Strain determination in MBE-grown $\mathrm{InAs}$ quantum wires on $\mathrm{InP}$

We have determined the strain in the three crystallographic directions in $\mathrm{InAs}$ quantum wires (QWr) grown by molecular beam epitaxy on $\left(001\right)\mathrm{InP}$ substrate. We used triple crystal x-ray diffraction to make scans along and perpendicular to the QWr direction in reciprocal space, around $\mathrm{InP}\left(220\right)$ reflections. We use the shape and strain sensitivity of the different scans to deconvolute both contributions. We used the ${\alpha }_f$ scan analysis in grazing incidence diffraction to measure the strain relaxation perpendicular to the QWr as a function of height in the wire. We finally compare these results with finite elements calculations of the strain tensor in $\mathrm{InAs}$ QWr on $\mathrm{InP}$.

Figure 1

(Color online) Atomic force microscopy (AFM) image of the surface of the studied sample showing the highly ordered arrays of self-assembled QWr-like structures of $\mathrm{InAs}$ on $\mathrm{InP}\left(001\right)$. The QWr are aligned parallel to the [1-10] direction. They are about $10–20\mathrm{\AA }$ high and can reach up to $1\mathrm{\mu }\mathrm{m}$ in length.

Figure 2

(Color) Reciprocal space scheme, showing some of the different scans performed in this work. Scans 1 and 2 are, respectively, angular scan perpendicular to the QWr radial scan parallel to the QWr, performed around the (2-20) reflection of the $\mathrm{InP}\left(001\right)$ substrate. Scans 3 and 4 are, respectively, angular scan parallel to the QWr and radial scan perpendicular to the QWr passing through the (220) reflection of the $\mathrm{InP}\left(001\right)$ substrate. The radial scan perpendicular to the QWr (scan 4) is simulated (Fig. 6) in order to obtain the strain profile along the [110] direction.

Figure 3

(Color online) Angular and radial scans near the (220) and (2-20) reflections of $\mathrm{InP}\left(001\right)$ substrate (scans 1–4 in Fig. 2). Angular scans are sensitive to the shape and correlation, while radial scans are sensitive also to the strain.

Figure 4

(Color) The angular scan passing through the (2-20) reflection is analyzed as the product of the correlation and the shape functions. The results of the fit are lateral $\mathrm{period}=190\mathrm{\AA }$, QWr $\mathrm{width}=138\mathrm{\AA }$, and lateral $\mathrm{correlation}=2490\mathrm{\AA }\sim 13$ periods.

Figure 5

Reciprocal space map around the $\mathrm{InP}\left(2\text{−}20\right)$ reflection. For clarity, we have shown only the diffracted intensity lower than 60% of the maximum of the $\mathrm{InP}$ peak. From this map we can obtain that: (a) all the $\mathrm{InAs}$ in the QWr has in the [1-10] direction the lattice parameter of the $\mathrm{InP}$ substrate, (b) the QWr have a length along the [1-10] direction of $2000\mathrm{\AA }$, obtained by the application of Scherer’s formula (Ref. 15) to the width of the correlation maxima to the sides of the $\mathrm{InP}$ peak.

Figure 6

(Color online) Comparison between the simulated and the experimental radial scan along the [110] direction passing by the $\left(220\right)\mathrm{InP}$ substrate reflection. From the simulation, a profile of the QWr strain along the [110] direction is obtained.

Figure 7

(Color) Shift of the maximum of the ${\alpha }_f$ while scanning along $q_r$ (scan 4 in Fig. 2d with recording of the ${\alpha }_{\mathrm{f}}$ scan). According to Kegel et al. Refs. 8, 9 the displacement of the maximum of ${\alpha }_f$ (normalized to the value of the critical angle ${\alpha }_c$) is related to the height $\left(z\right)$ for each $q_r$. On the other hand, $q_r$ is directly related to the lattice parameter along the [110] direction.

Figure 8

(Color online) Measurement of the lattice parameter along the [110] direction for increasing $z$ (height in the QWr), obtained from the shift of the ${\alpha }_{\mathrm{f}}$ scan [Kegel’s method (Refs. 8, 9)].

Figure 9

(Color) Strain distribution along QWr height $\left(E_{001}\right)$ obtained applying the elastic theory to the lattice parameters values along 110 and 1-10 $(E_{110},E_{1-10})$ obtained by measuring XRD at grazing incidence. This model assumes a diagonal strain tensor, which is obviously a simplification. For a more realistic model, it is possible to use finite elements simulations (Fig. 10)

Figure 10

(Color) Finite elements simulation of the vertical cross section of the QWr truncated triangle. The simulations yield a three-dimensional representation of the strain tensor, with nonzero off-diagonal strain values. We show only one of the two-dimensional maps for simplicity, with axis [110] (horizontal), and [001] (vertical). Note the important border and surface effects.