Surface segregation of Sn and Sb in the low index 
planes of Cu 
by 
 
Joseph Kwaku Ofori Asante 
MSc. 
 
This thesis is submitted in accordance with the requirements for the degree  
 
Philosophiae Doctor 
 
in the Faculty of Natural and Agricultural Sciences, 
 
 Department of Physics, 
at the  
 
University of the Free State 
 
Bloemfontein 
 
Republic of South Africa 
 
Promoter : Prof. W. D. Roos 
Co-Promoter: Prof. J.J. Terblans 
 
May 2005 
 
 
ACKNOWLEDGEMENTS 
 
 
The author wishes to express his sincere appreciation to the following: 
 
• Almighty GOD, for his expressed mandate to the author in Gen.1:28. 
• Prof. WD Roos, the author’s promoter, for his knowledge and great ideas in the field 
of this subject. 
• Prof. JJ Terblans, the author’s co-promoter, for his assistance in the running of the 
ternary computer programme and fruitful discussions  
• My wife, Sylvia, children: Kofi, Kobby, Akosua and Nhyira for their great support. 
• Prof. HC Swart, the Head of the Department of Physics (UFS), for his concerns and 
interest in this subject.. 
• The personnel of the Department of Physics (UFS), for their assistance and support. 
• The personnel of the Division of Instrumentation (UFS), for their assistance. 
• The personnel of the Division of Electronics (UFS), for their assistance. 
• Pastor At Boshoff, CRC, my great spiritual head for his teachings from the Word of 
God. 
• Pastor Andre Lombard, my zone pastor for his prayers. 
 
 1 
 
Summary 
 
In this study, the segregation parameters for Sn and Sb in Cu were determined for the first 
time using novel experimental procedures.  Sn was first evaporated onto the three low 
index planes of Cu(111), Cu(110) and Cu(100) and subsequently annealed at 920 oC for 
44 days to form three binary alloys of the same Sn concentration.  Experimental 
quantitative work was done on each of the crystals by monitoring the surface segregation 
of Sn.  Auger electron spectroscopy (AES) was used to monitor the changes in 
concentration build up on the surface by heating the sample linearly with time (positive 
linear temperature ramp, PLTR) from 450 to 900 K and immediately cooling it linearly 
with time (negative linear temperature ramp, NLTR) from 900 to 650 K at constant rates.  
The usage of NLTR, adopted for the first time in segregation measurements, extended the 
equilibrium segregation region enabling a unique set of segregation parameters to be 
obtained. 
 
The experimental quantified data points were fitted using the modified Darken model.  
Two supportive models – the Fick integral and the Bragg-Williams equations - were used 
to extract the starting segregation parameters for the modified Darken model that 
describes surface segregation completely.  The Fick integral was used to fit part of the 
kinetic section of the profile, yielding the pre-exponential factor and the activation 
energy.  The Bragg-Williams equations were then used to fit the equilibrium profiles 
yielding the segregation and interaction energies.  For the first time, a quantified value for 
interaction energy between Sn and Cu atoms through segregation measurements was 
determined (Ω.CuSn = 3.8 kJ/mol).  The different Sn segregation behaviours in the three Cu 
orientations were explained by the different vacancy formation energies (that make up the 
activation energies) for the different orientations.  The profile of Sn in Cu(110) lay at 
lowest temperature which implies that Sn activation energy was lowest in Cu(110). 
 
Sb was evaporated onto the binary CuSn alloys and annealed for a further 44 days 
resulting in Cu(111)SnSb and Cu(100)SnSb ternary alloys.  Sn and Sb segregation 
measurements were done via AES.  The modified Darken model was used to simulate Sn 
 2 
and Sb segregation profiles, yielding all the segregation parameters.  Guttman equations 
were also used to simulate the equilibrium segregation region that was extended by the 
NLTR runs to yield the segregation and interaction energies.  These segregation values 
obtained from the modified Darken model for ternary systems completely characterize the 
segregation behaviours of Sn and Sb in Cu.  For the ternary systems, it was found that Sn 
was the first to segregate to the surface due to its higher diffusion coefficient, which 
comes about mainly from a smaller activation energy (ESn(100) = 175 kJ/mol and ESb(100) = 
186 kJ/mol).  A repulsive interaction was found between Sn and Sb (Ω.SnSb = - 5.3 
kJ/mol) and as a result of the higher segregation energy of Sb, Sn was displaced from the 
surface by Sb.  This sequential segregation was found in Cu(100) (∆GSb(100) = 84 kJ/mol; 
∆GSn(100) = 65 kJ/mol) and in Cu(111) (∆GSb(111) = 86 kJ/mol; ∆GSn(111) = 68 kJ/mol).  It 
was also found that the profile of Sn in the ternary systems lay at lower temperatures due 
the higher pre-exponential factor (DoSn(binary) = 9.2 × 10-4 m2/mol and DoSn(ternary) = 3.4 × 
10-3 m2/mol) if compared to the binary systems. 
 
This study successfully and completely describes the segregation behaviour of Sn and Sb 
in the low index planes of Cu. 
 
 
 
 
 
 
 3 
 
 
 
 
Contents 
 
 
1. INTRODUCTION 7 
1.1 Segregation phenomenon 10 
1.2 The objectives of this work 12 
1.3 The outline 13 
2. THEORY 15 
2.1 Introduction 15 
2.2 The Fick theory for binary alloys 16 
2.3 The Bragg-Williams equation for binary alloys 19 
2.4 The modified Darken’s model 21 
2.4.1 The Darken rate equations for the binary system 22 
2.4.2 The Darken rate equations for the ternary system 24 
2.5  Guttman’s ternary equilibrium segregation equations 26 
2.6  Summary 27 
 
3. EXPERIMENTAL SETUP 28 
3.1 Introduction 28 
3.2 Sample Preparation 29 
3.3 Sample mounting and cleaning 34 
3.4 The AES/LEED system 37 
3.5 The AES measurements 39 
3.6 AES quantification from LEED patterns 40 
3.6.1 Cu(100) 42 
3.6.2 Cu(110) 43 
3.6.3 Cu(111) 45 
 
 4 
4. RESULTS 46 
4.1 Introduction 46 
4.2 The binary Cu-Sn system 47 
4.2.1 Cu(100)Sn  48 
4.2.1.1 The Fick integral fit 48 
4.2.1.2 The Bragg-Williams fit 50 
4.2.1.3 The modified Darken fit 51 
4.2.2 Cu(110)Sn 54 
4.2.2.1 The Fick integral and Bragg-Williams fit 54 
4.2.2.2 The modified Darken fit 55 
4.2.3 Cu(111)Sn 58 
4.2.3.1  The Fick integral and Bragg-Williams fit 58 
4.2.3.2 The modified Darken fit 59 
4.2.4 A summary of segregation parameters of Sn in CuSn binary alloy 62 
4.3 The ternary CuSnSb system 63 
4.3.1 Cu(100)SnSb ternary system 63 
4.3.1.1 Guttman fits 63 
4.3.1.2 The modified Darken fits 65 
4.3.2 Cu(111)SnSb ternary system 68 
4.3.2.1 Guttman fits 68 
4.3.2.2 The modified Darken fits 69 
4.3.3 A summary of Sn and Sb segregation parameters in Cu(100) and Cu(111) 
 72 
 
5. DISCUSSIONS AND CONCLUSIONS 73 
5.1 Introduction 73 
5.2 The binary CuSn system 74 
5.2.1 Sn segregation profile in Cu single crystal 74 
5.2.2 Sn segregation profile at different rates 77 
5.2.2.1 Sn in Cu(100) 77 
5.2.2.2 Sn in Cu(110) 77 
5.2.2.3 Sn in Cu(111) 78 
5.2.3 Sn segregation profile in the three orientations at the same rate 79 
 5 
5.2.4 Comparing published results to the present work 84 
5.3 The ternary CuSnSb system 85 
5.3.1 Sn and Sb segregation profiles Cu(100) 86 
5.3.2 Sn and Sb segregation profiles at different rates 89 
5.3.2.1 Cu(100) 89 
5.3.2.2 Cu(111) 90 
5.3.3 Sn and Sb segregation profiles in Cu(100) and Cu(111) at the same rate 
 91 
5.3.3.1 Crystal heating rate at 0.05 K/s 91 
5.3.3.2 Crystal heating rate at 0.075 K/s 93 
5.3.4 Comparing Sn in binary CuSn to Sn in ternary CuSnSb 94 
 
5.4 What has evolved in the course of this study 95 
 
A Flow chart for solving the Guttman equation 98 
 
Bibliography 99 
 
 6 
 
 
 
 
 
 
 
 
 
 
CHAPTER ONE 
 
INTRODUCTION 
 
 
Today, both simple and sophisticated metallurgical products are found in all aspects of 
modern life, both domestic and industrial.  There is still an ongoing search for better 
material properties for a great number of applications such as corrosion resistance, 
integrity of materials at high and low temperatures, wear resistance and weight reduction.   
 
With the limited world natural resources but growing demand for material (metallurgical) 
products, it is becoming imperative for material and surface science researchers to 
properly understand the behaviour of each material within its multi-parameter 
environment so that its best use could be defined.  Most material products come in the 
form of alloys.  From a metallurgical point of view, alloying elements could either be 
undesirable impurities or deliberate dopants in the alloying system.  It is also becoming 
imperative to seek possible alternatives for elements with a limited or uncertain source.  
The factors of high cost and time of production of material products must also be 
decreased.   
 7 
Introduction 8 
As materials are developed, inevitable problems associated with their usage under various 
conditions are coming to the fore and these demand scientific understanding and 
solutions.  A case in point is the well-known inter-granular fracture in the rotor of the 
Hinkley Point Power Station turbine generator [1,2].  During a routine test in 1969, one of 
the many 3Cr-0.5Mo steel rotors disintegrated and destroyed much of the turbine 
installation [3].  The other rotors were found to be safe, indicating that the disintegration 
of the odd one was not a characteristic of the steel type and its heat treatment.  Upon 
much scientific scrutiny of the broken parts, made possible with many surfaces and grain 
boundary techniques like Auger Electron Spectroscopy (AES), it was discovered that the 
segregation of impurities (mainly P) in the steel to the grain boundary sites caused the 
temper brittleness in the alloy and hence the rotor’s disintegration.  A large number of 
research investigation have been done on ferrous systems where high- and low-
temperature grain boundary fragility have been shown to be associated with the 
segregation of elements like As, Cu, Sn, Sb and S [4-7].  Knowledge of these impurities 
that caused embrittlement and their effects could be countered by the deliberate 
introduction of other elements (like rare earth metals such as La) that could also segregate 
to the grain boundaries to reject and neutralize these embrittling species [8].  Other 
problems associated with impurity segregation in alloys are inter-granular corrosion 
[9,10] and hydrogen embrittlement due to catalytic activity [11]. 
 
It is common practice in the field of microelectronics to coat Cu alloys contact with Sn, a 
process called “electrotinned” in order to minimise interface degradation [12].  It has also 
been found, however, that every tin-plated Cu alloy experiences the formation of copper-
tin inter-metallic compounds (Cu6Sn5 and Cu3Sn) at the interface of the tin and the base 
metal [13].  With time and/or increase in temperature, the inter-metallic compound moves 
Introduction 9 
towards the surface and can adversely affect contact resistance and solderability.  The 
inter-metallic growth could be retarded, however, by using a “barrier metal” (a metal that 
diffuses much, much more slowly with the base alloy and tin) [14].  Also, by allowing the 
segregation of Cu to the grain boundaries in Al thin film conductors electro-migration 
may be reduced considerably [15]. 
 
In the field of materials science and surface science, segregation of one or more 
components to interfaces and surfaces can influence both the physical and chemical 
properties of the alloy [16].  Some important areas that could be affected by grain 
boundary and surface segregation include crystal growth, catalysis, semi-conducting 
interfaces and the mechanical strength of solids.  Indeed, for multi-component alloys, 
segregation can induce the formation of two-dimensional compounds at the surface [17-
21].  These could be stabilised epitaxially and have different, better physical properties 
such as two-dimensional conductivity, superconductivity and magnetism compared to that 
of their individual constituents’ [22].  Surface and interfacial segregation plays a major 
role in the heat treatment of alloys and are therefore of great technological importance 
[23].   
 
Another field of study involving segregation is in nanoparticles.  One recent study [24] 
involved Monte Carlo simulations of the segregation of Ni in Pt-Ni nanoparticles.  Thin 
films of Pt-Ni inter-metallic alloy have been used as electro-catalysts in the lower 
temperature polymer electrolyte fuel cells [25].  However, besides the Pt-Ni nanoparticle 
having high surface-volume ratio, it has been found that Pt75Ni25 nanoparticle form a 
surface-sandwich structure with Pt atoms enriched in the outermost and third layers, while 
the Ni atoms are enriched in the second atomic layer as a result of surface segregation.   
1.2  The segregation phenomenon 10 
This nanoparticle elemental arrangement is very cost effective and places the Pt atoms at 
highly desirable position in its usage as electro-catalyst, as compared to that of a thin film.   
 
The above narration therefore point to a very important phenomenon, called the 
segregation of impurities in alloys.  The phenomenon has received the attention of surface 
scientists for over a century now [26].   
 
 
1.1 The segregation phenomenon 
 
Surface segregation is commonly regarded as the redistribution of solute atoms between 
the surface and the bulk of a material, resulting in a solute surface concentration that is 
generally higher than the solute bulk concentration.  The redistribution comes about so 
that the total energy of the crystal is minimised [27].  For a closed system, where pressure 
and temperature are the same for an interface and adjacent bulk, Gibbs free energy can be 
equated to the total energy.  The Gibbs free energy is the sum of chemical potentials of 
the various constituents in the system.  Equilibrium conditions may then be expressed as a 
function of the chemical potential terms instead of the total energy.  The change in 
chemical potential terms connects the energetic factors that are the segregation and the 
inter-atomic interaction energies.  In terms of these energetic factors therefore, surface 
phenomenon can also be regarded as the energy cost of transferring one impurity atom 
from the interior of a host crystal to its surface [28,29].  Surface structure and 
composition depend strongly on surface segregation energy.  Two distinct contributions 
responsible for surface segregation are the strain energy due to the atomic size mismatch 
between the solute and the solvent, as well as the differences in their surface energies 
1.2  The segregation phenomenon 11 
[30].  Thus the solute that has a different atomic size as well as lower surface energy than 
the solvent will therefore segregate and enrich the surface in the solid solution alloy so as 
to minimize the Gibbs free energy.   
 
When metal alloys are heated, the solute atoms of lower surface energy as compared to 
the solvent may move from within thousands of layers deep inside the bulk toward the 
surface.  The movement of solute atoms within the bulk-solvent-matrix also constitutes 
diffusion, which comprises the activation energy and the pre-exponential factor.  The pre-
exponential factor in tend, is made up of the vibrational frequency and entropy terms [31].  
The activation energy is made up of vacancy formation energy of the solvent and the 
solute atom migration energy [32].  Segregation parameters are then regarded as some 
energetics and diffusion factors that contribute in bringing about the phenomenon of 
surface segregation.  These are the segregation, activation and the interaction energies as 
well as the pre-exponential factor.  By measuring these solute enrichments on the surface 
as a function of temperature or time, their segregation parameters can be determined [33]. 
 
While there have been substantial efforts in examining surfaces of pure elements and the 
studies of the surface behaviours of binary alloys are rapidly developing, there is a gap in 
experimental knowledge of more complex multi-component alloys [34].  Yet a better 
understanding of alloy surface properties and more so, the knowledge of segregation 
parameters of the constituents in an alloy would be highly desirable and could even lead 
to advances in the ability to effectively design alloys for surface related applications.  The 
acquisition of segregation data on the various alloying elements, through surface and 
grain boundary segregation research works with further theoretical considerations and 
could lead to the manufacturing of super alloys. 
1.2  The objectives 12 
1.2 The objective of this work 
 
The main objective of this study was to establish an experimental procedure to determine 
segregation parameters in a binary and ternary all-metal-alloy systems. 
 
The procedure followed was to: 
 
1. Prepare binary alloys of Cu(100), Cu(110) and Cu(111) single crystals with the same 
Sn concentration. 
 
2. Measure and compare the segregation behaviour of Sn in each of the three Cu crystals 
using the AES technique with the method of linear temperature ramp (LTR). Do 
simulations of the experimental data via the modified Darken model. 
 
3. Extend the binary alloys to ternary by adding the same quantity of Sb concentration to 
Cu(100)Sn, Cu(111)Sn and Cu(110)Sn. 
 
4.  Measure and compare the segregation behaviour of alloying elements Sn and Sb in 
each of the ternary systems using AES with the method of positive (PLTR) as well as 
negative (NLTR). 
 
5.  Use the Guttman equilibrium segregation equations to fit the experimental data from 
the NLTR runs to yield the segregation energies of Sn and Sb as well as the interaction 
energies between the atoms of Sn, Sb and Cu. 
1 .4  The outline 13 
6. Extract the segregation parameters of Sn and Sb in the three low index Cu planes by 
fitting the modified Darken ternary segregation theory to the experimental results. 
 
7. Compare the solute Sn segregation behaviour in the binary to that of the ternary. 
 
 
1.3 The outline 
 
This thesis is divided into five chapters.  In chapter 2, the segregation theory and models 
for the binary and ternary systems that were used to interpret experimental results are 
given.  On the binary systems, the Fick integral and the Bragg-Williams equations will be 
given.  The shortcomings of these theories will be highlighted.  The Regular Solution 
Model that accounts for the interaction between the different atoms will be given in 
conjunction with the modified Darken model for the binary alloy system.  In the case of 
the ternary systems, the Guttman Ternary Regular Solution (TRS) model, also known as 
the equilibrium segregation equations, will be highlighted.  Finally, the modified Darken 
equations for the ternary alloy that explains the complete segregation profile will be 
given. 
 
In Chapter 3, the experimental set-up is given.  The sample preparation, apparatus and 
the experimental procedures are discussed.  Also given in this chapter is how the 
segregation measurements were conducted.  The surface enrichment measurements of a 
segregating species via AES are the Auger peak-to-peak heights (APPH).  These must be 
quantified to surface concentration in molar fractional terms.  The quantification approach 
1 .4  The outline 14 
involving the low energy electron diffraction (LEED) patterns will be given in this 
chapter. 
 
Experimental results follow in Chapter 4.  These include all the experimental data points 
and their calculated theoretical fits in graphical form.  Sn segregation in the binary 
systems: Cu(100)Sn, Cu(110)Sn and Cu(111)Sn will be treated first.  Quantitative 
assessment of the behaviour of Sn in the binary crystals in the form of segregation 
parameters will be made.  The behaviour of Sn and Sb in the ternary alloy systems will 
also be treated quantitatively.   
 
Chapter 5 will be for discussions and conclusions.  Segregation of Sn in the binary CuSn 
will be treated first.  Comparison of the rate of segregation of Sn will be given in the three 
orientations.  The progression study of Sn segregation from binary to ternary will be 
investigated.  Sn and Sb segregation in the Cu(100) and Cu(111) will be compared.  An 
attempt will be made to compare the segregation behaviour of Sn in the binary as well as 
the ternary alloy systems.  
 
 
 
 
 
CHAPTER TWO 
 
 
THEORY 
 
 
 
2.1 Introduction 
 
A total description of surface segregation embraces both the kinetic and the equilibrium 
processes [35].  Darken described the phenomenon as an uphill diffusion as far as the 
concentration gradient is concerned [36].  The measured intensity versus temperature or 
time from the AES technique gives a combined kinetic and equilibrium segregation 
profile of the surface.  From the kinetic region the diffusion parameters, pre-exponential 
 15 
2.1 Introduction 16 
factor Do, and the activation energy E could be extracted [37].  The data making up the 
equilibrium segregation profile can also be used to get the other segregation parameters, 
namely, the interaction coefficient between the atoms i and j, Ωij, and segregation energy 
∆Gi.  At the onset of segregation, the segregation energy is responsible for driving the 
solute atoms from the first bulk layer to the surface.  This creates a depleted layer and a 
concentration gradient between the depleted first layer and the rest of the bulk layers 
resulting in atomic flux toward the surface [38].  A number of models [39-42] are already 
in place to explain the segregation process.  These models can be classified into two: one 
which essentially consists of special solutions of the macroscopic transport equations and 
the other with models describing the transport processes at a microscopic scale via jump 
probabilities of atoms of neighbouring atom layers [43]. 
 
The segregation models used in the present study are based on the macroscopic transport 
equations.  These are the Fick, Bragg–Williams, Guttman and the Darken models.  This 
study involves segregation in binary as well as ternary systems.  In the following sections, 
the theories governing the binary systems will be treated first and will be followed by that 
on ternary systems.  
 
 
2.2 Fick theory for binary alloys 
 
One of the solutions to Fick’s second law of diffusion, 
 
 ∂X D ∂
2 X
=
∂t ∂x2
 (2.1) 
2.2 Fick theory for binary alloys 17 
where X is the concentration at the depth x after a time t and D is the coefficient of 
diffusion under the boundary condition:  Xφ = 0, at x = 0 for t > 0 and the initial condition:  
Xφ  = XB, at t = 0 for x > 0 is: 
 
 2 Dt 1/ 2  X φ = X B1+    

  (2.2) 
 d  π  
 
where Xφ is the solute surface concentration and d is the thickness of the segregated atom 
layer on the surface.  Equation 2.2 is appropriate for a binary alloy with segregating 
solute atoms of bulk concentration XB and valid under the following conditions: 
1) Xφ relates to short times t 
2) For a constant diffusion coefficient 
3) A homogenous concentration at t = 0. 
 
Quite a number of researchers [44-50] have used equation 2.2 to describe the time 
dependence of the segregated surface concentration at a constant temperature. 
 
Du Plessis and Viljoen [51] first introduce the method of LTR whereby the temperature of 
the sample is ramped linearly with time, at a constant rate.  They substituted t in equation 
2.2, for temperature T according to: 
 t T −T= o  (2.3) 
α
 
where To is the starting temperature, normally below a third of the melting point of the 
solute, so that sputtered-induced segregation can be neglected and α is the heating rate.  
2.2 Fick theory for binary alloys 18 
The diffusion coefficient D in equation 2.2 can be replaced with the pre-exponential 
factor, Do and the activation energy E according to the Arrhenius equation 2.4, below: 
 
 D = Do exp(− E / RT )  (2.4) 
 
where R is the universal gas constant.  Another concept, the enrichment factor, β, which is 
defined as  
 
φ
 β X − X
B
= B  (2.5) X
can be introduced into equation 2.2, to obtain the Fick integral equation: 
 
 1 β 2 2D
T
= o E2 ∫ e−E / RT dT  (2.6) 2 πα d To
 
where TE is the temperature at the end of the kinetic region of the segregation profile.  The 
solute surface concentration, Xφ, is then monitored by a surface sensitive technique such as 
Auger Electron Spectroscopy (AES) as a function of temperature or time.  One important 
advantage of equation 2.6 is that the increase in temperature is controlled and known at 
all stages as the solute segregates to the surface and it is thus possible to solve the 
diffusion equation for a given set of values of Do and E.  Equation 2.6 accounts only for a 
certain range of temperatures in the kinetic part of the segregation profile and could only 
be used to fit experimental data in this range to extract the diffusion parameters Do and E. 
2.3 The Bragg-Williams equation for binary alloys 19 
It is important, though, for the heating rate α to be very small so that more time is allowed 
for the atoms to move onto the surface [51-52]. 
 
After long times equilibrium segregation sets in and the Bragg-Williams equation can be 
used to describe this part of the process quite well. 
 
 
 
2.3 The Bragg-Williams equation for binary alloys 
 
The equilibrium conditions for a binary alloy, in terms of chemical potential terms µ vi is, 
 
 µφ1 − µ
B
1 − µ
φ
2 + µ
B
2 = 0  (2.7) 
 
where i = 1, 2 represent the solute and the solvent atoms respectively and v, the phase: 
surface φ  or the bulk B [35].  Each term in equation 2.7 can be expanded via the regular 
solution model that takes into account the interactions between the atoms and was first 
developed by Hildebrand [53].  
 
The regular solution model is based on three assumptions: 
1) Atoms are randomly distributed over positions in a three-dimensional lattice 
2) No vacancies exist 
3) The energy of the system may be expressed as the sum of pair wise interactions 
between neighbouring atoms. 
2.3 The Bragg-Williams equation for binary alloys 20 
The model proposes that the interaction coefficients Ωij, in a regular solution, where the 
components have atomic concentrations Xi are related via the chemical potential energy 
and the activity coefficient f [54,55].   
 
When equation 2.7 is therefore expanded in terms of the regular solution model, the 
Bragg-Williams equilibrium segregation equation is obtained as: 
 
X φ 1 X
B
1 ∆G + 2Ω
φ
12 (X 1 − X B= exp 1 )φ B    (2.8) 1− X 1 1− X 1  RT 
 
where ∆G = µ oB − µ oφ − µ oB + µ oφ  is the segregation energy. 1 1 2 2
 
Equation 2.8 can be used to fit the data of the equilibrium section of the segregation 
profile to yield both the segregation energy ∆G  and the interaction coefficient Ω12 [56]. 
 
Combining the Fick integral and the Bragg-Williams equations, however, do not 
completely describe the segregation process.  The all-embracing model that describes both 
the kinetic as well as the equilibrium segregation process adequately, however, is the 
modified Darken model. 
 
 
 
 
 
2.4 The modified Darken model 21 
2.4 The modified Darken model 
 
The modified Darken model considers the differences in the chemical potential energy 
between the multi-layers as the driving force behind segregation [57-59].  Atoms will 
move from the bulk, a region of high chemical potential, to the surface, a place of low 
chemical potential.  
 
The original model [57] proposed that the net flux of species i (Ji) through a plane at  
x = b is given by: 
 
 J = −M B ∂µi i i X i    (2.9) 
 ∂x x=b
 
where Mi is the mobility of the species i and X Bi  the supply concentration in between two 
layers (within the plane).  This supply concentration from within the planes has got no 
physical meaning and the first modification to the Darken model categorically associates 
the supply concentration to a specific layer as:   
 
( j+1, j)
 J ( j+1, j ) = M X ( j+1) ∆µii i i  (2.10) d
 
Equation 2.10 then indicates the flux of atoms from the (j + 1)-th layer to the j-th layer 
with the supply concentration X ( j+1)i  and the difference in the chemical potential between 
2.4 The modified Darken model 22 
the layers µ ( j+1, j )i .  The segregation system of surface φ  and bulk B is therefore described 
by: 
 ∂X
φ B1→φ
i Mi X B1i =  ∆µ (B1 ,φ )i   (2.11) ∂t  d 2 
and for the j-th layer, 
 
∂X ( j) M j+1→ j X ( j+1) M j→ j−1X ( j) i = i i ∆µ ( j+1, j) − i i ∆µ ( j , j−1)

 i i   (2.12) ∂t  d 2 d 2 
 
for i = 1,2, …, m – 1 and  j = φ ,B1,.. N.  
Now there are  (m – 1)( N + 1) rate equations for the  N + 1 layers. 
 
 
2.4.1 The Darken rate equations for binary systems 
 
For the binary alloy elemental composition m = 2, which implies that i = 1, is the solute in 
the alloy.  The two rate equations for the surface and the first subsurface layers are: 
 
∂X φ M B1 →φ B1 1 X

= 1 1 ∆µ (B1 ,φ ) 2 1   (2.13) ∂t  d 
 
∂X B1 M B2 →B1 X B2 B1 →φ B11 = 1 1 ∆µ (B2 ,B M1 ) − 1 X1 ∆µ (B1 ,φ )

 (2.14) 
∂t  d
2 1 d 2 1 
 
2.4 The modified Darken model 23 
where the mobility M B1 (= M
B2 →B1
1 = M
B1 →φ
1 ),  which for a dilute alloy 
D
= 1  [60]. 
RT
 
Equations 2.13, 2.14 and those of subsequent layers constitute a system of coupled non-
linear differential equations and they make up the modified Darken model.  Further, 
∆µ (B1 ,φ )1  is expanded using the regular solution model and equation 2.13 becomes: 
 
∂X φ M X B1  X B1 (1− X φ1 = 1 1 ∆G + RTIn 1 1 ) φ B 1
∂t d 2  X φ (1− X B1 ) + 2Ω12(X1 − X1 )  (2.15)  1 1 
 
and equation 2.14 becomes: 
 
∂X B1 M X B2  B2 B1 B1 φ1 = 1 12 RTIn
X1 (1− X1 ) + 2Ω (X B1 − X B X2 )− RTIn 1 (1− X1 )B ( B ) 12 1 1 φ ( − 2Ω (X
φ − X B1 )  
∂t d  X 11 1− X 21 X1 1− X
B1 ) 12 1 1 1 
 (2.16) 
 
The system of N + 1 differential equations are integrated for a given set of parameters, 
∆G1, Ω12 , M1 and X
B
1 .  In order to minimize boundary effects, N should be chosen as large 
as possible such as N + 1 = 300.  The change in concentration rate of the 300-th layer is 
then considered zero.  A rough estimate of the number of layers contributing to the flux of 
atoms to the surface is equal to the ratio of the maximum solute coverage to the bulk 
concentration times 100. 
 
At equilibrium, the change in concentration rates of all the layers become equal to zero 
and the modified Darken rate equations convert to that of Bragg-Williams. 
2.4 The modified Darken model 24 
 
 
 
2.4.2 The Darken rate equations for ternary systems 
 
In the case of a ternary alloy, m = 3, that is, there are two alloying elements besides the 
substrate and this yields two rate equations for each layer or cell of the crystal. 
 
(a) The Rate Equations for the Surface Layer (φ ) are given by: 
 
For solute 1, 
∂X φ M B1→φ X B1 1 = 1 1 (B

 ∆µ 1
,φ )  (2.13) 
∂t  d 2
1 

 
For solute 2, 
∂X φ M B1→φ X B1 2 = 2 2 ∆µ (B1 ,φ )

∂t  d 2 2 
 (2.17) 
 
 
According to the regular solution model, ∆µ (B1 ,φ )i  is a function of both the segregation 
energies ∆Gi and the interaction parameters Ω ij , between the alloying elements or species 
as shown in section 2.3. 
 
Selecting the equations of solute 1 for further analysis, from equation 2.13, the difference  
2.4 The modified Darken model 25 
in the chemical potential energy∆µ (B1 ,φ )1  between the surface φ , and the first bulk layer B1 
is given by: 
 
 ∆µ (B1 ,φ ) = µ B1 − µφ + µφ − µ B11 1 1 3 3  (2.18) 
Each of these chemical potential energy terms is further expanded according to the regular 
solution model to give the final analytical expression.  
 
At equilibrium, the rate of change in concentration in the layers equals zero and the 
equilibrium conditions become: 
 
 µφ B φ Bi − µi − µm + µm = 0  (2.19) 
 
For the ternary system, however, i = 1, 2 and m = 3 and the equilibrium equations, in terms 
of chemical potential terms are: 
 
 µφ − µ B − µφ + µ B1 1 3 3 = 0  (2.20) 
 
 µφ − µ B − µφ + µ B2 2 3 3 = 0  (2.21) 
 
Guttman [61] applied the regular solution model to the surface segregation in ternary 
alloys by way of expanding equations 2.20 and 2.21 in terms of surface concentrations. 
 
 
 
2.5 Guttman ternary equilibrium segregation 26 
2.5 Guttman ternary equilibrium segregation 
equations 
 
The expansion of each chemical potential term in equations 2.20 and 2.21 via the activity 
coefficient and the interaction coefficient yields 
 
X BX φ 1 exp(∆G1 / RT )1 =  (2.22) 1− X B1 + X B1 exp(∆G1 / RT ) − X B + X B2 2 exp(∆G2 / RT )
 
X φ X
B
2 exp(∆G= 2 / RT )2  (2.23) 1− X B1 + X B1 exp(∆G1 / RT ) − X B2 + X B2 exp(∆G2 / RT )
where 
∆G1 = ∆G
o + 2Ω B φ φ B1 13(X1 − X1 ) + Ω '(X 2 − X 2 )  (2.24) 
 
∆G2 = ∆G
o
2 + 2Ω23(X
B
2 − X
φ
2 ) + Ω '(X
φ
1 − X
B
1 )  (2.25) 
 
Equations 2.24 and 2.25 indicate that element i will segregate to the surface if ∆Gi > 0 . 
Further, according to equations 2.24 and 2.25, there are three driving forces in the 
segregation energy ∆Gi .  The first is the difference in standard chemical potentials 
between the surface and the bulk (∆Go1 ); the second is the term in Ω i3 which could be 
called the self-interaction term and lastly, the term Ω ' , which takes into account the 
interactions between the solute atoms.  The segregation energy ∆Gi  will thus be positive 
for Ω i3 < 0 and Ω
'  > 0. 
2.6 Summary 27 
Equations 2.22, 2.23, 2.24 and 2.25 indicate that equilibrium conditions are independent 
of the diffusion coefficients.  The equations can be used to get the segregation energies 
∆Gi  and the interaction coefficients Ω ij  mathematically by fitting to the equilibrium 
values of the measured data. (See annexure A for the flow chart). 
 
 
2.6 Summary 
 
The short time t constraint that is related to the temperature interval TE and To place on the 
Fick integral also shows its disadvantage.  The Fick integral can however be used to give 
the starting parameters to the Darken model if appropriate temperature intervals in the 
kinetic region could be selected.  The values extracted from the Bragg-Williams equations 
also serve as starting values for the modified Darken model in the case of the binary alloy 
of Cu and Sn. 
 
For the Cu, Sn and Sb ternary alloy, Guttman equilibrium segregation equations are also 
used to fit the segregation data (the high temperature region of the PLTR experimental 
values and also the data points for the NLTR) mathematically to yield the segregation 
energies of the solutes as well as the interaction coefficients of all the alloying elements.  
The advantage here is that the numbers of fit segregation parameters that are to be 
determined manually, in the solution of the modified Darken rate equations for the ternary 
alloy, are reduced to only diffusion coefficients and activation energies (but even here, the 
values obtained for Sn from the binary system could be used as starting values). 
 
 
 
 
CHAPTER THREE 
 
 
EXPERIMENTAL SETUP 
 
 
3.1 Introduction 
 
Sample preparation is a very important aspect of this segregation study.  In this section, 
an account of the sample preparation and the experimental procedures that were followed 
will be given.  The two surface techniques, Auger electron spectroscopy (AES) and low 
energy electron diffraction (LEED) will be described.  This will be followed by AES 
quantification using LEED. 
 
 28 
3.2 Sample Preparation 29 
3.2 Sample Preparation 
 
The Cu single crystals, all of 99.999% purity and cut along the (100), (110) and (111) 
planes and less than 1 degree orientation accuracy were ordered from Mateck, in Germany 
[62].  They were of the same diameter, 0.97 cm, and same thickness of 1.11 mm and 
polished below a roughness of 1 micron.  Polycrystalline Cu rod of 99.99% purity and 
standards of Sb (purity 99.995%) were also ordered from Mateck.  Sn (purity 99.995%) 
pellets, were obtained from Goodfellow Cambridge Limited [63].  Six dummy Cu 
polycrystalline samples were cut to similar sizes as the three crystals and mechanically 
polished up to 1µ m using a diamond suspended solution. 
 
All three Cu single crystals together with the six dummy polycrystalline Cu samples were 
mounted side-by-side on a carousel and introduced into an evaporation chamber, figures 
o
3.1 and 3.2.  A 50 k A  thick layer of Sn was deposited simultaneously onto the back face 
of the three Cu single crystals and the six dummy polycrystalline Cu samples  
 
 
 
 
 
 
 
 
3.2 Sample Preparation 30 
D C 
E 
B 
F 
G 
H I J A 
 
 
Figure 3.1 The evaporation system showing some of the external parts. A: electron gun 
filament current controller; B: Pirani gauge unit; C: Varian pressure gauge control unit; D: 
Inficon unit that indicates the evaporation rate and the evaporant thickness; E: glass dome 
cover; F: high voltage feed throughs connecting the electron gun filament; G: turbo-pump 
control unit; H: rotary pump; I: crucible manipulator; and J: stainless steel casing.  Other 
parts not in the picture include the turbo-pump and the ionisation pressure gauge. 
 
 
The block diagram of the inside components of the evaporation chamber is also shown in 
figure 3.2. 
 
3.2 Sample Preparation 31 
 
 
 
Cu crystal Quartz thickness monitor 
Carousel 
Shutter 
Evaporated Sn/Sb Glass shield 
Sn or Sb 
Crucible Electron beam 
W filament 
 
Figure 3.2 The evaporation system where Sn and Sb were evaporated onto the three Cu 
crystals and the six poly-crystals.  
 
Further detailed photograph of the crucible is given in figure 3.3.  
A: graphite pan in middle crucible  
A  
B: electron gun filament 
 
C: deflector 
 
D D: water cooling pipe 
B C 
3.2 Sample Preparation 32 
Figure 3.3  The three crucible compartments in the evaporation system.  The central 
crucible is aligned with the electron gun filament B.  The stream of electrons emitted are 
magnetically deflected and focused onto the substance to be evaporated in the graphite 
pan A. 
 
The crystals with the adhered evaporant were put in quartz tubes that had two protruding 
openings as in figure 3.4.  A steady but a slow flowing Ar gas source was connected to 
opening A.  When the entire tube was filled with Ar after a duration of 2 minutes, opening 
B was heated till it became soft and was clamped and sealed.  With the Ar gas still 
flowing but at a reduced rate, tube A was also quickly sealed.  The whole quartz tube was 
then immersed in a beaker of acetone to check for any leakage.   
 
A 
B Ar filled
samples 
 
Figure 3.4  The sealed (openings A and B) quartz tube filled with Ar gas with the sample 
ready for annealing at 920 oC. 
 
The tube was then transferred to the oven (see figure 3.5) and annealed at 920 oC for 44 
days for above 99 % homogeneous mixture (according to dissolution equation in Crank 
[64]) of Sn atoms in the Cu matrix.   
 
 
 
3.2 Sample Preparation 33 
 
 
 
A 
 
 
B B 
 
 
 
Figure 3.5 The annealing unit showing the thermo-couple junction A and the open-
ended quartz tube B into which is inserted the smaller sealed quartz tube with the crystals, 
as in figure 3.4.  
 
After annealing, atomic adsorption spectroscopy (AAS) measurements were done on two 
of the dummy samples and the bulk concentration of Sn found to be 0.145 ± 0.012 at. %. 
 
Both AES and LEED measurements were then taken on each of the three CuSn single 
crystals under the same experimental conditions. 
 
o
The next step was to evaporate 40 k A  Sb layer onto Cu(100)-Sn, Cu(110)-Sn and 
Cu(111)-Sn binary alloys as well as three other dummy samples simultaneously.  The 
crystals were again sealed in quartz tubes under Ar gas atmosphere and annealed at 920 oC 
for 44 days for a homogeneous mixture of Sn and Sb atoms in the Cu matrix.  
Unfortunately, the Cu(110)SnSb crystal was oxidised during the annealing process and 
3.3 Sample mounting and cleaning 34 
had to be discarded.  AAS measurements were done on two of the dummy samples and 
the bulk concentration of Sb found to be 0.121 ± 0.015 at. %.   
 
AES measurements were also carried out on the two ternary Cu(100)SnSb and 
Cu(111)SnSb crystals under the same experimental settings. 
 
 
3.3 Sample mounting and cleaning 
 
Each of the single Cu alloys was mounted side-by-side with the standard samples of Cu, 
Sb and Sn onto a carousel of the AES system as shown in figures 3.6 and 3.7.  
Carousel   
Standard Sn  
Standard C u  Faraday cup  
 Cu- S n-  Sb  Standard Sb  alloy  
 
Figure 3.6  The arrangement of the crystals onto the carousel in the AES system 
 
 
3.3 Sample mounting and cleaning 35 
 
 
 
 
 
 
 
 
 (a) (b) 
 
 
 
(a) (b) 
Figure 3.7  (a) The flange housing the carousel on a stand and the sample manipulators 
and (b) a detailed photograph of the sample holder with a Faraday cup to its right.  Notice 
also, the insulating wiring for the heater element as well as that for the thermocouple.   
 
 
A detailed sketch of how each of the single Cu alloys was mounted onto a resistance 
heater is given in figure 3.8. 
 
 
 
 
 
3.3 Sample mounting and cleaning 36 
Vacuum 
Cu crystal surface 
Steel screwed cap 
Ceramic casing 
Heating filament 
Thermocouple 
 
Figure 3.8  Section of the sample holder housing and the components for temperature 
measurements.   
 
The junction of the chromel-alumel thermocouple was embedded in a ceramic slab (see 
figure 3.8) upon which the back face of the crystal was mounted for the temperature 
measurements.  The heating filament was also put in the same ceramic slab. 
 
Before the AES and LEED measurements, the sample was first cleaned of surface 
contaminants (C, S, O) by using the following procedure: 
1. The sample was sputtered at room temperature for few minutes using Ar+ ions of 
energy 2 keV and ion current of 70 nA and rastered over an area of 3 mm × 3 mm. 
2. It was then heated to a higher temperature (550 oC) to de-absorbed trapped O and 
sputtered again for 5 minutes 
3. It was further heated to 650 oC for 10 minutes without sputtering so as to level off any 
concentration gradient [65]. 
4. The sample was then cooled down to 550oC and sputtered for 5 minutes. 
5. The cycle (steps 2 and 3) was repeated six times before a cleaned surface was 
obtained. 
3.4 The AES/LEED system 37 
3.4 The AES/LEED system 
 
The spectrometers consist of the following components as shown in Figure 3.9. 
10. LEED Control unit 
Ultra high Vacuum 
LEED optics, Varian VI 422 
Crystal 
1, 2 and 3 AES: Electron gun 
- e 
Ion gun 
6. Ion gun control unit 
8 and 9. Heater Control unit 4and 5. AES Control 
7. Computer 
 
Figure 3.9 A block diagram describing the AES / LEED system 
 
1. PHI 18-085 electron gun and control unit for providing the primary electron beam in 
the AES.  In this study, the primary electron beam energy and current were 4 keV 
and 10µ A respectively.  
3.4 The AES/LEED system 38 
2. PHI 15-110B single pass cylindrical mirror analyzer (CMA) for electron energy 
analysis.  It was used to measure the peak-to-peak height changes of Sn(M5N45N45), 
Sb(M5N45N45) and Cu(L3M45M45) peaks. 
3. PHI 20-075 electron multiplier (high voltage supply) for providing high voltage to 
the electron multiplier inside the CMA.  The voltage was 1150 V during 
measurements. 
4. PHI 20-805 analyser control for the Auger signal set to modulation amplitude of 1 
eV.  
5. PHI 32-010 Lock-in-amplifier differentiating the Auger signal with a sensitivity of 
10 mV and a time constant of 0.3 s. 
6. The Perkin Elmer 11-065 Ion gun control and the Perkin Elmer 04-303 differential 
Ion gun for cleaning the sample’s surface.  The ion beam current was approximately 
70 nA as measured with a Faraday cup, and accelerating voltage of 2 keV at a 
pressure of 5.2 ×10-5 torr. 
7. A Computer was used for controlling and data acquisitions in the case of the AES. 
8. A programmable temperature control unit capable of heating and cooling the sample 
at a set rate.   
9. A chromel-alumel thermocouple unit was used to measure the varying temperature 
of the sample. 
10.  The LEED system was a Varian Model VI 422.  LEED photos were taken of each of 
the samples before and after a LTR run.  The LEED optics had a Varian 981-2145 
electron gun unit and a Varian 981-2148 control unit. 
3.5 The AES measurements 39 
3.5 The AES measurements 
 
The AES was used to measure the peak-to-peak height changes of Sn(M5N45N45), 
Sb(M5N45N45) and Cu(L3M45M45) peaks in the derivative mode as a function of 
temperature.  Measurements on each crystal was done under the following instrumental 
settings: base pressure 4.6×10-9 Torr, primary electron beam of energy 4 keV and current 
10 µA, modulation energy 1 eV and a scan rate of 0.5 eV/s.  
 
So far, most of the studies of interface segregation of dilute systems have been restricted 
to a treatment of segregation under isothermal conditions [66-68].  The constant 
temperature measurements demand at least three experimental runs at different 
temperatures and the use of an Arrhenuis equation in order to determine the diffusion 
parameters.  It is not trivial to obtain exactly identical initial conditions for all 
measurements at the different temperatures.  Normally, because of time constraints, 
constant temperature runs are done at temperatures where diffusion is already active and 
significant concentration of solute that had not been monitored already on the surface.   
 
The above-mentioned problems were avoided by using the method of Linear Temperature 
Ramp (LTR) in the present segregation studies.  Only one run is sufficient to get all the 
segregation parameters.  Furthermore, the LTR run starts at low temperatures that 
correspond to low diffusion.  For the first time, the LTR run was made to follow with two 
ramping routines: the constant heating of the sample, called the positive LTR (PLTR) and 
then constant cooling of the sample, the negative LTR (NLTR).  In the PLTR runs, the 
computer was programmed to increase the crystal temperature from 150oC at a specified 
3.5 The AES measurements 40 
heating rate up to 630oC.  The NLTR runs followed immediately till the sample cooled to 
450oC.  The heating and cooling rates considered were: ± 0.05oC/s; ± 0.075oC/s and ± 
0.15oC/s respectively and are appropriate for dilute substitutional alloy systems [69].  At 
the onset of a run, the sample was sputter cleaned for 3 minutes at 150 oC and AES 
spectrum of the cleaned surface was taken.  The Sn (and Sb) surface concentration build- 
up as well as that of Cu were then monitored as a function of temperature for both PLTR 
and NLTR runs.  Segregation runs for the different heating and cooling rates were done 
on each of the crystals.  By cooling the sample slowly and linearly with time, the 
equilibrium segregation profile region was extended resulting in the attainment of more 
accurate set of equilibrium segregation parameters [70].   
 
AES spectra were taken at the end of each run, making sure that there were no other 
segregating elements except Sn (and Sb, for the ternary).  After a combined PLTR and 
NLTR runs, the crystal was heated again and remained at that temperature for more than 6 
hours to annul any concentration gradient before the next run. 
 
 
3.6 AES quantification from LEED patterns 
 
The measured Auger peak-to-peak height (APPH) in the derivative mode can be 
quantified to surface concentration in atomic percentage in two ways.  There is the 
approach that relates the APPH in the derivative mode of an element A, IA, to the atom 
density (in atoms/m3) of the element (NA(z)), at a depth z from the surface, apart from 
other parameters as [71]: 
3.6  AES quantification from LEED patterns 41 
∞  
 I A = Ioσ A(Eo )[secα]Rm (EA)T (EA)D(EA)∫  NA(z)× exp
z
− dz  (3.1) 0  λm (EA)cosθ 
 
where Io  is the primary electron current, σ A(Eo )  is the ionization cross section of atom A 
for electrons with energy Eo, α  is the angle of incidence of the primary electrons,  
Rm(EA) = 1 + rm(EA) and rm(EA) is the back scattering term dependent on both the matrix m 
and the binding energy for the core level electron involved in the transition, leading to an 
Auger electron with energy EA, T(EA) is the transmission efficiency of the spectrometer, 
D(EA) is the efficiency of the electron detector, λm (EA) is the inelastic mean free path in 
the matrix m and θ  = 42o, is the angle of emission. 
 
This approach demands AES spectra of the pure standards of all the alloying elements at 
the same experimental conditions, in order to find the correct sensitivity factor to 
correlate the molar fraction XA, to the AES signal intensity IA. 
 
The other approach is based on low energy electron diffraction (LEED) patterns and quite 
a number of researchers [72-74] have used this for AES quantification.  LEED photos of 
the cleaned sample were taken at room temperatures before a run and showed only the 
atomic patterns of the substrate.  After a run, the sample surface structure would be 
different as a result of segregation of the solute atoms from the bulk, and LEED 
photographs were taken again.  One then could classify the over-layer structure in terms 
of the substrate structure by the so-called Wood’s notation.  
 
 
 
 
3.6  AES quantification from LEED patterns 42 
3.6.1 Cu(100) 
 
       
 (a) (b) 
Figure 3.10. LEED patterns of cleaned Cu(100) substrate (a) and with Sn segregate (b) at 
the same electron beam energy of 117 eV. The additional spots show another surface 
structure attributed to the presence of Sn atoms. 
 
The observed LEED patterns however are (scaled) representations of the reciprocal net of 
 the pseudo 2-D surface structures as shown in figure 3.11 below. 
- Cu atom - Sn/Sb atom 
 (a) (b) 
Figure 3.11. The real space of (a) cleaned Cu(100) crystal surface structure and the same 
surface but different structure after Sn segregation (b). 
 
3.6  AES quantification from LEED patterns 43 
From figure 3.11, it is clear that Sn forms a (2×2) overlayer structure on Cu(100) surface. 
If one considers the unit cell of the overlayer of Sn, the ratio of the segregated atoms to 
that of the Cu substrate is 1 : 4.  The maximum Sn coverage is therefore 25 %. 
 
The surface concentration of Sn is then calculated from:  
 
 X φ (T ) RSn (T )Sn = max × 0.25  (3.2) Reqm
 
where RSn (T )
I
= Sn
(T )  the normalisation of which accounts for the possible shift in the 
ICu (T )
peaks as a result of sample and holder expansion.  Rmaxeqm  is the maximum value of RSn(T) 
in the  equilibrium region. 
 
3.6.2 Cu(111) 
 
      
 (a) (b) 
 
3.6  AES quantification from LEED patterns 44 
Figure 3.12. LEED patterns of cleaned Cu(111) substrate (a) and with segregated Sn (b) 
at the same electron beam energy of 117 eV.  The additional spots show another surface 
structure attributed to the presence of Sn atoms.  
 
 
 
 
 
 
 (a) (b) 
Figure 3.13. The real space of (a) cleaned Cu(111) surface and the same surface after (b) 
Sn segregation.   
 
From figure 3.13 (b), Sn forms a (√3×√3)R30o overlayer structure on Cu(111) surface.  
The unit cell of the overlayer Sn indicates the ratio of the number of the segregated atoms 
to Cu atoms as 1:3.  The maximum Sn coverage is therefore 33.3 %. 
 
The surface concentration of Sn was then calculated from:  
 
 X φ (T ) R (T )Sn = Snmax × 0.33  (3.3) Reqm
 
 
 
 
 
3.6  AES quantification from LEED patterns 45 
3.6.3 Cu(110) 
 (a)  (b) 
Figure 3.14. The real space of (a) cleaned Cu(110) surface and the same surface after (b) 
Sn segregation 
 
From figure 3.14, Sn forms a c(2×2) overlayer structure on Cu(110) surface.  The unit 
cell of the overlayer Sn, gives the ratio of the number of the segregted atoms to Cu atoms 
as 1: 2.  The maximum Sn coverage is therefore 50 %. 
 
The surface concentration of Sn is then calculated from:  
 
 X φSn (T )
R
= Sn
(T )
max × 0.50  (3.4) Reqm
 
Similar equations to 3.2, 3.3 and 3.4 hold for the surface concentrations of the alloying 
elements of Sn and Sb in the case of the ternary alloy, except that Rmaxeqm  in these cases is 
the maximum value of the sum of RSn(T) and RSb(T) at equilibrium. 
 
 
 
 
 
CHAPTER FOUR 
 
 
RESULTS  
 
 
4.1 Introduction 
 
In this chapter, four major results of which two have been published already [70,75] will 
be highlighted. The first deals with the consequences of the segregation behaviour of Sn 
in each of the three low index planes of Cu.  It involves the binary system of Cu-Sn and 
the interaction energy between the atoms of Cu and Sn.  The experimentally measured 
values will be given against the theoretical fittings that will embrace the Fick integral, the 
Bragg-Williams and the modified Darken equations.  Sn segregation parameters in the 
three Cu single crystals will then be determined and compared. 
 46 
4.2 The binary Cu-Sn system 47 
The second part will involve the surface concentration measurements of Sn and Sb in 
Cu(100) ternary systems.  The quantified experimentally measured values will be fitted 
with Guttman and the modified Darken equations and the segregation parameters of Sn 
and Sb will be extracted. The behaviour of the two alloying elements will be treated and 
compared. 
 
The segregation behaviours of Sn and Sb in the two ternary alloys of Cu(100)SnSb and 
Cu(111)SnSb will be compared and the necessary deductions given. 
 
Finally, the last part will see the progression study of Sn from binary Cu(111)Sn to 
ternary Cu(111)SnSb.  As a result of atomic interactions, the segregation profile of Sn in 
the binary CuSn will be affected when another impurity, in this case Sb, is introduced to 
the binary CuSn.  The extracted segregation parameters will be used to justify and explain 
the change in the segregation profile of Sn. 
 
 
4.2 The binary Cu-Sn system 
 
As was mentioned in Chapter Two, the Fick integral and the Bragg-Williams equations 
were used to fit some regions of the Sn segregation profile in the three Cu orientations.  
The extracted segregation parameters became the starting values for the main theory, the 
modified Darken equations that fit the complete segregation profile. 
 
The calculated values of activation energies (E) of Cu in Cu in the three low index Cu 
planes are: E(110) = 162.3 kJ/mol, E(100) = 182.8 kJ/mol and E(111) = 204.5 kJ/mol [76].  
4.2 The binary Cu-Sn system 48 
These activation energies were considered to be the minimum values (the basis) in the 
search for both E and Do in the three orientations. 
 
Sn segregation results in Cu(100) is considered first followed by that of Cu(110) and 
lastly, Cu(111). 
 
 
4.2.1 Cu(100)Sn binary system 
 
4.2.1.1 The Fick integral fit  
 
Figure 4.1 shows a measured PLTR run data comprising the kinetic segregation and 
equilibrium segregation profiles.  Part of the kinetic segregation profile below the dotted 
temperature line A at 765 K, shows the region where the Fick integral was used to fit the 
measured data points to yield E and Do as starting parameters for the modified Darken 
model.  
 
4.2 The binary Cu-Sn system 49 
0.3 
PLTR - measured 
Fick integral fit B 
A 
0.2 
0.1 
 
0.0  
400 500 600 700 800 900 1000 
Temperature (K)  
 
Figure 4.1 Measured Sn segregation in Cu(100) for PLTR run at heating rate of 0.05 K/s   
and the calculated Fick integral equation with Do = 6.2 × 10-6 m2/s and E = 189 kJ/mol. 
 
Temperatures above the dotted temperature line B at 815 K indicate the equilibrium 
segregation part of the profile, which is a very narrow region that could be extended by 
cooling the sample linearly with time as shown in figure 4.2. 
 
 
 
 
 
 
Sn fractional surface coverage  
4.2 The binary Cu-Sn system 50 
4.2.1.2 The Bragg-Williams fit  
 
0.3
0.2
PLTR - measured
0.1 NLTR - measured
Bragg-Williams fit
0.0
600 700 800 900 1000
Temperature (K)  
 
Figure 4.2  Measured Sn segregation in Cu(100) showing the equilibrium segregation 
profiles.  Note the narrow region from the PLTR run and the extended part from the 
NLTR run.  The calculated solid line is the Bragg–Williams fit for ∆G = 65 kJ/mol and 
ΩCuSn = 4.1 kJ/mol.  Sample’s heating and cooling rates were ± 0.05 K/s respectively. 
 
 
The segregation parameters E, Do, ∆G and the ΩCuSn obtained from the Fick integral and 
the Bragg-Williams equations were used as starting values for the modified Darken 
model.  Figures 4.3, 4.4 and 4.5 show measured PLTR runs at different rates and their 
corresponding modified Darken fits. 
 
Sn fractional surface coverage
4.2 The binary Cu-Sn system 51 
4.2.1.3 The modified Darken fits  
 
a) Sample heating rate: 0.05 K/s 
 
0.22
0.20 PLTR - measured
Darken fit
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
400 500 600 700 800 900 1000
Temperature (K)  
 
Figure 4.3 Measured Sn segregation in Cu(100) for PLTR run at heating rate of 0.05 K/s 
and the Darken model fit for segregation parameters: Do = 6.2 × 10-6 m2/s, E = 189 
kJ/mol, ∆G = 65 kJ/mol and ΩCuSn = 3.9 kJ/mol. 
 
 
 
 
 
Sn fractional surface coverage
4.2 The binary Cu-Sn system 52 
b) Sample heating rate: 0.075 K/s 
 
0.20 PLTR -measured
Darken fit
0.16
0.12
0.08
0.04
0.00
400 500 600 700 800 900 1000
Temperature (K)  
 
Figure 4.4 Measured Sn segregation in Cu(100) for PLTR run at heating rate of 0.075 K/s 
and the Darken model fit for segregation parameters: Do = 6.2 × 10-6 m2/s, E = 189 
kJ/mol, ∆G = 65 kJ/mol and ΩCuSn = 3.8 kJ/mol. 
 
 
 
 
 
 
 
 
Sn fractional surface covearge
4.2 The binary Cu-Sn system 53 
c) Sample heating rate: 0.15 K/s 
 
PLTR - measured
Darken fit
0.2
0.1
0.0
400 500 600 700 800 900 1000
Temperature (K)  
 
Figure 4.5 Measured Sn segregation in Cu(100) for PLTR run at heating rate of 0.15 K/s 
and the Darken model fit for segregation parameters: Do = 5.8 × 10-6 m2/s, E = 190 
kJ/mol, ∆G = 65 kJ/mol and ΩCuSn = 4.0 kJ/mol. 
 
For the rest of the results involving Cu(110)Sn and Cu(111)Sn, the fit from the auxiliary 
models of Fick and Bragg-Williams were combined on the same system of axes and 
considered only for rates of ± 0.05 K/s.  The main modified Darken model were however 
used to fit three different rates for each of the two orientations. 
 
 
 
Sn fractional surface coverage
4.2 The binary Cu-Sn system 54 
4.2.2 Cu(110)Sn binary system 
 
4.2.2.1 The Fick integral and Bragg-Williams fits  
 
0.5 
0.4 
PLTR - measured 
0.3 NLTR - measured 
Fick integral fit 
Bragg-Williams fit 
0.2 
0.1 
0.0 
400 500 600 700 800 900 1000 
Temperature (K)  
 
Figure 4.6 Measured Sn segregation in Cu(110) for PLTR and NLTR runs at heating rate 
and cooling rates of ± 0.05 K/s respectively as well as the calculated Fick integral 
equation (D  = 2.8 × 10-6 m2o /s and E = 168 kJ/mol) and the calculated Bragg-Williams 
equation (∆G = 62 kJ/mol and ΩCuSn = 3.8 kJ/mol.). 
 
 
 
 
    Sn fractional surface coverage  
4.2 The binary Cu-Sn system 55 
 
4.2.2.2 The modified Darken fits 
 
a) Sample heating rate: 0.05 K/s 
 
0.5
PLTR - measured
Darken fit
0.4
0.3
0.2
0.1
0.0
400 500 600 700 800 900 1000
Temperature (K)  
 
Figure 4.7 Measured Sn segregation in Cu(110) for PLTR run at heating rate of 0.05 K/s  
and the Darken model fit for segregation parameters: Do = 2.8 × 10-6 m2/s, E = 168 
kJ/mol, ∆G = 62 kJ/mol and ΩCuSn = 3.8 kJ/mol. 
 
 
 
 
 
Sn fractional surface coverage
4.2 The binary Cu-Sn system 56 
 
 
b) Sample heating rate: 0.075 K/s 
 
0.5
PLTR - measured
Darken fit
0.4
0.3
0.2
0.1
0.0
400 500 600 700 800 900
Temperature (K)
 
 
Figure 4.8 Measured Sn segregation in Cu(110) for PLTR run at heating rate of 0.075 K/s 
and the Darken model fit for segregation parameters: Do = 2.9 × 10-6 m2/s, E = 168 
kJ/mol, ∆G = 62 kJ/mol and ΩCuSn = 3.8 kJ/mol. 
 
 
 
 
 
 
Sn fractional surface coverage
4.2 The binary Cu-Sn system 57 
 
 
 
c) Sample heating rate: 0.15 K/s 
 
0.5
PLTR - measured
Darken fit
0.4
0.3
0.2
0.1
0.0
400 500 600 700 800 900 1000
Temperature (K)  
 
Figure 4.9 Measured Sn segregation in Cu(110) for PLTR run at heating rate of 0.15 K/s 
and the Darken model fit for segregation parameters: Do = 2.9 × 10-6 m2/s, E = 168 
kJ/mol, ∆G = 63 kJ/mol and ΩCuSn = 3.8 kJ/mol. 
 
 
 
 
 
Sn fractional surface coverage
4.2 The binary Cu-Sn system 58 
 
 
 
 
4.2.3 Cu(111)Sn binary system 
 
4.2.3.1 The Fick integral and Bragg-Williams fits  
 
0.3
0.2  PLTR - measured
 NLTR - measured
Fick integral fit
 Bragg-Williams fit
0.1
0.0
400 500 600 700 800 900
Temperature (K)
 
 
Figure 4.10 Measured Sn segregation in Cu(111) for PLTR and NLTR runs at heating 
rate and cooling rates of ± 0.05 K/s respectively as well as the calculated Fick integral 
equation (Do = 9.2 × 10-4 m2/s and E = 205 kJ/mol) and the calculated Bragg-Williams 
equation (∆G = 69 kJ/mol and ΩCuSn = 3.8 kJ/mol.). 
Sn fractional surface coverage
4.2 The binary Cu-Sn system 59 
 
 
 
 
 
4.2.3.2 The modified Darken fits 
 
a) Sample heating rate: 0.05 K/s 
 
PLTR - measured
0.3 Darken fit
0.2
0.1
0.0
400 500 600 700 800 900 1000
Temperature (K)
 
 
Figure 4.11 Measured Sn segregation in Cu(111) for PLTR run at heating rate of 0.05 K/s 
and the Darken model fit for segregation parameters: D -4 2o = 9.2 × 10  m /s, E = 205 
kJ/mol, ∆G = 70 kJ/mol and ΩCuSn = 3.8 kJ/mol. 
Sn fractional surface coverage
4.2 The binary Cu-Sn system 60 
 
 
 
 
 
b) Sample heating rate: 0.075 K/s 
 
0.4
PLTR - measured
Darken fit
0.3
0.2
0.1
0.0
400 500 600 700 800 900 1000
Temperature (K)  
 
Figure 4.12 Measured Sn segregation in Cu(111) for PLTR run at heating rate of 0.075 
K/s and the Darken model fit for segregation parameters: Do = 9.2×10-4 m2/s, E = 205 
kJ/mol, ∆G = 69 kJ/mol and ΩCuSn = 3.8 kJ/mol. 
 
 
 
Sn fractional surface coverage
4.2 The binary Cu-Sn system 61 
c) Sample heating rate: 0.10 K/s 
 
0.4
PLTR - measured
Darken fit
0.3
0.2
0.1
0.0
400 500 600 700 800 900
Temperature (K)
 
 
Figure 4.13 Measured Sn segregation in Cu(111) for PLTR run at heating rate of 0.10 K/s 
and the Darken model fit for segregation parameters: Do = 9.2 × 10-4 m2/s, E = 205 
kJ/mol, ∆G = 69 kJ/mol and ΩCuSn = 3.8 kJ/mol. 
 
 
 
 
 
 
Sn fractional surface coverage
4.2 The binary Cu-Sn system 62 
4.2.4 A summary of Sn segregation parameters in CuSn binary 
alloy 
 
Orientation Rate (K/s) Do (± 5%) E (± 1) ∆G (± 1) ΩCuSn (± 0.2) 
(m2/s) (kJ/mol) (kJ/mol) (kJ/mol) 
 0.050 6.2 × 10-6 189  65 3.9  
Cu(100) 0.075 6.2 × 10-6 189  65  3.8  
 0.150 5.8 × 10-6 190  65  4.0  
 0.050 2.7 ×10-6 168 62 3.8 
Cu(110) 0.075 2.9 × 10-6 168 62 3.8 
 0.150 2.8 × 10-6 168 63 3.8 
 0.050 9.2 × 10-4 205 70 3.8 
Cu(111) 0.075 9.2 × 10-4 205 69 3.8 
 0.100 9.2 × 10-4 205 69 3.8 
 
 Table 4.1. The segregation parameters of Sn in Cu(100), Cu(110) and Cu(111).  The 
errors indicated in the top row were not calculated statistically, but 
guessed from a change in the fit profile for a certain change in the specific 
segregation parameter. 
 
 
 
 
 
 
4.3. The ternary Cu-Sn-Sb system 63 
4.3 The ternary Cu-Sn-Sb system 
 
The solution of the Guttman equations was done using a MATLAB mathematical fitting 
routine, fmins based on Nelder-Mead Simplex Algorithm [77].  The equations were used 
to fit the experimental data for the NLTR runs.  See Appendix A for the flow chart.  
Segregation parameters from the Guttman fits were then used as starting parameters in the 
main model, the modified Darken equations. 
 
4.3.1 Cu(100)SnSb ternary system 
 
4.3.1.1 Guttman fits 
 
0.24 
0.20 
0.16 
Sb - measured NLTR 
 Sn - measured NLTR 0.12 Sn - Guttman fit 
Sb - Guttman fit 
0.08 
0.04 
0.00 
500 600 700 800 900 
Temperature (K)  
 
Figure 4.14 Measured Sn and Sb segregation in Cu(100) from NLTR run at –0.05 K/s. -- 
 
 Fractional surface coverage  
4.3. The ternary Cu-Sn-Sb system 64 
Calculated solid lines are fits using equations 2.29 to 2.32 with ∆GSn  = 66 kJ/mol, ∆GSb  
= 83 kJ/mol, ΩCuSn = 3.4 kJ/mol, ΩCuSb = 15.9 kJ/mol and ΩSnSb = - 5.4 kJ/mol. 
 
The energetic segregation parameters are the same, for a particular crystal orientation, 
irrespective of the sample “low” cooling rates for the profiles obtained from the 
equilibrium segregation runs (NLTR).  These five segregation parameters, obtained from 
the Guttman fits were then introduced as starting fit values in the modified Darken model.  
The other remaining starting parameters are the four from the kinetics segregation region, 
the Do and E’s, and these were chosen from the binary alloy values. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
4.3. The ternary Cu-Sn-Sb system 65 
4.3.1.2 The modified Darken fits  
 
a) Sample heating rate: 0.05 K/s 
 
0.24
Sn - measured PLTR
Sb - measured PLTR
0.20 Sb - Darken fit
Sn - Darken fit
0.16
0.12
0.08
0.04
0.00
400 500 600 700 800 900 1000
Temperature (K)  
 
Figure 4.15 Measured Sn and Sb segregation in Cu(100) for a PLTR run at a rate of 0.05 
K/s and the modified Darken model calculations for segregation parameters: Do(Sn) = 6.3 × 
10-6 m2/s, Do(Sb) = 2.8 × 10-5 m2/s, ESn = 175 kJ/mol, ESb = 186 kJ/mol, ∆GSn  = 65 kJ/mol, 
∆GSb  = 84 kJ/mol, ΩCuSn = 3.4 kJ/mol, ΩCuSb = 15.9 kJ/mol and ΩSnSb = - 5.4 kJ/mol. 
 
 
 
 
 
Fractional surface coverage
4.3. The ternary Cu-Sn-Sb system 66 
b) Sample heating rate: 0.075 K/s 
 
0.24 
Sn - measured PLTR 
 Sb - measured PLTR 0.20 Sb - Darken fit 
Sn - Darken fit 
0.16 
0.12 
0.08 
0.04 
0.00 
400 500 600 700 800 900 
Temperature (K)  
 
Figure 4.16 Measured Sn and Sb segregation in Cu(100) for PLTR run and the Darken 
model calculations for segregation parameters: D  = 6.3 × 10-6 m2o(Sn) /s, Do(Sb) = 2.8×10-5 
m2/s, ESn = 175 kJ/mol, ESb = 186 kJ/mol, ∆GSn = 64 kJ/mol, ∆GSb = 84 kJ/mol, ΩCuSn = 
3.4 kJ/mol, ΩCuSb = 15.9 kJ/mol and ΩSnSb = -5.4 kJ/mol. 
 
 
 
 
 
 
 
 
Fractional surface coverage  
4.3. The ternary Cu-Sn-Sb system 67 
c) Sample heating at 0.15 K/s 
 
0.24
Sn - measured PLTR
Sb - measured PLTR
0.20 Sn - Darken fit
Sb - Darken fit
0.16
0.12
0.08
0.04
0.00
400 500 600 700 800 900
Temperature (K)  
 
Figure 4.17 Measured Sn and Sb segregation in Cu(100) for PLTR run and the modified 
Darken model calculations for segregation parameters: Do(Sn) = 6.3 × 10-6 m2/s, Do(Sb) = 
2.8×10-5 m2/s, ESn = 175 kJ/mol, ESb = 186 kJ/mol, ∆GSn = 65 kJ/mol, ∆GSb = 84 kJ/mol, 
ΩCuSn = 3.4 kJ/mol, ΩCuSb = 15.9 kJ/mol and ΩSnSb = -5.1 kJ/mol. 
 
The modified Darken fits in the temperature region of 580 to 705 K for figures 4.15 to 
4.17 are not good.  This is perhaps experimental procedure problem due to the depleted, 
non-equilibrium near surface region, brought about as a result of the Ar+ ion sputtering of 
the sample surface prior to the segregation run. This procedure is unfortunately 
unavoidable. 
 
 
Fractional surface coverage
4.3. The ternary Cu-Sn-Sb system 68 
4.3.2 Cu(111)SnSb ternary system 
 
4.3.2.1 Guttman fits 
 
0.4
0.3
Sn - measured NLTR
0.2 Sb - measured NLTR
Sn - Guttman fit
Sb - Guttman fit
0.1
0.0
500 600 700 800 900
Temperature (K)  
 
Figure 4.18 Measured Sn and Sb segregation in Cu(111) from NLTR run at – 0.05 K/s. 
Calculated solid lines are the Guttman’s fit for ∆GSn = 68 kJ/mol, ∆GSb = 86 kJ/mol, 
ΩCuSn = 3.6 kJ/mol, ΩCuSb = 16.2 kJ/mol and ΩSnSb = - 5.3 kJ/mol. 
 
 
The above energetic parameters were then used as starting values in the modified Darken 
equations. 
 
 
Fractional surface coverage
4.3. The ternary Cu-Sn-Sb system 69 
4.3.2.2 The modified Darken fits 
 
a) Sample heating rate: 0.05 K/s 
 
0.4
Sn - measured PLTR
Sb - measured PLTR
Sn - Darken fit
0.3 Sb - Darken fit
0.2
0.1
0.0
400 500 600 700 800 900
Temperature (K)
 
 
Figure 4.19 Measured Sn and Sb segregation in Cu(111) for PLTR run and the Darken 
model calculations for segregation parameters: D  = 9.3 × 10-4 m2o(Sn) /s, Do(Sb) = 3.4 × 10-3 
m2/s ESn = 196 kJ/mol, ESb = 206 kJ/mol, ∆GSn = 68 kJ/mol, ∆GSb = 86 kJ/mol, ΩCuSn = 
3.6 kJ/mol, ΩCuSb = 16.2 kJ/mol and ΩSnSb = - 5.3 kJ/mol. 
 
 
 
 
 
Fractional surface coverage
4.3. The ternary Cu-Sn-Sb system 70 
b) Sample heating at 0.075 K/s 
 
0.4
Sn - measured PLTR
Sb - measured PLTR
Sn - Darken fit
0.3 Sb - Darken fit
0.2
0.1
0.0
400 500 600 700 800 900
Temperature (K)
 
 
Figure 4.20 Measured Sn and Sb segregation in Cu(111) for PLTR run and the Darken 
model calculations for segregation parameters: Do(Sn) = 9.3 × 10-4 m2/s, Do(Sb) = 3.4 × 10-3 
m2/s, ESn = 196 kJ/mol, ESb = 206 kJ/mol, ∆GSn = 68 kJ/mol, ∆GSb = 86 kJ/mol, ΩCuSn = 
3.6 kJ/mol, ΩCuSb = 16.2 kJ/mol and ΩSnSb = - 5.3 kJ/mol. 
 
 
 
 
 
 
 
Fractional surface coverage
4.3. The ternary Cu-Sn-Sb system 71 
c) Sample heating at 0.15 K/s 
 
0.4
Sn - measured PLTR
Sb - measured PLTR
Sn - Darken fit
0.3 Sb - Darken fit
0.2
0.1
0.0
400 500 600 700 800 900
Temperature (K)
 
Figure 4.21 Measured Sn and Sb segregation in Cu(111) for PLTR run and the Darken 
model calculations for segregation parameters: Do(Sn) = 9.1 × 10-4 m2/s, Do(Sb) = 3.4 × 10-3 
m2/s, ESn = 196 kJ/mol, ESb = 206 kJ/mol, ∆GSn = 68 kJ/mol, ∆GSb = 86 kJ/mol, ΩCuSn = 
3.6 kJ/mol, ΩCuSb = 16.2 kJ/mol and ΩSnSb = - 5.3 kJ/mol. 
 
 
 
 
 
 
 
 
 
Fractional surface coverage
4.3. The ternary Cu-Sn-Sb system 72 
4.3.3 A summary of Sn and Sb segregation parameters in 
Cu(100) and Cu(111). 
 
Crystal Segregating Rate Do  E  ∆G  ΩCu-i  ΩSnSb  
Species (K/s) (± 5%) (± 1) (± 1) (± 0.2) (± 0.3) 
(m2/s) (kJ/mol) (kJ/mol) (kJ/mol) (kJ/mol) 
  0.05 6.3 × 10-6 175 65 3.4  
 Sn 0.075 6.3 × 10-6 175 64 3.4  
  0.15 6.3 × 10-6 175 65 3.4  
Cu(100) -5 - 5.4  0.05 2.8 × 10  186 84 15.9 
  
Sb 0.075 2.8 × 10-5 186 84 15.9 
 
0.15 2.8 × 10-5 186 84 15.9 
  0.05 9.3 × 10-4 196 68 3.6  
 Sn 0.075 9.3 × 10-4 196 68 3.6  
  0.15 9.1 × 10-4 196 68 3.6  
Cu(111) -3 - 5.3  0.05 3.4 × 10  206 86 16.2 
  
Sb 0.075 3.4 × 10-3 206 86 16.2 
0.15 3.4 × 10-3 206 86 16.2 
 
Table 4.2.  The segregation parameters of Sn and Sb in Cu(100) and Cu(111). 
 
The average segregation parameters in Tables 4.1 and 4.2 will be used in the discussion 
chapter that follows. 
 
 
 
 
 
CHAPTER FIVE 
 
 
DISCUSSIONS AND CONCLUSIONS  
 
 
5.1 Introduction 
 
The discussions will follow the sequence as given in Section 4.1 in Chapter Four.  The 
segregation of solute Sn in the binary CuSn will be considered first and will be followed 
by discussions on Sn and Sb in the ternary CuSnSb system.  Both Sn and Sb segregation 
profiles in the two orientations, namely Cu(100) and Cu(111) will be compared.  Finally, 
the progression study of Sn segregation in both binary and ternary systems will then be 
discussed. 
 
 73 
5.2 The binary Cu-Sn system 74 
5.2 The binary Cu-Sn system 
 
Sn has a larger atomic size than Cu.  This size mismatch causes a lattice strain distortion 
that can only be minimized by the creation of vacancy-solute atom association.  There is 
also the electronic factor due to the differences in valency between the solute Sn and the 
solvent Cu atoms.  Vacancies in pure metals tend to have a negative electrical character, 
and are hence attracted to regions wherein the localized electron density is less than that 
of the matrix average [78].  A quadrivalent Sn will interact with the negatively charged 
vacancy by the electrostatic screened Coulomb attraction.  As a result of this binding 
energy, the vacancy formation energy is reduced and the vacancy concentration is 
enhanced.  This enhances solute Sn segregation in the sense that a diffusive jump occurs 
only via an exchange of position with an adjacent vacancy [79].  This behaviour of Sn, 
among others, is seen in the following segregation profiles. 
 
 
5.2.1. Sn segregation profile in Cu single crystal 
 
Following the method of PLTR, four main regions could be identified from the Sn 
segregation profile in figure 5.1.  In the region below the temperature line A, the 
temperature is low and therefore the diffusion coefficient D is small. However, the small 
surface concentration detected by the AES system, and called the surface dumping effect, 
is caused by the segregation energy.   
5.2 The binary Cu-Sn system 75 
0.2
PLTR-measured
NLTR-measured
0.1
B C 
A 
0.0
400 500 600 700 800 900 1000
Temperature (K)  
 
Figure 5.1 A typical measured Sn segregation profile in Cu(100) for both PLTR at 0.05 
K/s and NLTR at –0.05 K/s showing the kinetic as well as the equilibrium segregation 
regions. 
 
As the temperature increases, however, the bulk diffusion becomes the dominating effect.  
D values of the segregating species increase and the flux of Sn atoms increase from the 
bulk to the surface.  These occurrences give the kinetic segregation region profile 
identified to be between the temperature lines A and B.  The Fick integral was used to fit 
some of the data points in this region to yield the bulk diffusion parameters of Do and E.  
(See figure 4.1).  The region between the temperature lines B and C is governed by the 
differences in the chemical potential energy (∆µ) between the layers that are now rapidly 
decreasing and are approaching zero.  The next region could be identified as the 
equilibrium region, govern by the saturation effect [80] where ∆µ = 0 and for the PLTR, 
Sn fractional surface coverage
5.2 The binary Cu-Sn system 76 
is identified as the narrow region above the temperature line C.  In this region, the 
diffusion coefficient is large enough for a quasi-equilibrium description of the segregation 
process. 
 
Immediately after the PLTR run, the sample could be cooled linearly with time to give the 
negative linear temperature ramp (NLTR) profile.  Figure 5.1 clearly shows that all the 
data points from the NLTR runs follow after the equilibrium section of PLTR.  One could 
assert therefore that the equilibrium section of the PLTR run is extended be employing the 
NLTR run.  The NLTR region is further confirmed as equilibrium region with a repeat of 
PLTR run immediately after the NLTR run.  The advantage of the NLTR runs is that the 
equilibrium region is now extended and well defined and the use of the Bragg-Williams 
equation to fit the experimental data points will yield a more unique set of Ω12 and ∆G 
values than would be the case had the equilibrium section of the PLTR run only been used 
[81]. 
 
The role of the different heating and cooling rates are now visited in relation to Sn 
segregation in the three single Cu crystals.  Figures 5.2, 5.3 and 5.4 portray Sn 
segregation profiles in Cu(100), Cu(110) and Cu(111) respectively.  Each profile consists 
of the combination of PLTR and NLTR runs for a particular heating cum cooling rates. 
 
 
 
 
 
 
5.2 The binary Cu-Sn system 77 
5.2.2 Sn segregation profile at different rates 
 
5.2.2.1 Sn in Cu(100) 
0.2
Rates 
± 0.050 K/s
±  0.075 K/s
±  0.150 K/s
0.1
0.0
400 500 600 700 800 900 1000
Temperature (K)  
Figure 5.2 Measured Sn segregation profiles in Cu(100) at different heating rates 
 
5.2.2.2 Sn in Cu(110) 
 
0.5
0.4
Rates 
0.3 ±  0.050 K/s
±  0.075 K/s
±  0.150 K/s
0.2
0.1
0.0
400 500 600 700 800 900 1000
Temperature (K)  
Figure 5.3 Measured Sn segregation profiles in Cu(110) at different heating rates 
 
Sn fractional surface coverage Sn fractional surface coverage
5.2 The binary Cu-Sn system 78 
5.2.2.3 Sn in Cu(111) 
 
0.4 
0.3 
Rates 
±  0.050 K/s 
0.2 ±  0.075 K/s 
±  0.100 K/s 
0.1 
0.0 
400 500 600 700 800 900 1000 
Temperature (K)  
 
Figure 5.4 Measured Sn segregation profiles in Cu(111) at different heating rates 
 
The segregation profiles of Sn in figures 5.2, 5.3 and 5.4, in the three low index planes of 
Cu show the same trend.  Firstly, the curves shift to higher temperatures as the rate of 
heating the sample increases.  Viljoen and du Plessis [82] have also observed these shifts.  
At a particular temperature in the kinetic region of the segregation profile, say 705 K, as 
indicated in figure 5.4, a higher Sn coverage is seen on the surface for the slowest heating 
rate because the Sn atoms have had much more time to diffuse from the bulk to the 
surface at that temperature.  Secondly, for each of the Cu crystals, Sn segregation profiles 
(NLTR) obtained during cooling from high temperatures are within experimental error, 
the same irrespective of the “low” cooling rates.  This confirms the NLTR profiles to be 
equilibrium profiles. 
Sn fractional surface coverage  
5.2 The binary Cu-Sn system 79 
The next step in looking at the behaviour of Sn in binary CuSn alloy is to compare its 
segregation function in all three orientations at a particular heating and cooling rate.   
 
 
5.2.3 Sn segregation profile in the three low index planes 
of Cu at the same rate 
 
0.5
Cu(100)Sn
Cu(111)Sn
0.4 Cu(110)Sn
0.3
0.2
0.1
0.0
400 500 600 700 800 900 1000
Temperature (K)
 
Figure 5.5 Measured Sn segregation profiles in the three different Cu orientations at rates   
of ± 0.05 K/s 
 
The temperature at which the Sn surface concentration reaches, say 0.1 at.%, in figure 
5.5, is different for the different crystal orientations.  In Cu(110) the temperature is 669 K 
while that for Cu(111) and Cu(100) are 694 K and 740 K respectively.  This indicates 
Sn fractional surface coverage
5.2 The binary Cu-Sn system 80 
different Sn diffusion rates for the different orientations.  These different Sn segregation 
behaviours could be attributed to the number of vacancies that exist in the topmost layers. 
For the different Cu orientations, the number of vacancies and their corresponding 
vacancy formation energies are not the same.  An atom within the bulk has a coordination 
number of 12.  However, once it makes its way to the surface it has a new coordination 
number in relation to its Cu neighbouring atoms.  A surface impurity atom on Cu(110), 
Cu(100) and Cu(111) surfaces bonds with five, four and three nearest-neighbour Cu 
atoms, respectively. The surface cohesion energies, accordingly, is highest for Cu(110) 
and least for Cu(111) [83].  A similar surface energy trend was found for Pd, another fcc 
metal [84].  The vacancy formation energy can be defined as the difference between the 
bulk cohesion energy and the surface cohesion energy.  Thus, Cu(110) has the lowest 
vacancy formation energy, followed by Cu(100) while Cu(111) has the highest [85].  But 
the activation energy of diffusion in the bulk is the sum of the migration energy and the 
vacancy formation energy [86].  Flynn [87] has calculated the migration energy of a Cu 
atom in the bulk as 80.9 kJ/mol.  It is to be expected, therefore, that the activation 
energies (Ei) of a migrating atom in the different Cu orientations would differ because of 
the different vacancy formation energies, and these would indeed affect the diffusion rates 
accordingly.  Migrating atoms in a Cu(110) crystal will have the highest diffusion rate.  
The E values from Table 4.1 are repeated here as: ESn(110) = 167.8 kJ/mol, : ESn(100) = 
189.4 kJ/mol and : ESn(111) = 204.6 kJ/mol and they confirm the above fact that the 
diffusion rate of Sn in Cu(110) is the highest. 
 
However, from figures 5.5 and 5.6, it is established that the migrating atoms in Cu(111) 
have a higher diffusion rate than in Cu(100).  It is to be expected then that the pre-
exponential factor Do  (which consists of the frequency of vibration of the impurity atom 
5.2 The binary Cu-Sn system 81 
about its lattice position, the jump distance to a vacancy and an entropy of activation 
term) in Cu(111) would be higher than in Cu(100).  Actually, Do(111) is two orders of 
magnitude higher than Do(100)   The quantified values from Table 4.1 are: Do(111) = 9.2 × 
10-4 m2/s and D -6 2o(100) = 6.1 × 10  m /s and these justify the fact that Sn in Cu(111) 
segregation profile will lie at lower temperatures as compared to Cu(100).  The other 
result for heating and cooling rates of ± 0.075 K/s is shown in figure 5.6.  It shows similar 
trends for the different Cu orientations. 
 
0.5
Cu(100)Sn
Cu(111)Sn
0.4 Cu(110)Sn
0.3
0.2
0.1
0.0
400 500 600 700 800 900 1000
Temperature (K)  
Figure 5.6 Measured Sn segregation profiles in the three different Cu orientations at rates   
of ± 0.075 K/s 
 
Sn fractional surface coverage
5.2 The binary Cu-Sn system 82 
In trying to weigh the effect of the Do’s and E’s on the diffusion coefficient D of Sn in the 
bulk of the three low index planes of Cu, D as a function of temperature T could be 
written as follows: 
Cu(100)Sn, 
 D100 = 6.1×10
−6 exp 189.7 −   (5.1) 
 8.31×T 
Cu(110)Sn, 
 D −6  167.8 110 = 2.8×10 exp−   (5.2) 
 8.31×T 
Cu(111)Sn, 
 D111 = 9.2 10
−4 exp 204.6× −   (5.3) 
 8.31×T 
 
For comparison, the plots of D versus T are given in figure 5.7. 
10 -15  Cu(100) -16
10 Cu(110) 
- 17 10 Cu(111) 
 -18 10
-19 
10 
- 20 10
-21 
10 
 -22 10
-
10 23 
 -24 10
-25 
10  
10 -26
-27 
10 
-28 
10 
- 29 10
-30 
10 
400 500 600 700 800 900 
Temperature (K)  
Figure 5.7 The diffusion coefficient of Sn in the three low index planes of Cu as a 
function of temperature. 
D (m2 /s) 
5.2 The binary Cu-Sn system 83 
Figure 5.7 explicitly portrays the differences in Sn diffusion coefficient in the low index 
planes of Cu as function of temperature.  For all temperatures below 770 K, the bulk  
diffusion coefficient of Sn is highest in Cu(110).  As explained then, both the activation 
energy and the pre-exponential factor of impurity Sn in Cu(110) are lowest as compared 
to the other two.  Another significant point worth mentioning is the surprising result that 
Sn bulk diffusion coefficient in Cu(111) is higher than in Cu(100) despite the fact that the 
activation energy of Sn in Cu(111) is higher than in Cu(100).  But from equations 5.1 and 
5.3, the pre-exponential factor of Sn in Cu(111) is greater than in Cu(100) and that 
possibly overrides the activation energy factor making the diffusion coefficient of Sn in 
the former to be higher than that in the latter. 
 
For the first time in segregation measurements, the attractive interaction between Cu and 
Sn atoms has been determined as 3.8 kJ/mol.  See Table 5.1 for other published 
segregation parameters besides ΩCu-Sn in the literature. 
 
 
 
 
 
 
 
 
 
 
 
5.2 The binary Cu-Sn system 84 
5.2.4 Comparing published results to the present work  
 
Segregation Do E ∆G ΩCu-Sn Technique Reference 
parameter (m2/s) (kJ/mol) (kJ/mol) (kJ/mol) 
 7.0 × 10-6 168.0 64 ± 5  Const. temp. [89] 
Cu(100)Sn 1.4 × 10-2 203.0 63.0  LTR [83] 
(6.1 ± 0.3) 189 ± 1 65 ± 1 3.9 ± 0.2 LTR Present study 
× 10-6 
 5.0 × 10-5 178.0 72 ± 5  Const. temp. [89] 
Cu(110)Sn 3.0 × 10-3 205.0 72.0  LTR [83] 
(2.8 ± 0.3) 167.8 62.3 ± 1 3.8 ± 0.2 LTR Present study 
× 10-6 
 2.0 × 10-6 176.0 67.0  Const. temp. [45] 
 9.0 × 10-5 172.0 76 ± 5  Const. temp. [89] 
Cu(111)Sn 1.5 × 10-1 234.0 78.0  LTR [83] 
(9.2 ± 0.3) 204.6 69.5 ± 1 3.8 ± 0.2 LTR Present study 
× 10-4 
 
Table 5.1 Published results compared to the present study 
 
As was mentioned in Chapter One, the present study is unique on two counts.  Firstly, all 
the three samples were fabricated to have the same Sn bulk concentration.  The second 
point is that all the segregation measurements were taken under the same experimental 
conditions.  It is clear from Table 5.1 that the segregation parameters of Sn obtained for 
the present study compare fairly well with other published results. 
5.3 The ternary Cu-Sn-Sb system 85 
5.3 The ternary CuSnSb system 
 
Having seen the segregation behaviour of Sn in the three low index planes of Cu, the 
question arises what would happen if another low concentration alloying element was 
introduced to the binary Cu-Sn alloy systems to form ternary systems.  Sb was the 
element added to Cu(100)-Sn and Cu(111)-Sn binary alloys to convert them into ternary 
alloy systems.  As expected there was a change in the segregation behaviour of Sn 
brought about by the addition of Sb in both ternary alloys.  There were also differences in 
the combined segregation profiles of Sn and Sb in the two Cu crystals. 
 
Like Sn, Sb has a larger atomic size than Cu.  Sb is a pentavalent atom and as discussed 
before (section 5.2), that is, from the point of view of size mismatch and electronic 
factors, it can segregate to the surface of a thermally activated CuSb binary alloy system 
[88-91]. 
 
Earlier studies on Cu(111)-0.133 at% Sn-0.180 at. % Sb ternary alloy system [92] have 
shed some light on the segregation behaviours of Sn and Sb.  Sn is known to have a 
higher diffusion coefficient than Sb and as a result will segregate first to the surface of 
Cu(111) in a phenomenon called sequential segregation. 
 
Typical Sn and Sb segregation profiles in Cu are given in figure 5.8. 
 
 
 
5.3 The ternary Cu-Sn-Sb system 86 
5.3.1 Sn and Sb segregation profiles in Cu single crystal 
 
0.24
Sn - measured PLTR C 
Sb - measured PLTR
0.20
0.16 B 
0.12
0.08 A 
0.04
0.00
400 500 600 700 800 900
Temperature (K)  
Figure 5.8 A typical measured PLTR run of Sn and Sb segregation profile in Cu(100) at  
0.075 K/s rate showing four regions. 
 
Similar to figure 5.1, four main regions could be identified from the Sn and Sb 
segregation profiles in figure 5.8.  The region below the temperature line A has low 
temperatures and therefore the diffusion coefficients DSn and DSb are small and as a result 
the AES system does not detect any appreciable amount of Sn and Sb on the surface of 
the Cu crystal at these small time intervals (0.075 K/s).  In the region between A and B, 
however, the D values of the segregating species increase exponentially and the fluxes of 
Sn and Sb atoms increase from the bulk to the surface.  Sn starts with a higher surface 
concentration and build-up untill it reaches a maximum concentration at the temperature 
Fractional surface coverage
5.3 The ternary Cu-Sn-Sb system 87 
line B.  The possible reason lies in the fact that ESn = 175.2 kJ/mol is less than ESb = 186.3 
kJ/mol   
 
The interaction energy between the atoms of the three elements; ΩCu-Sn = 3.4 kJ/mol,  
ΩCu-Sb = 16.1 kJ/mol and ΩSn-Sb = -5.3 kJ/mol, coupled with the segregation energies have 
a profound effect on the segregation profiles of Sn and Sb, particularly toward and inside 
the equilibrium region, from higher temperatures above line B.  The repulsive interaction 
between the segregating Sn and Sb (ΩSn-Sb = -5.3 kJ/mol) causes the latter to desegregate 
as the temperature increases further at B.  Sb with a higher segregation energy, ∆GSb = 
84.3 kJ/mol as against ∆GSn = 64.5 kJ/mol will sequentially displace Sn from the surface 
in the region between lines B and C.  Site competition of Sn and Sb atoms could be 
another avid description in this temperature region and beyond.  The differences in the 
chemical potential energies between the surface and bulk layers that are due to the 
combined effect of the energetic factors, namely, the segregation and the atomic 
interaction energies and the surface concentrations, of both Sn and Sb are rapidly 
approaching zero as the temperature increases and approach the temperature line C.  Thus, 
the kinetic segregation region could be defined as the profiles occupying the temperature 
region between the temperature lines A and C.  Beyond the temperature line C, the 
differences in the changes in the chemical potential energy between the surface layer and 
the first bulk layer and between the rests of the bulk layers become zero.  The values of 
the surface concentrations of both Sn and Sb remain the same, implying that their net 
fluxes are zero.  An equilibrium segregation region could then be defined as the higher 
temperature region from the temperature line C for the PLTR segregation profile.  Figure 
5.9 shows these equilibrium segregation profiles for the PLTR and NLTR runs. 
 
5.3 The ternary Cu-Sn-Sb system 88 
0.24 
0.20 
0.16 
Sb - measured NLTR 
0.12 Sn - measured NLTR 
Sb - measured PLTR at equilibrium 
Sn - measured PLTR at equilibrium 
0.08 
0.04 
0.00 
500 600 700 800 900 
Temperature (K)  
 
Figure 5.9 A typical measured NLTR run (open symbols) of Sn and Sb segregation 
profiles in Cu(100) at -0.075 K/s rate.  The narrow regions of the PLTR run (closed 
symbols) are also shown. 
 
The proof of the NLTR runs as the equilibrium profile was ascertained with another 
PLTR run immediately after the end of the NLTR run.  The data points of the second 
PLTR run profiles are within experimental error the same as the NLTR.   
 
The set of fit values generated when the Guttman segregation equations were used to fit 
the narrow-equilibrium-temperature part of the PLTR profile were not unique because of 
the slowly changing, “stiff”, profile and narrow temperature range involved.  However, by 
extending the equilibrium region over a larger temperature range, by using a NLTR run, a 
much better set of segregation and interaction energies could be found mathematically as 
was shown in figure 4.14 of section 4.3.1. 
     Fractional surface coverage  
5.3 The ternary Cu-Sn-Sb system 89 
The effect of different heating and cooling rates on the segregation of Sn and Sb are given 
in figures 5.10 and 5.11 for Cu(100) and Cu(111) crystals respectively.  Like the 
segregation profiles of Sn in the binary CuSn alloy, (see figures 5.2, 5.3, and 5.4), the 
PLTR segregation profiles of Sn and Sb in the ternary alloys shift to higher temperatures 
as the heating rate of the crystal increases. 
 
5.3.2 Sn and Sb segregation profiles at different rates 
 
5.3.2.1 Cu(100) 
 
0.3
0.2
Rates 
±  0.05 K/s - Sb
±  0.05 K/s - Sn
±  0.075 K/s - Sn
±
0.1   0.075 K/s - Sb
0.0
400 500 600 700 800 900
temperature (K)  
 
Figure 5.10 Measured Sn and Sb segregation profiles in Cu(100) at different heating  
(PLTR) and cooling (NLTR) rates. 
Fractional surface coverage
5.3 The ternary Cu-Sn-Sb system 90 
5.3.2.2 Cu(111) 
 
0.4
0.3
Rates 
0.2 ±   0.05 K/s - Sb
±   0.05 K/s - Sn
±   0.075 k/S - Sn
±   0.075 K/s - Sb
0.1
0.0
400 500 600 700 800 900
Temperature (K)  
 
Figure 5.11 Measured Sn and Sb segregation profiles in Cu(111) at different heating and  
cooling rates. 
 
It should be noted that the cooling, NLTR profiles, are independent of the “low” cooling 
rates used.  Measurements were also taken from PLTR and NLTR runs at rates of ± 0.15 
K/s respectively and the PLTR profile lies to the right of 0.075 K/s PLTR profile at higher 
temperatures. Similar observations were found for Sn segregation in binary Cu(100)Sn 
alloy [93]. 
 
Fractional surface coverage
5.3 The ternary Cu-Sn-Sb system 91 
Having considered the segregation profiles of Sn and Sb in the two orientations namely, 
Cu(100) and Cu(111) crystals separately, they can further be compared at a chosen rate.  
This is done in the next section 5.3.3.  
 
5.3.3 Sn and Sb segregation profiles in Cu(100) and 
Cu(111) at the same rate 
 
5.3.3.1 Crystals heating rate at 0.05 K/s 
 
0.4
Sn - Cu(111)
Sb - Cu(111)
Sn - Cu(100)
0.3 Sb - Cu(100)
0.2
0.1
0.0
400 500 600 700 800 900
Temperature (K)
 
 
Figure 5.12 Measured Sn and Sb segregation profiles in Cu(100) and Cu(111) at the same  
heating rate of 0.05 K/s.  Notice the Sn and Sb pair in Cu(111) lying at lower 
temperatures. 
Fractional surface coverage
5.3 The ternary Cu-Sn-Sb system 92 
The segregates first appear on the surface of Cu(111) at relatively lower temperatures as 
compared to their appearance on Cu(100).  Interestingly, a similar trend was seen when 
the binary Cu(100)Sn and Cu(111)Sn alloys were compared (see figures 5.5 and 5.6).  It 
is to be expected then that the set of diffusion parameters of Sn and Sb in the two Cu 
orientations will differ.  Taking the segregation parameters of Sb as found for both 
Cu(100) as well as Cu(111) into consideration, the pre-exponential factors, DoSb(Cu(100) = 
2.8 × 10-5 m2s-1 is two orders of magnitude lower than DoSbCu(111) = 3.4 × 10-3 m2s-1.  This 
justifies the Sb segregation profile in Cu(111) lying at lower temperatures to that in 
Cu(100) notwithstanding the fact that ESbCu(100) = 186 kJ/mol is lower than ESb(111) = 206 
kJ/mol.  It simply means that the diffusion coefficient of Sb in Cu(111) is higher than in 
Cu(100) on the strength of higher Do.  Similar explanations also hold for Sn in the two Cu 
orientations.  The other result for the heating rate of 0.075 K/s is shown in figure 5.13. 
 
 
 
 
 
 
 
 
 
 
 
 
 
5.3 The ternary Cu-Sn-Sb system 93 
5.3.3.2 Crystals heating rate at 0.075 K/s 
 
0.4
Sn - Cu(111)
Sb - Cu(111)
Sn - Cu(100)
0.3 Sb - Cu(100)
0.2
0.1
0.0
400 500 600 700 800 900
Temeperature (K)  
 
Figure 5.13 Measured Sn and Sb segregation profiles in Cu(100) and Cu(111) at the same  
heating rate of 0.075 K/s. 
 
Figures 5.12 and 5.13 clearly show that Sn and Sb segregating set in Cu(111) lie at lower 
temperatures (to the left of the Cu(100) profiles).   
 
The next step is to follow the trend of Sn segregation in Cu(111) (binary alloy) and in 
Cu(111)Sb (ternary alloy).  Figure 5.14 shows the profiles of the two systems. 
 
 
Fractional surface coverage
5.3 The ternary Cu-Sn-Sb system 94 
5.3.4 Comparing Sn in binary CuSn to Sn in ternary 
CuSnSb 
 
0.4
Sn-binary
Sn-ternary
0.3
0.2
0.1
0.0
400 500 600 700 800 900 1000
Temperature (K)  
 
Figure 5.14.  Measured Sn segregation profiles in the binary (open circle) Cu(111)Sn and  
ternary (closed circle) Cu(111)SnSb.  The heating rate was 0.075 K/s. 
 
It is seen that the segregation profile of Sn in the ternary alloy lie at lower temperatures as 
compared to the profile in the binary.  Perhaps the first possible explanation lies in the 
differences in the interaction energies between the three atoms.  As explained earlier, both 
Sn and Sb segregate strongly in Cu and have attractive interactions with Cu atoms  
(Ω.Cu-Sn = 3.4 kJ/mol, Ω.Cu-Sb = 16.1 kJ/mol) but repulsive interaction between themselves 
(Ω.Sn-Sb = -5.3 kJ/mol).  Sn segregating first in the ternary could be attributed to its  
Fractional surface coverage
5. 4  What has evolved in the course of this study 95 
weaker attractive interaction with Cu and is further aided by the repulsive interaction 
between the atoms of Sn and Sb.  Sn in the ternary alloy is seen desegregating as a result 
of the stronger segregation energy of Sb and the repulsive interaction between the 
segregating elements.  The second explanation would lie in a change in the activation 
energy of Sn in the binary and in the ternary systems.  The addition of Sb solute that is 
also bigger in size than the matrix Cu atoms would elastically strain and expand the Cu 
lattice further.  Vacancies would then be attracted to the Sb and the Sn atoms and result in 
the decrease in the vacancy formation energy [94].  Also, the expansion of the lattice 
would imply that the energy required by an atom to “squeeze” through the saddle point 
would be reduced.  In these circumstances, the activation energy of the Sn solute that is 
made up of the migration and the vacancy formation energies would decrease and 
increase the diffusion rate of Sn in the ternary alloy.  The results in Table 4.2 show 
ESn(binary) = 205 kJ/mol as against ESn(ternary) = 196 kJ/mol and explains the kinetic 
segregation region of the two profiles.   
 
 
5.4 What has evolved in the course of this study  
 
1. In July 2001, at South African Institute of Physics (SAIP) annual conference at the 
University of Natal, Durban, a poster presentation was given entitled:  
“A Mathematical fit procedure to determine segregation parameters from 
experimental data of a linear temperature run”. 
Authors: Asante JKO; Terblans JJ; Roos WD. 
 
 
5. 4  What has evolved in the course of this study 96 
2. In September 2002, a poster entitled, “A reliable method for finding the equilibrium 
segregation parameters in a ternary system” was presented at SAIP annual conference 
at Potchefstroom University. 
Authors: JKO Asante; WD Roos; JJ Terblans. 
 
3. An oral presentation entitled, “Segregation parameters in a (111)CuSnSb crystal from 
constant temperature runs” was given at SAIP annual conference at University of 
Stellenbosch in June 2003. 
Authors: JKO Asante; WD Roos; JJ Terblans. 
 
4. Attended and presented a paper entitled “Finding the equilibrium segregation 
parameters in a Cu(111)SnSb ternary system” at the Fourth International 
Workshop on Surface & Interface Segregation in August 2003, at iThemba LABS, 
in Cape Town. 
Authors: JKO Asante; WD Roos; JJ Terblans. 
 
5. In June, 2004, an oral presentation was given at the annual SAIP conference title, 
“An AES study of Sn surface segregation in the low index planes of Cu single 
crystals” at the University of the Free State, Bloemfontein. 
Authors: JKO Asante; WD Roos; JJ Terblans. 
 
6. A paper entitled, “An AES study of Sn surface segregation in Cu single crystals” has 
been published in the Surface and Interface Analysis 2005; 37:517-521. 
Authors: JKO Asante, JJ Terblans and WD Roos. 
 
 
5. 4  What has evolved in the course of this study 97 
7. Another paper entitled, “Segregation of Sn and Sb in a ternary Cu(100)SnSb alloy” 
has been accepted for publication in Applied Surface Science on 4 March 2005. 
Authors: JKO Asante, JJ Terblans and WD Roos. 
 
8. A paper entitled, “Segregation of Sn in the binary Cu(111)Sn and ternary 
Cu(111)SnSb alloy systems” is in preparation. 
 
 
 
 
Guess : ∆G o1 ,∆G
o
2 , Ω12, Ω13, and Ω23 Input : ∆G
o ,∆G o1 2  Ω12, Ω13, Ω23, X(1) and X(2) 
Do : Temperatures T1 to TN 
f  = (X (1) – X(1))2 + (X (2) – X(2))2 Calculate Xn(1) and Xn(2) 1 n n equations (2.23) and (2.24) 
No 
f1 = min? Fmins : Change X(1) and X(2) 
Yes 
f2 = f2 + (X(1) – Xm(1))2 + (X(2) – X (2))2 Next temperature T m
          Fmins :  
No 
f  = min? Change ∆G
o
1 ,∆G
o
2   2
Ω12, Ω13, and Ω23 
Yes 
       Output 
∆G o1 ,∆G
o
2  Ω12,     
Ω13, and Ω23 
Appendix A:   The solution to Guttman equilibrium segregation equations 
 
 
 98 
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