Controlling the Propagation of X-Ray Waves inside a Heteroepitaxial Crystal Containing Quantum Dots Using Berry’s Phase

We study a new aspect of the Berry-phase effect as the collaborative x-ray translation by a crystal with undulated deformation. The macroscopic translation was observed around the interface of a heteroepitaxial crystal deformed by quantum dots of $4.1$ germanium monolayers on a silicon substrate. The quantum dots formed a large local gradient of deformation at the interface, which triggered the x-ray translation into two directions. This effect provides a new probe for investigating the interfacial strain, and leads to a single-crystal beam splitter with parallel exit beams.

Figure 1

Schematic diagrams of the fundamental settings. (a) Dispersion relation in two-dimensional momentum space for x-ray propagation inside crystals. A cross-sectional view on the fixed angular frequency, $\omega$, is shown. The $\alpha$ branch and $\beta$ branch correspond to the curves with the incidence angle slightly lower and higher than the Bragg angle. (b) The atomic displacement, from $\overrightarrow{r}$ to $\overrightarrow{r}+\overrightarrow{u}(\overrightarrow{r})$, in real space for the undulated crystal planes around the interface of heteroepitaxial crystals.

Figure 2

(a) Two pairs of points $P_1$ and $P_2$, and $P_3$ and $P_4$ are defined to model the x-ray translation effect on the undulated crystal plane. The reciprocal lattice vectors at these points are represented as ${\overrightarrow{G}}_1$ and ${\overrightarrow{G}}_2$. (b) and (c) Schematic reflected intensity profile as a function of the incident angle at $P_1–P_4$ assuming $\gamma =1/4$ (b) and $\gamma =3/4$ (c). The $\alpha$ and $\beta$ branches are simultaneously excited in the case of (c). Magnified views of the convex and concave regions of (a) are shown in (d) and (e), respectively, showing how the x-ray energy flows associated with the $\alpha$ and $\beta$ branches are deviated. For the case of $1/2<\gamma <1$, the x-ray translation due to the local CP depends on the sign of local gradient of deformation along the $y$ axis, $g_y(\overrightarrow{r})$ in Eq. (4). Expected trajectories crossing the four regions of the CP are exemplified.

Figure 3

Experimental setup, result, and model. (a) Schematic diagram of the experimental setup. For details, see Ref. 11. (b) Measured intensity profiles along the vertical axis on the x-ray imaging detector for seven offset angles. (c) Measured translation from the incident beam position, $y_d=0\mu \mathrm{m}$, as a function of the offset angle from the Bragg angle, $\Delta \theta$. Solid lines show the negative and positive translation on the $\alpha$ and $\beta$ branches, while dotted lines represents the theoretical results. (d) Model of the collaborative translation. Germanium and silicon atoms are schematically represented by circles with different colors with the graduation showing the amount of deformation. The x rays undergo repeated local translation when they pass through region (ii) with $\Delta \theta >0$ and region (iv) with $\Delta \theta <0$ as in Fig. 2d.