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Probing the√Buried √ Pb/Si(111) Interface with SPA LEED and STM on Si(111)-Pbr 3 3 M. Yakes† and M. C. Tringides* Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, United States ABSTRACT: High resolution spot profile analysis low energy electron diffraction (SPALEED) and variable temperature scanning tunneling microscopy (STM) have been used to √ √ observe the growth of Pb on the Pb/Si(111)-R 3 3 phase, which is driven by quantum size effects (QSE). A change in the rotation of the Pb grown islands with respect to the Si substrate has been observed with increasing coverage θ. At lower coverage, separated two-step islands are grown and are aligned with the [110] axis of the substrate. With increasing coverage above 1.5 ML, of the islands coalesce and form a bilayer, with additional islands grown on top. The preferred Pb island orientation changes to 5.6° with respect to the [110] direction. These changes at the metal/semiconductor buried interface are obtained both with SPA LEED and STM as changes to the period of the Moire pattern. The method of analysis of the corrugation period and rotation angle of the Moire pattern measured with diffraction and STM can be applied to obtain the structure of buried metal/substrate interfaces in other epitaxial systems.
I. INTRODUCTION Phenomena on the nanoscale can be very different from what is commonly observed in the bulk. In nanostructures with sufficiently small dimensions, electron confinement results in discrete energy levels whose position varies as the nanostructure size is changed. These affects are called quantum size effects (QSE) and have been predicted theoretically.1 Transport2 and photoemission3 experiments measuring the energy dispersion in epitaxially grown films of controllable dimensions have mapped out the dependence of the energy levels on film thickness. Unexpectedly, it was also found that the opposite can be true in some systems: in addition to the island height defining the confined electron levels, metallic islands can be grown of identical “magic” height because the energy of the confined electrons is minimized at the heights where the electron wave function “fits” optimally the confining well.4 The height selection based on QSE offers one of the most robust ways to control the geometry and dimensions of the grown structures. The Pb/ Si(111) system has shown such an intriguing growth mode that is made up of uniform height islands with flat tops and steep edges.5 First principles calculations on free-standing Pb films and mixed models have given more support to this conclusion.6,7 Prior to these experiments, some of the pioneering experiments showing the importance of QSE in film morphology and structure were performed by Prof. Toennies’ group on Pb/ Cu(111) with He diffraction. By monitoring the oscillations of the specular intensity in situ versus time, a modulated intensity pattern was observed, nominally corresponding to single and bilayer growth.8 Stronger intensity was found for bilayer than monolayer film completion and was attributed to the extent the confined electron wave function leaks out in the vacuum with film thickness. For thickness with low misfit, the specular intensity has maxima. Further scattering experiments9 at fixed r 2011 American Chemical Society
deposited θ as a function of the normal component of the momentum transfer (similar to the type of experiments used in ref 5 with SPA LEED) have shown that the extracted layer spacing oscillates as a result of variation of the electron density (and, therefore, Fermi wavelength λF) with film thickness. These pioneering QSE experiments have hinted that the discreteness of the grown nanostructures when their dimensions become comparable to λF can have dramatic effects on their electronic and structural properties. Pb/Cu(111) has also been shown later to have preferred heights with STS.10 Pb/Si(111) height selection have been observed with several experimental probes, that is, in real space, in reciprocal space, in electronic spectra, and experimental parameters were identified so the selected height becomes tunable. Stable heights oscillate with bilayer period, and in all substrates, a superstable height exists for the correct growth “window” of flux F, temperature T, and deposited amount θ. The superstable height (when measured from the wetting layer) depends on the type√ of Si(111) √ interface: 7-layer on Si/(7 7),5 5-layer on Si/Pb-β 3 3,11 and 4-layer on Si/In-(4 1)12). This demonstrates that QSE depend critically on the boundary conditions at the vacuum/ island and the island/substrate interfaces, which implies that accurate atomic and electronic information of the interface are crucial to a complete understanding of the growth mode. This paper focuses on how detailed information can be collected about the buried interface at the bottom of the islands that is not accessible by other techniques, especially the atom distribution Special Issue: J. Peter Toennies Festschrift Received: December 31, 2010 Revised: May 17, 2011 Published: May 17, 2011 7096
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The Journal of Physical Chemistry A within the coincidence supercell formed by the Si substrate layer and the bottom Pb(111) layer. This information is extracted from the period of the corrugation observed on top of the islands as a function of the azimuthal angle, and the results obtained with two complementary techniques, SPA LEED and STM, are in excellent agreement with each other. The island height selection holds up to a temperature of ∼250 K because of the relatively confined electron small energy differences (∼50 meV/atom) as a function of island height, which limits the thermal stability. Because substrate/island orientation changes with coverage can occur in other systems, these results are general and applicable in other epitaxial growth experiments. The corrugation √ √ has been observed on both Si(7 7) and SiPb R 3 3 (noted as R phase) interfaces,13,14 but it has been extensively studied on the latter.1517 The observed corrugation in STM originates from a combination of the geometric relaxation of the layers and an electronic effect related to QSE within the island. The relative contribution of each effect has been discussed in ref 17. Regardless of which effect is the one mapping interface information to the corrugation at the island top, it is a result of the Moire pattern derived from the lattice mismatch of the Pb(1 1) lattice and the Si(1 1) lattice (the 7 7 reconstruction is removed when the R phase is formed). The Moire pattern and observed corrugation of these islands has also been used as a template to grow self-organized Ag clusters on top of Pb islands.18 In previous experiments with SPA-LEED, the orientation of the Pb lattice relative to Si(111) has been observed as well as a change in the size and direction of the Moire pattern in STM.15,16 In STM experiments, it is difficult to obtain reliable statistics on the orientation of the Pb lattice, but the size and direction of the Moire pattern makes this easier. With the ability to illuminate and probe macroscopic distances (∼0.5 mm), diffraction is an ideal method for obtaining this type of data if the features of the diffraction pattern can be properly identified. A rotation of the Pb lattice relative to the Si substrate has been observed with LEED for islands grown in StranskiKrastanov growth mode at room temperature and coverage θ > 3 ML.19 The observed rotation was 6° for overlayers on Si(111)-7 7 √ Pb√ and 3° for overlayers on Pb- 3 √ 3-R √ phase. In ref 19, the dense (θ ∼ 4/3 ML) Si(111)-Pb 3 3 is referred to as the β phase; however, in recent literature, the convention is to refer to it as the R phase. For QSE islands grown on the 7 7, a rotation of 5.6° has been observed with SPA-LEED.20 For the Si(111)-7 7 phase, this rotation was explained19 by the geometric coincidence of a rigid Pb lattice and the underlying substrate. This coincidence lattice was found geometrically by comparing lengths of a 2-d vector constructed from integer multiples of the substrate unit cell vectors to the length of a vector that is an integer multiple of the Pb(111) lattice constant. If the length of the substrate vector matched the Pb vector within 3%, it was deemed a coincidence lattice and a possible candidate for the Pb rotation vector. Comparing with the experimental results, it yielded a match to the Si(111)-7 phase. However, the 3° √ 7√ overlayer rotation of the Pb- 3 3-R phase was not matched. Later, √ X-ray √ diffraction data indicate that Pb deposited on the 3 3 phase destroys the R phase and leaves an unreconstructed (1 1) Si surface,21,22 so the comparison should be made between the Si(1 1) and Pb(1 1) lattices. The coincidence of the Pb lattice with the substrate is an example of a coincident site lattice (CSL). CSLs are often seen at
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grain boundaries in polycrystalline materials. Many heteroepitaxial rotations occur such that a CSL is created between the substrate and the overlayer. The coincident site lattice is derived from two lattices that have different lattice constants, one that is fixed and one that is rotated. For a given angle, periodically, one lattice point of the rotated lattice is located close to a lattice point of the second lattice. These points of closest approach form a new lattice on the surface. For heteroepitaxial systems where the two lattices do not have the same lattice constant, in general, one lattice must be strained by a small amount to achieve perfect coincidence. Because these coincidence points are assumed to have lower strain energy than points far from coincidence, for small CSL cell sizes, the overall strain energy of the interface is reduced. While for any two given lattices there are an infinite number of possible CSLs, the observed lattice should have a small CSL period and small strain. Because CSL theory is a geometric theory and cannot account for atomic relaxations or interatomic potentials, it is not a universal theory and does not account for all overlayer rotations,2325 as seen in the mixed success of describing Pb/Si overlayers grown at room temperature.19 The rotation of the overlayer lattice relative to the substrate is a common effect in heteroepitaxial systems. Overlayer rotations often appear in systems with large lattice mismatches between the overlayer and the substrate. For example, overlayer rotations are seen in systems with rare gases physisorbed on graphite,26 metal on metal systems,27,28 and metal-oxides on metal.29 In these examples, rotations occur for both monolayer and multilayer surface coverages. However, in contrast to the case of Pb island growth that occurs at T ∼ 200 K, all of these examples are in systems with temperatures greater than 500 K. Complete overlayers are not required to see rotation. For Ag islands annealed to 480 K on H-terminated Si(111), it was observed that the rotation of the Ag lattice changed depending on the island size.30,31 As the coverage, size, and shape of the island changes, the rotation angle that gave the lowest interface energy also varied. With increasing Ag island size, the islands became better aligned with the CSL energy minimum. In this paper we identify the origin of the unusual diffraction patterns and STM images seen for Pb growth on the R phase and discuss their evolution with Pb coverage and film morphology. From these observations it is concluded that, with θ, the preferred island orientation increases and the initial island morphology changes from 2-layer islands to 2-layer islands plus 2-layer islands on top. The analysis is used to deduce the changes at the Pb/Si interface, which is not easily accessible with other techniques. The excellent agreement between the STM and SPA-LEED results points to the validity and universality of the effects discussed.
II. EXPERIMENT Experiments were performed in two separate UHV chambers with a base pressure of 2 1011 Torr. One chamber is equipped with an Omicron SPA-LEED system, Auger spectrometer, and mass spectrometer. The other chamber is equipped with an Omicron variable temperature STM and a conventional LEED system. In each experiment, the starting point was the Si(111) (7 7) phase prepared by cooling slowly after flashing the crystal to 1250 °C. The R-phase was formed by depositing slightly above 1.3 ML onto the clean Si 7 7 at low temperature and annealed for 15 min to 500 K so only 1.3 ML remains with the R-phase reconstruction. The sample was then cooled to 120 K followed by heating to 160200 K for additional Pb deposition for the Pb 7097
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Figure 1. Total of 0.6 ML deposited on the R-phase at 195 K. 2-D 40 40% BZ scans (a) centered on (00) beam showing Pb cluster spots. (b) Same experimental conditions as (a) centered on Si (10) beam showing no rotation of the Pb cluster spots at this coverage.
island growth and the SPA-LEED and STM experiments. Various coverages ranging from 0.6 to 3 ML were added to the R-phase at a flux rate of 0.2 ML/min.
III. RESULTS Figures 1 and 2 show 2-d diffraction patterns near the (00) and Pb(10) spots observed after Pb deposition on top of the R at 195 K. Each scan covers 40% of the Si Brillioun Zone (BZ). Figure 1a,b was obtained after 0.6 ML of Pb is deposited for a total of 1.9 ML. There are six weak spots located 15% BZ from the (00) spot along the six equivalent [110] directions. The 15% BZ value corresponds to a real space distance of 26 Å. It is known at this coverage and deposition temperature that Pb nanoclusters form32 and STM indicates that these clusters have triangular units of size 27 Å. The spots are broad, with a width of 6 ( 1% along [110], indicating an average domain size of 65 ( 10 Å. These nanoclusters are formed on R before island growth, consistent with the current observation of a very weak, nonrotated spot in Figure.1b. As more Pb is added, 2-layer islands begin to form. Figure 2a and c were taken for a surface with 1.3 ML deposited on R for 2.6 ML total. There are six spots located 11% BZ from the (00) spot along the [110] directions that are clearly distinct from the previous nanocluster spots at 15% BZ. This 11% spot position corresponds to a real space distance of 35 Å. We refer to these six spots as the hexagon pattern. These spots are elongated along the [112] direction and with their maximum on the [110] axis. With 1.8 ML deposited on R (3.1 ML total), the star pattern is seen in Figure 2b. In the star pattern, the sides of the hexagon are extended along the [112] direction to form a six-point star.16 The diffraction pattern has similar intensity at 11% BZ as Figure 2a, but the hexagon spots now appear to be extended along the [112] direction to form the six-point star pattern. Instead of a continuous distribution of intensity along the [112] direction, the intensity now has three lobes, one at the same position as the hexagon intensity and two new lobes near the corners of the star. Figure 2d shows the intensity distribution of the Pb(10) spot at the same 1.8 ML coverage.
STM images verify the island morphologies at coverages corresponding to the hexagon and star patterns. Two STM images at representative coverages are shown with Figure 3a over an area 800 800 Å2 with 1.0 ML of Pb deposited on R at 160 K. At this coverage, the hexagon pattern of Figure 2a is present. The surface is primarily covered with 2-layer islands. There are also small Si islands (seen as smooth dark orange patches) and small patches of Pb nanoclusters (seen as textured dark orange patches). There are two insets that have the contrast enhanced to show the corrugation patterns; Figure 3c,ds show the same images without the corrugation enhancement. There are two types of corrugation patterns on the surface that can be identified by the relative contrast of the holes in the Moire pattern, as has been discussed elsewhere.15 The corrugation lengths are measured as 35 and 36 Å, for the left and right insets, respectively. The orientation of the corrugation is measured as 4.5° away from the [110] direction for the left inset and 5.4° away from the [110] for the right inset. Figure 3b shows a 500 500 Å2 image with 1.5 ML of Pb deposited on R at 180 K. At this coverage, the star pattern appears in the diffraction images. The islands have grown together and now cover the surface, resembling large “continents”. On top of the second layer bilayer islands are grown. Corrugation patterns can be seen both on the second layer island as well as the islands nucleated on top. As in Figure 3a, portions of the second layer have enhanced contrast to show more clearly the corrugation and clear changes in length and orientation. The corrugation periods are measured as 29 Å for the left inset, 25 Å for the upper right inset, and 35 Å for the lower right inset. The rotation angles away from the [110] direction are 24.0° for the left inset, 3.6° for the upper right inset, and 0.5° for the lower right inset. The change with coverage of the diffraction pattern in Figure 2 and corrugation in Figure 3 are in excellent agreement with each other. The change is clearly visible for coverage larger than 1.5 ML. With SPA LEED, the surface morphology can be investigated using g(s) curves. The g(s) curve is derived from the intensity ratio of the sharp and broad components of the (00) spot as a function of normalized momentum transfer, s = Kn/(2π/d), where Kn is the normal component of the momentum transfer. This ratio oscillates as s changes from in-phase (the scattering 7098
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Figure 2. Change of the diffraction pattern from hexagon to star-like as a function of deposited Pb on the R-phase at 195 K: (ac) 1.3 ML and (bd) 1.8 ML; (a) centered on (00) beam showing hexagon spots; (b) the hexagon transforms to the star pattern close to (00); (c) centered on Si (10) beam showing Pb(10) spots with maximum intensity in the [110] direction; and (d) lobes developed of the Pb(10) spot indicating a preferred 5.6° rotation angle between the two lattices. The white arrows show the direction of the 1-d scans in Figures 5 and 6.
from the island top and the in between region leads to constructive interference) to out-of-phase conditions (the scattering from the island top and the in between region leads to destructive interference). The period of the oscillations of the g(s) curve is inversely proportional to the island height.5 The g(s) curve for a sample with 1.1 ML Pb on top of R is shown in Figure 4a. At this θ, the hexagon diffraction pattern is seen near the (00) spot. The primary oscillation period of 1/2 is clearly visible, meaning that the surface is predominantly covered with 2-layer islands, consistent with the STM results at this θ. Figure 4b shows the g(s) curve for a surface covered with 1.7 ML of Pb on top of R. At this θ, the star diffraction pattern is seen near the (00) spot. In Figure 4b, the 2-layer island oscillation has decreased in intensity as compared to Figure 4a, indicating that 2-layer islands are no longer the primary feature on the surface. The g(s) curve shows weaker oscillations of shorter period and amplitude corresponding to larger height islands in addition to the 2-layer ones. This is consistent with the STM results at this coverage, which show 2-layer islands coalescing into “continents”, with mostly bilayer islands growing on top of the “continents”.
Additional information can be determined using 1-D profiles of the (00) and Pb(10) spots. Figure 5 shows a series of 1-D scans taken at two coverages, again showing the major features of the 2-D diffraction patterns discussed earlier. 1-D scans provide more detail about the distribution of intensity along the sides of the hexagon and star, as well as the shape of the Pb(10) spots. Each scan was taken along the [112] direction centered at the maximum spot intensity on the [110] direction. These directions are shown by the white arrows in Figure 2a,c. Figure 5a and b are obtained from a surface with 1.2 ML on top of R (where the hexagon pattern is seen) and Figure 5c and d are from a surface with 1.9 ML on top of R (where the star pattern is seen). For the corrugation spots close to (00) in Figure 5a,c, the position of the satellite spots at 11% can be used to deduce the corrugation angle φ. The conversion is given by tan φ = (x)/(11.0), where x is the %BZ distance along the [112] direction away from the maximum of the hexagon spot. The reasons for converting the Brillioun Zone measurement to an angle will become clear in the discussion. Figure 5b and d (again corresponding to the hexagon and star patterns) are scans of the Pb (10) spot taken along the [112] 7099
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Figure 3. STM images showing island morphologies at different Pb coverage.(a,b) The contrast in the second layer has been enhanced in the inset sections to show the corrugation pattern more clearly. (a,c) 800 800 Å2 scan at 1.0 ML coverage on top of the R-phase deposited at 160 K. The island corrugation is visible on the 2-layer islands. There is a small fraction of Pb nanoclusters still present between the islands. The corrugation lengths are measured as 35 and 36 Å, for the left and right inset, respectively. (b,d) 500 500 Å2 image with 1.5 ML coverage deposited at 185 K. The 2-step islands have coalesced to form “continents”. The 2-layer islands have grown on top of the nearly complete bilayer. The corrugation lengths are measured as 29 Å for the left inset, 25 Å for the upper right inset, and 35 Å for the lower right inset. (a,b) The enhanced corrugation is used to measure the relative orientation angle φ with better accuracy than 0.4°.
centered maximum intensity on the [110] direction. To convert this % BZ measurement to the angle of rotation of the Pb lattice, we use the conversion tan θ = (x)/(109.7), where x is along the [112] direction of the Pb (10) spot. Unlike the spots closer to the (00) spot, which will be shown in the Discussion, to measure the size and orientation of the island corrugation, the Pb(10) spot can be used to directly measure the relative orientation of the Pb lattice with respect to the Si lattice. The evolution of the Pb(10) profile along the [112] direction with deposited coverage is shown in Figure 6. The evolution of island orientation with coverage can be seen by comparing the peak intensity at the center 0% BZ (nonrotated Pb islands) with the satellite spots on either side of the maximum (their intensity is a measure of the fraction of islands that have rotated). At the lowest coverage, θ = 0.6 ML, where only nanoclusters and small 2-layer Pb islands form (Figure.1b), the weak peak intensity of the centered spot is more than 8 times the satellite spots (indicating a
small fraction of nonrotated Pb islands coexisting with nanoclusters). As θ increases, the ratio of the peak intensities of the centered to the satellite spots decreases, indicating a larger number of Pb islands rotated off the [110] (a small shift of the wavevector position of the satellite spots shows a slight decrease of the rotation angle). At the highest deposited θ of 2.7 ML, the peak intensity of the satellite spots is approximately 50% higher than the spot at the center, consistent with many rotated islands (and the change of the corrugation angle and period seen in Figure 3b).
IV. DISCUSSION To connect the rotation of the Pb lattice with the period and orientation of the island corrugation, we created a number of hard-ball models, each with a different relative orientation of the Pb lattice with respect to the Si lattice. For these models, we used bulk lattice constants with either 1011 or 910 Si/Pb ratios, 7100
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Figure 4. G(s) curve showing 2-step intensity oscillations for Pb islands. (a) Surface prepared by depositing 1.1 ML onto the R-phase at 195 K. The beating period of 1/2 of s = Kn/(2π/d) indicates that the surface is covered primarily with 2-layer islands. (b) Surface prepared by depositing 1.7 ML onto the R phase at 195 K. The higher order periods and the reduction in the amplitude of the 2-step oscillations indicate that 2-layer islands form on top of the 2-layer islands.
which correspond to corrugation periods of 38.4 and 35 Å, respectively, using 3.5 Å for Pb and 3.84 Å for Si lattice constants. (A compressed Pb lattice constant of 3.46 Å was also used because the position of the Pb(10) spot can be slightly larger than the 109.7% BZ corresponding to the ideal 3.5 Å lattice constant.) Each angle of the rotation of the Pb lattice produces different Moire patterns and different corrugation lengths for the two overlaid lattices. The model only includes the beating between the Pb first layer and the Si lattice because the positions of the atoms in the additional Pb layers follow the same beating periodicity. For each model we define the angle θ to be the rotation of the Pb lattice away from the [110] direction (it will be clear from the context when θ refers to the angle and not the coverage). We measure two quantities, an angle φ, which is the angle of rotation of the corrugation lattice away from the [110] direction, and the corrugation length, which is the average over at least six coinciding locations in the Moire pattern, where the position of an atom in the Pb lattice coincides with position of an atom in the Si lattice. We estimate the error over the length of the measurement to be (0.2 Å for the corrugation length and (0.4° degree for φ. An example is shown in Figure 7, with the Pb lattice rotated by θ = 9° with respect to the Si lattice. Only part of the system used for the calculation is shown in the figure. The corrugation unit cell is highlighted to show the orientation of the Pb lattice with respect to the Si lattice. By visual inspection, one can see that the resulting Moire pattern has a corrugation length 20.0 Å (which is
Figure 5. 1-D scans along the [112] direction obtained after deposition at 195 K showing spot profiles of the corrugation spot close to (00) and the Pb spot near (10). Spot positions given in %BZ are centered at the [110] axis. Scans (a) and (b) show 1.2 ML is deposited on the R-phase (when the hexagon spots are present), while scans (c) and (d) are after 1.9 ML is deposited on the R-phase (when the star spots are present).
roughly half the length of the nonrotated coincidence lattice 38.5 Å for θ = 0° and is oriented at φ = 3.6° off [110]). Similar measurements of the beating periodicity and orientation were made with Pb lattice rotations of 0° < θ < 12° at 1° intervals. Due to the symmetry of the Pb and Si lattice in the first layer, the results for CCL and CL rotations of the Pb lattice yield the same results of corrugation length and angle but in the opposite sense. The results are tabulated in Table 1 for both Pb corrugation unit cell sizes (11 and 10 lattice constants). Reciprocal space lengths were calculated using Q = 2π/L, where Q is the length of the lattice vector for each corrugation unit cell in reciprocal space and L is the length of the corrugation length in the hard ball model. These values are then normalized to the Si Brillioun Zone, where 2π/aSi = 100% BZ and aSi = 3.84 Å. A corrugation pattern with period L and orientation φ generates a reciprocal space wavevector of length Q at the same orientation φ from the [110] reciprocal space direction. Because a range of rotation angles θ generates a range of periods L(θ) and rotation angles φ(θ) the intensity distribution in reciprocal space Q(φ) is a measure of the range of angles θ present. φ(θ) has the 7101
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Table 1. Measurements Done on Hard Ball Rotated Pb Models To Relate the Moire Pattern Parameters L, φ to the Rotation Angle θa θ, deg
L
0
38.4
10.0
0.0
0.0
1
37.6
10.2
11.4
11.4
2
35.9
10.7
15.2
15.2
3 4
33.7 30.9
11.4 12.4
29.6 21.7
30.4 38.3
5
28.6
13.4
14.7
45.3
6
25.8
14.9
9.7
50.3
7
24.0
16.0
5.2
54.8
8
21.7
17.7
0.0
60.0
9
20.0
19.2
3.6
63.6
10
18.5
20.7
6.7
66.7
11 12
17.2 16.1
22.3 23.8
9.4 11.9
69.4 71.9
0
34.5
0.0
0.0
1
34.2
11.2
10.3
10.3
2
33.3
11.5
19.6
19.6
3
30.8
12.4
27.3
27.3
4
29.2
13.2
24.1
35.9
5 6
26.8 24.6
14.3 15.6
18.0 11.8
42.0 48.2
7
22.4
17.1
6.9
53.1
8
20.9
18.4
3.7
56.3
9
19.5
19.7
1.7
61.7
10
18.0
21.3
3.7
63.7
11
16.6
23.2
6.6
66.6
12
15.6
24.7
9.3
69.3
100(as/L)(%BZ)
j, deg
j þ 60, deg
11 Pb Unit cells
Figure 6. Evolution of the Pb(10) spot with coverage 0.62.7 ML on top of the R-phase at 195 K; 40% BZ scans taken along [112] direction centered at the Pb spot maximum along the [110] direction. As coverage increases, the intensity at the satellite spots increases, indicating that the fraction of rotated islands is larger.
10 Pb Unit cells 11.1
a
Figure 7. Hard ball model of the rotated Pb lattice with respect to the Si substrate to relate the angle of rotation θ = 9° to the corrugation period and the rotation angle of the corrugation φ = 3.6°. The variation of L and φ is found by averaging several consecutive locations (at least six) where the Pb and Si lattice positions coincide.
6-fold symmetry of the substrate to give an equivalent rotation of the corrugation lattice for θ and θ þ 60°. For all rotations, the length in real space of the corrugation unit cell is longer for the 38.5 Å model (11 11) than the 34.6 Å model (10 10). It is also possible to determine the expected Moire pattern analytically.33 To calculate the expected Moire pattern of two hexagonal lattices with lattice constants as for the substrate and aover for the overlayer and with the relative rotation angle θ, the range 30° < θ < 30° is used. For each rotation the corrugation length L and orientation angle φ are determined . For aover ∼ as, the differences of the inverse two vectors corresponding to the two lattices are used to construct the Moire lattice period. In general if the two vectors are rotated by angles θover and θs (with respect to a reference axis) these differences are u = (1/aover) cos θover (1/as) cos θs, and v = (1/aover) sin θover (1/as) sin θs. The period and orientation of the Moire pattern are given as L1 = (u2 þ v2)1/2 and tan φ = (u/v). For the PbSi system taking θover = θ, θs = 0, aover = 3.5 Å, and
L refers to the average distance between coinciding atom positions of the Moire pattern and is used to determine the wavevector Q at corrugation angle φ. The angle φ is the angle of the corrugation unit cell direction relative to the [110] axis. The values over 0 < θ < 30° are used to repeat the L, φ parameters over the full range 0 < θ < 360° (denoted by φ*) because of the 6-fold symmetry.
as = 3.84 Å, we have ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2ffi 1 1 1 2 1 ¼ cosðθÞ þ sinðθÞ L aover as aover ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2 1 2 ¼ þ cosðθÞ aover as aover as 0 1 1 sinðθÞ B aover C C φ ¼ arctanB @ 1 1A cosðθÞ aover as As |ao as| becomes larger, higher order linear periods exist than the simple linear combination defining u and v. Therefore, this is not a general formula for any overlayer/substrate combination. However, for the case of Pb on Si, as will be seen, it matches our models and experimental data because the difference of the two lattice constants is 9%. 7102
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Figure 8. Results of rotation equations for Pb lattice rotations of 0° < θ < 30° using the analytical equation. (a) Corrugation angle φ vs θ. (b) Corrugation length L vs θ. The rotation angle φ increases and the corrugation length decreases as the lattice rotation θ increases.
Figure 8a shows the corrugation angle φ versus lattice rotation θ. When the Pb and Si lattices are aligned the corrugation pattern is also along the [110] direction. As the Pb lattice is rotated, the corrugation lattice rotates by a larger angle. The 11:10 PbSi corrugation pattern always rotates more than the 10:9 PbSi pattern. Figure 8b shows the corrugation length L versus lattice rotation θ. As θ increases, the corrugation length is reduced, and as expected, the 11:10 PbSi corrugation pattern always has a longer length than the 10:9 PbSi pattern (for the same θ). We compare the diffraction results of Figures 2 and 5 with the values calculated in the models. As noted before the angle φ in reciprocal space (for the vector Q) is the same as the real space orientation angle φ of the corrugation lattice. The calculated intensity must have 6-fold symmetry so rotations by 60° should produce the same diffracted intensity. For the results of Figure 9 (obtained for the 11:10 PbSi unit cell models), a uniform distribution is assumed such that each relative lattice orientation θ gives the same intensity at reciprocal space location (Q,φ). This analysis explains the origin of the hexagon and star patterns in Figure 2 and connects the SPA LEED and STM data. By limiting the allowed Pb rotation angles, the hexagon and star patterns are produced. Figure 9a shows only the models with 0° < θ < 3° and reproduces the hexagon pattern. Figure 9b shows the models from 0° < θ < 8° models and reproduces the star pattern. The models and the observed corrugation on the STM images can be used to determine the island orientation in STM images. For Figure 3a, the measured corrugations and angle φ were 35 Å and 4.5° for the left inset and 36 Å and 5.4° for the right inset. From the model of Figure 8 or Table 1, the rotation angle θ is less than 1° for each case. This is consistent with the hexagon distribution observed in the diffraction pattern. For Figure 3b, the measured
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Figure 9. Reciprocal space lattice positions of the corrugation wavector Q at rotation angle φ with respect to the Si BZ. (a) Pb model rotations 0° < θ < 3° degrees reproduce the hexagon pattern close to the (00) spot. (b) Pb model rotations 0° < θ < 8° degrees reproduce the star pattern close to the (00) spot.
corrugations and angle φ were 29 Å and 24° (or, equivalently, φ = 36°) for the left inset, 25 Å and 3.6° (or, equivalently, φ = 56.4°) for the upper right inset, and 35 Å and 0.5° for the lower right inset. Because the direction θ of the Pb lattice rotation cannot be easily determined from the image, φ is estimated from the model with much higher accuracy. The corrugation patterns from Figure 8b are consistent with lattice rotations θ of approximately 4, 6, and 0°, respectively. Because two patterns show θ larger than 3°, this is consistent with the star reciprocal space intensity distribution. The cause of rotation of the overlayers for many heteroepitaxial systems is the reduction of energy at the interface. The simplest explanation would be to look for a Coincidence Site Lattice (CSL) between the Si(1 1) and Pb(1 1) substrates with rotation near θ = 5.6° (the preferred orientation for the star), as done in ref 19. We performed similar calculations, shown in Table 2, to determine if the CSL was a viable explanation for the overlayer rotation. A direction in the substrate lattice is defined by two vectors a1 = n1aSi, a2 = n2aSi (with n1, n2 integers). The angle of the diagonal dg of the parallelogram with sides a1,a2 was tested as the candidate angle for the rotation θ of the Pb(111) overlayer, if (dg/aover) is close to an integer multiple N of aover the overlayer lattice constant. The strain was calculated from the fractional deficit sr = (dg N)/N. If the strain is less than (3% and the length dg is less than the unrotated length for coverage θ = 0 ML (dg = 11 for the 11 11 lattice), the model is allowed and included in Table 2. When this criterion is used, the smallest value of the rotation angle is θ = 7.6°, which is larger from the observed 5.6° rotation. The interface energy at the overlayer/substrate interface is a reasonable cause of the rotation with increasing coverage; the discrepancy with the simple estimate above reconfirms the simplicity 7103
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Table 2. Possible CSL Models Derived from the Overlapping Si(1 1) and Pb(1 1) Lattices at the Interfacea n2
θ, deg
dg/aover
N
strain (%)
10
0
0.00
10.97
11
0.26
6 5
1 1
7.59 8.95
7.19 6.11
7 6
2.78 1.81
4
1
10.89
5.03
5
0.55 0.22
n1
7
2
12.22
8.98
9
3
1
13.90
3.96
4
1.11
8
3
15.30
10.81
11
1.77
5
2
16.10
6.85
7
2.12
7
3
17.00
9.75
10
2.48
n1 and n2 are the candidate lattice vectors of Si, θ is the rotation angle away from [110], dg is the CSL period on the Si lattice, N is the CSL period as integer multiples of the overlayer lattice constant, aover, and strain is the fractional misfit sr = (dg N)/N. Only rotations with misfit less than |sr| < 3% (in absolute value) are shown with the smallest rotation angle found 7.6° (which is slightly larger than the 5.6° rotation observed with the star-like pattern close to the (00) spot). a
of the CSL model (and the limitations of the underlying hard ball assumption in CSL). More accurate theoretical first principles calculations of the strain energy with respect to island size and rotation are required. Clearly the variation of the Pb overlayer rotation with deposited Pb amount may provide additional tunable parameters for QSE driven nanostructure growth.
V. CONCLUSIONS A change in the orientation of the grown Pb islands relative to the Si substrate with increasing coverage has been observed for growth on the R phase. This rotation is probed from the Moire pattern generated by the two overlapping lattices at the interface observed with two complementary techniques:SPA LEED and STM. From the measured reciprocal diffracted intensity distribution that generates hexagonal or star like patterns close to the (00) and from the lobes of the Pb(10) spot, the rotation angle; θ is extracted. This is further confirmed from the corrugation (the period L and rotation angle φ) projected at the island top (generated by the confined electron states within the island) and the model developed, which relates the two measured quantities L,φ to the angle θ. For low Pb coverage, the Pb islands are mostly aligned with the [110] axis of the Si substrate. At higher Pb coverage, 2-layer islands coalesce to a continuous film with bilayer islands grown on top. At this coverage, the islands prefer a rotation of 5.6° relative to the substrate, the corrugation period in STM images decreases and the star-like pattern develops close to the (00) spot. Because buried interfaces are difficult to probe experimentally the success of the current investigation, the excellent agreement between the results obtained with two techniques and the model connecting measured parameters and rotation angles can be applied in other systems when information about the buried information is of interest . ’ AUTHOR INFORMATION
Present Addresses †
U.S. Naval Research Laboratory, 4555 Overlook Avenue SW, Washington, DC, 20375.
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