1. Introduction

Following the general trend of materials sciences towards nanosized materials, the thin-film community is increasingly concerned with layers with thicknesses that do not exceed a few nanometres. Such ultrathin layers exhibit fascinating properties, such as very high yield stresses (Arzt, 1998; Labat et al., 2000) or new magnetic properties (Sander et al., 1998; Bochi et al., 1996). The structure of these very thin layers is, however, expected to be highly inhomogeneous because of the presence of a surface and an interface in their close vicinity.

X-ray diffraction is the method of choice to study lattice parameters in materials because of its very high accuracy. One widely used method for studying elastic strains in bulk or thin film materials (Noyan & Cohen, 1987; Clemens & Bain, 1992; Kamminga et al., 2000) is the sin² method. It relies on the measurement of the position of X-ray lines diffracted from lattice planes with different orientations () with respect to the sample normal. If one considers a single-crystal thin film of a cubic material under an equal biaxial state of strain, the (strain) versus sin² plot is linear and yields the stress and the stress-free lattice parameter a₀. The proper use of this method in the case of more complex situations, like polycrystalline films with different texture components (Hauk et al., 1988; Dölle & Hauk, 1978; Zaouali et al., 1991; Leoni et al., 2001) or the presence of stress gradients (Lode & Peiter, 1981; Genzel, 1999), is well documented. It is known that the presence of different crystallographic textures in elastically anisotropic materials may yield oscillatory versus sin² plots. Another reason for non-linear versus sin² plots is the presence of stress gradients (Lode & Peiter, 1981; Genzel, 1999) on a scale larger than the absorption length of X-rays. Then the sampling of the gradient will depend on the angle and oscillations may occur in the versus sin² plot. One notes that such an effect is expected to disappear in thin films unless one uses a grazing incidence (Malhotra et al., 1995) in order to reduce considerably the penetration length of X-rays.

It is important to realise that the common approach in these works is to derive intensity-averaged quantities (such as the X-ray elastic constants). The purpose of this communication is to show that in very thin layers one should take into account coherency (in the sense of optics) effects, i.e. sum the amplitude of the diffracted waves (Fullerton et al., 1995; Shen & Kycia, 1997). Indeed we show that structure-factor calculations from thin layers with a lattice-spacing gradient yield shifted diffracted intensity maxima. We further show that applying blindly the sin² method to such layers may give erroneous results. These calculations are corroborated by a comparison with experimental results (Lallaizon, 2000) from Fe thin layers on GaAs.

2. Distance gradients and the sin² method

Let us consider a thin film grown along the z axis and assume that the interplanar spacings d(n) are a function of the plane number (see Fig. 1). Let us define q (q_x, q_y, q_z) as the scattering vector; then the intensity diffracted (neglecting any experimental corrections) in the q_z direction (symmetric diffraction) is simply the square modulus of the structure factor S(q_z):

where f_n is the scattering factor of plane number n (total number of planes N) and Z(n) is the distance between plane number n and plane 0:

where d(j) is the distance between plane number j - 1 and plane number j.

For the purpose of demonstration we use here a distance profile d(j) which was extracted from the refinement of the intensity diffracted from a thin (3  nm) Fe layer grown on GaAs. This distance profile was obtained from a joined fit on the 002 and 004 intensities. It is shown in Fig. 2. This gradient is attributed to Ga and As segregation during the room-temperature growth of the film (Lallaizon, 2000). Body-centred cubic (001) Fe grows epitaxially on (001) GaAs because the Fe lattice parameter (a_Fe = 0.28664  nm) is close to half that of zincblende GaAs (a_GaAs = 0.56538  nm), leading to a small lattice-parameter misfit: (a_GaAs - 2a_Fe)/2a_Fe = -1.37%. The 002 line, calculated using equation (1) and assuming that all the planes have the same scattering factor, is shown in Fig. 3, together with the low-q part of the experimental data (the high-q region is unusable because of the 004 GaAs line). The X-ray diffraction experiments were performed at the LURE synchrotron radiation facility on beamline DW22. The sample was at the centre of a four-circles diffractometer (with an angular resolution of 0.001° for the four angles). The wavelength of the beam ( = 0.8211  Å with a spectral resolution of 5 × 10^-4) was selected by a double-bounce Si (111) monochromator. Incident-beam divergence can be evaluated as less than 0.006°. One notes the clear asymmetry of the Laue fringes on each side of the Bragg peak. From the distance profile and equation (1), one can calculate the position (intensity maximum) of the different diffraction orders. This is shown in Fig. 4, where the solid line is a linear fit to the data. Interestingly, the q_z versus l plot is linear, but it has a non-zero contribution q₀ for l = 0. From the slope one can extract an interplanar distance of 0.14630  nm, to be compared with the average distance, Z(N - 1)/(N - 1), in the stack: 0.14819  nm. The linearity of the q_z versus l plot indicates that the stack almost obeys Bragg's law. A closer look, however, shows clearly important deviations. This is evidenced in Fig. 5 where the lattice parameter is reported as a function of the diffraction order l. This shows clearly that the average lattice parameter depends on the diffraction line that is used. A similar deviation from Bragg's law was reported in Appendix II of Thomas et al. (2002). From Fig. 5, one obtains a 1% difference between the lattice parameters extracted from the 002 and 008 diffraction lines. This may have important consequences for the determination of composition and strain in e.g. III-V alloyed epitaxic layers. Indeed, the composition of epitaxic III-V alloys is routinely performed through an analysis of Bragg peak positions combined with linear elasticity and Vegard's rule (Halliwell, 1997).

A sin² experimental plot for this 3  nm thick Fe layer is given in Fig. 5 (002, 004 and 006 reflections at = 0°; 224 reflections at = 35.26°; 303 and 404 reflections at = 45°). Here a_hkl is the cubic parameter calculated from d_hkl. A straightforward analysis (Clemens & Bain, 1992) of such a plot may be made using

where is the biaxial stress, a₀ is the stress-free lattice parameter (Noyan & Cohen, 1987) and S_ij are the elastic compliances for Fe. The use of bulk stiffness coefficients for a 3  nm thick layer is justified by a number of studies, which show that bulk elastic constants are valid down to a few monolayers (Müller & Thomas, 2000). The stress-free lattice parameter is 0.28767  nm, i.e. larger than the parameter for pure Fe (a_Fe = 0.28664  nm). This value is calculated by assuming the elastic constants of pure Fe. This expanded stress-free lattice parameter gives a hint for the occurrence of interfacial mixing. One should, however, be cautious with the results of such an analysis, as shown from the simulations. One notes that the extrapolation of such a plot to sin² = 1 gives an in-plane lattice parameter of 0.28190  nm, i.e. smaller than the substrate one (a_GaAs/2 = 0.28269  nm)! This is surprising since it is expected that such a thin film should be pseudomorphous. This data analysis should be improved by taking into account the variation of elastic constants with composition (Labat et al., 2000). This will modify the value of the strain-free lattice parameter but will not alter the in-plane lattice parameter. One notes, however, that the dependence of elastic constants on composition for an alloy of Fe (a 3d element) with a metalloid is probably not straightforward.

In a following step, we have calculated the expected sin² plot in the presence of the distance gradient shown in Fig. 2. This calls indeed for an assumption on the in-plane lattice parameter value. If the layer is pseudomorphous, i.e. if the in-plane lattice parameter is equal to a_GaAs/2, the expected positions of the diffraction lines may be readily calculated. Indeed, in a /2 experiment it is the length of the scattering vector that is measured. The corresponding sin² plot is shown in Fig. 7 for the same reflections as in Fig. 6. Qualitatively there is rather good agreement with the experimental plot of Fig. 6. The dispersion between the lattice parameters extracted at = 0° for the 002, 004 and 006 lines is directly related to Fig. 5. More quantitatively, an in-plane lattice parameter of 0.28297  nm and a stress-free lattice parameter of 0.28852  nm are obtained from the calculated sin² plot of Fig. 7. The discrepancies with the values extracted from the experimental plot can be attributed to experimental errors, such as goniometer misalignment, which could affect the experimental data. In an attempt to minimize experimental errors we have, however, corrected the diffraction peak positions by using the GaAs substrate as an internal standard. But a sizeable error still remains for the determination of Fe diffraction peak positions because this very weak diffraction peak is superimposed on a high sloping background arising from GaAs Bragg peak. Since GaAs is on the high-q side, this may introduce a slight shift towards large q values, in agreement with the trend of the observed discrepancy between calculated and measured values.

3. Conclusion

In conclusion we have shown that the presence of distance gradients in thin films may make the standard sin² analysis unusable. Indeed, for very thin films the X-ray scattered amplitudes should be added coherently. Under these conditions, because the number of diffracting planes is reduced and the interplanar distances are not constant, the long-range periodicity is lost. Hence the diffraction positions are no longer directly related to the interplanar spacings and a full structure calculation through the use of the diffracted intensity is necessary. These results have important practical consequences concerning the determination of strains by conventional XRD techniques in ultrathin films. Systematic calculations performed with the same distance profile and varying film thickness indicate that a relative difference of 0.1% between the average lattice parameter and that deduced from the position of the eighth diffraction order is reached when the graded zone is about 7% of the total film thickness.