The Ga diffusion length on the well-ordered GaAs(001)-(2 × 4) β2 surface was studied.

Figure 6 presents a schematic of the -(2 × 4)β2 surface. The migration barriers from

x to

x′ site and the Ga desorption energy from the

x site on the -(2 × 4)β2 surface are shown in

Table 3. In the MC random-walk simulation, the two-dimensional periodic boundary conditions were employed to the potential surface,

i.e., an extremely flat and defect-free surface was considered. Recently, a more precise Ga migration potential on GaAs(001)-(2 × 4)β2 was reported [64]. If their data matrix is applied to the MC simulation, more precise properties could be obtained. In the present MC simulations, we applied the coarse data matrix presented in

Figure 6 to confirm the feasibility of the simple model described in

Section 2.2. First, we compared the calculated surface lifetime,

τ, and diffusion coefficient,

D, with those obtained from experiments. Then, we discuss the diffusion length,

L, of Ga on the -(2 × 4)β2 surface because

L is generally given by

$L=\sqrt{2D\tau}$.

Figure 7a,b shows the Ga

τ and

D, respectively, as a function of the reciprocal temperature. The green solid and dashed lines in

Figure 7a represent the calculated Ga surface lifetime before desorption and before incorporation, respectively, as estimated by the ion-beam technique [

65]. The experiments were conducted using GaAs(001) that was misoriented by 2.3° ± 0.5° toward the (110) surface. If

τ is sufficiently long for Ga diffusion to reach the step edges, Ga would be incorporated into the crystal at the step edges or kink sites. We found that the Ga surface lifetime before desorption above ~860 K was shorter than that before incorporation. This result suggests that the Ga adatom would desorb from the terrace because it could not reach the step edges or kink sites due to the short

τ. Thus, the Ga incorporation–desorption transition temperature is estimated to be ~860 K, and this result agrees well with experimental results [

66]. Therefore, the decrease of the GaAs growth rate becomes significant above ~920 K and suggests that our computational method is feasible for predicting the Ga surface lifetime,

τ. As presented in

Figure 7b, the calculated Ga diffusion coefficients are represented by brown solid lines with open squares for the

$[1\overline{1}0]$ direction,

${D}_{[1\overline{1}0]}$, and with filled squares for the [110] direction,

D_{[110]}. The

${D}_{[1\overline{1}0]}$ is approximately five times larger than

D_{[110]} because the Ga adatom easily migrates along the missing As-dimer rows along the

$[1\overline{1}0]$ direction [

67]. This result agrees with experimental results [

68], and the Ga diffusion coefficient along the

$[1\overline{1}0]$ direction is approximately four times larger than that along the [110] direction. In addition, the diffusion coefficient lines calculated as a function of reciprocal temperature all lie between the lines obtained by the experiments [

65,

69]. These results confirm the validity of our computational method for predicting

τ and

D.

**Figure 6.**
Plane view of GaAs(001)-(2 × 4)β2. Adsorption sites for Ga are indicated by numbers. The migration barriers and desorption energies are presented in

Table 3.

Next, we calculated the Ga diffusion length,

L, on the -(2 × 4)β2 surface.

Figure 8 shows

L as a function of reciprocal temperature under the condition of

p_{Ga} = 1.4 × 10

^{−6} Torr. In

Figure 8, solid lines with open and filled squares show the calculated Ga diffusion length along the

$[1\overline{1}0]$ and [110] directions, respectively. The diffusion length decreases exponentially with temperature, even though the diffusion coefficient increases with temperature, as shown in

Figure 7b. This behavior is because the Ga surface lifetime decrease influences the diffusion length more effectively than the influence of the diffusion coefficient increase. As presented in

Figure 8, the extrapolated diffusion length value,

${L}_{[1\overline{1}0]}$, along the

$[1\overline{1}0]$ direction is approximately 700 nm at the incorporation-desorption transition temperature (

T = ~860 K).

Figure 8 presents experiments from [

70], where

${L}_{[1\overline{1}0]}$ = 250~1200 nm at 873 K. The results suggest that our computational method is appropriate for predicting the diffusion length on the surface.

**Figure 7.**
(

**a**) Ga surface lifetime,

τ; and (

**b**) diffusion coefficient,

D, as a function of reciprocal temperature. Green solid and dashed lines are the calculated

τ before desorption and

τ before incorporation [

66], respectively. Brown solid lines with open and filled squares are the

${D}_{[1\overline{1}0]}$ and

D_{[110]}, respectively. The experimental results for the Ga diffusion coefficient are also presented in the diagram by orange dotted (

) [

69] and dashed lines (

) [

65].

**Figure 8.**
Ga diffusion length, L, as a function of reciprocal temperature at p_{Ga} = 1.4 × 10^{−6} Torr.