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Finite-size effects in the magnetic properties of ferromagnetic clusters
Journal of Magnetism and Magnetic Materials 104- i1)7 (1992) 1577-1571~ North-Holland
~muw~c I r u m~mJs
Finite-size effects in the magnetic properties of ferromagnetic clusters P.V. H e n d r i k s e n
a, S. L i n d e r o t h
" a n d P.-A. Lindg'~rd t,
'~Laboratory of Applied Physics, Technical Unit'c,.sity of Denmark, DK-2800 Lyngby, Denmark t, Physics Department, Riso National Laboratot3.', DK-4800 Roskilde, Demnark Model calculations of the spontaneous magnetization of ferromagnetic clusters of various structm'es and sizes have been performed. The exchange interaction is modelled with the Heisenberg model, and a self-consistent spin-wave-spectrum is found by direct diagonalization of the equation of motion of S +. Mean-field calculations are performed tbr comparison and in order to infer high-temperature behaviour. The finite size is found to cause large deviation from the normal Bloch T3/2-1aw for the spontaneous magnetization at low temperatures and to lower the Curie temperature. In a recent study of small Fe clusters with 50-230 atoms [1] it was found that the average moment per atom increases with size, yet being below the bulk value. Some of these effects were addressed in a Monte Carlo study of lsing clusters [2]. We anticipate that the Heisenberg model is more approoriate and study in th,~s paper in particular the low-temperature behaviour. In deriving the magnetic properties of a bulk Heisenberg ferromagnet one exploits the translational invariance symmetry. This does not hold in the limit of very small systems, and therefore the predictions of the bulk-model, e.g. Bioch's T3/2-1aw, cannot bc extended to these. The limited number of degrees of freedom. on the other hand, allows a numerical approach to the problem. Consider a Heiscnbcrg Hamiltonian for a finite size cluster: I
~ ' = - S E J , jS, " Sj 6
(1)
where Jij is the exchange energy constant, S~ and Sj are the spin angular momenta on the sites i and j. The summation is extended over nearest neighbours only. The quantum mechanical equation of motion, describing the propagation of a spin deviation, can be written:
w S / = 2 J ~ , m , S 7 - Y'-Ji, i
(mi+mj) 2 S?,
The lack of full coordination at the surface of the cluster can bc expected to lead to larger spin deviations in this region than in the central part of the cluster. Therefore, rn; should be allowed to vary selfconsistently with position as well as with temperature. This is obtained in the following manner. The eigenvalues (Et,) and eigenvectors (qit, i) are found iteratively by diagonalizing eq. (2) starting from an initial assumption on the profile m,. The thermal mean value of the spin projection on the z-axis m, on a site i is then found by adding the statistically weighted deviations on this particular site (i) of all the cigcnstatcs (p):
m, = 1 - ~ I ~,,, I -'n,,,
where nt, is the Bose-weight of lhc state, ttereaftcr, the diagonalization of eq. (2) is repeated with the new values of m~, until the mean ,~aluc of the profile converges. We make sure that this convergence is double-sided, i.e. the final Z m, does not depend on whether the starting guess is smaller or larger than the accepted solution. In the mean-field calculations the magnetization on site i is given by: C H'/kaT-
(2)
(3)
P
e
tt'/kIJ
mi = 1 + e H ' / k d r + e tt,/~,(r"
(4)
i
where S+=Sx + iSy, and where m i and mj are lhc thermal averaged mean values of the spin projection on the z-axis at the sites i and j. In the translational invariant system, where m, is equal at all sites, cq. (2) can be diagonalized by a Fourier transformation to q-space. The proper statistical weighting of the eigenstates leads to the familiar T3/Z-law for the temperature dependence of the spontaneous magnetization. In the case of a cluster eq. (2) has to be solved in real space numerically; q is no longer a good quantum number.
where H, = ~,sJ,,sma is the mean-field on site i (6 runs over all nearest neighbours)_ In the me:m-field calculzttions the field distribution and magnetization profile arc calculated succcs:~ively until convergence in the mean magnetization is obtained. The temperature dependences of the calculated average magnetization of different sized fcc clusters are shown in fig. 1. The behaviour of the clusters is distinctly different from that of the infinite system. This can be easily understood as an effect of the limited number of degrees of freedom, which leads to an
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
P. I/. th'ndriksen et aL / bTnite size ctffects
1578
energy gap in the spin-wave spectrum, rcsultim,, in a llat curve at vcD, low temperatures. The magnetization decreases faster at higher temperatures in the clusters than in the hulk duc to lacking coordination at the surface. These remarks arc consistent wilh the facts that for the larger cluster sizes (i) the initial decrease is faster bccausc of a smallcr cnergy gap and (ii) the decrease at higher tcmpcraturcs is slower because of the smaller rclativc cffcct of the surfacc. The obscrvcd size dependence of the high-temperature bchaviour is in qualitative agreement ~;th recent Monte Carlo calculations [2]. it should bc noted that the temperature dependence of the magnetization of a small cluster, obtained from the spin-wave calcuhttion, turns out to bc close to an effective T 2 dcpcndcncc. The effect oi' tlac missing nearest ncighhours in the surface laycrs is further illustrated in fig. 2, where the magnetization o1' the different shells in an fcc cluster containing 225 spins arc shown. The magnetization is sccn to drop much faster in the outer regions than in the interior. Such a bchaviour has bccn reported for 4 nm t~-iron particles [3]. The spin-wavc calculation yields a much faster decrease in the surface magnetization than the mean-field calcuhttion. Thc Curic temperature, Tc, found from thc mcanfield calculation for this clustcr is approximately kB7 ~, = 7.0 J which should bc comparcd with 8.0 J for the infinite system. The lowering of the Curie temperature is ftmnd regardicss o f structurc (fcc or bcc), and gcts cvcn More pronoutlccd with decreasing cluster size (up to 25ck for an fcc cluster of 55 spins). Expcrinlcntal
1,020
0,980
":..-, ~
....
\\..~
<\'~"x~
,----g
",~,\
,~. 0.940
X ~
....
0.7
Spin-wave theory
°i!
0.65 O.
0"5"30 0:2 0;4 0:6 0'.8 i 112 1:4 1:6 118 2. 2:2 214 2.6 Temperature [J/k] 1 0.9" O.B" 0.7'~'
0.6-
:~ o.5.
M e a n field theory
0.4"
03" 0.2' 0.1 0
0
i
2
~
a
g
6
#
8
Temperature [J/k] Fig. 2. r h c temperature d e p e n d e n c e of the spoptancous magnclizalion of an fcc cluster having 11 shclls (225 spins) calculated from lhc spin-wave spectrum (upper) and from meanfield thcory (Iowcr). ('urvcs for different laycrs (shells), as well as the mcan-valuc, arc shown.
13
381 infinite
\\\ ',
°t
~. O.B
55
~, \ \
0,900
°:1
\\
0,860 ................................... ~......................k "~ __ ] O0 1.0 2.0 3,0 Fig. I. The Icnlpcraturc dependence of the spontaneous magnetization calculated from the spin-wave spectrum for three tliffclcnt sizctl fcc cluslcrs. For comparison is also shown the dcpcntlcncc for the infinite Icc-lalticc. The clusters arc spherical with at central spin.
observation of lowered Curic tcnlpcraturcs have bcen reported for other low-symmetry systems such as monolaycrs [4]. Although the spin-wave results are restricted to low temperature the same trend is observed when following the temperature Tu.75 at which the magnetization is reduced to M = 0.75M u. A study shows that T,.7s not only increases with cluster size, but also vary in a systematic way (magic numbers) with the magnetic 'softness' of the cluster, i.e. the rclativc number of ~tttl~ttlsU, t,t,tiU~--lllU~pL:LitiVU Ol SLIHC[Ul'C, (i'cc o r bcc). This leads us to suggest that the obtained results arc representative also for other, and less perfect, cluster structures. Prcliminary calculations, whcrc disorder or holes have bccn introduccd in the latticc supports this. In summary, bascd on a spin-wavc calculation for the tlciscnbcrg model wc have clucidatcd thc cffcct of l'inite-sizc fin small clusters. We havc neglected the effect of varying exchange interaction induced by struc-
P.V. th,mtriksen et al. / Finite size effects
rural relaxation in order to focus on the cffecis of the finite size. The results obtained are expressed in terms of magnetization profiles and in terms of the experimentally more direct observable mean magnetization. The trends of the simple model calculations are in agreement with experimental observations.
157'9
References [!] W.A. de itecr, P. Milani and A. ('hatclain, Phy~,. Roy. Left. 65 (1900) 488.
[21 J. Mcrikoski.
J. Timoncn. M. Mannincn and P. Jcna. Phy~,
Rev. Left. 6f) (1991) 93S. [3] F. Bodker, S. Morup, C.A. Oxbt)rrow, M.B. Madsen and J.W. Niemantsverdriet, tiffs conference. [4] M. Przybylski, J. Korecki and U. Gradmann, Appl. Phys. A 52 (1991) 33-47.
~muw~c I r u m~mJs
Finite-size effects in the magnetic properties of ferromagnetic clusters P.V. H e n d r i k s e n
a, S. L i n d e r o t h
" a n d P.-A. Lindg'~rd t,
'~Laboratory of Applied Physics, Technical Unit'c,.sity of Denmark, DK-2800 Lyngby, Denmark t, Physics Department, Riso National Laboratot3.', DK-4800 Roskilde, Demnark Model calculations of the spontaneous magnetization of ferromagnetic clusters of various structm'es and sizes have been performed. The exchange interaction is modelled with the Heisenberg model, and a self-consistent spin-wave-spectrum is found by direct diagonalization of the equation of motion of S +. Mean-field calculations are performed tbr comparison and in order to infer high-temperature behaviour. The finite size is found to cause large deviation from the normal Bloch T3/2-1aw for the spontaneous magnetization at low temperatures and to lower the Curie temperature. In a recent study of small Fe clusters with 50-230 atoms [1] it was found that the average moment per atom increases with size, yet being below the bulk value. Some of these effects were addressed in a Monte Carlo study of lsing clusters [2]. We anticipate that the Heisenberg model is more approoriate and study in th,~s paper in particular the low-temperature behaviour. In deriving the magnetic properties of a bulk Heisenberg ferromagnet one exploits the translational invariance symmetry. This does not hold in the limit of very small systems, and therefore the predictions of the bulk-model, e.g. Bioch's T3/2-1aw, cannot bc extended to these. The limited number of degrees of freedom. on the other hand, allows a numerical approach to the problem. Consider a Heiscnbcrg Hamiltonian for a finite size cluster: I
~ ' = - S E J , jS, " Sj 6
(1)
where Jij is the exchange energy constant, S~ and Sj are the spin angular momenta on the sites i and j. The summation is extended over nearest neighbours only. The quantum mechanical equation of motion, describing the propagation of a spin deviation, can be written:
w S / = 2 J ~ , m , S 7 - Y'-Ji, i
(mi+mj) 2 S?,
The lack of full coordination at the surface of the cluster can bc expected to lead to larger spin deviations in this region than in the central part of the cluster. Therefore, rn; should be allowed to vary selfconsistently with position as well as with temperature. This is obtained in the following manner. The eigenvalues (Et,) and eigenvectors (qit, i) are found iteratively by diagonalizing eq. (2) starting from an initial assumption on the profile m,. The thermal mean value of the spin projection on the z-axis m, on a site i is then found by adding the statistically weighted deviations on this particular site (i) of all the cigcnstatcs (p):
m, = 1 - ~ I ~,,, I -'n,,,
where nt, is the Bose-weight of lhc state, ttereaftcr, the diagonalization of eq. (2) is repeated with the new values of m~, until the mean ,~aluc of the profile converges. We make sure that this convergence is double-sided, i.e. the final Z m, does not depend on whether the starting guess is smaller or larger than the accepted solution. In the mean-field calculations the magnetization on site i is given by: C H'/kaT-
(2)
(3)
P
e
tt'/kIJ
mi = 1 + e H ' / k d r + e tt,/~,(r"
(4)
i
where S+=Sx + iSy, and where m i and mj are lhc thermal averaged mean values of the spin projection on the z-axis at the sites i and j. In the translational invariant system, where m, is equal at all sites, cq. (2) can be diagonalized by a Fourier transformation to q-space. The proper statistical weighting of the eigenstates leads to the familiar T3/Z-law for the temperature dependence of the spontaneous magnetization. In the case of a cluster eq. (2) has to be solved in real space numerically; q is no longer a good quantum number.
where H, = ~,sJ,,sma is the mean-field on site i (6 runs over all nearest neighbours)_ In the me:m-field calculzttions the field distribution and magnetization profile arc calculated succcs:~ively until convergence in the mean magnetization is obtained. The temperature dependences of the calculated average magnetization of different sized fcc clusters are shown in fig. 1. The behaviour of the clusters is distinctly different from that of the infinite system. This can be easily understood as an effect of the limited number of degrees of freedom, which leads to an
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
P. I/. th'ndriksen et aL / bTnite size ctffects
1578
energy gap in the spin-wave spectrum, rcsultim,, in a llat curve at vcD, low temperatures. The magnetization decreases faster at higher temperatures in the clusters than in the hulk duc to lacking coordination at the surface. These remarks arc consistent wilh the facts that for the larger cluster sizes (i) the initial decrease is faster bccausc of a smallcr cnergy gap and (ii) the decrease at higher tcmpcraturcs is slower because of the smaller rclativc cffcct of the surfacc. The obscrvcd size dependence of the high-temperature bchaviour is in qualitative agreement ~;th recent Monte Carlo calculations [2]. it should bc noted that the temperature dependence of the magnetization of a small cluster, obtained from the spin-wave calcuhttion, turns out to bc close to an effective T 2 dcpcndcncc. The effect oi' tlac missing nearest ncighhours in the surface laycrs is further illustrated in fig. 2, where the magnetization o1' the different shells in an fcc cluster containing 225 spins arc shown. The magnetization is sccn to drop much faster in the outer regions than in the interior. Such a bchaviour has bccn reported for 4 nm t~-iron particles [3]. The spin-wavc calculation yields a much faster decrease in the surface magnetization than the mean-field calcuhttion. Thc Curic temperature, Tc, found from thc mcanfield calculation for this clustcr is approximately kB7 ~, = 7.0 J which should bc comparcd with 8.0 J for the infinite system. The lowering of the Curie temperature is ftmnd regardicss o f structurc (fcc or bcc), and gcts cvcn More pronoutlccd with decreasing cluster size (up to 25ck for an fcc cluster of 55 spins). Expcrinlcntal
1,020
0,980
":..-, ~
....
\\..~
<\'~"x~
,----g
",~,\
,~. 0.940
X ~
....
0.7
Spin-wave theory
°i!
0.65 O.
0"5"30 0:2 0;4 0:6 0'.8 i 112 1:4 1:6 118 2. 2:2 214 2.6 Temperature [J/k] 1 0.9" O.B" 0.7'~'
0.6-
:~ o.5.
M e a n field theory
0.4"
03" 0.2' 0.1 0
0
i
2
~
a
g
6
#
8
Temperature [J/k] Fig. 2. r h c temperature d e p e n d e n c e of the spoptancous magnclizalion of an fcc cluster having 11 shclls (225 spins) calculated from lhc spin-wave spectrum (upper) and from meanfield thcory (Iowcr). ('urvcs for different laycrs (shells), as well as the mcan-valuc, arc shown.
13
381 infinite
\\\ ',
°t
~. O.B
55
~, \ \
0,900
°:1
\\
0,860 ................................... ~......................k "~ __ ] O0 1.0 2.0 3,0 Fig. I. The Icnlpcraturc dependence of the spontaneous magnetization calculated from the spin-wave spectrum for three tliffclcnt sizctl fcc cluslcrs. For comparison is also shown the dcpcntlcncc for the infinite Icc-lalticc. The clusters arc spherical with at central spin.
observation of lowered Curic tcnlpcraturcs have bcen reported for other low-symmetry systems such as monolaycrs [4]. Although the spin-wave results are restricted to low temperature the same trend is observed when following the temperature Tu.75 at which the magnetization is reduced to M = 0.75M u. A study shows that T,.7s not only increases with cluster size, but also vary in a systematic way (magic numbers) with the magnetic 'softness' of the cluster, i.e. the rclativc number of ~tttl~ttlsU, t,t,tiU~--lllU~pL:LitiVU Ol SLIHC[Ul'C, (i'cc o r bcc). This leads us to suggest that the obtained results arc representative also for other, and less perfect, cluster structures. Prcliminary calculations, whcrc disorder or holes have bccn introduccd in the latticc supports this. In summary, bascd on a spin-wavc calculation for the tlciscnbcrg model wc have clucidatcd thc cffcct of l'inite-sizc fin small clusters. We havc neglected the effect of varying exchange interaction induced by struc-
P.V. th,mtriksen et al. / Finite size effects
rural relaxation in order to focus on the cffecis of the finite size. The results obtained are expressed in terms of magnetization profiles and in terms of the experimentally more direct observable mean magnetization. The trends of the simple model calculations are in agreement with experimental observations.
157'9
References [!] W.A. de itecr, P. Milani and A. ('hatclain, Phy~,. Roy. Left. 65 (1900) 488.
[21 J. Mcrikoski.
J. Timoncn. M. Mannincn and P. Jcna. Phy~,
Rev. Left. 6f) (1991) 93S. [3] F. Bodker, S. Morup, C.A. Oxbt)rrow, M.B. Madsen and J.W. Niemantsverdriet, tiffs conference. [4] M. Przybylski, J. Korecki and U. Gradmann, Appl. Phys. A 52 (1991) 33-47.