KR20020059946A - Method for estimating and controlling the temperature distribution on the surface of a silicon single crystal for the purpose of minimizing microscopic defects present inside of the single crystal - Google Patents

Abstract

PURPOSE: Provided is a prediction method of the surface temperature distribution to minimize microscopic internal defects in the process of silicon single crystal growth by using s variational method. The prediction helps control of thermal environment during crystal growth, resulting in improving the quality of silicon single crystal. CONSTITUTION: The prediction and control method contain the sequences of: finding an objective function to minimize internal microscopic defects of silicon single crystal; setting up the Euler-Lagrange equation using the objective function, modeling formula and variational method; finding an optimized surface temperature distribution by numeric calculating of the Euler-Lagrange equation; deciding thermal environment of silicon single crystal growth from the optimized surface temperature distribution.

Description

METHOD FOR ESTIMATING AND CONTROLLING THE TEMPERATURE DISTRIBUTION ON THE SURFACE OF A SILICON SINGLE CRYSTAL FOR THE PURPOSE OF MINIMIZING MICROSCOPIC DEFECTS PRESENT INSIDE OF THE SINGLE CRYSTAL}

Silicon single crystal is a material of silicon wafer which is the basis for semiconductor production. Due to the characteristics of the highly integrated semiconductor industry, high quality of silicon single crystal is required and many efforts have been made to improve the quality of silicon single crystal. The factors that determine the quality of silicon single crystal can be divided into macro and micro defects. Among these, macro defects are currently controlled in commercial silicon single crystal production process, while research on micro defects is underway. .

In order to improve the quality of highly integrated semiconductors, many model equations have been proposed to explain the experimental results of micro-defects since the 1980's. Or, it helps to predict the distribution of microscopic defects inside the silicon only in the numerically calculated state. Therefore, there is a limitation that the distribution of microscopic defects can be known using model equation only when the thermal environment of the process is given. In the end, existing model equations did not tell us how to control the thermal environment of the process to optimize quality, but simply provided a basis for a given thermal environment to be compared with another given thermal environment.

In this way, the existing methods are only limited method studies to find the distribution of microscopic defects using model equations and compare them with each other through the case-by-case studies. It does not suggest the possibility of controlling the thermal environment of the process on which it is based.

In order to solve the above problems, the present invention predicts an optimal surface temperature distribution that minimizes microscopic defects in a silicon single crystal growth process by using a variation method, which is one of a model method and a mathematical methodology for finding an optimal solution, and uses the thermal process of the production process. The objective is to improve the quality of silicon single crystals by controlling the environment.

1 is a schematic diagram showing an analysis region of the present invention in a silicon single crystal process,

2 is a numerical analysis flowchart in the present invention,

3 is A graph showing (a) optimum temperature distribution in silicon single crystal and (b) optimum surface temperature distribution at phosphorus process conditions,

4 is A graph showing (a) optimum temperature distribution in silicon single crystal and (b) optimum surface temperature distribution at phosphorus process conditions,

FIG. 5 is a schematic diagram showing an optimal thermal environment proposed from the optimum surface temperature distribution of FIG. 3,

FIG. 6 is a schematic diagram showing an optimal thermal environment proposed from the optimum surface temperature distribution of FIG. 4,

7 is a conceptual diagram of temperature control using optimum surface temperature distribution data.

In order to achieve the above object of the present invention, in accordance with an aspect of the present invention, (1) construct an objective function that minimizes microscopic defects of silicon single crystals, and (2) use the constructed objective function, model equation, and Construct Euler-Lagrangian equations, (3) numerically calculate Euler-Lagrange equations and boundary conditions to obtain optimal surface temperature distributions that minimize microscopic defects in silicon single crystals, and (4) obtain optimal temperature distributions. An optimal surface temperature distribution prediction and control method for minimizing microscopic defects within a silicon single crystal, comprising determining the thermal environment of the silicon single crystal production process from the same.

The operation and features of the present invention will be described in more detail below.

The present invention predicts the surface temperature distribution to minimize microscopic defects in the production process of silicon single crystal, and based on this, it is possible to control the thermal environment in the actual silicon single crystal production process to improve the overall process and contribute to the improvement of the quality of silicon single crystal. Can be.

Features of the present invention

First, in order to minimize the microscopic defects, the methodology is based on mathematical model rather than experimental method for the prediction of optimal temperature distribution. Overcoming the limitations of existing case studies to find the optimal conditions,

Second, it can be used to control macroscopic defects as well as microscopic defects inside silicon single crystals.

Equations for modeling microscopic defects in conventional silicon single crystals have evolved from one-dimensional to two-dimensional models since the 1980s. In the case of the two-dimensional model there are many different types of models, but the basic form is shown in Equation 1 below. Microscopic defects at steady state can be thought of as being divided into interstitial and vacancy, and the equations dimensioned from the continuum assumption for each are as follows.

here,

I: interstitial, V: vacancy

R: radius of silicon single crystal

: Growth rate of silicon single crystal

C: dimensionless concentration (equilibrium concentration at actual concentration / melting point temperature)

Equilibrium concentration

D: diffusion coefficient

T: dimensionless temperature (actual temperature / melting point temperature of silicon)

: Melting point of silicon

The heat of formation

: Reaction coefficient

In addition, the equation regarding the temperature distribution inside the silicon is given by Equation 2 below.

Existing methodology first evaluates the thermal conditions of a given process through equations (1) and (2) to improve the quality of silicon single crystals, and then obtains a microscopic defect distribution based on the microscopic defect distribution between processes. It was. However, the existing methodology could compare and evaluate each case, but it was impossible to find the optimal condition.

Accordingly, an object of the present invention is to obtain an optimal surface temperature distribution that can minimize quality deterioration due to microscopic defects. According to the method of the present invention for this purpose,

(1) construct an objective function that minimizes the microscopic defects of silicon single crystals,

(2) construct Euler-Lagrangian equations using the constructed objective function, model equations, and

(3) numerically calculating Euler-Lagrange equations to find the optimal surface temperature distribution that minimizes microscopic defects in silicon single crystals,

(4) From the optimal temperature distribution, the thermal environment of the silicon single crystal production process is determined.

Specifically referred to herein as " model equation ", Equation 1 is microscopic with the assumption that vacancy and interstitial can be viewed as continuum and their behavior can be explained by a diffusion process. It refers to an equation modeled as a partial differential equation.

As used herein, the term "variance method" refers to a point (function that becomes zero when a variation of the objective function (I) is given when a condition that the optimal solution should satisfy the objective function is given, for example, in Equation 4). The minimum value of is the maximum value at the point where the derivative is zero.) This is a methodology for finding the optimal solution using the mathematical requirements, which is another form of partial differential equation that must be satisfied by solving the optimization problem given by the general partial differential equation called Euler-Lagrange equation. Finally, the variation method provides another form of partial differential equation that the solution we are trying to solve (minimizing the objective function) must satisfy (called the Euler-Lagrange equation) and the optimization problem is the partial differential equation analysis problem. In this way, the given partial differential equation can be solved to find the optimal solution.

As used herein, the term "finite difference method" One of the numerical methods is to differentiate partial differential equations in the area to be interpreted (that is, consider each point of the grid rather than dividing it into a lattice structure to consider all the continuous regions). The derivatives from the above Taylor series () Is a numerical method that finds an approximate solution that satisfies the equation converted to algebraic form at each given point by using

As used herein, the term "Euler-Lagrange equation" Another partial differential equation that must be satisfied by the optimal solution derived from the variation method, that is, the optimization problem first given is given by the objective function of the integral form, but the objective function that changes according to the theorem of the variation method needs to satisfy zero. Using the necessary conditions, the equations in the form of partial differential equations that must be satisfied by the optimal solution can be obtained and the optimal solution can be found by analyzing them using numerical methods for partial differential equations such as finite difference method. Equation 6 is the Euler-Lagrange equation here.

In the present invention, in consideration of the two-dimensional silicon single crystal system as shown in Figure 1, the objective function for minimizing the microscopic defect is configured as shown in Equation 3 below.

here,

: Melting Point Temperature of Silicon

: Concentration of interstitial

: Concentration of vacancy

= Equilibrium concentration of a gap at a given temperature

= Equilibrium concentration of the cavity at a given temperature

Mass variable

Given objective function In order to find the optimal solution that minimizes the problem, the present invention applies a variation method. The variation method gives a fact as in Equation 4 with respect to the general objective function.

Therefore, from the fact of Equation 4 provided by the variation method to find a solution that optimizes a given objective function, an optimal condition expressed by partial differential equation of the optimal solution called Euler-Lagrange equation can be obtained. However, for systems of interest in the present invention, the distribution of temperature distribution and microscopic defects is governed by Equations 1 and 2 below. Therefore, in order to find a solution satisfying Equations 1 and 2 while satisfying Equation 3, which is a given objective function, Lagrange multiplier is introduced and Equation 3 is changed to Equation 5 below.

here,

: Lagrangian multiplier of gap and cavity, respectively, temperature and micro defects

: Model of gap and cavity respectively

The Euler-Lagrangian equation, as shown in Equation 6 below, can be obtained by using the fact of Equation 4 of the variation method, and the solution satisfying the model equation can be numerically obtained by minimizing the given objective function.

(2)

(3)

(4)

(5)

(6)

here,

In order to interpret the six equations of Equation 6, the solution of Equation 6 was obtained according to the flow as shown in FIG. 2 using the finite difference method, which is one of numerical methods. The temperature distribution for the optimization of the single crystal production process can be obtained from Equation 6 obtained through the model equation, the objective function, and the variation method as shown in FIG. 3.

3 does not take into account the system variables of the actual process, but given any appropriate weighting factors, it is possible to find the optimal solution for the process environment under various conditions by adjusting the mass variables. 3 is radius , Production speed The optimum temperature distribution in the case of is shown. 4 is radius Production speed Is another case.

As such, according to the present invention, it is possible to find other optimal surface temperature distribution under various production conditions as shown in the results of FIGS. 3 and 4. The thermal environment of the process may be proposed as shown in, for example, FIG. 5 so as to have the optimum surface temperature distribution of FIG. 3, and the thermal environment of the process of FIG. 6 may be proposed from the optimal surface temperature distribution result of FIG. 4.

As a result, the analysis method of the present invention suggests an optimal temperature distribution by using existing model equations under various process conditions, and provides an optimal temperature distribution from the limit of comparison with each process. can do. In addition, it is possible to control the quality of silicon single crystals by using the optimal surface temperature distribution data of the thermal environment of the process.

FIG. 7 illustrates the concept of such a control process. As shown in FIG. 7, according to the present invention, the quality improvement of silicon is confirmed by confirming and maintaining an agreement with the optimum temperature distribution obtained from the optimum surface temperature measured in the actual process. Thermal environment control is enabled.

According to the present invention, the existing model equations are the same, but the Euler-Lagrangian equation is obtained by constructing the objective function mathematically representing the variance method and the quality optimization, which are optimization techniques, and numerically interpreted by the finite difference method. The surface temperature distribution of the silicon single crystal can be obtained for the optimal quality which minimizes the temperature, and the method of the present invention is applicable to various operating conditions of the silicon single crystal production. It is possible to control the thermal environment.

Claims (1)

(1) construct an objective function that minimizes the microscopic defects of silicon single crystals, (2) construct Euler-Lagrangian equation using the constructed objective function, model equation, and variation method, (3) numerically calculating Euler-Lagrange equations to find the optimal surface temperature distribution that minimizes microscopic defects in silicon single crystals, (4) determining the thermal environment of the silicon single crystal production process from the obtained optimal temperature distribution, Optimal surface temperature distribution prediction and control method to minimize microscopic defects inside silicon single crystal.