# Influence of Target-Substrate Distance on the Transport Process of Sputtered Atoms: MC-MD Multiscale Coupling Simulation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{ep}and V

_{eA}), but |V

_{eA}|/|V

_{ep}| was not equal to tanθ

_{com}(θ

_{com}is center-of-mass scattering angle). In most follow-up studies [10,11,12,13,14,15,16], the scattering angle of the sputtered particle after a collision is calculated by an approximate formula that is derived by supposing that the background gas atom is stationary before collision [17] and thus cannot precisely predict the post-collision velocity of the sputtered particle.

## 2. MC-MD Simulation Method

#### 2.1. Initial Status Data of Sputtered Atom

_{1}) are 12.5 mm and 15 mm, from which the mean (μ) of 12.5 and standard deviation (δ) of 2.5 in Gauss distribution can be obtained. Accordingly, the sputtering probability of an atom located at a distance of d from the target center can be evaluated. Then, the emission position coordinates of a sputtered atom were generated by a random number R

_{1}, which is distributed uniformly between 0 and 1.

_{1})

_{1})

_{b}is the binding energy of target material (Cu); E

_{Ar}is the bombarding energy of argon ion; γ = 4 m

_{g}m

_{s}/(m

_{g}+ m

_{s})

^{2}, m

_{g}, and m

_{s}stand for the masses of background gas atom and sputtered atom, respectively. Then, the initial energy E

_{0}of each sputtered atom was generated by the rejection algorithm [10,24].

_{t}is the total number of sputtered atoms, θ is the polar emission angle of the sputtered atom with respect to the target surface normal, and ξ is a fitting parameter according to the bombarding energy [25]. Due to symmetry, the azimuth emission angles of sputtered atoms were generated uniformly in [0, 2π].

#### 2.2. MC Simulation of the Free-Flight Processes of Sputtered Atoms in Gas Phase

_{m}|InR

_{2}|

_{m}is the mean free path; R

_{2}is a random number uniformly distributed between 0 and 1.

_{m}is introduced and can be expressed as below [14]:

_{g}is the concentration of background gas; σ is the collision cross-section; Pmax is the maximum impact parameter corresponding to the minimum of center-of-mass scattering angle; E

_{com}is the center-of-mass kinetic energy; v

_{p}is the most probable speed of background gas atoms; v

_{r}is the relative speed between the sputtered atom and background gas atom; K

_{B}is Boltzmann constant; T

_{g}is the temperature of background gas; m

_{s}and m

_{g}are the masses of sputtered and background gas atoms, respectively; r

_{s}and r

_{g}are the atomic radii of sputtered and background gas atoms, respectively.

_{s}) collides with a background gas atom (M

_{g}) with a collision parameter of p. The scattering angle of M

_{s}after collision is θ

_{com}, which can be determined as [26]:

_{3}is a random number uniformly distributed between 0 and 1. In our simulation, the interaction between the sputtered atom and background gas atom was described by Ziegler–Biersack–Littmark (ZBL) potential [27], which has also been adopted in other MC simulations of sputtered particle transport [28]. Once the U (r) is determined, for each specific E

_{com}, the function P

_{max}(E

_{com}) can be evaluated based on Equation (10) by letting θ

_{com}= 0.01 rad [13]. Then, the actual collision parameter p can be calculated by Equation (13), and the actual free path length λ can be determined by Equations (6)–(9).

#### 2.3. MD Model of the Collision between Sputtered Atom and Background Gas Atom

_{s}and M

_{g}in the center of mass coordinate system is the same as that in the laboratory coordinate system, the relative position relationship between the two colliding particles shown in Figure 2 can be used to determine the initial positions of the sputtered atom and background gas atom in the laboratory coordinate system during the MD simulation.

_{s}) and background gas atom (M

_{g}) in three-dimensional space when they collide with a collision parameter of p. As shown in Figure 3, the red ball at point o (0, 0, 0) represents M

_{s}, while the green ball at point m (x

_{g}, y

_{g}, z

_{g}) denotes M

_{g}. It can be seen that M

_{s}and M

_{g}are situated at the bottom surface center and the upper surface edge of an arbitrary spatial cylinder, respectively. The central symmetry axis of the spatial cylinder is along the direction of v

_{r}(the relative velocity between M

_{s}and M

_{g}), and the radius and height of the spatial cylinder are p (collision parameter) and L (the distance between M

_{s}and M

_{g}along the direction of v

_{r}), respectively. In the present simulation, L was set to the cutoff radius of the potential function U(r) in order to save computation time. Then, the initial location of M

_{g}can be expressed as:

_{x}, α

_{y}, and α

_{z}are the direction angles of v

_{r}(v

_{rx}, v

_{ry}, v

_{rz}) with respect to the X, Y, and Z coordinate axes, respectively. Φ is the azimuth angle of point m’ (m) with respect to the line oo

_{1}(o’o’

_{1}) on the bottom (upper) surface of the spatial cylinder shown in Figure 3, and line oo

_{1}represents the intersecting line between the XY coordinate plane and the bottom surface of the spatial cylinder. Without losing generality, Φ = 2πR

_{4}, where R

_{4}is a random number distributed uniformly between 0 and 1. Then, the initial position of the background gas atom can be obtained and expressed as:

_{g}) of the background gas atom can be chosen randomly from the Maxwellian distribution at 300 K, and the initial velocity (v

_{s}) of the sputtered atom can be selected from the Thompson distribution. Then, the elastic collision between the sputtered atom and the background gas atom was simulated by using the molecular dynamics code LAMMPS [29]. The size of the simulation domain was 100 × 100 × 100 Å

^{3}. Non-periodic and fixed boundary conditions were applied in all three directions in order to capture the entire collision process. The interaction between colliding atoms was described by the Ziegler-Biersack-Littmark (ZBL) potential [27] with a cut-off distance of 6.2 Å [30]. At the beginning of the simulation, the sputtered atom was placed at the geometric center of the simulation box (coordinate origin), while the background gas atom was put on the coordinates calculated according to Equations (20)–(22). The standard Velocity-Verlet algorithm was used for integrating Newton’s equation of motion. The time step was set to 0.1 fs [18]. The MD simulation was terminated when the force exerted on the sputtered atom went to zero. Then, the velocity data of the sputtered atom was written into a dump file, which would be read during the MC simulation.

#### 2.4. MC-MD Coupled Simulation Scheme

## 3. Reliability Verification of the MD Collision Model

_{s0}(v

_{sx}, 0, 0) and v

_{g0}(v

_{gx}, 0, v

_{gz}), respectively; the angle of initial relative velocity v

_{r0}with respect to the Z coordinate axis is β; the angle of v

_{g0}with respect to the X coordinate axis is γ; v

_{c}is the center-of-mass velocity.

_{1}Y

_{1}Z

_{1}was established, as shown in Figure 5b. In the coordinate system X

_{1}Y

_{1}Z

_{1}, the X

_{1}and Y

_{1}coordinate axes are along the directions of v

_{r0}and the Y axis of coordinate system XYZ, respectively. Consequently, the initial velocity of the Cu atom and the center-of-mass velocity in the coordinate system X

_{1}Y

_{1}Z

_{1}can be expressed by the following equations:

_{c1}Y

_{c1}Z

_{c1}, i.e., the coordinate system X

_{1}Y

_{1}Z

_{1}floating with the mass center of colliding atoms, the scattering process of the Cu atom after colliding with the Ar atom can be displayed in Figure 5c. Since angular momentum and energy are conserved and potential is just determined by the distance between the colliding atoms, the motion of the colliding atoms in the center of mass coordinate system has the following characteristics: (1) the relative velocity between Cu and Ar atoms in the center of mass coordinate system is identical to that in the laboratory coordinate system; (2) the movement of Cu atom is restricted in a plane [26]; (3) the velocity magnitude of Cu atom remains constant before and after collision. Accordingly, as shown in Figure 5c, the post-collision velocity of the Cu atom in the center of mass coordinate system X

_{c1}Y

_{c1}Z

_{c1}can be expressed as follows:

_{r0}| is the magnitude of v

_{r0}; Φ is the initial position azimuth angle of the Ar atom in the Y

_{c1}Z

_{c1}plane. Therefore, in the laboratory coordinate system X

_{1}Y

_{1}Z

_{1}, the post-collision velocity of the Cu atom can be expressed by:

_{g}/(m

_{g}+ m

_{s}); e denotes the charge of electron. Indeed, β is only determined by the initial velocity direction of the Ar atom (γ) as the initial velocity of the Cu atom is along the X axis, and θ

_{com}is only determined by the collision parameter p as E

_{com}is kept constant. Therefore, from Equation (27), it can be seen that θ

_{lab}is influenced by the initial velocity direction of the Ar atom (γ), collision parameter p, and the initial position azimuth angle of the Ar atom (Φ).

#### 3.1. Influence of the Initial Velocity Direction of Ar Atom on θ_{lab}

_{lab}with the initial velocity direction of the Ar atom when E

_{com}, p and E

_{Ar}are kept at 1 eV, 0 Å, and 0.0387 eV, respectively. As shown in Figure 6, the values of θ

_{lab}calculated by the MD collision model are consistent well with those calculated by Equation (27), which validates the reliability of the MD collision model. When E

_{com}= 1 eV and p = 0 Å, θ

_{lab}increases first and then decreases as γ increases from 0° to 180°, and the maximum value of θ

_{lab}is 36.12°. Accordingly, the initial velocity direction of the background gas atom significantly affects the scattering angle of the sputtered atom, but this effect cannot be considered in Equation (28).

#### 3.2. Influence of Collision Parameter p on θ_{lab}

_{lab}when Φ, E

_{com}, E

_{Ar}, and γ are kept at 90°, 1 eV, 0.0387 eV, and 90°, respectively. In Figure 7, the values of θ

_{lab}were calculated individually by Equation (27), Equation (28), and the MD collision model as p increased from 0 to 4 Å. As shown in Figure 7, as p increases from 0 to 4 Å, the values of θ

_{lab}calculated by the MD collision model agree well with the corresponding values of θ

_{lab}calculated by Equation (27), while significant errors exist in the values of θ

_{lab}(p < 2 Å, E

_{com}= 1 eV) calculated by Equation (28).

#### 3.3. Influence of the Initial Position Azimuth Angle of Ar Atom on θ_{lab}

_{lab}as a function of the initial position azimuth angle of the Ar atom (Φ) when E

_{com}, E

_{Ar}and γ are kept at 1 eV, 0.0387 eV, and 90°, respectively. In Figure 8, the values of θ

_{lab}were calculated individually by Equation (27) and MD collision model as Φ ranged from 0° to 360° and p increased from 0 to 3 Å. As shown in Figure 8, the values of θ

_{lab}calculated by the MD collision model are coincident with the corresponding values of θ

_{lab}calculated by Equation (27). As shown in Figure 8, when E

_{com}= 1 eV and 0.5 Å ≤ p ≤ 1.5 Å, the values of θ

_{lab}change significantly as Φ increases from 0° to 360°. Accordingly, the initial position azimuth angle of the background gas atom (Φ) exerts a significant influence on the scattering angle of the sputtered atom. However, this influence cannot be considered in Equation (28).

## 4. Results and Discussion

#### 4.1. Example of the Transport Process of a Sputtered Atom in Gas Phase

#### 4.2. Incident Energy Distribution

#### 4.3. Incident Angle Distribution

_{s}(θ) is the incident angle distribution of sputtered atoms as shown in Figure 12a; f

_{E}(θ) represents the distribution of differential deposition rate in unit solid angle. Herein, θ was set to 0.5° in the calculating f

_{E}(θ = 0°) such that sinθ is not equal to 0 in Equation (30).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Sketch of the erosion groove on the circular planar target (L

_{1}= 15 mm, and L

_{2}= 5 mm).

**Figure 3.**The relative position between the sputtered atom and background gas atom at the initial phase of elastic collision.

**Figure 5.**Scattering collision between sputtered Cu atom (M

_{s}) and background gas atom (M

_{g}) with specific initial velocities. (

**a**) Initial velocities and positions of M

_{s}and M

_{g}in the laboratory coordinate system XYZ; (

**b**) initial velocities and positions of M

_{s}and M

_{g}in the laboratory coordinate system X

_{1}Y

_{1}Z

_{1}; (

**c**) scattering collision between M

_{s}and M

_{g}in the center of mass coordinate system X

_{c1}Y

_{c1}Z

_{c1}when the coordinate system X

_{1}Y

_{1}Z

_{1}floats with the mass center of colliding atoms.

**Figure 6.**Influence of the initial velocity direction of the Ar atom on the scattering angle of the Cu atom (E

_{com}= 1 eV).

**Figure 7.**Influence of the collision parameter p on the scattering angle of Cu atom when Φ = 90° (E

_{com}= 1 eV).

**Figure 8.**Influence of the initial position azimuth angle of an Ar atom on θ

_{lab.}(E

_{com}= 1 eV).

**Figure 9.**Trajectories of a sputtered atom in the gas phase at the target-substrate distances of (

**a**) 30 mm, (

**b**) 90 mm, and (

**c**) 150 mm and the dynamic evolution of the elastic collisions between Cu and Ar atoms at the target-substrate distances of (

**d**) 90 mm and (

**e**,

**f**) 150 mm. In sub-graphs (

**a**–

**c**), red balls denote the locations of the Cu atom, and purple lines represent the free flight trajectories of the Cu atom. In sub-graphs (

**d**–

**f**), the positions of Cu and Ar atoms at different time steps are presented by two strings of colored balls marked by yellow and blue arrows, respectively.

**Figure 10.**Incident energy distribution of the deposited sputtered Cu atoms at different target-substrate distances plotted on (

**a**) logarithmic scale and (

**b**) linear scale.

**Figure 12.**Incident angle distribution of sputtered Cu atoms under different target-substrate distances, (

**a**) incident polar angle distribution; (

**b**) normalized differential deposited rate as a function of polar angle obtained by simulation and experiment [38].

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**MDPI and ACS Style**

Zhu, G.; Du, Q.; Xiao, B.; Chen, G.; Gan, Z.
Influence of Target-Substrate Distance on the Transport Process of Sputtered Atoms: MC-MD Multiscale Coupling Simulation. *Materials* **2022**, *15*, 8904.
https://doi.org/10.3390/ma15248904

**AMA Style**

Zhu G, Du Q, Xiao B, Chen G, Gan Z.
Influence of Target-Substrate Distance on the Transport Process of Sputtered Atoms: MC-MD Multiscale Coupling Simulation. *Materials*. 2022; 15(24):8904.
https://doi.org/10.3390/ma15248904

**Chicago/Turabian Style**

Zhu, Guo, Qixin Du, Baijun Xiao, Ganxin Chen, and Zhiyin Gan.
2022. "Influence of Target-Substrate Distance on the Transport Process of Sputtered Atoms: MC-MD Multiscale Coupling Simulation" *Materials* 15, no. 24: 8904.
https://doi.org/10.3390/ma15248904