The Curie temperature and the local spontaneous magnetization of ferromagnetic nanocomposites are investigated. The macroscopic character of the critical fluctuations responsible for the onset of ferromagnetic order means that there is only one Curie temperature, independent of the number of magnetic phases present. The Curie temperature increases with the grain size and is, in general, larger than predicted from the volume averages of the exchange constants. However, the Curie-temperature enhancement is accompanied by a relative reduction of the spontaneous magnetization. Due to the quadratic dependence of the permanent-magnet energy product on the spontaneous magnetization, this amounts to a deterioration of the magnets performance. The length scale on which an effective intergranular exchange coupling is realized (coupling length) depends on the Curie-temperature difference between the phases and on the spacial distribution of the local interatomic exchange. As a rule, it is of the order of a few interatomic distances; for much bigger grain sizes the structures mimic an interaction-free ensemble of different ferromagnetic materials. This must be compared to the magnetic-anisotropy coupling length, which is of the order of 10 nm. The difference is explained by the nonrelativistic character of the Curie-temperature problem

Sellmyer, David J.

Skomski, Ralph

English

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University of Nebraska - Lincoln 
DigitalCommons@University of Nebraska - Lincoln 
David Sellmyer Publications Research Papers in Physics and Astronomy 
May 2000 
Curie temperature of multiphase nanostructures 
Ralph Skomski 
University of Nebraska-Lincoln, rskomski2@unl.edu 
David J. Sellmyer 
University of Nebraska-Lincoln, dsellmyer@unl.edu 
Follow this and additional works at: https://digitalcommons.unl.edu/physicssellmyer 
 Part of the Physics Commons 
Skomski, Ralph and Sellmyer, David J., "Curie temperature of multiphase nanostructures" (2000). David 
Sellmyer Publications. 66. 
https://digitalcommons.unl.edu/physicssellmyer/66 
This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at 
DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in David Sellmyer Publications 
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Curie temperature of multiphase nanostructures
R. Skomskia) and D. J. Sellmyer
Department of Physics and Astronomy and Center for Materials Research and Analysis,
University of Nebraska, Lincoln, Nebraska 68588
The Curie temperature and the local spontaneous magnetization of ferromagnetic nanocomposites
are investigated. The macroscopic character of the critical fluctuations responsible for the onset of
ferromagnetic order means that there is only one Curie temperature, independent of the number of
magnetic phases present. The Curie temperature increases with the grain size and is, in general,
larger than predicted from the volume averages of the exchange constants. However, the
Curie-temperature enhancement is accompanied by a relative reduction of the spontaneous
magnetization. Due to the quadratic dependence of the permanent-magnet energy product on the
spontaneous magnetization, this amounts to a deterioration of the magnets performance. The length
scale on which an effective intergranular exchange coupling is realized ~coupling length! depends
on the Curie-temperature difference between the phases and on the spacial distribution of the local
interatomic exchange. As a rule, it is of the order of a few interatomic distances; for much bigger
grain sizes the structures mimic an interaction-free ensemble of different ferromagnetic materials.
This must be compared to the magnetic-anisotropy coupling length, which is of the order of 10 nm.
The difference is explained by the nonrelativistic character of the Curie-temperature problem.
© 2000 American Institute of Physics. @S0021-8979~00!63608-3#
I. INTRODUCTION
The onset of long-range order in ferromagnets has been
a long-standing and mind-stimulating problem. Since the
first third of the 20th century, when research focussed on the
basic relation between quantum mechanics and magnetic
order,1,2 interest has shifted towards critical phenomena3,4
and the behavior of particular magnetic features such as
surfaces5,6 and artificial structures.7–13 In the case of multi-
layers, mean-field Curie temperatures are obtained by diago-
nalizing tridiagonal exchange-interaction matrices,7–10
whereas Wang and Mills11 used a Landau–Ginzburg type
analysis to elaborate that the spontaneous magnetization var-
ies continuously across phase boundaries.
A problem of scientific and technological interest is the
Curie-temperature and spontaneous-magnetization behavior
of nanocomposites. There are attempts to use the concept of
exchange-field penetration known from multilayers to con-
struct effective coupling parameters,14 but there is no com-
prehensive theoretical understanding of nanocrystalline al-
loys. A particular question, important in the context of high-
temperature permanent magnetism, is whether the Curie
temperature TL of any magnetic material can be improved by
nanoscale exchange coupling to a phase having a Curie tem-
perature TH.TL . In the context of magnetic hysteresis, it
has been shown that interatomic exchange can be exploited
to improve the hard-magnetic performance of a magnet by
adding soft-magnetic material.15,16 In that case, the coupling
between the hard and soft regions is very effective on length
scales up to about 10 nm.
II. CALCULATION AND RESULTS
A. Curie temperature and spontaneous magnetization
Let us start with a mean-field analysis of ferromagnetic
nanocomposites. By definition, the spontaneous magnetiza-
tion vanishes at the Curie temperature, so that the mean-field
equations can be linearized and the spontaneous magnetiza-
tion M i5M z(ri) of the ith atom obeys
TM i5
1
kB (j J i jM j . ~1!
In this equation, the Ji j5J(ri2rJ) are local exchange pa-
rameters whose detailed meaning depends on the underlying
model. Diagonalization of Eq. ~1! yields a common Curie
temperature, which is equal to the highest eigenvalue of the
matrix Ji j /kB . For multilayers, it has been found that the
Curie temperature increases with increasing layer
periodicity.7,9 In the case of three-dimensional nanostruc-
tures, of arbitrary geometry, the matrix Ji j /kB is no longer
tridiagonal and the diagonalization is slightly more compli-
cated. Figure 1 shows cubic inclusions of a material with a
high ~bulk! Curie temperature TH in a matrix of a material
having a low ~bulk! Curie temperature TL,TH . The bars
show the Curie temperatures for TH51000 K, TL5400 K,
and an intermediate nearest-neighbor interface exchange of
Ji j /kB5700/6. As in the case of multilayers, TC increases
with the size of the TH and TL regions, and the composite
Curie temperature is close to TH .
The eigenmode belonging to TC gives the spacial varia-
tion of the order parameter in the vicinity of TC . @The other
eigenvalues and eigenmodes of Eq. ~1! have no physical
meaning, because the spontaneous magnetization is finite be-
low TC and the mean-field equations are no longer linear.#
Below TC , the Brillouin-type nonlinearity of the mean-fielda!Electronic mail: rskomski@unlserve.unl.edu
JOURNAL OF APPLIED PHYSICS VOLUME 87, NUMBER 9 1 MAY 2000
47560021-8979/2000/87(9)/4756/3/$17.00 © 2000 American Institute of Physics
Downloaded 17 Nov 2006 to 129.93.16.206. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
equations is important. For N nonequivalent sites, the local
spontaneous magnetization M i(T) is obtained by solving the
set of N nonlinear mean-field equations. Figure 2 shows a
typical result for N54 and classical (J5‘) Heisenberg
spins. The considered structure exhibits four nonequivalent
sites, one TH site and three TL sites, and each nonequivalent
site has a different spontaneous magnetization below TC . In
the TL phase, the spontaneous magnetization is largest for
sites which are nearest neighbors to TH sites. This is related
to the above-mentioned continuity of the order parameter at
phase boundaries and shows that the spontaneous magneti-
zation is not element specific but depends on the distance
from the TH – TL interfaces.
The lower single-phase Curie temperature TL does not
translate into any phase transition, although there are
resonance-like magnetization features at some nonsingular
temperature T*’TL ~Fig. 2!. This is reminiscent of the
many-sublattice behavior of garnets and rare-earth transition-
metal intermetallics:12,16,17 the rare-earth ~or ‘‘TL’’! atoms
are subject to the quasiparamagnetic exchange field of the
iron-series ~or ‘‘TH’’! atoms.
Above the thermodynamically not very well defined
temperature T*, the spontaneous magnetization for TL sites
far away from TH sites is very low, but it remains nonzero
between T* and TC . This smallness of the TL magnetization
is of practical importance. In particular, the quadratic depen-
dence of the permanent-magnet energy product on the spon-
taneous magnetization15,16 means that the exchange coupling
between TH and TL nanocrystallites is not accompanied by
an improvement of the material’s permanent magnetic per-
formance. In fact, the TH phase dilutes the magnet’s magne-
tization and reduces the energy product.
B. Range of exchange coupling
To analyze the coupling between TH and TL regions we
transform Eq. ~1! into a partial differential equation. Expand-
ing M j5M z@ri1(rj2ri)# in terms of rj2ri and taking into
account that urj2riu5a for simple-cubic lattices with
nearest-neighbor interactions we obtain after short calcula-
tion
a2
z
„@To~r!„M z#1@To~r!2T#M z50. ~2!
Here To(r)5zJo(r)/kB is a local Curie-temperature param-
eter so that To(r)5TL and To(r)5TH in the TL and TH
phases, respectively. For homogeneous magnets, M z(r)
5M s and TC5To5TL5TH , but in general Eq. ~2! has to be
interpreted as a generalized Landau–Ginzburg equation. It
differs from the corresponding expression in Ref. 11 by con-
taining the term „@To(r)„M z# rather than the „2M z
Landau–Ginzburg term. As analyzed in the context of two-
phase micromagnetism,15 the main effect of the inhomoge-
neous differential operator is to make the gradient of the
eigenfunction discontinuous at phase boundaries. This modi-
fies TC but leaves the behavior of the spontaneous magneti-
zation in a given phase unchanged.
Let us now consider the spontaneous magnetization in
the TL phase. The interesting temperature region is between
T* and TC , because above TC there is no ferromagnetism
and below TL the spontaneous magnetization is close to satu-
ration throughout the magnet. We ask to what extent the TH
phase is able to create an appreciable spontaneous magneti-
zation in the TL phase. In the considered regime, Eq. ~2!
yields correlations which decay exponentially with the decay
length ~correlation length!
FIG. 1. Mean-field Curie temperatures of three-dimensional composites.
The dark and bright areas correspond to high and low bulk Curie tempera-
tures TH51000 K and TL5400 K, respectively.
FIG. 2. Temperature dependence of the local spontaneous magnetization
~classical Heisenberg model!. There are four nonequivalent sites: one TH
site and three TL sites. The TL are classified by their position relative to the
TH site: nearest neighbors, neighbors connected by face diagonals, and
neighbors connected by cube diagonals. The Curie temperature is 611.9 K.
4757J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 R. Skomski and D. J. Sellmyer
Downloaded 17 Nov 2006 to 129.93.16.206. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
jL5
a
Az
A TLT2TL. ~3!
The correlation length diverges at TL , but it decreases with
increasing temperature and is of the order of
(a/Az)ATL /(TH2TL) in the vicinity of the Curie tempera-
ture. In other words, the coupling or ‘‘penetration’’ length is
large when TH’TL but does not exceed a few interatomic
distances for TL,TH’TC , that is for materials of interest in
two-phase permanent magnetism. The exponential decay of
the spontaneous magnetization M z(r) in TL regions quanti-
fies our finding that M z(r) is very small in the centers of
extended TL regions.
This scenario is not restricted to the mean-field approxi-
mation. Due to renormalization-group effects it is necessary
to replace Eq. ~3! in the vicinity of TL by18
jL’aS TLT2TLD
n
, ~4!
where n is equal to 0.63 and 0.71 for the three-dimensional
Ising and Heisenberg models, respectively. However, in the
present context Eq. ~4! yields only minor corrections to Eq.
~3!. For example,18 grain radii R of order 5 nm give rise to an
‘‘effective Curie-temperature enhancement’’ of about 1%.
III. DISCUSSION
The spacial variation of the spontaneous magnetization
and the existence of a unique Curie temperature TC mean
that there are neither element-specific magnetization curves
in inhomogeneous structures nor order parameters vanishing
at different temperatures. However, depending on the phases
and coupling mechanism involved, the site-dependent aver-
age spontaneous magnetization of the TL phase may be vir-
tually undetectable.13 In that case, an exteremely high tem-
perature resolution is necessary to distinguish a composite
structure from a superposition of single-phase structures.
It is important to emphasize that Eq. ~1! is model inde-
pendent, that is for appropriately rescaled exchange con-
stants it applies equally well to, e.g., Ising and Heisenberg
models. By contrast, due to the nontrivial involvement of the
Brillouin functions, the temperature dependence of the local
spontaneous magnetization is lattice- and model-dependent.6
However, this does not affect the characteristic involvement
of the different nonequivalent sites shown in Fig. 2.18
Aside from the secondary dependence on T2TL , the
coupling length scales as the interatomic distance, that is
ultimately as the Bohr radius a050.529 Å. This must be
compared with the ‘‘micromagnetic’’ coupling length ~ex-
change length! a0 /a57.52 nm, where a51/137 is Sommer-
feld’s fine structure constant.19 To explain this difference we
note that the inhomogenity To(r)2T in Eq. ~2! is analog to
the inhomogenity term 2K1(r)2m0HNM s in Ref. 15. How-
ever, anisotropies (K1) and nucleation fields ~coercivities!
HN are small relativistic corrections to the leading Coulomb
and exchange energies. ~In temperature units, they are of the
order of 1 K, that is much smaller than TC .16! Since both the
micromagnetic coupling in Ref. 15 and the present ‘‘Curie-
temperature’’ interaction are realized by interatomic ex-
change, Curie-temperature disorder is much more effective
in reducing the magnetic order than anisotropy disorder.
IV. CONCLUSIONS
In nanocomposites, there is only one Curie temperature
TC , independent of the bulk Curie temperatures of phases
involved. The Curie temperature is higher than the volume
average of the Curie temperatures of the individual phases.
Below TC , the spontaneous magnetization is site dependent
but nonzero throughout the magnet, because regions with
strong exchange coupling polarize all other sites. However,
in the middle of extended regions with low single-phase Cu-
rie temperatures the spontaneous magnetization may be very
low, and for large crystallite sizes the overall behavior of the
magnet is reminiscent of an interaction-free ensemble of two
~or more! ferromagnets with two ~or more! Curie tempera-
tures. Due to the nonrelativistic character of the interatomic
exchange, the Curie-temperature coupling range is atomic
rather than nanoscale, so that the comparatively high Curie
temperatures of nanocomposites do not translate into an en-
hanced permanent-magnet energy product.
ACKNOWLEDGMENT
This work is supported by DOE-DE-FG-03-98ER45703.
1 W. Heisenberg, Z. Phys. 49, 619 ~1928!.
2 E. Ising, Z. Phys. 31, 253 ~1925!.
3 L. Onsager, Phys. Rev. 65, 117 ~1944!.
4 K. G. Wilson, Rev. Mod. Phys. 55, 583 ~1983!.
5 K. Binder and P. C. Hohenberg, Phys. Rev. B 9, 2194 ~1974!.
6 R. Skomski, C. Waldfried, and P. A. Dowben, J. Phys.: Condens. Matter
10, 5833 ~1998!.
7 H.-R. Ma and Ch.-H. Tsai, Solid State Commun. 55, 499 ~1985!.
8 H. K. Sy, Phys. Lett. A 120, 203 ~1987!.
9 W. Maciejewski and A. Duda, Solid State Commun. 64, 557 ~1987!.
10 W. Maciejewski, IEEE Trans. Magn. 26, 213 ~1990!.
11 R. W. Wang and D. L. Mills, Phys. Rev. B 46, 11681 ~1992!.
12 R. Skomski, J. Appl. Phys. 83, 6724 ~1998!.
13 U. Bovensiepen, F. Wilhelm, P. Srivastava, P. Poulopoulos, M. Farle, A.
Ney, and K. Baberschke, Phys. Rev. Lett. 81, 2368 ~1998!.
14 I. Navarro, M. Ortun˜o, and A. Hermando, Phys. Rev. B 53, 11656 ~1996!.
15 R. Skomski and J. M. D. Coey, Phys. Rev. B 48, 15812 ~1993!.
16 R. Skomski and J. M. D. Coey, Permanent Magnetism ~Institute of Phys-
ics, Bristol, 1999!.
17 A. Herpin, The´orie du Magne´tisme ~Institut National des Sciences et
Techniques Nucle´aires, Saclay, 1968!.
18 R. Skomski ~unpublished!.
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4758 J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 R. Skomski and D. J. Sellmyer
Downloaded 17 Nov 2006 to 129.93.16.206. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp


Curie temperature of multiphase nanostructures

DigitalCommons@University of Nebraska

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln David Sellmyer Publications Research Papers in Physics and Astronomy May 2000 Curie temperature of multiphase nanostructures Ralph Skomski University of Nebraska-Lincoln, rskomski2@unl.edu David J. Sellmyer University of Nebraska-Lincoln, dsellmyer@unl.edu Follow this and additional works at: https://digitalcommons.unl.edu/physicssellmyer  Part of the Physics Commons Skomski, Ralph and Sellmyer, David J., "Curie temperature of multiphase nanostructures" (2000). David Sellmyer Publications. 66. https://digitalcommons.unl.edu/physicssellmyer/66 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in David Sellmyer Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Curie temperature of multiphase nanostructuresR. Skomskia) and D. J. SellmyerDepartment of Physics and Astronomy and Center for Materials Research and Analysis,University of Nebraska, Lincoln, Nebraska 68588The Curie temperature and the local spontaneous magnetization of ferromagnetic nanocompositesare investigated. The macroscopic character of the critical fluctuations responsible for the onset offerromagnetic order means that there is only one Curie temperature, independent of the number ofmagnetic phases present. The Curie temperature increases with the grain size and is, in general,larger than predicted from the volume averages of the exchange constants. However, theCurie-temperature enhancement is accompanied by a relative reduction of the spontaneousmagnetization. Due to the quadratic dependence of the permanent-magnet energy product on thespontaneous magnetization, this amounts to a deterioration of the magnets performance. The lengthscale on which an effective intergranular exchange coupling is realized ~coupling length! dependson the Curie-temperature difference between the phases and on the spacial distribution of the localinteratomic exchange. As a rule, it is of the order of a few interatomic distances; for much biggergrain sizes the structures mimic an interaction-free ensemble of different ferromagnetic materials.This must be compared to the magnetic-anisotropy coupling length, which is of the order of 10 nm.The difference is explained by the nonrelativistic character of the Curie-temperature problem.© 2000 American Institute of Physics. @S0021-8979~00!63608-3#I. INTRODUCTIONThe onset of long-range order in ferromagnets has beena long-standing and mind-stimulating problem. Since thefirst third of the 20th century, when research focussed on thebasic relation between quantum mechanics and magneticorder,1,2 interest has shifted towards critical phenomena3,4and the behavior of particular magnetic features such assurfaces5,6 and artificial structures.7–13 In the case of multi-layers, mean-field Curie temperatures are obtained by diago-nalizing tridiagonal exchange-interaction matrices,7–10whereas Wang and Mills11 used a Landau–Ginzburg typeanalysis to elaborate that the spontaneous magnetization var-ies continuously across phase boundaries.A problem of scientific and technological interest is theCurie-temperature and spontaneous-magnetization behaviorof nanocomposites. There are attempts to use the concept ofexchange-field penetration known from multilayers to con-struct effective coupling parameters,14 but there is no com-prehensive theoretical understanding of nanocrystalline al-loys. A particular question, important in the context of high-temperature permanent magnetism, is whether the Curietemperature TL of any magnetic material can be improved bynanoscale exchange coupling to a phase having a Curie tem-perature TH.TL . In the context of magnetic hysteresis, ithas been shown that interatomic exchange can be exploitedto improve the hard-magnetic performance of a magnet byadding soft-magnetic material.15,16 In that case, the couplingbetween the hard and soft regions is very effective on lengthscales up to about 10 nm.II. CALCULATION AND RESULTSA. Curie temperature and spontaneous magnetizationLet us start with a mean-field analysis of ferromagneticnanocomposites. By definition, the spontaneous magnetiza-tion vanishes at the Curie temperature, so that the mean-fieldequations can be linearized and the spontaneous magnetiza-tion M i5M z(ri) of the ith atom obeysTM i51kB (j J i jM j . ~1!In this equation, the Ji j5J(ri2rJ) are local exchange pa-rameters whose detailed meaning depends on the underlyingmodel. Diagonalization of Eq. ~1! yields a common Curietemperature, which is equal to the highest eigenvalue of thematrix Ji j /kB . For multilayers, it has been found that theCurie temperature increases with increasing layerperiodicity.7,9 In the case of three-dimensional nanostruc-tures, of arbitrary geometry, the matrix Ji j /kB is no longertridiagonal and the diagonalization is slightly more compli-cated. Figure 1 shows cubic inclusions of a material with ahigh ~bulk! Curie temperature TH in a matrix of a materialhaving a low ~bulk! Curie temperature TL,TH . The barsshow the Curie temperatures for TH51000 K, TL5400 K,and an intermediate nearest-neighbor interface exchange ofJi j /kB5700/6. As in the case of multilayers, TC increaseswith the size of the TH and TL regions, and the compositeCurie temperature is close to TH .The eigenmode belonging to TC gives the spacial varia-tion of the order parameter in the vicinity of TC . @The othereigenvalues and eigenmodes of Eq. ~1! have no physicalmeaning, because the spontaneous magnetization is finite be-low TC and the mean-field equations are no longer linear.#Below TC , the Brillouin-type nonlinearity of the mean-fielda!Electronic mail: rskomski@unlserve.unl.eduJOURNAL OF APPLIED PHYSICS VOLUME 87, NUMBER 9 1 MAY 200047560021-8979/2000/87(9)/4756/3/$17.00 © 2000 American Institute of PhysicsDownloaded 17 Nov 2006 to 129.93.16.206. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jspequations is important. For N nonequivalent sites, the localspontaneous magnetization M i(T) is obtained by solving theset of N nonlinear mean-field equations. Figure 2 shows atypical result for N54 and classical (J5‘) Heisenbergspins. The considered structure exhibits four nonequivalentsites, one TH site and three TL sites, and each nonequivalentsite has a different spontaneous magnetization below TC . Inthe TL phase, the spontaneous magnetization is largest forsites which are nearest neighbors to TH sites. This is relatedto the above-mentioned continuity of the order parameter atphase boundaries and shows that the spontaneous magneti-zation is not element specific but depends on the distancefrom the TH – TL interfaces.The lower single-phase Curie temperature TL does nottranslate into any phase transition, although there areresonance-like magnetization features at some nonsingulartemperature T*’TL ~Fig. 2!. This is reminiscent of themany-sublattice behavior of garnets and rare-earth transition-metal intermetallics:12,16,17 the rare-earth ~or ‘‘TL’’! atomsare subject to the quasiparamagnetic exchange field of theiron-series ~or ‘‘TH’’! atoms.Above the thermodynamically not very well definedtemperature T*, the spontaneous magnetization for TL sitesfar away from TH sites is very low, but it remains nonzerobetween T* and TC . This smallness of the TL magnetizationis of practical importance. In particular, the quadratic depen-dence of the permanent-magnet energy product on the spon-taneous magnetization15,16 means that the exchange couplingbetween TH and TL nanocrystallites is not accompanied byan improvement of the material’s permanent magnetic per-formance. In fact, the TH phase dilutes the magnet’s magne-tization and reduces the energy product.B. Range of exchange couplingTo analyze the coupling between TH and TL regions wetransform Eq. ~1! into a partial differential equation. Expand-ing M j5M z@ri1(rj2ri)# in terms of rj2ri and taking intoaccount that urj2riu5a for simple-cubic lattices withnearest-neighbor interactions we obtain after short calcula-tiona2z„@To~r!„M z#1@To~r!2T#M z50. ~2!Here To(r)5zJo(r)/kB is a local Curie-temperature param-eter so that To(r)5TL and To(r)5TH in the TL and THphases, respectively. For homogeneous magnets, M z(r)5M s and TC5To5TL5TH , but in general Eq. ~2! has to beinterpreted as a generalized Landau–Ginzburg equation. Itdiffers from the corresponding expression in Ref. 11 by con-taining the term „@To(r)„M z# rather than the „2M zLandau–Ginzburg term. As analyzed in the context of two-phase micromagnetism,15 the main effect of the inhomoge-neous differential operator is to make the gradient of theeigenfunction discontinuous at phase boundaries. This modi-fies TC but leaves the behavior of the spontaneous magneti-zation in a given phase unchanged.Let us now consider the spontaneous magnetization inthe TL phase. The interesting temperature region is betweenT* and TC , because above TC there is no ferromagnetismand below TL the spontaneous magnetization is close to satu-ration throughout the magnet. We ask to what extent the THphase is able to create an appreciable spontaneous magneti-zation in the TL phase. In the considered regime, Eq. ~2!yields correlations which decay exponentially with the decaylength ~correlation length!FIG. 1. Mean-field Curie temperatures of three-dimensional composites.The dark and bright areas correspond to high and low bulk Curie tempera-tures TH51000 K and TL5400 K, respectively.FIG. 2. Temperature dependence of the local spontaneous magnetization~classical Heisenberg model!. There are four nonequivalent sites: one THsite and three TL sites. The TL are classified by their position relative to theTH site: nearest neighbors, neighbors connected by face diagonals, andneighbors connected by cube diagonals. The Curie temperature is 611.9 K.4757J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 R. Skomski and D. J. SellmyerDownloaded 17 Nov 2006 to 129.93.16.206. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jspjL5aAzA TLT2TL. ~3!The correlation length diverges at TL , but it decreases withincreasing temperature and is of the order of(a/Az)ATL /(TH2TL) in the vicinity of the Curie tempera-ture. In other words, the coupling or ‘‘penetration’’ length islarge when TH’TL but does not exceed a few interatomicdistances for TL,TH’TC , that is for materials of interest intwo-phase permanent magnetism. The exponential decay ofthe spontaneous magnetization M z(r) in TL regions quanti-fies our finding that M z(r) is very small in the centers ofextended TL regions.This scenario is not restricted to the mean-field approxi-mation. Due to renormalization-group effects it is necessaryto replace Eq. ~3! in the vicinity of TL by18jL’aS TLT2TLDn, ~4!where n is equal to 0.63 and 0.71 for the three-dimensionalIsing and Heisenberg models, respectively. However, in thepresent context Eq. ~4! yields only minor corrections to Eq.~3!. For example,18 grain radii R of order 5 nm give rise to an‘‘effective Curie-temperature enhancement’’ of about 1%.III. DISCUSSIONThe spacial variation of the spontaneous magnetizationand the existence of a unique Curie temperature TC meanthat there are neither element-specific magnetization curvesin inhomogeneous structures nor order parameters vanishingat different temperatures. However, depending on the phasesand coupling mechanism involved, the site-dependent aver-age spontaneous magnetization of the TL phase may be vir-tually undetectable.13 In that case, an exteremely high tem-perature resolution is necessary to distinguish a compositestructure from a superposition of single-phase structures.It is important to emphasize that Eq. ~1! is model inde-pendent, that is for appropriately rescaled exchange con-stants it applies equally well to, e.g., Ising and Heisenbergmodels. By contrast, due to the nontrivial involvement of theBrillouin functions, the temperature dependence of the localspontaneous magnetization is lattice- and model-dependent.6However, this does not affect the characteristic involvementof the different nonequivalent sites shown in Fig. 2.18Aside from the secondary dependence on T2TL , thecoupling length scales as the interatomic distance, that isultimately as the Bohr radius a050.529 Å. This must becompared with the ‘‘micromagnetic’’ coupling length ~ex-change length! a0 /a57.52 nm, where a51/137 is Sommer-feld’s fine structure constant.19 To explain this difference wenote that the inhomogenity To(r)2T in Eq. ~2! is analog tothe inhomogenity term 2K1(r)2m0HNM s in Ref. 15. How-ever, anisotropies (K1) and nucleation fields ~coercivities!HN are small relativistic corrections to the leading Coulomband exchange energies. ~In temperature units, they are of theorder of 1 K, that is much smaller than TC .16! Since both themicromagnetic coupling in Ref. 15 and the present ‘‘Curie-temperature’’ interaction are realized by interatomic ex-change, Curie-temperature disorder is much more effectivein reducing the magnetic order than anisotropy disorder.IV. CONCLUSIONSIn nanocomposites, there is only one Curie temperatureTC , independent of the bulk Curie temperatures of phasesinvolved. The Curie temperature is higher than the volumeaverage of the Curie temperatures of the individual phases.Below TC , the spontaneous magnetization is site dependentbut nonzero throughout the magnet, because regions withstrong exchange coupling polarize all other sites. However,in the middle of extended regions with low single-phase Cu-rie temperatures the spontaneous magnetization may be verylow, and for large crystallite sizes the overall behavior of themagnet is reminiscent of an interaction-free ensemble of two~or more! ferromagnets with two ~or more! Curie tempera-tures. Due to the nonrelativistic character of the interatomicexchange, the Curie-temperature coupling range is atomicrather than nanoscale, so that the comparatively high Curietemperatures of nanocomposites do not translate into an en-hanced permanent-magnet energy product.ACKNOWLEDGMENTThis work is supported by DOE-DE-FG-03-98ER45703.1 W. Heisenberg, Z. Phys. 49, 619 ~1928!.2 E. Ising, Z. Phys. 31, 253 ~1925!.3 L. Onsager, Phys. Rev. 65, 117 ~1944!.4 K. G. Wilson, Rev. Mod. Phys. 55, 583 ~1983!.5 K. Binder and P. C. Hohenberg, Phys. Rev. B 9, 2194 ~1974!.6 R. Skomski, C. Waldfried, and P. A. Dowben, J. Phys.: Condens. Matter10, 5833 ~1998!.7 H.-R. Ma and Ch.-H. Tsai, Solid State Commun. 55, 499 ~1985!.8 H. K. Sy, Phys. Lett. A 120, 203 ~1987!.9 W. Maciejewski and A. Duda, Solid State Commun. 64, 557 ~1987!.10 W. Maciejewski, IEEE Trans. Magn. 26, 213 ~1990!.11 R. W. Wang and D. L. Mills, Phys. Rev. B 46, 11681 ~1992!.12 R. Skomski, J. Appl. Phys. 83, 6724 ~1998!.13 U. Bovensiepen, F. Wilhelm, P. Srivastava, P. Poulopoulos, M. Farle, A.Ney, and K. Baberschke, Phys. Rev. Lett. 81, 2368 ~1998!.14 I. Navarro, M. Ortun˜o, and A. Hermando, Phys. Rev. B 53, 11656 ~1996!.15 R. Skomski and J. M. D. Coey, Phys. Rev. B 48, 15812 ~1993!.16 R. Skomski and J. M. D. Coey, Permanent Magnetism ~Institute of Phys-ics, Bristol, 1999!.17 A. Herpin, The´orie du Magne´tisme ~Institut National des Sciences etTechniques Nucle´aires, Saclay, 1968!.18 R. Skomski ~unpublished!.19 R. Skomski, H.-P. Oepen, and J. Kirschner, Phys. Rev. B 58, 3223 ~1998!.4758 J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 R. Skomski and D. J. SellmyerDownloaded 17 Nov 2006 to 129.93.16.206. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

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D. J. Sellmyer

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Curie temperature of multiphase nanostructures

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