Ab initio modeling of optical functions after strain wave perturbation for defect detection

We model the change in the linear optical functions due to the propagation of a strain wave through a crystalline slab. Defects in the slab are introduced by arbitrarily displacing one of the atomic layers. The perturbing wave is assumed to be produced by laser light absorption; the properties of the propagating wave are explicitly related to the laser and material properties. We apply this model to a Si(111)($1\times 1$):H slab and find that the reflectance varies significantly at photon energies above 2.5 eV and that the disordered slab produces significant changes in the reflectance when compared to the unperturbed slab. We then induce a defect layer in the slab by uniformly translating an atomic layer. As the strain wave passes through the material, the behavior of the calculated reflectance difference spectrum changes according to the defect layer, allowing us to discern the depth of the defect.

Figure 1

Time snapshot of the (a) strain and (b) displacement profiles (as functions of the distance traveled) of a CAP pulse using parameters for silicon (Tables 1 and 2). These plots were calculated for a fluence $Q=0.3$ mJ/${\mathrm{cm}}^2$ and a pump photon energy $E_{\mathrm{pump}}=3.54$ eV, which corresponds to an absorption coefficient $\alpha =1.04\times {10}^6$ ${\mathrm{cm}}^{-2}$. (c) Cartoon representation of the slab unit cell, with two Si labeled atoms that will be moved in the direction indicated to create buried defect layers. The $x$ and $y$ axes are taken along the $\left[11\overline{2}\right]$ and $\left[1\overline{1}0\right]$ directions, respectively.

Figure 2

(a) $\mathrm{\Delta }R$ for maximum displacements of 40%, 30%, and 20% of the diagonal bond height $H_d=0.784$ Å, generated with three different fluence values ($Q=0.586$, 0.440, and 0.293 mJ ${\mathrm{cm}}^{-2}$). (b) $\mathrm{\Delta }R$ calculated from a maximum displacement of 40% of $H_d$ for the normal slab and five cases of slabs with defect layers at the atom indicated in the label for each curve; all curves are produced when the strain maximum (the wave front) is at the top surface of the material. The dashed box highlights the defect peak.

Figure 3

Three-dimensional representation of the reflectivity difference spectrum $\mathrm{\Delta }R$ as a function of the strain wave depth inside the slab. The red arrow indicates the direction of propagation of the wave, starting at a depth of 0 (top surface of the slab) to 151 Å (bottom surface of the slab). Colored energy regions will be referred to in the text. (a) Strain wave traveling through a slab with no defect. $\mathrm{\Delta }R$ goes from maximum to minimum, passing through zero in the middle of the slab. The thin black line indicates the center of the slab at which $\mathrm{\Delta }R$ is at a minimum. (b) Strain wave traveling through a slab with a defect layer at atom ${\mathrm{Si}}_{50}$. The defect peak goes from positive to negative at the exact position of the defect layer. The thin black line is a visual aid indicating the depth at which change in $\mathrm{\Delta }R$ occurs due to the defect layer.

Figure 4

Peak behavior for the normal and defective slabs over the (a) 2.5–3.0 eV, (b) 3.0–3.5 eV, and (c) 4.0–4.5 eV regions. These correspond to the light red, light green, and light purple regions from Fig. 3, respectively. Thin vertical colored dashed lines indicate the defect depth within the slab. The thick black vertical dashed line represents the middle of the slab.

Figure 5

The planar-integrated electron density ${\rho }_{n\mathbf{k}}\left(z\right)$ as a function of depth for the relaxed slab and the defective slab with a defect layer at ${\mathrm{Si}}_{13}$. The small black arrow indicates the $z$ position of ${\mathrm{Si}}_{13}$.