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Cr superlattices
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 320 (2008) 292–298 www.elsevier.com/locate/jmmm
Effect of bidimensional Fe clusters on magnetic properties of Fe/Cr superlattices N.S. Yartsevaa,, S.V. Yartsevb, C. Demangeatc, V.M. Uzdind a
Institute of Metal Physics UD of RAS, 18 S. Kovalevskoy Str., Ekaterinburg 620041, Russia ZAO NPO ‘‘Spektr’’, 14 Zapadnaya promzona, Berezovskiy, Sverdlovskaya oblast’ 623700, Russia c Institut de Physique et Chimie des Mate´riaux de Strasbourg, 23 rue du Loess, Strasbourg F-63037, France d St. Petersburg State University, 10 Linia V.O. 49, St. Petersburg 199178, Russia b
Received 15 December 2006; received in revised form 27 March 2007 Available online 12 June 2007
Abstract The effect of bidimensional Fe clusters and intermixing between the Fe and Cr atoms on the magnetic properties of Fe/Cr superlattices is studied in the framework of collinear and noncollinear periodic Anderson model. Self-consistent calculations performed for the superlattices with Fe layers thicknesses equivalent to 3 and 7 A˚ demonstrate the dependence of the magnetic moments distribution not only on the thickness of Fe layers but on the compositional structure of the interlayers due to different Cr surroundings of the Fe atoms. The superlattices with continuously smooth Fe layers of both thicknesses result in a collinear orientation of Fe and Cr atomic moments. The onset of the Fe clusters and switching on the intermixing of the atoms by a stochastic procedure do not destabilize the collinear orientation for the superlattices with Fe thickness equivalent to 7 A˚. On the contrary, the superlattices with Fe thickness equivalent to 3 A˚ results in noncollinear orientation of the magnetic moments for the intermixed continuous Fe layers as much as for the Fe clusters in the interlayers before and after intermixing. The clusters are shown have an arbitrary orientation of the average magnetic moments, so that the structure is considered to be a superparamagnetic system. r 2007 Elsevier B.V. All rights reserved. Keywords: Clusters of iron; Superlattice; Noncollinearity; Magnetic moment distribution
1. Introduction The Fe/Cr metallic superlattices are the subject of great interest because of giant magnetoresistance effect discovery [1]. It is known that magnetoresistance of the superlattices depends essentially on the roughness and thickness of the layers. The interfacial roughness results in a random variation of the deposited layers’ thickness and strongly affects the magnetic properties of the multilayered films [2,3]. At the beginning of the Fe deposition on the Cr film, the Fe clusters at the interface are formed, and only further coverage results in a continuous film. The experimental evidence of the clusters’ structure was obtained by means of scanning tunnel microscopy by Choi et al. [4]. The study Corresponding author. Tel.: +7 343 378 36 16; fax: +7 343 374 52 44.
E-mail address: [email protected] (N.S. Yartseva). 0304-8853/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2007.06.003
showed that the deposited Fe atoms can form the clusters of pure Fe as well as the ordered and disordered Fe/Cr alloys on the Cr(1 0 0) surface. In turn, Davies et al. [5] detected experimentally an intermixing of the Cr and Fe atoms in the interlayer. Reduction and reorientation of the Cr magnetic moments in Fe/Cr multilayers were observed by Almokhtar et al. [6] using a 119Sn Mo¨ssbauer probe. The observations suggest that the magnetic frustration resulting from the Fe–Cr interface imperfections is accommodated by reduction or reorientation of the Cr magnetic moments for the samples depending on the growth temperatures. The reorientation of the Fe magnetic moments was found in the Fe/Cr superlattices with ultrathin Fe layers by Ustinov et al. [7] through magnetometry. The authors supposed that a superparamagnetic behavior of the system is connected with the nanosized Fe clusters in the Fe layers, but the compositional structure of the interlayers was not thoroughly investigated.
ARTICLE IN PRESS N.S. Yartseva et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 292–298
Here we study the effect of small chaotically oriented bidimensional Fe clusters at the interlayer and intermixing of the atoms at the interfaces, on the distribution of the magnetic moments in the Fe/Cr superlattices with thin and ultrathin Fe layers. The self-consistent calculations of the magnetic moments of atoms are performed within the noncollinear approach of the periodic Anderson model (PAM). 2. Theoretical model The PAM has two approaches: collinear, where only parallel and antiparallel orientations of the magnetic moments exist, and noncollinear, with consideration of an arbitrary orientation of the magnetic moments relative to a global axis [8,9]. It is shown that PAM is effective in studying Fe/Cr [10] and Fe/V [11] multilayers with a large number of considered atoms in the supercell, and allows one to handle the nonhomogeneous structures at the surface and interface. Using a modification of zero and poles method in order to determine a mass operator and Green functions poles, the self-consistent numerical calculations for more than 1500 nonequivalent atomic sites turned out to be performed in a reasonable computation time. Numerical results for the Fe/Cr systems obtained in the framework of PAM are in reasonable agreement with ab initio [12] and tight binding [13] approaches, and have been successfully used in the interpretation of Mo¨ssbauer experiments [14,15]. The model assumes the existence of two bands: quasilocalized d-electrons and itinerant s-electrons. The PAM Hamiltonian is written as follows: X X X þ þ H¼ E k c^þ E ai d^ ia d^ia þ V ij d^ ia d^ ja ka c^ka þ ka
þ
X
i;a
iaj;a
V ki c^ka d^ia þ h:c.
ki;a þ cka Þ and d^ia ðd^ ia Þ refer to the creation (annihilaHere c^þ ka ð^ tion) operators of the s(p)-electrons with quasi-momentum k and spin a and of the d-electrons with spin a on the atomic site i, respectively. The Ek is the energy of s(p)electrons. The energy of d-electrons E ai ¼ E 0i þ U i na is i considered in Hartree–Fock approximation. Ui is the Coulomb repulsion between d-electrons, and na is the i number of d-electrons with spin a. Hopping parameters Vij and s–d hybridization potential, Vki, are supposedly spin-independent. The s–d interaction on a site is presupposed to be stronger than d–s–d interaction of d-electrons on the different sites. In this case one should construct the resonant d-states of a finite width Gi due to s–d interaction, X V ik V ki Gi ¼ Im o k k
and then introduce an electron hopping between the different sites.
293
In the noncollinear approach, the PAM Hamiltonian has been rewritten in terms of spin quantization along the global z-axis, which is the same for all the atoms [9]. To determine the direction of the moment relative to the quantization axis the polar angles yi and ji are introduced. After Hartree–Fock approximation, the additional on-site hoppings with inversion of spin projections V þ arise ii together with the intersite hoppings without change in the spin projection Vij. The magnetic moment and the number of d-electrons on each atom are determined through the imaginary part of ab the d-electron Green’s function Gab ij . The equations for G ij have the following form: ðo E ai ÞG ab ij ðoÞ
X
gb ab V ag il G lj ðoÞ ¼ d dij .
lg
Here the lower and upper indices enumerate the atomic sites and spin projection (+,), respectively; d is the Kronecker symbol. The energy E ai of the electron at site i with spin projection a and the hopping parameters V ab jj are expressed in terms of the d-electron number Ni, modulus of the magnetic moment Mi, and polar angles yi and ji. E i ¼ E 0i þ
U iNi U iMi cos yi , 2 2
þ V þ jj ¼ ðV jj Þ ¼
U j M j ijj e sin yj . 2
Spin-independent part of the energy (E0ieF)/Gi counted from the Fermi level eF, Coulomb repulsion on a site Ui/Gi and d-electrons hopping between the nearest neighbors Vij/ Gi are parameters of the model. We assume that d-electron level width Gi is independent of the site number and consider it as a normalizing parameter of the model G. The parameters are adjusted for consistency with the density of states and magnetic profiles of ab initio calculations for the considered structures. The self-consistent calculation of the magnetic moments distribution starts from the collinear model. The solution obtained in the collinear approach is taken as an original state for the self-consistent calculation in noncollinear PAM for the same structure. The initial angles ji are assigned to zero at each site. The initial angles yi are taken at either zero or p in accordance with the collinear state. At certain sites, the initial polar angles yi are changed on any suitable value at some sites as an excitation to deviate the starting configuration from the collinear state. Then selfconsistent d-electrons number and magnetic moments are calculated at the sites for slightly varying angles and a state with minimal energy is accepted for the next iteration. The energy of the fully converged noncollinear state is compared with the energy of the original collinear state, to evaluate which state is actually preferred. Generally, this procedure may guarantee a convergence only to a local configurational minimum of the total energy.
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3. Supercells modeling In order to study the effect of clusters and intermixing on the magnetic properties of Fe/Cr superlattices, we model first the supercells with the abrupt interfaces and afterwards perform the intermixing procedure to simulate the alloying process. The atoms intermixing procedure is carried out by a stochastic algorithm. This algorithm [16] assumes that during evaporation the exchange of the atoms with atoms of the substrate takes place only on the surface and there is no internal diffusion. Only some atoms from the monolayers (MLs) take part in the exchange procedure: the fraction of the exchanged atoms is chosen to simulate the temperature during the growing process. In each monolayer the exchanged atoms are chosen randomly and then layerwise, bottom-up, these atoms are exchanged with one of their four nearest neighbors from the preceding layer, also randomly chosen (see Fig. 1). This exchange leads to the floating of the atoms on the upper layers and the intermixing of atoms in the monolayers. The supercells are constructed as an ideal BCC lattice of 12 6 sites in the monolayers and 9 and 12 successive
monolayers for the Fe thicknesses of 2 MLs (about 3 A˚) and 5 MLs (about 7 A˚), respectively, (see Tables 1 and 2). These two types of supercells, for all the samples, have 7 MLs (about 10 A˚) of Cr atoms to provide the ferromagnetic ordering of the Fe slabs for the superlattices with smooth interfaces, and to be close to the chemical composition of the experiments reported in Ref.[7]. The Fe layers are modeled by two geometrical configurations: either the ideally smooth continuous MLs, or the solid Fe clusters separated by Cr atoms. The supercells with clusters are shown in Figs. 2(a) and 3(a). Fig. 2(a) depicts the Fe clusters of 4 MLs height with the number of atoms equivalent to the deposition of 3 A˚. For deposition equivalent to 7 A˚ of Fe, 2 separated clusters of 4 MLs height are placed on the top of the 3 continuous Fe MLs (see Fig. 3(a)). The intermixing of the Fe–Cr atoms is carried out for the fraction of exchanged atoms k equal to 0.5. In Tables 1 and 2, the number of the Fe and Cr atoms in the successive layers for the structures with continuous Fe MLs and Fe clusters, before and after intermixing, are presented. Figs. 2 and 3 depict a cutting of the supercells along the (0 0 1) direction with clusters of Fe equivalent to 3 and 7 A˚ before (a) and after (b) intermixing; the odd atomic lines correspond to the atoms placed on the cube-side sites and the even ones to the atoms of the central sites of BCC lattice.
4. Distribution of the magnetic moments
Fig. 1. Scheme of atoms exchange between the nearest neighbors from two successive monolayers. Black and gray circles correspond to the Fe and Cr atoms.
The self-consistent calculations of the magnetic moments at each atomic site are performed in the collinear and noncollinear PAM approaches. The parameters of the PAM for Fe, VFeFe/G ¼ 0.9, UFe/G ¼ 13.0, (EdFeeF)/ G ¼ 11.5, are fitted to reproduce the magnetic moments and particle numbers for the bulk Fe; the parameters for Cr, VCrCr/G ¼ 0.9, UCr/G ¼ 6.75, (EdCreF)/G ¼ 3.37, are fitted to reproduce the magnetic moments of the bulk antiferromagnetic chromium. The intersite Fe–Cr hoppings
Table 1 Distribution of Fe and Cr atoms before and after intermixing in successive monolayers of the supercell with 2 MLs of Fe Number of layer Supercell with continuous 2 Fe MLs
1 2 3 4 5 6 7 8 9
Supercell with clusters of 4 MLs height
Number of atoms before intermixing
Number of atoms after intermixing
Number of atoms before intermixing
Number of atoms after intermixing
Fe
Cr
Fe
Cr
Fe
Cr
Fe
Cr
0 0 0 0 0 0 72 72 0
72 72 72 72 72 72 0 0 72
8 0 1 0 4 15 46 55 15
64 72 71 72 68 57 26 17 57
0 0 0 0 30 32 40 42 0
72 72 72 72 42 40 32 30 72
0 1 0 9 22 31 35 36 10
72 71 72 63 50 41 37 36 62
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Table 2 Distribution of Fe and Cr atoms before and after intermixing in successive monolayers of the supercell with 5 MLs of Fe Number of layer Supercell with continuous 5 Fe MLs
1 2 3 4 5 6 7 8 9 10 11 12
Supercell with clusters of 4 MLs height on the 3 MLs of Fe
Number of atoms before intermixing
Number of atoms after intermixing
Number of atoms before intermixing
Number of atoms after intermixing
Fe
Cr
Fe
Cr
Fe
Cr
Fe
Cr
0 0 0 0 0 0 72 72 72 72 72 0
72 72 72 72 72 72 0 0 0 0 0 72
1 1 0 1 6 14 48 72 70 70 57 20
71 71 72 71 66 58 24 0 2 2 15 52
0 0 0 0 30 32 40 42 72 72 72 0
72 72 72 72 42 40 32 30 0 0 0 72
2 2 0 5 21 36 37 51 59 70 57 20
70 70 72 67 51 36 35 21 13 2 15 52
Fig. 2. The transverse cuttings of the supercells with clusters of Fe equivalent to 3 A˚ before (a) and after (b) intermixing. The odd and even MLs correspondingly depict the cube side sites and central sites of the BCC lattice.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi are taken as V FeCr =G ¼ V FeFe V CrCr =G. A typical value of G is about 1 eV. First, the calculations for the considered supercells are carried out in the collinear model. In all directions, the periodic boundary conditions are used. To compare the results for the superlattices with continuous and clusters structure of the Fe interlayers as well as to study the influence of the thickness of Fe layers, we plot the statistical distribution histograms of the Fe magnetic moments. The histograms could be regarded as a hyperfine field distribution obtained by Mo¨ssbauer spectroscopy [17,18]. The magnetic moments histogram for the supercell with continuously smooth Fe layers of 3 A˚ thickness shows a single peak of 1.9mB caused by the surroundings of two Fe MLs by the antiferromagnetically ordered Cr MLs. Each Fe atom has 4 nearest and 2 next Cr neighbors which reduce the Fe moment when compared with the bulk value of 2.2mB. The histogram for the supercell with continuously smooth Fe layers of 7 A˚ thickness shows 3 peaks: 1.7mB, typically corresponds to the surface value of abrupt interface with 4 nearest and 1 next Cr neighbors, bulk Fe magnetic moment of 2.2mB and enhanced moments of 2.3mB. The intermixing of atoms modifies substantially the magnetic moments distribution. The histogram becomes widely distributed with a large number of peaks and shifted to the lower moments, which correspond to the large number of the Cr atoms in the Fe surroundings. The supercell with 3 A˚ thickness of Fe after intermixing has a strongly pronounced peak at the 1.7mB as well as the peaks at smaller and larger magnetic moments (see Fig. 4(a)). The supercell with Fe of 7 A˚ thickness after intermixing has 4 very explicit peaks at the magnetic moments near 1.7, 1.9, 2.2 and 2.3mB with the highest value near the bulk moment of 2.2mB as well as a very weak maximum at 1.1mB (see Fig. 4(b)).
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Fig. 3. The transverse cuttings of the supercells with clusters of Fe equivalent to 7 A˚ before (a) and after (b) intermixing. The odd and even atomic lines correspondingly depict the cube side sites and the central sites of the BCC lattice.
The self-consistent calculations for the supercell with Fe clusters of the thickness equivalent to 3 A˚ before intermixing give the explicit peaks at 1.2mB corresponding to a large number of Cr atoms in the Fe surroundings, at 1.7mB corresponding to 4 nearest and 1 next nearest neighbors, and peak at magnetic moments near 2.3mB. The intermixing of atoms widens the distribution and shifts the maxima (Fig. 5(a)). The self-consistent calculations for the supercell with separated Fe clusters of the thickness equivalent to 7 A˚ before intermixing give 4 explicit maxima at 1.7, 1.9, 2.1 and 2.3 mB, which are close to the histogram of the supercell with continuous Fe MLs of 7 A˚ after
Fig. 4. Histogram of magnetic moments distribution for continuous Fe layers of 3 A˚ thickness (a) and of 7 A˚ thickness (b) in the collinear model after intermixing of atoms.
intermixing. However, the intermixing in the supercell with clusters widens the peaks and merges two of them about the bulk value (Fig. 5(b)); so it makes the histogram different from the similar one for the supercell with the intermixed continuous Fe MLs. Second, we perform the noncollinear calculations for the superlattices with continuous Fe MLs and with the cluster structures before and after intermixing, taking the initial in-plane angles on the sites of the Fe atoms rotated on 901 , and assigning zero to the angles on all of the Cr atoms. The out-of-plane angles are taken equal to zero for all the atoms. The calculations for the supercells with continuously smooth Fe layers of both 3 and 7 A˚ thickness lead to a ground state with collinear magnetization. Intermixing
ARTICLE IN PRESS N.S. Yartseva et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 292–298
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modifies the results of the calculations drastically for the superlattices with Fe thickness of 3 A˚: noncollinear orientation of the magnetic moments presents a gain in energy of about 23 K per site. Besides, the superlattices with Fe thickness of 7 A˚ prefer to keep the collinear orientation of the moments. The calculations for the supercell with two Fe clusters after intermixing of the Fe and Cr atoms are performed for the initial in-plane angles on the sites of one of the clusters rotated by 45, 90 or 1801 , assigning zero to the angles on the sites of another cluster and on the Cr atoms. The selfconsistent procedure for the supercell with thickness equivalent to 3 A˚ of Fe results in various orientations of the magnetic moments at the Fe and Cr atoms. Magnetic
Fig. 5. Histogram of magnetic moments distribution for the Fe clusters equivalent to 3 A˚ thickness (a) and 7 A˚ thickness (b) in the collinear model after intermixing of atoms.
Fig. 6. Distribution of magnetic moments and in-plane angles at the Fe and Cr atoms (black and gray circles) for the Fe clusters equivalent to 3 A˚ of thickness in the 7th monolayer of the supercell (see Table 1).
Fig. 7. Histogram of magnetic moments distribution for supercells with the Fe clusters equivalent to 3 A˚ (a) and 7 A˚ (b) thickness after intermixing of atoms obtained in the noncollinear model counted on all considered initial orientation of the clusters.
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moments values and in-plane angles on the atoms of the 7th monolayer (see Table 1) are shown in Fig. 6 for the selfconsistent solution with initial angles on the Fe atoms of one of the cluster deviated by 901. The energies of the noncollinear decisions for the supercells with initial angles of 0, 45, 90 and 1801 are found to be close to each other, while their values gain about 20 K per site in comparison with the collinear energy. The calculations show that the supercells with each initial angle prefer to keep the mutual orientation of the average clusters moments close to the initial ones, besides the moments orientation on the sites is different. Because of this fact, one can conclude that the Fe clusters have a variety of orientations of the average magnetic moments, and the superlattices have to be considered as a superparamagnetic system. Calculations for the supercells, consisting of the clusters on 3 continuous Fe MLs with Fe thickness equivalent to 7 A˚, result in the collinear orientation of the magnetic moments with the energy close to the energy of the collinear approach. The magnetic moments histograms counted on all considered initial orientations of the clusters are shown in Fig. 7. One can see that the histograms differ by the number of peaks and its values for the supercells with the thickness of Fe equivalent to 3 and 7 A˚. The histogram for the supercell with Fe layer of 7 A˚ forms the typical sextet measured by Mo¨ssbauer spectroscopy for the Fe with a large number of Fe atoms. 5. Conclusion The self-consistent calculations of magnetic moments distribution in the Fe/Cr superlattices performed in PAM using the collinear and noncollinear approaches show that the superlattices of 3 and 7 A˚ thicknesses with continuously smooth Fe layers have the collinear orientation of magnetic moments. The intermixing of Fe and Cr atoms applied to these structures leads to noncollinearity only for the superlattice with Fe thickness equivalent to 3 A˚. The separated clusters in the Fe interlayers also lead to noncollinear orientation of the magnetic moments for the Fe layers equivalent to 3 A˚ after the intermixing, as before. The average moments of Fe clusters are found to keep the mutual orientation preset by the initial angles of 0, 45, 90 and 1801 with the total energies for all considered
configurations close to each other. It means that every orientation can take place in the superlattice and such a structure could be considered as a superparamagnetic system. The superlattices with the thicker Fe layers equivalent to 7 A˚ always keep the collinear magnetic moments orientation. The magnetic moments histograms can be compared with experiments to show how the intermixing and onset of the Fe clusters at the interlayers influence on the hyperfine fields distribution measured by Mo¨ssbauer spectroscopy. Acknowledgment This work is partially supported by RFBR 07-02-01289, 06-02-16722, Scientific School of RF 5869.2006.2 and the CNRS-RAS program.
References [1] S.S.P. Parkin, N. More, K.P. Roche, Phys. Rev. Lett. 64 (1990) 2304. [2] B. Heinrich, J.F. Cochran, T. Monchesky, R. Urban, Phys. Rev. B 59 (1999) 14520. [3] H. Hasegawa, J. Phys.: Condens. Matter 4 (1992) 169. [4] Y.J. Choi, I.C. Jeong, J.Y. Park, S.-J. Kahng, J. Lee, Y. Kuk, Phys. Rev. B 59 (1999) 10918. [5] A. Davies, J.A. Stroscio, D.T. Pierce, R.J. Celotta, Phys. Rev. Lett. 76 (1996) 4175. [6] M. Almokhtar, K. Mibu, T. Shinjo, Phys. Rev. B 66 (2002) 134401. [7] V.V. Ustinov, L.N. Romashev, M.A. Milayev, A.V. Korolev, T.P. Krinitsina, A.M. Burkhanov, J. Magn. Magn. Mater. 300 (2005) 148. [8] A.K. Kazansky, V.M. Uzdin, Phys. Rev. B 52 (1995) 9477. [9] V.M. Uzdin, N.S. Yartseva, Comput. Mater. Sci. 10 (1998) 211. [10] N.S. Yartseva, S.V. Yartsev, V.M. Uzdin, Phys. Met. Metallogr. 93 (2002) 42. [11] N.S. Yartseva, S.V. Yartsev, N.G. Bebenin, C. Demangeat, Phys. Rev. B 71 (2005) 039514. [12] M. Alden, H.L. Skriver, S. Mirbt, B. Johansson, Surf. Sci. 315 (1994) 157. [13] L.E. Klebanoff, R.H. Victora, L.M. Falicov, D.A. Shirley, Phys. Rev. B 31 (1985) 1997. [14] V. Uzdin, W. Keune, Phys. Met. Metallogr. 91 (Suppl. 1) (2001) S82. [15] V.M. Uzdin, W. Keune, H. Schro¨r, M. Walterfang, Phys. Rev. B 63 (2001) 104407. [16] V. Uzdin, N. Yartseva, C. Demangeat, Phase Trans. 76 (2003) 377. [17] H. Akai, M. Akai, S. Blugel, B. Drittler, H. Ebert, K. Terakura, R. Zeller, P.H. Dederichs, Suppl. Prog. Theor. Phys. 101 (1990) 11. [18] A.K. Arzhnikov, L.V. Dobysheva, Phys. Rev. B 62 (2000) 5324.
Journal of Magnetism and Magnetic Materials 320 (2008) 292–298 www.elsevier.com/locate/jmmm
Effect of bidimensional Fe clusters on magnetic properties of Fe/Cr superlattices N.S. Yartsevaa,, S.V. Yartsevb, C. Demangeatc, V.M. Uzdind a
Institute of Metal Physics UD of RAS, 18 S. Kovalevskoy Str., Ekaterinburg 620041, Russia ZAO NPO ‘‘Spektr’’, 14 Zapadnaya promzona, Berezovskiy, Sverdlovskaya oblast’ 623700, Russia c Institut de Physique et Chimie des Mate´riaux de Strasbourg, 23 rue du Loess, Strasbourg F-63037, France d St. Petersburg State University, 10 Linia V.O. 49, St. Petersburg 199178, Russia b
Received 15 December 2006; received in revised form 27 March 2007 Available online 12 June 2007
Abstract The effect of bidimensional Fe clusters and intermixing between the Fe and Cr atoms on the magnetic properties of Fe/Cr superlattices is studied in the framework of collinear and noncollinear periodic Anderson model. Self-consistent calculations performed for the superlattices with Fe layers thicknesses equivalent to 3 and 7 A˚ demonstrate the dependence of the magnetic moments distribution not only on the thickness of Fe layers but on the compositional structure of the interlayers due to different Cr surroundings of the Fe atoms. The superlattices with continuously smooth Fe layers of both thicknesses result in a collinear orientation of Fe and Cr atomic moments. The onset of the Fe clusters and switching on the intermixing of the atoms by a stochastic procedure do not destabilize the collinear orientation for the superlattices with Fe thickness equivalent to 7 A˚. On the contrary, the superlattices with Fe thickness equivalent to 3 A˚ results in noncollinear orientation of the magnetic moments for the intermixed continuous Fe layers as much as for the Fe clusters in the interlayers before and after intermixing. The clusters are shown have an arbitrary orientation of the average magnetic moments, so that the structure is considered to be a superparamagnetic system. r 2007 Elsevier B.V. All rights reserved. Keywords: Clusters of iron; Superlattice; Noncollinearity; Magnetic moment distribution
1. Introduction The Fe/Cr metallic superlattices are the subject of great interest because of giant magnetoresistance effect discovery [1]. It is known that magnetoresistance of the superlattices depends essentially on the roughness and thickness of the layers. The interfacial roughness results in a random variation of the deposited layers’ thickness and strongly affects the magnetic properties of the multilayered films [2,3]. At the beginning of the Fe deposition on the Cr film, the Fe clusters at the interface are formed, and only further coverage results in a continuous film. The experimental evidence of the clusters’ structure was obtained by means of scanning tunnel microscopy by Choi et al. [4]. The study Corresponding author. Tel.: +7 343 378 36 16; fax: +7 343 374 52 44.
E-mail address: [email protected] (N.S. Yartseva). 0304-8853/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2007.06.003
showed that the deposited Fe atoms can form the clusters of pure Fe as well as the ordered and disordered Fe/Cr alloys on the Cr(1 0 0) surface. In turn, Davies et al. [5] detected experimentally an intermixing of the Cr and Fe atoms in the interlayer. Reduction and reorientation of the Cr magnetic moments in Fe/Cr multilayers were observed by Almokhtar et al. [6] using a 119Sn Mo¨ssbauer probe. The observations suggest that the magnetic frustration resulting from the Fe–Cr interface imperfections is accommodated by reduction or reorientation of the Cr magnetic moments for the samples depending on the growth temperatures. The reorientation of the Fe magnetic moments was found in the Fe/Cr superlattices with ultrathin Fe layers by Ustinov et al. [7] through magnetometry. The authors supposed that a superparamagnetic behavior of the system is connected with the nanosized Fe clusters in the Fe layers, but the compositional structure of the interlayers was not thoroughly investigated.
ARTICLE IN PRESS N.S. Yartseva et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 292–298
Here we study the effect of small chaotically oriented bidimensional Fe clusters at the interlayer and intermixing of the atoms at the interfaces, on the distribution of the magnetic moments in the Fe/Cr superlattices with thin and ultrathin Fe layers. The self-consistent calculations of the magnetic moments of atoms are performed within the noncollinear approach of the periodic Anderson model (PAM). 2. Theoretical model The PAM has two approaches: collinear, where only parallel and antiparallel orientations of the magnetic moments exist, and noncollinear, with consideration of an arbitrary orientation of the magnetic moments relative to a global axis [8,9]. It is shown that PAM is effective in studying Fe/Cr [10] and Fe/V [11] multilayers with a large number of considered atoms in the supercell, and allows one to handle the nonhomogeneous structures at the surface and interface. Using a modification of zero and poles method in order to determine a mass operator and Green functions poles, the self-consistent numerical calculations for more than 1500 nonequivalent atomic sites turned out to be performed in a reasonable computation time. Numerical results for the Fe/Cr systems obtained in the framework of PAM are in reasonable agreement with ab initio [12] and tight binding [13] approaches, and have been successfully used in the interpretation of Mo¨ssbauer experiments [14,15]. The model assumes the existence of two bands: quasilocalized d-electrons and itinerant s-electrons. The PAM Hamiltonian is written as follows: X X X þ þ H¼ E k c^þ E ai d^ ia d^ia þ V ij d^ ia d^ ja ka c^ka þ ka
þ
X
i;a
iaj;a
V ki c^ka d^ia þ h:c.
ki;a þ cka Þ and d^ia ðd^ ia Þ refer to the creation (annihilaHere c^þ ka ð^ tion) operators of the s(p)-electrons with quasi-momentum k and spin a and of the d-electrons with spin a on the atomic site i, respectively. The Ek is the energy of s(p)electrons. The energy of d-electrons E ai ¼ E 0i þ U i na is i considered in Hartree–Fock approximation. Ui is the Coulomb repulsion between d-electrons, and na is the i number of d-electrons with spin a. Hopping parameters Vij and s–d hybridization potential, Vki, are supposedly spin-independent. The s–d interaction on a site is presupposed to be stronger than d–s–d interaction of d-electrons on the different sites. In this case one should construct the resonant d-states of a finite width Gi due to s–d interaction, X V ik V ki Gi ¼ Im o k k
and then introduce an electron hopping between the different sites.
293
In the noncollinear approach, the PAM Hamiltonian has been rewritten in terms of spin quantization along the global z-axis, which is the same for all the atoms [9]. To determine the direction of the moment relative to the quantization axis the polar angles yi and ji are introduced. After Hartree–Fock approximation, the additional on-site hoppings with inversion of spin projections V þ arise ii together with the intersite hoppings without change in the spin projection Vij. The magnetic moment and the number of d-electrons on each atom are determined through the imaginary part of ab the d-electron Green’s function Gab ij . The equations for G ij have the following form: ðo E ai ÞG ab ij ðoÞ
X
gb ab V ag il G lj ðoÞ ¼ d dij .
lg
Here the lower and upper indices enumerate the atomic sites and spin projection (+,), respectively; d is the Kronecker symbol. The energy E ai of the electron at site i with spin projection a and the hopping parameters V ab jj are expressed in terms of the d-electron number Ni, modulus of the magnetic moment Mi, and polar angles yi and ji. E i ¼ E 0i þ
U iNi U iMi cos yi , 2 2
þ V þ jj ¼ ðV jj Þ ¼
U j M j ijj e sin yj . 2
Spin-independent part of the energy (E0ieF)/Gi counted from the Fermi level eF, Coulomb repulsion on a site Ui/Gi and d-electrons hopping between the nearest neighbors Vij/ Gi are parameters of the model. We assume that d-electron level width Gi is independent of the site number and consider it as a normalizing parameter of the model G. The parameters are adjusted for consistency with the density of states and magnetic profiles of ab initio calculations for the considered structures. The self-consistent calculation of the magnetic moments distribution starts from the collinear model. The solution obtained in the collinear approach is taken as an original state for the self-consistent calculation in noncollinear PAM for the same structure. The initial angles ji are assigned to zero at each site. The initial angles yi are taken at either zero or p in accordance with the collinear state. At certain sites, the initial polar angles yi are changed on any suitable value at some sites as an excitation to deviate the starting configuration from the collinear state. Then selfconsistent d-electrons number and magnetic moments are calculated at the sites for slightly varying angles and a state with minimal energy is accepted for the next iteration. The energy of the fully converged noncollinear state is compared with the energy of the original collinear state, to evaluate which state is actually preferred. Generally, this procedure may guarantee a convergence only to a local configurational minimum of the total energy.
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3. Supercells modeling In order to study the effect of clusters and intermixing on the magnetic properties of Fe/Cr superlattices, we model first the supercells with the abrupt interfaces and afterwards perform the intermixing procedure to simulate the alloying process. The atoms intermixing procedure is carried out by a stochastic algorithm. This algorithm [16] assumes that during evaporation the exchange of the atoms with atoms of the substrate takes place only on the surface and there is no internal diffusion. Only some atoms from the monolayers (MLs) take part in the exchange procedure: the fraction of the exchanged atoms is chosen to simulate the temperature during the growing process. In each monolayer the exchanged atoms are chosen randomly and then layerwise, bottom-up, these atoms are exchanged with one of their four nearest neighbors from the preceding layer, also randomly chosen (see Fig. 1). This exchange leads to the floating of the atoms on the upper layers and the intermixing of atoms in the monolayers. The supercells are constructed as an ideal BCC lattice of 12 6 sites in the monolayers and 9 and 12 successive
monolayers for the Fe thicknesses of 2 MLs (about 3 A˚) and 5 MLs (about 7 A˚), respectively, (see Tables 1 and 2). These two types of supercells, for all the samples, have 7 MLs (about 10 A˚) of Cr atoms to provide the ferromagnetic ordering of the Fe slabs for the superlattices with smooth interfaces, and to be close to the chemical composition of the experiments reported in Ref.[7]. The Fe layers are modeled by two geometrical configurations: either the ideally smooth continuous MLs, or the solid Fe clusters separated by Cr atoms. The supercells with clusters are shown in Figs. 2(a) and 3(a). Fig. 2(a) depicts the Fe clusters of 4 MLs height with the number of atoms equivalent to the deposition of 3 A˚. For deposition equivalent to 7 A˚ of Fe, 2 separated clusters of 4 MLs height are placed on the top of the 3 continuous Fe MLs (see Fig. 3(a)). The intermixing of the Fe–Cr atoms is carried out for the fraction of exchanged atoms k equal to 0.5. In Tables 1 and 2, the number of the Fe and Cr atoms in the successive layers for the structures with continuous Fe MLs and Fe clusters, before and after intermixing, are presented. Figs. 2 and 3 depict a cutting of the supercells along the (0 0 1) direction with clusters of Fe equivalent to 3 and 7 A˚ before (a) and after (b) intermixing; the odd atomic lines correspond to the atoms placed on the cube-side sites and the even ones to the atoms of the central sites of BCC lattice.
4. Distribution of the magnetic moments
Fig. 1. Scheme of atoms exchange between the nearest neighbors from two successive monolayers. Black and gray circles correspond to the Fe and Cr atoms.
The self-consistent calculations of the magnetic moments at each atomic site are performed in the collinear and noncollinear PAM approaches. The parameters of the PAM for Fe, VFeFe/G ¼ 0.9, UFe/G ¼ 13.0, (EdFeeF)/ G ¼ 11.5, are fitted to reproduce the magnetic moments and particle numbers for the bulk Fe; the parameters for Cr, VCrCr/G ¼ 0.9, UCr/G ¼ 6.75, (EdCreF)/G ¼ 3.37, are fitted to reproduce the magnetic moments of the bulk antiferromagnetic chromium. The intersite Fe–Cr hoppings
Table 1 Distribution of Fe and Cr atoms before and after intermixing in successive monolayers of the supercell with 2 MLs of Fe Number of layer Supercell with continuous 2 Fe MLs
1 2 3 4 5 6 7 8 9
Supercell with clusters of 4 MLs height
Number of atoms before intermixing
Number of atoms after intermixing
Number of atoms before intermixing
Number of atoms after intermixing
Fe
Cr
Fe
Cr
Fe
Cr
Fe
Cr
0 0 0 0 0 0 72 72 0
72 72 72 72 72 72 0 0 72
8 0 1 0 4 15 46 55 15
64 72 71 72 68 57 26 17 57
0 0 0 0 30 32 40 42 0
72 72 72 72 42 40 32 30 72
0 1 0 9 22 31 35 36 10
72 71 72 63 50 41 37 36 62
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Table 2 Distribution of Fe and Cr atoms before and after intermixing in successive monolayers of the supercell with 5 MLs of Fe Number of layer Supercell with continuous 5 Fe MLs
1 2 3 4 5 6 7 8 9 10 11 12
Supercell with clusters of 4 MLs height on the 3 MLs of Fe
Number of atoms before intermixing
Number of atoms after intermixing
Number of atoms before intermixing
Number of atoms after intermixing
Fe
Cr
Fe
Cr
Fe
Cr
Fe
Cr
0 0 0 0 0 0 72 72 72 72 72 0
72 72 72 72 72 72 0 0 0 0 0 72
1 1 0 1 6 14 48 72 70 70 57 20
71 71 72 71 66 58 24 0 2 2 15 52
0 0 0 0 30 32 40 42 72 72 72 0
72 72 72 72 42 40 32 30 0 0 0 72
2 2 0 5 21 36 37 51 59 70 57 20
70 70 72 67 51 36 35 21 13 2 15 52
Fig. 2. The transverse cuttings of the supercells with clusters of Fe equivalent to 3 A˚ before (a) and after (b) intermixing. The odd and even MLs correspondingly depict the cube side sites and central sites of the BCC lattice.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi are taken as V FeCr =G ¼ V FeFe V CrCr =G. A typical value of G is about 1 eV. First, the calculations for the considered supercells are carried out in the collinear model. In all directions, the periodic boundary conditions are used. To compare the results for the superlattices with continuous and clusters structure of the Fe interlayers as well as to study the influence of the thickness of Fe layers, we plot the statistical distribution histograms of the Fe magnetic moments. The histograms could be regarded as a hyperfine field distribution obtained by Mo¨ssbauer spectroscopy [17,18]. The magnetic moments histogram for the supercell with continuously smooth Fe layers of 3 A˚ thickness shows a single peak of 1.9mB caused by the surroundings of two Fe MLs by the antiferromagnetically ordered Cr MLs. Each Fe atom has 4 nearest and 2 next Cr neighbors which reduce the Fe moment when compared with the bulk value of 2.2mB. The histogram for the supercell with continuously smooth Fe layers of 7 A˚ thickness shows 3 peaks: 1.7mB, typically corresponds to the surface value of abrupt interface with 4 nearest and 1 next Cr neighbors, bulk Fe magnetic moment of 2.2mB and enhanced moments of 2.3mB. The intermixing of atoms modifies substantially the magnetic moments distribution. The histogram becomes widely distributed with a large number of peaks and shifted to the lower moments, which correspond to the large number of the Cr atoms in the Fe surroundings. The supercell with 3 A˚ thickness of Fe after intermixing has a strongly pronounced peak at the 1.7mB as well as the peaks at smaller and larger magnetic moments (see Fig. 4(a)). The supercell with Fe of 7 A˚ thickness after intermixing has 4 very explicit peaks at the magnetic moments near 1.7, 1.9, 2.2 and 2.3mB with the highest value near the bulk moment of 2.2mB as well as a very weak maximum at 1.1mB (see Fig. 4(b)).
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Fig. 3. The transverse cuttings of the supercells with clusters of Fe equivalent to 7 A˚ before (a) and after (b) intermixing. The odd and even atomic lines correspondingly depict the cube side sites and the central sites of the BCC lattice.
The self-consistent calculations for the supercell with Fe clusters of the thickness equivalent to 3 A˚ before intermixing give the explicit peaks at 1.2mB corresponding to a large number of Cr atoms in the Fe surroundings, at 1.7mB corresponding to 4 nearest and 1 next nearest neighbors, and peak at magnetic moments near 2.3mB. The intermixing of atoms widens the distribution and shifts the maxima (Fig. 5(a)). The self-consistent calculations for the supercell with separated Fe clusters of the thickness equivalent to 7 A˚ before intermixing give 4 explicit maxima at 1.7, 1.9, 2.1 and 2.3 mB, which are close to the histogram of the supercell with continuous Fe MLs of 7 A˚ after
Fig. 4. Histogram of magnetic moments distribution for continuous Fe layers of 3 A˚ thickness (a) and of 7 A˚ thickness (b) in the collinear model after intermixing of atoms.
intermixing. However, the intermixing in the supercell with clusters widens the peaks and merges two of them about the bulk value (Fig. 5(b)); so it makes the histogram different from the similar one for the supercell with the intermixed continuous Fe MLs. Second, we perform the noncollinear calculations for the superlattices with continuous Fe MLs and with the cluster structures before and after intermixing, taking the initial in-plane angles on the sites of the Fe atoms rotated on 901 , and assigning zero to the angles on all of the Cr atoms. The out-of-plane angles are taken equal to zero for all the atoms. The calculations for the supercells with continuously smooth Fe layers of both 3 and 7 A˚ thickness lead to a ground state with collinear magnetization. Intermixing
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modifies the results of the calculations drastically for the superlattices with Fe thickness of 3 A˚: noncollinear orientation of the magnetic moments presents a gain in energy of about 23 K per site. Besides, the superlattices with Fe thickness of 7 A˚ prefer to keep the collinear orientation of the moments. The calculations for the supercell with two Fe clusters after intermixing of the Fe and Cr atoms are performed for the initial in-plane angles on the sites of one of the clusters rotated by 45, 90 or 1801 , assigning zero to the angles on the sites of another cluster and on the Cr atoms. The selfconsistent procedure for the supercell with thickness equivalent to 3 A˚ of Fe results in various orientations of the magnetic moments at the Fe and Cr atoms. Magnetic
Fig. 5. Histogram of magnetic moments distribution for the Fe clusters equivalent to 3 A˚ thickness (a) and 7 A˚ thickness (b) in the collinear model after intermixing of atoms.
Fig. 6. Distribution of magnetic moments and in-plane angles at the Fe and Cr atoms (black and gray circles) for the Fe clusters equivalent to 3 A˚ of thickness in the 7th monolayer of the supercell (see Table 1).
Fig. 7. Histogram of magnetic moments distribution for supercells with the Fe clusters equivalent to 3 A˚ (a) and 7 A˚ (b) thickness after intermixing of atoms obtained in the noncollinear model counted on all considered initial orientation of the clusters.
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moments values and in-plane angles on the atoms of the 7th monolayer (see Table 1) are shown in Fig. 6 for the selfconsistent solution with initial angles on the Fe atoms of one of the cluster deviated by 901. The energies of the noncollinear decisions for the supercells with initial angles of 0, 45, 90 and 1801 are found to be close to each other, while their values gain about 20 K per site in comparison with the collinear energy. The calculations show that the supercells with each initial angle prefer to keep the mutual orientation of the average clusters moments close to the initial ones, besides the moments orientation on the sites is different. Because of this fact, one can conclude that the Fe clusters have a variety of orientations of the average magnetic moments, and the superlattices have to be considered as a superparamagnetic system. Calculations for the supercells, consisting of the clusters on 3 continuous Fe MLs with Fe thickness equivalent to 7 A˚, result in the collinear orientation of the magnetic moments with the energy close to the energy of the collinear approach. The magnetic moments histograms counted on all considered initial orientations of the clusters are shown in Fig. 7. One can see that the histograms differ by the number of peaks and its values for the supercells with the thickness of Fe equivalent to 3 and 7 A˚. The histogram for the supercell with Fe layer of 7 A˚ forms the typical sextet measured by Mo¨ssbauer spectroscopy for the Fe with a large number of Fe atoms. 5. Conclusion The self-consistent calculations of magnetic moments distribution in the Fe/Cr superlattices performed in PAM using the collinear and noncollinear approaches show that the superlattices of 3 and 7 A˚ thicknesses with continuously smooth Fe layers have the collinear orientation of magnetic moments. The intermixing of Fe and Cr atoms applied to these structures leads to noncollinearity only for the superlattice with Fe thickness equivalent to 3 A˚. The separated clusters in the Fe interlayers also lead to noncollinear orientation of the magnetic moments for the Fe layers equivalent to 3 A˚ after the intermixing, as before. The average moments of Fe clusters are found to keep the mutual orientation preset by the initial angles of 0, 45, 90 and 1801 with the total energies for all considered
configurations close to each other. It means that every orientation can take place in the superlattice and such a structure could be considered as a superparamagnetic system. The superlattices with the thicker Fe layers equivalent to 7 A˚ always keep the collinear magnetic moments orientation. The magnetic moments histograms can be compared with experiments to show how the intermixing and onset of the Fe clusters at the interlayers influence on the hyperfine fields distribution measured by Mo¨ssbauer spectroscopy. Acknowledgment This work is partially supported by RFBR 07-02-01289, 06-02-16722, Scientific School of RF 5869.2006.2 and the CNRS-RAS program.
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