Masaryk university
Faculty of science
Study materials
Characterization of surfaces and thin films
doc. Mgr. Lenka Zajíčková, Ph.D
Brno 2016
2
Contents
1 Structure of Condensed Matter 7
1.1 Amorphous and Crystalline Materials . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 Description of Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 Bravais lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.3 Index systems for crystal directions and planes . . . . . . . . . . . . . . . . 12
1.2 Bonds in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.1 Ionic bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.2 Covalent bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2.3 Van der Waals bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2.4 Metallic bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.5 Summary of Properties Associated with Interatomic Bonds . . . . . . . . . 27
1.3 Types of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.1 Ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.2 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.3.3 Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.3.4 Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.3.5 Metals and Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.3.6 Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.3.7 Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Physics at surfaces 33
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Thermodynamics of Clean Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.1 Surface tension and surface stress . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 Electronic Structure of Clean Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 Hartree-Fock Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.2 Electron states at surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4 Work function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.1 Contact potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Work function in semiconductors and dielectrics . . . . . . . . . . . . . . . 37
2.4.3 Work function measuring techniques . . . . . . . . . . . . . . . . . . . . . 38
2.4.4 Change of work function with temperature . . . . . . . . . . . . . . . . . . 38
2.4.5 Change of work function with electrical field . . . . . . . . . . . . . . . . . 39
2.5 Thermionic emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3
4 CONTENTS
2.5.1 Comparison of the various measuring techniques . . . . . . . . . . . . . . . 40
2.6 Chemisorption and physisorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.7 Diffusion at surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.8 Surface relaxation and reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.9 Preparation of Clean Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.9.1 Thermal desorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.9.2 Cleavage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.9.3 Sputtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.9.4 Desorption in strong electric field . . . . . . . . . . . . . . . . . . . . . . . 46
2.9.5 Electron bombardment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.9.6 Laser beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.9.7 Surface reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Interaction of Photons with Matter 49
3.1 Interaction of general radiation with matter . . . . . . . . . . . . . . . . . . . . . 49
3.2 Introduction to photon interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Very short wavelengths (< 10−12
m, γ-rays) . . . . . . . . . . . . . . . . . . . . . . 52
3.3.1 Mössbauer effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4 Short wavelengths (10−12
to 10−9
m, including X-rays and ultraviolet) . . . . . . . 58
3.4.1 Fowler‘s theory for metal/vacuum interfaces . . . . . . . . . . . . . . . . . 58
3.4.2 Measurement of the photoelectric work function . . . . . . . . . . . . . . . 59
3.4.3 Interaction of excited electrons with matter . . . . . . . . . . . . . . . . . 60
3.4.4 Photoemission from semiconductors . . . . . . . . . . . . . . . . . . . . . . 60
3.4.5 Photocathode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5 Intermediate wavelengths (including visible and ultraviolet light) . . . . . . . . . . 61
3.6 Long wavelengths (infrared) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.7 Very long wavelengths photons (≥ 1mm) - including radio and microwaves . . . . 61
4 Interaction of Electrons with Matter 67
4.1 Electron interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Penetration depth of electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.1 Electron emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Electron spectroscopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 Electron spectroscopy conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5 Sources of primary particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6 Electron energy analysers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6.1 Analysers with retarding field . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.7 Ultraviolet photoelectron spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 78
4.8 Spectroscopic photoemission of electronic states . . . . . . . . . . . . . . . . . . . 81
4.8.1 L-S (Russell-Saunders) coupling . . . . . . . . . . . . . . . . . . . . . . . . 81
4.8.2 jj coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.9 Auger electron spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.9.1 Auger effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.9.2 Fine Auger structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
CONTENTS 5
4.9.3 Quantitative analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.10 X-ray photoelectron spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.10.1 Energy scale calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.10.2 Primary structure information . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.10.3 Final state effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.10.4 Quantitative analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.10.5 Angular effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.10.6 Numerical data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.11 Electron energy loss spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.12 Threshold techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5 Interaction of Ions with Matter 95
5.1 Ion interactions and penetration depth . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1.1 Basic ion penetration concepts and the energy transfer mechanism . . . . . 96
5.1.2 Secondary ion-electron emission . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Ion sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2.1 Electron impact ionisation sources . . . . . . . . . . . . . . . . . . . . . . . 98
5.2.2 Plasma sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2.3 Surface ionization sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2.4 Field ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3 Ion scattering spectroscopy - ISS . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3.1 Qualitative analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4 Rutherford backscattering spectroscopy - RBS . . . . . . . . . . . . . . . . . . . . 103
5.4.1 Elemental composition determination . . . . . . . . . . . . . . . . . . . . . 104
5.4.2 Energy loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.5 Detection of ejected atoms - ERDA . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.6 Secondary ion mass spectrometry - SIMS . . . . . . . . . . . . . . . . . . . . . . . 105
5.6.1 Ion sputtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.6.2 Ion sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.6.3 Mass analysis system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6 CONTENTS
Literature
1. Ch. Kittel, Úvod do fyziky pevných látek, Academia, Praha 1985
2. L. Eckertová, Elektronika povrchu, SPN, Praha 1983
3. J. Hlávka, Za´klady fyziky povrchu pevných látek, SPN, Praha 1988 (skripta Univerzity J.
E. Purkyně v Brně)
4. L. Eckertová (ed.), Metody analýzy povrchu - elektronová spektroskopie, Academia, Praha
1990
5. L. Frank, J. Král (ed.), Metody analýzy povrchu - iontové, sondové a speciální metody,
Academia 2002
6. A. Zangwill, Physics at Surfaces, Cambridge University Press 1988
7. P. E. J. Flewitt, R. K. Wild, Physical Methods for Materials Characterisation (SE), IOP
Publishing, Bristol 2003
8. J. A. Venables, Introduction to Surface and Thin Film Processes, Cambridge University
Press, Cambridge 2000
9. W. Martienssen and H. Warlimont, Springer Handbook of Condensed Matter and Materials
Data, Springer, Berlin 2005
10. V. Milata, Aplikovaná molekulová spektroskopia, STU, Bratislava 2008
11. P. Griffiths, J.A. De Haseth, Fourier Transform Infrared Spectrometry, Wiley-Interscience,
New Jersey 2007
12. B.C. Smith, Fundamentals of Fourier Transform Infrared Spectroscopy, Second Edition, CRC
Press, 2011
13. P.Y. Yu, Fundamentals of Semiconductors: Physics and Materials Properties (Graduate
Texts in Physics), Springer, 2010
14. J.J Laserna, Modern Techniques in Raman Spectroscopy, Wiley, 1996
15. P.Larkin, Infrared and Raman Spectroscopy; Principles and Spectral Interpretation, Elsevier,
2011
7
8 CONTENTS
16. E.Smith, Modern Raman Spectroscopy: A Practical Approach, Wiley, 2005
17. E.B. Wilson, Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra
(Dover Books on Chemistry), Dover Publications, 1980
18. R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, Springer, 2011
Chapter 1
Structure of Condensed Matter
1.1 Amorphous and Crystalline Materials
Atoms can be aggregate through a number of different processes which determine, at least initially,
whether this collection will take the form of gas, liquid or solid. The state usually changes as its
temperature or pressure is modified. Melting is the process most often used to form an aggregate
of atoms. When the temperature of a melt is lowered to a certain point, the liquid will form either
a crystalline or amorphous solid.
Most crystals are solids, so the atoms, ions or molecules from which they are composed, with
various types of long-range order, form 3-D structures. A more modern definition is that all material
which show sharp diffraction peaks are crystalline. In this sense, aperiodic or quasicrystalline
materials, as well as periodic materials, are crystals. A real crystal is never a perfect arrangement.
Defects in the form of vacancies, dislocations, impurities, and other imperfections are often very
important for the physical properties of the crystal. This aspect has been largely neglected in
the classical crystallography but is becoming more and more a topic of modern crystallographic
investigations.
The difference between crystalline and amorphous materials can be demonstrated as on an
example of silicon oxide SiO2, which can form both the forms. Silicon atoms are surrounded by
three oxygen atoms, which is a short-range ordering. In crystalline polymorph, quartz, oxygen
atoms are arranged in a hexagonal structure (see fig. 1.1).
1.1.1 Description of Crystal Structure
An ideal crystal is constructed by the infinite repetition of identical groups of atoms (a motif). A
group is called the basis. The set of mathematical points to which the basis is attached is called
the lattice. This principle is demonstrated for 2-D planar patterns in figure 1.2.
The lattice in three dimensions is defined by three translation vectors a1, a2, a3, such that the
arrangement of atoms in the crystal looks the same when viewed from the point r as when viewed
from every point r translated by an integral multiple of a1,2,3
r = r + u1a1 + u2a2 + u3a3 (1.1)
where u1, u2 and u3 are arbitrary integers.
9
10 CHAPTER 1. STRUCTURE OF CONDENSED MATTER
Figure 1.1: Schematic view of SiO2: a) amorphous SiO2 with short-range ordering b) crystalline
SiO2 with long-range ordering.
The lattice is said to be primitive if any two points from which the attomic arrangement looks
the same always satisfy eq.(1.2) with a suitable choice of the integers ui. This statement defines
primitive translation vectors ai. An example of their possible selections for 2D lattice is shown in
figure 1.3.
The parallelepiped defined by the primitive axes ai is called a primitive cell (figure 1.5). A
primitive cell is type of unit cell (or just cell). There is no cell of smaller volume than the
primitive cell that can serve as a building block for the crystal structure. Its volume is calculated
as |a1.a2 × a3|.
Primitive translation vectors ai are often used to define the crystal axes which form three
adjacent edges of the primitive parallelepiped. Nonprimitive axes are used as crystal axes when
they have a simple relation to the symmetry of the structure (figure 1.4).
1.1.2 Bravais lattices
In order to describe the crystal structure it is necessary to answeer three important questions:
1. what lattice we have (for a particular structure can be more than one),
2. what translational vectors a1, a2, a3 are we using to describe the lattice (more sets of translational
vectors can be selected for a given lattice) and
3. what is the basis (which is choose after the lattice and translational vectors are selected).
Crystal lattice can be transformed into themselves by the lattice translation T
T = u1a1 + u2a2 + u3a3 (1.2)
1.1. AMORPHOUS AND CRYSTALLINE MATERIALS 11
Figure 1.2: Application of the lattice and the basis (motif) for creation of 2D crystalline patterns.
Figure 1.3: Lattice points in 2D. All pairs of vectors a1, a2 are translational vectors but a,,,
1 , a,,,
2
are not primitive because the lattice translation T cannot be formed by integral combinations of
a,,,
1 and a,,,
2 .
12 CHAPTER 1. STRUCTURE OF CONDENSED MATTER
Figure 1.4: Lattice points for 2D centered rectangular Bravais lattice with the selection of primitive
translation vectors a1 and a2 and nonprimitive translational vectors c1 and c2 that exhibits the
rectangular symmetry more clearly.
Figure 1.5: Translational vectors forming a parallelepiped in 3 dimensional scheme.
1.1. AMORPHOUS AND CRYSTALLINE MATERIALS 13
Figure 1.6: 2D Bravais lattices
and by various other symmetry point operations. A typical symmetry point operation is that of
rotation about an axis that passes through a lattice point. Lattices can be found such that one-,
two-, three-, four- and six-fold rotation axes carry the lattice into itself, corresponding to rotations
by 2π, 2π/2, 2π/3, 2π/4 and 2π/6 radians and by integral multiples of these rotations. Another
symmetry operations are mirror reflections about a plane through a lattice point. The collection
of symmetry point operations which, applied about a lattice point, carry the latice into itself is
called lattice point group.
In two dimensions, there is a general lattice known as oblique lattice and four special lattices.
The oblique lattice is invariant only under rotation of π and 2π about any lattice point but special
lattices can be invariant under rotation 2π/3, 2π/4 and 2π/6 or under mirror reflection. Bravais
lattice is the common phrase for a distinct lattice type and we say that in 2D there are five Bravais
lattices: oblique, rectangular, centered rectangular (rhombic), hexagonal, and square (figure 1.6).
In three dimensions, there are 7 unique unit-cell shapes that can fill the space (triclinic,
monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal and cubic). These are the 7
crystal systems. The Bravais lattices are obtained by combining one of the seven lattice systems
with one of the lattice centerings. The lattice centerings are:
• simple - lattice points on the cell corners only.
• body-centered - one additional lattice point at the center of the cell.
14 CHAPTER 1. STRUCTURE OF CONDENSED MATTER
• face-centered - one additional lattice point at center of each of the faces of the cell.
• base-centered - one additional lattice point at the center of each of one pair of the cell faces.
There are in total 7 × 6 = 42 combinations but it can be shown that several are equivalent to
each other. Therefore, the number is reduced to 14 conventional Bravais lattices described
in table 1.1. The unit cells related to 14 Bravais lattices are shown in figure 1.7. From them only
simple cubic (sc) is a primitive cell (figure 1.8).
Primitive translation vectors of the body-centered cubic (bcc) lattice (in units of lattice parameter
a) are a1 = (1/2, 1/2, −1/2); a2 = (−1/2, 1/2, 1/2); a3 = (1/2, −1/2, 1/2) (figure 1.9
left). The primitive cell is the rhombohedron. The packing ratio is 0.68, defined as the maximum
volume which can be filled by touching hard spheres in atomic positions. Each atom has 8 nearest
neighbors. The conventional unit cell is a cube based on vectors a1 = (0, 0, 1); a2 = (0, 1, 0);
a3 = (0, 0, 1). It is twice big compared to the primitive unit cell and has two atoms in it with
coordinates r1 = (0, 0, 0) and r2 = (1/2, 1/2, 1/2) (figure 1.9 right). The bcc lattice have alkali
metals such as Na, Li, K, Rb, Cs, magnetic metals such as Cr and Fe, and and refractory metals
such as Nb, W, Mo, Ta.
Primitive translation vectors of the face-centered cubic (fcc) lattice (in units of lattice parameter
a) are a1 = (1/2, 1/2, 0); a2 = (0, 1/2, 1/2); a3 = (1/2, 0, 1/2). The primitive cell is the
rhombohedron. The packing ratio is 0.74. Each atom has 12 nearest neighbors. The conventional
unit cell is a cube based on vectors a1 = (0, 0, 1); a2 = (0, 1, 0); a3 = (0, 0, 1). It is 4 times bigger
than the primitive unit cell and has 4 atoms in it with coordinates r1 = (0, 0, 0); r2 = (1/2, 1/2, 0);
r3 = (0, 1/2, 1/2); r4 = (1/2, 0, 1/2). The fcc lattice have noble metals such as Cu, Ag, Au,
common metals such as Al, Pb, Ni and inert gas solids such as Ne, Ar, Kr, Xe.
1.1.3 Index systems for crystal directions and planes
Crystal directions
Any lattice vector can be written as that given by eq. (1.2). The direction is then specified by
the three integers [u1 u2 u3]. If the numbers u1, u2, u3 have a common factor, this factor is
removed. For example, [111] is used rather than [222], or [100], rather than [400]. When we speak
about directions, we mean a hole set of parallel lines, which are equivalent due to transnational
symmetry. Opposite orientation is denoted by the negative sign over a number. An example is in
figure 1.11.
Crystal planes
The crystal plane is defined by three points in the plane provided they are not collinear. The
orientation of a plane in a lattice is specified by Miller indices:
• Find the intercepts on the axes in terms of the lattice constants a1, a2 and a3. The axes
may be those of a primitive or nonprimitive cell.
• Take the reciprocals of these numbers and then reduce to three (usually smallest) integers
having the same ratio (see figure 1.11).
1.1. AMORPHOUS AND CRYSTALLINE MATERIALS 15
Figure 1.7: 3D Bravais lattices
Figure 1.8: Simple cubic Bravais lattice - primitive cell.
16 CHAPTER 1. STRUCTURE OF CONDENSED MATTER
Figure 1.9: Body-centered Bravais lattice with depicted primitive translational vectors defining
primitive cell (left) and conventional selection of cell expressing better the symmetry (right).
Figure 1.10: Face-centered Bravais lattice with depicted primitive translational vectors defining
primitive cell (left) and conventional selection of cell expressing better the symmetry (right).
1.1. AMORPHOUS AND CRYSTALLINE MATERIALS 17
The result enclosed in parentheses (hkl) is called the index of plane. Examples for a cubic system
are shown in figure 1.12.
18 CHAPTER 1. STRUCTURE OF CONDENSED MATTER
Crystal Centering Axial Distances Axial Angles Examples
System (edge lengths)
Cubic simple a = b = c α = β = γ = 90o
NaCl,
body-centred Zinc Blende,
face-centred Cu
Tetragonal simple a = b = c α = β = γ= 90o
White tin,
body-centered SnO2, TiO2,
CaSO4
Orthorhombic simple a = b = c α = β = γ = 90o
Allotropes
body-centered of sulfur,
face-centered KNO3,
base-centered BaSO4
Hexagonal simple a = b = c α = β = 90o
,γ = 120o
Graphite,
ZnO, CdS
Rhombohedral simple a = b = c α = β = γ =90o
CaCO3,
(trigonal) HgS
Monoclinic simple a = b = c α = γ = 90o
, β =90o
Monoclinic Sulphur,
base-centered Na2SO4 ∗10H2O
Triclinic simple a = b = c α = β = γ =90o
K2Cr2O7, CuSO4
∗5H2O, H3BO3
Table 1.1: Description of 14 Bravais lattices in 3D.
Figure 1.11: Example of crystal directions (left) and indexing of crystal plane (right).
1.1. AMORPHOUS AND CRYSTALLINE MATERIALS 19
Figure 1.12: Example of crystal directions.
20 CHAPTER 1. STRUCTURE OF CONDENSED MATTER
1.2. BONDS IN SOLIDS 21
1.2 Bonds in Solids
Interatomic bonds in solids are following
• ionic
• covalent
• metalic
• Van der Waals
1.2.1 Ionic bonds
Ionic bonds are formed between particles which have a net electrical charge. Positive ions are
called cations, since they are attracted to the negative cathode during electrolysis, negative ions
are called anions and migrate to the anode on electrolysis. Ionic bond occurs when atoms with
low ionization energy (lose electrons easily, i.e. they become positive ions) interact with the
atoms with high electron affinity (easily accept electrons, i.e., become negative ions). They are
held together by the Coulomb interaction between positive cations or negative anions. Large
atoms or molecules have a low ionization energy, while small molecules tend to have higher
ionization energies. Alkaline metals have low ionization energy. The electron affinity of an
atom or molecule is the amount of energy released when an electron is added to a neutral atom
or molecule to form a negative ion. Halogens have high electron affinity.
In ionically bonded crystals ions form a lattice in which each cation (anion) is surrounded
by as many anions (cations) as possible, in a symmetric arrangement. It is thus not possible to
distinguish electrically neutral molecules within the lattice. The most important contribution to
Figure 1.13: Ionization energy of atoms as a function of atomic number.
22 CHAPTER 1. STRUCTURE OF CONDENSED MATTER
Figure 1.14: Model of the coordination of Na and Cl inll NaCl.
Figure 1.15: Model of the coordination of Cs and Cl inl CsCl.
the cohesive energy of crystals is due to the Coulomb interactions between ions. Because of their
long range, one must take into account the attraction and repulsion between arbitrarily distant
ions when calculating the energy of the system. There are two typical ionic crystals, NaCl and
CsCl. In the sodium chloride (or rock-salt) crystal the sites of a regular cubic lattice are occupied
by Na+
and Cl−
ions alternately, in a 3D checkerboard pattern. In the CsCl crystal, Cs+
ions
make up a regular cubic lattice, and Cl−
ions occupy the center of each elementary cube.
The lattice constant a of the rock-salt crystal (NaCl) is customarily defined as the shortest
distance between two similar ions along one of the lattice axes. The nearest neighbours of each
Na+
ion are then six Cl−
ions at a distance d1 = a/2 because of the difference between atom size.
In the case of CsCl the atoms have similar size and thus each atom of Cs is surrounded by 8 Cl
ones. The cohesive energy is the crystal energy which is released when individual neutral atoms
are created. Coulomb potential for ions. Let us consider Na+
in NaCl. It is surrounded by six
Table 1.2: Electron affinity in eV.
Fluorine 3.45
Chlorine 3.61
Bromine 3.36
Iodine 3.06
1.2. BONDS IN SOLIDS 23
Cl−
at the distance r:
V1 = −
6e2
4πε0r
. (1.3)
Another neighbour is 12 Na+
each at the distance
√
2r :
V2 = +
12e2
4πε0
√
2r
. (1.4)
When is summed for the entire crystal, we obtain:
VCoulomb = −
e2
4πε0r
6 −
12
√
2
+ . . . (1.5)
= −1, 748
e2
4πε0r
= −α
e2
4πε0r
.
The constant α is called Madelung crystal constant and has a values between 1.6 and 1.8 for
simple crystals. The contribution of the repulsive force to the total potential energy can be written
as:
Vrepulsive =
B
rn
. (1.6)
Total potential energy in the crystal is:
V = VCoulomb + Vrepulsive (1.7)
= −
αe2
4πε0r
+
B
rn
.
In the steady case the separation of ions at a distance r0 must be minimum V :
dV
dr r
= r0 = 0, (1.8)
so
B =
αe2
4πε0n
rn−1
0 (1.9)
and the total potential energy will be:
V = −
αe2
4πε0r0
1 −
1
n
(1.10)
where n is approximately 9, r0 = 2.81 in NaCl, thus V = −1, 27 × 10−18
J = −7, 97 eV. We
must also calculate the energy required to transfer an electron between Na and Cl, which is the
difference between the ionization energy 5.14 eV for Na and the electron affinity of -3.61 eV for Cl
, i.e. 1.53 eV. Each atom is contributing with half of the value, so the overall cohesive energy per
atom Ecohesive = (−3, 99+0, 77) eV/atom = −3, 22 eV/atom. Ionic compounds conduct electricity
when molten or in solution, but not as a solid. They generally have a high melting point and tend
to be soluble in water. Many ionic crystals are hard material due to the strength of their bonds.
24 CHAPTER 1. STRUCTURE OF CONDENSED MATTER
1.2.2 Covalent bond
Covalent bond is formed when one or more electrons are shared by two atoms. The electron density
reaches its maximum in the region between the two atoms. Covalent solids are held together by
networks of directional covalent bonds between neighbouring atoms, where the wavefunctions of
covalent bonds are constructed from pairs of atomic states.
If each bonding state is occupied by two electrons of opposite spins – i.e., each bond is saturated
– then the electrons that participate in bonding are localized in space and cannot contribute to
electrical conductivity. That is why covalent materials are usually insulators or semiconductors.
This is the case for diamond and for two further elements of the carbon group: germanium and
silicon, which have a diamond structure. Here each atom has four sp3
hybrid states, and the bonds
formed by them make up a tetrahedral network. The same structure is seen in semiconducting
compounds formed by elements in groups 13 (IIIA) and 15 (VA) of the periodic table. As it was
mentioned in the previous subsection, besides sp3
wavefunctions, other hybrid states may also
give rise to covalent bonding – however, their spatial directionality depends on which states are
hybridized. The orientation of the bonds plays a crucial role in determining the crystal structure.
The development of short–range order in the amorphous state of covalently bonded solids is also
related to the directionality of the bonds. The cohesive energy of covalent crystals is given by the
sum of the binding energies of individual bonds.
The binding energies of some typical covalent bonds are listed in table below. Much larger
than their counterparts for molecular crystals, these values are comparable to the energies of ionic
crystals.
1.2.3 Van der Waals bond
All atoms and molecules, even inert-gas atoms such as those of helium and argon, exhibit weak,
short-range attractions for one another due to van der Waals forces. These forces were proposed a
century ago by the Dutch physicist Johannes van der Waals to explain observed departures from
the ideal-gas law. The explanation of the actual mechanism of the force, of course, is more recent.
Van der Waals forces are responsible for the condensation of gases into liquids and the freezing
of liquids into solids in the absence of ionic, covalent, or metallic bonding mechanisms. Such
familiar aspects of the behaviour of matter in bulk as friction, surface tension, viscosity, adhesion,
Table 1.3: Binding energy of some typical covalent bonds
Bond eV kJ/mol Bond eV kJ/mol
H–H 4.48 432 C–H 4.28 413
N–N 1.65 159 C–N 3.16 305
P–P 2.08 201 N–H 4.03 389
C–C 3.58 346 Al–P 2.13 205
Si–Si 2.30 222 Si–C 3.17 306
Ge–Ge 1.95 188 Ga–As 1.63 157
O–O 1.47 142 Ga–P 1.78 172
1.2. BONDS IN SOLIDS 25
cohesion, and so on, also arise from these forces. As we shall find, the van der Waals attraction
between two molecules r apart is proportional to r−7
, so that it is significant only for molecules
very close together.
Figure 1.16: Polar-polar attraction
Figure 1.17: Polar-nonpolar attraction
Polar-polar attraction
We begin by noting that many molecules, called polar molecules, have permanent electric dipole
moments. An example is the H2O molecule (Fig. 1.19), in which the concentration of electrons
around the oxygen atom makes that end of the molecule more negative than the end where the
hydrogen atoms are. Such molecules tend to align themselves so that ends of opposite sign are
adjacent, as in Fig. 1.16. In this orientation the molecules strongly attract each other.
Polar-nonpolar attraction
A polar molecule is also able to attract molecules which do not normally have a permament dipole
moment. The process is illustrated in Fig. 1.17. The electrical field of the polar molecule causes a
separation of charge in the other molecule, with the induced moment the same in direction as that
of the polar molecule. The result is an attractive force. The effect is the same as that involved in
the attraction of an unmagnetized piece of iron by a magnet.
26 CHAPTER 1. STRUCTURE OF CONDENSED MATTER
Figure 1.18: Nonpolar-nonpolar attraction
It is not difficult to determine the characteristics of the attractive force between a polar and a
nonpolar molecule. The electric field E at a distance r from a dipole of moment p is given by
E =
1
4πε0
[
p
r3
−
3(pr)
r5
r] (1.11)
We recall from vector analyses that pr = pr cos θ, where θ is the angle between p and r. The field
E induces in the other, normally nonpolar molecule an electric dipole moment p proportional to
E in magnitude and ideally in the same direction. Hence
p = αE (1.12)
where α is a constant called the polarizability of the molecule. The energy of the induced dipole
in the electrical field E is
E = −p E = −
α
(4πε0)2
(1 + cos2
θ)
p2
r6
(1.13)
The mutual energy of the molecules that arises from their interaction is thus negative, signifying
that the force between them is attractive, and is proportional to r−6
. The force itself is equal to
dE/dr and so proportional to r−7
, which means that it drops rapidly with increasing separation.
Doubling the distance between two molecules reduces the attractive force between them to only
0.8 % of its original value.
Nonpolar-nonpolar attraction
More remarably, two nonpolar molecules can attract each other by the above mechanism. Even
though the electron distribution in a nonpolar molecule is symmetric on the average, the electrons
1.2. BONDS IN SOLIDS 27
Figure 1.19: Water molecule
themselves are in the constant motion and at any given instant one part or another of the molecule
has an excess of them. Instead of the fixed charge asymmetry of a polar molecule, a nonpolar
molecule has a constantly shifting asymmetry. When two nonpolar molecules are close enough,
their fluctuating charge distributions tend to shift together, adjacent ends always having oposite
sign (Fig. 1.18) and so always causing an attractive force.
Hydrogen bond
An especially strong type of van der Waals bond called a hydrogen bond occurs between certain
molecules containing hydrogen atoms. The electron distribution in such an atom is so distorted
by the affinity of the “parent” atom for electrons that each hydrogen atom is essence has donated
most of its negative charge to the parent atom, leaving behind a poorly shielded proton. The
result is a molecule with a localized positive charge which can link up with the concentration of
negative charge elsewhere in another molecule of the same kind. The key factor here is the small
effective size of the poorly shielded proton, since electric forces vary as r−2
.
1.2.4 Metallic bond
A great part of the chemical elements have fewer electrons on the incomplete shells than what
is necessary for having saturated covalent bonds between each pair of neighbouring atoms in the
solid state. Electrons participating in unsaturated bonds are not localized. One may also say that
the atoms lose these outermost (in general, s or p) electrons. While the positively charged ions
left behind are arranged in a more or less regular pattern, the freed electrons fill the region among
the ions almost evenly. This moving cloud of electrons gives rise to metallic bonding. In transition
metals, where incomplete d-shells are also found under the outermost shell, further electrons may
participate in metallic bonding. The same kind of bonding may appear in materials built up of
molecules with incomplete shells.
28 CHAPTER 1. STRUCTURE OF CONDENSED MATTER
The wavefunction of such an electron system cannot be written as the product of the wavefunctions
of pairs of electrons forming bonds. It has to be chosen in such a way that it should
show explicitly the antisymmetry with respect to the interchange of the coordinates of any two
electrons. The analysis of such systems, the determination of electronic energies and states, and
the study of those properties of solids that are due to electrons will be among our most important
tasks. The determination of the total energy of metals is a difficult problem of solid-state physics.
The cohesive energy per atom is usually 1–5 eV.
Electrons in metals
In particular, this problem is solved using the one electron approximation. However, it is necessary
to introduce further approximations, such as
• approximation of free electrons - can correctly explain many properties of metals, such as
specific heat, thermal conductivity, electrical conductance.
• approximation of weakly bound electrons - explain other important phenomena such as the
difference between metals, insulators and metalloid, the relationship between conductivity
and valence electrons in the metal
In both cases, we consider a system of electrons in a ideal gas of fermions (max 1 particle in a
given state). Then the distribution of electron energies n( )
n( ) =
dn
d
= fFD g( ), (1.14)
where g( ) is the energy distribution of states and fFD is the Fermi-Dirac distribution function for
free electrons:
fFD =
1
exp( −µ
kT
) + 1
. (1.15)
The chemical potential µ can be determined from the normalization of the electron numbers N
N =
∞
0
fFD g( )d . (1.16)
is a function of T. For particles with m = 0 so = p2
/2m
g( )d = g0
+d
dqdp
h3
, (1.17)
where g0 contains information about the spin. If we take in account the two possible orientations
of the electron we obtain
g( )d =
4πL3
h3
(2m)3/2 1/2
d (1.18)
At T = 0 the fFD is the highest energy to which states are filled = µ = EF; it corresponds
also to the electrochemical potential of the system. The Fermi energy is only a very slow function
of temperature, and for many effects not dependent directly on the rate of change of fFD with
1.3. TYPES OF MATERIALS 29
temperature. It can be shown that the temperature dependence of the Fermi energy can be
expressed as
µ = µ0

1 −
π2
12
kT
µ0
2
. . .

 , (1.19)
where µ0 is the chemical potential at absolute zero. The solution for Schrödinger equation in
electric field for periodic ionic crystals shows the existence of a separate area of the energy bands.
These areas are called forbidden bands.
1.2.5 Summary of Properties Associated with Interatomic Bonds
Property Ionic Covalent Metallic Van der Waals
Non-directional; Direcional; Non-directional; Analogous to
Structures of Structures of Structures of metallic bonds
high coordination low coordination high coordination
and low density and high density
Mechanical Strong, hard Strong, hard Variable Weak, soft
crystals crystals crystals crystals
Thermal High melting High melting Range of Low melting
point, low point, low melting points point
expansion expansion extended large expansion
coefficient coefficient liquidus range coefficient
Electrical Weak insulator, Insulator in Conduction by Insulator
conduction by solid and electron
ion transport liquid state transport
when liquid
Optical Absorption and High refractive Opaque, with Properties
other properties index, absorption similar properties of individual
mainly of the different in in liquid state molecules
individual ions solid and gas
1.3 Types of Materials
1.3.1 Ceramics
The constitution of a ceramic is usually a combination of one or more metals with a non-metallic
element, usually oxygen. As a result, the atoms in a ceramic crystal are linked by a combination
of ionic and covalent bonds. The combination of oxygen atoms with the metal atoms provides
a strong ionic bond because oxygen, with two vacancies in the outer electron shell, effectively
borrows two electrons from the neighbouring metal atoms.
30 CHAPTER 1. STRUCTURE OF CONDENSED MATTER
Table 1.4: Classification of materials based on nature and applications by Bever (1986)
Nature Applications
Ceramics Industrial Materials
Glasses Electrical Materials
Metals and Alloys Electronic Materials
Other Inorganic Materials Superconducting Materials
Polymers Magnetic Materials
Elastomers Nuclear Materials
Fibres Material for other
Energy Applications
Composite Materials Optical Materials
Wood Biomedical Applications
Paper and Paperboard Dental Materials
Other Biological Materials Building Materials
(*) M. B. Bever ed., Encyclopedia of Materials, Science and Engineering, Vol. 1 ed. R. W. Cahn,
Oxford (1986), Pergamon
If ceramic crystals were of perfectly organised structure and uniform microstructure, these
materials would have mechanical properties that exceed those achieved. Indeed, failure of a
ceramic is generally a consequence of a microstructural defect, or combination of defects, such as
inclusions, pores, voids and distribution of irregular size grain.
1.3.2 Semiconductors
Many of the traditional semiconductor materials have crystal structures that are related to the
simple diamond cubic lattice where each atom is tetrahedrally coordinated, but the local atomic
environment is not identical for all atoms. Table below gives the values of basic physical parameters
of some commonly encountered semiconductor materials. Most semiconductor compounds and
alloys are designed to keep the average electron to atom ratio to a value of four. The simplest
illustration is given by the range of AB-type semiconductor materials formed between Group III
and Group V elements in the Periodic Table of Elements. These so-called III to V semiconductors
include GaAs and have a sphalerite (zinc-blend) superlattice structure, whereas the Types II and
VI and IV and VI compounds usually have crystal structures of the sphalerite and rock salt
respectively. Here the lattice constants of these materials lie in the range 0.50 to 0.65 nm and
are generally larger than for metallic elements. This makes it generally easier to obtain details
of the crystalline defects. Moreover the binary semiconductor compounds have band gaps in
the range 0 to 3 eV. Although covering the band gap range for applications to microelectronic
devices it is not possible to prepare such devices from this limited range of materials. This is
due to (i) difficulties in preparing suitable defect-free pure materials and (ii) the need to have
precisely controlled band gaps to optimise the performance of particular devices. Hence the need
to create materials by a combination of binary semiconductor compounds to give ternary and even
quaternary semi-conductors (Pollock et al, 1982).
1.3. TYPES OF MATERIALS 31
Table 1.5: Crystal structure and basic electrical properties of some important semi-conductor
elements and compounds.
Element Crystal structure Lattice spacing Band-gap width
(nm) (at 300K) (eV)
IV elemental and compounds
Si D 0.5431 1.12
Ge D 0.5646 0.66
C (diamond) D 0.3566 5.47
SiC S 0.4359 2.36
SiC W 0.3073 3.23
SiGe W 0.5431 0.67/1.11
D is diamond cubic, S is zinc-blend, W is wurtzite and R is the rocksalt lattice.
1.3.3 Superconductors
Superconductors are characterized by anomalous temperature dependence of the electrical resistivity.
Below a critical temperature T0, their resistivity drops by more than a factor of 1010
.
In superconductors the magnetic flux density B = µrBa induced by an externally applied field
Ha is zero, like in ideal diamagnets with µr = 0 (Meissner–Ochsenfeld effect). If Ha exceeds a
critical value Hc the superconductor becomes normal conducting. But the magnetic induction B
decreases from Ba at the free surface to B = 0 in the interior through a layer of finite thickness
characterized by the Landau penetration depth λ. Superconducting substances of fundamental as
well as practical interest may be divided into 3 main groups:
1. Metallic superconductors;
2. Superconducting oxides;
3. Superconducting carbides, nitrides and borides.
1.3.4 Glasses
Glass is a class of materials that does not crystallise when cooled from the molten state and,
therefore, does not have long range periodicity within the atomic structure. A pure oxide glass
consists of a random three-dimensional network of atoms where each oxygen atom is bonded to
two atoms of metals, such as boron and each metal atom is bonded with three oxygen atoms.
However, there are many type of glasses and in the case of silica glass, each metal atom is bonded
with four oxygen atoms producing a more complex atomic configuration. The addition of fluxing
atoms such as sodium reduces the number of bond cross links.
The major constituents of glasses are in two separated regions of the Periodic Table, Group
VI (O, Si, Se and Te) and Groups I and II (used primarily as fluxes).
32 CHAPTER 1. STRUCTURE OF CONDENSED MATTER
1.3.5 Metals and Alloys
Metals and alloys are opaque, lustrous and relatively heavy, easily fabricated and shaped, have
good mechanical strength and high thermal and electrical conductivity. All these properties are a
consequence of the metallic bonds. In general, they form one of the face centred cubic (fcc), body
centred cubic (bcc) or hexagonal close packed (hcp) structures.
The overall mechanical properties of metals and alloys are controlled by the crystal lattice
defects, such as dislocations and vacancies. Mechanical and chemical properties can be modified
by the addition of alloying elements in varying proportion.
1.3.6 Polymers
Polymers are by definition materials composed of long-chain molecules, typically 10 to 20 nm,
that have been developed as a consequence of the linking of many smaller molecules, monomers.
The combination of tensile strength and flexibility make these materials attractive.
If the molecular chains are packed side by side, the molecules form an array with a crystalline
structure. Natural polymers have complex microstructure comprising a mixture of crystalline and
amorphous material. In the case of polymeric materials, the interatomic bonds between molecular
chains are the weak van der Waals forces, but in the crystalline structures, the chains are closer
together over comparatively large distances so that the contribution of intermolecular forces has
the effect of producing a more rigid material. To develop stronger, more rigid, polymers:
1. production of a crystalline structure (polyethylene, nylon),
2. formation of a strong covalent bond between the molecular chains by cross linking (vulcanising
raw rubber by heating with the controlled addition of sulphur atoms).
1.3.7 Composite Materials
A composite material was originally considered to be a combination of two materials but now this
class of material is regarded as any combination which has particular physical and mechanical
properties. The concept of composite materials has led to the design and manufacture of a
new range of structural materials that are generally lighter, stiffer and stronger than anything
previously manufactured.
• natural composites: wood - cellulose fibres provide tensile strength and flexibility and lignin
provides the matrix for binding and adds the property of stiffness; bone - strong, but soft,
protein collagen and the hard, brittle mineral apatite,
• synthetic composites: combining individual properties such as strong fibres of a material
(for example carbon) in a soft matrix (such as an epoxy resin).
1.3. TYPES OF MATERIALS 33
Element Crystal structure Lattice spacing Band-gap width
(nm) (at 300K) (eV)
III-V compounds
GaAs S 0.5653 1.42
GaP S 0.5451 2.26
GaSb S 0.6096 0.72
GaN W 0.3186 3.4
InAs S 0.6058 0.36
InP S 0.5869 1.35
InSb S 0.6479 0.17
InN W 0.3536 0.65
InGaAs S 0.5699 0.36/1.43
InGaP W 0.3536 0.65
BAs S 0.4777 1.5/3.47
BP S 0.4537 2.0
BN S 0.3615 5.5/6.4
AlAs S 0.5661 2.16
AlSb S 0.6136 1.58
AlP S 0.5463 2.5
AlN S/W 0.436 6.2
AlGaAs S 0.5653 1.42/2.16
AlGaN S 0.316 3.44/6.28
AlGaP S 0.5537 2.26/2.45
II-VI compounds
CdS S/W 0.5832 2.42
CdSe S 0.605 1.7
CdTe S 0.6482 1.56
CdZnTe S 0.6442 1.4/2.2
HgCdTe S 0.648 0/1.5
HgZnTe S 0.6459 0/2.25
HgTe S 0.644 0
ZnS S/W 0.542 3.68
ZnSe S 0.5669 2.7
ZnTe S 0.6089 2.2
ZnO S/W 0.352 3.3
Chalcopyrite
CuInSe2 S 0.5782 1.04
CuCl S 0.541 3.4
Cu2S W 0.4142 1.2
IV-VI compounds
PbS R 0.594 0.41
PbSe R 0.612 0.27
PbTe R 0.646 0.31
SmTe R 0.632 0.18
D is diamond cubic, S is zinc-blend, W is wurtzite and R is the rocksalt lattice.
34 CHAPTER 1. STRUCTURE OF CONDENSED MATTER
Chapter 2
Physics at surfaces
2.1 Introduction
The physical surface is obviously something different than the mathematical surface (an infinitely
thin layer at the interface between solid and vacuum or gas). When we speak about the physics
at surfaces, we usually mean a surface layer from a substance whose properties are different than
those of volumes.
It is not possible to determine the thickness of the surface layer. It is different for different
substances and it also varies depending on the properties that we consider, but is only a few
atomic layers.
At the surface there are physical properties modifications and new phenomena, because the
surface atoms are surrounded by asymmetric atoms.
• surface tension,
• surface space charge,
• work function,
• phase conditions,
• surface oscillations - the oscillations amplitude for surface atoms is higher for surface atoms
than for the atoms in volume,
• lattice relaxation (atoms are slightly shifted from the equilibrium position),
• reconstruction (creation of new regular structures),
• adsorption.
2.2 Thermodynamics of Clean Surfaces
It introduces the concepts of surface free energy and surface stress. The essential features of bulk
thermodynamics can be stated very succinctly (Callen, 1985). In equilibrium, a one-component
35
36 CHAPTER 2. PHYSICS AT SURFACES
system is characterized completely by internal energy, U, which is a unique function of entropy,
volume and particle number of the system:
U = U(S, V, N), (2.1)
dU = |
∂U
∂S
|V,N dS + |
∂U
∂V
|S,N dV + |
∂U
∂N
|S,V dN, (2.2)
dU = TdS − PdV + µdN. (2.3)
These equations define the temperature, pressure and chemical potential of the bulk. The
extensive property of the internal energy,
U(λS, λV, λN) = λU(S, V, N), (2.4)
together with the combined first and second laws of entropy, lead to the Euler equation,
U = TS − PV + µN. (2.5)
Differentiating the Euler equation and using the entropy, we arrive at a relation among the intensive
variables, the Gibbs-Duhem equation:
SdT − V dP + Ndµ = 0 (2.6)
2.2.1 Surface tension and surface stress
We create a surface of area A from the infinite solid by cleavage process. Since the bulk doe not
spontaneously cleave, the total energy of the system must increase by amount proportional to A.
The constant of proportionality, γ, is called surface tension:
U = TS − PV + µN + γA. (2.7)
In equilibrium at any finite temperature and pressure, the semi-infinite solid coexists with its
vapor. A plot of the particle density as a function of distance normal to the surface is shown in
the figure below.
Gibbs recognized that is convenient to be able to ascribe definite amounts of the extensive
variables to a given area of surface. Accordingly, the vertical lines in the figure indicate a portion
of space into a bulk solid volume, a bilk vapor volume and a transition, or surface, volume. The
remaining extensive quantities can be partitioned likewise:
S = S1 + S2 + S3, (2.8)
V = V1 + V2 + V3,
N = N1 + N2 + N3.
2.3. ELECTRONIC STRUCTURE OF CLEAN SURFACES 37
In this formulae, the bulk quantities are defined by
i = 1, 2
Si = siVi;
Ni = ρiVi.
where ρi and si characterize the uniform bulk phases. Once the surface volume is chosen, the
other surface quantities are defined as excesses. Note that changes in the surface excess quantities
are completely determined by changes in the bulk quantities:
∆Ss = −∆S1 − ∆S2, (2.9)
∆Vs = −∆V1 − ∆V2,
∆Ns = −∆N1 − ∆N2.
Evidently, there is nothing unique about the particular choice of the boundary positions illustrated
in the figure. Nevertheless, it will emerge that one always can be choose a subset of the
surface excesses that are perfectly well-defined quantities with values that are independent of any
such conventional choices. Now consider the effect of small variations in the area of the system,
e.g., by stretching. We assume that the energy change associated with this process is described
adequately by linear elasticity theory (Landau & Lifshitz, 1970). Accordingly, the entropy should
be replaced by
dU = |
∂U
∂S
|V,N,AdS +
∂U
∂V
|S,N,AdV +
∂U
∂N
|S,V,AdN + A
i,j
|
∂U
∂ i.j
|S,V,N d i.j
dU = TdS − PdV + µdN + A
i,j
σi,jd i.j
where σi,j and i.j are the surface stress and strain tensor.
2.3 Electronic Structure of Clean Surfaces
2.3.1 Hartree-Fock Approximation
The wave function describing the set of I electrons has the general form
Ψ(r1, Sz1, r2, Sz2, . . . , ri, Szi, . . . rN , SzN )
where ri is the position of electron number i, and Szi its spin in a chosen z-direction, with
measurable values 1
2
¯h and −1
2
¯h. Of course, what answer you get for the wave function will also
depend on where the nuclei are, but in this section, the nuclei are supposed to be at given positions,
so to reduce the clutter, the dependence of the electron wave function on the nuclear positions
will not be explicitly shown.
Hartree-Fock approximates the wave function in terms of a set of single-electron functions,
each a product of a spatial function and a spin state where stands for either spin-up, ↑, or
38 CHAPTER 2. PHYSICS AT SURFACES
spin-down, ↓. (By definition, function ↑(Sz) equals one if the spin Sz is 1
2
¯h, and zero if it is −1
2
¯h,
while function ↓(Sz) equals zero if Sz is 1
2
¯h and one if it is −1
2
¯h.) These single-electron functions
are called “orbitals” or “spin orbitals.” The reason is that you tend to think of them as describing
a single electron being in orbit around the nuclei with a particular spin. Wrong, of course: the
electrons do not have reasonably defined positions on these scales. But you do tend to think of
them that way anyway.
The spin orbitals are taken to be an orthonormal set. Note that any two spin orbitals are
automatically orthogonal if they have opposite spins: spin states are orthonormal so ↑|↓ = 0. If
they have the same spin, their spatial orbitals will need to be orthogonal.
Single-electron functions can be combined into multi-electron functions by forming products
of them of the form
where n1 is the number of the single-electron function used for electron 1, n2 the number of
the single-electron function used for electron 2, and so on, and an1,n2,...,nI
is a suitable numerical
constant. Such a product wave function is called a “Hartree product.”
2.3.2 Electron states at surfaces
Electronic surface states were first postulated by Tamm based on a crude one dimensional model.
The solid was represented as a one-dimensional potential well with a periodic potential in the
form of positive delta-functions. The general solutions for that potential are plane waves. Shockley
considered a more realistic, yet still one-dimensional potential and Bloch-wave solutions . Shockley
showed that the surface states found by Tamm were due to an incompleteness of the potential at
the surface, i.e. due to a surface potential that was altered compared to the bulk. In the later
literature, the distinction between surface states caused by a changed potential vs. surface states
caused by the finiteness of the lattice was frequently made by referring to "Tamm-states" and
"Shockley states", respectively. Some authors also refer to Tamm-states and Shockley-states
in the sense that the former should be localized, tight-binding type surface states and the latter
free-electron type surface states, a distinction that has little foundation in the original work of
Tamm and Shockley. Early experimental evidence for the existence of surface states came from
the electric properties of space charge layers at semiconductor surfaces. A high density of surface
states pins the Fermi-level at the surface to a fixed position with respect to the valence and
conduction band edges.
2.4 Work function
The work function is the energy, typically a few electronvolts, required to move an electron from
the Fermi Level, EF, to the vacuum level, E0, as shown in figure below
The work function, φ, depends on the crystal face hkl and rough surfaces typically have lower
φ. Electron affinity, χ, and ionization potential Φ would be the same for a metal, and equal
to φ. The electron affinity χ is the difference between the vacuum level E0, and the bottom of
the conduction band EC, as shown in figure above. The ionization potential Φ is E0-EV, where
EV is the top of the valence band. These terms are not specific to surfaces: they are also used for
2.4. WORK FUNCTION 39
Figure 2.1: Schematic diagrams of (a) the work function φ; (b) the electron affinity χ and ionization
potential Φ, both in relation to the vacuum level E0, the Fermi energy EF, and conduction and
valence band edges EC and EV.
atoms and molecules generally, as the energy level which (a) the next electron goes into, and (b)
the last electron comes from.
2.4.1 Contact potential
When two metals are electrically isolated from each other, an arbitrary potential difference may
exist between them. However, when two different metals are brought into contact, electrons
will flow from the metal with a lower work function to the metal with the higher work function
until the electrochemical potential of the electrons in the bulk of both phases are equal. The
actual numbers of electrons that passes between the two phases is small, and the occupancy of
the Fermi levels is practically unaffected. In the case of semiconductors, the space charge give
rise to a contact potential difference between the two sides of the junction. The presence of
the contact potential difference makes possible the smooth transition of the Fermi level from its
position near the acceptor level on the p side to its position near the donor level on the n side.
This is the reason why the contact potential difference if often called band bending. It leads to a
reduction in the electron affinity. Some materials (e.g. Cs/p-type GaAs) can even be activated to
negative electron affinity, and such materials form a potent source of electrons, which can also be
spin-polarized as a result of the band structure.
2.4.2 Work function in semiconductors and dielectrics
The work function is the minimum energy that must be given to an electron to liberate it from
the surface of a particular substance. In the photoelectric effect, electron excitation is achieved
by absorption of a photon. If the photon’s energy is greater than the substance’s work function,
photoelectric emission occurs and the electron is liberated from the surface. Excess photon energy
results in a liberated electron with non-zero kinetic energy. The work function is also important
in the theory of thermionic emission. Here the electron gains its energy from heat rather than
photons. According to the Richardson-Dushman equation the emitted electron current density.
40 CHAPTER 2. PHYSICS AT SURFACES
2.4.3 Work function measuring techniques
A related method for determining the zero-electric-field condition between two parallel plates is
by measuring directly the tendency for charge to flow as a result of changing the work function
of one of the surfaces such as might occur during adsorption. In this static method, called static
capacitor, the tendency for charge to flow through the external circuit is detected electronically
and used to establish very quickly the correct value for the compensating potential.
In the vibrating capacitor method (Kelvin-Zisman) the contact potential difference between
two metal connected electrically is just equal to the difference in their respective work functions.
The contact potential difference arises from the transfer of electrons from the metal of lower work
function to the second mater until their Fermi levels line up. If the two metals are arranged as
two plates of a capacitor, then in the absence of an applied electric field, an electric field appears
in the region between plates. By the application of a compensating potential, the field between
the plates can be reduced to zero. In other words, the applied potential necessary to reduce the
electric field is just equal to the contact potential difference.
2.4.4 Change of work function with temperature
For metals the work function has a linear relation with the temperature change:
φ(T) = φ(T0) + α(T − T0), (2.10)
where α has values between 10−4
–10−5
eV/K. For semiconductors and insulators the chemical
potential varies strongly with temperature:
φ(T) = Eaf +
Eg
2
+
kT
2
ln
Nc
Nv
, (2.11)
where Eg is the band gap, Eaf is the electron affinity, Nc a Nv are the densities of the conduction
and valence band.
For n-type semiconductors at lower ionisations:
φ(T) = Eaf +
∆ED
2
+
kT
2
ln
Nc
ND
, (2.12)
where ∆ED is the activation energy and ND is the density of the donor states. At high temperatures
it is change to:
φ(T) = Eaf +
kT
2
ln
Nc
ND
. (2.13)
In the case of p-type semiconductors:
φ(T) = Eaf + Eg −
∆EA
2
−
kT
2
ln
Nv
NA
(2.14)
and
φ(T) = Eaf + Eg −
kT
2
ln
Nv
NA
. (2.15)
2.5. THERMIONIC EMISSION 41
2.4.5 Change of work function with electrical field
The work function is actually the binding energy of an electron to a metal surface. Neglecting the
external electric field, this is work:
φ =
e2
(4πε0)4α2
(2.16)
against the attractive image force (between the electron and its mirror image inside metal)
φ =
e2
(4πε0)(2r)2
(2.17)
Here, α is a typical interatomic distance in metal. In the presence of an external electric field E,
extracting electrons from cathode, the total electric field applied to the electron can be expressed
as a function of it‘s distance from the metal surface:
F0 =
e2
16πε0x2
− eE (2.18)
As can be seen, the extraction takes place if electron distance from the metal exceeds the critical
one: rcr = e/16πε0E when attraction to the surface is changed to repulsion. Thus, the work
function can be calculated as the integral:
φ =
rcr
a
F(r)dr ≈
e2
16πε0α
−
1
√
4πε0
e3/2
√
E (2.19)
The Schottky relation above returns in the absence of electric field E = 0 to the previously
mentioned expression W = W0 for the work function. For practical calculation of the work function
decrease in an external electric field, the Schottky relation can be rewritten in the following
numerical way:
φ, eV = φo − 3.8 · 10−4
· E, V/cm (2.20)
The decrease of the work function is relatively small at reasonable values of electric field.However,
the Schottky effect can result in a major change of the thermionic current because of it‘s strong
exponential dependence on the work function in accordance with the Sommerfeld formula.
2.5 Thermionic emission
The Richardson – Dushman equation, dating from 1923, describes the current density emitted
by a heated filament, as
J(T) = AT2
exp(−φ/kT) (2.21)
so that a plot of log(J/T2
) versus 1/T yields a straight line whose negative slope gives the work
function φ. This value of φ is referred to as the "Richardson" work function, since there is an
intrinsic temperature dependence of the work function, whose value dφ/dT is of order 10−4
to
10−3
eV/K, with both positive and negative signs (H¨olzl and Schulte 1979). When data is taken
over a limited range of T, this temperature dependence will not show up on such a plot, but
42 CHAPTER 2. PHYSICS AT SURFACES
will modify the pre-exponential constant. This constant, A, can be measured in principle, but
is complicated in practice by the need to know the emitting area independently, since what is
usually measured is the emission current I rather than the current density, J. The form of this
equation can be derived readily from the free electron model, by considering the Fermi function,
and integrating over all those electrons, moving towards the surface, whose "perpendicular energy
" is enough to overcome the work function. In this calculation, ignoring reflection at the surface
by low energy electrons, the value of A is 4πmk2
e/h3
= 120A/cm2
/K2
. Where absolute values of
current densities have been measured, values of this order of magnitude have been found.
2.5.1 Comparison of the various measuring techniques
The measurement of surface work function means determination of the energy difference between
the Fermi level and the top of the potential energy barrier. We expect to measure different apparent
values of φ by various techniques due to changes in the barrier height, band structure
effects, reflection problems and so forth. For example, the Schottky effect must be corrected for in
thermionic emission measurements, and thermal effects must be considered in the field emission
and photoemission measurements. Emission measurements on a polycrystalline surface will yield
geometrically weighted average values.
For a single crystal surface, the capacitance methods of work function measurement should yield
true values of contact potential difference between the sample under study and the reference surface.
These methods do not depend upon detailed knowledge of electron emission mechanisms,
nor are they affected by surface reflection of slow electrons or other similar properties. They
simply measure the difference in surface potential between two electrodes whose Fermi energies
are equalized. Because this type of measurement is usually made at room temperature, thermal
effects are generally not serious. There is a major drawback, however, in these methods. The
work function of the reference surface must be precisely known id absolute determination of the
sample work function is to be made, Even when only relative changes are to be measured, as in
adsorption experiments, the reference surface work function must be stable, either unaffected by
adsorption or cleanable before each measurement.
The various emission techniques depend upon precise knowledge of the emission mechanism involved.
The theories of thermionic emission, field emission and photoemission are based upon
idealized models and do not take into account the detailed band structure in the region of the
Fermi level (field emission) or near the top of the emission barrier (thermionic emission and photoemission).
The emission may correspond exactly with the theory only in the free-electron limit.
Deviation from the free-electron case, as observed to a greater or lesser extent in all metals, can
result in an apparent work function greater than the true barrier heigh when measured by emission
methods.
The retarding-potential methods have perhaps the broadest applicability of all. A variety of experimental
arrangements may be used, including the scanning low-energy probe technique (SLEEP)
which can be used to study work function variation over a non uniform surface. When the field
emission retarding potential (FERP) technique is used, absolute work function values may be determined.
IN general, the emission properties of the electron source may be checked independently
to verify cleanliness, but , to first order, this is unnecessary in the FERP case.
2.6. CHEMISORPTION AND PHYSISORPTION 43
2.6 Chemisorption and physisorption
A qualitative distinction is usually made between chemisorption and physisorption, in terms of
the relative binding strengths and mechanisms. In chemisorption, a strong "chemical bond " is
formed between the adsorbate atom or molecule and the substrate. In this case, the adsorption
energy, Ea, of the adatom is likely to be a good fraction of the sublimation energy of the substrate,
and it could be more. It was found that in a nearest neighbour pair bond model, Ea = 2eV for an
adatom on an FCC (100) surface when the sublimation energy L0 = 3eV . In that case the atoms
of the substrate and the "adsorbate" were the same, but the calculation of the adsorption stay
time, τa, would have been valid if they had been different. Energies of 1 – 10 eV/atom are typical
of chemisorption.
Physisorption is weaker, and no chemical interaction in the usual sense is present. But if there
were no attractive interaction, then the atom would not stay on the surface for any measurable
time – it would simply bounce back into the vapor. In physisorption, the energy of interaction
is largely due to the (physical) van der Waals force. This force arises from fluctuating dipole
(and higher order) moments on the interacting adsorbate and substrate, and is present between
closed-shell systems. Typical systems are rare gases or small molecules on layer compounds or
metals, with experiments performed below room temperature. Physisorption energies are 50 –
500 meV/atom; as they are small, they can be expressed in kelvin per atom, via 1eV ≡ 11604K,
omitting Boltzmann’s constant in the corresponding equations. These energies are comparable to
the sublimation energies of rare gas solids.
Adsorption of reactive molecules may proceed in two stages, acting either in series or as alternatives.
A first, precursor, stage has all the characteristics of physisorption, but the resulting
state is metastable. In this state the molecule may reevaporate, or it may stay on the surface
long enough to transform irreversibly into a chemisorbed state. This second stage is rather dramatic,
usually resulting in splitting the molecule and adsorbing the individual atoms: dissociative
chemisorption. The adsorption energies for the precursor phase are similar to physisorption of rare
gases, but may contain additional contributions from the dipole, quadrupole, and higher moments,
and from the anisotropic shape and polarizability of the molecules. The dissociation stage can
be explosive-literally. The heat of adsorption is given up suddenly, and can be imparted to the
resulting adatoms.
The material in the adsorbed state is called adsorbate. The substance to be adsorbed (before
it is on the surface) is called the adsorpt or adsorptive. The substance, onto which adsorption
takes place, is the adsorbent.
Density of the molecules hitting the surface:
j =
p
√
2πkTm
, (2.22)
where p is gas pressure. At normal conditions (atmospheric pressure and room temperature) it falls
1 cm2
at about 1024
molecules per second, at 10−4
Pa is in order 1015
, which is approximatively the
number of atoms in the surface layer.An important question is how much of a material is adsorbed
to an interface. This is described by the adsorption function Γ = f(P, T), which is determined
experimentally. It indicates the number of adsorbed moles per unit area. In general, it depends on
the temperature. A graph of Γ versus P at constant temperature is called an adsorption isotherm.
44 CHAPTER 2. PHYSICS AT SURFACES
Figure 2.2: Schematics diagram - explanation of terms: adsorbate, adsorbent and adsorpt.
For a better understanding of adsorption and to predict the amount adsorbed, adsorption isotherm
equations are derived. They depend on the specific theoretical model used. For some complicated
models the equation might not even be an analytical expression.
Table 2.1: Adsorption coefficient for oxygen
material (111) (100)
Si 10−1
10−2
Ge 10−3
–10−4
10−3
InSb 10−5
–
A useful parameter to characterize adsorption is the adsorption time. Let us first assume
that no forces act between the surface and a gas molecule. Then, if a molecule hits the surface,
it is reflected elastically with the same energy. An energy transfer between the surface and gas
molecule does not take place. As a consequence "hot" molecules do not cool down even when they
hit a cold surface. The residence time in proximity to the surface can be estimated by:
τ =
2∆x
¯vx
≈
2∆x
kbT/m
, (2.23)
where ∆x is the thickness of the surface region and ¯vx is the mean velocity normal to the surface.
An attractive force between the gas molecule and the surface increases the average residence
time of the molecule at the surface to
τ = τ0 exp(Ea/kT), (2.24)
2.7. DIFFUSION AT SURFACES 45
Where τ0 is the bond vibration frequency and Ea is the heat of adsorption.
Another useful parameter is the accommodation coefficient α. The accommodation coefficient
is defined by the temperature of the molecules before the impact T1, the surface temperature T2,
and the temperature of the reflected molecules T3
α =
T3 − T1
T2 − T1
(2.25)
For an elastic reflection, the mean velocity of the molecules before and after hitting the surface
are identical and so are the temperatures: T1 = T3. Then α = 0. If the molecules reside a long
time on the surface they have the same temperature, after desorption, as the surface: T2 = T3
and α = 1. Thus, the accommodation coefficient is a measure of how much energy is exchanged
before a molecule leaves the adsorbent again.
2.7 Diffusion at surfaces
Diffusion phenomena at surfaces range from the random walk of single atoms to mass transport on
macroscopic length scales which involves hundreds, even thousands of different individual processes
with merely a few of them being rate determining. Their nature remains frequently unknown. A
theoretical description of diffusive mass transport has to take the various length scales and the
different levels of knowledge into account.
One of the most direct applications of the field emission microscopy is the study of adsorbate
diffusion on clean metal surfaces as a function of surface coverage. The usual method relies on
observing changes in the intensity of a well-defined temperature T. To minimize the possibility of
the electric field affecting the measurement, diffusion is usually initiated at elevated temperatures
in the absence of a field.
Gomer (1953) demonstrated that three distinct types of diffusion could occur for hydrogen and
oxygen in tungsten. The particular type of diffusion depends on the morphology of the surface. If
more than a monolayer of adsorbate was deposited, uniform diffusion with a sharp boundary was
observed below 20 K for hydrogen and below 30 K for Oxygen. Gomer explained the observations
by assuming that mobile physisorbed layer was diffusing on top of an immobile chemisorbed
layer. When the molecules in the physisorbed layer reached the edge of the chemisorbed layer,
they precipitated onto the clean surface, where they chemisorbed, thereby extending the sharp
boundary. Gomer characterized this type of diffusion by an average diffusion constant
D =
¯x2
t
(2.26)
where ¯x is the average distance traversed by the boundary at the time t. He calculated an
activation energy for diffusion by assuming a diffusion coefficient of the form
D = A2
ν exp(−Ed/kT) (2.27)
where ν ≈ 1012
s−1
and A ≈ 0.3.
46 CHAPTER 2. PHYSICS AT SURFACES
2.8 Surface relaxation and reconstruction
An ideal crystal is characterized by a regular arrangement of its constituent atoms. Its microscopic
structures possesses a three-dimensional translation invariance which mean that a crystal can be
constructed by an infinite repetition or a unit cell along three independent directions, A surface
eliminates this invariance along one of the directions. It retains a symmetry with respect to twodimensional
translation along the surface itself.
Upon crystal cleavage the surface atomic layer, as well a few adjacent layers, have a different local
environment compared with the atomic layers deep inside the crystal. The forces from the atoms
in the crystal interior acting on the surface atoms are no longer compensated by the removed part
of the crystal. As a consequence, the net force applied to the surface atoms tends to rearrange their
equilibrium positions. The possible types of surface rearrangements fall into two main classes. A
modifications if the distances between the atomic planes near the surface is called relaxation. It
is not accompanied by changes in the topmost surface layer structure compared with the structure
in the bulk. Under a surface reconstruction, however, the surface atoms are shifted along the
surface with respect to their positions in the bulk. As a result, the surface unit cell differs from
the one which would exist if there were no displacements of the surface atoms.
Silicon, with a diamond-like FCC structure, usually cleaves along (111) and (100) planes. If the
surface is (100), the as-cleaved surface without lateral relaxation has a 2 × 2 square unit cell, and
each atom has two dangling bonds from diamond structures. After relaxation, two neighbouring
Si atoms reunite with two dangling bonds to form a new bond. Hence, the number of dangling
bonds is increased by a factor of 2, leading to a much lower surface energy. Due to the formation
of the new bond, two Si atoms form a pair. They are closer to each other than the column of Si
atoms on the left or the right side of the pair. Hence, the symmetry of the 2 × 2 unit cell in not
longer held. Instead, the lower energy surface plane has reconstructed itself into a 2×1 symmetry
with full and empty columns of atoms alternating with each other.
2.9 Preparation of Clean Surfaces
To prepare crystalline surfaces, usually the starting material is a suitable, pure, three-dimensional
single crystal. From this crystal, a slice of the desired orientation is cut. Therefore the crystal
must be oriented. Orientation is measured by X-ray diffraction. Hard materials are then grounded
and polished. Soft materials are cleaned chemically or electrochemically. The surfaces are still
mechanically stressed, contaminated, or chemically changed, e.g. oxidized. In principle, electrochemical
processes in liquid can be used to generate clean crystalline surfaces. The problem is
that an electrochemical setup is not compatible with an UHV environment. There are, however,
combined ultrahigh vacuum/electrochemistry instruments (UHV-EC), where the transfer of electrochemically
treated samples into the UHV chamber without contact to air, is possible. Still the
sample surfaces may undergo structural or compositional changes during the transfer. Usually in
situ methods of surface preparation are preferred that can be applied within the UHV chamber.
2.9. PREPARATION OF CLEAN SURFACES 47
2.9.1 Thermal desorption
Heating of the material may cause desorption of weakly bound species from the surface and can
therefore be used to clean surfaces. A positive side effect is that annealing reduces the number of
surface defects since it increases the diffusion rates of surface and bulk atoms. There can also be
some unwanted side effects: surface melting and other types of phase transitions may occur well
below the bulk melting point, leading to other than the desired surface structure
vdes = ns
1
τ0
exp(−
Qads
kT
), (2.28)
where ns is the surface concentration of the adsorbed species,τ0 is the oscillation, Qads is adsorption
energy and T is the temperature. For each system is possible to determine the temperature
necessary for a perfect cleaning of the surface from the adsorbed species. The thermal desorption
can be done:
1. current flow
2. radiation
3. electron bombardment
There are also some advantages of this method: the simplicity and the efficiency.
Some disadvantage are:
• high temperature required (≥ 1000 K),
• slow and prolonged heating can cause diffusion of impurities,
• sample stoichiometry can be altered,
• cannot remove particles (dirt).
2.9.2 Cleavage
Cleavage of bulk crystals to expose clean, defined lattice planes is possible for brittle substances.
Some materials, like mica or highly oriented pyrolytic graphite (HOPG) that exhibit a layered
structure, are readily cleaved by just peeling off some layers or using a razor blade. For other
materials the so-called "double-wedge technique " can be used. Cleavage is a quick way to produce
fresh surfaces from brittle or layered materials. There are, however, limitations: Cleavage
may produce metastable surface configurations that are different from the equilibrium structure.
Depending on the material, only certain cleavage planes can be realised. Crystals usually cleave
along non-polar faces, so that positive and negative charges compensate within the surface. GaAs,
for example, can only be cleaved along the non-polar 110 planes, whereas the polar 100 and 111
faces cannot be obtained.
48 CHAPTER 2. PHYSICS AT SURFACES
2.9.3 Sputtering
A universal method for cleaning surfaces is sputtering. In sputtering the surface is bombarded with
noble gas ions. Thereby contaminants – and in most cases the first few layers of the substrate
as well – are removed. In sputtering, an inert gas (usually argon) at a pressure of typically 1
Pa is inserted into a vacuum chamber. A high electric field is applied. In any gas and at any
given moment a certain number of atoms are ionized (although very few), for example by cosmic
background radiation. The electrons which are set free in the ionization process are accelerated
in the electric field. They hit and ionize further gas atoms. More electrons are liberated so that
eventually a plasma is created. The ions are accelerated by the applied voltage of some kV towards
the sample. After sputtering, annealing is often necessary to remove adsorbed or embedded noble
gas atoms and to heal defects of the crystal surface created by the bombardment.
Sputtering is an excellent and versatile cleaning technique for elemental materials. Care has to be
taken when applying it to composite materials like, e.g., alloys. Sputtering rates will depend on the
component and lead to changes in surface stoichiometry. In these cases, cleavage may be the better
choice. The necessary annealing process can also be critical and even induce surface contamination.
A prominent example is iron, that will usually contain some sulphur that segregates at the surface
during annealing. With such materials, consecutive cycles of annealing and sputtering may be
required to obtain a clean crystalline surface.
2.9.4 Desorption in strong electric field
The metallic samples in the positive pole. If a strong electric field (108
V/cm) is applied, the
valence electron level of the adsorbed atom is shifted just above the Fermi level. In this case, it is
possible to tunnel electrons into the bulk of the metal. The atom becomes a positive ion, which is
repelled by the electrostatic forces. In order to easily use this technique, the adsorbate should be
electropositive. If greater electric fields are provided, electronegative atoms can be removed from
the surface and also may as well pull out atoms from the sample.
2.9.5 Electron bombardment
If the surface is bombarded by low energy electrons (50-200 eV), the heating is negligible, but the
probability that adsorbed species would be excited and thus resulting in a weakly surface bound
is increased.
2.9.6 Laser beam
Irradiation with intense pulses of UV laser from an excimer laser can result in clean surfaces. The
sample is inserted in a UHV chamber and the laser beam is directed through a window. In this
case, the fast heating if the upper surface regions or the explosive evaporation of the uppermost
layer, which has been penetrated by the laser, causes the cleaning.
Advantages:
• vacuum conditions, no foreign particles
• only surface heating
2.9. PREPARATION OF CLEAN SURFACES 49
Disadvantages:
• ablation of material
• low area cleaned
• high cost
2.9.7 Surface reactions
The organic contaminants can be oxidized using oxygen, some impurities can be converted to
volatile compounds by heating in hydrogen atmosphere.
50 CHAPTER 2. PHYSICS AT SURFACES
Chapter 3
Interaction of Photons with Matter
3.1 Interaction of general radiation with matter
Interaction of radiation with the matter is necessary in order to characterize the microstructures:
• photons
• electrons
• neutrons
• protons
• ions/atoms
Penetration depth is a measure of how deep light or any electromagnetic radiation can penetrate
into a material. It is defined as the depth at which the intensity of the radiation inside the material
falls to 1/e of its original value at (or more properly, just beneath) the surface. The penetration
depth or mean free path of the incident beam determines the depth and volume of material that
will be sampled. In many cases, the material is irradiated with one type of radiation but a second
type is detected. This occurs in X-ray photoelectron spectroscopy (XPS) where the incident probe
is a beam of X-ray photons but emitted electrons are detected, whereas in energy dispersive X-ray
(EDX) analysis it is reversed. Generally the particle or radiation which has the shortest mean free
path in the material will determine the volume analyzed.
Whatever beam is selected we must be aware of
• its interaction with the material (loss processes), i. e. what radiation (photons, electrons etc.)
is ejected or other particles are ejected and
• and how the internally ejected radiation interact with the material.
Only in this way we can use the emitted signals to gain an understanding of the material being
examined.
Each radiation can cause a specific material damage and it has to be considered when selecting
the analytical methods.
51
52 CHAPTER 3. INTERACTION OF PHOTONS WITH MATTER
• photons
Regarded in general as the least harmful, the photons will damage the target nevertheless.
Damage can occur by heating the target, which depends on the radiant energy, photon flux
and the depth of penetration. Moreover X-rays can induce surface oxidation and lasers can
burn holes into material. In general the most photon radiation causes very little damage ⇒
it is used at the beginning of target characterization.
• electrons
Although electrons have a dual nature, their weight leads to a relatively large transfer angular
momentum if they are accelerated to several hundreds keVs. The resulted damage depends
on the heat transferred and the thermal conductivity of the material. For metals and alloys
the damage is minimum, but in polymers and oxides the damage is worse. It is possible to
cover the surface with a conductive material (usually gold) but the disadvantage is that the
composition can not longer be investigated.
• ions and atoms
When atoms and ions penetrate the target, they can react with it or not cause any damage
at all ( e.g. Ion Scattering Spectroscopy, where ions interact elastically with the target) or
they cause significant damage:
– ejection of atoms form their normal lattice position
– breaking the bonds (requires 10x more energy)
For low ion fluxes the damage areas are isolated (the amorphous material is surrounded by
intact one) but for high fluxes, the ions create an amorphous layer.
Mean free path determines the depth resolution, which has influence in the spatial one (defined
as the perpendicular to the direction of the incident beam). Generally, the spatial resolution is
influenced by the beam size and the wavelength. The image can be obtain by:
• illumination of the sample and use of lens for focusing the reflected or emitted radiation.
The spatial resolution depends on the lens system used and the wavelength of emitted or
reflected radiation. Optical, X-ray and some ion microscopes are using this method.
• a narrow beam is directed on the sample and absorbed or reflected radiation is detected. The
sample surface is scanned by a narrow beam. In this case the spatial resolution depends in
the wavelength, beam size and the scattered radiation in the sample. Most of the equipment
are using electrons and ions.
Nowadays, laser microscopes are used for sample characterization.
3.2 Introduction to photon interactions
Photons are discrete quanta of electromagnetic radiation. The photon is identified by the wavelength
λ, energy E, and frequency ν as
hν = E = hc/λ (3.1)
3.2. INTRODUCTION TO PHOTON INTERACTIONS 53
where h is the Planck constant and c the velocity if light.
The electromagnetic spectrum spans a vast range with wavelengths varying from 106
m down
to 10−14
m (see Figure 3.1). If we are to use electromagnetic radiation for microstructural characterisation
of materials a photon wavelength is needed that is of comparable size to the features
being studied ⇒ between 10−4
–10−10
m.
The long wavelength infrared radiation is used to characterise materials by determining how
specific wavelengths are absorbed, visible light is used in a variety of instruments mainly to obtain
a visual image of the surface while at the shorter wavelength ultraviolet radiation is often used to
obtain information concerning the electron distribution in the surface atoms. After visible light,
X-rays are probably the most utilized photon source for investigation of material microstructure.
When electromagnetic radiation is incident on the surface of a material, it may be (partly)
reflected from that surface and there will be a field containing energy transmitted into the material.
This electromagnetic field interacts with the atoms and electrons inside the material. Depending
on the nature of the material, the electromagnetic field might travel very far into the material, or
may die out very quickly. For a given material, penetration depth will generally be a function of
wavelength.
• IR radiation: characterization of materials, which is depending of the absorbed wavelengths;
• VIS radiation: used often to see the surface. Some material are opaque while others are
transparent. Even for high reflectivity opaque materials the penetration depth is in order of
Figure 3.1: The electromagnetic spectrum illustrating the relationship between wavelength, frequency
and energy of photons.
54 CHAPTER 3. INTERACTION OF PHOTONS WITH MATTER
50–300 nm.
• UV radiation: reveals information about the electron distribution for surface atoms. Most
substances absorb in this region.
• X-ray radiation: X-ray penetration is easier to predict than visible radiation. The penetration
depth is typically µm. The intensity of radiation which is transmitted through sample
thickness d is
I = I0 exp(−αd), (3.2)
where I0 is the intensity of incident radiation and α, the absorbance, increases with the
atomic number.
• Gamma (γ) radiation: has energy between 50 keV–50 MeV ≈ 10−2
nm. Absorbance is
governed by the same relation for X-rays, but here α is inversely proportional with the
atomic number. Gamma radiation penetrate almost all the laboratory samples.
3.3 Very short wavelengths (< 10−12
m, γ-rays)
The method that uses γ-rays to characterize materials is called Mössbauer spectroscopy.
Key to the success of the technique is the discovery of recoilless γ-ray emission and absorption,
now referred to as the Mössbauer effect, after its discoverer Rudolph Mössbauer, who first observed
the effect in 1957 and received the Nobel Prize in Physics in 1961 for his work.
3.3.1 Mössbauer effect
Nuclei in atoms undergo a variety of energy level transitions, often associated with the emission or
absorption of a gamma ray. These energy levels are influenced by their surrounding environment,
both electronic and magnetic, which can change or split these energy levels. These changes in the
energy levels can provide information about the atom’s local environment within a system and
ought to be observed using resonance-fluorescence. There are, however, two major obstacles in
obtaining this information:
• the ’hyperfine’ interactions between the nucleus and its environment are extremely smaller,
• the recoil of the nucleus as the gamma-ray is emitted or absorbed prevents resonance.
Just as a gun recoils when a bullet is fired, conservation of momentum requires a free nucleus
(such as in a gas) to recoil during emission or absorption of a gamma ray.
• If a nucleus at rest emits a gamma ray, the energy of the gamma ray is slightly less than the
natural energy of the transition (difference is equal to the recoil energy ER),
• but in order for a nucleus at rest to absorb a gamma ray, the gamma ray’s energy must be
slightly greater than the natural energy.
This means that nuclear resonance (emission and absorption of the same gamma ray by identical
nuclei) is unobservable with free nuclei, because the shift in energy is too great and the
emission and absorption spectra have no significant overlap.
3.3. VERY SHORT WAVELENGTHS (< 10−12
M, γ-RAYS) 55
Figure 3.2: Recoil of free nuclei in emission or absorption of a gamma-rays
Figure 3.3: Resonant overlap in free atoms. The overlap shown shaded is greatly exaggerated.
56 CHAPTER 3. INTERACTION OF PHOTONS WITH MATTER
In order to achieve resonance in emission and absorption the loss of the recoil energy must be
overcome in some way.
Nuclei in a solid crystal are not free to recoil because they are bound in place in the crystal
lattice. When a nucleus in a solid emits or absorbs a gamma ray, some energy can still be lost
as recoil energy, but in this case it always occurs in discrete packets called phonons (quantized
vibrations of the crystal lattice). Any whole number of phonons can be emitted, including zero,
which is known as a "recoil-free" event. In this case conservation of momentum is satisfied by the
momentum of the crystal as a whole, so practically no energy is lost.
Mössbauer found that a significant fraction of emission and absorption events will be recoilfree,
which is quantified using the Lamb–Mössbauer factor. This fact is what makes Mössbauer
spectroscopy possible, because it means gamma rays emitted by one nucleus can be resonantly
absorbed by a sample containing nuclei of the same isotope, and this absorption can be measured.
Figure 3.4: Atoms fixed in lattice.
3.3. VERY SHORT WAVELENGTHS (< 10−12
M, γ-RAYS) 57
In its most common form, Mössbauer absorption spectroscopy,
• a solid sample is exposed to a beam of gamma radiation,
• and a detector measures the intensity of the beam transmitted through the sample.
The atoms in the source emitting the gamma rays must be of the same isotope as the atoms in
the sample absorbing them.
If the emitting and absorbing nuclei were in identical chemical environments, the nuclear
transition energies would be exactly equal and resonant absorption would be observed with both
materials at rest. The difference in chemical environments, however, causes the nuclear energy
levels to shift in a few different ways. Although these energy shifts are tiny (often less than a microelectronvolt),
the extremely narrow spectral linewidths of gamma rays for some radionuclides make
the small energy shifts correspond to large changes in absorbance. To bring the two nuclei back
into resonance it is necessary to change the energy of the gamma ray slightly, and in practice this
is always done using the Doppler effect.
During Mössbauer absorption spectroscopy, the source is accelerated through a range of velocities
using a linear motor to produce a Doppler effect and scan the gamma ray energy through a
given range. A typical range of velocities for 57Fe, for example, may be ±11 mm/s (1 mm/s =
48.075 neV).
In the resulting spectra, gamma ray intensity is plotted as a function of the source velocity.
At velocities corresponding to the resonant energy levels of the sample, a fraction of the gamma
rays are absorbed, resulting in a drop in the measured intensity and a corresponding dip in the
spectrum. The number, positions, and intensities of the dips (also called peaks; dips in transmitted
intensity are peaks in absorbance) provide information about the chemical environment of the
absorbing nuclei and can be used to characterize the sample.
Mössbauer spectroscopy is limited by the need for a suitable gamma-ray source. Usually, this
consists of a radioactive parent that decays to the desired isotope. For example,
• the source for 57Fe consists of 57Co, which decays by electron capture to an excited state
of 57Fe,
• then subsequently decays to a ground state emitting the desired gamma-ray.
• The radioactive cobalt is prepared on a foil, often of rhodium.
Ideally the parent isotope will have a sufficiently long half-life to remain useful, but will also have
a sufficient decay rate to supply the required intensity of radiation.
Mössbauer spectroscopy has an extremely fine energy resolution and can detect even subtle
changes in the nuclear environment of the relevant atoms. Typically, there are three types of
nuclear interactions that are observed
• isomer shift (or chemical shift) - is a relative measure describing a shift in the resonance
energy of a nucleus due to the transition of electrons within its s orbital. The whole spectrum
is shifted in either a positive or negative direction depending upon the s electron charge
density. This change arises due to alterations in the electrostatic response between the
non-zero probability s orbital electrons and the non-zero volume nucleus they orbit.
58 CHAPTER 3. INTERACTION OF PHOTONS WITH MATTER
• quadrupole splitting - reflects the interaction between the nuclear energy levels and surrounding
electric field gradient (EFG). Nuclei in states with non-spherical charge distributions,
i.e. all those with angular quantum number (I) greater than 1/2, produce an asymmetrical
electric field which splits the nuclear energy levels. This produces a nuclear quadrupole
moment.
Figure 3.5: Chemical shift and quadrupole splitting of the nuclear energy levels and corresponding
Mössbauer spectra
3.3. VERY SHORT WAVELENGTHS (< 10−12
M, γ-RAYS) 59
• hyperfine splitting (or Zeeman splitting) - is a result of the interaction between the nucleus
any surrounding magnetic field. A nucleus with spin, I, splits into 2I + 1 sub-energy levels
in the presence of magnetic field. For example, a nucleus with spin state I= 3/2 will split
into 4 non-degenerate sub-states with mI values of +3/2, +1/2, -1/2 and −3/2. Each split is
hyperfine, being in the order of 10−7
eV. The restriction rule of magnetic dipoles means that
transitions between the excited state and ground state can only occur where mI changes by
0 or 1. This gives six possible transitions for a 3/2 to 1/2 transition.
Figure 3.6: Magnetic splitting of the nuclear energy levels and the corresponding Mössbauer
spectrum
60 CHAPTER 3. INTERACTION OF PHOTONS WITH MATTER
3.4 Short wavelengths (10−12
to 10−9
m, including X-rays and
ultraviolet)
If photon has enough energy to excite electron to high energy levels, two effects can occur:
• internal photoeffect (e.g. photoconductivity)
• external photoeffect or the photoemission.
The kinetic energy is:
1
2
mv2
= EF ± δE − Eaf + hν − ∆E. (3.3)
At 0 K the kinetic energy is maximum
(
1
2
mv2
)max = hν − (Eaf − EF) = hν − χ = h(ν − ν0), (3.4)
where ν0 is the frequency of photoelectric effect.
In the case of photoemission induced absorption (X-ray), the photon has enough energy to
excite an electron from some inner layers of solid.
1
2
mv2
= hν − χ − EB, (3.5)
where EB is the binding energy of the inner layers. The binding energy is characteristic for each
substance, so the kinetic electrons emitted bear information related to the electronic structure
and thus, the type of material.
3.4.1 Fowler‘s theory for metal/vacuum interfaces
Fowler proposed a theory in 1931 which showed that the photoelectric current variation with the
light frequency could be accounted for by the effect of temperature in the number of electrons
available for emission, in accordance with the distribution law of Sommerfeld‘s theory for metals.
Sommerfeld‘s theory had resolved some of the problems surrounding the original models for
electron in metals. In classical Drude theory, a metal had been envisaged as a three-dimensional
potential well (box) containing a gas of freely mobile electrons. This adequately explained their
high electrical and thermal conductivities. However, because experimentally it was found that
metallic electrons do not show a gas-like heat capacity, the Boltzman distribution law is inappropriate.
A Fermi-Dirac distribution function is required, consistent with the need that the electrons
obey the Pauli exclusion principle and this distribution function has the form
P(E) =
1
1 + exp[(E − EF )/kT]
(3.6)
At ordinary temperatures, the Fermi energy EF kT. Fowler‘s law, referred to above as the
"square law", is readily tested in practice since it predicts at T=0 K
i ∝ (ν − ν0)2
(3.7)
3.4. SHORT WAVELENGTHS (10−12
TO 10−9
M, INCLUDING X-RAYS AND ULTRAVIOLET)61
where ν0 is the photoelectric threshold frequency and ν refers to a range of energies from threshold
to a few kT above threshold. To a good approximation, the photoelectric yield per unit light
intensity near ν0 is proportional simply to the number of electrons incident per unit time. The
energy condition can be written
(1/2m)p2
+ hν > φ (3.8)
where p is the initial momentum of the electron normal to the surface. The escaping electrons
have been termed the "available electrons".
Fowler‘s derivation for the single photon does not explicitly involve the quantum mechanical form
of current; instead, a semi-classical flux of electrons arriving at the metal surface is used. The
electron gas in the metal will obey Fermi-Dirac statistics, and the number of electrons per unit
volume having velocity components in the ranges u, u + du, ν, ν + dν and w + dw is given by the
formula
n(u, ν, w)dudνdw = 2(
m
h
)2 dudνdw
1 + exp[1
2
m(u2 + ν2 + w2) − EF ]/kT
(3.9)
where u is the velocity normal to the surface and m is the electron mass. The number of electrons
per volume, n(u)du, with their velocity component normal to the surface in the range u, u + du is
given by:
n(u)du =
4πkT
m
(
m
h
)3
log[1 + exp(EF −
1
2
mu2
)/kT]du (3.10)
The solution for the photoelectric current resulted in the form
i ∝
(hν − φ)3/2
(φ0 − hν)1/2
. (3.11)
A method for determining the threshold frequency of the illuminated surface ν0 is that which
makes use of the complete photoelectric emission. When a metal surface is exposed to the total
thermal radiation from a black body with a temperature T, the total photoelectric current is given
by:
i = A‘T2
e−
hν0
kT (3.12)
were A‘ is a constant.
3.4.2 Measurement of the photoelectric work function
• Fowler isotherm: measurement of the photocurrent depending on the frequency of incident
radiation at constant temperature, the ln if/T2
is expressed as hν/kT and compared with
the theoretical curve
ln
if
T2
= ln(αA0) + ln f(x) = B + Φ[
h
kT
(ν − ν0)]. (3.13)
From the Fowler equation, giving the relation between temperature and photoelectric emission,
ν0 can be determinate by plotting the ln f(∆) as a function of ∆. A universal curve is
obtained. Then ln i
T2 is plotted against hν
kT
. A curve us obtained of the same shape as the
universal curve, which can be made to coincide with it by a parallel shift. The vertical shift
is measure of B, the horizontal is equal to hν0
kT
.
62 CHAPTER 3. INTERACTION OF PHOTONS WITH MATTER
• duBridge isochromatic method: measurements at one frequency ν but different temperatures.
For this purpose ln f(∆) is plotted against ln |∆|, and again a universal curve is
obtained. From the experimental data ln if/T2
is plotted against ln 1/T. For different frequencies
the temperature dependence of the photoemission differs
• for ν = ν0 is if = αA0 = π2
/12T2
• for x 1; ν > ν0, is f(x) = π2
/6 + x2
/2 and
if =
αA0
2
h2
(ν − ν0)2
k2
+
π2
3
, (3.14)
where the parabolic dependence is applied obly when the first term in the brackets is sufficiently
small.
• for x 1, ν < ν0, is f(x) = ex
and
if = αA0T2
exp(−
h(ν0 − ν)
kT
). (3.15)
3.4.3 Interaction of excited electrons with matter
The hot-electron problem represents the study of the deviations from the linear response regime
due to heating of charge carriers above the thermal equilibrium. Studies have been carried out in
1930s, mainly by Russian physicist Landau and Davidov. This means that electron temperature
Te which , in the presence of an external high electric field, is higher that the lattice temperature
T.
Nonlinear transport deals with problems that arise when a sufficiently strong electric field is
applied to a semiconductor sample, so that the current deviates from the linear response. This
effect is typical of semiconductors and cannot be seen on metals since, owing to their large
conductivities, Joule heating would destroy the material before deviations from the linearity
could be observed.
Considering
v(i)
(t) = v
(i)
0 + a∆t(i)
(3.16)
where v(i)
(t) is the instantaneous velocity if the i-th electron at time t, ∆t(i)
is the same time elapsed
after its last scattering event, and a = eE/m is the electron acceleration, due to a constant and
uniform applied electric field E. The region of field strengths where transport starts to deviate
from linearity is called warm-electron region.
3.4.4 Photoemission from semiconductors
There are some differences when we speak about the photoemission from metals and semiconductors.
All high quantum yield photocathodes are semiconductor ones. The absorption of light is
affected by:
• internal absorption due to their band gap
3.5. INTERMEDIATE WAVELENGTHS (INCLUDING VISIBLE AND ULTRAVIOLET LIGHT)63
• external absorption from the external layer
The Fermi level is influenced by the surface layers and the type of semiconductor used. Electron
coming from different depths have different space charge and work function ⇒ as they would have
different work function for different photoelectrons excited at different wavelengths.
The density of stated around Fermi level increases strongly and we cannot assume that photon
absorption is independent from electron‘s energy state. The quantum yield increases with the
temperature faster than in metals.
3.4.5 Photocathode
• IR region: Ag-O-Cs, also called S-1. This was the first compound photocathode material,
developed in 1929. Sensitivity from 300 nm to 1200 nm. Since Ag-O-Cs has a higher dark
current than more modern materials photomultiplier tubes with this photocathode material
are nowadays used only in the infrared region with cooling.
• visible region: Sb-Cs has a spectral response from UV to visible and is mainly used in
reflection-mode photocathodes. and also bi-alkali (antimony-rubidium-caesium Sb-Rb-Cs,
antimony-potassium-caesium Sb-K-Cs). Spectral response range similar to the Sb-Cs photocathode,
but with higher sensitivity and lower dark current than Sb-Cs. They have sensitivity
well matched to the most common scintillator materials and so are frequently used
for ionizing radiation measurement in scintillation counters.
• UV region: metals and Cs-Te (Cs-I). These materials are sensitive to vacuum UV and UV
rays but not to visible light and are therefore referred to as solar blind. Cs-Te is insensitive
to wavelengths longer than 320 nm, and Cs-I to those longer than 200 nm.
3.5 Intermediate wavelengths (including visible and ultraviolet
light)
• Absorption/emission by isolated atoms - spectroscopy of gaseous samples (ICP used for solid
materials)
• Raman and Rayleigh scattering
3.6 Long wavelengths (infrared)
3.7 Very long wavelengths photons (≥ 1mm) - including radio
and microwaves
All nucleons, that is neutrons and protons, composing any atomic nucleus, have the intrinsic
quantum property of spin. The overall spin of the nucleus is determined by the spin quantum
64 CHAPTER 3. INTERACTION OF PHOTONS WITH MATTER
Figure 3.7: Atomic emission spectroscopy in inductively coupled discharge
Figure 3.8: AES-ICP with laser ablation
3.7. VERY LONG WAVELENGTHS PHOTONS (≥ 1MM) - INCLUDING RADIO AND MICROWAVES65
Figure 3.9: Demonstration of different energy levels in molecules.
Figure 3.10: Different spectroscopies involving vibrational(-rotational) states.
66 CHAPTER 3. INTERACTION OF PHOTONS WITH MATTER
number I. Some nuclei have integral spins (e. g. I = 1, 2, 3 . . .), some have fractional spins (e.g.
I = 1/2, 3/2, 5/2....), and a few have no spin, I = 0 (e.g. 12C, 16O, 32S, ....). Isotopes of particular
interest and use to organic chemists are 1H, 13C, 19F and 31P, all of which have I = 1/2.
A non-zero spin is always associated with a non-zero magnetic moment µ
µ = γS (3.17)
where γ is the gyromagnetic ratio.
It is this magnetic moment that allows the observation of nuclear magnetic resonance (NMR)
absorption spectra caused by transitions between nuclear spin levels in the presence of external
magnetic field (spin states splitting).
http://www2.chemistry.msu.edu/faculty/reusch/VirtTxtJml/Spectrpy/nmr/nmr1.htm
Electron spin resonance (ESR) or electron paramagnetic resonance (EPR) is a related technique
in which transitions between electronic spin levels are detected rather than nuclear ones. The basic
principles are similar but the instrumentation, data analysis, and detailed theory are significantly
different. Moreover, there is a much smaller number of molecules and materials with unpaired
electron spins that exhibit ESR absorption than those that have NMR absorption spectra. ESR
has much higher sensitivity than NMR does.
http://www.uottawa.ca/publications/interscientia/inter.2/spin.html
https://en.wikipedia.org/wiki/Electron_paramagnetic_resonance
3.7. VERY LONG WAVELENGTHS PHOTONS (≥ 1MM) - INCLUDING RADIO AND MICROWAVES67
Figure 3.11: Position of peaks in Raman scattering.
68 CHAPTER 3. INTERACTION OF PHOTONS WITH MATTER
Chapter 4
Interaction of Electrons with Matter
4.1 Electron interactions
• desorption of neutral particles
• excitation of photons (x-ray, chatodoluminescence)
• reality of material: cleaning, activation energy for chemical and physical processes, electron
lithography
• change of material
• emission of electrons: (a) without interaction, (b) elastic scattering (diffraction on passage),
(c) elastic reflection (diffraction on reflection), (d) inelastic reflection (characteristic energy
loss), (e) backscatter electron, (f) electron transmission with the possibility to emit others
and (g) electron transmission with secondary electron emission.
When a sample is bombarded with charged particles, the strongest region of the electron
energy spectrum is due to secondary electrons. The secondary electron yield depends on
many factors, and is generally higher for high atomic number targets, and at higher angles
of incidence. There is a lot of information in this secondary electron "background", but,
unlike Auger and other electron spectroscopies, it is not directly chemical or surface specific
in general.
Total electron current from a substance is the current of secondary electrons on reflection
or transmission.
4.2 Penetration depth of electrons
Penetration depth changes significantly with the electron energy and the atomic number of the
material:
• for stainless steel: around one tenth of a µm for energy 10 keV and 2 µm for 30 keV,
69
70 CHAPTER 4. INTERACTION OF ELECTRONS WITH MATTER
Figure 4.1: Electron interaction with the solid state.
• for energies higher than 10 keV: elements with atomic number under 20 have a penetration
depth up to 10 µm while the elements with Z higher than 40 below 2 µm
• for energies between 0–2 keV: the penetration depth is reduces to 0.4–300 nm
4.2.1 Electron emission
The ration between this current Is and primary Ip is called secondary emission coefficient
σ = Is/Ip. By selecting the secondary electrons by their energies we can identify individual
interactions:
• the highest maximum of primary energy level ⇒ elastic reflection of e−
interaction with the atom like a whole, there is no change in energy but, the electron momentum
can be changed dramatically. For electron energy higher than (≥ 5 keV)the effect
is called Rutherford Backscattering (RBS)
dQ(θ)
dΩ
=
1
4
e2
(Z − F(θ))
4π 0 mred g2
2
1
sin4
(θ/2)
, (4.1)
where F(θ) is the screening effect of electrons. F(θ) is insignificant at high angles θ and
strong scattering can occur only in heavy atoms (used for TEM contrast). When the target
is a crystalline material diffraction and channelling occurs. When electron scattering
from crystal lattice is coherent, amplification in those direction occurs, which is represents
Bragg law:
nλ = 2dhkl sin θ, (4.2)
4.2. PENETRATION DEPTH OF ELECTRONS 71
Figure 4.2: Penetration depth for different particles.
where n is a integer dhkl is is the spacing between the planes in the atomic lattice expressed
with Miller indexes. This method is used to examine the bulk crystalline structure (E ≈
10 keV) or the surface structure using low-energy electron diffraction (LEED) or very small
angle relection in reflection high-energy electron diffraction (RHEED).
• the smaller peaks to the left show the shift in maximum elastic reflection of e−
⇒ electrons
which suffer energy losses.
Electrons undergo a variety of interactions:
• adsorbed molecules are excited on vibrational state ∆E ≈ 50–500 meV;
• individual interaction with valence e−
, ∆E ≈ 3–20 eV but without a sharp peak because the
valence band has a width of several eV;
• multiple interactions with the valence e−
which is bombarded by an electron, vibrating with
frequency
ωp =
1
0
n0e2
m
(4.3)
for plasmon in volume, or
ωs =
ωp
√
1 + r
=
ω
√
2
(4.4)
for excitation of the surface plasmon. ∆E is between 5 and 60 eV.
Inelastic interactions include phonon excitations, inter and intra band transitions, plasmon
excitations, inner shell ionizations are studied by electron energy loss spectroscopy (EELS).
72 CHAPTER 4. INTERACTION OF ELECTRONS WITH MATTER
• smaller peaks which are not shifted by energy ⇒ Auger e−
which have for each substance
a characteristic position.
Figure 4.3: Electron energy distribution.
Primary e−
removes a core e−
(e.g. from K shell). The core state electron is removed
leaving behind a hole. As this is an unstable state, the core hole can be filled by an outer shell
electron, whereby the electron moving to the lower energy level loses an amount of energy equal
to the difference in orbital energies. The transition energy can be coupled to a second outer shell
electron which will be emitted from the atom if the transferred energy is greater than the orbital
binding energy. The Auger energy released
Ek = EK − EL − EV − Eaf. (4.5)
The EV is not equal with the original level energy EV, because the e−
on the level V moves in a
potential and the e−
on the level L is missing. The Auger requires the existence of at least two
energy levels to three electrons. That is why it can not measure H and He. The elements with
atomic number between 3 and 14 have KLL line, elements between 14 and 40 have LMM line,
between 40 and 79 are MNN line and for heavier elements NOO line exists.
4.3 Electron spectroscopies
• UPS - Ultraviolet Photoelectron Spectroscopy, VUV radiation, especially using synchrotron
radiation;
• XPS - X-ray Photoelectron Spectroscopy (ESCA - Electron Spectroscopy for Chemical Analysis),
soft X-rays;
4.3. ELECTRON SPECTROSCOPIES 73
Figure 4.4: Overwiev of measuring methods using emitted electrons.
• AES - Auger Electron Spectroscopy
• ELS or EELS - Electron Energy Loss Spectroscopy, used in the case of internal excitation
of electron levels, also known as IS - Ionization Spectroscopy;
• INS - Ion Neutralization Spectroscopy
• FES - Field Electron Spectroscopy, energy distribution measurements of field-emitted elec-
trons
Figure 4.5: Overwiev of different dispersive electron spectroscopy methods
74 CHAPTER 4. INTERACTION OF ELECTRONS WITH MATTER
• XXAPS, XEAPS - X-ray X-ray Appearance Potential Spectroscopy, X-ray Electron Appearance
Potential Spectroscopy. Methods in which X-rays are generated from the sample. They
are practically not used since electron generation is much easier.
• SXAPS - Soft X-ray APS, electrons impact the sample and emitted X-rays are detected.
• DAPS - Disappearance Potential Spectroscopy, used to measure the surface electrons current.
• AEAPS - Auger Electron APS, records the secondary electrons current.
Figure 4.6: Overwiev of different APS methods
4.4 Electron spectroscopy conditions
All methods except FES require a monochromatic source - more or less focused, a analyser
and a detector (electron multiplier). All this system must be kept under high vacuum conditions
(≤ 10−7
Pa). The surface sensitivity is greater so the depth profile is small. The most stringent
conditions for surface are for UPS and not so for XPS. The device is equipped with a surface
cleaning systems (Ar+
sputtering and baking) or for preparation of clean surfaces in situ (cleaving
or applying of layers).
4.5 Sources of primary particles
X-rays: soft X-rays are generated by bombardment of a suitable material (Al,Mg, Mo, Co, W)
with sufficient e−
energy ⇒ characteristic X-rays.
• when beam width may limit the resolution (Ek = hν − EB − eφ). The best e−
energy is
≈ 1.0 eV ⇒ characteristic X-ray width s narrower.
• The energy must be high enough in order to obtain emission of photoelectrons from the bulk
of material.
4.5. SOURCES OF PRIMARY PARTICLES 75
Figure 4.7: Schematic drawing of method’s differences.
• In addition, the anode material needs to be capable of working even under e−
bombardment.
⇒ Mg Kα (E = 1253.6 eV, δE = 0.7 eV), Al Kα (E = 1486.6 eV, δE = 0.85 eV).
Photoemission starts at e−
energies equal with electron ionization threshold for the inner shell and
growing rapidly with increasing energy. Therefore, devices allow e−
acceleration up to 15 keV.
The e−
source is the closest to the anode, which is often composed of two anodes. For
continuous operation at 1–8 kW, a rotating anode is required. Beside the characteristic lines
(Kα for Mg and Al), satellite, Kβ and continuous spectrum (Bremsstrahlung) lines exists ⇒
sometime monochromatic radiation is used: Bragg diffraction from crystal.Rowland focusing is
approximated by aligning the platform of the multi-crystal segmented analyser at a tangent to
the Rowland circle (0.5 m).
Synchrotron radiation if the charged particles are accelerated, they radiate energy which
covers an area of electromagnetic radiation from hard X-rays up to IR. The synchrotron is a
particular type of cyclic particle accelerator in which the magnetic field (to turn the particles so
they circulate) and the electric field (to accelerate the particles) are carefully synchronised with
the travelling particle beam. Synchrotron radiation has many advantages: a broad spectrum,
natural collimation, high intensity, coherence, the source is under vacuum. Maximum intensity
is at the critical wavelength (0,1–0,4 nm), which is the curving path of the e−
. The intensity
decreases rapidly for lower wavelengths.
Sources of electrons thermionic or field-emission guns. AES monochromatic e−
source. Only
the electrons with sufficient energy are able to ionize the inner layer. Beam size is important
for EELS max. 0,3–0,5 eV, which is to large for HREELS which requires special monochromatic
76 CHAPTER 4. INTERACTION OF ELECTRONS WITH MATTER
Figure 4.8: Scheme of X - ray source.
electron sources.
4.6 Electron energy analysers
The electron spectroscopy uses electrostatic analysers. The measured values can be divided into
• analysers with retarding field - recording of the total current of particles,
• dispersion analysers - select only certain e−
energies by changing the trajectories.
When choosing the analyser some parameters are assessed:
• resolution R = E/∆E - indicates the energy;
• intensity (brightness) -affects the intensity of the output signal;
• analyser size - needs to be under vacuum;
• complexity - economic factor
The final solution is a compromise because the individual requirements have a contradictory
character.
4.6. ELECTRON ENERGY ANALYSERS 77
4.6.1 Analysers with retarding field
The operation principle is based on potential barrier high Vb. The flow of particles over the barrier
is devided in fast and slow components. Only the faster ones are usually recorded:
J(E) =
∞
E
N(E )dE , (4.6)
where N(E ) is the energy distribution of particles, E = eVb is the lower energy limit and the
symbol ∞ upper limit (maximum energy of particles). The energy distribution function can be
obtained from the derivative of the above equation.
Planar analyser
The analysed beam, which is crucial for the accuracy of measurement is usually selected by
modifying the slit. Around the threshold energy E = eVb, the resolution is:
Rr0 =
16d2
r2
0
. (4.7)
The basic parameter is the diameter of the solid angle Ω, under which the beam enters the
detector (effective area). Impute increases with the slit and Ω.
Figure 4.9: Scheme of planar analyser
Spherical analysers
Compact electrode system often use a system of 3 or 4 grids (collectors). Total energy
analysis is carried out in a radial trajectory of the particles. It can be used for LEED if the
system in equipped with a fluorescent screen or a small movable detector. Retarding fields are
generated between the 1st and 2nd (or 3rd and 4th) grids. The resolution is given by the field
inhomogeneity(mesh size, number of grids). Analyser throughput is high because it can detect
78 CHAPTER 4. INTERACTION OF ELECTRONS WITH MATTER
Figure 4.10: Scheme of spherical analyser
particles from a large solid angles (120◦
or 180◦
).
Dispersion analyser (deflectors)
In this case are detected either the coordinates of the particle impact or the energy E0. Analysed
properties reveal the dispersion D(E) = E ∂x/∂E, which along the field focus properties
determines the resolution.In all cases of analyser type, the energy determination is inaccurate due
to overlapping trajectories of different energy E0 and incorrect angle input. It is used the following
approximation:
R
.
=
2D
Ms1 + s2 + Σ
, (4.8)
where s1 and s2 are the slit sizes for input and output, M is the magnification (∼1) and Σ is the
blur due to the input divergence. From this expression is observed the decrease of R with the
increase of slit size directly (Ms1 + s2), and indirectly with Σ. Dispersion analysers are divided
into:
• mirror analyser: the beam enters through field lines at an different angle than π/2; the
velocity component perpendicular to the field direction is maintained, v is gradually changed
to the opposite direction ⇒ symmetrical axis trajectory to the field. The input and output
slits are places in the positive potential electrode. The entrance angle is chosen with regard
to the position of the source and the detector.
– planar - two parallel electrodes focusing only the deflection plane a. Two arrangements
(a) φ0 = π/4, source and detector in the entrance and exit slits, (b) φ0 = 30◦
, source
and detector tangent to field.
– cylindrical (CMA) - it is frequently used because allows the construction of high resolution
systems. It consists of to concentric cylinders, with a positive potential between.
4.6. ELECTRON ENERGY ANALYSERS 79
Figure 4.11: Scheme of dispersion analyser
• sectors: the beam enters ⊥ to the field. The main trajectory is a circle, where the centrifugal
force associated with the kinetic energy compensates the centripetal force. The incident and
reflected angles is determined by the focus and position of the source and detector. It is
mainly used in the differentiated angular spectroscopy:
– cylindrical - two cylindrical sector analyser and to pairs of planar electrodes (in front
and lateral); in the frontal one are located the slits. Deflection potential V1 − V2 is
inserted between the cylinders (inner potential is positive), both planar electrodes are
isolated and maintained on the potential V0 = (V1 + V2)/2. The analyser is focusing
only in one direction. Sector angle is Φ = π/
√
2 = 127, 3◦
therefore ⇒ the name of
127◦
analyser.
– spherical (CHA - concentric hemispherical analyser) - two concentric spherical segments;
input and output electrodes are radial slices are maintained at median surface
potential V0. The spherical deflector is capable of spatial focusing. Angle or the sector
is usually Φ = 180◦
.
Conclusion
Analysing the energy of electrons has some other associated phenomena such as reflection and
secondary emission. It is advantageous to use electron optics in order to increase the input angle,
to obtain the image in a suitable place in the optimal regime area of the analyser, finely tune
the input angle for slow electrons. The relative resolution of the analyser is constant ⇒ absolute
intensity will decrease with the decrease of energy.
80 CHAPTER 4. INTERACTION OF ELECTRONS WITH MATTER
4.7 Ultraviolet photoelectron spectroscopy
• if the photon energy is < 3 eV (VIS radiation), photoemission can occur only for substances
with very low φ and the energy spectrum is narrow because e−
are emitted only from the
highest occupied level ⇒ information only about φ.
• if the photon energy is higher (UV radiation) electrons from the valence band can be emitted
(UPS) ⇒ distribution of densities of states in valence band can be obtained.
• in addition to energy distribution is possible to measure the angular distribution (ARUPS).
In an isolated atom, the electrons occupy certain energy levels. In a solid, the levels involved
in bonding ’the valence states’, become blurred, and electrons can occupy a range of energies
called bands. The energy of an electron in the solid depends on its momentum. Hence, by
detecting photoelectrons emitted from a surface at different emission angles, the energy of
the electrons as a function of the momentum vector may be determined. This process is
known as "band mapping" and is a powerful probe of the electronic structure of crystalline
materials.
• The mean free path in the case of UPS is only 0,5–2 nm ⇒ surface characterization method.
Several modes exist for the evaluation of the emitted electrons:
• for a fixed position of source, surface elementary cell and the photos energy is measured by
the kinetic energy of electrons EDC (energy distribution curve)by integrating or differentiating
the angle.
• for constant photon energy and the electrons change the position in the detector ADC (angle
distribution curve).
• for fixed geometries and fine tuned energies of photons - the kinetic energy detected is
constant CIS (constant-initial-state spectra).
• for fixed geometries and constant kinetic energy CFS (constant-final-state spectra).
Photoemission
The first paper of Burglund and Spicer (1964) worked out what is called the three-step model for
photoemission, and the second applied it to new data on Cu and Ag. The three-step model uses
this picture in considerable detail to predict the spectrum of photoelectrons from a given band
structure. This spectrum sometimes resembles the density of occupied electronic states weighted
by a function that varies smoothly with energy. This model assumes the photoemission process
can be described by three sequential processes:
• photoexcitation of an electron (photoabsorbtion) - general theory based on Fermi‘s golden
rule, which determines the transition probability between two electron states of the same
Hamiltonian HN
, if the disturbance ∆ is small.
w =
2π
¯h
| < ψN
f |∆|ψN
i >2
δ(EN
f − EN
i − ¯hω) (4.9)
4.7. ULTRAVIOLET PHOTOELECTRON SPECTROSCOPY 81
• transport of that electron to the surface without inelastic scattering
• scape of the electron into the vacuum
The law of energy conservation must be applied (Ekin = Eu
f − eΦ = Eu
i + ¯hω − eΦ) and the
wavelength vector. Overall, the wave-vector is not maintained because the emitted electron escapes
into vacuum, and the system achieve three-dimensional symmetry. The surface has only twodimensional
translation symmetry of the surface reciprocal lattice vector GS:
kvn
= ku
f + GS (4.10)
Inside the crystal the wave-vector is preserved:
ku
f = ku
i + GB, (4.11)
where GB is the reciprocal lattice volume vector. Dispersion relation for free e−
:
Ekin =
¯h2
2m
(k2
+ k2
⊥). (4.12)
If we don‘t know the dispersion relation within the crystal we are able to directly measure the band
structure, but if we know the initial dispersion or the final states, we can limit the dispersion or
the initial conditions. For surface conditions these restrictions do not apply ⇒ layered materials
with two-dimensions periodicity exists: TaS2, GeS or graphite, where the bulk band structure
E(k ) will be explored.
k =
2m
¯h2 Ekin sin θ, (4.13)
where θ is the angle between the surface normal and the axis of the detector. The UPS method
can be applied to absorb on substrate system like CO on Ni
EDC
The spectrum actually measured in a photoemission experiment involves not only the electron
excitation out of the atom but also the probability that this electron can escape from the solid.
This conversion from the energy distribution of electrons inside the solid produced by the optical
excitation to the experimental distribution of electrons measured outside the solid, termed the
energy distribution curve. The EDC measured experimentally are directly proportional to the
rate of photo-excitation and hence the density of states. Since the density of states can be
determined analytically from the atomic wave-function and the band theory and the response of
the experimental system can be factored out, one can use the spectral feature measured versus
calculated to measure a wide range of features.
In the case of EDC that results from exciting an arbitrary initial density of states (DOS) with the
photon hν, the photo-excitation creates a replica of the density of occupies states distributed in
energy below the vacuum level, shifted up by the photon energy. Electrons that scatter inelastically
are termed secondary electrons and form a continuum with the lowest density at the highest EDC
energy and highest density at near-zero kinetic energy, corresponding to the vacuum level cutoff.
CFS
CFS spectroscopy uses the tunability of synchrotron radiation to measure the density of empty
82 CHAPTER 4. INTERACTION OF ELECTRONS WITH MATTER
states above the vacuum level into which electrons are photoexcited. First, one selects a photon
energy range from slightly below to slightly above the energy difference between a relatively
narrow core level energy Ecore and an empty surface state at or below the conduction band edge
Ec. Second sets the electron analyser energy to an energy Ek several eV above the cutoff energy
for photoemission. Finally, one scans hν across the range to obtain partial yield spectra. These
spectra exhibit peaks in the emission of secondary electrons when electrons are photoexcited from
the core level to an empty state either in the conduction band or within the bandgap. The energy
gained by refilling the core hole indirectly increases the emission of secondary electrons.
The electron emission versus photon energy curves at a constant final kinetic energy reveal changes
with surface condition that can be attributed to the formation of new surface state. Even though
complicated by the lattice relaxation energy correction factor, the CFS or partial yield spectroscopy
demonstrates the creation of new gap states as new atoms absorb.
CIS
The converse of CFS is CIS spectroscopy. In this technique, the incident photon energy is again
scanned, but Eg is scanned in energies to match. In other words, Efinal −hν = Ei (initial) energy.
Densities of final states are useful in determining filled state densities since the former must
be known to determine the latter acording to Fermi‘s golden rule. It is also known as inverse
photoemission spectroscopy and provides useful empty state information for semiconductors and
organic electronic materials.
4.8. SPECTROSCOPIC PHOTOEMISSION OF ELECTRONIC STATES 83
4.8 Spectroscopic photoemission of electronic states
Molecules can possess several types of angular momentum. Electrons possess orbital and
spin angular momenta, rotation of the molecule generates angular momentum, degenerate
vibrations may give rise to vibrational angular momentum, and nuclei may also have spin angular
momentum. Interactions (coupling) between these various types of angular momentum can have
important implications for interpreting spectra, particularly high resolution spectra, and so it
is important to be familiar with some basic results from the quantum theory of angular momentum.
The angular momentum is a vector quantity and a simple vector model provides a useful
visual recipe for assessing how different angular momenta interact. There are more powerful
and rigorous mathematical approaches to angular momentum coupling than that described here.
These more comprehensive treatments deal explicitly with the angular momenta as mathematical
operators, and the coupling behaviour then follows from the properties of these operators when
added vectorially. The simple vector picture is use to emphasize the physical principles underlying
the coupling of angular momenta, focussing on electronic angular momentum.
The magnitude of the orbital angular momentum for a single electron in an atom is given by
the expression ¯h l(l + l) where l is the orbital angular momentum quantum number with allowed
values 0, 1, 2, . . . Similarly, the magnitude of the spin angular momentum vector is given by
¯h s(s + l) where s is the spin quantum number which, for a single electron, can only take the value
1/2. For an electron with both orbital and spin angular momenta, the total angular momentum
vector j will be the sum of the two constituent vectors j = l + s. The magnitude of this vector is
¯h j(j + l), where j is the corresponding quantum number, and according to the quantum theory
of angular momentum this can only take on the values j = l + s, l + s − 1, ..., |l − s|. Series such as
this are called Clebsch – Gordan series and arise when the total angular momentum results from a
system composed of any two sources of quantized angular momentum. In the general case, if two
angular momenta, say j1 and j2, can interact then the allowed values for the angular momentum
quantum number for the resultant total angular momentum will be given by the Clebsch – Gordan
series j = j1 + j2, j1 + j2 − 1, ..., |j1 − j2|
4.8.1 L-S (Russell-Saunders) coupling
In the Russell – Saunders limit the coupling between the orbital angular momenta is strong.
The source of the coupling is the electrostatic interaction between two electrons. Electron
spins can also couple together, although spin is a magnetic phenomenon and therefore the
coupling is via magnetic fields,which tends to be a weaker effect than electric field coupling. In
Russell – Saunders coupling the interaction between the orbital and spin angular momenta of
a given electron is assumed to be small compared with the coupling between orbital angular
momenta. In this limit the principal torque causes the orbital angular momenta to precess
about a common direction, the axis of the total orbital angular momentum of all electrons.
The spin angular momenta also couple together through the magnetic interaction of electron spins.
The vector model described above can be employed to see the effect of coupling on atoms.
84 CHAPTER 4. INTERACTION OF ELECTRONS WITH MATTER
Figure 4.12: Forming the total angular momentum.
For illustration, consider an atom with two electrons outside its closed shell. The orbital angular
momenta of these electrons are represented by l1 and l2, and the corresponding spin momenta by
s1 and s2. The coupling of orbital angular momenta dominates and they form a resultant total
orbital angular momentum L = l1 + l2. In the same way, the spin momenta also couple to form
S = s1 + s2. When this coupling case is valid, the individual orbital angular momenta l1 and l2
precess rapidly around L, and s1 and s2 precess rapidly around S. L and S can themselves couple
with each other, but this coupling, known as spin – orbit coupling, is assumed to be relatively weak.
As a result, L and S precess slowly around the resultant total angular momentum J = (L + S).
4.8.2 jj coupling
The Russell–Saunders scheme describes the electronic states of light atoms rather well but breaks
down for heavier atoms, particularly for the lanthanides and actinides. This is due to increasing
coupling between the orbital and spin angular momenta of individual electrons to the point where
it is no longer negligible, as was assumed in the Russell – Saunders case. In the limit of very strong
coupling between orbital and spin angular momenta, the appropriate coupling scheme is known
as jjcoupling. A dominant spin – orbit torque will first couple the spin and orbital momenta of
each electron to form resultants, j1 = l1 + s1 and j2 = l2 + s2. The vectors j1 and j2 interact more
weakly, forming the total angular momentum J = j1 + j2. In jj coupling, l1(l2) and s1(s2) precess
rapidly around j1(j2), while jj1 and j2 precess slowly around their resultant J. As a result, only
j1, j2, J and the projection of J(MJ ) are good quantum numbers, i.e. the quantum numbers L
and S in the Russell – Saunders scheme have no significance in jj coupling. Of course it is also
possible that the spin–orbit coupling is neither weak enough for Russell–Saunders to be applicable,
nor strong enough for true jj coupling.
4.9. AUGER ELECTRON SPECTROSCOPY 85
4.9 Auger electron spectroscopy
4.9.1 Auger effect
Auger electrons were named after Pierre Auger who, together with Lise Meitner, discovered Auger
electron emission in 1920s. When a high-energy electron (or X-ray photon) strikes an inner shell
electron of an atom, the energy of the incident particle can be high enough to knock out the inner
shell electron. Thus, the atom becomes ionized and in an excited state. The atom will quickly
return to its normal state after refilling the inner electron vacancy with an outer shell electron.
In the meantime, the energy difference between the outer shell electron and the inner shell may
cause emission of either an Auger electron from an electron shell or a characteristic X-ray photon.
The kinetic energy of an Auger electron is approximately equal to the energy difference between
binding energies in the electron shells involved in the Auger process. For example, the kinetic
energy of an Auger electron is approximated by the following equation:
EKL1L2,3 = EK − EL1 − E∗
L2,3, (4.14)
where Ei is the binding energy and E∗
L2,3 is mark with asterisk because the energy levels L∗
2,3
differs slight from the energy levels of L2,3.
The notation for kinetic energy of Auger electron describes its origin, but the nomenclature is
rather complicated. For example, the kinetic energy of an Auger electron is from a following
process: an incident electron knocks out a K shell electron, a L1 shell electron refills the K shell
vacancy, and a L2,3 shell electron is ejected as the Auger electron. Auger electron spectroscopy
(AES) identifies chemical elements by measuring the kinetic energies of Auger electrons. In an
AES spectrum, an individual kinetic energy peak from an Auger electron is marked with an
elemental symbol and subscripts indicating the electron shells or sub shells involved, for example,
in aluminium oxide case, AlKLL, OKLL. . .
A typical AES spectrum is a plot of intensity versus kinetic energy; most commonly,it is a plot
of the first derivative of intensity versus the kinetic energy. The primary electrons ejected from
a solid surface by inelastic scattering comprise the background of an AES spectrum in the region
of high kinetic energies while the secondary electrons comprise the background in the region of
low kinetic energies. The number of Auger electrons ejected from a sample is much less than that
of scattered primary electrons and secondary electrons. Thus, the direct-mode spectrum is not
favoured except for quantitative analysis.
4.9.2 Fine Auger structure
A recent interest is concerned with the fine structures in the neighbourhood of Auger lines,
denoted by EXAFS (extended X-ray absorption fine structure). It is a powerful nondestructive
technique that provides structural information on metal or metal oxide phases highly dispersed
on inorganic supports under reaction conditions. Bond lengths and coordination numbers of
X-ray absorbing material can be determined. When single crystals are employed as supports,
characterization of metal sites on them can be achieved separately in two different bonds parallel
86 CHAPTER 4. INTERACTION OF ELECTRONS WITH MATTER
and normal to the surface by polarized X-ray radiation.
In transmission mode, the EXAFS signal of surface species on solid substrates can not be
detected because of the large adsorption by the substrate. To determine surface structures, it is
possible to measure EXAFS spectra by detecting the reflected X-rays from sample, but usually
the sensitivity is worse.
4.9.3 Quantitative analysis
Quantitative elemental analysis in electron spectroscopy is similar to that in X-ray spectroscopy.
Analysis quantifies the concentrations of chemical elements on a sample surface from the peak
intensities of the spectra. In theory, the quantitative relationship between the intensities of electron
signals and atomic fractions of elements can be calculated. In practice, for quantification in both
XPS and AES, most parameters for calculations are not available.
Sensitivity factor method: the Auger current is described by
Iα,XYZ = Ipγα,XYZ T(Eα,XYZ)D(Eα,XYZ)
Ep
Eα,X
σtotal
α,X (E)dE
∞
0
nα(z)e
− z
λM(Eα) cos θ
dz (4.15)
where Ip is the excitation current, γα,XYZ is the Auger efficiency, σtotal
α,X (E) is the ionization cross
section X the element α, Ep is the energy of the primary electron nα(z) is the density of element α
at depth z. The transmittance of the spectrometer is T(Eα,XYZ) and the sensitivity of the detector
is D(Eα,XYZ).
Figure 4.13: The physical processes occuring when an electron strikes a solid surface.
4.9. AUGER ELECTRON SPECTROSCOPY 87
Ionization is caused not only by the electrons but also by the backscattered energy which has
the distribution n(E)
σtotal
α,X = σα,X(Ep) +
Ep
Eα,X
σα,X(E)n(E)dE (4.16)
This relation can be approximated as:
σtotal
α,X = σα,X(E)[1 + rM
(Eα,XYZ, ψ] (4.17)
which can be simplified for homogeneous material
Iα,XYZ = Ipγα,XYZ T(Eα,XYZ)D(Eα,XYZ)σα,X(E)[1 + rM
(Eα,XYZ, ψ]nαλM
(Eα) cos θ (4.18)
The sensitivity factor Sα,XYZ is defined as:
Iα,XYZ = Ipnα cos θ Sα,XYZ (4.19)
and the atomic concentration α in the M matrix
cα =
nα
α
nα
=
Iα,XYZ
Sα,XYZ α
Iα ,XYZ
Sα ,XYZ
(4.20)
The sensitivity factor must be determined either theoretically or directly experimentally. Common
practice is to use the sensitivity factors from published handbooks with certain corrections
according to the instrument characteristics, because it is not feasible to compile a set of in-house
sensitivity factors.
Standard method: For simplicity, a binary alloy αβ is selected. No contamination on the
surface or segregation phenomena exists:
Iα
Is
α
Is
β
Iβ
=
nα
nβ
ns
α
ns
β
λαλs
β
λβλs
α
(1 + rα)(1 + rs
β)
(1 + rβ)(1 + rs
α)
=
nα
nβ
K (4.21)
and the elemental concentration α is
cα = 1 +
IβIs
α
IαIs
β
1
K
−1
. (4.22)
The matrix correction factor K describes the influence of the αβ alloy. When K = 1, we assume
that the influence on matrix is negligible.
88 CHAPTER 4. INTERACTION OF ELECTRONS WITH MATTER
4.10 X-ray photoelectron spectroscopy
The X-ray photoelectron is an electron ejected from an electron shell of an atom when the atom
absorbs an X-ray photon. An incident X-ray photon can have sufficient energy (a value of hν)
to knock out an inner shell electron, for example from the atom’s K shell. In such a case, the
K-shell electron would be ejected from the surface as a photoelectron with kinetic energy EK.
Knowing the kinetic energy EK, we can calculate the binding energy of the atom’s photoelectron
(EB) based on the following relationship:
EB = hν − EK − φ (4.23)
The value of φ depends on both the sample material and the spectrometer. The binding energies
(EB) of atomic electrons have characteristic values, and these values are used to identify elements,
similar to the way that characteristic X-ray energy is used in X-ray spectroscopy. X-ray photoelectron
spectroscopy (XPS) identifies chemical elements from the binding energy spectra of X-ray
photoelectrons.
Inner shell
• depending on how deep is, we obtain atomic number and the energy of X-rays
• doublet separations in photoelectron lines ⇐ spin-orbital coupling jj. Splitting distance is
directly proportional with < 1/r3
> for a given orbit, so it increases for higher Z or lower l
(at constant n).
subshell value of j area ratio
s 1
2
–
p 1
2
,3
2
1:2
d 3
2
,5
2
2:3
f 5
2
,7
2
4:4
• Relative intensities are given by the photoemission cross section, analyser transmission and
very little by the X-ray energy
• peak width (FWHM) is determined by
∆E = (∆E2
n + ∆E2
p + ∆E2
a )1/2
, (4.24)
where ∆En is the natural width of the inner surface, ∆Ep is the width of the X-rays ∆Ea is
the analyser resolution (in all cases Gaussian distribution).
– ∆En associated with the lifetime τ
Γ =
h
τ
. (4.25)
where τ is related with the photoemission of X-rays, emission of Auger electrons or
special type Auger electron (Coster-Kronig transition), which prefers l and is very fast.
Width for light elements (1s, 2p) increases with the atomic number.
4.10. X-RAY PHOTOELECTRON SPECTROSCOPY 89
– ∆Ea is constant for all peaks in "constant analyser energy" (CAE) mode, but changes
in "constant retard ratio" (CRR), because ∆E/E ratio remains constant.
Valence shell
Electrons with low binding energy (0–20 eV). They are very close to each other ⇒ band structure.
At high resolution spectrum scanning can show the band structure and a sharp decrease of
intensity for EF.
Auger series
Kinetic e−
energy from Coster-Kronig transition is very low; four basic series exists:
• KLL - from boron to magnesium(with Mg Kα) or aluminium (with Al Kα),
• LMM - from sulphur to germanium (with Mg Kα) or selenium (with Al Kα),
• M4,5N4,5N4,5 - from molybdenum to neodymium (with Mg Kα),
• M4,5N6,7N6,7 - only for high energy X-rays (e.g. Ag Lα, 2984 eV).
4.10.1 Energy scale calibration
For meaningful measurements it is important to ensure the instrument is calibrated. For example,
to identify chemical constituents from the peak energy correctly, the energy scale needs to be
calibrated. To provide quantitative information, use sensitivity factors and compare with other
instruments, while the intensity scale needs to be linear and also corrected for the intensity
response function. The intensity response function (IRF) includes the angular acceptance of the
analyser to electrons, the transmission efficiency of electrons and the detection efficiency.
• for metals is possible to observe directly the Fermi level in the valence lines;
• the semiconductors are sufficiently conductive in order to offset the EF. It is yet difficult to
determine the exact location of line since the Fermi level depends on doping, surface states
and defects;
• in the analysis of dielectrics the surface charge and energy shift occurs. The most commonly
used method is called internal standard - a reference line from photoemission with a known
binding energy from other measurements. It is based on the fact that often the sample is
contaminated with hydrocarbons (C1s = 284,6–285,2 eV).
Cu and Au samples set with their angle of emission ≤ 56◦
are sputtered clean, and the
measured energies of the peaks are compared with the energy values given in the table. As
for AES, the peak energy for the calibration is evaluated from the top of the peak without
background subtraction. For gold line 4f7/2 the energy is 83.95 eV. The lineshapes and energies
of the Kα1 and Kα2 X-rays that characterize hν appear to be the same in all unmonochromated
instruments. However, the lineshapes and energies of the Kα X-rays, when monochromated
vary significantly and depend on the set-up of the monochromator and its thermal stability. By
90 CHAPTER 4. INTERACTION OF ELECTRONS WITH MATTER
altering the monochromator settings, the measured energies of the peak may be moved over a
kinetic energy range of 0.4 eV without too much loss of intensity.
Use of the standard with a modern, well-maintained spectrometer should result in calibration
within tolerance limits of ±0.2 eV over four months or ±0.1 eV over one month before recalibration
is required. For many purposes ±0.2 eV is satisfactory.
4.10.2 Primary structure information
Inner shell line
The binding energy is influenced by interactions with other electrons → chemical shift. The exact
calculation of the binding energy is very difficult and limited to small molecules → approximate
methods, such as electrostatic model - the model potentials(charge Potential model)
EB = E0
B + kqi +
i=j
qj
4π 0rij
, (4.26)
where E0
B has significant calibration factor , k s a constant indicating an average repulsion
between inner shell and valence e−
, qi is the atom i charge and sum of potentials of neighbouring
atoms j, also called Madelung potential, where rij are the internuclear distances.
Valence band peak
The valence electrons are directly involved in bond formation and molecular interactions, so the
intensity and energy of the valence band peaks depend on their bonding environment. Thus,
quantitative interpretation of the valence band spectrum for most materials requires a full
molecular orbital calculation. For multi-component materials or those containing a large number
of atoms per structural unit (e.g. polymers) this requires computer calculations.
Valence band analysis is worth pursuing because it can provide electronic structure information
that cannot be obtained from typical core level analysis. Also, it is sometimes possible to extract
useful structural information from valence band spectra.
In metals the information about the valence band and the Fermi energy give the opportunity to
observe the transformation of the material from dielectric to metal (VO2 non-conductor x metal
at 65◦
, NaxWO3 non-conductor x metal when changing x).
Auger chemical shift
To identify the oxidation state of an element is often preferable to use the Auger parameter:
α = EK(jkl) − EK(i) = EK(jkl) + EB(i) − hν, (4.27)
where EK(jkl) and EK(i) is the kinetic Auger energy. α does not depends on the reference surface
or the sample charge. The Auger parameter is modified
α = α + hν = EK(jkl) + EB(i). (4.28)
Auger line shapes
Auger lines can be readily distinguished from photoemission lines by changing X-ray sources (e.g.
4.10. X-RAY PHOTOELECTRON SPECTROSCOPY 91
using a Mg Kα source instead of an Al Kα source). If the Auger process causes the formation of
at least one hole, then the distribution of Auger lines intensities strongly depends on the type of
molecule.
4.10.3 Final state effects
X-ray satellites and ghost peaks
Satellites appear together with the less intensive X-ray characteristic peaks. The main peak
Kα1,2 corresponding to the 2p3/2,1/2 → 1stransition, satellites are Kβ, valence bands → 1s or
multi ionized atoms.
Ghosts appear due to excitation of surface contaminants. Often, Al Kα1,2 from the Mg Kα (due
to the aluminium window), followed by Cu Lα and O Kα.
Multiplet splitting
A requirement for this splitting of the s photoemission peak into a doublet is that there be unpaired
orbitals in the valence shells. Complex peak splitting can be observed in transition metal ions and
rare earth ions when multiplet splitting occurs in p and d levels.
∆E = (2Sv
+ 1)Ksd a
I(Sv
+ 1/2)
I(Sv − 1/2)
=
Sv
+ 1
Sv
, (4.29)
where Sv
is the total spin of a unpaired valence electron and Ksd is the s-d exchange integral.
Shake-up satellites
It represents photoelectrons that have lost energy through promotion of valence electrons from
an occupied energy level (e.g. a π level) to an unoccupied higher level (e.g. a pi∗
level). Shake-up
peaks (also called "loss peaks" because intensity is lost from the primary photoemission peak) are
most apparent for systems with aromatic structures, unsaturated bonds or transition metal ions.
In contrast to the continuum of reduced energies seen in the inelastic scattering tail, shake-up
peaks have discrete energies (∼ 6.6eV higher binding energy than the primary peak in C1s spectra
of aromatic-containing molecules) because the energy loss is equivalent to a specific quantized
energy transition (i.e., the π → π∗
transition).
Asymmetric metal lines
The metal lines may change with the degree of surface charging during data acquisition. With
asymmetric metal lines as encountered with many transition metals, this effect may mimic the
presence of metal ions, The line broadening is , most likely, due to varying degree of differential
charging, since it is paralleled in the widths of the support elements.
Asymmetric non-metal lines
Appears only at high resolution. The asymmetry occurs due to fine vibrational structure of
molecules. After ionization there is a change in interatomic distances and narrowing the potential
curves of the molecule.
92 CHAPTER 4. INTERACTION OF ELECTRONS WITH MATTER
Shake-off satellites
If the departing photoelectron transfers sufficient energy into the valence electron to ionize it into
the continuum, the photoemission loss peak is called a "shake-off" peak. The shake-off satellite
peaks of the photoemission peak can have a wide range of possible energies (of course, always with
a lower KE than the photoemission peak). This energetically broad feature is typically hidden
within the background signal and is usually not detected or used analytically.
4.10.4 Quantitative analysis
The intensity of the photoemission line i
Ii
A ∼ QA cAσi
Aλi
Ti
Li
Af(φ, θ), (4.30)
where Q is the photon flux [cm−1
s−1
], A effective are of the sample, cA concentration of A, σi
A is
the partial ionization cross section, λi
mean free path, Ti
the transmission, Li
A angular asymmetry
coefficient and f(φ, θ) function, depending on the geometry of the experiment. The above equation
does not detect the absolute concentration of elements. The intensities of the lines for individual
measured elements under the same conditions can determine the relative atomic composition.
When a solid is homogeneous within the analysed depth for all elements of interest (A and B for
example), the uncertainties can be partially cancelled by considering relative atomic values. We
obtain:
Ii
A
Ij
B
=
cAσi
Aλi
Ti
Li
A
cBσj
BλjTjLj
B
=
cAσi
ALi
A
cBσj
BLj
B
R. (4.31)
If the kinetic energy differs by more than ∼ 400 eV, correction of T and λ should be performed.
the correction used for transmission:
T ∼ 1/Ek for constant energy (CAE) (4.32)
and
T ∼ Ek for constant resolution (CRR). (4.33)
To calculate the mean free path, the typically used equation is
λ = aE−2
k + bE
1/2
k . (4.34)
One of the major problem in quantitative analysis is the integral of photoemission intensity
lines. When using the theoretical values σ we have determine the overall intensity of lines including
the contributions of all satellites and inelastic electron scattering. For practical purposes we can
use the equation:
Ii
A
Ij
B
=
cASi
A
cBSj
B
. (4.35)
4.10. X-RAY PHOTOELECTRON SPECTROSCOPY 93
4.10.5 Angular effects
1. Enhancement of system sensitivity and depth profile
Almost all electrons (95 %) come on a path equal with 3λ. If e−
are detected at angle θ
different from normal, the thickness of the analysed layer is:
d = 3λ sin θ (4.36)
For the substrate (s) with homogeneous layer (v), the change of intensity in the ideal case
is:
Is
(d) = Is
0 e−d/λ sin θ
and Iv
(d) = Iv
0 (1 − e−d/λ sin θ
). (4.37)
In practice, however, the geometry of the system also exhibits a angular response dependence
function ⇒ measured values for Iv
/Is
2. Single-crystal studies
Measuring the absolute core electron intensity from a single-crystal surface, depending on
the angle θ gives modulated intensity. Similar modulation is observed for fixed θ and rotation
of surface plane. ⇒ X-ray photoelectron diffraction XPD
4.10.6 Numerical data analysis
Simple operation with data:
(a) integration and determination of area, (b) noise removal, (c) satellites removal, (d) background
subtraction, (e) addition and subtraction of spectra, (f) search for lines maxima, (g) conversion
to the binding energy after calibration.
background subtraction
• linear background: straight line between the first and last point of the spectrum
• integral background: the integral is calculated over the hole line area and a calibrated curve
thus obtained is subtracted from the peak area
• background based on the elastic and inelastic processes
Signal to noise ratio
The noise is characterised in three different ways: peak-to-peak (p.t.p.), root-mean-square (r.m.s.),
standard deviation. If the spectrometer is limited to statistical noise, the signal to noise ratio
(S/N) can be defined like:
S/N =
S + B
F
, (4.38)
where S is the height of the line above background B, is the background B and F is the intensity
factor
F =
(S/B + 1)(S/B + 2)
(S/B)2
. (4.39)
94 CHAPTER 4. INTERACTION OF ELECTRONS WITH MATTER
If we desire S/N constant
t2/t1 = F2/F1. (4.40)
Smoothing
• fitting the data with a suitable smooth function (polynomial, fractional polynomial, exponential,
Fourier and splines)
• convolution data using a suitable algorithm, which leads to a smoothing
ysm
r =
m
r=−m
Cryr
NORM
, (4.41)
where Cr is the convolution number and NORM is the normalizing factor. Savitsky-Golay
convolution is recursive. the convolution function represented by lest squares fitting of an
polynomial(cubic-quadratic) to m points on either side of each data point.
• frequency filter for Fourier transformation (mathematical equivalent of convolution algo-
rithm).
Analysis of overlapping spectral lined
• spectra derivation (background removal, separation of overlapping lines), negative lines in
the second derivative correspond to approximately the position of overlapping lines ⇒ noise
problems ⇒ smoothing with derivation
• deconvolution, mostly for the shape of the line
ym
j =
N
i=1
yt
i gj−1 + nj, (4.42)
where nj is the noise. The instrument function g can be determined by line measurement of
Ag 4 eV from the Fermi level with a monochromatic X-ray source.
• fitting with Gaussian or Lorentz function. It can either be used the Voigt convolution or a
mix Gauss-Lorentz function
f(E) =
A
[1 + M(E − E0)2/β2] exp([1 − M][ln 2(E − E0)2]/β2)
, (4.43)
wheree M is mixing ration (1 for pure Lorentz shape) and β is a parameter, which is nearly
0.5 FWHM.
4.11. ELECTRON ENERGY LOSS SPECTROSCOPY 95
4.11 Electron energy loss spectroscopy
The EELS experiment was developed from gas-phase electron scattering experiments, which
probed electronic states within molecules. For the analysis of surface vibrations, the technique
utilizes the interaction of very low energy electrons (1 – 10 eV) with the surface electric fields
produced by adsorbate molecules and substrate atoms.
Two types of scattering of electrons may be considered, "elastic" and "inelastic" (a third mechanism
in which the incoming electron is trapped for a finite time within the surface forming a so
called "negative ion resonance" is not treated here). Electron scattering is directly analogous to
neutron diffraction. This analogy with a diffraction experiment is a useful one, since it illustrates
why the best energy resolution in EELS, as measured by the full width at half maximum height
(FWHM) of the elastically scattered electron peak, is obtained when using a single crystal substrate.
Both elastic and inelastic scattering are broadened in energy terms when the scattering
potential of the surface is poorly defined, e.g. a polycrystalline sample.
The laws of energy and impulse conservation forms
E (k ) = E0(k) − ¯hω, (4.44)
k = k − q ± g , (4.45)
where E0(k) is the incident e−
energy with momentum ¯hk, E (k ) is the backscattered e−
energy
with momentum ¯hk , ¯hω is e−
energy loss in the transmission, q is the transferred momentum g is
the lattice reciprocal vector. EELS modifications: AREELS - Angle Resolved EELS, HREELS High
Resolution EELS.
Experimental
If we consider the experimental requirements for EELS or HREELS, low energy electrons, single
crystal substrates (usually), then it is clear that this is a high vacuum technique. In practice the
working pressure limit for most EELS instruments is 10−6
mbar due to a gas-phase scattering
processes.
Usually at least one unit, the monochromator or analyzer, has some ability to rotate about an
axis perpendicular to the plane of the experiment while the sample may also rotate. In this way
it is possible to vary the angle of incidence over a wide angular range and still maintain the ability
to detect at angles away from the specular position. The monochromators and analysers are
electrostatic selection units based upon either 127-degree cylindrical capacitors or hemispherical
capacitors. The potentials applied to these capacitors are such that only electrons having specific
pass energies may reach first the sample surface and second the detector. The working surfaces of
the spectrometer are coated with an inert material such as graphite in order to minimize changes
in spectrometer work function that may occur upon exposure to various gases.
The design of an EEL spectrometer is such that the energy spread of the incident low-energy electron
beam is minimized while maintaining a reasonable electron flux. This primary consideration
means that the area of surface sampled by the beam is of secondary importance.
Three independent ways for measuring spectra can be used:
1. measurements at constant angle θ, θ and constant energy loss value ¯hω depending on the
incident energy loss profile
96 CHAPTER 4. INTERACTION OF ELECTRONS WITH MATTER
2. measurement of the loss energy ¯hω at constant k in accordance with the scattering angle θ .
3. measurements at constant k electron dispersion direction depending on the energy loss
spectrum.
The basic problem of the measurements is the assignment scattering mechanism for measured energy
loss. First the plasma loss should be determined and then distinguish between the remaining
peak volume and the surface processes. They are often represented as the corresponding maximum
ionization of deeper levels.
4.12 Threshold techniques
An alternative form of core level spectroscopy is based on the detection of the onset on ionization
as the exciting beam energy passes through such a threshold (equal to the binding energy of the
core level relative to the Fermi level of the solid). A technique of this kind involves sweeping the
exciting energy beam and detecting the onset of emission of photons or Auger electrons associated
with the refilling of the core hole thus created. These different modes of detection form a group
under the general heading of APS (Appearance Potential Spectroscopy). In principle, either incident
electrons or photons could be used but the photon calls for a tunable light source in the
50–2000 eV range, only possible using synchrotron radiation.
All three modes of the incident electron technique have been used, however, the thresholds either
being detected by fall in the elastically scattered electron flux (Disappearance Potential
Spectroscopy - DAPS), by the onset of photons emission (Soft X-ray Appearance Potential Spectroscopy
- SXAPS) or by the onset of the new Auger electron emission channel (Auger Electron
Appearance Potential Spectroscopy - AEAPS). Note that in all of these techniques the surface
sensitivity is guaranteed by inelastic scattering of the incident electron beam.
Primary energy Ep is determined by:
Ep = eVp + φ + kT. (4.46)
By applying the conservation of energy
Ep − E1 = EB + E2, (4.47)
where E1 is the primary electron energy level, EB is the binding energy of the electron on the
surface E2. At the threshold energy can be captures just above the Fermi level i.e.E1 = E2 = 0 and
Ep=EB. If instead of thermionic cathode we have tunnel cathode, the eVp = EB. The probability
of excitation of the inner layers is:
W(Ep) ≈
Ep
0
ψ(E − EB)
Ep+E
0
(4.48)
Chapter 5
Interaction of Ions with Matter
5.1 Ion interactions and penetration depth
Differences exists in the interaction of ions and electrons with matter. Ions are much heavier, have
bigger dimensions and they can have higher charges. At ion impact may occur the emission of:
• secondary electrons (secondary ion-electron emission),
• ions (ion scattering, ion-ion secondary emission),
• neutral particles (cathode sputtering),
• photons (characteristic x-rays and visible radiation - ionoluminescence).
Also, changes in the electronic shell may occur and also ion implantation in material.
Penetration depth of protons: Protons interactions are similar to electron ones, but some peculiarities
can be observed. Being 1836 times the mass of electron, has a higher angular momentum
and loses only a small portion of eat at each collision ⇒ penetration depth is greater than for
electrons. For 2.5 MeV proton, the penetration depth is around 55 µm for carbon and 28 µm for
silver; it decreases with the the proton energy and Z of the material.
Penetration depth depends on the number of protons in the ion, its energy and the target. For
low energies (several eVs) it is reflected from the target and part of its energy is transferred to the
surface atom, causing ejection of atoms, ions or clusters. It can also create cascade collisions.
The range R - the total distance that the ion (projectile) travels in coming to rest, is longer than
the penetration depth x. The projected range Rp is defined as the total path length of the
projectile with energies 0.002 ≤ ≤ 0.1 keV measured along the direction of incidence.
Rp = C1(µ)M2




Z
2/3
1 + Z
2/3
2
Z1Z2

 E


2/3
, (5.1)
where M2 is the atomic mass of the target , Z is the atomic number while C1(µ) is experimentally
determined. for projectiles with energies 0.5 ≤ ≤ 10 keV is
Rp = C1(µ)M2




(Z
2/3
1 + Z
2/3
2 )1/2
Z1Z2

 E


2/3
. (5.2)
97
98 CHAPTER 5. INTERACTION OF IONS WITH MATTER
Ion channelling is possible along crystallographic axes with small critical angles. In this case
the penetration depth is substantially increased.
Figure 5.1: Calculation of total distance of ion.
5.1.1 Basic ion penetration concepts and the energy transfer mechanism
Because mion ∼ mtarget, it leads to considerable energy loss due to elastic collisions. The energy
loss is defined like ∆E/∆x → dE/dx, where ∆x → 0. Often
S ≡
dE
dx
(5.3)
is called stopping power or specific energy loss. The stopping cross section can be defined
as:
=
1
N
dE
dx
or =
1
ρ
dE
dx
, (5.4)
where N represent [atoms/m3
] and ρ is the density [kg/m3
]. Sometimes also represents stopping
power.
The relative importance of interactions between ions and matter depends on the ion speed, their
charge and the atoms of the target
• for vion v0, where v0 is the Bohr velocity of electrons in the atom. Ions will carry their
electrons which tend to be neutralized by electron capture. At these speeds, the elastic
collisions dominates and the losses are called nuclear energy losses.
• for higher velocities, the importance of nuclear energy losses decreases with a factor of 1/E
and the inelastic collisions with electrons because important. Above 200 keV/amu, the
contribution of nuclear energy losses is typically less than 1 % than electronic one. Between
0.1v0 ≤ vion ≥ Z
2/3
1 v0 electron collisions are proportional with the velocity (E1/2
).
5.1. ION INTERACTIONS AND PENETRATION DEPTH 99
• for v v0 the ion charge increases till it completely losses all electrons. The Bethe-Bloch
Formula for the electron energy is
dE
dx
= NZ2(Z1e2
)2
f(E/M1), (5.5)
where Z1 is the ion atomic number.
5.1.2 Secondary ion-electron emission
According to the adiabatic principle, heavy ions are not efficient for transferring energy to light
electrons to provide ionization. This general statement can be referred to the direct electron
emission from solid surfaces induced by ion impact.
The secondary electron emission coefficient γ (electron yield per ion) becomes relatively high only
at very high ion energies exceeding 1 keV, when the Massey parameter becomes low. Although
the secondary electron emission coefficient γ is low at lower ion energies, it is not negligible and
remains almost constant at ion energies below the kilovolt range.
γ =
ie
ii
. (5.6)
Total γ coefficient is the sum of
γ = γp + γk, (5.7)
where γp is the potential emission and γk for the kinetic one. The Penning mechanism of secondary
ion-electron emission, also called the potential mechanism defines the ion approaching
the surface which extracts an electron from there because the ionization potential Φ exceeds the
work function φ. The defect of energy Φ − φ is usually large enough (Φ − φ > φ) to enable the
escape of more than one electron from the surface. Such a process is non-adiabatic and its probability
is not negligible. The secondary ion-electron emission coefficient γ can be estimated
using the empirical formula:
Φ ≥ 2φ, (5.8)
⇒ potential emissions are observed especially when bombing the surface with ions of inert gases
that have high Φ. γp is higher for higher ion charge.
Ion-neutralization spectroscopy (INS) is an emission electron spectroscopy. In INS the
externally applied agent is a slowly moving positive ion, such as He+
, presented to the solid
surface. Near the surface a non radiative, Auger type electronic transition process occurs which
simultaneously neutralizes the ion to the ground state of the parent atom and excites a second
electron, which may be ejected into vacuum. The kinetic-energy distribution of these ejected
electrons contains spectroscopic information concerning the electronic structure in the surface
region of the solid.
• When the incoming ions is just outside the metal surface, two electrons in the filled valence
band of the metal interact, exchanging energy and momentum.
• One electron, the neutralizing electron, tunnels through the potential barrier into the potential
well presented by the ion, and drops to the vacant atomic ground level.
100 CHAPTER 5. INTERACTION OF IONS WITH MATTER
• The energy released in this transition is taken up by the second interacting election which
now may have sufficient energy to escape from the metal
• These Auger type transition can take place anywhere within the filled valence band so that
the ejected electrons have a range of energies rather than one specific energy. Outside the
metal surface the electron energy distribution can be measured quite straightforwardly.
5.2 Ion sources
In case of surface and thin films analysis can ion sources can be used. Suitable sources provide
proper ion beams for surface cleaning and for surface layer sputtering in order obtain a depth
profile (such as XPS or AES), primary beam for static and dynamic SIMS. Another sources are
represented by the particles accelerators used for energetic ion analysis (such as RBS, PIXE, NRA,
ERDA etc.)
The ionization methods can be classified according to:
• electron impact
• fast atom bombardment
• electro-spray ionization
• gas discharge ion sources
• solid state surface atom ionisation sources
• high electric field desorption
Ion beam focus:
• wide ion beam sources
• narrow focused ion beam sources
5.2.1 Electron impact ionisation sources
The principle of ionization source is similar as a ionisation gauge. The only modification include
a geometry of gauge and ion collector must be replaced by the system which is able to extract
the produced ions. Ionisation space is inside the cylinder - screened by anode with the ion trap
at the end. Electrons are accelerated from a thermionic emission cathode by high voltage (100 -
500 V). The electrons travel through the ionisation space and ionize gas molecules around them.
The resulting ions are collected at a negative trap electrode - thus resulting in an ion beam. The
operating pressure is lower 0.01 Pa and the source provides high focused ion beam with beam
current to 10 µA. Due to its properties it is possible another manipulation in ion optics.
Another source modification - double anode source. Two parallel bars - anodes - are placed
along the cylindrical cathode (voltage up to 10 kV). Electrostatic field with a saddle point is form
on anode axis. Electrons oscillate around this point and they are effectively ionise the gas. The
5.2. ION SOURCES 101
produced ions are accelerated along the plane symmetry to cathode with the outlet. Operation
pressure is tenths of Pa, ion current is tens of µA. Electrons are ejected in the form of diverging
beam with wide energy distribution → they are inappropriate to another manipulation in ion
optics. It causes a high chromaticism. In consequence, it is use to surface sputtering.
5.2.2 Plasma sources
Plasma sources generate plasma - excitation of the gas requires ionization of neutral atoms and
molecules. High intensity electric field extracts ions from the plasma surface (negative charged
electrode). The dependency of extracts ion current I and electrode surface voltage Uex on the
thickness d is described by Child-Langmuir law
I ≈ U3/2
ex /d2
. (5.9)
Two types of plasma sources are most often use to surface and thin films analyses. In van
der Graff accelerators, high frequency discharge are the ion sources generally use to analysis with
ion energy in order MeV. The duoplasmatron is a type of ion source compresses the plasma by
a magnetic field in front of the extraction system. The plasma is so dense that matching to the
extraction field strength requires an expansion cup to lower the current density. Most often is
used as a source of Ar and O ions.
High frequency ion source
Ions can be created in an inductively or capacitively coupled plasma. The electric field has a
frequency in order tenth of MHz, the power is tenth to hundreds of W, the pressure is units of Pa,
current is from tenth of µA to tenth of mA. The ionization efficiency can be enhanced by magnetic
field.
Two types of inductively coupled RF structures are most common. In one type, the induction coil
is placed outside a glass or quartz cylindrical tube in which the plasma is generated. The other
type has a metallic discharge chamber with an RF antenna inside the discharge plasma.
Although capacitively coupled RF sources are widely used in plasma etching processes, they are
seldom employed for the production of ion beams. Capacitively coupled discharges form a very
high positive plasma potential. As a result, material is sputtered from the chamber wall to form
impurity ions in the extracted beam.
Duoplasmatron
The duoplasmatron operates as follows: a cathode filament emits electrons into a vacuum chamber,
then the gas is introduced in very small quantities into the chamber, where it becomes charged
or ionized through interactions with the free electrons from the cathode, forming a plasma. The
plasma is then accelerated through a series of at least two highly charged grids, and becomes
an ion beam, moving at fairly high speed from the aperture of the device. Duoplasmatrons are
characterized by high efficiency - usually above 90 %.
The duoplasmatron components are:
102 CHAPTER 5. INTERACTION OF IONS WITH MATTER
• cathode (K)- generally the heated cathode (thermionic emission), sometimes (for oxygen
ions) is use the hollow cold cathode
• intermediate electrode (IE),
• anode (A).
The intermediate electrode and anodes are ferromagnetic and create the part of magnetic circuit.
Demirkhanov installed a ferromagnetic anticathode electrode to ensure oscillation of electrons in
the anode region of the duoplasmatron. The anticathode was located behind a copper anode and
had a potential close to the cathode potential. Electrons oscillating in the magnetic field (1500
G) efficiently ionize the gas at a low pressure of ∼5–10˘3
Torr. When the anticathode negative
bias relative to the anode was increased from 0 to 100 V, the ion current increased by a factor of
five. The source, which had an outlet aperture 6 mm in diameter and an expansion cup 50 mm in
diameter, provided a pulsed hydrogen ion current of up to 1.5 A at a discharge current of 20 A.
Figure 5.2: Duoplasmatron scheme.
Duoplasmotrons can have different ion sources:
• Penning-type electrode system (PIG) to maintain a high current discharge by electrons
supplied through a double electric layer from a plasma cathode, has been embodied most
completely in the DuoPIGatron type ion source. The magnetic field diverges toward the
ion optics, ensuring oscillation of primary electrons between the intermediate and screen
electrodes and formation of a large, uniform plasma surface. To increase the cathode lifetime,
an inert gas (argon) is fed into the cathode stage and oxygen into the anode stage of the
source. An ion optics system with 13 holes of diameter 5 mm provided an ion beam current
up to 170 mA at an accelerating voltage of up to 50 kV. After mass separation of the beam
and acceleration of ions to energy of 200 keV, a beam of O+
ions with current 100 mA was
obtained. The filament cathode lifetime is over 25 hours.
5.2. ION SOURCES 103
• periplasmatron-type ion source has a unique electrode system design, which decreases heating
of the screen electrode by thermal radiation of the cathode, reduces the effect of secondary
electron back flow from the accelerating gap, and ensures a high uniformity of plasma flow
to the screen grid. A rectangular variant of this source provided a hydrogen ion current of
96 A from an extractor 40 x 16 cm in size.
• magnetic-field-free source for use in neutral beam injectors has been developed in which
the plasma is generated by a diffuse low-pressure high-current discharge, with a distributed
thermionically emitting cathode. Twenty hairpin filaments, 0.5 mm in diameter, are installed
around the periphery of the discharge chamber, 14 cm in diameter, near the cylindrical wall,
which serves as the anode. The peripheral placement of the cathode provides generation of
uniform (± 6 % ), oscillation-free plasma of 12 cm diameter. For a discharge current of 1000
A the deuterium ion beam current density is 0.5 A cm˘2
and the beam current 15 A at an
extraction voltage of 15 kV.
5.2.3 Surface ionization sources
Surface ionization sources are commonly used for producing a Cs+
ion beam. The surface ionization
source emits ions from a hot plate. This configuration is ideal for designing beam optics
with minimum aberration because the emitting surface is a solid boundary. In fact, the surface
ionization source can be designed to produce a beam with very large aperture (and therefore high
current per beam) because of the rigid emitting surface. The ion temperature from a surface
ionization source is typically low, of the order of a fraction of an eV; therefore the beam emittance
is kept low, even when the source diameter is large. The two main types of surface ionization
sources are contact ionizers and aluminosilicate sources.
The ration of density of emitted positive ions flux j+ to density of emitted atoms flux ja is
defined
j+/ja =
g+
ga
e−
Vi−φ
kT , (5.10)
where g+ a ga are statistical weight of ionized and atomic state of adsorbed atom, Vi is ionization
energy of adsorbed atom, φ represents work function of solid, T is the temperature of solid and k
is Boltzmann constant.
5.2.4 Field ionization
Field emission from sharp needle points at which very intense electric fields are created can be
used to extract either electrons or ions from the solid (or liquid) state:
• liquid metal ion sources (LMIS) - is used in liquid metal ion sources to produce beams of
species of low melting point metals such as Ga, In, Bi, Al, Sn, ... with extraordinarily high
brightness
• gas field ion source (GFIS)
In both types the electric field with intensity 10 V/nm is use.
104 CHAPTER 5. INTERACTION OF IONS WITH MATTER
5.3 Ion scattering spectroscopy - ISS
Sometimes called Low Energy Ion Scattering Spectroscopy (LEIS) is a surface sensitive analytical
technique used to characterize the chemical and structural properties of materials. The range of
ion energies is from hundreds of eV to units keV. When a beam of ions hits a solid surface part
of the projectiles will be scattered back into the vacuum after one or more collisions with target
atoms of the layer. Measurement of the energy of the backscattered particles is use to identify
chemical structure of solid. primary ion energies of 0.5 – 5 keV are used with noble gas ions (He,
Ne, Ar) and also alkali ions (Li, Na, K). With this method, information is obtained from the
topmost atomic layer, under certain circumstances also from the second or third layer.
Figure 5.3: Comparison of RBS and ISS techniques.
5.3.1 Qualitative analysis
From the applied conservation and momentum laws in the case of two-body collisions it is possible
to calculate the ion kinetic energy after a collision with a surface atom without knowing the
interaction potential.
If the incident ion has energy E0, the mass M1 and the surface atom M2, we obtain the energy
and mass conservation relation for the backscatter energy E1:
E1
E0
=
cos θ ± [(M2/M1)2
− sin2
θ]1/2
M2/M1 + 1
2
, (5.11)
5.4. RUTHERFORD BACKSCATTERING SPECTROSCOPY - RBS 105
When this relation is derived, the kinetic energy of atom should be neglected
For 90◦
and 180◦
: backscattering angles the relation is simplified:
E1
E0
=
M2 − M1
M2 + M1
for θ = 90◦
(5.12)
E1
E0
=
(M2 − M1)2
(M2 + M1)2
for θ = 180◦
(5.13)
The ration K = E1/E0 is named kinematic factor. The above relations show that the energy
of ion after collision is given by the projectile and the target atom weight and the backscattered
angle (for constant E0) =⇒ determination of target atom mass M2. The corresponding expression
for the recoiling target atom is:
E2
E0
=
4A
(1 + A)2
cos2
θ2 (5.14)
Scattered-ion energy spectra are transformed into mass spectra by the backscattered relation for
θ = 180◦
. Consequently, the mass resolution also can be calculated from this equation for the
special case of θ = 90◦
:
M2
∆M2
=
E
∆E
2A
A2 − 1
(5.15)
Assuming a constant relative energy resolution of the detector of E/∆E = 100 the mass resolution
is better for large scattering angles and about equal ion and target atom masses. So the primary
projectile mass has to be selected accordingly if mass resolution is important.
5.4 Rutherford backscattering spectroscopy - RBS
In Rutherford Backscattering Spectroscopy (RBS) the primary ion energy ranges from about 100
keV (for H+
) to several MeV (for He+
and heavier ions). The ion-target atom interaction can
be described using the Coulomb potential from which the Rutherford scattering cross-section is
derived,which allows absolute quantification of the results. Information in principle arises from a
thickness of the order of 100nm (10−5
cm), but analysis of surface layers is also possible by using
channelling/blocking techniques. Scattering of H+
with energies around 100 keV is sometimes
referred to as MEIS (Medium Energy Ion Scattering), probably because it only needs a smaller
type of accelerator, but physically it is within the RBS regime.
The physical principles are the same for both techniques (ISS and RBS): an ion beam is directed
onto a solid surface, a part of the primary projectiles is backscattered from the sample and
the energy distribution of these ions is measured. Since the ion–target atom interaction can be
described by two-body collisions, the energy spectra can be easily converted intomass spectra. The
difference between ISS and RBS arises from the difference in the cross-sections and the influence of
electronic excitations and charge exchange processes, which result indifferent information depths.
In both cases, structural information is obtained from crystalline samples by varying the angles
between beam and sample. Deduction of structural information from the data is straightforward,
since both techniques are "real space" methods which are based on fairly simple concepts. With
ion scattering only the individual atoms of an element can be detected and no information on
compounds or molecules can be gained.
106 CHAPTER 5. INTERACTION OF IONS WITH MATTER
In terms of analytical characteristics, is important the variable cross section for elastic scattering,
which integrates a number of dispersed particles with the number of atoms in the substance
to be examined. Differential scattering cross section at the angle θ in the laboratory system is
then:
dσR
dΩ
=
Z1Z2e2
2E0
2
(M2
2 − M2
1 sin2
θ)1/2
+ M2 cos θ
2
M2 sin4
θ(M2
2 − M2
1 sin2
θ)1/2
, (5.16)
where Z1 and Z2 are the atomic numbers of the ion and the target.
Analysis using heavier particles and lower energies can play a significant role in electron screening.
In such cases the first correct approximation of the cross section is multiply by the correction
factor F in the form:
F =
σ
σR
= 1 −
0, 049Z1Z
4/3
2
ECM
, (5.17)
where ECM is the energy of particles expressed in keV. In a conventional design RBS the correction
is below 5 %. Significant deviation of the cross section appear when the energy of the particles is
increased , or particles with lower atomic number are used. The effect of nuclear forces appears
when the scattered are scattered at large angles and energies E0 > Ecritical
Ecritical = 103
Z1Z2M
−1/3
2 . (5.18)
For example, nuclear effects occur when the α particles (Z1 = 2, M1 = 4)on Si atoms (Z2 = 14,
M2 = 28) sat energiesEkrit > 9 MeV are scattered.
5.4.1 Elemental composition determination
The basic relationship is
Y = QN
∆ω
dσ
dΩ
dΩ, (5.19)
where Y is the number of the dispersed particles registered, Q is the number of incident particles,
N is the area density in the layer (atoms/cm2
) and ∆ω is the interval of the solid angle from which
is detected the scattered particle. The equation can be applied in a layer containing multi-elements
or a thin layer on a stronger substrate.
5.4.2 Energy loss
Only a small fraction of the primary ions come close enough to a target nucleus (impact parameters
of the order of 10−12
cm) to undergo an elastic nuclear collision which is described by the kinematics
given in the previous section. If such an ion is backscattered, its final energy is determined by the
elastic nuclear collision in a certain depth of the sample and the additional inelastic energy loss
to electrons on its way in and out of the target.
The relation for energy loss per unit length, − − dE/dx given in eV/Åand commonly called
stopping power:
S = −
dE
dx
(keV.cm−1
), (5.20)
5.5. DETECTION OF EJECTED ATOMS - ERDA 107
The stopping cross section (eV/(atoms/cm2
)) relates this quantity to the atomic density N and
is therefore more specific for a
=
1
N
S (keV.cm2
), (5.21)
The stopping power curve exhibits a broad maximum around 1 MeV and that is the operational
regime of RBS: here, the stopping power does not depend very much on the ion energy and hence
can be assumed to be constant as a sufficient approximation in many cases; the stopping power
there has its maximum value and therefore RBS its best depth resolution and in this energy range
the nuclear interaction is exactly given by the Coulomb potential, giving RBS the advantage of
an absolute analytical method. Stopping cross section for a particle with energy E and a mixture
of AmBn elements is
AmBn
= m A
(E) + n B
(E), (5.22)
where A
(E) ans B
(E) are the stopping cross sections for the elements A and B.
5.5 Detection of ejected atoms - ERDA
Detection of light elements by RBS is difficult due to the low effective elastic scattering crosssections
and often because the signal lies on a high background originated by scattering on heavier
elements. Therefore, the method used is ERDA (Elastic recoil detection analysis).
In this method, atoms or ions are detected which are removed from the surface by one single
collision with an incoming projectile. They can therefore be identified by their kinetic energy.
Directly recoiling particles are therefore different from those secondary particles that originate
from a collision cascade in a sputtering process and have a broad energy distribution around one
to two eV. They are used in secondary ion or neutral mass spectroscopy.Recoil cross-sections are
generally of the same order as scattering cross-sections. They are largest towards θ2 = 90◦
, but
then the recoil energy approaches zero. A useful energy range is obtained between 30◦
and 60◦
.
ERDA is also often done using a relatively low energy (2 MeV) He beam specifically to depth
profile hydrogen. In this technique multiple detectors are used, at backscattering angles to detect
heavier elements by RBS and a forward (recoil) detector to simultaneously detect the recoiled
hydrogen (the cross-section for which reaction is strongly non-Rutherford at these energies). The
recoil detector has to have a range foil: a thin film (typically 6 micrometres of PET film) to
preferentially stop the incident He beam scattered into the forward direction.
5.6 Secondary ion mass spectrometry - SIMS
Secondary Ion Mass Spectrometry (SIMS) is a relatively new technique for surface chemical analysis
compared with Auger enectron spectroscopy (AES) and X - ray photoelectron spectroscopy
(XPS). SIMS examines the mass of ions escaped from a solid surface to obtain information on
surface chemistry.
SIMS uses energized primary particles, usually ions such as Ar+
, Ga+
and Cs+
, to bombard
a solid surface in order to induce sputtering of secondary particles from an area. The interactions
108 CHAPTER 5. INTERACTION OF IONS WITH MATTER
Figure 5.4: SIMS instumentation.
between primary ions and the solid are rather complicated. First, the secondary particles include
electrons, neutral species of atoms or molecules, and ions. The majority of the secondary particles
are neutral and not useful in SIMS. Only the secondary ions generated by the bombarding process
carry chemical information. Second, the interactions are often more than a simply one-to-one
knock-out of a surface ion by a primary ion. Commonly, the primary ions induce a series of
collisions (collision cascade) in a solid because the energy of a primary ion is transferred by
collisions between atoms in a solid before secondary ions on the surface are emitted.
The ionization probability is strongly affected by the electronic properties of the sample matrix.
Ionization directly affects the signal intensity of secondary ions as shown in the basic equation of
secondary ion yield.
J±
s = JpcY β±
f, (5.23)
where Jp is the primary ion flux, Y is the sputter yield, c is concentration of species m in the
surface layer, β±
represents the probability for positive ions and f is the transmission of the
detection system. The transmission is defined as the ratio of the ions detected to ions emitted,
and it varies from 0 to 1 depending on the analyser. The yield of elemental secondary ions can
vary by several orders of magnitude across the periodic table.
5.6.1 Ion sputtering
Emission of secondary particles can result from collision sputtering or from other processes such
as thermal sputtering. Collision sputtering includes direct collision sputtering and slow collision
sputtering. The former can be considered as a direct impact between a primary ion and a surface
atom. Direct collision sputtering is extremely fast, and occurs in the range of 10−15
to 10−14
seconds after the primary ion strikes a surface. Slow collision sputtering, represents the case that
a primary ion never has chance to collide with a surface atom; instead, atoms in the solid transfer
the impact energy to surface atoms after a series of collisions. The time scale of slow collision
sputtering is in the range of 10−14
to 10−12
seconds.
5.6. SECONDARY ION MASS SPECTROMETRY - SIMS 109
Only a small portion of secondary particles (∼ 1% of total secondary particles) are ionized and
become the secondary ions that are analysed in SIMS. A sputtered particle faces competition
between ionization and neutralization processes when it escapes a sample surface. Ionization
probability represents the chance of a sputtered particle being an ion.
5.6.2 Ion sources
The commonly used primary ions include argon (Ar+
), xenon (Xe+
), oxygen (O2+
), gallium (Ga+
)
and caesium (Cs+
) ions. Heavier metal ions such as bismuth (Bi+
) are also available in modern
static SIMS instruments. The ions of elements normally occurring in gaseous phases, such as
oxygen and argon, are produced by electron bombardment sources or plasma ion sources. Electron
bombardment sources use a circular filament cathode surrounding a cylindrical grid anode. The
gas to be ionized is injected into the open space of the grid, and the gas molecules are bombarded
by the electrons emitted from the cathode. The electrons travel in orbit inside the grid in order to
increase their chances of striking gas molecules. An extraction field inducted in the grid will draw
the gas ion beam through an opening in the center of the extractor. The electron bombardment
sources provide moderate brightness of primary ions (∼ 105
Am−2
per solid angle).
5.6.3 Mass analysis system
The mass analysis system collects and analyses the ion masses to produce mass spectra with the
assistance of a computer. The extractor filter extracts secondary ions from the surface, selects a
mass range of ions to analyse, and eliminates scattered primary ions from a mass spectrum. The
mass analyser, the critical component for secondary ion mass analysis, can be one of the following
types:
• magnetic sector analyser,
Figure 5.5: Secondary particle generation by an energetic primary particle.
110 CHAPTER 5. INTERACTION OF IONS WITH MATTER
• quadrupole analyser,
• time-of-flight analyser.
Magnetic sector analyser
This analyser was the oldest type used for mass spectroscopy. Secondary ions are accelerated by an
extraction potential of 4 kV before entering a magnetic field. The magnetic field in the magnetic
sector will impose a field force in a direction orthogonal to the direction in which the ions travel.
The ions with a given kinetic energy will change their travel path to a circular trajectory. The
radius of path curvature R has the following relationship with the ion mass.
R =
(2V )1/2
B
m
z
1/2
(5.24)
where the extraction potential (V ) is constant. We can adjust the magnetic field strength B, to
select ions with a certain mass-to-charge ratio (m/z) with a fixed radius of magnetic sector R.
Quadrupole mass analyser
The quadrupole mass analyser selects ions with a certain mz−1
by generating unstable oscillation
Figure 5.6: The sputtering proces: (a) direct collision sputtering;(b) collision cascade; and (c)
thermal sputtering.
5.6. SECONDARY ION MASS SPECTROMETRY - SIMS 111
travels for the non-selected ions. Oscillations of ion trajectories are created by an electric field
combining a constant direct current (DC) and alternating current with a radio frequency (RF).
The secondary ions are accelerated by an extraction field, and then the ions travel through the
center of four circular rod electrodes. Combined DC and RF voltages are applied to one pair
of rods and equal but opposite combined voltages are applied to another pair of rods. Such
an arrangement of electric fields generates ion oscillation. The oscillation can be so severe
that ion trajectories become unstable and ions strike the rods. Ions with unstable trajectories
cannot travel through the exit slit of the analyser to reach the mass detector. Only the ions
with a certain mz−1
have stable trajectories and can pass through the exit slit under a given
ratio of DC:AC voltages. The analyser is a sequential type that only allows the ions with
single mz−1
values to reach the detector. Thus, the quadrupole analyser is a device with low
transmission, as less than 1% of ions can reach the detector at any time. To obtain a whole
mz−1
spectrum, the analyser increases the voltages but keeps the ratio of DC:AC voltages constant.
Time-of-Flight analyzer
The time-of-flight (ToF) analyser is the most widely used analyser in static SIMS. As its name
indicates, for this analyser the flight time of an ion is the parameter for measurement. When ions
are obtained with a constant kinetic energy from an acceleration potential (V ) of 3 – 8 kV, the
flight time of ions through a distance (L) of flight tube to reach a detector is calculated.
t = L(2V )−1/2 m
z
1/2
(5.25)
Thus, the mz−1
of ions is analysed by measuring their flight time in the analyser. Heavier ions
will have longer flight times in the tube. To measure the time of flight, precisely pulsed primary
ions should be used. The pulse is controlled by a highly accurate clock. The pulse periods are
typically in the order of 10 ns. The flight time of ions to the detector is electronically measured
and converted to mz−1
. In a pulse period, all the ions can be measured and a whole mass spectrum
can be obtained almost simultaneously. Again, the initial kinetic energy of secondary ions will
affect the resolution of mass analysis, because the velocity could be different when ions with the
same mz−1
enter the flight tube.