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Surface structure analysis of metal adsorbed Si(1 1 1) surfaces by Patterson function with LEED I–V curves
Applied Surface Science 254 (2008) 7824–7826
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Surface structure analysis of metal adsorbed Si(1 1 1) surfaces by Patterson function with LEED I–V curves T. Kuzushita, A. Murata, A. Yamamoto, T. Urano * Faculty of Engineering, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan
A R T I C L E I N F O
A B S T R A C T
Article history:
Wu and Tong proposed the calculation method of Patterson function obtained directly from the LEED I–V curves which shows the relative position of surface atoms as an image. We have made the calculation program of Patterson function and applied to the structural analysis of the Si(1 1 1)1 1-Fe surface. Surface structure was able to be expressed almost correctly by the Patterson function obtained from the theoretical I–V curves for the model structure. In the Patterson function obtained from the experimental I–V curves, the locational relation between the atoms of subsurface layer was in agreement with the CsCl type structure. More over, because the faint peak, by which we can distinguish the model, can be seen, it seems that the model B8 is preferable to the model A8. This result is consistent with the model shown by Walter et al. ß 2008 Elsevier B.V. All rights reserved.
Available online 29 February 2008 Keywords: LEED I–V curves Patterson function Si(1 1 1)1 1-Fe
1. Introduction
2. Theory
Low energy electron diffraction (LEED) is the leading technique to analyze the surface structure. In conventional LEED analysis, experimental I–V curves are compared with theoretical I–V curves obtained from the model structure, and the most preferable model structure is regarded as a true surface structure. However, because many model structures must be assumed usually, it is not easy to determine the surface structure actually. Wu et al. proposed the calculation method which obtains the image of the relative location of surface atoms by the direct Fourier transform of LEED I–V curves [1–4]. This method is expected to achieve the complemental role to determine the surface structure, and, for example, some works for the structure analysis of Si(1 1 1)4 1-In have done successfully [4,5]. We constructed the calculation program and applied this method to the theoretical I–V curves to check the reproducibility of the model structure. Then, the experimental I–V curves acquired from the observed LEED pattern was analyzed to verify whether surface atomic structure can be determined.
The Patterson function [6] proposed by Wu et al. is shown in Eq. (1) [1]. 2 X X Z i~ q~ r ~ ~ ~ ~ Pðr Þ ¼ (1) Iðki ; g jj þ q ? ez Þ e dq ? kˆi ~gjj
* Corresponding author. Tel.: +81 78 803 6079; fax: +81 78 803 6079. E-mail address: [email protected] (T. Urano). 0169-4332/$ – see front matter ß 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2008.02.092
where I is the diffraction intensity, ~ r is the positional vector of a surface atom, ~ ki is the wave vector of an incidence wave, ~ g jj is a two-dimensional reciprocal lattice vector, ~ q is a scattering vector, q? is a surface normal component of ~ q, and ~ ez is a unit vector of normal to the surface. Diffraction intensity I is shown in Eq. (2), 2 0 eikr13 i~ ~0 Ið~ ki ; ~ k Þ / f 1 ei~q~r1 þ f 2 ei~q~r2 þ f 1 f 3 eik ~r1 e ki ~r3 þ r 13 (2) 0 ~ r 2 are the positional vector of surface atoms, k is the where ~ r1 , ~ wave vector of outgoing wave, and fi is a scattering factor of atom i. The first two terms express the single scattering from atoms located at ~ r 1 and ~ r 2 , respectively. The third term expresses the multiple scattering which takes place along the scattering path between atoms located at ~ r 1 and ~ r 3 , with r 13 ¼ j~ r1 ~ r 3 j. One of
T. Kuzushita et al. / Applied Surface Science 254 (2008) 7824–7826
the cross term of the single scattering of Eq. (2) is shown in Eq. (3) ~0 Þ / f f ei~q~r12 I1;2 ð~ ki ; k 1 2
(3) i~ q~ r12
i~ q~ r
is conjugate to the phase of e in At ~ r ¼~ r 12 , the phase of e Eq. (1). Then, the maximum of Patterson function is given at ~ r ¼~ r 12 . Thus, by changing ~ r , we can know the relative positional vector between atoms in which the Patterson function gives the maximum. On the other hand, the terms including the multiple scattering are cancelled with the sum of many beams with several incidence angles. 3. Experimental The experiments are carried out in an ultrahigh-vacuum chamber with a base pressure of 3 108 Pa with LEED optics. The substrate sample is an n-type Si (1 1 1) wafer with the resistivity of about 1 V cm. The sample is flashed at 1200 8C for 15 s and annealed at 700 8C for 10 min to obtain a clean Si(1 1 1)7 7 surface. The Si(1 1 1)1 1-Fe surface was prepared by deposition of about 1 ML Fe onto the Si(1 1 1)7 7 surface followed by annealing at 350 8C for 10 min. The LEED pattern was taken between 30 and 350 eV at several incidence angles of u = 08, 118, 228 inclined along [1 1 0] axis parallel to the surface. Experimental I–V curve about each diffraction spot was obtained from the intensity every 2 eV. 4. Result and discussion As shown in previous section, by calculating the Patterson function using the theoretical or experimental I–V curves, image of the locational relation between atoms is obtained. The Patterson image for theoretical I–V curves has been compared with the model structure to check the reproducibility at first. Then the Patterson image for experimental I–V curves has been compared with those for theoretical I–V curves to intend which model is desirable.
7825
For the Si(1 1 1)1 1-Fe surface, many models has been proposed [7–9]. Among those a CsCl type model is considered to be most possible. The assumed model structure is shown in Fig. 1, (a) is X–Y plane parallel to the surface, (b) and (c) are Y–Z plane normal to the surface along h2 1 1i axis. These models are for 1 ML coverage of Fe on Si(1 1 1) surface. The number of ligand around Fe atom is 8 and 2 types of stacking are considered. One is the structure in which stacking is not faulted at the interface defined as A-type, and another is the structure in which stacking is faulted at the interface defined as B-type. In this research, it is intended to confirm which models is better for the Si(1 1 1)1 1-Fe surface structure using the Patterson function constructed from LEED I–V curves. The theoretical I–V curves in normal incidence are calculated for these model structures using the Tensor LEED program named ‘‘A. Barbieri/M.A. Van Hove Symmetrized Tensor LEED package’’, and the Patterson images are shown in Fig. 2. The number of the used I– V curves is 84 in normal incidence. The image size is 10 A˚ 10 A˚, and the peak which appears in the images shows the relative position from the reference atom located at the origin. In Fig. 2(b), (c), by removing a reference peak of self-correlation which shows the largest intensity, the other locational relation in a plane normal to the surface is emphasized. The distance and the direction of the relative position vector in Fig. 2 are in agreement with the model structures as shown by broken arrows in Fig. 1. Here, although both models in X–Y plane have a same structure, the image in Y–Z plane can be used to distinguish the models. If the Patterson images of A8 model and B8 model are compared each other in Y–Z plane, the peaks which are not seen in common can be seen at the position surrounded by a dotted circle a, b, g in Fig. 2. The surface structure of Si(1 1 1)1 1-Fe can be distinguished by whether these peaks exist in the Patterson image obtained from the experimental I–V curves, or not. The Patterson images obtained from the experimental I–V curves is shown in Fig. 3, (a) X–Y plane image, (b) Y–Z plane image obtained from the I–V curves of normal incidence only, (c) Y–Z
Fig. 1. The assumed model structures for the Si(1 1 1)1 1-Fe surface, (a) in X–Y plane parallel to the surface, (b) A8 model structure in Y–Z plane normal to the surface along h2 1 1i. (c) B8 model structure in Y–Z plane. The broken arrows show that the locational relation shown in common for both models, and the solid arrows show the relation which is different each other.
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T. Kuzushita et al. / Applied Surface Science 254 (2008) 7824–7826
Fig. 2. The Patterson images obtained from the theoretical I–V curves of normal incidence for model structures in Fig. 1. (a) The Patterson image in X–Y plane. (b) The Patterson image for A8 model in Y–Z plane. (c) The Patterson image for B8 model in Y–Z plane. The number of the I–V curves is 84 and the total energy range of the I–V curves is about 30,000 eV at normal incident angle (u = 08). By removing a reference peak of self-correlation which shows the largest intensity, the other locational relation in a plane normal to the surface is emphasized. The positions surrounded by dotted circles a, b, g show the characteristic peaks to distinguish the model.
Fig. 3. The Patterson images obtained from the experimental I–V curves. (a) The Patterson image in X–Y plane. (b) The Patterson image in Y–Z plane only for the I–V curves of normal incidence. (c) The Patterson image in Y–Z plane obtained after adding all I–V curves of several incident angles (u = 08, 118, 228). In (a), (b), the number of the I–V curves is 34 and the total energy range of the I–V curves is about 6500 eV at normal incident angle (u = 08). In (c), the number of the I–V curves is 147 and the total energy range of the I–V curves is about 32,000 eV at several incident angles (u = 08, 118, 228).
plane image obtained from all I–V curves of several incident angles (u = 08, 118, 228). In the case of normal incidence only, the peak which is not in the model structure appears in Fig. 3(b) as surrounded collectively by dotted ellipse. These must be ‘‘the ghost’’. Actually, the similarity between experimental and theoretical I–V curves at normal incidence by the conventional LEED I–V curve analysis for both models is not good. The r-factor value is not so small. Therefore, we considered that we could determine which model is prefer or not, at least. The ghost is reduced when the I–V curves of all incidence angles are included. We consider that by including more numbers of I–V curves, i.e., ~ g jj, to calculate the Patterson function, the influence of multiple scattering is much reduced and locational relations can be expressed more clearly. The Patterson images obtained from the experimental I–V curves were compared with the Patterson images obtained from the theoretical I–V curves in Y–Z plane. Since the locational relation between atoms is in agreement with two model structures in X–Y plane parallel to the surface in Fig. 3(a) and in Y–Z plane normal to the surface by broken arrows in Fig. 3(c), it is considered that the structure of the Si(1 1 1)1 1-Fe surface is a CsCl type. Moreover, weak peaks surrounded by the dotted circle are seen in Fig. 3(c). These peaks seem to be more likely with the peak b of B8 model in Fig. 2(c) than with the peak a of A8 model in Fig. 2(b). Therefore, we consider that the B8 model is suitable for the structure of Si(1 1 1)1 1-Fe surface than the A8 model. This result is consistent with the model shown by Walter et al. [9]. Another distinctive peak g of the B8 model could not be seen clearly. One reason of this is that the bigger the relative distance between atoms in the depth direction becomes, the more difficult to find the spot is. The distance g is a little bit bigger than the distance b.
5. Conclusion The Patterson function was calculated using LEED I–V curves for the structural analysis of the Si(1 1 1)1 1-Fe surface. Surface structure was able to be expressed almost correctly by the Patterson function obtained from the theoretical I–V curves for the model structure. When Patterson function was applied for the experimental I–V curves, the ghost has been reduced by including the I–V curves of several incidence angles and increasing the number of I–V curves and the relative atomic position in the plane parallel to the surface and in the subsurface of the plane normal to the surface has been shown correctly. Moreover, because the faint peak by which we can distinguish the model can be seen, it seems that the model B8 is preferable to the model A8. It is difficult to determine surface structure correctly by the analysis of the Patterson function only. However, it will be useful as the supplemental method of conventional LEED analysis.
References [1] [2] [3] [4] [5] [6] [7]
H. Wu, S.Y. Tong, Phys. Rev. B 87 (2001) 036101. H. Wu, S. Xu, S. Ma, M.H. Xie, S.Y. Tong, Phys. Rev. Lett. 89 (2002) 216101. S.Y. Tong, H. Wu, J. Phys. Condens. Matter 14 (2002) 1231. J. Wang, H. Wu, R. So, Y. Liu, M.H. Xie, S.Y. Tong, Phys. Rev. B 72 (2005) 245324. T. Abukawa, T. Yamazaki, S. Kono, e-J. Surf. Sci. Nanotechnol. 4 (2006) 661. A.L. Patterson, Phys. Rev. 46 (1934) 372. W. Weiss, M. Kutschera, U. Starke, M. Mozaffari, K. Reshoft, U. Kohler, K. Heinz, Surf. Sci. 377–379 (1997) 861. [8] S. Walter, F. Blobner, M. Krause, S. Muller, K. Heinz, U. Starke, J. Phys.: Condens. Matter 15 (2003) 5207. [9] S. Hajjar, G. Garreau, S. Pelletier, D. Bolmont, C. Pirri, Phys. Rev. B 68 (2003) 033302.
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Surface structure analysis of metal adsorbed Si(1 1 1) surfaces by Patterson function with LEED I–V curves T. Kuzushita, A. Murata, A. Yamamoto, T. Urano * Faculty of Engineering, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan
A R T I C L E I N F O
A B S T R A C T
Article history:
Wu and Tong proposed the calculation method of Patterson function obtained directly from the LEED I–V curves which shows the relative position of surface atoms as an image. We have made the calculation program of Patterson function and applied to the structural analysis of the Si(1 1 1)1 1-Fe surface. Surface structure was able to be expressed almost correctly by the Patterson function obtained from the theoretical I–V curves for the model structure. In the Patterson function obtained from the experimental I–V curves, the locational relation between the atoms of subsurface layer was in agreement with the CsCl type structure. More over, because the faint peak, by which we can distinguish the model, can be seen, it seems that the model B8 is preferable to the model A8. This result is consistent with the model shown by Walter et al. ß 2008 Elsevier B.V. All rights reserved.
Available online 29 February 2008 Keywords: LEED I–V curves Patterson function Si(1 1 1)1 1-Fe
1. Introduction
2. Theory
Low energy electron diffraction (LEED) is the leading technique to analyze the surface structure. In conventional LEED analysis, experimental I–V curves are compared with theoretical I–V curves obtained from the model structure, and the most preferable model structure is regarded as a true surface structure. However, because many model structures must be assumed usually, it is not easy to determine the surface structure actually. Wu et al. proposed the calculation method which obtains the image of the relative location of surface atoms by the direct Fourier transform of LEED I–V curves [1–4]. This method is expected to achieve the complemental role to determine the surface structure, and, for example, some works for the structure analysis of Si(1 1 1)4 1-In have done successfully [4,5]. We constructed the calculation program and applied this method to the theoretical I–V curves to check the reproducibility of the model structure. Then, the experimental I–V curves acquired from the observed LEED pattern was analyzed to verify whether surface atomic structure can be determined.
The Patterson function [6] proposed by Wu et al. is shown in Eq. (1) [1]. 2 X X Z i~ q~ r ~ ~ ~ ~ Pðr Þ ¼ (1) Iðki ; g jj þ q ? ez Þ e dq ? kˆi ~gjj
* Corresponding author. Tel.: +81 78 803 6079; fax: +81 78 803 6079. E-mail address: [email protected] (T. Urano). 0169-4332/$ – see front matter ß 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2008.02.092
where I is the diffraction intensity, ~ r is the positional vector of a surface atom, ~ ki is the wave vector of an incidence wave, ~ g jj is a two-dimensional reciprocal lattice vector, ~ q is a scattering vector, q? is a surface normal component of ~ q, and ~ ez is a unit vector of normal to the surface. Diffraction intensity I is shown in Eq. (2), 2 0 eikr13 i~ ~0 Ið~ ki ; ~ k Þ / f 1 ei~q~r1 þ f 2 ei~q~r2 þ f 1 f 3 eik ~r1 e ki ~r3 þ r 13 (2) 0 ~ r 2 are the positional vector of surface atoms, k is the where ~ r1 , ~ wave vector of outgoing wave, and fi is a scattering factor of atom i. The first two terms express the single scattering from atoms located at ~ r 1 and ~ r 2 , respectively. The third term expresses the multiple scattering which takes place along the scattering path between atoms located at ~ r 1 and ~ r 3 , with r 13 ¼ j~ r1 ~ r 3 j. One of
T. Kuzushita et al. / Applied Surface Science 254 (2008) 7824–7826
the cross term of the single scattering of Eq. (2) is shown in Eq. (3) ~0 Þ / f f ei~q~r12 I1;2 ð~ ki ; k 1 2
(3) i~ q~ r12
i~ q~ r
is conjugate to the phase of e in At ~ r ¼~ r 12 , the phase of e Eq. (1). Then, the maximum of Patterson function is given at ~ r ¼~ r 12 . Thus, by changing ~ r , we can know the relative positional vector between atoms in which the Patterson function gives the maximum. On the other hand, the terms including the multiple scattering are cancelled with the sum of many beams with several incidence angles. 3. Experimental The experiments are carried out in an ultrahigh-vacuum chamber with a base pressure of 3 108 Pa with LEED optics. The substrate sample is an n-type Si (1 1 1) wafer with the resistivity of about 1 V cm. The sample is flashed at 1200 8C for 15 s and annealed at 700 8C for 10 min to obtain a clean Si(1 1 1)7 7 surface. The Si(1 1 1)1 1-Fe surface was prepared by deposition of about 1 ML Fe onto the Si(1 1 1)7 7 surface followed by annealing at 350 8C for 10 min. The LEED pattern was taken between 30 and 350 eV at several incidence angles of u = 08, 118, 228 inclined along [1 1 0] axis parallel to the surface. Experimental I–V curve about each diffraction spot was obtained from the intensity every 2 eV. 4. Result and discussion As shown in previous section, by calculating the Patterson function using the theoretical or experimental I–V curves, image of the locational relation between atoms is obtained. The Patterson image for theoretical I–V curves has been compared with the model structure to check the reproducibility at first. Then the Patterson image for experimental I–V curves has been compared with those for theoretical I–V curves to intend which model is desirable.
7825
For the Si(1 1 1)1 1-Fe surface, many models has been proposed [7–9]. Among those a CsCl type model is considered to be most possible. The assumed model structure is shown in Fig. 1, (a) is X–Y plane parallel to the surface, (b) and (c) are Y–Z plane normal to the surface along h2 1 1i axis. These models are for 1 ML coverage of Fe on Si(1 1 1) surface. The number of ligand around Fe atom is 8 and 2 types of stacking are considered. One is the structure in which stacking is not faulted at the interface defined as A-type, and another is the structure in which stacking is faulted at the interface defined as B-type. In this research, it is intended to confirm which models is better for the Si(1 1 1)1 1-Fe surface structure using the Patterson function constructed from LEED I–V curves. The theoretical I–V curves in normal incidence are calculated for these model structures using the Tensor LEED program named ‘‘A. Barbieri/M.A. Van Hove Symmetrized Tensor LEED package’’, and the Patterson images are shown in Fig. 2. The number of the used I– V curves is 84 in normal incidence. The image size is 10 A˚ 10 A˚, and the peak which appears in the images shows the relative position from the reference atom located at the origin. In Fig. 2(b), (c), by removing a reference peak of self-correlation which shows the largest intensity, the other locational relation in a plane normal to the surface is emphasized. The distance and the direction of the relative position vector in Fig. 2 are in agreement with the model structures as shown by broken arrows in Fig. 1. Here, although both models in X–Y plane have a same structure, the image in Y–Z plane can be used to distinguish the models. If the Patterson images of A8 model and B8 model are compared each other in Y–Z plane, the peaks which are not seen in common can be seen at the position surrounded by a dotted circle a, b, g in Fig. 2. The surface structure of Si(1 1 1)1 1-Fe can be distinguished by whether these peaks exist in the Patterson image obtained from the experimental I–V curves, or not. The Patterson images obtained from the experimental I–V curves is shown in Fig. 3, (a) X–Y plane image, (b) Y–Z plane image obtained from the I–V curves of normal incidence only, (c) Y–Z
Fig. 1. The assumed model structures for the Si(1 1 1)1 1-Fe surface, (a) in X–Y plane parallel to the surface, (b) A8 model structure in Y–Z plane normal to the surface along h2 1 1i. (c) B8 model structure in Y–Z plane. The broken arrows show that the locational relation shown in common for both models, and the solid arrows show the relation which is different each other.
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T. Kuzushita et al. / Applied Surface Science 254 (2008) 7824–7826
Fig. 2. The Patterson images obtained from the theoretical I–V curves of normal incidence for model structures in Fig. 1. (a) The Patterson image in X–Y plane. (b) The Patterson image for A8 model in Y–Z plane. (c) The Patterson image for B8 model in Y–Z plane. The number of the I–V curves is 84 and the total energy range of the I–V curves is about 30,000 eV at normal incident angle (u = 08). By removing a reference peak of self-correlation which shows the largest intensity, the other locational relation in a plane normal to the surface is emphasized. The positions surrounded by dotted circles a, b, g show the characteristic peaks to distinguish the model.
Fig. 3. The Patterson images obtained from the experimental I–V curves. (a) The Patterson image in X–Y plane. (b) The Patterson image in Y–Z plane only for the I–V curves of normal incidence. (c) The Patterson image in Y–Z plane obtained after adding all I–V curves of several incident angles (u = 08, 118, 228). In (a), (b), the number of the I–V curves is 34 and the total energy range of the I–V curves is about 6500 eV at normal incident angle (u = 08). In (c), the number of the I–V curves is 147 and the total energy range of the I–V curves is about 32,000 eV at several incident angles (u = 08, 118, 228).
plane image obtained from all I–V curves of several incident angles (u = 08, 118, 228). In the case of normal incidence only, the peak which is not in the model structure appears in Fig. 3(b) as surrounded collectively by dotted ellipse. These must be ‘‘the ghost’’. Actually, the similarity between experimental and theoretical I–V curves at normal incidence by the conventional LEED I–V curve analysis for both models is not good. The r-factor value is not so small. Therefore, we considered that we could determine which model is prefer or not, at least. The ghost is reduced when the I–V curves of all incidence angles are included. We consider that by including more numbers of I–V curves, i.e., ~ g jj, to calculate the Patterson function, the influence of multiple scattering is much reduced and locational relations can be expressed more clearly. The Patterson images obtained from the experimental I–V curves were compared with the Patterson images obtained from the theoretical I–V curves in Y–Z plane. Since the locational relation between atoms is in agreement with two model structures in X–Y plane parallel to the surface in Fig. 3(a) and in Y–Z plane normal to the surface by broken arrows in Fig. 3(c), it is considered that the structure of the Si(1 1 1)1 1-Fe surface is a CsCl type. Moreover, weak peaks surrounded by the dotted circle are seen in Fig. 3(c). These peaks seem to be more likely with the peak b of B8 model in Fig. 2(c) than with the peak a of A8 model in Fig. 2(b). Therefore, we consider that the B8 model is suitable for the structure of Si(1 1 1)1 1-Fe surface than the A8 model. This result is consistent with the model shown by Walter et al. [9]. Another distinctive peak g of the B8 model could not be seen clearly. One reason of this is that the bigger the relative distance between atoms in the depth direction becomes, the more difficult to find the spot is. The distance g is a little bit bigger than the distance b.
5. Conclusion The Patterson function was calculated using LEED I–V curves for the structural analysis of the Si(1 1 1)1 1-Fe surface. Surface structure was able to be expressed almost correctly by the Patterson function obtained from the theoretical I–V curves for the model structure. When Patterson function was applied for the experimental I–V curves, the ghost has been reduced by including the I–V curves of several incidence angles and increasing the number of I–V curves and the relative atomic position in the plane parallel to the surface and in the subsurface of the plane normal to the surface has been shown correctly. Moreover, because the faint peak by which we can distinguish the model can be seen, it seems that the model B8 is preferable to the model A8. It is difficult to determine surface structure correctly by the analysis of the Patterson function only. However, it will be useful as the supplemental method of conventional LEED analysis.
References [1] [2] [3] [4] [5] [6] [7]
H. Wu, S.Y. Tong, Phys. Rev. B 87 (2001) 036101. H. Wu, S. Xu, S. Ma, M.H. Xie, S.Y. Tong, Phys. Rev. Lett. 89 (2002) 216101. S.Y. Tong, H. Wu, J. Phys. Condens. Matter 14 (2002) 1231. J. Wang, H. Wu, R. So, Y. Liu, M.H. Xie, S.Y. Tong, Phys. Rev. B 72 (2005) 245324. T. Abukawa, T. Yamazaki, S. Kono, e-J. Surf. Sci. Nanotechnol. 4 (2006) 661. A.L. Patterson, Phys. Rev. 46 (1934) 372. W. Weiss, M. Kutschera, U. Starke, M. Mozaffari, K. Reshoft, U. Kohler, K. Heinz, Surf. Sci. 377–379 (1997) 861. [8] S. Walter, F. Blobner, M. Krause, S. Muller, K. Heinz, U. Starke, J. Phys.: Condens. Matter 15 (2003) 5207. [9] S. Hajjar, G. Garreau, S. Pelletier, D. Bolmont, C. Pirri, Phys. Rev. B 68 (2003) 033302.