*4.1. Formation Process of Deep Nanoholes*

Figure 18 schematically shows the structural change in the electron exit surface at the initial stage of irradiation. At first, surface vacancies are formed in the first layer of the metal by sputtering. Then, the surface vacancies agglomerate to produce monolayer islands of surface vacancies. Under continuous irradiation, surface vacancies are formed in the area of islands, and finally, multilayered pits are generated. The growth process of the pits has been theoretically given by Cherns [11]. Nonetheless, his model does not predict the formation of the aligned nanogrooves and the deep nanoholes found in the present work.

In contrast with what has been said above for the formation of nanogrooves, the deepening of the nanoholes is not believed to involve self-organization. As may be seen from Figure 3, the diameters of nanoholes do not vary significantly along their depth, indicating one-dimensional growth along the irradiation direction. We can estimate that more than 2000 atoms must be removed to produce a nanohole of 1.5 nm width and 20 nm depth. We guess that irradiation-induced diffusion of surface vacancies plays a significant role,

although there may be other processes leading to the deepening of nanohole. Note that the nanoholes are generated on the rear surface, growing opposite to the irradiation direction. Irradiation-induced diffusion of surface vacancies on the side walls of the holes can result in a vacancy migration opposite to the irradiation direction, which is towards the bottom of the nanohole. Thus, nanoholes can be deeper by this process. Note that the nanoholes are generated on the rear surface, growing opposite to the irradiation direction. Irradiation-induced diffusion of surface vacancies on the side walls of the holes can result in a vacancy migration opposite to the irradiation direction, which is towards the bottom of the nanohole. Thus, nanoholes can be deeper by this process. How does the nanohole grow so deep? First, one can consider that a geometrical fac-

Figure 18 schematically shows the structural change in the electron exit surface at the initial stage of irradiation. At first, surface vacancies are formed in the first layer of the metal by sputtering. Then, the surface vacancies agglomerate to produce monolayer islands of surface vacancies. Under continuous irradiation, surface vacancies are formed in the area of islands, and finally, multilayered pits are generated. The growth process of the pits has been theoretically given by Cherns [11]. Nonetheless, his model does not predict the formation of the aligned nanogrooves and the deep nanoholes found in the present

In contrast with what has been said above for the formation of nanogrooves, the deepening of the nanoholes is not believed to involve self-organization. As may be seen from Figure 3, the diameters of nanoholes do not vary significantly along their depth, indicating one-dimensional growth along the irradiation direction. We can estimate that more than 2000 atoms must be removed to produce a nanohole of 1.5 nm width and 20 nm depth. We guess that irradiation-induced diffusion of surface vacancies plays a significant role, although there may be other processes leading to the deepening of nanohole.

*Quantum Beam Sci.* **2021**, *5*, x FOR PEER REVIEW 15 of 20

*4.1. Formation Process of Deep Nanoholes* 

work.

How does the nanohole grow so deep? First, one can consider that a geometrical factor would be responsible for the deepening of pits. Surface vacancies created in a terrace with a width w, as shown in Figure 19, can contribute to the erosion of the descending step but not of the ascending step at temperatures below which the layer-by-layer removal starts [28]. We can easily see that the area A in the terrace becomes larger near the center if the length of the descending step and the width of the terrace w are the same. This means that the erosion rate of the descending step near the center should be faster in this case due to the large amount of incoming surface vacancies for the unit length of the descending step. Under prolonged irradiation, however, the pit profile should reach a steady state after a critical dose, because a faster erosion of the descending step would lead to a decrease in the terrace width, thereby leading to a decrease in the erosion rate of the descending step. tor would be responsible for the deepening of pits. Surface vacancies created in a terrace with a width w, as shown in Figure 19, can contribute to the erosion of the descending step but not of the ascending step at temperatures below which the layer-by-layer removal starts [28]. We can easily see that the area A in the terrace becomes larger near the center if the length of the descending step and the width of the terrace w are the same. This means that the erosion rate of the descending step near the center should be faster in this case due to the large amount of incoming surface vacancies for the unit length of the descending step. Under prolonged irradiation, however, the pit profile should reach a steady state after a critical dose, because a faster erosion of the descending step would lead to a decrease in the terrace width, thereby leading to a decrease in the erosion rate of the descending step.

**Figure 19.** Oblique and cross-sectional views of a circular pit. Surface vacancies created in a terrace with width w can contribute to the erosion of the descending step but not of the ascending step at temperatures below that at which the layer-by-layer removal starts [28]. **Figure 19.** Oblique and cross-sectional views of a circular pit. Surface vacancies created in a terrace with width w can contribute to the erosion of the descending step but not of the ascending step at temperatures below that at which the layer-by-layer removal starts [28].

Here, we discuss the pit profile in the steady state. The growth rate of the circular step with an inner radius of R<sup>p</sup> in Figure 19 is given [11] in the continuum approximation by

$$\text{dB}\_{\text{P}}/\text{dt} = \text{P}(\text{R}\_{\text{P}-1}\,^2 - \text{R}\_{\text{P}}\,^2)/2\text{N}\_0\text{R}\_{\text{P}}\,(\text{R}\_{\text{P}} > \text{a}\_0),\tag{1}$$

where P and N<sup>0</sup> are the production rate of surface vacancies and the number of atoms per unit area of the surface, respectively, and a<sup>0</sup> is the atomic size.

As Rp−<sup>1</sup> = R<sup>p</sup> + w, Equation (1) can be expressed as

$$\begin{array}{l} \text{dR}\_{\text{P}}/\text{dt} = \text{P}[(\text{R}\_{\text{P}} + \text{w})^2 - \text{R}\_{\text{P}}^2]/2\text{N}\_0\text{R}\_{\text{P}}\\ = \text{Pw}/\text{N}\_0 + \text{Pw}^2/2\text{N}\_0\text{R}\_{\text{P}'} \end{array} \tag{2}$$

When a pit grows steadily, Equation (2) should be constant (=k).

In the case where the terrace is far from the center (R<sup>p</sup> > > w), the second term of Equation (2) is negligibly small. Then, the width of the terrace is approximately

$$\mathbf{w} = \mathbf{k} \mathbf{N}\_0 / \mathbf{P} \tag{3}$$

by

In the case where the terrace is located beside the center (R<sup>p</sup> = w), the width of the terrace is estimated, by applying R<sup>p</sup> = w and dRp/dt = k, as w = kN0/P, (3) In the case where the terrace is located beside the center (Rp = w), the width of the terrace is estimated, by applying Rp = w and dRp/dt = k, as

In the case where the terrace is far from the center (Rp > >w), the second term of

Here, we discuss the pit profile in the steady state. The growth rate of the circular step with an inner radius of Rp in Figure 19 is given [11] in the continuum approximation

where P and N0 are the production rate of surface vacancies and the number of atoms per

dRp/dt = P[(Rp + w)2 − Rp2]/2N0Rp

*Quantum Beam Sci.* **2021**, *5*, x FOR PEER REVIEW 16 of 20

unit area of the surface, respectively, and a0 is the atomic size. As Rp−1 = Rp + w, Equation (1) can be expressed as

When a pit grows steadily, Equation (2) should be constant (=k).

Equation (2) is negligibly small. Then, the width of the terrace is approximately

$$\mathbf{w} = (\mathbf{2}/\mathbf{3})\mathbf{k}\mathbf{N}\_0/\mathbf{P} \tag{4}$$

= Pw/N0 + Pw2/2N0Rp, (2)

dRp/dt = P(Rp−12 − Rp2)/2N0Rp (Rp > a0), (1)

The difference in the values of w in Equations (3) and (4) means that the width of the terrace near the center becomes narrower; i.e., the inclination of the face of the pit near the center becomes steeper. The shape of the steadily growing pit can be obtained by the numerical calculation for pit growth for finite intervals given by Cherns [11]. However, the result of these calculation does not reveal the significant deepening of pits observed in the present experimental study. Therefore, the deepening of the nanoholes should be due to some factors other than the geometrical factor. The difference in the values of w in Equations (3) and (4) means that the width of the terrace near the center becomes narrower; i.e., the inclination of the face of the pit near the center becomes steeper. The shape of the steadily growing pit can be obtained by the numerical calculation for pit growth for finite intervals given by Cherns [11]. However, the result of these calculation does not reveal the significant deepening of pits observed in the present experimental study. Therefore, the deepening of the nanoholes should be due to some factors other than the geometrical factor.

Two factors can be mentioned as being responsible for the deepening, which are preferential sputtering at growth point and irradiation-induced diffusion. The sputtered surface quickly develops a high proportion of surface ledge sites, especially near the pit center, as seen in Figure 20. The sputtering rate would increase at the ledge sites due to the lower number of bonding. Therefore, it is probable that the preferential sputtering at the growth point leads to the deepening of pits. Irradiation-induced diffusion of surface vacancies and adatoms on the side walls of nanoholes, on the other hand, may play an important role. Surface vacancies and adatoms will move opposite to and along the irradiation direction, respectively, both movements enhancing the deepening of the holes. Two factors can be mentioned as being responsible for the deepening, which are preferential sputtering at growth point and irradiation-induced diffusion. The sputtered surface quickly develops a high proportion of surface ledge sites, especially near the pit center, as seen in Figure 20. The sputtering rate would increase at the ledge sites due to the lower number of bonding. Therefore, it is probable that the preferential sputtering at the growth point leads to the deepening of pits. Irradiation-induced diffusion of surface vacancies and adatoms on the side walls of nanoholes, on the other hand, may play an important role. Surface vacancies and adatoms will move opposite to and along the irradiation direction, respectively, both movements enhancing the deepening of the holes.

**Figure 20.** Growth process of nanohole. Surface vacancies on the wall of nanohole move to the tip of nanohole, but adatoms are evacuated from the nanohole. Thus, deep nanoholes can be formed.
