Modeling and Simulation

Abstract

This chapter is devoted to recent developments in mathematical modeling and computer simulation of copper electrodeposition. We focus our attention on continuum models and kinetic Monte Carlo simulations for shape evolution and the effects of additives on copper deposition, especially the filling of small features in microelectronics. The modeling, mathematical treatments, and simulation results are reviewed with brief summaries of efficient numerical algorithms. Fast computing and prospects of simulation research are also discussed.

Appendices

Appendix A Level Set Method

LSM is a tracking method of moving boundaries, which is commonly used in recent numerical studies of electrodeposition. The level set function $\phi_{\text{L}}$ is a continuous function of space and time, defined in the whole area of liquid–solid interface. The surface is defined by $\phi_{\text{L}} = 0$, and the inner space (electrode) by $\phi_{\text{L}} < 0$ and the outer space (solution) by $\phi_{\text{L}} > 0$. The time derivative of $\phi_{\text{L}}$ is

$$ \frac{{\partial \phi_{\text{L}} }}{\partial t} = {\mathbf{v}} \cdot \nabla \phi_{\text{L}} , $$

(A.1)

where ${\mathbf{v}}$ is the surface front velocity. The normal vector on the surface is:

$$ {\mathbf{n}} = \frac{{\nabla \phi_{\text{L}} }}{{\left| {\nabla \phi_{\text{L}} } \right|}} , $$

(A.2)

and the normal velocity is defined as:

$$ v = {\mathbf{v}} \cdot {\mathbf{n }}. $$

(A.3)

Using Eqs.(A.2) and (A.3), Eq.(A.1) becomes

$$ \frac{{\partial \phi_{\text{L}} }}{\partial t} = v \left| {\nabla \phi_{\text{L}} } \right| . $$

(A.4)

The curvature of the surface in LSM is given by $\kappa = \nabla \cdot {\mathbf{n}}$. Equation (A.4) is the basic equation of LSM which should be solved numerically. Since $v$ is defined not only at the interface but outside the interface, the extension velocity $v_{\text{ext}}$ is defined by:

$$ \nabla v_{\text{ext}} \cdot \nabla \phi_{\text{temp}} = 0 , $$

(A.5)

where $v = v_{\text{ext}}$ at $\phi_{\text{L}} = 0$ and $\phi_{\text{temp}}$ is calculated by using the condition

$$ \left| {\nabla \phi_{\text{L}} } \right| = 1 . $$

(A.6)

Using $v_{\text{ext}}$ the LSM equation is written as:

$$ \frac{{\partial \phi_{\text{L}} }}{\partial t} = v_{\text{ext}} \left| {\nabla \phi_{\text{L}} } \right| $$

(A.7)

which is solved with the equations representing the electrodeposition reactions on the surface which give the boundary conditions. The equations are discretized and the quantities are evaluated on the grid points. The method of discretization and numerical procedures of LSM in combination with the FV code are found in the literature [9, 22].

Appendix B Rejection-Free Algorithm for KMC Simulation

Here we describe the application of the rejection-free algorithm to the 2D SOS and SBS models. Three events are assumed to occur on the surface. The rate constants for adsorption $k_{n}^{ + }$, desorption $k_{n}$ , and surface diffusion $k_{nm}$ are dependent upon the number of nearest neighbor solid atoms $n,m$ at the sites, where $1 \le n,m \le 4$. The rates of the creation (adsorption), annihilation (desorption), and surface diffusion are defined as

$$ k_{c} = \mathop \sum \limits_{n = 1}^{3} k_{n}^{ + } N_{c(n)} , k_{a} = \mathop \sum \limits_{n = 1}^{3} k_{n} N_{a(n)} , k_{d} = \mathop \sum \limits_{n,m = 1}^{3} k_{n} N_{d(nm)} , $$

(B.1)

respectively, where $N_{c(n)}$, $N_{a(n)}$, $N_{d(nm)}$ are the numbers of candidate atoms (sites) for the events. The rate at which one of the three events occurs is given by

$$ k_{t} = k_{c} + k_{a} + k_{d} . $$

(B.2)

The KMC algorithm for this model is as follows.

• Search and tabulate the candidate surface atoms (sites) for the events.

• Choose one of the events using a random number $R$ on [0,1].

  $$ R < \frac{{k_{c} }}{{k_{t} }}\,:{\text{adsorption}} $$
  $$ \frac{{k_{c} }}{{k_{t} }} \le R < \frac{{k_{c} + k_{a} }}{{k_{t} }}\,:{\text{desorption }} $$
  $$ \frac{{k_{c} + k_{a} }}{{k_{t} }} \le R\, : {\text{surface diffusion}} $$

• Choose the type of events (the number of bonds) by generating another random number $R'$. In the case of adsorption,

  $$ R' < \frac{{k_{1 }^{ + } N_{c(1)} }}{{k_{t} }}\,: n = 1 $$
  $$ \frac{{k_{1 }^{ + } N_{c(1)} }}{{k_{t} }} \le R' < \frac{{k_{1}^{ + } N_{c\left( 1 \right)} + k_{2}^{ + } N_{c(2)} }}{{k_{c} }}\,: n = 2 $$
  $$ \frac{{k_{1}^{ + } N_{c\left( 1 \right)} + k_{2}^{ + } N_{c(2)} }}{{k_{c} }} \le R'\,: n = 3 $$

• Select one atom (site) from the table of the candidates for the event chosen in 2 and 3, and realize the reaction.

• Renew the table and go to 2

This cycle defines one KMC step and the average time corresponding to this cycle is $1/k_{t}$. Additives and their reactions are incorporated in this algorithm as additional events. The numbers of the candidates for the reactions are calculated using the concentrations, which are passed from the continuum code or the results of CGRW in solution.

Appendix C Algorithm for Searching Surface Atoms

The crucial point in the extension of the SOS model to the SBS model is an efficient and accurate algorithm for searching surface atoms. Figure A.1 illustrates an example of the algorithm for a two-dimensional model. Black and white squares denote solid and liquid sites, respectively. Vacancies are denoted by white squares. The algorithm consists of successive numbering of the squares.

Fig. A.1

Searching algorithm of surface atoms. Successive numbering of the squares is illustrated. Black squares are solid atoms, white squares with numbers are liquid atoms, and white squares without numbers are vacancies. The surface solid atoms are the black squares adjacent to the numbered white squares

• 1. From the top of the liquid sites, put “1” to the white squares successively moving down in –y direction.

• 2. Then, put “2” to the white squares adjacent to the white squares numbered as “1”.

• 3. Put “3” to the white squares adjacent to the white squares numbered as “2”.

• 4. Repeat numbering the white squares adjacent to the already numbered squares.

• 5. Stop the numbering if there is no white square without numbers around the numbered white squares.

• 6. The solid squares adjacent to the “numbered” white squares are surface solid atoms. White squares adjacent to the surface solid squares are surface liquid sites.

Since the white squares surrounded by solid squares are not numbered, the liquid sites and vacancies are distinguished. It is straightforward to extend this method to a 3D model.