- Email: [email protected]
Cu(001 ) interfaces
Nuclear Instruments and Methods in Physics Research B 193 (2002) 359–364 www.elsevier.com/locate/nimb
A molecular dynamics study of Ni/Cu(0 0 1) interfaces Jose C. Jimenez-S aez a, Javier Domınguez-V azquez b, A. Mari Carmen Perez-Martın b,*, Jose J. Jimenez-Rodrıguez a
b
Departamento de Fısica y Quımica Aplicadas a la T ecnica Aeron autica, EUIT Aeron autica, Universidad Polit ecnica, E-28040 Madrid, Spain b Departamento de Electricidad y Electr onica, Facultad de Ciencias Fisicas, Universidad Complutense, E-28040 Madrid, Spain
Abstract This work is focused mainly on the analysis of effects related to a lattice misfit at a metallic interface. The system studied is the Ni/Cu(0 0 1) which exhibits a misfit of 2.6%. For this structure, the adjustment between the lattice parameters of a Ni crystal layer over Cu(0 0 1) substrate is analysed. To avoid edge effects a large enough substrate is taken while the Ni crystal set on top has smaller dimensions than the substrate. We have studied structures of one, two, four and ten monolayers of Ni set on top of the Cu substrate. It is shown how the stabilisation of different interface structures on an atomic scale is achieved; especially, the type of processes that help to accomplish a gradual change in the atomic distances. The main conclusion is the anisotropy of the coupling provokes that a cubic becomes a tetragonal lattice. The rearrangement of atoms and the strain field induced by the coupling are studied in detail. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Molecular dynamics; Metallic interfaces; Ni/Cu interfaces
1. Introduction The knowledge of the physical properties in metallic junctions is very important from the point of view of technological applications such as metallic contacts in integrated circuits and multilayered magnetic devices. Recent technological advances have allowed to research experimentally quantitative structural deformation and formation of defects near the interface [1,2]. The aim of these studies is to achieve a better understanding of the correlation between the size of the grown overlayer and the formation of defects. The main magnetic
*
Corresponding author. Tel./fax: +34-91-394-5196. E-mail address: cperez@fis.ucm.es (A.M.C. Pe´rez-Martı´n).
properties exhibited by metal/metal interfaces are the oscillatory magnetic interlayer exchange coupling and the giant magnetoresistance [3]. Recently, it has been shown that the interface plays an important role in the magnetic coupling between monolayers and in the formation of magnetic states [4]. Moreover, it can be said that a line of progress in the new magnetic systems is connected to the possibility of producing systems such as clusters on top of metallic surfaces [5]. For the last three decades computer simulations have allowed to improve the knowledge of the physical properties of metals and alloys. In particular, due to the development of empirical interatomic potentials [6,7], it has become possible to describe by the molecular dynamics (MD) technique a great number of solid properties [8,9]. All
0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 0 8 0 5 - 4
360
J.C. Jimenez-Saez et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 359–364
these processes are rather difficult to study by other methods due to the enormous number of involved atoms. The MD technique has been used in this work to explain the formation of a metal/ metal interface, in particular a Ni/Cu(0 0 1) interface. Ni/Cu is a magnetic system that exhibits a strong tendency to segregation near the temperature of 0 K, i.e. it tends to form abrupt interfaces [10] unlike other systems whose interfaces tend to be pseudo-amorphous alloys [11]. The lattice mismatch between these two different magnetic elements gives rise to magnetoelastic anisotropy. It is now well established that the perpendicular magnetic anisotropy (PMA) in Ni/Cu systems [12,13] is caused by strain. This work is devoted to studying structures consisting of a substrate of Cu(0 0 1) on top of which Ni crystals of different thicknesses were put down. The objective of the simulations performed has been to understand the causes that cause the modification of the atomic positions at the interface and effects induced by temperature and the number of grown monolayers. The fundamental bases of the calculations are described in Section 2, the results are presented and discussed in Section 3 and Section 4 is devoted to the summary and conclusions.
2. Calculation method The method of calculation is based on the MD technique. As interatomic potential, we have chosen the embedded atom method (EAM) that has been successfully used in the study of bulk properties of metal and alloys, surface relaxation and reconstruction features [6,14]. The classical equations of motion were solved according to the algorithm of Smith and Harrison [15]. The maximum time step value was 7:8 104 ps and in order to reduce the computer time, a linked cell neighbour list algorithm was utilised [16]. Our interest is focused on the processes involved in the coupling of two crystalline structures with an initial misfit of 2.6% in their lattice parameters. To avoid the effects associated to the growth kinetics [17], we start with a system close to the final stage of a cluster/substrate structure in
which the growth processes are over. Specifically, we placed a Ni crystal (a few monolayers) on the surface of a Cu crystal (substrate). The substrate was larger than the Ni cluster in all directions. Fixed walls were considered for the substrate, except at the surface in contact with Ni. The Ni crystal was left with all its walls free. The calculations have been performed for a Cu(0 0 1) substrate containing 27 613 atoms distributed in (23 23 12) unit cells. At the centre of the free Cu surface, a Ni(0 0 1) crystal is set down. At the interface two different arrangements are simulated: AA and BA. To explain what we mean by AA or AB arrangements let us consider the well known A and B arrangement of atomic planes, for instance in a Cu crystal. Let us think about a couple of such planes and suppose that one of the planes (any) is replaced by a Ni plane with its corresponding atomic spacing. Due to the misfit in the lattice parameters there will be now regions that will be strictly speaking AB type, others will be AA type and the rest will be something in between. To differentiate these cases the surface area of the Ni/Cu interface is chosen as large as possible but avoiding including more than one configuration. For both cases, AA and BA, the Ni crystal surface size was (10 10) unit cells. Its thicknesses were one, two, four and ten monolayers. Initially, the separation between both crystals . This value corresponds to the miniwas 2.16 A mum of the potential energy as a function of the separation distance between two ideal crystals of Ni and Cu, both (0 0 1) oriented, whose interface surfaces are (26 26) unit cells. The results shown in this work do not depend on this distance, although a proper initial value saves computer time. Once a Ni crystal is set down on top of a Cu substrate, the whole system (Ni over Cu) is allowed to evolve dynamically for 15.6 ps (relaxation time). The appearance of phonons makes the quantitative analysis of the atomic distances at the interface more difficult. In order to eliminate this effect, we used the method of conjugate gradients [18] after the relaxation to minimise the potential energy of the system. Then, we proceeded to study the physical properties of the interface: strain and structural coupling.
J.C. Jimenez-Saez et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 359–364
All systems were analysed without any initial temperature. Nevertheless, the initial potential energy at the interface produced after relaxation a mean increase in the temperature of the system between 30 and 40 K. At the interface, the initial local temperature of the relaxation process was about 100 K. The effect of the system temperature on the relaxation process was also analysed. All systems were relaxed at 300 K as well. During the relaxation time, a thermal bath was applied to the Cu substrate in order to control the temperature of the system. After the relaxation of the system, the conjugate gradient method was also applied. To avoid border effects, the analysis was performed in a parallelepiped centred in both structures. The base size (parallel to the interface) is always chosen excluding the outermost cell of the Ni crystal and the height is enough to embrace all the planes of both crystals.
3. Results First of all, let us describe the mechanisms by which a Ni/Cu(0 0 1) system achieves the equilibrium starting from the initial situations described above. Lateral movement of the Ni atoms tries to adjust the in-plane lattice parameter to minimise the initial stress at the interface, this movement is linked to an upward flux of the Ni atoms from a layer to the upper one, originating from the initial overlapping of the in-plane position of the atoms of the interface layers (AA). The upward flux can develop an adlayer on top of the Ni crystal, which is only present for low coverage (1–4 monolayers). In the case of 10 monolayers the pressure over the interface prevents the vertical flux, and the lateral movement prevails. Next, we explain in detail these effects in the case of four monolayers in the AA configuration. Fig. 1 shows the xz-projection at different time steps of the relaxation process of an AA system made up of a Cu substrate and four Ni monolayers. The result is the formation of an adlayer, leaving all the Ni monolayers with approximately the same number of atoms. The splitting of the planes at the centre makes it possible to transfer atoms upwards from one layer to the next. Atoms
361
from the Ni monolayer next to the Cu rise to the upper monolayer and arrange adequately. At the same time, a number of atoms from this monolayer must go up to the monolayer above and move into fcc sites and so on. Once the transfer of atoms is over the fcc structure is recovered. It has also been observed that Ni atoms start to form this adlayer along the h1 1 0i directions at the surface. Fig. 2 displays the final positions of the atoms in each layer: the upper Cu layer (Fig. 2(a)), the four Ni monolayers (2(b)–(e)) and the top adlayer (Fig. 2(f)). Note that in the figure open symbols indicate a different layer of origin. In this figure, it can be seen how the atoms along the [1–10] diagonal have been pushed upwards (see Fig. 2(c)) making room for other atoms to be displaced laterally from an A-type to a B-type plane position and vice versa. The number of atoms moving from a plane to the plane above steadily increases when moving away from the interface. This process gives rise to an adlayer shown in Fig. 2(f), where the atoms make a pattern very similar to that represented in Fig. 2(e) which is formed by the atoms coming from the lower layer. The upward displacements of the atoms is almost vertical, that is, their x and y coordinates correspond nearly to those in the next plane. The A planes become B planes in the centre for a better lattice coupling, therefore, an atom that goes up to the next plane keeps its original A or B configuration. Note also that, as part of the same relaxation process, some adatoms appear on the lateral surfaces, a few Cu atoms occupy positions in the Ni layer and some vacancies have been generated in the upper Cu layer. Similar processes have been observed for the case of one or two initial Ni monolayers and AA interface, respectively. These simulations clearly show the tendency of Ni(0 0 1) to form adlayers along h1 1 0i directions parallel to the interface. This directionality also appears in magnetic phenomena. In fact, the directions of the easy magnetization axes in Ni are along the h1 1 0i directions [12]. This anisotropic arrangement of Ni has also been experimentally observed in other structures such as Au/Ni(1 1 0) [19]. It is remarkable that systems with one, two and four monolayers at room temperature recover the fcc structure and form an adlayer as well.
362
J.C. Jimenez-Saez et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 359–364
Fig. 1. Dynamic evolution of a (10 10 2) Ni crystal () on a (23 23 12) Cu substrate () with AA interface. The images correspond to the xz-projection at different times. Only the upper Cu plane is shown.
For a 10 monolayer system with an AA interface, only a lateral relaxation is observed, which gives rise to adatoms at the lateral surfaces. Ten Ni monolayers produce a sufficiently high pressure over the interface which prevents the upward flux of atoms. For BA structures of one, two, four and ten monolayers, the formation of adlayers has not been observed, independently of the temperature of the system (0 or 300 K). This result is due to a better initial arrangement of the atoms, which allows a coupling of the interface only by a lateral relaxation and without the appearance of structural defects. To quantify the deformation processes we plot, in Fig. 3, the lattice strain [2] as a function of the
number of the layer in a 10 Ni monolayer system. The strain is defined as ðb0 bÞ=b, where b0 and b are, respectively, the lattice parameter parallel to the interface in a perfect crystal (either Ni or Cu) and in a Ni/Cu(0 0 1) relaxed system as obtained from our MD calculations. In this figure, we have also plotted the ratio of the lattice parameters b=c, c being the lattice parameter perpendicular to the interface. Hereafter, we shall refer to this ratio as the lattice deformation. It can be seen, in Fig. 3, how the strain and the deformation run parallel along all the layers. In Cu, at the interface the strain has a change of sign although its value is rather close to zero and the deformation takes a value close to one corresponding to an almost
J.C. Jimenez-Saez et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 359–364
363
Fig. 3. Strain and lattice deformation (b=c) after the relaxation of a (10 10 5) Ni crystal on a (23 23 12) Cu substrate for AA (j) and BA (D) interfaces.
Fig. 2. xy-projection of the atoms in the four Ni monolayer system after the relaxation of the structure described in Fig. 1. We distinguish atoms of the same monolayer () from those coming from the plane below () or the plane above (M).
ideal lattice. The deformation and the strain decrease at deeper layers (c < b), until both magnitudes reach a minimum. After this, the deformation and strain increase until they adopt their ideal values of 1 and 0, respectively. In Ni, the strain and the deformation decrease as the number of the layer increases. Far from the interface, their values are above the corresponding ideal values and near the interface they are below these. The minimum value is at the interface, which means a more distorted structure there (c < b). Although the Ni structure distorts to fit the Cu substrate, it is remarkable that the unit cell volume does not change, while it becomes smaller for Cu. Experimental results for Ni [1] confirm this result.
In short, the connection between (2 0 0) and (0 2 0) planes between Ni and Cu causes a strain in the lattice. This strain gives rise to a deformation, which is approximately proportional to the strain. Moreover, the lattice deformation shows the collapse of the Ni crystal along the direction normal to the interface (c < b) so that the lattice becomes tetragonal. This compression causes for the (0 0 2) (2 Ni planes a minimum separation of 1.67 A . The layer), while the initial separation was 1.76 A reason is the expansion of (0 2 0) and (2 0 0) Ni planes to connect with their corresponding Cu planes (pseudo-morphic growth). This can lead to a stress-induced magnetic anisotropy due to inverse magnetostriction and, this anisotropy may be the major contribution to the observed PMA [12].
4. Summary and conclusions Ni clusters with different sizes set on top of a Cu(0 0 1) substrate have been studied. To avoid
364
J.C. Jimenez-Saez et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 359–364
effects of the growth kinetics, the Ni cluster was put down over the Cu-substrate, in such a way that the resultant system was close to an epitaxial growth. The dynamic evolution and the conjugate gradient method were used to lead the system to the state of minimum energy. We have observed in systems whose initial state was far from fcc coupling between the two crystals the appearance of a series of structures implying long-range atomic rearrangements such as adlayers. Systems of one or two monolayers eject atoms and form an adlayer along h1 1 0i directions at the surface. Systems of four monolayers are in the limit between the upward ejection of atoms and the lateral relaxation and still form an adlayer. This adaptation of Ni/Cu structure makes the initially cubic system evolve to a tetragonal system (a ¼ b 6¼ c) at the interface. This result reflects the anisotropy of the relaxation process along the transverse and parallel directions to the interface. We have analysed in detail the strain and the lattice deformation in the cluster/substrate system observing that both magnitudes are proportional everywhere in the system and have a maximum in absolute value at the Ni layer of the interface. Despite this deformation of the structure, it has been determined that the size of the unit cell in Ni remains constant, as observed experimentally.
Acknowledgements This work has been partially supported by DGICYT (Project no. MAT98-0859).
References [1] W. Platow, U. Bovensiepen, P. Poulopoulos, M. Farle, K. Baberschke, L. Hammer, S. Walter, S. M€ uller, K. Heinz, Phys. Rev. B 59 (1999) 12641. [2] S.A. Chambers, H.W. Chen, I.M. Vitomirov, S.B. Anderson, J.H. Weaver, Phys. Rev. B 33 (1986) 8810. [3] S.S.P. Parkin, N. More, K.P. Roche, Phys. Rev. Lett. 64 (1990) 2304. [4] G. Panaccione, F. Sirotti, E. Narducci, G. Rossi, Phys. Rev. B 55 (1997) 389. [5] V.S. Stepanyuk, W. Hergert, P. Rennert, K. Kokko, A.F. Tatarchenko, K. Wildberger, Phys. Rev. B 57 (1998) 15585. [6] S.M. Foiles, M.I. Baskes, M.S. Daw, Phys. Rev. B 33 (1986) 7983. [7] G.J. Ackland, V. Vitek, Phys. Rev. B 41 (1990) 10324. [8] J. Domınguez-Vazquez, E.P. Andribet, A.M.C. PerezMartın, J.J. Jimenez-Rodrıguez, Radiat. Effects Defects Solids 142 (1997) 115. [9] M.J. Caturla, T. Dıaz de la Rubia, G.H. Gilmer, J. Appl. Phys. 77 (1995) 3121. [10] A.M.N. Niklasson, I.A. Abrikosov, B. Johansson, Phys. Rev. B 58 (1998) 3613. [11] P. Schieffer, M.-H. Tuilier, M.-C. Hanf, C. Krembel, G. Gewinner, D. Chandesris, H. Magnan, Phys. Rev. B 57 (1998) 15507. [12] R. Naik, C. Kota, J.S. Payson, G.L. Dunifer, Phys. Rev. B 48 (1993) 1008. [13] G. Lauhoff, C.A.F. Vaz, J.A.C. Bland, J. Lee, T. Suzuki, Phys. Rev. B 61 (2000) 6805. [14] M.S. Daw, Surf. Sci. 166 (1986) L161; S.M. Foiles, Surf. Sci. 191 (1987) L779. [15] R. Smith, D.E. Harrison Jr., Comp. Phys. 3 (1989) 68. [16] M.P. Alien, D.J. Tildesley, Computer Simulation of Liquids, Oxford Science Publications, 1996. [17] H. Lei, Q. Hou, M. Hou, Nucl. Instr. and Meth. B 164–165 (2000) 537. [18] M.J.D. Powell, Math. Program. 12 (1977) 241. [19] L. Pleth Nielsen, F. Besenbacher, I. Stensgaard, E. Laegsgaard, C. Engdahl, P. Stoltze, K.W. Jacobsen, J.K. Norskov, Phys. Rev. Lett. 71 (1993) 754.
A molecular dynamics study of Ni/Cu(0 0 1) interfaces Jose C. Jimenez-S aez a, Javier Domınguez-V azquez b, A. Mari Carmen Perez-Martın b,*, Jose J. Jimenez-Rodrıguez a
b
Departamento de Fısica y Quımica Aplicadas a la T ecnica Aeron autica, EUIT Aeron autica, Universidad Polit ecnica, E-28040 Madrid, Spain b Departamento de Electricidad y Electr onica, Facultad de Ciencias Fisicas, Universidad Complutense, E-28040 Madrid, Spain
Abstract This work is focused mainly on the analysis of effects related to a lattice misfit at a metallic interface. The system studied is the Ni/Cu(0 0 1) which exhibits a misfit of 2.6%. For this structure, the adjustment between the lattice parameters of a Ni crystal layer over Cu(0 0 1) substrate is analysed. To avoid edge effects a large enough substrate is taken while the Ni crystal set on top has smaller dimensions than the substrate. We have studied structures of one, two, four and ten monolayers of Ni set on top of the Cu substrate. It is shown how the stabilisation of different interface structures on an atomic scale is achieved; especially, the type of processes that help to accomplish a gradual change in the atomic distances. The main conclusion is the anisotropy of the coupling provokes that a cubic becomes a tetragonal lattice. The rearrangement of atoms and the strain field induced by the coupling are studied in detail. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Molecular dynamics; Metallic interfaces; Ni/Cu interfaces
1. Introduction The knowledge of the physical properties in metallic junctions is very important from the point of view of technological applications such as metallic contacts in integrated circuits and multilayered magnetic devices. Recent technological advances have allowed to research experimentally quantitative structural deformation and formation of defects near the interface [1,2]. The aim of these studies is to achieve a better understanding of the correlation between the size of the grown overlayer and the formation of defects. The main magnetic
*
Corresponding author. Tel./fax: +34-91-394-5196. E-mail address: cperez@fis.ucm.es (A.M.C. Pe´rez-Martı´n).
properties exhibited by metal/metal interfaces are the oscillatory magnetic interlayer exchange coupling and the giant magnetoresistance [3]. Recently, it has been shown that the interface plays an important role in the magnetic coupling between monolayers and in the formation of magnetic states [4]. Moreover, it can be said that a line of progress in the new magnetic systems is connected to the possibility of producing systems such as clusters on top of metallic surfaces [5]. For the last three decades computer simulations have allowed to improve the knowledge of the physical properties of metals and alloys. In particular, due to the development of empirical interatomic potentials [6,7], it has become possible to describe by the molecular dynamics (MD) technique a great number of solid properties [8,9]. All
0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 0 8 0 5 - 4
360
J.C. Jimenez-Saez et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 359–364
these processes are rather difficult to study by other methods due to the enormous number of involved atoms. The MD technique has been used in this work to explain the formation of a metal/ metal interface, in particular a Ni/Cu(0 0 1) interface. Ni/Cu is a magnetic system that exhibits a strong tendency to segregation near the temperature of 0 K, i.e. it tends to form abrupt interfaces [10] unlike other systems whose interfaces tend to be pseudo-amorphous alloys [11]. The lattice mismatch between these two different magnetic elements gives rise to magnetoelastic anisotropy. It is now well established that the perpendicular magnetic anisotropy (PMA) in Ni/Cu systems [12,13] is caused by strain. This work is devoted to studying structures consisting of a substrate of Cu(0 0 1) on top of which Ni crystals of different thicknesses were put down. The objective of the simulations performed has been to understand the causes that cause the modification of the atomic positions at the interface and effects induced by temperature and the number of grown monolayers. The fundamental bases of the calculations are described in Section 2, the results are presented and discussed in Section 3 and Section 4 is devoted to the summary and conclusions.
2. Calculation method The method of calculation is based on the MD technique. As interatomic potential, we have chosen the embedded atom method (EAM) that has been successfully used in the study of bulk properties of metal and alloys, surface relaxation and reconstruction features [6,14]. The classical equations of motion were solved according to the algorithm of Smith and Harrison [15]. The maximum time step value was 7:8 104 ps and in order to reduce the computer time, a linked cell neighbour list algorithm was utilised [16]. Our interest is focused on the processes involved in the coupling of two crystalline structures with an initial misfit of 2.6% in their lattice parameters. To avoid the effects associated to the growth kinetics [17], we start with a system close to the final stage of a cluster/substrate structure in
which the growth processes are over. Specifically, we placed a Ni crystal (a few monolayers) on the surface of a Cu crystal (substrate). The substrate was larger than the Ni cluster in all directions. Fixed walls were considered for the substrate, except at the surface in contact with Ni. The Ni crystal was left with all its walls free. The calculations have been performed for a Cu(0 0 1) substrate containing 27 613 atoms distributed in (23 23 12) unit cells. At the centre of the free Cu surface, a Ni(0 0 1) crystal is set down. At the interface two different arrangements are simulated: AA and BA. To explain what we mean by AA or AB arrangements let us consider the well known A and B arrangement of atomic planes, for instance in a Cu crystal. Let us think about a couple of such planes and suppose that one of the planes (any) is replaced by a Ni plane with its corresponding atomic spacing. Due to the misfit in the lattice parameters there will be now regions that will be strictly speaking AB type, others will be AA type and the rest will be something in between. To differentiate these cases the surface area of the Ni/Cu interface is chosen as large as possible but avoiding including more than one configuration. For both cases, AA and BA, the Ni crystal surface size was (10 10) unit cells. Its thicknesses were one, two, four and ten monolayers. Initially, the separation between both crystals . This value corresponds to the miniwas 2.16 A mum of the potential energy as a function of the separation distance between two ideal crystals of Ni and Cu, both (0 0 1) oriented, whose interface surfaces are (26 26) unit cells. The results shown in this work do not depend on this distance, although a proper initial value saves computer time. Once a Ni crystal is set down on top of a Cu substrate, the whole system (Ni over Cu) is allowed to evolve dynamically for 15.6 ps (relaxation time). The appearance of phonons makes the quantitative analysis of the atomic distances at the interface more difficult. In order to eliminate this effect, we used the method of conjugate gradients [18] after the relaxation to minimise the potential energy of the system. Then, we proceeded to study the physical properties of the interface: strain and structural coupling.
J.C. Jimenez-Saez et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 359–364
All systems were analysed without any initial temperature. Nevertheless, the initial potential energy at the interface produced after relaxation a mean increase in the temperature of the system between 30 and 40 K. At the interface, the initial local temperature of the relaxation process was about 100 K. The effect of the system temperature on the relaxation process was also analysed. All systems were relaxed at 300 K as well. During the relaxation time, a thermal bath was applied to the Cu substrate in order to control the temperature of the system. After the relaxation of the system, the conjugate gradient method was also applied. To avoid border effects, the analysis was performed in a parallelepiped centred in both structures. The base size (parallel to the interface) is always chosen excluding the outermost cell of the Ni crystal and the height is enough to embrace all the planes of both crystals.
3. Results First of all, let us describe the mechanisms by which a Ni/Cu(0 0 1) system achieves the equilibrium starting from the initial situations described above. Lateral movement of the Ni atoms tries to adjust the in-plane lattice parameter to minimise the initial stress at the interface, this movement is linked to an upward flux of the Ni atoms from a layer to the upper one, originating from the initial overlapping of the in-plane position of the atoms of the interface layers (AA). The upward flux can develop an adlayer on top of the Ni crystal, which is only present for low coverage (1–4 monolayers). In the case of 10 monolayers the pressure over the interface prevents the vertical flux, and the lateral movement prevails. Next, we explain in detail these effects in the case of four monolayers in the AA configuration. Fig. 1 shows the xz-projection at different time steps of the relaxation process of an AA system made up of a Cu substrate and four Ni monolayers. The result is the formation of an adlayer, leaving all the Ni monolayers with approximately the same number of atoms. The splitting of the planes at the centre makes it possible to transfer atoms upwards from one layer to the next. Atoms
361
from the Ni monolayer next to the Cu rise to the upper monolayer and arrange adequately. At the same time, a number of atoms from this monolayer must go up to the monolayer above and move into fcc sites and so on. Once the transfer of atoms is over the fcc structure is recovered. It has also been observed that Ni atoms start to form this adlayer along the h1 1 0i directions at the surface. Fig. 2 displays the final positions of the atoms in each layer: the upper Cu layer (Fig. 2(a)), the four Ni monolayers (2(b)–(e)) and the top adlayer (Fig. 2(f)). Note that in the figure open symbols indicate a different layer of origin. In this figure, it can be seen how the atoms along the [1–10] diagonal have been pushed upwards (see Fig. 2(c)) making room for other atoms to be displaced laterally from an A-type to a B-type plane position and vice versa. The number of atoms moving from a plane to the plane above steadily increases when moving away from the interface. This process gives rise to an adlayer shown in Fig. 2(f), where the atoms make a pattern very similar to that represented in Fig. 2(e) which is formed by the atoms coming from the lower layer. The upward displacements of the atoms is almost vertical, that is, their x and y coordinates correspond nearly to those in the next plane. The A planes become B planes in the centre for a better lattice coupling, therefore, an atom that goes up to the next plane keeps its original A or B configuration. Note also that, as part of the same relaxation process, some adatoms appear on the lateral surfaces, a few Cu atoms occupy positions in the Ni layer and some vacancies have been generated in the upper Cu layer. Similar processes have been observed for the case of one or two initial Ni monolayers and AA interface, respectively. These simulations clearly show the tendency of Ni(0 0 1) to form adlayers along h1 1 0i directions parallel to the interface. This directionality also appears in magnetic phenomena. In fact, the directions of the easy magnetization axes in Ni are along the h1 1 0i directions [12]. This anisotropic arrangement of Ni has also been experimentally observed in other structures such as Au/Ni(1 1 0) [19]. It is remarkable that systems with one, two and four monolayers at room temperature recover the fcc structure and form an adlayer as well.
362
J.C. Jimenez-Saez et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 359–364
Fig. 1. Dynamic evolution of a (10 10 2) Ni crystal () on a (23 23 12) Cu substrate () with AA interface. The images correspond to the xz-projection at different times. Only the upper Cu plane is shown.
For a 10 monolayer system with an AA interface, only a lateral relaxation is observed, which gives rise to adatoms at the lateral surfaces. Ten Ni monolayers produce a sufficiently high pressure over the interface which prevents the upward flux of atoms. For BA structures of one, two, four and ten monolayers, the formation of adlayers has not been observed, independently of the temperature of the system (0 or 300 K). This result is due to a better initial arrangement of the atoms, which allows a coupling of the interface only by a lateral relaxation and without the appearance of structural defects. To quantify the deformation processes we plot, in Fig. 3, the lattice strain [2] as a function of the
number of the layer in a 10 Ni monolayer system. The strain is defined as ðb0 bÞ=b, where b0 and b are, respectively, the lattice parameter parallel to the interface in a perfect crystal (either Ni or Cu) and in a Ni/Cu(0 0 1) relaxed system as obtained from our MD calculations. In this figure, we have also plotted the ratio of the lattice parameters b=c, c being the lattice parameter perpendicular to the interface. Hereafter, we shall refer to this ratio as the lattice deformation. It can be seen, in Fig. 3, how the strain and the deformation run parallel along all the layers. In Cu, at the interface the strain has a change of sign although its value is rather close to zero and the deformation takes a value close to one corresponding to an almost
J.C. Jimenez-Saez et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 359–364
363
Fig. 3. Strain and lattice deformation (b=c) after the relaxation of a (10 10 5) Ni crystal on a (23 23 12) Cu substrate for AA (j) and BA (D) interfaces.
Fig. 2. xy-projection of the atoms in the four Ni monolayer system after the relaxation of the structure described in Fig. 1. We distinguish atoms of the same monolayer () from those coming from the plane below () or the plane above (M).
ideal lattice. The deformation and the strain decrease at deeper layers (c < b), until both magnitudes reach a minimum. After this, the deformation and strain increase until they adopt their ideal values of 1 and 0, respectively. In Ni, the strain and the deformation decrease as the number of the layer increases. Far from the interface, their values are above the corresponding ideal values and near the interface they are below these. The minimum value is at the interface, which means a more distorted structure there (c < b). Although the Ni structure distorts to fit the Cu substrate, it is remarkable that the unit cell volume does not change, while it becomes smaller for Cu. Experimental results for Ni [1] confirm this result.
In short, the connection between (2 0 0) and (0 2 0) planes between Ni and Cu causes a strain in the lattice. This strain gives rise to a deformation, which is approximately proportional to the strain. Moreover, the lattice deformation shows the collapse of the Ni crystal along the direction normal to the interface (c < b) so that the lattice becomes tetragonal. This compression causes for the (0 0 2) (2 Ni planes a minimum separation of 1.67 A . The layer), while the initial separation was 1.76 A reason is the expansion of (0 2 0) and (2 0 0) Ni planes to connect with their corresponding Cu planes (pseudo-morphic growth). This can lead to a stress-induced magnetic anisotropy due to inverse magnetostriction and, this anisotropy may be the major contribution to the observed PMA [12].
4. Summary and conclusions Ni clusters with different sizes set on top of a Cu(0 0 1) substrate have been studied. To avoid
364
J.C. Jimenez-Saez et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 359–364
effects of the growth kinetics, the Ni cluster was put down over the Cu-substrate, in such a way that the resultant system was close to an epitaxial growth. The dynamic evolution and the conjugate gradient method were used to lead the system to the state of minimum energy. We have observed in systems whose initial state was far from fcc coupling between the two crystals the appearance of a series of structures implying long-range atomic rearrangements such as adlayers. Systems of one or two monolayers eject atoms and form an adlayer along h1 1 0i directions at the surface. Systems of four monolayers are in the limit between the upward ejection of atoms and the lateral relaxation and still form an adlayer. This adaptation of Ni/Cu structure makes the initially cubic system evolve to a tetragonal system (a ¼ b 6¼ c) at the interface. This result reflects the anisotropy of the relaxation process along the transverse and parallel directions to the interface. We have analysed in detail the strain and the lattice deformation in the cluster/substrate system observing that both magnitudes are proportional everywhere in the system and have a maximum in absolute value at the Ni layer of the interface. Despite this deformation of the structure, it has been determined that the size of the unit cell in Ni remains constant, as observed experimentally.
Acknowledgements This work has been partially supported by DGICYT (Project no. MAT98-0859).
References [1] W. Platow, U. Bovensiepen, P. Poulopoulos, M. Farle, K. Baberschke, L. Hammer, S. Walter, S. M€ uller, K. Heinz, Phys. Rev. B 59 (1999) 12641. [2] S.A. Chambers, H.W. Chen, I.M. Vitomirov, S.B. Anderson, J.H. Weaver, Phys. Rev. B 33 (1986) 8810. [3] S.S.P. Parkin, N. More, K.P. Roche, Phys. Rev. Lett. 64 (1990) 2304. [4] G. Panaccione, F. Sirotti, E. Narducci, G. Rossi, Phys. Rev. B 55 (1997) 389. [5] V.S. Stepanyuk, W. Hergert, P. Rennert, K. Kokko, A.F. Tatarchenko, K. Wildberger, Phys. Rev. B 57 (1998) 15585. [6] S.M. Foiles, M.I. Baskes, M.S. Daw, Phys. Rev. B 33 (1986) 7983. [7] G.J. Ackland, V. Vitek, Phys. Rev. B 41 (1990) 10324. [8] J. Domınguez-Vazquez, E.P. Andribet, A.M.C. PerezMartın, J.J. Jimenez-Rodrıguez, Radiat. Effects Defects Solids 142 (1997) 115. [9] M.J. Caturla, T. Dıaz de la Rubia, G.H. Gilmer, J. Appl. Phys. 77 (1995) 3121. [10] A.M.N. Niklasson, I.A. Abrikosov, B. Johansson, Phys. Rev. B 58 (1998) 3613. [11] P. Schieffer, M.-H. Tuilier, M.-C. Hanf, C. Krembel, G. Gewinner, D. Chandesris, H. Magnan, Phys. Rev. B 57 (1998) 15507. [12] R. Naik, C. Kota, J.S. Payson, G.L. Dunifer, Phys. Rev. B 48 (1993) 1008. [13] G. Lauhoff, C.A.F. Vaz, J.A.C. Bland, J. Lee, T. Suzuki, Phys. Rev. B 61 (2000) 6805. [14] M.S. Daw, Surf. Sci. 166 (1986) L161; S.M. Foiles, Surf. Sci. 191 (1987) L779. [15] R. Smith, D.E. Harrison Jr., Comp. Phys. 3 (1989) 68. [16] M.P. Alien, D.J. Tildesley, Computer Simulation of Liquids, Oxford Science Publications, 1996. [17] H. Lei, Q. Hou, M. Hou, Nucl. Instr. and Meth. B 164–165 (2000) 537. [18] M.J.D. Powell, Math. Program. 12 (1977) 241. [19] L. Pleth Nielsen, F. Besenbacher, I. Stensgaard, E. Laegsgaard, C. Engdahl, P. Stoltze, K.W. Jacobsen, J.K. Norskov, Phys. Rev. Lett. 71 (1993) 754.