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Theory of RHEED and application to surface structure studies
Vltramicroscopy 48 (1993) 425-432 North-Holland
~
~
V
Theory of R H E E D and application to surface structure studies A. I c h i m i y a , S. K o h m o t o , H. N a k a h a r a a n d Y. H o r i o Department of Applied Physics, School of Engineering, Nagoya University, Chikusa-ku, Nagoya 464-01, Japan Received 17 August 1992; in revised form 10 October 1992
A dynamical theory of reflection high-energy electron diffraction (RHEED) by a multi-slice transfer matrix method is described briefly. According to the theory we give optimum thickness of slices and effects of higher-order Laue zones for the dynamical calculations. Examples of surface structural analyses for the S i ( l l l ) " l × l " surface and structure during homoepitaxial growth on Si(111)7 × 7 surface are also described.
1. Introduction
2. Method of RHEED dynamical calculation
For surface structure determination by reflection high-energy electron diffraction ( R H E E D ) , intensity rocking curves are analyzed by R H E E D dynamical calculations [1-6]. In the case of R H E E D , it is easy to analyze a rocking curve at the one-beam condition which is measured at the incident beam direction chosen at a certain azimuthal angle with respect to a crystal zone axis [6]. Under the one-beam condition, the R H E E D intensity is a function of interlayer distances and atomic densities of surface parallel layers. The surface normal components of the atomic positions are determined by analysis of the one-beam rocking curve with dynamical calculations [3]. Using the results at the one-beam condition, lateral components of atomic positions are determined by analysis of rocking curves at many-beam conditions, where the direction of the incident beam is chosen along a low-order crystal zone axis. In the present paper, a theoretical treatment of a R H E E D dynamical calculation is shown briefly. According to the theory, we showed optim u m thickness of slices and effects of higherorder Laue zones for the dynamical calculations. Then the phase transition of the S i ( l l l ) surface at high temperature and homoepitaxial growth processes are discussed on the basis of the results of the structure analysis.
2.1. Theory of RHEED Since the details of the method of R H E E D dynamical calculation have been described earlier [3,7], an outline of the method is given briefly. The crystal surface is assumed to be periodic parallel to the surface. Crystal potential V(r) and electron wave function &(r) are expanded in a two-dimensional(2D) Fourier series as h2
V(r) = ~2m~-S~ U,.(z) exp(iBm • rt),
(1)
m
~b(r) = ~;'~Cm(Z) exp(i(Kot+B,, ) . r t ) ,
(2)
m
where m, e, and h are usual physical constants, z and r t the surface-normal and the surface-parallel components of a position vector respectively, B m a vector in the 2D reciprocal space and K0t the parallel component of the incident electron wave vector. Substituting eqs. (1) and (2) into the Schr6dinger equation, we obtain d2
dz2cm(Z) +
+ E Um_n(z)cn(z) = O, tt
(3) where Fm is the surface-normal component of the
0304-3991/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
A. lchimiya et al. / ITwory qf R H E E D and application to su(face structure s'tudies
426
diffracted wave vector for the ruth rod. Solving eq. (3) with asymptotic boundary conditions at - = _+zc results in
)
( 60,,, exp( - i I ~ l z
c,,,(z) ~ ~
+R,,, exp(iF,,,z),
/
IT,,, e x p ( - i / ; , , z ) ,
z--.~c,
(4)
z-~ -~c
from which we can obtain R H E E D intensities I R,,, [ 2 Here, ~30,. is Kronecker's delta which is the incident wave amplitude. R,,, and T,,, are the amplitudes of reflected and transmitted waves, respectively. In order to solve eq. (3), the Cm(Z) terms are put as
where the elements of q r consist of ?;~,,, and R .... the elements of qt ~ are T,,, and 0, and M is the transfer matrix. When the matrix elements of M are obtained, qt~ can be calculated. In order to calculate the matrix M, an atomic layer parallel to the crystal surface is divided into slices with thickness Az. In the region at the jth slice, zj,~
A ( z , ) = ~ 1 f=,_ A(z) dz. ~l+[
and l
2,
U"(zi)= . A z z f ' U,,(z) dz.
(7)
The transfer matrix of thc ./th slice M expressed as
G~-z- il;,, c , . ( z ) = - i p , . ( z ) .
can bc
Then cq. (3) can be reduced to the following matrix formula referred to as the semi-reciprocal equation,
M,=exp[-iA(z,)
d d~qr(z) = iA(z)~(z).
for m _< N. The matrix elements of A(z) are
where 7(zj) is an eigen value of A(zj) and Oi a matrix for diagonalization. From the transfer matrix M = FIjM i, the amplitude of the reflected wave, R .... is calculated by eq. (5). Using the transfer matrix M, the numerical computation of R H E E D intensities diverges quickly, when thc matrix includes the terms of evanescent waves. Therefore a column vector R is calculated successively as
(A(z))
.......... :v
R = Po~,,
+ C , 6 ......
where ~ 0 is the column vector for incident wave and matrix P~) is obtained from P1, P~," " ", P/ by
(5)
The elements of a column vector qr(z ) in N-beam case are ( q ' ( Z )),,, = ~',,,( Z ).
( ~ ( z) ),,,+,~,=t,,,,( z),
..... = - ( A ( z ) )
u,,, .(z) -
2/;,
( A ( Z)),,,+N.,, : - ( A ( z ) )
u, . . . .
...... +.~
= O,{exp[-iT(z,)/~Z]
~,
(8)
(z) x(q,l(Zl
for m, n _< N. Using the boundary conditions of eq. (4), eq. (5) is solved formally as t
}Q/ I
Pl~: (q21( z,) + q22( z,)Pl)
2C,
@~=Mq
Az]
(6)
Pi-I
)
(q2,(Z,) X (q,,(zi)
+q]2(z,)pl ) i +q22(Zi) P) +q,2(zi)P,.)
PL = q 2 1 ( Z t . ) q l l l ( Z L ) .
l
A. Ichimiya et al. / Theory of RHEED and application to surface structure studies at the bottom slice ( L t h slice). H e r e qmn(gj) (m, n = 1, 2) is a partial matrix of Mj written as
Mj=
'q,l( zj)
q,2( Zj) )
q2,( zj )
q22(Zj ) .
the ruth 2D Fourier coefficient of the real potential. A Fourier extention of Vm°(Z) is given as oo
V°(z) = f
VmO(() exp(2~-i~'z) d~',
(12)
oo
where V°(() is a Fourier coefficient of the real potential. Substituting eqs. (11) and (12) into eq. (10), we obtain
The R H E E D beam amplitude,
Rm = ( ro/rm)( R)m, can be obtained from eq. (8). For a surface with finite area, the R H E E D intensity I m is obtained as [8]
Im= (Fm/K)IR~ I 2,
427
Vm(~') = L ~
×
• j=
2. 2. Thickness of slice
[1 -- exp( -2~ri~" Az)]
× [1 -- exp(2~ri~" ± z ) ]
(9)
where K is the wave number of the incident wave in vacuum. The absorption of electrons in a crystal is given phenomenologically by an imaginary potential iV'(r), and the crystal potential is rewritten as the sum of real and imaginary potentials V(r) + iV'(r). Therefore Urn(z) in eq. (7) is replaced by Um(Z) + iU'(z). When the absolute value of V'(r) is very small compared with the real potential value, i.e. V'(r)<< V(r), the imaginary parts of the eigen values y(zj) can be obtained by a perturbation method described earlier [3].
4~---~Z
exp{2~ri(~"-- ()zj} d~".
(13)
--c~
Assuming z o = 0, +oc
£
exp{2rri((' - ~') z~}
j=--oe +o¢
=
e x p [ 2 7 r i { ~ " - ( - ( n / A z ) } j Az]
£ j=--m
=
-
-
(,,/Az)}/Az,
for n = - ~ , . . . , - 1, 0, 1. . . . . + % where 6(() is Dirac's delta function. Therefore eq. (13) becomes
Vm(() ~- Vm°(() sin2(~-~ " A z ) / ( r r ( AZ) 2
In order to calculate R H E E D intensities by the multi-slice method, it is necessary to know the optimum thickness of slices. In the calculation we use step-wise potentials. For the step-wise potential according to eq. (7), a Fourier coefficient of one-dimensional potential Vm(Z) is given as
x
'''
.
(14) If the overlaps of V°'s are neglected, i.e. V°(~") >> Vm°(( + (n/Az)) for n :# 0, eq. (14) is reduced as
ac
z
Vm(() = ~ j=
f~f ~zVmj e x p ( - Z ~ - i ( z ) dz,
--oc
V,,(~') --- Vm°(~") sin2(~-( Az)/(rr c Az) z
.--
(lO)
where
Vm~=
1
AZ
zj
f=
-Az
VmO(Z ) dz
(11)
from eq. (7) using zj+ 1 = zj - Az, where V~°(z) is
(is)
Comparing the exponential term of the above equation and the Debye-WaIler factor, e x p ( - B ( 2 / 4 ) , which is included in V°(~'), it is required that the exponent (~r( Az)Z/3 is small enough, e.g. less than one tenth, compared to B(2/4, in order that the effect of division into
428
A. lchim(va et aL / Theory ()f R H E E D attd application to su([~lcc xtructure studies
slices is negligible. We set the criterion of the thickness of the slices, Az, as
sponding reciprocal rods shown with encircling lines A - E in fig. 1. The numbers of beams are 61 for A and 11 for D. The beam set of E is taken by including only the positive Laue zone for propagating waves. Fig. 2 shows rocking curves of (00) and (10) rods of the beams sets A, D and E in fig. 1. The rocking curves for set D shown in figs. 2b and 2e, which are calculated with the beams in only the zeroth Laue zone, are in very good agreement with those for set A shown in figs. 2a and 2d, while the curves for set E shown in figs. 2c and 2f, a set of 36 beams in the positive Laue zones, are very different from the curves fl)r set A. This can be explained by Bethe's dynamical theory [9] as pointed out by Meyer-Ehmsen [10]. According to Bethe's correction methods for R H E E D [11], the corrected potential for the zeroth Laue zone becomes
6 z -- 0.(0.3B3B/2~-. This relation is very similar to the form of the thermal vibration amplitude, u = 0.~f~B/2~-. Since crystal potentials spread by thermal vibrations, it is reasonable that the slice thickness is chosen as Az ~ u, the thermal vibration amplitude.
2.3. Effect of higher-order Laue zones Zhao et al. [5] have pointed out that the role of evanescent waves is very significant for manybeam dynamical calculations. In order to test the effect of evanescent waves, calculations of R H E E D intensity rocking curves were carried out for a Ag(001) surface for several sets of beams. The set of beams used in the dynamical calculations are indicated by the sets of corre-
U,, /J~_,,,
~3rdLaue
"~~
o
o
o
0
0
0
CoD B o 0
~ 2Km-B'+B ~ ,
U,~-°',',,(z) = U , .... ( : ) -
4th
o
o
o
0
O
0
o
o
(16)
zones
O/~/~_,2 n
× "/
/
-1st
d
-3rd
Fig. 1. An upper view of the sets of reciprocal rods of Ag(001) for the dynamical calculations. The arrow indicates the incident beam direction. Solid and open circles are reciprocal rods in the positive and negative Laue zones, respectively. Set A: 61 beams, set B: 43 beams, set C: 29 beams, set D: 1 I beams, set E: 36 beams.
A. Ichimiya et al. / Theory of RHEED and application to surface structure studies (a)
~
429
term of eq. (16) becomes negligibly small compared with the first term. This means that the effects due to the positive and negative Laue zones are canceled out by each other for the zeroth Laue zone. On the other hand, it is clear that R H E E D dynamical calculations with only propagating beams give inaccurate results for surface structure analysis.
(d)
(b)
2.4. O n e - b e a m condition
(f)
(c)
u) zua z
/'t (g)
(h)
["! 0
1
2
3
4
5
6
•
]
2
3
4
5
6
7
8
GL ANCING ANGLE (DEG)
Fig. 2. (00)- and (01)-rod rocking curves for beam sets A, D and E in fig. 1. (00)-rod curves for (a) 61-beam (set A), (b) l 1-beam (set D) and (c) 36-beam (set E), and (01)-rod curves for (d) 61-beam, (e) ll-beam and (f) 36-beam calculations. Curves (g) for (00)-rod and (h) for (01)-rod are obtained by 7-beam calculation of the zeroth Laue zone.
where l in the suffix is taken for the reciprocal rod indices in higher-order Laue zones. Since the terms in the summation take opposite signs for positive and negative Laue zones, the second
Fig. 3. Projection of a crystal lattice into incident direction at the one-beam condition.
As shown in the previous section, R H E E D dynamical calculation with a beam set of only the zeroth Laue zone gives accurate results for rocking curves. Therefore it is considered that R H E E D intensity rocking curves depend on the lattice arrangement of the projection of crystal into the direction of the electron incidence, but hardly depend on displacements of lattices parallel to the incident direction. When a crystal is rotated around the surface-normal axis, incident electrons see the crystal as shown in fig. 3. At this condition, called the one-beam condition at which the main diffraction beam is simply the specular one, a rocking curve of the specular reflection intensity is a function of surface-normal components of atomic positions, but scarcely depends on lateral components of them. Therefore surface-normal components of atomic positions of surface layers are determined by dynamical calculation analysis of a one-beam rocking curve with short computation times. For the S i ( l l l ) surface the incident beam direction at the one-beam condition for 10 keV electrons was experimentally set at near 7° off the [112] direction [6]. In order to examine effects of simultaneous reflections on one-beam rocking curves, 37-beam calculations with the beam set shown in fig. 4 were carried out at the one-beam condition for the S i ( l l l ) surface. Fig. 4 shows rocking curves obtained by the one-beam and 37-beam calculations at one-beam condition from Si(111)7×7 surfaces with the d i m e r - a d a t o m stacking-fault (DAS) structure [12]. In the calculation the surface-normal components of atomic positions used were those obtained previously [6], and the lateral components were those obtained by Tong et al. [13]. The curves calculated with 37
A. lchimiya et al. / Theory of RHEED and application to surface structure studies
430
i • °~ o~
, O
]
2
3
4
5
(~lanc:Lng angLe(d~'cl)
6
O
/',,,
/i
7-\./ "~/ I, ]
2
3
.
"i,
|
/I.....I"I d 4
5
6
G]~<~c:ii~cT an 9]~,(d~9)
Fig. 4. Set of reciprocal rods of Si(l 11 ) surface for dynamical calculations of the rocking curves, and rocking curves from S i ( l l l ) T x 7 surface at one-beam conditions. (a) One-beam rocking curve, (b) 37-beam rocking curve at 7.2 ° off [112.], (c) 37-beam rocking curve at 7.5 ° off [112] and (d) experimental rocking curve at 7.5 ° off.
beams (the 37-beam rocking curves) are very similar to the one-beam one. The peak and shoulder positions of the 37-beam curves are the same as those of the one-beam curve, and there are no extra peaks appearing in the curves of figs. 4b and 4c. Peak and shoulder heights of the 37-beam curves are not significantly different from those of the one-beam one. Since we analyze a surface structure using peak and shoulder positions of rocking curves, peak-height differences at a glancing angle of 4 ° in fig. 4 is not so significant. The curve at 7.5 ° off [112] is in very good agreement with an experimental curve shown in fig. 4d obtained at the same azimuthal angle.
structure with random adatoms (adatom model) or a random dimer-adatom-stacking-fault [12] structure (random DAS model). Fig. 5a shows an experimental rocking curve of the R H E E D intensity under the one-beam condition for "1 × 1" at 900°C [14]. The best-fit curves obtained by R H E E D dynamical calculations for the above four models are shown in fig. 5 with the experimental curve. In the figure, it can be seen that the curve for the adatom model and the random DAS model are in very good agreement with the experimental curve. For the vacancy model, the peak position at the 222 reflection of the curve shifts about 0.2 ° to a lower angle compared with the peak of the experimental one indicated by the arrow in the figure. Under the one-beam condition, we cannot determine either the adatom model or the random DAS model. In order to distinguish between the two models, a (00)-rod rocking curve at [011]
(a)
333
222 /
444
(b)
cd
b,) f t] t-q Ca O I ~D co
3. Structure of S i ( l l l ) " l × 1" at high temperatares In a R H E E D pattern from the S i ( l l l ) " l x 1" surface, diffuse ~ - × ~ - and 2 × 1 streaks are observed [14]. Therefore it is considered that this surface includes disordered structures such as a randomly relaxed bulk-like structure (bulk-like model), a relaxed bulk-like structure with random vacancies (vacancy model), a relaxed bulk-like
0].23456 GLAI,ICI[NG ANGLE ( d e g ) Fig. 5. Experimental and calculated rocking curves at onebeam condition from the Si( 111 ) surface at 900°C. (a) Experimental curve, (b) calculated curve for bulk-like model, (c) that for a vacancy model, (d) that for an adatom model, and (e) that for a random DAS model. The arrows indicate the peak position of the 222 reflection in the experimental curve (a).
431
A. lchimiya et al. / Theory of RHEED and application to surface structure studies
ratio, H / T 4, the energy difference between the H 3 and T 4 sites is estimated as the energy of the T 4 site being about 0.2 eV lower than that of the H 3 site.
&
4. Homoepitaxial growth process on the S i ( l l l ) surface
>-~ k-q u]
Z H A O I O O
0 I 2 3 4 5 6 GLANCING ANGLE (deg) Fig. 6. Experimental and calculated (00)-rod rocking curves for [(111] incidence. (a) Experimental curve, (b) calculated curve for a vacancy model, (c) that for a random DAS model, (d) that for the adatom model at H 3 site, and (el that for the adatom model at T4 site.
incidence was analyzed by a 7-beam dynamical calculation• C o m p a r i n g with the experimental curve and calculated curves shown in fig. 6, it is concluded that the curve for the a d a t o m model is in good a g r e e m e n t with the experimental curve. For the a d a t o m model, it is considered that the adatoms distribute randomly on T 4 and H 3 sites. By calculations for several mixing ratios of H 3 and T4, a rocking curve for a mixing ratio, H3/T4, of 1 / 4 is in the best a g r e e m e n t with the experimental curve [14]. In this structure a d a t o m coverage is 0.25 m o n o l a y e r (ML), a value which is nearly equal to the a d a t o m coverage of the D A S structure [12]. This value is also in very good a g r e e m e n t with a result of reflection electron microscopy at the phase transition from 7 x 7 to 1 x 1 at high t e m p e r a t u r e obtained by Latyshev et al. [15]. T h e atoms just below the adatoms are pushed down, and the remaining atoms not bonding with the adatoms are pushed up from the bulk position. These features are very similar to those of the D A S structure [6]. F r o m the mixing
At temperatures of the Si(ll 1) substrate higher than 300°C, R H E E D intensity oscillations during homoepitaxial growth are very stable [16]. For the beginning 5 M L deposition up to 500°C, the oscillation amplitudes are irregular, but after that the oscillations b e c o m e stable and periodic• At this stable region, profiles of rocking curves c h a n g e d periodically with the coverage [16] as shown in fig. 7. T h e r e f o r e it is concluded that no defects and stacking faults remain in the growing layers• In a R H E E D pattern during growth, a mixed
(1) 4J < H
_ '
0 2 4 6 8 l0 Coverage (monolayer) a
c A
4~
<
__
~ '
__ ~ W
~J'
0 l 2 Glancing
-~13'
/~'\~AL
<~
/
"---,~\
17_
3 4 5 angle (deg)
Fig. 7. RHEED intensity oscillation and rocking curves under one-beam condition, measured at the position indicated by the corresponding numbers. The oscillation was measured at a glancing angle of 3°.
432
,4. h:himiya et al. / Theory of RHEED and application to surface structure studies
gion is dissolved into the normal stacking due to the metastable structure with stimulation by further deposited atoms. Successive epitaxial growth at this temperature is assisted by formation of such metastable structure. At higher temperature it is proposed that the metastable structure is formed at an intermediate stage of the layer growth.
Acknowledgments
4
5
6
Glanc~ng angle(deg) Fig. 8. R l t E E D intensity rocking curves at one-beam condition during homoepitaxial growth. (a) At 28(1°C, (b) at 4/)0°C, (c) at 6011°C and (d) at 7110°(".
This work was carried out under the support of a Grant-in-Aid to the Scientific Research on Priority Areas by the Ministry of Education, Science and Culture (Nos. 03243219 and (14227216).
References phase of 5 × 5 and 7 × 7 structures was observed. At the maximum intensity of the oscillation, the one-beam rocking curves at higher than 400°C are very similar to the curve from the Si(lll)7 x 7 DAS structure as shown in fig. 8. Therefore the structures of 5 × 5 and 7 × 7 at the surface arc nearly the same as the usual DAS structure. On the other hand, at 280°C the one-beam rocking curve is very different from that of the DAS structure observed from growing a surface at higher temperatures. The rocking curve is very similar to the curve from the 87 × 7 structure [17] which is observed at the initial stage of adsorption of H, Li, Na, Ag and Si atoms at room temperature [18]. Since R H E E D intensities near the half-order spots are stronger than those of other spots in the pattern from growing a surface at 280°C, the surface structure is a little different from the 87 × 7 one, and 2a (a 3.84 ~,) periodicity remains on the surface such as a pyramidal cluster-like structure [19]. Since R H E E D intensity oscillation is very stable at this temperature, it seems that there are no defects remaining in the growth layer, and that the stacking fault re-
[1] N. Masud and J.B. Pendry, J, Phys. C 9 11976) 1833. [2] P.A. Maksym and J.L. Beeby, Surf. Sci. 1111(1981) 423. [3] A. lchimiya, Jpn. J. Appl. Phys. 22 119831 176:24 (1985) 1365.
[4] L.-M. Peng and J.M. Cowley, Acta C~'sl. A 42 119861 545. [5] T.('. Zhao, H.C. Poon and S.Y. Tong, Phys. Roy. 13 38 119881 1172. [6] A. lchimiya, Surf. Sci. 192 119871 L893: Structure of Surfaces IIl (1990) 162. [7] A. lchimiya, Surf. Sci. 235 11990) 75. [8] A. lchimiya, Surf. Sci. 187 (19871 194; 219 119891 352. [9] H. Bethe, Ann. Phys. (Leipzig) 87 11928) 55. [111] G. Meyer-Ehmsen, Surf. Sci. 219 119891 177. [11] A. Ichimiya, Acta Cryst. A 44 91988) 11142. [12] K. Takayanagi, Y. Tanishiro, M. Takahashi and S. Takahashi, Surf. Sci. 164 11985) 367. [13] S.Y. Tong, tt. Huang, C.M. Wei, W.E. Packard, F.K. Men, G. Glanden and B. Webb, J. Vac. Sci. Technol. A 6 11988) 615. [14] S. Kohmoto and A. lchimiya, Surf. Sci. 223 (19891 400. [15] A.V. Latyshev, A.L. Aseev, A.B. Krasilnikov and S.1. Stenin, Surf. Sci. 254 (19911 911. [16] H. Nakahara and A. lchimiya, ]. Cryst. Growth 95 119891 472. [17] 11. Nakahara and A. lchimiya, Surf. Sci. 241 1199l) 124. [18] A. Ichimiya, Mater. Res. Soc. Symp. Proc. 2118 (1991) 3. [19] H. Nakahara and A. lchimiya, Surf. Sci. 242 (1991) 162.
~
~
V
Theory of R H E E D and application to surface structure studies A. I c h i m i y a , S. K o h m o t o , H. N a k a h a r a a n d Y. H o r i o Department of Applied Physics, School of Engineering, Nagoya University, Chikusa-ku, Nagoya 464-01, Japan Received 17 August 1992; in revised form 10 October 1992
A dynamical theory of reflection high-energy electron diffraction (RHEED) by a multi-slice transfer matrix method is described briefly. According to the theory we give optimum thickness of slices and effects of higher-order Laue zones for the dynamical calculations. Examples of surface structural analyses for the S i ( l l l ) " l × l " surface and structure during homoepitaxial growth on Si(111)7 × 7 surface are also described.
1. Introduction
2. Method of RHEED dynamical calculation
For surface structure determination by reflection high-energy electron diffraction ( R H E E D ) , intensity rocking curves are analyzed by R H E E D dynamical calculations [1-6]. In the case of R H E E D , it is easy to analyze a rocking curve at the one-beam condition which is measured at the incident beam direction chosen at a certain azimuthal angle with respect to a crystal zone axis [6]. Under the one-beam condition, the R H E E D intensity is a function of interlayer distances and atomic densities of surface parallel layers. The surface normal components of the atomic positions are determined by analysis of the one-beam rocking curve with dynamical calculations [3]. Using the results at the one-beam condition, lateral components of atomic positions are determined by analysis of rocking curves at many-beam conditions, where the direction of the incident beam is chosen along a low-order crystal zone axis. In the present paper, a theoretical treatment of a R H E E D dynamical calculation is shown briefly. According to the theory, we showed optim u m thickness of slices and effects of higherorder Laue zones for the dynamical calculations. Then the phase transition of the S i ( l l l ) surface at high temperature and homoepitaxial growth processes are discussed on the basis of the results of the structure analysis.
2.1. Theory of RHEED Since the details of the method of R H E E D dynamical calculation have been described earlier [3,7], an outline of the method is given briefly. The crystal surface is assumed to be periodic parallel to the surface. Crystal potential V(r) and electron wave function &(r) are expanded in a two-dimensional(2D) Fourier series as h2
V(r) = ~2m~-S~ U,.(z) exp(iBm • rt),
(1)
m
~b(r) = ~;'~Cm(Z) exp(i(Kot+B,, ) . r t ) ,
(2)
m
where m, e, and h are usual physical constants, z and r t the surface-normal and the surface-parallel components of a position vector respectively, B m a vector in the 2D reciprocal space and K0t the parallel component of the incident electron wave vector. Substituting eqs. (1) and (2) into the Schr6dinger equation, we obtain d2
dz2cm(Z) +
+ E Um_n(z)cn(z) = O, tt
(3) where Fm is the surface-normal component of the
0304-3991/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
A. lchimiya et al. / ITwory qf R H E E D and application to su(face structure s'tudies
426
diffracted wave vector for the ruth rod. Solving eq. (3) with asymptotic boundary conditions at - = _+zc results in
)
( 60,,, exp( - i I ~ l z
c,,,(z) ~ ~
+R,,, exp(iF,,,z),
/
IT,,, e x p ( - i / ; , , z ) ,
z--.~c,
(4)
z-~ -~c
from which we can obtain R H E E D intensities I R,,, [ 2 Here, ~30,. is Kronecker's delta which is the incident wave amplitude. R,,, and T,,, are the amplitudes of reflected and transmitted waves, respectively. In order to solve eq. (3), the Cm(Z) terms are put as
where the elements of q r consist of ?;~,,, and R .... the elements of qt ~ are T,,, and 0, and M is the transfer matrix. When the matrix elements of M are obtained, qt~ can be calculated. In order to calculate the matrix M, an atomic layer parallel to the crystal surface is divided into slices with thickness Az. In the region at the jth slice, zj,~
A ( z , ) = ~ 1 f=,_ A(z) dz. ~l+[
and l
2,
U"(zi)= . A z z f ' U,,(z) dz.
(7)
The transfer matrix of thc ./th slice M expressed as
G~-z- il;,, c , . ( z ) = - i p , . ( z ) .
can bc
Then cq. (3) can be reduced to the following matrix formula referred to as the semi-reciprocal equation,
M,=exp[-iA(z,)
d d~qr(z) = iA(z)~(z).
for m _< N. The matrix elements of A(z) are
where 7(zj) is an eigen value of A(zj) and Oi a matrix for diagonalization. From the transfer matrix M = FIjM i, the amplitude of the reflected wave, R .... is calculated by eq. (5). Using the transfer matrix M, the numerical computation of R H E E D intensities diverges quickly, when thc matrix includes the terms of evanescent waves. Therefore a column vector R is calculated successively as
(A(z))
.......... :v
R = Po~,,
+ C , 6 ......
where ~ 0 is the column vector for incident wave and matrix P~) is obtained from P1, P~," " ", P/ by
(5)
The elements of a column vector qr(z ) in N-beam case are ( q ' ( Z )),,, = ~',,,( Z ).
( ~ ( z) ),,,+,~,=t,,,,( z),
..... = - ( A ( z ) )
u,,, .(z) -
2/;,
( A ( Z)),,,+N.,, : - ( A ( z ) )
u, . . . .
...... +.~
= O,{exp[-iT(z,)/~Z]
~,
(8)
(z) x(q,l(Zl
for m, n _< N. Using the boundary conditions of eq. (4), eq. (5) is solved formally as t
}Q/ I
Pl~: (q21( z,) + q22( z,)Pl)
2C,
@~=Mq
Az]
(6)
Pi-I
)
(q2,(Z,) X (q,,(zi)
+q]2(z,)pl ) i +q22(Zi) P) +q,2(zi)P,.)
PL = q 2 1 ( Z t . ) q l l l ( Z L ) .
l
A. Ichimiya et al. / Theory of RHEED and application to surface structure studies at the bottom slice ( L t h slice). H e r e qmn(gj) (m, n = 1, 2) is a partial matrix of Mj written as
Mj=
'q,l( zj)
q,2( Zj) )
q2,( zj )
q22(Zj ) .
the ruth 2D Fourier coefficient of the real potential. A Fourier extention of Vm°(Z) is given as oo
V°(z) = f
VmO(() exp(2~-i~'z) d~',
(12)
oo
where V°(() is a Fourier coefficient of the real potential. Substituting eqs. (11) and (12) into eq. (10), we obtain
The R H E E D beam amplitude,
Rm = ( ro/rm)( R)m, can be obtained from eq. (8). For a surface with finite area, the R H E E D intensity I m is obtained as [8]
Im= (Fm/K)IR~ I 2,
427
Vm(~') = L ~
×
• j=
2. 2. Thickness of slice
[1 -- exp( -2~ri~" Az)]
× [1 -- exp(2~ri~" ± z ) ]
(9)
where K is the wave number of the incident wave in vacuum. The absorption of electrons in a crystal is given phenomenologically by an imaginary potential iV'(r), and the crystal potential is rewritten as the sum of real and imaginary potentials V(r) + iV'(r). Therefore Urn(z) in eq. (7) is replaced by Um(Z) + iU'(z). When the absolute value of V'(r) is very small compared with the real potential value, i.e. V'(r)<< V(r), the imaginary parts of the eigen values y(zj) can be obtained by a perturbation method described earlier [3].
4~---~Z
exp{2~ri(~"-- ()zj} d~".
(13)
--c~
Assuming z o = 0, +oc
£
exp{2rri((' - ~') z~}
j=--oe +o¢
=
e x p [ 2 7 r i { ~ " - ( - ( n / A z ) } j Az]
£ j=--m
=
-
-
(,,/Az)}/Az,
for n = - ~ , . . . , - 1, 0, 1. . . . . + % where 6(() is Dirac's delta function. Therefore eq. (13) becomes
Vm(() ~- Vm°(() sin2(~-~ " A z ) / ( r r ( AZ) 2
In order to calculate R H E E D intensities by the multi-slice method, it is necessary to know the optimum thickness of slices. In the calculation we use step-wise potentials. For the step-wise potential according to eq. (7), a Fourier coefficient of one-dimensional potential Vm(Z) is given as
x
'''
.
(14) If the overlaps of V°'s are neglected, i.e. V°(~") >> Vm°(( + (n/Az)) for n :# 0, eq. (14) is reduced as
ac
z
Vm(() = ~ j=
f~f ~zVmj e x p ( - Z ~ - i ( z ) dz,
--oc
V,,(~') --- Vm°(~") sin2(~-( Az)/(rr c Az) z
.--
(lO)
where
Vm~=
1
AZ
zj
f=
-Az
VmO(Z ) dz
(11)
from eq. (7) using zj+ 1 = zj - Az, where V~°(z) is
(is)
Comparing the exponential term of the above equation and the Debye-WaIler factor, e x p ( - B ( 2 / 4 ) , which is included in V°(~'), it is required that the exponent (~r( Az)Z/3 is small enough, e.g. less than one tenth, compared to B(2/4, in order that the effect of division into
428
A. lchim(va et aL / Theory ()f R H E E D attd application to su([~lcc xtructure studies
slices is negligible. We set the criterion of the thickness of the slices, Az, as
sponding reciprocal rods shown with encircling lines A - E in fig. 1. The numbers of beams are 61 for A and 11 for D. The beam set of E is taken by including only the positive Laue zone for propagating waves. Fig. 2 shows rocking curves of (00) and (10) rods of the beams sets A, D and E in fig. 1. The rocking curves for set D shown in figs. 2b and 2e, which are calculated with the beams in only the zeroth Laue zone, are in very good agreement with those for set A shown in figs. 2a and 2d, while the curves for set E shown in figs. 2c and 2f, a set of 36 beams in the positive Laue zones, are very different from the curves fl)r set A. This can be explained by Bethe's dynamical theory [9] as pointed out by Meyer-Ehmsen [10]. According to Bethe's correction methods for R H E E D [11], the corrected potential for the zeroth Laue zone becomes
6 z -- 0.(0.3B3B/2~-. This relation is very similar to the form of the thermal vibration amplitude, u = 0.~f~B/2~-. Since crystal potentials spread by thermal vibrations, it is reasonable that the slice thickness is chosen as Az ~ u, the thermal vibration amplitude.
2.3. Effect of higher-order Laue zones Zhao et al. [5] have pointed out that the role of evanescent waves is very significant for manybeam dynamical calculations. In order to test the effect of evanescent waves, calculations of R H E E D intensity rocking curves were carried out for a Ag(001) surface for several sets of beams. The set of beams used in the dynamical calculations are indicated by the sets of corre-
U,, /J~_,,,
~3rdLaue
"~~
o
o
o
0
0
0
CoD B o 0
~ 2Km-B'+B ~ ,
U,~-°',',,(z) = U , .... ( : ) -
4th
o
o
o
0
O
0
o
o
(16)
zones
O/~/~_,2 n
× "/
/
-1st
d
-3rd
Fig. 1. An upper view of the sets of reciprocal rods of Ag(001) for the dynamical calculations. The arrow indicates the incident beam direction. Solid and open circles are reciprocal rods in the positive and negative Laue zones, respectively. Set A: 61 beams, set B: 43 beams, set C: 29 beams, set D: 1 I beams, set E: 36 beams.
A. Ichimiya et al. / Theory of RHEED and application to surface structure studies (a)
~
429
term of eq. (16) becomes negligibly small compared with the first term. This means that the effects due to the positive and negative Laue zones are canceled out by each other for the zeroth Laue zone. On the other hand, it is clear that R H E E D dynamical calculations with only propagating beams give inaccurate results for surface structure analysis.
(d)
(b)
2.4. O n e - b e a m condition
(f)
(c)
u) zua z
/'t (g)
(h)
["! 0
1
2
3
4
5
6
•
]
2
3
4
5
6
7
8
GL ANCING ANGLE (DEG)
Fig. 2. (00)- and (01)-rod rocking curves for beam sets A, D and E in fig. 1. (00)-rod curves for (a) 61-beam (set A), (b) l 1-beam (set D) and (c) 36-beam (set E), and (01)-rod curves for (d) 61-beam, (e) ll-beam and (f) 36-beam calculations. Curves (g) for (00)-rod and (h) for (01)-rod are obtained by 7-beam calculation of the zeroth Laue zone.
where l in the suffix is taken for the reciprocal rod indices in higher-order Laue zones. Since the terms in the summation take opposite signs for positive and negative Laue zones, the second
Fig. 3. Projection of a crystal lattice into incident direction at the one-beam condition.
As shown in the previous section, R H E E D dynamical calculation with a beam set of only the zeroth Laue zone gives accurate results for rocking curves. Therefore it is considered that R H E E D intensity rocking curves depend on the lattice arrangement of the projection of crystal into the direction of the electron incidence, but hardly depend on displacements of lattices parallel to the incident direction. When a crystal is rotated around the surface-normal axis, incident electrons see the crystal as shown in fig. 3. At this condition, called the one-beam condition at which the main diffraction beam is simply the specular one, a rocking curve of the specular reflection intensity is a function of surface-normal components of atomic positions, but scarcely depends on lateral components of them. Therefore surface-normal components of atomic positions of surface layers are determined by dynamical calculation analysis of a one-beam rocking curve with short computation times. For the S i ( l l l ) surface the incident beam direction at the one-beam condition for 10 keV electrons was experimentally set at near 7° off the [112] direction [6]. In order to examine effects of simultaneous reflections on one-beam rocking curves, 37-beam calculations with the beam set shown in fig. 4 were carried out at the one-beam condition for the S i ( l l l ) surface. Fig. 4 shows rocking curves obtained by the one-beam and 37-beam calculations at one-beam condition from Si(111)7×7 surfaces with the d i m e r - a d a t o m stacking-fault (DAS) structure [12]. In the calculation the surface-normal components of atomic positions used were those obtained previously [6], and the lateral components were those obtained by Tong et al. [13]. The curves calculated with 37
A. lchimiya et al. / Theory of RHEED and application to surface structure studies
430
i • °~ o~
, O
]
2
3
4
5
(~lanc:Lng angLe(d~'cl)
6
O
/',,,
/i
7-\./ "~/ I, ]
2
3
.
"i,
|
/I.....I"I d 4
5
6
G]~<~c:ii~cT an 9]~,(d~9)
Fig. 4. Set of reciprocal rods of Si(l 11 ) surface for dynamical calculations of the rocking curves, and rocking curves from S i ( l l l ) T x 7 surface at one-beam conditions. (a) One-beam rocking curve, (b) 37-beam rocking curve at 7.2 ° off [112.], (c) 37-beam rocking curve at 7.5 ° off [112] and (d) experimental rocking curve at 7.5 ° off.
beams (the 37-beam rocking curves) are very similar to the one-beam one. The peak and shoulder positions of the 37-beam curves are the same as those of the one-beam curve, and there are no extra peaks appearing in the curves of figs. 4b and 4c. Peak and shoulder heights of the 37-beam curves are not significantly different from those of the one-beam one. Since we analyze a surface structure using peak and shoulder positions of rocking curves, peak-height differences at a glancing angle of 4 ° in fig. 4 is not so significant. The curve at 7.5 ° off [112] is in very good agreement with an experimental curve shown in fig. 4d obtained at the same azimuthal angle.
structure with random adatoms (adatom model) or a random dimer-adatom-stacking-fault [12] structure (random DAS model). Fig. 5a shows an experimental rocking curve of the R H E E D intensity under the one-beam condition for "1 × 1" at 900°C [14]. The best-fit curves obtained by R H E E D dynamical calculations for the above four models are shown in fig. 5 with the experimental curve. In the figure, it can be seen that the curve for the adatom model and the random DAS model are in very good agreement with the experimental curve. For the vacancy model, the peak position at the 222 reflection of the curve shifts about 0.2 ° to a lower angle compared with the peak of the experimental one indicated by the arrow in the figure. Under the one-beam condition, we cannot determine either the adatom model or the random DAS model. In order to distinguish between the two models, a (00)-rod rocking curve at [011]
(a)
333
222 /
444
(b)
cd
b,) f t] t-q Ca O I ~D co
3. Structure of S i ( l l l ) " l × 1" at high temperatares In a R H E E D pattern from the S i ( l l l ) " l x 1" surface, diffuse ~ - × ~ - and 2 × 1 streaks are observed [14]. Therefore it is considered that this surface includes disordered structures such as a randomly relaxed bulk-like structure (bulk-like model), a relaxed bulk-like structure with random vacancies (vacancy model), a relaxed bulk-like
0].23456 GLAI,ICI[NG ANGLE ( d e g ) Fig. 5. Experimental and calculated rocking curves at onebeam condition from the Si( 111 ) surface at 900°C. (a) Experimental curve, (b) calculated curve for bulk-like model, (c) that for a vacancy model, (d) that for an adatom model, and (e) that for a random DAS model. The arrows indicate the peak position of the 222 reflection in the experimental curve (a).
431
A. lchimiya et al. / Theory of RHEED and application to surface structure studies
ratio, H / T 4, the energy difference between the H 3 and T 4 sites is estimated as the energy of the T 4 site being about 0.2 eV lower than that of the H 3 site.
&
4. Homoepitaxial growth process on the S i ( l l l ) surface
>-~ k-q u]
Z H A O I O O
0 I 2 3 4 5 6 GLANCING ANGLE (deg) Fig. 6. Experimental and calculated (00)-rod rocking curves for [(111] incidence. (a) Experimental curve, (b) calculated curve for a vacancy model, (c) that for a random DAS model, (d) that for the adatom model at H 3 site, and (el that for the adatom model at T4 site.
incidence was analyzed by a 7-beam dynamical calculation• C o m p a r i n g with the experimental curve and calculated curves shown in fig. 6, it is concluded that the curve for the a d a t o m model is in good a g r e e m e n t with the experimental curve. For the a d a t o m model, it is considered that the adatoms distribute randomly on T 4 and H 3 sites. By calculations for several mixing ratios of H 3 and T4, a rocking curve for a mixing ratio, H3/T4, of 1 / 4 is in the best a g r e e m e n t with the experimental curve [14]. In this structure a d a t o m coverage is 0.25 m o n o l a y e r (ML), a value which is nearly equal to the a d a t o m coverage of the D A S structure [12]. This value is also in very good a g r e e m e n t with a result of reflection electron microscopy at the phase transition from 7 x 7 to 1 x 1 at high t e m p e r a t u r e obtained by Latyshev et al. [15]. T h e atoms just below the adatoms are pushed down, and the remaining atoms not bonding with the adatoms are pushed up from the bulk position. These features are very similar to those of the D A S structure [6]. F r o m the mixing
At temperatures of the Si(ll 1) substrate higher than 300°C, R H E E D intensity oscillations during homoepitaxial growth are very stable [16]. For the beginning 5 M L deposition up to 500°C, the oscillation amplitudes are irregular, but after that the oscillations b e c o m e stable and periodic• At this stable region, profiles of rocking curves c h a n g e d periodically with the coverage [16] as shown in fig. 7. T h e r e f o r e it is concluded that no defects and stacking faults remain in the growing layers• In a R H E E D pattern during growth, a mixed
(1) 4J < H
_ '
0 2 4 6 8 l0 Coverage (monolayer) a
c A
4~
<
__
~ '
__ ~ W
~J'
0 l 2 Glancing
-~13'
/~'\~AL
<~
/
"---,~\
17_
3 4 5 angle (deg)
Fig. 7. RHEED intensity oscillation and rocking curves under one-beam condition, measured at the position indicated by the corresponding numbers. The oscillation was measured at a glancing angle of 3°.
432
,4. h:himiya et al. / Theory of RHEED and application to surface structure studies
gion is dissolved into the normal stacking due to the metastable structure with stimulation by further deposited atoms. Successive epitaxial growth at this temperature is assisted by formation of such metastable structure. At higher temperature it is proposed that the metastable structure is formed at an intermediate stage of the layer growth.
Acknowledgments
4
5
6
Glanc~ng angle(deg) Fig. 8. R l t E E D intensity rocking curves at one-beam condition during homoepitaxial growth. (a) At 28(1°C, (b) at 4/)0°C, (c) at 6011°C and (d) at 7110°(".
This work was carried out under the support of a Grant-in-Aid to the Scientific Research on Priority Areas by the Ministry of Education, Science and Culture (Nos. 03243219 and (14227216).
References phase of 5 × 5 and 7 × 7 structures was observed. At the maximum intensity of the oscillation, the one-beam rocking curves at higher than 400°C are very similar to the curve from the Si(lll)7 x 7 DAS structure as shown in fig. 8. Therefore the structures of 5 × 5 and 7 × 7 at the surface arc nearly the same as the usual DAS structure. On the other hand, at 280°C the one-beam rocking curve is very different from that of the DAS structure observed from growing a surface at higher temperatures. The rocking curve is very similar to the curve from the 87 × 7 structure [17] which is observed at the initial stage of adsorption of H, Li, Na, Ag and Si atoms at room temperature [18]. Since R H E E D intensities near the half-order spots are stronger than those of other spots in the pattern from growing a surface at 280°C, the surface structure is a little different from the 87 × 7 one, and 2a (a 3.84 ~,) periodicity remains on the surface such as a pyramidal cluster-like structure [19]. Since R H E E D intensity oscillation is very stable at this temperature, it seems that there are no defects remaining in the growth layer, and that the stacking fault re-
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[4] L.-M. Peng and J.M. Cowley, Acta C~'sl. A 42 119861 545. [5] T.('. Zhao, H.C. Poon and S.Y. Tong, Phys. Roy. 13 38 119881 1172. [6] A. lchimiya, Surf. Sci. 192 119871 L893: Structure of Surfaces IIl (1990) 162. [7] A. lchimiya, Surf. Sci. 235 11990) 75. [8] A. lchimiya, Surf. Sci. 187 (19871 194; 219 119891 352. [9] H. Bethe, Ann. Phys. (Leipzig) 87 11928) 55. [111] G. Meyer-Ehmsen, Surf. Sci. 219 119891 177. [11] A. Ichimiya, Acta Cryst. A 44 91988) 11142. [12] K. Takayanagi, Y. Tanishiro, M. Takahashi and S. Takahashi, Surf. Sci. 164 11985) 367. [13] S.Y. Tong, tt. Huang, C.M. Wei, W.E. Packard, F.K. Men, G. Glanden and B. Webb, J. Vac. Sci. Technol. A 6 11988) 615. [14] S. Kohmoto and A. lchimiya, Surf. Sci. 223 (19891 400. [15] A.V. Latyshev, A.L. Aseev, A.B. Krasilnikov and S.1. Stenin, Surf. Sci. 254 (19911 911. [16] H. Nakahara and A. lchimiya, ]. Cryst. Growth 95 119891 472. [17] 11. Nakahara and A. lchimiya, Surf. Sci. 241 1199l) 124. [18] A. Ichimiya, Mater. Res. Soc. Symp. Proc. 2118 (1991) 3. [19] H. Nakahara and A. lchimiya, Surf. Sci. 242 (1991) 162.