Dynamical RHEED from MBE growing surfaces

Surface

Science Letters 238 (1990) L&K&L452 North-Scotland

Surface Scicace Letters

Dynamkal L.-M.

Peng

RHEED and

M.J.

from MBE growing

surfaces

Whelan

Dqmrtment of hrateriaIs, Unruersity of Oxford, Purks Roud, Oxford OXI 3PN, Received

4 May 1990; accepted

for publication

UK

18 July 1990

A practical I:omputing procedure has been developed for calculating reflection high energy electron diffraction (RHEED) from molecular beam epitaxy (MBE) growing surfaces. A birth-death model is employed to describe the epitaxy growth on the surfaces, and the diffrac ion is treated dynamically. A typical calculation of RHEED intensity from a C&&(001) MBE growing system consisting of a lmlk substrate crystal and up to ten growing surface layers takes five seconds on a VAX 8800 system.

Reflectior. high energy electron diffraction (RHEED) has become well established as one of the most powerful and versatile techniques for growth and :;urface studies of semiconductor films prepared by molecular beam epitaxy (MBE). In particular t le technique of RHEED intensity oscillations leas been used routinely, amongst other things, to citlibrate beam fluxes and alloys composition ancl to control the thicknesses of quantum wells arid superlattice layers [l]. Several interpretations [2-5] have been proposed for the strong oscillations in RHEED intensity during MBE growth. In the first place the Philips group [2,6] has used a dynamical argument, treati-lg diffraction as a multiple scattering process, am attributes the oscillations in RHEED intensity during steady state growth to a periodically varyin,: contribution of inelastic/or incoherent processes. On the other hand, Cohen and co-workers at the University of Minesota [3,7] employ a kinematic theory, treating diffraction as a single sea ttering process, and put much emphasis on the interference of beams reflected from terraces on the crystal surface. A deficiency of this type of theory, which is also inherent in present dynamical treatment, is that it is unable to account for diffracltion effects dependent on azimuth, which frorr experiment are known to exist [8,9]. 0039-6028/90,/$03.50

0 1990 - Elsevier Science Publishers

Full dynamical calculations have also been done for an artificial surface consisting of a periodic array of surface steps [4]. However, the enormous amount of computation involved prevents the theory being applied to any more realistic expitaxially growing system. In this Letter we report a simple and practical method for calculating dynamical RHEED from MBE growing surfaces. The MBE growth on a low-index surface is simulated by using a birthdeath model as proposed by Cohen et al. [lo], and the dynamical diffraction processes are treated within the framework of the general matrix formulation of Peng and Whelan [ll]_ The dynamical diffraction calculations reported in this Letter utilize the so-called “systematic reflection” case in RHEED, in which the atomic potential in planes parallel to the surface are projected on the surface normal, so that the results are insensitive to the atomic arrangement (reconstruction, etc.) in layers parallel to the surface and hence cannot explain diffraction effects dependent on azimuth. We shall first consider epitaxial growth on an As-stable GaAs(001) surface. Since on a low-index surface there is no natural spatial origin, a mean field description is possible. If we go further and use a solid-on-solid (SOS) appro~mation [12], i.e. exclude overhangs, the coverage S,, of the n th

B.V. (North-Holland)

L.-M. Peng, M. J. Whelan / RHEED

surface layer then satisfies equations de,,/dt=

a set of differential

(i,b)(e,_,-en) +k(%+,

-e,+,)(%-I

0)

during the growth, and de,/dt = k(B,,+, - en+2)(en-1 - k(e, - e,+i)(enP2

- 6) - e,-i),

(2)

after the growth is interrupted. Here r is the time to deposit a monolayer and k is a filling parameter that measures the mobility of uncovered atoms. The initial and boundary conditions of the sets of equations are e,(t)= 1, e,(o)= 0 (n = 1, 2, . . . ), and B,(t) = 0. A perfect layer growth corresponds to an infinite k in eqs. (1) and (2). The layer coverages are given by

e,(t)=

0,

t+-l)r,

t+(n-l),

(n--1)71t1n7,

i 1,

been tally years more

referred to as an “idealized case”, it is practiachievable as has been done nearly sixty ago by Kikuchi and Nakagawa [13], and recently by Ichimiya [14] and Cohen et al.

171.

-et7j

-k(e,-8,+,)(8,-,-e,-,),

from MBE growing surfaces

To simulate the structural variations of growing surface layers along the surface normal direction, we divide the surface layers into an assembly of thin slices parallel to the surface (in the case of a GaAs(001) surface, one hundred slices per monolayer, i.e. a double layer, one of Ga and the other of As atoms), and assume that each slice has a constant potential { U,(zi)} normal to the surface. Within each thin slice the electron wave function has the form: I/J,(Z) = T, exp(ik,z)

+ R, exp( -ik,z),

(4)

and its surface normal derivative: q:(z)

= ik,T, exp(ik,z)

- ik,R,

exp(-ik,z). (5)

t>nr. (3)

We now consider the dynamical diffraction from an epitaxially growing system consisting of a bulk substrate crystal and several growing surface layers as described by the set of equations (1). For an adequate treatment of this system, the theoretical framework should be able to incorporate the non-periodic variations of the scattering potential in the surface normal direction, and non-periodically distributed surface disorders within the surface layers parallel to the low-index substrate surface. There exist many dynamical approaches nowadays which meet the first requirement (see references in ref. [l]). However a full and reliable treatment of surface disorders is unavoidably tedious and impracticable. To avoid the complexity of treating the non-periodic surface disorders on the growing layers and the unknown atomic details of the reconstructed surface, in what follows we make a so called systematic reflection approximation, considering only the reflection by planes parallel to the substrate surface. Although the systematic reflection case in RHEED has often

In eqs. (4) and (5) the exponential term exp(ik, . rp) has been omitted (here subscript p denotes the tangential component), since it is a common phase term for all slices, and k, depends on zi (the depth of the ith slice) through the relation

kz(z;) = /xI+uD(Li) where x is the electron wave vector in the vacuum and U,(z,) the inner potential of the i th thin slice. In matrix notation we have

(6) where A,(k,l

z) exp(ik,z)

=

i

ik, exp(ik,z)

exp( -ik,z) -ik,

exp(-ik,z)

i

’

On the upper surface of the thin slice, z = z,_i, the boundary conditions give

(7)

L.-M. Peng, M.J. Whelan / RHEED

By combining (6) and (7) and eliminating transmission and reflection coefficients T, R,,, we obtain

the and

(8) in which M, =‘&(k.,

z,M,‘(k,>

z,-1) sin( k,L )/k,

cos(kA?)

=

sin( k,t,)

-k,

i

co4kztn)

tion effects, k,(z,) in the scattering matrix (11) will become complex, giving a non-trivial divergent problem when a large absorption is used [15]. In this case the substrate bulk crystal needs to be treated separately. For details see Peng and Whelan [ll]. The scattering potential for high energy electrons is usually expressed in terms of a sum of contributions of individual atoms centred at r, [Ml:

(9) i>

and

or, in terms the G,(r):

K’(k,,

=-

of the Fourier

transforms,

U,(S), of

z)

I 2

(

exp( -ik,z)

exp( -ik,z)/ik,

exp(ik,z)

- exp(ik,z)/ik,

V(s) i’

by cos[&(z,)t,l ,(

- k,(6)

= Cui(s>

exp( -is.r,).

(15)

(10)

and t, is the thickness (z,, - z,_ i) of the thin slice. The scattering matrix M for relating the electron wave function and its surface normal derivative on the upper and lower surfaces of an assembly of thin slices, each having thickness tj is given

M=n

from MBE growing surfaces

sin[k,(z,)r,I/kZ(z,)

In (14) * denotes convolution, and in (15) V(s) is the Fourier transform of the crystal potential: V(s)

=/V(r)

exp( -is-r)

dr.

variation

along

The potential mal is given by U,(Z)

=

T i

sintk,(z,)t,l

/+mV(O,

06) the surface

0, s,) exp(is,z)

nor-

ds,

-02

01) Since the vacuum region above the crystal slab contains only the incident and specular reflected beam, and in the vacuum region below the slab only a transmitted beam exists, we have

To exp(ik,t) ix,T,

and therefore tudes R,:

R”=-

(12)

exp(ik,t) the specular

[Ml1 [Ml, -

Wl/ixrl MZl/ixrl

reflected

beam

ampli-

X exp( - is,z,)]

When an imaginary potential component is added to take into account the anomalous absorp-

ds,,

(17)

where S, is the area of a two dimensional unit cell parallel to the surface. Since for isolated atoms of spherical symmetry u(s) can be expressed by the Doyle-Turner analytical approximation to the electron scattering factor [17], we have

- IM22- ix2421 + [M2, - ixM121 .

(13)

exp(is,z)

ak

and also, following U,(z)

= CU@Yz), n

eXP[ -bh/2)2]>

(17) for the GaAs(001)

(18) system, (19)

L-M. Peng M.J. Whelan / RHEED from MBE growing surfaces

a j: j i

1.0”

2 0 0.8al

; 1 ; :; j

0) E

5 06-

0” $

0.4 -

I

/

:

z?

: : i’’ ‘ ; f ) ; i’ I ; i : / j : ; : ; / j ; / j i : j

I:

/

/

:

/

; ! j i’ ! / i ! i i ; !’ i i i’ ! / /

1

-

-t.~’ ... --------

1st layer 2nd layer 3rd layer 4th layer 5th layer 6th layer 7th layer 8th layer

-

1st layer 2nd layer 3rd layer . e 4th layer ---- 5th layer --- 6th layer -7th layer -8th layer --- 9th layer -- 10th layer

.._ ..+.

! .’

Monolayer

Deposited

..... ...s

1st layer 2nd layer 3rd layer . . . 4th layer ---- 5th layer --- 6th layer -7th layer -8th layer --- 9th layer -10th layer

1

2

honiaye:

Depisitelf

*

9

10

Fig. 1. Surfaceccwxagesof ten growing layers for (a) a perfect layer growth model and (b, c) a diffusive growth model with k = 15. III (a) and (b) ten monolayers

are deposited,

and in (c) the growth

is interrupted

after the deposition

of 6 monolayers.

L.-M. Peng M.J. Whelan / RHEED from MBE growrng surfaces

xexr [ -4m2(z

- z,,)‘/h,(As)]

r,

xexp-4n’(i-z.,-o,,4)‘,b,(Ga)]/.

(20)

Monolayer

deposited

In eqs. (- 8)-(20) ak and b, are the Gaussian parameters If Doyle and Turner [17], a, is the lattice cons ant of GaAs, 0, is the surface layer coverage as determined by (l), m and m, are the relativistic ; nd rest electron masses respectively, and the sun mation is over atom layers. Shown i 1 fig. 1 are surface layer coverages during epitz xy growth on an As-stable GaAs(OO1) surface. In fig. la a perfect layer growth model and in figs. lb and lc a diffusive growth model are used. :or the diffusive growth the filling parameter I has been chosen to be 15 so that the growth mot e is intermediate between perfect layer growth and three-dimensional cluster growth. Ten monolayers of Ga and As atoms are deposited in figs. la and lb. In fig. lc the growth is interrupted after six n lonolayers are deposited. The corresponding a Jeraged scattering potential just after the growth is interrupted has been plotted in fig.

Monolayer

deposited

Fig. 3. RHEED intensity oscillations for 20 keV fast electrons incident on a GaAs(OO1) surface, for perfect layer by layer growth (k = co). The incident glancing angles are (a) 53 mrad (at the kinematical 200 Bragg angle), (b) 30 mrad (at the low angle side of the 400 dynamical Bragg reflection peak), and (c) 54.23 mrad (close to the 400 dynamical Bragg reflection peak). 5 55

11’30

1695

22 60

28 25

Z-Axis (in angstrom) Fig. 2. The xriation of the crystal potential, averaged along the (001) su.face normal of a GaAs growing system, correspondin ; to fig. lb just after the growth is interrupted.

2. Fast electrons having 20 keV are incident at a perfect layer by layer growing surface and a diffusive growing surface in figs. 3a-3c and figs. 4a-4c

L.-M. Peng, M. J. Whelan / RHEED

Monolayer

0

1

2

3

4

Monolayer

Notably in figs. 3 and 4 the intensities of reflection minima are not zero for all incidence ranges from non-Bragg to Bragg incident conditions, and the reflected beam intensity does not necessarily decrease immediately after the growth is started, in contrast to the predictions of kinematic theory. Also it is amply demonstrated in these figures that the phase shift (see, for example, ref. [18]) acts as a rule rather than an exception. Detailed analysis and comparisons with other theories and experimental results will be given elsewhere. In conclusion, a practical method for analyzing diffraction data from epitaxy growing surfaces is developed. In our present model the MBE growing system consists of an As-stable GaAs(OO1) substrate and up to ten growing layers, and a typical calculation of RHEED intensity takes five seconds on a VAX 8800 system.

deposited

5

5

from MBE growing surfaces

7

8

9

deposited

The authors wish to thank Professor Sir Peter Hirsch, FRS, for the provision of laboratory facilities and for his encouragement of this work. One of us (L.M.P.) is supported by the UK Science and Engineering Research Council and Wolfson College, Oxford.

References Ul P.K. Larsen and P.J. Dobson,

Monolayer

deposited

Fig. 4. RHEED intensity evolution during MBE growth and after the growth is interrupted at six monolayer depositions, corresponding to fig. lc (k = 15). Fast electrons having 20 keV are incident at an As-stable GaAs(OO1) growing surface at (a) 53 mrad, (b) 30 mrad and (c) 54.23 mrad.

respectively. The incident glancing angles are 53 mrad in fig. 3a and fig. 4a, 30 mrad in fig. 3b and fig. 4b, and 54.23 mrad in fig. 3c and fig. 4c.

Eds., RHEED and Reflection Electron Imaging of Surfaces (Plenum, New York, 1988). 121J.J. Harris, B.A. Boyce and P.J. Dobson, Surf. Sci. 103 (1981) L90. 131 PI. Cohen, P.R. Pukite, J.M. Van Hove and C.S. Lent, J. Vat. Sci. Technol. A 4 (1986) 1251. T. Natori, T. Sakamote and P.A. Maksym, [41T. Kawamura, Surf. Sci. 181 (1987) L171. [51A. Ichimiya, Surf. Sci. 187 (1987) 194. 161 B.A. Joyce, J.H. Neave, J. Zhang and P.J. Dobson, in: RHEED and Reflection Electron Imaging of Surfaces, Eds. P.K. Larsen and P.J. Dobson (Plenum, New York, 1988) p. 397. [71 PI. Cohen, P.R. Pukite and S. Batra, in: Thin Film Growth Techniques for Low-Dimensional Structures, Eds. R.F.C. Farrow et al. (Plenum, New York, 1986) p. 69. J.H. Neave, B.A. Joyce, B. PI P-K. Larsen, P.J. Dobson, Bolger and J. Zhang. Surf. Sci. 169 (1986) 176. [91J.H. Neave, B.A. Joyce and P.J. Dobson, Appl. Phys. A 34 (1984) 179.

15.-hi. Peng, M.J. Whelan / RHEED [IO]

P.I. Cohen, G.S. Petrich, P.R. Pukite, G.J. Whaley and A.S. Arrott, Surf. Sci. 216 (1989) 222. [I l] L.-M. Peng and M.J. Whelan, Proc. R. Sot. A, in press. [12] J.D. Weeks and G.H. Gilmer, Adv. Chem. Phys. 40 (1979) 157. [ 1i] S. Kikuchi and S. Nakagawa, Sci. Pap. Inst. Phys. Chem. Res. Jpn. 21 (1933) 80, 256.

fromMBE

growing surfaces

[14] A. Ichimiya, Jpn. J. Appl. Phys. 24 (1985) 1579. [15] T.C. Zhao and S.Y. Tong, Ultramicroscopy 26 (1988) 1.51. [16] J.M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1975). [17] P.A. Doyle and P.S. Tuner, Acta Cryst. A 24 (1968) 390. [18] J. Zhang, J.H. Neave, P.J. Dobson and B.A. Joyce. Appl. Phys. A 42 (1987) 317.