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Strain relaxation in thin films of Cu grown on Ni(0 0 1)
Physica B 248 (1998) 34—38
Strain relaxation in thin films of Cu grown on Ni(0 0 1) Frank Berg Rasmussen!,*, Jeff Baker!, Mourits Nielsen!, Robert Feidenhans’l!, Robert L. Johnson" ! Condensed Matter Physics and Chemistry Department, Ris~ National Laboratory, DK-4000 Roskilde, Denmark " II Institut fu( r Experimentalphysik, Universita( t Hamburg, Luruper Chausee 149, D-22761, Germany
Abstract Surface X-ray diffraction and kinematical model calculations are used to determine the strain relaxation of embedded wedges with internal (1 1 1) facets formed in thin Cu films when grown on Ni(0 0 1). We show the wedges to be inhomogenously strained with a large lateral relaxation near the Cu/Ni interface which decays rapidly away from the interface. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Strain relaxation; Nanoclusters; X-ray diffraction
1. Introduction In a recent scanning tunnelling microscopy (STM) study on Cu growth on Ni(0 0 1) characteristic stripes along S1 1 0T direction were observed on the surface [1]. The stripes have widths increasing linearly with coverage and they protrude about 0.6 A_ above the level of the surface for all coverages. It was suggested that the misfit between the larger Cu lattice and the Ni lattice (2.6% misfit) could be partly relaxed through the formation of wedgeshaped buried nanoscale Cu-clusters as shown in Fig. 1. The atoms inside the wedges are displaced half a nearest-neighbour distance along the wedge direction and a little, 0.4—0.6 A_ , upwards (Fig. 1). * Corresponding author. fax: (#45) 4677 4790; e-mail: [email protected].
We have recently performed surface X-ray diffraction (SXRD) measurements on Cu films grown on Ni(0 0 1) and presented direct evidence for wedge formation in the Cu film as described above [2]. The buried surfaces of the wedges are (1 1 1) facets and give rise to a very distinct scattering response (facets peaks) near the positions of truncation rods [3] from extended (1 1 1) facets (Fig. 2). Due to the displacement of all atoms within the wedges by half a nearest-neighbour distance, these will scatter in or out of phase with the Cu atoms outside for the corresponding scattering vector being an odd or even integer [2]. This is illustrated in Fig. 2 and represents the fingerprint of the wedges. In this paper we detail how information on the strain distribution in the Cu film is obtained by direct simulation of the measured intensity profiles.
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 1 9 9 - 9
F. Berg Rasmussen et al. / Physica B 248 (1998) 34—38
35
Fig. 1. Wedge-shaped nanoclusters with internal (1 1 1) facets (light grey atoms) formed in thin Cu films grown on Ni(0 0 1). The dark-grey Cu atoms are placed at the (pseudomorphic) 4-fold hollow sites with respect to the Ni substrate (black atoms).
2. Experimental details The SXRD experiments were performed at BW2 in HASYLAB (DESY). The sample was prepared according to the prescriptions of Ref. [1] at the FLIPPER II beam line. Then it was transferred to a portable UHV chamber with a hemispherical Be window which was mounted on the diffractometer. The photon energy was chosen to be either 8.1 or 9.6 keV, the surface of the crystal was aligned by total external reflection of the X-ray beam and the angle of incidence was kept constant and equal to the critical angle a "0.41°. In the rest of this paper # we shall use the LEED notation for the in-plane momentum transfer, see Ref. [2]. We shall discuss results obtained from a 9 monolayer (ML) film but the results are consistent with measurements on several films with coverages ranging from 5 to above 20 ML. Typical scans in reciprocal space are shown in Figs. 3 and 4.
3. Analysis
Fig. 2. k-scans in reciprocal space through the points (h, 0, 0.5) (LEED notation). When h is odd two extra ‘facets peaks’ are observed in addition to the central rod. For h even only the central rod is observed.
We shall in this paper simply assume the model of Ref. [1] (Fig. 1). Hence, we assume that the Cu atoms surrounding the wedges are laterally pseudomorphic with the substrate and that the wedges as a whole are displaced half a nearest-neighbour distance in the wedge direction and upwards
36
F. Berg Rasmussen et al. / Physica B 248 (1998) 34—38
Fig. 3. Calculated and measured transverse scans through the (10) CTR, the scan direction is indicated in the insert. The circles represent the experimental data points and the solid lines are the calculated profiles as described in the text.
Fig. 4. Calculated and measured longitudinal scans through the (10) CTR, the scan direction is indicated in the insert. The circles represent the experimental data points and the solid lines are the calculated profiles as described in the text.
by an amount h . We allow all Cu atoms inside C6 and outside the wedges to have a common vertical lattice constant c . Finally, for the atoms inside the 0 wedge we set the lattice parameter in the wedge direction equal to that of Ni but allow the in-plane lattice parameter b(n) along the short side of the wedges to depend on the height (in atomic layers) above the interface. We then perform model calculations using kinematical theory and compare these with the experimental scattering profiles. The main objective is not to simulate the central crystal truncation rods (CTR), where a more sophisticated treatment may be needed [2,4], but to reproduce the facet peaks which generally are well separated from the central rods. All experimental effects as Lorentz and polarisation factors, rod interception, detector acceptance, area corrections, etc., are taken into ac-
count in the simulations using standard methods [5]. As discussed in Ref. [2] we may regard the wedges as monodisperse at a given coverage and as no interference effects between the individual wedges were observed we may adopt a simple model consisting of a slab of Cu containing one wedge placed on top of a Ni crystal. The contribution from the Ni substrate is taken as the CTRscattering from a perfect Ni crystal of infinite extent along [0 0 1] [3]. The Ni CTRs do not, in general, contribute to the scattering intensity at the positions of the facet peaks but are included to get reasonable agreement between experimental and simulated rod scans (l scans) used to determine the actual coverage. We first consider a (1 0 l) rod scan (not shown). The best agreement between the simulated and experimental intensity distribution is obtained
F. Berg Rasmussen et al. / Physica B 248 (1998) 34—38
with a substrate—film distance equal to 1.83 A_ , a Debye—Waller factor B"2 A_ 2 and a coverage of 9.2 ML in reasonable agreement with the coverage determined by means of a crystal thickness monitor during growth. Simulations for a 9.2 ML film is obtained by adding the intensities calculated for a 9 ML and a 10 ML film with proper weights. The optimum parameters found here depend only weakly on the lateral strain distribution and are kept constant in the following. We then turn to simulations of the transverse and longitudinal scans (Figs. 3 and 4) and more specifically to the strain distribution inside and outside the wedge in terms of the parameters c and 0 h as well as the functional b(n). We have tried C6 a number of different functional forms of b(n) and obtain the best fit to the experimental data with a large lateral expansion at the bottom of the wedge following a Gaussian dependence with distance away from the interface. b(n)"b #e e~(n@n#)2. = 3%In this expression b is the lattice parameter at the = top of thick wedges (large n), e is the extra lattice 3%relaxation at the bottom of the wedge (n"0) and n determines the decay of the exponential. A satis# factory fit to all the measured data is obtained with the parameters of Table 1 and a common scale factor. The simulated intensity profiles are shown as solid lines in Figs. 3 and 4 and generally the agreement between simulated and experimental profiles is good. For the common vertical lattice constant, c , we 0 obtain a value of 3.67 A_ which corresponds to a vertical lattice expansion of 4% with respect to the Ni lattice. Assuming the whole Cu film to be
Table 1 Strain parameters obtained for a 9.2 ML Cu film on Ni(0 0 1), see the text (A_ )
Lattice units c 0 h C6 b = e 3%n #
1.04 0.14 1.00 0.07 7 layers
3.67 0.5 2.49 0.17
37
Fig. 5. In-plane lattice parameter along the short side of the wedges as a function of distance away from the interface measured in atomic layers, see the text.
laterally pseudomorphic with the substrate, i.e. neglecting wedge formation, a 4% lattice expansion is precisely what is expected from elasticity theory using the elastic constants of bulk Cu. The protrusion h is here found to be 0.5 A_ in C6 good agreement with the 0.4 A_ expected from a hard-sphere model and the 0.6 A_ quoted in the STM study [1]. 4. Discussion and conclusions The height dependence of the lateral lattice parameter within the wedges is plotted in Fig. 5 using the values given in Table 1. The obtained strain distribution is inhomogenous with a large relaxation near the apex of the wedges that rapidly decays away from the interface. For a coverage approaching 20 ML the lateral lattice parameter of the topmost layer is close to that of Ni offering an explanation to the observed change in growth above 20 ML [1]. In conclusion it has been possible to determine the strain relaxation in thin Cu films grown on Ni(0 0 1) using SXRD and kinematical model calculations revealing the unusual strain relaxation present in this system. Acknowledgement This project was supported by DANSYNC and BMBF (05 622GUA1).
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F. Berg Rasmussen et al. / Physica B 248 (1998) 34—38
References [1] B. Mu¨ller, B. Fischer, L. Nedelmann, A. Fricke, K. Kern, Phys. Rev. Lett. 76 (1996) 2358. [2] F. Berg Rasmussen, J. Baker M. Nielsen, R. Feidenhans’l, R.L. Johnson, Phys. Rev. Lett. 79 (1997) 4413.
[3] I.K. Robinson, Phys. Rev. B 33 (1986) 3830. [4] A.J. Steinfort, P.M.L.O. Scholte, A. Ettema, F. Tuinstra, M. Nielsen, E. Landemark, R. Feidenhans’l, G. Falkenberg, L. Seehofer, R.L. Johnson, Phys. Rev. Lett. 77 (1996) 2009. [5] E. Vlieg to be published; C. Schamper, H.L. Meyerheim, W. Moritz, J. Appl. Crystallogr. 26 (1993) 687.
Strain relaxation in thin films of Cu grown on Ni(0 0 1) Frank Berg Rasmussen!,*, Jeff Baker!, Mourits Nielsen!, Robert Feidenhans’l!, Robert L. Johnson" ! Condensed Matter Physics and Chemistry Department, Ris~ National Laboratory, DK-4000 Roskilde, Denmark " II Institut fu( r Experimentalphysik, Universita( t Hamburg, Luruper Chausee 149, D-22761, Germany
Abstract Surface X-ray diffraction and kinematical model calculations are used to determine the strain relaxation of embedded wedges with internal (1 1 1) facets formed in thin Cu films when grown on Ni(0 0 1). We show the wedges to be inhomogenously strained with a large lateral relaxation near the Cu/Ni interface which decays rapidly away from the interface. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Strain relaxation; Nanoclusters; X-ray diffraction
1. Introduction In a recent scanning tunnelling microscopy (STM) study on Cu growth on Ni(0 0 1) characteristic stripes along S1 1 0T direction were observed on the surface [1]. The stripes have widths increasing linearly with coverage and they protrude about 0.6 A_ above the level of the surface for all coverages. It was suggested that the misfit between the larger Cu lattice and the Ni lattice (2.6% misfit) could be partly relaxed through the formation of wedgeshaped buried nanoscale Cu-clusters as shown in Fig. 1. The atoms inside the wedges are displaced half a nearest-neighbour distance along the wedge direction and a little, 0.4—0.6 A_ , upwards (Fig. 1). * Corresponding author. fax: (#45) 4677 4790; e-mail: [email protected].
We have recently performed surface X-ray diffraction (SXRD) measurements on Cu films grown on Ni(0 0 1) and presented direct evidence for wedge formation in the Cu film as described above [2]. The buried surfaces of the wedges are (1 1 1) facets and give rise to a very distinct scattering response (facets peaks) near the positions of truncation rods [3] from extended (1 1 1) facets (Fig. 2). Due to the displacement of all atoms within the wedges by half a nearest-neighbour distance, these will scatter in or out of phase with the Cu atoms outside for the corresponding scattering vector being an odd or even integer [2]. This is illustrated in Fig. 2 and represents the fingerprint of the wedges. In this paper we detail how information on the strain distribution in the Cu film is obtained by direct simulation of the measured intensity profiles.
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 1 9 9 - 9
F. Berg Rasmussen et al. / Physica B 248 (1998) 34—38
35
Fig. 1. Wedge-shaped nanoclusters with internal (1 1 1) facets (light grey atoms) formed in thin Cu films grown on Ni(0 0 1). The dark-grey Cu atoms are placed at the (pseudomorphic) 4-fold hollow sites with respect to the Ni substrate (black atoms).
2. Experimental details The SXRD experiments were performed at BW2 in HASYLAB (DESY). The sample was prepared according to the prescriptions of Ref. [1] at the FLIPPER II beam line. Then it was transferred to a portable UHV chamber with a hemispherical Be window which was mounted on the diffractometer. The photon energy was chosen to be either 8.1 or 9.6 keV, the surface of the crystal was aligned by total external reflection of the X-ray beam and the angle of incidence was kept constant and equal to the critical angle a "0.41°. In the rest of this paper # we shall use the LEED notation for the in-plane momentum transfer, see Ref. [2]. We shall discuss results obtained from a 9 monolayer (ML) film but the results are consistent with measurements on several films with coverages ranging from 5 to above 20 ML. Typical scans in reciprocal space are shown in Figs. 3 and 4.
3. Analysis
Fig. 2. k-scans in reciprocal space through the points (h, 0, 0.5) (LEED notation). When h is odd two extra ‘facets peaks’ are observed in addition to the central rod. For h even only the central rod is observed.
We shall in this paper simply assume the model of Ref. [1] (Fig. 1). Hence, we assume that the Cu atoms surrounding the wedges are laterally pseudomorphic with the substrate and that the wedges as a whole are displaced half a nearest-neighbour distance in the wedge direction and upwards
36
F. Berg Rasmussen et al. / Physica B 248 (1998) 34—38
Fig. 3. Calculated and measured transverse scans through the (10) CTR, the scan direction is indicated in the insert. The circles represent the experimental data points and the solid lines are the calculated profiles as described in the text.
Fig. 4. Calculated and measured longitudinal scans through the (10) CTR, the scan direction is indicated in the insert. The circles represent the experimental data points and the solid lines are the calculated profiles as described in the text.
by an amount h . We allow all Cu atoms inside C6 and outside the wedges to have a common vertical lattice constant c . Finally, for the atoms inside the 0 wedge we set the lattice parameter in the wedge direction equal to that of Ni but allow the in-plane lattice parameter b(n) along the short side of the wedges to depend on the height (in atomic layers) above the interface. We then perform model calculations using kinematical theory and compare these with the experimental scattering profiles. The main objective is not to simulate the central crystal truncation rods (CTR), where a more sophisticated treatment may be needed [2,4], but to reproduce the facet peaks which generally are well separated from the central rods. All experimental effects as Lorentz and polarisation factors, rod interception, detector acceptance, area corrections, etc., are taken into ac-
count in the simulations using standard methods [5]. As discussed in Ref. [2] we may regard the wedges as monodisperse at a given coverage and as no interference effects between the individual wedges were observed we may adopt a simple model consisting of a slab of Cu containing one wedge placed on top of a Ni crystal. The contribution from the Ni substrate is taken as the CTRscattering from a perfect Ni crystal of infinite extent along [0 0 1] [3]. The Ni CTRs do not, in general, contribute to the scattering intensity at the positions of the facet peaks but are included to get reasonable agreement between experimental and simulated rod scans (l scans) used to determine the actual coverage. We first consider a (1 0 l) rod scan (not shown). The best agreement between the simulated and experimental intensity distribution is obtained
F. Berg Rasmussen et al. / Physica B 248 (1998) 34—38
with a substrate—film distance equal to 1.83 A_ , a Debye—Waller factor B"2 A_ 2 and a coverage of 9.2 ML in reasonable agreement with the coverage determined by means of a crystal thickness monitor during growth. Simulations for a 9.2 ML film is obtained by adding the intensities calculated for a 9 ML and a 10 ML film with proper weights. The optimum parameters found here depend only weakly on the lateral strain distribution and are kept constant in the following. We then turn to simulations of the transverse and longitudinal scans (Figs. 3 and 4) and more specifically to the strain distribution inside and outside the wedge in terms of the parameters c and 0 h as well as the functional b(n). We have tried C6 a number of different functional forms of b(n) and obtain the best fit to the experimental data with a large lateral expansion at the bottom of the wedge following a Gaussian dependence with distance away from the interface. b(n)"b #e e~(n@n#)2. = 3%In this expression b is the lattice parameter at the = top of thick wedges (large n), e is the extra lattice 3%relaxation at the bottom of the wedge (n"0) and n determines the decay of the exponential. A satis# factory fit to all the measured data is obtained with the parameters of Table 1 and a common scale factor. The simulated intensity profiles are shown as solid lines in Figs. 3 and 4 and generally the agreement between simulated and experimental profiles is good. For the common vertical lattice constant, c , we 0 obtain a value of 3.67 A_ which corresponds to a vertical lattice expansion of 4% with respect to the Ni lattice. Assuming the whole Cu film to be
Table 1 Strain parameters obtained for a 9.2 ML Cu film on Ni(0 0 1), see the text (A_ )
Lattice units c 0 h C6 b = e 3%n #
1.04 0.14 1.00 0.07 7 layers
3.67 0.5 2.49 0.17
37
Fig. 5. In-plane lattice parameter along the short side of the wedges as a function of distance away from the interface measured in atomic layers, see the text.
laterally pseudomorphic with the substrate, i.e. neglecting wedge formation, a 4% lattice expansion is precisely what is expected from elasticity theory using the elastic constants of bulk Cu. The protrusion h is here found to be 0.5 A_ in C6 good agreement with the 0.4 A_ expected from a hard-sphere model and the 0.6 A_ quoted in the STM study [1]. 4. Discussion and conclusions The height dependence of the lateral lattice parameter within the wedges is plotted in Fig. 5 using the values given in Table 1. The obtained strain distribution is inhomogenous with a large relaxation near the apex of the wedges that rapidly decays away from the interface. For a coverage approaching 20 ML the lateral lattice parameter of the topmost layer is close to that of Ni offering an explanation to the observed change in growth above 20 ML [1]. In conclusion it has been possible to determine the strain relaxation in thin Cu films grown on Ni(0 0 1) using SXRD and kinematical model calculations revealing the unusual strain relaxation present in this system. Acknowledgement This project was supported by DANSYNC and BMBF (05 622GUA1).
38
F. Berg Rasmussen et al. / Physica B 248 (1998) 34—38
References [1] B. Mu¨ller, B. Fischer, L. Nedelmann, A. Fricke, K. Kern, Phys. Rev. Lett. 76 (1996) 2358. [2] F. Berg Rasmussen, J. Baker M. Nielsen, R. Feidenhans’l, R.L. Johnson, Phys. Rev. Lett. 79 (1997) 4413.
[3] I.K. Robinson, Phys. Rev. B 33 (1986) 3830. [4] A.J. Steinfort, P.M.L.O. Scholte, A. Ettema, F. Tuinstra, M. Nielsen, E. Landemark, R. Feidenhans’l, G. Falkenberg, L. Seehofer, R.L. Johnson, Phys. Rev. Lett. 77 (1996) 2009. [5] E. Vlieg to be published; C. Schamper, H.L. Meyerheim, W. Moritz, J. Appl. Crystallogr. 26 (1993) 687.