Kalliopi Trohidou

Computational Condensed Matter Physics Group

IAMPPNM, Department of Material Science, NCSR “Demokritos”,

Athens, Greece

Modeling of Bi-Magnetic Nanoparticle

assemblies

IMS

Understanding of Physical Properties

Finite size effects

Surface and interface effects

Interparticle Interactions effects

Applications

Magnetic recording media

Biomedical applications

Magnetic Behaviour of Nanoparticles

Outline of the talk

Introduction

Core/shell nanoparticles, basic characteristics

Interparticle interaction effects in assemblies

Granular Solids

Concluding remarks

Dilute Dipolar Interactions

Dense Interplay Dipolar + Exchange

Interactions

Magnetic Materials

Ferromagnets

Magnetic Domains

Energy Minimisation

Nanoparticles are single domain

N

Μ= Σ mi

i=1

Critical size : for a spherical Fe nanoparticle radius

~15 nm

How big?

100 atoms Co – 1.3 nm diameter

500 atoms Co – 2.2 nm diameter

1000 atoms Co – 2.8 nm diameter

5000 atoms Co – 4.7 nm diameter

Co-fcc structure

Finite Size Effects

Τcurie is not defined

Specific heat has a finite value at Τcurie

Behaviour of Finite Systems

Nanoparticles Assemblies

Langevin Function

μ=μ L(μ.Η/ΚΤ)=cot(μ.Η)-ΚΤ/μ.Η

For μ.Η << ΚΤ

μ=μ2 Η/ 3ΚΤ

For μ.Η >> ΚΤ

μ=μ(1-ΚΤ/μΗ)

The ensemble becomes superparamagnetic above the

Blocking Temperature (TB)

μ>> μatomic

Superparamagnetism

Α. Modified Bulk Properties •Temperature dependence of the Magnetization

•Higher anisotropies (at least an order of magnitude) than the bulk materials

•Different anisotropy strength and type on the surface because of the reduced

symmetry

Influence on the Coercive field

Β. Properties Non-observable in bulk materials •Finite magnetization in antiferromagnetic particles.

•High coercive fields in antiferromagnetic particles

•Shifted Hysteresis Loops and High coercive fields on oxide coated particles

Magnetic Nanoparticles

“Monte Carlo studies on surface and interface effects of magnetic nanoparticles”

Chapter in ‘Surface Effects in Magnetic Nanoparticles’. Editor D. Fiorani,

Springer Verlang Series on Nanostructure Science and Technology (2005)

Storage Densities on Single

Particles

20 nm

600 Gb/in2

10 nm

2.4 Tb/in2

Unstable at room

temperature

< 5 nm

>10 Tb/in2

Required

Scientific challenges

Storage medium (hard disk) Areal density > 100 Gbits/in2

Need to overcome magnetic instability

(superparamagnetic limit) – high anisotropy

Ordered arrays (self-organisation)

Calculated SAR values as a function of size for nanoparticles of maghemite (currently available) Fe and

FeCo (available by gas-phase synthesis) for different values of the anisotropy constant K for an applied

field amplitude, H, of 4850 Am-1 and a frequency, f, of 100 kHz.

Specific Absorption Rate (SAR) of MNPs

beyond the current state

Requirement of High Anisotropy and High Mr

Exchange anisotropy

The interaction between the ferromagnetic (FM) core and the antiferromagnetic

(AF) shell, leads to a unidirectional anisotropy which is called exchange

anisotropy.

The exchange anisotropy results in :

1. Enhanced coercivity.

2. Shifted hysteresis loops after a field cooling procedure.

3. Reversal in the temperature dependence of coercivity with particle size

Core/Shell Nanoparticles

Assymetric Hysteresis Loops

Exchange Field Coercive Field

Hex= (H1-H2)/2 Hc= (H1+H2)/2

Atomic-scale modeling

Ferromagnetic Core

and an

Antiferromagnetic

shell

, ,

,

( )

( )

FM i j iFM i i i jSH

i j FM i FM i j SH

iSH i i IF i j i

i SH i FM j SH

Ε J S S K S e J S S

K S e J S S H S

i

²

²

Energy minimization method

Metropolis Monte Carlo

System Morphology

Metropolis Monte Carlo Algorithm

Ei

Ej

P~exp [- β(Ej-Ei)]

β=1/KT

MC simulations correspond to the canonical ensemble

Numerical Modelling of magnetic nanoparticles

with core/shell morphology in an atomic scale

Origin of the exchange bias effects

Effect of the core/shell ratio

Critical shell thickness for the appearance of Hex

Effect on the exchange bias properties of JSH and JIF

Origin of the vertical (DM) shift

Training Effect

Aging Effect

Phys. Rev B71, 134406 (2005)

Phys. Rev B74 , 144403 (2006)

Modern Phys.Lett. B21, 1169(2007)

J. Phys. Cond.Matt.19, 225007(2007)

Phys. Stat. Sol. (A), Vol. 205, 1865(2008)

Modification of the magnetic and blocking behavior

with the interparticle distance due to interparticle

interactions

To study the effect of the interparticle dipolar and exchange interactions,

on the coercive (HC) and the exchange bias field (Hex) in granular solids

consisted of nanoparticles with FM core/AFM shell morphology

Magnetic Behavior of Assemblies of Nanoparticles

Aim of the Work

System Morphology

Multiscale Modeling :

Beyond the single-spin (SW) model

AFM Shell (2 sublattices)

AFM Interface (2 sublattices)

FM Interface

FM Core

Atomic Mesoscopic Assembly

Scaling parameters for the magnetisation in

Atomic-scale modeling

AFM Shell (2 sublattices-2 spins)

AFM Interface (2 sublattices-2 spins)

FM Interface

FM Core

Numerical Modeling

AFM Shell

AFM Interface (alloying with the FM interface)

FM Interface (alloying with the AFM interface)

FM Core

PR B 71, 134406 (2005)

Numerical Modeling

AFM Shell (2 sublattices)

AFM Interface (2 sublattices)

FM Interface

FM Core

Magnetic state of a nanoparticle : Four regions and six spins

E (Intra-particle) = Anisotropy + Exchange + Zeeman

∑ ˆˆ-∑ ˆˆ-∑ ˆˆ

2

ii

iii

iii

HmhemJemKE

e

H

1S2S

3S

5S

6S

4S

Numerical Modeling

(Dilute Assemblies)

E (Inter-particle) = Dipole-Dipole Interactions (DDI)

Granular

film

∑

ˆˆˆˆ3-ˆˆ3

iij

ijjijiji

R

rmrmmmgE

Total Energy

Total E = E (Intra-particle) + E (Inter-particle)

Energy Minimisation by Metropolis Monte Carlo algorithm

The magnetic field and the anisotropy constant are expressed in units JFM/gμΒ

and the temperature in units JFM/k.

JFM=1

JAFM=0.5

JFM=0.5

KIF=0.5

KAFM=1.0

KFM=0.1

Exchange parameters Anisotropy parameters

Ms2/K

Dipolar Strength

System initially at positive saturation

Positive external field reduced gradually

Magnetization recorded at H=0 : Remanence

Negative applied field reduced gradually

Field recorded when M=0 : Coercivity

Modeling of Isothermal Hysteresis

-8 -6 -4 -2 0 2 4 6 8

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

c Hc

10% 1.0599

5% 0.4953

1.5% 0.3226

Co/Mn

c=10%

c=5%

c=1.5%

M/M

s

H(JFM

/gµB)

MC simulations on the Hysteresis behaviour of FM core and

AFM shell assemblies of nanoparticles

-2000 0 2000

-0,1

0,0

0,1

Nanospin Samples

T = 5K

VFF=9,8%

VFF=4,7%

VFF=1,5%

m/

ms

at (

a.u

.)

H (T)

Co in Mn

Increase of Hc with VF

Co in Mn:

J. Phys D Appl. Phys.(2011)

C=4.7%

i j

ij

jiijjiji

iii

R

mmRmRmgemkHmhE

3

2

1

ˆˆˆˆˆˆ3ˆˆˆˆ

Single-spin nanoparticle assemblies

Granular film

Energy parameters Dipolar

Anisotropy

Zeeman

g=m2/D3

k=K1V

h=mH

Mesoscopic scale model

Coherent rotation model (Stoner-Wolfarth)

-8 -6 -4 -2 0 2 4 6 8

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

M/M

s

H(J)

Co/Mn

c=10%

c=5%

c=1.5%

Co/Ag

c=5%c Hc Hex

10% 1.0599 0.2020

5% 0.4953 0.08702

1.5% 0.3226 0.0379

-4 -2 0 2 4

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

M/M

s

H(JFM

/gµB)

Co/Mn

Co/Ag

C=5%

Comparison of Co/Mn and Co/Ag Hysteresis Behaviour

0 20 40

0,0

0,1

0,2

0,3

0,4

0,5

Hc

Hex

H

ex (

JF

M/g

µB)

H

c(J

FM/g

µB)

T(J/KB)

Coercivity and Exchange Bias

Temperature dependence

C=5%

Exponential decay of Hex and Hc with temperature for the

core/shell nanoparticles

0 20 40

0,0

0,1

0,2

0,3

0,4

0,5

Co/Mn

Hc

Hex

Co/Ag

Hc

He

x(J

FM/g

µB)

H

c(J

FM/g

µB)

T(J/KB)

Exchange Bias

Thermal dependence of Hc Hex in Co/Mn

5 10 15 20 25 30 350

500

1000

1500

2000

2500

Co@Mn

Hc ZFC

Hc FC

Heb

Co@Ag

Hc ZFC

HC(O

e),

HE

B(O

e)

T (K)

Hi (T) = Hi0·e (-T/T0

)

Hi0 T0

Hc ZFC (2550 50) Oe (11.2 0.3)

Hc FC (3690 40) Oe (10.1 0.1) K

Hex (2030 75) Oe (5.0 0.1) K

Co/Mn

C=4.7%

Trainning Effect in Hex

~45% decrease after

the first loop, in

agreement with

experiments on

Co/Mn

0 1 2 3 4 5 60

200

400

600

800

1000

He

b (

Oe

)

n (number of cycles)

T = 5 K

0 1 2 3 4 5 6 70,00

0,02

0,04

0,06

0,08

0,10

n ( number of cycles)

He

x(J

FM/g

MC Simulations EXPERIMENT Co/Mn

C=4.7% C=5%

Co in Mn: ZFC/FC Measurements:

Superspin glass behavior under Tg = 62 K (TN (Mn)= 95 K)

0 100 200 3000,0

3,0x10-7

VFF= 9,8%

VFF= 4,7%

VFF= 1,5%

(

em

u/O

e)

T (K)

H = 100 Oe

Co in Mn TMax

S1 9.2% 64 K

S2 4.7% 65 K

S3 1.3% 58 K

Superspin glass behavior of the FC;

Random freezing of the magnetic moments of

the nanoparticles

System initially at high-T and H=0 (M=0)

Temperature lowered gradually to zero limit

(Ground state)

Weak field applied & temperature raised gradually

- MZFC (T)

Field remains & temperature lowered gradually

– MFC(T)

Modeling of Temperature Dependent Magnetization

(ZFC/FC)

0 20 40 60 80 100 120

0,00

0,02

0,04

0,06

0,08

0,10

0,12M

T(J/KB)

c=10%

c=5%

c=1.5

SIMULATED ZFC MAGNETIZATION CURVES for Co@Mn

Co @ Ag Concentration Effects

0 50 1000.0

1.5

3.0

H = 100 Oe

VFF = 9,8%

VFF = 4,7%

VFF = 1,5%

(

em

u/ cm

3

Co)

T (K)

VFF TB

8.9% 18 K

4.7% 17 K

1.5% 9.5 K

0 5 10 15 20 25 30 350.00

0.05

0.10

0.15 Co@Ag Co@Mn

Mto

t

T(J/KB)

Left:Simulated ZFC curves for Co@Mn(full circles) and Co@Ag (open circles)

nanoparticle assembliesm. Right: Measurered ZFC/FC curves.

Comparison of Co/Mn and Co/Ag Magnetisation curves

0 50 100 150 200 250 3000.0

5.0x10-3

1.0x10-2

1.5x10-2

0.0

0.5

1.0

1.5

2.0

2.5

(

em

u/c

m3)

Co

@A

g

FC

Co@Mn

(

em

u/c

m3)

Co

@M

nT (K)

FC

ZFC

ZFC

Co@Ag

C~5% Co@Mn has higher TB

J. Nogues, V. Skumryev, J. Sort, S.

Stoyanov, and D. Givord PRL 97,

157203 (2006)

Co/CoO

S1 xv=0.08

S3 xv=0.33

Jcsh1=0.32

Jcsh2=0.3

Jsh1sh2=-6

Intra-particle

Exchange coupling

constants

Ιcishj1=2.0

Icishj2=0.5

Ishishj=2.5

Inter-particle

Exchange coupling

constants

Anisotropy

constants

Ksh=8.0

Kc=0.1 g=0.1

Dipolar Strength

Μesoscopic Model of a Dense assembly of core/shell nanoparticles

Adv. Mater. 24, 4331-6, (2012)

Experimental evidence Co/CoO *

Spherical shape

core Dc=4nm

Shell tsh=1nm

Total size Dp=Dc+2tsh=6nm

* J. Nogues, V. Skumryev, J. Sort, S. Stoyanov,

and D. Givord PRL 97, 157203 (2006)

1 1

2 2

intra inter

h an ex ex ddiE E E E E E

0 , 1 1, 2 2,

2 2 2

, , 1 1, , 2 , ,

1 , 1, 2 , 2, 1 2 1, 2,

1 1 1

( )

( ( ) ( ) ( ) )

h c c i sh sh i sh sh i h

i

an c c i c i sh sh i sh i sh sh i sh i

i

N N Nintra

ex csh c i sh i csh c i sh i sh sh sh i sh i

i i i

inter

ex

E H m s m s m s e

E K s e K s e K s e

E J s s J s s J s s

E

1, 1, 2, 2, 1 , 1, 2 , 2,

, , , ,

, , 1, 1, 2, 2, , , 1, 1, 2, 2,

, 1

( ) ( )

sh sh i sh j sh sh i sh j csh c i sh j csh c i sh j

i j i j i j i j

N

ddi c i c i sh i sh i sh i sh i ij c j c j sh j sh j sh j sh j

i ji j

I s s I s s I s s I s s

E g m s m s m s D m s m s m s

Atomic scale Model Mesoscopic Model Nanoparticles

Assembly

Shell

Core

Μesoscopic Model of a Dense assembly of core/shell nanoparticles

Magnetisation Curves

Increase of HC ,HE for xv=0.33

For xV = 0.08 HE < HC

for xV = 0.33 HE > HC,

TB , increases with

concentration

In agreement with the

experimental findings

Exchange Interactions play the dominant role

The Role of the Interactions

A. Exchange Interactions

single h an exE E E E

0

2 2 2

int

, , ,

, , ,

( ) ( ) ( )

h i i h

i

an c i i if i i sh i i

i core i erface i shell

i j core i core j shell i j shell

ex c i j if i j sh i j

i j i j i j

E H m s e

E k s e k s e k s e

E j s s j s s j s s

Microscopic Model

, ,

,

single i inter ij

i i j

E E E ,

( ) ( )

,

[ , ]

k iparticlel jparticle

i j

inter ij kl k l

k l

E I s s

jC=1.0

jIF=0.7

jSH=-6

Intra-particle

Exchange coupling

constants

Iij=0.7

Inter-particle

Exchange coupling

constants

Anisotropy

Constants

kC=1.0

kIF=0.7

kSH=-6

a) 1 nanoparticle

b) Cluster of 4 nanoparticles

Comparison between

Micro-Meso scopic models

Microscopic Model Mesoscopic Model

1 particle 4 particles 1 particle 4 particles

HC 0.027 0.035 0.0016 0.0068

Hex 0.0045 0.017 0.0652 0.0004

The mesoscopic and microscopic models have the same trends

Dependence on the neighbors’ arrangement in the small clusters

Increase of HC with g

Decrease of Hex by increasing g

Weak effect on TB, at xV=0.08

xV=0.33, TB remains almost ct for g = 0 and 0.l and increases with increasing g

The Role of the Interactions

B. Dipole-Dipole Interactions

Increase of HC and Hex with xV.

dominance of Hex for a wide range of xV

In the 3D lattice HC και Hexincrease above the percolation

threshold and for high xV they decrease

In the 2D lattice HC και Hexincrease later above the

percolation threshold and the increase insits for higher xV

2D PBC Number of Nearest neighbors

xV p zavg z=0 z=1 z=2 z=3 z=4

0.08 0.15 0.594 0.525 0.368 0.096 0.011 0.000

0.33 0.63 2.518 0.019 0.128 0.327 0.370 0.157

Comparison between the 2D and 3D assemblies

p=6xv/π

Concluding Remarks

Random Assemblies of bi-magnetic nanoparticles exhibit:

Higher Hc than the SW type nanoparticle assemblies and

appearance of Hex

Increase in Hc and Hex with the particle density.

Higher TB than the SW type nanoparticle assemblies

Collaborators Modeling

Computational Condensed Matter Physics Group, IMS E. Eftaxias

M. Vasilakaki

G. Margaris

J.A. Blackman (University of Reading, UK)

Experiments C. Binns (Leicester, UK)

D. Fiorani (Rome, Italy)

N. Domingo(Barcelona, Spain)

J. Nogues(Barcelona, Spain)

The work was supported by

EU STREP project No. NMP CT-013545

NANOSPIN

(SELF-ORGANISED COMPLEX-SPIN MAGNETIC NANOSTRUCTURES)

GSRT ARISTEIA I Project No 22

COMANA

(COMPLEX MAGNETIC NANOSTRUCTURES)