2.
230 A.J. Steinfort et al. / Surface Science 409 (1998) 229–240
described with a pair correlation function. Good much smaller than unity, the intensity I(q) in the
agreement is found with electron diﬀraction data scattered beam is given by
[10–14]. However, X-ray reﬂectivity data from
KP K
multilayers are poorly described [15], as refraction 2
I(q)= f (r)p(r) e−iqr dr , (1)
eﬀects are ignored. Recent work by Holy et al.´
[16 ] presented an intensity calculation for multi- r
layers with vicinal interfaces in the distorted-wave with the scattering vector q taken in the vacuum
Born approximation. Comparison with reﬂectivity (i.e. outside the sample). The interfaces are
data obtained from miscut GaAs/Ga In As/ described with p(r) being deﬁned by the interface
1−x x
GaPAs multilayers resulted in a good agreement. proﬁles functions p (r) like
This conﬁrms the large impact of refraction eﬀects h
G
on the line proﬁle of X-ray scattering distributions 1 if r at interface h
p(r)=∑ p (r)= (2)
at small scattering angles. h 0 if r not at interface h
In this paper, we present a general expression h
for scattering from multilayers in the Born approx- The scattering factor f =f(r) for r in layer h is
h
imation, including refraction eﬀects. This approach related to the layer-dependent refractive index n
h
diﬀers from the kinematic scattering approach in by
that including refraction eﬀects, its validity still
l2e2
remains at small scattering angles. Compared to n =1−d =1− Nf . (3)
the distorted-wave Born approximation, the h h 2pmc2 h
expression is simpliﬁed as no position-dependent
The scattering vector q is expressed in the in-plane
electric wave amplitudes are included. However,
and the out-of-plane components. The in-plane
this also implies that the expression is not valid
component q is independent of the height position
for small scattering angles with incidence angles x
in the multilayer. The out-of-plane component of
and exit angles close to the critical angle for total
the scattering vector qh has to be evaluated in
external reﬂection. The expression has been formu- z
every layer, and is expressed in the layer-dependent
lated for two general cases. First, the case of
incidence and exit angles and the refractive index
interfaces containing two levels will be treated.
This description is applicable, for example, to according to
interfaces with an islanded morphology. The for- qh =n k (sin vh +sin vh ), (4)
mation of ripples or islands at multilayer interfaces z h 0 out in
has been observed for the case of SiGe multilayers where k is the wave vector in vacuum. The local
0
[17,18]. It has been shown to play an role in the incident angle vh and the exit angle vh from the
in out
relaxation of misﬁt-induced strain [18]. Next, an electric wave in layer h are calculated from Snell’s
expression for the scattering from miscut interfaces law; starting at the surface (h=0) with n =1 and
0
will be given. In the calculation, the layer-depen- v0 +v0 =2h, the scattering angle for given q is
out in
dent scattering vector is included, as well as a ﬁnite (q , q0 ). In this two-dimensional description, the
x z
in-plane and out-of-plane correlation length. The interface positions are given by z and the lateral
h
results are compared with experimental data direction is denoted as x. The roughness at each
obtained from Si/Ge Si multilayers. interface is described by steps with step heights ma
x 1−x
where m is an integer, as shown in Fig. 1a for the
case of m=1. The intensity in Eq. (1) can now be
2. Calculations rewritten as a summation over the interfaces h, k,
i.e.
The incident and exit angles are assumed to be
larger than the critical angle, so the relative ampli-
tude of the electromagnetic wave at each interface
I(q)=∑ f f e−iwhk
h,k
h k P ∑ C (u , u )
uz
hk x z
is approximately equal to unity. Under the assump- ux
tion that the amplitude in the reﬂected beam is ×exp{−i[q u +a(qh m−qk n)]} du , (5)
x x z z x
3.
A.J. Steinfort et al. / Surface Science 409 (1998) 229–240 231
Fig. 1. (a) Schematic representation of a multilayer system with two-level interfaces. The step height is a and the island lengths are
denoted by L . The expression for the correlation is evaluated in diﬀerent directions. (b) A grey-scale level representation of the
n
scattering intensity from a multilayer system as shown in (a) as a function of the in-plane and out-of-plane q and q .
y z
with u =a(m−n). The phase factor w results ability of ﬁnding two scatterers at interfaces h and
z hk
from the interface distance z −z . After correction k separated by a vector (u , u +z −z ). It can be
k h x z k h
for the optical path length, the expression becomes expressed as
G
k
w =
∑ qp d
p=h+1
h
z p
if h+1≤k
, (6)
hk x z
m
h P
C (u , u )= ∑ p (x, ma)p (x+u , ma+u ) dx.
k x z
hk ∑ −qp d if k+1≤h x
z p (7)
p=k+1
0 if h=k
In the following, an expression for the pair
where d =z −z is the thickness of layer p. The correlation function in a multilayer system with
p p+1 p
pair correlation function C (u , u ) gives the prob- rough interfaces will be derived. We start from the
hk x z
4.
232 A.J. Steinfort et al. / Surface Science 409 (1998) 229–240
expression for the pair correlation function from With the relative occupation of the lower level
a single surface derived by Lent et al. [10] which given by c, the dependencies between the in-plane
will be extended for multilayers. First, the rough- partial correlation functions c (u ) are given by
mn x
ness at the interfaces is described by two levels.
Next, scattering from a descending stepped inter- c (u )+c (u )=c,
00 x 10 x
face is given. The expressions are derived under c (u )+c (u )=1−c. (12)
the assumptions that no roughness is present on 01 x 11 x
the terraces. The step height is assumed to be With Eq. (12), the summation over the step levels
smaller than the thickness of the layer. m,n=0,1 in Eq. (9) is carried out. The resulting
The multilayer system is deﬁned by M interfaces, expression for the scattered intensity is split into
each having a two-level step distribution, as shown the specular part I (0, q0 ) and the diﬀuse compo-
spec z
in Fig. 1a. In this two-level description, the levels nent I (q), i.e.
diff
at each interface are indicated by m and n equal
I (q)+I (0, q0 )32 ∑ f f Y
to 0 and 1 for the lower level and the top level, diff spec z h k hk
h,k
respectively. The partial correlation functions
C (u ) are deﬁned by the probability of having h≤k
P
hk,ll+D x
a scatterer at interface h, level l and a scatterer at × du e−iqxux C (u )+I (0, q0 )d(q ), (13)
interface k, level l+D separated over u in the x hk,01 x spec z x
x
lateral direction. It is related to the pair correlation
where d(q ) is the Kronecker delta, and
function by x
C (u , Da)=∑ C (u ). (8) Y =cos(Q )+cos(Q )
hk x hk,ll+D x hk hk,01 hk,10
l −cos(Q )−cos(Q ). (14)
The intensity in Eq. (5) is now rewritten as hk,00 hk,11
The expression for the specular part of the intensity
P
2 is given by
I(q)=2 du e−iqxux ∑ ∑ f f
x h k
−2 h,kh m,n I (0, q0 )32 ∑ f f [2c cos(Q )
spec z h k hk,00
h≤k h,k
×[2 cos(Q )]C (u ), (9) h≤k
hk,mn hk,mn x
with +(1−c) cos(Q )]. (15)
hk,11
Q =a(qh m−qk n)+w . (10) The in-plane pair correlation function c (u ) is
hk,mn z z hk mn x
given by the sum over all step conﬁgurations and
In Eq. (1)a the correlation function C (u ) terrace lengths over a distance u . The number of
hk,mn x x
in the multilayer system is evaluated in diﬀerent steps over a distance u is denoted as n . An
directions. Laterally for u =0, the correlation x x
z example of a step conﬁguration with terrace
function C (u )=c (u ) is assumed to be inde- lengths L with n =6 is shown in Fig. 1a.
hh,mn x mn x 0…nx x
pendent of h (i.e. identical at every interface). This Evaluation of C (u ) results for positive values
is not unlikely, as the surface forms a template to hk,01 x
of u in [10]
the following deposited layer. In the z direction, x
PPP
the correlation function H with the correlation 2
hk C (u )3H G (u ) ∑ …
length j is deﬁned by C (u =0)=c (0)H . hk,01 x hk hk x
z hk,mn x mn hk nx=0
For u ≠0 and u ≠0, the correlation function
x z nx odd
C (u ) is assumed to be decomposable, as in
hk,mn x
C (u )=c (u )H G (u ),
hk,mn x mn x hk hk x
where c and H are as deﬁned above. G (u )
(11)
P
× T (L )T (L )T (L ) … T (L )
0,o 0 1 1 0 2 1,f nx
mn hk hk x
A B
represents the lateral distance j over which the nx
x ×d u − ∑ L dL dL dL … dL . (16)
proﬁles at interface h and interface k are replicated. x i 0 1 2 nx
i=0
5.
A.J. Steinfort et al. / Surface Science 409 (1998) 229–240 233
T (L) is the probability of ﬁnding a scatterer at exponential decay. The calculations are performed
0,o
level 0 at the origin a distance L before the ﬁrst on a multilayer system with four bilayers with
step, and T (L) is the probability of ﬁnding a island formation on the interfaces. The islands
1,f
scatterer at level 1 a distance L after the last step. have an average length of 700 nm and a step height
All T (L)s in between give the probability of ˚
of 4 A, and the correlation length in the surface
m
ﬁnding a scatterer at level m a distance l away normal direction is taken to be equal to the
from the preceding step. T (L) and T (L) are multilayer thickness. The in-plane correlation
0,o 1,f
written in terms of T (L) and T (L) as length is chosen to be 10 mm.
0 1
Because of the interfacial correlation, the diﬀuse
P
c 2 scattering proﬁle forms sheets in reciprocal space
T (L)= T (x) dx,
0,o L 0 through the multilayer Bragg positions located at
0 L
the specular line through q =0. The bending of
P
2 x
T (L)= T (x) dx, (17) the intensity sheets results from the position depen-
1,f 1 dence of the scattering vector, i.e. inclusion of the
L
layer-dependent refractive index. In the intensity
with
sheets, two maxima at each side of the specular
P
2 direction are visible. From their position and the
L = xT (x) dx, (18) in-plane width, the average size and deviation of
i i
0 the terrace width can be deduced.
with i=0,1 being the average terrace size of level In the same way, the scattered intensity from a
i. For negative u , the in-plane correlation function misoriented surface can also be described. The step
x height is given by m a with the corresponding step
is i
height distribution Z(m a). The pair correlation
C (u <0)=C* (u >0). (19) i
hk,mn x hk,mn x function which is valid for multilayers is derived
Taking an exponential decay as the expression for from the pair correlation function obtained for a
the in-plane correlation function G (u ), the total single interface [10], resulting in
hk x
diﬀuse scattering distribution ( Eq. (13)) is analyti-
cally solvable. With the modiﬁed terrace length
distributions
C (u , u )3G (u )H
hk x z hk x hk
T (u )d
G
0,f x ux,0
T ∞ (L)=T (L)G (u ), (20) 2 2
P P
hk,i i hk x 2 2 2
+ ∑ ∑ … ∑ …
the expression for diﬀuse scattering becomes:
nx=1 m1=−2 mn =−2
x L0=0 Ln =0
×T (L )Z(m a)T(L )Z(m a)T(L ) x
C
8c …
I (q)3∑ f f Y H 0 0 1 1 2 2
diff
q2 L h k hk hk
x 1 h,k ×Z(m a)T(L )d[u −L (n )]
nx nx x lat x
h≤k
×Re hk,0 x
G
[1−T ∞ (q )][1−T ∞ (q )]
hk,1 x
1−T ∞ (q )T ∞ (q ) HD , (21)
×d[u −L (n )] dL dL dL … dL ,
z vert x 0 1 2 nx H (23)
hk,0 x hk,1 x
with with
P
2 L Snx L + ma Snx m a
T∞ = T ∞ (L) e−iqxL dL. L (n )= i=0 i i=1 i , (24)
hk,i hk,i lat x
0 L 2+ ma 2
In Fig. 1b, an example of diﬀuse scattering
according to Eq. (21) is shown. The interface-to- ma Snx L − L Snx m a
L (n )= i=0 i i=1 i . (25)
interface correlation function H is taken as an vert x L 2+ ma 2
hk
6.
234 A.J. Steinfort et al. / Surface Science 409 (1998) 229–240
Inserting this expression into Eq. (5) results in and
C
1 2
I (q)3 ∑ f f H 2 cos(w ) Z(q∞ )= ∑ Z(m a) e−iq∞ mja,z (28)
diff q∞2L h,k h k hk hk z j
x j=−2
h≤k where q∞ and q∞ are the reciprocal vectors in the
z x
x∞ and z∞ directions, as shown in Fig. 2a.
G HD
[1−T ∞ (q∞ )][1−Z(q∞ )]
×Re hk x z , (26) In Fig. 2b, an example of scattering is shown in
1−T ∞ (q∞ )Z(q∞ ) the case of a multilayer including correlated vicinal
hk x z
with interfaces. The terrace width is 900 nm and the
˚
step height is 10 A. In contrast to scattering from
P
2 the islanded multilayer structure as shown in
T ∞ (q∞ )= T ∞ (L) e−iq∞ L dL,
x (27)
hk x hk Fig. 1b, the miscut structure gives rise to an asym-
0
Fig. 2. (a) Schematic representation of a descending staircase as present on a vicinal surface. The x, z and x∞, z∞ directions correspond
to the directions parallel and perpendicular to the physical and to the crystallographic surface, respectively. (b) A grey-scale level
representation of the intensity distribution from a multilayer system with interfaces, as shown in the upper ﬁgure, as a function of
q and q .
x z
7.
A.J. Steinfort et al. / Surface Science 409 (1998) 229–240 235
metric intensity distribution along the q direction, geometry, the surface normal is oriented in the
x
caused by the asymmetry at the interfaces. scattering plane and the anisotropic shape of the
In the description of the surfaces, no restrictions instrumental resolution function does not result in
are imposed on the size of the islands. In the limit a change in resolution for diﬀerent scan directions.
of the island or terrace size to inﬁnity, the pair The measurements were performed on a
correlation function C (u , u ) is a constant. The 4×(Si/Ge Si ) multilayer system on a vicinal
hk x z x 1−x
diﬀuse scattering for q ≠0 is zero and the intensity Si(001) substrate. The multilayers were grown by
x
is localised at q =0. In the other extreme, where molecular beam epitaxy under ultrahigh vacuum.
x
the in-plane correlation length reaches zero, the The substrate had a miscut of 0.45±0.02°, as
diﬀuse scattering intensity has a constant value for determined by a combination of optical alignment
all q . In practice, only a limited range in q is of the surface and the orientation of the Si(001)
x x
accessible because of shading eﬀects from the direction by X-ray diﬀraction. First, a 250 nm
sample. This range will set the lower limit to which thick Si buﬀer layer was deposited on the Si
spatial frequencies can be measured. The upper substrate. Alternate Si and Ge Si layers were
x 1−x
limit of spatial frequencies to be measured is given deposited at a deposition temperature of 550°C.
by the instrumental broadening or, in the case of In the case of a miscut surface, the surface and
high-resolution measurements, the spatial coher- interface characteristics cannot be regarded as
ence length of the X-rays, which is of the order of invariant under azimuthal rotation. The reciprocal
micrometers. q and q directions were deﬁned as being perpen-
x y
dicular and parallel to the steps on the surface.
The q direction was deﬁned as being perpendicular
z
to the optical surface. Scans were performed in
3. Measurements two planes in reciprocal space with q =0 and
x
q =0. The reciprocal planes contain full informa-
y
3.1. Experimental tion about the in-plane characteristics, both paral-
lel and perpendicular to the steps.
In Section 2 we showed that the in-plane charac-
teristics of the surface and interfaces can be 3.2. Presentation of the data
deduced from the diﬀuse scattering proﬁle. The
observed intensities are normally orders of magni- The layer thicknesses and Ge concentration, the
tude smaller in intensity than the intensity in the refractive indexes as well as the root mean square
specular direction. Because of the non-periodic (RMS) interface roughness were estimated from
nature of the interface roughness, the intensity is the out-of-plane specular reﬂectivity proﬁle. The
not as localised as the multilayer Bragg peaks in experimental proﬁle was compared with a theoreti-
the specular direction. cal description [19] to obtain the out-of-plane layer
The measurements were performed with a four- parameters. In Fig. 3, the data are presented which
circle diﬀractometer. The Cu K X-ray beam was were collected in the specular direction (circles).
a
taken from a standard X-ray source ﬁtted with a The theoretical proﬁle is shown as a solid line.
commercial tube running at 40 kV and 20 mA. The average RMS interface roughness is
The divergence of the primary beam was deﬁned ˚
7.4±0.1 A and the RMS surface roughness is
by slits and was set to 0.03°. The resolution was ˚
11.2±0.2 A. The Ge concentration x is 0.40 and
˚
set to dq=4.2×10−3 A−1 using a 0.25 mm slit in the Si and Ge Si ˚
layer thicknesses are 120 A
0.40 0.60
˚ , respectively.
front of the detector. The height of the slit was and 49 A
1 mm. A Ni ﬁlter was placed in front of the In Figs. 4 and 5 the experimental data in the
detector to eliminate the K contribution in the two q =0 and q =0 planes are shown. The data
b x y
wavelength spectrum. All scans were performed in were collected in the in-plane direction in 50 steps
the parallel scan mode by adjusting the rotational ˚
with a step size of 2.2×10−4 A−1. In the out-of-
angles v and w of the diﬀractometer. In this plane direction, the data were collected on 50
8.
236 A.J. Steinfort et al. / Surface Science 409 (1998) 229–240
multilevel interfaces. Parallel to the steps, a two-
level system is assumed at every interface to
describe the islanded proﬁle. The intensity proﬁles
are calculated using Eq. (21). In the direction
perpendicular to the steps, the interfaces are
described as descending staircases, with the result-
ing intensity calculated from Eq. (26).
The terrace widths are assumed to have a geo-
metric (exponential ) distribution according to
C D
1 −(L−L )
T (L)=T (L)= exp islands
0 1 dL dL
islands islands
for L>L (29)
islands
for the two level description, and
C D
1 −(L−L )
T(L) exp steps for L>L
dL dL steps
steps steps
(30)
for the miscut-induced stepped morphology. The
step height h is chosen to be constant (h ), leading
Fig. 3. The specular reﬂectivity curve from a 0
to the step-height distribution function
4×(Si/Ge Si ) multilayer. From the position of the
0.40 0.60
maxima and the relative intensity, the bilayer thicknesses and Z(h)=d(h−h ). (31)
the Ge concentration as well as the interface roughness are 0
established. The calculated intensities are convoluted with
the instrumental reponse function and corrected
˚
points separated by 1.4×10−3 A−1. The experi- for geometric factors arising from diﬀerent inci-
mental data show the clear sheet-like appearance dence angles. A satisfactory ﬁt of the calculated
of the intensity distribution. This indicates that intensity to the measured data is obtained by
the interface-to-interface correlation length is large varying only the distribution function parameters
compared to the bilayer thickness. The sheets are and a scale factor.
concentrated around the multilayer Bragg peak In Fig. 6, two line scans are shown parallel
positions. In the q direction the intensity is asym- and perpendicular to the steps at a height
x ˚
metrical around the specular direction. Only at q =0.163 A, which is through the fourth Bragg
z
one side next to the specular direction is an addi- peak of the proﬁle in Fig. 3. The data are repre-
tional maximum observed. This is in agreement sented as open circles. The central intensity
with the model calculations presented in Fig. 2b. maxima correspond to the specular intensity, and
In the perpendicular q direction, a symmetric the intensity distribution for q ≠0 and q ≠0 con-
y x y
intensity distribution is observed around q =0. tains information about the lateral roughness dis-
x
Next to the specular direction, two minor side tribution on the interfaces and the surface. The
peaks are visible. This indicates an islanded proﬁle symmetric proﬁle, shown as a solid line in Fig. 6a,
at all interfaces in the direction parallel to the steps. is the calculated intensity distribution according
to Eqs. (21) and (29), describing scattering from
3.3. Discussion stepped surfaces and interfaces. The optimal values
for the parameters as used in the calculations are
For further analysis of the data, line scans at given in Table 1. The island height h cannot be
0
constant q are used. The data are described with estimated from this proﬁle, since variation of this
z
9.
A.J. Steinfort et al. / Surface Science 409 (1998) 229–240 237
Fig. 4. Logarithmic intensity distribution from a 4×(Si/Ge Si ) multilayer with a miscut angle of 0.45° as a function of q and
0.40 0.60 x
q . The q direction is parallel to the miscut-induced steps. Note the sheet-like intensity distribution at multilayer Bragg positions
z x ˚
with a side maximum at q =0.00062 A−1.
x
Fig. 5. Logarithmic intensity distribution from a 4×(Si/Ge Si ) multilayer with a miscut angle of 0.45° as a function of q and
0.40 0.60 y
q . The q direction is perpendicular to the miscut-induced steps. The sheet-like intensity distribution at the multilayer Bragg positions
z y ˚ −1.
is symmetrical around q =0 with side maxima at q =0.001 A
x x
parameter only results in a uniform ampliﬁcation mainly inﬂuences the lateral width of the specular
of all points at constant q . In Fig. 6b, the scatter- reﬂection. No additional broadening is found apart
z
ing intensity distribution perpendicular to the steps from instrumental broadening. This means that
is shown: a clear side peak appears at the interfacial correlation length is equal or larger
˚
q =0.00062 A−1. This maximum is caused by scat- than the correlation length of the X-rays, and the
x
tering from terraces arranged as a descending interfaces are highly conformal. From line scans
staircase. The asymmetry is caused by the asym- in the q direction, the value for the out-of-plane
z
metric step geometry. The average size of the correlation length j has been determined to be
z
terraces determines the position of the maximum. 65±5 nm, which is equal to the total thickness of
The optimal parameters used to calculate the inten- the multilayer system (68 nm). The estimated
sity distribution are given in Table 1. The solid island height equals an RMS roughness of 7.5 A. ˚
line in Fig. 6b is the calculated intensity with the This is equal to the RMS interface roughness
optimal parameters. The correlation length j estimated from the specular proﬁle. Only at the
x
10.
238 A.J. Steinfort et al. / Surface Science 409 (1998) 229–240
˚
Fig. 6. Intensity distribution from a 4×(Si/Ge Si ) multilayer taken from Figs. 4 and 5 at q =0.163 A−1. The open circles are
0.40 0.60 z
the measured data and the solid lines are the best ﬁt. The directions correspond to (a) parallel to the steps and (b) perpendicular to
the steps.
Table 1
Optimal values for the Si/Ge Si multilayer interface parameters for the in-plane directions x in the step direction and y perpendicu-
0.40 0.60
lar to the step
x-direction perpendicular to the steps
L (nm) dL (nm) j (nm) j (nm) h (nm)
steps steps x 0
855±36 484±36 >104 65±5 1.3±0.3
y-direction parallel to the steps
L (nm) dL (nm) j (nm) j (nm) h (nm)
island island x 0
675±28 194±11 >104 65±5 –
The characteristic length scales are given by L in the x-direction and L in the y-direction. The correlation length in the
steps islands
x-direction and the z-direction are denoted by j and j , respectively. The height of the islands is denoted by h .
x z 0
˚
surface is a higher RMS value of 11.2 A found. As the step height of these terraces is probably of
This means that apart from island formation at the order of monoatomic distances, scattering from
the interfaces, no additional interface roughness is these terraces is not observed since the intensity
present. The islands are highly interface-to-inter- decreases rapidly with decreasing step height.
face correlated throughout the whole multilayer. The results were also compared with an atomic
Only at the surface is additional uncorrelated force microscopy (AFM ) image from the surface,
roughness present. which is shown in Fig. 7. It is clear that the AFM
In the case of ﬂat terraces the vicinal angle is image only provides information about the surface,
calculated from the terrace length and the step as the X-ray data are a result of all interfaces. The
height, resulting in a miscut angle of 0.1°. This is x and y directions as deﬁned by the diﬀraction
not in agreement with the observed value of 0.45°. experiments are indicated in Fig. 7. The average
This implies the presence of steps on the terraces, step direction is from the lower left to the upper
resulting in a vicinal angle of 0.35° on the terraces. right corner. In this image, 2D islands are clearly
11.
A.J. Steinfort et al. / Surface Science 409 (1998) 229–240 239
between the experimental data and the calculated
intensity proﬁles. Experimental data are presented
from a Si/Ge Si multilayer system. The sample
x 1−x
has a miscut of 0.45°. The interfaces of the miscut
sample contain islands with typical length scales
which are diﬀerent along and perpendicular to the
steps. Typical island sizes are 855 nm×675 nm,
with the short side parallel to the steps. The
standard deviation of the average island size in the
direction perpendicular to the steps is signiﬁcantly
larger (484 nm) than the deviation along the steps
(194 nm). This is also seen in the AFM image of
the multilayer surface. The morphology of the ﬁrst
interface (substrate–multilayer) is repeated at each
following interface, resulting in a large correlation
length in the surface normal direction. A large
part of the roughness at each interface is corre-
Fig. 7. AFM image from the surface of a 4×(Si/Ge Si ) lated. Only at the surface is uncorrelated rough-
0.40 0.60
multilayer. The surface miscut is 0.45°. In the inset, the miscut- ness present.
induced average step direction (x) is indicated. The grid in the
inset shows the avreage island-to-island distance as determined
by X-ray diﬀraction.
Acknowledgements
visible. The mean distance between the islands in We wish to acknowledge the expert technical
the y direction is about 660 nm, and the average assistance of C.W. Laman and R.F. Staakman.
˚
height diﬀerence is 6 A. In the perpendicular direc- This work was supported by the Netherlands
tion, the island size is about 900 nm with an Foundation for Fundamental Research ( FOM ).
˚
average height diﬀerence of 9 A. With the given
miscut angle of 0.45°, this conﬁrms the presence
of steps on the islands. The overall RMS roughness References
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