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Dynamical diffraction effect for RHEED intensity oscillations: phase shift of oscillations for glancing angles
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surface science ..
298 (1993) 261-272
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Dynamical diffraction effect for RHEED intensity oscillations: phase shift of oscillations for glancing angles Y. Horio and A. Ichimiya Department of Applied Physics, School of Engineering, Nagoya University, Nagoya 464-01, Japan Received
16 March
1993; accepted
for publication
6 June
1993
Reflectivity of high-energy electron beams at an MBE growing surface is calculated for varying layer thicknesses and glancing angles of the electron beam. In order to investigate some fundamental behaviors of RHEED intensity oscillations during MBE growth, we introduce a “simple potential model”, which is a step-like potential for each uncompleted growing layer. The depth of the growing layer potential is assumed to be in direct proportion to the coverage of the layer. We also introduce a realistic crystal potential in the calculation to check the “simple potential model”. A birth-death growth model is assumed in these calculations. It is found that the potential of the growing surface layer plays an important role in the phase-shift and frequency-doubling phenomena which are observed in experimental RHEED intensity oscillation curves. These phenomena can be explained by the averaged potential in the growing layer instead of the realistic potential.
1. Introduction
Reflection high-energy electron diffraction (RHEED) is one of the most powerful tools for the study of molecular beam epitaxy (MBE). Oscillations of specular beam intensity were observed for the first time during MBE growth by Harris et al. [ll. Cohen et al. [2] explained the oscillations by kinematic diffraction theory and showed the change of intensity of the diffracted beam for various terrace distributions. Ichimiya [3] used Kirchhoff s diffraction theory and showed that integrated intensities scarcely depend on step distributions, but the intensities on reciprocal rods depend sensitively on the step densities and terrace areas. Kawamura and Maksym [4] calculated the effects of a periodic array of surface steps by dynamical theory and obtained intensity oscillations during MBE growth. Peng and Whelan [5] developed a practical computing procedure in the one-beam case and obtained RHEED oscillations with a phase shift. Recently, Mitura et al. [6,7] reported frequency-doubling phenomena, during the growth of a lead-indium alloy, which have two maxima in a period of RHEED oscillations at 0039-6028/93/$06.00
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Science
Publishers
very low glancing angles of 0.35” and reproduced the phenomena by dynamical calculation. The main aim of this paper is to make the origin of the phase-shift [8,9] and frequency-doubling phenomena [6,8] clear, where these phenomena cannot be explained by a kinematic approach in the condition of layer-by-layer growth. At the one-beam condition [lo], at which the azimuth of the incident beam direction is turned away from a certain crystallographic direction, high-energy electrons are mainly affected by the averaged potentials of atomic layers parallel to the surface. Then, a one-dimensional (z-direction) averaged potential is used in the calculation of reflectivity during MBE growth. For example, a Si/Si(lll) homo-epitaxial growing surface was chosen as a subject of the study. We calculated a reflectivity at the growing surface in the one-beam condition, using a 10 kV electron beam, where the surface structure was treated as an ideal surface. In order to study some fundamental behaviors on the reflectivity, a simple step-like potential, “simple potential model”, was introduced for the growing layer potential. Taking advantage of the simplicity of
B.V. All rights reserved
262
Y. Horio, A. Ichimiya / Dynamical diffraction effect for RHEED intensity oscillation
the above potential model, it is possible to understand the physics of the characteristic behaviors of RHEED oscillations. To check the validity of the “simple potential model”, a realistic potential, “realistic potential model”, was also used in the calculation. Features and origins of oscillation curves and also of rocking curves calculated by these two potential models are described on the basis of the interference of electron waves from the growing layer potential.
2. Calculation of reflectivity Fig. 1 shows the atomic arrangements of a Si(ll1) surface in the direction of (a): 11121 and
Fig, 1. Atomic
arrangements
of Si(lll).
Directions
(b): 7” turned away from [112]. It is seen from fig. lb that the three-dimensional crystal lattice can be regarded as a one-dimensional array of lattice planes parallel to the surface. This azimutha direction corresponds to the one-beam condition. In this condition, the crystal potential can be treated as a one-dimensional potential in the surface normal direction, as shown in fig. 2. The “simple potential model” is indicated by a broken line. We assumed the potential V, of the nth growing layer to be in direct proportion to the nth layer coverage 0, as V, = O,V, [4]. For the substrate, the uniform mean inner potential V, of the Si crystal was used. The real part of V, was assumed to be Vi = 12 V and the imagina~ part to be vi = 1 V. Since a bilayer of Sic11 1) is one
of view are (a): [113] and (b): 7” from [ll?], which corresponds to the one-beam condition.
Y. Horio, A. khimiya
/ Dynamical diffraction effect for RHEED intensity oscillation
unit in the epitaxial growth of Si/Si(lll), the terrace width of the step-like potential was set as d = 3.14 A. The number of steps depends on the growth mode. After fulfilling one bilayer, the completed growth layer is incorporated into the substrate potential VO. The dynamic change of the potential in growing layers during MBE growth gives rise to the reflectivity change. On the other hand, the “realistic potential model” is indicated by a solid line in fig. 2. The potential is cut into 30 slices in one bilayer and averaged within each thin slice. Thirty bilayers of the Si(ll1) substrate are taken into account. It has been confirmed that the number of slices and of the substrate layers is enough for accurate calculations. The wave functions of fast electrons at the one-beam condition are generally given as q’(r) =A exp(iK-r)
+B exp(iK’-r)
in vacuum, &n(r) = a, exp( ik, * r) +
b,
263
from the back surface is considered. Here one layer means one bilayer in the case of Si/Si(lll). The reflectivity is calculated as square of amplitude ratio of the reflected wave A to the incident wave B. The reflectivity R for the case of the “simple potential model” can be analytically deduced from two boundary conditions at each interface of the growing layers, that is the continuities of the wave functions and of their differential functions. 2
=
~-a~exp(-i~+d~)~~+b~e~(-i~-d~) i
PaN
exp(ir-d,)+
T-b,
exp(iI’+d,)
’
where P=r+y, and T-=T-y,,,. N is the number of the last growing layer. All notations are indicated in fig. 3. a, and b,, are represented as series of numbers,
exp( ik; * r)
in the nth sliced layer, and (crO(r) = a0 exp(ik, *r) in the last sliced layer, at which no reflection
-d
2d
glowing
Fig. 2. Potential in direction of Si(lll)
--
layers
of Si’
Si(lll1
-2d
-3d
substrate
surface normal. Broken and solid lines indicate the “simple potential model” and the “realistic potential model”, respectively.
Y. Horio, A. Ichimiya / Dynamical diffraction effect for RHEED intensity oscillation
264
and Yl + Yo
a1=-xb 1 = -a Y1 -Y” 2Yl
01
where r,i+,=Y,,+~
+ Y,
and -Y;+~=~,,+l
-Y,.
r
and y,, are the surface normal components of K and k,, respectively. y,, is a complex number as yn = y,’ + iy$ y,’ and yi are the real and imaginary parts of yny,,respectively. They are given by the conditions of tangential continuity of the wave at interfaces as
d (P
Y,’=
( r2 +
+ un’) +
u;), f (u$
0
exP(iyi,, AZ) c
-1
Pin
’ ri,n
i Pi,n
3 =
i
substrate
1st layer
1
according to the multi-slice RHEED theory [lO,ll], where ?j,n = r + Yiyi.,9
4
dynamical
AZ is the width of the slice, zi -.z~_.~. YE= 1, 2, 3... N are growing Iayers and n = 0 substrate layers. One layer is sliced at 1= 30. Oj is the coverage of the jth growing layer and 0, = 1 for the substrate. V(z,> is the averaged potential in AZ at z = zi and is a complex number. The real part of VCzi) is caIculated by using Doyle and Turner’s analytical formula [121. The imaginary part is assumed to be 0.1 of the real part. Then the transfer matrix for the substrate is
where m = 30 are the layers of the substrate considered. The transfer matrix, on the other hand, for growing layers is N I I \
where N is the number of growing layers. Finally, the amplitudes A and B of the wave function in vacuum are expressed as
Pi,n = r - Yi,n 5
l-=
i--
$E;, 2me
yj,&
=
~
r2 + -pnV(Zi)
di
4
9
7
‘in
n-th Iayer
Fig. 3. Notations of incident and reflected electron waves in each growing layer potential.
Here UL = 2meVi/k2 and Ud = 2meV,‘/R2. In the case of the “realistic potential model”, on the other hand, the calculation of reflectivity was carried out by the multi-slice method. The transfer matrix Pi,+ at the ith slice in the nth layer is
X’
growing layers
Then, reflectivity is calculated as .
265
I’. Ho&, A. Ichimiya / Dynamical diffraction effect for RHEED intensity oscillation
0 Thickness (EL)
(b)
Thickness (BL)
(4
2
1
3
4
5
Thickness (BL)
Fig. 4. Surface growth modes by birth-death model. Solid and broken lines indicate growing layer coverage and roughness, respectively. (a) Nearly perfect layer-by-layer growth mode of D = 3500. (b), Cc)Multi-layer growth modes of D = 100 and D = 10, respectively.
:~~“~,~~~ ,‘Y2.7
1
2
3
THICKNESS (BL)
4
W
THICKNESS (BL)
Fig. 5. RHEED intensity oscillation curves under the growth mode of I) = 3500. (a) Calculated by the “simple potential model”; (b) by the “realistic potential model”.
Y. Horio, A. Ichimiyn / Dynamical diffruction effect for MEED
266
=-----I~_.___
For MBE growth, growing layer coverage was simulated by using the birth-death model [13,2]. da,, 1 = ;(o,-, dt
t(
n -
I.
.I,~
1st
-
@,+,J(%-1 - %>
with a growth rate of l/r monolayers per second. We chose the following three types as growth mode: (1) diffusion parameter D =i 3500 (such a large value makes a nearly perfect layer-by-layer growth mode and only one growing layer always exists), (2) D = 100, and (3) D = 10 (such a small value makes a multi-Iayer growth mode and three or four uncompleted growing layers always exist). The layer coverage and roughness of the three growth modes are shown in fig. 4. The roughness [2] 4’ is given by A2=
--.___ --.___ ..__-.__ -.__
.-I;)\--n,
+D(%+,
t/+pn - On+,).
3. Calculated results and discussion The oscillation curves of reflectivity, that is RHEED intensity oscillations of the specular beam, are shown in fig. 5; (a>: calculated by the “simple potential model”, and (b): calculated by the “realistic potential model”. Their growth modes are simulated as a nearly perfect layer-bylayer growth mode of D = 3500. The intensities of the re~ectivi~ are normalized to 1 at the ma~mum for each curve. At first, it is found that behaviors of the oscillation curves calculated by the “simple potential model” are rather similar to those by the “realistic potential model”. This means that the approximation of the “simple potential model” is appropriate. Detail features of the oscillations of the “simple potential model” are as follows: (i> a frequency-doubling phenomena is seen at low glancing angles such as 0 = OS”, (ii) a phase-shift phenomenon appears when the glancing angle is varied, but in this case the period of the oscillation is kept at one bilayer,
intensity oscillation
....
growing layer “,..._
d
Fig. 6. Interference between electron beams reflected from top and bottom faces of the first growing bilayer. Broken lines indicate each path of the electron beams giving rise to the phase difference.
(iii) the reflectivity always shows a maximum at the integer number of thickness, (iv) anomalous behavior is seen at around the integer number of thickness in the curve of 8 = 2.0”, (v) The oscillation curves are sinusoidal at 8 = 1” and 2.5”, but not at the other glancing angles. These features can be explained by the interference of electron beams, which is caused by the phase difference between the electron beams reflected from the top and bottom faces of the first growing bilayer as shown in fig. 6. The phase difference is represented by 2yrd, which depends on the potential depth of the first growing layer, i.e. on the coverage of the first growing layer, and on the glancing angle. Her, 2y, is the momentum transfer concerned with the OO-rod. Fig. 7 shows the phase values for the coverage of the first growing layer when varying the glancing angle from 8 = 0.5“ to 3.0”. The phase value of 2nrr (n is an integer) corresponds to the in-phase condition (this means constructive interference) and that of (2n + 1)~ to the out-of-phase condition (destructive interference). Closed and open circles indicate the in-phase and out-of-phase conditions, respectively. At lower glancing angle such as 0 = 0.5”, two out-of-phase conditions and one in-phase condition are satisfied in the region of layer coverage 0 = O-l. At higher glancing angles such as @= 2.5 or 3.0“, only one phase condition (in-phase or out-of-phase) is satisfied. In order to see the relation between the phase conditions and the reflectivity, arrows are introduced
267
Y. Horio, A. Ichimiya / Dynamical diffraction effect for RHEED intensity oscillation
in fig. 5. Upward and downward arrows indicate in-phase and out-of-phase conditions, respectively. It is found that peaks and dips in the oscillation curves nearly agree with the upward and downward arrows, respectively. At 8 = 0.5”, two out-of-phase conditions and one in-phase condition give rise to the frequency-doubling phenomenon of feature 6). Such phase conditions are shifted with the change of the glancing angle. This is the reason for the phase-shift phenomenon of feature (ii>. A flat surface appears at the integer number of thickness in the case of a perfect layer-by-layer growth mode, and then the potential at the top surface becomes most abrupt in the “simple potential model”. Considering potential scattering, reflectivity increases when the potential at the top surface becomes abrupt. This is the reason for feature (iii). In the “realistic
67~
5lr
01 : ; : : : : :
I
:
-1
1.0
0.5
0.0
Layer coverage Fig. 7. Dependence of the phase difference on the coverage of the growing layer at each glancing angle, (a): 0 = OS”, (b): l.O”, cc): 1.5”, cd): 2.0”, (e) 2.5” and (f): 3.0”.
I\
97
0i---~1 Fig. 8. RHEED
intensity
2
3
4
THICKNESS CEIL)
(a) oscillation
curves
5
(b)
THICKNESS CEIL)
under the growth mode of D = 100. (a) Calculated (b) by the “realistic potential model”.
by the “simple
potential
model”;
Y. Horio, A. Ichirniya / Dynamical diffraction effect for RHEED intensity oscillation
268
0 = 1 and 2.5”, the out-of-phase condition is located nearly at the middle point of one bilayer. In this case, the two factors are cooperative and sinusoidal curves arise. This is the reason for feature (v). RHEED oscillations are also calculated under the growth modes of D = 100 and D = 10 as shown in figs. 8 and 9. The number of uncompleted growing layers increases as the value of diffusion parameter D decreases, which corresponds to a multi-layer growth mode. In this case, the reflectivity is affected by the increased number of electron beams reflected from many uncompleted growing layers. Superposition of these increased waves makes the curves sinusoidal, as seen in fig. 9. The phase consideration based on the growth mode of D = 3500 is also supported,
potential model”, in fig. 5b, however, reflectivity is not a maximum at the thickness of the integer number, which is considered to be affected by a diffraction effect from the substrate. Anomalous behavior is seen at around the integer number of thickness in the curve of f3= 2.0” in fig. 5a. Although the out-of-phase condition is satisfied at this point (indicated by a downward arrow), a peak arises in the curve. The contradiction is due to competition between a negative factor (out-ofphase condition) and a positive factor (abruptness of the top surface potential), and the positive factor is superior to the negative factor. This is the reason for feature (iv>. The relation between the phase condition and the abruptness of the potential determines whether the oscillation curve is sinusoidal or not. In the oscillation curves at
I
1
oL
i
2
3
THICKNESS (BL) Fig. 9. RHEED
intensity
oscillation
curves
L
-il_
_.__~~___
4
5
30
3
t
0 (b;l
,1
1
?
3
4
253.9
5
THICKNESS (BL)
under the growth mode of D = 10. (a) Calculated (b) by the “realistic potential model”.
by the “simple
potential
model”;
Y Horio, A. Ichimiya / Dynamical diffraction effect for RHEED intensity oscillation
1
2
3
Thickness
Fig. 10. ~nematically
4
269
5
(BL)
calculated RHEED intensity oscillation curves under the growing mode of (a): D = 3500, (bf: D = 100 and Glancing angles are (A): 19= 0.5”, (B): I.O”, (C): 1.5”, (D): 2.0”, (E): 2.5” and (F): 3.0”.
CC): D = 10.
even in D = 100 and D = 10, because upward and downward arrows nearly correspond to the peaks and dips in the curves, respectively. The similar phase-shift phenomenon is also seen in the experimental result [9]. In order to compare these one-beam dynamical calculation results, kinematically calculated intensities of the specular beam at glancing an-
(al
Glancing
Angle
(deg.)
gles from 8 = 0.5 to 3.0” are shown in fig. 10. The intensities [21 are calculated by the single scattering from each uncovered terrace as 2
la
f (0, -On+,) n=O
exp( -i2rnd)
.
It is found that the oscillation amplitude depends
Glancing
Angie
(deg.)
Fig. il. Rocking curves under the growth mode of D = 3504 at each growing stage from 8 = 0 to 0.9. (a) Calculated by the “simple potential modet”; (b) by the “realistic potential model”.
on the glancing angle, namely on the degree of satisfaction of the Bragg condition (the maximum amplitude of oscillation is obtained just on the conditions of off-Bragg and no oscillation just on the condition of on-Bragg). RHEED oscillation curves do not have any phase shifts varying the glancing angle in the kinematic approach. To see another property of the reflectivity, the dependence of the reflectivity on the glancing angle, rocking curves of the specular beam were also calculated at many growing stages. Fig. 11 shows the rocking curves; (a): cafcufated by the “simple potential model” and (b): caIcuIated by the “realistic potential model”. In the rocking curves, the reflectivities are multipIied by sin 8 because of the change of the incident beam intensity at the surface depending on the glancing angle. These rocking curves are calculated at ten growing stages from 0 = 0 to 0.9 BL, under the growth mode of D = 3500, Rocking curves (a) of
Glancing
the “‘simpfe potential model” show only interference of eIectron waves from the step-like potential and no diffraction effect from bulk Si crystal. Especially at 0 = 0 BL, the potential shape is a uniform distribution of mean inner potential V,, and reflectivity shows a monotonous decay as the glancing angle is increased. Advancing the Iayer coverage to 0 = 0.5 BL, Bragg peaks gradually appear in the curve, due to the potential step of the first growing layer, and gradually disappear close to 0 = 1 BL, due to the potential becoming uniform again, The Bragg peak positions are shifted to an angle a little higher, because the step-like potential of the uncompleted layer is shallower than the mean inner potential P’a of the substrate Si crystal. On the other hand, the rocking curves (b) of the “realistic potential model” reveal sharp Bragg peaks and can be compared with experimental curves. It is found that both rocking curves of (a) and Cb) reveal a similar
Angie (deg.1
Fig. 12. Rocking curves under the growth mode of D = 10 at each growing stage from 0 = 3.0 to 3.9. fa) Cafeulated by the “simple potential model” and (h) by the “realistic potential model”.
I’. Horio, A. Ichimiyu / L$vzamical diffraction effect for RHEED intensity oscillation
fluctuation at a lower glancing angle (0 < 2”). The rocking curves at lower glancing angles are sensitively affected by the surface growth stage, namely the potential change of the first growing layer. Checking the behavior in detail, the dips, indicated by arrows, shift to lower angles as the layer coverage becomes one bilayer. This feature is also seen in the experimental rocking curves of Si/Si(lll) [9]. It is considered that the fluctuation at lower glancing angles is due to the interference effect, because the dip positions, indicated by arrows in fig. lla, correspond to the out-of-phase condition. Fig. 12 shows rocking curves calculated from 0 = 3.0 to 3.9 BL under the growth mode of D = 10. In the rocking curves (a) of the “simple potential model”, Bragg peaks appear more clearly than those of the growth mode D = 3500 in fig. lla. This is because many uncompleted growing layers always exist at such a multi-layer growth mode of D = 10, and many steps of the potential make the Bragg peak intensity stronger. In the rocking curves (b) of the “realistic potential model”, the behavior of the dip at a low glancing angle is the same as in the case of D = 3500, except for a difference of the shape. In conclusion, we have tried to analyze the RHEED oscillation with the “simple potential model” and can reproduce the phase-shift and frequency-doubling phenomena on the assumption of the birth-death growth. These phenomena cannot be reproduced by the kinematic approach. It is found that these phenomena can be explained by the interference of electron waves from the potential of the top growing layer. It has been confirmed that the behaviors of the oscillation curves of the “simple potential model” are similar to those of the “realistic potential model”. As a result, it is considered that the RHEED oscillation is not due to the potential shape so much, but to the dynamic change of the potential depth of the growing layer. This means that the phase-shift phenomena can be explained by the averaged potential in the growing layer, instead of by the complicated potential. The RHEED oscillation becomes sinusoidal, which shape is generally observed in experiment, as the growth mode changes to multi-layer growth. We have
271
also calculated the rocking curves at MBE growing stages by the two potential models. The rocking curves of the “simple potential model” merely show the interference of waves from the step-like potential; then strict comparison with experiment cannot be done. The intensity profiles of the rocking curves are rather different between the two potential models, however, the similar fluctuation in the low glancing angle region (0 < 2.0°) can be seen. The behavior in the low glancing angle region is also observed in experimental rocking curves of Si/Si(lll) MBE growth [9] and can be explained by interference of electron waves from the growing layer potential. AI1 calculations in this paper were carried out on the one-beam condition with a 10 kV electron beam. In this condition, the approximation using averaged potential parallel to the surface becomes more correct as the glancing angle becomes lower. At very low glancing angle, the reflectivity of high-energy electron beam increases considerably and surface potential change becomes more important for RHEED oscillation curves as well as for rocking curves. It is considered that our “simple potential model”, with a relatively short calculation time, is a usefu1 interpretation method for RHEED oscillations.
Acknowledgements The authors thank Professor J.L. Beeby for valuable discussions. This work was carried out under the support of Grants-in-Aid to the Scientific Research on Priority Areas by the Ministry of Education, Science and Culture (Nos. 03243219 and 04227216) and an international collaboration program of the Japan Society for the Promotion of Science.
References 111 J.J. Harris, (1984) 121. I21 P.I. Cohen, A.S. Arrott, [3] A. Ichimiya, [4] T. Kawamura
B.A. Joyce G.S. Surf. Surf. and
and P.J. Dobson,
Surf. Sci. 139
Petrich, P.R. Pukite, C.J. Whaley and Sci. 216 (1989) 222. Sci. 187 (1987) 194. P.A. Maksym, Surf. Sci. 161 (1985) 12.
272
Y Horio, A. Ichimiya / Dynamical diffraction
[5] L.-M. Perry and M.J. Whelan, Surf. Sci. Lett. 238 (1990) L446. [6] Z. Mitura, M. Stroiak and M. Jaiochowski, Surf. Sci. Lett. 276 (1992) L15. [7] Z. Mitura and A. Daniluk, Surf. Sci. 277 (1992) 229. [81 P.J. Dobson, B.A. Joyce, J.H. Neave and J. Zhang, J. Cryst. Growth 81 (1987) 1. (93 H. Nakahara and A. Ichimiya, .I. Cryst. Growth 95 (1989) 472.
effectfor RHEED inter&
oscillation
[lOI A. Ichimiya, The Structure of Surfaces III, Vol. 24 of Springer Series in Surface Sciences (Springer, Berlin, 1991) p. 162. [ll] A. Ichimiya, Jpn. J. Appl. Phys. 22 (1983) 176; 24 (1985) 1365. [12] P.A. Doyle and P.S. Turner, Acta Cryst. A 24 (1968) 390. [131 J.D. Weeks, G.H. Giimer and K.A. Jackson, J. Chem. Phys. 65 11976) 712.
“.‘.‘.-~~~.1.........,.......,..,...,,,,,,,,,,,, . . . .. :.:.:::::::::::.:.,:.:.:.:.?.:.:.:.:.> . . . . . .. . . . . :.:.:.: . .. . . . ?”
.....>> :;.:; .............. Surface Science North-Holland
,.:.:.:.:.:
‘.)
surface science ..
298 (1993) 261-272
..,.:.:.: ...... ‘.‘.‘.‘.:... .,., ., .:.>:.:.::: . ,.,., .:..>..+:::::j:::&:.:.:> :.:.:.:::, ‘:‘f”.l: ?3: ...(,.‘.‘.‘. .......,.,.,_, ,,,, ‘::‘i::‘.‘V. .,.. :.,,,,_ ,,,,,. ,,,,(,,,,, ““:::.::i::.:.:.:.:.: .:.:.:.:.. .....i. ........‘..‘...)... :,:: ,.;, >. ,‘p...‘.
Dynamical diffraction effect for RHEED intensity oscillations: phase shift of oscillations for glancing angles Y. Horio and A. Ichimiya Department of Applied Physics, School of Engineering, Nagoya University, Nagoya 464-01, Japan Received
16 March
1993; accepted
for publication
6 June
1993
Reflectivity of high-energy electron beams at an MBE growing surface is calculated for varying layer thicknesses and glancing angles of the electron beam. In order to investigate some fundamental behaviors of RHEED intensity oscillations during MBE growth, we introduce a “simple potential model”, which is a step-like potential for each uncompleted growing layer. The depth of the growing layer potential is assumed to be in direct proportion to the coverage of the layer. We also introduce a realistic crystal potential in the calculation to check the “simple potential model”. A birth-death growth model is assumed in these calculations. It is found that the potential of the growing surface layer plays an important role in the phase-shift and frequency-doubling phenomena which are observed in experimental RHEED intensity oscillation curves. These phenomena can be explained by the averaged potential in the growing layer instead of the realistic potential.
1. Introduction
Reflection high-energy electron diffraction (RHEED) is one of the most powerful tools for the study of molecular beam epitaxy (MBE). Oscillations of specular beam intensity were observed for the first time during MBE growth by Harris et al. [ll. Cohen et al. [2] explained the oscillations by kinematic diffraction theory and showed the change of intensity of the diffracted beam for various terrace distributions. Ichimiya [3] used Kirchhoff s diffraction theory and showed that integrated intensities scarcely depend on step distributions, but the intensities on reciprocal rods depend sensitively on the step densities and terrace areas. Kawamura and Maksym [4] calculated the effects of a periodic array of surface steps by dynamical theory and obtained intensity oscillations during MBE growth. Peng and Whelan [5] developed a practical computing procedure in the one-beam case and obtained RHEED oscillations with a phase shift. Recently, Mitura et al. [6,7] reported frequency-doubling phenomena, during the growth of a lead-indium alloy, which have two maxima in a period of RHEED oscillations at 0039-6028/93/$06.00
0 1993 - Elsevier
Science
Publishers
very low glancing angles of 0.35” and reproduced the phenomena by dynamical calculation. The main aim of this paper is to make the origin of the phase-shift [8,9] and frequency-doubling phenomena [6,8] clear, where these phenomena cannot be explained by a kinematic approach in the condition of layer-by-layer growth. At the one-beam condition [lo], at which the azimuth of the incident beam direction is turned away from a certain crystallographic direction, high-energy electrons are mainly affected by the averaged potentials of atomic layers parallel to the surface. Then, a one-dimensional (z-direction) averaged potential is used in the calculation of reflectivity during MBE growth. For example, a Si/Si(lll) homo-epitaxial growing surface was chosen as a subject of the study. We calculated a reflectivity at the growing surface in the one-beam condition, using a 10 kV electron beam, where the surface structure was treated as an ideal surface. In order to study some fundamental behaviors on the reflectivity, a simple step-like potential, “simple potential model”, was introduced for the growing layer potential. Taking advantage of the simplicity of
B.V. All rights reserved
262
Y. Horio, A. Ichimiya / Dynamical diffraction effect for RHEED intensity oscillation
the above potential model, it is possible to understand the physics of the characteristic behaviors of RHEED oscillations. To check the validity of the “simple potential model”, a realistic potential, “realistic potential model”, was also used in the calculation. Features and origins of oscillation curves and also of rocking curves calculated by these two potential models are described on the basis of the interference of electron waves from the growing layer potential.
2. Calculation of reflectivity Fig. 1 shows the atomic arrangements of a Si(ll1) surface in the direction of (a): 11121 and
Fig, 1. Atomic
arrangements
of Si(lll).
Directions
(b): 7” turned away from [112]. It is seen from fig. lb that the three-dimensional crystal lattice can be regarded as a one-dimensional array of lattice planes parallel to the surface. This azimutha direction corresponds to the one-beam condition. In this condition, the crystal potential can be treated as a one-dimensional potential in the surface normal direction, as shown in fig. 2. The “simple potential model” is indicated by a broken line. We assumed the potential V, of the nth growing layer to be in direct proportion to the nth layer coverage 0, as V, = O,V, [4]. For the substrate, the uniform mean inner potential V, of the Si crystal was used. The real part of V, was assumed to be Vi = 12 V and the imagina~ part to be vi = 1 V. Since a bilayer of Sic11 1) is one
of view are (a): [113] and (b): 7” from [ll?], which corresponds to the one-beam condition.
Y. Horio, A. khimiya
/ Dynamical diffraction effect for RHEED intensity oscillation
unit in the epitaxial growth of Si/Si(lll), the terrace width of the step-like potential was set as d = 3.14 A. The number of steps depends on the growth mode. After fulfilling one bilayer, the completed growth layer is incorporated into the substrate potential VO. The dynamic change of the potential in growing layers during MBE growth gives rise to the reflectivity change. On the other hand, the “realistic potential model” is indicated by a solid line in fig. 2. The potential is cut into 30 slices in one bilayer and averaged within each thin slice. Thirty bilayers of the Si(ll1) substrate are taken into account. It has been confirmed that the number of slices and of the substrate layers is enough for accurate calculations. The wave functions of fast electrons at the one-beam condition are generally given as q’(r) =A exp(iK-r)
+B exp(iK’-r)
in vacuum, &n(r) = a, exp( ik, * r) +
b,
263
from the back surface is considered. Here one layer means one bilayer in the case of Si/Si(lll). The reflectivity is calculated as square of amplitude ratio of the reflected wave A to the incident wave B. The reflectivity R for the case of the “simple potential model” can be analytically deduced from two boundary conditions at each interface of the growing layers, that is the continuities of the wave functions and of their differential functions. 2
=
~-a~exp(-i~+d~)~~+b~e~(-i~-d~) i
PaN
exp(ir-d,)+
T-b,
exp(iI’+d,)
’
where P=r+y, and T-=T-y,,,. N is the number of the last growing layer. All notations are indicated in fig. 3. a, and b,, are represented as series of numbers,
exp( ik; * r)
in the nth sliced layer, and (crO(r) = a0 exp(ik, *r) in the last sliced layer, at which no reflection
-d
2d
glowing
Fig. 2. Potential in direction of Si(lll)
--
layers
of Si’
Si(lll1
-2d
-3d
substrate
surface normal. Broken and solid lines indicate the “simple potential model” and the “realistic potential model”, respectively.
Y. Horio, A. Ichimiya / Dynamical diffraction effect for RHEED intensity oscillation
264
and Yl + Yo
a1=-xb 1 = -a Y1 -Y” 2Yl
01
where r,i+,=Y,,+~
+ Y,
and -Y;+~=~,,+l
-Y,.
r
and y,, are the surface normal components of K and k,, respectively. y,, is a complex number as yn = y,’ + iy$ y,’ and yi are the real and imaginary parts of yny,,respectively. They are given by the conditions of tangential continuity of the wave at interfaces as
d (P
Y,’=
( r2 +
+ un’) +
u;), f (u$
0
exP(iyi,, AZ) c
-1
Pin
’ ri,n
i Pi,n
3 =
i
substrate
1st layer
1
according to the multi-slice RHEED theory [lO,ll], where ?j,n = r + Yiyi.,9
4
dynamical
AZ is the width of the slice, zi -.z~_.~. YE= 1, 2, 3... N are growing Iayers and n = 0 substrate layers. One layer is sliced at 1= 30. Oj is the coverage of the jth growing layer and 0, = 1 for the substrate. V(z,> is the averaged potential in AZ at z = zi and is a complex number. The real part of VCzi) is caIculated by using Doyle and Turner’s analytical formula [121. The imaginary part is assumed to be 0.1 of the real part. Then the transfer matrix for the substrate is
where m = 30 are the layers of the substrate considered. The transfer matrix, on the other hand, for growing layers is N I I \
where N is the number of growing layers. Finally, the amplitudes A and B of the wave function in vacuum are expressed as
Pi,n = r - Yi,n 5
l-=
i--
$E;, 2me
yj,&
=
~
r2 + -pnV(Zi)
di
4
9
7
‘in
n-th Iayer
Fig. 3. Notations of incident and reflected electron waves in each growing layer potential.
Here UL = 2meVi/k2 and Ud = 2meV,‘/R2. In the case of the “realistic potential model”, on the other hand, the calculation of reflectivity was carried out by the multi-slice method. The transfer matrix Pi,+ at the ith slice in the nth layer is
X’
growing layers
Then, reflectivity is calculated as .
265
I’. Ho&, A. Ichimiya / Dynamical diffraction effect for RHEED intensity oscillation
0 Thickness (EL)
(b)
Thickness (BL)
(4
2
1
3
4
5
Thickness (BL)
Fig. 4. Surface growth modes by birth-death model. Solid and broken lines indicate growing layer coverage and roughness, respectively. (a) Nearly perfect layer-by-layer growth mode of D = 3500. (b), Cc)Multi-layer growth modes of D = 100 and D = 10, respectively.
:~~“~,~~~ ,‘Y2.7
1
2
3
THICKNESS (BL)
4
W
THICKNESS (BL)
Fig. 5. RHEED intensity oscillation curves under the growth mode of I) = 3500. (a) Calculated by the “simple potential model”; (b) by the “realistic potential model”.
Y. Horio, A. Ichimiyn / Dynamical diffruction effect for MEED
266
=-----I~_.___
For MBE growth, growing layer coverage was simulated by using the birth-death model [13,2]. da,, 1 = ;(o,-, dt
t(
n -
I.
.I,~
1st
-
@,+,J(%-1 - %>
with a growth rate of l/r monolayers per second. We chose the following three types as growth mode: (1) diffusion parameter D =i 3500 (such a large value makes a nearly perfect layer-by-layer growth mode and only one growing layer always exists), (2) D = 100, and (3) D = 10 (such a small value makes a multi-Iayer growth mode and three or four uncompleted growing layers always exist). The layer coverage and roughness of the three growth modes are shown in fig. 4. The roughness [2] 4’ is given by A2=
--.___ --.___ ..__-.__ -.__
.-I;)\--n,
+D(%+,
t/+pn - On+,).
3. Calculated results and discussion The oscillation curves of reflectivity, that is RHEED intensity oscillations of the specular beam, are shown in fig. 5; (a>: calculated by the “simple potential model”, and (b): calculated by the “realistic potential model”. Their growth modes are simulated as a nearly perfect layer-bylayer growth mode of D = 3500. The intensities of the re~ectivi~ are normalized to 1 at the ma~mum for each curve. At first, it is found that behaviors of the oscillation curves calculated by the “simple potential model” are rather similar to those by the “realistic potential model”. This means that the approximation of the “simple potential model” is appropriate. Detail features of the oscillations of the “simple potential model” are as follows: (i> a frequency-doubling phenomena is seen at low glancing angles such as 0 = OS”, (ii) a phase-shift phenomenon appears when the glancing angle is varied, but in this case the period of the oscillation is kept at one bilayer,
intensity oscillation
....
growing layer “,..._
d
Fig. 6. Interference between electron beams reflected from top and bottom faces of the first growing bilayer. Broken lines indicate each path of the electron beams giving rise to the phase difference.
(iii) the reflectivity always shows a maximum at the integer number of thickness, (iv) anomalous behavior is seen at around the integer number of thickness in the curve of 8 = 2.0”, (v) The oscillation curves are sinusoidal at 8 = 1” and 2.5”, but not at the other glancing angles. These features can be explained by the interference of electron beams, which is caused by the phase difference between the electron beams reflected from the top and bottom faces of the first growing bilayer as shown in fig. 6. The phase difference is represented by 2yrd, which depends on the potential depth of the first growing layer, i.e. on the coverage of the first growing layer, and on the glancing angle. Her, 2y, is the momentum transfer concerned with the OO-rod. Fig. 7 shows the phase values for the coverage of the first growing layer when varying the glancing angle from 8 = 0.5“ to 3.0”. The phase value of 2nrr (n is an integer) corresponds to the in-phase condition (this means constructive interference) and that of (2n + 1)~ to the out-of-phase condition (destructive interference). Closed and open circles indicate the in-phase and out-of-phase conditions, respectively. At lower glancing angle such as 0 = 0.5”, two out-of-phase conditions and one in-phase condition are satisfied in the region of layer coverage 0 = O-l. At higher glancing angles such as @= 2.5 or 3.0“, only one phase condition (in-phase or out-of-phase) is satisfied. In order to see the relation between the phase conditions and the reflectivity, arrows are introduced
267
Y. Horio, A. Ichimiya / Dynamical diffraction effect for RHEED intensity oscillation
in fig. 5. Upward and downward arrows indicate in-phase and out-of-phase conditions, respectively. It is found that peaks and dips in the oscillation curves nearly agree with the upward and downward arrows, respectively. At 8 = 0.5”, two out-of-phase conditions and one in-phase condition give rise to the frequency-doubling phenomenon of feature 6). Such phase conditions are shifted with the change of the glancing angle. This is the reason for the phase-shift phenomenon of feature (ii>. A flat surface appears at the integer number of thickness in the case of a perfect layer-by-layer growth mode, and then the potential at the top surface becomes most abrupt in the “simple potential model”. Considering potential scattering, reflectivity increases when the potential at the top surface becomes abrupt. This is the reason for feature (iii). In the “realistic
67~
5lr
01 : ; : : : : :
I
:
-1
1.0
0.5
0.0
Layer coverage Fig. 7. Dependence of the phase difference on the coverage of the growing layer at each glancing angle, (a): 0 = OS”, (b): l.O”, cc): 1.5”, cd): 2.0”, (e) 2.5” and (f): 3.0”.
I\
97
0i---~1 Fig. 8. RHEED
intensity
2
3
4
THICKNESS CEIL)
(a) oscillation
curves
5
(b)
THICKNESS CEIL)
under the growth mode of D = 100. (a) Calculated (b) by the “realistic potential model”.
by the “simple
potential
model”;
Y. Horio, A. Ichirniya / Dynamical diffraction effect for RHEED intensity oscillation
268
0 = 1 and 2.5”, the out-of-phase condition is located nearly at the middle point of one bilayer. In this case, the two factors are cooperative and sinusoidal curves arise. This is the reason for feature (v). RHEED oscillations are also calculated under the growth modes of D = 100 and D = 10 as shown in figs. 8 and 9. The number of uncompleted growing layers increases as the value of diffusion parameter D decreases, which corresponds to a multi-layer growth mode. In this case, the reflectivity is affected by the increased number of electron beams reflected from many uncompleted growing layers. Superposition of these increased waves makes the curves sinusoidal, as seen in fig. 9. The phase consideration based on the growth mode of D = 3500 is also supported,
potential model”, in fig. 5b, however, reflectivity is not a maximum at the thickness of the integer number, which is considered to be affected by a diffraction effect from the substrate. Anomalous behavior is seen at around the integer number of thickness in the curve of f3= 2.0” in fig. 5a. Although the out-of-phase condition is satisfied at this point (indicated by a downward arrow), a peak arises in the curve. The contradiction is due to competition between a negative factor (out-ofphase condition) and a positive factor (abruptness of the top surface potential), and the positive factor is superior to the negative factor. This is the reason for feature (iv>. The relation between the phase condition and the abruptness of the potential determines whether the oscillation curve is sinusoidal or not. In the oscillation curves at
I
1
oL
i
2
3
THICKNESS (BL) Fig. 9. RHEED
intensity
oscillation
curves
L
-il_
_.__~~___
4
5
30
3
t
0 (b;l
,1
1
?
3
4
253.9
5
THICKNESS (BL)
under the growth mode of D = 10. (a) Calculated (b) by the “realistic potential model”.
by the “simple
potential
model”;
Y Horio, A. Ichimiya / Dynamical diffraction effect for RHEED intensity oscillation
1
2
3
Thickness
Fig. 10. ~nematically
4
269
5
(BL)
calculated RHEED intensity oscillation curves under the growing mode of (a): D = 3500, (bf: D = 100 and Glancing angles are (A): 19= 0.5”, (B): I.O”, (C): 1.5”, (D): 2.0”, (E): 2.5” and (F): 3.0”.
CC): D = 10.
even in D = 100 and D = 10, because upward and downward arrows nearly correspond to the peaks and dips in the curves, respectively. The similar phase-shift phenomenon is also seen in the experimental result [9]. In order to compare these one-beam dynamical calculation results, kinematically calculated intensities of the specular beam at glancing an-
(al
Glancing
Angle
(deg.)
gles from 8 = 0.5 to 3.0” are shown in fig. 10. The intensities [21 are calculated by the single scattering from each uncovered terrace as 2
la
f (0, -On+,) n=O
exp( -i2rnd)
.
It is found that the oscillation amplitude depends
Glancing
Angie
(deg.)
Fig. il. Rocking curves under the growth mode of D = 3504 at each growing stage from 8 = 0 to 0.9. (a) Calculated by the “simple potential modet”; (b) by the “realistic potential model”.
on the glancing angle, namely on the degree of satisfaction of the Bragg condition (the maximum amplitude of oscillation is obtained just on the conditions of off-Bragg and no oscillation just on the condition of on-Bragg). RHEED oscillation curves do not have any phase shifts varying the glancing angle in the kinematic approach. To see another property of the reflectivity, the dependence of the reflectivity on the glancing angle, rocking curves of the specular beam were also calculated at many growing stages. Fig. 11 shows the rocking curves; (a): cafcufated by the “simple potential model” and (b): caIcuIated by the “realistic potential model”. In the rocking curves, the reflectivities are multipIied by sin 8 because of the change of the incident beam intensity at the surface depending on the glancing angle. These rocking curves are calculated at ten growing stages from 0 = 0 to 0.9 BL, under the growth mode of D = 3500, Rocking curves (a) of
Glancing
the “‘simpfe potential model” show only interference of eIectron waves from the step-like potential and no diffraction effect from bulk Si crystal. Especially at 0 = 0 BL, the potential shape is a uniform distribution of mean inner potential V,, and reflectivity shows a monotonous decay as the glancing angle is increased. Advancing the Iayer coverage to 0 = 0.5 BL, Bragg peaks gradually appear in the curve, due to the potential step of the first growing layer, and gradually disappear close to 0 = 1 BL, due to the potential becoming uniform again, The Bragg peak positions are shifted to an angle a little higher, because the step-like potential of the uncompleted layer is shallower than the mean inner potential P’a of the substrate Si crystal. On the other hand, the rocking curves (b) of the “realistic potential model” reveal sharp Bragg peaks and can be compared with experimental curves. It is found that both rocking curves of (a) and Cb) reveal a similar
Angie (deg.1
Fig. 12. Rocking curves under the growth mode of D = 10 at each growing stage from 0 = 3.0 to 3.9. fa) Cafeulated by the “simple potential model” and (h) by the “realistic potential model”.
I’. Horio, A. Ichimiyu / L$vzamical diffraction effect for RHEED intensity oscillation
fluctuation at a lower glancing angle (0 < 2”). The rocking curves at lower glancing angles are sensitively affected by the surface growth stage, namely the potential change of the first growing layer. Checking the behavior in detail, the dips, indicated by arrows, shift to lower angles as the layer coverage becomes one bilayer. This feature is also seen in the experimental rocking curves of Si/Si(lll) [9]. It is considered that the fluctuation at lower glancing angles is due to the interference effect, because the dip positions, indicated by arrows in fig. lla, correspond to the out-of-phase condition. Fig. 12 shows rocking curves calculated from 0 = 3.0 to 3.9 BL under the growth mode of D = 10. In the rocking curves (a) of the “simple potential model”, Bragg peaks appear more clearly than those of the growth mode D = 3500 in fig. lla. This is because many uncompleted growing layers always exist at such a multi-layer growth mode of D = 10, and many steps of the potential make the Bragg peak intensity stronger. In the rocking curves (b) of the “realistic potential model”, the behavior of the dip at a low glancing angle is the same as in the case of D = 3500, except for a difference of the shape. In conclusion, we have tried to analyze the RHEED oscillation with the “simple potential model” and can reproduce the phase-shift and frequency-doubling phenomena on the assumption of the birth-death growth. These phenomena cannot be reproduced by the kinematic approach. It is found that these phenomena can be explained by the interference of electron waves from the potential of the top growing layer. It has been confirmed that the behaviors of the oscillation curves of the “simple potential model” are similar to those of the “realistic potential model”. As a result, it is considered that the RHEED oscillation is not due to the potential shape so much, but to the dynamic change of the potential depth of the growing layer. This means that the phase-shift phenomena can be explained by the averaged potential in the growing layer, instead of by the complicated potential. The RHEED oscillation becomes sinusoidal, which shape is generally observed in experiment, as the growth mode changes to multi-layer growth. We have
271
also calculated the rocking curves at MBE growing stages by the two potential models. The rocking curves of the “simple potential model” merely show the interference of waves from the step-like potential; then strict comparison with experiment cannot be done. The intensity profiles of the rocking curves are rather different between the two potential models, however, the similar fluctuation in the low glancing angle region (0 < 2.0°) can be seen. The behavior in the low glancing angle region is also observed in experimental rocking curves of Si/Si(lll) MBE growth [9] and can be explained by interference of electron waves from the growing layer potential. AI1 calculations in this paper were carried out on the one-beam condition with a 10 kV electron beam. In this condition, the approximation using averaged potential parallel to the surface becomes more correct as the glancing angle becomes lower. At very low glancing angle, the reflectivity of high-energy electron beam increases considerably and surface potential change becomes more important for RHEED oscillation curves as well as for rocking curves. It is considered that our “simple potential model”, with a relatively short calculation time, is a usefu1 interpretation method for RHEED oscillations.
Acknowledgements The authors thank Professor J.L. Beeby for valuable discussions. This work was carried out under the support of Grants-in-Aid to the Scientific Research on Priority Areas by the Ministry of Education, Science and Culture (Nos. 03243219 and 04227216) and an international collaboration program of the Japan Society for the Promotion of Science.
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Y Horio, A. Ichimiya / Dynamical diffraction
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effectfor RHEED inter&
oscillation
[lOI A. Ichimiya, The Structure of Surfaces III, Vol. 24 of Springer Series in Surface Sciences (Springer, Berlin, 1991) p. 162. [ll] A. Ichimiya, Jpn. J. Appl. Phys. 22 (1983) 176; 24 (1985) 1365. [12] P.A. Doyle and P.S. Turner, Acta Cryst. A 24 (1968) 390. [131 J.D. Weeks, G.H. Giimer and K.A. Jackson, J. Chem. Phys. 65 11976) 712.