# Modeling of Reactive Diffusion: Mechanism and Kinetics of the Intermetallics Growth in Ag/Ag Interconnections

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## Abstract

The phenomenological model describing the growth of intermetallic phases in multi-component systems is presented. Full time-dynamics approach is applied without the often-used simplifications such as flux constancy. General form of the species flux is considered, which consists of chemical potential gradient as a driving force for diffusion with additional drift term. Stefan-type (moving) boundary conditions are applied. In the present form, the model assumes local equilibrium at each interface and that the process of growth of intermediate phases is controlled by diffusion of reagents through the layers and/or chemical reactions at the boundaries. The model is solved in its full generality. Numerical method for the solution of the problem has been developed. Specially selected change of dependent variables transforms the moving boundary problem into an equivalent fixed boundary problem. Such problem has been treated using the method of lines which converts partial differential equations into a system of ordinary differential equations, which is subsequently solved numerically. The obtained solution was tested and compared with analytic ones available in special cases, showing satisfactory agreement. The growth of intermetallic phases in Ag/Sn/Ag system has been modeled and compared with experimental results.

### Keywords

intermetallics joining modeling processes## Introduction

Diffusion soldering technology is an effective method of obtaining stable metal/metal interconnections using solder which forms intermetallic phase with the joining materials. The joint microstructure, chemical composition, and the sequence of appearance of the intermetallic phases are important factors that influence its stability. The important parameters determining diffusion soldering technology are the thickness of the solder, duration, and temperature of the process. Understanding of the mechanism of this process is necessary for prediction of growth kinetics of phases and, finally, optimization of the diffusion soldering technology.

In this article mathematical model of the process is formulated. Method of solving moving-boundary problem is developed. Numerical solution of the problem is verified against analytic solutions. Comparison of the model with experimental results is presented.

## Mathematical Model

A model of reactive diffusion describing the growth of intermediate phases in multi-component systems in one-dimensional geometry is presented below. It includes data, physical laws of transport processes, initial conditions, boundary conditions, and unknowns.

### Data

- 1.
Diffusion coefficients of components (

*i*= 1,…,*r*) in different phases (α = 1,…,*f*) as functions of the molar fraction:*D*_{i}^{α}(*N*_{1},…,*N*_{r}), where*r*and*f*denote the number of components (species) and phases, respectively. - 2.
Initial positions of the phase boundaries:

*s*_{0}(0),…,*s*_{α}(0),…,*s*_{f}(0). - 3.
Activities of the components in all phases as functions of the molar fractions of components:

*a*_{i}^{α}(*N*_{1},…,*N*_{r}), where*i*= 1,…,*r*; α = 1,…,*f*. - 4.
Duration (time) of the process: \( \hat{t} \).

### Physical Laws

*c*

_{i}

^{α}and

*J*

_{i}

^{α}denote concentration (mole/m

^{3}) and flux (mole/m

^{2}s) of the

*i*th component in the phase α, respectively.

*J*

_{i}

^{α}in Eq 1 will be expressed as a sum of diffusive flux (

*J*

_{i}

^{α,d}) and drift flux (

*c*

_{i}υ). Hence,

*B*

_{i}

^{α}is the mobility of the

*i*th component in the phase \( \upalpha , \) and \( \sum\nolimits_{j} {F_{j} } \) is the sum of thermodynamic forces causing diffusion. In the isothermal-isobaric conditions a diffusion flow is generated by the gradient of chemical potential, μ

_{i}

^{α}(Ref 4). Thus, we have the following form of the diffusion flux:

*D*

_{ij}

^{α}in phases α = 1,…,

*f*are defined by the formulas:

*J*

_{i,max}

^{d}(Ref 7):

_{i}denotes the characteristic distance for the diffusion (jump distance of a defect). The phase exists if and only if the diffusion flux in this phase does not exceed

*J*

_{i,max}

^{d}. Thus, this condition can be written as

The constraints on the maximum value of the flux were earlier applied in the hydromechanics problems (Ref 8, 9), plasma physics modeling (Ref 10), and for charge transport processes in microelectronic systems (Ref 11, 12). The kinetic constraint (8) for inter-diffusion problems of multi-component systems was applied also by Danielewski and Wakihara (Ref 7).

### Initial and Boundary Conditions

#### Initial Conditions

The model permits the use of any initial concentration \( c_{i}^{\upalpha } (0,x) = c_{i}^{\upalpha ,0} (x) \) in each phase α. In diffusion soldering, the initial compositions are usually simple, for example, constant concentrations in the substrate and in the solder, *c*_{i}^{α,0} and *c*_{i}^{γ,0}, respectively.

#### Boundary Conditions

*s*

_{α}(

*t*) is the moving position of the phase boundary (α − 1|α) at time

*t*;

*c*

_{i}

^{j}(

*s*

_{k}(

*t*),

*t*), and

*J*

_{i}

^{j}(

*s*

_{k}(

*t*),

*t*) are the concentration and flux, respectively, of the

*i*th species in

*j*th phase at the

*k*th boundary. The boundary conditions may be imposed for each species at each phase boundary.

By using Eq 11, the equilibrium concentrations at the interfaces for the boundary conditions (10) can be determined. Subsequently, the diffusion problem given by Eq 1-11 can be solved for each single-phase region. The velocities and positions of the interfaces are determined by solving the flux-balance equations (10).

### Unknowns

- 1.
Positions of phase boundaries as functions of time:

*s*_{0}(*t*),…,*s*_{f}(*t*) for \( t \in [0,\hat{t}] \). - 2.
Concentrations profiles

*c*_{i}^{α}(*x*,*t*),*i*= 1,…,*r*in each phase α = 1,…,*f*for \( x \in [s_{\upalpha - 1} (t),s_{\upalpha } (t)] \) and \( t \in [0,\hat{t}]. \)

#### Solution of the Model

Equations describing the growth of multi-component layers form a system of nonlinear partial differential equations. Moreover, situation is complicated further because the positions of interfaces change with time, giving rise to the Stefan-like problems. Hence, we will look for a suitable numerical solution using the finite difference method.

### Numerical Solution

Theoretical treatment of growth of phases in diffusion soldering process involves the solution of the so-called Stefan problem (Ref 13). The growth rate is calculated from a mass balance equation at the moving interface and suitable mass balance equations in each phase. The problem is complicated by the fact that the position of phase interface is also an unknown. With the exception of simple cases—where analytic solutions are known (Ref 13)—numerical methods must be applied. Extensive and detailed overview of these methods is provided, for example, in Ref 14. In connection with our development, we mention here only the Murray-Landis and enthalpy methods.

*s*′ is obtained from the flux balance at the interface, (

*c*

^{α+1}−

*c*

^{α})

*s*′ =

*J*

^{α+1}−

*J*

^{α}using one-sided finite differences, for example,

*enthalpy method*uses a fixed grid. At each time step, we keep information about the grid points between which the interface is located. Other points belong to different adjacent phases. After advancing in time the concentration profile, a set of rules is applied to assign anew the grid points to one or second phase, or to the interface region. Because at each step we know only two adjacent points between which the interface is located, some interpolation formula is used to obtain the approximate position of the interface. For example, if

*i*is the grid point closest to the interface, the position may be calculated as

*h*is distance between grid points,

*c*

^{α+1}and

*c*

^{α}are the concentrations close to the interface.

The aim of our method was simplicity of presentation. From what has just been described, we can see that above methods are relatively obscure because they address the moving interface in its original form. Thus, we devised a simple change of variables that transforms the moving boundary problem to fixed boundary problem which in turn may be dealt with by virtually any standard numerical method.

The idea of the method is presented below. Let us assume that a system has *f* phases [β_{α−1}, β_{α}], α = 1,…,*f*. Concentration of *i*th component in α phase will be denoted by *u*_{i}^{α} = *u*_{i}^{α}(*x*, *t*). In this section an idea of numerical solution of Stefan-like problem given by Eq 12 is described.

### Equations

*a*

_{ij}

^{α−1/α}and

*a*

_{ij}

^{α/α+1}are known coefficients.

### Boundary Conditions

Jumps of concentrations of an *i*th component at αth boundary are denoted by δ_{i}^{α}.

### Initial Conditions

Such formulated problem is a moving boundary problem. For numerical computations, it is more convenient to transform the problem with moving boundaries into a problem with the fixed boundaries. This will be done by introducing suitable new variables.

^{α}:

_{x}, φ

_{t}denote derivatives with respect to

*x*and

*t*) :

### Multi-Phase Binary System

*c*

^{α}(

*x*,

*t*), and one diffusion coefficient, \( \tilde{D}^{\upalpha } , \) in each phase (the

*chemical diffusion coefficient*).

*f*.

In general, the chemical diffusion coefficient may be variable. However, to verify the model with experiment, we assumed \( \tilde{D}^{\upalpha } = {\text{const}}. \)

*v*

^{α}(

*x*,

*t*) =

*c*

^{α}(φ

^{α}(

*x*,

*t*),

*t*), and following virtually the same calculations as in (17)-(20), we arrive at

### Numerical Solution

The system (25) has been numerically solved using the Radau II method for stiff system of equations (Ref 19).

### Test Problem with Analytic Solution

*c*

^{∞}. The local mass balance (10) now reads

*J*

^{α}= 0. The flux

*J*

^{γ}is governed by the standard Fick’s law, \( J^{\upgamma } = - D^{\upgamma } (\partial^{2} c^{\upgamma})/(\partial x^{2}). \) The analytic solution to this problem can be found, for example, in Ref 4. It states that the phase boundary

*s*=

*s*(

*t*) is moving according to parabolic law:

*K*is the solution of the following nonlinear equation:

## Experiment

Production conditions of diffusion joints of the Ag/Sn/Ag type

Temperature, °C | Time, min |
---|---|

230 | 10, 30, 60, 90, 120, 150 |

243 | 10, 20, 30, 40, 60, 80 |

250 | 10, 20, 60, 120, 180 |

258 | 20, 30, 40, 50, 60, 80 |

265 | 10, 20, 30, 60, 90, 120 |

The obtained Ag/Sn/Ag interconnectors were characterized using light and scanning electron microscopy (SEM) techniques, while the Sn concentration profiles across the joints were determined using the energy x-ray dispersive spectroscopy (EDX) (for details see also Ref 20).

### Kinetics

*d*(

*t*), for diffusion soldering process, the following equation has been used:

If the value of the kinetic parameter is *n* = 0.5, then this corresponds to the volume diffusion, while for range *n* < 0.5, one can expect some contribution from the grain boundary diffusion.

In order to determine the kinetic parameter (exponential factor) *n*, the thickness of Ag_{3}Sn intermetallic phase was measured after various periods of time (see Table 1) applying the following procedure.

Four to six measurements were carried out for each intermetallic phase on both sides of the joint in different and representative areas of the sample keeping both sides parallel. Hence, no correction of the measurements due to the slope of the sample with respect to the optical axis of a microscope was necessary. In addition, the locations where the intermetallic phases grew in the form of scallops were not taken into account which helped us to avoid larger deviations for individual measurements. Hence, each calculated result is the average from many measurements, and the error produced by the program may have slightly different values even for the same series of samples.

_{3}Sn phase thicknesses

*d*(

*t*) vs. the annealing time

*t*for different temperatures. The resulting values of kinetic parameter

*n*in the Eq 30 were determined on the basis of log

*d*vs. log

*t*plots, and are listed in Table 2.

The values of the kinetic parameter, *n*, for Ag_{3}Sn phase growth for different temperatures of the Ag/Sn/Ag joint

Temperature, °C | Parameter, |
---|---|

230 | 0.67 ± 0.10 |

243 | 0.38 ± 0.07 |

250 | 0.17 ± 0.02 |

258 | 0.45 ± 0.07 |

265 | 0.55 ± 0.02 |

In the temperature range of 230-243 °C, a clear decrease of the *n* value is visible indicating the change of the diffusion mechanism, which can be explained by the growing contribution of the grain boundary diffusion and the decrease of the phase boundary reaction contribution up to it, leading to the Ag_{3}Sn phase formation. At the temperature of 250 °C, an anomalous decrease of the *n* value was noticed, which, however, cannot be clearly explained by the present research. At 265 °C, the *n* value increases again, and it may indicate the volume diffusion as the rate controlling factor.

### Modeling of Reactive Diffusion in Ag/Sn/Ag Interconnectors

Modeling of intermetallic phase (Ag_{3}Sn) growth for different temperatures in Ag/Sn/Ag joints have been performed based on the presented model. Proposed numerical method allowed us to solve effectively the problem. For calculations, the following data have been used:

Data

- (1)
\( c_{{{\text{Sn}},{\text{R}}}}^{{({\text{Ag}})}} = 0.1012\;{\text{mol/mol}} \)

- (2)
\( c_{{{\text{Sn}},{\text{L}}}}^{\upvarepsilon } = 0.2295\;{\text{mol/mol}} \)

- (3)
\( c_{{{\text{Sn}},{\text{R}}}}^{\upvarepsilon } = 0.2476\;{\text{mol/mol}} \)

- (4)
\( c_{{{\text{Sn}},{\text{L}}}}^{\text{Liq}} = 0.95\;{\text{mol/mol}} \)

- (5)
β

_{1}= 0 , β_{1}= 1 × 10^{−9}m - (6)
Calculated \( \tilde{D}^{\upvarepsilon } = \left\{ {\begin{array}{*{20}c} {230^\circ {\text{C}}} & {3.9 \times 10^{ - 13} \;{\text{m}}^{2} / {\text{s}}} \\ {243^\circ {\text{C}}} & {6.3 \times 10^{ - 13} \;{\text{m}}^{2} / {\text{s}}} \\ {250^\circ {\text{C}}} & {1.1 \times 10^{ - 13} \;{\text{m}}^{2} / {\text{s}}} \\ {258^\circ {\text{C}}} & {8.2 \times 10^{ - 13} \;{\text{m}}^{2} / {\text{s}}} \\ {265^\circ {\text{C}}} & {1.4 \times 10^{ - 12} \;{\text{m}}^{2} / {\text{s}}} \\ \end{array} } \right. \)

- (7)
*N*= 200—number of nodes in the approximation scheme.

*n*was found to be 0.17. Such a value corresponds to the significant contribution coming from grain boundary diffusion (Ref 21). The calculated activation energy equals 79 ± 13 kJ/mol. Similar value of 70.3 kJ/mol was obtained by Su et al. (Ref 22). Flanders et al. (Ref 23) obtained activation energy in Cu/Sn-Ag/Cu system for phase Cu

_{3}Sn to be 70.7 kJ/mol.

_{3}Sn intermetallic phase width is calculated as a function of time for different temperatures and compared with experimental results showing satisfactory agreement. The calculated kinetics of intermetallic phase growth shows parabolic behavior. This is the consequence of simplifying assumption that the thickness of outer phases being in contact with intermetallic Ag

_{3}Sn phase is semi-infinite. It ensures that concentrations at the boundaries are constant. This is not the limitation of the model, but we selected this semi-infinite configuration based on experimental results—see, EDX concentration profiles taken across the joint (Fig 3).

- (1)
Initial positions of the boundaries

*s*_{i}(0) = β_{i}= {0, 0.01, 0.02, 30} × 10^{−6}m - (2)
Diffusion coefficients in each layer [

*s*_{i}(*t*),*s*_{i+1}(*t*)],*D*_{i}= {5, 1, 0.1} × 10^{−13}m^{2}/s

Figure 8 shows evolution of moving boundaries *s*_{i}(*t*), and Fig. 9 shows thicknesses of layers, *d*_{i}(*t*) = *s*_{i+1}(*t*) − *s*_{i}(*t*), vs. time in a parabolic plot.

It can be seen that thicknesses of the layers exhibit initial non-parabolic growth. In particular, the first layer, *d*_{1}(*t*), is monotonically increasing, the third one, *d*_{3}(*t*), is monotonically decreasing, and the second, *d*_{2}(*t*), shows non-monotonic evolution.

## Conclusions

The mathematical model describing reactive diffusion in multi-component system with moving boundaries has been presented. Effective numerical method to solve the problem has been developed and applied. In the numerical approach, a moving boundary problem was converted into the equivalent fixed boundary problem. The resulting partial differential equations system was treated by the method of lines which gives a system of ordinary equations, which was subsequently solved numerically by the Radau II method. The investigation of the growth kinetics of intermetallic Ag_{3}Sn phase during diffusion soldering revealed different diffusion mechanisms. The Ag_{3}Sn phase grew as a product of the volume diffusion when kinetic parameter *n* is close to 0.5. In this case, a good agreement between calculated and experimental results has been obtained. When *n* = 0.17 (*T* = 250 °C), the significant deviation from the slope of Arrhenius plot was observed which indicates that the model cannot be applied. In the case of strong contribution of grain boundary diffusion, generalized boundary conditions including non-planar geometry of the interface (2D and/or 3D models) are necessary to describe formation of “scallops.” Such model will be subject of a separate article.

## Acknowledgment

This study has been supported by the Polish Ministry of Higher Education and Science—AGH grant no. 11.11.160.800.

### Open Access

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