A Combined Experimental and Modeling Study for the Growth of
SixGel, Single Crystals by Liquid Phase Diffusion (LPD)
Mehmet YILDIZ
B.A.Sc., Yildiz Technical University, 1996 M.A.Sc., Istanbul Technical University, 2000
A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of
DOCTOR OF PHILOSOPHY
in the Department of Mechanical Engineering
O Mehrnet Yildiz, 2005 University of Victoria
All rights resewed. This dissertation may not be produced in whole or in part by photocopy or other means, without the permission of the author.
Supervisor: Dr.S.Dost
Abstract
Si,Ge,-, alloy is an emerging semiconductor material with many important potential applications in the electronic industry due to its adjustable physical, electronic and optical properties. It has been scrutinized for the fabrication of high-speed micro electronics (e.g., SiGe heterojunction bipolar transistors (HBT) and high electron mobility field effect transistors) and thermo-photovoltaics (e.g., photodetectors, solar cells,
thermoelectric power generators and temperature sensor). Other applications of Si,Ge,-,
include tuneable neutron and x-ray monochromators and y-ray detectors. In these applications, SixGel-, alloy is generally used in the form of epilayers that have to be
deposited on a lattice-matched substrate (wafer). Therefore, SixGel-, bulk single crystals
with a specific composition ( x ) are needed for the extraction of such wafers.
LPEE (Liquid Phase Electroepitaxy) was considered as a technique of choice for the growth of single crystals. However, LPEE growth process needs a single crystal seed with the same composition as the crystal to be grown. Yet, such a seed substrate with particularly higher composition is not commercially available. In order to address this important issue in LPEE, a crystal growth technique, which is named "Liquid Phase Diffusion" (LPD), was developed and used to produce the needed seed substrate materials. This was the main motivation of the present research.
This thesis presents a combined experimental and modelling study for LPD growth of compositionally graded, germanium-rich single crystals of 25 rnrn in diameter for use as lattice-matched seed substrates. The experimental part focuses on the design and development of a complete LPD grow system. The experimental set-up was tested by
growing ten Si,Ge,-, single crystals. Grown crystals were characterized by macroscopic
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Abstract 111
single crystallinity and growth striations. Compositional mapping of selected crystals were performed by using Electron Probe Micro Analysis (EPMA) as well as Energy Dispersive X-ray analysis (EDX). It was shown that the LPD technique can be successfully utilized to obtain Si,Ge,-, single crystals up to 6-8 % at.Si with uniform radial composition distribution.
The modeling part presents a rational continuum mixture model developed to study transport phenomena (heat and mass transfer, fluid flow) occurring during the LPD
growth of Si,Ge,-,
.
Based on the continuum model developed, two and three-dimensionaltransient numerical simulations were carried out. The numerical simulation models presented account for some important physical features of the LPD growth process
ofSi,Ge,-,
,
namely (1) a growth zone design on the thermal field, (2) the structure of thebuoyancy induced convective flow and its effect on the growth and transport mechanisms, (3) the shape and evolution of the initial and progressing growth interfaces, and (4) the spatial and time variation of the crystal growth velocity. It was numerically
shown that, as the name LPD implies, the growth of Si,Ge,-, by LPD is mainly a
diffusion driven process except the initial stages of the growth process during which the natural convection in the solution zone is prominent and has significant effects on the composition of the grown crystal. The simulated evolution of the growth interface agrees with experimental observations. In addition, the numerical growth velocities are in good agreement with those of experiments. The numerical model developed can be used to study other crystal growth processes such as LPEE, Traveling Heater Method THM, and vertical Bridgman with slight modifications.
Table of Contents
Abstract Table of Contents List of Figures List of Tables Nomenclature Acknowledgements Dedication1 Introduction
1.1 Motivation and Goals..
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1.2 Outline of the Thesis.. . .
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2 Structural Properties of SixGel, Alloy System
2.1 Introduction..
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2.2 Crystal Structure and Equilibrium Phase Diagram of SixGel-,.. . . .
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2.3 Band Gap Structure of Bulk Si,Gel-, Alloys..
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Crystal Growth Methods for SixGel_,
Alloys
3.1 Introduction
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3.2 Melt Growth Methods..
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3.2.1 Crystal Pulling..
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3.2.2 Vertical Bridgrnan Technique (VB). . .
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9 9 16 19 19 19 2 1 2 8Table of Contents v
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3.2.3 Floating Zone Technique (FZ)
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3.2.4 Zone Melting (ZM)
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3.2.5 Difficulties in Melt Growths
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3.3 Solution Growth
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3.3.1 Liquid Phase Epitaxy (LPE)
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3.3.2 Liquid Phase Electroepitaxy (LPE)
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3.4 Vapour Phase Growth
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3.4.1 Chemical Vapour Deposition (CVD)
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3.4.2 Molecular Beam Epitaxy (MBE)
3.5 Strained and Relaxed Si,-,Ge, Layers in Si,,Ge, / Si System
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Applications of SixGel.
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4.1 Introduction 51...
4.2 Photodetectors 51...
4.2.1 PIN Photodetector 52...
4.2.2 HIP Infrared Detector 53
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4.3 Heterojunction Bipolar Transistor 54
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4.3.1 SiGeISi HBT for RF Applications 55
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4.3.2 High Electron Mobility Transistor 56
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4.4 Solar Cells 57
4.5 Summary
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A Rational Continuum Mixture Model for LPD Growth of
SiXGel_,
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5.1 Introduction 59
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5.2 The Rational Mixture Model 63
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5.2.1 Kinematics of Mixtures 63
5.2.2 Axioms of Therrnomechanics and the Field Equations
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5.2.2.1 Balance of Mass 74
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Table of Contents
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5.2.2.3 Balance of Moment of Momentum
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5.2.2.4 Balance of Energy...
5.2.2.5 Entropy Inequality...
5.2.3 Constitutive Equations...
5.2.3.1 The Liquid Phase
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5.2.3.2 The Solid Phase
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5.2.3.3 Interface Conditions
Governing Equations and Boundary Conditions for the Computational
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Model
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5.3.1 The Liquid Phase
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5.3.2 The Solid Phase
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Physical Parameters of the LPD Si,Gel., System
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Summary
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Numerical Simulation for the Growth of SixGel+ Single Crystals
by LPD
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Introduction
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Order of Magnitude Analysis for the LPD Growth System
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Discretisation Method and Solution Algorithms
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Finite Volume Mesh
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Grid Modification and Interface Movement
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Results of Two-Dimensional Numerical Simulation
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6.6.1 Temperature Field
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6.6.2 Flow and Concentration Fields
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6.6.3 Growth Velocity
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Outcomes of Three-Dimensional Numerical Simulation
Summary
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The Growth of Compositionally Graded SixGel. Single Crystals
by Liquid Phase Diffusion (LPD)
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Table of Contents vii 7.2 LPD Crystal Growth System
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7.3 LPD Growth Steps...
7.3.1 Temperature Profile
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7.3.2 Preparation of Growth Charges...
7.3.2.1 Cutting and Core-drilling Charge Materials
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7.3.2.2 Cleaning and Chemical Treatment
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7.4 LPD Growth Principles
7.4.1 A Typical Procedure for an LPD Crystal Growth Experiment
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7.5 Experimental Results and Characterization
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7.6 Summary
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Conclusion and Outlook
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8.1 Conclusions...
8.2 Contributions...
8.3 Future WorkBibliography
Appendix A Summary of Literature Review on Crystal Growth
Methods for SiGe Alloy System
Appendix B Derivations of Model Equations for Rational
Continuum Mixture Model
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B
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1 Introduction...
B.2 Transport Theorem for an Arbitrary Field
B.3 Derivations of Thermomechanical Balance Laws and Associated Jumps
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B.3.1 Conservation of Mass
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3.2 Balance of Linear Momentum...
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B.3.3 Balance of Moment of Momentum
B.3.4 Balance of Energy
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Table of Contents viii
B.3.5 Entropy Inequality
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23 3 B.3.6 Open Form of Jump Balances... . .
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242List of Figures
Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 Figure 5.1 Figure 5.2 Figure 5.3a-) The sketch of the diamond unit cell unit, b-) projection of the cube face (numbers indicate displacements of atoms normal to the paper plane.).
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Equilibrium phase diagram of Si,Ge,-, system...
A representative equilibrium phase diagram of Si,Ge,-, system... . .
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Lattice parameter of Si,Ge,-,system as a function of Ge
composition: circles indicate experimentally measured lattice parameters and square symbols indicate lattice parameters calculated using Vegard's law..
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Band gap energy versus composition x for Si,Ge,-, as determined from absorption measurements (cross) and low temperature PL spectra (filled-in black squares).. . .
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A schematic drawing of a radio frequency heated Cz furnace..
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Schematic drawing of a standard and detached VB technique... . . .
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A representative diagram for a zone melting processRepresentative figure for the multi compartment sliding boat technique.
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Formation of strained coherent epitaxial layer...
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Misfit dislocation formation at the interface..
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The schematic view of the LPD growth system...
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Coordinate system...
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Mapping defined for a mixture of two constituents, (i.e. 8 = 2 ).. . .
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List of Figures Figure 5.4 Figure 6.1 Figure 6.2 Figure 6.3 Figure 6.4 Figure 6.5 Figure 6.6 Figure 6.7 Figure 6.8 Figure 6.9 Figure 6.10 Figure 6.1 1 Figure.6.12 Figure 6.13
The schematics of the vertical cross section of the LPD growth system with the representative thermal profile where t is the unit
tangential vector to the growth interface, while n is the unit normal
to the growth interface pointing into the liquid
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Flow diagram of the solution procedure... . .
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Sample finite volume mesh used in the current simulation, a-) a typical mesh for 3-D simulation, b-) mesh for 2-D simulation... . .
Representative vertical cross-section and top view of the 3-D computational domain... . .
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Grid plane perpendicular to the axial direction..
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Computed thermal field within entire computational domain (2-D) for varying growth times (Temperature in the labels are given in Kelvin).
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Computed thermal field within entire computational domain (2-D) for varying growth times (Temperature in the labels are given in Kelvin).. . .
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Flow (left column) and concentration (right column) fields for the growth time 0.5 hours. Flow field is given in terms of magnitude ofthe flow velocity, U =
JR
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and solute concentration is given interms of silicon mass fraction..
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Flow and concentration fields at various hours of the growth..
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Flow and concentration fields at various hours of the growth...
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Decrease in the magnitude of flow velocity as a function of silicon mass fraction... . .
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Flow and concentration fields for different hours of growth wherepc
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Interface position as a function of growth time for r = 0..
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List
of Figures xi Figure 6.14 Figure 6.15 Figure 6.16 Figure 6.17 Figure 6.18 Figure 6.19 Figure 7.1 Figure 7.2 Figure 7.3 Figure 7.4 Figure 7.5 Figure 7.6 Figure 7.7 Figure 7.8 Figure 7.9 Figure 7.10The time evolution of the growth interface for the half geometrical domain is shown on the left (the time interval between each line is three hours, and total simulated growth time is 39 hours), while the cross section of an LPD grown crystal on the right. Agreement between the experimental and the simulation results are quite good.
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Computed thermal field within entire computational domain (3-D) at the growth time t=l h. Temperature in the labels is given in Kelvin....
Flow (up) and concentration fields (down) at = 0.5 h.....
Flow and concentration fields at growth time, t = 3.5 h.....
Flow and concentration fields at growth time t = 5.5 h.....
Flow fields at different growth time: 0.5 2.5,3.5,4.5 and 5.5 hours.The horizontal plane is at a distance of 14 mm from the bottom of
the substrate.
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A view of the LPD growth set-up used in this thesis work..
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Schematic illustration of LPD crystal growth platform..
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The detailed drawing for the experimental LPD growth set up..
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Top view of the furnace..
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Picture of quartz components.....
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Gas distribution network..
Thermal profile measured along the quartz ampoule (square) and the axis of a center drilled silicon dummy load (circle), and the
schematic diagram of the LPD growth system..
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A representative figure for the growth mechanism of LPD growth
system.
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Typical heating and cooling cycle for the growth experiment..
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Cutting configuration for the characterization of grown crytals (up), and pictures of two mm thick vertical slices for several grown single crystals (down).
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of Figures xii Figure 7.1 1 Figure 7.12 Figure 7.13 Figure 7.14 Figure 7.1 5 Figure B. 1 Figure B.2Several LPD grown compositionally graded Si,Gel-, single crystals.. 180 Vertical cross-section of four grown single crystals for determining
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the growth velocity.. 184
Interface displacement versus growth time for the center and edge
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regions of the grown crystals.. 185
Radial silicon distribution for LPD-5 along with corresponding axial
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distribution.. 186
Silicon concentration distribution for LPD- 19; radial silicon distribution at various axial steps (left), corresponding axial silicon distribution for different radial locations..
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Discontinuity surface. 2 1 1
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XlllList of Tables
Table 5.1 Table 5.2 Table 5.3 Table 6.1 Table 6.2 Table 7.1 Table A.
1 Table A.2 Table A.3 Table A.4SiGe phase diagram coefficients
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Design parameters for LPD growth system
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Physical properties of growth charges and the quartz crucible
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Non-dimensionless numbers and their characteristic values calculated for the growth of germanium-rich SiGe single crystals by LPD
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Solver parameters...
Dimensions of various components of the LPD growth platform
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S~~nnnary of SiGe growth by Cz technique
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Summary of SiGe growth by VB technique
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Summary of SiGe growth by FZ technique
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xiv
Nomenclature
Latin
Letters
Description Phenomenological coefficients Material bodiesMaterial body of the
a
th constituent Phenomenological coefficientsGravitational body force per unit mass, acting on the
a
th constituentGravitational body force per unit mass, acting on the
a
th constituent in component formGravitational body force per unit mass, acting on the mass averaged particle
Binormal unit vector (refer to appendix B) Mass fraction increment for solute
Mass fraction of the
a
th constituent Mass fraction of soluteReference mass fraction of solute Dissipative part of the stress tensor
Nomenclature
Concentration gradient related diffusion coefficient for the liquid phase
Molecular diffusion coefficient
Temperature gradient related diffusion coefficient for the liquid phase (Soret coefficient)
Symmetric deformation rate tensor
Trace of symmetric deformation rate tensor Area element
Length element Volume element
Euclidian space (Cartesian space) Internal energy for the
a
th constituent Permutation symbolDeformation gradient for the
a
th constituentDeformation gradient for thea th constituent in component form
The resultant external force acting upon thea th constituent
The resultant external force acting upon thea th constituent in the component form
Other external forces acting upon the
a
th constituentInternal heat generation rate per unit mass for the
a
thconstituent
Internal heat generation rate per unit mass for the mixture
The height of liquid phase Tangential unit normal Identity tensor rn2 / s m 2 / s m 2 / S K 1 /s l/s m2 m m J N N
N
J
/ kgsJ
/ kgs mNomenclature xvi
Base vector in the reference state Base vector in the deformed state Jacobian
Mass flux for the
a
th constituent, with respect to a stationary coordinate frameDiffusion flux for the
a
th constituent Kinetic energy of thea
th constituentConcentration gradient-related material coefficient for the liquid phase (Dufour coefficient)
Temperature gradient-related material coefficient for the liquid phase
Combined concentration gradient-related conductivity for the liquid phase
Combined temperature gradient-related conductivity for the liquid phase
Molecular weigh of the
a
th constituentMolecular weigh of the mixture Moment of momentum supply vector
Moment of momentum supply vector in component form
Mass of the
a
th constituentThe number of constituents within the mixture Unit normal
Projection tensor
Projection tensor in component form Material point
A particle of the
a
th constituent in the reference state Spatial point kg / m2s kg / m 2 s J J / sm J / smK J / sm J / smK kg / mol kg / mol N / m 2 N / m 2 kgNomenclature xvii Unknown pressure function
Modified pressure
Momentum production rate per unit volume for the
a
th constituentMomentum production rate per unit volume for the
a
th constituent in component formHeat energy supply per unit time for the
a
thconstituent
Heat influx vector of the
a
th constituent in componentform
Inner part of the heat influx vector in component form Heat influx vector for the mixture
Modified heat influx vector for the ath constituent Modified heat influx vector for the mixture
Radial coordinate direction
Mass production rate per unit volume for the a t h constituent
Viscous portion of the interface stress tensor for the
a
th constituentEntropy of a thermodynamic system
Entropy of a thermodynamic system per unit volume Entropy of a thermodynamic system per unit mass Surface bounding the material volume V
Boundary of the volume 'V
Mass production rate per unit area for the
a
thconstituent
Reference temperature Temperature increment
Nomenclature xviii
Partial surface traction per unit area
Partial stress tensor surface traction per unit area Partial stress tensor in component form
Partial interface stress vector Partial interface stress tensor Magnitude of flow intensity
Internal energy of a thermodynamic system per unit volume
Internal energy of a thermodynamic system per unit mass
Other energies per unit time
Velocity of the discontinuity surface
Velocity of the discontinuity surface in component form
Volume per unit mass (Specific volume) Volume of the material body
Volume of a body in the deformed state (spatial volume)
The material velocity of the material particle X"
The material velocity of the material particle X" in
component form
Mass averaged velocity of the mixture
Mass averaged velocity of the mixture in component form
Diffusion velocity for the
a
th constituentDiffusion velocity for the
a
th constituent in componentNomenclature xix Surface velocity vector for the
a
th constituentVelocity component in radial direction
Velocity component in circumferential direction Velocity component in vertical direction
Mechanical energy per unit time for the a t h constituent
Position vector of a particle of t h e a th constituent or body
Position vector of a particle of t h e a th constituent or body in component form
Position vector for a mass centered particle (fictitious particle)
Position vector of the particle X" in spatial
configuration
Position vector of the particle X" in spatial
configuration in component form
Material velocity of the material particle X"
Molar fraction of the
a
th constituent Molar fraction of the mixtureVertical coordinate direction
Greek Letters
Symbol Description
PC
Solutal expansion coefficientPT
Thermal expansion coefficientza
Deformation function for the ath constituentm / s
mole /mole mole / mole m
Nomenclature
Deformation function for the ath constituent in index notation
Inverse deformation function for the ath constituent Inverse deformation function for the ath constituent in index notation
Internal energy per unit mass for the
a
th constituent J / k gInner part of the internal energy per unit mass J / k g
Internal energy per unit mass for the mixture J / k g
Energy supply density to the
a
th constituent per unit J / m3stime
An
arbitrary field (scalar, vector or tensor)An
arbitrary material function (i.e. scalar, vector or tensor quantity)An
arbitrary spatial function (i.e. scalar, vector or tensor quantity)An
arbitrary spatial function for the ath constituent (i.e. scalar, vector or tensor quantity)Material time derivative of the function
r
,
following the motion of the particle X" of thea
th constituent Material time derivative of the function lr,
following the motion of the mass averaged particle XAn
arbitrary field (scalar, vector or tensor) Specific heat for the liquid phaseSurface tension for the
a
th constituent Surface tension for the mixtureEntropy per unit mass for the
a
th constituentEntropy per unit mass for the mixture
Nomenclature xxi Difference in chemical potentials between solute and
solvent
Coefficient related to the chemical potential differences between solute and solvent
Viscosity
Temperature related viscosity Concentration related viscosity Kinematic viscosity
Thermodynamic pressure
Absolute temperature for the
a
th constituentAbsolute temperature for the mixture Mass density of the ath constituent Mass density of the mixture
Discontinuity surface Mean curvature
Molar density of the ath constituent Molar density of mixture
An arbitrarily field per unit volume An arbitrarily field per unit mass
Specific Helmholtz free energy for the mixture
Operators
Symbol Description m2 /s ~ / m ~ K K kg / m 3 kg / m 3 m2 l / m mole / m 3 mole / m 3(
) / m 3(
) / k g J / k gTotal time derivative dt
Nomenclature xxii Material time derivative, following the motion of the
particle X" of the
a
th constituentMaterial time derivative, following the motion of the mass centered particle X
Partial time derivative
Partial spatial derivative Surface gradient operator
34
grad 4 =
-
i k = e k i k , gradient of a scalar field axkgrad u =*ilik = u k l i l i k , gradient of a vector field 3x1
d ~ k -
divu =
-
- u ~ , ~ , divergence of a vector field axkC U ~ Z U = dU'Eklrnirn = u ~ , ~curl o f a vector field E ~ ~ ~ ~ ~ , axk -
4,kk
,
Laplacian operator on a div( grad$) =-
- axkaxk scalara4
u
-
V 4 = uk - = u , ~ @ , ~ , compound nabla operator axku
.
V v = uk * i l = u k q p i l , compound nabla axkoperator
Nomenclature xxiii
Indices and Symbols
Symbol Description
a
th constituent,(a
=1
,.
. .
,ill)P
th additional forcePrime indicates the material time derivative, following the motion of the particle of X" of the
a
th constituent Dot indicates the material time derivative, following the motion of the mass averaged particle XThe jump of the enclosed quantities Variable evaluated at reference state Liquid phase
Solid phase
Coefficients associated with temperature
xxiv
Acknowledgements
I would like to thank all the people who did not leave me alone in this four-year journey of hardship by their endless support and encouragement.
First and foremost, I would like to express my deepest gratitude to my supervisor Dr. Sadik Dost for giving me the opportunity to work with him and for introducing me to a challenging and exciting research field, crystal growth. His guidance throughout the course of my study played a significant role to conclude this work. It was a privilege for me to work with such an expert in the field of crystal growth.
I also would like to thank my lovely wife for her continuous support and help in spite of many lonely hours. This dissertation would not have been possible without her.
I would like to thank my little son for giving me peace and comfort with his smiley face and hugs.
I wish to express my thanks to Mr. Brian Lent and Dr. Yongcai Liu for sharing their expertise in the field of crystal growth.
Many thanks to Dr. Hamdi Sheibani, Mrs. Sema Dost, and all my fellow graduate students for their kind helps and friendship.
Most of all, I would like to thank my parents, and my brothers and sister for their unconditional support, encouragement and prays in all these years to achieve my goals.
I thank GOD, the most merciful, the most compassionate, for his blessings and love. The financial support provided by BL Consulting Ltd. of Victoria, DL Crystals Inc. of Victoria, and the Microgravity Science Program of the Canadian Space Agency is gratefully acknowledged. The financial support for the LPD equipment and its operation is provided by NSERC and CFI.
Dedication
Dedicated To My Beloved Parents, Caring Wife, Lovely Son (Emir Berk), All My Brothers and Sister in recognition of their contribution to this dissertation.
CHAPTER 1
Introduction
1 . Motivation and Goals
Research on the SixGel-, alloy system dates back to as early as 1954 [I]. However, an impressive body of research associated with the growth of high quality SixGel-, single
crystals, epitaxial techniques leading to thin Si,Ge,-,layers and development of
SixGel-,
-
based microelectronic semiconductor devices started coming into existenceapproximately two decades ago. The Si,Ge,, alloy system offers very promising material
features such that the band structure and thus the effective mass and mobility of both electrons and holes are significantly affected by alloy composition, temperature and strain. Therefore, the electrical and optical properties of this material can be tailored according to the needs of device applications. Furthermore, other very important aspects of SixGel-, alloy system are its ease of integration with the well-developed and long- existing silicon technology, and its low-cost production compared to 111-V group semiconductor materials.
The above-enumerated features make the SixGel-, system a very promising candidate for a variety of micro electronic and optoelectronic device applications. For instance,
Chapter I-Introduction 2
Si,Ge,-, has been attempted for use as a base in SiISiGe heterojunction bipolar transistors (HBT) [2,3,4,5] and high electron mobility field effect transistors [6,7,8] due to its higher carrier mobility, amendable band gap energy at any value between silicon
and germanium and adjustable band gap offset at the junction between Si,Ge,-, and
silicon.
Si, Gel-, has been utilized for photodetector [9,10] and solar cell [ 1 1,121 applications because of its enhanced sensitivity to the detection of light and high solar cell response in
the infrared region of solar spectrum. The band structure of SixGel-, alloys enables Auger
generation processes, thereby providing solar cell efficiency of approximately 40% [13].
SixGel-, can be employed as a substrate to fabricate exactly lattice-matched
GaAs / SiGe heterostructures for higher efficiency solar cell applications [14, 151.
Other applications of Si,Ge,-, include thermoelectric power generators (due to its
low thermal conductivity and high Seebeck coefficients [ 16,17,1 S]), tuneable neutron and
x-ray monochromators based on the large lattice parameter variation of the Ge-Si solid solution 119-201 and high speed temperature sensor for the range of 20-400 OC based on conductivity change, and y-ray detectors [2 11.
Si,Gel-, single crystal substrates with a specific composition ( x ) are needed for the
applications mentioned above. Si,Ge,-, single crystals for microelectronic and
optoelectronic device applications have been generally prepared in the form of thin films grown on a silicon substrate by different epitaxial growth techniques such as Molecular Beam Epitaxy (MBE) [22,23], Liquid Phase Epitaxy (LPE) [24,25], Rapid Thermal Chemical Vapour Deposition (RTCVD) [26], Chemical Vapour Deposition (CVD) [27], Ultra High Vacuum Chemical Vapour Deposition (UHVICVD) [28,29,30]. On the other
hand, when a Si,Ge,-, alloy is epitaxially deposited on a silicon substrate, the lattices of
the deposited layer try to embrace in-plane lattice parameters to form a coherent interface with the underlying substrate. As a result, the alloy layer will be compressively strained since the freestanding lattice parameter of the silicon is smaller than those of SixGel-,
Chapter I-Introduction 3
alloys and pure germanium. When the thickness of the strained layer exceeds a so-called critical thickness, a high density of misfit dislocations at the interface of the Si,Ge,-, and
Si and many threading dislocations, traversing the SixGel-, layer, are invariably created to
relieve the built-in compressive strain. Upon relief of the strain, the layer relaxes to its freestanding lattice parameter. The existence of misfit and threading dislocations severely reduces the mobility and electronic quality of the material [2]. The critical layer thickness decreases significantly with increasing germanium content. Nevertheless, most of the
applications require a much thicker SixGel-, layer with high germanium content than that
imposed by the critical thickness limit and thus, a large number of researches are still underway to reduce the dislocation density. One extensively researched method is to
form fully relaxed and compositionally graded SixGel-, layers on a silicon substrate,
increasing the germanium content at each step up to the composition of interest and hence less lattice mismatched interfaces will be created, resulting in lower dislocation densities confined in graded layers. Such a graded structure is known as a "virtual substrate" and is
used as a fully or partially lattice matched substrate on which active SixGel-, layers can
be grown for device applications. Nevertheless, virtual substrates are also accompanied by other problems such as surface undulation or roughness created between layer interfaces, which are harmful to device performances as well. In addition, preparation of virtual substrates is expensive, time consuming and not an easy process;
Considering these problems associated with the virtual substrates, the need for high
quality and relatively defect free and compositionally uniform SixGel-, substrates can
easily be appreciated. Availability of such substrates will allow the deposition of thicker lattice matched thin film alloys with high germanium content, which have been highly needed for development of high performance microelectronic and optoelectronic devices.
In order to achieve high quality SixGel-, crystals with different germanium contents, a
variety of melt crystal growth techniques, such as Czochralski [3 1-36], floating zone [37], Bridgman [38-401, multi component [14] and liquid encapsulated zone melting [41] and others [42], have been tried.
Chapter 1-Introduction 4
It is however very difficult to grow single crystals of uniform composition and low defect
densities from a binary system of SixGe,-, since SixGel-, has a large miscibility gap and
there are significant differences in physical properties of the constituent elements, such as density, melting temperature, and lattice parameter. Because of the large miscibility gap, which gives rise to segregation coefficients far from unity, (k, = c, 1 c, =5.5 for silicon),
(k, = c, 1 c, =0.3 for germanium) in SixGe,-, alloy, any small changes in the solidification
rate will lead to significant composition variation. Upon the formation of a SixGel-, solid
solution, depletion of silicon atoms in the melt just ahead of the solid and liquid interface
takes places. This is because silicon atoms are preferentially consumed by the growing .
crystal from the limited amount of growth melt. In the mean time, germanium atoms are rejected into the melt. This increases the germanium content and while decreases the silicon content in the melt along the growth direction. Consequently, composition of the grown crystal varies in the growth direction. Various types of defects may occur during growth. For instance, strong segregation of constituents may cause constitutional supercooling in the melt near the crystallization front, which is the main mechanism for transformation from a single crystal structure to a polycrystalline one. Other main defects in melt grown crystal can be grown-in dislocations (electrically active defects) in the order of lo4-lo6 ~ m ' ~ depending on growth methods and concentric circles (so called striation marks) caused by the variation in composition and lattice parameters in the radial direction. Striation marks in the crystal are because of fluctuations in temperature. If the concentration gradient in a single crystal exceeds a critical limit, it may result in crack formation in the structure.
Despite the fact that Cz is one of the most used growth techniques to grow single crystals of large diameter, it is difficult with this technique to obtain single crystals with uniform
compositions and without striations [3 1-36]. Similarly, there are non-homogeneous
concentration distributions along the growth direction and radial striations in SixGe,-, single crystals, which are grown by Bridgrnan [38,39] and floating zone [37]
Chapter 1-Introduction 5
and high quality compatible with the requirements of device applications is a challenging task.
The problems encountered in growth of SixGel-,single crystals by either the aforementioned deposition techniques or high temperature melt growth techniques may be successfully circumvented by employing a low temperature solution growth technique such as Liquid Phase Electroepitaxy (LPEE) and Traveling Heater Method (THM). LPEE was considered as a technique of choice for the growth of Si,Gel-, single crystals of high quality unattainable by melt growth because it offers, in principle, better controllability of defects than other techniques discussed previously. LPEE has been proven to be successful in the growth of 111-V alloy semiconductor materials [43,44]. In LPEE, single crystals are grown from metallic solutions at lower growth temperatures than melt growth. Thermal gradients developed in the LPEE growth cell (liquid solution zone) are generally sufficiently low so that the amount of grown-in defects such as dislocations and point defects in the final structure can be controlled and reduced.
Growth in LPEE is initiated and sustained by the passage of an electric current through a substrate-solution-source system by maintaining the growth temperature constant. The application of the electric current gives rise to two well-known mechanisms, namely Peltier coolinglheating, and electromigration. Electromigration takes places due to the momentum-exchange between electrons and atoms, leading to the transport of solute species towards to the seed crystal (substrate). Peltier cooling reduces the temperature at the crystallization front in the order of approximately 1 O C and in so doing, it results in
supersaturation of the advancing growth interface and thus growth.
LPEE growth process requires a Si,Ge,-, single crystal seed with the same composition as the crystal to be grown (a small composition difference within the limit of acceptable lattice mismatch is tolerable). In order to address this issue, a crystal growth technique, which is called in this work "Liquid Phase Diffusion, (LPD)", has been utilized. Two
Chapter 1-Introduction 6
objectives were in mind in initiating the present study. One was the growth of
compositionally graded bulk Si,Gel-, single crystals fi-om which the seed substrates with
required compositions can be extracted for LPEE. At the time this work was commenced,
no successful growth of bulk Si,Ge,-, single crystals of large size for whole composition
ranges in the phase diagram was reported, and such crystals were not commercially available. The second objective was the development of the first stage of a hybrid growth technique that combines LPD and LPEE in a single process. In this hybrid technique in mind, a compositionally graded single crystal would be grown by LPD up to the composition of interest, and then at this stage, the LPEE process would be initiated by passing an electric current through the growth system at a uniform temperature, leading to the growth single crystal with desired uniform composition. This hybrid growth process would eliminate the adverse effects of growing crystals in two stages. Such a growth process will be developed in the near future.
LPD growth technique was selected over other melt growth methods due to its simplicity
and low cost of growth equipment. LPD was used by Nakajima et al. under the name of
"Multicomponent Zone Melting" [14,45] for the growth of compositionally uniform
germanium-rich Si,Ge,-, single crystals of 15 rnm in diameter. Their crystals were
reported to be single in compositionally graded region and partially polycrystalline in compositionally uniform region due to the pulling process. In LPD technique, the solvent material (Ge) is sandwiched between single crystal substrate (seed, Ge) and polycrystalline source material (feed, Si). Initially, all these three layers of materials (the growth charge) are solid. On locating the growth charge (or growth zone) in an axial thermal gradient, solid germanium in the middle totally melts to form the liquid solvent for the growth. The germanium substrate partially melts. The silicon source, on the other hand, remains in the solid state due to its higher melting point. The molten germanium dissolves the silicon source according to thermodynamic equilibrium. Dissolved silicon
species are incorporated into the germanium liquid, thereby forming a binary Si,Gel-,
Chapter I -Introduction 7
growth interface by combination of several driving forces. When the mixture at the growth interface is supersaturated, solidification takes place due to the presence of constitutional supercooling. Since the growth solution (SiGe binary mixture) is prepared through the dissolution of solute (Si) by the solvent (Ge), LPD technique utilized here resembles a solution growth technique.
The composition range required from a SixGel-, alloy system is very wide depending on
the device application. Our literature survey for determining the target composition has shown that the entire compositional spectrum is feasible. Therefore, in this work, we focused on the germanium rich region of the phase diagram since the other end of the phase diagram brings some operational difficulties due to the high melting temperature of silicon.
In conclusion, the objective of this work is to perform a combined experimental and theoretical study to show the applicability of LPD to produce relatively large seed crystals of desired compositions. The objective has been successfully realized by growing several Si,Ge,-, single crystals of 25 rnm in diameter up to 6-8 % at. Si and by developing a rational mixture model and performing two-and three-dimensional numerical simulations to study transport phenomena in detail during the growth process. The seed materials for LPEE growth can be extracted from the composition of interest by cutting a horizontal slice from the LPD grown crystals.
To the best of our knowledge, there has been no attempt to grow Si,Gel-, single crystals
in 25 mm in diameter by LPD. As well, no comprehensive two -and three-dimensional
simulations of the LPD growth system have been performed. These significant contributions to the field of crystal growth make this work original.
Chapter I -Introduction
1.2
Outline
of
the Thesis
Following a brief introduction on the subject matter, Chapter 2 presents the crystal
structure, phase diagram and the band gap structure of the SixGel-, alloy system. Chapter
3 introduces short reviews on most crystal growth techniques including melt, solution and
deposition methods, which have been used in the production of Si,Ge,-, single crystals.
It also addresses the current state of such studies, and examines their status in the growth of SixGel-, alloy systems. Chapter 4 is devoted to a literature survey to find out the
possible applications of Si,Ge,-, along with the compositional requirements for the
application of interest. Chapter 5 presents to the development of a rational macroscopic
mixture model for the LPD growth system. Chapter 6 introduces two-and three-
dimensional transient simulations of the LPD growth system with moving boundary by
employing the model developed in this thesis. Chapter 7 addresses the design,
construction and testing of the LPD growth system. It also introduces experimental results and characterization of grown crystals. Chapter 8 concludes the work presented in this thesis with a short summary followed by the contributions to the field of crystal growth and a discussion about future works. Appendix-A summarizes the results of the
studies for the growth of Si,Ge,-, by various growth methods published in the literature.
CHAPTER 2
Structural Properties of SixGel-, Alloy System
2.1
Introduction
This chapter provides relatively comprehensive information on the crystal (or the lattice structure), lattice parameters, equilibrium phase diagram, electronic band structure, thermal and physical properties of SixGe,-, solid solution since it forms the basis for understanding of forthcoming discussions about possible device applications and growth of SixGe,-, systems by different techniques, particularly by Liquid Phase Diffusion
(LPD)
.
2.2 Crystal Structure and Equilibrium Phase Diagram of
SixGel-,
Lattice structure of a crystal is formed by the periodic reoccurrence of a unit cell in three- dimensional space, which is defined as the smallest building block of the crystal, so that the unit cell completely describes the lattice structure. Thus, to visualize the lattice structure of a crystal, it is adequate to examine the unit cell of the crystal.
Chapter 2 - Structural Properties of SixGel, Alloy System 10 Silicon and germanium are the first two elemental semiconductor materials discovered. Both belong to the IV-group of the periodic table. These two elements and their mixture (hereafter referred to as Si,Ge,-, solid solutions) crystallize with the diamond lattice structure under the atmospheric pressure. The diamond lattice structure is a generic name, which characterizes all crystals crystallizing in the same pattern as the diamond. The arrangement of atoms in the diamond unit cell is pictured in figure 2. la. The diamond unit cell is a cubic structure with atoms at each comer and at each face of the cube akin to the well-known face-centered cubic (fcc) cell. Unlike the fcc unit cell, the diamond cell hosts four additional interior atoms represented by yellow filled-in circles in figure 2.1 a. Each interior atom is located on one of the four body diagonals of the cube. All interior atoms are displaced one-quarter of the body diagonal with respect to associated corners along the corresponding diagonals. The numbers inside the circles in diamond unit cell denote the height of the atoms with reference to base as a fraction of the cell dimension, denoted by ' a
'.
Each atom in the unit cell is covalently bonded with its four nearest neighbours forming a tetrahedral structure as illustrated by a dashed tetrahedral in the interior of the diamond unit cell. The bonds between nearest-neighbour atoms are represented by lines.An alternative and easier way of visualizing the diamond space lattice might be to
consider a lattice formed by the interpenetration of two fcc unit cells. Namely, the second fcc unit cell is displaced a quarter of the body diagonal relative to the first cell along the diagonal direction. In this case, the corner and face atoms of the diamond unit cell can be viewed as if they belong to first fcc unit cell, whereas atoms totally confined within the diamond unit cell belong to the second fcc cell. As each unit cell is brought adjacent to each other in the crystal, then each cell contains one-eighth of each corner atom (8
*
1 / 8 ) and one-half of each face atom ( 6*
1 / 2 ). Therefore, with four interior atoms, each diamond unit cell contains eight complete atoms.Chapter 2 - Structural Properties of SixGel, Alloy System
-
Base of the cube
Figure 2.1 : a-) The sketch of the diamond unit cell unit, b-) projection of the cube face (numbers indicate displacements of atoms normal to the paper plane.)
The equilibrium phase diagram for Si,Gel-, system (refer to figure 2.2) consists of three distinct regions, namely the upper region describing the liquid state, the lower region describing the solid state and an intermediate region where the solid and liquid states coexist. The curves that separate intermediate region of the phase diagram from the liquid state and solid state are called the liquidus and solidus curve, respectively. The region between the liquidus and solidus curves is usually referred to as the miscibility gap. Silicon and germanium have complete miscibility in each other both in solid and liquid states over the entire composition range as can be seen from equilibrium phase diagram. Upon solidification, the mixture of silicon and germanium forms a substitutional solid solution. For a given composition in the phase diagram, the concentration of the silicon in the liquid and solid states can be determined by drawing a tie line intercepting with the liquidus and solidus line at a given temperature. It can easily be noticed from the phase diagram that the concentration of silicon in the solid is always higher than that in the
Chapter 2 - Structural Properties of SixGel, Alloy System 12
liquid state, which implies that silicon solidifies or crystallizes faster than germanium since the melting temperature of silicon is higher than that of germanium.
Weight Percent Silicon
800
80 m 80 90 100
Ge Atomic Percent Silicon Si
Figure 2.2: Equilibrium phase diagram of SixGe,-, system [46]
To be more particular, solidification steps of the SixGe,-, system are explained by using the "alloy-L" that is indicated on the representative phase diagram given in figure 2.3. When the "alloy-L" is cooled to the temperature
T,
from liquid state, the formation of a solid solution starts. The composition of the first solidified solid solution can be obtainedby drawing a vertical line from the meeting point of the tie line at T, with the solidus
curve to the composition axis. In this case, the composition of the first solidified solid solution is :x
.
Chapter 2 - Structural Properties of SixGel, Alloy System
Temperature
Atomic percent Si
+
Figure 2.3: A representative equilibrium phase diagram of the SixGe,-, system
The percentage of Si at :x composition is obviously greater than that in initial melt
composition x,S'. On decreasing the temperature of the alloy fin-ther to T,, the
composition x,S' moves to xf along the liquidus line because silicon in the melt is
preferentially consumed during the formation of a crystal with composition x:.
Therefore, the composition of the crystal layer to solidify next changes to xfi. As the solidification progresses, the germanium concentration increases in both solid and liquid at the solidification front due to the rejection of germanium atoms into the liquid, and silicon concentration decreases due to its preferential consumption, thereby resulting in continuous compositional variations along the solidification direction. The intensity of the compositional non-homogeneity, the so-called macrosegregation, in a solidified crystal depends on the location in the phase diagram where the alloy is solidified. The
macrosegregation is usually characterized by a coefficient k, that is known as
equilibrium segregation coefficient. " k, " is defined as the ratio of the concentration of the solute in the solid to that in the liquid at the solid-liquid interface, namely,
Chapter 2 - Structural Properties of SixGel, Alloy System 14
k, = c, / e l
.
It is a function of temperature. If k, = l , the alloy is known to be of aconcurrent melting temperature. For the Si,Ge,-, alloy system, k, values are far from
unity. For germanium-rich alloys, where germanium is a background fluid or solvent, and silicon is the solute, k, is bigger than unity and takes values up to 5.5. Regarding the
silicon-rich alloys, k, is less than one and varies between one and about 0.3. Since the
segregation coefficient is further away from unity for germanium rich-alloys than for silicon rich-alloys, it is relatively easy to obtain a single crystal of a uniform composition from silicon-rich side of the phase diagram. To conclude, SixGel-, is a good example for an alloy system having a strong tendency towards the formation of a composition gradient along the solidification direction due to the large miscibility gap between the solid and liquid phase and the high temperature difference in melting points of silicon and germanium.
The lattice parameter of the SixGel-, system at room temperature varies linearly from
0.543 10 nm to 0.56575 nm [47] as the germanium content increases from x = 0 to x = 1
as indicated in figure 2.4. Figure 2.4 also indicates that experimentally measured lattice parameters represented by circles can be approximated closely by taking a linear interpolation between the lattice parameters of pure silicon and germanium. This approximation is known as Vegard's rule [48], and mathematically expressed as
asj,-xGex = a , +[a, -a,]x, where
,
a,, aGe are lattice parameters of silicon-germanium alloy, silicon and germanium, respectively and x is the atomic fraction of
Ge. The lattice constant difference between pure silicon and germanium is 4.17 %,
Chapter 2 - Structural Properties of SixGel, Alloy System
Composition, at. % Ge
Figure 2.4: Lattice parameter of the SixGel, system as a function of Gecomposition; circles indicate experimentally measured lattice parameters [47] and square symbols
indicate lattice parameters calculated using Vegard's law.
Having introduced the lattice structure of SixGe,-, alloy, the focus of the discussion can be turned to the microscopic ordering of constituents within the SixGe,-, solid solution. No experimental observation of any ordered structure has been reported for both silicon
and germanium rich bulk SixGel-, crystals of various compositions and for Si0.5Ge0.5 bulk
crystals grown from the melt by vertical Bridgman technique [49]. On the other hand, existence of long range ordering was reported on Si,Ge,-, thin films grown on a silicon
(001)' germanium (001) and hlly relaxed SiXGel,buffer layer having (100)
crystallographic orientation by Molecular Beam Epitaxy (MBE) at lower temperature than the usual growth temperature (550 "C) [49]. The ordering of silicon and germanium is limited to thin films grown on substrates of (100) orientation, and received considerable interest since it may affect the electronic band structure and the material's adjustable band gap.
Chapter 2 - Structural Properties of Si,Ger, Alloy System
2.3 Band Gap Structure of Bulk SixGel, Alloys
Silicon-germanium alloys are in the category of indirect band gap semiconductors like their constituents. Optical studies confirm that in pure silicon and germanium, the
maximum in the valence band occurs at the centre of the energy (E) versus k = p / A
diagram wherek=O, whereas the minimum in the conduction band is located at k # 0 [50]. Here, k is the wave number, p is the momentum and A = h / 2 z is the
modified Plank's constant. A semiconductor whose maximum in the valence band and
minimum in the conduction band do not occur at the same k value is called an indirect
band gap semiconductor. In silicon and germanium, the conduction band minima occur along <loo> and <1 1 1> crystallographic directions, respectively.
In the indirect band semiconductors, when an electron jumps down from the conduction band to the valence band to recombine with a hole in a semiconductor material, it must release some of its energy to embrace the energy level of the hole. According to the law of conservation of energy the energy released must be absorbed by some outside sources. This energy release process may happen in two different ways; firstly, energy can be released in the form of radiation creating photons, in which case the recombination process is called radiative, or secondly, it can be transferred directly to a lattice in the form of vibration forming a phonon. In addition, because the maximum in the valance band and minimum in the conduction band occur at different k values, the momentum of the electron has to be conserved during the transition fi-om the conduction band into the valence band. This dictates the interaction of the electron with either phonons or impurities in the semiconductor. This indirect nature of the electron transition has significant effect on the optical properties of the semiconductor material. Therefore, indirect band gap semiconductors do not show as good optical properties as direct band gap semiconductors.
Chapter 2 - Structural Properties of Si,Gel, Alloy System 17
In 1954, by performing optical absorption investigations on SixGel-, alloys for 0 < x < 1
,
Johnson and Christian [I] proposed that the indirect band structure depends only on the alloy composition. They observed a sudden change in the slope of energy band gap
versus composition curve. The abrupt change in the curve occurring at about 15 at % Si
was interpreted as a switch over from germanium like L -band structure to the silicon like
A - band structure [51]. Four years later from the study of Johnson and Christian,
Braunstein et al. [50] have performed absorption measurements as a function of
composition, somewhat more detailed than the one performed previously [I], in order to gather more accurate information about the band structure of the alloy system. They
observed that the indirect energy band gap of the SixGel-, system does not depend only on
composition but also on temperature, and determined the values of band gap by fitting the absorption data at low absorption levels to the Macfarlane-Roberts expression. Their
work [50] confirmed the discontinuity in the curve obtained by Johnson et al. However,
while Johnson et al's curve consists of two linear portions with different slopes, the first
portion of Braunstein et al's curve is linear in the 0- 15 mole percent germanium and the
second potion of the curve is quadratic in the remaining composition range as can be seen from figure 2.5. The discrepancy between earlier data [ l ] and their data [50] was largely attributed to the lower material purity of samples used in the study [I]. Recently, Weber and Alonso [52] measured the composition dependence of the energy band gap of SixGel-, system at 4.2 K by employing photoluminescence (PL). They found that the band gap varies with the increasing germanium content from pure silicon energy gap at
1.155 eV to the pure germanium band gap at 0.740 eV, as shown in figure 2.5.
All three studies indicate that in the Si,Ge,-, system, the crossover from "the lowest lying conduction band minimum" along 4 11> crystallographic direction in reduced Brillion zone, to "the lowest lying conduction band minimum" along 4 0 0 > directions takes places at about 15 atomic percent silicon.
Chapter 2 - Structural Properties of SixGel, Alloy System
Atomic percent germanium-+
Figure 2.5: Band gap energy versus composition x for SixGel-, as determined from
absorption measurements (cross [50]) and low temperature PL spectra (filled-in black
squares [52]).
Experiments have shown that the presence of tensile or compressive strain in a Si,Ge,-, layer, induced by the lattice mismatch, gives rise to drastic changes in the band structure of the SixGel-, system by shifting and splitting the valence and conduction band
edges [53,54]. In addition to the band gap reduction caused by increasing germanium
content, the built-in compressive strain further reduces the energy gap of theSixGe,-, alloy for a coherent SixGe,-, layer on a silicon substrate [55, 561.
Consequently, concentration, temperature and strain dependency of the band structure of Si,Ge,-, alloys can be used to tailor the band structure according to the needs for a particular application.
