A 3D atomic deposition simulation model written entirely in Python. PytOpenGL library is used for 3D visualization of individual atoms being deposited on a flat surface. This was done as a course project (EE216: Nanomaterials and Nanometer-scale Device) at UCSC and was done with a group of three PhD students:

• Yucheng Li
• Md Nafiz Amin
• Vahid Ganjalizadeh (https://github.com/vganjali)

Basic Physics: L-J potential

The Lennard-Jones potential (L-J potential) is a mathematical model to approximate the interaction between a pair of neutral atoms or molecules. $V_{LJ}=4\epsilon[(\frac{\sigma}{r})^{12}-(\frac{\sigma}{r})^{6}]=\epsilon[(\frac{r_{m}}{r})^{12}-2(\frac{r_{m}}{r})^{6}]$ where

• $\epsilon$: Depth of the potential well
• $\sigma$: Distance at wich $V_{LJ}=0$
• $r_m$: Distance at wich $V_{LJ}$ is minimum ($=-\epsilon$)
• ($r_m=2^{1/6}\sigma\approx1.122\sigma$)
• $r^{-12}$: The repulsive term, describes Pauli repulsion at short ranges due to overlapping electrin orbitals
• $r^{-6}$ The attractive term, describes attraction at long ranges (for example, Van de Waals force)
• The same equation is used in many-particle environment

Algorithm Flowchart

Key Considerations

Bottom Layer

• A well-arranged bottom layer is assumed to be pre-formed when the simulation starts
• Exerts attractive/repulsive force on incoming particles according to L-J potential
• Provides a general model for the simulation

Adaptive Time-step

• Particle slows down when L-J potential changes sharply
• Increases the stablity around the equilibrium point
• $r_{k+1}=r_{k}+\Delta t \nu_{k}+{\Delta t}^2\frac{F_{r_k}}{m}$
• $r_{k+1}=\nu_{k}+\frac{\Delta t}{2m}(F_{r_k}+F_{r_{k+1}})$
• $\Delta t_{k+1}={1}/{[\nu_k \times exp(r_c-r_n)]}$
• $r_c$ is critical distance to look for neighbors
• $r_n$ is the distance from the closest neighbor

Timeout

• A counter is the triggered whenever the first change in the sign of the force $F=-\frac{\partial V_{LJ}}{\partial r}$ is detected
• After a reasonable number of time-steps, timeout happens and the particle reaches the equilibrium
• Simplifies complexity due to large force from the bottom surface

Velocity Damping

• Velocity of the particle is dampled if there is a change in sign of velocity or force
• Velocity sign change indicates collision, and damping emulates inelastic collision
• Force sign change indicates crissing the equilibrium and damping velocity around it simplifies the motion

Grids

• 1/0 indicate presence/absence of old atoms
• Submatrix dimension $(2 critical distance+1) \times (2 critical distance+1)$

Force Submatrix

• At the beginning, components of F are calculated for each point within the submatrix
• A new particle uses a look-up table (LUT) of pre-calculated force values
• This increases the computation speed even when number of particles is large

Real Physical Parameters

• For Cu atoms:

Snapshots taken while running simulation