Reflection High Energy Electron Diffraction Studies of the Dynamics of Molecular Beam Epitaxy

8

Reflection High Energy Electron Diffraction Studies of the Dynamics of Molecular Beam Epitax Philip L Cohen, Gale S. Petrich, and Gregory J. Whaley

1.0

INTRODUCTION

The success of molecular beam epitaxy is largely due to its compatibility with in-situ surface characterization. Most other crystal growth environments require ambient pressures or fields too hostile to apply the electron probes developed for the study of surfaces. The vacuum requirements and geometry of MBE, however, conveniently permit the use of reflection high-energy electron diffraction (RHEED). Only an electron gun and a phosphor-covered viewport are needed for these measurements. With this simple apparatus, it is easy to obtain important information about the character of the surface and the nature of the growth mode. From the symmetry and separations of the diffracted beams, one can determine the surface reconstruction and lattice constant. From the disappearance of reconstructions, one can determine flux ratios. From phase transitions, one can calibrate temperature. From the qualitative appearance of the pattern, cluster growth or layer growth can be distinguished. But far more information can be obtained by careful analysis of the diffraction. A difficulty is that, at this next level of interpretation, the patterns are quite complicated and not completely understood. The purpose here is to describe current progress in the use of quantitative electron diffraction measurements to understand the dynamics of the microscopic processes of epitaxial growth.

669

670

Molecular Beam Epitaxy

RHEED is primarily sensitive to surface atomic structure. The absolute diffracted intensities depend on the positions of the surface atoms. The relative intensities and the shape of the diffracted beams depend more upon surface morphology, i.e., the surface order and distribution of atomic steps. The first of these is just beginning to be attacked[ 2]-[s][2~] with success. It is based on the first dynamical calculations appropriate to RHEED.[ s] It is a hard problem to which there is little data that theory can compare. The second is the main focus of this chapter. Diffraction suffers from the disadvantage that it is not as direct as microscopy. Other techniques, such as reflection electron microscopy, [7][8] scanning tunneling microscopy[ 9] and low-energy electron microscopyP ~ are able to image the same steps that can cause streaks in a RHEED pattern. Diffraction has the distinct advantage that it measures statistics and does not follow an individual step. To understand epitaxial growth, the latter is required, though both would be exceedingly useful. Unfortunately, heroic measures are required to apply these high resolution imaging techniques to the growth of a complicated film at elevated temperatures. Since the main application of RHEED has been to the growth of semiconductors, this dominates our discussion. Two kinds of surfaces will be distinguished: low-index or singular surfaces, and vicinal surfaces. In the next two sections, the fundamentals of electron diffraction from these two types of surfaces are summarized. The special diffraction features particular to RHEED are explained. In the fourth section, measurements during the growth of GaAs and AIGaAs are discussed in terms of these fundamentals. To avoid artifacts due to multiple scattering, it are essential to compare to trends at several incident angles. In the fifth section, RHEED measurements on a lattice mismatched system are reviewed. Finally these measurements are discussed in terms of simple mathematical models of epitaxy. Our main theme is that at the growth rates important to molecular beam epitaxy, all of these processes demand consideration of the role of both 2D cluster formation and step propagation.

2.0

DIFFRACTION GEOMETRY

In a typical MBE, application a 10-20 keV beam strikes a sample at a glancing angle of less than 5~ so as not to block the Knudsen cells or gas

RHEED Studies of the Dynamics of MBE

671

sources used to provide the incident fluxes. At these high energies, stray magnetic fields from the Knudsen cells and sample heater have little effect on the electron trajectories. Figure I shows the scattering geometry. The incident and final glancing angles, Oi, ~ and the incident azimuthal angle, q~ are indicated. Three electron trajectories are also shown, the specular beam, the part of the incident beam that misses the sample, and the shadow edge that corresponds to the locus of low-energy secondaries that can just leave the surface. We use the convention that all glancing angles, 8, are measured with respect to the low-index plane and not the macroscopic surface plane. This distinction will be important for diffraction from a vicinal surface. For now, it suffices to recognize that, by measuring the position of the primary and specular beams, the angle of incidence can be determined. RHEED gun I0 keY

U

Li

ovens

sample motion

'~~

8f

J

conetic shield

....... phosphor scr xy = = st age

I

_

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J::j B fringi ng field

~] slit lensaperture photom uIt i plier

data acquisition Figure 1. Schematic of a diffraction apparatus used during MBE. The specular, shadow edge, and primary electron trajectories are shown. Angles are defined with respect to the low-index surface plane.

672

Molecular Beam Epitaxy

There has not been a systematic effort to determine the relative merits of increasing or reducing the electron energy. It might be that kinematic arguments are more successful at lower energy. Higher energy is better suited to systems with e-beam deposition. Energies as low as 3 keV can more easily be combined with simultaneous Auger spectroscopy.[1~] At this point, all we can say is that reducing the energy below about 5 keV degrades the operation of our electron gun. Going to higher energy spatially compresses the pattern, as the square root of the energy. The resulting diffraction pattern is shown in Fig. 2 for a GaAs(100) surface that has been annealed in an As 4 flux. This is the characteristic pattern from a well-ordered surface and consists of a set of sharp beams arrayed along a circle of radius k sin Oi. The primary beam is also evident, though its intensity has been reduced through partial occlusion by a shutter. This is the fourfold pattern of the 2 x 4 GaAs(100) reconstruction. Every fourth beam would be present even without the reconstruction and are termed integral order beams. The additional beams due to the super period are termed fractional orderbeams. Because reconstructions do not always cover entire terraces formed by steps, there can be relative displacements between reconstructed domains and hence scattering phase differences between the reconstructions over the surface. The direct consequence is that the intensities, rather than amplitudes, diffracted from the domains must be added incoherently. Thus the behavior of the beam shape and the time development of the intensities of the fractional order beams and integral order beams can be very different.[ ~2] In this discussion, only the behavior of the integral order beams is described. If growth is initiated by opening the Ga shutter, then the sharp diffraction pattern of Fig. 2 becomes the pattern shown in Fig. 3. The sharp beams are now elongated perpendicular to the surface with only slight changes in their width parallel to the surface. The latter is more evident if the intensity is measured. The intensity at the peak positions also changes. In fact, the intensity of the specular and other integral order beams often oscillates in time with a period corresponding to the growth of a monolayer of GaAs.[ 13][14] These changes in shape and intensity are the main subject of this discussion. Other broadening mechanisms are also observed. For example, in the twofold GaAs(100) pattern, antiphase mistakes elongate the half order streaks[~S][16] and, with some multiple scattering, the integral order streaks.

RHEED Studies of the Dynamics of MBE

673

Figure 2. Photograph of the four-fold diffraction pattern of a well annealed 2 x 4GaAs(100) surface. The sample is at growth temperature (600~ in an As 4 flux.

Figure 3. After growth is initiated on GaAs(100), the sharp beams of the four-fold pattern of the 2 x 4 reconstruction elongate into streaks perpendicular to the surface. The elongation is due to the diffraction from random atomic steps. Though not perfect, the surface is still relatively smooth.

674

Molecular Beam Epitaxy

There are several choices of detecting schemes. In our laboratory, the intensity of the diffraction pattern is measured with a photomultiplier. The distribution of the intensity versus scattering angle is determined by scanning with a magnetic deflection system. A schematic of the apparatus is shown in Fig. 1. After striking the sample, the 10 keV beam forms a diffraction pattern on the ITO coated phosphor (JEOL P15) covered screen. A photomultiplier lens assembly is positioned with micrometer stage.[ TM] Then the intensity in the desired region is measured by deflecting the pattern with an external magnetic field. Note that the detector in this mode is fixed. The output is then measured with a fast preamplifier and sampled with a 12 bit analog-to-digital converter. This system has several advantages. First, it is fast with high spatial resolution and wide dynamic range. Second, it can be positioned with a minimal software interface. Most importantly, after positioning the detector over a given phosphor grain, it is insensitive to non-uniformities of the phosphor screen. This is especially necessary in an MBE environment where the electron beam catalytically decomposes the As 4 background gas onto the phosphor. The disadvantage of this arrangement is that simultaneously following the intensity in different portions of the pattern at once is cumbersome, though two detectors are still quite practical.[~7][~8] For the measurements discussed here, the scanning system is mainly used to measure the diffracted intensity along the length of the streak, perpendicular to the sample surface. To measure the intensity across the width of the streak, parallel to the surface, or to measure the separation of two streaks (for lattice constant determination), a different set of scanning coils must be used. Alternative video systems are, of course, practical. Larsen has measured the intensity along the length of the streaks[19]-[21] and Grunthaner has made intensity measurements.[ 22] Unfortunately, the spatial resolution of CCD or SIT video systems is limited to between 500 and 1000 pixels and are relatively slow because of the enormous amount of data that must be transferred. Image dissectors with custom software should be faster, not spending the time required to measure unwanted portions of the pattern. For some measurements, it is useful to combine the scanning arrangement with the video system. In general, the video system is somewhat more convenient.

RHEED Studies of the Dynamics of MBE 3.0

DIFFRACTION FUNDAMENTALS

3.1

Kinematic Approximation

675

Care must be taken when applying kinematic or single scattering theory to the analysis of electron diffraction from surfaces. Strictly speaking, kinematic theory is applicable if the inelastic mean free path is less than the elastic mean free path. It is, then, unlikely that any elastically scattered electron has scattered more than once. Unfortunately, this condition does not hold in RHEED. In the RHEED geometry from surfaces, the path traversed by the electron beam is comparable in length to the inelastic mean free path so that the incident and exiting electrons can undergo multiple elastic scattering. A calculation of the absolute diffracted intensity must include all such processes. It is a dynamic calculation in the sense that it requires consideration of the scattering potential. From this calculation one expects to be able to extract the atomic positions in a reconstructed surface.[Z][6] A calculation of the positions of the diffracted beams, however, need only consider the surface symmetry. These positions are determined by momentum and energy conservation, i.e., the kinematics of the scattering process. If this symmetry is slightly reduced due to disorder or due to the finite lateral dimensions of the surface, then the diffracted beams will broaden. The essence of our argument is that the broadening of the diffracted beams is determined by the relaxation of momentum conservation and, to first order, is given by kinematic calculation.[23][24] This approximation is admittedly limited, as discussed below, and some[ 27] even argue inappropriate. It should only be appropriate when two-dimensional islands on a surface are large enough that coherent scattering between islands is not important and when island edges do not represent a large fraction of the scattering. Other approximations have been attempted, such as relating the diffraction to the step density on the surface during epitaxy.[ 28] This approximation must also be incomplete. Though it gives some agreement to Monte Carlo calculations at a fixed incident angle, it is difficult to believe that very small clusters scatter the same as a large cluster or that the angle of incidence dependence is so simple. The discussion of the kinematic treatment below is offered as a reference point to illustrate what a dynamic treatment must ultimately deal with and because it has been found to give agreement with experiment in some cases. However, it is crucial that its predicted angular dependence be checked. The discussion is probably more correct for low energy

676

Molecular Beam Epitaxy

electron diffraction, where the normal incidence geometry makes step edge scattering a smaller fraction of the total and coherent scattering between islands less important. The treatment of disorder is probably the most important impediment to the application of diffraction to understanding epitaxial growth. It is being pursued by a number of workers[S][29][3~ and progress is being made. Our expectation is that some modification of the kinematic result presented here will be practicable. The application of the kinematic approximation is equivalent to the column approximation of electron microscopy. The important points are that (1) several layers near the surface are important to the diffraction, so that shadowing of lower layers by step edges on an upper layer is ignored, (2) the full dynamical scattering under a finite region of a surface is included, and (3)though the subsurface atoms are included, only the toplayer coverages appear in the result. The full application of dynamic theory to calculate the diffraction from a disordered surface is not yet possible, though a start has been made.[3][ 3~] To apply the column approximation of electron microscopy, consider a surface with random steps with edges at R i = di~ + 1~. If each terrace is large enough, the diffracted amplitude is the kinematic sum of the full dynamically scattered amplitude from each block, A i, i.e.,

Eq. (1)

A(S, ~, q~)= ~ Ai(S, ~, qo)eis'R' i

This neglects multiple scattering between the blocks. Here O and q~are the set of appropriate polar and azimuthal scattering angles. As the blocks become small, it is not expected to work well. For example, if one were to try to calculate the diffraction from a periodically stepped surface with a repeat that was several unit-ceils, then this should not be a good approximation. In that case, one needs to do a full dynamic calculation[ 31] since the surface is more like a reconstruction than a stepped surface. Our second assumption[ 23] is that the most rapid variation of A i with S is due to the finite size of the block. This is not always the case, especially when taking data near conditions of high symmetry or if the intensity varies rapidly with Sz. In the latter case, some correction can be made by including the measured Sz dependence of the diffracted intensity from a more perfect surface.[ 32] With this assumption on the variation of Ai, we let f(O,q~) be the dynamic amplitude from a perfect surface and redefine Ai(S) to only contain the kinematic size effects of a block, including the number of scatterers. Then in this approximation, the diffracted intensity is

RHEED Studies of the Dynamics of MBE Eq. (2)

677

A(S, O, q~)= f(O, q~) ~ Ai(S)eis'R' I

with the recognition that one should keep the scattering angles O ,q~fixed. Because of this last point, we will, for the most part, set f = 1. Kinematic shape effects are most easily understood by performing the sum indicated in Eq. (2) for a two-dimensional rectangular net of dimensionsNla 1 andN2a 2. This is a special case in which a single block is considered to be a plane of scatterers. (It is important to realize that, more generally, by including a three dimensional block in Eq. (2), one is able to include single and multiple scattering within a given column.) But with only one plane, there is no kinematic dependence on Sz and the famous result [33] is that for one block Eq. (3)

sin2(SxNl a l/2) I(S='Sy)='~AI2= sin2(S=aj2)

sin2(SyN2a2/2) sin2(Sya2/2)

In the Sx direction, the FWHM of this function is approximately 2:rt/Nla 1. The broadening is similar for the Sy direction. Figure 4 illustrates two cases in diffraction experiments in which the incident wavevector, k;, is fixed and the final wavevector, kf is uncertain due to this broadening. In Fig. 4a, the incident electron is in the J direction, along q~ = 0, making a glancing angle Oi to the surface plane. For near forward scattering as shown, the range of final angles, 8q~ythat correspond to the uncertainty in momentum transfer satisfies k6q~y = 2zt/Ly where Ly = N2a 2. It is important to note that this broadening will not be observable if the range of angles in the incident beam is greater than this. Alternatively, if ~)q~is the range of angles in the incident beam, then order over a distance less than about 2zt/k6~ will broaden the beam measurably. This distance is the resolvable distance or transfer width of the diffractometer in the direction perpendicular to the incident beam. Figure 4b illustrates the same argument for the direction parallel to the direction of the incident beam. In this direction, because of the small glancing angle, Oi the diffraction is much more sensitive to disorder on the surface. Here the range of final glancing angles, 8Of, that correspond to the uncertainty in momentum transfer is given by k b OsinO = 2zt/L x, where Lx = Nla I. If the range of angles in the incident beam is greater than ~)Of, then this will not be observed. Hence order over distances less than 2:t/k sinOytiO, where 50 is the range of angles in the incident beam, will broaden the diffracted beam. At an angle of incidence

678

Molecular B e a m Epitaxy

of @i = ~f = 2~ disorder over the crystal will broaden the beam in the .~ direction about 30 times that of the ~ broadening. This is the fundamental cause of streaks in RHEED. In subsequent sections, the impact of a particular class of disorder, atomic steps, are emphasized. 9 i n t e r s e c t i o n with 9 "

-" i

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't

Ewald here

broad~ned reciprocal lattice rod

o

i

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v

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Rgure 4. An Ewald construction showing the conservation of energy and momentum for the case in which the sharp reciprocal lattice lines of the 2D surface are broadened into rods by some disorder. (a) A given amount of disorder greatly broadens the beam in Of. (b) The same amount of disorder has less effect on the width of a streak, i.e., q~. The asymmetry in the instrumental sensitivity to disorder is purely a result of the low glancing angle of incidence, 0 i. The transfer width or distance over which order is resolvable is sometimes referred to as (certainly incorrectly) coherence length. The important point is that the length is asymmetric. For our instrument, distance over about 10,000 .~ in the direction of the incident beam can be resolved. The main impact is that when studying a misoriented surface, i.e., one in which there is a staircase of steps, it is possible to be in a situation that, when the beam is pointing down the staircase, more than one terrace contributes coherently to the diffraction (add amplitudes). But

RHEED Studies of the Dynamics of MBE

679

when the beam is parallel to the step edges on the same surface, the steps (if straight and parallel) could be so large that one must add intensities diffracted from each. 3.2

Disorder on Low-Index Surfaces

During perfect layer-by-layer growth, single-layer islands first form on an otherwise featureless plane, and then fill in until the layer is smooth and complete. During this process, the two-dimensional islands that form could have a range of sizes and shapes. In the diffraction pattern, there will be interference because of the difference in the path length traversed by electrons scattering off the different levels of the surface. Depending upon the scattering geometry and the distribution of islands, the diffracted beams will broaden, giving rise to the long streaks typically observed during growth. From the shape of the beams, one hopes to determine the distribution of island sizes. For fixed glancing and azimuthal angles of incidence, Oi and q~i, the intensity along a streak can be measured versus the final glancing angle Of. For the specular streak, for example, scattering angle can be related to momentum transfer via Eq. (4)

s . = k cos o s - k cos o i

s, = k ( ~ + o s)

With this connection, the intensity along the specular streak is expected to have a simple general form if only a few layers contribute to the diffraction. For example, suppose there are islands on top of a flat surface, with the coverage of the islands 0. Then there will be interference between the top layer with coverage 0 and the layer below with exposed coverage 1 - 0. This two-level case (see the Appendix), shows that, for a Markov distribution of island sizes, the diffracted intensity is given by Eq. (5)

l(Sx, Sz) = [02 + (1- 0) 2 + 20(1- 0) cos Szd] 2~6(Sx) + 20(1- 0)(1 - cos Szd) [2X/(;Lz+ S~)]

This says that, for a two-level system, the diffracted intensity along the streak can be separated into a broad part and a central spike. The FWHM of the broad part is 2/X or twice the reciprocal of the sum of the average hole and island size.[34] This should be compared to the width of about 2~/L

680

Molecular Beam Epitaxy

given by Eq. (3). One can think of Sz as roughly determined from Oi and S= determined from Of - Oi, but these must be calculated more exactly using Eq. (4). The two terms of Eq. (5) are: a central spike that results from the long range order, and a broad part that is due to the disorder. The first term is written as a delta function, but there is a range of angles in the instrument and so this term is broadened. As shown in Eq. (3), the width determines the size of the coherently scattering region. The second term depends upon the form of the distribution of island sizes. A form more general than the simple exponential distribution given here can be derived[32] but this will suffice. Further, more than two levels can be considered by including additional terms in Eq. (5). To observe this distribution of intensity, we deposited a submonolayer coverage of Ge onto a Ge(111) surface.[ 35] The diffusion of Ge is low so that this system gives an ideal compromise between continuous epitaxy and the time required to make a measurement. Figure 5 shows the results of experiment and a calculation based on Eq. (5). The left panel shows a plot of the intensity versus Of along the specular streak for several incident angles Oi. The right panel is a fit using Eq. (5) after converting to angle with Eq. (4). In both there is a central spike and a broad component. The spike dominates at in-phase conditions in which S, cl is an integer times 2= and the broad part dominates at out-of-phase conditions half way in between. These conditions are crucial to the formation of streaks in RHEED. At in-phase conditions, the extra scattering path length between different terraces is an integral number of electron wavelengths. The diffraction is insensitive to the step disorder and only the central spike, the long range order, is seen. At out-of-phase conditions, the diffraction is maximally sensitive to steps. If the coverage were one half layer, the central spike would vanish. However, and this is an important point, because of the second term in Eq. (5), the total intensity does not go zero. To our minds, this shape is the dominant feature of the diffraction. During layer-by-layer epitaxy, the diffracted intensity is traded between these two components as growth proceeds. How each component contributes to the measured peak intensity determines the RHEED intensity oscillations to be discussed in Sec. 4.1. The right panel is a one parameter fit to the data. Note that the same asymmetries in the data are present in the calculation and are primarily due to the cut that the Ewald sphere makes through the reciprocal lattice rod. A better fit could be obtained by using a third level in the calculation. These measurements give us a great deal of confidence in the kinematic analysis of beam shape and the interpretations used in the remainder of this chapter.

RHEED Studies of the Dynamics of MBE

9

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Figure 5. The left panel shows scans of the intensity of the specular streak of a smooth Ge(111) surface with 0.4 monolayers of Ge deposited. The intensity is measured versus Of at several different glancing incident angles Op It is compared to a calculation[ 3s] in the right panel that shows that at in-phase Oi, the beam is sharp, while at out-of-phase Oi, the beam exhibits a broad component under a sharp spike.

3.3

Vicinal Surfaces

A wafer that is cut and polished from a boule is never perfectly oriented parallel to a low-index plane. This misorientation results in a series of steps separated by low-index terraces. Typically these steps are one atomic layer in height, though double-layer and higher have been reported.[ 36]-[39] Due to the exceedingly long transfer width of RHEED, interference between waves scattered from different terraces is often the strongest component to the shape of the diffracted beams. In this section, the diffraction from a regular staircase of steps is considered. Later, the complication of disorder will be added. An important point is that, because

682

Molecular Beam Epitaxy

a staircase has a specific direction, the resulting modification in the shape of the diffracted beams will depend markedly on azimuthal angle of incidence % An ordered staircase will cause the specular beam to split into two components. We calculate this angle here. Consider a regular, onedimensional staircase of monatomic steps in the ~ direction. Let each terrace have length L and the steps have height d. Then the staircase consists of blocks with corners at R m = mL.~- md~.. Further, following the arguments in Sec. 3.1, let the full dynamically scattered intensity from the block under one terrace be f(S,O, qp). Then the diffracted amplitude from the staircase is, following Eq. (2) Eq. (6)

A(S) = ,~ fm eis'R" m

where for generality the f's might depend upon m. If the terraces are all identical, then with the beam in the ~ direction (along q0 = 0) this becomes Eq. (7)

A($., $~) = f ~ d (S~t - s~4,, /1'/

There is constructive interference when there is both energy conservation and S,L - Szd = even =. For example, the specular beam will satisfy this at two final angles, Of, 1 and Of 2 corresponding to two values of Sx and two values of Sz. Taking differences, the separation AOf = Of. 2 - Of, 1 is given by Eq. (8)

AS,L- ASzd = 2~

Further, for the specular beam Eq. (9)

s, = k cos o s - k cos oi

s, = k

+ o s)

so that Eq. (10)

Z~, = - k sin Of AOf and &$z = k AOf

Combining Eqs. 8 and 10, one has for the beam pointing down the staircase misoriented by Oc = d/L Eq. (11)

AOf = (2~/#a) Oc/(sin Of + Oc)

RHEED Studies of the Dynamics of MBE

683

Thus given d and measuring Aef, one can determine the misorientation of the staircase, Oc, with great precision. Alternatively, one can measure the misorientation with O- 29 x-ray goniometry[4~ 41] and then determine the step height, d, using RHEED. This can easily be extended[ 42] for other azimuthal angles of incidence. The main points to be mentioned are (1)in order to distinguish between Kikuchi lines, the incident glancing and azimuthal angular dependence must sometimes be checked,[ 43] and (2) the separation AOf depends upon the disorder in the staircase as described in Sec. 3.4.[ 32] Summarizing, the signature of a staircase in the diffraction pattern is a split diffracted beam. The amount of splitting depends upon scattering angles in a characteristic way. 3.4

Disorder on Vicinal Surfaces

For a regular staircase, a staircase in which each terrace length is identical, the specular beam will be split into two components. The width of these components will be equal to the range of angles in the incident beam. The separation of the components will be determined by the misorientation as described in the previous section. If, however, there is disorder, then one needs a model for the distribution of terrace lengths to calculate the diffraction. The main result is that the width of the components is broadened and the separation of the components is decreased.[ 32] As a specific example, assume that the probability of a terrace of length L >Lo is given by Eq. (12)

P(L) = c~e-(,(L-Lo)

and zero for L < L o. The order of the terrace lengths is assumed to be uncorrelated. For this distribution of terrace lengths, the mean terrace length is given by L = L o + 1/(~ and the rms deviation from the mean is 1/(~. When the diffracted intensity is measured versus Of, both S= and S, are changed. Hence it is a bit simpler to express the intensity at constant Sz when considering the width of the components and the separation of the components. With this in mind, the intensity for the shifted exponential distribution of Eq. (12) is obtained atSzd = ~ from Eq. (19) of Ref. 32 to be w

m

4/L Eq. ( 1 3 ) I ( S = , S z = = / d ) =

2(~2+ 2o~2 cos2~TJc+ (2=/Lo)2X 2- (4~(~/Lo)x s i n 2 ~

684

Molecular Beam Epitaxy

where x = S=Lo/2m This is illustrated in Fig. 6 for several values of o~. When (~ is not too large (1/(~ is small compared to L ), one can also show for this distribution that the separation between the peaks AS. is AS. = 2~/ L . Using o = 1/o~ as the rms deviation from the mean terrace length, the width of one component, normalized to the separation of the peaks, 8S./ &S., can then be calculated as" m

Eq. (14)

..&S.

= 1.5 ,o~q,__=..)2 "L"

To convert to angular width, 6t~f = &S./k sinOi. As discussed in Sec. 4.2, by following the shape of the diffraction from a vicinal surface, one can observe step bunching versus growth.

160 Lo= IOOA

"~

120

"~

40

o

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2

__c

0

-0.10

-0'.06

-0'.02

0.02

0.C)6

0.10

0

Sx(A-I) Rgure 6. Diffracted intensity exhibiting the split components from a vicinal surface. The intensity is plotted versus S. at Sz = =/d for the case in which the distribution of terrace lengths is given by the shifted exponential distribution of Eq. (12). Results for several values of o/L o are plotted. The separation of the components is 2~ / E.

RHEED Studies of the Dynamics of MBE 4.0

685

DIFFRACTION MEASUREMENTS

The time evolution of the diffracted intensity during epitaxial growth is in a sense a record of the development of surface atomic structure and morphology. Changes in surface reconstruction, increases in the density of point defects, the agglomeration of islands, step bunching, lattice plane bowing near dislocations~all of these contribute to the diffraction. An important example is the periodic intensity oscillations observed during layer-by-layer growth. Our goal is to understand these diverse contributions and to deduce the rates of the important microscopic processes. Ultimately a coherent, quantitative, and predictive picture of epitaxy should emerge. This section focuses on the diffraction from GaAs and related materials since these have been the object of most of the experimental work. Indeed, using RHEED, the first intensity oscillations were first reported on GaAs(100).[ 13] Recent work on metals[l~ 45] and elemental semiconductors[46]-[ 48] shows that much of the same phenomena can be observed there. This section is organized as follows: first, two general classes of observations are distinguished, reflecting the qualitative differences observed in the diffraction from low-index surfaces as compared to vicinal surfaces. The main difference in growth on these two surfaces is the change that occurs when the incorporation of adatoms at the built-in steps of a vicinal surface competes with island growth on flat terraces. For the case of the low-index surfaces, the underlying mechanisms giving rise to RHEED intensity oscillations are described. Evidence supporting the kinematic view is presented along with results that as yet are not explained so simply. In addition, related intensity oscillations that give the energetics of the sublimation of GaAs are discussed. Second, the development of the diffracted intensity during growth on vicinal surfaces is presented. A model discussing the damping of intensity oscillations on vicinal surfaces is given. The two main topics are diffusion and step bunching. For both, the roles of the two different types of steps on zincblende crystals are important. Third, the dynamics of lattice mismatched growth is discussed. Strain and dislocation formation strongly influence the evolution of the diffraction pattern. Precise lattice constant measurements are compared to measurements of intensity oscillations.

686

Molecular Beam Epitaxy

The main results are that upon initiation of growth on a smooth GaAs(100) surface, the diffracted beams usually broaden into the streaks shown in Fig. 3 and the diffracted intensity can oscillate with period equal to a monolayer completion time. The separation of diffracted beams can also change. Our main message is that the intensity at the peak of the specular beam and the angular distribution of the intensity contain important complementary information on the growth process. The peak intensity is simpler to measure versus time, but to a large extent ignores the lateral structure of the surface. In this section, the time development of both kinds of measurements for the two classes of surfaces is emphasized.

4.1

Low-Index Surfaces

Intensity Oscillations. Intensity oscillations observed at the start of MBE growth on a smooth GaAs(100) surface are shown in Fig. 7. These are measurements of the peak intensity of the specular beam versus time and occur simultaneously with periodic changes in the shapes of the diffracted beams. For these data, the period of the oscillations corresponds to the time required for the deposition of a layer of GaAs. In this case in which there is excess As 4, it is also the time required to deposit a layer of Ga, since that is the rate-limiting step[5~ of the growth. The exact form of the oscillations--their amplitude, damping, phase, and sometimes additional frequency components--depends on the scattering geometry (scattering angles Oi and r as well as growth parameters, so that care must be taken before growth information can be extracted. It is generally agreed that these intensity oscillations indicate a layer-by-layer growth mode characterized by two-dimensional island formation as opposed to pure step propagation or step flow. It is also generally agreed that the oscillations result from an alternation in surface roughness. Little else is well understood. There is no quantitative theory that accounts for all of the features observed. One can try to consider models in which the step edges scatter out of the diffracted beams[5~][52] but not too much data has been compared to these. The kinematic treatment is presented here. As will be seen, kinematic theory is only partially successful and one goal could be to choose experimental conditions and methods to enhance its chance of success. But another goal should be to develop theories that can account for the demonstrated sensitivities of this measurement to scattering geometry.

RHEED Studies of the Dynamics of MBE

687

Qe

t,t)

btl

,4-" om

C::

J3

13

-I--,

CO

I,.-I

d.

Time Figure 7. Examples of the peak intensity of the specular beam measured after the initiation of growth on GaAs(100). The period of the growth is the time to deposit a monolayer. The envelope, phase, and magnitude depend on scattering geometry as well as growth parameters. (a) Beats can be observed on a small sample with non-uniform illumination by the electron beam. The incident beam is on a symmetry azimuth. Note that the initial intensity increases. (b,c,d) The glancing angle was ei = 33 mrad, the azimuthal angle ~)i = 7~ from a [011 ]direction. (b) T = 580~ and As4/Ga is 6, (c) T = 550~ (d) T = 580~ and As4/Ga = 80.

688

Molecular

Beam Epitaxy

In Fig. 8, the sequence of specular intensity oscillations is measured during a growth in which the macroscopic surface plane was parallel to the (100) to within 1 mrad. For these measurements, the flux variation over the surface is less than one percent so that beats due to a range of periods do not contribute to the envelope.[53] Intensity oscillations on GaAs(100) surfaces can be observed under a wide range of growth conditions, and for the data in panels 8(a)-(d) the growth rate is changed from 0.01 to 0.2 layers per second while the substrate temperature was maintained at 580~ The temperature could be varied between 450 ~ and 700 ~ C with oscillations still observable. By contrast on a GaAs(110) surface, intensity oscillations can be observed only under a very narrow range of substrate temperatures and growth fluxes. For this (100) surface, the intensity oscillations shown in Fig. 8 exhibit trends that are commonly observed in other systems. After the initiation of growth, there is a sudden change that, depending upon scattering geometry, can be an increase or decrease. After this initial transient, the higher the growth rate, the larger the initial oscillation amplitude and the lower the baseline. Though the period corresponds to the time to deposit a monolayer, the maxima and minima do not necessarily correspond to the times at which integral numbers of layers are deposited. For these low-index surfaces, the baseline is nearly constant and the maxima damp slowly. Similar intensity oscillations in many epitaxial systems are now routinely used to measure growth rates during MBE.

r

o !

I

o

I

I

if) Z t.~ I-Z_

9

o

o

so

~oo

I

o

I

so

I

uoo

TIME (sec) Figure 8. The peak intensity of the specular beam versus time on a low-index surface for a variety of growth parameters. The surface misorientation is less than 1 mrad, the incident azimuthal angle was 7~ from the (010) and the incident glancing angle was Oi = 30 mrad. This figure is to be compared to results on a vicinal surface shown in Fig. 17.

RHEED Studies of the Dynamics of MBE

689

The basic mechanism[ TM] of these oscillations lies in the cyclic variation in surface roughness during layer-by-layer growth. This is illustrated in Fig. 9 where, starting with a smooth surface, islands are nucleated causing a dramatic decrease in the diffracted intensity. As growth proceeds, the surface diffusion and the preference for adatoms to bond at steps causes the islands to become larger until the layer is completed. During this process, the kinematic diffraction reaches a minimum at maximum roughness and then a second maximum as the surface becomes smooth. Depending upon the diffusion of the adatoms, it is likely that islands will nucleate on large terraces before the first layer is completed so that the intensity never recovers to its initial value and, in fact, exhibits an envelope which is damped. Ultimately, depending upon the growth of the particular materials, a steady state surface is reached in which there is no net change in roughness. The constant baseline should correspond to the situation in which the first term in Eq. (5) is small, with the second term dominating. Even with more than two layers, this kinematic result holds.[ ~~o] If the steady state intensity equals the baseline intensity, it suggests that the disorder scattering is dominating at both the minima and the long time result. One steady state could be a case in which new islands are formed at the same rate as old islands are assimilated into larger ones.[ 54] The original surface can be recovered if growth is interrupted and kept at a temperature sufficient for significant adatom diffusion. For GaAs, an incident arsenic flux must also be maintained in order to replace arsenic that desorbs from the surface. The length of time required for the surface to become smooth depends on the same mechanisms that give rise to the intensity oscillations. The time, which is of the order of tens of minutes here, depends upon sample history, misorientation, and As4 flux. For these well-aligned surfaces at 580 ~ C, maximum smoothness could take as long as one hour. An important point is that these intensity oscillations result from the competition between cluster formation and step propagation. If the surface mobility of adatoms were such that all adatoms could migrate and attach at a step, then the built-in steps of the surface would just walk across the surface and there would not be a change in surface roughness~there would not be oscillations in the diffraction. On the other hand, if the adatoms did not diffuse, then the surface would just become randomly rough, giving at best very weak intensity oscillations.[ 11~ Layer-by-layer growth requires both step propagation and surface adatom migration.

690

Molecular Beam Epitaxy

?

Figure 9. Schematic of island growth and coalescence during layer-by-layer growth, giving rise to intensity oscillations. Several processes are illustrated, including step propagation, nucleation on terraces and islands, coalescence of islands, and island growth. One expects that maxima intensity corresponds approximately to minimum roughness.

The kinematic interpretation of these changes in surface roughness gives a simple correspondence to the measured diffraction. In fact, the interpretation includes multiple scattering for large terraces within the limits of the column approximation, for the same reasons discussed further in Sec. 3.1. Complete multiple scattering calculations have not progressed to the point of realistically including random island growth.[2][3][49] The intensity oscillations can be understood in terms of the shape of the diffracted beam shown in Fig. 5. There the distribution of intensity along a streak is composed of two parts: a central spike due to long range order, and a broad part due to the step disorder. As seen later in Eq. (43), this spike depends mainly on the coverages of each layer. Starting with a smooth surface, the intensity consists only of the central spike and the diffraction is maximum. As islands are formed, the central spike is reduced in magnitude and the broad disorder scattering is increased. As the islands grow in size, the intensity returns to the central spike until a maximum is once again obtained. If third layers start to form before the second layer is finished growing, the intensity will not regain the initial value. The intensity then continues to oscillate as growth proceeds layerby-layer. Finally in steady state, some number of layers on average are always present and though locally the growth may be similar, the diffracted intensity does not vary. The magnitude of the intensity oscillations in this

RHEED Studies of the Dynamics of MBE

691

model depends upon the scattering angle corresponding to the dependence shown in Eq. (5). Simply stated, when the scattering angles are such that the path length difference between different terraces are an integral number of electron wavelengths, the scattering is insensitive to the steps; when the angles are such that the path length differences between different terraces is a half of a wavelength, the diffraction is maximally sensitive to the steps. For example, when the angles are such that scattering from different steps is in phase, the intensity should not oscillate. How well this interpretation works can be seen qualitatively in Fig. 10, where intensity oscillations during the growth of GaAs on GaAs(100) are shown for several glancing angles ei. These particular data were taken with the incident electron beam directed along an azimuth that was 7 ~ from the [010] direction. At this azimuth there are a minimum of diffracted beams that are strongly excited[ 55] and the diffracted intensity of the specular beam is strong. One hopes that multiple scattering in the form of a sharing of intensity with other less strongly excited beams is therefore minimized. The main point is that the oscillations are weak at 43, 66, and 83 mrad, which are near in-phase conditions; and they are strong at angles corresponding to out-of-phase conditions. A more quantitative view is given in Fig. 11 where the ratio of the first minimum to second maximum is plotted versus ei. This removes the importance of the initial transient. When this ratio is near unity, the oscillations are weak, and when the ratio is small, the oscillations are strong. Figure 11 shows again that the qualitative explanation works well. Further, if we assume that only two layers are involved in the diffraction (perfect layer growth), then from Eq. (5) at Sx = 0, the ratio of a minimum (half coverage or e = 1/2)to a maximum (full coverage or 0 = 1) is = 1 -cos(2kdOi). This ratio, with the constant of proportionality fit at one point, is plotted as the solid curve in Fig. 11. The fit is remarkably good for such a simple calculation, though at small angles there is a shift from the data. For comparison, a similar plot for Ge on Ge(110) shows no such shift. From our perspective, this agreement indicates that kinematic diffraction from islands plays a large role in the diffraction. There are also serious discrepancies with kinematic theory, even at azimuthal angles at which there is coupling to only a few diffracted beams. Note that after growth is initiated the intensity will either increase or decrease, depending upon the glancing angle Oi. In the kinematic analysis, the central spike cannot increase. For this to be a kinematic effect within the assumptions made so far, the diffuse component due to the step

692

Molecular Beam Epitaxy

disorder would have to dominate the diffraction. Alternatively, there may be a reconstruction change once small islands are nucleated. This is, in fact, seen in the growth when the fractional order beams become weaker after growth is begun. To explain the shift of the ratio measurement from calculation, one could require that the scattering phase of a small island be different than that of the substrate, so that the simple path length argument is modified. However, it is difficult to explain the shift in the position of the maxima of the intensity oscillations that can be seen to some extent in Fig. 10 and which also depend on (Pi. These are described in detail by Joyce and coworkers.P 7] The main difficulty is that the maxima should corre, spond to a smooth surface in nearly any reconstruction-independent model that has moderate diffusion. Shadowing or a significant role of step scattering should produce strongly asymmetric oscillations, which are not usually observed. The strong Oi dependencies are being investigated, and we note that they are evidently not seen in LEED.[ 5s] Recently, Horio and Ichimiya[ 57] have shown that dynamical considerations can at least qualitatively account for the phase shift. They treated a growth front with a potential that was proportional to coverage. The phase shift is seen to come from both the surface roughness and the coverage dependent potential.

85

74 Figure 10. Specular intensity oscillations measured during the growth of GaAs on lowindex GaAs(100) surface. The in-phase angles correspond to Oi = 43, 66, and 83 mrad where the oscillations are weak. The out-of-phase conditions are in between and the oscillations are much stronger. The amplitudes are compared to calculation in Fig. 11.

64

,~ "O 53 O

E ~43 (It>

32

0

0

TIME

0

0

0

( arb. units)

RHEED Studies of the Dynamics of MBE

693

I 0.8

0 k-<~ rr

0.6 0.4 0.2 0 ! I0

", =30

, 50

, 70

l 90

e i(mrad) Figure 11. Oscillation strength versus 0 i from data like that in Fig. 10. The ratio of the first minimum to the first peak is plotted to show that interference between deposited layers is the cause of the intensity oscillations. At the in-phase angle of 65 mrad, the oscillations are weak; at the out-of-phase angle of 76 mrad, the oscillations are strong. At low angles the maximum and minimum are shifted, though the alternating trend is still clear. The solid curve is a calculation for a twolevel system from Eq. (5).

If the surfaces are sufficiently flat and if the growth does not produce defects, then the oscillations damp very slowly. Three cases are illustrated in Fig. 12, GaAs on GaAs(100), Ge on GaAs(110), Fe on Fe(100). The first two are MBE-prepared substrates; the last is a whisker. In all cases, the lateral dimensions of the surfaces were sufficiently small that the effects due to flux variation over the surface were minimized. For example, Fig. 7 (a) shows a case in which there is a 10% variation over the surface. The surface is small enough that the illumination of the incident electron beam is uniform over the surface. In this case, beats are observed. If the surface is large, giving both a flux variation and variation of electron intensity, then the oscillations are just damped. This must be removed from the problem if growth information is to be extracted from the damping.

694

Molecular Beam Epitaxy

(a) GaAs(100) at 580 ~ (5 x time)

.~_---" E ::3 <

.~ ._ m c

(b) Ge(110) at 400 ~ (4 x time)

..._c s LU LU "1-

(c) Fe(100)

0

100

200 Time (sec)

300

Rgure 12. Intensity oscillations on very well-oriented substrates. (a) GaAs on GaAs(100) oriented to better than 0.7 mrad. (b) Ge on Ge(110) lattice-matched to GaAs(110). (c) Fe on an Fe(100) whisker. Note that in (b) the nucleation of Ge clusters is affected by residual arsenic.

Layer-by-Layer Sublimation. Not only can cyclic variations in surface roughness be observed during growth, but intensity oscillations during the sublimation of GaAs are seen.[ 58][59] From the temperature dependence of these oscillations, one should be able to learn about surface binding energies and the relative importance of equilibrium and kinetic processes. Figure 13 shows oscillations seen in the specular beam intensity during the growth and sublimation of GaAs at moderately high temperatures. By first growing GaAs and then interrupting the Ga flux, both growth and sublimation could be observed without changing the sample temperature. (This procedure also obviates the need to compensate for temperature transients.) In Fig. 13, GaAs is first grown at 1 monolayer per 7.2 sec; then growth is interrupted and there is a short time during which the surface anneals and the intensity increases. After this anneal, the intensity oscillates corresponding to the time required to

RHEED Studies of the Dynamics of MBE

695

remove a layer. We interpret the annealing process to produce terraces large enough for sublimation from the middle of terraces to cause a change in surface roughness. If the temperature is increased, then the period of the sublimation is reduced. Or) l--

A

J

g

Z rn rr <~

l-O9 Z LLI l-Z J

7

I0I

0

I

I00

,

I

200

I

I

1.10

1

1.20 ! ....

300

TIME (sec) Figure 13. Intensity oscillations during both growth and sublimation of GaAs on GaAs(100). With the sample held at 665~ GaAs is first grown at 1 monolayer per 7.2 s; then growth is interrupted, and after a short anneal period, sublimation oscillations are observed with a 25 s period. Sublimation is only observed for certain surface structures. Figure 13 shows the transition line between the 1 x 1 and c(2 x 4) surface reconstructions, plotted versus substrate temperature, and As 4 flux under conditions of no growth. These data were determined on a low-index surface by measuring the intensity of the quarter order reflection versus substrate temperature while the temperature was raised. For low-index surfaces only, the intensity is observed to decrease sharply, over 2 ~, at the transition temperatures shown. The sublimation oscillations could be induced by crossing into the 1 x I region, either by raising the temperature or lowering the As 4 flux. The latent heat of transition is 4.5 eV. This value has been disputed by Newstead et al.[ 6~ as being too high; their value is 3.9 eV.

696

M o l e c u l a r B e a m Epitaxy

15 PAs

10 3xl

PGa / I 750

700

l

+x6 ! C(8x2) I ! 800 850

4xl I 900

950

Substrate Temperature (K) Figure 14. Observed reconstructions on GaAs(100) versus As4 flux and substrate temperature for the case of zero Ga flux. A mass action analysis gives agreement with these measurements. Equilibrium between vapor and solid could be described by the reaction Eq. (15)

Ga

+ 1/2As2 <

>

GaAs

so that the pressures are related according to Eq. (16)

PGa = lip (PAs2)v~

By detailed balance, if the equilibrium Ga vapor is removed, Eq. (16) should give the sublimation rate of Ga.[ 61] The agreement of the data with this mass action analysis is shown in Fig. 15 where the Ga sublimation rate is plotted versus arsenic flux. For these measurements, the sample temperature was determined by a thermocouple calibrated at the AISi eutectic. The arsenic flux was determined from an ion gauge calibrated against intensity oscillations in which an excess of Ga was present on the surface so that the supply of arsenic was rate-limiting.[ s3] At these substrate temperatures, identical results were obtained with either As 4 or As2; the slope of the curve in Fig. 15 was 0.5. Similarly, one can make an Arrhenius plot of the sublimation rate of Ga versus 1/T at several different As pressures; from this, one determined that the enthalpy of formation in Kp is 4.6 + 0.2 eV, in good agreement with the bulk data.

RHEED Studies of the Dynamics of MBE

As INCORPORATION

697

PERIOD (S)

I0

6

4

2

I

0.6

0.4

02

,

l

|

l

I

,

,

,

o_ s

~

-o.e

TGo

oC TAs

Z

o_9_ o

201-

o

0 I 0 "T

2

4

6

I0""

PRESSURE

2

As 4

4

6

I0 "=

( "rORR )

Figure 15. From measurements shown in Fig. 13, the Ga sublimation rate is plotted versus arsenic flux. A similar analysis can be used to determine the mole fraction of AIxGal_~As at temperatures where sublimation of Ga occurs. Treating GaAs as a constituent with activity 1-x, the sublimation rate of Ga is now Eq. (17)

~subl _ Kp(1 - x) ~rGa

-

(PAs2) '/= As a result, the mole fraction, x, of A! will depend upon the sublimation rate. Using Eq. (17), the normalized growth rate, 1 - ~ = / T ~ b~, can be calculated and the results for particular growth rates are shown in Fig. 16 to have excellent agreement with measured rates using RHEED. A main result of these studies is that above 580~ no differences between As 2 or As 4 fluxes have been observed. This means that As 4 is quickly equilibrated at the surface. The success of the mass action analysis also calls into question rate equation models of the growth kinetics. For example, it is tempting to argue that if 0 is the surface coverage of As adatoms and 1 - e is the fraction of Ga sites available, that the change in As coverage during growth is Eq. (18)

at

- 2]As4( 1 "0) 2" 2k~ 02 "JGa + k2(1 " 0)

698

Molecular Beam Epitaxy

Here the probability of As 4 dissociating is proportional to the availability of two adjacent Ga sites, each As 4 molecule leaves behind two As adatoms, and the kiare temperature dependent rate constants. Further, the desorption of As is assumed to be a second order process and all Ga that arrives sticks. The problem is that at equilibrium, this equation does not satisfy mass-action. For mass-action to be satisfied, the incorporation term for Ga would need to be replaced by ./Ga0. Tsao has suggested[ ss] that the difficulty can be fixed by formulating the kinetic equations in terms of chemical potentials.

e

1.0

0

E

s GeAs

lid

r

\

~

AIGaAs

0.5

~D O Ikl

J P

O

E tO

Z

O.0

600

650

700

Temperature

7.'$0 eC

Figure 16. Measured growth rate (reciprocal intensity oscillation period) versus substrate temperature during the growth of AIxGal.xAS divided by the rate at 550~ As the temperature is increased, the sublimation increases in quantitative agreement with Heckingbottom'sIs1] mass-action analysis. 4.2

Vicinal Surfaces

Intensity Oscillations. Commercial GaAs(100) surfaces are typically misoriented from the (100) by about 1/3 ~ or 5 mrad. The terraces that make up this macroscopic misorientation are on the average approximately 500 ,~, in length and contain about 150 atomic rows. Since intensity

RHEED Studies of the Dynamics of MBE

699

oscillations are easily observed on such surfaces at 600~ the adatom mobility at this temperature must be such that cluster formation competes with step propagation. At higher temperatures or higher misorientations, one finds conditions at which step propagation dominates, with the result that no intensity oscillations are observed. When step propagation of the built-in staircase contributes to the growth, the qualitative form of the intensity oscillations changes. This is illustrated in Fig. 17 where the results from a sequence of growths on a 5 mrad surface, using the parameters as in Fig. 8, are shown. The striking change is that the oscillations are now strongly damped, with minima and maxima rising and falling together. For this surface, the strength of the damping and the qualitative form of the oscillations are independent of the incident beam direction, whether down the staircase or parallel to the step edges.

(t)

:3 JO L O

c

z

o

; o

' IOO

TIME

i 50

I IOO

(sec)

Figure 17. Sequence of intensity oscillations measured on a GaAs(100) surface misoriented by 5 mrad. The growth parameters are close to those used in Fig. 8. There is only a quantitative difference between growth on vicinal and low-index surfaces, but when significant numbers of adatoms can reach the step edges of a vicinal surface, variations in adatom and adcluster density on an individual terrace becomes the focus. As growth proceeds, this terrace moves across the surface. The notion of a layer coverage oscillating between zero and complete coverage does not mean

700

Molecular Beam Epitaxy

as much since, in a monolayer completion time, a terrace just moves a distance equal to its length. In fact, on a staircase, even half-coverage becomes difficult to reach. The more mobile the adatoms, the less change in roughness there can be. As the temperature is raised, the layer-by-layer nature is enhanced by the step propagation at the expense of cluster formation. The minima which correspond to maximum surface roughness are then not as deep. Step Structures on Zincblende. There are two types of steps on GaAs(100) and these might be expected to affect the growth differently. The distinction arises for the same crystallographic reason that GaAs(111) is either As- or Ga-terminated. If the (100) surface is misoriented in the (011 ) direction, then (111) steps will be forced. This is illustrated in Fig. 18. For the same reason, on a low-index surface, each cluster will have two sides bounded by Ga-terminated steps and two sides bounded by Asterminated ones. One expects that these distinct steps might have different kink densities and different probabilities of Ga attachment. Other processes, such as the step-catalyzed dissociation of As4 postulated by Singh,[ 64] might depend on type of step.

-6-~-//~/ 6 / 6

A

~--6--6--6.

6

-

6

/

D0o] ~[oTO =[01I]

6

6

0

As

9 Ga

--6--6~

--6--6--

6

6

,/~a.

o

-~. ~

F

~

~

,'~xf

/I

omo-

w

Rgure 18. Ball and stick model of Ga-terminated and As-terminated GaAs(100) staircases, assuming bulk termination, formed by miscutting from the perfect (100) in the (011 ) and (001) directions. One expects that the real surface will have reconstructed terraces and reconstructed steps. Real steps will meander depending upon the thermodynamics of kink formation.

RHEED Studies of the Dynamics of MBE

701

As described in Sec. 3.3, the signature of a vicinal surface is a specular beam that is split into several components. The angular separation of the components must follow Eq. (11). Figure 26a shows a measurement of the split specular beam for GaAs(100) surfaces misoriented by 2 ~ toward the (011) and ( 0 1 i ) directions. The misorientation-induced splitting, as well as the disorder (broadening) among the terrace lengths, is seen. In Fig. 27a and b, the beam is directed parallel to the ledges and the width corresponds to the kink density. These measurements were taken without growth but in an As 4 flux while the surface was in the c(2 x 4) reconstruction. One can see that the Ga-terminated steps are straight with a large variation in terrace lengths while the As-terminated steps have a high kink density but a narrow distribution of terrace lengths. Upon increasing the temperature or lowering the As flux to go into the 1 x 1 reconstruction, this difference disappears. If growth is initiated, this difference is more difficult to observe. Surface Diffusion. To study surface diffusion, one makes use of the observation that the strength of the intensity oscillations on vicinal surfaces is reduced if the mean terrace length is decreased[ 53] or if the surface temperature is increased. For vicinal GaAs(100) surfaces, Neave and coworkers[TM] fixed the Ga flux, then raised the substrate temperature until the intensity oscillations were extinguished. They then associated the mean terrace length with a diffusion length and extracted a diffusion coefficient at this critical temperature using the relation that D~ = L 2. For they chose the time to deposit a monolayer. The rationale is that if adatoms have sufficient mobility to reach a step edge, then the growth is by step flow. There is no change in the surface roughness during growth. On the contrary, if the adatoms do not have sufficient mobility to reach a step edge, then nucleation occurs, the roughness of the surface changes, and intensity oscillations are observed. After one critical temperature was determined for one Ga flux, the Ga flux was then changed and the measurement repeated. Knowing ~ and L, a plot of the diffusion coefficient, D, versus reciprocal substrate temperature, 1/7, was obtained. For the MBE growth of GaAs, they found thatD = 5.3 x 10 12 exp[0.3(eV)/ksT ]. This result has now been used in several Monte Carlo calculations.[Tz][73] The interpretation of the experiments even within the rationale of the measurements is not clear-cut for several reasons. Mainly, it is not obvious what to take as the diffusion time 9 or diffusion length. Note that there are several characteristic times for the problem. There is the time

702

Molecular Beam Epitaxy

required to deposit a monolayer, the residence time of a Ga adatom, the mean time between collisions of adatoms, and the time for an adatom which after adsorption, diffuses to, and after some number of attempts, incorporates at a step. A second difficulty with the approach (actually diffusion on this surface in general) is that the diffusion of Ga, as well as the Ga sublimation rate,[ sl][58] might depend upon the As flux and on the surface reconstruction.F4] The surface reconstruction, in turn, depends upon the As concentration. When the temperature is raised to extinguish oscillations at the different growth rates, the relative As to Ga fluxes will have changedEthe Ga sublimation rate and the As concentration will also have changed. One diffusion coefficient might be insufficient to describe the process. Finally, it is difficult to determine the probability with which an adatom incorporates at a step. This means that both an adatom that has a short diffusion length and one that has a high diffusion length, but low probability of incorporation at steps will likely nucleate a 2D island, giving rise to RHEED intensity oscillations. It will be difficult to separate the two effects. For example, one would like to use this method to study diffusion anisotropy on GaAs. Figure 19 shows the results of changing the surface temperature at a fixed As flux for 2 ~ misoriented (100) surfaces. The left panel shows intensity oscillations for the case in which the steps are Gaterminated. The right panel exhibits results for As-terminated steps. As can be seen, these measurements give very different results. On the Gaterminated surface, the intensity oscillations are well behaved and the one can usually determine a clear temperature at which the oscillations are extinguished. On the As-terminated surface, however, this determination is not always so straightforward. This is seen in Fig. 19 which shows intensity oscillations for various temperatures near the transition between step flow and 2D cluster formation. The growth rate is fixed. There is a strong difference on the two surfaces but one can not say whether adatoms diffuse more easily in the [01 1 ] direction or whether incorporation at Asterminated steps has a high probability. In determining diffusion over a wide range of growth conditions, one must be especially wary of changes in surface reconstruction. At low growth rates, away from the 2 x 4 to 1 x 1 phase transition, it may be possible to apply Neave's method, but near the appearance of the 1 x 1 reconstruction, some other bond-breaking process appears to be operating.I"8][TM]

RHEED Studies of the Dynamics of MBE

575~

540~

..=. r-

..o tb=

..9 ~

L_._

>I. - =z LLJ

600~

55o~c

Z

610~

703

Figure 19. Intensity oscillations on 2~ misoriented GaAs(100) A and B surfaces. The behavior of the oscillations is very different indicating that diffusive and/or step incorporation processes are different on these two surfaces. On the 2~B misorientation, the method of determining the transition between step flow and 2D cluster nucleation is difficult.

-~_.__~o%

a.

TIME

TIME

In fact, the concept of diffusion during MBE is problematical because of the nonequilibrium nature of the process. [75][7s] Monte Carlo calculations indicate that adatoms adsorbed at the start of a layer migrate very large distances, while those adsorbed near the completion of a layer travel on average only short distances. This strong concentration dependence seems to indicate that it makes more sense to examine the microscopic hopping and cluster probabilities more than macroscopic diffusion. One expects that diffusion in which there are multiple types of sites available for hopping will be equilibrated only if all such sites are visited in the correctly weighted way. Vvedensky [72] has used a diffusion coefficient with a hopping probability that is weighted (exponentially) by the number of near neighbor bonds of an atom at a particular site in its diffusive motion. The argument described in the Sec. 4.2 corresponds to the limiting case of the absence of adatom-adatom collisions and cluster formation so that equilibrium diffusion notions are more likely appropriate. Continuum Analysis. To put this measurement of diffusion on a firmer foundation, Nishinaga and coworkers [78] have associated the appearance and disappearance of intensity oscillations with the transition between step flow and nucleated growth in a BCF formulation. The main feature of their analysis is the solution of the BCF equation, carefully considering the importance of desorption of adatoms versus incorporation

704

Molecular Beam Epitaxy

at step edges. The advantage of their method is that by focusing on the step flow side of the transition, they are able to avoid the intricacies of the coarsening of the 2D islands and the complicated nature of diffusion between island edge sites, adatom sites, and steps. By treating the limit of no nucleation, adatom-adatom collisions can be ignored and they can deal only with incorporation at steps. The method is to solve the diffusion equation, as described below, and then calculate (1) the condition at which islands form and (2) the growth rate on a vicinal surface. In terms of understanding surface diffusion, the second is important since it supplies an independent check on the relevant lengths of the problem. First, we treat the determination of the condition for the disappearance of intensity oscillations in the growth of GaAs. We describe how Nishinaga's model accounts for differences in incorporation at step edges. Simplifying Nishinaga's analysis slightly, write the surface diffusion equation for growth on a staircase of steps, each of equal length. Examining growth at typically low MBE temperatures, we assume that there is no sublimation. Then the adatom density is just determined by a balance between diffusion towards step edges and adsorption. Assuming that the density is too low for nucleation, the time rate of change of the mobile adatom population n(x, t) is given in one dimension by Eq. (19)

On(x,t)

=D

o~t

,92n(x,t) ~t 2

+J

where D is the surface diffusion coefficient and J is the incident flux of Ga atoms. For the coordinate system, choose the origin as the center of a step of length L. Initially solve Eq. (19) in steady state subject to the boundary condition that n(+ L/2 ) = 0. This assumption implies that the steps are perfect sinks, always incorporating incident adatoms. This assumption will be removed shortly, but for now it helps to emphasize the main features of the method. Solving Eq. (19) in steady state, the mobile adatom density on the surface is given by Eq. (20)

nr

J

L2

RHEED Studies of the Dynamics of MBE

705

In this symmetric problem, n(x) has its maximum at z = 0 and is given by n = jL2/8D. The main assumption of Nishinaga[ 78] is that nucleation will occur when this is as large as some critical value

Eq. (21)

ncrit =

o~n~. o

where n,o is the equilibrium adatom population of the surface held at a temperature T and the factor c~ (also slightly temperature dependent) is given by classical nucleation theory.F{}] Then nucleation will occur when Eq. (22)

JGaL2 = 8Dczns

To make this slightly more general, the boundary condition at the step should be n( + L / 2) = nso. This can be seen by first considering equilibrium, where there is no growth. The surface is bathed in a flux of Ga and As that just match desorption. There will be some mobile adatom population on the surface and since there is no growth, the gradient (the diffusive flux to the steps) must be zero. This means that there will be a uniform adatom density nso over the entire terrace. If the Ga flux is increased so that there is growth, this adatom population will increase, except that, at the steps, there can be sufficient adsorption sites to be able to maintain the equilibrium. Any nonequilibrium distribution at the steps is suppressed. With this more reasonable boundary condition, the condition for 2D cluster formation is slightly modified, becoming Eq. (23)

./GaL2 = 8D((z - 1)nso

This condition of perfect equilibration will be relaxed again shortly. In Eq. (23) the flux and surface concentration can now be taken as a unit width of surface, or more simply, as two-dimensional fluxes and concentrations. To make it dimensionless, divide each side by no = 1/a 2, the density of atomic sites in a monolayer of a square lattice with cell side a. Then since J = no/z, this becomes Eq. (24)

L 2 = 8Dz(cz- 1)Oso

where eso is the equilibrium coverage of mobile adatoms and z is the time to deposit a monolayer of Ga or GaAs. For a givens and coverage Oso, the density of mobile adatoms will be sufficiently high that nucleation is the

706

Molecular Beam Epitaxy

preferred growth mode. At lower steady state coverages, nuclei will not be stable, allowing step flow to dominate. This equation should be compared to the model of Neave[TM] in which they took L 2 = D~. The complication in determining the diffusion coefficient, D, is that both D and 0so are exponentially dependent on temperature. It is not possible to separately measure these quantities. The best one can do is to factor out the temperature dependence of the equilibrium flux as follows. Under equilibrium conditions for GaAs(100), the desorption rate of Ga is nso/'r,s where x s is the surface lifetime of Ga. Assuming unity sticking coefficients, this equals the incident equilibrium flux. Using kinetic theow to convert pressure to flux Eq. (25)

nso =

~GaXs ~/2nm k T

or using the law of mass action, Eq. (26)

nso =

Ts

PA'~~f2=mkT

Combining this last equation with Eq. (24), one obtains the condition for the disappearance of intensity oscillations to be Eq. (27)

L2= 8x(c~ - 1)D~ s

PA/"$ d2~mk T

Nishinaga takes Kp = 4.3 x 10 is exp(-4.6/kT). The measurement procedure is once again to, first, fix the growth rate and As flux and choose the substrate misorientation; then vary the substrate temperature until the intensity oscillations disappear. Assuming that the exponential temperature dependence in Kp given here is correct and that mass action applies, one can determine the exponential dependence of the length ~s = (D~s)'~. At higher temperatures this is easily modified to include sublimation in Eq. (23). How well this entire approach works will depend on how clearly the disappearance of the intensity oscillations can be determined and how well the condition for 2D cluster formation is followed. For GaAs(100), Nishinaga[78] finds ~s = 4.0 x 104] exp (Ed/k ~ cm, where Ed = 0.3 eV.

RHEED Studies of the Dynamics of MBE

707

The reasonableness of this measurement can be evaluated by exam-ining the component quantities in the calculation. Using x s = vexp(Ed/k7 ) wherev =, 1012 andE d = 1.7 eV, one obtains at an As pressure of 4 x 10 -s torr and, at a substrate temperature of 550~ a surface lifetime of 2.5 x 10 -3 s and a surface coverage of 0so = 9.2 x 10 8. The latter seems very low, though the ratio gives a reasonable sublimation rate. On a 1~ surface, this equilibrium surface coverage corresponds to one adatom in a terrace of length 160 A separated by other atoms by about 10 s ./~. Especially if the steps meander, this might not be appropriate. Nishinaga calculates (~ to be of the order of 10. At the point of 2D cluster formation, the atoms would, on the average, be separated by 105,A,. These values are smaller than one might intuitively expect for critical cluster formation. To include partial incorporation at the different types of steps that are created when GaAs(100) is misoriented to either the [011] or [011] directions, the boundary condition in the preceding analysis must be changed. We assume that the critical nucleation condition is unchanged. Once again following Nishinaga[ 79] and Burton, Cabrera, and Frank,[ 77] let the equilibration time at a step bex k. Without growth and in equilibrium (an incident flux Jo), the surface adatom concentration on a terrace, nso, must again be uniform. Then, at an edge, the flux to the edge must balance the flux leaving the edge, i.e. n

Eq (28) 9

0

=

anso

jstep v

O

"[K

Here the first term on the right represents the equilibrium flux of nso atoms moving at a velocity of a/x k toward the step. The possibility of reflections is included in the value of x k. The second term represents the flux desorbing from the step and is assumed to hold even away from equilibrium. Away from equilibrium, the density of adatoms next to the step is then given by Eq. (29)

Js-

anstep xk

anso

tk

where Js is the growth flux. In the case of minimal sublimation the growth flux is just the density of atoms in a layer, no 1/a 2, times the velocity of a step, s The growth flux must be divided by two since we are only considering the flux on one side of a step. The boundary condition that =

708

Molecular Beam Epitaxy

must be applied to solving Eq. (19) is then that the adatom density at x = +LI2 is Eq. (30)

nstep --nso +

L 2a 3

Repeating the arithmetic followed before this gives Eq. (31)

L 2 (1 +

4 D Tk

aL

) = 8Dx(ot- 1)0so

Thus, adding to the complication that the exponential temperature dependence of Oso must be determined, one needs to correct the measured diffusion coefficient for differences in ~k. Further, if ~k ==~s, then sublimation must be incorporated into the analysis.[78] This analysis is able to handle (1) the transition between step flow and cluster formation, (2) sublimation rates comparable to growth rates, (3) an average step incorporation time, and (4) a variable As 2 flux. The last is especially important since one expects that, at low As 2 to Ga flux ratios (or As4), the Ga mobility over the surface should be exceptionally high. This last can be seen in the widths of the RHEED beams and in the success of the technique of migration enhanced epitaxy (MEE).[8~ In Nishinaga's analysis, the diffusion coefficient is independent of the As flux. Instead, the As flux enters by controlling the equilibrium surface concentration, "so. This has two effects: the first and main one is that at reduced As flux, the critical concentration for 2D cluster formation is increased because the equilibrium surface concentration, "so, and hence o~nso is increased. This more difficult requirement on 2D cluster formation means that Ga adatoms are more likely to reach a step edge and incorporate. The second effect of a reduced As flux and increased nso is an increased sublimation rate. These are somewhat indirect. There is not yet quantitative consideration in changes in step incorporation[ 8~] or As dependent changes in the defect densities. Nishinaga has also examined the validity of the diffusion times and diffusivities in a beautiful consideration of the variation in mole fraction of InGaAs growth on vicinal surfaces, that doesn't rely on a calculation of densities for critical nucleation. He compared, measured, and calculated mole fractions versus misorientation angle. The main effect is obtained by noting that for mobile adatoms that have large sublimation rates, the chances of their incorporating at steps increases if the mean distance to a

RHEED Studies of the Dynamics of MBE

709

step edge is reduced. The method is similar to the preceding analysis, also solving the diffusion equation to find the incorporation rate of mobile adatoms in the step flow regime. For the case in which the terrace length L is much less than the diffusion length ;ks, the incorporation rate is found to be approximately JGa" nGa/Ts, i.e., the excess above the equilibrium sublimation rate. For the case in which the terrace length is several times the diffusion length, the incorporation rate is Eq. (32)

(3in- n,o 4

2~. s

L

If the equilibrium sublimation terms are included when calculating the expected mole fraction x, then the measured mole fraction of In, Xmeas, is given by Eq. (33)

Xmeas =

L/~.s + x(1 - L/;k s)

This gives results close to the exact formulation of Nishinaga at high values of L/X s. Note that when considering the sublimation, the As flux affects both the Ga and In equilibrium adatom densities and that stoichiometric InxGal_xAs is required. How well this formulation works suggests that the step flow model of growth and diffusion lengths are quite reasonable. Knowing the onset of intensity oscillations is then important only for ensuring that the growth is in the step flow regime. Collision Time Analysis. Kuroda[ 82] has suggested that the collision time between adatoms is a more appropriate choice to determine the condition at which intensity oscillations disappear. In this model, the assumption is that a dimer constitutes a smallest stable nucleus. Substrate temperatures are assumed to be sufficiently low that sublimation does not occur. The rate at which dimers form is proportional to the square of the adatom density so that the time dependence of the adatom (monomer) density is Eq. (34)

a,,(t) ot

= J-

On, 2

where n(t) is the adatom population and J is the incident flux. This has solution, assuming that n(0) = 0, of

710

Molecular Beam Epitaxy

Eq. (35)

n(t) = n ( = ) tanh(t/Xc)

where x c = 1 / ~ / J D and n(=)= ~ is the steady state population of adatoms. The time xc is interpreted as the mean collision time as adatoms are deposited. The density of dimers is easily seen to be n(oo)(x - tanh x), where x = t/x c. The argument is that at a time t = x c the monomer density is roughly comparable to the dimer density of the order of ]~J~D. At this point the density of nuclei will not increase since collisions with nuclei that are already established is likely. When this critical density of nuclei have formed, the mean area per nucleus is the reciprocal of the density or This is the capture area of each nucleus and subsequent deposition is incorporated into these nuclei without the formation of new ones. The diameter of this area is approximately Eq. (36)

;kc = ( D / I ) v"

and Kuroda argues that RHEED intensity oscillations will damp strongly when half a terrace length equals ;kc. The condition that should be compared to that of Neave is

[,4 Eq. (37)

L)~-

16a2

where L is the mean terrace length of the vicinal surface, x is the monolayer time, and a is the side of the unit square mesh (J = 1/xa2). This third estimate will give the same activation energy for diffusion as Neave but will give a larger pre-exponential factor. Comparison. The determination of diffusion from the point of rapid damping of RHEED intensity oscillations is thus seen to be largely dependent on the choice of the size of the stable nucleus. The surprise is that the dependence on the terrace length size is very different. In the Neave model and the Nishinaga model, the point at which intensity oscillations are extinguished depends on the square versus Kuroda's fourth power of the terrace length. Especially for the latter, at a given substrate temperature, the strength of intensity oscillations will quickly decrease as the terrace length is reduced. Step Bunching. The growth of AIxGal_xAs shows a very clear case of step train disordering. This was observed by Tsui and coworkers[ a3] who found that AIGaAs grown on GaAs(100) exhibited a textured surface

RHEED Studies of the Dynamics of MBE

711

morphology depending upon the direction of the substrate misorientation. This macroscopic disordering can be followed by RHEED during growth. As shown in Sec. 3.4, the characteristic diffraction pattern of a vicinal surface consists of split peaks. When the misorientation is large,[ 32] the separation of the peaks is determined by the misorientation and the shape of the components is determined by the disorder. If the terrace lengths in the staircase become more disordered, the components will broaden. Our method is to measure the shape of the split beams versus time during growth on the two types of zincblende step terminations. As a function of the AI mole fraction and substrate temperature, the step train order as described by RHEED shows a strong variation with time during growth. Both the Schwoebel effect[ 84] or step pinning by impurities[85]-[87] could be involved. Figure 20 shows the RHEED profile during growth of AIxGal_xAS with x = 0.25 on a GaAs(100) substrate that was misoriented by 2 ~ toward the (111)A. For this growth, the ratio of column V to column III flux was about 2, the growth rate was 1 pm/hr, and the substrate was held at 675~ This diffraction profile was measured with the incident beam pointing down the staircase of steps on the surface. First, one should note that though the two components forming the split specular streak were of nearly equal intensity and width at 580~ at this relatively high temperature where one can grow smooth AIGaAs, the relative intensity of the two peaks are very different. This is a reversible change with temperature[ 8s] and could be due to a reconstruction change from a 2 x 4 to the 1 x 1 that exists at these high temperatures. Second, upon growth there is not much of a change and, for this case, after an hour there might even be a slight ordering. By contrast, for growth on a GaAs(100) surface misoriented toward the (111)B, at very close to the same conditions, the staircase disordered. This is seen in Fig. 21 where after starting with split components that were sharp, growth for 40 min left a surface with only a weakly resolved, characteristic step diffraction pattern. This is reversible in the sense that if GaAs is grown on top of this surface, the original ordered step train is developed. The results[85][8s] for a variety of temperatures are shown in Fig. 22 for both A and B step terminations. Here the width of the components of the split peak corresponds to the variance in the terrace length distribution as described in Eq. (14). For the data, one sample was used; after each growth a buffer of GaAs was grown and the surface annealed in an As 4 flux to obtain a sharp diffraction pattern. Differences in the initial peak width reflect this procedure. The main point is that the amount of disordering is

712

Molecular Beam Epitaxy

reduced if the substrate temperature is raised or if the A step termination is used. Further, there is some disordering even on surfaces with Gaterminated steps. Finally, at some temperatures the disordering begins immediately, while at others there is a slow initial delay before the disordering begins in earnest. For both surfaces and at all temperatures, the disordering is more rapid as the AI mole fraction is increased.

I jJ/hr AIo,6Gao.TsAs A surface ...~.V=

peak half

675~

width

Figure 20. RHEED profile during growth of AIxGal.xAS with x = 0.25 on a GaAs(100) substrate that was misoriented by 2 ~ toward the (111)A. For this growth the ratio of column V to column III flux was about 2, the growth rate was 1 /Jm/hr, and the substrate was held at 675~

0 min .'~_. r .a t.

o >I-

Z W IZ

2.5 mrad

60m,n

) ef

Figure 21. RHEED profiles of the split specular beam during the growth of AIxGal.xAS on a GaAs(100) substrate that was misoriented by 2 ~ toward the (111)B. For this experiment, the sample was mounted next to the sample used in Fig. 20, so that the growth conditions are essentially identical for the two experiments.

AIo.tsGa=zoAs, As steps

/t II IJ

I jJ/hr

i i

i

e i = 72 mrad

i

"'1 I""

FWHM (mrad)

IO

20

50

2

60

70

ef (mrad)

80

90

R H E E D Studies of the D y n a m i c s of M B E

B ~

V =2.5

I~/hr

'"

6

o

635 ~

5-

61501655 ~ / / / 675~

"-~*' =

Al~

surface

713

3

A

surface

~

660 ~

~

2

~....-~------'='-'-----"~

~"

"

"

635 ~

It 0

0

I

I

I

10

20

30

time

I

40

(min)

Figure 22. The peak halfwidths of one component of the split specular beam versus time during growth for several substrate temperatures for both A and B misorientations (2~ From Eq. (14) the halfwidth corresponds to the variance of the distribution of terrace lengths.

Gilmer[ 88] has developed a simple rate equation model to describe the step bunching process in terms of one parameter. This parameter reflects the asymmetry in the ease at which an adatom can cross a step from different directions. Later the model was reworked by Tokura et al. to obtain a relation for the time dependence of the variance in the terrace length from the mean. The basic mechanism is illustrated in Fig. 23. Here a staircase of steps is shown with terrace length T,, for the ,th step. The growth rate is one monolayer in 9 seconds and one assumes that there is only step flow. The parameter~l describes the asymmetry in attachment to the neighboring steps. If adatoms striking the nth terrace could attach at the (n-1)th terrace as easily as the (n+l)th terrace then ~1= O. If, however, there is an asymmetric barrier for crossing a step from the right over the left, then, as shown, there might be (1+ ~i)/2 enlarging the (,-l)th terrace and (1- Xl)/2 enlarging the ,th terrace. There are four such terms which, when summed, yield a rate equation describing change in the length of the nth terrace, %,:

714

Molecular Beam Epitaxy

Eq. (38)

dT, ~

1 + ~1 =

dt

2x

1 - TI

(Tn+1 - 7".) + - - - - - ( T n - Tn.1) 2x

To solve this,[ 89] expand in a finite series and assume periodic boundary conditions. Set T, = ~ T q e iqn

q = 0,1, ...,

N-I N

2~[

to obtain the variance of the step terrace length distribution Eq. (39)

AT,, = olT.(O)12 exp[-4~l(~)sinZq/2]

Depending upon the initial distribution of terrace lengths and on the asymmetry parameter, one can determine the subsequent behavior. If this asymmetry parameter Xl is positive, then the staircase orders; but if it is negative, the fluctuations diverge. For ordering, one requires that there is a feedback mechanism by which small steps grow faster than larger steps. One should note that the initial distribution is important since if there is no initial disorder, then there is no reason for the growth to prefer one step over another so that no change will take place. If however there is an initial difference between steps, then an asymmetry can take over. To illustrate this, assume that initially the T,, = T + d, with a random distribution. Then the Fourier components are identical and a calculation of Eq. (39) for several values of ~1is shown in Fig. 24. Only slight values of asymmetry are needed to achieve modest agreement with the data. For some experimental curves, there is an initial delay that cannot be fit with this assumed random initial terrace length distribution. Finally we should note that impurities could also be important. They could adsorb randomly, pinning some steps at the expense[ 89] of others, producing terrace length disorder.

RHEED Studies of the Dynamics of MBE

715

1/r monolayers / sec

1 ~+~

!'--

1

Figure 23. Definition of the asymmetry parameter 11 that gives either step ordering or step bunching.

TERRACE WIDTH ROUGHENING/ORDERING 6000

~/=-0.001

5000

~=-0.0005

4000 b

3000 2000

1/=-0.0001

1000 7=0.0001

o

1'o

2'o

3'o

4o

TIME (minutes) Figure 24. A calculation of the terrace length roughening or ordering from Eq. (39). The small values of ~1 give modest agreement with experiment. The long delay before disorder initiation in some of the data of Fig. 22 cannot be fitted with the assumed random initial distribution of terrace lengths.

716

Molecular Beam Epitaxy

Step Meandering. Though we do not know the detailed structure of the Ga-terminated or As-terminated steps, it does appear from simple analysis of the diffraction data that Ga-terminated steps are much straighter than the As-terminated ones. The picture of the GaAs(100) surface that is deduced from the data is schematically illustrated in Fig. 25. This result is similar to the difference in kink density of the two types of steps that can form on Si(100).[7][9] The diffraction patterns from surfaces with these two types of step termination are strikingly different. If two samples are placed side by side on the holder, and the beam switched from one to the other even the eye can distinguish the degrees of order. Note that the A and B misorientations can be prepared by taking a wafer that is polished on both sides, cleaving it into two, and then mounting both but with one turned upside down. Examination of the 2 x 4 reconstruction and determination of the staircase direction from RHEED[ 9~ or from x-ray diffraction quickly gives the step termination.

A

B

Rgure 25. Schematic diagram showing the order (disorder) in the two types of stepped surfaces that can be created on GaAs(100). The data to be presented in the next figures show that Ga-terminated steps are straight while there is severe terrace length disorder. For As-terminated steps, the step edges meander though the variation in terrace lengths is less. This holds for GaAs(100) at 600~ with the 2 x 4 reconstruction and not the 1 x 1 reconstruction.

RHEED Studies of the Dynamics of MBE

717

To determine the terrace length order, the incident beam is directed down the staircase of steps and the diffracted beam intensity is measured along the length of the streaks. This is described in Sec. 3.3. The results for the two types of step termination are shown in Fig. 26. For these data two GaAs wafers were mounted side by side on the sample holder to minimize differences in sample history, incident flux, and temperature. A GaAs buffer was grown by MBE and the measurements were made under an As 4 flux and with the surface in a 2 x 4 reconstruction. Note that in the top curve of the figure, the cut off peak is an artifact. The main result is that the components of the peaks for the Ga-terminated surface are much broader than those for the surface with As-terminated steps. This broadening is removed if the temperature is raised to cause the higher temperature 1 x 1 reconstruction and can be recovered reversibly. Based on the analysis of Sec. 3.4, we estimate that the rms deviation in terrace lengths twice as high for the Ga-terminated step structure.

e i = 7 2 mrad

/(~

stGeaps

/4.6 mrad

3/ W...

t-

L_

>..

F-

A z__

"

....

il

I

60

I

s, e s / i 1"6 mrad

I

I

70

I

80

90

Gf (mrad) Figure 26. Intensity profiles along the specular beam for GaAs(100) misoriented by 2~ The flat top is an artifact. The Ga-terminated stepped surface gives broad components corresponding to terrace length disorder. The As-terminated stepped surface exhibits much less disorder in terrace lengths.

718

Molecular Beam Epitaxy

To examine the meandering, the electron beam is directed parallel to the steps. Once again the difference between the two step terminations is striking. The data are shown in Fig. 27 and are from the same samples as in Fig. 26. The top curve corresponds to the As-terminated surface (the beam is in the [011] direction parallel to the steps) and the bottom curve corresponds to the Ga-terminated step surface. For these data, a slit detector was used to integrate over one of the split components. At this point, these measurements have not been evaluated over the range of scattering geometries, terrace lengths, and substrate temperatures to make quantitative statements about observed trends. Nonetheless, the results are reproducible for a few misorientations and many sample preparations, indicating that the Ga-terminated steps meander less than those that are As-terminated. Recently, Pashley[ 91] has confirmed these results with scanning tunnelling microscope images of A and B GaAs(100) surfaces. In addition, he has given an electron counting argument describing the relative stability of the various structures that are observed and not observed.

|

!

i

d

i

I

"

Ga

E)i=30mra A a.

steps .

mrad

D

As steps

z__

b.

---

I0

20

*-- 6.5 mrad

50

40

ef (mrad)

50

60

Figure 27. Intensity profile of one of the split components with the incident beam directed parallel to the step edges. A slit detector is used. The Ga-terminated steps meander less.

RHEED Studies of the Dynamics of MBE 4.3

719

Strained Layer Growth

Recent interest in device applications for strained epitaxial layers has focused investigations on the influence of coherent strain in the kinetics of epitaxial growth.[ 92]-[94] Besides devices such as MODFETs and quantum well lasers that utilize coherently strained epitaxial films in active regions of the device, one can use strain to tailor the band structure of materials or to combine optical materials with Si technology. In III-V MBE, an important system is the growth of In=Ga1_~,s on GaAs(100). The lowest energy state of an epitaxial film with a lattice parameter that is slightly different than the substrate is coherently strained to accommodate the mismatch. In this case, known as pseudomorphic growth, the in-plane strain in the film is equal to the lattice mismatch. Since the lattice parameter of InAs is approximately 7% larger than that of GaAs, strains of between 0 and 7% can be induced by varying the mole fraction of In contained in the alloy. A major consequence of the very large strains used in these films is the generation of dislocations at the film/substrate interface that can relieve the misfit strain and allow the film to relax toward its bulk value. The thickness at which dislocations are created is termed the critical thickness and is an inverse function of strain and film thickness. At this point, the effect of strain on nucleation and growth of clusters, surface reconstruction, step propagation, or on surface adatom mobility is not well understood. We cannot predict which lattice-mismatched systems will give pseudomorphic growth or, even, epitaxy. Intensity Oscillations. The main difference between the growth of InxGa l_xAs and GaAs is that the former contains two components with very different surface mobilities. This could partially be due to strain but is also due to the weaker bonding of In. This can be seen to some extent in the conditions under which intensity oscillations are observed during homoepitaxy on InAs(100) substrates. Like the case for GaAs(100), on InAs(100) the surface mobility behaves as if it is very As-dependent. Intensity oscillations are observed under growth conditions giving the Asrich c(4 x 4) surface reconstruction at around 430 ~ C and cannot be observed in the In-rich C(8 x 2) on a surface with a misorientation as small as 0.50.[ 18] In comparison with Ga on GaAs(100), the In adatoms are quite mobile. InxGal_xAS grown on GaAs(100) has a lattice mismatch from 0 to 7% depending on the In mole fraction in the film. The measured specular intensity during growth of InxGa ~oxAS at 510~ as shown in Fig. 28 exhibits

720

Molecular Beam Epitaxy

RHEED intensity oscillations over the entire range of x, thus yielding the film growth rate and the mole fraction, assuming the GaAs growth rate is known. Knowledge of the mole fraction allows calculation of the misfit strain via Vegard's law, which states that the alloy lattice parameter is a linear function of the respective mole fractions. For the data in Fig. 28, the growth rate was 0.3 layers per second, x = 0.33, the (100) substrate was misoriented by less than 1 mrad, and a GaAs buffer was grown at 580~ prior to the InGaAs growth. The intensity oscillations in Fig. 28 are different from those in Fig. 27 in several respects. First, and most striking, the envelope of the diffracted intensity oscillations from the pseudomorphic film decreases more rapidly than in the case of homoepitaxy. The fast damping is followed by relatively weak but sustained oscillations. The change in the envelope appears to be different for the case of compressive versus tensile strain,[ sg] which we speculate has to do with a changing surface Debye-Waller factor. Berger[s7] and coworkers have suggested that the rapid decay is due to enhanced adatom nucleation, causing increased surface roughness. There is also an increase in the diffuse background between the diffracted streaks, indicating a rising point defect density. Finally, at a strain-dependent thickness, there are new diffraction features appearing as inverted V's or chevrons, at points of the bulk diffraction beams. These are interpreted to correspond to transmission diffraction and refraction through 3D clusters on the surface.[ ~8] For InGaAs these clusters have 114 facets. 100 A

,'t:t K::

t~

KZ ,e..m

c m

E

GI m A

0 0 O

|

0

10

20

30

40

i

50

Time in Seconds

Figure 28. Intensity oscillations during the growth of InGaAs on GaAs(100). The In mole fraction was 0.33 and the substrate temperature was 510~

R H E E D Studies of the D y n a m i c s of M B E

721

A simple picture that might explain this behavior is that after some critical thickness, the strain energy in the film is so great that misfit dislocations form at the interface. The strain field around these dislocations is now such that at the surface, in the vicinity of the dislocations, there is partial relaxation. Adatoms on the surface, seeking to find the least costly adsorption site in terms of having to distort its bonding, will nucleate in these regions, producing three-dimensional clusters. This is consistent with results on the (100), but recent work on GaAs(111),[ 95] indicates that dislocation formation does not necessarily preclude strong layer-by-layer growth, as indicated by persistent RHEED intensity oscillations during InxGa+_xAs growth on that surface. Though diffraction techniques are not very sensitive to the onset of dislocation formation, it is useful to plot the film thickness at which chevrons are observed versus misfit strain. This is shown in Fig. 29 for InxGal_xAs on GaAs(100). [18][96][98][99] For comparison, the MatthewsBlakeslee (MB) critical thickness is plotted as the solid curve.[ 68] These measurements delineate two regimes of 3D cluster formation. For those films with strain greater than 2%, the measured "critical" thickness is less than the MB prediction; for those films with strain less than 2% the critical thickness is larger than the MB value. The latter does not contradict the MB model since sluggish relaxation due to a variety of kinetic limitations can affect the measured critical thickness.[ 93] For the former, Price [97] has included a surface energy term to account for the discrepancy. Alternatively, a different mechanism could be responsible for the relaxation and 0.06

.~

0.04

N

0.02

0.00

0

|

J

0

50

0

,

a

o

0 0

I

l

1O0

150

Film Thickness in A Figure 29. The measured thickness at which 3D features are observed in the diffraction pattern versus mole fraction of In. The solid line is a calculation of the Matthews-Blakeslee critical thickness for single kink relaxation.

722

Molecular Beam Epitaxy

decay of the RHEED intensity oscillations. Orr[ ~~176 and Srolovitz[ TM] have suggested that at strains larger than about 2%, a surface instability is responsible for roughening the surface at some critical thickness. This thickness is dependant on surface free energies as well as on surface kinetics. Lattice Relaxation. A more direct measurement of the generation of dislocations during lattice mismatched epitaxy is to follow the surface inplane lattice parameter during growth and when growth is interrupted. When dislocations form to relieve strain in the pseudomorphic film, the lattice constant should revert toward its unstrained, bulk value. Since RHEED senses only the last few atomic layers, in-plane lattice constants can be measured without averaging over the entire film. This was done first for the case of InGaAs growth on GaAs(100) by Whaley[ 17] and then confirmed by Berger.[ 7~ The method relies on the most basic diffraction measurement: in any diffraction technique the separation between two beams is inversely proportional to the lattice parameter. To avoid having to know the distance from the sample to the screen and to know the energy of the beam accurately, measurements are performed relative to the GaAs substrate. Figure 30 shows the apparatus for measuring the angular separation between two diffracted beams. The positions of two beams must be measured to eliminate errors due to drift. (In principle, the position of one beam relative to the fixed specular beam could be measured.) In this method, two diffracted beams are focused on the entrance slits of two separate detectors. The slits are oriented parallel to the diffraction streaks. Then the beams are magnetically deflected across the stationary slits to obtain the two intensities versus applied field. As each beam is swept across the detectors, a signal is measured that near each maximum approximates a parabola. These data are then fit to a parabolic function and the center determined. Knowing the deflection produced by the field, one can measure relative changes in the separation of the beams. The biggest source of error in the measurement is due to changes in shape of the diffracted beams during growth, i.e., the parabolas become broader. Likely because of a combination of multiple scattering and disorder,[ l~ the beams are broadened asymmetrically, mimicking changes in lattice parameter. Though precise measurements are difficult, if sufficient care is taken it is possible to use RHEED to measure the lattice parameter to about 0.003 .~. j-

RHEED Studies of the Dynamics of MBE

Phosphor\ Screen " ~

,

GaAs Sample '

723

Detectors

'|I

~

Gun

diffracted beamssimultaneously.

Aperture

Figure 30. Schematic of the apparatus used to measure the separation between two diffracted beams. The beams are magnetically deflected across slit apertures. The asymmetry of the beams is the limit of the measurement.

Figure 31 shows the results of measurements of the surface lattice parameter for the growth of InxGa l_xAs with x = 0.33 for various substrate temperatures. At 510~ smooth layer-by-layer growth is observed until, after about 13 layers, the lattice parameter abruptly begins to increase. At this point, one also observes chevrons in the diffraction pattern. Hence we associate 3D cluster formation with the measurable change in lattice parameter. For the measurements, care was taken to only measure the separation between diffraction streaks, away from the positions of the diffuse, bulk-like 3D chevrons. At lower temperatures the lattice relaxation is suppressed; intensity oscillations are observed to continue as well-both of these indicate that kinetic factors limit the formation of dislocations. A surprise was that if growth were interrupted while maintaining the As 4 flux, then even the higher temperature data ceased to relax. This further suggests that surface properties strongly affect the kinetics of the dislocation generation and motion. Current work has attempted to use adsorbates to modify the relaxation. For example, Sn which rides the surface during growth might be expected to act as a surfactant and change the critical thickness, relaxation rate, and growth mode. But no effect was observed.[ 95] By contrast, in the growth of SiGe films, Sb has been observed to modify the transition to 3D cluster formation. [1~

724

Molecular Beam Epitaxy

5.705 510 ~ .< ,.m

5.695

c m r

c 0 0

5.685

500~

0

_.~ 5.675 470~ A,

5.665 0

I

10

-

i

,

a

,

I

,

|

20 30 40 50 Film Thickness in Monolayers

-

J

60

-

70

Rgure 31, Surface lattice parameter versus measured film thickness during growth of InxGal.xAS with x = 0.33 at several substrate temperatures. Strain relaxation is suppressed at only slightly reduced growth temperatures.

5,0

SIMPLE GROWTH MODELS

Our goal is to apply electron diffraction to understand the time evolution of surface structure during MBE growth. We would like to understand changes in the long range order of step trains on vicinal surfaces, the agglomeration of islands, the preferential evaporation of one of the components of an alloy, and the diffusion or migration of adatoms. The purpose of this section is to develop an intuition for the requirements of oscillatory behavior in layer-by-layer growth. For this it is useful to have some simple limits in mind. This will give two important capabilities: first, if the model is not too crude and contains some elements of the data, then one can calculate the diffraction. In our case, this will illustrate some of the difficulties of comparing calculation to measurements from real instruments. Second, one can appreciate the difficult job that many-parameter Monte Carlo calculations face in distinguishing among growth modes. In this section, simple growth models with varying amounts of surface diffusion will be developed. Unfortunately an essential ingredient, cluster formation and dissolution, has not yet been incorporated into the models. We consider growth on a low-index surface first. As shown in Fig. 32, imagine a (100) surface with atoms or scatterers distributed in such a way as to produce clusters and clusters on top of clusters. Let the vector

RHEED Studies of the Dynamics of MBE

725

L,.., I Q

|

e

' n

en+l

en-1

Figure 32. Model of growth on a low-index surface showing the layer coverages 0,,. In addition, growth and interdiffusion processes are indicated.

from an origin to an atom at r to be given by r = x + nd~ where d is the interlayer spacing and n is an integer. Two useful quantities are the layer coverage of the nth level, 0,,, and the exposed coverage, c,. The layer coverage is the total number of atoms on the nth level divided by the total number of sites. The exposed coverage is the fraction of sites on which there is an atom that is a topmost surface atom. A simple relation exists between them: Eq. (40)

c,, = 0,,- 0,,+1

Then the singly scattered diffracted amplitude from top layer scatterers only is just Eq. (41)

A(S) = ~ f(E,O)e is'r

where f is the atomic scattering factor, the momentum transfer is S = kfk i where k = 2n/;k. is the electron wavevector, E is the electron energy, and is a scattering angle. Here the sum only includes those x and n that correspond to scatterers occupying top layer sites. As in Sec. 3.1, take f = 1 so that Eq. (42)

A(S) = ~ eiS,,"x e iS:''d X~/ll

If there are N sites per level, then at S ii = 0 this is further simplified[ 1~ to yield

726

M o l e c u l a r B e a m Epitaxy

oo

Eq. (43)

oo

A(S II = 0'Sz) = N ~ c,,e/s='M= N ~ (On - 0,,+1) e/s`M n=O

n=O

The diffracted intensity l(S~) (more precisely, the interference function)DO~] is calculated by taking the square modulus. For a perfect instrument, this would be the peak intensity of, for example, the specular beam. It is essential to note that by performing the sum at S II = 0, one completely neglects the intensity due to the lateral distribution of adatoms. It is difficult to extract the peak intensity from a real measurement. The main modification will be that the total intensity never drops to zero. In what follows we will calculate 0,,(t) for several models of growth. The diffracted intensity given by the square modulus of Eq. (43), apart from a factor of N, will be calculated and compared to the rms roughness. With a growth rate of 1/~ monolayers per second, the roughness is given by:[ TM] oo

Eq. (44)

A2 = ~ (n- t/z) 2 (0,,- 0,+1) /1=0

5.1

Perfect L a y e r - G r o w t h

First consider the extreme limit that every atom deposited goes into the topmost unfilled layer until that layer is completed. For example, for a growth rate of 1/~ monolayers per second, one has during the interval 0 =; ts~that Eq. (45)

0o = 1 01 = t/~ O,, = 0 n > 1

This gives a diffracted intensity at 3'zd = ~ of [1 - (2t mode)] 2 and an rms roughness of A2= (tmod~)[1- (tmod~)]. The coverages, diffracted intensity, and rms roughness are plotted in Fig. 33, with each repeated every monolayer. In the figure, the straight line segments are the coverages, O,,, increasing with slope 1/~. The calculated peak diffracted intensity decreases from unity to zero at half coverage and then increases back to unity. This cusp-like behavior has been observed by Van Hove et al.[ TM] The rms roughness is also periodic, being roughest when the intensity is zero.

RHEED Studies of the Dynamics of MBE

727

1.2

W

0.8

(.9 <~ Of W 0.4

> 0

0.0

0

0

.................... 2 4

6

8

10

1.0

Z

~0.5JI

/

/

1111

o.o!V

0

2

0

2

U') U') 1.0 m Z T O Z) 0.5

.............. 4

6

8

10

4

6

8

10

o rY o3 s

o.o

TIME ( t / ' r ) Figure 33. The expected coverages, diffracted intensity, and rms roughness for perfect layer growth.

5.2

Nondiffusive Growth on a Low-index Surface

A more realistic limit would be to assume that once an atom impinges onto an exposed portion of a layer, the adatom is confined to that layer. It is not allowed to cross a boundary defined by a step edge. This model might approximately hold for growth at low temperature or at high rates. After a cluster is formed, a smaller cluster could grow on top, and then on top of that, until the surface becomes very rough. With these severe restrictions, a simple recursion can be derived. If the net growth rate is once again 1/~ monolayers per second, then the fundamental equation describing this growth mode is

728

Molecular Beam Epitaxy

d01t

Eq. (46)

dt %(0

- (l/x)(0,,.~ - 0,,) = 1

0,,(o) = o

Here 0,_1 - 0, is the fraction of the area of the nth layer that is unfilled. Like solid-on-solid models, no overhangs are allowed. This is a birth-death model[l~ 11~ since the growth on an unfilled layer is rapid, while the growth of a nearly completed layer is slow. These equations are easy to solve beginning first with n = 1. The solutions are Eq. (47)

01(t ) = 1 - e"t/x

Eq. (48)

0z(t) = 1 -

e-W

or in general,

Eq. (49)

n

O.(t) = 1 - e't/x.~ (t/-[)J/j.l j=o

The first few solutions, for a growth rate of one monolayer per second are shown in Fig. 34. As expected, the coverages are qualitatively similar to the perfect layer-growth model, but before one layer is complete, another begins. The important change here is that neither the rms roughness nor diffracted intensity is cyclic. Instead, the intensity at Sii = 0 and Sz = n/d, shown as the solid line, decreases rapidly to zero. Substituting Eq. (49) into Eq. (43) gives Eq. (50)

I(Sll = O, S, =n/d) = e "4t/x

This falls off very quickly with the number of layers deposited, t/~. In this model, where there is no inducement for layers to fill in, the surface becomes progressively rougher with ~2 = t/~. Notice that in all models the intensity will decrease as in Eq. (50) during the initial nucleation, since only the first layer is being filled and there is no transfer between layers. Recently, Evans has shown that if site exclusion is included, then very weak oscillations can be obtained.

RHEED Studies of the Dynamics of MBE

729

1.2 ILl 0.8 < n," i,i 0.4 > O O 0.0 o

2

4

6

8

10

8

10

1.0

Nc Z ~ 0.511

N~s n -& 0.0

03 4.0 (/3 Ld Z -1(_9 ~D 2.0 0 n,"

0

2

4 .

.

.

.

.

6 .

.

.

U') ~,

0.o

TIME ( t / T )

Figure 34. Solution given by Eq. (49) of coverages with growth without diffusion. 5.3

Diffusive Growth on a Low-index Surface

Adding the possibility that an atom can jump to a lower level gives a model intermediate between perfect layer-growth and non-diffusive growth. Schematically the differential equations would be Eq. ( 5 1 )

dO,, = (l/z) (0,. 1 - 0,,) + dumps from n + l to n) dt - (jumps from n to n-l)

A variety of schemes could be used for the last two terms.[ 1~176 One could also allow jumping, with different probabilities, from lower to upper levels. In keeping with the birth-death approach, we assume a jump rate in going from, for example, n+l to n that is proportional to the product of the available space on level n and the uncovered area on level n+l. The reasoning is that once an atom is covered, it is not able to diffuse to an edge. Then Eq. (51) becomes

730

Molecular Beam Epitaxy

Eq. (52)

dO,,

dt

= (1/~)(0,,_1 - 0,,) + k(e,,+l - 0,,+2)(0,. 1 - e,,) - k(0,, - e,+l)(e,,. 2 - 0,.1)

The coverages, diffracted intensity, and rms roughness of this particular nonlinear model are remarkably easy to evaluate numerically, subject to the conditions eo(t ) = 1 and 0,,(0) = 0. The results are shown in Fig. 35 with k = 100. Several points are worth noting. First, the calculated peak intensity drops rapidly as soon as growth begins. This is similar to what is observed experimentally, though there, one expects reconstruction changes and perhaps multiple scattering to be more important. Second, there is a phase shift so that, though the period of the intensity is ~, the peak positions do not correspond to integral average film thicknesses. Third, both the rms roughness and diffracted intensity continue undamped, more similar to Ge or Si(100) than GaAs on GaAs(lO0). Finally, as expected, the coverage plots are intermediate between non-diffusive growth and perfect layer-growth. a~ 1.2j I,I (.9 n," i,i > 0 0

O.8 0.4 0.0

0

1

. . . . . . . . . . . . . . . . . . 2 3 4 5 6 7 8 9

10

1.0

~

0.0: U3 O3 IM Z -r

t// v

0

1

2

3

4

5

6

7

.8

9

1.0

0 ,

:Z) 0.5 0 n,"

:~ 13::

0.0

0

1

2

3

4

5

6

7

8

9

0

TIME (t/'r) Figure 35. Solution of Eq. (52) with k = 100. The curves are intermediate between perfect layer growth and non-diffusive growth. The slopes of the coverages are just the growth rates. How the layers are completed determines the form of the intensty oscillations.

RHEED Studies of the Dynamics of MBE

5.4

731

Diffusive Growth on a Vicinal Surface

It is difficult to apply the birth-death approach to a staircase of steps because there is no distinction between the statistics of different levels. Instead we examine the solution to a one-dimensional diffusion equation for the simple case in which there are no clusters. Parts of the analysis should carry forward to that more complicated case. The main result here is to see how moving steps modify the classic Burton-Cabrera-Frank solution. The solution to this differential equation will show that step velocity oscillations are associated with RHEED oscillations and that the envelope shown in Fig. 17 arises in a natural way. As a model appropriate to vicinal surfaces, consider a one-dimensional staircase with equal terrace lengths. Describe the adatom concentration on a terrace by the probability n(x) that a site x is covered by a mobile adatom. Here, x is measured with some origin that is fixed with respect to the bulk. We assume no clustering and no evaporation. Further we assume that the upper and lower edges of terraces are perfect sinks for mobile atoms and we assume periodic boundary conditions. In the fixed reference frame, the diffusion equation is Eq. (53)

On(x,t)

=D

Ot

02n(x,t)

+

OX2

1 at

where D is the diffusion coefficient of non-interacting adatoms, a is the inter-row separation, and 9 is the time required to deposit a monolayer. This equation says that adatoms arrive at the surface at a prescribed rate and diffuse to the top and bottom step boundaries. A gradient of mobile adatoms is set up, thus driving the diffusion. The steps are moving at the same rate, by periodicity, so that the terrace, which has lengthL, is moving across the surface with a velocity v(t). In order to apply the boundary conditions that the mobile adatom concentration be zero at the steps edges, change to a reference frame that is moving at the step velocity, v(t), via

Eq. (54)

u = x - ~oV(q)dq

so that Eq. (53) becomes Eq. (55)

On(u,t) Ot

a2n(u,t) = D~----+ On2

v(t)

02n(u,t) 0U

+

1 "r a

732

Molecular Beam Epitaxy

In the moving reference frame this can be solved subject to the boundary condition that n(u,t) = 0 at u = 0J.. For the numerical solution, one uses v(t) = aD[an(o,t)lau- an (L , t) / a u]. The resulting solution is a distribution, n(u,t), of mobile adatoms that is skewed by the moving step. At t = 0, n(u,O) starts at zero and then increases symmetrically across the step, keeping the boundaries fixed at zero. Since n(u,t) = 0 at the step edges, there is a gradient that feeds the steps, causing them to move. As the steps move, there is a pile up of mobile adatoms near u = 0 and a low adatom density near u = L, where new surface is being created. This continues until a steady state is reached, which can be solved exactly.[ TM] The steady state distribution, for moderate diffusion, approaches 1/a near u = 0 and falls linearly to zero at u = L. At this point, convective motion of the steps is balanced by surface diffusion of the mobile adatoms.[ ~~ The Fourier transform of this adatom distribution ,(u,=) at the appropriate diffraction geometry gives the steady state intensity. To arrive at this value, the distribution ,(u,t) undergoes a transient behavior that gives intensity oscillations. Figure 36 illustrates the solution to Eq. (55) for L2/D~ = 20. Initially n(u,t) is relatively flat. At a later time, it is skewed toward u = 0, oscillating about its final steady state value.

= I.i1 ~

o t=O.gs

.o.~ ~ . [ / ~

9 t:l.Os

"1 ~f//f~-"',, '~"~-"%,

-I////

o t: I. Is

-,,, ,,,,,. -,,,,-,,'%

, t=l.Zs

w

-.

0 I

position on terrace

'~ ] tl

0:'o h

~

I

stop

f'{

~

1

20

-i

20 ,"

At=O.OIs

9

O

~

,

----~

I

~

,

~

~

~

10 lime (s)

Figure 36. (a) Time variation of the position dependence of the density of mobile adatoms on a terrace of a vicinal surface. (b) The velocity of a step calculated in the modified BCF model. The velocity exhibits oscillations since a step edge will move very quickly in regions of high step density (or islands); then slow after the adatoms are used up; finally, speed up when the step density recovers. Here L 2/D~ = 20.

RHEED Studies of the Dynamics of MBE

733

To determine the diffracted intensity, write Eq. (7) with the scattering amplitude of a terrace written explicitly as a Fourier transform. We use a continuum approximation and evaluate the result at Sz = :rid and Sx = ~L, which corresponds to the peak of one of the components of the split specular beam. We add the scattering between two layers separated by step height d by using n(u,t) as the probability there is a scatterer at u on the top level and 1 - n(u,f) as the probability there is a scatterer at u on the lower level. At this scattering geometry, the diffracted intensity is

Eq. (56)

I=

~L

+

Jo [l "n(u't)]e isxu - iS'ddu

I

which using the values of Su and Sz becomes

Eq. (57)

/=

!

The resulting intensity for L2/Dr. = 45 is shown in Fig. 37. As can be seen in this figure, like experiment, the diffracted intensity starts at an initial value and decreases rapidly depending upon the dimensionless parameter L2/D~. In contrast to the low-index model discussed earlier, here the maxima fall and the minima rise until a nonzero steady state value is reached. Like our experimental results, the calculated oscillations are independent of whether the incident beam is directed parallel to the step edges or down the staircase. However, as discussed in Sec. 2, because of the asymmetric instrumental response of RHEED, it might be possible to find a situation in which amplitudes from each terrace are added in one direction and intensities in the other. In this case, one expects different results. One result that has yet to be confirmed is the prediction that, at low values of L2/D~, the period of the intensity oscillations is slightly longer than the expected monolayer completion time. This occurs because the step edges rob the mobile adatoms on the terraces of some fraction of the incident flux. Proper inclusion of clustering might eliminate this shift, and as of yet we have not been able to separate this small effect from nonuniformities of the incident flux.

734

Molecular B e a m Epitaxy

1.0

Z

-

::}0.5

,

. . . . .

J i

Z~"

-1 0.0 . . . . . . . . . . . . . . . . . .

o

20

TIME ( t / ' r ) Figure 37. Calculated diffracted intensity on a vicinal surface using the modified BCF solution with /_.2/D~= 45. Though no cluster growth is incorporated, this calculation mimics many features observed in experiment. In growth on the vicinal surface, the intensity oscillations are seen to be associated with periodic oscillations of the step velocity. Similar conclusions are obtained from Monte Carlo calculations.[72] Just before an intensity maxima, the adatom concentration on a terrace is such that the step velocity begins to increase dramatically. The steps move rapidly across the surface, creating something approaching the starting staircase and giving maximum diffracted intensity. After the essentially fresh surface is created, the step velocity slows until the next period. The oscillations reach steady state when the profile is able to feed the steps without these transient readjustments. At this point, the steps move with a constant velocity of L/r.. If clustering is included, then one expects a similar enhancement of the cluster density near the trailing edge and a reduction of cluster density on the newly created part of terrace. If clustering is significant, then the low-index methods may be a better description of the growth process. Like the low-index model, the growth term in Eq. (55) can be removed and the recovery of the intensity calculated. From this model there is no evidence of the two-process form postulated by Joyce. [1~ Unlike the low-index model, the intensity always recovers to the starting value because the steps that act as perfect sinks are included in the calculation.

RHEED Studies of the Dynamics of MBE

6.0

735

CONCLUSION

Reflection high-energy electron diffraction is sensitive to many of the microscopic processes that occur during crystal growth. The diffracted beams broaden and the peak exhibits intensity variations due to changes in the defect and morphological structure of the surface during epitaxy. Contained in these changes are the microscopic processes of surface diffusion, island growth, dislocation formation, step bunching, and step disorder. These changes have been analyzed in terms of kinematic diffraction theories that neglect of multiple scattering between steps, and which assume that the scattering from isolated islands is essentially identical to the underlying substrate. Simple models can easily describe the main features of the data. In this incomplete discussion we presented rate equation methods. The main point is that since it doesn't take too much to obtain qualitative agreement, the mere existence of intensity oscillations does not need to indicate the reliability of a model. More realistic models are, of course, needed, though from our perspective the analysis of the diffraction data should improve in parallel. Otherwise the many more parameters and interactions that the models could include would be lost in the comparison to the data. As described in the discussion of the broadening of the diffracted beams for both low-index and vicinal surfaces, a major advantage of diffraction over direct imaging is that the statistics of the island structure are measured. But the complexity of non-equilibrium crystal growth processes, as well as uncertainties of the role of two dimensions and multiple scattering, limit the quantitative impact that has so far been felt. Despite progress in understanding the diffraction patterns and its evolution during crystal growth, the subject needs the combined attack of more than one in-situ technique. Ways to apply scanning tunnelling microscopy, lowenergy electron microscopy, or reflection electron microscopy are sorely needed. Nonetheless, there is still much one can say. With advances in analysis and direct imaging to calibrate the diffraction pattern, more definitive answers to fundamental questions of epitaxy should be forthcoming.

ACKNOWLEDGMENTS

This work was partially supported by grants from the National Science Foundation (DMR-93-07852) and from the Air Force Office of Scientific Research (AFOSR F49620-93-1-0080).

736

Molecular Beam Epitaxy

APPENDIX: TWO-LEVEL DIFFRACTION To calculate the diffraction from a surface with a random distribution of islands or two-dimensional clusters, the simple sums of the previous sections no longer suffice. A correlation function approach must be used. This approach has been treated exhaustively elsewhere.[ 34] In this discussion we will give the simplest, "back of the envelope" treatment of a very special case and then just state how it is generalized. Hopefully, the generality of the method won't go unnoticed. The two main goals at this point are (1) to obtain the kinematic angular dependence of the intensity oscillations and (2) to describe how to extract the calculated peak intensity of Sec. 3.2. We treat the case of scatterers distributed among two levels in onedimension. The kinematic intensity from this surface is

Eq. (58)

l(S) = [~, ,(r) e'iS'r[ P

where n(x,z) is equal to I if there is a surface scatterer at coordinates (x,z) and 0 otherwise, and S = 3"~ + S~. It is important to note that though this sum is over top layer scatterers only, the same column approximation arguments of Sec. 3.1 apply, so that the dynamically scattered intensity from a cluster and all of the atoms underneath it are included. Since the clusters are, apart from lateral size, the same, we just need to include the path length difference of the origins. If there are No total lattice sites in a layer, then this can be rewritten as Eq. (59)

/(S) =

E eiS'r n(r) r

~,eiS'r'n(r ") = N o E e-iS'uC(u) P

u

where we define the correlation function Eq. (60)

C(u) = (1/No) ~,,(r) n(r + u) r

The correlation function C(u) is the probability of finding two scatterers on the surface separated by a vector u. Our procedure is to evaluate the correlation function on the surface and then take its Fourier transform to obtain the diffracted intensity.

RHEED Studies of the Dynamics of MBE

737

To make this simpler, we use the result that it is sufficient [34] to transform Eq. (59) to an integral over x, convolving the result with the onedimensional reciprocal lattice. In the discrete case the correlation function C(u) = C(na, ld) where I = -1, O, 1. Instead we now write na = x but retain the discrete two levels. Our method is to break C(x,lo) into partial correlation functions, Cq, q+l(X), which are straightforward to calculate. Following the definition in Eq. (60), Eq. (61)

C(x,z = ld) = ~ Cq, q+ I (x) q

where Cq, q+I (x) is the probability that there is a scatterer at the origin on level q and also on level q + ! at a distance x away. These continuum partial correlation functions will now be dealt with for a very simple case. Following the notation of Sec. 5, let the exposed coverage of the top level be c so that the exposed coverage of the second level is 1 - c. Next, let f(x) be the probability that there is a top layer scatterer at x, hence the probability that there is a second layer scatterer at x is 1 - f(x). We then make a Markov assumption to give the probability that there is a step in going from x to x + a in terms of the jump probabilities Pd and Pu as Eq. (62)

f(x+a) = (1 - Pd) f(x) + pu[1 - f(x)]

Since we only want the continuum distribution, expand about x to obtain

Eq. (63)

a

d: dx

+ (Pd + P,) f (x) - Pu = 0

To find Cll(x), we calculate the probability that there is an atom anywhere on level one, i.e., c, times the probability that, given an atom at the origin, there is onex away. This second factor is f(x) given by the equation above subject to the boundary conditions Eq. (64)

:(oo) = c f(o) = I

The solution is, applying some symmetry, f(x) = (1 - c) exp(-~.lxl) + c, where ~ = (Pd + Pu)/a and c = Pu/(Pd + Pu). Thus the first partial correlation function is

738

Molecular Beam Epitaxy

Eq. (65)

Cll (x) = c(1 - c) e-Xlxl + c2

Similarly C21 (x) is found by finding the probability that given no atom at the origin on the first level that there is an atom at x on the first level. The result is (1 - c)f(x) where now f(x) is a solution to Eq. (63) subject to the boundary conditions that Eq. (66)

.f(oo) = c = o

f(o)

This gives Eq. (67)

C21 (x) = (1 - c) c(1 - e-Xlxl)

Then because there is no distinction between left and right on a two-level surface, C12(x) = C12(-x ) = C21(x ). Finally, C22(x ) is found similarly to Cll to be Eq. (68)

C22(x) = c(1 - c) e-Xlxl + (1 - c) 2

Putting these partial correlation functions together in Eq. (61), one obtains the actual correlation function to be C(x,O) = Cll + C22 and C(x,1) = C21, so that Eq. (69)

C(x,0) = c2 +(1 - c) 2 + 2c(1 - c) eXl-'4

C(x, + 1) = c(1 - c) (1 - e-XN) These correlation functions could be compared to Eq. (26) of Ref. 34. At last we can calculate the diffraction profile. Putting the correlation function into Eq. (59) we have oo

Eq. (70)

I(S) = X e~"~ f dx e%= C(x, ld) |

oo

Eq. (71)

-~

60

I(S) = f dx e-;sxx C(x,O) + 2 cos s ~ f dx e"is~x C(x,1) --GO

-GO

RHEED Studies of the Dynamics of MBE

Eq. (72)

739

I(S,,S~) = [c2 + (1 - c)2 + 2c(1 - c) cosSzd]2=8(Sx) + 2c(1 - c)(1 - cosSzd) [2;k./(;k.2 + $2)]

This last equation says that the diffracted beam versus Sx can be separated into a broad part and a central spike. Note that the delta function is broadened by the transfer width of a real instrument. Further since, for example, the specular beam, Sz = 2/~, the relative size of these two components will depend upon the angle of incidence, 0 i. When Szd = 2~, an in-phase angle, the second term will vanish, indicating that the diffraction is insensitive to steps. Similarly, at half coverage and at 8,d = =, the first term vanishes. However, and in contrast to our calculation in Sec. 5, if there is step disorder, the total intensity does not drop to zero!

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