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Influence of roughness profile on reflectivity and angle-dependent X-ray fluorescence
D.K.G. de Boer, A. Leenaers, W.W. van den Hoogenhof
To cite this version:
D.K.G. de Boer, A. Leenaers, W.W. van den Hoogenhof. Influence of roughness profile on reflectivity and angle-dependent X-ray fluorescence. Journal de Physique III, EDP Sciences, 1994, 4 (9), pp.1559- 1564. �10.1051/jp3:1994222�. �jpa-00249205�
Classification Physics Abstracts
61.10D 68.358 78.70E
Influence of roughness profile on reflectivity and angle-dependent X-ray fluorescence
D-K-G- de Boer(~), A-J-G- Leenaers(~) and W.W. van den Hoogenhof(~)
(~ Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands (~) Philips Analytical X-Ray B-V-, Lelyweg 1, 7602 EA Almelo, The Netherlands
(Received 19 November 1993, accepted 3 March 1994)
Abstract. The theory of X-ray scattering from rough interfaces using the second-order distorted-wave Born approximation is reviewed. For specular reflectivity
a new expression is
given which smoothly connects the Ndvot-Croce and Debye-Waller expressions. The shape
of the reflectivity curve depends not only on the average roughness, but also
on the lateral correlation of the roughness profile. Using the
same theory, the several ways in which roughness
can influence angle-dependent X-ray fluorescence are evaluated. Finally, possible shapes for the
roughness correlation function
are discussed.
Introduction.
X-ray reflectivity (both specular and diffuse) is a well-known method for the analysis of surfaces and thin layers 11, 2]. A complementary technique is angle-dependent X-ray fluorescence (AD- XRF) [3], with which the compositional depth profile of layered materials can be determined.
The combination of both techniques in a single instrument is called glancing-incidence X-ray analysis (GIXA) [4].
This paper deals with the effect of rough interfaces on measurements performed with those
techniques. Interface roughness gives rise to diffuse reflection, reduces the specular reflectivity and influences AD-XRF in various ways.
The intensity of diffuse scattering can be calculated using the distorted-wave Born approx- imation (DWBA) [2]. It depends on the lateral correlation function of the roughness profile.
The reduction of the specular reflectivity can also be determined with the aid of the DWBA.
Recently, we showed that for a general case it is necessary to pursue the calculation up to second order [5]. This means that the specular reflectivity contains twc-step scattering contri-
butions via intermediate states which are sensitive to the lateral correlation of the roughness.
To calculate the influence of roughness on AD-XRF, second-order perturbation theory has to be used too. Below, we will outline the method of calculation and discuss the results.
Finally, we will give some comments on the shape of the correlation function.
JOURNAL DE PHhSIQUE III -T 4 N'~ ~EPTEMBER lW4 ,~
1560 JOURNAL DE PHYSIQUE III N°9
~ ~~~~~~~
~ll'~~~o)
j
lP~,Pi)
Fig. 1. Wave vectors for reflection and transmission at a rough interface. The thick arrows indicate
specular reflectivity, the thin arrows indicate diffuse scattering.
Distorted-wave Born approximation.
In the DWBA, a calculation is done first for the situation with flat interfaces and then the
roughness is introduced as a perturbation. If the incident X-ray field is a plane wave, the reflectivity and transmissivity are found with the aid of the Fresnel coefficients [6]. For the rough interface, the scattered field can be expressed as a sum (or integral) of outgoing plane
waves [5]:
§~k~~~ ~ / ~~
~(( ~~P[~ (P(( ~ P°~~j ~(P'~) IF0 (1)
outside the sample and
# k(r) = ~
~
d~pjj exp[I (pjj x + pi z)] T(p, k)/ p1 (2)
87r
/
inside the sample. The wave vectors (all having a length k) of the incident and scattered X-rays
are shown in figure I. The scattering amplitude depends on the T-matrix element T(p,k), I-e-
an integral over all space of the perturbation potential sandwiched between a state with wave
vector p and a perturbed state with wave vector k. (The bar in p denotes a wave starting
inside the material, I-e- a "time-reversed" state.)
The diffuse-scattering cross section is obtained from a configurational average of (T(p,k)(~.
Details can be found in the literature [2]. The change in the specular reflection coefficient is obtained from a configurational average of T(k, k) and the change in the transmission coefficient from a configurational average of T(k,k). The total field due to the X-rays can be written as
a perturbation series:
§~k(~) " §~~~(~) + §~i~(~) + §~i~(~) + (3)
where #f~(r) is the solution for the flat interface, ii (r) involves first-order T-matrix elements,
etc.
Now it is possible to calculate specular reflectivity and AD-XRF within the DWBA.
Specular reflectivity of rough samples.
Using the second-order perturbation theory we find the following expression for the specular reflection coefficient:
f~ = r[°~
i 2kokia2 1/(2gr2 kok2(1 n]) / d
2pjj/(po + pi ©(pjj kjj)1(4)
H- nm
~
~m
Fig. 2. Reflection and transmission at an interface with a small value off (top) and with
a large
value of ( (bottom). (After Ref. [7].)
where rf~
= (ko ki)/(ko + ki) is the Fresnel coefficient for reflection at the flat interface, a
is the root-mean-square (r.m.s.) interface roughness, ni is the refractive index of the sample
and ©(qjj e f d ~X exp(iqjj X) C(X). Here C(X) is the correlation function of the roughness profile (cf. [2]), which in general is a function decaying with a characteristic length (, the lateral correlation length (see below). The last term in equation (4) is second order in the
perturbation potential and can be regarded as a sum of diffuse scattering contributions. Like the first-order term, it is, however, of order k(a2 and hence both terms should be included.
As a generalized result for arbitrary koa we propose to use
fk t rf~ exp -2kokia~
1/(27r~)ko k~(1- n() d~pjj /(po + pi d(pjj kjj) (5)
To justify this extrapolation, we will compare the results for extreme values of the lateral correlation length ( with those obtained by Ndvot and Croce (NC) [7]. If ( < k/k(, we have fk Ci rf~ exp(-2kokia~). NC obtained the same expression for, in their terminology, "predom-
inance of high spatial frequencies" (Fig. 2, top). If ( » k/k(, we have fk ci rf~ exp(-2k(a~),
I-e- an expression which has the form of a Debye-Waller (DW) factor. Indeed, this is expected
for "predominance of low spatial frequencies" [7] (Fig. 2, bottom). Hence, equation (5) is valid for these extreme cases. For intermediate values off, it interpolates smoothly between the two
extremes.
In figure 3 we show the results of calculations for the reflectivity (fk(~ of CuKoi radiation
on a gold sample with a
= 1.5 nm, using the form C(X)
= a~ exp(-(X(/() for the correla-
1562 JOURNAL DE PHYSIQUE III N°9
1
~ l
_£
l$(
~ t = 0 (NC)
( = 100 nm
( =1000 nm
( = ~ low)
i-o 1.5 z-o
2k~(nm~')
Fig. 3. Specular reflectivity as a function of wave-vector transfer 2ko, calculated for Cu Kcvi radiation on a gold sample with
an r-m-s- roughness a
= 1.5 nm and four values of the latpal correlation
length (.
tion function. As is seen, the reflectivity depends on the lateral correlation length ( and the difference between small and large values of ( can be a factor of two.
We note that in an analogous way the transmission coefficient can be found [5]. If ( < k/k(,
we find ik t tf~exp[(ko ki)~a~/2]; if ( » k/k(, we find ik t tf~ exp[-(ko ki)~a~/2], where
tf~ = 2ko/(ko + ki is the Fresnel transmission coefficient for the flat interface. It is interesting
that for ( < k/k( the result of interface roughness is an enhancement of transmissivity by the
same amount as the reduction of the reflectivity, whereas the diffuse scattering vanishes in that limit.
Angle-dependent X-ray fluorescence of rough samples.
In AD-XRF the X-ray fluorescence intensities of the elements present in a sample are measured
as a function of the incidence angle. From sucl~ a measurement the compositional depth profile
of the sample can be obtained [3, 4]. Figure 4 shows an example of calculations carried out for impurity atoms at the top surface of a sample. If no roughness is present, the AD-XRF is
simply proportional to the intensity of the incident X-rays at the surface, I-e- proportional to the transmissivity (Fig. 4, solid line). To obtain the correct compositional depth profile from such a measurement, it is important to describe the influence of interface roughness correctly.
The intensity of X-ray fluorescence emitted by an atom at a position r in the sample is
proportional to the intensity due to the incident X-rays at that position, I-e- proportional to
I#k(r)l~
= 1#?~(r)l~ + 2Re1#?~"(r)#?~(r) + #?~"(r)#?~(r)I +1#[~~(r)l~ + (6)
To obtain the total AD-XRF intensity, this has to be integrated over the whole sample. Up
to order k(a~ the above four terms contribute to X-ray fluorescence from inside the sample.
Two more terms arise from the fact that the real interface deviates from the flat one, an effect to which only the first two terms contribute (up to order k(a~). In this order, the six terms
contributing to the AD-XRF intensity are: (1) absorption of the directly transmitted beam as if no roughness was present; (2) enhancement of transmissivity due to roughness with a small
0 2 3
angle of incidence (mrad)
Fig. 4. Mo Ka-induced AD-XRF, calculated for impurity atoms at a silicon surface with a roughness
a = 2 nm. Solid line: no roughness, or roughness with a large value of (. Dotted line: including
correction (2). Dashed line: including corrections (2) and (5), I-e- ( < k/k( (see text).
value off (< k/k(); (3) reduction of transmissivity due to roughness with a large value off; (4) absorption of diffusely scattered radiation, only appreciable for large values off; (5) absorption of direct beam in the rough interfacial layer; (6) second-order correction to (5), reducing the
absorption in the case of large values off- It is found that if ( » k/k( all corrections (2) to (6)
are cancelled and the same curve is found as for the case a
= 0 (Fig. 4, solid line). If ( « k/k(,
however, only corrections (2) and (5) have an effect (Fig. 4, dashed line). The dotted line in
figure 4 shows the effect of correction (2) only.
The correlation function.
Now we will make some more remarks on the correlation function C(X). We have found
that for specular reflectivity and AD-XRF its exact shape is not important. However, the diffuse scattering distribution provides more details. At lateral length scales smaller than the lateral correlation length ( a rough interface can often be described as being self-affine fractal [2, 8], characterized by a parameter H which is associated with the fractal dimension D of the
interface (D
= 3 H). Sinha et al. [2] propose to use
C(X) = a~ exPl-(lXl/f)~~i
Indeed, this expression gives the right fractal behaviour for (X( < (, but the cross-over to non-fractal behaviour is not always well-defined. An altimative expression, which shows the
right overall behaviour, is (cf. Ref. [9]):
C(X) = P f~ lXl~ KH(lXl/f)
where KH is a modified Bessel function of order H and P is a constant. In this model a
+~
(~,
as expected for fractal film growth [8]. For H
= o the Edwards-Wilkinson growth model [10] is obtained (and a cut-off has to be introduced to prevent divergence for (X( - 0). For
H = 1/2 both formulas for C(X) are equivalent and give the exponential form used above in the calculation of the specular reflectivity.
1564 JOURNAL DE PHYSIQUE III N°9
Discussion and conclusion.
Using the method outlined above, the influence of interface roughness on both specular and diffuse reflectivity, as well as on AD-XRF can be calculated. For specular reflectivity and AD- XRF from samples with a single interface, the relevant roughness parameters are the r,m.s.
roughness a and the lateral correlation length (. An additional parameter H is needed to describe the shape of the diffuse scattering distribution. Above, we have given two expressions for the roughness correlation function which can be used in calculations. Experience will have to show which is the most suitable form.
For layered materials the situation is more complicated, since then interference between X- rays scattered from different interfaces has to be taken into account, as well as correlation
between the roughness of the various layers [2, 11, 12].
In conclusion, it should be possible to obtain details on the roughness profile from the
glancing-incidence X-ray methods: reflectivity and AD-XRF.
References
[1] Plotz W. and Lischka K., X-ray reflectivity, these proceedings.
[2] Sinha S-K-, Sirota E-B-, Garof S. and Stanley H-B-, X-ray and neutron scattering from rough surfaces, Phys. Rev. B 38 (1988) 2297;
Sinha S-K-, X-ray diffuse scattering as a probe for thin film and interface structure, these pro- ceedings.
[3] de Boer D-K-G-, Glancing-incidence X-ray fluorescence of layered materials, Phys. Rev. B 44
(1991) 498.
[4] van den Hoogenhof W-W- and de Boer D-K-G-, Glancing-incidence X-ray analysis, Spectrochim.
Acta 488 (1993) 277.
[5] de Boer D-K-G-, Influence of the roughness profile on the specular reflectivity of X-rays and neutrons, Phys. Rev. B 49 (1994) 5817.
[6] Born M. and Wolf E., Principles of Optics (Pergamon Press, Oxford, sth ed., 1975).
[7] Ndvot L. and Croce P., Caractdrisation des surfaces par rdflexion rasante de rayons X. Application
I l'dtude du polissage de quelques verres silicates, Rev. Phys. Appt. 15 (1980) 761;
Croce P. and Ndvot L., #tude des couches minces et des surfaces
par rdflexion rasante, spdculaire
ou diffuse, de rayons X, Rev. Phys. Appl. ll (1976) 113.
[8] Meakin P., Fractal structures, Progr. Solid State Chem. 20 (1990) 135.
[9] Church E-L- and Takacs P-Z-, Statistical and signal processing concepts in surface metrology, SATE 645 (1986) 107.
[10] Edwards S-F- and Wilkinson D-R-, The surface statistics of a granular aggregate, Proc. Roy. Sac.
London A 381 (1982) 17.
[ll] Holf V., Kubdna J., Ohhdal I., Lischka K. and Plotz W., X-ray reflection from rough layered systems, Phys. Rev. B 47 (1993) 15896;
Holf V. and Baumbach T., Non-specular X-ray reflection from rough multilayers, Phys. Rev. B 49 (1994) 10668.
j12j de Boer D-K-G., Leenaers A.J.G. and van den Hoogenhof W-W-, The profile of layered materials reflected by glancing-incidence X-ray analysis, Appt. Phys. A 58 (1994) 169.
