Fredrick Michael et al- Size dependence of ferromagnetism in gold nanoparticles: Mean field results

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Size dependence of ferromagnetism in gold nanoparticles: Mean field results

Fredrick Michael*Computational Chemistry Group, NIST Center for Theoretical and Computational Nanosciences,

National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA

and Interdisciplinary Network of Emerging Science and Technologies (INEST) Group Postgraduate Program,

Philip Morris, Richmond, Virginia 23234, USA

Carlos GonzalezComputational Chemistry Group, NIST Center for Theoretical and Computational Nanosciences,

National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA

Vladimiro MujicaComputational Chemistry Group, NIST Center for Theoretical and Computational Nanosciences,

National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA;

Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, USA;

Argonne National Laboratory, Center for Nanoscale Materials, Argonne, Illinois 60439-4831, USA;

and Interdisciplinary Network of Emerging Science and Technologies (INEST), Philip Morris, Richmond, Virginia 23234, USA

Manuel MarquezInterdisciplinary Network of Emerging Science and Technologies (INEST) Research Center, Philip Morris,

4201 Commerce Road, Richmond, Virginia 23234, USA

Mark A. Ratner

Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, USA

Received 25 April 2007; published 12 December 2007

In this paper, a simple spin-spin Ising interaction model for the surface ferromagnetism is combined with thebulk Au diamagnetic response to model the size dependence of the magnetization of a Au nanoparticle. Usingthe maximum entropy formalism, we obtain the average temperature dependent magnetization within a meanfield model. Our results qualitatively reproduce recent experimental observations of size-dependent magneti-zation of Au nanoparticles in which the ferromagnetic moment of thiol-capped nanoparticles is seen to increasefor diameters larger than 0.7 nm, peaking at approximately 3 nm, and subsequently decreasing as the particlediameter increases further.

DOI: 10.1103/PhysRevB.76.224409 PACS numbers: 75.70.Rf, 75.75.

a, 75.50.Tt, 75.90.

w

I. INTRODUCTION

The experimental observation of ferromagnetic momentformation at the nanoscale in Au nanoparticles is an interest-ing departure from the bulk behavior of gold, which is dia-magnetic. The measured magnetization is strongly size de-pendent, increasing with particle diameter at the smallernanoparticle sizes, peaking at approximately 3 nm for Au-thiol nanoparticles, and subsequently decreasing with in-creasing nanoparticle size.1 The latter observation is consis-tent with the expected behavior that as the nanoparticle sizeincreases, the Au atomic configuration approaches that of thebulk gold lattice. This has been confirmed by using x-rayphotoemission spectroscopy2,3 measurements of the nanopar-ticle size dependence of the Au cluster electron binding en-ergies relative to bulk.

There have been several ideas set forth as to the cause ofthe observed ferromagnetic moment in Au nanoparticles.These range from the Fermi hole charge imbalance at thesurface1,4 to Hunds rule electron filling at the smallerscales.5 Also, it has been acknowledged that the large surfaceto volume ratio, and thereby surface coordination number,plays an important role in the magnetic moment size depen-

dency, given that magnetic moments arising from a net spinpolarization in surface atoms are not fully quenched, in con-trast to the quenching observed in core lattice arrangements.Another possible cause is the increased charge localizationand the metal-insulator transition postulated at the extremesmall scale, reported for Au nanoparticles to be 3 nm indiameter.1 This effect shows up at intermediate scales in atight binding model as a band narrowing.6

The most important quantum-mechanical aspect of theproblem is to characterize the underlying physical mecha-nism that accounts for the ferromagnetic behavior of goldnanoparticles. The causes for the ferromagnetic moment for-mation will be discussed in more detail in a forthcomingarticle.7 In a related work, we have performed detailed elec-tronic structure calculations, reported elsewhere,8 of goldclusters of different sizes that support a model wherechanges in cluster geometry, compared to that of bulk gold,are coupled to the appearance of localized magnetic mo-ments. As cluster size increases, the onset of a distinct core-shell separation is observed, where the magnetic momentsare essentially localized on the surface and there is a prefer-ential ferromagnetic interaction.

PHYSICAL REVIEW B 76, 224409 2007

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In this paper, we present a statistical-mechanical model ofthe size dependence of the ferromagnetic moment, itself oftheoretical and experimental interests, which depends on afew key assumptions about the microscopic behavior of thesystem. A statistical model that accounts for the most rel-evant features of the system must describe the core and sur-face effects in a consistent way. The size-dependent magne-tization in gold should exhibit a transition from the

nanodomain, where ferromagnetic behavior has been ob-served, to the bulk diamagnetic behavior. In nanoparticles,core atoms are characterized by a relatively high coordina-tion, while surface atoms exhibit lower coordination as wellas reconstruction effects. Due to the existence of these twodifferent atomic environments, the numeric core to surfaceatomic ratio becomes a key parameter. As this ratio in-creases, the core should resemble the geometric and elec-tronic properties of the bulk, with the magnetization becom-ing essentially associated with the surface atoms. Accordingto this simple argument, the size dependency of magnetiza-tion observed in gold nanoparticles is the result of a delicatebalance between the number of atoms in the surface and the

core.Based on our results and the experimental observations,we propose an Ising Hamiltonian model that describes sur-face and core spin-spin interactions, where the total magne-tization includes a diamagnetic contribution that will dependon the ferromagnetic long-range magnetic field. This effec-tive magnetic field created by the surface atoms will induce adiamagnetic response of the core atoms in a way analogousto the effect of the Weiss field. Thus, we systematically ac-count for the surface and core effects and interactions be-tween the two regions. For ease of calculations and in orderto obtain analytical expressions, we apply mean field aver-aging techniques to obtain temperature dependent magneti-zation. The use of statistical-mechanical methods in the lit-erature to study pure and bimetallic nanoparticles is welldocumented, e.g., Refs. 912. It reproduces in a semiquanti-tative way the most striking experimental result, i.e., thesize-dependent magnetization in gold nanoparticles.

Section II of this paper presents the model Hamiltonianand the mean field statistical treatment that we use to calcu-late the magnetization of nanoparticles with a dominant fer-romagnetic exchange interaction within a frozen-core ap-proximation. Section III considers more general models ofnanoparticles where variable exchange mechanisms ofsurface-core spin interactions are considered. Section IVstates the conclusions of our work.

II. THEORY: MEAN FIELD MODEL

A. Model Hamiltonian

A model Hamiltonian for the nanoparticle must includethe ferromagnetic exchange interaction of the spins at thesurface layer. The range of the spin-spin ferromagnetic inter-action for Au has been postulated from the Fermi hole modelto be kF, or approximately 1.5 nm.

1 The surface spins arethen assumed to be in a shell near the surface.

Based on the physical assumptions described in the Intro-duction, we consider as our starting point a semiclassical

Ising spin-spin interaction model such as the one consideredin Ref. 13, with the additional assumption that there is noexternal magnetic field, so that the Zeeman interaction termis absent. The total Hamiltonian can be partitioned into asurface and a core term H=Hs +Hc, where

Hs =

i

Ns

j

Ns

Jijs

Si Sj 1

2i

Ns

j

Nc

Jijcs

Si Sj +

i

Ns

ksnr Si2,

Hc = i

Nc

j

Nc

Jijc

Si Sj 1

2i

Ns

j

Nc

Jijsc

Si Sj i

Nc

kcSzi

2 , 1

where the symbol indicates a restriction ij in the indi-ces, and g is an appropriated gyromagnetic factor, such that

the magnetic moment per atom is obtained as i = gSi.This Hamiltonian contains interactions between core and

surface spins and both surface and core anisotropy terms.13

The anisotropy terms represent the competition between lin-

ear and radial orientations of the core and surface spinswhich are assumed to coexist in the nanoparticle; moreover,kc and ks are the anisotropy constants of the core and surfacespins, respectively, and nr is a unit vector specifying the localradial direction on the surface of the nanoparticle. The inter-action terms are partitioned into NsNs 1 surface atom in-teractions, NsNc core atom interactions with the surface at-oms, and NcNc 1 core atom interactions with couplingfunctions Jij

s , Jijcs, Jij

sc, and Jijc , respectively. The total number

of spins is N=Ns +Nc, and we assume that the ferromagneticinteraction parameters Jij

x0, where x= s , c , cs , sc are con-

stants.The atomic core to surface ratio can comprise two cases:

NsN

cN and N

cN

s. The latter case N

cN

s, valid for

extremely small nanoparticles a few tens of atoms, will notbe explored in this paper. Following Ref. 14, the surfaceanisotropy terms in Eq. 1 are assumed to be aligned alongthe radial nr, whereas the bulk anisotropy is considered tobe unidirectional and chosen to lie along the z axis in thiswork. Based on the Hamiltonian given by Eq. 1, we nowfocus on developing expressions for the magnetization in thepartitioned regions of the surface and the core, with the in-clusion of the respective anisotropies.

B. Surface and total magnetic moment

In this section, we focus in the treatment of the magneti-zation arising from the surface atoms neglecting any interac-tion with the core atoms. The surface-core interactions rep-resented by the exchange terms Jsc and Jcs of Eq. 1 will beconsidered in subsequent sections. The calculation of distri-bution functions and observables obtained from the mo-ments, such as the magnetization and the energy, can be per-formed by utilizing the maximum entropy method oranalogously the information theoretic approach.16 Utilizingthis maximum entropy approach, the resulting spins at eachsite are being taken as S = = 1. The Gibbs-Boltzmannform of the entropy to be maximized for the surface spins is

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= kBij

Ns

nijs i,jln nij

s i,j , 2

where nijs are joint probability distributions for the two par-

ticle interaction Hamiltonian in Eq. 1. This entropy ismaximized given the constraints on the energy of the systemas represented by the interaction Hamiltonian. Mean field

approximations of the type we use here are extensively de-scribed in the literature, e.g., in Ref. 17, so we present only abrief account of the method including the surface terms thatare absent in lattice models.

To obtain the mean field result, we first average the firstterm the interaction term of Eq. 1, which becomes

iNsj

NsJsSi Sj, and assuming that the surface spin average

Sj, it can be written in terms of a constant surface magnetic

moment per atom. With s as Sj =gs, we obtain the fol-lowing expression for the mean field surface Hamiltonian inthe preferred z direction ignoring the interaction with thecore:

HSMF =

i

Ns 1

gJsSiNss +

i

Ns

ksnr Si2. 3

In the above expression, Ns is the number of magnetic sur-face atoms. Maximization of the single spin entropy =kBi

Nsnisiln ni

si including the mean field Hamiltonianas a constraint leads to the following expression:

+ HSMF 0. 4

Solving Eq. 4 using Eq. 3 and the single spin entropy of Eq. 2 leads to the following expression for the leastbiased distribution ni

s:

nis = exp1/gJsiNss + ghzi 1 +

2/821/g2

ksNs2s

2 , 5

where the inverse temperature is = 1kBT

and the last term1+

2

82ksNs

2s2 is obtained from averaging the anisotropy

term nr Si2, as shown in the following. The averaged sur-face anisotropy can be projected into the applied externalfield direction under consideration. Rewriting the radial nor-mal vector as nr= nz cos1i, the z component of the spin isobtained by

nr Si2 = cos1i Szi2. 6

The average of the squared term cos1i Szi2 in the

previous equation is related to the variance var by the fol-lowing expression:

var= cos1i Szi2 cos1i Szi

2, 7

where var scales as the inverse temperature . At low tem-peratures, this will lead to a comparatively negligible contri-bution, and we can write approximately

cos1i Szi2 cos1i Sz

i2

=0

0

2 cos1i

4sinididiSzi

2

.

8

At strong enough external fields or interactions, the angles iwill be small, making it possible to approximate the inverse

cosine as cosi1 1+

i2

4 after truncation to lowest orderof a series expansion in i. Performing the integration withthese expressions substituted gives the averaged anisotropyin the distribution of Eq. 5.

The surface ferromagnetic polarization can be obtainedfrom the spin distributions directly, given that the magnetic

moment is defined as s =ni

snis

nis+ni

sB, where B is the Bohr

magneton. We are then led to the following ferromagnetic

mean field result:

s = tanh1g

JsNss + ghz , 9which is to be solved self-consistently or alternately by pa-rametrizing the argument of the hyperbolic function,

s = tanh,

= 1g

JsNss + ghz , 10and plotting the two equations. The value of the ratio JS /kBTis chosen to give the order of magnitude observed experi-mentally for the magnetic moment.15 All calculations aredone with g=1, and the number of surface atoms is related tothe radius of the nanoparticle as explained below. The inter-section of the two curves gives the self-consistent magneti-zation moment, as shown in Fig. 1. Note that this plot cor-responds to a zero external magnetic field and that theaveraged anisotropy term has been factored out. This wouldalso occur if we had not averaged the anisotropy term, as any2 terms will not contribute to the polarization given that

-2 -1 0 1 2

-1

-0.5

0

0.5

1

FIG. 1. Magnetic moment vs =JsNsMs. The saturation mag-netization is the solution of the intersection of the two parametrizedcurves. For size dependence calculations and varying number ofatoms, a root finder is used to solve the self-consistent mean filedmagnetization equation.

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= 1, and 2 = +1 can also be factored from the resultantdistributions.

The total magnetization can be obtained if one considersthat the nanoparticle consists of the surface shell of ferro-magnetically ordered spins and the core contribution, whichin bulk Au is diamagnetic. The diamagnetic response of bulkgold is induced by an external field, but in the case of ananoparticle, it is a response to the Weiss field created by the

surface spins. The effective field induces a core diamagneticcontribution to the total magnetization provided that the fer-romagnetism persists in the outer surface shell of spins.This picture of ferromagnetic outer shell and an internal core,which behaves as the bulk diamagnetic gold, is supported bydetailed ab initio density functional calculations performedon 38 atom and 68 Au atom nanoparticles.8 As the size of thenanoparticle increases, the core contribution becomes domi-nant and the entire particle is diamagnetic.

We can write the total magnetization Mt as the sum of theindividual magnetic contributions Mt=Ms +Mc from the sur-face and the core magnetizations. The surface magnetizationMs =Nss is given by the mean field result in Eq. 9. The

core magnetization can be obtained from the effective fieldheffective =Ms = J

sNsMs and the diamagnetic susceptibilityper unit volume D,

Mc = Dheffective = DJsNsMs,

D = e2

12mc23Nc

1/3, 11

where and are two phenomenological constants withunits ofD

1 and DE1, respectively, and the gold suscep-tibility is approximated as that of an electron gas given byEq. 11.15 The total magnetization is now,

Mt = Ms + DJsNsMs , 12

and it can be seen to depend only on the surface magnetiza-tion which is ferromagnetic. The total magnetic moment peratom t=

Mt

Ntis dependent on the number of surface and core

atoms Nt=Ns +Nc. Using Eqs. 11 and 12, it can be writtenas

t =Ns

Nts1 J

sNsNt Ns1/3 , 13

where we have explicitly written the negative diamagneticsusceptibility in terms of Nt and Ns and combined the con-

stants of the diamagnetic susceptibility into a different con-stant = 3

1/3e2

124/3mc2, which has units of E1 and an approxi-

mate value of1 =1106 KJ/mol for the parameters usedin this work.

The total number of atoms is related to the size of thenanoparticle as Nt=

R

rs3, where R is the nanoparticles radius

and the parameter rs is obtained from the density relation

43rs

3 =1. The number of surface atoms is then related by thedensity to the total number of atoms approximately as Ns4Nt

2/3. The ratio of surface to core atoms as well as Au-Aubond lengths approximately 2.88 for our Au nanopar-

ticles can also be used to estimate the radial parameter. Asan example, for Au nanoparticles with a 1.4 nm diameter,75% of the atoms lie on the surface, giving a value for theradial parameter rs of 0.2107 nm. The total magnetic mo-ment per atom Eq. 13 can be plotted as a function of thenumber of atoms or diameter. The results depicted in Fig. 2show how our model reproduces, in a qualitative manner, the

size dependence observed experimentally. Figure 2 indicatesa net increase in magnetic moment at the very small scale orsmall total number of atoms in the nanoparticle, a pro-nounced peak with increasing number, and a subsequent de-crease as the number of atoms increases into the hundreds ofatoms. The effect in the size dependence of the inclusion of acore diamagnetic response due to the surface magneticWeiss field is shown in Fig. 3 for varying values of, thefield-combined constants of the diamagnetic susceptibility.

III. COUPLED MODELS OF SURFACE AND CORE

INTERACTIONS

In several types of nanoparticles such as ferrites and fer-romagnetic single domain core particles as well as bulk para-magnetic particles, differing ferromagnetic and antiferromag-netic exchange mechanisms are assumed to couple thesurface and core spin interactions, giving different magneticbehaviors for different material nanoparticles.18 Examples ofthe need to include surface-core interactions through an ex-change mechanism are found in single domain ferromagneticnanoparticles in which the partitioning into surface and core

FIG. 2. Magnetic moment per atom as a function of nanoparticlediameter nm. Js = 0.01 and =0.0025 in arbitrary units.

FIG. 3. Magnetic moment per atom as a function of nanoparticlediameter nm for different values of. Js =0.01 and =0.0025 asin Fig. 2. The curves correspond to =1106 solid line, = 5107 dashed line, and =1107 dotted line.


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interaction terms allows the modeling of surface effects sepa-rately from core or bulk effects. Also, oxidized surface nano-particles have been described utilizing antiferromagnetic ex-change interactions between the surface and the core.19

Interesting magnetic behavior in nanoparticles of differentmetallic materials can be described by partitioned surface-core Hamiltonians such as the full Hamiltonian of Eq. 1with the exchange terms explicitly represented by the

surface-core interactions.Small gold nanoparticles similar to the ones considered in

this study have been reported to exhibit ferromagnetic mo-ments independent of the external field applied at Refs. 1 and20. It is interesting to examine the resulting expressions forthe magnetization obtained upon inclusion of an antiferro-magnetic surface and core interaction term in the Hamil-tonian. In order to achieve this, it is necessary to modify theHamiltonian with terms corresponding to surface-core inter-actions Jsc, Jcs as in Eq. 1. Making use of the simplifiedmean field approach as before and performing the corre-sponding maximization procedure, the following surface andcore least biased distribution functions are obtained,

nis = exp1/gJsiNss + 1/gJ

csiNcc + ghzi

1 + 2/8ksNs2s

2/g2 , 14

nis = exp1/gJsiNss + 1/gJ

csiNcc + ghzi

kscos1i

2i2 , 15

nic = exp1/gJci

cNcc + 1/gJ

scicNss + ghzi

c

+ kcic2 , 16

where the unaveraged surface and core anisotropy terms

have been included in the distributions given by Eqs. 15and 16, respectively. In Eq. 16, ic denotes the spin vari-

able.The surface s and core c polarizations can be obtained

from the spin distributions as before,

s = tanh1g

JsNss +1

gJcsNcc + ghz ,

c = tanh1g

JcNcc +1

gJscNss + ghz . 17

The sign of the Jc interaction term determines the magneticbehavior of the core. A positive value Jc0 corresponds toa ferromagnetic core nanoparticle, while a negative valueJc0 would represent an antiferromagnetic interaction inthe core sublattice region. Given that, in our treatment, thenanoparticles Hamiltonian has been partitioned into a sur-face ferromagnetic region and a core that behaves more likediamagnetic bulk gold, we can assume a zero valued corespin-spin interaction such that Jc =0. This approximation isappropriate as long as the surface comprises all the layersof spins that exhibit the ferromagnetic interaction. If, how-ever, the surface is considered to consist of only the surface

layer of atoms, a nonzero interaction constant Jc0 wouldhave to be included in order to treat the outer layers of corespins. The general expression for the surface magnetic mo-ment per atom can be rewritten as

s = tanh1g

JsNss + JcsNc tanh 1

gJcNcc

+1

gJscNss + ghz + ghz . 18

This expression can be simplified by truncation of the expan-sion of the hyperbolic tangent tanhxx x

3

3 + definingc in Eq. 13 to first order for small values of the argument.Assuming Jc =0 and no external fields such that we are solv-ing for the spontaneous magnetization, the following self-consistent mean field result for the surface magnetization isobtained,

s = tanh1g

JsNss +1

gJcsNt NsJ

scNss ,19

which can be solved as before. The diamagnetic response ofthe core can be accounted for by specifying that the interac-tion of the surface with the core be negative Jsc0. Usingthe definition for the total magnetization moment per atomt=

Ns

Nts +

Nc

Ntc, and substituting for c and Nc as before, we

obtain

t =Ns

Nts1 J

scNsNt Ns , 20

which is to be compared with Eq. 13. Note that in thisderivation, the terms corresponding to the diamagnetic termsin Eq. 13 have an explicit temperature dependence, whichis more in keeping with a paramagnetic susceptibility. Thiscan be accounted for by the way that the effective field wasdefined in our treatment and our definition of the propor-tionality factor.

IV. CONCLUSIONS

In this work, we report on a simple mean field theoreticalmodel to describe the size dependence of magnetization inAu nanoparticles. The model incorporates mean field resultsfor the surface ferromagnetism and accounts for the diamag-netic bulklike behavior of the core atoms due to the influenceof the internal magnetic field of the surface shell of atoms.This surface magnetic field is the Weiss field and acts as aneffective applied field as seen by the core atoms. This is ageneral effect and our results should be applicable in allcore-shell nanoparticles of diamagnetic metals where surfacemagnetization results as a symmetry breaking effect that canbe described in terms of a Heisenberg Hamiltonian. The re-sulting expression for the magnetization per atom is plottedas a function of the number of atoms in the nanoparticle, andit reproduces the experimentally observed peak in magneti-zation and is reported in Ref. 1, the connection between theexperimental results and theory being made by considering

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the relationship between the number of Au atoms in thenanoparticle and the nanoparticle diameter.

The results obtained are statistical in nature, and as suchthey assume a certain microscopic behavior that in our caseis reflected in the appearance of surface ferromagnetism. Aquantum description that explicitly supports this kind of be-havior will be presented elsewhere.

ACKNOWLEDGMENTS

M.A.R. thanks the NSF-MRSEC program for supportthrough the Northwestern MRSEC. M.A.R thanks the Chem-istry Division of the NSF for support. V.M. would like toacknowledge the Center for Nanoscale Materials of ArgonneNational Laboratory for support.

*[email protected]@[email protected]@[email protected]

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Fredrick Michael et al- Size dependence of ferromagnetism in gold nanoparticles: Mean field results

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Transcript of Fredrick Michael et al- Size dependence of ferromagnetism in gold nanoparticles: Mean field results