Measuring crystal orientation from etched surfaces via directional reflectance microscopy

The DRM apparatus consisted of an Olympus SZ6145 microscope equipped with an industrial monochrome CMOS camera, a custom-built motorized stage providing precise control over the incoming light direction, and a collimated white LED light source. The apparatus is shown in Fig. 1a. We collected a sequence of micrographs (2448 pixels × 2048 pixels) of the etched sample surface from different incoming light directions, \( \varvec{I} \), defined by its corresponding azimuth (\( \varphi \)) and elevation \( (\theta \)) coordinates (Fig. 1b). We set the camera exposure time to ensure that most of the pixels are below the saturation limit of the CMOS sensor throughout the DRM dataset. One DRM dataset comprises a total of 1584 DRM micrographs, taken by varying \( \varphi \) from \( 0^\circ \) to \( 355^\circ \) using step increments of \( 5^\circ \), and \( \theta \) from \( 15^\circ \) to \( 67.5^\circ \) using step increments of \( 2.5^\circ \). For each DRM micrograph of the sample, we captured a set of corresponding background micrographs of a diffuse white surface to normalize DRM micrographs.

This operation is performed in MATLAB and is similar, in principle, to the background subtraction process described in Ref. [12]. Here, however, the operation involves a division as opposed to a subtraction. The background division process is exemplified in Fig. 2.

We acquired EBSD measurements using a step size of 21.4 μm and 6 μm to assess the mean grain orientation and the intragranular grain orientation distribution, respectively. We set the acceleration voltage for both measurements to 20 kV and enable pattern center correction to minimize distortions in the large area scans. We processed the EBSD raw data using MTEX 5.0.3 [17, 18] under MATLAB.

Figure 3 illustrates the sequence of operations required to determine grain orientation from DRM measurements. First, we segment the sample surface (Fig. 3a) by identifying areas where changes in the directional reflectance occur [12]. The segmented image contains a network of grain boundaries that separate all grains that can be resolved by DRM (Fig. 3b). Using this image as a mask, we plot the directional reflectance profile (DRP) from each grain (Fig. 3c). A DRP is a stereographic projection of the reflection intensity measured as a function of the illumination angle, \( \varvec{I} \) [12]. Each DRP contains 1584 intensity values—one for each DRM micrograph—computed as the average reflection intensity over the grain area in that micrograph and normalized between 0 and 1. Because of the faceted surface topography induced during the chemical etching process (Fig. 3d), grain reflectance changes with \( \varvec{I} \). For a certain \( \varvec{I} \), the etch-induced facets reflect light toward the microscope, making the grain appear bright in the corresponding micrograph, and causing the DRP to exhibit a reflectance peak at that illumination direction (Fig. 3c). Via this relationship, DRPs can be analyzed to infer the geometry of the etch-induced surface facets across the grain surface. By determining the relationship between facet geometry and crystallography, we calculate the grain orientation in terms of Euler angles (Fig. 3e).

Grain directional reflectance originates from the specular reflection of light at etch-induced surface facets [12]. Therefore, it should be possible to uniquely derive the orientation and tilt of such facets by identifying the \( \varphi \) and \( \theta \) coordinates corresponding to the center of reflectance peaks in DRPs. Specifically, the peaks \( \varphi \)-coordiante describes the in-plane orientation of the facet, while the \( \theta \)-coordinate is used to compute the facet out-of-plane tilt, \( t \), following [12]

To connect surface facet geometry with the underlying grain orientation, we compute the crystallography of facets by examining the angles between reflectance peaks. The DRP in Fig. 3c shows three reflectance peaks that originate from three sets of facets on the surface of a (111) out-of-plane oriented grain. We hypothesize that the symmetry axis of these three peaks must lie along the [111] crystallographic direction because that is the only threefold symmetry axis in cubic crystals. The facets on the grain surface causing the reflections share the same threefold symmetry and must have Miller indices of the form {x11} where x takes some value larger than 1. Following Eq. 1, we measure facet tilts of ~ 35°, which are consistent with the angles between (111) and the three adjacent {411} surfaces. Grains with orientations other than (111) exhibit different facet geometry. However, all (111) grains give rise to DRPs with threefold symmetry. To confirm this relationship between surface structure and directional reflectance, we measured the topography of another (111) grain using atomic force microscopy (AFM) [19]. The AFM contour map and 3D reconstruction of the surface topography are shown in Fig. 4a, b, respectively. We find that the facet tilt in this grain is approximately 20°, which is consistent with {211} crystallographic planes. We speculate that the surface evolves during etching from flat, to increasingly angled facets as the depth of etching increases (as shown schematically in Fig. 4c). After sufficient time, we would expect the fully etched surface to be dominated by {100} facets, although none of the surfaces investigated here reached that state.

The different facet crystallography found in the two (111) grains suggests that the facet crystallography may change from grain to grain and sample to sample as different surface structures evolve during the etching process, resulting in reflectance peaks with different \( \theta \)-coordinates. However, peaks \( \varphi \)-coordinates should remain relatively constants as {x11} surfaces with changing x share the same in-plane direction. This property provides the opportunity to develop a simple mathematical relationship between grain orientation and directional reflectance for grains around (111) orientation.

Since crystal orientation is defined by three independent angles, we select DRPs that exhibit threefold symmetry and compute the orientation of the corresponding (111) oriented grains using exclusively the three \( \varphi \)-coordinates of reflectance peaks. Figure 4d, e shows an electron micrograph of the surface structure on such a grain and its DRP, respectively. \( \varphi_{x} \), \( \varphi_{y} \), and \( \varphi_{z} \) describe the in-plane orientation of the three facet groups covering the surface of the grain.

To compute the Euler angles, \( \phi_{1} , \Phi , \phi_{2} \) [20], we must express them in terms of these three \( \varphi \)-coordinates. Thus, we write a system of equations to relate Euler angles to \( \varphi \)-coordinates. The rotations matrices used in the Euler angle parametrization are:

Here, the rotation matrix \( R_{i} \left( \alpha \right) \) describes a passive rotation of \( \alpha \) degrees about the Cartesian axis, \( i \), where \( \alpha \) can be \( \phi_{1} \),\( {{\Phi }} \), or \( \phi_{2} \), and \( i \) is either \( X \) or \( Z \). Any grain orientation, \( O \), can be parametrized using these rotation matrices:

Substituting Eq. (2) into Eq. (3), we obtain

Here, \( c \) and \( s \) refer to cosine and sine, respectively. The \( x \) and \( y \) components of the rotated Cartesian basis vectors, \( \left[ {\begin{array}{*{20}c} {X_{x}^{'} } & {Y_{x}^{'} } & {Z_{x}^{'} } \\ \end{array} } \right] \) and \( \left[ {\begin{array}{*{20}c} {X_{y}^{'} } & {Y_{y}^{'} } & {Z_{y}^{'} } \\ \end{array} } \right] \), define the projections of the \( X^{'} \), \( Y^{'} \), and \( Z^{'} \) vectors onto the sample surface. These projections can be also expressed in terms of \( \varphi_{x} \), \( \varphi_{y} \), and \( \varphi_{z} \) as:

The solution of this system is rather involved since \( {{\Phi }} \) and \( \phi_{2} \) are coupled. To simplify the solution, we first impose \( \varphi_{z}^{'} = 90^\circ \) for all DRPs (which corresponds to an in-plane rotation of the crystal and a rotation about the center of the DRP), solve the system of equations as a function of a rotated set of \( \varphi ' \)-coordinates, and then rotate the in-plane orientation of the crystal back by an angle \( \varphi_{z} - 90^\circ \). Setting \( \varphi_{z}^{'} = 90^\circ \) yields \( \phi_{1} \) = 180° (from Eq. 5), which results in the new system of equations

Multiplying the first two expressions in Eq. (6) yields:

We note that \( \varphi_{x}^{'} \) and \( \varphi_{y}^{'} \) can also be written as a function of \( \varphi_{z} - 90^\circ \) as:

Thus, by substituting Eq. (8) into Eq. (7) and correcting the in-plane orientation by \( \varphi_{z} - 90^\circ \), we obtain:

We use Eq. (9) to derive the Euler angles from the measured \( \varphi \)-coordinates of directional reflectance peaks in the DRPs of (111) oriented grains.

To compute \( \varphi_{x} \), \( \varphi_{y} \), and \( \varphi_{z} \) from DRPs, we determine the center of reflection peaks in DRPs by applying an intensity threshold to binarize the reflectance distribution and calculating the centroids of the resulting peaks. Figure 5 shows comparison an example DRP before and after binarization using a threshold of 0.5. We find that the actual threshold value does not affect the accuracy of the \( \varphi \)-coordinates computation significantly. One problem associated with this method is that peaks occurring in proximity of \( \varphi = 0^\circ \) may be counted twice, yielding errors in the computation of the \( \varphi \)-coordinate. With reference to Fig. 5a, the peak that intersect the \( \varphi = 0^\circ \) is split in two halves, each one with a distinct—and incorrect—centers of mass (see Fig. 5c). To solve this issue, we impose periodic boundary conditions along the \( \varphi \)-axis and avoid a miscount of peaks (Fig. 5b, d).

Using binarized DRPs is a simple, computationally inexpensive process to estimate the centroids of reflectance peaks. The centroid of the high-reflectance peak area may not always coincide with the actual peak center (i.e., the position of the intensity maximum). However, re-computing the crystal orientation using a peak-fitting algorithm that identifies the peak local maximum, increased the average measurement error (not shown here), suggesting that simple binary peak estimation is sufficient.