Does the Nonmagnetic Surface Layer Exist Throughout Ferromagnetic Nanoparticles?

H. M. Lu, C. L. Zhao, S. C. Tang, and X. K. Meng*Department of Materials Science and Engineering, National Laboratory of Solid State Microstructures, NanjingUniVersity, Nanjing 210093, P. R. China

ReceiVed: July 21, 2007; In Final Form: October 8, 2007

The analytic models for size-dependent Curie temperatureTc and saturation magnetization at room temperatureMs of Fe3O4 nanoparticles have been proposed in terms of the size-dependent melting temperature. TheTc(r)andMs(r) values decrease with a decreasing of the particle radiusr. Agreement between model predictionsand the corresponding experimental results can be found, which enable us to determine the size dependencesof the thickness of nonmagnetic surface layerδ(r) andt(r) in respectively describingTc(r) andMs(r) functions.It is found that bothδ(r) and t(r) increase with a decreasing ofr and t(r) is always twice as large asδ(r).Moreover, the surface layer should be an intrinsic property of ferromagnetic crystals since the nonmagneticsurface layer exists in the whole size range rather than vanishes asr approaches infinite.

Introduction

Magnetite (Fe3O4) is an intensively studied and stronglycorrelated transition metal oxide, which is ordered below arelatively high transition temperature, i.e., the Curie temperatureTc ) 851 K.1 This material has recently received renewed andtremendous interest because of its potential technologicalapplication in giant magneto-resistive devices, catalysis, andspin-valve considering the theoretically predicted 100% spinpolarization and its high Curie temperature.2

The Curie temperature is one of the most important magneticproperties to describe the phase stability of Fe3O4. Tc offerromagnetic nanoparticles becomes tunable, which allows usto adjust theTc value for switches functioning in a designedtemperature range. A number of outstanding theoretical modelshave been developed to explain the size dependence ofTc(r)with r being the radius of ferromagnetic nanoparticles.3-8

Among them, Nikolaev and Shipilin proposed that the ferro-magneticTc suppression originated from nothing more than thesurface layer that contained atoms with only half the numberof exchange bonds (i.e., the coordination number imperfection)per unit volume compared with the bulk.5 Under the assumptionthat the Curie temperature for a ferromagnetic nanoparticle isproportional to the mean number of exchange bonds, theTc

suppression can be described as5

where∞ denotes the bulk value,∆ is the difference, andδ isdefined as the thickness of the surface layer that is half-depletedof exchange bonds. However, the attempts to fit the theoreticalcurves calculated in terms of eq 1 with a constantδ to theexperimental results taken from literature9 were unsuccessful.The deviation unambiguously indicates that the quantityδ is aparameter characterizing the influence of the surface layer onthe Curie temperature rather than the thickness of the layerdepleted of exchange bonds5 and suggests that theδ value varies

with the particle size. Nevertheless, the size dependence ofδ-(r) has not yet been established.5,6

On the other hand, the saturation magnetizationMs, definedas the maximum of the magnetization value achieved in asufficiently large magnetic field, is also one of the mostimportant magnetic properties.Ms is a function of measuringtemperatureT. Ms at room temperature is found to decreasesharply with decreasing size, which was first proposed byBerkowitz et al. in 1968.10 Subsequently, numerous models havebeen developed for theMs suppression.11-17 Hereinto, when acore-shell structure is assumed where the shell layer (nonmag-netic surface layer) has lowerMs value than the correspondingbulk oneMs(∞),12 Tang et al. derived an empirical relation forsize-dependent saturation magnetizationMs(r)12

wheret is a fitting parameter from the experiments, which isdefined as the constant thickness of nonmagnetic surface layer,similar to δ in eq 1. However, the theoretical attempt todetermine thet value is also rare.

In eqs 1 and 2, bothδ andt denote the thickness of the surfacelayer, which have contributions on the depressions of thecorresponding magnetic properties. However, when eqs 1 and2 are employed to fit experimental data ofTc(r) and Ms(r)functions of Fe3O4 nanoparticles, the parametersδ and t willreduce to zero whenr is large enough. For example,δ is thoughtto approximately equal zero whenr g 20 nm.5 Does such anonmagnetic surface layer exists throughout, and if so, is it anintrinsic property of ferromagnetic nanocrystals? Moreover,there is no work done to study the relationship betweenδ andt to the best of our knowledge. In this contribution, the analyticmodels for theTc(r) andMs(r) functions are established, whichenable us to determine the size dependences ofδ and t andfurther study the relationship between these two quantities.

Model

In terms of the bond order-length-strength (BOLS) correlationmechanism and the Ising premise, bothTc andMs are determined

* To whom correspondence should be addressed. Tel.:+86-025-8368-5585. Fax: +86-025-8359-5535. E-mail: mengxk@nju.edu.cn.

Tc(r)

Tc(∞)) 1 - 3δ

2ror

∆Tc(r)

Tc(∞)) - 3δ

2r(1)

Ms(r)

Ms(∞)) 1 - 3t

ror

∆Ms(r)

Ms(∞)) - 3t

r(2)

18585J. Phys. Chem. C2007,111,18585-18588

10.1021/jp075733l CCC: $37.00 © 2007 American Chemical SocietyPublished on Web 11/21/2007

by the spin-spin exchange interaction energyEexc(r,T), whichis the sum of a portion of the cohesive energyEc(r) and thethermal vibration energyEv(T),6 i.e., Eexc(r,T) ) AEc(r) + Ev-(T) with A (0 < A < 1) being a coefficient. Based on the meanfield approximation and Einstein’s relation,Ev(T) ) kBT, wherekB is the Boltzmann constant.6 Moreover,Ev(T) required fordisordering the exchange interaction is a portion ofEc whenT) Tc.6 Under the consideration that the above energeticrelationship remains even whenr f ∞, there is

Recently, Zhong et al. have developed a model where theMs

suppression was thought to result from the decrease of theatomic cohesive energy due to the coordination numberimperfection of surface atoms.16,17In terms of BOLS correlation,Ms(r) at room temperature for ferromagnetic nanoparticles canbe expressed as16

Using this equation, good agreement between predictions andexperimental or Monte Carlo simulations results for a numberof specimens was shown.16,17

Although the size dependence ofEc(r) functions of elementalnanocrystals has been described by Jiang et al.,18 the necessaryphysical quantity, the solid-vapor transition entropySb, cannotbe directly determined. According to the liquid-drop model andthe BOLS mechanism,19,20 Tm(∞) ∝ Ec(∞) has also beenproposed withTm being the melting temperature since bothquantities characterize the bond strength. When the nature ofthe chemical bond does not change, the above relationship canbe extended to the nanometer range as a first order approxima-tion, namelyTm(r)/Tm(∞) ) Ec(r)/Ec(∞).

Combining theTm(r) function described in the literature21 andthe above considerations, a universal relation can be describedas

whereh is the nearest ionic distance between different ions,Ris the ideal gas constant, andSvib denotes the vibrationalcomponent of the melting entropySm atTm(∞). It is known thatthe bond of Fe3O4 is metallic when the temperature is largerthan the Verwey temperatureTv of 122 K.22 SinceTm(∞) ofFe3O4 is far aboveTv and the nature of chemical bonds doesnot vary on melting, the electronic entropy can be neglected,23

namelySvib ≈ Sm-Spos with Spos being the positional entropy.Spos has been given bySpos ) -R(xA ln xA + xB ln xB) with xA

andxB respectively denoting the molar fractions of the crystalsand vacancies.23 For a melting process,xA ) 1/(1 + ∆V/V) andxB ) 1 - xA where∆V/V is the normalized volume change onmelting.

Combining eq 5 with eqs 1 and 2, there is

Results and Discussion

Figure 1 shows the calculatedTc(r)/Tc(∞) function (the solidline) of Fe3O4 nanoparticles as a function of the particle radiusin terms of eq 5. It is evident thatTc(r) decreases with decreasingradius and the drop forms dramatically once the radius decreasesbelow 5 nm. As a comparison, available experimental results(the closed symbols) are also listed.9 As shown in Figure 1, themodel predictions correspond well to the available experimentalresults.

Figure 2 compares theMs(r)/Ms(∞) function of Fe3O4

nanoparticles between the model predictions in terms of eq 5(the solid line) and the corresponding experimental results (b,9, and2).27-29 It is obvious that the model predictions of 76%(r ) 6.2 nm), 63% (r ) 4.8 nm), and 48% (r ) 3.0 nm) are inagreement with the corresponding experimental results of 73%,27

67%,29 and 54%,28 whereas the other two experimental data(shown as2) give a large difference of about 50% from thepredictions of eq 5. The sharp drops in|∆Ms(r)/Ms(∞)| of 71%and 67% at 4 and 4.2 nm reflected by these two experimentaldata29 are surprising and larger than those in other oxides. Forexample, the|∆Ms(r)/Ms(∞)| value of Fe2O3 nanoparticles is inthe range of 30%∼50% when the size ranges from 2.4 to 5.0nm. The reason for the sharp drop remains unknown. Thus, thisdisagreement between the predictions of eq 5 and these twoexperimental data may hardly illustrate the incorrectness of eq5. In terms of the above discussion, we believe that our modelpredictions are at least quantitatively correct in the size range.As shown in the figure,Ms(r) also declines with decreasing ofthe particle radius and the drop forms dramatically once theradius decreases below about 10 nm.

Figure 1. Comparison ofTc(r)/Tc(∞) function described by variousmodels and the corresponding experimental results.9 h ) 0.21 nm.2

Since the∆V/V value of Fe3O4 is unavailable, the mean value of thoseof Al2O3 (20%) and FeO (22%) is taken.24,25The melting enthalpyHm

) 19.7 kJ/g atom,26 Tm ) 1870 K,26 and thusSvib ≈ 6.40 J/g atom K.

Tc(r)

Tc(∞))

Eexc(r)

Eexc(∞))

Ec(r)

Ec(∞)(3)

∆Ms(r)

Ms(∞)≈ 4

∆Eexc(r)

Eexc(∞)(4)

∆Tc(r)

Tc(∞))

∆Ms(r)

4Ms(∞))

∆Ec(r)

Ec(∞))

∆Tm(r)

Tm(∞))

exp(-2Svib

3R1

r/3h - 1) - 1 (5)

Figure 2. Comparison ofMs(r)/Ms(∞) function described by variousmodels and the corresponding experimental results.27-29

t(r) ) 2δ(r) ) 4r3 [1 - exp(-

2Svib

3R1

r/3h - 1)] (6)

18586 J. Phys. Chem. C, Vol. 111, No. 50, 2007 Lu et al.

The agreement shown in Figures 1 and 2 indicates that eq 5can satisfactorily describe the size dependence of the Curietemperature and the saturation magnetization of Fe3O4 nano-particles. Since it is claimed that eqs 1 and 2 can also beemployed to model theTc(r) andMs(r) functions, eq 6 derivedfrom eqs 1, 2, and 5 should also be used to determine the sizedependences ofδ and t.

Figure 3 shows the calculatedδ(r)/h (the solid line) andt(r)/hfunctions (the dash line) of Fe3O4 nanoparticles in terms of eq6. At the same time, the simulatedδ(r) values are also listedfor comparison.5 As shown in Figure 3, the model predictionsfor δ(r) values correspond to the simulation results atr < 10nm, whereas a large deviation happens atr > 10 nm. In contrastwith Tc(r) and Ms(r) functions, δ(r) and t(r) decrease withincreasing of the particle radius and then reach a plateau whenthe particle radius rises above 10 nm, which is different fromthe claim of the simulation that the surface layer vanishes whenr g 20 nm.5 We think that the nonmagnetic surface layer shouldexist throughout in terms of eq 6 and Figure 3, and it is anintrinsic property of ferromagnetic crystals, similar to thedielectric dead layer in capacitors.30

It is known that exp(-x) ≈ 1 - x whenx is small enough,e.g.,x < 0.1. Whenr f ∞, with this first-order approximationand 4Svib/(3R) ≈ 1 for Fe3O4, there is

where the subscript “min” is the abbreviation of minimal. It isevident that the size dependences ofδ(r) and t(r) stronglydepend on the value ofSvib.

With δ ) δmin ≈ h, the curve of theTc(r)/Tc(∞) function interms of eq 1 (the dash line) is plotted in Figure 1 forcomparison. It is found that these two lines described by eqs 1and 5 are nearly overlapped atr > 5 nm, whereas the differencebetween them appears atr < 5 nm and increases with decreasingof the radius. Similarly, the curve of theMs(r)/Ms(∞) functionin terms of eq 2 witht ) tmin ≈ 2h (the dash line) is also plottedin Figure 2 for comparison. It is also discernible that the resultsfrom these two equations are close to each other atr > 10 nm,whereas they start to separate atr < 5 nm and increase with adecreasing of the radius.

As shown in Figures 1 and 2, the size effects of the magneticproperties described by eqs 1 and 2 are weaker than those ofeq 5. Equations 1 and 2, or the 1/r correlation mechanism,consider the magnetic properties of interior atoms to be the sameas those of the corresponding bulk crystal, and the depressionof magnetic properties solely originates from the increase ofthe thickness of surface layer asr drops. However, similar tothe surface atoms, the magnetic properties of the interior atoms

should also decline with a decrease in the radius. A similar caseoccurs in the vibration amplitude where the vibration amplitudesof surface and interior atoms increase asr decreases and theratio between them keeps constant at all size ranges.20,31,32Whenit is noted that increasing thermal vibrations tend to counteractthe dipole coupling forces in ferromagnetic materials and thenreduce the magnetic properties, the interior atoms of nanocrystalsshould also have contribution on the depression of the magneticproperties, which makes the depressions described by eq 5stronger than those in terms of eqs 1 and 2 where only theinfluence of the surface layer is considered.

As indicated by eq 7 and also shown in Figure 3, when theradius approaches infinity,t describing the influence of surfacelayer on theMs(r) function is twice as large asδ reflecting thecontribution of the surface layer on theTc(r) function. Why doesthe thickness of the surface layer in describing theMs(r) andTc(r) functions have different values? As shown in the deductionof eq 7, the relationship oft(r)/δ(r) ) 2 results from the factthat the suppression ofMs(r) at room temperature is four timesof that of Tc(D), which can be qualitatively explained by thefollowing: On one hand, the absolute value ofEc(r) drops dueto the increase in the portion of lower-coordination atoms innanocrystals, which leads to the weakening of the interspininteraction and thus the suppression ofMs(r);16 On the otherhand, with rising temperature, increased thermal vibrations tendto counteract the dipole coupling forces in ferromagneticmaterials. Consequently,Ms gradually diminishes with increas-ing T. SinceMs reduces to near zero up toTc,33 Ms seems to beproportional to (Tc/T - 1)â whereâ denotes an exponent. Infact, a similar expression ofMs ∝ (Tc/T - 1)1/2 for theferromagnetic case has been found by Burns.34 BecauseTc

decreases when the size is decreased while the concernedtemperature forMs(r) here has been fixed at room temperature,Tc/T reduces. In other words, the effect of decreasing size isequivalent to that of rising temperatureT. Thus, the both effectsbring out stronger suppression ofMs(r) at room temperaturethan that ofTc(r) where the latter is only induced by reducingsize. As result, the different thickness values of the surface layerin describingMs(r) and Tc(r) functions should originate fromthe different influencing factors.

Interestingly, when theTc(r) value is also set as the roomtemperature, the correspondingr value is about 1.0 nm in termsof eq 5 andδ(r ≈ 1.0 nm)≈ 2δmin in terms of eq 6, which isjust equal totmin. Namely, the thickness of the surface layer indescribing theTc(r) andMs(r) functions has the same values inthis case. The above considerations imply that the thickness ofthe surface layer for Fe3O4 is about two atomic monolayers atroom temperature and decreases to one atomic monolayer whenthe temperature increases to the Curie temperature.

Conclusion

In summary, the analytic models for size-dependentTc(r) andMs(r) at room temperature of Fe3O4 nanoparticles have beenestablished where bothTc and Ms decrease with decreasingrand ∆Ms(r)/Ms(∞) is four times ∆Tc(r)/Tc(∞). The modelpredictions correspond to the corresponding experimental results.These agreements enable us to determine the size dependencesof δ(r) andt(r) and to study the correlation between them. It isfound thatδ(r) andt(r) increase with decreasingr andt(r), whichis always twice as large asδ(r). Moreover,t(r) varies with thechanges of the size and temperature, whereasδ(r) is onlyaffected by the size. In the whole size range, the nonmagneticsurface layer exists throughout withtmin ≈ 2h andδmin ≈ h asshown in Figure 3, rather than vanishes atr f ∞,5 which implies

Figure 3. δ(r)/h and t(r)/h as a function ofr.

tmin ) 2δmin )8Svibh

3R≈ 2h (7)

Existence of the Nonmagnetic Surface Layer J. Phys. Chem. C, Vol. 111, No. 50, 200718587

that the surface layer should be an intrinsic property offerromagnetic crystals. Since no special assumption is made inthe deductions of the models, the above conclusions should bealso applicable for other ferromagnetic crystals.

Acknowledgment. Financial support from the State KeyProgram for Basic Research of China (Grant No. 2004CB619305),the Natural Science Foundation of Jiangsu Province (Grant No.BK2006716), the Postdoctoral Science Foundation of China(Grant No. 20070410326), and the Jiangsu Planned Projects forPostdoctoral Research Funds (Grant No. 0701029B) is acknowl-edged.

References and Notes

(1) Dedkov, Y. S.; Fonin, M.; Vyalikh, D. V.; Hauch, J. O.; Molodtsov,S. L.; Rudiger, U.; Guntherodt, G.Phys. ReV. B 2004, 70, 073405.

(2) Li, Y. L.; Yao, K. L.; Liu, Z. L. Surf. Sci.2007, 601, 876 andreferences therein.

(3) Zhang, R.; Willis, R. F.Phys. ReV. Lett. 2001, 86, 2665.(4) Zhong, W. H.; Sun, C. Q.; Tay, B. K.; Li, S.; Bai, H. L.; Jiang, E.

Y. J. Phys.: Condens. Matter2002, 14, L399.(5) Nikolaev, V. I.; Shipilin, A. M.Phys. Solid State2003, 45, 1079.(6) Sun, C. Q.; Zhong, W. H.; Li, S.; Tay, B. K.; Bai, H. L.; Jiang, E.

Y. J. Phys. Chem. B2004, 108, 1080 and references therein.(7) Yang, C. C.; Jiang, Q.Acta Mater.2005, 53, 3305.(8) Lang, X. Y.; Zheng, W. T.; Jiang, Q.Phys. ReV. B 2006, 73, 224444.(9) Sadeh, B.; Doi, M.; Shimizu, T.; Matsui, M. J.J. Magn. Soc. Jpn.

2000, 24, 511.(10) Berkowitz, A. E.; Shuele, W. J.; Flanders, P. J.J. Appl. Phys.1968,

39, 1261.(11) Coey, J. M. D.Phys. ReV. Lett. 1971, 27, 1140.(12) Tang, Z. X.; Sorensen, C. M.; Klabunde, K. J.; Hadjipanayis, G.

C. Phys. ReV. Lett. 1991, 67, 3602.(13) Parker, F. T.; Foster, M. W.; Margulies, D. T.; Berkowitz, A. E.

Phys. ReV. B 1993, 47, 7885.

(14) Kodama, R. H.; Berkowitz, A. E.; McNiff, E. J.; Foner, S.Phys.ReV. Lett. 1996, 77, 394.

(15) Mamiya, H.; Nakatani, I.; Furubayashi, T.Phys. ReV. Lett. 1998,80, 177.

(16) Zhong, W. H.; Sun, C. Q.; Li, S.Solid State Commun.2004, 130,603.

(17) Zhong, W. H.; Sun, C. Q.; Li, S.; Bai, H. L.; Jiang, E. Y.ActaMater. 2005, 53 3207.

(18) Jiang, Q.; Li, J. C.; Chi, B. Q.Chem. Phys. Lett. 2002, 366, 551.(19) Nanda, K. K.; Sahu, S. N.; Behera, S. N.Phys. ReV. A 2002, 66,

013208.(20) Sun, C. Q.Prog. Solid State Chem.2007, 35, 1.(21) Jiang, Q.; Tong, H. Y.; Hsu, D. T.; Okuyama, K.; Shi, F. G.Thin

Solid Films1998, 312, 357.(22) Rozenberg, G. Kh.; Pasternak, M. P.; Xu, W. M.; Amiel, Y.;

Hanfland, M.; Amboage, M.; Taylor, R. D.; Jeanloz, R.Phys. ReV. Lett.2006, 96, 045705.

(23) Regel, A. R.; Glazov, V. M.Semiconductors1995, 29, 405.(24) Ansell, S.; Krishnan, S.; Weber, J. K. R.; Felten, J. J.; Nordine, P.

C.; Beno, M. A.; Price, D. L.; Saboungi, M. L.Phys. ReV. Lett. 1997, 78,464.

(25) Smithells, C. J.Metals Reference Book, 5th ed.; Butterworth:London, 1976.

(26) Weast, R. C.Handbook of Chemistry and Physics, 69th ed.;Chemical Rubber Co.: Boca Raton, FL, 1988-1989.

(27) Panda, R. N.; Gajbhiye, N. S.; Balaji, G.J. Alloy Compd.2001,326, 50.

(28) Liu, Z. L.; Liu, Y. J.; Yao, K. L.; Ding, Z. H.; Tao, J.; Wang, X.J. Mater. Synth. Process.2002, 10, 83.

(29) Amulevicius, A.; Baltrunas, D.; Bendikiene, V.; Daugvila, A.;Davidonis, D.; Mazeika, K.Phys. Stat. Sol. A2002, 189, 243.

(30) Stengel, M.; Spaldin, N. A.Nature2006, 443, 679.(31) Shi, F. G.J. Mater. Res.1994, 9, 1307.(32) Jiang, Q.; Zhang, Z.; Li, J. C.Chem. Phys. Lett.2000, 322, 549.(33) Callister, W. D. In Fundamentals of Materials Science and

Engineering, 5th ed.; Anderson, W., Ed.; John Wiley & Sons: New York,2001; p S-291.

(34) Burns, G.Solid State Physics; Academic Press: Orlando, 1985; p621.

18588 J. Phys. Chem. C, Vol. 111, No. 50, 2007 Lu et al.