- Email: [email protected]
Interpretation of reflection high-energy electron diffraction oscillation phase
Journal of Crystal Growth 198/199 (1999) 905—910
Interpretation of reflection high-energy electron diffraction oscillation phase Z. Mitura *, S.L. Dudarev , M.J. Whelan Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK Institute of Physics, Marie Curie-Sk!odowska University, pl. M. Curie-Sk!odowskiej 1, 20-031 Lublin, Poland
Abstract The use of reflection high energy electron diffraction (RHEED) became very popular after the discovery of reflected electron beam intensity oscillations in the early 1980s. Although a number of theoretical explanations of the effect have already been given, many aspects of the question as to why RHEED oscillations appear still remains open. We propose a new mechanism which may lead to the appearance of RHEED oscillations. This mechanism can be briefly described as the periodic variation of diffraction peak positions due to changes in the average scattering potential near the crystal surface during the deposition of a new layer. We conclude that in a general case one can explain qualitatively the origin of RHEED oscillations as the superposition of effects due to: (1) periodic variations in surface roughness, (2) changes in interference conditions for waves scattered from different surface terraces and (3) periodic variations in the average potential in the topmost layer. 1999 Elsevier Science B.V. All rights reserved. PACS: 61.14.Hg; 68.35.Bs; 68.55.Jk Keywords: RHEED; Oscillations; MBE
From early 1980s the use of reflection high energy electron diffraction (RHEED) to monitor epitaxial growth of thin films has become very popular [1]. This is largely a consequence of the discovery RHEED intensity oscillations [2,19,20]. It was experimentally found that if epitaxial growth follows a layer-by-layer mode then regular oscillating changes in the intensity of specularly reflected beam
* Corresponding author. Tel.: #44 1865 273714; fax: 44 1865 273764; e-mail: [email protected].
occur and the period of such oscillations corresponds to the deposition of one monolayer. However, it has turned out that the theoretical explanation of this effect is not quite so simple. This is because a fully quantitative description of electron diffraction from crystals can be achieved only if dynamical diffraction theory is employed. In fact, even for the case of perfectly flat surfaces, dynamical theoretical analysis is rather complicated. The case of a growing surface is far more difficult to deal with. Due to permanent theoretical progress in this research area, we may expect that some day we will
0022-0248/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 8 ) 0 1 0 4 0 - 9
906
Z. Mitura et al. / Journal of Crystal Growth 198/199 (1999) 905–910
be able to compute RHEED oscillations precisely. However, because RHEED oscillations are generally the result of a number of simultaneous diffraction processes it seems to be impossible to develop both a simple and a quantitative description of them. Nevertheless, it is reasonable to attempt to find the most important mechanisms leading to the periodic changes in the intensity which could be used in qualitative work. Until now two such mechanisms have been recognised. Researchers from the Phillips group (later at the Imperial College) pointed out that during the deposition of a new layer there occur periodic changes in the roughness at the surfaces and it implies periodic changes in the electron intensity [3,21]. Another mechanism was recognised by researchers from the University of Minnesota. They pointed out that during growth there occur changes in interference conditions of electron waves reflected from different surface terraces [4,22]. In this paper we show another, new mechanism. We suggest that in some situations RHEED oscillations may be successfully explained as a consequence of periodic changes in the average scattering potential near the crystal surface during the deposition of a new layer. It is known from experiment that if RHEED oscillations are observed for different conditions they usually differ in phase [5]. The phase of the oscillations can be determined experimentally in the following way. One needs to measure the time t of the occurrence of the minimum in the second period of the oscillations [5]. The experimental phase can be defined as follows " 2p(t /¹!1.5), where ¹ is the period of the oscil lations. Theoretically it is more convenient to use a similar definition employing Fourier analysis [6]. Both the Phillips and the University of Minnesota approaches cannot explain why oscillations actually measured differ in phase. According to both of these approaches the phase should always be the same. Very recently, the one-dimensional proportional model of the scattering potential has been used successfully by us [6] in the interpretation of the RHEED oscillation phase determined experimentally [7]. In this paper, we show in detail why the phase may be a rapidly varying function of the angle of the electron beam incidence.
Let us assume that intensity of elastically scattered electrons can be calculated using the Schro¨dinger equation
W(r)!v(z)W(r)#KW(r)"0,
(1)
where
"q "º 2m »(z), (2) v(z)" 1# m c "q "º 2m "q "º. K" 1# (3) 2m c In Eqs. (1)—(3) »(z) is the one-dimensional scattering potential of a crystal and º is the absolute value of the accelerating voltage of the electron gun, m , q and are the electron rest mass, the electron charge and Planck’s constant. Above the crystal (z'z ), we can write the solution W(r) in the 2 form W(r)"exp(i k ) q)[exp(!ik(z!z )) , 2 #R exp(#ik(z!z ))], (4) 2 where k and q are parallel components of K and r. , Further, k""K"sin0, where 0 is the glancing angle of the incident beam. The specular beam intensity I is given by I""R". The scattering potential » (z) can be determined using electron scattering factors tabulated by Doyle and Turner [8]. The contribution to »(z) coming from the jth layer can be expressed as 4 p » (z)"!H (1#0.1i) H H m X a 4p L exp ! ; (z!z ) , (5) H b b L L L where H is the layer coverage, X is the area of the H two-dimensional surface unit cell and z is the layer H position along the axis perpendicular to the surface. The coefficients a , b are given by Doyle and L L Turner [8]. To interpret actual experimental data it is important to take account also of the effect of lattice thermal vibrations by replacing the [8] and Turner coefficients b in Eq. (5) by b #8pu, L L where u denotes the mean square displacement of atoms along the z-axis. It is worth emphasizing the fact that in our model the potential from the
Z. Mitura et al. / Journal of Crystal Growth 198/199 (1999) 905–910
partially completed layer is proportional to its coverage [see the multiplier in Eq. (5)]. The potential of the whole crystal is determined by summing contributions from all layers, i.e. »(z)" » (z). H H Finally, we are able to solve Eq. (1), using numerical codes developed within the framework of the two-dimensional Bloch wave approach (for review see Ref. [9]). The model considered in this paper can be used to describe diffraction from surfaces experimentally grown only under special conditions. Namely, the azimuth of the incident beam should be set few degrees off the main crystal direction [10]. Furthermore, during the crystal growth the average distance between nucleation centres at the surface should not exceed a few atomic radii. We have found that to understand the diffraction from a one-dimensional crystal it is useful to carry out a series of calculations as follows. For a moment, let us consider theoretically a crystal composed of Au atoms for which the lattice and atom parameters are taken carefully from crystallographic tables. Let us further introduce some modification: let us preserve the geometry of the crystal but artificially replace all atoms by weaker scatterers. Numerically, we require that the potential arising from each layer (both fully and partially filled) be multiplied by some factor a. Fig. 1 shows results of calculations for such crystals for a equal to 0.01, 0.50 and 1.00 (the first two crystals are pure artificial ones, the last one is realistic). We determined numerically the intensity from growing surfaces assuming the perfect layer-by-layer growth. Each part of Fig. 1 shows four rocking curves (the specular beam intensities versus the glancing angle) computed for different coverages of the topmost layer. 䉴 Fig. 1. RHEED calculations for crystals composed of atoms for which the scattering potential is taken to be the potential of a Au atom multiplied by some factor a. The value of a is equal to: (a) 0.01, (b) 0.50, (c) 1.00. In all three cases the geometry of a Au (1 1 1) crystal is assumed and the energy of the incident electron beam is taken to be 15 keV. It is assumed that the crystals are grown in the perfect layer-by-layer mode. RHEED rocking curves are computed for different stages of the growth, namely for four different coverages of the topmost layer.
907
For the case of very small values of the scattering potential (see Fig. 1a), we can clearly recognise a number of regularities. First, we observe that both for a flat surface (H"1) and rough ones
908
Z. Mitura et al. / Journal of Crystal Growth 198/199 (1999) 905–910
(H(1) the positions of Bragg peaks are exactly the same. Second, between Bragg peaks the intensity is always smaller for rough surfaces. In more detailed theoretical work [11] we showed that the following analytical formula is valid for the case of a crystal composed of very weak scatters: I(H, 0)"I(1, 0)[1#2H(1!H) ;(cos(2"K"d sin 0)!1)],
(6)
where I(H, 0) and I(1, 0) means respectively the intensity for rough and flat surfaces and d is the growing layer thickness. In fact, it is surprising that this formula coincides with that developed by researchers from Minnesota University [4,22] for their terrace model of scattering form growing crystals. If we analyse cases shown in Fig. 1b and Fig. 1c we can still recognise the tendency described analytically by Eq. (6). However, the situation has become more complicated. We observe now that all Bragg peaks are shifted towards lower angles in comparison with those shown in Fig. 1a. Moreover, detailed analysis shows that the positions of peaks for flat and rough surfaces are no longer the same (see Fig. 2a). Bragg peaks for rough surfaces always appear at slightly higher angles in comparison with those observed for flat surfaces. This regularity has important implications for the appearance of RHEED oscillations. Let us recall that RHEED oscillations are changes in the intensity of the specular beam intensity which are observed during the deposition of the material, for a fixed glancing angle. If we compare the shapes of oscillations determined for angles slightly lower and slightly higher than those assigned to flat surface Bragg peak maxima, we can recognise that the respective difference in phase is equal to about n (see Fig. 2b and Fig. 2c). In other words, for 䉴 Fig. 2. (a) Rocking curves for different coverages of the topmost layer for Au (1 1 1) surface in the vicinity of the Bragg peak 333 shown in detail and on a linear scale (this peak is also shown in Fig. 1c). (b) The RHEED oscillation phase shown as a function of the glancing angle corresponding to the situation presented in part (a). (c) Plots of RHEED oscillations for glancing angles of 2.6° (condition (i)) and 3.0° (condition (ii)) for crystal growth as in (a) and (b).
lower angles than those for flat surface Bragg reflections the intensity initially decreases after starting growth, while for higher angles the intensity increases. Thus, we can say that due to the periodic variations in peak positions during growth we can
Z. Mitura et al. / Journal of Crystal Growth 198/199 (1999) 905–910
expect that in the peak vicinities RHEED oscillations should be relatively large in amplitude and their phase should be a strongly varying function of the glancing angle. To give a qualitative explanation of this behaviour we can say that the precise positions of Bragg peak maxima are dependent on the value of the average scattering potential near the surface. The lower value of the volume average (this quantity is always negative in our notation), the larger shift of the peak is observed. For growing crystals the lowest potential average taken over the volume of the topmost layer occurs for a flat surface. We should say that similar models as considered in this paper have been the subject of much theoretical speculation in the past. Kawamura and Maksym [12] first introduced the concept of the proportional model of the scattering potential. Further Peng and Whelan [13,23] did extensive research work for the one-dimensional case. They in fact did an initial analysis of the oscillation phase. Next the one-dimensional proportional model has been used [14,15] to investigate the effect of double maxima in a period. The possible extension of the proportional model to include diffuse scattering has been theoretically discussed by Dudarev et al. [16]. Experimental verifications of predictions of the plots of oscillations calculated using the proportional model of the potential are shown in Refs. [17,18]. The recent successful interpretation [6] of the experimental oscillation phase is perhaps the strongest confirmation that simple models, ignoring the detailed topography of the surface and taking into consideration only the average potential, may be indeed quite useful. Nevertheless, more generally one should admit that in some situations (as discussed in Refs. [3,4,21,22]) consideration of the detailed shapes of islands at the surface may be of crucial importance. In conclusion, for a general case we can point out three simultaneous mechanisms leading to the appearance of RHEED oscillations: variations in roughness at the surface [3,21], changes in terrace distribution [4,22] and variations in the average scattering potential at the surface (see Ref. [6] and this paper). At present the question of which mechanism is dominant in experimental practice is still open (for a detailed discussion see Ref. [6]). But
909
however, we believe that it could be answered in the near future if common experimental and theoretical research work devoted to this particular problem is undertaken. One of us (Z.M.) gratefully acknowledges financial support from the Leverhulme Trust (for the award of a Research Fellowship). Numerical computations have been performed using the facilities of the Materials Modelling Laboratory at the Department of Materials of the University of Oxford. We gratefully acknowledge funding for these facilities from the EPSRC and the HEFCE/OST, under the joint research equipment initiative with matching funding from Hewlett-Packard. We also wish to acknowledge stimulating discussions with Professor P.I. Cohen, Professor A. Ichimiya, Professor B.A. Joyce and Dr. J.H. Neave.
References [1] M.A. Herman, H. Sitter, Molecular Beam Epitaxy: Fundamentals and Current Status, Springer, Berlin, 1989. [2] J.J. Harris, B.A. Joyce, P.J. Dobson, Surf. Sci. 103 (1981) L90. [3] J.H. Neave, B.A. Joyce, P.J. Dobson, N. Norton, Appl. Phys. A 31 (1983) 1. [4] J.M. Van Hove, C.S. Lent, P.R. Pukite, P.I. Cohen, J. Vac. Sci. Technol. B 1 (1983) 741. [5] J. Zhang, J.H. Neave, P.J. Dobson, B.A. Joyce, Appl. Phys. A 42 (1987) 317. [6] Z. Mitura, S.L. Dudarev, M.J. Whelan, Phys. Rev. B 57 (1998) 6309. [7] G.E. Crook, K.G. Eyink, A.C. Campbell, D.R. Hinson, B.G. Streetman, J. Vac. Sci. Technol. A 7 (1989) 2549. [8] P.A. Doyle, P.S. Turner, Acta Crystallogr. A 24 (1968) 390. [9] P.A. Maksym, Surf. Rev. Lett. 4 (1997) 513. [10] A. Ichimiya, Surf. Sci. 192 (1987) L893. [11] Z. Mitura, S.L. Dudarev, M.J. Whelan, Surf. Rev. Lett. 5 (1998) 101. [12] T. Kawamura, P.A. Maksym, Surf. Sci. 161 (1985) 12. [13] L.-M. Peng, M.J. Whelan, Surf. Sci. 238 (1990) L446. [14] Z. Mitura, A. Daniluk, Surf. Sci. 277 (1992) 229. [15] Y. Horio, A. Ichimiya, Jpn. J. Appl. Phys. 33 (1994) L377. [16] S.L. Dudarev, L.-M. Peng, M.J. Whelan, Surf. Sci. 279 (1992) 380. [17] Z. Mitura, M. Stro´z ak, M. Ja"ochowski, Surf. Sci. 276 (1992) L15. [18] A. Daniluk, P. Mazurek, P. Miko"ajczak, Surf. Sci. 369 (1996) 91.
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[19] J.J. Harris, B.A. Joyce, P.J. Dobson, Surf. Sci. 108 (1981) L444. [20] C.E.C. Wood, Surf. Sci. 108 (1981) L441. [21] T. Shitara, D.D. Vvedensky, M.R. Wilby, J. Zhang, J.H. Neave, B.A. Joyce, Appl. Phys. Lett. 60 (1992) 1504.
[22] P.R. Pukite, P.I. Cohen, S. Batra, in: P.K. Larsen, P.J. Dobson (Eds.), RHEED and Reflection Imaging of Surfaces, Plenum Press, New York, 1988, p. 427. [23] L.-M. Peng, M.J. Whelan, Proc. Roy. Soc. London, Ser. A 432 (1991) 195.
Interpretation of reflection high-energy electron diffraction oscillation phase Z. Mitura *, S.L. Dudarev , M.J. Whelan Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK Institute of Physics, Marie Curie-Sk!odowska University, pl. M. Curie-Sk!odowskiej 1, 20-031 Lublin, Poland
Abstract The use of reflection high energy electron diffraction (RHEED) became very popular after the discovery of reflected electron beam intensity oscillations in the early 1980s. Although a number of theoretical explanations of the effect have already been given, many aspects of the question as to why RHEED oscillations appear still remains open. We propose a new mechanism which may lead to the appearance of RHEED oscillations. This mechanism can be briefly described as the periodic variation of diffraction peak positions due to changes in the average scattering potential near the crystal surface during the deposition of a new layer. We conclude that in a general case one can explain qualitatively the origin of RHEED oscillations as the superposition of effects due to: (1) periodic variations in surface roughness, (2) changes in interference conditions for waves scattered from different surface terraces and (3) periodic variations in the average potential in the topmost layer. 1999 Elsevier Science B.V. All rights reserved. PACS: 61.14.Hg; 68.35.Bs; 68.55.Jk Keywords: RHEED; Oscillations; MBE
From early 1980s the use of reflection high energy electron diffraction (RHEED) to monitor epitaxial growth of thin films has become very popular [1]. This is largely a consequence of the discovery RHEED intensity oscillations [2,19,20]. It was experimentally found that if epitaxial growth follows a layer-by-layer mode then regular oscillating changes in the intensity of specularly reflected beam
* Corresponding author. Tel.: #44 1865 273714; fax: 44 1865 273764; e-mail: [email protected].
occur and the period of such oscillations corresponds to the deposition of one monolayer. However, it has turned out that the theoretical explanation of this effect is not quite so simple. This is because a fully quantitative description of electron diffraction from crystals can be achieved only if dynamical diffraction theory is employed. In fact, even for the case of perfectly flat surfaces, dynamical theoretical analysis is rather complicated. The case of a growing surface is far more difficult to deal with. Due to permanent theoretical progress in this research area, we may expect that some day we will
0022-0248/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 8 ) 0 1 0 4 0 - 9
906
Z. Mitura et al. / Journal of Crystal Growth 198/199 (1999) 905–910
be able to compute RHEED oscillations precisely. However, because RHEED oscillations are generally the result of a number of simultaneous diffraction processes it seems to be impossible to develop both a simple and a quantitative description of them. Nevertheless, it is reasonable to attempt to find the most important mechanisms leading to the periodic changes in the intensity which could be used in qualitative work. Until now two such mechanisms have been recognised. Researchers from the Phillips group (later at the Imperial College) pointed out that during the deposition of a new layer there occur periodic changes in the roughness at the surfaces and it implies periodic changes in the electron intensity [3,21]. Another mechanism was recognised by researchers from the University of Minnesota. They pointed out that during growth there occur changes in interference conditions of electron waves reflected from different surface terraces [4,22]. In this paper we show another, new mechanism. We suggest that in some situations RHEED oscillations may be successfully explained as a consequence of periodic changes in the average scattering potential near the crystal surface during the deposition of a new layer. It is known from experiment that if RHEED oscillations are observed for different conditions they usually differ in phase [5]. The phase of the oscillations can be determined experimentally in the following way. One needs to measure the time t of the occurrence of the minimum in the second period of the oscillations [5]. The experimental phase can be defined as follows " 2p(t /¹!1.5), where ¹ is the period of the oscil lations. Theoretically it is more convenient to use a similar definition employing Fourier analysis [6]. Both the Phillips and the University of Minnesota approaches cannot explain why oscillations actually measured differ in phase. According to both of these approaches the phase should always be the same. Very recently, the one-dimensional proportional model of the scattering potential has been used successfully by us [6] in the interpretation of the RHEED oscillation phase determined experimentally [7]. In this paper, we show in detail why the phase may be a rapidly varying function of the angle of the electron beam incidence.
Let us assume that intensity of elastically scattered electrons can be calculated using the Schro¨dinger equation
W(r)!v(z)W(r)#KW(r)"0,
(1)
where
"q "º 2m »(z), (2) v(z)" 1# m c "q "º 2m "q "º. K" 1# (3) 2m c In Eqs. (1)—(3) »(z) is the one-dimensional scattering potential of a crystal and º is the absolute value of the accelerating voltage of the electron gun, m , q and are the electron rest mass, the electron charge and Planck’s constant. Above the crystal (z'z ), we can write the solution W(r) in the 2 form W(r)"exp(i k ) q)[exp(!ik(z!z )) , 2 #R exp(#ik(z!z ))], (4) 2 where k and q are parallel components of K and r. , Further, k""K"sin0, where 0 is the glancing angle of the incident beam. The specular beam intensity I is given by I""R". The scattering potential » (z) can be determined using electron scattering factors tabulated by Doyle and Turner [8]. The contribution to »(z) coming from the jth layer can be expressed as 4 p » (z)"!H (1#0.1i) H H m X a 4p L exp ! ; (z!z ) , (5) H b b L L L where H is the layer coverage, X is the area of the H two-dimensional surface unit cell and z is the layer H position along the axis perpendicular to the surface. The coefficients a , b are given by Doyle and L L Turner [8]. To interpret actual experimental data it is important to take account also of the effect of lattice thermal vibrations by replacing the [8] and Turner coefficients b in Eq. (5) by b #8pu, L L where u denotes the mean square displacement of atoms along the z-axis. It is worth emphasizing the fact that in our model the potential from the
Z. Mitura et al. / Journal of Crystal Growth 198/199 (1999) 905–910
partially completed layer is proportional to its coverage [see the multiplier in Eq. (5)]. The potential of the whole crystal is determined by summing contributions from all layers, i.e. »(z)" » (z). H H Finally, we are able to solve Eq. (1), using numerical codes developed within the framework of the two-dimensional Bloch wave approach (for review see Ref. [9]). The model considered in this paper can be used to describe diffraction from surfaces experimentally grown only under special conditions. Namely, the azimuth of the incident beam should be set few degrees off the main crystal direction [10]. Furthermore, during the crystal growth the average distance between nucleation centres at the surface should not exceed a few atomic radii. We have found that to understand the diffraction from a one-dimensional crystal it is useful to carry out a series of calculations as follows. For a moment, let us consider theoretically a crystal composed of Au atoms for which the lattice and atom parameters are taken carefully from crystallographic tables. Let us further introduce some modification: let us preserve the geometry of the crystal but artificially replace all atoms by weaker scatterers. Numerically, we require that the potential arising from each layer (both fully and partially filled) be multiplied by some factor a. Fig. 1 shows results of calculations for such crystals for a equal to 0.01, 0.50 and 1.00 (the first two crystals are pure artificial ones, the last one is realistic). We determined numerically the intensity from growing surfaces assuming the perfect layer-by-layer growth. Each part of Fig. 1 shows four rocking curves (the specular beam intensities versus the glancing angle) computed for different coverages of the topmost layer. 䉴 Fig. 1. RHEED calculations for crystals composed of atoms for which the scattering potential is taken to be the potential of a Au atom multiplied by some factor a. The value of a is equal to: (a) 0.01, (b) 0.50, (c) 1.00. In all three cases the geometry of a Au (1 1 1) crystal is assumed and the energy of the incident electron beam is taken to be 15 keV. It is assumed that the crystals are grown in the perfect layer-by-layer mode. RHEED rocking curves are computed for different stages of the growth, namely for four different coverages of the topmost layer.
907
For the case of very small values of the scattering potential (see Fig. 1a), we can clearly recognise a number of regularities. First, we observe that both for a flat surface (H"1) and rough ones
908
Z. Mitura et al. / Journal of Crystal Growth 198/199 (1999) 905–910
(H(1) the positions of Bragg peaks are exactly the same. Second, between Bragg peaks the intensity is always smaller for rough surfaces. In more detailed theoretical work [11] we showed that the following analytical formula is valid for the case of a crystal composed of very weak scatters: I(H, 0)"I(1, 0)[1#2H(1!H) ;(cos(2"K"d sin 0)!1)],
(6)
where I(H, 0) and I(1, 0) means respectively the intensity for rough and flat surfaces and d is the growing layer thickness. In fact, it is surprising that this formula coincides with that developed by researchers from Minnesota University [4,22] for their terrace model of scattering form growing crystals. If we analyse cases shown in Fig. 1b and Fig. 1c we can still recognise the tendency described analytically by Eq. (6). However, the situation has become more complicated. We observe now that all Bragg peaks are shifted towards lower angles in comparison with those shown in Fig. 1a. Moreover, detailed analysis shows that the positions of peaks for flat and rough surfaces are no longer the same (see Fig. 2a). Bragg peaks for rough surfaces always appear at slightly higher angles in comparison with those observed for flat surfaces. This regularity has important implications for the appearance of RHEED oscillations. Let us recall that RHEED oscillations are changes in the intensity of the specular beam intensity which are observed during the deposition of the material, for a fixed glancing angle. If we compare the shapes of oscillations determined for angles slightly lower and slightly higher than those assigned to flat surface Bragg peak maxima, we can recognise that the respective difference in phase is equal to about n (see Fig. 2b and Fig. 2c). In other words, for 䉴 Fig. 2. (a) Rocking curves for different coverages of the topmost layer for Au (1 1 1) surface in the vicinity of the Bragg peak 333 shown in detail and on a linear scale (this peak is also shown in Fig. 1c). (b) The RHEED oscillation phase shown as a function of the glancing angle corresponding to the situation presented in part (a). (c) Plots of RHEED oscillations for glancing angles of 2.6° (condition (i)) and 3.0° (condition (ii)) for crystal growth as in (a) and (b).
lower angles than those for flat surface Bragg reflections the intensity initially decreases after starting growth, while for higher angles the intensity increases. Thus, we can say that due to the periodic variations in peak positions during growth we can
Z. Mitura et al. / Journal of Crystal Growth 198/199 (1999) 905–910
expect that in the peak vicinities RHEED oscillations should be relatively large in amplitude and their phase should be a strongly varying function of the glancing angle. To give a qualitative explanation of this behaviour we can say that the precise positions of Bragg peak maxima are dependent on the value of the average scattering potential near the surface. The lower value of the volume average (this quantity is always negative in our notation), the larger shift of the peak is observed. For growing crystals the lowest potential average taken over the volume of the topmost layer occurs for a flat surface. We should say that similar models as considered in this paper have been the subject of much theoretical speculation in the past. Kawamura and Maksym [12] first introduced the concept of the proportional model of the scattering potential. Further Peng and Whelan [13,23] did extensive research work for the one-dimensional case. They in fact did an initial analysis of the oscillation phase. Next the one-dimensional proportional model has been used [14,15] to investigate the effect of double maxima in a period. The possible extension of the proportional model to include diffuse scattering has been theoretically discussed by Dudarev et al. [16]. Experimental verifications of predictions of the plots of oscillations calculated using the proportional model of the potential are shown in Refs. [17,18]. The recent successful interpretation [6] of the experimental oscillation phase is perhaps the strongest confirmation that simple models, ignoring the detailed topography of the surface and taking into consideration only the average potential, may be indeed quite useful. Nevertheless, more generally one should admit that in some situations (as discussed in Refs. [3,4,21,22]) consideration of the detailed shapes of islands at the surface may be of crucial importance. In conclusion, for a general case we can point out three simultaneous mechanisms leading to the appearance of RHEED oscillations: variations in roughness at the surface [3,21], changes in terrace distribution [4,22] and variations in the average scattering potential at the surface (see Ref. [6] and this paper). At present the question of which mechanism is dominant in experimental practice is still open (for a detailed discussion see Ref. [6]). But
909
however, we believe that it could be answered in the near future if common experimental and theoretical research work devoted to this particular problem is undertaken. One of us (Z.M.) gratefully acknowledges financial support from the Leverhulme Trust (for the award of a Research Fellowship). Numerical computations have been performed using the facilities of the Materials Modelling Laboratory at the Department of Materials of the University of Oxford. We gratefully acknowledge funding for these facilities from the EPSRC and the HEFCE/OST, under the joint research equipment initiative with matching funding from Hewlett-Packard. We also wish to acknowledge stimulating discussions with Professor P.I. Cohen, Professor A. Ichimiya, Professor B.A. Joyce and Dr. J.H. Neave.
References [1] M.A. Herman, H. Sitter, Molecular Beam Epitaxy: Fundamentals and Current Status, Springer, Berlin, 1989. [2] J.J. Harris, B.A. Joyce, P.J. Dobson, Surf. Sci. 103 (1981) L90. [3] J.H. Neave, B.A. Joyce, P.J. Dobson, N. Norton, Appl. Phys. A 31 (1983) 1. [4] J.M. Van Hove, C.S. Lent, P.R. Pukite, P.I. Cohen, J. Vac. Sci. Technol. B 1 (1983) 741. [5] J. Zhang, J.H. Neave, P.J. Dobson, B.A. Joyce, Appl. Phys. A 42 (1987) 317. [6] Z. Mitura, S.L. Dudarev, M.J. Whelan, Phys. Rev. B 57 (1998) 6309. [7] G.E. Crook, K.G. Eyink, A.C. Campbell, D.R. Hinson, B.G. Streetman, J. Vac. Sci. Technol. A 7 (1989) 2549. [8] P.A. Doyle, P.S. Turner, Acta Crystallogr. A 24 (1968) 390. [9] P.A. Maksym, Surf. Rev. Lett. 4 (1997) 513. [10] A. Ichimiya, Surf. Sci. 192 (1987) L893. [11] Z. Mitura, S.L. Dudarev, M.J. Whelan, Surf. Rev. Lett. 5 (1998) 101. [12] T. Kawamura, P.A. Maksym, Surf. Sci. 161 (1985) 12. [13] L.-M. Peng, M.J. Whelan, Surf. Sci. 238 (1990) L446. [14] Z. Mitura, A. Daniluk, Surf. Sci. 277 (1992) 229. [15] Y. Horio, A. Ichimiya, Jpn. J. Appl. Phys. 33 (1994) L377. [16] S.L. Dudarev, L.-M. Peng, M.J. Whelan, Surf. Sci. 279 (1992) 380. [17] Z. Mitura, M. Stro´z ak, M. Ja"ochowski, Surf. Sci. 276 (1992) L15. [18] A. Daniluk, P. Mazurek, P. Miko"ajczak, Surf. Sci. 369 (1996) 91.
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Z. Mitura et al. / Journal of Crystal Growth 198/199 (1999) 905–910
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