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Layer-by-layer growth of GaAs(001 ) studied by in situ synchrotron X-ray diffraction
Surface Science 525 (2003) 126–136 www.elsevier.com/locate/susc
Layer-by-layer growth of GaAs(0 0 1) studied by in situ synchrotron X-ray diffraction Wolfgang Braun *, Bernd Jenichen, Vladimir M. Kaganer, Alexander G. Shtukenberg, Lutz D€ aweritz, Klaus H. Ploog Paul-Drude-Institut f€ur Festk€orperelektronik, Hausvogteiplatz 5–7, D-10117 Berlin, Germany Received 29 April 2002; accepted for publication 28 October 2002
Abstract We investigate the time-dependent surface evolution during molecular beam epitaxy of GaAs(0 0 1) using synchrotron X-ray diffraction at a newly-built dedicated beamline at the synchrotron BESSY II. The crystal truncation rods analyzed at growth temperature agree with the room-temperature bð2 4Þ reconstruction published in the literature. The layer coverage evolution during growth is analyzed by fitting the oscillating intensity along crystal truncation rods. Our results show that the structure of the reconstructed surface unit cell does not change during growth. Using numerical simulations, we determine the terrace size distribution on the surface at growth temperature and verify the validity of our analysis for multi-level initial surfaces, as long as the mean terrace size is larger than the nucleation distance during growth. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Molecular beam epitaxy; Growth; X-ray scattering, diffraction, and reflection
1. Introduction Molecular beam epitaxy (MBE) is a versatile method to grow single crystals and heteroepitaxial structures with atomic layer precision. This makes it an outstanding tool both for fundamental studies of growth mechanisms and the production of artificial structures for various device applications. In many cases, the layer-by-layer growth mode is preferred in MBE. Layer-by-layer growth is based on the existence of large, atomically flat
*
Corresponding author. Tel.: +49-30-20377-366; fax: +4930-20377-201. E-mail address: [email protected] (W. Braun).
terraces. Then, the atoms arriving from the incoming flux diffuse on the surface until they nucleate monolayer-high islands. These islands grow by attaching further adatoms, coalesce and form a complete crystal layer. Ideal growth, with one layer being fully completed before the next one nucleates, is not realized in our experiments. When one layer is almost completed, the atomic steps at the island edges are no longer sufficiently strong sinks for adatoms, and the next layer starts to nucleate and grow on top of the islands. However, layer-by-layer growth fairly close to the ideal behavior, with very few incomplete layers at any given moment, can be experimentally realized for hundreds of layers, if island nucleation and adatom attachment to the island edges dominate
0039-6028/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 2 ) 0 2 5 5 1 - 7
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over the attachment of adatoms to existing steps. This growth mode thus requires large, atomically flat terraces on the surface, a sufficiently large atomic deposition flux and sufficiently low surface diffusion. Layer-by-layer growth in MBE is commonly monitored by reflection high-energy electron diffraction (RHEED) through periodic modulations of the scattered intensity [1–4]. The period of one oscillation corresponds to the deposition of one layer. Although RHEED intensity oscillations are widely used to control growth and measure growth rates, a quantitative interpretation of the measured intensities remains difficult due to the strong interaction of electrons with the surface, leading to strong multiple (dynamical) scattering. Dynamical calculations of RHEED intensities require long computation times even when strongly simplifying assumptions about incomplete layers are applied [5]. The detailed structure of the surface, in particular island shapes and shadowing of part of the surface at grazing illumination, is not treated dynamically. On the other hand, the single-scattering approximation of radiation–matter interaction (kinematical approximation) directly relates surface structure and scattered intensity. The RHEED studies stimulated the development of a kinematical theory of surface scattering [6,7], but the relation of the kinematically calculated intensities to actual RHEED measurements is always questionable. Due to the weak X-ray–matter interaction, surface X-ray scattering can be quantitatively interpreted using kinematical theory, thus providing information on the structure and morphology of the surface. Whereas there exist a substantial number of static surface X-ray scattering studies, the list of time-resolved studies of MBE growth is quite short. Published works include the homoepitaxy of only one semiconductor material, Ge [8–10], and the metals Ag [11,12], Cu [13], and Au [14]. The organometallic vapor-phase epitaxy of GaAs was also studied using time-resolved X-ray scattering [15–18]. The present paper reports on surface X-ray scattering studies of GaAs(0 0 1) MBE growth. The experiments were performed at the PHARAO surface diffraction beamline U125/2-KMC, dedi-
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cated to the in situ analysis of MBE growth of III– V-based structures, at the synchrotron BESSY II (Berlin, Germany). The beamline is designed to perform diffraction experiments on surfaces and interfaces in an ultrahigh vacuum (UHV) environment during crystal growth by MBE. A detailed description of the experiment is published as a separate article [19]. We performed time-resolved X-ray intensity measurements at different points L along the crystal truncation rods (CTRs) 11L, 13L, and 31L. Due to the bulk atomic structure of GaAs, antiphase conditions are realized at L ¼ 0; 2; . . ., in contrast to the frequently treated case [6,7] of a simple cubic structure, where integer L provide inphase scattering of subsequent atomic layers and half-integer L give rise to destructive interference between layers (antiphase condition). From the measured time-dependent intensities, we obtain the layer coverages during growth and conclude that no more than three layers are partially occupied at any time. By comparing time-dependent intensities at different points L along a CTR, we are able to show that the atomic structure of the (2 4) reconstructed unit cell does not change during growth. This is accomplished first by fitting whole sets of intensity curves measured at different L by one and the same coverage evolution and second by showing that the structure factors of the reconstructed unit cell do not change during growth. The oscillation model assumes a perfectly flat surface prior to growth. Using numerical simulations, we demonstrate that the time evolution of the oscillation signal at the peak maximum is practically identical for a wide range of terrace size distributions, allowing its application to typical multi-level surfaces as long as the mean terrace width is larger than the nucleation distance during growth.
2. Experiment The GaAs epi-ready substrates used in this study were held on the sample holders by springloaded clamps. The substrates were first degassed at 100 °C in the load-lock chamber, then baked for 1 h at 500 °C in an intermediate preparation
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chamber before being introduced into the growth chamber. The oxide was desorbed under RHEED control while heating the sample at a rate of 10 °C/ min. As common in MBE, the oxide desorption temperature was taken as 580 °C, serving as a temperature reference for the subsequent measurements [20]. With the new sample holders used, drifts of up to 20 °C may occur. The temperature values given below therefore have to be taken with some caution. The As4 /Ga or V/III flux ratio was determined from As-limited RHEED oscillations [21]. From run to run, the As4 flux was monitored and adjusted by means of a Bayard–Alpert vacuum gauge close to the sample position. The As4 flux reaching the substrate is in part due to As4 scattered by the chamber walls. When growth starts, Ga hitting the chamber walls traps some of this As, thereby reducing the total As flux reaching the sample. In the figure captions the V/III flux ratios are given before and during the growth. The growth conditions were optimized first by scanning the substrate temperature through the stability range of the bð2 4Þ surface reconstruction and then fine-tuning the growth temperature around the center of this temperature interval by minimizing the damping of RHEED oscillations. Following the usual convention, the [1 1 0] axis runs along the long edge of the 2 4 surface unit cell, the ½ 1 1 0 axis is parallel to the short edge. The 2 4 surface unit cell is rotated by 45° with respect to the cubic zincblende unit cell of the bulk GaAs. Direction vectors, crystal planes and diffraction indices refer to the bulk unit cell.
3. Surface morphology Fig. 1 shows an atomic force microscopy (AFM) image of the surface of one of the GaAs(0 0 1) samples used in the present study. The AFM image was obtained after a few tens of growth pulses carried out for the X-ray scattering measurements described below. The surface consists of flat terraces separated by steps of elementary height (0.28 nm). The mean terrace width obtained from a series of AFM images taken in different places of the sample is 500 nm.
Fig. 1. AFM image of the GaAs(0 0 1) sample surface after a few tens of growth pulses performed for the X-ray scattering measurements. The lines in the image are steps of GaAs bilayer height (0.28 nm). The last series of growth pulses before the sample cooling was carried out at the substrate temperature of 560 °C and V/III ratio of 9 decreasing to 5 during growth.
Fig. 2 presents x scans (rotation of the sample about its normal) obtained on the same sample as shown in Fig. 1 in two subsequent days of the Xray measurements. The splitting of the peaks in Fig. 2(a) is the result of an ordered staircase of steps [7,22,23]. From the angular peak separation Dx ¼ 0:029°, one can determine [23] the average terrace pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi width lT ¼ aðsin w þ cos w tan hÞ=Dx h2 þ k 2 , where a is the lattice constant, h is the Bragg angle, w is the angle between the scattering vector and the miscut direction in the surface plane, h ¼ 3, k ¼ 1 are the in-plane reflection indices. We obtain lT ¼ 366 nm, somewhat smaller than the result of the AFM measurements. The peak widths in Fig. 2(a) are 0:0142° 0:0016°, just above the resolution limit of 0.01°, which indicates a small amount of disorder in the step staircase. The peak in Fig. 2(b) is not split, indicating stronger disorder in the terrace widths [7]. The disorder is evident from the AFM image in Fig. 1. The integrated intensities of the peaks in Fig. 2(a) and (b) differ by less than 3%, which directly confirms the consistency of our treatment and the validity of the kinematical theory used.
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Fig. 2. Omega scans of the 3 1 0.05 reflection performed on the sample shown in Fig. 1. Measurement (b) was carried out one day after measurement (a). Solid lines are fits to a sum of two Lorentzians. The substrate temperature is 560 °C and the V/III ratio is 9 based on hypothetical growth with a period of 7 s per layer.
4. X-ray intensity oscillations during layer-by-layer growth Fig. 3 shows the time dependence of X-ray intensities recorded at different L along the CTR 13L (open circles). The background intensity is obtained from sufficiently wide x scans of the sample prior to growth and subtracted from the measured intensity. The period of the intensity oscillations s ¼ 7 s is the time required to complete one bilayer of the GaAs crystal. The solid lines in the intensity plots of Fig. 3 are calculated with the layer coverages hn ðtÞ presented in the top panel. The mathematical model for the coverages hn ðtÞ is described below in Section 5.2. The fit of the intensity curves is obtained simultaneously, i.e. with one and the same layer coverages hn ðtÞ, for all values of L. At small L, the
Fig. 3. X-ray intensity oscillations measured during the MBE of GaAs(0 0 1) at different points along the crystal truncation rod 13L (open circles). The fit (solid curves) is obtained simultaneously for all curves with the layer coverage shown in the top panel of the figure. The substrate temperature was 570 °C and the V/III ratio 13 at the start of the growth interval, 8 at the end of it.
intensity decreases to zero at the minimum of the first oscillation, indicating that only one layer is partially covered at that moment. However, the following maxima do not reach the initial intensity, which means that the next layer starts to grow somewhat before the previous one is completed. The growth is rather close to the ideal, with the next layer containing less than 10% when the previous one is filled to about 90%.
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Fig. 4. X-ray intensity oscillations measured during the MBE of GaAs on GaAs(0 0 1) at different points on the crystal truncation rod 3 1L (open circles). The fit (solid curves) is obtained simultaneously for all curves with the layer coverage shown in the top panel. The substrate temperature was 560 °C, and the V/III ratio was 27, decreasing to 21.
Fig. 4 presents X-ray intensity measurements performed on the CTR 3 1L. The intensities measured at different L are simultaneously fitted to the same layer coverages like in the previous figure. In this case, the growth is two times slower compared to Fig. 3 (the growth period is s ¼ 15 s) and the As coverage is higher. The calculated layer coverages (top part of Fig. 4) show that the second layer starts to grow when the first one is covered to about 80%, while, say, the ninth layer starts to grow when the previous one is only 30% covered. Hence, the damping of the oscillations is fairly sensitive to a continuous roughening of the growing surface. Fig. 5 shows measurements on the CTR 11L (the growth period is s ¼ 6:8 s). One can clearly see the symmetry of the curves with respect to the inphase condition at L ¼ 1 DL, giving identical oscillations with increasing amplitude towards L ¼ 0 and 2 (antiphase conditions) and decreasing towards L ¼ 1 (in-phase condition).
Fig. 5. X-ray intensity oscillations measured during growth of GaAs(0 0 1) at different points on the crystal truncation rod 11L (open circles). The fit (solid curves) is obtained simultaneously for all curves with the layer coverage shown in the top panel. The substrate temperature was 565 °C and the V/III ratio was 14.
Fig. 6 compares the intensities along the CTRs obtained prior to a growth run (completely covered surface layer) and at the first intensity minimum during growth (half-covered surface layers). The central panel collects the results of independent measurements on two equivalent CTRs, 31L and 31L, performed on different samples. The full symbols are the initial intensities obtained before growth by integrating over the x scan. This is the quantity commonly measured and analyzed in surface structure analysis [24,25]. The solid lines are the CTRs calculated by using the atomic coordinates of the GaAs(0 0 1) bð2 4Þ unit cell determined by room-temperature X-ray surface structure analysis [26], and assuming a flat surface without steps. We used the room-temperature coordinates [26] and the bulk Debye–Waller factors for Ga and As (there is no information on Debye– Waller factors for surface atoms in [26]). From the good agreement between the measured and calculated CTRs, we conclude that the positions of
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calculations, which confirms the flatness of the surface. Note, however, that this conclusion is based on the analysis of only four rods, which is a relatively small subset of the typical number of rods used in an X-ray structure analysis. The open symbols in Fig. 6 represent the intensity at the minimum of the first oscillation during growth, scaled to the pre-growth intensity. The first intensity minimum corresponds to the deposition of half a monolayer. Further intensity minima can be deeper because several incompletely covered layers are present simultaneously, cf. Figs. 3–5. The dashed curves in Fig. 6 are calculated with the same assumptions regarding surface flatness and atomic positions in the reconstructed cell as above, but with a 50% covered second layer. The agreement between the measurements and the calculations confirms that the structure of the bð2 4Þ surface reconstruction cell does not change during growth.
5. Theory 5.1. Antiphase conditions for a stepped GaAs(0 0 1) surface
Fig. 6. Measured and calculated intensities along crystal truncation rods prior to growth (full symbols and solid lines) and at the first minimum of the growth intensity oscillations (open symbols and dashed lines).
the atoms in the reconstructed unit cell do not change noticeably between room temperature and the growth temperature. This agreement is reached without introducing a roughness parameter in the
The experimental results presented above show that the maximum sensitivity of the X-ray diffracted intensity to surface steps is achieved at L ¼ 0, when both the incident and the diffracted beams are in the surface plane. On one hand, the sensitivity to surface morphology under these conditions is expected: for example, a bulk-forbidden reflection 310 lies halfway between two bulk reflections 311 and 311 and corresponds to an intensity minimum along the CTR 31L, where the sensitivity to surface roughness is best [25]. On the other hand, this sensitivity is surprising since, at L ¼ 0, the scattering wave vector does not have a component normal to the surface and cannot probe the steps by sampling their height. Hence, it is worth to analyze how the surface steps become visible in an in-plane diffraction geometry. GaAs(0 0 1) grows by incorporating Ga–As bilayers. The bulk GaAs unit cell is formed by two such bilayers, being shifted by a relative translation t ¼ a½0 12 12. Here, a is the bulk lattice period. A
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single surface step is one bilayer in height and hence involves an in-plane translation of half the in-plane lattice period. When two columns (normal to the surface) of bulk unit cells are terminated at adjacent terraces separated by a single step, they are related by the same translation t. The same applies to the reconstructed cells on these terraces. Then, for a reflection Q ¼ ð2p=aÞ½hkL, the structure amplitudes of the two terraces differ by a phase factor expðiUÞ, where U ¼ Q t ¼ pðk þ LÞ:
ð1Þ
Let us consider the reflections with odd h and k, for example 11L, 13L, etc. Then, the phase factor U ¼ p is reached at L ¼ 0 and causes destructive interference of waves diffracted from two adjacent terraces, resulting in the highest sensitivity to surface steps. In other words, the steps are detected due to a lateral displacement of the atoms of half the lattice period, rather than due to the vertical displacement. One could just as well transform one terrace into the other by a translation t0 ¼ a½12 0 12 instead of t, resulting in a phase factor exp ½ipðh þ LÞ. The two phase factors are identical along all CTRs connecting bulk reflections of GaAs, since all bulk reflections hkL have both odd or both even indices h and k. On a given terrace, the (2 4) unit cell of the surface reconstruction can nucleate in eight different positions with respect to a chosen origin. Fractional-order rods are sensitive to these relative shifts and allow the determination of reconstruction domain sizes. For integer order rods, however, the signal from all eight surface unit cell positions adds up in phase, making these rods insensitive to the relative placement of the surface unit cells [27]. This means that integer order rods sample only the terraces, without being sensitive to the size, shape and arrangement of the surface reconstruction domains on them. 5.2. Kinetics of the layer coverages The average surface shape during growth can be described in terms of the coverages hn ðtÞ of successive layers. Growth starts from a perfectly flat surface with h0 ¼ 1 and hn ¼ 0 ðn P 1Þ. During growth, the coverages hn ðtÞ run from 0 to 1, with
hnþ1 < hn . The fraction of the surface at the nth layer level is hn hnþ1 and hence the intensity diffracted by a crystal with the stepped surface is 2
SðqÞ ¼ jFhkL j jðh0 h1 Þ þ ðh1 h2 ÞeiU 2
þ ðh2 h3 Þe2iU þ j ;
ð2Þ
where FhkL is the structure amplitude of a given reflection and the phase U is given by Eq. (1). The validity of the kinematical formula (2) for X-ray scattering is well established, in contrast to RHEED where it may be used only qualitatively. Eq. (2) describes the coherent part of the intensity. We do not consider diffuse intensity here [6,7], since Eq. (2) is sufficient for a complete description of the experimental data presented in the following section. We thus conclude that typical nucleation distances are small, so that the diffuse scattering of the island or hole structure during layer formation is distributed in a wide angular range. This means that a negligible part of diffuse intensity falls within the detector window together with the coherent intensity. When the top layer covers one half of the surface, Eq. (2) gives an intensity minimum equal to Imin =I0 ¼ ð1 þ cos UÞ=2; 2
ð3Þ
where I0 ¼ jFhkL j . The intensity minimum Imin becomes zero when the antiphase condition U ¼ p is fulfilled. It follows from Eq. (1) that on the CTRs hkL with h, k odd, the antiphase condition is reached at L ¼ 0, which provides the most appropriate condition for a surface scattering experiment. We have therefore chosen the rods 11L, 13L, and 31L (the latter two are not equivalent because of the polarity of the GaAs(0 0 1) surface) for the experiments. Models for the time dependence of the layer coverages hn ðtÞ were proposed by Cohen et al. [28]. However, these models are not suitable to fit our experimental data. These mean-field models cannot describe the smooth continuous decay of the observed intensity oscillations. Rather, after a few initial oscillations, they give oscillations with a constant amplitude. The development of an adequate physical model of the layer-by-layer growth process remains a demanding task and is not considered here. Modifications proposed to fit
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X-ray scattering data [11,12] involve the introduction of an explicit dependence of the coefficients in the equations by Cohen et al. on the layer number [11] or time [12], which has no obvious motivation in the MBE deposition process. Trying to separate the analysis of the oscillations from the underlying growth model, we obtain the layer coverages hn ðtÞ from the following fit, which attempts to avoid any assumptions about the physics of the growth process. The time derivative of the layer coverage dhn =dt can be generally written as a difference dhn =dt ¼ ðJn 1 Jn Þ=s
ð4Þ
with Jn depending on the coverages and time. Growth starts at time t0 with hn ¼ 0 and J0 ¼ 1, Jn ¼ 0 (n P 1). Summation of Eq. (4) over all n shows that P1 these equations obey mass conservation: dð n¼1 hn Þ=dt ¼ 1=s, i.e., the total change of the coverage per unit time is equal to the deposition flux 1=s. The simplest growth model, nondiffusive growth [28], assumes that mass transfer between the layers is absent and the adatoms are incorporated into the layer following the one they land onto without moving in the surface plane. This is described by Jn ¼ hn and leads to an exponential intensity decay without oscillations. Ideal layer-by-layer growth is described by dhn =dt equal to the growth rate 1=s in the time interval ðn 1Þs < t < ns and zero elsewhere. In this case, Jn is equal to the Heaviside function H ðt nsÞ. We model the intermediate case of imperfect layer-by-layer growth by a smoothed Heaviside function, Jn ðtÞ ¼ tanh½ðt nsÞ=bn :
ð5Þ
Different coefficients bn 1 and bn allow a different behavior for the onset of the nth layer and its completion. However, an unrestricted fit of all the coefficients bn is not reliable due to the noise of the data. We therefore choose bn ¼ anb , with two fit parameters, a and b, both varying between 0.1 and 0.5 for the data presented below. The fits of the experimental data in Figs. 3–5 use the constant intensity measured 30 s before growth as a normalization factor to the calculated intensity IðqÞ. We conclude from the simultaneous fit of all curves that the structure amplitudes FhkL in
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Eq. (2) do not change during growth. The structure amplitude FhkL includes contributions from the bulk crystal and the reconstructed layer on top of it. A constant FhkL during growth means that the structure of the reconstructed cell is preserved during growth. Remember that on integer order rods, the domain size of the reconstruction or any disorder in the relative positions of the surface unit cells is irrelevant, since all surface unit cells add up in phase. The simultaneous fit of the intensity curves for different L also proves that the reconstruction is established on top of a nucleated layer in a time much smaller than the growth period. 5.3. Layer-by-layer growth on terraced surfaces The model of Eq. (2) is based on the assumption that during growth, the surface is flat with twodimensional islands forming a partially covered layer, possibly with smaller islands on top of them and one-layer deep pits. In particular, it assumes that growth initially starts from a perfectly flat and uniform plane. This is in apparent contradiction with the surface morphology described in Section 3: we have concluded from Figs. 1 and 2 that the surface is vicinal, with either ordered or disordered step trains. Several terraces are coherently illuminated by X-rays, which follows from the observation of the peak splitting due to the ordered staircase in Fig. 2(a) and the peak broadening due to a disordered step train in Fig. 2(b). In such a system, described by the average layer distribution of Eq. (2), the deposition of a fraction of a monolayer in the form of two-dimensional islands is equivalent to the attachment of the same amount of material to nearby steps in the stepflow growth mode. The latter process, however, does not produce intensity oscillations. Why are intensity oscillations characteristic for layerby-layer growth observed on vicinal surfaces with both ordered and disordered steps? To analyze this apparent contradiction, we performed simple numerical X-ray scattering simulations of such surfaces with two-dimensional islands. Scattering from a vicinal surface can be described analytically [7], but the presence of islands makes the problem too complicated for an analytical analysis.
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An appropriate model for a terrace width distribution with variable disorder is the gamma distribution function P ðxÞ ¼ xM 1 e x =CðMÞ, where P ðxÞ is the probability distribution, M is the parameter governing the disorder; and CðMÞ is the gamma function [7]. The parameter M spans the range from a regular staircase of steps, M ¼ 1, to complete disorder (where the width of a terrace is independent of the widths of the adjacent ones), M ¼ 1. Fig. 7(a) shows vicinal surfaces simulated by the gamma distribution with different degrees of ordering: a regular staircase (M ¼ 1, curve 1), a partially disordered staircase (M ¼ 3, curve 2), and a completely disordered staircase (M ¼ 1, curve 3). In a second pass, we cover these surfaces with two-dimensional islands, Fig. 7(b), by creating random steps up followed by steps down, so that a 50% island coverage is obtained. The island
Fig. 7. Vicinal surfaces with the same mean misorientation, but different degrees of terrace order (a), magnified parts of these surfaces with monolayer-height islands deposited to a mean coverage of 50% (b). Diffraction patterns from these surfaces at the antiphase condition L ¼ 0 are shown in (c) and for an intermediate value L ¼ 0:5 in (d). The signal calculated from the terraces is shown by thin lines and the one from half-covered terraces by thick lines. The terrace widths follow the gamma distribution with the parameter M ¼ 1 (regular staircase, curve 1), M ¼ 3 (curve 2), and M ¼ 1 (curve 3).
sizes and distances are also defined by a gamma distribution with M ¼ 3. Fig. 7(c,d) present the diffraction peaks obtained from the terrace distributions of Fig. P 7(a,b) using the Fourier transform IðqÞ ¼ j x exp 2 ½iQhðxÞ þ iqxj , with a subsequent convolution with the resolution function RðqÞ. Here, Q ¼ 2pL=a is the diffraction vector corresponding to a given position L on the diffraction rod. We choose a Gaussian resolution function RðqÞ. The mean terrace width and the width of the resolution function are chosen so that the calculated peaks agree with the experiment, cf. Figs. 7(c) and 2. Thin lines in Fig. 7(c,d) show diffraction peaks obtained from the terrace structures of Fig. 7(a). For the antiphase condition L ¼ 0, the regular staircase gives rise to a split peak (curve 1) [7, 22,23]. When the regularity in the terrace widths decreases, the peaks broaden (curve 2). An exponential distribution of terrace widths gives rise to a Lorentzian peak (curve 3) [7]. The calculated peaks agree with the measured ones, Fig. 2. For L significantly different from zero, only one peak is present, Fig. 7(d). Its width increases and the height decreases with increasing terrace width disorder. Thick lines in Fig. 7(c,d) show the diffraction peaks of the island-covered terraces. The coverage is 50%, a small part of the surface is shown in Fig. 7(b). For L ¼ 0, the islands give rise to diffuse scattering, with the scattered intensity transferred from the coherent peaks to broad diffuse peaks. Then, the intensity at the position of the coherent peak, which is detected by the fixed detector during the growth, decreases almost to zero. For L 6¼ 0, the peak intensity does not decrease to zero. The ratio of the peak intensities for the flat terraces I0 and the terraces half-covered by the islands Imin follow Eq. (3). In particular, for the case L ¼ 0:5, Eq. (3) gives the ratio Imin =I0 ¼ 0:5, which coincides with the ratio of the peak intensities in Fig. 7(d). One can see, by comparing the curves 1, 2 and 3, that the ratio of peak intensities does not depend on the degree of ordering of the terraces. This result shows that the calculation of the intensity diffracted by a surface with partially covered top layers can be performed by Eq. (2) even if the surface is not flat but consists of terraces. Simul-
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taneous coherent illumination of several terraces does not influence the applicability of this equation derived for a flat surface. This conclusion is obviously valid as long as the sizes of the twodimensional islands are small compared with the mean terrace width. Fig. 8 presents a similar simulation for a surface which does not have a mean miscut but consists of terraces defined by random up and down steps. The surfaces in Fig. 8 have uncorrelated up and down steps of equal height and the terrace widths given by the gamma distribution with M ¼ 3 (curve 1) and M ¼ 1 (curve 2). Fig. 8(b) shows magnified pieces of the surfaces with single step high islands deposited on them to a coverage of 50%. Again, the island sizes follow the gamma
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distribution with M ¼ 3, in the same way as in the previous figure. The calculated diffraction peaks, Fig. 8(c,d) are similar to these in Fig. 7(c,d). One can see that the intensity ratio at the position of the coherent peak follows Eq. (3): at the antiphase condition, L ¼ 0, the intensity from the half-covered surface is close to zero while for L ¼ 0:5 the ratio Imin =I0 is 0.5, which again coincides with Eq. (3). We conclude, therefore, that the presence of terraces, the mean width of which is smaller than the X-ray coherence length, does not affect the calculation of the scattered intensities from the layer coverages hn by making use of Eq. (2). This conclusion is valid for ordered or disordered terraces on vicinal or low-index surfaces. 6. Conclusions
Fig. 8. Surfaces with different degrees of terrace order without a mean misorientation (a), magnified parts of these surfaces with monolayer-height islands deposited to a mean coverage of 50% (b), and diffraction patterns from these surfaces at the antiphase condition L ¼ 0 (c) and for an intermediate value L ¼ 0:5 (d). Diffraction patterns from the terraces are shown by thin lines and patterns from terraces half-covered by monolayer height islands are shown by thick lines. The terrace widths follow the gamma distribution with the parameter M ¼ 1 (regular staircase, curve 1), M ¼ 3 (curve 2), and M ¼ 1 (curve 3).
The first results obtained at our combined synchrotron X-ray diffraction/MBE beamline show submonolayer sensitivity of surface diffraction experiments, which allows us to perform a detailed time-resolved analysis of GaAs(0 0 1) homoepitaxial growth. We measure X-ray intensities during growth at different points L along the CTRs 11L, 13L, and 31L. This allows us to extract the time evolution of the layer coverage without assuming a specific model for layer-by-layer growth. The results indicate that no more than three layers are partially covered at any moment. By comparing the intensity evolution at different points L along a CTR, we show that the atomic structure of the bð2 4Þ surface reconstruction unit cell is preserved during growth. No differences to the bð2 4Þ structure determined at room temperature are observed. Using numerical simulations, we determine the terrace size distributions of the surface under typical growth conditions and show that the layer coverage fit during growth is also valid for multilayer starting surfaces, as long as the mean terrace widths are larger than the typical island separations occurring during growth. References [1] J.J. Harris, B.A. Joyce, P.J. Dobson, Surf. Sci. 103 (1981) L90.
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[2] C.E.C. Wood, Surf. Sci. 108 (1981) L441. [3] J.M. van Hove, C.S. Lent, P.R. Pukite, P.I. Cohen, J. Vac. Sci. Technol. B 1 (1983) 741. [4] J.H. Neave, B.A. Joyce, P.J. Dobson, N. Norton, Appl. Phys. A 31 (1983) 1. [5] P.A. Maksym, U. Korte, J.M. McCoy, H.J. Gotsis, Surf. Rev. Lett. 5 (1998) 873. [6] C.S. Lent, P.I. Cohen, Surf. Sci. 139 (1984) 121. [7] P.R. Pukite, C.S. Lent, P.I. Cohen, Surf. Sci. 161 (1985) 39. [8] E. Vlieg, A.W.D. van der Gon, J.F. van der Veen, J.E. Macdonald, C. Norris, Phys. Rev. Lett. 61 (1988) 2241. [9] R.G. van Silfhout, J.W.M. Frenken, J.F. van der Veen, S. Ferrer, A. Johnson, H. Derbyshire, C. Norris, J.E. Macdonald, Colloque de Physique 50 (1989) C7. [10] R.G. van Silfhout, J.W.M. Frenken, J.F. van der Veer, S. Ferrer, A. Johnson, H. Derbyshire, C. Norris, J.E. Macdonald, J. Phys.: Condens. Matter 1 (1989) SB213. [11] H.A. van der Vegt, H.M. van Pinxteren, M. Lohmeier, E. Vlieg, J.M.C. Thornton, Phys. Rev. Lett. 68 (1992) 3335. [12] J. Alvarez, E. Lundgren, X. Torrelles, S. Ferrer, Phys. Rev. B 57 (1998) 6325. [13] H.A. van der Vegt, J. Alvarez, X. Torrelles, S. Ferrer, E. Vlieg, Phys. Rev. B 52 (1995) 17443. [14] M.V.R. Murty, T. Curcic, A. Judy, B.H. Cooper, A.R. Woll, J.D. Brock, S. Kucia, R.L. Headrick, Phys. Rev. B 60 (1999) 16956. [15] F.J. Lamelas, P.H. Fuoss, P. Imperatori, D.W. Kisker, G.B. Stephenson, S. Brennan, Appl. Phys. Lett. 60 (1992) 2610.
[16] D.W. Kisker, G.B. Stephenson, P.H. Fuoss, F.J. Lamelas, P. Imperatori, S. Brennan, J. Cryst. Growth 124 (1992) 1. [17] P.H. Fuoss, D.W. Kisker, F.J. Lamelas, G.B. Stephenson, P. Imperatori, S. Brennan, Phys. Rev. Lett. 69 (1992) 2791. [18] D.W. Kisker, G.B. Stephenson, J. Tersoff, P.H. Fuoss, S. Brennan, J. Cryst. Growth 163 (1996) 54. [19] B. Jenichen, W. Braun, V.M. Kaganer, A.G. Shtukenberg, L. D€aweritz, C.-G. Schulz, K.H. Ploog, A. Erko, Rev. Sci. Instr., March 2003 issue. [20] A.J. Springthorpe, S.J. Ingrey, B. Emmerstorfer, P. Mandeville, W.T. Moore, J. Vac. Sci. Technol. B 5 (1987) 739. [21] B.F. Lewis, R. Fernandez, A. Madhukar, F.J. Grunthaner, J. Vac. Sci. Technol. B 4 (1986) 560. [22] P.R. Pukite, J.M.V. Hove, P.I. Cohen, J. Vac. Sci. Technol. B 2 (1984) 243. [23] G. Renaud, P.H. Fuoss, J. Bevk, B.S. Freer, Phys. Rev. B 45 (1992) 9192. [24] R. FeidenhansÕl, Surf. Sci. Reports 10 (1989) 105. [25] I.K. Robinson, D.J. Tweet, Rep. Progr. Phys. 55 (1992) 599. [26] Y. Garreau, M. Sauvage-Simkin, N. Jedrecy, R. Pinchaux, M. Veron, Phys. Rev. B 54 (1996) 17638. [27] E. Vlieg, J.F. van der Veen, S.J. Gurman, C. Norris, J.E. MacDonald, Surf. Sci. 210 (1989) 301. [28] P.I. Cohen, G.S. Petrich, P.R. Pukite, G.J. Whaley, A.S. Arrot, Surf. Sci. 216 (1989) 222.
Layer-by-layer growth of GaAs(0 0 1) studied by in situ synchrotron X-ray diffraction Wolfgang Braun *, Bernd Jenichen, Vladimir M. Kaganer, Alexander G. Shtukenberg, Lutz D€ aweritz, Klaus H. Ploog Paul-Drude-Institut f€ur Festk€orperelektronik, Hausvogteiplatz 5–7, D-10117 Berlin, Germany Received 29 April 2002; accepted for publication 28 October 2002
Abstract We investigate the time-dependent surface evolution during molecular beam epitaxy of GaAs(0 0 1) using synchrotron X-ray diffraction at a newly-built dedicated beamline at the synchrotron BESSY II. The crystal truncation rods analyzed at growth temperature agree with the room-temperature bð2 4Þ reconstruction published in the literature. The layer coverage evolution during growth is analyzed by fitting the oscillating intensity along crystal truncation rods. Our results show that the structure of the reconstructed surface unit cell does not change during growth. Using numerical simulations, we determine the terrace size distribution on the surface at growth temperature and verify the validity of our analysis for multi-level initial surfaces, as long as the mean terrace size is larger than the nucleation distance during growth. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Molecular beam epitaxy; Growth; X-ray scattering, diffraction, and reflection
1. Introduction Molecular beam epitaxy (MBE) is a versatile method to grow single crystals and heteroepitaxial structures with atomic layer precision. This makes it an outstanding tool both for fundamental studies of growth mechanisms and the production of artificial structures for various device applications. In many cases, the layer-by-layer growth mode is preferred in MBE. Layer-by-layer growth is based on the existence of large, atomically flat
*
Corresponding author. Tel.: +49-30-20377-366; fax: +4930-20377-201. E-mail address: [email protected] (W. Braun).
terraces. Then, the atoms arriving from the incoming flux diffuse on the surface until they nucleate monolayer-high islands. These islands grow by attaching further adatoms, coalesce and form a complete crystal layer. Ideal growth, with one layer being fully completed before the next one nucleates, is not realized in our experiments. When one layer is almost completed, the atomic steps at the island edges are no longer sufficiently strong sinks for adatoms, and the next layer starts to nucleate and grow on top of the islands. However, layer-by-layer growth fairly close to the ideal behavior, with very few incomplete layers at any given moment, can be experimentally realized for hundreds of layers, if island nucleation and adatom attachment to the island edges dominate
0039-6028/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 2 ) 0 2 5 5 1 - 7
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over the attachment of adatoms to existing steps. This growth mode thus requires large, atomically flat terraces on the surface, a sufficiently large atomic deposition flux and sufficiently low surface diffusion. Layer-by-layer growth in MBE is commonly monitored by reflection high-energy electron diffraction (RHEED) through periodic modulations of the scattered intensity [1–4]. The period of one oscillation corresponds to the deposition of one layer. Although RHEED intensity oscillations are widely used to control growth and measure growth rates, a quantitative interpretation of the measured intensities remains difficult due to the strong interaction of electrons with the surface, leading to strong multiple (dynamical) scattering. Dynamical calculations of RHEED intensities require long computation times even when strongly simplifying assumptions about incomplete layers are applied [5]. The detailed structure of the surface, in particular island shapes and shadowing of part of the surface at grazing illumination, is not treated dynamically. On the other hand, the single-scattering approximation of radiation–matter interaction (kinematical approximation) directly relates surface structure and scattered intensity. The RHEED studies stimulated the development of a kinematical theory of surface scattering [6,7], but the relation of the kinematically calculated intensities to actual RHEED measurements is always questionable. Due to the weak X-ray–matter interaction, surface X-ray scattering can be quantitatively interpreted using kinematical theory, thus providing information on the structure and morphology of the surface. Whereas there exist a substantial number of static surface X-ray scattering studies, the list of time-resolved studies of MBE growth is quite short. Published works include the homoepitaxy of only one semiconductor material, Ge [8–10], and the metals Ag [11,12], Cu [13], and Au [14]. The organometallic vapor-phase epitaxy of GaAs was also studied using time-resolved X-ray scattering [15–18]. The present paper reports on surface X-ray scattering studies of GaAs(0 0 1) MBE growth. The experiments were performed at the PHARAO surface diffraction beamline U125/2-KMC, dedi-
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cated to the in situ analysis of MBE growth of III– V-based structures, at the synchrotron BESSY II (Berlin, Germany). The beamline is designed to perform diffraction experiments on surfaces and interfaces in an ultrahigh vacuum (UHV) environment during crystal growth by MBE. A detailed description of the experiment is published as a separate article [19]. We performed time-resolved X-ray intensity measurements at different points L along the crystal truncation rods (CTRs) 11L, 13L, and 31L. Due to the bulk atomic structure of GaAs, antiphase conditions are realized at L ¼ 0; 2; . . ., in contrast to the frequently treated case [6,7] of a simple cubic structure, where integer L provide inphase scattering of subsequent atomic layers and half-integer L give rise to destructive interference between layers (antiphase condition). From the measured time-dependent intensities, we obtain the layer coverages during growth and conclude that no more than three layers are partially occupied at any time. By comparing time-dependent intensities at different points L along a CTR, we are able to show that the atomic structure of the (2 4) reconstructed unit cell does not change during growth. This is accomplished first by fitting whole sets of intensity curves measured at different L by one and the same coverage evolution and second by showing that the structure factors of the reconstructed unit cell do not change during growth. The oscillation model assumes a perfectly flat surface prior to growth. Using numerical simulations, we demonstrate that the time evolution of the oscillation signal at the peak maximum is practically identical for a wide range of terrace size distributions, allowing its application to typical multi-level surfaces as long as the mean terrace width is larger than the nucleation distance during growth.
2. Experiment The GaAs epi-ready substrates used in this study were held on the sample holders by springloaded clamps. The substrates were first degassed at 100 °C in the load-lock chamber, then baked for 1 h at 500 °C in an intermediate preparation
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chamber before being introduced into the growth chamber. The oxide was desorbed under RHEED control while heating the sample at a rate of 10 °C/ min. As common in MBE, the oxide desorption temperature was taken as 580 °C, serving as a temperature reference for the subsequent measurements [20]. With the new sample holders used, drifts of up to 20 °C may occur. The temperature values given below therefore have to be taken with some caution. The As4 /Ga or V/III flux ratio was determined from As-limited RHEED oscillations [21]. From run to run, the As4 flux was monitored and adjusted by means of a Bayard–Alpert vacuum gauge close to the sample position. The As4 flux reaching the substrate is in part due to As4 scattered by the chamber walls. When growth starts, Ga hitting the chamber walls traps some of this As, thereby reducing the total As flux reaching the sample. In the figure captions the V/III flux ratios are given before and during the growth. The growth conditions were optimized first by scanning the substrate temperature through the stability range of the bð2 4Þ surface reconstruction and then fine-tuning the growth temperature around the center of this temperature interval by minimizing the damping of RHEED oscillations. Following the usual convention, the [1 1 0] axis runs along the long edge of the 2 4 surface unit cell, the ½ 1 1 0 axis is parallel to the short edge. The 2 4 surface unit cell is rotated by 45° with respect to the cubic zincblende unit cell of the bulk GaAs. Direction vectors, crystal planes and diffraction indices refer to the bulk unit cell.
3. Surface morphology Fig. 1 shows an atomic force microscopy (AFM) image of the surface of one of the GaAs(0 0 1) samples used in the present study. The AFM image was obtained after a few tens of growth pulses carried out for the X-ray scattering measurements described below. The surface consists of flat terraces separated by steps of elementary height (0.28 nm). The mean terrace width obtained from a series of AFM images taken in different places of the sample is 500 nm.
Fig. 1. AFM image of the GaAs(0 0 1) sample surface after a few tens of growth pulses performed for the X-ray scattering measurements. The lines in the image are steps of GaAs bilayer height (0.28 nm). The last series of growth pulses before the sample cooling was carried out at the substrate temperature of 560 °C and V/III ratio of 9 decreasing to 5 during growth.
Fig. 2 presents x scans (rotation of the sample about its normal) obtained on the same sample as shown in Fig. 1 in two subsequent days of the Xray measurements. The splitting of the peaks in Fig. 2(a) is the result of an ordered staircase of steps [7,22,23]. From the angular peak separation Dx ¼ 0:029°, one can determine [23] the average terrace pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi width lT ¼ aðsin w þ cos w tan hÞ=Dx h2 þ k 2 , where a is the lattice constant, h is the Bragg angle, w is the angle between the scattering vector and the miscut direction in the surface plane, h ¼ 3, k ¼ 1 are the in-plane reflection indices. We obtain lT ¼ 366 nm, somewhat smaller than the result of the AFM measurements. The peak widths in Fig. 2(a) are 0:0142° 0:0016°, just above the resolution limit of 0.01°, which indicates a small amount of disorder in the step staircase. The peak in Fig. 2(b) is not split, indicating stronger disorder in the terrace widths [7]. The disorder is evident from the AFM image in Fig. 1. The integrated intensities of the peaks in Fig. 2(a) and (b) differ by less than 3%, which directly confirms the consistency of our treatment and the validity of the kinematical theory used.
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Fig. 2. Omega scans of the 3 1 0.05 reflection performed on the sample shown in Fig. 1. Measurement (b) was carried out one day after measurement (a). Solid lines are fits to a sum of two Lorentzians. The substrate temperature is 560 °C and the V/III ratio is 9 based on hypothetical growth with a period of 7 s per layer.
4. X-ray intensity oscillations during layer-by-layer growth Fig. 3 shows the time dependence of X-ray intensities recorded at different L along the CTR 13L (open circles). The background intensity is obtained from sufficiently wide x scans of the sample prior to growth and subtracted from the measured intensity. The period of the intensity oscillations s ¼ 7 s is the time required to complete one bilayer of the GaAs crystal. The solid lines in the intensity plots of Fig. 3 are calculated with the layer coverages hn ðtÞ presented in the top panel. The mathematical model for the coverages hn ðtÞ is described below in Section 5.2. The fit of the intensity curves is obtained simultaneously, i.e. with one and the same layer coverages hn ðtÞ, for all values of L. At small L, the
Fig. 3. X-ray intensity oscillations measured during the MBE of GaAs(0 0 1) at different points along the crystal truncation rod 13L (open circles). The fit (solid curves) is obtained simultaneously for all curves with the layer coverage shown in the top panel of the figure. The substrate temperature was 570 °C and the V/III ratio 13 at the start of the growth interval, 8 at the end of it.
intensity decreases to zero at the minimum of the first oscillation, indicating that only one layer is partially covered at that moment. However, the following maxima do not reach the initial intensity, which means that the next layer starts to grow somewhat before the previous one is completed. The growth is rather close to the ideal, with the next layer containing less than 10% when the previous one is filled to about 90%.
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Fig. 4. X-ray intensity oscillations measured during the MBE of GaAs on GaAs(0 0 1) at different points on the crystal truncation rod 3 1L (open circles). The fit (solid curves) is obtained simultaneously for all curves with the layer coverage shown in the top panel. The substrate temperature was 560 °C, and the V/III ratio was 27, decreasing to 21.
Fig. 4 presents X-ray intensity measurements performed on the CTR 3 1L. The intensities measured at different L are simultaneously fitted to the same layer coverages like in the previous figure. In this case, the growth is two times slower compared to Fig. 3 (the growth period is s ¼ 15 s) and the As coverage is higher. The calculated layer coverages (top part of Fig. 4) show that the second layer starts to grow when the first one is covered to about 80%, while, say, the ninth layer starts to grow when the previous one is only 30% covered. Hence, the damping of the oscillations is fairly sensitive to a continuous roughening of the growing surface. Fig. 5 shows measurements on the CTR 11L (the growth period is s ¼ 6:8 s). One can clearly see the symmetry of the curves with respect to the inphase condition at L ¼ 1 DL, giving identical oscillations with increasing amplitude towards L ¼ 0 and 2 (antiphase conditions) and decreasing towards L ¼ 1 (in-phase condition).
Fig. 5. X-ray intensity oscillations measured during growth of GaAs(0 0 1) at different points on the crystal truncation rod 11L (open circles). The fit (solid curves) is obtained simultaneously for all curves with the layer coverage shown in the top panel. The substrate temperature was 565 °C and the V/III ratio was 14.
Fig. 6 compares the intensities along the CTRs obtained prior to a growth run (completely covered surface layer) and at the first intensity minimum during growth (half-covered surface layers). The central panel collects the results of independent measurements on two equivalent CTRs, 31L and 31L, performed on different samples. The full symbols are the initial intensities obtained before growth by integrating over the x scan. This is the quantity commonly measured and analyzed in surface structure analysis [24,25]. The solid lines are the CTRs calculated by using the atomic coordinates of the GaAs(0 0 1) bð2 4Þ unit cell determined by room-temperature X-ray surface structure analysis [26], and assuming a flat surface without steps. We used the room-temperature coordinates [26] and the bulk Debye–Waller factors for Ga and As (there is no information on Debye– Waller factors for surface atoms in [26]). From the good agreement between the measured and calculated CTRs, we conclude that the positions of
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calculations, which confirms the flatness of the surface. Note, however, that this conclusion is based on the analysis of only four rods, which is a relatively small subset of the typical number of rods used in an X-ray structure analysis. The open symbols in Fig. 6 represent the intensity at the minimum of the first oscillation during growth, scaled to the pre-growth intensity. The first intensity minimum corresponds to the deposition of half a monolayer. Further intensity minima can be deeper because several incompletely covered layers are present simultaneously, cf. Figs. 3–5. The dashed curves in Fig. 6 are calculated with the same assumptions regarding surface flatness and atomic positions in the reconstructed cell as above, but with a 50% covered second layer. The agreement between the measurements and the calculations confirms that the structure of the bð2 4Þ surface reconstruction cell does not change during growth.
5. Theory 5.1. Antiphase conditions for a stepped GaAs(0 0 1) surface
Fig. 6. Measured and calculated intensities along crystal truncation rods prior to growth (full symbols and solid lines) and at the first minimum of the growth intensity oscillations (open symbols and dashed lines).
the atoms in the reconstructed unit cell do not change noticeably between room temperature and the growth temperature. This agreement is reached without introducing a roughness parameter in the
The experimental results presented above show that the maximum sensitivity of the X-ray diffracted intensity to surface steps is achieved at L ¼ 0, when both the incident and the diffracted beams are in the surface plane. On one hand, the sensitivity to surface morphology under these conditions is expected: for example, a bulk-forbidden reflection 310 lies halfway between two bulk reflections 311 and 311 and corresponds to an intensity minimum along the CTR 31L, where the sensitivity to surface roughness is best [25]. On the other hand, this sensitivity is surprising since, at L ¼ 0, the scattering wave vector does not have a component normal to the surface and cannot probe the steps by sampling their height. Hence, it is worth to analyze how the surface steps become visible in an in-plane diffraction geometry. GaAs(0 0 1) grows by incorporating Ga–As bilayers. The bulk GaAs unit cell is formed by two such bilayers, being shifted by a relative translation t ¼ a½0 12 12. Here, a is the bulk lattice period. A
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single surface step is one bilayer in height and hence involves an in-plane translation of half the in-plane lattice period. When two columns (normal to the surface) of bulk unit cells are terminated at adjacent terraces separated by a single step, they are related by the same translation t. The same applies to the reconstructed cells on these terraces. Then, for a reflection Q ¼ ð2p=aÞ½hkL, the structure amplitudes of the two terraces differ by a phase factor expðiUÞ, where U ¼ Q t ¼ pðk þ LÞ:
ð1Þ
Let us consider the reflections with odd h and k, for example 11L, 13L, etc. Then, the phase factor U ¼ p is reached at L ¼ 0 and causes destructive interference of waves diffracted from two adjacent terraces, resulting in the highest sensitivity to surface steps. In other words, the steps are detected due to a lateral displacement of the atoms of half the lattice period, rather than due to the vertical displacement. One could just as well transform one terrace into the other by a translation t0 ¼ a½12 0 12 instead of t, resulting in a phase factor exp ½ipðh þ LÞ. The two phase factors are identical along all CTRs connecting bulk reflections of GaAs, since all bulk reflections hkL have both odd or both even indices h and k. On a given terrace, the (2 4) unit cell of the surface reconstruction can nucleate in eight different positions with respect to a chosen origin. Fractional-order rods are sensitive to these relative shifts and allow the determination of reconstruction domain sizes. For integer order rods, however, the signal from all eight surface unit cell positions adds up in phase, making these rods insensitive to the relative placement of the surface unit cells [27]. This means that integer order rods sample only the terraces, without being sensitive to the size, shape and arrangement of the surface reconstruction domains on them. 5.2. Kinetics of the layer coverages The average surface shape during growth can be described in terms of the coverages hn ðtÞ of successive layers. Growth starts from a perfectly flat surface with h0 ¼ 1 and hn ¼ 0 ðn P 1Þ. During growth, the coverages hn ðtÞ run from 0 to 1, with
hnþ1 < hn . The fraction of the surface at the nth layer level is hn hnþ1 and hence the intensity diffracted by a crystal with the stepped surface is 2
SðqÞ ¼ jFhkL j jðh0 h1 Þ þ ðh1 h2 ÞeiU 2
þ ðh2 h3 Þe2iU þ j ;
ð2Þ
where FhkL is the structure amplitude of a given reflection and the phase U is given by Eq. (1). The validity of the kinematical formula (2) for X-ray scattering is well established, in contrast to RHEED where it may be used only qualitatively. Eq. (2) describes the coherent part of the intensity. We do not consider diffuse intensity here [6,7], since Eq. (2) is sufficient for a complete description of the experimental data presented in the following section. We thus conclude that typical nucleation distances are small, so that the diffuse scattering of the island or hole structure during layer formation is distributed in a wide angular range. This means that a negligible part of diffuse intensity falls within the detector window together with the coherent intensity. When the top layer covers one half of the surface, Eq. (2) gives an intensity minimum equal to Imin =I0 ¼ ð1 þ cos UÞ=2; 2
ð3Þ
where I0 ¼ jFhkL j . The intensity minimum Imin becomes zero when the antiphase condition U ¼ p is fulfilled. It follows from Eq. (1) that on the CTRs hkL with h, k odd, the antiphase condition is reached at L ¼ 0, which provides the most appropriate condition for a surface scattering experiment. We have therefore chosen the rods 11L, 13L, and 31L (the latter two are not equivalent because of the polarity of the GaAs(0 0 1) surface) for the experiments. Models for the time dependence of the layer coverages hn ðtÞ were proposed by Cohen et al. [28]. However, these models are not suitable to fit our experimental data. These mean-field models cannot describe the smooth continuous decay of the observed intensity oscillations. Rather, after a few initial oscillations, they give oscillations with a constant amplitude. The development of an adequate physical model of the layer-by-layer growth process remains a demanding task and is not considered here. Modifications proposed to fit
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X-ray scattering data [11,12] involve the introduction of an explicit dependence of the coefficients in the equations by Cohen et al. on the layer number [11] or time [12], which has no obvious motivation in the MBE deposition process. Trying to separate the analysis of the oscillations from the underlying growth model, we obtain the layer coverages hn ðtÞ from the following fit, which attempts to avoid any assumptions about the physics of the growth process. The time derivative of the layer coverage dhn =dt can be generally written as a difference dhn =dt ¼ ðJn 1 Jn Þ=s
ð4Þ
with Jn depending on the coverages and time. Growth starts at time t0 with hn ¼ 0 and J0 ¼ 1, Jn ¼ 0 (n P 1). Summation of Eq. (4) over all n shows that P1 these equations obey mass conservation: dð n¼1 hn Þ=dt ¼ 1=s, i.e., the total change of the coverage per unit time is equal to the deposition flux 1=s. The simplest growth model, nondiffusive growth [28], assumes that mass transfer between the layers is absent and the adatoms are incorporated into the layer following the one they land onto without moving in the surface plane. This is described by Jn ¼ hn and leads to an exponential intensity decay without oscillations. Ideal layer-by-layer growth is described by dhn =dt equal to the growth rate 1=s in the time interval ðn 1Þs < t < ns and zero elsewhere. In this case, Jn is equal to the Heaviside function H ðt nsÞ. We model the intermediate case of imperfect layer-by-layer growth by a smoothed Heaviside function, Jn ðtÞ ¼ tanh½ðt nsÞ=bn :
ð5Þ
Different coefficients bn 1 and bn allow a different behavior for the onset of the nth layer and its completion. However, an unrestricted fit of all the coefficients bn is not reliable due to the noise of the data. We therefore choose bn ¼ anb , with two fit parameters, a and b, both varying between 0.1 and 0.5 for the data presented below. The fits of the experimental data in Figs. 3–5 use the constant intensity measured 30 s before growth as a normalization factor to the calculated intensity IðqÞ. We conclude from the simultaneous fit of all curves that the structure amplitudes FhkL in
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Eq. (2) do not change during growth. The structure amplitude FhkL includes contributions from the bulk crystal and the reconstructed layer on top of it. A constant FhkL during growth means that the structure of the reconstructed cell is preserved during growth. Remember that on integer order rods, the domain size of the reconstruction or any disorder in the relative positions of the surface unit cells is irrelevant, since all surface unit cells add up in phase. The simultaneous fit of the intensity curves for different L also proves that the reconstruction is established on top of a nucleated layer in a time much smaller than the growth period. 5.3. Layer-by-layer growth on terraced surfaces The model of Eq. (2) is based on the assumption that during growth, the surface is flat with twodimensional islands forming a partially covered layer, possibly with smaller islands on top of them and one-layer deep pits. In particular, it assumes that growth initially starts from a perfectly flat and uniform plane. This is in apparent contradiction with the surface morphology described in Section 3: we have concluded from Figs. 1 and 2 that the surface is vicinal, with either ordered or disordered step trains. Several terraces are coherently illuminated by X-rays, which follows from the observation of the peak splitting due to the ordered staircase in Fig. 2(a) and the peak broadening due to a disordered step train in Fig. 2(b). In such a system, described by the average layer distribution of Eq. (2), the deposition of a fraction of a monolayer in the form of two-dimensional islands is equivalent to the attachment of the same amount of material to nearby steps in the stepflow growth mode. The latter process, however, does not produce intensity oscillations. Why are intensity oscillations characteristic for layerby-layer growth observed on vicinal surfaces with both ordered and disordered steps? To analyze this apparent contradiction, we performed simple numerical X-ray scattering simulations of such surfaces with two-dimensional islands. Scattering from a vicinal surface can be described analytically [7], but the presence of islands makes the problem too complicated for an analytical analysis.
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An appropriate model for a terrace width distribution with variable disorder is the gamma distribution function P ðxÞ ¼ xM 1 e x =CðMÞ, where P ðxÞ is the probability distribution, M is the parameter governing the disorder; and CðMÞ is the gamma function [7]. The parameter M spans the range from a regular staircase of steps, M ¼ 1, to complete disorder (where the width of a terrace is independent of the widths of the adjacent ones), M ¼ 1. Fig. 7(a) shows vicinal surfaces simulated by the gamma distribution with different degrees of ordering: a regular staircase (M ¼ 1, curve 1), a partially disordered staircase (M ¼ 3, curve 2), and a completely disordered staircase (M ¼ 1, curve 3). In a second pass, we cover these surfaces with two-dimensional islands, Fig. 7(b), by creating random steps up followed by steps down, so that a 50% island coverage is obtained. The island
Fig. 7. Vicinal surfaces with the same mean misorientation, but different degrees of terrace order (a), magnified parts of these surfaces with monolayer-height islands deposited to a mean coverage of 50% (b). Diffraction patterns from these surfaces at the antiphase condition L ¼ 0 are shown in (c) and for an intermediate value L ¼ 0:5 in (d). The signal calculated from the terraces is shown by thin lines and the one from half-covered terraces by thick lines. The terrace widths follow the gamma distribution with the parameter M ¼ 1 (regular staircase, curve 1), M ¼ 3 (curve 2), and M ¼ 1 (curve 3).
sizes and distances are also defined by a gamma distribution with M ¼ 3. Fig. 7(c,d) present the diffraction peaks obtained from the terrace distributions of Fig. P 7(a,b) using the Fourier transform IðqÞ ¼ j x exp 2 ½iQhðxÞ þ iqxj , with a subsequent convolution with the resolution function RðqÞ. Here, Q ¼ 2pL=a is the diffraction vector corresponding to a given position L on the diffraction rod. We choose a Gaussian resolution function RðqÞ. The mean terrace width and the width of the resolution function are chosen so that the calculated peaks agree with the experiment, cf. Figs. 7(c) and 2. Thin lines in Fig. 7(c,d) show diffraction peaks obtained from the terrace structures of Fig. 7(a). For the antiphase condition L ¼ 0, the regular staircase gives rise to a split peak (curve 1) [7, 22,23]. When the regularity in the terrace widths decreases, the peaks broaden (curve 2). An exponential distribution of terrace widths gives rise to a Lorentzian peak (curve 3) [7]. The calculated peaks agree with the measured ones, Fig. 2. For L significantly different from zero, only one peak is present, Fig. 7(d). Its width increases and the height decreases with increasing terrace width disorder. Thick lines in Fig. 7(c,d) show the diffraction peaks of the island-covered terraces. The coverage is 50%, a small part of the surface is shown in Fig. 7(b). For L ¼ 0, the islands give rise to diffuse scattering, with the scattered intensity transferred from the coherent peaks to broad diffuse peaks. Then, the intensity at the position of the coherent peak, which is detected by the fixed detector during the growth, decreases almost to zero. For L 6¼ 0, the peak intensity does not decrease to zero. The ratio of the peak intensities for the flat terraces I0 and the terraces half-covered by the islands Imin follow Eq. (3). In particular, for the case L ¼ 0:5, Eq. (3) gives the ratio Imin =I0 ¼ 0:5, which coincides with the ratio of the peak intensities in Fig. 7(d). One can see, by comparing the curves 1, 2 and 3, that the ratio of peak intensities does not depend on the degree of ordering of the terraces. This result shows that the calculation of the intensity diffracted by a surface with partially covered top layers can be performed by Eq. (2) even if the surface is not flat but consists of terraces. Simul-
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taneous coherent illumination of several terraces does not influence the applicability of this equation derived for a flat surface. This conclusion is obviously valid as long as the sizes of the twodimensional islands are small compared with the mean terrace width. Fig. 8 presents a similar simulation for a surface which does not have a mean miscut but consists of terraces defined by random up and down steps. The surfaces in Fig. 8 have uncorrelated up and down steps of equal height and the terrace widths given by the gamma distribution with M ¼ 3 (curve 1) and M ¼ 1 (curve 2). Fig. 8(b) shows magnified pieces of the surfaces with single step high islands deposited on them to a coverage of 50%. Again, the island sizes follow the gamma
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distribution with M ¼ 3, in the same way as in the previous figure. The calculated diffraction peaks, Fig. 8(c,d) are similar to these in Fig. 7(c,d). One can see that the intensity ratio at the position of the coherent peak follows Eq. (3): at the antiphase condition, L ¼ 0, the intensity from the half-covered surface is close to zero while for L ¼ 0:5 the ratio Imin =I0 is 0.5, which again coincides with Eq. (3). We conclude, therefore, that the presence of terraces, the mean width of which is smaller than the X-ray coherence length, does not affect the calculation of the scattered intensities from the layer coverages hn by making use of Eq. (2). This conclusion is valid for ordered or disordered terraces on vicinal or low-index surfaces. 6. Conclusions
Fig. 8. Surfaces with different degrees of terrace order without a mean misorientation (a), magnified parts of these surfaces with monolayer-height islands deposited to a mean coverage of 50% (b), and diffraction patterns from these surfaces at the antiphase condition L ¼ 0 (c) and for an intermediate value L ¼ 0:5 (d). Diffraction patterns from the terraces are shown by thin lines and patterns from terraces half-covered by monolayer height islands are shown by thick lines. The terrace widths follow the gamma distribution with the parameter M ¼ 1 (regular staircase, curve 1), M ¼ 3 (curve 2), and M ¼ 1 (curve 3).
The first results obtained at our combined synchrotron X-ray diffraction/MBE beamline show submonolayer sensitivity of surface diffraction experiments, which allows us to perform a detailed time-resolved analysis of GaAs(0 0 1) homoepitaxial growth. We measure X-ray intensities during growth at different points L along the CTRs 11L, 13L, and 31L. This allows us to extract the time evolution of the layer coverage without assuming a specific model for layer-by-layer growth. The results indicate that no more than three layers are partially covered at any moment. By comparing the intensity evolution at different points L along a CTR, we show that the atomic structure of the bð2 4Þ surface reconstruction unit cell is preserved during growth. No differences to the bð2 4Þ structure determined at room temperature are observed. Using numerical simulations, we determine the terrace size distributions of the surface under typical growth conditions and show that the layer coverage fit during growth is also valid for multilayer starting surfaces, as long as the mean terrace widths are larger than the typical island separations occurring during growth. References [1] J.J. Harris, B.A. Joyce, P.J. Dobson, Surf. Sci. 103 (1981) L90.
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