Crystal Growth Technology

Semiconductors and Dielectrics

Hans J. Scheel, Peter Capper, Peter Rudolph

• Publisher: Wiley-VCH
• Year: 2011
• Print ISBN:
  9783527325931
• Edition: 1
• eBook ISBN:
  9783527632893
• Topic: Technology & Engineering
• Subtopic: Materials Science
• Index: Technology & Engineering

Semiconductors and dielectrics are two essential materials found in cell phones and computers, for example, and both are manufactured by growing crystals.
Edited by the organizers of the International Workshop on Crystal Growth Technology, this ready reference is essential reading for materials scientists, chemists, physicists, computer hardware manufacturers, engineers, and those working in the chemical and semiconductor industries. They have assembled an international team of experts who present the current challenges, latest methods and new applications for producing these materials necessary for the electronics industry using bulk crystal growth technology.
From the contents:
* General aspects of crystal growth technology
* Compound semiconductors
* Halides and oxides
* Crystal growth for sustaining energy
* Crystal machining

Part I

Basic Concepts in Crystal Growth Technology

1

Thermodynamic Modeling of Crystal-Growth Processes

Eberhard Buhrig, ManfredJurisch, Jürgen Korb, and Olf Pätzold

1.1 Introduction

Single-crystal growth from the melt is a very complex process that can be described, analyzed and controlled by various models. Relevant models [1-9] focus on:

• temperature distribution in melt, growing crystal and growth equipment, convective flow in the melt, stress/strain field in the growing crystal including the formation of defects like dislocations;
• temperature - dependent defect inventory of the crystal;
• behavior ofmelt/crystal interface and atomic growth mechanisms;
• thermochemical reactions in the growth chamber.

This chapter shows current possibilities and limitations of thermodynamic/thermochemical modeling in bulk single- crystal growth. In view of the highly complex subject matter the choice of the examples makes no claim to be complete, but shall consider very different modeling applications.

The fundamentals of thermodynamics/thermochemistry in crystal-growth processes have been described by Jacobs [5] in a very comprehensive way and are assumed to be known as a basis for the following.

The complex software package ChemSage and its successor FactSage [10] that are based on a minimization of the Gibbs free reaction energy are used for the thermodynamic calculations. This software offers solutions for the entire procedure from data gathering up to process simulation. To avoid confusion with other definitions and to facilitate our own activities the ChemSage nomenclature is used in this chapter [11].

It should, however, be mentioned that there are also available other program packages with similar capabilities, for example, Thermo-Calc [12] or HSC Chemistry [13]. As the general approach of thermodynamic modeling does not depend on the software the application of ChemSage/FactSage is not compulsory.

Thermodynamic/thermochemical models are simplified versions of the real processes, that is, they may only reproduce aspects included in the model. The resulting limitations have to be taken into consideration for any interpretation of results. In particular, it should be emphasized that thermodynamic models only describe thermodynamic equilibrium conditions and do not include reaction kinetics.

1.2 General Approach of Thermodynamic Modeling

1.2.1 Basics

Thermodynamic modeling is a highly efficient tool that may be used to describe, develop, optimize and control crystal- growth technologies. It has, however, also the following shortcomings and limitations:

• only equilibrium conditions can be described that very seldom occur under conditions of practical relevance;
• a complete data set of all relevant species is necessary for a comprehensive description of the respective thermodynamic system;
• a highly efficient software is required for calculations.

Therefore, thermodynamic modeling is mainly applied to make statements of trend.

1.2.1.1 State Variables for the Description of Equilibrium Conditions

Thermochemical and thermodynamic modeling are regarded as synonyms with thermodynamic modeling being the preferred term in this chapter.

Thermodynamics defines thermodynamic state variables that are both independent influencing parameters and measures at the same time. The most important state variables are given in Table 1.1.

Table 1.1 State variables.

Temperature	T	In K or°C
Volume	V	In m^3, l, dm^3
Amount	n	In mole
Concentration	c,x	Mole fraction
Pressure	P	In bar
Partial pressure	p	For ideal systems
Fugacity	f	For real systems
Surface	O	In m^2/mole, relevant for very small particles
Energy	E	In J (magnetic, mechanical, electromagnetic, etc.)

Reactions between the components of a thermodynamic system can be characterized by thermodynamic state functions (thermodynamic potentials) depending on state variables. The most important state function in thermodynamic modeling is the Gibbs free energy (G) that defines

• the equilibrium state of a system;
• the capacity for doing chemical work;
• the chemical potential of substances;
• the direction of chemical reactions;
• the magnitude of the driving force for chemical reactions.

The Gibbs free energy is given by

(1.1)

with H, enthalpy, T, temperature and S, entropy.

It can be split into several terms:

(1.2)

with the following meanings:

• G^ref[reference values of pure substances at T = 298.5 K and P = 1 bar taken from databases;
• G^ld, Gibbs free energy of an ideal system in dependence on state variables;
• G^ex, deviation from ideal behavior in a mixed system.

For the excess Gibbs free energy G^ex the model of a sub-regular solution according to Redlich-Kister-Muggianu [14] is used in this chapter. It is valid for binary i-j interactions in multi-component systems.

(1.3)

L^(ν), interaction parameter

x, concentration

If necessary, a fourth member, the so-called constrained power operator, can be added to Equation 1.2 by which additional influencing parameters like magnetic or electric fields, electrolytic potentials as well as time-dependent aspects of single reactions may be integrated into the model [15].

1.2.1.2 The ChemSage Software Package

The commercial ChemSage software package has been developed by Gesellschaft für Technische Thermodynamik und -physik GmbH [16]. This package allows for the complete thermodynamic modeling of complex crystal- growth processes. A schematic overview of the ChemSage family is given in Figure 1.1.

Figure 1.1 The ChemSage family.

The actual thermodynamic calculations for all program members take place in the central module (solver), the Gibbs free energy minimizer. This module is not accessible to users.

ChemApp [17] is a freely accessible programming platform which allows for programming of peripheral applications and online connection to external programs of global models, for example, crystal- growth processes.

In the following, the FactSage program is preferentially used. With this program it is possible to realize numerical calculations, graphical representations and input/output procedures, as well. FactSage also allows for connections with international database systems by which the standard values of the Gibbs free energy can be obtained for all relevant species of the investigated system.

In addition, the following modules belong to the ChemSage family:

• SimuSage, a software for interactive simulation ofprocesses;
• ChemSheet, a software for handling input/output operations including all functions of the Excel software. Even externally programmed processes like diffusion or other time-dependent procedures can be integrated by ChemSheet.

Furthermore, the ChemSage family comprises an effective module for optimization of thermodynamic data by which missing initial data can be added or adjusted. Finally, a thermodynamic model of a reactor is available that allows for simulations of pseudo- 3D and time - dependent processes.

In the following, there are given some examples for the calculation of realistic thermodynamic functions of single-species and stoichiometric reactions as well as calculations of many - component, many - phase equilibria.

The first example will be considered in detail in order to demonstrate the single steps of thermodynamic modeling, whereas the remaining ones are illustrated in an abbreviated form, as the general approach is always the same.

1.3 Crystal Growth in the System Si-C-O-Ar (Example 1)

The first step of modeling is the detailed formulation of the task and the selection of the components relevant for the system.

In the following, a special aspect of thermal silicon technology will be shown. The system components are Si, C, O and Ar chosen because of their practical relevance:

• me...

Table of contents

• Cover
• Title
• Copyright
• Foreword
• Preface
• List of Contributors
• Part I: Basic Concepts in Crystal Growth Technology
• Part II: Semiconductors
• Part III: Dielectrics
• Part IV: Crystal Machining
• Index