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RHEED oscillations at special diffraction conditions
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Surface Science 298 (1993) 293-298 North-Holland
RHEED
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oscillations at special diffraction conditions
P.A. Maksym a, Z. Mitura b and M.G. Knibb a a Department of Physics and Astronomy, Uniuersity of Leicester, Uniuersity Road, ’ Department of Ex~r~enta~
Physics, Marie Cute-Sk~~owska
Leicester LEl 7RH, UK ~n~uersi~, pL M. Curie-~k~odowsk~ej I, 20-031 Lubi~n~Poland
Received 16 March 1993; accepted for publication 23 April 1993
It is suggested that analysis of RHEED intensity oscillations can be simplified when diffraction conditions are chosen appropriately and two cases are presented as supporting evidence. The first is an out-of-phase condition where the specular reflection from the substrate is very weak. At this condition the oscillations can be described by a kinematic calculation which is modified to take account of refraction. The second case occurs when electrons are incident at glancing angles less than about 1”. Under this condition the oscillations can have a double-minimum structure with one minimum at very small coverages of the growing layer and a second minimum at higher coverages. The minima occur because the growing layer becomes near-transparent at certain critical strengths of the laterally averaged scattering potential. This is illustrated with numerical results for lead and silver.
1. Introduction RHEED intensity oscillations during MBE growth have been known for some time although the mechanism that causes them is not yet properly understood. A number of different approaches have been proposed fl-51, for example, diffraction by arrays of steps and terraces or oscillations in step density, whose general validity is not well established. The situation is compiicated by the dynamical nature of the diffraction process and the disordered nature of a growing surface. This makes it very difficult to calculate exactly the diffraction pattern for an arbitrary growing surface at an arbitrary diffraction condition. Nevertheless, there are indications that the oscillations can be understood simply at certain special diffraction conditions. The search for these conditions was initiated by Knibb [6] who showed that oscillations at the out-of-phase condition could be understood by means of a modified kinematic theory. Subsequently, Mitura et al. [7,81 reported that oscillations have extra minima and maxima when the glancing angle of the incident electrons is very small and showed that 0039-6028/93/$06.00
these features could be reproduced by a calculation where the scattering potential is averaged in the lateral direction. In either case, the reason why the oscillations can be understood simply is that there is one simple physical process which dominates. The purpose of this work is to detail the physics of each case and to argue that the special diffraction conditions have some general validity.
2. WEED
intensity oscillations
The first systematic examination of dynamical effects in RHEED intensity oscillations was made by Knibb E6l who compared kinematic and dynamic results for a model of GaAs growth. The positions of the surface atoms were obtained from a Monte Carlo simulation on a 7 X 7 supercell which, although not large enough to provide a realistic mode1 of the growth, was perfectly adequate to study the diffraction processes that occur in RHEED by growing surfaces. Knibb found
0 1993 - Eisevier Science Publishers B.V. All rights
reserved
294
P.A. Maksym et al. / RHEEB oscillations at special diffraction conditions
that there was very little agreement between the kinematic and dynamic results for most diffraction conditions. However, it turned out that the oscillations in the specular beam intensity at an out-of-phase diffraction condition were very well reproduced by a modified kinematic theory. Oscillations in the intensity of other beams at the same diffraction condition could not be reproduced in this way. The kinematic nature of the specular oscillations appears to be connected with the weakness of the specular reflection from the substrate at the out-of-phase condition. This suggests that the only multiple scattering that is relevant to the specuIar beam intensity at this diffraction condition is multiple forward scattering. Physically, this just affects the refractive index of the scattering system. The refraction is easily taken into account by means of a distorted wave approximation in which the refracted waves are taken to be scattered kinematically. Since a growing surface contains incomplete layers its refractive index is depth dependent and this must be taken into account to get good agreement with the dynamical results. This point is illustrated in fig. 1 which shows results obtained by Knibb for the specular beam oscillations. The oscillations are not smooth because of statistical fluctuations introduced by the small scale of the Monte Carlo
simulation. Each part of the figure shows the same dynamical result (solid line) compared with kinematic results for a different model of the refractive index. In the calculations leading to fig. la, the depth dependence of the refraction was ignored. The kinematic results were calculated by taking the wavevector to be the same as in the vacuum (dotted line) or the buIk (chained Iine) throughout the growing part of the surface. In the latter case, absorption of electrons was taken into account by adding an imaginary component to the wavevector. Clearly, there is very poor agreement with the exact result in both cases. The kinematic curve for fig. lb was computed with the bulk wavevector but absorption was ignored. In this case the minima of the kinematic oscillations are too low. Fig. lc shows results computed with a depth-dependent refractive index. This does the best job of matching both the minima and the maxima of the exact oscillations. The calculations with the modified kinematic theory are very economical on computer time. If the specular beam oscillations at the out-of-phase condition are generally near-kinematic it should be possible to use the modified kinematic approach to compare the predictions of various growth models with experiment in a realistic way. The fact that this approach was found to work when the substrate
Time (Sec.) Fig. 1. Comparison of RHEED oscillations for simulated growth of GaAs as calculated from dynamic and kinematic theories. Each part of the figure shows specular beam oscillations calculated dynamically and with one of the kinematic models described in the text. The incident beam azimuth is (710) and the diffraction condition is the out-of-phase condition for bilayers.
P.A. Maksym et ai. / RHEED oscillations at special diffraction conditions
0
2
4
THICKNESS
Fig. 2. Low glancing angle Pb-35%In(lll). The electron dent beam azimuth is (Zii). cated in
6
8
10
(ML)
oscillations for the growth of energy is 20 keV and the inciThe glancing angles are indithe figure.
reflection is very weak suggests that it should be valid for systems other than the one studied numerically by Knibb. 2.2. Oscillations at low glancing angle Mitura et al. [7,8] were the first to recognise that RHEED intensity oscillations have interesting features when the glancing angle of the incident beam is very low. They considered oscillations that occur during the growth of Pb-In alloys on substrates of the same material and found that the specular beam oscillations have extra structure. Similar structure has recently been discussed by Horio and Ichimiya 191.It is illustrated in fig. 2 which shows experimental and theoretical oscillations found by Mitura et al. for 20 keV electrons at glancing angles less than 0.5”. Each cycle of the oscillation contains a minimum and there is a further, very sharp minimum at the start of each cycle. A detailed description of experimental conditions under which the effect
295
occurs is given by Stroiak et al. (101. Here it is argued that these features occur because the growing layer becomes transparent to electrons when its potential reaches a critical strength. Similar features are likely to be found in most systems when the glancing angle of the incident electrons is very low. Understanding these effects requires knowledge of the limiting behaviour of the reflection coefficient of a single atomic layer at very low glancing angles. Mitura and Daniluk [71 have shown that the double-minimum structure occurs even when all processes other than propagation in the laterally averaged potential are neglected. In other words, the physics is dominated by the propagation of the specular beam in this potential. Therefore, the coupling to other beams does not need to be taken into account to obtain a qualitative explanation of the observed effects. When this coupling is omitted the electron wave function has the form exp(ik,, . p) ~/AZ>, where p is a position vector parallel to the surface and k,, is the parallel component of the incident electron wavevector. The function #(z) obeys the one-dimensional Schrodinger equation
(1) and in this work the potential, I’, is taken to be real. The quantity k, is the magnitude of the perpendicular component of the incident and reflected electron wavevectors. It is related to the electron energy, E, and the parallel component of the incident electron wavevector, k,,, by the equation ki = 2mE/h2 - ki;‘. The intensity reflection and transmission coefficients, R and T, can be written in terms of the two linearly independent solutions of eq. (1). Detailed analysis of the resulting expressions (which is too long to be given here) shows that the limiting behaviour of R and T is normally that R approaches 1 and T approaches 0 as k, approaches zero, i.e. as the glancing angle approaches zero. The exceptional case occurs when the laterally averaged potential has a zero-energy bound state. In this case, either both R and T are non-zero at zero glancing angle, or R = 0 and T = 1. A zero-energy bound
296
oscillations at speciai diffraction conditions
P.A. Maksym et al. ,/ RHEED
state can only occur at certain strengths of the potential and this leads to drastic changes in the reflection coefficient with potential strength. The effect is particularly strong when R vanishes at zero glancing angle. It is illustrated in fig. 3 which gives the reflection and transmission coefficients for 30 keV electrons incident on a single Ag(ll1) layer at a glancing angle of lo-’ degrees. The curves were calculated using methods described in ref. [ll]. Each curve shows how R or 7’ depends on the coverage, c, which in this case is simply a scaling factor that changes the strength of the potential. (The laterally averaged potential of a growing layer is proportional to its coverage.) There is a minimum in I? which is so sharp that a coverage change of about 10e7 ML is sufficient to change R by more than an order of magnitude. The minimum in R is accompanied by an equally
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Coverage parameter Fig. 3. Specular reflection and transmission coefficients for 30 keV electrons incident on a single AAl 11) layer as a function of potential strength. The potential strength scales with coverage as explained in the text and the coverage parameter is defined to be (c - 0.591293) X 106. The glancing angle is lo-’ degrees.
5 Glancing
angle (degrees)
Fig. 4. Rocking curves for a single Ag(lll) layer when the covergage is just above Cc = 0.6) and just below (c = 0.5) the critical value for adding a bound state. The electron energy is 30 keV.
sharp rise in T - there is 100% transmission even at this extremely low gIancing angle! The idea that this transparency is associated with the addition of a new bound state to the potential can be checked by plotting the phase of the transmission coefficient. For 3D-scattering, Levinson’s theorem gives a relation between the zero-energy phase shift and the number of bound states, and a similar relation can be shown to hold for the phase of the transmission coefficient in one dimension. According to this theorem the phase increases by 7 when the potential strength is increased by an amount sufficient to bind an extra state. The phase change shown in fig. 3 has precisely this behaviour. The drastic change in the reflection coefficient at the critical coverage is not merely confined to glancing angles close to zero. Numerical results show that the reflection coefficient is strongly affected at glancing angles up to around 1”. This is illustrated in fig. 4 which shows rocking curves for a single Ag(l11) layer for a coverage just above the critical value for transparency (0.6 ML) and for a coverage such that the reflection coefficient behaves in the normal way (0.5 ML). Clearly, there is a large drop in the reflection coefficient when the coverage is just above the critical value and this drop occurs over a wide range of glancing angles. Indeed there appears to be a transmission resonance at B = 0.2”. The drop in the
PA Maksym et al. / RHEED oscillations at special diffraction conditions
reflection causes the second minimum in the oscillations shown in fig. 2, that is the minimum that occurs at a coverage of about 0.7 ML: when the potential of the growing layer reaches the critical value for binding a new state the layer becomes near-transparent and the reflection drops to a minimum. It remains to explain the minimum that occurs at very low coverage, that is at the start of the growth cycle. This has no counterpart in the reflection coefficient of a single atomic layer. The reason is that any one-dimensional potential always has at least one bound state and the reflection minima for a single layer occur when further states bind. To understand the first minimum it is necessary to consider the substrate as well as the growing layer. The relevant question is then whether a surface state can bind in the growing layer. When it does, this layer becomes transparent and the reflection coefficient goes through a minimum in the same way as the reflection coefficient for a single layer. The situation can be modelled by considering an atomic layer embedded in an asymmetric potential which approaches zero above the layer and approaches the mean substrate potential below it. Physically, a state can localise in the growing layer only if the layer centre potential is sufficiently deep relative to the mean substrate potential. A simple model of this, which can easily be solved analytically, is the case of a &potential of strength V, on the edge of a potential step of depth - U. A criterion for the surface state to localise can be found if it is assumed that both parts of the potential scale in the same way with a parameter that represents the coverage. In other words, the potential of the growing system is modelled by c(V,$(z) Ue(-z>>, where 0(z) is the unit step function. The criterion for the surface state to localise is
297
square of potential strength appears in the denominator it is entirely reasonable that the critical coverage is so small. A similar analysis can be done for a general potential and it is found that a similar criterion results except that the strength of the a-potential is replaced by an integral that involves the product of the potential and the wave function. Again, the square of this integral appears, suggesting that the low glancing angle RHEED oscillations generally have a minimum when coverage reaches some low critical value. The results of this section show that the extra structure Mitura et al. found in RHEED oscillations results from the tendency for the growing layer to be transparent to the incident electrons when the laterally averaged potential reaches a critical strength. This effect should be quite general although the details could depend on the strength of the potential for individual cases. In particular, occurrence of the second minimum may be sensitive to the atomic composition of the growing layer although the first minimum is likely to occur generally. As emphasised by Mitura et al. [7,81, the fact that this minimum occurs at very low coverage could be exploited to identify the point at which a new layer starts growing.
3. Conclusion The study of RHEED intensity oscillations could be aided by the use of diffraction conditions specially chosen so that one aspect of the diffraction physics dominates. Particular cases are the out-of-phase condition and the condition of low glancing angle. In each case the dynamical calculations described here have helped to pinpoint the relevant physics and this should serve as the starting point for further investigations.
h2U ”
2mV,2’
(2) Acknowledgements
When values for U and V,, which are consistent with a realistic scattering potential are substituted into this inequality it is found that the critical coverage is a few percent which is in remarkable agreement with fig. 2. Since the
We thank Professor J.L. Beeby for valuable discussions. Financial support from the UK Science and Engineering Research Council, the Royal Society, and Grant No. 2 03829101 of the
298
P.A. Maksym et al. / RHEED oscillations at special diffraction conditions
Polish Scientific Research fully acknowledged.
Committee
is grate-
References (11 J.H. Neave, B.A. Joyce, P.J. Dobson and N. Norton, Appl. Phys. A 31 (1983) 1. [2] S. Clarke and D.D. Vvedensky, Phys. Rev. Lett. 58 (1987) 2235. [3] J.M. Van Hove, C.S. Lent, P.R. Pukite and P.I. Cohen, J. Vat. Sci. Technol. B 1 (1983) 741. [4] T. Kawamura and P.A. Maksym, Surf. Sci. 161 (1985) 12.
El
L.-M. Peng and M.J. Whelan, Surf. Sci. Lett. 238 (1990) L446. [61M.G. Knibb, Surf. Sci. 257 (1991) 389. 171 Z. Mitura and A. Daniluk, Surf. Sci. 277 (1992) 229. and M. Jabchowski, Surf. Sci. [81 Z. Mitura, M. Stroiak Lett. 276 (1992) L15. [91 Y. Horio and A. Ichimiya, Surf. Sci. 298 (1993) 261. and M. Subotow[lOI M. Stroiak, Z. Mitura, M. Jalochowski icz, Vacuum, in press. [ill P.A. Maksym and J.L. Beeby, Surf. Sci. 110 (1981) 423; P.A. Maksym, in: Thin Film Growth Techniques for Low Dimensional Structures, Eds. R.F.C. Farrow, S.S.P. Parkin, P.J. Dobson, J.H. Neave and A.S. Arrott (Plenum, New York, 1987) p. 95.
..J.‘.‘.:-:. in ....“i<.~ ~:~:~.~.~~~~~~,~~~:~~~~.:::.~
.
. .
.
. ..
*
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..:.... :.:.:.: . ...:. :.:,:.:.:;:
:,:,:.:, ~.::~
::,::,
::~.~:.:!~.:.~~::~.:::::~~:~,i~.~:~~,~~
;g+:.:...
scienc6 .surface .
Surface Science 298 (1993) 293-298 North-Holland
RHEED
).,... :::.:$g “:~:::~.:.::.:.:vi “i,.,...,,., .,,., .~ .:,:,.:;,:,..:.,,, ...,., I,.. .,.,.. ::.::‘:‘:‘:.~,~,~:::~~~.:~ ““‘*‘:‘.-~~~“..~. *.‘.‘A c..* .,:,: ,,,,,, _,~~,,.~~,,,,,,,,. ,,,,,,., ““‘.:~:.::“....... ..,.,‘..,. > ,....,,.~,,~,~~,,,,,,:,, “’ “:‘f-.:.:.: .A .,:,:.:,..,. ..........~.._.~.....,~,,, .‘:‘;.~~i....~.:.:~.:.:,:.:.;::; 1;.,:
oscillations at special diffraction conditions
P.A. Maksym a, Z. Mitura b and M.G. Knibb a a Department of Physics and Astronomy, Uniuersity of Leicester, Uniuersity Road, ’ Department of Ex~r~enta~
Physics, Marie Cute-Sk~~owska
Leicester LEl 7RH, UK ~n~uersi~, pL M. Curie-~k~odowsk~ej I, 20-031 Lubi~n~Poland
Received 16 March 1993; accepted for publication 23 April 1993
It is suggested that analysis of RHEED intensity oscillations can be simplified when diffraction conditions are chosen appropriately and two cases are presented as supporting evidence. The first is an out-of-phase condition where the specular reflection from the substrate is very weak. At this condition the oscillations can be described by a kinematic calculation which is modified to take account of refraction. The second case occurs when electrons are incident at glancing angles less than about 1”. Under this condition the oscillations can have a double-minimum structure with one minimum at very small coverages of the growing layer and a second minimum at higher coverages. The minima occur because the growing layer becomes near-transparent at certain critical strengths of the laterally averaged scattering potential. This is illustrated with numerical results for lead and silver.
1. Introduction RHEED intensity oscillations during MBE growth have been known for some time although the mechanism that causes them is not yet properly understood. A number of different approaches have been proposed fl-51, for example, diffraction by arrays of steps and terraces or oscillations in step density, whose general validity is not well established. The situation is compiicated by the dynamical nature of the diffraction process and the disordered nature of a growing surface. This makes it very difficult to calculate exactly the diffraction pattern for an arbitrary growing surface at an arbitrary diffraction condition. Nevertheless, there are indications that the oscillations can be understood simply at certain special diffraction conditions. The search for these conditions was initiated by Knibb [6] who showed that oscillations at the out-of-phase condition could be understood by means of a modified kinematic theory. Subsequently, Mitura et al. [7,81 reported that oscillations have extra minima and maxima when the glancing angle of the incident electrons is very small and showed that 0039-6028/93/$06.00
these features could be reproduced by a calculation where the scattering potential is averaged in the lateral direction. In either case, the reason why the oscillations can be understood simply is that there is one simple physical process which dominates. The purpose of this work is to detail the physics of each case and to argue that the special diffraction conditions have some general validity.
2. WEED
intensity oscillations
The first systematic examination of dynamical effects in RHEED intensity oscillations was made by Knibb E6l who compared kinematic and dynamic results for a model of GaAs growth. The positions of the surface atoms were obtained from a Monte Carlo simulation on a 7 X 7 supercell which, although not large enough to provide a realistic mode1 of the growth, was perfectly adequate to study the diffraction processes that occur in RHEED by growing surfaces. Knibb found
0 1993 - Eisevier Science Publishers B.V. All rights
reserved
294
P.A. Maksym et al. / RHEEB oscillations at special diffraction conditions
that there was very little agreement between the kinematic and dynamic results for most diffraction conditions. However, it turned out that the oscillations in the specular beam intensity at an out-of-phase diffraction condition were very well reproduced by a modified kinematic theory. Oscillations in the intensity of other beams at the same diffraction condition could not be reproduced in this way. The kinematic nature of the specular oscillations appears to be connected with the weakness of the specular reflection from the substrate at the out-of-phase condition. This suggests that the only multiple scattering that is relevant to the specuIar beam intensity at this diffraction condition is multiple forward scattering. Physically, this just affects the refractive index of the scattering system. The refraction is easily taken into account by means of a distorted wave approximation in which the refracted waves are taken to be scattered kinematically. Since a growing surface contains incomplete layers its refractive index is depth dependent and this must be taken into account to get good agreement with the dynamical results. This point is illustrated in fig. 1 which shows results obtained by Knibb for the specular beam oscillations. The oscillations are not smooth because of statistical fluctuations introduced by the small scale of the Monte Carlo
simulation. Each part of the figure shows the same dynamical result (solid line) compared with kinematic results for a different model of the refractive index. In the calculations leading to fig. la, the depth dependence of the refraction was ignored. The kinematic results were calculated by taking the wavevector to be the same as in the vacuum (dotted line) or the buIk (chained Iine) throughout the growing part of the surface. In the latter case, absorption of electrons was taken into account by adding an imaginary component to the wavevector. Clearly, there is very poor agreement with the exact result in both cases. The kinematic curve for fig. lb was computed with the bulk wavevector but absorption was ignored. In this case the minima of the kinematic oscillations are too low. Fig. lc shows results computed with a depth-dependent refractive index. This does the best job of matching both the minima and the maxima of the exact oscillations. The calculations with the modified kinematic theory are very economical on computer time. If the specular beam oscillations at the out-of-phase condition are generally near-kinematic it should be possible to use the modified kinematic approach to compare the predictions of various growth models with experiment in a realistic way. The fact that this approach was found to work when the substrate
Time (Sec.) Fig. 1. Comparison of RHEED oscillations for simulated growth of GaAs as calculated from dynamic and kinematic theories. Each part of the figure shows specular beam oscillations calculated dynamically and with one of the kinematic models described in the text. The incident beam azimuth is (710) and the diffraction condition is the out-of-phase condition for bilayers.
P.A. Maksym et ai. / RHEED oscillations at special diffraction conditions
0
2
4
THICKNESS
Fig. 2. Low glancing angle Pb-35%In(lll). The electron dent beam azimuth is (Zii). cated in
6
8
10
(ML)
oscillations for the growth of energy is 20 keV and the inciThe glancing angles are indithe figure.
reflection is very weak suggests that it should be valid for systems other than the one studied numerically by Knibb. 2.2. Oscillations at low glancing angle Mitura et al. [7,8] were the first to recognise that RHEED intensity oscillations have interesting features when the glancing angle of the incident beam is very low. They considered oscillations that occur during the growth of Pb-In alloys on substrates of the same material and found that the specular beam oscillations have extra structure. Similar structure has recently been discussed by Horio and Ichimiya 191.It is illustrated in fig. 2 which shows experimental and theoretical oscillations found by Mitura et al. for 20 keV electrons at glancing angles less than 0.5”. Each cycle of the oscillation contains a minimum and there is a further, very sharp minimum at the start of each cycle. A detailed description of experimental conditions under which the effect
295
occurs is given by Stroiak et al. (101. Here it is argued that these features occur because the growing layer becomes transparent to electrons when its potential reaches a critical strength. Similar features are likely to be found in most systems when the glancing angle of the incident electrons is very low. Understanding these effects requires knowledge of the limiting behaviour of the reflection coefficient of a single atomic layer at very low glancing angles. Mitura and Daniluk [71 have shown that the double-minimum structure occurs even when all processes other than propagation in the laterally averaged potential are neglected. In other words, the physics is dominated by the propagation of the specular beam in this potential. Therefore, the coupling to other beams does not need to be taken into account to obtain a qualitative explanation of the observed effects. When this coupling is omitted the electron wave function has the form exp(ik,, . p) ~/AZ>, where p is a position vector parallel to the surface and k,, is the parallel component of the incident electron wavevector. The function #(z) obeys the one-dimensional Schrodinger equation
(1) and in this work the potential, I’, is taken to be real. The quantity k, is the magnitude of the perpendicular component of the incident and reflected electron wavevectors. It is related to the electron energy, E, and the parallel component of the incident electron wavevector, k,,, by the equation ki = 2mE/h2 - ki;‘. The intensity reflection and transmission coefficients, R and T, can be written in terms of the two linearly independent solutions of eq. (1). Detailed analysis of the resulting expressions (which is too long to be given here) shows that the limiting behaviour of R and T is normally that R approaches 1 and T approaches 0 as k, approaches zero, i.e. as the glancing angle approaches zero. The exceptional case occurs when the laterally averaged potential has a zero-energy bound state. In this case, either both R and T are non-zero at zero glancing angle, or R = 0 and T = 1. A zero-energy bound
296
oscillations at speciai diffraction conditions
P.A. Maksym et al. ,/ RHEED
state can only occur at certain strengths of the potential and this leads to drastic changes in the reflection coefficient with potential strength. The effect is particularly strong when R vanishes at zero glancing angle. It is illustrated in fig. 3 which gives the reflection and transmission coefficients for 30 keV electrons incident on a single Ag(ll1) layer at a glancing angle of lo-’ degrees. The curves were calculated using methods described in ref. [ll]. Each curve shows how R or 7’ depends on the coverage, c, which in this case is simply a scaling factor that changes the strength of the potential. (The laterally averaged potential of a growing layer is proportional to its coverage.) There is a minimum in I? which is so sharp that a coverage change of about 10e7 ML is sufficient to change R by more than an order of magnitude. The minimum in R is accompanied by an equally
I,. 0.8 3 ‘2
0.6.
g
0.4.
;
i 0.2 .
,/’,:
,.,I 0 1 --..-.-..---------“; __/ 0 0.2 0.4 0.6
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., I :,, “.., ......-.....___ __.__.._.. 1 1.2 1.4
0.8
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4.5 . ,_,’
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./
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Coverage parameter Fig. 3. Specular reflection and transmission coefficients for 30 keV electrons incident on a single AAl 11) layer as a function of potential strength. The potential strength scales with coverage as explained in the text and the coverage parameter is defined to be (c - 0.591293) X 106. The glancing angle is lo-’ degrees.
5 Glancing
angle (degrees)
Fig. 4. Rocking curves for a single Ag(lll) layer when the covergage is just above Cc = 0.6) and just below (c = 0.5) the critical value for adding a bound state. The electron energy is 30 keV.
sharp rise in T - there is 100% transmission even at this extremely low gIancing angle! The idea that this transparency is associated with the addition of a new bound state to the potential can be checked by plotting the phase of the transmission coefficient. For 3D-scattering, Levinson’s theorem gives a relation between the zero-energy phase shift and the number of bound states, and a similar relation can be shown to hold for the phase of the transmission coefficient in one dimension. According to this theorem the phase increases by 7 when the potential strength is increased by an amount sufficient to bind an extra state. The phase change shown in fig. 3 has precisely this behaviour. The drastic change in the reflection coefficient at the critical coverage is not merely confined to glancing angles close to zero. Numerical results show that the reflection coefficient is strongly affected at glancing angles up to around 1”. This is illustrated in fig. 4 which shows rocking curves for a single Ag(l11) layer for a coverage just above the critical value for transparency (0.6 ML) and for a coverage such that the reflection coefficient behaves in the normal way (0.5 ML). Clearly, there is a large drop in the reflection coefficient when the coverage is just above the critical value and this drop occurs over a wide range of glancing angles. Indeed there appears to be a transmission resonance at B = 0.2”. The drop in the
PA Maksym et al. / RHEED oscillations at special diffraction conditions
reflection causes the second minimum in the oscillations shown in fig. 2, that is the minimum that occurs at a coverage of about 0.7 ML: when the potential of the growing layer reaches the critical value for binding a new state the layer becomes near-transparent and the reflection drops to a minimum. It remains to explain the minimum that occurs at very low coverage, that is at the start of the growth cycle. This has no counterpart in the reflection coefficient of a single atomic layer. The reason is that any one-dimensional potential always has at least one bound state and the reflection minima for a single layer occur when further states bind. To understand the first minimum it is necessary to consider the substrate as well as the growing layer. The relevant question is then whether a surface state can bind in the growing layer. When it does, this layer becomes transparent and the reflection coefficient goes through a minimum in the same way as the reflection coefficient for a single layer. The situation can be modelled by considering an atomic layer embedded in an asymmetric potential which approaches zero above the layer and approaches the mean substrate potential below it. Physically, a state can localise in the growing layer only if the layer centre potential is sufficiently deep relative to the mean substrate potential. A simple model of this, which can easily be solved analytically, is the case of a &potential of strength V, on the edge of a potential step of depth - U. A criterion for the surface state to localise can be found if it is assumed that both parts of the potential scale in the same way with a parameter that represents the coverage. In other words, the potential of the growing system is modelled by c(V,$(z) Ue(-z>>, where 0(z) is the unit step function. The criterion for the surface state to localise is
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square of potential strength appears in the denominator it is entirely reasonable that the critical coverage is so small. A similar analysis can be done for a general potential and it is found that a similar criterion results except that the strength of the a-potential is replaced by an integral that involves the product of the potential and the wave function. Again, the square of this integral appears, suggesting that the low glancing angle RHEED oscillations generally have a minimum when coverage reaches some low critical value. The results of this section show that the extra structure Mitura et al. found in RHEED oscillations results from the tendency for the growing layer to be transparent to the incident electrons when the laterally averaged potential reaches a critical strength. This effect should be quite general although the details could depend on the strength of the potential for individual cases. In particular, occurrence of the second minimum may be sensitive to the atomic composition of the growing layer although the first minimum is likely to occur generally. As emphasised by Mitura et al. [7,81, the fact that this minimum occurs at very low coverage could be exploited to identify the point at which a new layer starts growing.
3. Conclusion The study of RHEED intensity oscillations could be aided by the use of diffraction conditions specially chosen so that one aspect of the diffraction physics dominates. Particular cases are the out-of-phase condition and the condition of low glancing angle. In each case the dynamical calculations described here have helped to pinpoint the relevant physics and this should serve as the starting point for further investigations.
h2U ”
2mV,2’
(2) Acknowledgements
When values for U and V,, which are consistent with a realistic scattering potential are substituted into this inequality it is found that the critical coverage is a few percent which is in remarkable agreement with fig. 2. Since the
We thank Professor J.L. Beeby for valuable discussions. Financial support from the UK Science and Engineering Research Council, the Royal Society, and Grant No. 2 03829101 of the
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P.A. Maksym et al. / RHEED oscillations at special diffraction conditions
Polish Scientific Research fully acknowledged.
Committee
is grate-
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