Quantitative analyses of RHEED patterns from MBE grown GaAs(001)−2 × 4 surfaces

Quantitative analyses of RHEED patterns from MBE grown GaAs(001)−2 × 4 surfaces

surface science ::+.... _,,_:,&:g “‘+:“‘.=A’“~ ... .,.,../.,.... ......,......., . _,,_ \:::~~,::~::~j::?.~,:: ,.,,,, .,,,, Surface Science 289 (1993...

2MB Sizes 1 Downloads 53 Views

surface science ::+.... _,,_:,&:g “‘+:“‘.=A’“~ ... .,.,../.,.... ......,......., . _,,_ \:::~~,::~::~j::?.~,:: ,.,,,, .,,,,

Surface Science 289 (1993) 47-67 North-Holland

Quantitative analyses of RHEED GaAs( 001) -2 x 4 surfaces

” ‘.‘.““‘. ‘“4 .d. .i./....,.. :<.:.:‘>..>:,y.;,x: _,,,,,_ ..‘a.. ‘-:‘.-” ‘.A... ...i. .......i._.,, ..,.,‘.~.:.~.,,,‘,,~;~., .,‘“~. -‘-~‘y.n.........: ‘-.‘.i’~.:::.:.:‘:.....‘ ....5. :.,.. _,_::(,,_,:: ..,...A ..i....:.in

patterns from MBE grown

Y. Ma a, S. Lordi a, P.K. Larsen b and J.A. Eades a a Materials Research Laboratory University of Illinois at Urbana-Champaign,

Urbana, IL 61801, b Philips Research Laboratories, P.O. Box 80000, 5600 JA Eindkouen, Netherlands

USA

Received 26 October 1992; accepted for publication 6 January 1993

A muitisiice formahsm of Cowley and Moodie (Acta Crystallogr. 10 (19.57) 6091with a recently developed edge-patching method has been applied to quantitative analyses of RHEED patterns from MBE grown GaAs(OOl)-2 x 4 surfaces. The analyses are based on the ordering of visually estimated spot intensities of several observed RHEED patterns from the GaAs(OOH-2 X 4 surfaces. The surface structure is proved to be a dimerized vacant 2 x 4 reconstruction missing one dimer of every four along the [llOl direction, which is consistent with previous STM observations. The relaxation of the top layer is found to be about 6 = + 8.0% (0.113 &. The results give an overall Debye-Wailer factor of B = 0.35 A2 and a crystal absorption of a = 0.1 for the conditions at which the observed RHEED patterns were acquired. The effects of dimer twist on the intensities of RHEED patterns are briefly assessed. This indicates that twist of dimers on the GaAs(OOl)-2 X 4 surface is unlikely. Errors involved in the calculations are also discussed.

1. Int~duction Quantitative analyses of reflection high-energy electron diffraction patterns (RHEED) are increasingly getting attention as a surface analytical technique for they may generate detailed surface structural information in circumstances convenient for crystal growth by molecular beam epitaxy (MBE) [l]. Since RHEED started in the 1930’s [2], it has been used mainly to characterize surface periodicity (surface recrystallization or reconst~ction) for which only the locations rather than the intensities of spots in the patterns are relevant [3-61. Since the correlation between tempora1 RHEED intensity oscillations and monolayer MBE growth was discovered [7-91, the use of intensity information from RHEED to study various structure-related surface phenomena (growth dynamics during MBE, surface phase transformation and surface relaxation etc.) has increasingly attracted the interest of surface scientists [lo-151. The GaAs(001) surface is the most widely used gallium arsenide face in the MBE growth of ~39-602~/93/$06.00

muItilayer electronic device structures. Various reconstructions on this face have been reported [16]. They range from the As-rich (2 X 4) to the Ga-rich (4 x 2). These surface structures and their stoichiometry have been studied in detail for several years using LEED [16,17], RHEED [12, 14,151, STM [l&20], AES [16,17,21], XPD [221 and core-level photoemission measurement 123,241. The As-rich (2 X 4) surface is the most important one of these, since MBE growth usually starts and ends with this surface. The use of RHEED as a quantitative analytical technique for studying this surface has recently been attempted [l&25--281. Accurate quantitative measurements of RHEED patterns from near-ideal MBE grown GaAs(OOl)-2 x 4 surfaces have been achieved [15,25,26]. A dynamical simulation method has also been used for quantitatively analyzing the measured RHEED intensities 127,281. In the present paper, the results of quantitative analyses of RHEED patterns from GaAs(0011-2 X 4 surfaces in the [ilO] and fllO] azimuths are reported. The e~erimental results

Q 1993 - Elsevier Science Publishers B.V. All rights reserved

48

Y. Mu et al. / ~u~ntitatiue analyses of WEED

used are those published in refs. [14,15,25]. The computations were made using the multislice formalism of Cowley and Moodie with a recently developed edge-patching method [29-311. A comparison of visually estimated and dynamically calculated intensities is used to analyze the data. This is similar to the approach used in early X-ray structure determinations. The results show that the relaxation of reconstructed atoms in the top surface layer can be refined to an accuracy of N 0.05 A. The correct model of the 2 x 4 reconstruction is easily identified using the RHEED patterns. A possible twist of the As-As dimers in the 2 x 4 vacancy model is discussed based upon the same observed RHEED patterns. A quantitative determination of the length of the As-As dimers using an observed RHEED pattern acquired in the [IlO] azimuth is attempted. Errors involved in the calculations are also discussed. The principle aim of the work is to give the first experimental test for the multislice-edge-patching method in its application to quantitative analyses of RHEED patterns.

2. Background Three dynamical calculation methods for RHEED are currently under intense scrutiny: (1) Bethe’s method (or BIoch wave method) which dates back to the 1930’s [32,331. (2) A transfer of LEED (low-energy electron diffraction) calculation methods to RHEED. This method was developed by Maksym and Beeby [34] and independently by Ichimiya 1351in which the crystal is sliced parallel to surface and the surface is treated as a series of 2D periodic layers. (3) The multislice formalism of Cowley and Moodie with an edge-patching method, in which the crystal is diced normal to the surface and to the incident beam. The slice potentials do not have to be periodic normal to the surface or along the incident beam [29-311. Two points here should be made clear: (1) These different methods should be subjected to quantitative tests against experimental data. (2) They should be consistent with each other quantita-

patterns from MBE grown Gds(OOIj-2

x 4 surfaces

tively. For the transmission case, such comparison between Bethe’s method and the multislice approach has been made [36,37]. So far, experimental tests have been done in very few cases in reflection, partially because quantitatively measured RHEED patterns from ideal surfaces are not often available. The most extensive experimental analyses using Maksym and Beeby’s method were recently done by McCoy et al. [28], Stock and Meyer-Ehmsen 1381 and Korte and Meyer-Ehmsen [39], in which the intensities of several reflections were used for fitting rocking curve. However, each reflection was normalized separately. As far as the consistency between different calculation methods for RHEED is concerned, a critical test has been undertaken to check for numerical consistency between Bethe’s method and the multislice approach [30]. This revealed that in the case of an Au(llO) surface the planar boundary match for Bethe’s method in the RHEED geometry failed. This is because the surface is not flat and it is not adequate to match the bulk (Bloch wave) solution with the vacuum (plane wave) solution on a plane. By contrast, the methods agree for an Au(100) surface. High-energy electron diffraction is very sensitive to potential variation in the plane nearly perpendicular to the incident electron beam. This sensitivity is what makes it possible to achieve atomic resolution in the transmission case and also gives the possibility for high sensitivity to surface structure in RHEED. On the other hand, the problem is that the calculations must model the potential accurately in certain dimensions if good results are to be obtained. The advantages of the multislice method as used here are its high sampling rate (- lo-15 pts/A) of the crystal potential in the plane nearly perpendicular to the beam (i.e., the slice plane) and the free arrangement of scatterers in the crystal slices. Sampling along the incident beam (i.e., perpendicular Lo the slices) can be much coarser (0.5-2 pts/A) since the HOLZ (higher order Laue zone) diffraction is relatively insignificant in RHEED. This approach with the edge-patching method can now be applied to quantitative simulation of RHEED patterns from both perfect and imperfect surfaces,

49

Y. Ma et al. / Quantitative analyses of RHEED patterns from MBE grown GaAs(OOl)-2 x 4 surfaces

3. Numerical approach and models of GaAs(OOl)2 X 4 surface The principle of the edge-patching method and its implementation in the multislice formalism of Cowley and Moodie were explained in previous publications [29-311. The multislice program for quantitative RHEED simulation and the software for image display used in the present paper were developed by one of us (S.L.) at the University of Illinois at Urbana-Champaign. They were based on the EMS package developed by Stadelmann, in which the Doyle-Turner potential was used [40,41]. The GaAs(OOl)-2 X 4 surface reconstruction has been known for many years [16], but the model of its atomic structure is still under intensive investigation. There are two basic types of atomic models for the surface: (1) asymmetric dimer models, in which the four-fold periodicity along [llO] shown in LEED and RHEED patterns is formed by asymmetrical As-As dimers with [42] and without [43] twist after the dimerization of an As-terminated surface and (2) dimer vacancy models, in which the four-fold periodicity is explained by missing As-As dimers [14,18]. The former was proposed for a complete Asterminated surface (0,, = 1) and was extensively examined by rocking curve fitting of RHEED [27]. However, the latter was favored by the results of surface energy calculations [44] and XPD analyses 1221and confirmed by STM observations [18,19], although the electronic structures of both were predicted to be metallic, contrary to the observed non-metallic behavior of the GaAs(0002 X 4 surface. The observed variation of the stoichiometry of the surface (0.5 I 0,~ 1) also supports the vacancy model since different numbers of missing dimers can explain the stoichiometry variation [14]. The analyses in the present paper therefore are based on the dimer vacancy models. Top views of the dimer vacancy models of the GaAs(OOl)-2 x 4 surface are illustrated in fig. 1, which shows three basic types of missing dimer reconstructions for the surface. The unit cell in the analyses is taken as aa x b, x co = u x &?a X aa, where a = 5.65 A is the magnitude of the primitive unit-cell vector of

GaAs (001)

-2x4

0 Surface AS 0 Second layer Ga l

Third layer As

0

FourIh layer

loo11 Ii101 ?-Ill01

Ga u

Fig. 1. Top views of the dimer vacancy models of the As terminated GaAs(OOl)-2x4 surfaces, which show three basic types of dimer missing reconstructions for the surface.

GaAs; a, = a is set along [OOll; b, = fia along [ilO] and co = fia along [llOl. It is the smallest unit which can represent periodicity of the GaAs(OOl)-2 x 4 surface dimerized along [llOl. Fig. 2a shows a 3D diagram of the super-cell for the simulation of RHEED. The cell size is 16a, x 2b, x 2c,. Its projections along the [ilO] and [llO] directions are shown in figs. 2b and 2c. They were used for the RHEED simulations in the corresponding azimuths. The cells are comprised of l/4 crystal (from A to B) and 3/4 vacuum (from B to D>. The patched-edge area is from C to D l/3 of the vacuum. The data for a stationary solution of the reflected wave fields are collected in the region from B to C. The missing of one As-As dimer in every four on the surface is seen in the super-cell shown in fig. 2b. The super-cell is sliced into eight slices along [ilO] (fig. 2b) and sixteen slices along [llO] (fig. 2~). The size of sampling array of each slice is 1024 x 256 for fig. 2b and 1024 X 128 for fig. 2c. In other words, the potential sampling rates in the plan: normal to the incident beam are about 16.0 pts/A in [OOll and 11.3 pts/A in [llOl (fig. 2b) or [liO] (fig. 2~). It is 1.0 pt/A in the direction along the incident beam (i.e., AZ = 1.0 A). To match experimental conditions, RHEED patterns were simulated for 12.5 keV electrons incident upon the GaAs(OOl)-2 x 4 surfaces at glancing angles of 2.85”, 3.10” and 3.30” [15,25]. The accuracy of the simulated angles of incidence is limited by the sampling rate along [OOl] in

50

Y. Ma et al. / Quantitative analyses of RHEED patterns from MBE grown GaAs(OOl)-2 x 4 surfaces

reciprocal space, which is Ag[,,] = 1/16a, = 0.0111 A-‘. The actual simulated incidence angles are: sin- ’ (nAgl,,dk,) = 2.834, 3.111”, 3.319”, where k, = 9.172 A is the k-vector of the incident electrons and IZ= 41, 45, and 48, respectively. This accuracy is compatible with the precision of the experimental measurements of the incident angle ( f 0.03 - 0.05’) [14,15,26]. Three types of parameters were adjusted for the fitting to the experimental patterns: (1) geometrical locations of the outermost reconstructed surface atoms, including surface relaxation (61, the twist angle (~3) and length (Id) of surface As-As dimers; (2) crystal absorption C(Y),which was taken into account by taking the imaginary component of the crystal potential as a fraction of its real component (set to 0.1 as is conventional); (3) overall Debye-Waller factor (B). 6 is defined as the percentage of the interplanar distance in bulk which is a/4 in this case. The outermost reconstructed atoms were relaxed normal to (001) for both expansion (6 > 0) and contraction (6 < 0). The possible relaxation of sub-layers was not taken into account, although it may also affect the intensities of spots in RHEED patterns. Id is defined as the percentage of the As-As bond length before dimerization. The calculations were conducted on a series of IBM RS/6000’s available at University of Illinois at Urbana-Champaign. A stationary solution for the reflected wave (excluding patched area) was achieved after - 2400 iterations of the multislice routines. However, the solution of the 3200th iteration of each simulation was used for quantitative analysis in order to ensure full convergence under various simulation conditions. The CPU time for each pattern simulated was - 7 h, but now it can be reduced to - 1 h with a slightly modified version of the software.

4. Results 4.1. Effects of surface relaxation (6) Since intensities of spots in RHEED patterns are highly sensitive to surface relaxation [34,35, 451, this parameter (6) was subjected to pattern

fitting first. Three RHEED patterns from the GaAs(OOl)-2 X 4 surfaces, quantitatively measured by Larsen et al. [15,25] with the incident beam in the [ilO] azimuth and three different incidence angles (0,) are shown in fig. 3, 2.85” (a) [151, 3.1” (b) [25] and 3.3” (c) [15l. The super-cell shown in fig. 2b was used for the simulation of these RHEED patterns. This is the 2 x 4 vacancy model shown in fig. lc: one As-As dimer in every four is missing along the [ 1101 direction. The length of t)ese surface dimers
D

(b)

W

Fig. 2. 3D diagram of the super-cell with the size of lba, X 2b, X co for the simulation of the 2x4 missing dimer reconstruction in the [ilO] and [llOl azimuths (a) and its projections along the [ilO] (b) and [1101 Cc) directions. Small dots in the diagram denote the As atoms, while the large dots denote the Ga atoms. Three pairs of small black dots represent three surface As-As dimers.

I’. Mu et al. / Quantitative analyses of RHEEDpatterns

52

lated RHEED patterns corresponding to the experimental conditions of figs. 3a-3c. The intensities of individual spots in each calculated pattern were integrated for the sampling points within the spots along [loo] and normalized to the strongest reflection in the pattern for comparison. Intensity integration along El101 (fig. Zb) or [liO] (fig. 24 is not necessary, since the resolution in reciprocal space is equal to the spacing between the nearest spots in the vertical dimen-

from MBE grown GaAs(OOl)-2

x

4 surfaces

sion in the analyses. The integrated intensities are listed in table 1, together with the visually estimated intensities of experimental patterns for comparison. A reliability factor R is defined as:

The R-values were calculated using the numerically scaled experimental and calculated intensities for each pattern. The R-value evaluation

Table 1 Integrated intensities of individual spots of each calculated RHEED pattern shown in fig. 5, listed together with those visually estimated intensities of individual spots of corresponding observed RHEED patterns shown in fig. 3; the visually estimated intensities of individual spots, both letter and numerically scaled intensities are listed in the two left columns in the table, where “m” denotes “medium”, “s” strong, and “w” weak 8

2.85”

Vision intensities

Calculated

(0, k)

- 20%

- 10%

0%

10%

20%

30%

0

ss

l/4 l/2 3/4 1

W

5/4 3/2 7/4 2

W

1.000 0.250 0.200 0.400 0.750 0.250 0.050 0.500 0.010

1.0000 0.0417 0.0082 0.1716 0.0095 0.4289 0.1663 0.3045 0.0061

1.0000 0.0468 0.0099 0.1528 0.0841 0.4275 0.0923 0.2964 0.0050

1.0000 0.0577 0.0049 0.2212 0.2847 0.3686 0.0692 0.4079 0.0037

1.0000 0.0906 0.0002 0.4319 0.7654 0.2612 0.1290 0.8449 0.0041

0.3887 0.0612 0.0076 0.3899 0.8456 0.0357 0.1803 1.0000 0.0055

0.0155 0.0316 0.2231 0.5740 0.0502 0.2375 1.0000 0.0079

0.36

0.31

0.16

0.09

0.36

0.60

1.0000 0.0064 0.0363 0.0534 0.0458 0.1535 0.1454 0.6241 0.1268

1.0000 0.0033 0.0164 0.0747 0.1586 0.1385 0.2335 0.8834 0.1464

0.7237 0.0414 0.0165 0.0827 0.3936 0.0859 0.2872 1.0000 0.1041

0.4295 0.1136 0.0419 0.0839 0.6736 0.0507 0.3062 1.0000 0.0463

0.2500 0.2137 0.0906 0.0913 1.0000 0.0480 0.3129 0.9838 0.0221

0.1681 0.2602 0.1213 0.0783 1.0000 0.0418 0.2041 0.6611 0.0585

0.36

0.26

0.17

0.17

0.25

0.32

0.1411 0.0332 0.0349 0.0884 0.2291 0.0250 0.4266 0.5417 1 .oooo

0.0228 0.0529 0.0481 0.1281 0.4265 0.0336 0.3987 0.5464 1.0000

0.0119 0.0595 0.0508 0.1353 0.4593 0.0257 0.2487 0.3836 1.0000

0.0627 0.0472 0.0357 0.1049 0.3185 0.0116 0.1065 0.1860 1.0000

0.1001 0.0352 0.0229 0.0768 0.1964 0.0063 0.0458 0.0819 1 .oOOo

0.1267 0.0275 0.0151 0.0556 0.1165 0.0070 0.0229 0.0301 1.0000

0.30

0.22

0.09

0.06

0.12

0.18

W

m s WW

m ww

0

S

l/4 l/2 3/4 1

m

5/4 312 7/4 2

W

m S

m W ss ww

0.800 0.300 0.100 0.250 0.750 0.500 0.100 1.000 0.050

R-value 3.30”

(relaxation)

Numer.

R-value 3.10”

intensities

Lett.

0

WW

l/4 l/2 3/4 1

W

5/4 312 7/4 2

W

R-value

WW

m m WW W ss

0.030 0.050 0.010 0.300 0.500 0.050 0.010 0.200 1 .ooo

53

Y. Ma et al. / Quantitative analyses of RHEED pattems from MBE grown GaAs(OOl)-2 X 4 surfaces

-10 %

-20%

Fig. 5. Simulated

RHEED

patterns

0

10%

corresponding to the observed patterns in figs. 3a-3c, 6 = - 20%, - lo%, 0, lo%, 20% and 30%.

using visually estimated intensities makes the comparison more evident and convenient, although it may not improve the quantification of

Vision intensities

Calculated

30%

with different

relaxation

parameters:

the refinement procedure very much. The R-value of each pattern is also given in table 1. Comparing the data listed in the table:

Table 2 Integrated intensities of individual spots of RHEED patterns calculated a smaller interval of 6: 0.5%; they are listed together with those corresponding observed RHEED patterns shown in fig. 3

2.85”

-20%

at the same conditions as those shown in fig. 5, except with visually estimated intensities of individual spots of the

intensities

(relaxation)

(0, k)

Lett.

Numer.

6%

8%

10%

12%

14%

0

ss

l/4 l/2 3/4 1

W

5/4 312 7/4 2

W

1.000 0.250 0.200 0.400 0.750 0.250 0.050 0.500 0.010

1.0000 0.0721 0.0015 0.3128 0.4931 0.3162 0.0880 0.5820 0.0034

1.0000 0.0791 0.0007 0.3572 0.5937 0.2945 0.1018 0.6755 0.0036

1.0000 0.0906 0.0002 0.4319 0.7654 0.2612 0.1290 0.8449 0.0041

0.9812 0.0991 0.0004 0.4935 0.9144 0.2281 0.1557 1 .oOOo 0.0047

0.7993 0.0903 0.0013 0.4721 0.9125 0.1602 0.1600 1.oooo 0.0047

0.08

0.06

0.09

0.17

0.19

R-value

W

m S

WW

m W

54

I’. Ma et al. / Quantitative

analyses of

RHEEDpatterns from MBE grown GaAs(aOl)-2

(a> R-values for the three incidence angles (0,) are well behaved and show a smooth functional relation with 6. A single minimum occurs in each case. (b) The R-value minima for the three 8, fall into the same range of relaxation variation around 6 = 10%. (c) For S = lo%, the ordering of spot intensities of each calculated pattern closely follows that of the corresponding observed pattern for all three 0,, especially for strong spots.

x 4 surfaces

Cd) The sensitivity of the RHEED patterns to the variation of 6 increases with a decrease of 19,,, which is physically expected. As one can see, the intensity ordering of neighboring spots in the case of 8, = 2.85” changes with 6 more significantly than it does in the case of 8, = 3.30”. This means that surface relaxation can be refined more accurately at the lower angles. (e) The calculated pattern with 8, = 2.85” and 6 = 10% visually distinguishes itself from the one with 8, = 2.85” and S = 0% by the change of the

Table 3 Data corresponding to those in table 1, while they were calculated with a different surface model: the 2 missing two dimers of every four

x

4 vacancy model of

e

Vision intensities (0, k)

Lett.

Numer.

- 20%

- 10%

0%

10%

20%

30%

2.85”

0

ss

l/4 l/2 3/4 1

W

5/4 312 7/4 2

W

1.000 0.250 0.200 0.400 0.750 0.250 0.050 0.500 0.010

0.4566 0.2051 0.0112 0.1728 0.0747 1.0000 0.1312 0.4742 0.0045

0.5332 0.2412 0.0106 0.2058 0.3229 1.0000 0.1042 0.1468 0.0047

0.7524 0.2673 0.0151 0.3030 0.7788 1.0000 0.1851 0.1493 0.0046

0.6912 0.1756 0.0213 0.3214 1.0000 0.5571 0.3666 0.9951 0.0030

0.1905 0.0318 0.0117 0.1092 0.3717 0.0702 0.2382 1.0000 0.0010

0.0586 0.0060 0.0089 0.0427 0.1514 0.0166 0.1907 1.0000 0.0007

0.67

0.55

0.39

0.31

0.60

0.84

1.0000 0.1155 0.0043 0.0634 0.1225 0.4692 0.0242 0.8403 0.0850

1.0000 0.0417 0.0040 0.0345 0.0571 0.2968 0.0421 0.9287 0.0792

0.8800 0.0371 0.0045 0.0118 0.0569 0.1486 0.0764 1.0000 0.0514

0.7592 0.0903 0.0056 0.0024 0.0842 0.0508 0.1172 1.oooo 0.0202

0.7806 0.1897 0.0073 0.0098 0.1299 0.0093 0.1662 1.oooo 0.0062

1.oooo 0.3568 0.0099 0.0372 0.2046 0.0012 0.2247 0.9113 0.0324

0.21

0.26

0.28

0.29

0.27

0.25

0.4374 0.0239 0.0033 0.1127 0.1342 0.0928 0.0768 1.oooo 0.4542

0.3185 0.0419 0.0124 0.1953 0.2267 0.0660 0.0755 1.0000 0.5447

0.5580 0.1735 0.0093 0.3513 0.3801 0.0735 0.1354 1.0000 0.9651

0.3643 0.1022 0.0080 0.1904 0.1816 0.0288 0.0768 0.3299 1.oooo

0.2624 0.0573 0.0058 0.1078 0.0807 0.0247 0.0368 0.1115

1.oooo

0.2032 0.0362 0.0035 0.0692 0.0329 0.0294 0.0143 0.0350 1.oooo

0.92

0.74

0.70

0.18

0.20

0.24

W

m S

WW

m WW

Calculated intensities (relaxation)

R-value 3.10”

0 l/4 l/2 3/4 1 5/4 312 7/4 2

S

m W

m S

m S ss WW

0.800 0.300 0.100 0.250 0.750 0.500 0.100 1.000 0.050

R-value 3.30”

0 l/4 l/2 3/4 1 5/4 3/2 7/4 2 R-value

WW W WW

m m W WW W SS

0.030 0.050 0.010 0.300 0.500 0.050 0.010 0.200 1.000

Y. Ma et al. / ~u~nt~tatitle ana~~es of RHEED latterly from MB.Egrown GaAs@Of)-2 x 4 surfaces

intensity ordering between spots (0, 1) and (0, 5/4), while from the one with B0 = 2.85” and S = 20% by the shift of the intensity ordering between spots (0, 0) and (0, 7/4), etc. In order to get a more accurate 6 value, a similar fitting procedure was applied to the experimental pattern with eoO=2.85” for smaller steps of 6: (0.5% = 0.028 A). The results are shown in table 2. The R-value minimum (6%) is reached for the calculated pattern with S = 8%, in which the calculated intensity of spot (0, 7/4) is closer to the observed medium intensity level. It should be noted that only one parameter (6) was involved in the fitting procedure, while the rest of adjustable parameters, such as the thermal parameter B, crystal absorption LYand slice thickness etc. were set to values conventional in the multislice calculations. The over-determination ratio (number of reflections/number of parameters) is 9. The R-value of N 6.0% then should be considered as a reasonably good agreement between the observed and calculated data.

4.2. Effects

of the

2 X 4 vacancy modei

55

oftwo-

dimer-missing

The 2 X 4 vacancy model of the stabilized GaAs(OOl)-2 x 4 surface shown in fig. 3c, missing one dimer of every four, has been confirmed by STM images [18-201. The previous fitting for determining 6 and in which a good agreement between the observed and calculated data was reached was based on the same model. However, this does not indicate whether the model is unique. In order to demonstrate that the three observed RHEED patterns reject the alternative model shown in fig. 3a, missing two dimers of every four, the same kind of calculation as previously explained was conducted for this model. This was done numerically by taking one more surface As-As dimer away from the three appearing in the unit cell in fig. 2b. The results are listed in table 3. As one can see by comparing observed data with calculated data in table 3, there is no reasonable match between the two.

Fig. 6. Wave fields calculated for the GaAs(OOl)-2 X 4 surfaces in the [ilO] azimuth, with B, = 2.85”, B = 0.35 AZ, 6 = 8%, I,, = 50% and different absorption parameters: cy= 0 (a), 0.05 (b), 0.075 Cc),0.1 Cd), 0.15 Ce), 0.2 (f) and 0.25 (g).

Y. Ma et al. / Quantitative analyses of RHEED patterns from MBE grown GaAs(OOl)-2 x 4 surfaces

56

r

[ilO]

PO11

I*~01

Fig. 7. RHEED patterns corresponding to fig. 6, i.e., Fourier transforms of reflected wave fields (from B to C) in fig. 6.

This is shown not only by higher R-values for all the calculated patterns, but also by comparison of the intensity ordering of neighboring spots between observed and calculated patterns. For example, none of the calculated patterns shows that (0, 0) is the strongest reflection in the case of f& = 2.85”, while the observed pattern (fig. 3a) does. For 8, = 3.30”, the strongest reflection is (0, 2) in the three patterns calculated with 6 = lo%, 20% and 30%, respectively, which is consistent with the observed pattern (fig. 3b), but all of them present (0,O) as the second-strongest beam, which is contrary to the observed pattern. However, this rejection may not be entirely certain. Some small domains of the second model may

still exist on the surface to accommodate the variation of surface stoichiometry in the range of OS-O.75 [14], but the above RHEED analyses show that the mode1 shown in fig. 2c represents the dominant structure of the stabilized GaAs(OOl)-2 x 4 surface. 4.3. Effects of the absorption parameter (cd Crystal absorption is the second parameter which significantly affects the intensity of a RHEED pattern. It physically determines how effectively the atoms in sub-surface layers contribute to the surface scattering and diffraction process and, therefore, the effect they have on

Table 4 Integrated intensities of individual spots of each calculated RHEED pattern shown in fig. 7, listed together with visually estimated intensities of individual spots of the corresponding observed RHEED pattern shown in fig. 3 B

2.85”

Calculated intensities (abso~tion)

Vision intensities (0, k)

Lett.

Numer.

0

0.05

0.075

0.10

0.15

0.20

0.25

0 l/4 l/2 3/4 1 5/4 312 7/4 2

ss

1.000 0.250 0.200 0.400 0.750 0.250 0.050 0.500 0.010

0.4635 0.0508 0.0736 0.3398 0.3684 1.0000 0.0234 0.2337 0.0091

1.0000 0.1406 0.0056 0.7257 0.7001 0.7377 0.0805 0.8338 0.0054

1.0000 0.1080 0.0011 0.5162 0.6676 0.4442 0.1011 0.7787 0.0043

1.0000 0.0791 0.0007 0.3572 0.5937 0.2945 0.1018 0.6755 0.0036

1.0000 0.0526 0.0036 0.2149 0.4930 0.1903 0.1036 0.5668 0.0031

1.0000 0.0441 0.0079 0.1667 0.4474 0.1654 0.1183 0.5459 0.0032

1.0000 0.0417 0.0125 0.1491 0.4262 0.1636 0.1432 0.5646 0.0034

0.53

0.24

0.10

0.06

0.09

0.11

0.13

R-value

W

w m s w WW m WW

57

Y. Ma et al. / Quantitative analyses of RHEED patterns from MBE grown GaAs(UOl)-2 X 4 surfaces

the intensities of spots in a RHEED pattern. Fig. 6 shows the wave field calculated for the GaAs(OOl)-2 X 4 surfaces with 0, = 2.85”, B = 0.35 A*, 6 = 8%, Id = 50% and different absorption parameters (a). LYwas adjusted from 0 to 0.25 with an interval of 0.05, except that 0.075 was inserted between 0.05 and 0.1. As one can see, the wave intensities inside the crystal decay more quickly as (Y is increased. The corresponding RHEED patterns are displayed in fig. 7. The integrated intensities of individual spots of the patterns shown in fig. 7 are listed in table 4, together with the corresponding visually estimated intensities. It can be seen that compared to 6, the spot intensities are less sensitive to changes in cx. This is reasonable since (Y only changes the percentage of the contribution of each scatterer to the elastic surface diffraction process, while 6 changes the positions of the scatterers in the top layer. Nonetheless, the Rvalue shows a well-behaved functional correlation with (Y. The minimum of R is located in the range around 0.1, which justifies the use of 0.1 in the calculation of table 1.

the toplayer and sub-layers. The Debye-Waller factor B in surface diffraction should also be more anisotropic than in the bulk diffraction since surface atoms will vibrate in the direction normal to the surface a lot more easily than in other directions. However, for the sake of time, only an overall B was refined at this stage. The calculations were conducted for the GaAs(OOl)-2 X 4 surfaces in the [ilO] azimuth with 8, = 2.85”, (Y= 0.1, S = 8%, I,, = 50% and different values of B. B was adjusted from 0.0 to 0.65 A* with an interval of 0.1. The integrated intensities of individual spots of the calculated patterns are listed in table 5, together with those visually estimated intensities. The intensities of RHEED are not very sensitive to the overall B, although the Rvalues again show a well-behaved functional correlation with B. The minimum of R is located in the range around 0.35 A*. This justifies the use of a preset B-value conventional in multislice calculations, although the match between the observed and calculated patterns for the incidence of 2.85” in the [ilO] azimuth has not been improved by refining an overall B parameter.

4.4. Effects of the Debye-Waler

4.5. Effects of the length of As-As surface (Zd)

factors (B)

The Debye-Waller factor (B) is a parameter representing the level of thermal vibration of atoms in a crystal. For surfaces, there should be considerable differences in B between atoms in

dimers on the

The RHEED patterns in the [ilO] azimuth will not be sensitive to the dimerization of As atoms on the surface since the incident high energy

Table 5 Integrated intensities of individual spots of RHEED patterns calculated for the GaAs(OOl)-2 X 4 surfaces in the [ilO] azimuth with B. = 2.85”, LY= 0.1, 6 = 8%, I,, = 50% and different B: 0.0 I B < 0.65 AZ, listed together with visually estimated intensities of individual spots of the corresponding observed RHEED pattern shown in fig. 3 I9

Vision intensities (0, k)

2.85”

0 l/4 l/2 3/4 1 5/4 3/2 7/4 2 R-value

Lett. ss W W

m S W

ww m ww

Calculated intensities (D-W factor) Numer. 1.000 0.250 0.200 0.400 0.750

0.250 0.050 0.500 0.010

0.0

0.15

0.25

0.35

0.45

0.55

0.65

1.0000 0.0908 0.0072 0.5133 0.7041 0.3746 0.1475 0.9964 0.0038

0.7201 0.0739 0.0210 0.5141 0.5915 0.3386 0.1462 1.oOOo 0.0029

0.5410 0.0620 0.0391 0.5239 0.5137 0.3173 0.1447 1.0000 0.0024

0.17

0.21

0.28

1.0000 0.0437 0.0232 0.0605 0.2860 0.0936 0.0192 0.0964 0.0022

1.0000 0.0580 0.0126 0.1505 0.3941 0.1673 0.0407 0.2503 0.0031

1.0000 0.0681 0.0043 0.2384 0.4895 0.2261 0.0662 0.4277 0.0033

1.0000 0.0791 0.0007 0.3572 0.5937 0.2945 0.1018 0.6755 0.0036

0.28

0.16

0.08

0.06

58

Y. Ma et al. / Quantitative analyses of RHEED patterns from MBE grown GuAs(OO1)-2

Fig. 8. RHEE,D

patterns calculated for the GaAs(OOl)-2 X

X 4

surfaces

4 surfaces in the [I‘iO] azimuth, with @a= 2.85”, (Y= 0.1, 6 = 8% and different dimer lengths: Id = 50% (a), 62.5% (b), 75% (cl, 87.5% (d) and 100% (e).

electrons basically do not clearly “see” any crystal potential modulation along the beam path. However, the length of the As-As dimers (1,) will manifest itself in the RIIEED patterns in the [llO] azimuth in a quantitatively tractable manner, since the crystal potential moduIation caused by dimerization is in the plane nearly perpendicular to the incident beam along the [llO] direction. Fig. 8 shows RHEED patterns in the [110] azimuth calculated with 8, = 2.85”, LY= 0.1, 6 = 8%, B = 0.35 AZ and different 1,. The integrated in-

tensities of individual spots in each pattern are listed in table 6. In the case without dimerization (Id = lOO.O%), the intensities of those halfordered spots are zero as expected. They generally increase with decreasing 1,. The intensity ordering between reflections (0, 0) and (0, 1) may characterize a particular range of 1,. It may be difficult to tell the differences between patterns calculated with ld = 62.5% and 1, = 75.0% in terms of the intensities of (0, O), (0, l/2) and (0, 11, but the differences in term of the intensi~ of

Table 6 Integrated intensities of individual spots of RHEED patterns calculated for the GaAsfOOl)-2 @a= 2.85” and 1.80”, (Y= 0.1, 6 = 8%, B = 0.35 .k2 and different Id

X

4 surfaces in the [ilO] azimuth with

B

(0, k)

Calculated intensities (dimer length) 50.0%

62.5%

75.0%

87.5%

100.0%

2.85”

0 t/2 1 3/2 2

0.8044 0.2856 1.0000 0.3493 0.0028

1.0000 0.0327 0.2490 0.1443 0.0006

1.oooo 0.0070 0.2309 0.0323 0.0017

0.6603 0.0164 1.oOOo 0.0692 0.0032

0.3590 O.WOO 1.oOOo 0.~~ 0.0024

50.0%

62.5%

75.0%

87.5%

100.0%

0 l/2 1

25.0% 1.oooo 0.0402 0.0396

37.5%

1.80

1.oooo 0.0440 0.1762

1.oooo

0.6471 0.0666 1.oooo

0.3267 0.1253 1.oooo

0.8909 0.2125 1.oooo

1.00000.0000 0.3854

0.0996 0.6053

59

Y. Ma et al. / Quantitativeanalysesof RHEED patternsfrom MBE grown GaAs(OOl)-2X 4 surfaces

(0, 3/2) must be visible in observed RHEED patterns. The intensi~ ordering between (0, 0) and (0, 1) is reversed when the dimer length is changed from I, = 50.0% to I, = 62.5%. Since there were no RHEED patterns at the El101 azimuth reported with a quality similar to those shown in fig. 3, a decisive quantitative refinement using experimental data seems to be difficult. However, a semi-quantitative determination of Id, based on a comparison between the observed pattern appearing in figs. 9 and lc of ref. [14] and the patterns calculated with different Z, has been attempted for a small incidence angle (0, = 1.80”). The calculated data are also listed in table 6. Compared to the reflection intensities of the pattern shown in fig. 9, Id = 62.5%, 75.0% and 87.5% can be easily excluded since the intensity ordering between (0, 0) and (0, 1) in the patterns calculated with these Z, values is clearly opposite to that observed in fig. 9. Id = 100.0% is also excluded because the (0, l/2) reflection has been observed in fig. 9. it may be difficult to determine the closeness to the experimental data of the patterns calculated with id = 37.5% and I, = 50.0% without digitized intensities of the observed patteofn shown in fig. 9. However, & = 37.5% (1.5 A) is physically unrealistic since it is considerably smaller than two times the covaIent radius of As (2.40 A). The intensity of (0, 0) is

-0i Ill01

I-

w11

-00 -01

[iio]

Fig. 9. Observed RHEED pattern from the GaAs@Ol)-2x4 surface in the [110] azimuth, at the incidence angle of 1.80”, published in ref. [14].

also visually lower than five times that of (0, 1) in fig. 9. Therefore, one can give the range of Id as 2.0 A I I,, < 2.5 A. 4.6. Egects of dimer twist (@ The model of full As-coverage with twisted As-As dimers for the GaAs(OOl)-2 x 4 surface has been proposed 142,431 and investigated by RHEED and other techniques [27]. However, it

Table 7 Integrated intensities of individual spots of RHEED patterns calculated for the GaAs(OOl)-2 x 4 surfaces in the [ilO] azimuth with 0, = 2.85”, cy= 0.1, 6 = 8%, I3 = 0.35 AZ, Id = 50% and different twist angles /k l”, 5”, 10” for both asymmetrical and symmetrical models, listed together with visually estimated intensities of individual snots of the corresponding observed RHEED pattern shown in fig. 3

e

Vision intensities (0, k)

2.85”

0 */4 l/2 3/4 1 5/4 312 7/4 2 R-value

Lett.

Calculated intensities Number.

0"

a~rnet~

twist

1”

5”

10”

symmet~ twist 1”

5”

10”

SS

1.000

I .oooo

1.0000

1.0000

1.0000

0.250 0.200 0.400 0.750 0.250 0.050 0.500 0.010

0.0791 0.0007 0.3572 0.5937 0.2945 0.1018 0.6755 0.0036

0.0787 0.0008 0.3582 0.5976 0.2940 0.1004 0.6828 0.0033

1.0000 0.0689 0.0024 0.3635 0.5924 0.2955 0.0902 0.7372 0.0025

1 .oooo

W

0.0396 0.0169 0.3659 0.5134 0.3026 0.0688 0.8168 0.0022

0.0792 0.0008 0.3561 0.5969 0.2965 0.1007 0.6774 0.0034

0.0742 0.0020 0.3396 0.5922 0.3185 0.0937 0.7063 0.0027

1.0000 0.0582 0.0094 0.2972 0.5372 0.3739 0.0839 0.7962 0.0023

0.06

0.07

0.08

0.11

0.06

0.07

0.11

W

m s W WW

m ww

Y. Ma et al. / Quantitative analyses of RHEED patterns from h4BEgrown Gds(OOl)-2 x 4 surfaces

60

Asymmetrical

Symmetrical

Fig. 10. Diagrams of asymmetrical twist and symmetrical twist models. The twist angle /3 here is varied to be 1”. 5” and LO”, respectively.

was excluded later by STM observations [l&19], which clearly supported the 2 x 4 vacancy model of one-dimer-missing. But the resolution of STM 0.08 ; 0.07

in this case was still not high enough to resolve the issue of dimer twist. An investigation of this issue, based on comparison between observed and calculated patterns has been attempted. in order to avoid massive CPU consumption, just a Iimited number of search calculations were conducted. Two basic dimer twist models were tested: namely the asymmetrical twist and symmetrical twist models. Their diagrams are shown in fig. 10. The twist angle p has been set to l”, 5” and lo”. The calculated results for B0 = 2.85” are listed in table 7. As one can see, the R-value generally becomes worse for both models when p increases. More importantIy, the intensi~ of (0, l), the second-strongest reff e&ion in calculated patterns tends to decrease with increasing & which is contrary to the observed pattern (fig. 3a). It therefore can be suggested that dimer twist in the 2 x 4 vacancy model of one-dimer-missing is un-

(O,O) n

J

Fig. 11. Rocking curves of nine reflections calculated for the GaAs(OOl)-2 x 4 surface in the [ilO] azimuth, for the case of the best fitted parameters, i.e., S = +8.00/G, B = 0.35 ,&‘, Q = 0.1 and 1, = 50%, where the range of the incidence angle is: 0.97” I H,,5 3.67”.

61

Y Ma et al. / Quantitative analyses of RHEED pattems from MBE grown GaAs(OOl)-2 x 4 surfaces 0.016

0.025

A

(0,1/2) -

(0,3/2) -

Fig. 11. Continued.

likely. However, this is not a decisive conclusion since the intensities of RHEED patterns are not very sensitive to /3. 4.7. Calculated rocking curves from the GaAs(OOl)-2 X 4 surfaces A rocking curve is a plot of the intensity of a particular reflection versus the angle of the inci-

dent beam. An investigation based on the information in each rocking curve is similar to the structure refinement based on the intensity information in each disc in convergent beam electron diffraction (CBED). The difference is that here the surface is involved. This makes quantitative investigation by fitting a rocking curve more vulnerable to experimental uncertainties since sev-

62

Y. Ma et al. / Quantitative

analyses of

RHEEDpattems from MBE grown GaAs(OOL)-2 x 4 surfaces

era1 crucial elements affecting reflection intensities in RHEED are incidence-angle dependent. These include the effects of surface morphology, surface illumination, instrumentation-related errors (measurement of incidence angle, convergence etc.) and surface inelastic scattering. These elements are almost impossible to model into an elastic diffraction theory at this stage. The reproducibility of observed rocking curves in RHEED has not been carefully examined so far. In some limited cases, the poor reproducibility has already led to unphysical discrepancies in results [28]. On the other hand, rocking curves do give an overview of the surface reflection process for a large range of incidence angles and they are sensitive to a variety of surface phenomena in RHEED. Combined with the information from the relative intensities of individual spots in RHEED patterns, further investigation based on rocking curves or convergent beam RHEED (CBRHEED) [46-491 may not only be informative as some work has demonstrated [28,38,39,50], but also reliable, which is crucially important to a technique under development. One set of rocking curves for nine reflections has been calculated and plotted in fig. 11 for the case of the best fitted parameters, i.e., 6 = + 8.0%, B = 0.35 A’, (Y= 0.1 and Z, = 50%. The range of the incidence angle is: 0.97” I 8, s 3.67”. The azimuth is the same as for the calculation of table 1. The threshold incidence angle for each reflection (i.e., the minimum incidence angle at

Table 8 Threshold incidence angles for individual reflections of RHEED patterns from the GaAs(OOl)-2x4 surface in the RlO] azimuth, when the incident energy E, = 12.5 keV

which the elastically scattered electrons in each reflection can manage to escape from the surface) is listed in table 8. One of the important features of these curves is that the threshold incidence angle of the reflection (0, k) indicated by each curve (k 2 3/4) is accurately consistent with its predicted value in table 8. As one can see, most reflections experience a large intensity hike right after they “escape” from the surface. If a calculated rocking curve of one particular reflection does not present its threshold incidence angle correctly, it must mean that some unphysical treatments or approximations are involved in the calculation which violates basic rules of elastic scattering (fig. 3 in ref. [28]). For observed rocking curves, a systematic error in the threshold incidence angle is an indication of either instrumentation errors in measurements of incidence angles or surface refraction effects. This could be used to correct the angle of incidence in a fitting procedure, if the inner potential value has been assessed simultaneously. 4.8. Limitation RI?EED

Threshold angle (degree)

(0, 0)

0.0 0.391

and

0.782 1.173 1.564 1.955 2.346 2.733 3.129

Z,(g) = I F(g)

(0,3/4) (0, 1) (0,5/4) (0,3/2) (0,7/4) co,21

kinematic

approximation

in

In order to evaluate the validity of the kinematic approximation in RHEED, kinematic intensities (Z,(g) N I F’ I *I of RHEED patterns from the GaAs(OOl)-2 X 4 surfaces with different incidence angles were calculated. The two top atomic layers were included in the calculations. The results are listed in table 9, which corresponds to table 1, except that the kinematic instead of dynamic intensities are listed. The kinematic intensity of each reflection in RHEED was calculated by the following formulas:

(0, k)

(0, l/4) (0, l/2)

of

m

I 2>

where g= kg- k,, 8, is the angle between k, and k, and f, the atomic scattering factor of the mth atom in the unit cell. k, and k, are the wave vectors of the reflection g and the incident beam, respectively. Comparing table 9 to table 1:

Y. Ma et al. / ~~ntiiatiue

63

analyses of RISEED patterns from MBE grown GaAdOOl)-2 x 4 surfaces

Table 9 Kinematic intensities of individual spots of RHEED patterns from the GaAs@Olj-2 x 4 surfaces in the [ilO] azimuth calculated for different S and three incidence angles: 2.8.5”, 3.1” and 3.3*, listed together with visually estimated intensities of indi~dua1 spots of the corresponding observed RHEED patterns shown in fig. 3

8

Vision intensities (0, k)

2.85

0 l/4 l/2 3/4 1 5/4 312 7/4 2

Calculated intensities (relaxation)

Lett.

Numer.

- 20%

- 10%

0%

10%

20%

30%

SS

1.000 0.250 0.200 0.400 0.750 0.250 0.050 0.500 0.010

1.0000 0.0273 0.0273 0.0273 0.3091 0.0272 0.0272 0.0272 0.9967

1.0000 0.0359 0.0359 0.0358 0.7165 0.0357 0.0357 0.0357 0.9993

0.6334 0.0342 0.0341 0.0341 l.oooO 0.0340 0.0340 0.0340 0.6352

0.2675 0.0265 0.0265 0.0265 l.OOOQ 0.0264 0.0264 0.0264 0.2699

0.0883 0.0228 0.0228 0.0227 1.0000 0.0227 0.0227 0.0226 0.0901

0.01% 0.0213 0.0213 0.0213 1.~~ 0.0213 0.0212 0.0212 0.0201

0.78

0.68

0.50

0.54

0.66

0.72

1 .OOoo 0.0340 0.0340 0.0339 0.6406 0.0339 0.0338 0.0338 0.9841

1.0000 0.0259 0.0259 0.0259 0.2508 0.0258 0.0258 0.0257 0.9869

1.0000 0.0222 0.0222 0.0222 0.0723 0.0221 0.0221 0.0221 0.9893

1.0000 0.0211 0.0210 0.0210 0.0167 0.0210 0.0210 0.0209 0.9914

1.oOOo 0.0220 0.0220 0.0219 0.0607 0.0219 0.0219 0.0219 0.9936

1.ooOO 0.0254 0.0254 0.0253 0.2229 0.0253 0.0252 0.0252 0.9960

0.84

0.94

1.03

1.06

1.04

0.96

0.6315 0.0338 0.0338 0.0338 1.0000 0.0337 0.0336 0.0336 0.6339

0.2295 0.0255 0.0255 0.0255 1.0000 0.0254 0.0254 0.0253 0.2322

0.0571 0.0219 0.0219 0.0219 1.0000 0.0218 0.0218 0.0218 0.0587

0.0183 0.0211 0.0211 0.0211 1.0000 0.0210 0.0210 0.0210 0.0178

0.0944 0.0227 0.0227 0.0226 1.0000 0.0226 0.0226 0.0225 0.0907

0.3226 0.0274 0.0274 0.0273 1.0000 0.0273 0.0272 0.0272 0.3140

0.61

0.71

0.90

0.96

0.86

0.66

W W

m S W WW

m ww

R-value 3.10

0

S

l/4 l/2 3/4 1

m

5/4 3/2 7/4 2

W

m S

m W SS Ww

0.800 0.300 0.100 0.250 0.750 0.500 0.100 1.000 0.050

R-value 3.30”

1

Ww

l/4 l/2 3/4 1

W

5/4 312 7/4 2 R-value

WW

m m W WW W SS

0.030 0.050 0.010 0.300 0.500 0.050 0.010 0.200 1.000

(a) The kinematic intensities of calculated whole-order reflections do vary with changing relaxation parameters, but are not as sensitive as the dynamical values in table 1. (b) The kinematic intensities of calculated quarter-order reflections are all low and not sensitive to the relaxation parameter 6. Cc) The kinematic intensities of calculated whole-order reflections do vary with incidence angle. An extremely low kinematic intensity of (0, 0) is obtained with 8, = 3.3” and 6 = lo%, which is consistent with the observed value, but as whole

the variation does not correlate to observed data at all. Cd) In general, there is not even a qualitative match between kinematically calculated and observed intensities, which confirms that RHEED in principle is dynamical.

5. Discussions The most important point throughout the above investigation (which can be called “pattern

64

Y Ma et al. / Quantitative analyses of RHEED patterns from MBE grown GaAs(OOl)-2 x 4 surfaces

fitting”, as opposed to “rocking curve fitting”) is that the intensity characteristics of each RHEED pattern is sensitive to changes in various parameters. This makes it possible to derive quantitative resuhs from visually estimated intensities of several observed RHEED patterns. We feel that it is physically unreasonable to ignore the important information carried by relative intensities of reflections in RHEED patterns when the relative intensities form the basis of structure determination for most diffraction techniques. Compared to rocking curve fitting, pattern fitting is less affected by the in~idence-angle-dependent errors caused by surface imperfections, instrumentation inaccuracy and surface inelastic scattering etc. This is because for pattern fitting, only the relative intensities of reflections in each pattern are important. They are calculated in one calculation and experimentally recorded at a fixed angle of incidence. This also significantly reduces the time for both calculations and experiments. Most RHEED cameras currently installed on MBE facilities were not designed for rocking curve recording. This gives additional motivation for using pattern fitting for surface structure dete~ination. In order to raise the reliability of structure determination or increase the over-determination ratio, several patterns at different incidence angles couId be fitted simultaneously, as done in the first part of section 4, but it is not necessary to fit RHEED patterns over the whole range of incidence angles. The calculation here is quite straightforward. No new computation approach was involved, except the edge patch method which is a simple numerical modification to the multislice formalism of Cowley and Moodie. The multislice method and its codes in various versions have been widely tested in transmission. In all the above analyses, there were only four parameters involved: toplayer relaxation (61, dimer length (I,), absorption parameter (a), and overall Debye-Waller factor (B). The analyzed result of (Y= 0.1 is consistent with the long-time tested value in the transmission case and B = 0.35 A2 is physically reasonable. The intensities are most sensitive to 6. 6 = +8.0% (+0.113 A) is compatible with the result of surface energy calculation for a relaxed

GaAs(OOl)-2 X 1 surface (21%) [44]. The analyzed results also reject the ho-dimer-missing model, in agreement with the STM observations. The semi-quantitative resuh of dimer length, 2.0 I Z, I 2.5 A is in broad agreement with the result of XPD analyses ( - 2.2 A> [22], but is a little smaller than th: result of surface energy calculations (- 2.63 A) [44]. All these show that the parameters involved in the analyses are physically well defined and their results are physically reasonable. The mean inner potential V, was not adjusted as a correction to the Doyle-Turner potential in the above analyses as suggested by some references 128,39,51]. This is a problem related to the refraction of incident electrons at a crystal surface. Adding an arbitrary constant to V,,, introduces the problem of making this additional potential inside the crystal decay to zero at the surface. This may introduce additional artifacts relating to the structure of the surface potential into the fitting procedure. To avoid dealing with this problem we chose not to vary V,. An error in V, would introduce an error in the effective angle of incidence, but for the e~erimental conditions relevant here the change in angle will be smaI1: N 0.04” for 1 eV error in Y0 for 12.5 keV eIectrons and the angIe of incidence near 3”, which is in the range of experimental precision for the angle of incidence. The uncertainty range of V, previously discussed was - 0.8-1.3 eV for the 12.5 keV electrons [28]. The testing calculations around each angle of incidence relevant to those whose results were presented in table 1 were also carried out by setting fz’ = n rt All\n, where IZ= 41, 45, 48 and An = 1, 2. The results showed that the R-values increased. Therefore, we consider it unlikely that this has a significant effect on our analyses. Compared to the RHEED pattern in fig. 3a, the intensities of two weak reflections [CO,l/4) and (0, l/2>] of the corresponding calculated patterns shown in figs. 5 to 7 and tables 1 to 5 are inherently lower than those observed. Although in terms of intensity ordering they are still compatible with experiment: Zto,1,4) > I(,,, 1,2). This is not the result of possible errors in experimental measurement of the angle of incidence since the

Y. Ma et al. / Quantitative analyses of RHEED patterns from MBE grown GaAs(OO1)-2x 4 surfaces

calculated rocking curves of these two reflections shown in fig. 11 give the same low intensities in the range around 8, = 2.85”. The reason for the discrepancy is not clear yet. The analyses presented here are still quite primitive; they correspond to early X-ray diffraction work for bulk structure determination, The GaAs(OOl)-2 X 4 surfaces observed by STM [l&19] are much more complicated than the models in the analyses. There are all kinds of imperfections on the real surfaces: kinks, steps, dislocated domains and vacancies etc. The volume fraction of imperfections here is much higher than that in bulk diffraction. The sublayer relaxation and possible “buckling” of the surface layers have not been taken into account in the analyses. We also do not know what role the surface inelastic scattering plays in RHEED. It is supposed to have stronger effects on weak reflections. The variation of As surface coverage, discussed by some references [14,28,44], has not been incorporated into the current model. All these facts may play roles in causing the discrepancies between experiments and calculations. The key point is that without driving into these complexities, the determined values of the major parameters (6, B, ar and Id) are quantitatively or semi-quantitatively consistent with values determined using different techniques and the basic surface model was verified. It is not a good strategy to introduce more and more nonphysical or arbitrary parameters into calculations in order to obtain a better fit. In the case of RHEED, the experimental data and theoretical understanding are not yet at a level that makes it sensible to pursue further elaboration of the model. The HOLZ diffraction is another non-major fact in RHEED. The calculated intensities of the first Laue zone reflections in RHEED are normally two or three orders lower than the zerozone reflections, which is related to the energy of incident electrons and the angle of incidence. The intensities of reflections in higher-order zones are even lower. The errors caused by an inaccurate treatment of HOLZ diffraction are considered to be insignificant. They may partially contribute to the discrepancies between calculations and experiments for those weak reflections,

65

such as (0, l/4) and (0, l/Z). The largest excitation error for the incidence of 3.3” in the current analyses is 0.061 A-‘, which is still smaller than half of the reciprocal iattice spacing along the incident beam, 0.125 A-‘. In the calculations, HOLZ diffraction was approximately taken into account by sampling the crystal potential aloqg the incident beam at a rate of 1.001 pts/A. Doubling this sampling rate in the calculation improves the results only insignificantly. The limitation of this multislice-edge-patching approach is its slow speed and large amount of CPU consumption. It cannot, at present, be used as the basis for a general fitting procedure, because of its slowness. However, with the development of computing power and the improvement of understanding and data acquisition in RHEED, the current approach can well be further developed into a general and reliable method for the quantitative analyses of RHEED patterns.

6. Conclusions A method of multislice-edge patching has been used for quantitative analyses of observed RHEED patterns from MBE-grown GaAs(OOl)2 X 4 surfaces [14,15,25]. The surface structure has been proved to be a dimerized vacant 2 x 4 reconstruction with one dimer of every four missing. This is consistent with previous STM observations [18,191. The results showed that the relaxation of the top layer (6) was the parameter most sensitive to the intensities of observed RHEED patterns. Although only 6 was taken into account in the !nalyses, the result obtained, S = i-8.0% (0.113 A), is compatible with the one obtained in surface energy calculations in which the relaxation of sublayers was considered and a 2 x 1 unit cell was employed [44]. The results gave that the overall Debye-Waller factor B = 0.35 A2 and the crystal absorption (Y= 0.1 for the conditions in which the published RHEED patterns were acquired. The dimer length (1,) was also determined semi-quantitatively as 2.0 I Id i 2.5 A, with the help of the observed RHEED pattern acquired in the fllO] azimuth [14]. The determined range of 1, is smaller than the result obtained in

66

I’SMa et al. / Quantitative

analyses of RHEED patterns from MBE grown GaAs(OOl)-2

total-energy calculations [44], but consistent with previous XPD observation [22]. The effects of dimer twist on the intensities of RHEED patterns were also briefly assessed, which indicated that twist of dimers on the GaAs(OOl)-2 X 4 surface is unlikely. The effects of kinematic scattering for the problem were also examined, which demonstrated again that the RHEED from the GaAs(OOl)-2 X 4 surface is basically a dynamical process. The rocking curve of each reflection of the RHEED patterns in the [ilO] azimuth for the case of the best fitted parameters (i.e., 6 = +8.0%, B = 0.35 A’, (Y= 0.1 and I, = 50%) was calculated and plotted, which correctly indicated the threshold incident angle for each reflection. Various errors possibly involved in the calculations have been discussed. All these results strongly demonstrate that the multislice-edgepatching method is a useful and physically comprehensible approach to the quantitative analyses of RHEED patterns. However, the method needs to be further improved, various conditions for data acquisition in RHEED should be reassessed and the acquired data in RHEED should be carefully evaluated. Special caution must be exercised both theoretically and experimentally in a quantitative analysis of RHEED from crystal surfaces.

Acknowledgement

This work was supported by the Department of Energy of USA through Materials Research Laboratory, University of Illinois at UrbanaChampaign, Grant No. DEFG02-91ER45439.

References [l] M.A. Herman and H. Sitter, Molecular Beam Epitaxy (Springer, Berlin, 1989). [2] VS. Kikuchi and S. Nakagawa, Sci. Rep. Inst. Phys. Chem. 21 (1933) 256. [3] L. Trepte, Chr. Menzel-Kopp and E. Menzel, Surf. Sci. 8 (1967) 223. [4] G.J. Russell, Surf. Sci. 19 (1970) 217. [5] S. Ino, Jpn. J. Appl. Phys. 16 (1977) 891. [6] S. Ino, Jpn. J. Appl. Phys. 19 (1980) 1277.

x 4 surfaces

[7] J.J. Harris, B.A. Joyce and P.J. Dobson, Surf. Sci. 103 (1981) L90. [B] C.E.C. Wood, Surf. Sci. 108 (1981) L441. [9] J.J. Harris, B.A. Joyce and P.J. Dobson, Surf. Sci. 108 (1981) L444. DOI J.H. Neave, B.A. Joyce, P.J. Dobson and N. Norton, Appl. Phys. A 31 (1983) 1. [ill J.H. Neave, P.K. Larsen, B.A. Joyce, J.P. Gowers and J.F. van der Veen, J. Vat. Sci. Technol. B 1 (1983) 668. il.21P.K. Larsen, P.J. Dobson, J.H. Neave, B.A. Joyce, B. Bolger and J. Zhang, Surf. Sci. 169 (1986) 176. [I31 J. Aarts and P.K. Larsen, Surf. Sci. 188 (1987) 391. [141 P.K. Larsen and D.J. Chadi, Phys. Rev. B 37 (1988) 8282. Surf. Sci. 240 (1990) iI51 P.K. Larsen and G. Meyer-Ehmsen, 168. [I61 P. Drathen, W. Ranke and K. Jacobi, Surf. Sci. 77 (1978) L162. J.E. Crombeen and T.G.J. van [I71 A.J. van Bommel, Oirschot, Surf. Sci. 72 (1978) 95. [181 M.D. Pashley, K.W. Haberern, W. Friday, J.M. Woodall and P.D. Kirchner, Phys. Rev. Lett. 60 (1988) 2176. and J.M. Gaines, Appl. I191 M.D. Pashley, K.W. Haberern Phys. Lett. 58 (1991) 406. DO1 M.D. Pashley, K.W. Haberern and J.M. Gaines, Surf. Sci. 267 (1992) 153. WI J. Massies, P. Etienne, F. Dezaly and N.T. Linh, Surf. Sci. 99 (1980) 121. w S.A. Chambers, Surf. Sci. Lett. 248 (1991) L274. D31 R.Z. Bachrach, R.S. Bauer, P. Chiaradia and G.V. Hansson, J. Vat. Sci. Technol. 18 (1981) 797. [241 P.K. Larsen, J.H. Neave, J.F. van der Veen, P.J. Dobson and B.A. Joyce, Phys. Rev. B 27 (1983) 4966. [251 B. Bolger and P.K. Larsen, Rev. Sci. Instrum. 57 (1986) 1363. B. Bolger and A.-J. L’61 P.K. Larsen, G. Meyer-Ehmsen, Hoeven, J. Vat. Sci. Technol. A 5 (1986) 611. [271 M.G. Knibb and P.A. Maksym, Appl. Phys. A 46 (1988) 25. Dl J.M. McCoy, U. Korte, P.A. Maksym and G. MeyerEhmsen, Surf. Sci. 261 (1992) 29. D91 Y. Ma, Acta Crystallogr. A 47 (1991) 37. I301 Y. Ma and L.D. Marks, Acta Crystallogr. A 47 (1991) 707. [311 Y. Ma and L.D. Marks, Micro. Res. Tech. 20 (1992) 371. 1321H. Bethe, Ann. Phys. 87 (1928) 55. [331 K. Shinohara, Inst. Phys. Chem. Res. (Tokyo) 18 (1932) 223. [341 P.A. Maksym and J.L. Beeby, Surf. Sci. 110 (1981) 423. I351 A. Ichimiya, J. Appl. Phys. Jpn. 22 (1983) 176. [361 P.G. Self, M.A. O’Keefe, P.R. Buseck and A.E.C. Spargo, Ultramicroscopy 11 (1983) 35. [371 Y. Ma and L.D. Marks, Acta Crystallogr. A 46 (1990) 11. Surf. Sci. Lett. 226 [381 M. Stock and G. Meyer-Ehmsen, (1990) L59. Surf. Sci. 271 (1992) [391 U. Korte and G. Meyer-Ehmsen, 616. Ultramicroscopy 21 (1987) 131. [401 P.A. Stadelmann,

Y. Ma et al. / Quantitative analyses of RHEED patterns from MBE grown GaAs(OOlh2 X 4 surfaces [41] P.A. Doyle and P.S. Turner,

Acta Crystallogr. A 24 (1968) 390. [42] B.A. Joyce, J.H. Neave, P.J. Dobson, P.K. Larsen and J. Zhang, J. Vat. Sci. Technol. A 5 (1987) 834. [43] P.K. Larsen, J.F. van der Veen, A. Mazur, J. Pollmann, J.H. Neave and B.A. Joyce, Phys. Rev. B 26 (1982) 3222. [44] Guo-Xin Qian, R.M. Martin and D.J. Chadi, Phys. Rev. B 38 (1988) 7649. [45] J.E. Bonevich, Y. Ma and L.D. Marks, Acta Crystallogr. A 47 (1991) 789. [46] M.D. Shannon, J.A. Eades and M.E. Meichle, Ultramicroscopy 16 (1985) 175.

67

[47] J.A. Eades, A.E. Smith and D.F. Lynch, Proceedings of the 45th Annual Meeting of the Electron Microscopy Society of America, Ed. G.W. Bailey (San Francisco Press, San Francisco, 1987) p. 30. [48] A.E. Smith and D.F. Lynch, Acta Crystallogr. A 44 (1988) 780. [49] A.E. Smith, Ultramicroscopy 31 (1989) 431. [50] T.C. Zhao, H.C. Poon and S.Y. Tong, Phys. Rev. B 38 (1988) 1172. [51] M.G. Knibb, Surf. Sci. 257 (1991) 389.