Diffuse scattering of x rays from nonideal layered structures
M. Kopeck$@
Institute of Physics, Czech Academy of Sciences, Na Slovance 2, I80 40 Prague, Czech Republic
(Received 8 July 1994; accepted for publication 22 November 1994)
A new theory of nonspecular x-ray scattering from layered systems with random rough interfaces
based on the distorted-wave Born approximation is presented. Calculations of the diffuse scattering
from a single gold layer and two W/Si multilayer mirrors has been carried out. The theory explains
the existence of maxima and minima in the angular distribution of diffusely scattered intensity
resulting from standing-wave-enhanced scattering and other dynamical effects. The influence of the
mutual correlation between individual interface profiles on x-ray scattering is discussed. 0 1995
American Institute of Physics.
I. INTRODUCTION The generalization of a scattering theory for layered
structures with rough interfaces is a very complicated prob-
Nowadays, thin layers and synthetic layered structures lem. The scattered field is a superposition of many plane
are of an increasing significance in applied physics. The waves (modes) and it is impossible to take the multiple scat-
physical properties of layered systems depend not only on tering of every mode within the system into account. There-
the composition and thickness of sublayers but can be fore a simplified model of interaction between electromag-
strongly affected by the quality of interfaces. netic radiation and stratified medium has to be considered.
Recent progress in the controlled deposition of ultrathin The development of the BA for the case of x-ray scat-
films has made it possible to fabricate nanometer-period tering from nonideal multilayer structures has been carried
multilayer structures which may be used as optical elements out by Stearns.' His solution is based on the so-called
for soft x rays.' It is well known that the specular reflectivity "specular field approximation." In this approach, the coher-
can be considerably modified by boundary imperfections. ent (specular) field within the multilayer is treated dynami-
Moreover, interfacial roughness can lead to a loss of contrast cally, including multiple reflection and extinction, and the
between specularly reflected and diffusely scattered radiation incoherent (diffuse) field is treated kinematically, i.e., the
which is detrimental for imaging systems. For these applica- total incoherent field is approximated by the sum of diffuse
tions, both the root-mean-square (rms) value and the lateral scattering from each interface.
characteristics of interfacial profiles are important. The de- More recently, the extension of the DWBA for layered
termination of interfacial roughness is therefore a necessary systems with rough interfaces has been performed by Holy
assumption for predicting scattering properties of multilayers et al.Y In their approach, the specular field within the layered
and, prospectively, for improvement of deposition technol- system with smooth interfaces has been used for calculating
the diffusely scattered intensity.
ogy- X-ray methods provide tools for investigating surfaces, tn this paper, a new theory of x-ray diffuse scattering
thin layers, and layered structures. There is a large body of from layered systems based on the DWBA is described. The
real specular field and the real transmission coefficient of the
work discussing the spectdarly reflected component of x-ray radiation diffusely scattered from inner interfaces are consid-
scattering.2*3 The reflectivity curves yield information on ered. By means of this theory, the previously measured
densities and thicknesses of sublayers and are frequently modulations in the diffusely scattered intensity are explained.
used for a rms roughness evaluation. Considerably less atten- The influence of mutual correlation between individual inter-
tion has been devoted to the diffuse component of the scat- face profiles on the x-ray diffuse scattering is discussed.
tering, which depends on the rms value as well as on the
correlation function.4-8
Several theories of nonspecular scattering from a single II. THEORY
random rough surface have been developed. The Born ap-
proximation (BA) and the distorted-wave Born approxima- Let us consider a plane wave ET with wave vector `kr
tion (DWBA) are mainly used in the x-ray region.4 The BA incident on a system of N- 1 layers deposited on a thick
is valid at glancing angles much greater than the critical substrate. This wave is specularly reflected and refracted (co-
angle 8,. whereas it breaks down as we approach the total herent field) and simultaneously diffusely scattered (mcoher-
external reflection region. On the contrary, the DWBA is ent field) from each interface.
suitable for glancing angles close to tic. However, at larger A, The coherent field
glancing angles, the expressions for the differential cross We start by determining the coherent field within the
section derived in the DWBA reduce to the results in the layered structure which may be in each medium i expressed
BA." as the superposition of two plane waves ET and Ef (Fig. 1).
The continuity of the tangential components of the electric
a)E-mail: kopecky@fzu.cz and magnetic vectors at the boundary i may be written as"
2380 J. Appl. Phys. 77 (6), 15 March 1995 0021-8979/95/77(6)/2380/8/$6.00 Q 1995 American Institute of Physics
Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
Ri=EPIET is the reflection coefficient of the system below
air (vacuum I 2 the boundary i. The reflection coefficient ~i,i+l of this rough
medium interface interface is related to the Fresnel reflection coefficient Ti,i+,
of a smooth interface byi
ri,i+ 1 =ri,i+l Xi(J&ZTL). (5)
?+=ll `x xi denotes the one-dimensional characteristic function of the
rough interface i. Vector q,(O,O,2'&) represents the differ-
=2 ence of wave vectors of waves EF, ET. The set of equations
(4) may be solved by starting at the bottom medium N+ 1,
where RN+ i = 0 (since the thickness of the substrate is as-
sumed to be infinite).
Further, the coherent field may be determined using
formulas13
E;(8A=Ri(8,W:(8,), (6)
i+2 `i*l
Eir+O,j= l+Ri~~f:$f$)l,e,, goi+1(4.WT(ed.
N E;: v E::
\ N- (7)
ZN
N+l The calculation is now carried out from top to bottom.
' ET
N+l
substrate
8. The diffuse x-ray scattering from a single rough
interface
FIG. 1. Scheme of the coherent field within stratified medium. As the second step, the scattering of the coherent field
from each rough interface has to be evaluated. Let us con-
sider rough interface (surface) i between two homogeneous
1 media i and i+ 1. The deviation of this interface from the
z$(el)+Ef(elj= @Au average reference plane z = zi is described by the profile
9i+l(ed function h,(x,y).
(14 The electric field for x rays poIarized perpendicular to
the plane of incidence satisfies the wave equation
V2~(r>+k~~(r)-V(r)~,(r)=0, (8)
where r is the position vector and the scattering potential
V(r) is related to the refractive index n(r) by
V(r) =k$l -n'(r)]. (91
where `ki is the wave vector of the wave ET, 4oi the ampli- The polarization of x rays in the plane of incidence is not
tude factor corresponding to the perpendicular depth d, , discussed since at grazing incidence the results are the same
for both the polarization components.
qi(O1)=e-iLki~di for i=Z,...,N, (2) Let us assume that the real rough interface represents a
and ~+++,=l. small perturbation from the smooth interface, for which the
According to SnelI's law, the wave vectors `ki are deter- exact eigenfunctions are known. Then it is convenient to split
mined by the refractive indices ni of individual layers, wave- the scattering potential into two parts:
length A, and the glancing angle 0i of the incident wave: V(r) = VI (4 + V2(rL (10)
Ikix=`kl, 9 (34 where V1 is the scattering potential of the system with a
smooth interface and V, is regarded as the perturbation due
`kiv=lkly, CW to the roughness:
`ki,=-ko@-c0S2 Bl=-`<i, (34 for Z>Q, (11)
where k,,= 27~0~ is the magnitude of the vacuum wave vec- for :<zi
tor. By dividing the difference of Eqs. (la) and (lb) by their and
sum, the well-known recursion formula'O*" may be obtained: k$+n;+&
for zi<z<zi+hi(x,y),
Ri+lCe,)cp~+,l(el)+Fi,i+l(el)
&(4)= v,= -kz(nF-nF+,), for zi+hi(x,y)<z<zi, (12)
Ri+1(e1)50i2+1(e1)vi,i+l(e,)+
1 ' (4) 0, elsewhere.
J. Appt. Phys., Vol. 77, No. 6, 15 March 1995 M. KopecG 2381
Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
incident
i
i+l
medium i+l
ni+l 2i;I. ,'
,;:, `\
/-
/
FIG. 2. X-ray scattering from random rough interface i. The incident wave
(`ki) is scattered into the same half space cki). FIG. 3. X-ray scattering from random rough interface i. The incident wave
(`k;) is scattered into the opposite half space (`kLki+ 1).
Let the plane wave
14=ei'kir (13) (21/11V211~)=k~(~~-~~+l)ti,i+l(1ki)ti,i+1(-2k~j
with the wave vector `ki fall on the boundary i. According to xpi(%+l)* (17)
the Fresnel theory, the eigenstate for the smooth interface is The functions F, are given by
e'lki'+ri,i+l(lki)e i'k!r 1 , for Z>Zir
`*w= (14) & dy(e-iPi+lzktx~Y)- 1)
ti,i+l(lkijei'kf+~`, for Z<Zi)
where `k' xe--i(w+qyy)
i 7 `k+ 1 are the wave vectors of the specularly 08)
reflected and transmitted waves. The amplitudes of these
waves are given by the Fresnel reflection and transmission and the wave-vector transfers
coefficients Ti,i+ 1 and ti,i+ 1, respectively. qi=2ki-1ki. 09)
Sinha et al." define another eigenstate for the smooth
interface, Note that the tangent components of all the vectors qi are the
same (qir=qxr gi,=q,). The integration in Eq. (18) is car-
~~zki'+r~i+l(-2ki)ei2k~r, for z>z~, ried out over the illuminated area SO of the size L,L, .
"i&r) = * (19 In a similar way, the field diffusely scattered from the
i t&+`( -`kijeiLki+lr, for ZCZi, interface i into the medium i+ 1 may be derived. Then the
which is a time reversed state for the incident plane wave eigenstate `4 is chosen as the time reversed state for the
with the wave vector --`ki (Fig. 2). wave incident on the interface i from the medium i+ 1 with
The T matrix for scatteriilg between states `ki and `ki the wave vector `ki+l (Fig. 3):
(proportional to the scattered electric field) has in the DWBA i'ki', for z>zi,
the form'" 2$crj= ti*,l,i(-`ki+l)e
e'2ki+~rfr~+l,i( -2ki+l)ei2k[+l', for z<zi,
(21T11)=(2~1V111~)+(2~1V211(//). (16) i w
The diffuse scattering is the consequence of the perturbation leading now to the perturbation matrix element13
potential V2 so it is related to the second term on the right-
hand side of Eq. (16) only. wlv2111cr)
Substituting expressions (12), (14), and (15) for V,,
' @, and `I$, respectively, the perturbation matrix element can =~~(~~-~~+~jti,i+l(`ki)ti+l,i(-2ki~~jFi(qi+1).
be evaluated as (21)
2382 J. Appl. Phys., Vol. 77, No. 6, 15 March 1995 M. KopecG
Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
C. The diffuse x-ray scattering from a layered system ruys." Further, the field outside the layered system produced
For the evaluation of the diffuse scattering from layered by these "SOUX~S" has to be determined. In contrast to the
systems, the scattering of the waves ET(&), kinematical approximation, the total incoherent field is not
EF+,,(I~I)~~+~(
01) from each interface i both into the me- expressed as the sum of the waves scattered from each inter-
dium i and i+ 1 has to be taken into account. The wave face but the multiple reflections and thus the real transmis-
vectors ski of the scattered waves are again determined by sion are included in the calculation. Consequently, besides
the observation angle 19,. Let us denote the pertinent matrix YTR and YTR, the terms ..YT', YfT have to be considered.
elements (2$/V211(/1> by .YFR, YT' (the scattering of the In the medium i, the sources Yi'_`, , YfT,, Y"TR, .55fR
wave ET) and YPR -i"f' (the, scattering of the wave produce the electromagnetic field, which is a superposition
EP+, (pi+:). It follows'from Eqs. (17) and (21) that of two specularly related waves 99:, @. The boundary con-
ditions for the interfaces i- 1 and i yield
~~R(ql)=ET(Bl)k~(a~-n~+l)
Xt i,i+l(e,)t,,i+l(e2)Fi(15i+l+2~i+1),
(224
2 2
.Y;T(qJ=ET(e*)kO( I
n. -4+,)
-tYiT_T*(q& (25)
Xti,*+l(b)l)ti+l,i (e2)Fi(-`~i+1+25i+l)r
(22b) where Qi are the reflection coefficients of layered systems
above the interface i. We can determine Ri( 0,) from Eq. (4)
~~~ql)=--EP+1(~l~cpi+l(e1)ko2~~~+l-~?) and Qi( 0,) from the similar recursive formula
xti+*,i(~l)ti,i+1(B2)~i(15i-2~i), (224 Q,(e )= Qi-l(e2)(P~-l(e2)+Y"i,i-l(e2)
~~T~~l)=-~~+l~elj~i+l~eljk~(~~+l-~~~ 1 2 Qi-l(e2)cp~-l(e2)Ti,i--l(e2j+l. (26)
xt~+l,i(e,>ti+l,i(e2jFi(-1~i-2~ij, (224 The values of Qj are calculated in the direction from top
where the tangential components qX, q, of vectors qi , qi+ , (Q, = 0) to bottom.
are omitted in the argument of functions Fi. The variables If we express the term ti from Eqs. (24) and (25),
`ci are defined analogically to `& as multiply it by the transmission coefficient Ti( 0,) for the sys-
"ci=ko tem above the interface i, and carry out the summation over
(23) all interfaces, the final matrix element Z?& corresponding to
If the multiple diffuse scattering is neglected, the vari- the total diffusely scattered field outside the layered structure
ables Yi may be considered as "independent sources of x may be obtained:
I
@i(ql)=; Ti(B&Z9f(ql)=$ Ti( 0,) ~i(~2)~i(~2)~~_Tl(ql)+~~~~l(q~)l+~R(ql)+.~(ql)
i=l i=l 1 -Ri(e2)Qi-l(e2j~o?(e2j (27)
The transmission coefficients Ti( 0,) may be determined cients Ti( O,), jGi, has to be performed starting by the wave
in the following way: Let the plane wave I$+ t (6,) in the not in the substrate but in a layer closer to the surface.
substrate pass through the system of layers toward the sur- By substituting Eqs. (22a)-(22d) for .Y<? the term ?&u
face. By analogy-f0 Eq. (7), the amplitudes of the waves may be rewritten in the form
Ef( 0,) in each medium may be expressed using the recur-
sive formula
%(qlj=~ [ai(q1)Fi(`~i+l+2~i+l)
~$7 0,) = l+Qi(e,j i=l
l+Qi-1(e2)p~(e2) 4oi+lm~~+lw. 63)
From Eq. (28), all the transmission coefficients T~( e,) may
be calculated as
where CLi, pi, yi, and Si are the following coefficients:
(29)
If the amplitude of the wave Ef( 0,) in a medium i is too Tit 4) (31aj
small, the calculation of the remaining transmission coeffi- X1-Ri(e2)Qj-l(e2j(P~(e2)'
J. Appl. Phys., Vol. 77, No. 6, 15 March 1995 M. KopecG 2383
Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
1 describe smooth hills and valleys, whereas values of H
approaching 0 characterize extremely jagged interfaces. The
correlation function is exponential for H = i and Gaussian for
(3 lb) H= 1. The parameters cij and 7ij are defined by means of
the rms roughness oI and the correlation length 7i of indi-
vidual interfaces as
CT;+CT,"
(3lc) 2
(J-..=------- e -1zpjll7,
`I 2
and
7ij' 7i7j.
r (39)
(31d) The parameter T; describes the tendency of individual layers
to replicate the substrate surface (vertical correlation). For
The diffuse component of the differential cross section4 simplicity, rz is supposed to be the same for all layers of the
system.
c%s%) - woM%)
1695-z ' (32)
Ill. RESULTS AND DISCUSSION
is then In order to demonstrate results of the scattering theory
presented in the previous section, the diffuse x-ray scattering
(Cu K, radiation) from a single layer and periodical multi-
layers has been calculated. The Gaussian random rough in-
XFij(1Si+l+25i+l,`~j+1+2~j+1)+...l, terfaces and the exponential shape of correlation functions
have been assumed.
(33) Three scanning modes were considered to map out the
distribution of the diffusely scattered intensity in the recip-
where the symbol ( ) denotes the process of averaging across rocal space.t6 In the 8,
the random rough interfaces. The term in parenthesis consists mode (transverse scan), the sample
rotates and the scattering angle et+ cl,=20 is kept constant.
of 16 items, which differ only in the multiplicative factors The angle w of the sample rotation is given by w=(e2- &)/2.
and in the arguments of the functions Fi, defined as In the 0, mode, the angle of incidence remains fixed and the
Fij(K1 ,K*)=(Fi(KI)F~(KZ))-(Fi(KI))(Fj*(K2)). angular distribution of the scattered intensity is scanned. Fi-
(34) nally, in the offset (0,2t9) scan the sample and analyzer are
moved in such a way that the difference of angles 0, and 0,
Equation (33) is the general formula describing the rela- is conserved.
tionship between the diffuse component of x-ray scattering The theoretical 8t scans calculated for a gold layer on a
from the layered system and the topography of the interfaces. silicon substrate are shown in Fig. 4. As follows from de-
The shape of function Fij depends on the distribution of tailed numerical analysis, the contribution from the upper
interface heights h,(x,y). interface predominates in the nonspecularly scattered inten-
If h,(x,y) are Gaussian random variables with standard sity and the correlation between the substrate and gold sur-
deviations oi, the function Fij may,be expressed in the form face profiles has no appreciable effect on these scans. It is
LL, noteworthy that the diffuse scattering exhibits local maxima
-(KfO;+K2 %312
Eiq(Kl,K2)= - C? (known as Yoneda peaks)t7 if either 0i or 0, is equal to 8,.
K,K; This effect is caused by a standing wave with a maximum
located at the surface, resulting in an enhanced diffuse scat-
X dX dy(eKIK,*CdX,Y)- 1 jei(qd+qyY), tering. Moreover, subsidiary maxima and minima may occur
in the scattered intensity. These fringes arise due to the varia-
(35) tion of the surface coherent field produced by the interfer-
ence of the incident wave and the wave specularly reflected
from the substrate. The differential cross sections are calcu-
C*j(X~Y)~(hi(x+X9Y+
Y)hjix,Y)) (36) lated in two ways. The former approach includes the influ-
ence of interfacial roughness on the local coherent field ET,
is the correlation function between height fluctuations of the Ef which affects the diffusely scattered field (30) through the
interfaces i and j. We assume isotropic self-affine coefficients (31). On the contrary, the latter approach corre-
interfacesi described by the correlation functions sponds to the use of exact eigenfunctions for the ideal lay-
Cij(R) = (+fje-(R17ij)2H, 07) ered structure for working out the perturbation theory
(DWBA).' The difference between these two curves is more
where the exponent H determines the texture of the rough- significant at larger angles [Fig. 4(b)]. Here, the reflection
ness and takes values between 0 and 1. Values of H close to from the substrate is suppressed by the roughness, the modu-
2384 J. Appl. Phys., Vol. 77, No. 6, 15 March 1995 M. Kopeck$
Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
When the wave vector of the incident or scattered wave ful-
(a> a1 fils the Bragg condition," the sharp maxima arise. However,
satellite minima may sometimes occur in these positions.
-g IO3 +02 = 1.50 deg.
This somewhat surprising but already experimentally ob-
-2.
6 IO2 served phenomenon13V16*21
can be explained within the
3z framework of the theory presented above. When one of the
2 IO' angles 0,) a, approaches the Bragg angle, not only are the
G interference effects and the coherent field modulations im-
2
- IO0 portant but, in addition, the transmission of the incident wave
or the waves scattered from individual boundaries exhibits a
10-l 4 local extreme. Whether the increase or the decrease of the
-0.6 -0.3 0.0 0.3 0.6 transmission occurs it depends particularly on the ratio of W
and Si in the bilayer and on the number of periods. That is
Q [deg.] why no decision-which effect is dominant and whether the
IO' -3 scattering in a given direction is enhanced or attenuated-
(b) may be done without carrying out the numerical calculations.
91 +*2 = 3.00 deg.
2 IO0 For uncorrelated roughness, rZ-+O, CijiO if i#j and
-is the sum in the relation (33) for the diffuse component of the
I IO" differential cross section has only N nonzero terms. On the
32 other hand, if the roughness is fully conformal, r,--+m and
$ lo-2 all terms contribute to the total diffuse scattering. Therefore
the nonspecular intensity is usually in the former case much
8 smaller. That is why the vertical correlation is very undesir-
,o-3 able at layered systems used as image elements, because it
causes the contrast degradation of the final image. But the
0.0 opposite situation may also occur and the incoherent scatter-
01 [deg.] ing from the layered structure having uncorrelated interfaces
may be comparable or even higher [Fig. 6(c)] than for the
FIG. 4. Theoretical transverse scans for the gold layer (dAu=20 nm) on the identical system. The reason why the multilayer with corre-
silicon substrate. The differential cross sections calculated using the coher- lated interface profiles exhibits a very low diffusely scattered
ent field within the stratified medium with rough (solid line) and smooth intensity at the third-order Bragg maxima is the fact that the
(dashed line) interfaces are compared. The rms roughness ~~2.0 nm and the third order is nearly structure factor forbidden. Another in-
correlation length ~0.1 ym are assumed at both interfaces. teresting feature is also that incoherent scattering from sys-
tems with completely uncorrelated roughness may show fine
lation of the coherent field is therefore negligible, and the structure but this is mostly less apparent. This can be ex-
fringes disappear. Only the calculations starting from the co- plained by the effect of primary extinction in the vicinity of
herent field within nonideal stratified medium give realistic Bragg angles22 (reduced or enhanced transmission if the
results in this case. nodes of the standing wave pass through the W or Si layers).
Figure 5 depicts the influence of the vertical
correIation16~`s~`g on the theoretical 0, scan. If the roughness
is correlated (identical layered systems), the maxima and
minima periodically alternate in the angular distribution of IO4 6.. s ' ' s * - I * Q r s *. ' *. I * * * - - * = * - I * - = * *. * - `4
the scattered intensity. Similar to the Kiessig structure of
reflectivity curves, these oscillations result from the interfer-
ence between the waves diffusely scattered from the top and
bottom interfaces. However, if the interface profiles are com-
pletely uncorrelated, nonspecular scattering monotonously
decreases with the increasing angle 0,.
For the demonstration of the diffuse scattering from
more complicated structures, two, W/Si multilayers have
been chosen. These x-ray mirrors differ only in the thickness
ratio of layers composed of heavy and light elements. The
rms roughness rr=OS nm and the correlation length r= 1 ,um IO" F.
0.0 1.0 2.0 3.0 4.0
are considered at each interface. 0, Meg.1
The series of theoretical transverse scans at 28 values
corresponding to the first-, second-, and third-order Bragg
maxima are plotted in Figs. 6 and 7. The x-ray scattering FIG. 5. Theoretical Ci, scans for the gold layer (dA,=20 nmj with correlated
from multilayers with correlated and uncorrelated roughness (solid line) and uncorrelated (dashed line) interface profiles. The rms rough-
ness (r= 1.0 nm and the correlation length ~0.1 pm are considered at both
is compared. These pIots show several interesting features. interfaces.
J. Appl. Phys., Vol. 77, No. 6, 15 March 1995 M. Kopeclj 2385
Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
IO5 b' ' ' ' * - * * ' I - ' 8 ' ' ' 1' * b ' ' `. * ' ' - - I ' * - * g ' *. `j
F (a) 0, +e,=I88deg. 1
lo2'....,,...1......~~~~~~~~~~~~~~~,.*~~~~~'
-0.6 -0.3 0.0 0.3 0.6 ' --0.6 -0.3 0.0 0.3 0.6
0 [deg.1 0 Wg.1
~~.~,~~~`~~"~)~.`.~~"~J~.~.
(b) 01 + 92 = 3.60 deg. $1 + 92 = 3.60 deg.
0.0
co [deg.] 0 Wg.1
~"~.~`,"`~~.*","~`~`.`.I.`~~`~"`,~~
0, + e2 = 5.36 deg.
y IO3
-g IO2
g IO'
iz IO0
3
2. IO"
1 o-2 -2.0 -1.0 0.0 1.0 2.0 -2.0
-2.0 -1.0
-1.0 0.0
0.0 1.0
1.0 2.0
2.0
w Peg.1 0
0 Wg.1
Wg.1
FIG. 6. Transverse
scans calculated for the multilayer WEi @,=I.7 nm, FIG. 7. Transverse
scans calculated for the multilayer W/S (d,=1.2 nm,
dsi=3.3 nm, 40 periodsj at (a) the first-, (b) second-, and (c) third-order ~&~=3.8 nm, 40 periods) at (a) the first-, (bj second-, and (c) third-order
Bragg maxima. Bragg maxima.
in the specular reflection. For the parameter 7z = 0.2 pm, not
The off (0,2 19) scan measurement appears to be a very only the profiles of adjoining boundaries are correlated but a
promising method for determining vertical correlation. The certain degree of correlation also exists between the rough-
numerical calculations were carried out for the muhilayer ness of the substrate and of the surface. This again results in
consisting only of 10 pairs of Fe/C (Fig. 8). A smaller num- fast oscillations equivalent to the Kiessig ones.
ber of periods was chosen in order to reach a better resolu- The presented results show that the structure of layered
tion of subsidiary interference fringes. This scan exhibits no systems has a strong impact on their scattering properties.
fine structure for completely uncorrelated interfaces. With
increasing vertical correlation, the interference maxima start IV. SlJttlMARY
to appear in the regions where the angle of scattering 20 is
equal to twice the Bragg angle. These peaks correspond to In this paper, the theory of diffuse x-ray scattering from
the constructive interference of waves diffusely scattered rough layered systems based on the distorted-wave Born ap-
from individual bilayers in a full analogy to the Bragg peaks proximation has been presented. To author's knowledge, this
2366 J. Appl. Phys., Vol. 77, No. 6, 15 March 1995 M. Kopeck$
Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
may be quantified. Different scattering properties of systems
with correlated and uncorrelated roughness enables one to
specify the level of vertical correlation.
A paper devoted to the characterization of layered struc-
tures by coherent and incoherent scattering of x rays is being
prepared.
ACKNOWLEDGMENTS
This research has been supported by the Grant Agency
of the Czech Republic under Grant No. 202/93/1023 and by
the Grant Agency of Czech Academy of Sciences under
Grant No. 110427.
0, Wg.1
`See. for example, P Chakraborty, Int. J. Mod. Phys. B 5, 2133 (1991).
FIG. 8. Theoretical offset (0.28) scans for multilayers Fe/C (dFe= 1.77 nm, "A. Horn&up, F. E. Christensen, J. Polny, and H. W. Schnopper, Proc.
&=3.28 nm, 10 periods) with different vertical correlation. The same val- SPIE 982,409 (1988).
ues of u=OS MI and 7= 1.0 ym are considered at each interface. "0. Renner, M. Kopecky, E. Krousky, F. Schafers, B. R. Miiller, and N. I.
Chkhalo, Rev. Sci. Instrum. 63, 1478 (1992).
4S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, Phys. Rev. B 38,
is the first model valid also at small glancing angles, where 2297 (1988).
`D. G. Stearns, J. Appl. Phys. 65, 491 (1989).
the calculation of the diffusely scattered intensity respects 6R A. J. de Korte and R. Lain&, Appl. Opt. 18, 236 (1979).
both the real coherent field within a nonideal stratified me- 7E. L. Church and P. Z. Takacs, Proc. SPIE 640, 126 (1986).
dium and a real transmission of waves scattered from inner *W. Weber and B. Lengeler, Phys. Rev. B 46, 7953 (1992).
boundaries. `V. Holi, J. Kub&ra, I. Ohlidal, K. Lischka, and W. Plotz, Phys. Rev. B 47,
15896 (1993).
By using the above-mentioned approach, the Yoneda t"L. G. Parratt, Phys. Rev. 95, 359 (1954).
anomalous scattering of x rays can be described. Further, the "J. H. Underwood and T. W. Barbee, Jr., Appl. Opt. 20, 3027 (1981).
modulation in the nonspecularly scattered intensity can be `*L. Nevot and P. Croce, Rev. Phys. Appl. 15, 761 (1980).
explained as the consequence of dynamical effects. The scat- 13M. Kopecky, Doctoral thesis, Technical University, Prague, 1993.
14L. I. Schiff, Qtrantum Mechanics (McGraw-Hill, New York, 1968).
tering from periodical multilayers exhibits satellite maxima 15B. B. Mandelbrodt, The Frucral Geometry qf Nature (Freeman, New York,
and minima when 8, or ~9, approaches the Bragg angle. 1982).
These features depend on the mutual correlation between in- 16D E Savage, J. Kleiner, N. Schimke, Y.-H. Phang, T. Jankowski, J. Ja-
dividual boundaries but they can be observed for completely cdbs,.R. Kariotis, and M. G. Lagally, J. Appl. Phys. 69, 1411 (1991).
17Y. Yoneda, Phys. Rev. 131, 2010 (1963).
correlated as well as for uncorrelated interface profiles. "D E Savage, H. Schimke, Y.-H. Phang, and M. G. Lagally, J. Appl. Phys.
This scattering theory may be also used for fitting ex- 7;, ;283 (1992).
perimental data. In this way, basic statistical parameters 19E. Spiller, D. G. Stearns, and M. Kmmrey, J. Appl. Phys. 74, 107 (1993).
(root-mean-square roughness, correlation function) of ran- "A. E. Rosenblutb and P Lee, Appl. Phys. Lett. 40, 466 (1982).
"J. B. Kortright, J. Appl. Phys. 70, 3620 (1991).
dom rough surfaces and interfaces within layered structures "B. W. Batterman and H. Cole, Rev. Mod. Phys. 36, 681 (1944).
J. Appl. Phys., Vol. 77, No. 6, 15 March 1995 M. Kopeck$ 2387
Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp