Grazing-incidence diffraction anomalous fine structure: Application to the structural investigation of group-III nitride quantum dots

The relevance of grazing-incidence anomalous diffraction as a tool to investigate the strain and structure of small-size embedded nano-objects is examined. Multiple scattering effects, originating from the grazing-incidence setup, are analyzed with a special emphasis on the cusp of the diffraction anomalous spectrum and the extended diffraction anomalous fine-structure oscillations. It is shown that even for grazing-incidence angle, a Born approximation treatment is justified for quantum dots (QDs) on top of a thin wetting layer. The discussion focuses on the overgrowth of AlN on top of GaN QDs. Both the in-plane and out-of-plane strains in the dots can be specifically determined, by extracting the Ga partial scattering amplitude from measurements of the scattered intensity along both the in- and out-of-plane directions, close to the $\left(30\overline{3}0\right)$ and $\left(30\overline{3}2\right)$ reflections, at several energies across the Ga $K$ edge. The study is complemented by the analysis of the local environment of Ga atoms in the dots through the measurement of the fine-structure oscillations in diffraction condition. The oscillations are found almost insensitive to the grazing-incidence multiple-scattering effects. Accordingly, the out-of-plane strain and possible intermixing specifically in the dots can be deduced. The QDs are shown to remain pure GaN all along the capping process. The QDs strain state exhibits a larger strain relaxation than expected from an elastic model, suggesting the presence of a plastic strain relaxation, possibly through dislocations at the vicinity of the QDs. Finally, the influence of the substrate as regards strain relaxation in the QDs is discussed by comparing our results to those we previously obtained for a series of samples grown on AlN/sapphire.

Figure 1

(Color online) $1\mu {\mathrm{m}}^2$ AFM images of uncapped GaN QDs.

Figure 2

(Color online) Grazing-incidence geometry for the (a) $\left(30\overline{3}0\right)$ and (b) $\left(30\overline{3}2\right)$ reflections. The two reflections are denoted by their scattering vectors, ${\mathbf{Q}}_{\left(30\overline{3}0\right)}$ and ${\mathbf{Q}}_{\left(30\overline{3}2\right)}$. The scattering plane is vertical. The x-ray polarization vectors of the incident and exit beams are ${\sigma }_i$ and ${\sigma }_f$, perpendicular to the scattering plane. Note that for the out-of-plane $\left(30\overline{3}2\right)$ reflection (b), the exit angle is nongrazing. (c) Scattering angle, incident and exit angles, and detector slits setup for summing the scattering contributions along ${\alpha }_f$ for an in-plane reflection.

Figure 3

(Color online) Fluorescence spectrum measured close to the $\left(30\overline{3}0\right)$ reflection, at $h=3.03$, $k=0$, and $l=0$, for 11 ML AlN capping, ${\alpha }_i=0.3°$.

Figure 4

(Color online) Wave vectors of the incident and exit waves (${\mathbf{k}}_i$ and ${\mathbf{k}}_f$), transmitted waves (${\mathbf{k}}_i^t$ and ${\mathbf{k}}_f^t$), and reflected waves (${\mathbf{k}}_i^r$ and ${\mathbf{k}}_f^r$).

Figure 5

(Color online) Fluorescence yield, measured for the $M_0$ sample, with the x-ray polarization vector perpendicular to the sample surface and simulated for isolated QDs. The WL fluorescence yield was vertically shifted for clarity.

Figure 6

Multiple-scattering paths in DWBA.

Figure 7

(Color online) (a) Specular reflectivity $(E=10.3\mathrm{keV})$ simulated for $M_0$ as a function of the incident angle, compared with the experiment. (b) Reflectivity simulated for various ${\alpha }_i$ angles as a function of the energy across the Ga $K$ edge. (c) Simulated and experimental anomalous specular reflectivities simulated for $M_0$ at ${\alpha }_i=0.3°$, across the Ga $K$ edge.

Figure 8

Scattered intensity simulated for $40\times 3.5{\mathrm{nm}}^2$ GaN QDs on $M_0$ (see text for details) along the $\left[30\overline{3}0\right]$ direction, at $10.3\mathrm{keV}$, for each of the four scattering paths, as a function of the exit angle $\left({\alpha }_f\right)$, for (a) ${\alpha }_i=0.15°$ and (b) ${\alpha }_i=0.3°$. The solid line represents the total intensity.

Figure 9

(Color online) Overall scattered intensity simulated for $40\times 3.5{\mathrm{nm}}^2$ GaN QDs on $M_0$ (see text for details) close to the $\left(30\overline{3}0\right)$ reflection, at $10.3\mathrm{keV}$, as a function of the incident ${\alpha }_i$ and exit ${\alpha }_f$ angles.

Figure 10

Scattered intensity simulated for $40\times 3.5{\mathrm{nm}}^2$ GaN QDs on $M_0$ (see text for details), at (a) ${\alpha }_i={\alpha }_f=0.15°$ and (c) ${\alpha }_i={\alpha }_f=0.3°$ in the DWBA, and for the first $\left(1^{\mathrm{st}}\right)$, second $\left(2^{\mathrm{nd}}\right)$, third $\left(3^{\mathrm{rd}}\right)$, and fourth $\left(4^{\mathrm{th}}\right)$ paths. The second and third paths have equal contributions provided that ${\alpha }_i={\alpha }_f$. Constant scale factors were applied to the less intense paths for clarity. Corresponding extracted fine-structure oscillations for (b) ${\alpha }_i={\alpha }_f=0.15°$ and (d) ${\alpha }_i={\alpha }_f=0.3°$. The oscillations were vertically shifted for clarity.

Figure 11

Scattered intensity simulated for $40\times 3.5{\mathrm{nm}}^2$ GaN QDs on $M_0$ (see text for details), summed up over ${\alpha }_f$, at (a) ${\alpha }_i=0.15°$ and (c) ${\alpha }_i=0.3°$ in the DWBA, and for the first $\left(1^{\mathrm{st}}\right)$, second $\left(2^{\mathrm{nd}}\right)$, third $\left(3^{\mathrm{rd}}\right)$, and fourth $\left(4^{\mathrm{th}}\right)$ paths. Constant multiplying scale factors were applied to the less intense path for clarity. Corresponding extracted fine-structure oscillations for (b) ${\alpha }_i=0.15°$ and (d) ${\alpha }_i=0.3°$. The oscillations were vertically shifted for clarity.

Figure 12

(Color online) Scattered intensity measured at ${\alpha }_i=0.15°$ along the $h$ reciprocal space direction close to the $\left(30\overline{3}0\right)$ reflection of SiC, (a) as a function of the AlN deposit, at $10.267\mathrm{keV}$, and (b) below ($-100$ and $-10\mathrm{eV}$), above $(+100\mathrm{eV})$, and at the Ga $K$ edge $\left(10.367\mathrm{keV}\right)$ for 11 ML AlN deposit. (c) Corresponding extracted partial scattering contributions (see text for details) compared to the square root of the scattered intensity at $10.267\mathrm{keV}$, for 11 ML AlN deposit. Note that the extraction procedure is not reliable close to $h=3$, where the intensity variations are mainly due to the sharpness of the SiC peak. (d) Partial scattering contributions and square root of the scattered intensity for 4 ML AlN deposit.

Figure 13

(Color online) Schematic representation in the complex plane of the structure factor $F$ as a function of $F_T$, $F_{\mathrm{Ga}}$ and $F_{\mathrm{N}}$ (see text).

Figure 14

In-plane lattice parameter $a_{\mathrm{Ga}\mathrm{N},\mathit{MAD}}$ and strain (relative to bulk GaN) in GaN deduced from the position of the $F_{\mathrm{Ga}}$ peak maximum. Bulk GaN gives ${\epsilon }_{xx,\mathit{MAD}}=0\%$ with $a_{\mathrm{Ga}\mathrm{N},\mathit{\text{bulk}}}=0.3189\mathrm{nm}$, while bulk AlN gives ${\epsilon }_{xx,\mathit{MAD}}=-2.4\%$ with $a_{\mathrm{Al}\mathrm{N}}=0.3112\mathrm{nm}$. Note that the uncertainties refer to the $a_{\mathrm{Ga}\mathrm{N},\mathit{MAD}}$ values.

Figure 15

(Color online) Scattering contributions (a) $\parallel F_T\parallel$ and (b) $\parallel F_{\mathrm{Ga}}\parallel$ mapped along the in- and out-of-plane $h$ and $l$ directions, close to the $\left(30\overline{3}2\right)$, extracted from MAD measurements at ${\alpha }_i=0.15°$ at three energies across the Ga $K$ edge for 11 ML AlN deposit. The cross indicates the maximum of $\parallel F_{\mathrm{Ga}}\parallel$.

Figure 16

Diffracted intensity measured as a function of the incident angle ${\alpha }_i$ at maximum Ga contribution $\left(h_{\mathrm{Ga}}\right)$ as deduced by MAD, for the sample with 2 ML of AlN capping.

Figure 17

(Color online) DAFS spectrum measured at maximum Ga $\left(30\overline{3}0\right)$ contribution, at ${\alpha }_i=0.3°$, for the sample with 4 ML AlN capping, and fit of the line shape according to the MAD principles. The fit was vertically shifted for clarity. See text for details.

Figure 18

(Color online) Experimental EDAFS (open circles) for the samples with (a) 0, (b) 2, (c) 4, (d) 8, (e) 11, and (f) 18 ML AlN capping as compared with the best-fit result (solid lines).

Figure 19

Scheme of GaN wurzite structure. The most relevant virtual phoelectron scattering paths used for the EDAFS simulation are represented as follows: 1, out-of-plane I shell Ga-N; 2, nearly in-plane I shell Ga-N; 3, out-of-plane II shell Ga-Ga (Al); 4, III shell Ga-N along [0001]; and 5, IV shell Ga-N. Ga atoms are represented by white spheres, N by black ones.

Figure 20

(Color online) Out-of-plane lattice parameter $a_{\mathrm{Ga}\mathrm{N}}$ and strain (relative to bulk GaN) in GaN deduced from the EDAFS oscillations analysis and from the position of the $F_{\mathrm{Ga}}$ maximum (MAD). Bulk GaN gives ${\epsilon }_{zz}=0\%$ with $c_{\mathrm{Ga}\mathrm{N},\mathit{\text{bulk}}}=0.5185\mathrm{nm}$, while bulk AlN gives ${\epsilon }_{xx}=-3.9\%$ with $c_{\mathrm{Al}\mathrm{N}}=0.4982\mathrm{nm}$. Note that the uncertainties refer to the $c_{\mathrm{Ga}\mathrm{N}}$ values.

Figure 21

(Color online) GaN QD strain ${\epsilon }_{xx}=(a_{\mathrm{Ga}\mathrm{N},\mathit{MAD}}-a_{\mathrm{Ga}\mathrm{N},\mathit{\text{bulk}}})∕a_{\mathrm{Ga}\mathrm{N},\mathit{\text{bulk}}}$ versus ${\epsilon }_{zz}=(c_{\mathrm{Ga}\mathrm{N},\mathit{EDAFS}}-c_{\mathrm{Ga}\mathrm{N},\mathit{\text{bulk}}})∕c_{\mathrm{Ga}\mathrm{N},\mathit{\text{bulk}}}$ values for all the samples studied compared with elastic biaxial strain of a GaN thin film.