single domain limit: size L for which the flower and vortex states have equal energy.

http://www.ctcms.nist.gov/~rdm/mumag.org.html

Fast Monte-Carlo method µMag group (NIST)

L  8 lex

where

lex = (2A/µ0 M0²)1/2

A = exchange stiffness constant α J Co:

lex = 3.6 nm

a0 = 0.2 nm

J0 =2.35x103 kOe/µB

 L  29 nm and N  3x106 atoms !

Fast Monte-Carlo method For given L’ (small): Monte-Carlo  J’

x = J’/J0

? 

L = L’/ xη

Calculations: L’1 = 5 a0  x1 = 2.240×10-3 L’2 = 7 a0  x2 = 4.125×10-3 L’3 = 9 a0  x3 = 6.509×10-3



η

η

η

Fast Monte-Carlo method Results L = 8.02 lex η = 0.551

Conclusion It is possible to combine the Monte-Carlo method with the scaling technique  fast and reliable method for investigating the equilibrium properties of magnetic nanoparticles