Coherent x-ray diffraction from quantum dots

Coherent x-ray diffraction is a new experimental method for studying perfect and imperfect crystals. Instead of incoherent averaging, a coherent sum of amplitudes produces a coherent diffraction pattern originating from the real space arrangement of the sample. We applied this method for studying quantum dot samples that were specially fabricated GeSi islands of nanometer size and in a regular array embedded into a Si substrate. A coherent beam was focused by special Kirkpatric–Baez optics to a micrometer size. In the experiment it was observed that such a microfocused coherent beam produced coherent diffraction pattern with Bragg spots and broad diffuse maxima. The diffuse peak breaks up into a fine speckle pattern. The grazing incidence diffraction pattern has a typical shape resulting from the periodic array of identical islands. We used this diffraction pattern to reconstruct the average shape of the islands using a model independent approach.

Figure 1

AFM micrographs of 2D ordered Ge islands.

Figure 2

(Color online) Coherent diffraction pattern from a sample in Fig. 1a. Diffraction pattern measured near the (022) reflection of Si with an incidence angle ${\alpha }_i=1\mathrm{deg}$. Reciprocal space coordinates of the center of the diffraction pattern is at (0, 1.99, 1.96), top edge is at ($-0.0218$, 1.9819, 2.0169), the right edge is at ($-0.0101$, 1.9373, 1.9639) in the Si reciprocal lattice. (b) Diffraction pattern in the grazing angle geometry measured near the $(-111)$ reflection of Si with an incidence angle ${\alpha }_i=12.5\mathrm{deg}$ and exit angle ${\alpha }_f=1\mathrm{deg}$. Reciprocal space coordinates of the center of the diffraction pattern is at ($-0.9794$, 0.9800, 0.9696), top edge is at ($-0.9771$, 0.9762, 1.0216), the right edge is at ($-1.0216$, 1.0048, 0.9734) in the Si reciprocal lattice. (c) Calculated diffraction pattern from a sample with a periodic array of quantum dots (from Ref. 26).

Figure 3

(Color online) Coherent diffraction from a sample in Fig. 1 in GISAXS geometry. (a) Geometry with incidence angle ${\alpha }_i$ higher than the critical angle ${\alpha }_c$ of the substrate $({\alpha }_i>{\alpha }_c)$ and CCD detector covering angles smaller than the specularly scattered incident beam; (b) geometry with very small incidence angle $({\alpha }_i<{\alpha }_c)$ and CCD detector covering angles higher than the specular reflected beam. (c) Diffraction patern corresponding to geometry (a), here ${\alpha }_f={\alpha }_c=0.199\mathrm{deg}$ and ${\alpha }_f={\alpha }_i=0.77\mathrm{deg}$. (d) Diffraction pattern corresponding to geometry (b).

Figure 4

Ewald construction for a scattering on a periodic array of quantum dots. Periodic array of dots with the period $d_x$ in the $x$ direction along the beam propagation will give rise to a periodic array of intensities in reciprocal space with the period of $2\pi ∕d_x$. If $N_x$ dots are illuminated by the elongated beam this will give rise to an additional fine structure with spacing in reciprocal space $2\pi ∕N_x{d}_x$. The finite size of the dots in vertical direction $s_z$ will produce additional modulation of intensity in ${\mathbf{Q}}_z$ direction with spacing $2\pi ∕s_z$. Ewald sphere with the radius $\mid k_i\mid =\mid k_f\mid$ cuts side maximuma adjacent to the strong one and produce a fine variation of intensity on the CCD detector [see Fig. 3d].

Figure 5

(Color online) (a) Reconstructed data in the reciprocal space from GISAXS experiment [Fig. 3d] used for reconstruction; (b) reconstructed image of the quantum dot array in real space obtained with the support in the form of elongated window.

Figure 6

(Color online) (a) Experimental data from GISAXS experiment [Fig. 3d] used for reconstruction (log scale); (b) reconstructed data in the reciprocal space; (c) support used for reconstruction; (d) final image of the reconstructed quantum dot array in real space, the period of the array is $0.43\mu \mathrm{m}$ and the average size of the Ge islands in the array are $128\left(h\right)\times 45\left(v\right)\mathrm{nm}$.