Crystal Growth For Beginners: Fundamentals Of Nucleation, Crystal Growth And Epitaxy (Third Edition)

Fundamentals of Nucleation, Crystal Growth and Epitaxy

Ivan V Markov

• Publisher: WSPC
• Year: 2016
• Edition: 3
• eBook ISBN:
  9789813143869
• Topic: Scienze fisiche
• Subtopic: Materia condensata

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The processes of new phase formation and growth are of fundamental importance in numerous rapidly developing scientific fields such as modern materials science, micro- and optoelectronics, and environmental science. Crystal Growth for Beginners combines the depth of information in monographs, with the thorough analysis of review papers, and presents the resulting content at a level understandable by beginners in science. The book covers, in practice, all fundamental questions and aspects of nucleation, crystal growth, and epitaxy.

This book is a non-eclectic presentation of this interdisciplinary topic in materials science. The third edition brings existing chapters up to date, and includes new chapters on the growth of nanowires by the vapor–liquid–solid mechanism, as well as illustrated short biographical texts about the scientists who introduced the basic ideas and concepts into the fields of nucleation, crystal growth and epitaxy. All formulae and equations are illustrated by examples that are of technological importance. The book presents not only the fundamentals but also the state of the art in the subject.

Crystal Growth for Beginners is a valuable reference for both graduate students and researchers in materials science. The reader is required to possess some basic knowledge of mathematics, physics and thermodynamics.

--> --> Contents: Crystal–Ambient Phase Equilibrium;Nucleation;Crystal Growth;Epitaxial Growth;The Shoulders on Which We Stand; --> -->
Readership: Professionals and graduate students in materials science, dealing with crystals; with some basic knowledge of mathematics, physics and thermodynamics.
-->Nucleation, Epitaxy, Crystal Surfaces, Surface Structure, Crystal Growth, Misfit Dislocations, Thin Solid Films, Wetting, Surface Diffusion, Surfactanr Epitaxy, Nanowires, Ehrlich-schwoebel Effect, Silicon0

Chapter 1

Crystal–Ambient Phase Equilibrium

1.1Equilibrium of Infinitely Large Phases

The equilibrium between two infinitely large phases α and β is determined by the equality of their chemical potentials μ_α and μ_β. The latter represent the derivatives of the Gibbs free energies with respect to the number of particles in the system at constant pressure P and temperature T, μ = (∂G/∂n)_P,T, or, in other words, the work which has to be done in order to change the number of particles in the phase by unity. In the simplest case of a single-component system, we have

The above equation means that the pressures and the temperatures in both phases are equal. The requirement P_α = P_β = P is equivalent to the condition that the boundary dividing both phases is flat or, in other words, that the phases are infinitely large. This question will be clarified in the next section where the equilibrium of phases with finite sizes will be considered.

Let us assume now that the pressure and the temperature are infinitesimally changed in such a way that the two phases remain in equilibrium, i.e.,

It follows from (1.1) and (1.2) that

Recalling the properties of the Gibbs free energy (dG = VdP – SdT), we can rewrite (1.3) in the form

where s_α and s_β are the molar entropies, and v_α and v_β are the molar volumes of the two phases in equilibrium with each other.

Rearranging (1.4) gives the well-known Equation of Clapeyron

where Δs = s_α − s_β, Δv = v_α − v_β, and Δh = h_α − h_β is the enthalpy of the corresponding phase transition.

Let us consider first the case when the phase β is one of the condensed phases, say, the liquid phase and the phase α is the vapor phase. Then the enthalpy change Δh will be the enthalpy of evaporation Δh_ev = h_v − h_l and v_l and v_v will be the molar volumes of the liquid and the vapor phases, respectively. The enthalpy of evaporation is always positive and the molar volume of the vapor v_v is usually much greater than that of liquid v_l. In other words, the slope dP/dT will be positive. We can neglect the molar volume of the crystal with respect to that of the vapor and assume that the vapor behaves as an ideal gas, i.e., P = RT/v_v. Then Eq. (1.5) attains the form

which is well known as the Equation of Clapeyron–Clausius. Replacing Δh_ev with the enthalpy of sublimation Δh_sub, we obtain the equation which describes the crystal–vapor equilibrium.

Assuming Δh_ev (or Δh_sub) does not depend on the temperature, Eq. (1.6) can be easily integrated to

where P₀ is the equilibrium pressure at some temperature T₀.

In the case of the crystal–melt equilibrium, the enthalpy Δh is equal to the enthalpy of melting Δh_m which is always positive and the equilibrium temperature is the melting point T_m.

Figure 1.1: Phase diagram of a single-component system in P–T coordinates. O and O′ denote the triple point and the critical point, respectively. The vapor pressure becomes supersaturated or undercooled with respect to the crystalline phase if one moves along the line AA′ or AA″. The liquid phase becomes undercooled with respect to the crystalline phase if one moves along the line BB″. ΔP and ΔT are the supersaturation and undercooling, respectively.

As a result of the above considerations, we can construct the phase diagram of our single-component system in coordinates P and T (Fig. 1.1). The enthalpy of sublimation of crystals Δh_sub is greater than the enthalpy of evaporation Γh_ev of liquid, and hence the slope of the curve in the phase diagram giving the crystal–vapor equilibrium is greater than the slope of the curve of the liquid–vapor equilibrium. On the other hand, the molar volume v_l of the liquid phase is usually greater than that of the crystal phase v_c (with some very rare but important exceptions, for example, in the cases of water and bismuth), but the difference is small so that the slope dP/dT is great, in fact, much greater than that of the other two cases and is also positive with the exception of the cases mentioned above. Thus, the P–T space is divided into three parts. The crystal phase is thermodynamically favored at high pressures and low temperatures. The liquid phase is stable at high temperatures and high pressures, and the vapor phase is stable at high temperatures and low pressures. Two phases are in equilibrium along the lines and the three phases are simultaneously in equilibrium at the so-called triple point 0. The liquid–vapor line terminates at the so-called critical point 0′ beyond which the liquid phase does not exist any more because the surface energy of the liquid becomes equal to zero and the phase boundary between both phases disappears.

1.2Supersaturation

When moving along the dividing lines, the corresponding phases are in equilibrium, i.e., Eq. (1.1) is strictly fulfilled. If the pressure or the temperature is changed in such a way that we deviate from the lines of the phase equilibrium one or another phase becomes stable. This means that its chemical potential becomes smaller than the chemical potentials of the phases in the other regions. Any change of the temperature and/or pressure which lea...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Dedication
• Preface to the Third Edition
• Preface to the Second Edition
• Preface to the First Edition
• About the Author
• Contents
• Chapter 1. Crystal–Ambient Phase Equilibrium
• Chapter 2. Nucleation
• Chapter 3. Crystal Growth
• Chapter 4. Epitaxial Growth
• Chapter 5. The Shoulders on Which We Stand
• Appendix A. Units, Conversion Factors and Constants
• References
• Index