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Interpretation of RHEED oscillations during MBE growth
Surface Science Letters 245 (1991) L159-L162 North-Holland
L159
Surface Science Letters
Interpretation of RHEED oscillations during MBE growth G. L e h m p f u h l a, A. I c h i m i y a b a n d H. N a k a h a r a b ~ Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 1000 Berlin 33, Germany h Department of Applied Physics, School of Engineering, Nagoya University, Nagoya 464-01, Japan
Received 29 October 1990; accepted for publication 20 December 1990
The intensity oscillations of RHEED reflections during molecular beam epitaxial growth are explained in terms of refraction and total reflection. Experimental observations of integrated intensity oscillations on silicon are in good agreement with calculations based on the simple model.
R H E E D intensity oscillations during crystal growth are used for controlling the preparation of well defined solid state devices. There are m a n y discussions concerning the reason for the Braggintensity oscillations [1], but, to our knowledge, the effect of refraction is neglected in these considerations. Reflection electron microscope (REM) investigations of atomically flat surfaces with only a few monatomic steps have clearly shown the effect of refraction from such monatomic steps [2]. In these investigations atomic steps on (111) and (100) facets of a Pt single-crystal sphere have been imaged. For imaging the steps with high intensity and contrast, an intensity-enhanced Bragg reflection was used. Since the conditions for intensity enhancement depend most sensitively on the direction of the incident electron beam, effects of refraction have a strong influence on the image intensity. In the investigation [2] it was shown that the intensity of a monatomic step can be changed from dark to light (with respect to the neighbouring area) by a slight change of the direction of the incident beam. This was demonstrated for both monatomic "up"-steps and monatomic "down"steps, as e.g. in fig. 5 of ref. [2], indicating the effect of different refraction through a flat surface and through a step shown schematically in fig. 1.
Typical experimental conditions for the measurement of R H E E D oscillations are: (1) Low energy of the incident electrons, of the order of 10 keV; (2) grazing incidence of the electrons of the order of 1.5 °; and (3) small detector aperture for intensity recording [31. These conditions together with the effect of refraction explain the measured intensity oscillations. /
(1)
\\
\ -
"" \ I
(1)~ (21\
~.(3)
[] DETECTOR
Fig. 1. Schematic refraction of ray paths by a step.
0039-6028/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
L160
G, Lehmpfuhl et a L / Interpretation of RHEED oscillations during MBE growth
(112
(1)(2)
2d.cose
Fig. 2. Schematic condition for total reflection.
The influence of refraction on the direction of the scattered electrons is also shown schematically in fig. 1 in which the steps are represented by a strongly simplified square form. Due to refraction by a step. Intensity is scattered away from the detector. Effects of this kind can be seen in diffraction patterns by Zhang et al. [4]. The observation of out-of-phase oscillations can also be explained by this simple model. The beams (1) or (3) in fig. 1 should be 180 ° out of phase. The smaller the detector aperture, the more pronounced the oscillations. From kinematic approximation one would expect that the integrated intensity recorded with a large aperture would not show the oscillations. However, there exist reflection conditions where the integrated intensity also shows the oscillations. This effect is due to total reflection of
Y / Fig. 3. Refraction on a surface.
Fig. 4. Reflection and total reflection from the area of a step and by the step itself.
electrons entering the crystal through a monatomic step with reduced refraction and being kept in the crystal when totally reflected on the inner side, as shown schematically in fig. 2. In order to estimate the effect of refraction we calculate the angular deviation of the incident electron beam. The refraction is shown schematically in fig. 3: cos 0 = n c o s ( 0 + A0 ),
(1)
with n = v'(E + V ) / E , E = energy of the electrons and V = mean inner potential of the crystal. This is equivalent to 1 / n = V/1 - sin 2 A0 - tan 0 sin A0.
(2)
If 0 = 0 we have total reflection. This corresponds to an angle A O = vl(n 2 - 1 ) / n 2 : ~'V/( E + V ) , and for 10 keV electrons and a mean inner potential of 10 eV about 1.81 °. The experiments are usually carried out at smaller angles, so that total reflection may occur even at a smaller effect of refraction from a step which is in reality not so sharp as in the schematic drawings of figs. 1 and 2. The oscillation: amplitude of the integrated intensity due to total reflection depends on the angle
L161
G. Lehmpfuhl et aL / Interpretation of RHEED oscillations during MBE growth
Table 1 Amplitudes of the integrated intensity oscillations, experimental and theoretical values at two temperatures T (deg)
L + L' (ref. [5])
0(exp) (deg)
2d/
A1/I
((L + L') tan 0)
(fig. 5)
600
3000
1.12
0.11
0.12
5
0.07 0.51
0.09 0.29
"~
500
1.68 1.4
500
(A)
(a)
|
~ I
start
AI
c
of incidence 0 (smaller than the critical angle of total reflection), and can be estimated by a simple approximation. Let us assume that each step of height d is surrounded by a " l o w e r " surface area of size L and a " h i g h e r " area of size L ' shown schematically in fig. 4. Such a step leads to a fraction of the incident beam being totally reflected within the area 2 d cos O. The integrated intensity is reduced by such a fraction from each effective step I = 1 - N 2 d cos O,
E} : 1.12*
s
&I/I:0.12
time/min
10
I
I
c
Al
(b)
start
(4)
where N is the density of steps given by N = 1 / ( ( L + L ' ) sin 0). L + L ' corresponds to the size of the domains. The m a x i m u m n u m b e r of effective steps depends on the angle of incidence. Inserting N in relation (4) for the intensity we find the simple expression I = 1 - 2 d / ( ( L + L ' ) tan 0).
T: 600 *C
o
T : 6 0 0 *C
0= 1.68"
AIII=0.09
time / min
(5)
The right-hand part in this equation is the amplitude of oscillation which can be c o m p a r e d with experimental observations. U n d e r the assumption that the d o m a i n size ( L + L ' ) is constant at a distinct temperature [5], L decreases linearly during evaporation while L ' increases, b o t h periodically. The effective size of L has to be larger than d / t a n 0 because of shadowing. The temperature dependence of the Si(111)5 × 5 d o m a i n size has recently been determined by two of us [5] in agreement with H o r n van Hoegen et al. [6]. With this assumption the amplitude of the integrated intensity oscillation can be estimated for different experimental conditions and c o m p a r e d with observations shown in fig. 5. Observations have been m a d e at 500 and 6 0 0 ° C . The table shows the observed data A I / I together with the calculated oscillation amplitude 2 d / ( ( L + L ' ) tan 0) for d =
(c) &I c
2 start
[
T : 5 0 0 *C
e : 1.t,o
10
AIII:0.29
time / min
20
Fig. 5. Oscillation of the integrated intensity from a Si(lll) surface during Si evaporation under different conditions for temperature T and angle of incidence 0. (a) T = 600 ° C, 0 = 1.12 ° . (b) T=600°C 0=1.68 ° . (c) T=500°C, 0=1.4 ° . (In (c) is an incubation region indicated after which a regular oscillation starts.)
L162
G. Lehrnpfuhl et al. / Interpretation of RHEED osciUations during MBE growth
3.14 A. The agreement between experiment a n d an estimate using the simple a p p r o x i m a t i o n is quite good. With this description R H E E D intensity oscillations can be explained by step density oscillations. Very recently Vvedensky a n d Clarke [7] have shown that step density evolution calculated by M o n t e Carlo simulation is very similar to experimental R H E E D intensity oscillations d u r i n g epitaxial growth on a GaAs(001) surface [8]. This shows that the refraction effect is significant for R H E E D oscillations. Besides this effect of refraction there m a y be other reasons for intensity variation, as discussed in different papers, b u t they seem to be of m i n o r influence. We thank Dr. K. K a m b e and Dr. Y. Uchida for m a n y discussions and suggestions.
References [1] P.K. Larsen and P.J. Dobson in: Reflection High-Energy Electron Diffraction and Reflection Electron Imaging of Surfaces, Nato ASI Series, Series B: Physics Vol. 188 (Plenum Press, New York, 1987). [2] G. Lehrnpfuhl and Y. Uchida, Ultramicroscopy 26 (1988) 177; T. Hsu and G. Lehmpfuhl, Ultramicroscopy 27 (1989) 359. [3] H. Nakahara and A. Ichimiya, J. Cryst. Growth 95 (1989) 472. [4] Z. Zhang, J.H. Neave, P.J. Dobson and B.A. Joyce, Appl. Phys. A 42 (1987) 317. [5] H. Nakahara and A. lchimiya, J. Cryst. Growth 99 (1990) 514; H. Nakahara and A. lchimiya, Surf. Sci. 241 (1991) 124. [6] M. Horn yon Hoegen, J. Falta and M. Henzler, Thin Solid Films 183 (I989) 220. [7] D. Vvedensky and S. Clarke, Surf. Sci. 225 (1990) 375. [8] J.H. Neave, P.J. Dobson, B.A. Joyce and J. Zhang, Appl. Phys. Len. 47 (1985) 100.
L159
Surface Science Letters
Interpretation of RHEED oscillations during MBE growth G. L e h m p f u h l a, A. I c h i m i y a b a n d H. N a k a h a r a b ~ Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 1000 Berlin 33, Germany h Department of Applied Physics, School of Engineering, Nagoya University, Nagoya 464-01, Japan
Received 29 October 1990; accepted for publication 20 December 1990
The intensity oscillations of RHEED reflections during molecular beam epitaxial growth are explained in terms of refraction and total reflection. Experimental observations of integrated intensity oscillations on silicon are in good agreement with calculations based on the simple model.
R H E E D intensity oscillations during crystal growth are used for controlling the preparation of well defined solid state devices. There are m a n y discussions concerning the reason for the Braggintensity oscillations [1], but, to our knowledge, the effect of refraction is neglected in these considerations. Reflection electron microscope (REM) investigations of atomically flat surfaces with only a few monatomic steps have clearly shown the effect of refraction from such monatomic steps [2]. In these investigations atomic steps on (111) and (100) facets of a Pt single-crystal sphere have been imaged. For imaging the steps with high intensity and contrast, an intensity-enhanced Bragg reflection was used. Since the conditions for intensity enhancement depend most sensitively on the direction of the incident electron beam, effects of refraction have a strong influence on the image intensity. In the investigation [2] it was shown that the intensity of a monatomic step can be changed from dark to light (with respect to the neighbouring area) by a slight change of the direction of the incident beam. This was demonstrated for both monatomic "up"-steps and monatomic "down"steps, as e.g. in fig. 5 of ref. [2], indicating the effect of different refraction through a flat surface and through a step shown schematically in fig. 1.
Typical experimental conditions for the measurement of R H E E D oscillations are: (1) Low energy of the incident electrons, of the order of 10 keV; (2) grazing incidence of the electrons of the order of 1.5 °; and (3) small detector aperture for intensity recording [31. These conditions together with the effect of refraction explain the measured intensity oscillations. /
(1)
\\
\ -
"" \ I
(1)~ (21\
~.(3)
[] DETECTOR
Fig. 1. Schematic refraction of ray paths by a step.
0039-6028/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
L160
G, Lehmpfuhl et a L / Interpretation of RHEED oscillations during MBE growth
(112
(1)(2)
2d.cose
Fig. 2. Schematic condition for total reflection.
The influence of refraction on the direction of the scattered electrons is also shown schematically in fig. 1 in which the steps are represented by a strongly simplified square form. Due to refraction by a step. Intensity is scattered away from the detector. Effects of this kind can be seen in diffraction patterns by Zhang et al. [4]. The observation of out-of-phase oscillations can also be explained by this simple model. The beams (1) or (3) in fig. 1 should be 180 ° out of phase. The smaller the detector aperture, the more pronounced the oscillations. From kinematic approximation one would expect that the integrated intensity recorded with a large aperture would not show the oscillations. However, there exist reflection conditions where the integrated intensity also shows the oscillations. This effect is due to total reflection of
Y / Fig. 3. Refraction on a surface.
Fig. 4. Reflection and total reflection from the area of a step and by the step itself.
electrons entering the crystal through a monatomic step with reduced refraction and being kept in the crystal when totally reflected on the inner side, as shown schematically in fig. 2. In order to estimate the effect of refraction we calculate the angular deviation of the incident electron beam. The refraction is shown schematically in fig. 3: cos 0 = n c o s ( 0 + A0 ),
(1)
with n = v'(E + V ) / E , E = energy of the electrons and V = mean inner potential of the crystal. This is equivalent to 1 / n = V/1 - sin 2 A0 - tan 0 sin A0.
(2)
If 0 = 0 we have total reflection. This corresponds to an angle A O = vl(n 2 - 1 ) / n 2 : ~'V/( E + V ) , and for 10 keV electrons and a mean inner potential of 10 eV about 1.81 °. The experiments are usually carried out at smaller angles, so that total reflection may occur even at a smaller effect of refraction from a step which is in reality not so sharp as in the schematic drawings of figs. 1 and 2. The oscillation: amplitude of the integrated intensity due to total reflection depends on the angle
L161
G. Lehmpfuhl et aL / Interpretation of RHEED oscillations during MBE growth
Table 1 Amplitudes of the integrated intensity oscillations, experimental and theoretical values at two temperatures T (deg)
L + L' (ref. [5])
0(exp) (deg)
2d/
A1/I
((L + L') tan 0)
(fig. 5)
600
3000
1.12
0.11
0.12
5
0.07 0.51
0.09 0.29
"~
500
1.68 1.4
500
(A)
(a)
|
~ I
start
AI
c
of incidence 0 (smaller than the critical angle of total reflection), and can be estimated by a simple approximation. Let us assume that each step of height d is surrounded by a " l o w e r " surface area of size L and a " h i g h e r " area of size L ' shown schematically in fig. 4. Such a step leads to a fraction of the incident beam being totally reflected within the area 2 d cos O. The integrated intensity is reduced by such a fraction from each effective step I = 1 - N 2 d cos O,
E} : 1.12*
s
&I/I:0.12
time/min
10
I
I
c
Al
(b)
start
(4)
where N is the density of steps given by N = 1 / ( ( L + L ' ) sin 0). L + L ' corresponds to the size of the domains. The m a x i m u m n u m b e r of effective steps depends on the angle of incidence. Inserting N in relation (4) for the intensity we find the simple expression I = 1 - 2 d / ( ( L + L ' ) tan 0).
T: 600 *C
o
T : 6 0 0 *C
0= 1.68"
AIII=0.09
time / min
(5)
The right-hand part in this equation is the amplitude of oscillation which can be c o m p a r e d with experimental observations. U n d e r the assumption that the d o m a i n size ( L + L ' ) is constant at a distinct temperature [5], L decreases linearly during evaporation while L ' increases, b o t h periodically. The effective size of L has to be larger than d / t a n 0 because of shadowing. The temperature dependence of the Si(111)5 × 5 d o m a i n size has recently been determined by two of us [5] in agreement with H o r n van Hoegen et al. [6]. With this assumption the amplitude of the integrated intensity oscillation can be estimated for different experimental conditions and c o m p a r e d with observations shown in fig. 5. Observations have been m a d e at 500 and 6 0 0 ° C . The table shows the observed data A I / I together with the calculated oscillation amplitude 2 d / ( ( L + L ' ) tan 0) for d =
(c) &I c
2 start
[
T : 5 0 0 *C
e : 1.t,o
10
AIII:0.29
time / min
20
Fig. 5. Oscillation of the integrated intensity from a Si(lll) surface during Si evaporation under different conditions for temperature T and angle of incidence 0. (a) T = 600 ° C, 0 = 1.12 ° . (b) T=600°C 0=1.68 ° . (c) T=500°C, 0=1.4 ° . (In (c) is an incubation region indicated after which a regular oscillation starts.)
L162
G. Lehrnpfuhl et al. / Interpretation of RHEED osciUations during MBE growth
3.14 A. The agreement between experiment a n d an estimate using the simple a p p r o x i m a t i o n is quite good. With this description R H E E D intensity oscillations can be explained by step density oscillations. Very recently Vvedensky a n d Clarke [7] have shown that step density evolution calculated by M o n t e Carlo simulation is very similar to experimental R H E E D intensity oscillations d u r i n g epitaxial growth on a GaAs(001) surface [8]. This shows that the refraction effect is significant for R H E E D oscillations. Besides this effect of refraction there m a y be other reasons for intensity variation, as discussed in different papers, b u t they seem to be of m i n o r influence. We thank Dr. K. K a m b e and Dr. Y. Uchida for m a n y discussions and suggestions.
References [1] P.K. Larsen and P.J. Dobson in: Reflection High-Energy Electron Diffraction and Reflection Electron Imaging of Surfaces, Nato ASI Series, Series B: Physics Vol. 188 (Plenum Press, New York, 1987). [2] G. Lehrnpfuhl and Y. Uchida, Ultramicroscopy 26 (1988) 177; T. Hsu and G. Lehmpfuhl, Ultramicroscopy 27 (1989) 359. [3] H. Nakahara and A. Ichimiya, J. Cryst. Growth 95 (1989) 472. [4] Z. Zhang, J.H. Neave, P.J. Dobson and B.A. Joyce, Appl. Phys. A 42 (1987) 317. [5] H. Nakahara and A. lchimiya, J. Cryst. Growth 99 (1990) 514; H. Nakahara and A. lchimiya, Surf. Sci. 241 (1991) 124. [6] M. Horn yon Hoegen, J. Falta and M. Henzler, Thin Solid Films 183 (I989) 220. [7] D. Vvedensky and S. Clarke, Surf. Sci. 225 (1990) 375. [8] J.H. Neave, P.J. Dobson, B.A. Joyce and J. Zhang, Appl. Phys. Len. 47 (1985) 100.