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X-ray diffuse scattering as a probe for thin film and interface structure
S. Sinha
To cite this version:
S. Sinha. X-ray diffuse scattering as a probe for thin film and interface structure. Journal de Physique III, EDP Sciences, 1994, 4 (9), pp.1543-1557. �10.1051/jp3:1994221�. �jpa-00249204�
Classification Physic-s Ahsn.acts 61. IO
X-ray diffuse scattering as a probe for thin film, and interface structure
S. K. Sinha
Exxon Research and Engineering Company. Corporate Research, Route 22 East, Annandale.
NJ 08801, U-S-A-
jReceii,ed19 Noi'emher1993, iei,ised l0Maic.h 1994, accepted 30Maich 1994)
Abstract. The structure of thin films and interfaces can be probed by X-ray specular and off-
specular (diffuse) scattering. As is well-known, the former yields the ai>era,ge density profile
across the film or interface. Diffuse scattering as treated here is the analogue (for interfacesl of
small-angle scattering from bulk materials, but with the ability to probe much larger length-scales.
We shall discuss how the diffuse scattering yields information regarding the detailed morphology
of the interface roughness, the conformality of the roughness between successive interfaces, the
morphology of the erosion or pit-structure shall illustrate with results on several systems studied
using synchrotron radiation at the National Synchrotron Light Source.
1. Introduction.
X-rays and neutrons have proved to be a marvelous non-destructive probe Ii for the study of
the structure and morphology of thin films and interfaces. While not yielding direct imaging
information, as obtained from the complementary techniques of electron microscopy, scanning tunneling microscopy or atomic force microscopy, X-ray scattering is capable of yielding quantitative global statistical information about interfaces over an enormous range of length
scales (angstroms to microns). In addition, buried interfaces can be probed with ease.
Consequently, the use of this technique is becoming increasingly popular. In this article, we shall not deal with some of the beautiful results obtained by X-ray surface diffraction in areas such as surface reconstruction, monolayer adsorption, etc., which we might term « surface
crystallography ». Instead, we shall concentrate on the characterization of disorder, fluctu- ations and inhomogeneities at interfaces which typically occur at the mesoscopic scale (I nm-
l ~). This is a matter of considerable importance for technological applications and issues such
as film growth.
2. Roughness at a single interface.
One of the most ubiquitous examples of surface disorder is surface roughness. The problem is (a) how to characterize the roughness and its morphology quantitatively, (b) how to relate it to
Fig. I.-TEM picture of an amorphous Si-Nb multilayer showing the conformal and increasing roughness with layer number (from Ref. [2]).
the X-ray scattering, and (c) how to extract quantitative information about the surface morphology from scattering experiments. Figure shows a TEM picture of a section through a
Nblamorphous Si multilayer [2] where interface roughness is clearly observable. Note that there is a degree of confoimalit_v to the roughness (I.e. a correlation of the height fluctuations
between different interfaces). In addition, the uppermost layers show a gradual increase in roughness. The roughness may be characterized by a height-height correlation function
(&=(0)6=(R)) where 6z(R) is the height fluctuation at lateral position R in the (,<, y) plane. We represent it in the form
C(R)m j&t(0) &=(R)j = «~exp(- iRlii~~') (I)
where
« is the rms value of the surface roughness, h is a roughness exponent and
f is a roughness correlation length. At small R, equation ) yields for the mean-square height
difference
g(R)m ([&z(0)- &z(R)]~) 2 «~(R/f)~~' (2)
which is the scaling form for a self-afline surface, as defined by Mandelbrodt and Voss [3].
The variables 6z (R) are assumed to be Gaussian random variables, and the surface is said to be
generated by « fractional Brownian motion » [3]. Not all roughness is Gaussian, of course, and
we shall discuss specific examples later. However, the above representation has been used in computer simulations to generate remarkably realistic « landscapes» and also in fitting roughness data on real surfaces. An important exception to equation (I) is the special case
where g(R) is logarithmic rather than power-law, as for instance for liquid surfaces [4] or surfaces at the roughening transition [5]. The advantage of equation is that it characterizes the morphology of the roughness in terms of only 3 parameters, which can usually be fit to
experimental data. A critical discussion of alternative forms for C (R) is given by Palasantzas
& Krim [6].
The scattering of X-rays (or neutrons) by such a surface can be discussed within the framework of the Born Approximation, in the case where the scattering is weak, I-e-, it will not be valid in the vicinity of the regions of total reflection. The result for the scattering function
S(q) (q being the wavevector transfer) is [7, 8] for a single rough interface S(q
=
~~~i~~ e~~ " jj d-i dy e~- ~~~ e~''~' ' ~ ~' '~
(3) qj
where A is the illuminated area, and (Ap is the scattering length density contrast between the media on either side of the surface. We have assumed that the media have no structure e~cept
at the interface. We shall discuss later how to generalize the result to include internal structure of the media, but for (q~ ' » typical atomic length scales in the media, this is a reasonable
approximation. Since C(R)-0 as R
- cc, the integral in equation (3) contains a delta- function part in jq,, q,) which corresponds to the specular reflectivity which can be explicitly separated out by subtracting from the integrand. Thus we have
Sd,ttu,e(q
=
~
e ~ " jj d-i dy[e~- ~ '~ e~'~~' ' ~ ~' '~ (4)
q=
and
~'PeLuldr(~)
~ ~P ~-q ,r ,
~) ~ ~ ~(~, 6 (q' (~
which can be shown [7, 8] to reduce to the formula for the specular reflectivity R(q=)
=
~~ ~j~~~e~~-"~ ~ (6)
which is well-known in the Born Approximation [9j. Equations (4) and (5) work well for most
rough surfaces in the region of validity of the Born Approximation and can be used to derive.
«, f, h by fitting to the scattering data. Often the X-ray apparatu~ integrates over one direction of q in the plane of the surface (e.g. q,) as that equation (4) then becomes
sziu,~(q
=
~ ~~
[~P~~ e "
'~'
di ie~ ~ ~'' e'« ' (7)
q
so that the experiment measures correlations along the surface in the plane of scattering. From
equation ), it can be seen that transverse diffuse scans (I.e., q,-scans or rocking curves) are a
function of (q, f only so that the width in q,-space scales inversely as f. Longitudinal diffuse
scans (I.e. just under or close to the specular condition q, = q, = 0) depend on f only through a
scale factor but are sensitive to h. In fact the asymptotic form of S~,~~~,~(q) for large q= ar'd q< ~ q, ~
° 'S
s~,~~~,~(q = ? (~~((~(~j~ ds .v exp (- s"~ (8)
q)~~ ~' ~''~ ' 0
and for the q,.-integrated diffuse scattering
2 flrA Ap ~ f j" ds exp (- s~~) ~~~
S~)fu~e(~)
~ (2 + "h
~ Ill'
0
so that h can be extracted from the asymptotic power law for this scattering. Note that in order
to extract the true specular part, given by the delta function part (Eq. (5)) broadened by
instrumental resolution, one must substract the diffuse background in order to get a true
measure of the ,global roughness, otherwise the roughness will be considerably underestimated.
From equations (8) and (9) it can be seen that the diffuse scattering dies much more slowly than the specular, which decreases like a Gaussian.
In order to describe the scattering in the vicinity of the critical angle and below, one must
take into account that the wavefunction in the vicinity of the interface is no longer well
described by a single plane wave. If one employs the distorted Wave Born Approximation
(DWBA) [7, 10-12], equation (4) is repaced by
Sd,ttu,e(q = T(a )1~ T(P )1~
fi(
~ e
' ""~
~ ~~~ jj d-~dj, je '~= ~ ~~~' i 1-'~~,' +~, "' q=
lo)
were T(a is the transmission coefficient for the interface for grazing angle of incidence
a (p is the corresponding angle for the outgoing beam).
@= is the value of the wavevector transfer inside the reflecting medium and may be purely imaginary (e.g. below the critical
angle of incidence).
The most dramatic effect of the DWBA is to show that the diffuse scattering is enhanced at
a or p equal to the critical angle, yielding the wings in the diffuse scattering at low angles known as « Yoneda wings » [I I]. Figure 2 shows this effect for a transverse diffuse scan
(rocking curve) for a silver film deposited on Si at q= = 0. I h~ ' [14]. Equation lo) works well in fitting measured scattering data, although there is still some debate about what to use for the T(a ), T(p ). Pynn [10] has suggested that T(a should be the transmission coefficient for the
iou,qh surface, rather than the Fresnel coefficient (smooth surface) originally suggested [7], although there is no rigorous way of calculating this. A self-consistent DWBA calculation in
the above spirit for the specular reflectivity [10] yields the well-known Nevot-Croce
expression [15]
R(q=)
=
R~e~~°~'"~ (ll)
which is more generally valid (for Gaussian roughness) than equation (6).
One application of the above formalism is to the study of film growth. In principle one must consider the scattering from both the top and bottom surface of the film. The latter may in certain circumstances be neglected if either the film material is identical to that of the
substrate [16] or if the roughness of the top surface is much larger than that of the
substrate [14]. For tie growth of films, several theories and computer simulations predict that the growing surface should be of the self-affine form characterized by a universal roughness
exponent (corresponding to our h above). They also predict that
« should scale as
t~, where p is a universal exponent and t is the time of deposition (proportional to the average film thickness). There is another
« dynamic » scaling exponent z
= Alp which yields the
dependence of the correlation length f ~t"° Recent electron scattering [16] and X-ray
Aq/Si in situ deposition
io~~
10~~
o~
o~
o~
oo oo
8~
coo°o~ o°°°°o~
o°° o
~
o o~
o° oo
o°°oO°° oo°°Oo~
o o
~° ° ° °o
Q o °o
~ 10~ o° °°
fi2 o
~
°°
-
i~ Oo° °o
10~' °
o
-0.75 -0.50 -0.25 0
0.25 0.50 0.75
q~ [xl 0"~ A~~]
Fig. 2. Diffuse scattering scan in transverse (q,) direction at q= 0.I l~' for
a silver film vapor deposited on a silicon substrate. The sharp peak at q, 0 is the specular reflection and the Yoneda wings
are clearly visible on either side (from Ref. [14]).
scattering experiments [14, 17] have addressed this problem. Figure 3 shows specular reflectivity curves for a Ag film vapor deposited on a Si-substrate [14] with progressively increasing film thicknesses which have been fit with a rigorius multi-interface model [18] to obtain the value of « (shown in the inset) as a function of thickness. This yields the value
p =0.26= 0.05 for the growth exponent. The longitudinal diffuse scattering at large
q= values (where the contribution from the much smoother substrate interface is negligible) was
used to extract the exponent h which yielded h
=
0.63. On the other hand, direct STM
measurements of the surface yielded a reasonable fit to the form (I with h
=
0.78. Thus we
may put h = 0.70 = 0.08. The correlation length obtained from the diffuse scattering was
consistent with the scaling relation f
=
19.9 I t "~ ? These exponents are consistent with those
reported for vapor deposited iron films by Chevrier et al. [19] (p
= 0.25-0.32) and He et al. [16] (h
=
0.79 = 0.05, p = 0.22 = 0.02) and are close to those obtained from the model of Villain, das Sarma and others [20] (h
= 0.67, p
= 0.2). The Kardar-Parisi-Zhang theory [21]
yields h
=
0.33, p
=
0.25. The values reported by You et al. [17] for vapor deposited gold
films (h
= 0.42, p = 0.4) are however quite different.
10~
ios
~~4 fl 30
~ fi
~~~ b 2.o °
lo p=026+0.05
o iol
( ~~o iol io2
j <h> (nm)
tJ' 10~
~ )~_~ (A)
10~~ (B)
io-5 io-6 10~~
~~~
)~jg (D)
io-10
_~~ ~~~
~~
0 O-1 0.2 0.3 0A 0.5
q~ (i~~)
Fig. 3. -Specular reflectivity data for progressively thicker Ag films (together with fit~. The inset depicts a log-log plot of (r iei.<tt.< film thicknew with dope p 0.26 ± 0.15 (from Ref. II 4]).
3. Multiple interfaces.
Turning now to multiple interfaces, we recognize that a degree of conformal roughness implies
a non-vanishing value of the correlation function
C,~(R) (6z,(0) 6=~(R)) (12)
which is a generalization of equation (1), 6=,, 6=~ are now height fluctuations of the I-th and j-
th interfaces. This effect has been recognized for some time [?2-25] and yields as the
generalization of equation (4) in the Born Approximation,
~ N (q l'T/ +'T/
+ 6' 1' j
,,~ ~-
Sd,ttu,e(q
= m I e 6P, 6Pj* e ~' ' E,j(q '3)
qi
, j =1
where
E,~ (q
= d-i dv [e~~~" '~ ' l e~''~' ' ~ ~' '' 4)
where «, is the rms roughness of the I-th interface, Ap, is the scattering contrast across it,
=, is its average height and 6 is the rms deposition error in the layer spacing, which is
cumulative.
Equation (13) yields « ridges » in the diffuse scattering at q~ values which correspond to the specular Bragg peaks from the multilayer. In fact in the limit of perfect conformality C,~(R) = C (R) (independent of (I, j )), the expression for the specular reflectivity
j6
g~
N qij<r,~+WI+ 3 ii j
R(q= = ~ jj e ~ 6p, 6p~*e'~°~_~' ' (i5)
qi
, j
looks very similar. Figure 4 shows specular and longitudinal diffuse (q, = 0, q, integrated
over) scans [25] far a highly perfect GaAs/AlAs multilayer on a GaAs substrate which. away from the critical angle region, are well fitted by equations (14) and (15), assuming perfect
conformal roughness. The transverse diffuse scans show an interesting anisotropy as shown in
figure 5 where scans are shown for two positions of the sample rotated 90° in-plane. This is because the substrate has a ° miscut, resulting in step-roughness perpendicular to the miscut
direction. For q, scans parallel to the steps, the diffuse scattering is very sharply peaked around q, =
0 yielding a correlation length f
= 6 400 I (h
= 0.4), while for scans normal to the step direction the correlation length is rather shorter (f
=
200 I, h
= 0.46). It is of course not at all obvious that a stepped surface has a height correlation function which can be represented by
an anisotropic generalization of equation I), although it appears so empirically. This has been discussed elsewhere [25. 26].
Going beyond the Bom Approximation and using the DWBA formalism for multilayers rapidly becomes very complicate [I1, 27, 28]. The z-component of the incident and scattered
wavevectors are different for each medium and given by
where (k[~ is the normal component of the incident wavevector in vacuo, and q~ is the critical
wavevector transfer for that medium. One must first solve the problem for the specular
reflection using the matrix or iterative methods [18] separately for the incident wavevector
kj and for the scattered wavevector k~ (or more correctly its time-reversed state), yielding
coefficients qj, b~ and a~, hi for the amplitudes of their respective wavefunctions for waves propagating in the positive and negative z directions (z is defined positive into the multilayers from the surface) in each medium. Then one must define two wavevector transfers for each
medium I,
q, = kj _(i) k~-(1) (17a)
qi
= ki -(I + (~ =(i ). (17b)
One may the define for each interface (I a set of 4 coefficients A~(I
aj a, e~'~'° hi b, e'~'~' hi a, e~'~~ ~'
Aj= ; A2= Ai= ",
q, q, qi
~ fi ~'q'~,
A~ = ~ l8)
~~
The index (I) is implicit for the A~, aj, a~, hi, b~. z, is the average height of the I-th interface
10~
. jeculoritted
10~~
io-3
##
~~-5
n~
#=
io-7
'
io~' . . :
.
~~-ii
0 0.2 0.4 0.6 0.8 1.0
q~ lA-~l
10~'
. Diffuse Fitted
io-5
j# 10~
c
~
_X( 10~~
O
~ ,
10~'
10~~
0.4 0.7 1.0
q~ IA 'I
Fig. 4. Specular reflectivity (a) and longitudinal diffuse scattering (b) for a 77 bilayer GaAs/AlAs multilayer prepared on a loo) single crystal GaAs substrate. The fit to this specular corresponds to a
periodicity of 122.9 A, a GaAs/AlAs thickness ratio of 0.684, an rms interface roughness of 2.8 1 and
a
thickness fluctuation of 1.071. The diffuse scattering is fitted in h
=
0.4. (From ref. [25].)
10~
. Experi«ental
-Fitted
io-5
io~
10~~
-0. 010 -0. 005 0 0. 005 0. 010
q~
lr~l
10~~
. Measured fitted 10~~
10~ n=5
a~10~~ n = s
fim
~ 10~~
10~~ ° ~~
o=t4
. . ~
io-8 n=17 .
10~'
-0.010
q~ lA~~l
ig. 5.
specular
ragg in the orientation where the surface steps are parallel to the direction
q,. The diffuse is fitted with f = 6 400 A and h = 0.4.
cattering makes it hard to di~tinguish from
the eak.
(b) diffuse the
orientation where
the surface steps are
perpendicular to the direction of q,. The diffuse is
broad, and the specular peaks are evident. The curves correspond to
scans across various order
Bragg peaks, as indicated. The fitted values are f = 1 2001 and h = 0.64.
(the top interface being taken as I
= at z =
0 into medium I below it). One also must take conformal roughness into account and define generally
lq) <r,' + iq,* ~ <T/, jj ~ ~~ c (R -, (q , + q i
F~~
j~ = e dx dy[e ' ' e ' '
19) where
~',l ~~i
qj,2 = qj (20)
q',i"q~
q,,4 " ql (21)
Finally, we obtain
~ N 4
Sd,ftu,e(q)
= ~
z z (P, P, (Pi Pi >* A
~
(i )Am g F,,,, (I, j (22)
16 ~
,j=i,, -1
where
p, =
k((I n)) (23)
n, being the (complex) refractive index in medium I, and ko is the magnitude of the incident
wavevector in free space. Unfortunately, the above formulation is computationally demanding
as it involves the evaluation (in principle) of a large number of integrals. We can simplify this
as follows. We assume that the experiment integrates over q, so as before only integrals over
>. are involved. We may write the integral
°~
i~ (e~, "/ "~~~P ~'~~ ~~
l e~ '~' '
=
2 f ~~~
"
~'~'~~' ~
"~~"
fl~(q, fIn~~~*l
~ ,,
= ~ f ql[ v(qj)v'
~~ ( U,~)~
n,
fl~(q, f/n"~")
where
" ' (24)
r~p)
= dx e~'~~
cos ~px). (25)
~
In general this series needs be taken up to n s lo only. We then assume perfect conformality (only one «, f, h involved). Equation (22) may then be written
where
Y,, =
( (
(Pi P, A ~(i ql',
~
e
~~ "~
(27)
,-1,=1
which is much faster computationally.
