- Email: [email protected]
Magnons in Co dot
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 305 (2006) 182–185 www.elsevier.com/locate/jmmm
Magnons in Co dot J.-C.S. Levya,, M. Krawczykb, H. Puszkarskib a
Materiaux et Phenomenes Quantiques, UMR 7162-CNRS, Universite Paris 7, case 7021, 75251 Paris, Cedex 05, France b Surface Physics Division, Faculty of Physics, UAM ul Umultowska 85, Poznan 61-614, Poland Received 11 July 2005; received in revised form 21 October 2005 Available online 17 January 2006
Abstract Eigen spin wave frequencies and profiles of a cobalt hexagonal dot with exchange and anisotropy energies are derived. The lowest mode frequency is shown to be a linear function of edge anisotropy, so edge anisotropy controls the whole dot magnetization reversal and can be measured from spin wave resonance. The low-temperature dependence of cobalt dot magnetization is shown to be driven by edge anisotropy as well. r 2006 Elsevier B.V. All rights reserved. Keywords: Spin wave resonance; Magnetic dots; Anisotropy
Introduction Low-dimensionality magnetic materials become more and more common in nowadays physics. Dilute magnetic semiconductors [1] where magnetic clusters are more or less ordered according to the annealing processes are an example of low-dimension magnetism. This is also the case of magnetic ultrathin films [2] and of magnetic dots [3] which are often deposited as array of dots [4] or periodic arrays of dots [5]. The recently improved spin wave resonance resolution and the additive contribution of arrays of dots makes spin wave resonance of dots possible [3,5]. So the question of spin waves in low-dimensionality magnetic materials arises since spin wave resonance and thermal variations of magnetization can be observed in dots and arrays of dots. A good example could be cobalt dot over gold (1 1 1) since due to the herringbone gold reconstruction [6] regular arrays of cobalt dots can be produced [7]. The new Hamiltonian term to be introduced to account for a dot is just the giant edge anisotropy of border atoms [8] which is observed from X-ray magnetic circular dichroism and has been estimated from ab initio computations to be Corresponding author. Tel.: +33 1 44 27 43 77; fax: +33 1 44 27 28 52.
E-mail address: [email protected] (J.-C.S. Levy). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.12.010
quite large. The distances between dots are assumed to be large enough, so dipole–dipole interaction between dots can be neglected. Dipole–dipole interactions within the dot are weak and can be neglected. So the spectrum of magnetic excitations as well as the eigenmode profiles can be deduced from an estimated value of edge anisotropy. This enables us to predict the result of spin wave resonance on extended arrays of independent dots as the sum of spin wave resonance effects on individual dots. From the profile of the lowest frequency modes, the magnetization reversal of an individual dot and of a set of independent dots is easily understood when neglecting the contribution of higher modes. The knowledge of the spin wave spectrum enables us to deduce the magnetization behavior as a function of temperature at least, at low enough temperature. So the study of spin wave resonance in dots and magnetization measurements induces two indirect measurements of edge anisotropy in dots. This defines the main goal of this work. A secondary goal is to evidence the practical symmetries in a finite sample and their consequence on the spin wave spectrum. This could be compared to experimental results [9]. Section 1 gives the general method to derive normal modes of dots, while the specific points of the practical determination are reported in Section 2. A short conclusive comparison to recent observations is done in Section 3.
ARTICLE IN PRESS J.-C.S. Levy et al. / Journal of Magnetism and Magnetic Materials 305 (2006) 182–185
183
1. Method
1.2. Resonance condition and magnetization
We assume that the static external magnetic field H 0 is normal to the dot plan with the following Hamiltonian [10]: 1X 1X H¼ J f;g; S af Sag I f ðS zf Þ2 2 f;g;a 2 f X gmB H eff S zf , ð1Þ
Resonance fields, in those formulae are calculated from eigenvalues O, Eq. (3):
f
where sites f and g are nearest neighbors with the local single ion anisotropy I f which takes the value I b in the bulk of the dot and I s at the edge of the dot, and the effective magnetic field H eff ¼ H 0 4pN z M S . A uniform static demagnetizing field with the factor N z is assumed to occur. This equation of motion reads in RPA or Tyablikov approximation: ! X d þ S ¼ SJ i_ j f;g þ if S þ f dt f g X þ SJ j f;g S þ ð2Þ g þ gmB H eff S f , g
where normalized exchange and anisotropy integrals, j f;g ; if are introduced. J f;g ¼ Jj f;g ¼ Jdf;g ;
I f ¼ Jif
and
S ¼ hSz i.
g
(3)
g
_o OJS þ 4pN z M S . gmB
The dynamic matrix is defined element by element. The eigenvectors define the spin wave profiles. And the dot magnetization as a function of temperature reads [11] 1 X X _O M¼N nðOÞ ¼ N exp (4) 1 . kT O In the usual case at low temperature where just one lowfrequency mode drives the magnetization behavior it writes: 1 _Omin M ¼ N exp (5) 1 . kT 2. Practical calculations For practical visualization the case n ¼ 5 with N ¼ 91 atoms and 33% of border atoms is shown, but calculations were done from n ¼ 1 up to 10. The reduced bulk anisotropy as deduced for bulk cobalt is: K b ¼ 0:004 [12]. The selected values of material parameters are:
Fourier transformation in time is introduced: Z ðtÞ ¼ eiot Sþ Sþ f f ðoÞ do, and the eigenvalue problem reads: X X þ þ ¼ j S þ i S j f;g S þ OSþ f f;g g f f f
H0 ¼
frequency of the rf field i.e.: o ¼ 2p9:96 GHz; value of exchange integral for bulk cobalt: JS can take values in the range: h1:522:6i1014 (erg). The admitted value is JS ¼ 1:5 1014 (erg) [12]. g factor for cobalt is 2:2 [12]; Bulk spontaneous magnetization for cobalt: M S 1:44 kG [12]; O is an eigenvalue taken from calculations.
with O ¼ _o gmB H eff =JS. With those values : H 0 ¼ 4:4 735:3 OðkGÞ: 1.1. Hexagonal dot 2.1. Results For a hexagonal monolayer of cobalt of edge size n þ 1, there are N ¼ 3n2 þ 3n þ 1 cobalt atoms of which 6n 6 are edge atoms and six are corner atoms so three types of diagonal elements of the dynamic matrix occur: for bulk spins: 6j þ i and i ¼ K b the reduced bulk anisotropy; for edge spins: 4j þ i and i ¼ K s the reduced edge anisotropy; and for corner spins: 3j þ i and i ¼ K s the reduced edge anisotropy, i.e. edge and corner anisotropy are not distinguished here, so there are N s ¼ 6n effective edge atoms. Off diagonal elements (for neighbors spins only) are equal to 1.
The DC field for resonance as a function of the edge anisotropy K s is shown in Fig. 1 for the first mode with n ¼ 5, i.e. a 91 atoms hexagon. It shows a linear variation of the first mode resonant field with the edge anisotropy as observed for thin films [10], with surface anisotropy instead of edge anisotropy. The anisotropy acts as a local field. So the reduced frequency O is zero when the total anisotropy field is zero, i.e. when N s K s þ ðN N s ÞK b ¼ 0, i.e. K s;0 ¼ ððN N s Þ=N s ÞK b ¼ 0:008133 as observed in Fig. 1. More precisely for a hexagonal dot of n concentric layers, i.e. of edge size n, the reduced frequency O is zero when K s;0 ¼ ðð3n2 3n þ 1Þ=6nÞK b an hyperbolic function of n. This law is perfectly followed for no9. Of course, for this value K s;0 of edge anisotropy there is an instability of the magnetization orientation.
ARTICLE IN PRESS 184
J.-C.S. Levy et al. / Journal of Magnetism and Magnetic Materials 305 (2006) 182–185
Fig. 1. The resonance field and reduced frequency of the first spin wave mode as a function of edge anisotropy K s for a cluster of 91 Co atoms. Note this is a straight line with a zero frequency for K s ¼ 0:0081: The spin wave profile of the first mode is reported in inset (a) when K s ¼ 0:01, (b) when K s ¼ 0:004 and (c) when K s ¼ 0:005. Note the uniform mode in (b) and the edge mode in (c).
Fig. 3. The profiles of the first ten modes for 91 atoms with K s ¼ 0:01. Note the sixfold symmetry and the low value of wavevectors.
Fig. 2. The full spectrum of spin wave modes for 91 Co atoms with K s ¼ 0:01. Note the degeneracies and the finite gaps. For the lowest modes the density of states is constant, OðmÞ ¼ lm.
Fig. 2 gives us the spectrum of eigenfrequencies when K s ¼ 0:01. The lowest frequency mode is a single mode while gaps separate higher modes. Many modes are degenerate. At this scale the frequency spectrum does not show any effect when submitted to a change of the edge anisotropy. This is due to the fact that anisotropy is quite negligible in front of exchange. The profiles of the first ten modes are reported in Fig. 3 with evidence for sixfold symmetry and lowest wavevectors. These calculations are reported in the realistic case where K s ¼ 0:01 which is not so far from anisotropy observations [8]. Not shown spectra for other values of edge anisotropy confirm this mode classification and even the gap values. Modes 2 and 3 are degenerate as observed in Fig. 2. Modes 4 and 5 are also degenerate. The only profile which is sensitive to the edge anisotropy is the first mode profile, as well known for thin films where this mode can become a surface mode [11]. The effect of
edge anisotropy variation on the lowest spin wave mode profile is shown in the three insets of Fig. 1. Three different values of edge anisotropy K s are introduced. When edge anisotropy is equal to bulk anisotropy, i.e. K s ¼ 0:004, the first mode is perfectly uniform, so the magnetization reversal consists in a perfect Stoner–Wolfarth uniform precession. When edge anisotropy K s ¼ 0:01 is a few times the bulk anisotropy, edge spins are more fastened than bulk spins, the first mode is pinned at the surface. So the dot magnetization reversal follows this first mode profile, i.e. induces a strong rotation of bulk spins and a smaller rotation of border spins. This is a ‘‘hard’’ magnetic edge. When edge anisotropy is lower than bulk anisotropy, here K s ¼ 0:005, edge spins are more loosely held than bulk spins. There is an ‘‘edge’’ spin wave, a spin wave localized close to the edge. This singular behavior can be observed by comparing spin wave spectra for different dot sizes [11]. In this case, edge magnetization is more easily rotated than the bulk one. This is a ‘‘soft’’ magnetic edge. As a consequence of these remarks the variation of magnetization with temperature given by Eq. (5) is driven
ARTICLE IN PRESS J.-C.S. Levy et al. / Journal of Magnetism and Magnetic Materials 305 (2006) 182–185
by the first mode frequency and thus strongly depends on the edge anisotropy. This is strictly true when the temperature is lower than T 1 , the temperature deduced from the difference between the frequency values of the first and second mode. In the case of cobalt T 1 ¼ 20 K when n ¼ 5 and T 1 ¼ 30 K when n ¼ 4. When dot size decreases, the energy gap between successive modes is increased. These temperature ranges can be easily reached experimentally. And for temperature lower than T 1 the variation of magnetization with temperature is exponential. The exponent is a linear function of edge anisotropy K s . Of course when the temperature is quite large in front of T 1 , the sum over all modes must be done as in Eq. (5). As a result there is a final linear decrease of magnetization with temperature as usual for 2D magnetic samples [11]. So finally edge anisotropy can be measured either from spin wave resonance or from low-temperature magnetization measurements. 3. Conclusion In the considered case where dipolar (i.e. magnetostatic) interactions are completely negligible since dots are small and distant, the influence of edge anisotropy, i.e. border anisotropy on basic magnetic properties is studied. It is shown that edge anisotropy can be deduced in a linear way from the first resonance field, and so with accuracy. Then the compared value of edge and bulk anisotropy determines the nature of magnetization reversal mechanism with ‘‘hard’’ or ‘‘soft’’ edge behaviors. Finally edge anisotropy has a crucial part in the magnetization variation with temperature at low temperature. In the possible case of a strong negative edge anisotropy which can occur in magnetic dots, the first mode is an edge mode. The edge anisotropy can still be deduced from spin wave resonance. Then the magnetization reversal starts from the border and it is necessary to take into account a first bulk mode to obtain a complete magnetization reversal. In presence of an external field which stabilizes the magnetic state of the sample, the magnetization variation with temperature is once more driven by the edge anisotropy. More generally, similar calculations can be applied to more complex magnetic clusters which appear in low-
185
dimensional magnetism. The present results indicate that spin wave resonance in such clusters must be quite sensitive to geometric details as here to edge anisotropy and to dot size. Similar calculations as reported here give the electronic excitation energy of nanoclusters when using a tight bonding model, or the phonon excitation spectrum of quantum dots, with a sensitivity to edge coupling constants. Acknowledgements The authors are glad to acknowledge fruitful discussions with Drs S. Rohart and V. Repain from MPQ, Universite´ Paris 7, who brought us the basic material of this work and confirmed the magnetization reversal mechanism by micromagnetism simulations. References [1] S.T.B. Goennenwein, T. Graf, T. Wassner, M.S. Brandt, M. Stutzmann, J.B. Philipp, R. Gross, M. Krieger, K. Zu¨m, P. Ziemann, A. Koeder, S. Frank, W. Schoch, A. Waag, Appl. Phys. Lett. 82 (2003) 730; T.G. Rappoport, P. Redlinski, X. Liu, G. Zarand, J.K. Furdyna, B. Janko, Phys. Rev. B 69 (2004) 125213. [2] R. Zdyb, A. Pavlovska, A. Locatelli, S. Heun, S. Cherifi, R. Belkhou, E. Bauer, Appl. Surf. Sci. 249 (2002) 40. [3] G. Gubbioti, M. Conti, G. Carlotti, P. Candeloro, E. Di Fabrizio, K.Yu. Guslienko, A. Andre, C. Bayer, A.N. Slavin, J. Phys.: Condens. Matter 16 (2004) 7709. [4] M. Spasova, U. Wiedwald, R. Ramchal, M. Farle, M. Hilgendorff, M. Giersig, J. Magn. Magn. Mater. 240 (2002) 40. [5] S. Jung, B. Watkins, L. Delong, J.B. Ketterson, V. Chandrasekhar, Phys. Rev. B 66 (2002) 132401. [6] J.V. Barth, H. Brune, G. Ertl, R.J. Behm, Phys. Rev. B 42 (1990) 937. [7] S. Rohart, G. Baudot, V. Repain, Y. Girard, S. Rousset, H. Bulou, C. Goyhenex, L. Proville, Surf. Sci. 559 (2004) 47. [8] P. Gambardella, S. Rusponi, M. Veronese, S.S. Dhesi, C. Grazioli, A. Dallmeyer, I. Cabria, R. Zeller, P.H. Dederichs, K. Kern, C. Carbone, H. Brune, Science 300 (2003) 1130. [9] R.D. McMichael, M.D. Stiles, J. Appl. Phys. 97 (2005) 101901. [10] J.-C.S. Le´vy, E. Ilisca, J.-L. Motchane, Phys. Rev. B 5 (1972) 1099. [11] J.-C.S. Le´vy, Surf. Sci. Rep. 1 (1981) 39; H. Puszkarski, Surf. Sci. Rep. 20 (1994) 45. [12] B. Heinrich, J.F. Cochran, M. Kowaleski, J. Kirchner, Z. Celinski, A.S. Arrott, K. Myrtle, Phys. Rev. B 44 (1991) 9348.
Journal of Magnetism and Magnetic Materials 305 (2006) 182–185 www.elsevier.com/locate/jmmm
Magnons in Co dot J.-C.S. Levya,, M. Krawczykb, H. Puszkarskib a
Materiaux et Phenomenes Quantiques, UMR 7162-CNRS, Universite Paris 7, case 7021, 75251 Paris, Cedex 05, France b Surface Physics Division, Faculty of Physics, UAM ul Umultowska 85, Poznan 61-614, Poland Received 11 July 2005; received in revised form 21 October 2005 Available online 17 January 2006
Abstract Eigen spin wave frequencies and profiles of a cobalt hexagonal dot with exchange and anisotropy energies are derived. The lowest mode frequency is shown to be a linear function of edge anisotropy, so edge anisotropy controls the whole dot magnetization reversal and can be measured from spin wave resonance. The low-temperature dependence of cobalt dot magnetization is shown to be driven by edge anisotropy as well. r 2006 Elsevier B.V. All rights reserved. Keywords: Spin wave resonance; Magnetic dots; Anisotropy
Introduction Low-dimensionality magnetic materials become more and more common in nowadays physics. Dilute magnetic semiconductors [1] where magnetic clusters are more or less ordered according to the annealing processes are an example of low-dimension magnetism. This is also the case of magnetic ultrathin films [2] and of magnetic dots [3] which are often deposited as array of dots [4] or periodic arrays of dots [5]. The recently improved spin wave resonance resolution and the additive contribution of arrays of dots makes spin wave resonance of dots possible [3,5]. So the question of spin waves in low-dimensionality magnetic materials arises since spin wave resonance and thermal variations of magnetization can be observed in dots and arrays of dots. A good example could be cobalt dot over gold (1 1 1) since due to the herringbone gold reconstruction [6] regular arrays of cobalt dots can be produced [7]. The new Hamiltonian term to be introduced to account for a dot is just the giant edge anisotropy of border atoms [8] which is observed from X-ray magnetic circular dichroism and has been estimated from ab initio computations to be Corresponding author. Tel.: +33 1 44 27 43 77; fax: +33 1 44 27 28 52.
E-mail address: [email protected] (J.-C.S. Levy). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.12.010
quite large. The distances between dots are assumed to be large enough, so dipole–dipole interaction between dots can be neglected. Dipole–dipole interactions within the dot are weak and can be neglected. So the spectrum of magnetic excitations as well as the eigenmode profiles can be deduced from an estimated value of edge anisotropy. This enables us to predict the result of spin wave resonance on extended arrays of independent dots as the sum of spin wave resonance effects on individual dots. From the profile of the lowest frequency modes, the magnetization reversal of an individual dot and of a set of independent dots is easily understood when neglecting the contribution of higher modes. The knowledge of the spin wave spectrum enables us to deduce the magnetization behavior as a function of temperature at least, at low enough temperature. So the study of spin wave resonance in dots and magnetization measurements induces two indirect measurements of edge anisotropy in dots. This defines the main goal of this work. A secondary goal is to evidence the practical symmetries in a finite sample and their consequence on the spin wave spectrum. This could be compared to experimental results [9]. Section 1 gives the general method to derive normal modes of dots, while the specific points of the practical determination are reported in Section 2. A short conclusive comparison to recent observations is done in Section 3.
ARTICLE IN PRESS J.-C.S. Levy et al. / Journal of Magnetism and Magnetic Materials 305 (2006) 182–185
183
1. Method
1.2. Resonance condition and magnetization
We assume that the static external magnetic field H 0 is normal to the dot plan with the following Hamiltonian [10]: 1X 1X H¼ J f;g; S af Sag I f ðS zf Þ2 2 f;g;a 2 f X gmB H eff S zf , ð1Þ
Resonance fields, in those formulae are calculated from eigenvalues O, Eq. (3):
f
where sites f and g are nearest neighbors with the local single ion anisotropy I f which takes the value I b in the bulk of the dot and I s at the edge of the dot, and the effective magnetic field H eff ¼ H 0 4pN z M S . A uniform static demagnetizing field with the factor N z is assumed to occur. This equation of motion reads in RPA or Tyablikov approximation: ! X d þ S ¼ SJ i_ j f;g þ if S þ f dt f g X þ SJ j f;g S þ ð2Þ g þ gmB H eff S f , g
where normalized exchange and anisotropy integrals, j f;g ; if are introduced. J f;g ¼ Jj f;g ¼ Jdf;g ;
I f ¼ Jif
and
S ¼ hSz i.
g
(3)
g
_o OJS þ 4pN z M S . gmB
The dynamic matrix is defined element by element. The eigenvectors define the spin wave profiles. And the dot magnetization as a function of temperature reads [11] 1 X X _O M¼N nðOÞ ¼ N exp (4) 1 . kT O In the usual case at low temperature where just one lowfrequency mode drives the magnetization behavior it writes: 1 _Omin M ¼ N exp (5) 1 . kT 2. Practical calculations For practical visualization the case n ¼ 5 with N ¼ 91 atoms and 33% of border atoms is shown, but calculations were done from n ¼ 1 up to 10. The reduced bulk anisotropy as deduced for bulk cobalt is: K b ¼ 0:004 [12]. The selected values of material parameters are:
Fourier transformation in time is introduced: Z ðtÞ ¼ eiot Sþ Sþ f f ðoÞ do, and the eigenvalue problem reads: X X þ þ ¼ j S þ i S j f;g S þ OSþ f f;g g f f f
H0 ¼
frequency of the rf field i.e.: o ¼ 2p9:96 GHz; value of exchange integral for bulk cobalt: JS can take values in the range: h1:522:6i1014 (erg). The admitted value is JS ¼ 1:5 1014 (erg) [12]. g factor for cobalt is 2:2 [12]; Bulk spontaneous magnetization for cobalt: M S 1:44 kG [12]; O is an eigenvalue taken from calculations.
with O ¼ _o gmB H eff =JS. With those values : H 0 ¼ 4:4 735:3 OðkGÞ: 1.1. Hexagonal dot 2.1. Results For a hexagonal monolayer of cobalt of edge size n þ 1, there are N ¼ 3n2 þ 3n þ 1 cobalt atoms of which 6n 6 are edge atoms and six are corner atoms so three types of diagonal elements of the dynamic matrix occur: for bulk spins: 6j þ i and i ¼ K b the reduced bulk anisotropy; for edge spins: 4j þ i and i ¼ K s the reduced edge anisotropy; and for corner spins: 3j þ i and i ¼ K s the reduced edge anisotropy, i.e. edge and corner anisotropy are not distinguished here, so there are N s ¼ 6n effective edge atoms. Off diagonal elements (for neighbors spins only) are equal to 1.
The DC field for resonance as a function of the edge anisotropy K s is shown in Fig. 1 for the first mode with n ¼ 5, i.e. a 91 atoms hexagon. It shows a linear variation of the first mode resonant field with the edge anisotropy as observed for thin films [10], with surface anisotropy instead of edge anisotropy. The anisotropy acts as a local field. So the reduced frequency O is zero when the total anisotropy field is zero, i.e. when N s K s þ ðN N s ÞK b ¼ 0, i.e. K s;0 ¼ ððN N s Þ=N s ÞK b ¼ 0:008133 as observed in Fig. 1. More precisely for a hexagonal dot of n concentric layers, i.e. of edge size n, the reduced frequency O is zero when K s;0 ¼ ðð3n2 3n þ 1Þ=6nÞK b an hyperbolic function of n. This law is perfectly followed for no9. Of course, for this value K s;0 of edge anisotropy there is an instability of the magnetization orientation.
ARTICLE IN PRESS 184
J.-C.S. Levy et al. / Journal of Magnetism and Magnetic Materials 305 (2006) 182–185
Fig. 1. The resonance field and reduced frequency of the first spin wave mode as a function of edge anisotropy K s for a cluster of 91 Co atoms. Note this is a straight line with a zero frequency for K s ¼ 0:0081: The spin wave profile of the first mode is reported in inset (a) when K s ¼ 0:01, (b) when K s ¼ 0:004 and (c) when K s ¼ 0:005. Note the uniform mode in (b) and the edge mode in (c).
Fig. 3. The profiles of the first ten modes for 91 atoms with K s ¼ 0:01. Note the sixfold symmetry and the low value of wavevectors.
Fig. 2. The full spectrum of spin wave modes for 91 Co atoms with K s ¼ 0:01. Note the degeneracies and the finite gaps. For the lowest modes the density of states is constant, OðmÞ ¼ lm.
Fig. 2 gives us the spectrum of eigenfrequencies when K s ¼ 0:01. The lowest frequency mode is a single mode while gaps separate higher modes. Many modes are degenerate. At this scale the frequency spectrum does not show any effect when submitted to a change of the edge anisotropy. This is due to the fact that anisotropy is quite negligible in front of exchange. The profiles of the first ten modes are reported in Fig. 3 with evidence for sixfold symmetry and lowest wavevectors. These calculations are reported in the realistic case where K s ¼ 0:01 which is not so far from anisotropy observations [8]. Not shown spectra for other values of edge anisotropy confirm this mode classification and even the gap values. Modes 2 and 3 are degenerate as observed in Fig. 2. Modes 4 and 5 are also degenerate. The only profile which is sensitive to the edge anisotropy is the first mode profile, as well known for thin films where this mode can become a surface mode [11]. The effect of
edge anisotropy variation on the lowest spin wave mode profile is shown in the three insets of Fig. 1. Three different values of edge anisotropy K s are introduced. When edge anisotropy is equal to bulk anisotropy, i.e. K s ¼ 0:004, the first mode is perfectly uniform, so the magnetization reversal consists in a perfect Stoner–Wolfarth uniform precession. When edge anisotropy K s ¼ 0:01 is a few times the bulk anisotropy, edge spins are more fastened than bulk spins, the first mode is pinned at the surface. So the dot magnetization reversal follows this first mode profile, i.e. induces a strong rotation of bulk spins and a smaller rotation of border spins. This is a ‘‘hard’’ magnetic edge. When edge anisotropy is lower than bulk anisotropy, here K s ¼ 0:005, edge spins are more loosely held than bulk spins. There is an ‘‘edge’’ spin wave, a spin wave localized close to the edge. This singular behavior can be observed by comparing spin wave spectra for different dot sizes [11]. In this case, edge magnetization is more easily rotated than the bulk one. This is a ‘‘soft’’ magnetic edge. As a consequence of these remarks the variation of magnetization with temperature given by Eq. (5) is driven
ARTICLE IN PRESS J.-C.S. Levy et al. / Journal of Magnetism and Magnetic Materials 305 (2006) 182–185
by the first mode frequency and thus strongly depends on the edge anisotropy. This is strictly true when the temperature is lower than T 1 , the temperature deduced from the difference between the frequency values of the first and second mode. In the case of cobalt T 1 ¼ 20 K when n ¼ 5 and T 1 ¼ 30 K when n ¼ 4. When dot size decreases, the energy gap between successive modes is increased. These temperature ranges can be easily reached experimentally. And for temperature lower than T 1 the variation of magnetization with temperature is exponential. The exponent is a linear function of edge anisotropy K s . Of course when the temperature is quite large in front of T 1 , the sum over all modes must be done as in Eq. (5). As a result there is a final linear decrease of magnetization with temperature as usual for 2D magnetic samples [11]. So finally edge anisotropy can be measured either from spin wave resonance or from low-temperature magnetization measurements. 3. Conclusion In the considered case where dipolar (i.e. magnetostatic) interactions are completely negligible since dots are small and distant, the influence of edge anisotropy, i.e. border anisotropy on basic magnetic properties is studied. It is shown that edge anisotropy can be deduced in a linear way from the first resonance field, and so with accuracy. Then the compared value of edge and bulk anisotropy determines the nature of magnetization reversal mechanism with ‘‘hard’’ or ‘‘soft’’ edge behaviors. Finally edge anisotropy has a crucial part in the magnetization variation with temperature at low temperature. In the possible case of a strong negative edge anisotropy which can occur in magnetic dots, the first mode is an edge mode. The edge anisotropy can still be deduced from spin wave resonance. Then the magnetization reversal starts from the border and it is necessary to take into account a first bulk mode to obtain a complete magnetization reversal. In presence of an external field which stabilizes the magnetic state of the sample, the magnetization variation with temperature is once more driven by the edge anisotropy. More generally, similar calculations can be applied to more complex magnetic clusters which appear in low-
185
dimensional magnetism. The present results indicate that spin wave resonance in such clusters must be quite sensitive to geometric details as here to edge anisotropy and to dot size. Similar calculations as reported here give the electronic excitation energy of nanoclusters when using a tight bonding model, or the phonon excitation spectrum of quantum dots, with a sensitivity to edge coupling constants. Acknowledgements The authors are glad to acknowledge fruitful discussions with Drs S. Rohart and V. Repain from MPQ, Universite´ Paris 7, who brought us the basic material of this work and confirmed the magnetization reversal mechanism by micromagnetism simulations. References [1] S.T.B. Goennenwein, T. Graf, T. Wassner, M.S. Brandt, M. Stutzmann, J.B. Philipp, R. Gross, M. Krieger, K. Zu¨m, P. Ziemann, A. Koeder, S. Frank, W. Schoch, A. Waag, Appl. Phys. Lett. 82 (2003) 730; T.G. Rappoport, P. Redlinski, X. Liu, G. Zarand, J.K. Furdyna, B. Janko, Phys. Rev. B 69 (2004) 125213. [2] R. Zdyb, A. Pavlovska, A. Locatelli, S. Heun, S. Cherifi, R. Belkhou, E. Bauer, Appl. Surf. Sci. 249 (2002) 40. [3] G. Gubbioti, M. Conti, G. Carlotti, P. Candeloro, E. Di Fabrizio, K.Yu. Guslienko, A. Andre, C. Bayer, A.N. Slavin, J. Phys.: Condens. Matter 16 (2004) 7709. [4] M. Spasova, U. Wiedwald, R. Ramchal, M. Farle, M. Hilgendorff, M. Giersig, J. Magn. Magn. Mater. 240 (2002) 40. [5] S. Jung, B. Watkins, L. Delong, J.B. Ketterson, V. Chandrasekhar, Phys. Rev. B 66 (2002) 132401. [6] J.V. Barth, H. Brune, G. Ertl, R.J. Behm, Phys. Rev. B 42 (1990) 937. [7] S. Rohart, G. Baudot, V. Repain, Y. Girard, S. Rousset, H. Bulou, C. Goyhenex, L. Proville, Surf. Sci. 559 (2004) 47. [8] P. Gambardella, S. Rusponi, M. Veronese, S.S. Dhesi, C. Grazioli, A. Dallmeyer, I. Cabria, R. Zeller, P.H. Dederichs, K. Kern, C. Carbone, H. Brune, Science 300 (2003) 1130. [9] R.D. McMichael, M.D. Stiles, J. Appl. Phys. 97 (2005) 101901. [10] J.-C.S. Le´vy, E. Ilisca, J.-L. Motchane, Phys. Rev. B 5 (1972) 1099. [11] J.-C.S. Le´vy, Surf. Sci. Rep. 1 (1981) 39; H. Puszkarski, Surf. Sci. Rep. 20 (1994) 45. [12] B. Heinrich, J.F. Cochran, M. Kowaleski, J. Kirchner, Z. Celinski, A.S. Arrott, K. Myrtle, Phys. Rev. B 44 (1991) 9348.