Combined experimental and theoretical study of fast atom diffraction on the ${\beta }_2(2\times 4)$ reconstructed GaAs(001) surface

A grazing incidence fast atom diffraction (GIFAD or FAD) setup, installed on a molecular beam epitaxy chamber, has been used to characterize the ${\beta }_2(2\times 4)$ reconstruction of a GaAs(001) surface at $530{}^{\circ }\mathrm{C}$ under an ${\mathrm{As}}_4$ overpressure. Using a 400-eV ${}^4\mathrm{He}$ beam, high-resolution diffraction patterns with up to eighty well-resolved diffraction orders are observed simultaneously, providing a detailed fingerprint of the surface structure. Experimental diffraction data are in good agreement with results from quantum scattering calculations based on an ab initio projectile-surface interaction potential. Along with exact calculations, we show that a straightforward semiclassical analysis allows the features of the diffraction chart to be linked to the main characteristics of the surface reconstruction topography. Our results demonstrate that GIFAD is a technique suitable for measuring in situ the subtle details of complex surface reconstructions. We have performed measurements at very small incidence angles, where the kinetic energy of the projectile motion perpendicular to the surface can be reduced to less than 1 meV. This allowed the depth of the attractive van der Waals potential well to be estimated as $-8.7$ meV in very good agreement with results reported in literature.

Figure 1

Schematic representation of the experimental setup showing a GIFAD ion source, consisting of an ion gun and neutralization chamber mounted by a flexible bellows onto an MBE growth chamber, shown here in side view. The angle of incidence of the atom beam on the sample is ${\varphi }_{\mathrm{in}}<1^{\circ }$. The imaging detector is mounted on the opposite side of the chamber, and consists of two microchannel plates, a phosphor screen and a CDD camera.

Figure 2

Schematic representation of the ${\beta }_2\mathrm{(2}\times 4)$ reconstruction of the GaAs(001) surface. The atomic arrangement plotted in the bottom is the one probed when the beam is aligned along the $\left[1\overline{1}0\right]$ direction. The left gray square shows the projected bulk unit cell while the gray rectangle shows the reconstructed surface unit cell. Topmost atoms are represented with larger radii.

Figure 3

Sketch of the scattering geometry. A helium atom of momentum $k$ impinges the ${\beta }_2\mathrm{(2}\times 4)$ reconstructed GaAs(001) surface. $\varphi$ stands for the polar incidence angle and $\Gamma$ stands for the azimuthal incidence angle. The $\Gamma$ angle is measured with respect to the low-index $\left[1\overline{1}0\right]$ direction. The diffracted beams leave the surface at exit angles ${\varphi }_j$ and ${\Gamma }_j$ detected in the momentum space $(k_x,k_z)$. The index $j$ stands for the diffraction order. Note that to be consistent with the convention used in the quantum scattering formalism, $j=0$ is taken to be the diffraction spot in the axial channeling direction, rather than the specular spot. The blue Laue circle corresponds to the case where the low-index direction at the surface is perfectly aligned the incident beam, whereas the black one corresponds to a misalignment angle $\Gamma$ of two lattice vectors $G$ through the relation sin($\Gamma )=2G/k$.

Figure 4

Diffraction image recorded for 400-eV helium atoms impinging at the ${\beta }_2(2\times 4)$ reconstructed GaAs(001) surface. The incident beam is oriented with an incidence angle $\varphi =0.{65}^{\circ }$ ($E_{\perp }=51.8\mathrm{meV})$, and with a misalignment angle $\Gamma =0.{06}^{\circ }$ with respect to the $\left[1\overline{1}0\right]$ direction. The diffraction spots are located along the Laue circle of radius $R$, centered at $(x_c,z_c)$. The vertical white line linking the specular and the direct beam spots is the intercept with the scattering plane.

Figure 5

The raw intensity profile taken along the Laue circle from the diffraction image (Fig. 4) is given by the blue dashed line. The intensity after the inelastic contribution has been interpolated and substracted is given by the red solid line.

Figure 6

The ab initio results $V_{\mathrm{DFT}}(x_n,y_n,z)$ (dots) and the parametric fit given by Eqs. (12) and (13) (lines) for the interaction potential between He projectile and the ${\beta }_2(2\times 4)$ reconstructed GaAs(001) surface. Results are shown as a function of the projectile surface distance $z$ measured from the central As layer of the 13-layer-thick GaAs slab as used in the DFT calculations. Prior to the fit, the attractive part of the DFT data has been scaled as explained in the main text so that the theoretical diffraction charts match the experimental ones at low $E_{\perp }$ [see Sec. 2b2]. Different panels of the figure correspond to the different impact points $(x_n,y_n)$ at the surface as indicated in Fig. 7.

Figure 7

(a) The contour plot of the averaged projectile-surface interaction potential $V_{2\mathrm{D}}(x,z)$. The label of the contour lines give the potential in meV. Vertical dashed lines separated by $L_x$ delimit a unit cell. Atoms used in the ab initio studies are sketched with blue (As) and red (Ga) dots. (b) Surface corrugation function $\mathcal{Z}(x,y)$ as function of the $x$ and $y$ coordinates parallel to the ${\beta }_2(2\times 4)$ reconstructed GaAs(001) surface. $\mathcal{Z}(x,y)$ is defined such that the full 3D potential $V(x,y,\mathcal{Z})=120\mathrm{meV}$. The surface unit cell is shown with dashed lines. Crosses indicate the impact points $(x_n,y_n)$ used for the DFT calculation of the potential $V_{\mathrm{DFT}}(x_n,y_n,z)$. The numbered impact points correspond to the panels of Fig. 6.

Figure 8

Deconvoluted diffracted intensities recorded for 400 eV He atoms at $\varphi =0.{72}^{\circ }$, i.e., $E_{\perp }=63\mathrm{meV}$ (red line), and $\varphi =0.{69}^{\circ }$, i.e., $E_{\perp }=58\mathrm{meV}$ (black line), incidence close to the $\left[1\overline{1}0\right]$ direction of the ${\beta }_2(2\times 4)$ reconstruction of GaAs(001). The azimuthal misalignment angle $\Gamma =-0.{06}^{\circ }$ corresponds to the transverse momentum $k_x$ given by twice the primitive surface reciprocal lattice vector $k_x\approx -2G_x$.

Figure 9

Diffraction charts for ${}^2\mathrm{F}$ atoms incident along the $\left[1\overline{1}0\right]$ direction of the ${\beta }_2(2\times 4)$ phase of GaAs(001). The azimuthal misalignment angle $\Gamma$ is such that the momentum component of the incident beam $k_x\approx -2G_x$, where the reciprocal lattice parameter across the axial channel $G_x=0.39$ Å${}^{-1}$. The intensity of the diffracted beams is displayed as function of the diffraction order (horizontal axis) and perpendicular to the surface momentum of the incident beam $k_z$ (vertical axis). The momentum $k_z$ is measured in Å${}^{-1}$. On the right vertical axis we give the corresponding perpendicular energy $E_{\perp }=k_z^2/2M$. The panels of the figure correspond to (a) theoretical $\mathbb{S}$-matrix calculations and (b) experimental data obtained by changing the incident angle at a constant total energy of 400 eV. The dashed lines delimit the region between closed and open diffraction channels. The green lines show the position of the classical rainbow due to the shallow potential valley between the two topmost As dimers.

Figure 10

Sketch of the semiclassical analysis based on quasispecular reflection from the flat sections of the corrugation function $\mathcal{Z}\left(x\right)$ calculated for $E_{\perp }=100\text{meV}$. (a), (b), and (c) show the diffraction charts associated with the color points depicted in the respective insets where vertical bars are separated by $L_x/4$. In (a), the low-frequency modulation originating from the isolated secondary valleys is outlined. Taking ${\Delta }_{13}={\Delta }_{24}$, the same diffraction chart is obtained either from the top valley [points (1), (2), and (1′)] or from the bottom one [points (2), (4), and (2′)]. The diffraction chart in (b) outlines the high frequency components resulting from the topmost and bottommost sections. In (c), all six flat sections are included reproducing the two-link chain pattern.

Figure 11

Diffraction charts at lower perpendicular energy compared to Fig. 9, for the same GaAs surface. The He perpendicular energy is now below 30 meV and the misalignment angle is here negligible ($\Gamma \sim 0$). The panels (a) and (c) of the figure display theoretical results and panel (b) displays the experimental data. The coupled channel calculations have been performed with atom-surface interaction potential: (a) directly obtained from the ab initio DFT study; (c) obtained by applying a 0.4 scaling factor to the attractive part of the DFT potential. The dashed lines delimit the region between closed and open diffraction channels.