Waveguide-enhanced grazing-incidence small-angle x-ray scattering of buried nanostructures in thin films

X-ray standing waves generated by the interference of the scattered x rays from parallel surfaces of a thin film, the so-called waveguide effect, can be used to enhance or reduce the scatterings from certain depths of the film. Used in combination with grazing-incidence small-angle x-ray scattering, this resonance effect provides depth sensitivity to extract buried structures in thin films of polymer and polymer/nanoparticle nanocomposite, which are not readily accessible by most surface techniques, such as scanning probe microscopy. We developed a rigorous theory of the diffuse scattering in the framework of the distorted-wave Born approximation using a discretization method analogous to Parratt’s recursive formalism. In such a case, the distortion of the electric field of the unperturbed state from the nanostructures of interest is considered in a self-consistent manner. This theory allows a quantitative determination of the buried nanostructures when the x-ray waveguide enhancement is present or the size of the nanostructures of interest is comparable to or larger than the spatial frequency of electric-field intensity modulation. A unique capability afforded by this theory is that a nanometer or even subnanometer spatial resolution can be achieved in the depth information of the buried nanostructures, along with the in-plane correlation of the structures.

Figure 1

GISAXS geometry from a multilayer film of embedded nanostructures. The incident and exit wave vectors are ${\mathit{k}}^i$ and ${\mathit{k}}^f$. The incident and exit angles are ${\alpha }^i$ and ${\alpha }^f$. The angle out of the scattering plane is $\chi$.

Figure 2

Schematic of the wave vector transfers ${\mathit{q}}_j^m$ in the $j$th layer. They are defined as the vector differences between the wave vectors ${\mathit{k}}^i$ (incident) and ${\mathit{k}}^{\prime i}$, and those for the time-reversed state ${\mathit{k}}^f$ (exit) and ${\mathit{k}}^{\prime f}$.

Figure 3

Calculated EFI profiles extended above a silver mirror in air for two incident angles.

Figure 4

(a) Schematic of the slicing of a spherical nanoparticle buried at the middle of a Si-supported polystyrene film. Particle size is not drawn to scale. (b) Normalized EFI plotted as a two-dimensional (2D) map as a function of incident angle and film depth. (c) EFIs for three incident angles. The shadow region corresponds to the location of the nanoparticle. (d) EFIs at the particle region [shadow in (c)] rescaled to EFI $(z=d/2)$.

Figure 5

Calculated differential cross sections at an incident angle of (a) 0.140${}^{\circ }$, (b) 0.159${}^{\circ }$, and (c) 0.171${}^{\circ }$, respectively, using four models described in the text for a gold nanoparticle buried at the middle of a Si-supported polystyrene film. The curves are shifted vertically for a better comparison.

Figure 6

(a)–(c) Calculated normalized EFI with an incident-beam divergence 0.001${}^{\circ }$, 0.005${}^{\circ }$, and 0.01${}^{\circ }$ for the sample shown in Fig. 4a. (d)–(f) Corresponding EFI profiles for three incident angles, 0.140${}^{\circ }$, 0.159${}^{\circ }$, and 0.171${}^{\circ }$. (g)–(i) EFIs rescaled to the EFI values at $z=d/2$.

Figure 7

Effect of the incident-beam divergence on the differential cross section at a nominal incident angle of (a) 0.159${}^{\circ }$ and (b) 0.171${}^{\circ }$, using the multilayer DWBA model with sliced form factor.

Figure 8

(a) and (b) are the reflectivity and the normalized EFI for a sandwiched film schematically shown in the inset of (a). Critical angles for PtBA and Ag are shown as red dotted lines in (a). The white dotted line in (b) indicates the position of the Au nanoparticle monolayer. (c) Electron density profile normal to the surface. (d) and (e) Calculated GISAXS patterns with incident angle at the first resonant angle and a nonresonant angle marked by arrows in (a). (f)–(j) Same film but with the Au nanoparticles evenly dispersed throughout the PtBA layer.

Figure 9

(a) Schematic of the nanoparticle-block copolymer composite film. (b) and (c) Calculated GISAXS patterns at a grazing-incidence angle of 0.15${}^{\circ }$ when the depth of the nanoparticles is at $z_{\mathrm{np}}=2$ nm, 10 nm, 50 nm and uniformly distributed in P2PV, respectively. (d) Comparison of the vertical linecuts along the (10) diffractions.

Figure 10

(a) Transmission electron microscopy (TEM) image of the nanoparticle-polymer composite film after solvent and thermal annealing. (b) GISAXS pattern taken with x rays of energy 7.35 keV at an incident angle of 0.18${}^{\circ }$. (c) Circles correspond to the vertical linecut along the (20) lamellar diffraction peak marked by the dotted line in (b). Solid line is the best fit to the model described in the text. Inset is the fitted electron-density profile showing an aggregation of nanoparticles near the surface (shadow region).

Figure 11

Scanning electron microscopy (SEM) image of a hexagonally patterned silicon substrate. (b) Schematic of the ordered embedded P2VP micellar monolayer within in a well defined by the pattern. (c) GISAXS image at an incident angle of 0.195${}^{\circ }$. (d) Circles correspond to the vertical linecut along the (10) micellar diffraction peak. Solid line: Best fit to the model described in the text with the micellar center located at a depth of 11.9 nm, near the middle of the PS matrix film. Dashed line: Calculated values assuming the depth of the micellar center is 8 nm, near the top of the PS film. Inset: Fitted electron-density profile and corresponding normalized EFI at 0.195${}^{\circ }$.

Figure 12

(a) and (b) Reflectivity and normalized EFI for Si-supported Ge nanoislands of dimensions displayed in the inset in (a). The arrows in (a) indicate the three incident angles for GISAXS calculations, with corresponding EFIs shown in (c). (d)–(f) and (g)–(i) Calculated GISAXS patterns using the multilayer DWBA and island DWBA theories described in the text, respectively.