1
`
`UTC 2017
`General Electric v. United Technologies
`IPR2016-01289
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`

`ENGINEERING
`
`MATERIALS
`
`SCIENCE
`
`Milton 0hring
`Department of Materials Stience and Engineering
`Stevens Institute of Tethnolog'y
`Hoboi:en. New Jersey
`
`ok Belongs
`or DR. Clarke.
`3%“:3 8’ £2agh
`Barony It.
`
`
`
`ACADEMIC PRESS
`
`San Diego New York Boston London Sydney Tokyo Toronto
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`2
`
`

`

`
`
`Fm»! Cover Photograph: Single crystal superalloy turbine blades used in a jet
`aircraft engine (courtesy of Howmet Corporation).
`Back Cover Photograph: Pentium microprocessor integrated circuit chip
`(courtesy of INTEL Corporation).
`Both products rank among the greatest materials science and engineering
`triumphs of the 20th century.
`
`This book is printed on acid-free paper.
`
`Copyright © 1995 by ACADEMIC PRESS, INC.
`
`All Rights Reserved.
`
`No part of this publication may be reproduced or transmitted in any form or by any
`means. electronic or mechanical, including photocopy, recording, or any information
`storage and retrieval system, without permission in writing from the publisher.
`
`Academic Press, Inc.
`_
`A Division of Harcourt Brace 3c Company
`525 B Street, Suite 1900, San Diego, California 92101—4495
`
`United Kingdom Edition published by
`Academic PreSS Limited
`24—28 Oval Road, London NW1 TDX
`
`Library of Congress Cateloging-in—Publication Data
`
`Uhring, Milton, date.
`Engineering materials science 1' by Milton Gluing.
`p.
`cm.
`Includes index.
`ISBN 0-12—524995-0
`1. Materials science.
`Tit-103.037
`1995
`620. l ' l --dc20
`
`2. Materials.
`
`1. Title.
`
`9440228
`CIP
`
`PRINTED IN THE UNITED STATES OF AMERICA
`95969i‘989900DO98765432]
`
`3
`
`

`

`
`
`
`
`ELECTRONS IN ATOMS AND
`
`SOLIDS: BONDING
`
`
`
`It is universally accepted that atoms influence materials properties, but which
`subatomic portions of atoms (e.g., electrons, nuclei consisting of protons,
`neutrons) influence which properties is not so obvious. Before addressing this
`question it is necessary first to review several elementary concepts introduced
`in basic chemistry courses. Elements are identified by their atomic numbers
`and atomic weights. Within each atom is a nucleus containing a number of
`positively charged protons that is equal to the atomic number (Z ). Circulating
`about the nucleus are Z electrons that maintain electrical neutrality in the
`atom. The nucleus also contains a number of neutrons; these are uncharged.
`Atomic Weights _(M) of atoms are related to the sum of the number of
`protons and neutrons. But this number physically corresponds to the actual
`weight of an atom. Experimentally, the Weight of Avogadro’s number (N, =
`6.023 X 1033) of carbon atoms, each containing six protons and six neutrons,
`equals 12.00000 g, where 12.00000 is the atomic 'weight. One also speaks
`about atomic mass units (amu): 1 amu is one-twelfth the mass of the most
`common isotope of carbon, 12C. On this basis the weight of an electron is
`5.4858 X 10‘4 amu and protons and neutrons weigh 1.00728 and 1.00867
`amu, respectively. Once the atomic weight of carbon is taken as the standard,
`M values for the other elements are ordered relative to it. A mole of a given
`element weighs M grams and contains 6.023 X 1023 atoms. Thus, if we had
`
`29
`
`4
`
`

`

`30
`
`CHM‘TER 2
`
`ELECTHD'HS IN MUHS MlD SOLIDS: BUNDIHG
`
`only 1033 atoms of copper, by a simple proportionality they would weigh
`1!6.023 X 63.54 = 10.55 g (0.01055 kg). Note that the atomic weight of Cu,
`as well as most of the other elements in the Periodic Table including carbon,
`is not an integer. The reason for this is that elements exist as isotopes (some
`are radioactive, most are not}, with nuclei having different numbers of neutrons.
`These naturally occurring isotopes are present in the earth’s crust in differing
`abundances, and when a weighted average is taken, nonintegral values of M
`result. If compounds or moleCules (e.g., SiOl, GaAs, N1) are considered, the
`same accounting scheme is adopted eXCept that for atornic quantities we substi—
`tute the corresponding molecular ones.
`
`
`a. What weights of gallium and arsenic should be mixed together for the
`purpose of compounding 1.000 kg of gallium arsenide (GaAs) semiconductor?
`b. If each element has a purity of 99.99999 at.%, how many impurity atoms
`will be introduced in the GaAs?
`
`Note:.M(;, = 69.72 gimol, MA, = 74.92 gfmol, Mom = 144.64 gfmol.
`
`ANSWER a. The amount of Ga required is 1000 >< (69721144154) = 482
`g. This corresponds to 432169.72 or 6.91 mol Ga or, equivalently, to 6.91 X
`6.023 X 1013 _= 4.16 X 1034 Ga atoms. Similarly, the amount of As needed
`is also 6.91 mol, or 518 g. The equiatornic stoichiometry of GaAs means that
`4.16 X 1014 atoms of As are also required.
`b. Impurity atoms introduced by Ga + As atoms number 2 X (0.00001!
`100) X 6.91 X 6.023 X 1013 = 8.32 x10”. Because the total number 'of
`Ga + As atoms is 8.32 X 1024, the impurity concentration corresponds to
`10—7, or 1 part in 10 million.
`
`Returning to the subatomic particles, we note that electrons carry a negative
`charge of -1.602 X 10‘” coulombs (C); protons carry the same magnitude
`of charge, but are positive in sign. Furthermore, an electron weighs only
`9.108 X 10‘” g, whereas protons and neutrons are about 1840 times heavier.
`In a typical atom in which M = 60, the weight of the electrons is not quite
`0.03 “A: of the total weight of the atom. Nevertheless, when atoms form solids,
`it is basically the electrons that control the nature of the bonds between the
`atoms, the electrical conduction behavior, the magnetic effects, the optical
`properties, and the chemical reactions between atoms. In contrast, the sub-
`uuclear particles and even nuclei, surprisingly, contribute very little to the story
`of this book. Radioactivity, the effects of radiation, and the role of high-energy
`ion beams in semiconductor processing (ion implantation} are exceptions. One
`reason is that nuclear energies and forces are enormous compared with what
`
`5
`
`

`

`
`
`1.1. ATUHIC ELEETROHS lll SINGLE “OHS
`
`3|
`
`A
`
`Electron orbit
`
`Bohr radius
`(0.059 nm)
`(A) Model of a hydrogen atom showing an electron executing a circular orbit around a proton. (B) De
`Broglie standing waves in a hydrogen atom for an electron orhlt corresponding to n = 4.
`
`atoms experience during normal Processing and use of materials. Another
`reason is that the nucleus is so very small compared with the extent to which
`electrons range. For example, in hydrogen, the smallest of the atoms (Fig.
`2-1A),
`the single electron circulates around the proton in an orbit whose
`radius, known as the Bohr radius, is 0.059 nm long {1 nm 2 10—9 m = 10 A
`(angstroms)]. The radius of a proton is 1.3 X 10—5 nm, whereas nuclei, typically
`"ell/l“3 times larger, are still very much smaller than the Bohr radius. Before a
`pair of atomic nuclei move close enough to interact, the outer electrons have
`long since electrostatically interacted and repelled each other. The preceding
`considerations make it clear why the next topic addressed is the atomic elec-
`trons.
`
`Before the role of electrons in solids can be appreciated it is necessary to
`understand their behavior in single isolated atoms. The simplest model of an
`atom assumes it to be a miniature solar system at whose center is a positively
`charged nucleus (the sun) surrounded by a cloud of orbiting negatively charged
`atomic electrons (the planets). Charge is distributed so that the atom is electri-
`cally neutral. We now know these atomic electrons display a complex dynamical
`behavior (the larger the Z, the more cornplex the behavior) governed by the
`laws of quantum mechanics. The underlying philosophy and mathematical
`description of quantum theory are quite involved. Nevertheless, the resulting
`concepts and laws that derive from them can be summarized for our purposes
`in terms of a few relatively simple equations and rules that are discussed in turn.
`
`2.1L WavelParticle Duality
`
`The term wavefpariicle duality suggests that both particles and waves have
`a dual nature. For the most part particles (a baseball, a stone, an electron)
`obey the classical Newtonian laws of mechanics. Occasionally, however, they
`
`
`
`
`
`
`Nucleus (proton)
`
`a-
`319mm“
`
`6
`
`

`

`
`
`32
`
`CHhPTER 1
`
`ELEC'TRDNS IN ATDMS AND SBLIDS: BGNDIHG
`
`reveal another, mysterious side of their character and behave like waves. An
`electron speeding down the column of an electron microscope produces such
`wavelike diffraction effects upon interaction with materials. Similarly, waves
`that exhibit standard optical effects like refraction and diffraction can surpris-
`ingly dislodge electrons from metals by impinging on them under certain tendi-
`tions. Our first instinct is to attribute this photoelectric effect to a mechanical
`collision, attesting to the particle-like nature of waves. Wavefparticle duality
`is expressed by the well-known de Broglie relationship
`
`A = Mp = Mme.
`
`(2-1)
`
`In this important formula A is the effective wavelength, andp is the momentum
`of the associated particle whose mass is m and velocity v. Planck’s constant b
`(is = 6.62 X 10‘“ J-s) hears witne55 to the fact that quantum effects are at
`play here.
`The de Broglie relationship provides a way to rationalize the stability of
`electron orbits in atoms. Classically, circulating electrons ought to emit electro—
`magnetic radiation, lose energy in the process, and spiral inward finally to
`collapse into the nucleus. But this does not happen and atoms survive in so-
`called stationary states. Why? If the de Broglie waves associated with the
`electron in hydrogen were standing waves, as shown in Fig. 2-1B, they would
`retain their phase, and orbits would persist intact despite repeated electron
`revolutions. But, if the waves did not ciose on themselves, they would increas-
`ingly interfere destructiver with one anorher and, with each revolution, move
`more and more out of phase. Electron orbit disintegration would then be
`inevitable. This same notion of standing waves is used again later in the chapter
`(Section 2.4.3.2) to derive the energies of electrons in metals.
`
`
`
`251,2, Quantited Energies
`
`We know from experience that both particles (objects) and waves possess
`energy. The kinetic energies of gas molecules and the heating effects of laser
`light are examples. It outwardly appears that the energies can assume any
`values whatever. But this is not true. A fundamental law of quantum theory
`holds that energies of particles and waves, or more appropriately photons, can
`assume on I}; certain fixed or quantized values. For photons, the energy is given
`by Planck’s formula
`
`E = by = bent.
`
`{2.2)
`
`Here n is the frequency of the photon which travels at the speed of light 5,
`where c = wt (6 = 2.998 X 108 mils). Interestingly, the vibrations of atoms
`in solids give rise to waves or phonons, whose energies are given by a simi-
`lar formula:
`
`E=(n+&)bv,
`
`n=1,2,3,....
`
`(2-3}
`
`7
`
`

`

`1.1. ATUHIE ELECIEONS Ill SINGLE MOMS
`
`3]
`
`In this expression :1 is a quantum number that assumes integer values. This
`means that solids absorb and emit thermal energy in discrete quanta of by
`when their atoms vibrate with frequency v. In solids, v is typically 1013 Hz.
`More relevant to our present needs is the fact that the energies of electrons
`in atoms are also quantized, meaning that they can assume only certain discrete
`values. The case of the single electron in the hydrogen atom may be familiar
`to readers. Here the energy levels (E) are given by the Bohr theory as
`
`—27rzm,q’l g -13.6
`hlnl
`n1
`
`(2—4)
`
`E:
`
`eV,
`
`n=1,2,3,....
`
`The electron mass and charge are m, and g, respectively. Again, it is an integer
`known, in this case, as the principal quantum number. The closer they are to
`the nucleus, the lower the energies of the electrons. And because the electron
`energy inside the atom is less than that outside it {where the zero-energy-level
`reference is assumed}, a negative sign is conventioually used. Resulting energy
`levels are enumerated in Fig. 2—2. At any given time only one level can be
`occupied. In an unexcited hydrogen atom the electron resides in the ground
`state (it = 1) while the other levels or states are vacant. Absorption of 13.6
`eV of energy (1 eWatom = 1.60 X 10’15‘]i’atom = 96,500 jimol) will excite
`the electron sufficiently to eject it and thus ionize the hydrogen atom.
`For other atoms a very crude estimate of the electmn energies is given by
`
`E = —13.6Z3inl
`
`eV,
`
`n = 1, 2, 3, ...,
`
`{2-5)
`
`Where Z, the atomic number, serves to magnify the nuclear charge. The effect
`of core electron screening of the nuclear charge is neglected in this formula.
`
`2.2.3. The Pauli Principle
`
`In reality, multielectron atoms and hydrogen as well are more complicated
`than the simple Bohr model of Fig. 2-1. Electrons orbiting closer to the nucleus
`shield outer electrons from the ,pull of the nuclear charge. This complicates
`their motion sufficiently that different electrons are not at the same energy
`level; furthermore, electron energies are not easily calculated. In general, more
`advanced theories indicate that the electron dynamics within all atoms is charac-
`terized by four quantum numbers n, l, m, and 5. These arise from solutions to
`the celebrated Schrodinger equation, a cornerstone of the modern quantum
`description of atoms. Specifically, the threeedimensional motion of electrons
`is embodied in the quantum numbers n, I, and m.
`The principal quantum number is still it and it can assume only integer
`values 1, 2, 3, ..., as before- Electrons are now organized into shells. When
`n = 1 we speak of the K electron shell, While for the L and M shells, in = 2
`and n = 3, respectively. As will become evident after introduction of the other
`quantum numbers, there are 2 electrons in the K shell, 8 in the L shell, 18 in
`the M shell, and so on.
`
`8
`
`

`

`34
`
`[HAPlEll
`
`2
`
`ELEEIRONS Ill ATUHS MID SOLIDS: BONDING
`
`
`n~oo
`. Vacuum
`
`level
`
`
`
`
`Paschen
`
`‘1
`
`0.85eV
`
`—1.5 eV
`
`n=5
`
`n=4
`
`=3
`
`n=2
`
`
`Balmer
`
`~34 eV
`
`>.
`E’to1:
`LL]
`
`FIGURE 2-2
`
`
`
`n=1
`
`—13.B EV
`(Ground state)
`Lyman
`Electron energy levels in the hydrogen atom. The ground state corresponds to the electron level
`n = II with E = - |3.6 eV. For the n = I: level, corresponding to E = U. the electron has gained the
`|3.6 eV energy needed to ionize hydrogen Electron transitions from upper levels to the n = 1
`level
`are known to spectroscopiscs as the Lyman series; to the n = 2 level. the Balmer series; to the n = 3
`level. the Paschen series.
`
`The angular momentum arising from the rotational motion of orbiting
`electrons is also quantized or forced to assume specific values which are in the
`ratio of whole numbers. Recognition of this fact is taken by assigning to the
`orbital quantum number 1 values of 0, 1, 2, ..., n - 1. The shape of electron
`orbitals is essentially determined by the l quantum number. When I = 0 We
`speak of s electron states. These electrons have no net angular momentum,
`and as they move in all directions with equal probability, the charge distribution
`is spherically symmetrical about the nucleus. For I = 1, 2, 3, ..., we have
`corresponding p, d, f,
`states.
`A third quantum number, m, specifies the orientation of the angular momen-
`
`9
`
`

`

`
`
`2.2. AlflHlE ELECTRONS Ill SINGLE NONE
`
`35
`
`turn along a specific direction in space. Known as the magnetic quantum
`number, m takes on integer values between +l and “i, that is —l, —i + 1, ...,
`+l — 1, +l.
`Lastly there is the spin quantum number, s, in recognition of the fact that
`electrons spin as they simultaneously orbit the nucleus. Because there are only
`two orientations of spin angular momentum, up or down, in assumes —%
`and +i values. We return to electron spin in Chapter 14 in connection with
`ferromagnetism.
`The Pauli principle states that no two electrons in an atom can have the
`some four quantum numbers.
`Let us applyr the Pauli principle to an atom of sodium. As Z = 11 we have
`to specify a tetrad of quantum numbers (n, l, m, s) for each of the 11 electrons:
`
`15 states {K shell)
`Zs states (L shell)
`2;; states it shell)
`
`35 state
`
`(M shell)
`
`(1, 0, 0, +é) and (1, 0, U, —%l;
`(2, 0, O, +1§l and (2, O, 0, —%);
`(2, 1, 0, +s), (2, 1,0,~s),(2,1,1,+s),(2,1, 1, er),
`(2) 1: ‘13 +95 and [2: 1: _]:—31)5
`(3, 0, O, +i).
`
`Another way to identify the electron distribution in sodium is 153 233 2,06
`35' and similarly for other elements.
`in shorthand notation the integers and
`letters are the principal and orbital quantum numbers, respectively, and the
`superscript number tells how many electrons have the same 11 and i values.
`
`2.1.4. Electron Energy Level lransitions
`
`The atomic model that has emerged to this point includes a set of electrons,
`each having a unique set of numbers that distinguishes it from others of the
`same atom. Furthermore, the energies of the electrons depend primarily on n
`and, to a lesser extent, on i. This means that an electron energy level scheme
`like that for hydrogen (Fig. 2-2) exists for every element. In the case of hydrogen
`the electron in the ground state (it = 0) can make a transition to an excited
`state provided (1) the state is Vacant, and (2) the electron gains enough energy.
`The energy needed is simply the difference in energy between the two states.
`Likewise, energy is released from a hydrogen atom when it is deexcited, a
`process that occurs when an electron descends from an occupied excited state
`to fill an unoccupied lower-energy state. Photons are frequently involved in
`both types of transitions as schematically indicated in Fig. 2-3- If the energy
`levels in question are E1 and E1, such that E3 > El, then
`
`As a computational aid, if A5 is expressed in electron volts and A in microme-
`ters, then
`
`as (eV) = 1.24m (pm).
`
`{2—7)
`
`10
`
`10
`
`

`

`36
`
`CHAPTER 2
`
`ELECT-RUNS IN MOMS MID SOLIDS: BONDING
`
`A
`
`Before
`
`After
`
`52— _-L
`
`hvjzm
`
`Excitation
`
`E1 _IL
`
`B
`
`E_
`EZ—J—
`
`Lie-excitation
`
`|WhV1a
`
`e
`
`Sd1ematic representation of electron transitions between two energy levels: (A) Excitation. (B) Deexci-
`Latina.
`
`EXIMFLE 2-1
`
`What is the wavelength of the photon emitted from hydrogen during an
`n = 4 to it = 2 electron transition?
`
`ANSWER Using Eq. 24, AB = E, — E; = —13.6(1i41 - 1f22) = 2.55 eV.
`From Eq. 2-7, it = 1.24IAE = 1.24i’2‘55 = 0.486 pm. This wavelength of
`486 nm, or 4860 it, falls in the blue region of the visible Spectrum.
`
`2.2.5. Spatial Distribution of Atomic Electrons
`
`It makes a big difference whether electrons move freely in a vacuum as
`opposed to being confined to orbit atoms. For one thing, in the former case
`electron motion is usually viewed in classical terms. This means that the electron
`can have any energy; there is no quantum restriction that limits energies to
`only certain fixed values as is the case for electrons confined to atoms. We also
`imagine that the actual location of classical electrons is very precisely limited
`to coincide within their intrinsic dimensions. According to quantum mechanics
`and, in particular, the Heisenberg uncertainty principle,
`
`Ap - ex = M217,
`
`(2-3)
`
`this view is too simplistic. Rather, Eq. 2-8 suggests that the more precisely the
`electron momentum p is known, the less we know about its position x, and
`vice versa. The A’s in Eq. 2-8 represent the uncertainties in p and x.
`A cardinal feature of the wave mechanics approach in quantum theory is
`the notion that electron waves can have a presence or time-averaged charge
`
`11
`
`11
`
`

`

`1.3. FlNGEHPRINl’ING MOMS
`
`3?
`
`density that extends in space, sometimes well beyond atomic dimensions. We
`speak of wave functions that mathematically describe the regions where the
`electronic charge can most probably be found. If they could be observed with
`a microscope, the picture would be strange and bewildering For example,
`pictorial representations of the geometrically complex charge distributions in
`hydrogen-like s, p, and (3 states are shown in Fig. 2-4. It takes some doing to
`imagine that a single electron can split its existence between two or more
`distinct locations simultaneously.
`
`
`
`The previously described electronic theory of atoms is not only the subsrance
`of textbooks and the Culture of science. A number of very important and
`sophisticated commerical instruments have appeared in recent years that capi—
`talize on the very cancepts just introduced. Their function is to perform qualita—
`tive as well as quantitative analysis of atoms in very localized regions of
`solids. Developed largely for the semiconductor industry to characterize the
`composition of electronic materials and devices, their use has spread to include
`the analysis of all classes of inorganic and organic, as well as biological,
`materials. In this section we focus on the emission of both X rays and electrons
`and learn how to fingerprint or identify the excited atoms from which they
`originate. Optical spectrosmpy, a science that has been practiced for a long
`time, also fingerprints atoms but by measuring the wavelength of visible light
`derived from low-energy, outer—electron-level transitions. In contrast X-ray
`spectroscopy, which we consider next, relies on emission of X rays from the
`deeper, high-energy core electron levels.
`
`1.31. X~Ray Spectroscopy
`
`There is hardly a branch of scientific and engineering research and develop—
`ment activity today that does not make use of a scanning electron microscoPe
`(SEM), shown in Fig. 2-5; many examples of remarkable SEM images are
`found throughout this book and others have been widely reproduced in the
`printed and electronic media. The SEM is more fully discussed in Section 3.6.3.
`Our interest in the SEM here is as a source of finely focused electrons that
`impinge on a specimen under study. This causes electronic excitations within
`a volume that typically extends 1000 nm deep and measures about 108 nm3.
`Typical bacteria and red blood cells are considerably larger. Within the volume
`probed the chemical composition of impurity particles, structural features, and
`local regions of the matrix can be analyzed. The incident (~30-keV) electrons
`are sufficiently energetic to knock out atomic electrons from previously occu-
`pied levels as schematically indicated in Fig. 2-6. An electron from a more
`
`12
`
`12
`
`

`

`38
`
`CHAPTER I
`
`ELEE-TRDHS lH MOMS hill] SDLlDS: BONDFHG
`
`Z
`
`5 states
`
`
`
`“GU RE 2_4
`
`Pictorial representation of the charge distribution in hydrogen-like s. p. and d wavefunctions. sstates are
`spherically symmetrical, whereas p states have two charge lobes, or regions of high electron density
`extending along the axes of a rectangular coordinate system. d states typically have four charge lobes.
`
`0‘ states
`
`13
`
`13
`
`

`

`2.3. FINGEHPREHTIHG MOMS
`
`-
`
`39
`
`«Ana—1.4.
`
`: '|
`
`4—
`
`- __. v.-
`l-
`
`.'I'
`
`it
`
`I
`
`Photograph of a modern scanning electron microscope with X-ray energy dispersive analysis detector.
`The left- and right-hand monitors display the X-ray spectrum and specimen image. respectively. Operation
`of the SEMI is controlled by a personal computer. Courtesy of Philips Electronic Instrumenu Company.
`a Division of Philips Electronics North America Corporation.
`
`energetic level can now fall into the vacant level, a process accompanied by
`the emission of a photon.
`To see how this works in practice, consider the electron energy level diagram
`for titanium metal depicted in Fig. 2-7A, where values for the energies of K,
`L, and M electrons are quantitatively indicated.
`
`
`
`If an electron vacancy is created in the K shell of Ti and an L3 electron fills
`it, what are the energy and wavelength of the emitted photon?
`
`ANSWER This problem is identical in Spirit to Example 2-2. The energy of
`the photon is EL} - BK = [—4555 - ("4966.4” = 4510.9 eV. By Eq. 2-7,
`this corresponds to a wavelength of 2.75 X 10—4 p.111, or.0.275 rim. Therefore,
`the photon is an X ray spectrally identified as Kl1 . The X-ray emission spectrum
`diagram for Ti is shown in Fig. 2-7B.
`
`
`
`
`
`14
`
`14
`
`

`

`
`
`40
`
`CHi‘il’iEll
`
`2
`
`ELECTNDNS IN MOMS AND SOLIDS: BONDING
`
`B
`
`
`
`A V
`
`acuum
`
`M
`
`L23 :lIIZEEI:
`
`L1 ——o—o—
`
`K v—O—I—-—
`
`lnilial state
`
` 28
`
`“We
`
`.z'r
`
`Electron ejected
`
`C
`
`ho = EK— EL1
`
`Em = EK— EL1— EL!
`
`
`
`X-ray emission
`
`
`
`—o—o———
`
`Auger electron
`emission
`
`FIG U R E L6 Schematic representation of electron energy transitions: (A) Initial state. (B) Incldent photon or electron
`eject: K-shell electron. (C) X-ray emission when 25 electron fills vacant
`level. (D) Auger electron
`emission process
`
`15
`
`15
`
`

`

`2.3‘ FINGERFRINTIHG MOMS
`
`'
`
`4i
`
`
`500
`
`387 418
`_I___.L—|__
`300
`400
`
`Electron energy, eV
`
`
`
`EDX
`
`Kn
`
`_|_.___l—L____
`4.0
`4.5
`5.0
`
`X-ray energy, keV
`
`Fl 2;} Electron excitation processes in titanium. (A) Energy level scheme. All electron energies are negative in
`magnitude (B) X-ray emission spectrum of Ti. Only the K, and K1: lines are shown. (C) Auger electron
`spectral lines for Ti.
`
`16
`
`16
`
`

`

`42
`
`[HAPTEE I
`
`ELECTRON!
`
`IN MOMS AND SOLIDS: BGHDIHG
`
`During analysis, the electron beam actually generates electron vacancies in
`all levels. Therefore, many transitions occur simultaneously, and instead of a
`single X—ray line there is an entire spectrum of lines. Each element has a
`unique X—ray spectral signature that can be used to identify it unambiguously.
`Importantly, high~speed pulse processing electronics enables multielement anal-
`ysis to be carried out simultaneously and in a matter of seconds. X rays emitted
`from the sample are sensed by a nearby cryogenically cooled semiconductor
`photon detector attached to the evacuated SEM column and analyzed to dctEr-
`mine their energy spectrum. Results are displayed as a spectrum of signal
`intensity versus X-ray energy, hence the acronym EDX (Energy Dispersive
`X ray) for the method. For multielectron atoms, Eq. 2—5 suggests that core
`levels for many useful elements can be expected to have X—ray energies of tens
`of kilo-electron volts. Extensive tables of the X-ray emission spectra of the
`elements exist to aid in materials identification.
`
`An allied technique makes use of incident energetic X-ray or gamma ray
`photons, rather than electrons. They also induce identical electron transitions
`and X—ray emission, so that from the standpoint of identification of atoms,
`there is no difference. This so—called X-ray fluorescence technique is used when
`the specimen cannot withstand electron bombardment or when it is not feasible
`to place it into the vacuum chamber of an SEM. Chemical analyses of pigments
`in oil paintings and inks in paper currency have been made this way to ex-
`pose forgeries.
`
`2.3.2. Auger Electrori Spectroscopy
`
`Electron transitions within and between outer and core electrons are involved
`in the technique of Auger electron spectroswpy (AE5), as indicated ianig.
`2~6D. But, unlike EDX where X~ray photons are emitted, so called Auger
`(pronounced oh—zhay) electrons are released in ABS. A low—energy electron
`beam (~2 keV) impinges on the specimen and creates the initial vacancy that
`is filled by an outer electmn; however, the photon that is normally created
`never exits the atom. Instead, it transfers its energy to another electron, the
`Auger electron. A collection of Auger electrons with varying intensity and
`unique energies does emerge to establish the Wiggly spectrum that is the unam-
`biguous signature of the atom in question. The AES spectrum for Ti is shown
`in Fig. 2-7C. Because core and not valence (or chemical bonding) electrons are
`involved in both EDX and AES, it makes little difference in which chemical
`state the Ti atoms exist: pure Ti, TiAl3, TiOl, ((1351-15)l Till, and so on. Titanium
`is detected independently of the other elements to which it is bonded.
`What makes ABS so special is that the low—energy Auger electrons can
`penetrate only ~1—2 mm (10—20 A) of material. This means that AES is limited
`to detection of atoms located in the uppermost 1- to Z-nm-thiek surface layers
`of the specimen. Because it is a surface-sensitive technique, extreme cleanliness
`and a very high vacuum environment are required during analysis to prevent
`
`17
`
`17
`
`

`

`2.4. ELE£TROHS IN HOLHULES AND SOLIDS
`
`'
`
`43
`
`
`
`m Photograph of an Auger electron spectrometer. Courtesy of Perkin—Elmer Physical Electronics Division.
`
`further surface contamination. The atomic detection limit of both EDX and
`ABS techniques is about 1%. In the case of ABS where the incident electron
`beam spot size is ~100 nm, the volume sampled for analysis is [77 (100)2f4] X
`1 = 7850 nm3, which corresponds to about 400,000 atoms. Thus, under
`optimum conditions it is possible to detect about 4000 impurity atoms in the
`analysis. This astounding capability does not come cheaply, however. The
`commercial AES spectrometer shown in Fig. 2—8 costs more than half a million
`dollars. Auger analysis is used to determine the composition of surface films
`and contaminants in semiconductors, metals, and ceramics. An example of its
`use is shown in Fig. 2—9.
`Despite the fact that sophisticated analytical methods have been introduced
`here, both EDX and ABS rely on energy transitions involving core electron
`levels. In closing, our debt to the framers of the quantum theory of electrons
`in atoms should be appreciated.
`
`1:4. "ELECTIIOHS. illi'MOLECUlES MID "SOLIDS
`
`2.4.l. Forming a Hydrogen Molecule
`
`What happens to the atomic electrons when two or more widely separated
`atoms are brought together to form a molecule? This is an important question
`
`18
`
`18
`
`

`

`44
`
`CHAPTER 2
`
`ELEETROHS ill ATDHS AND SOLlDS: BONDING
`
`POINT 1
`
`ALUMINUM
`BOUNDARY
`
`
`SAW CUT
`
`
`6.00
`
`4.00
`
`AESsignal
`
`2.00
`
`0.00
`30.00
`
`430.00
`
`830.00
`
`1230.00
`
`1630.00
`
`2030.0!3
`
`Electron energy. eV
`
`I: I G U RE 2_9
`
`Case study involving AES. In hot-dip galvanizing. steel acquires a protective zinc coating after being dipped
`into a molten Zn bath. The iron and zinc react to form a compound that sometimes flakes off during
`fabrication. To prevent Fe—Zn reaction. 0. l0% aluminum is added to the Zn bath. The very thin Al-rich
`layer that forms prevents the two metals from reacting. (A) The test specimen in this application shown
`schematically.
`(B) A magnified image of the Al-rich layer is analyzed at point
`I. (C) The resulting
`AES spectrum detected at point
`I revealsthe presence of Al. Courtesy of Perkin—Elmer Physical
`Electronics Division.
`
`19
`
`
`
`19
`
`

`

`
`
`2.4. ELEURDHS IH MOLECULES Mil] SOLIDS
`
`45
`
`because by extension to many atoms, we can begin to understand condensation
`to the solid state. Whatever else happens, however, there must be a final overall
`reduction in the total energy of individual atoms when they interact and form
`stable molecules or solids. The quantum theories involved are very complex.
`One approach stresses replacement of atomic orbitals or wavefunctions by an
`equal number of molecular orbitals. To see how this works, let us consider
`two distant atoms of hydrogen (1-1) that are brought together to form a hydrogen
`molecule (H3). Focusing only on the spherically symmetric ls charge distribu—
`tion, we note little change in each atom as long as they are far apart. But when
`they move close enough so that the two is charge clouds overlap, the rules of
`quantum chemistry suggest two new molecular charge distributions. In the
`first of these, the electron density is enhanced in the region between the nuclei
`(Fig. Z-IOA). This preponderance of negative charge binds the positively
`charged nuclei together; an energy reduction occurs in this so-called bonding
`orbital. In Fig. 2—103 the second possibility is shown. Here the electron charge
`density is enhanced on the side of each nucleus away from the other nucleus.
`Because little negative charge is left between the positive nuclei, the latter
`strongly repel each other on approach, and the energy rises. This is the case
`of the antibonding orbital. The overall electron energy versus distance of ap-
`proach for both orbitals is shown in Fig. 2—1 0C. Ofnote are the energy minimum
`and stable molecular configuration for the bonding orbital. But when the
`nuclei move too close, their mutual electrostatic repulsion begins to exceed the
`attraction caused by the electron density between them, and the energy rises.
`
`
`
`I -
`
`A
`
`Bonding
`orbital
`
`Aniibonding
`orbital
`
`C \:5
`
`2'IDC
`UJ
`
`
`
`Distanoe(nn1} —+
`
`HGURE 2-10
`
`(A) Representation of electron cloud contours in the two-hydrogen-atorn bonding orblt'al. There is a
`high electron density between the protons. (B) Representation of electron cloud contours in the two—
`hydrogen-atom antibonding orbital. The electron density between the protons is low. (C) Electron
`energies of the bonding and antibonding orbitals as a lFunction of the internuclear distance.
`
`20
`
`20
`
`

`

`46
`
`[HAPTER 1
`
`ELEC‘TROHS IH ATBHS AND SOLIDS: BONDING
`
`In Summary, recurring themes in the remainder of this chapter are:
`
`1. The splitting of one energy level into two (or more generally, N levels
`for N electrons)
`2. The minimum in the energy at an interatomic distance corresponding
`to the stable molecule (or solid)
`3. The rise in energy when the nuclei approach too closely.
`
`1.4.2. Electron Transfer in [unit Molecules
`
`
`
`At a higher level of