PHYSICAL REVIEW B VOLUME 54, NUMBER 11 15 SEPTEMBER 1996-I

Diffuse scattering from interface roughness in grazing-incidence x-ray diffraction

S. A. Stepanov,* E. A. Kondrashkina,* M. Schmidbauer, R. Ko¨hler, and J.-U. Pfeiffer
MPG-AG ``Ro¨ntgenbeugung,'' Hausvogteiplatz 5-7, Berlin D-10117, Germany

T. Jach
National Institute of Standards and Technology, Gaithersburg, Maryland 20899

A. Yu. Souvorov
European Synchrotron Radiation Facility, Boi te Postale 220, Grenoble Cedex F-38043, France
Received 12 March 1996
A theory of x-ray diffuse scattering from interface roughness in grazing-incidence diffraction GID is
presented. The theory assumes dynamical diffraction of x rays from perfect multilayers with the diffuse
scattering from roughness calculated in the distorted-wave Born approximation. This permits the calculation of
scattering due to roughness at all points on the diffraction curves, including the vicinity of the Bragg peaks. It
is shown that the measurements of diffuse scattering in GID can provide information on atomic ordering at
crystal interfaces which is not accessible by usual x-ray specular reflection and nonspecular x-ray scattering.
The theory is found to be in good agreement to the two GID experiments carried out with an etched Ge surface
and an AlAs/GaAs superlattice at the Cornell High-Energy Synchrotron Source and European Synchrotron
Radiation Facility, respectively. In the case of the etched Ge surface, an anti-Yoneda dip in the diffuse
scattering pattern at the Bragg peak and two symmetrical shoulders on the Bragg curve wings have been found
and explained. In the case of the AlAs/GaAs superlattice, the diffuse scattering has been separated from GID
by means of high-resolution measurements. A comparison between diffuse scattering in GID and diffuse
scattering in grazing incidence far from the diffraction conditions has shown that the atomic ordering was
preserved in the interface roughness, while it was partially destroyed in the surface roughness.
S0163-1829 96 05035-7

I. INTRODUCTION observed, while in Ref. 21 an attenuation of the tails with an
exponent similar to that found by Ne´vot and Croce24,25 for
The combination of Bragg diffraction and total external the tails of x-ray specular reflection curves was predicted.
reflection TER effects in grazing-incidence x-ray diffrac- We have assumed that discrepancies between Refs. 15
tion GID opens up a wealth of possibilities in the study of and 21 were due to the diffuse scattering of x rays from
thin surface layers of crystals.14 GID has been applied with surface roughness that was measured in Ref. 15 in combina-
success to investigations of the surface treatment and surface tion with the diffracted intensity of GID. Therefore, the ef-
oxidation of semiconductor wafers,5,6 to studies of structure fect of roughness on GID has become the subject of our
transformations during ion implantation,79 and to analysis study.
of strain relaxation in epitaxial layers1,10,11 and multilayers.12 In Sec. II a theory of diffuse scattering arising from sur-
The measurements of diffuse scattering DS in GID can face roughness as well as correlated and uncorrelated inter-
provide additional information on crystal lattice defects in face roughness in multilayers ML's is presented for GID.
surface layers.13,14 However, along with diffuse scattering Our model is based on the dynamical theory of GID in
caused by lattice defects, strong scattering due to surface and multilayers2628 and the distorted-wave Born approxima-
interface roughness can be expected in GID by analogy with tion.2932 The results are general for GID and extremely
x-ray TER studies, since the angle of incidence of the x rays asymmetric x-ray diffraction, and are applicable in the case
is equally small in both these cases.1519 Roughness can also of normal lattice strains in multilayers.
give rise to a change in the diffracted intensities.2022 Thus In Sec. III we derive expressions for the diffuse scattering
one has to distinguish between the effects of roughness and observed in GID with different experimental setups. In Sec.
crystal lattice defects on GID. IV the theory is applied to the explanation of Ref. 15, where
The effect of roughness on x-ray Bragg diffraction has not GID curves were taken from a sample with strong surface
been adequately explored. As shown in Ref. 23, in normal- roughness produced by etching.
incidence diffraction geometries the effect is relatively weak. In Sec. V some numerical examples demonstrating the
In GID, there are several theoretical predictions21,22 of a role of diffuse scattering in GID studies of semiconductor
strong effect of roughness on the coherent reflection. How- multilayers are given. The effect of the correlation between
ever, only a few experimental results20,15 are known, and roughness of different interfaces is discussed. Calculations
they are not in good agreement with the theory. In particular, are carried out for different experimental geometries.
in Ref. 15 an enhancement of the intensity of the tails of the In Sec. VI we describe a high-resolution GID experiment
diffracted beam of GID measured from a rough surface was carried out with an AlAs/GaAs superlattice at the optical

54 DIFFUSE SCATTERING FROM INTERFACE ROUGHNESS . . . 8151

and 35 or dynamical3 diffraction theory. The kinematical
theory is applicable i far from the Bragg peaks from perfect
crystals, ii for imperfect mosaic crystal structures, or iii
for GID from very thin layers 1-10 monolayers . The dif-
fracted intensities in these cases are weak and given by the
DWBA. Then the scattering from interface roughness can be
calculated in the kinematical theory either simultaneously
with the diffraction from layers1719 or in the second-order
DWBA. The dynamical theory is applicable at all points on
GID curves corresponding to bulk reflections from perfect
structures. In this case, a strong diffraction intensity at the
Bragg peak and the effect of diffraction on the incident and
specular waves are taken into account. That is just the case
for the both experiments discussed in Secs. IV and VI, where
the measurements are taken near the Bragg peaks of bulk
reflections. Therefore, we use the dynamical diffraction
theory in order to explain the peculiarities of diffuse scatter-
ing near the Bragg peaks of GID. For the reader's conve-
nience, a short outline of the dynamical theory is given be-
low.For the sake of simplicity, we neglect the changes in x-ray
FIG. 1. Schematic layouts of grazing-incidence x-ray diffraction polarization by diffuse scattering in GID. These changes can
for a a real incident wave Ein0 and b for an ``imaginary'' incident be simply added to the model, but they are too weak of the
wave Eout
0 which is inverted with respect to one of diffuse scattered order of noncoplanarity of GID, i.e., 10 3) to be of an
waves shown in a by a dashed line. The cones in a schematically interest for present-day experiments. Our derivations below
illustrate DS along the diffracted and reflected waves of GID. For are carried out for polarization. The equations are simply
the explanation of other vectors, see the text. extended for polarization by incorporating cos(2 B) in the
x-ray polarizabilities
beamline of the European Synchrotron Radiation Facility h and h¯ .
ESRF . These direct measurements of diffuse scattering in A. Some results from the dynamical theory
GID are compared to calculations based on our model. We of GID in multilayers
conclude with some possible uses of diffuse scattering in
GID and their distinction from similar scattering in total ex- In the framework of the dynamical diffraction theory, the
ternal reflection. wave fields Ein(r) and Eout(r) can be found as solutions to
the system of the dynamical diffraction equations in
II. THEORY multilayers.2628 Let us consider x-ray Bragg diffraction in a
multilayer see Fig. 1 . It is assumed to be a stack of N
In GID, the diffuse scattering can be observed along the perfect crystalline layers with laterally matched lattice spac-
directions of both diffracted and reflected beams see Fig. ing, and every layer can possess its own lattice spacing in the
1 a . We restrict our consideration to the scattering along direction normal to the surface: an n
z az az , where
the diffracted beam, since the majority of measurements are anz az , and index n denotes the number of the layer in
made in this manner.1,514,20,22 The analysis of the scattering the stack counted from the surface. This model corresponds
along the reflected beam of GID is analogous. to a so-called unrelaxed multilayer containing no misfit dis-
The x-ray diffuse scattering is due to the deviations locations.
(r) of the polarizability of the scatterer from an ``ideal'' At the first step, we assume flat interfaces between the
distribution id(r). The most effective way for its calcula- layers. For Bragg diffraction from a reciprocal-lattice vector
tion, in the lowest-order perturbation (r), is to apply the h approximately parallel to the surface, the polarizability can
reciprocity theorem.32 The amplitude of DS can be repre- be presented as the following sum over the polarizabilities of
sented as the layers:

N
f 2/4 Eout r r Ein r d3r, 1 flat r n r H z zn H zn 1 z , 2
n 1
where Ein(r) and Eout(r) are the wavefields in the ideal ob-
ject with flat interfaces produced by the incident x-ray n r n n n
0 h ehn*r e hn*r. 3
h¯
wave and the wave backprojected to the object from the ob- n n
servation point, respectively see Fig. 1 . The parameter is Here H(z) is the steplike Heaviside function, and zn are the
the magnitude of the wave vector of these waves in vacuum. coordinates of interfaces. Reciprocal-lattice vectors hn in the
The approximation 1 is commonly referred to as the nth layer differ slightly from the mean vector h because of
distorted-wave Born approximation DWBA . the variations in the normal lattice spacing between layers:
In case of GID, the wave fields Ein(r) and Eout(r) can be hn h hznZ, where hzn h, and Z is a unit vector
found with the help of either kinematical33 see also Refs. 34 along the internal surface normal.

8152 S. A. STEPANOV et al. 54

In dynamical Bragg diffraction, the x-ray wave field in The substitution of Eq. 10 into Eq. 9 adds the phase
each layer can be expanded over the sum of the transmitted factor to Dhnj corresponding to the expansion of the x-ray
and diffracted Bloch waves with wave vectors k0n and wave field over h instead of hn in 4 . Thus we can transfer
khn k0n hn , and amplitudes D0n and Dhn , this phase factor from 9 to 4 , and proceed to the expan-
respectively:3638 sion over h for the whole multilayer. At that, the dispersion
equations 8 remain formally unchanged because the expo-
En r D0neik0n*r Dhneikhn*r. 4 nents on the right side cancel each other.
The amplitudes D
The amplitudes D 0n j can be found with the help of the
0n and Dhn can be treated as constants boundary conditions for the x-ray waves and their deriva-
satisfying the dynamical diffraction equations in each layer: tives at each interface.24 It is convenient to present the
k2 2 boundary conditions in a (4 4) matrix form,9,2628
0n 0 nD n D
k2 D0n 0 0n h¯ hn , U
0n n SvEv S1F1 D1 ,
5
k2 2 S L D U D
hn h n 1F1 1 S2F2 2 ,
D nD
k2 Dhn h 0n 0 hn . 11
hn n . . .
Making use of the fact that the lateral components of all L U
vectors k SN 1FN 1DN 1 SNFN DN .
0n and khn coincide because they remain unchanged
at refraction and specular reflection, and substituting see Here Ev (E0 , 0, Es , Eh) and Dn (D0n1, D0n2, D0n3,
Fig. 1 a : 0z sin 0 , hz sin h , k0zn un , and D0n4) are the four-component vectors composed by un-
h n
zn n (1 az/a), we can rewrite equations 5 as known x-ray amplitudes E0 , Es , and Eh are the amplitudes
of the incident, specularly reflected and diffracted waves in
u2 n n vacuum; see Fig. 1 a . Parameters S
n sin2 0 0 D0n D
h¯ hn , v , Sn , and Fn denote
n (4 4) characteristic matrices of layers:
6
u n n
n n 2 sin2 h 0 Dhn h D
n 0n .
The condition 2 2 1 0 1 0
0 1 0 1
h 0 presuming the elastic scattering of Sv , 12
x rays, gives sin 0 0 sin 0 0
0 sin
sin2 h 0 sin h
h sin 0 2 , 7
where (2 0h h2)/ 2 is the standard parameter for the
angular deviation of the incident wave from the Bragg con- 1Vnj
dition in the dynamical theory of diffraction. Snj , 13
u
The values of u n j
n are determined by the dispersion equa-
tion which is the condition for the existence of a solution of Vnj unj
Eq. 6 : and F(U,L)
ni j ijexp(iunj z(U,L)), where indices (L) and (U)
indicate that the exponents are evaluated at the lower upper
u 2 n n n n
n sin2 0 0 un n 2 sin2 h 0 .
h¯ h boundary of layer, (L)
n n n (U)n 1.
8 The solution to Eq. 11 is straightforward:
Equation 8 is a fourth-order polynomial equation for un , E 1S 1S 1 S U D
and has four solutions. As shown in Ref. 39, there are always v Sv 1Ft1S1 2Ft2 * * * SN 1 NFN N , 14
two solutions corresponding to x-ray waves damping with where Ftnij ijexp( iunj tn) and tn z(L) z(U). After cal-
z Im(un) 0 , and two other solutions corresponding to the culating the matrix product in the right-hand side of Eq. 14 ,
waves growing with z Im(un) 0 . The latter waves are we obtain four linear equations for four amplitudes: Es ,
usually treated as being specularly reflected from the lower Eh , D0N1, and D0N2. The other amplitudes are given by
interfaces of the layers. For each of the solutions, Eq. 6 Eq. 11 . For the details of the numerical procedure, the
gives ( j 1, . . . ,4): reader is referred to Refs. 2628.

u2 sin2 n B. Amplitude of diffuse scattering from individual fluctuations
D n j 0 0
hn j n D0nj VnjD0nj . 9 in interface positions
h¯n Following the approach developed for x-ray reflection by
The polarizabilities n rough surfaces,2932 we describe rough interfaces between
h correspond to vectors h
n n which layers as local fluctuations z
vary from layer to layer. Therefore, it is more convenient to n( ) in the interface positions
( is the coordinate vector along interfaces . This approach
use the expansion over h see, e.g., Ref. 40 : is obviously valid at length scales much greater than the
interatomic distances. Then instead of 2 the polarizability
n n n
h exp h z . 10
n h z of the layers is given by

54 DIFFUSE SCATTERING FROM INTERFACE ROUGHNESS . . . 8153

N
rough r n r H z zn zn
n 1

H zn 1 zn 1 z , 15
Respectively, the fluctuations of polarizability can be
written as a sum over the interfaces,
N
r rough r flat r Pn r n r , 16
n 1

where P(r) is the steplike function introduced by analogy
with Refs. 30 and 31:
1, z zn,zn zn , zn 0
Pn r 1, z zn zn ,zn , zn 0
0 elsewhere
17
and
n r n n 1 n n 1
0 0 h h eih*r
n n 1 e ih*r. 18
h¯ h¯ FIG. 2. Four different diagrams of x-ray scattering from a fluc-
tuation of interface position shaded area in GID as given by the
Thus the perturbation of the crystal polarizability due to four terms in Eq. 25 . Coherent and diffuse scattered waves are
roughness possesses the form shown by thick and thin vectors, respectively. Horizontal lines
present the Bragg planes in crystal.
N
n r P n n n
n r 0 heihr e ihr . 4 4
n 1 h¯ 2 exp iQnijzn
19 f n 4 Enij
i 1 j 1 iQnij
Substituting Eq. 4 and Eq. 19 into Eq. 1 we also
assume2932 that the wave fields do not change considerably d2 eiQnij zn 1 eq* , 24
on the scale of the roughness. Then the wave fields of one of
two layers forming an interface e.g., of the lower one can where it is denoted
be used at both sides of the interface, and we obtain the
scattering amplitude E in out n in out n
ni j D0niD0n j h DhniDhn j h¯
N 2 N 4 4 z in out in out n
n zn D D D D , 25
f f hni 0n j 0ni hn j 0
n d2 dz
n 1 4
n 1 i 1 j 1 zn Q in out
ni j uni un j . 26
ei in out in out
0 0 i uni un j z Din in
0ni Dhnieih*r The four terms in Eq. 25 can be treated as shown in
out out Fig. 2.
0 heih*r h¯e ih*r D0n j Dhn jeih*r . i Term 1 corresponds to the Bragg diffraction of x rays
20 by the fluctuations of interfaces Fig. 2 a .
We are interested in diffuse scattering about the direction ii Term 2 corresponds to the triple Bragg diffraction:
of the grazing diffracted beam. Therefore, out in the incident wave is diffracted by ML, then diffracted at
0 ( 0 h),
and it is convenient to write h in the fluctuations, and diffracted by the multilayer again
Fig. 2 b .
in in iii Term 3 corresponds to the Bragg diffraction of the
0 0Br qin, 21 incident wave by the multilayer, and the small-angle scatter-
out in ing of the diffracted wave by the fluctuations Fig. 2 c .
0 0Br h qout, 22 iv Term 4 corresponds to the small-angle scattering of
q qin qout, 23 the incident wave by the fluctuations and the Bragg diffrac-
tion of the scattered wave by the multilayer Fig. 2 d .
where in0Br is the vector in the incidence direction, exactly One can see that, in all four cases, x rays are scattered
satisfying the Bragg condition. The terms in Eq. 20 con- through the large angle 2 B due to the diffraction from the
taining exp( ih*r) oscillate with at an atomic scale and Bragg planes. That is, the momentum transfer due to scatter-
can be neglected. Then, after carrying out the integration ing from roughness is small and the intensity of DS must be
over z, Eq. 20 is transformed to much greater than that in Refs. 41 and 42. In those cases,

8154 S. A. STEPANOV et al. 54

x-ray scattering through large angles corresponding to large
momentum transfers was measured from amorphous multi-
layers with rough interfaces in a geometry formally similar
to GID.

C. Cross section of diffuse scattering
from statistical roughness
Proceeding from Eq. 24 to the cross section of diffuse
scattering, we obtain

d 4 N 4 4
*
d f 2 16 2 EnijEn i j
n,n 1 i,i 1 j,j 1
ei Q *
ni j Qn i j zn Q *
ni jQn i j 1

d2 d2 eiq
eiQ *
ni j zn iQ
n i j zn 1 , 27

where . . . denotes averaging over random functions
zn( ) assumed to be Gaussian.
The application of the general formula43 exp( j jxj)
exp( FIG. 3. Different degrees of collimation in the scattering plane
jk j k xjxk /2), where j are constants and x j are
Gaussian random variables, to Eq. 27 gives of grazing-incidence x-ray diffraction: a , b , and c show single-,
double-, and triple-crystal experimental schemes, respectively. Co-
herent and diffuse scattered waves are shown by thick and thin
d 4 N 4 4
* * vectors, respectively. The incident and scattered beams are colli-
d S 16 2 CnijEnijCn i j En i j mated and analyzed in a direction normal to the plane of the figure.
n,n 1 i,i 1 j,j 1

d2 eQ * ( j 1 and 2 . This procedure can be called ``the specular-
ni jQn i j Knn 1 eq* , 28 reflection approximation.'' Below, we show that it has more
applications.
where C 2 2
ni j exp(iQnijzn nQnij/2)/Qni j , n is the rms
height of roughness, 2 2
n zn(0) , and Knn ( )
z III. INTEGRATED DIFFUSE SCATTERING
n(0) zn ( ) is a correlation function. The same cor-
relation functions proposed for diffuse scattering in x-ray IN DIFFERENT TYPES OF GID EXPERIMENT
TER should be valid here.29,31,32,44,45 In the majority of cases the incidence and the exit angles
Thus expression 28 for diffuse scattering in GID is for- of diffracted beam are controlled in GID experiments be-
mally similar to that in TER compare Refs. 3032 and 44 cause these angles determine the penetration depth of GID
47 . However, there are two essential differences. The physi- inside the samples. However, the angles in the Bragg plane
cal difference is that diffuse scattering in TER originates are not always controlled see Fig. 3 , and that provides
from 0 the fluctuations of the mean target density , while some averaging of the pattern and simplification of Eq. 28 .
in GID it is mainly due to h the crystal structure of the The general formula 28 is applicable in the case of
fluctuations . That means that diffuse scattering in GID pro- triple-crystal measurements only, when the incident beam is
vides information about the degree of crystal structure per- collimated and the scattered beam is analyzed in two planes
fection at interfaces. The mathematical difference consists of Fig. 3 c . That would be the most informative case, but the
the dependence of Enij and En i j on q, which is not the case low intensity of GID might impose serious experimental
in TER. Hence the DS pattern is far more complicated in limitations.
GID than in TER, and depends on four diffraction angles: the
angles of the incident and the scattered x-ray waves with
respect to the surface and their deviations from the Bragg A. Single-crystal scheme: No angular resolution
angle in the surface plane. An analytical integration of Eq. in the Bragg plane
28 over one of the components of q, like that used to re- Some authors48,27,12 have performed measurements where
duce the calculations in the case of diffuse scattering in the incident beam is collimated in 0 , but not in , and the
specular reflection, is thus not possible. diffracted waves are separated over their exit angles by a slit
One simplification occurs when DS is studied far from the or a position-sensitive detector PSD . This single-crystal
direction of propagation of the coherent diffracted beam scheme49 is based on Eq. 7 , where 2sin(2 B)( B)
where the Bragg diffraction of waves scattered by roughness 10 5 is proportional to the deviation of the in-plane angle
can be neglected. Then Dout out out2 n
hn j 0 and un j ( 0 0)1/2, from the kinematic Bragg angle B . The large width of
giving Dout
0n j as solutions to the specular reflection problem Bragg peaks for 0 and h 10 210 3 makes these

54 DIFFUSE SCATTERING FROM INTERFACE ROUGHNESS . . . 8155

measurements very convenient. However, the coherent re- in the Bragg plane, but the acceptance of the scattered beam
flection and diffuse scattered radiation are all counted to- is limited by an analyzer crystal.
gether see Fig. 3 a .
If neither the in-plane angle of the incident or diffracted IV. ANALYSIS OF A GID EXPERIMENT
beam is collimated, then waves with large Bragg deviations
dominate in the incident and scattered fans, and the solutions WITH AN ETCHED Ge SURFACE
to the specular reflection problem can be used for both The theory given in Secs. II and III has been applied to
Ein(r) and Eout(r). Thus the dynamical diffraction problem interpreting the results of the experiment in Ref. 15. In this
of GID need not be considered for the calculations of DS at study, carried out at the Cornell High-Energy Synchrotron
all. The experiment integrates over both the in-plane compo- Source CHESS , the grazing-incidence diffraction was mea-
nents of q, providing (x) (y)/ 2 in integral 28 . Thus one sured from 220 Ge planes at 1.55 Å. A double-crystal
finds scheme of measurements was used. The incident beam was
collimated in the incidence angle
d N 2 2 0 within 0.25 mrad, and
1 2 n in out in the diffraction angle within 0.0014 mrad. The sample was
d S16 2 Cnij hD0niD0nj
n,n 1 i,i 1 j,j 1 rocked through the Bragg angle B at fixed 0 4 mrad,
and the entire scattered intensity was collected over the take-
C n in out
n i j h D0n i D0n j * off angle h and over the in-plane exit angle.
The sample surface consisted of two different parts: a
eQ *
ni jQn i j Knn 0 1 . 29 high-quality polished part and an etched quadrant provided a
As follows from Eq. 29 , the DS measured in the single- spectrum of surface roughness. The measured GID curves
crystal scheme is completely determined by K for these two parts are presented in Figs. 4 a and 4 b . The
nn (0). In the
case where the roughness of different interfaces is com- curve taken from a smooth surface coincides well with the
theoretical calculations for the perfect case. A peculiarity of
pletely uncorrelated, K 2
nn (0) n nn , the measurements of both the experimental and theoretical curves in the perfect
integral DS provide the rms roughness height. case is the zero reflection coefficient to the left of the Bragg
In the case of small completely correlated roughness, peak, where the diffracted wave becomes surface trapped
where Knn (0) n n and the exponent in 29 can be ex- the angle
panded ( h becomes imaginary due to Eq. 7 . Contrary to
nQni j 1), formula 29 for the diffuse scattering the smooth surface case, the experimental curve for a rough
becomes very similar to that for the intensity of coherent surface exhibits two nearly symmetrical shoulders at both
GID calculated in the DWBA see, e.g., Ref. 22 : sides of the Bragg peak. These shoulders are obviously due
d N 2 to diffuse scattering at surface roughness, because the effect
1 S n in out of roughness on the coherent beam would appear to be a
d 16 2 n hD0niD0nj
n 1 i, j 1 Debye-Waller attenuation of the intensity on the wings.21
The diffuse scattering has been calculated with the help of
ei uin out 2 in out
ni un j zn n 2 uni un j 2 2. 30 Eq. 31 . In the case of only one interface and 0, it is
greatly simplified:
The expression for GID differs from 30 by the substitu- 2 2 Q*2 /2
tion of n d 2 3 e 2 Qi j
h instead of n out in in
h . The consequences of this 2 D *
analogy are discussed in Sec. V, where we give some nu- d S 16 2 D0 j
i, j 1 QiQj* Di
merical examples.
dx eQiQ*jK x 1 eqxx, 32
B. Double-crystal scheme: Partial angular resolution
in the Bragg plane where Dout 2sin out/(sin out uout), Q in uout),
In some more advanced double-crystal experiments, the 0 0 0 i (ui
uout (sin2 out in are the two solutions to the
incident beam is collimated in the Bragg plane, while the 0 0)1/2, and ui in
entire in-plane spread of the scattered beam is accepted Fig. dispersion equation 8 with Im(ui ) 0.
3 b . Then the ``specular reflection'' approximation can be Equation 32 was integrated over h , and renormalized
applied to Eout(r) and, additionally, one can average Eq. 28 to the reflectivity see Eq. 2.13 in Ref. 29 :
over the components of q normal to h . This procedure re- 1 d
duces the integral in 28 to a one-dimensional one, R 2 2
DS Ssin 0 0 d d h . 33
d N 4 2
2 3 in out
d S 16 2 CnijDniD0nj The correlation function was chosen in the Gaussian form:
n,n 1 i,i 1 j,j 1 K( ) 2exp( 2/ 2), where is the rms height and is
C in out the lateral correlation length of roughness. The integration in
n i j Dn i D0n j * 33 was carried out numerically.
The normalized calculated flux of DS for different is
dx eQ *
ni jQn i j Knn x 1 eqxx, 31 presented in Fig. 4 c . The shape of DS curves strongly de-
pends on the lateral correlation length of roughness: at
where q n n
x q* h / , and Dni D0ni h Dhni 0 The same greater correlation lengths DS is concentrated closer to the
situation takes place when the incident beam is uncollimated coherent diffracted beam of GID. The dependence of calcu-

8156 S. A. STEPANOV et al. 54

lated DS on is presented on Fig. 4 d . The intensity of DS
quickly grows with , while the shape of the curves is prac-
tically independent of in a wide range up to 10 Å. At
10 Å, the DWBA starts to diverge at B 0, where
the diffracted wave of GID is surface trapped. This diver-
gence is due to a small penetration depth of surface-trapped
x-rays see Fig. 4 e . The DWBA breaks down when the
x-ray wave fields undergo large changes on the scale of the
height of the roughness.
The roughness height for the case in Fig. 4 b measured
with a 3- m profilometer tip was 200 Å. This value was
obviously beyond the applicability of the DWBA. In particu-
lar, the DWBA fails to explain the attenuation of maximum
reflectivity at the Bragg peak in Fig. 4 b . However, taking
into account the weak dependence of the shape of DS curves
on , we can fit the shape of the curves at small , and then
extrapolate the data to higher where the DWBA diverges.
The fitting procedure was carried out in two steps: first,
the shape of the DS curve was fitted at B 0 where the
coherent reflection is zero see Fig. 4 a . Then the calculated
DS was added to the calculated coherent reflectivity attenu-
ated by some empirical factor ch in order to fit the maximum
of the reflection coefficient: R 2 2 2
total ch R coherent R DS .
The factor ch was introduced to account for the relative con-
tributions of coherent diffracted and scattered radiation when
the roughness was great.
The parameters of the fit presented in Fig. 4 b are
1600 Å and ch 0.4. The roughness rms height fitted at
B 0, where the DWBA does not diverge is 43 Å.
This value is consistent with the profilometer data, since the
rms roughness at 1600 Å need not be as great as the rms
roughness at 30 000 Å corresponding to the horizontal
resolution limit of the profilometer. The long-wavelength
roughness measured with the profilometer could not cause
the DS on the tails of the curve Fig. 4 b because the DS
corresponding to the long-wavelength roughness is strongly
concentrated near the Bragg peak. This roughness was prob-
ably responsible for the broadening of the experimental
Bragg peak in Fig. 4 b . If, on the other hand, we consider
the possibility that etching causes the crystal structure disor-
dering, then, the parameter h in Eq. 32 would be re-
duced by a static Debye-Waller factor, and the same intensity
of DS would be achieved with a greater .
Both the experiment and theory show a dip in the DS
pattern at the Bragg peak. In Ref. 15 this dip was supposed
to be due to a cutoff of the maximum roughness wavelength
to which the experiment should be sensitive because of the
limited coherence length of the diffracted beam ( 3 m .
The theory presented here contains no assumptions on the
coherence length of the source, and the interpretation of this
FIG. 4. Fitting of GID data measured in Ref. 15 Ge, 220 effect is different. As follows from Eqs. 31 and 32 , the
Bragg planes, 1.55 Å, and 0 4 mrad . Data are represented intensity of DS in the double-crystal scheme is approxi-
by dots, and fits by solid lines. a Smooth surface fit assumes GID mately proportional to the total intensity of the x-ray wave
with no diffuse scattering . b Rough surface fit is GID plus DS field illuminating the interface:
calculated with 1600 Å and 43 Å . c Calculated shape of
DS curves vs correlation length of roughness the dashed line indi-
cates the total x-ray intensity at crystal surface . d Intensity of DS d 2 2
2 in in
vs height of roughness the DWBA diverges at * in in
B 0 for d Di Di 0 D0i Dhi 2
i,i
20, 30, and 40 Å . e Penetration depth inside the crystal cal- 1 i 1
culated for two wave fields Din1,2 of GID at 0 4 mrad the
decrease in penetration depth at B 0 causes the divergence of in in in
0 E0 Es Eh 2.
the DWBA in d . 34

54 DIFFUSE SCATTERING FROM INTERFACE ROUGHNESS . . . 8157

The right side of Eq. 34 exhibits a dip at 220 , pro-
viding the minimum in DS see the dashed line in Fig. 4 c .
Thus, the observed effect has the same origin as the Yoneda
peaks50,29 in x-ray DS and the secondary emission yield ap-
pearing near the critical angle for TER. Yoneda peaks are
due to the enhancement of x-ray intensity at the surface near
the critical angle for total external reflection. Here we have a
dip in intensity at the atomic planes near the Bragg angle,
which is matched in this instance with the threshold angle for
total internal reflection with the angle , where h becomes
an imaginary quantity due to Eq. 7 and the diffracted wave
becomes surface trapped . This dip provides a minimum in
DS in Fig. 4 b as well as a minimum in the fluorescence
yield from lattice sited atoms which was observed in GID
standing wave experiment.51,52 Therefore, our case can be
referred to as an ``anti-Yoneda effect.''
At 0, the critical angle for total internal reflection
given by Eq. 7 may not coincide with the Bragg angle.
Then two dips in DS are predicted by our theory, the dip at
the critical angle for total internal reflection being stronger
than that at the Bragg angle. FIG. 5. Calculated x-ray reflectivity curves vs the takeoff angle
for the single-crystal scheme of GID. The curves are for a 20-period
AlAs/GaAs superlattice (t
V. DIFFUSE SCATTERING IN MULTILAYERS AlAs 125 Å and tGaAs 95 Å on a 001
GaAs substrate. The parameters of the calculations are as follows:
In the case of multiple and periodic rough interfaces, the 220 reflection, 1.5 Å; 0 0.3°, 5 Å, and 2000 Å.
effects of roughness become much more prominent. We have Curves 1 and 2 show DS for uncorrelated ( z 0) and completely
carried out calculations for a sample similar to that discussed correlated ( z ) roughnesses of interfaces, respectively. Curves 3
in Refs. 27 and 31: an AlAs/GaAs superlattice consisting of and 4 present the coherent GID reflection for flat and rough inter-
20 periods of 125 Å AlAs and 95 Å GaAs on a 001 - faces, respectively.
oriented GaAs substrate. The calculations assumed a 220
Bragg reflection of polarized incident x rays with GID measurements is due to fluctuations in the crystal struc-
1.5 Å and ture h , while the parameters of roughness given by x-ray
0 0.3°. The correlation function was cho-
sen in the form suggested by Ming et al.,44 reflectivity refer to the fluctuations of material density
0 . In the case where the crystal structure is destroyed by
K roughness, the data of TER and GID may disagree: the latter
nn n n e / 2e zn zn / z, 35 may exhibit a reduced intensity corresponding to a smaller
with a rms roughness height n 5 Å and a lateral correla- . Then, to isolate the diffuse component of GID, one has to
tion length 2000 Å. select scattered radiation in double- or triple-crystal measure-
The calculated intensity of DS for the single-crystal case ments.
of GID renormalized to a reflectivity as in Eq. 33 is pre- A comparison between the diffuse scattering in TER and
sented in Fig. 5 as a function of exit angle h . Curves 1 and GID can be used for investigating the crystal structure of
2 correspond to the uncorrelated ( z 0) and completely cor- rough interfaces. In our model, one can add a Debye-Waller
related ( z ) roughness of interfaces, respectively. One factor for h and h¯ in Eq. 25 describing the attenuation
can see that in both the cases the curves exhibit multilayer of polarizabilities due to disordering of the crystal structure
Bragg peaks at the same angular positions as the peaks of at rough interfaces. The value of this Debye-Waller factor
coherent GID curves 3 and 4 . Therefore, it is difficult to can be found from the difference in given by TER and
separate the diffracted and diffuse intensities. In the case of GID. Thus the diffuse scattering in GID can deliver a mea-
correlated roughness, the DS about the multilayer Bragg sure of crystal structure ordering at rough interfaces. This
peaks becomes much more prominent, and the intensity at information is not accessible by conventional x-ray scatter-
large h is comparable to the intensity of coherent GID. The ing techniques.
shape of the DS curve is very similar to that of coherent The calculations for the double-crystal scheme of GID are
GID, as given by 30 and well seen in Fig. 5, but the de- presented in Fig. 6 for noncorrelated a and completely cor-
crease in DS intensity with h is slower. These results may related b interface roughness in multilayers. Due to the
explain the discrepancies between the theory and experiment in-plane angular collimation of the incident x rays, for each
at high angles observed in the single-crystal measurements B the diffracted wave exits the crystal at the certain
of GID in multilayers in Ref. 27. angle h , as given by Eq. 7 . The exit angle of the diffracted
To distinguish the effect of DS on single-crystal GID wave at different B is traced on the maps by thick solid
curves, one might measure the parameters of roughness by stripes. The intensity at all other points on the maps can be
x-ray reflectivity and diffuse scattering in TER, substitute the attributed to DS. Thus a separation of coherent and diffuse
roughness value into Eq. 29 , and subtract the calculated DS scattering is possible.
from the measured GID curves. However, as follows from The maps in Fig. 6 clearly show the bunching of the DS
Eq. 29 and Fig. 2, the DS contributing to the single-crystal into resonance diffraction sheets RDS's for correlated in-

8158 S. A. STEPANOV et al. 54

h can exit the crystal at each B . Diffuse and coherent
scattering could be separated out in this case by means of the
triple-crystal scheme. The calculations of DS for relaxed su-
perlattices could be performed using Eq. 1 and the wave
fields of GID found in Ref. 53.

VI. MEASUREMENTS OF DIFFUSE SCATTERING
FROM AlAs/GaAs MULTILAYER
In order to provide a comprehensive test of our theory, we
have undertaken high-resolution measurements of GID from
a 20-period AlAs/GaAs superlattice SL . The GID experi-
ment has been carried out at the optics beamline BL10 of
ESRF.
The superlattice grown by molecular-beam epitaxy on a
001 GaAs substrate was characterized in the laboratory by
x-ray Bragg diffraction, x-ray specular reflection, and non-
specular x-ray scattering. A -2 scan near the 004 Bragg
peak did not reveal any strain relaxation, indicating that the
superlattice possessed a laterally matched crystal structure.
The thickness of the layers obtained from the fitting of this
scan was tAlAs (154 1) Å, and tGaAs (73 1) Å. The
x-ray-diffraction data also indicated a sample surface miscut
of ( 0.38 0.02)° along 110 . The x-ray specular reflec-
tion -2 scans confirmed the thickness of AlAs and GaAs
layers and revealed a surface transition layer with a thickness
FIG. 6. Calculated maps of diffuse scattering for the double- of (18 2) Å, probably due to natural oxidation. A fit to the
crystal scheme of GID experiments. The parameters of the calcula- specular reflection curve gave the rms height of interface
tions are the same as in Fig. 5. The equal-intensity map a is for roughness i (4 0.5) Å. Finally, the longitudinal and
uncorrelated interface roughness, and b is the same for completely transverse scans of nonspecular x-ray scattering at grazing
correlated roughness. Thick stripes marked by arrows show the po- incidence ( -2 scans with offsets of the sample and
sition of the coherent wave of GID. scans at fixed 2 positions of the detector, respectively were
carried out in order to determine the vertical and lateral cor-
terface roughness. This effect is completely analogous to the relation lengths of roughness. The longitudinal scans exhib-
formation of RDS's ``Holy bananas'' in DS during x-ray ited very sharp RDS maxima corresponding to a complete
specular reflection.31 The vertical black fringe at 0 cor- correlation between the roughness of different interfaces.
responds to the anti-Yoneda minimum in DS which we have Therefore, the data were fitted with the simple correlation
discussed above for scattering from a rough surface. function 35 assuming z . The transverse scan along
Another interesting peculiarity displayed by Fig. 6 b is RDS-8 (2 3.172°) and the corresponding fit are shown in
the appearance of RDS's at negative B where the dif- Fig. 7. The fit consists of two independent contributions of
fracted wave of GID is surface trapped and cannot exit the interface and surface roughness, which explain the central
crystal. This effect is surprising because the surface-trapped part of the curve and the Yoneda peaks at the wings, respec-
wave has a small penetration depth inside the crystal see tively. The fitted parameters are 4 Å, 3000 Å for the
Fig. 4 e , and one could expect only a few interfaces con- interface roughness, and 9 Å, 500 Å for the surface
tributing to DS at these angles. However, two types of x-ray roughness. The height of the surface roughness corresponds
wave fields are generally excited in the crystal under GID: to half of the thickness of the surface transition layer found
one wave can be roughly connected to the diffracted wave in by reflectometry. The lower-order RDS indicated a more
vacuum, and the other one to the incident wave. The angle of complicated spectrum of roughness. For example, the trans-
the latter wave to the surface is not small in our example, verse scan at RDS-5 (2 2.033°) was better explained by a
thus providing a greater penetration depth. This effect can be combination of the same surface roughness and a sum of
used for the experimental measurements of DS because the interface roughness with 4 Å, 4500 Å, 3 Å,
separation of the diffracted beam is unnecessary. We note a 3000 Å and 2Å, 1500 Å. However, in the first
small difference in the positions of RDS's at positive and approximation the parameters 4 Å, 3000 Å found at
negative B . RDS-8 were applied to modeling the results of the GID ex-
As long as the coherent and diffuse scattering can be dis- periment.
criminated with the double-crystal scheme of the GID ex- The experimental configuration of the GID experiment is
periment, the triple-crystal scheme Fig. 3 c is not of par- shown in Fig. 8. The grazing-incidence diffraction was mea-
ticular interest. The situation might change in the case of sured from the 220 AlAs/GaAs planes in the double-crystal
relaxed superlattices containing a distribution of lattice spac- scheme corresponding to Fig. 3 b , and similar to the experi-
ings along the lateral direction. In this case, Eq. 7 becomes ment at CHESS.15 However, in contrast to Ref. 15, the an-
inapplicable,53 and a fan of diffracted waves with different gular spectrum of the diffracted beam was analyzed with a

54 DIFFUSE SCATTERING FROM INTERFACE ROUGHNESS . . . 8159

FIG. 7. Transverse scan through the eighth-order resonance
sheet (2 3.172°) of x-ray diffuse scattering taken from an AlAs/
GaAs superlattice far from diffraction conditions. Experimental FIG. 9. Double-crystal rocking curve of GID taken from the
data are represented by circles, and the theoretical fit by the solid AlAs/GaAs superlattice at 0 0.5°. Dots represent experimental
line. The fit is the sum of contributions of completely correlated data, and the solid line is a theoretical simulation convoluted with
interface roughness ( 4 Å, 3000 Å and surface roughness the rocking curve of the five-reflection monochromator. The top
( 9Å, 500 Å . line is the magnified reflection coefficient for the specular x-ray
beam. Vertical marks indicate the sample positions where PSD
spectra of diffracted intensity were taken see Fig. 11 .
linear PSD, and the experimental setup was optimized in
order to provide a good separation of GID and DS. from a Ge wafer: the width of the Bragg peak corresponded
A Si 111 double-crystal monochromator was tuned to to the calculated value. The angular resolution of the PSD
1.40 Å. The crystals in the monochromator were slightly over the takeoff angle was 15 sec of arc.
offset in angle to suppress the third and higher harmonics.54 The front of the beam impinging on the sample was re-
The divergence of the x-ray beam in the vertical plane over stricted by the output slits S3 with vertical and horizontal
0 was mainly determined by the monochromator since the sizes of 0.04 and 0.2 mm, respectively. This provided an
vertical divergence of synchrotron radiation at ESRF is 1 illuminated area smaller than the sample size, and eliminated
sec of arc. The collimation of the beam in the horizontal edge reflections. The primary and the secondary slits S1 and
plane over was provided by five 220 reflections in a S2 restricted the beam front to 1 2 mm2 and 0.1 1
channel-cut Si crystal. The five-reflection collimator was mm2, respectively, and reduced the background in the ex-
used to suppress the tails of the x-ray beam and provide a perimental hutch.
better discrimination of GID and DS at the position-sensitive In the first step of the experiment, the PSD was replaced
detector. by a scintillation counter. The diffracted beam was recorded
The estimated parameters of the incident beam at the while scanning as in Ref. 15 with no separation of the
sample were: / 1.2 10 4, 0 6 sec of arc, and coherent and diffuse components. A second counter Si pho-
4 sec of arc. The last parameter is given with account todiode simultaneously recorded the rocking curves of the
of the dispersion effect caused by the difference in the Bragg specularly reflected x-ray beam. Figure 9 presents the mea-
angles of Si and GaAs 220 reflections. The horizontal di- sured and calculated rocking curves for the incidence angle
vergence was examined by recording a 220 rocking curve 0 0.5°. The parameters for the calculations are taken from
the laboratory data presented at the beginning of this section
and the algorithm is described in Sec. II and elsewhere.21
The experiment clearly demonstrates the high quality of the
superlattice: i the half-widths of the Bragg peak and the
superlattice peaks correspond to the calculated parameters;
and ii the rocking curve of the specular beam exhibits a
maximum at the first superlattice Bragg peak, thus proving
that the diffraction is dynamical.
At the same time, the reflectivity at the wings of measured
GID curve is noticeably higher than expected and is probably
due to DS. Figure 10 shows the map of DS calculated for the
FIG. 8. The scheme of the high-resolution GID experiment car-
ried out on the optical beamline of ESRF. The linear position- experimental conditions according to 31 . In contrast to the
sensitive detector PSD provided an angular analysis of radiation example given in Fig. 6 b , the DS is concentrated along the
scattered along the diffracted beam of GID. The intensity of the diffracted beam of GID because of a greater lateral correla-
specular x-ray beam was recorded integrally using a Si photodiode. tion length. However, a characteristic pattern of DS with
For the integral GID measurements presented in Fig. 9, the PSD periodic superlattice peaks in two directions is well devel-
was replaced by a scintillation counter. oped. The SL peaks parallel to the -axis are especially

8160 S. A. STEPANOV et al. 54

FIG. 10. Calculated map of diffuse scattering in GID
( 0 0.5°) for the AlAs/GaAs superlattice studied at ESRF. The
thick stripe marked by the arrow shows the position of the coherent
wave of GID. The theoretical curves in Fig. 11 are the sections of
this map drawn along the vertical axis at different B .

strong, and should be clearly seen in PSD observations
which cut the map along the h axis.
The PSD spectra have been taken at different
-positions of the sample as marked in Fig. 9. The results are
shown in Fig. 11. The curves in the figure are vertically
shifted with respect to each other as explained in the figure
caption. The experiment is compared to the theoretical
curves, which consist of four parts

I IF
Total h , IGID h , IDS h ,

ISurf
DS h , IBackgr . 36

Here IGID( h , ) RGID ( h) R5 ( h) ( )] is
the product of x-ray Bragg reflections from the sample and
the five-reflection collimator, respectively; ( h) is the
x-ray Bragg deviation for the sample given by Eq. 7 , and
( ) 2 sin(2 B) is the difference in for collima-
tor and sample due their angular misalignment corre-
sponding to a given PSD spectrum. The terms IIF Surf
DS and IDS
are the DS intensities for the interface and surface roughness,
respectively, calculated according to Eq. 31 with the pa-
rameters found from the laboratory measurements. Finally,
IBackgr is a constant term equal to the experimental back-
ground.
Different terms in Eq. 36 explain different peculiarities
of the experimental spectra in Fig. 11. The coherent reflec-
tion provides the floating peaks marked by arrows. The po-
sition of these peaks is given by Eq. 7 with ( ).
The strongest effect of coherent reflection is observed at
3 arc sec, where two coherent peaks are found. At
0, the contribution of coherent reflection is small and
invisible on the spectra.
DS due to interface roughness forms regular superlattice FIG. 11. PSD spectra of GID intensity vs exit angle at
peaks, with a maximum intensity around the position of the 0 0.5° and different B for the AlAs/GaAs superlattice. Dots
coherent reflection, as expected from the map presented in present experimental data, and solid lines are calculations with pa-
Fig. 10. At 0 and 65 arc sec, the integrated inten- rameters taken from the laboratory studies of the sample. Arrows
sity of DS peaks exceeds the integrated intensity of the dif- mark coherent peaks. The curves are successively shifted by 103 for
fraction peak itself, thus explaining the high-intensity wings clarity. The central curve is then reduced by a factor of 20 because
of the experimental curve in Fig. 9. It is worth noting that su- of its high intensity.

54 DIFFUSE SCATTERING FROM INTERFACE ROUGHNESS . . . 8161

perlattice peaks are observed for 0. This proves that the ness in crystals can give rise to x-ray diffuse scattering about
x-ray wave field in GID possesses greater penetration inside the diffraction beams of GID. The theoretical model is based
the superlattice, even though the diffracted wave is surface on the distorted-wave Born approximation29 and the dynami-
trapped. cal theory of GID by multilayers.2628 The model can be
Finally, the surface roughness provides a maximum in DS applied to asymmetric x-ray diffraction and to the diffraction
near the critical angle of TER at of neutrons. Expressions have been obtained for GID experi-
h 0.3°) for the data
corresponding to 169 and 313 arc sec, and 0. ments in different geometries. The effect of interface-
Here we have found that the intensity of DS calculated due interface roughness correlations on x-ray diffuse scattering in
to the surface roughness with 9 Å was a factor of 4 GID has been taken into account. It has been demonstrated
greater than observed in the experiment. The theoretical that in the case of periodic interfaces, interface-interface cor-
curves presented in Fig. 11 correspond to 4.5 Å, half of relations give rise to the formation of resonance sheets in DS
the value determined from the laboratory measurements. We similar to ``Holy bananas''31 in small-angle x-ray scattering.
suggest that the difference in the observed values is due to When applied to the analysis of GID data taken from a Ge
the distinction between diffuse scattering in TER and GID: crystal etched to provide a roughened surface, the theory
the diffuse scattering in TER is proportional to 2 obtains the observed diffracted intensity and diffuse scatter-
0 , while ing. The intensity shoulder on the experimental curve at
that in GID is mainly given by 2h , a measure of the atomic
ordering in roughness. The crystal structure at the surface B 0, where the GID intensity must otherwise be zero,
has been explained by diffuse scattering from the surface
could be partially destroyed by the oxidation, giving rise to roughness. A dip in the diffuse scattering near the Bragg
weaker DS in GID. Thus our experiment confirms the con- peak can be attributed to an ``anti-Yoneda'' effect.
clusion derived in Sec. V that the measurements of DS in High-resolution measurements of GID have been re-
GID may provide information on atomic ordering at inter- ported. This approach allowed the diffracted flux from an
faces which is not accessible by DS in TER. AlAs/GaAs superlattice to be resolved into GID and diffuse
While the agreement between our theory and experiment scattering due to correlated interface roughness. The experi-
is reasonable, some discrepancies can be attributed to the mental results are in good agreement with the theory.
simplified model of the correlation function used in the cal- Measurements of diffuse scattering in GID are sensitive to
culations. It is worth noting that no diffuse scattering due to atomic ordering in roughness, thereby providing information
crystal structure defects in the superlattice was observed. As which is not accessible by conventional small-angle x-ray
shown in Refs. 55 and 56, a peak of DS due to pointlike scattering.
defects could be expected at h 0 0.24°. Some discrep-
ancies between the theory and experiment at small h near ACKNOWLEDGMENTS
the critical angle of TER might be due to this kind of scat-
tering, but the scattering from interface roughness was the One of us S.A.S. is pleased to thank S. Sinha Argonne
major contribution. National Laboratory and V. Kaganer Institute of Crystal-
lography, Moscow for stimulating discussions. We also
VII. CONCLUSIONS wish to acknowledge the valuable advice from T. Baumbach
Institut Laue-Langevin, Grenoble and J. Ha¨rtwing and U.
We have presented a theory and experimental results on Lienert ESRF concerning the experimental scheme, and we
the effects of roughness in grazing-incidence diffraction by are especially grateful to R. Hey Paul-Drude Institute, Ber-
crystals. It has been shown that surface and interface rough- lin for the preparation of the AlAs/GaAs superlattice.

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