(1996) - Acta Cryst-Dynamical - RHEED - Calculations - From - The - Surface - of - A - Semi-Infinite - Crystal
(1996) - Acta Cryst-Dynamical - RHEED - Calculations - From - The - Surface - of - A - Semi-Infinite - Crystal
(1996) - Acta Cryst-Dynamical - RHEED - Calculations - From - The - Surface - of - A - Semi-Infinite - Crystal
net/publication/244628605
CITATIONS READS
6 18
3 authors, including:
Some of the authors of this publication are also working on these related projects:
Carbon Nanotube high performance optoelectronic devices and integrated circuits View project
All content following this page was uploaded by Lian-Mao Peng on 23 February 2017.
aBeijing Laboratory of Electron Microscopy, Chinese Academy of Sciences, PO Box 2724, Beijing 100080, People's
Republic of China, and bDepartment of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, England.
E-mail: Impeng @Implab.blem.ac, cn
(Received 2 November 1995; accepted 25 January 1996)
in which ~'c(z) is the diffracted-beam amplitude associ- in which T¢a is the reflected-beam amplitude associated
ated with the Gth rod of the reciprocal lattice and ~ is with the Gth reciprocal-lattice rod.
its surface normal derivative. The matrix M(c) appearing For a semi-infinite crystal, since all physically allowed
in (1) is called the scattering matrix and the constant c is quantities must have a finite amplitude, the lower half of
the lattice constant along the surface-normal direction. the column vector C - 1 M s ~ ' (0) on the right-hand side of
For a Bloch wave b(r) in a crystal, it can be readily (8) must vanish. In terms of the two lower submatrices
shown that both the Gth component b 6 and its surface- M~I and M22 of the matrix M ~ = C-1Ms and the
normal derivative b~ = d(bo)/dz satisfy the Bloch surface reflected-beam-amplitude vector { ~ 6 } , we can
theorem write the condition as
b(J3(z
G + c) = exn(",-,(J) ~ G
(j) (Z),
~.,t. c~,b (9)
b~.J3(z + c) = exp(i@J)c)b~J)(z) (3)
which gives
and that they are therefore both Bloch waves. We define
a super vector b (J~ by
+
Since (1) holds for a general wave function, it also holds from the surface of a semi-infinite crystal, the scattering
for b (J~ defined by (4), i.e. matrix M(c) associated with a repeating unit slab must
first be calculated and then diagonalized by a similarity
bO3(z + c) = exp(iq,(J)c)b(J)(z) = M(c)b(J)(z), (5) transformation as in (6). However, to achieve a conver-
gent dynamical RHEED calculation, some evanescent
in which "7(j) is the jth eigenvalue associated with the
beams must be included. The inclusion of evanescent
jth super vector b(J)(z).
beams, in particular positive higher-order Laue-zone
For a dynamical RHEED calculation involving n (HOLZ) beams lying outside the Ewald sphere, usually
reciprocal-lattice rods, the scattering matrix M is a gives rise to a divergent scattering matrix M(c) for
2n x 2n matrix. In general, this 2n × 2n matrix will a repeating unit slab, putting an upper limit on the
give a total of 2n Bloch waves, of which n propagate number of evanescent beams that may be included in
downwards into the crystal slab, and in the presence of the calculation and consequently on the accuracy of the
absorption decay in amplitude downwards. The other calculation.
n propagate and decay in the reverse direction. By The situation can be improved to a certain extent if
introducing a 2n × 2n matrix C = (b (1), . . . , b (2n)) and the repeating unit slab of the substrate crystal consists
a diagonal matrix T(z) = {exp(i3,(J~z)}, we then have of an assembly of identical layers of atoms, such as a
M(z) = C ' r ( z ) C - ' . (6) monolayer of atoms, each having a thickness c and a
relative shift R t with respect to each other. The scattering
For a crystal slab consisting of m repeating unit slabs, matrix associated with the ith layer is then related to the
substitution of (6) into (1) gives ( i - 1)th layer by the relation
'~(z + mc) = CT(mc)C-I'~(z). (7) M i - Q - I M i _ I Q -- ( Q - l ) i - l M 0 ( Q ) i - l , (11)
Without loss of generality, we can assume that among
the total 2n Bloch waves the first n Bloch waves in which the matrix Q is a diagonal matrix with {Q}c =
are evanescent waves and the remaining waves are e x p ( i G . R t ) , and M 0 is the scattering matrix associated
anti-evanescent. For a semi-infinite crystal, these anti- with the first layer. Following the general relation (1),
evanescent Bloch waves are not physically allowed and we have
must be discarded. Assmning that the interface between
the selvedge and the bulk crystal is at z = Zs and that
~(Z + mc) = MmI,~[Z"q- (m-- 1)c]
the scattering matrix associated with the selvedge is M s, = MmMm_ 1 ... M0~I'(z )
we then have for a crystal slab consisting of m repeating = (Q-I)m(QMo)m,~(z). (12)
unit slabs
By a similarity transformation of the matrix Q M 0,
• (z + mc) = c"r(mc)C-lMs~(O)
= C ( {exp(i@i)mc)} {exp(iT(i+n)mc)} ) QM o = CTC -l, (13)
x { )
C-IMsk{kGz(6GO- TO.G)} ' (8)
we obtain
This expression is similar to (7) and, following a similar In principle, the problem of dynamical RHEED in-
argument to that leading to (10), we obtain an identical volves an infinite number of rods of the reciprocal
expression for the reflected-beam-amplitude vector (10). lattice. However, since the scattering power of the crystal
Since the thickness of a layer of atoms is smaller is effectively band-width limited, only a finite set of
than that of a repeating unit slab, the validity of the reciprocal-lattice rods, say N rods, are needed for the
above procedure is improved compared with that using description of the main features of RHEED. Artificially,
a repeating unit slab. we may divide the N rods of the reciprocal lattice
For a selvedge with a simple structure, the scattering into two groups, one referring to the strong beams and
matrix M s associated with the surface selvedge can be the other to the weak beams, based on the quantity
simple and convergent if the scattering matrix associated /-2 = Ke _ [K/+ HIe associated with the beams, where
with the bulk slab M(c) is convergent. However, for K is the incident electron wave vector corrected for
a complicated surface structure, the thickness of the the mean inner potential and the subscript t denotes
selvedge could be larger than that of a bulk slab and the tangential component of the wave vector. For weak
its scattering matrix M s could be divergent even if beams, we have the inequality IF2 - K2I >> IUmax[,
that associated with the bulk slab is convergent. In this where IUmax[ is the maximum of all Iu..I for weak
case we can propagate a so-called R matrix (Ichimiya, beams, while for strong beams this inequality is not
1983; Zhao et al., 1988) rather than the scattering matrix satisfied. If both groups of beams are treated fully in
through the selvedge. Assuming that the whole crystal the dynamical RHEED theory, then, roughly speaking,
system consists of a total of m bulk slabs and the for weak beams there exists a one-to-one correspondence
selvedge interface is at z = zs, we have then the between Bloch waves and the reciprocal-lattice rods, and
following expression for relating the super vector at the for these Bloch waves we have approximately ,if(i)2 ,x,
bottom of the whole crystal system to that at the selvedge /-2. For evanescent beams, we have ,if(i) r~ +i[/-Hi" From
interface (8), it is clear that the scattering matrix M associated
with a crystal slab then contains terms of the form
• (z s + mc) = C T ( m c ) C - l f f ~ ( Z s ) . (15) exp(I/-HIc). For a given value of the slab thickness
c, we then have an upper limit /-max on I/-.1. i.e. all
By writing weak beams with Ir.I > /"max cannot be included in
the calculation. Extensive computations confirmed that
the inclusion of beams with I/"cl </"max is usually not
C -I = C / , (16)
= C21 C22 sufficient to produce convergent results.
At this stage, we note that, in the method of Bethe
and following the same procedure leading to (9), we potentials (Bethe, 1928; Ichimiya, 1988; Peng, Dudarev
obtain & Whelan, 1996), beams with sufficiently large values
of [/"HI may be treated as a perturbation. Justification for
! • !
C21 { ~ G ( Z s ) ) + C22{--l~b(Zs) ) = 0, (17) the use of the Bethe potential method is that for an N-rod
RHEED case, among the total of 2N Bloch waves only a
which gives the R matrix at the selvedge interface, smaller set of 2n (n < N) Bloch waves are important and
have appreciable excitation amplitudes. Although all N
R = -(C~2) -x (C21),
' (18) rods of the reciprocal lattice are needed to give correct
eigenvalues and eigenvectors for these 2n important
that relates the surface-normal derivative of the wave- Bloch waves, only n reciprocal-lattice rods need to be
function vector {k~ } and the wave-function vector {~c } treated fully, while the effect of the remaining N - n rods
may be taken into account as a perturbation. We have
{ - i ~ ( z ) } = R{k~(z)}. (19) found that the method of Bethe potentials works well in
this respect. In the next section, we will show that the
This R matrix can then be propagated upwards to upper limit imposed on I/-.1 for evanescent beams by the
the surface to give the required surface reflected-beam condition that the scattering matrix M must be finite and
amplitudes (Peng, 1994). the low limit set by the Bethe approximation overlap.
When the solid is composed of moderately strong The divergence problem encountered in diagonalizing
scattering atoms, the scattering matrix associated with the scattering matrix M is then solved by the use of the
even a monolayer of atoms can be divergent. The prob- Bethe potential method.
lem arises from the inclusion of the evanescent beams
lying outside the Ewald sphere. It has now been well
3. Numerical results
established that, although each of these strong evanes-
cent waves contributes little to the resulting RHEED In this section, we will present some results for the Ag
rocking curves, in general their collective contribution (001) surface. The high-energy electrons are incident on
is not negligible. the surface along the [110] azimuth. For convenience,
474 DYNAMICAL RHEED CALCULATIONS FROM A SEMI-INFINITE CRYSTAL
we will use an index system that the conventional figure with the key '16 rod + Bethe potentials' was
[110] beam azimuth corresponds to [10]. Using this calculated using the basic set of 16 rods as in other
indexing notation, the zero-order Laue-zone (ZOLZ) curves, and the remaining 21 rods were treated using
beams (n, ~, 0) can be simply written as (0, n) and the the method of Bethe potentials (see Peng et al., 1996, for
HOLZ beams (n + m, ~ + m, 0), where m refers to the details). Fig. 2 shows that, while the 16-rod calculation
order of the HOLZ zones, is written as (m, n). The differs substantially from the full 33-rod calculation, the
primary-beam energy used in the following calculation use of Bethe potentials produces almost perfect results.
is 20 keV and the complex atomic scattering factors are To be more quantitative, we have calculated a so-called
taken from Dudarev, Peng & Whelan (1995). R factor defined as
Shown in Fig. 1 is a calculated kinematic RHEED
pattern using CERIUS 2 of MSI. While it is well known RAB -- ~ IIA -- 4I/IA, (20)
that for RHEED geometry the kinematic theory does not i
give the right intensity, this kinematic diffraction pattern
nevertheless gives the correct diffraction geometry. The in which I a and 18 denote intensities of curves A and
two diffraction rings consist of diffraction spots of the B, and the index i refers to the ith data point. For
type (0, n) and (1, n) are ZOLZ and first negative HOLZ the specular (00) beam, R = 0.53 between curves of
zones, respectively, and all positive HOLZ beams are the full 33-rod calculation and the 16-rod calculation,
evanescent and not visible in the RHEED pattern. R = 0.0085 between the full 33-rod calculation and
Shown in Fig. 2 are dynamical RHEED rocking the 16-rod curve with Bethe potentials. For the (01)
curves for the specular (00) and (01) beams. The curve rod, the values for the two cases are 0.90 and 0.015,
with the key '16 rod calculation' was calculated using respectively. These figures demonstrate clearly that the
nine ZOLZ beams (0,0), (0,+1), (0,+2), (0,+3), use of Bethe potentials greatly improves the accuracy of
( 0 , + 4 ) and seven HOLZ beams ( - 1 , 0), (-1,-4-1), the approximate calculation using only a limited set of
( - 1 , +2), (-1,-t-3) and using (10). When higher-order beams and equation (10).
ZOLZ beams and other HOLZ beams are included, the
scattering matrix M(c) becomes divergent. However, the
curve with the key 'full 33 rod calculation' in the figure 4. Conclusions
shows clearly that the 16-rod calculation is not conver- In this paper, a slab method is developed for calculating
gent. This 33-rod calculation uses the same 16 rods of dynamical RHEED from the surface of a semi-infinite
the reciprocal lattice as in the earlier curve, but includes crystal. This method combines the usual slab method
an additional 17 HOLZ beams and uses the conventional of RHEED with the method of Bethe potentials. Our
RHEED slab method (Peng & Whelan, 1990) rather than results show that this method is efficient and convergent
the method discussed in §2. These additional beams are and they agree well with those calculated using the
( - 2 , 0 ) , ( - 2 , + 1 ) , ( - 2 , + 2 ) , (1,0), (1,+1), (1,±2), conventional slab method.
(1, ±3), (2,0), (2, ±1), (2,4-2). The third curve in the
This work was supported by the Chinese Academy
of Sciences and National Natural Science Foundation
.7_
of China (LMP), the Engineering and Physical Science
.6_
- - Full 33 rod calculation
0,1 , , 16 rod + Bethe potentials
.5_ ............ 16 rod calculation
0.0
.4_ 0:0
i7 i7 "
.9._
~ 0.4
9_ 19. ~ • 04 i9 2
.1_ • • • i • • 0.2
• •
0.0 t . . . . . _ . . . . . . .
0.0 10.0 20.0 30.0 40.0 50,0 60.0 70.0 80.0
Angle of incidence (mrad)
I 1_.9. I I L I I
-.4 .0 .2 .4
Fig. 2. DynamicalRHEED rocking curves calculated from an Ag (100)
x surface. The calculations are made for 20 keV primary-beamenergy
Fig. 1. Kinematic RHEED pattern calculated for 20 keN primary-beam and 293 K, and the two graphs are for the specular (00) beam and
energy and an Ag (001) surface for the [110] beam azimuth. a side (01) beam.
L.-M. PENG, S. L. DUDAREV AND M. J. WHELAN 475
Research Council (EPSRC) (grant no. GR/H96423) and Peng, L.-M. (1989). Surf. Sci. 222, 296-312.
the Royal Society (Joint Project no. Q711). Peng, L.-M. (1994). Advances in Imaging and Electron Physics,
Vol. 90, edited by P. W. Hawkes, pp. 205-351. New York:
Academic Press.
References Peng, L.-M. & Cowley, J. M. (1986). Acta Cryst. A42,
Bethe, H. (1928). Ann. Phys. (Leipzig), 87, 55-129. 545-552.
Colella, R. (1972). Acta Cryst. A28, 11-15. Peng, L.-M., Dudarev, S. L. & Whelan, M. J. (1996). Surf.
Dudarev, S. L., Peng, L.-M. & Whelan, M. J. (1993). Proc. R. Sci. Lett. In the press.
Soc. London Ser. A, 440, 567-588. Peng, L.-M., Gjcnnes, K. & Gjcnnes, J. (1992). Microsc. Res.
Dudarev, S. L., Peng, L.-M. & Whelan, M. J. (1995). Surf. Tech. 20, 360-370.
Sci. 330, 86-100. Peng, L.-M. & Whelan, M. J. (1990). Proc. R. Soc. London
Ichimiya, A. (1983). Jpn. J. Appl. Phys. 22, 176-180. Ser. A, 431, 111-124, 125-142.
Ichimiya, A. (1988). Acta Cryst. A44, 1042-1044. Rez, P. (1995). Acta Cryst. A51, 38--47.
Ma, Y. & Marks, L. D. (1989). Acta Cryst. A45, 174-182. Smith, A. E. & Lynch, D. F. (1988). Acta Cryst. A44, 780-
McRae, E. G. (1968). Surf. Sci. 11, 479--491. 788.
Maksym, P. A. & Beeby, J. L. (1981). Surf. Sci. 110, 423--438. Stampfl, C., Kambe, K., Riley, J. D. & Lynch, D. F. (1992).
Meyer-Ehmsen, G. (1989). Surf. Sci. 219, 177-188. J. Phys. Condens. Matter, 4, 8461-8476.
NAG (1993). NAG Fortran Library Manual. NAG Ltd, Oxford, Zhao, T. C., Poon, H. C. & Tong, S. Y. (1988). Phys. Rev. B,
England. 38, 1172-1195.