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OCR for page 127
6
Defects ant! Diffusion
INTRODUCTION
The field of defects and diffusion in solid materials is concerned with
the structure of possible Haws in otherwise homogeneous materials and
with their observable consequences in the properties of materials. Of
particular interest are point defects, which are fairly localized on an
atomic scale; line defects, such as dislocations; and boundary defects,
such as surfaces, stacking faults, and grain boundaries. Although it is
one of the older fields in condensed-matter physics and materials
sciences, it remains an attractive arena for the observation and
description of new physical phenomena and therefore maintains an
enduring interest of physicists, in addition to researchers with other
scientific orientations.
An indication of the vigor that defect concepts still possess is that
they mold the physical understanding of many phenomena in appar-
ently unrelated fields that are of interest in condensed-matter physics
today. Several examples serve to make this point:
Solitons made their appearance in defect physics as lattice disloca-
tions. Coupled electron-lattice complexes have long been exemplified
by self-trapped holes and other polarons. These ideas have recently
been combined into the unexpected form of the coupled electron-lattice
127
OCR for page 128
128 A DECADE OF CONDENSED-MA TTER PHYSICS
modes that form the soliton charge carriers that have excited interest in
connection with the transport properties of conducting polymers such
as polyacetylene.
Two-dimensional structures have become the focus of much recent
activity for their unusual characteristics with regard to melting and
phase transitions. The melting of adsorbed overlayers, or equivalently
of impurity planes intercalated in layered compounds, to form a hexatic
"floating raft" phase, is described in current theory by the thermally
activated dissociation of dislocation dipole pairs. Similarly, the theo-
retical building blocks of the roughening transitions of surfaces and
interfaces are just the steps and jogs of classical model surfaces.
Textures in liquid crystals, which led to dislocation descriptions of
defective crystals, echo in the topical description of structure both in
solids that support incommensurate charge-density waves, through
discommensuration structures, and in the structural characteristics of
the magnetic phases of liquid The that exist only below 1 mK.
Local tunneling systems have long been the model for inversion of
molecules and for the pocket states of centers tunneling among
equivalent configurations in crystals. They return in recent advances as
the central characteristic causing the linear specific heat and dynamical
echo phenomena of apparently all amorphous solids (metals, ceramics,
and polymers) and many disordered solids (e.g., the ~ aluminas).
Internal fric lion from stress-induced changes of defect structures,
first understood for such classic systems as C in Fe and the damping of
brass reeds, is now applied to the analysis of both backbone and
side-chain effects in polymeric materials and even in natural
carbonaceous materials such as coal and amber.
While chosen primarily for their critical importance in research fields
of current interest in solid-state physics, these examples do indicate the
flow of seminal ideas back and forth between areas that are clearly
physics and those that are not and the way in which new subfields have
emerged from areas of defect physics. Because it links directly to
practical materials, the field of defects and diffusion exhibits these
interconnections to a high degree, and accordingly presents the great-
est problems for concise summary.
NEW FIELDS FROM OLD: AN EXAMPLE
A wide variety of choice is available to exemplify the growth of new
subfields from areas of defect physics. For example, the surfaces of
crystals, and the faults, reconstruction, steps, and other configurations
adopted by them, now form a separate field discussed in Chapter 7.
OCR for page 129
DEFECTS AND DIFFUSION 129
The evolution of new vigorous subfields will be exemplified here
through brief descriptions of three fertile subfields of particle-beam
irradiation of materials.
Phase Microstructure and Phase Generation in Radiation Fields
In the past few years it has become apparent that when energetic
radiation produces atomic displacements in solid materials it not only
generates defects, defect aggregates, dislocations, and voids but may
also give rise to phases that are not present without prior exposure to
radiation. Here we mean phases in the thermodynamic sense of
spatially bounded regions with distinct compositions and/or physical
properties and with abrupt interphase boundaries. These phases are
truly radiation induced, in the sense that they are thermally unstable in
the absence of radiation. They thus differ from radiation-enhanced
phases, which are merely thermodynamically stable phases that un-
dergo more rapid formation when assisted by radiation-induced mix-
~ng.
Radiation-induced phases have been widely observed in binary and
multicomponent alloys, including semiconductors. They occur also in
insulating solids in the form of metal colloids in NaC1 or amorphous
islands in crystalline quartz, for example. Several possible causes for
their formation can be suggested. One is that the excess point defects
produced and maintained by the radiation field may favor a state or
solid phase different from the one that is stable in the absence of
radiation. Examples of this are the formation of amorphous silicon, of
highly supersaturated crystalline solid solutions, and of the disordered
state of ordered alloys, all during irradiation. These states are retained
after irradiation when the thermal reordering is sufficiently slow. It is
not yet known whether systems exist for which a transformation from
one crystalline phase to a different phase is made energetically favor-
able solely because excess defects are present in either or both of the
phases. An alternative mechanism involves the way that radiation-
induced segregation typically redistributes the components of a solid
on a microstructural scale of 10-3 to 10 ~m. There are two causes for
this redistribution: first, persistent defect fluxes are set up during
irradiation, and, second, certain components couple preferentially to
the defect fluxes. Changes of composition then shift the system locally
into a region of the phase diagram that differs from that occupied by the
overall homogeneous alloy. A new radiation-modified equilibrium
phase may then precipitate locally, or an existing equilibrium phase
may dissolve.
OCR for page 130
130 A DECADE OF CONI>ENSED-MA TTER PHYSICS
Surface and Near-Surface Probes
Recent years have seen the development of important methods for
the microstructural and microchemical analysis of the surface and
near-surface constitution of solid materials. Particle-beam methods
have so revolutionized analytical science in these areas that the first
o
10,000 A (1 ~m) of a material structure can now be analyzed with a
chemical sensitivity often approaching 1 part in 107, with a depth
resolution of 100 A, or with a lateral resolution of 0.1 Em or better.
Different techniques are complementary in depth, spatial, or chemical
resolution; experts have therefore learned to use an arsenal of powerful
new particle-beam methods to obtain detailed near-surface chemical
analysis of materials structures.
Ions with energy greater than 100 eV incident on a crystal penetrate
the surface and dissipate their energy and momentum. By detecting the
x rays caused by the collisions it is possible to detect trace impurities
at the level of 1 in 108 in favorable cases. Collisions also eject surface
atoms from the material as secondary atoms or charged ions. By
detecting the surface chemical species as the material is sputtered away
it is then possible to determine the original microchemistry of the
material. The detection can be performed by secondary-ion mass
spectrometry (SIMS), in which the secondary species are fed into an
isotope-imaging mass spectrometer, or by Auger spectroscopy of core
levels excited by an auxiliary electron beam. The resolution limit
perpendicular to the surface is ~100 A. Lateral resolution is limited at
present to about 1 lam in SIMS and 0.05 lam in Auger probes. In
chemical sensitivity SIMS can often achieve a level of 1 part in 106 or
better. By Auger methods the sensitivity is reduced to 1 part in 103, but
the depth resolution may be improved to a few atomic layers since the
Auger electrons from deeper layers are scattered and lost. Sputtering
then allows a three-dimensional chemical map of the surface region to
be acquired.
The mixing that takes place through the top 100 A or so while atoms
are being sputtered away causes a steady-state nonuniform subsurface
concentration profile to develop even in a chemically uniform bulk
material.
In a second category of technique, the incident and detected fluxes
involve the same species. The incident beam, usually H+ or He+,
impinges on the surface. Part is reflected by Rutherford backscattering
from atoms in the subsurface region and with an energy transfer that
depends first on the target mass and second on the depth to which the
OCR for page 131
DEFECTS AND DIFFUSION 13 1
particle penetrates. The spectrum of reflected particle energies con-
tains information about both depth and chemical structure that can be
separated to obtain accurate chemical information with a depth reso-
lution of about 100 A. Alternatively, channeling methods can be used
to probe the position of an atomic species in the lattice structure. When
the beam is directed along a crystallographic axis so that the particle
range is long, atoms located off crystal lattice sites scatter particles into
other channels and into the bulk material. In this way, the location of
off-site atoms can be determined with some precision. It is also
possible to measure the misfit at the heterojunctions between different
crystals by this method.
Ion-Beam Microfabrication
As described in Appendix C, new types of crystals, compounds,
alloys, microstructures, and other materials can bring with them
formerly unimagined opportunities for scientific and technological
advancement. This is particularly true of microfabrication, in which
components are chemically and structurally tailored for particular
application on a microscopic scale. Several important methods are
based principally on ion-beam methods. Here we mention three areas
of major effort:
Ion implantation is the process in which foreign dopant or alloying
ions are accelerated to energies of typically 1-300 keV and implanted
into the near-surface regions of target materials. Implantation depths
are typically of the order of 1 ~m, depending on the implantation
energy, and the profiles are reasonably well understood. The compo-
sitions achieved by implantation are not constrained by usual thermo-
dynamic or kinetic limitations. As ions penetrate the solid they create
lattice displacements, and the material is damaged by the implanting
beam. Further manipulation of the resulting nonequilibrium structure is
then often desirable.
Ion-beam mixing often involves a surface layer, typically a few
hundred angstroms thick, of one material being mixed with a bulk
substrate of a second material by means of a penetrating ion beam. The
advantage of this procedure is that higher concentrations of new alloy
phases may be achieved using only a small fraction of the irradiation
fluence normally needed for ion implantation. The effects of sputtering,
cascade mixing, radiation-induced segregation, and radiation-
enhanced diffusion remain important but are as yet imperfectly under-
stood.
OCR for page 132
132 A DECADE OF CONDENSED-MA TTER PHYSICS
Lithography has been used in the semiconductor industry for many
years. It proceeds by damaging a chemically resistive material using a
photon, electron, or ion beam and then preferentially etching away
either the damaged or undamaged region and the substrate beneath it.
Complex patterns with resolution of about 0.1 Am may be inscribed
into semiconductors for device fabrication by these methods.
CALCULATIONS OF DEFECT STRUCTURE
A decade ago no reliable procedures were available by which
accurate calculations of cohesion could be carried out for most solids.
The exceptions to this statement were highly ionic solids and, to some
degree, simple metals. It is now possible to make calculations that
correctly indicate the small differences of relative energy among
alternative bulk crystal structures of metals and covalent compounds.
The energy of a surface and its electronic structure can be calculated
quite well.
A mainstay of defect calculations for two decades has been the
modeling of crystal configurations using energies derived from model
interactions among atoms. A large crystallite with appropriate bound-
ary conditions is used. These methods have played a major role in the
development of fundamental ideas about defect structure in simple
materials. The practice for metallic systems has been to sum pairwise
potentials, and this has some measure of validity although the model
clearly lacks a rigorous basis. Nevertheless, qualitatively useful studies
even of such complicated defects as impurities bound in mixed-
dumbbell interstitials have been forthcoming. For ionic materials,
elaborate codes have been developed to add appropriate treatments of
core polarizability to the coulombic and core-core interactions, in
order to simulate the total energy of a configuration more accurately.
By whatever method, the energy as a function of configuration is finally
computed, and the result provides the input for calculations of molec-
ular dynamics and properties of relaxed point defects or surfaces, for
example.
If not highly precise, these methods can nevertheless often repro-
duce systematic trends in data such as F-center excitation energies and
Schottky pair energies with absolute values within 20 percent of those
observed. Long experience, the systematic elimination of errors, and
fine tuning of the codes have made the procedures reliable for ionic
materials such as alkali halide and alkaline earth fluorides, which have
large excitation energies and hence stable polarizabilities. Advances
over the past decade include reasonable description of defect volumes
OCR for page 133
DEFEC TS A ND DI FF USI ON 1 3 3
(errors formerly led to large discrepancies) and entropies. A good
understanding has been achieved for important model impurity prob-
lems involving dopants, both with and without effective charges, in
halide and fluoride structures. The case of rare-gas impurity properties
in these lattices warrants special mention. Ongoing efforts seek to
broaden applications of this general approach to less-ionic materials
such as oxides, where added effects of covalency must be simulated.
A powerful approach to the properties of metallic and covalent
systems, mentioned in Chapter 1, is provided by methods derived from
the density functional formalism. Normally this involves iteration of
the one-electron Green's function to obtain a self-consistent electron
density and hence the energy by integration over position. Various
paths to these results employ frozen cores, pseudopotentials, or other
strategies to avoid the atomic core. In applications to total energies of
extended solids these methods have been highly successful: relative
energies, and hence stabilities, of different structures are predicted
with a precision that often is better than 0.1 eV. Note that the problem
of calculating the total energy of the electron liquid from first principles
is circumvented rather than solved in these approaches. Variational
and Green's function Monte Carlo approaches to the specific problem
of the electron liquid appear to offer feasible future routes.
The difficulty in representing excited configurations, particularly
those containing inhomogeneities such as charge localization, probably
places serious limitations on the applications of the density functional
method to areas of excited-state spectroscopy. At the time of writing,
applications to the ground state of dilute impurity systems, both in
metals and semiconductors, and including transition metal centers,
have been completed to provide good insight into the local structure.
Typically the approach is to use a small cluster consisting of the atom
and its neighbors as a perturbation on the electronic structure of the
perfect solid; this is then iterated through to self-consistency and the
properties derived. Magnetic systems are treated using an Ansatz for
the spin-polarized density functional. The impurity ground state is
discussed more successfully than the excitation energy, particularly for
deep levels. Vacancies have been treated as relatively simple defect
centers in both semiconducting and metallic crystals. The problem of
incorporating lattice relaxation into the calculation must be solved
before the energies obtained can usefully be compared with experi-
ment.
An alternative approach, which has developed mostly in connection
with insulators, is now finding important applications to metals and
covalent materials. This is the unrestricted Hartree-Fock approxima
OCR for page 134
134 A DECADE OF CONDENSED-MATTER PHYSICS
tion. A critical recent breakthrough has been the development of
many-body perturbation theory to correct the Fock results for corre-
lation by an expansion in pair excitations. Almost 90 percent of the
correlation energy is returned by these methods in many problems, so
practical calculations can be completed with chemical accuracy, or 0.1
eV.
One important advantage of the Hartree-Fock approach is its ability
to deal accurately with excited configurations. This is particularly the
case for excited configurations that differ in symmetry from the ground
state, so that the two remain unconnected by pair excitations in the
many-body perturbation theory.
FUNDAMENTALS OF ATOMIC MOBILITY
Until recently, computer simulation using molecular dynamics has
provided the sole method by which jump rates can be calculated
quantitatively for a given defect in a model crystal in which atoms
interact through a specified potential function. This method can be
employed for model crystallites of 102-103 atoms, using periodic
boundary conditions that minimize surface erects. Computer runs of
104-105 iterative steps necessarily involve only ~103 vibrational periods
in order that the iteration can mimic smooth dynamics. Therefore, the
method has been useful only when a number of jump events take place
in 103 periods. This has limited investigations to fast-diffusing species
and to high-temperature properties. Topics on which attention has
focused include diffusion in liquids, in superionic conductors, and on
surfaces. Each of these will be mentioned further in what follows.
It should be emphasized that existing evaluations of statistical
theories describing atomic jump rates have disagreed with the results of
correct molecular-dynamics calculations for the identical model sys-
tems. This has been a consequence of the insufficient accuracy in the
statistical evaluation afforded, for example, by absolute rate theory
(ART). For the jump frequencies encountered in real crystals it has
been possible to perform dynamical simulations only at the highest
temperatures. Consequently there has been an almost complete ab-
sence of accurate theoretical information for ordinary materials about
the way atomic jump processes are determined by the potential energy
of interaction among the atoms. Much discussion has arisen over the
past decade about the possible dependence of the energies, entropies,
and volumes of migration on temperature and pressure, since they are
sensitive to theoretically intractable derivatives of the jump rate with
respect to these variables. The meaning of the isotope effect and its
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DEFECTS AND DIFFUSION 135
dependence on thermodynamic coordinates have remained equally
obscure.
One recent advance in this area is to correct ART predictions for
those nonrandom return jumps that are inherent in the dynamical
system and that therefore cause an intrinsic error in the rate theory
formulation. It turns out that in most crystals these amount to only 10
percent of the jumps, even at high temperature, so the correction is
small. Apparently absolute rate theory alone is a relatively sound first
. . .
approximation In many cases.
ART is nevertheless valid only for a limited range of jump problems.
A second formulation, Brownian rate theory (BRT), is potentially
useful under different circumstances from those for ART. In ART it is
supposed that the system randomizes completely once a jump occurs;
therefore ART requires correction for subsequent dynamical events
that occur before randomization. In BRT the inertia of the jumping
system is retained but introduces random motion into the remaining
system, which causes a viscosity. No clear synthesis of these two
approaches has appeared. The physical sense of BRT becomes appar-
ent for problems such as adatom diffusion on a smooth surface where
the adatom may move many lattice spacings before undergoing a
significant collision with the lattice. Under these circumstances the
assumption in ART of immediate randomization is incorrect. Diffusion
occurs instead by long flights broken by collisions in a way that could
conceivably be described by viscosity. No first principles evaluation of
the damping has, however, been possible as yet for atomic migration.
One striking application of the BRT has been to the breakaway of
dislocations in a stress field. Here, thermal activation through pinning
points plays a role, as does the inertia of the dislocation at its terminal
velocity in the viscous field. An attractive feature of the phenomenon
is that the viscous drag from the electronic system can be modified to
an observable extent by the superconducting transition, which thus
affects mechanical properties. For this example the model holds
together in a semiquantitative way.
The influence of quantum constraints on atomic jump processes has
been the source of a large theoretical literature that has dwarfed the
incidence of verifiable experimental reports citing observations of
quantum effects. Well-established observations exist for tunneling of
even quite heavy atoms among pocket states of asymmetric defect
configurations; these results include, for example, off-center Cu sub-
stituted in salts and, more recently, Zn-A1 mixed dumbbell interstitials
in Al. These systems undergo tunneling transitions among equivalent
configurations and have unmistakable signatures of symmetry and
OCR for page 136
136 A DECADE OF CONDENSED-MATTER PHYSICS
tunnel splitting of the energy-level structure. An elegant system that
still lacks adequate model treatment is the motion of exceedingly heavy
self-interstitials in certain metals. In Pb, for example, the interstitial is
observably mobile below 1 K, so it may in fact delocalize rapidly in the
perfect crystal at O K. A possible complication in metals, not currently
well understood, is coupling to the conduction electron excitation
continuum, with excitation density characteristically proportional to
excitation energy in the Fermi liquid.
In the 1970s, the migration of light impurity atoms through crystals
was modeled using polaron concepts to find regimes of multiphonon
(thermally activated) hopping at high temperature, power-law few-
phonon hopping at intermediate temperature, and (perhaps attainable)
propagation at the lowest temperatures. Feynman path integral meth-
ods are now thought to present a viable route for more accurate
calculations of transition rates, particularly in combination with Monte
Carlo treatments of the classical lattice modes. Extensive studies of the
most promising systems, principally H interstitials in bee refractory
metals, have seemed consistent with this modeling to some workers,
although this is not fully accepted. Certainly, large isotope effects are
measurable in mechanical relaxation and specific heat, for example.
The situation is complicated by H trapping in tunneling levels at other,
heavier interstitials and also at dislocations. True migration of a light
particle at low temperatures in these materials has proved elusive.
Quantum crystals over the best opportunities for examination of
atomic mobility modified by quantum-mechanical requirements. Solid
HI and Do are complicated by rotational transitions, and Ne, Ar, and
CH4 are too heavy; the focus therefore falls almost entirely on the
isotopes 3He and 4He. It is possible that exchange delocalizes vacan-
cies in these materials and mixes them into the crystal ground state so
that they are never absent in equilibrium, even at O K. Explicit
measurements have revealed that the vacancy content of hop 4He and
bee 3He is below 1 part in 104 as T ~ 0. The defect density is
nevertheless sufficient to promote diffusion with a diffusion constant
D ~ 10-6 cm' s-', characteristic of the melting temperature of more
ordinary solids. It is further observed that the vacancy formation
energy is equal to the activation energy for diffusion over a wide range
of volumes in bee 3He. Evidently this structure has little or no barrier
to migration, although the hop structure at lower molar volume does
exhibit activated hopping. With the use of variational and Green's
function Monte Carlo methods with model pairwise forces it is cur-
rently possible to reproduce cohesive and structural properties of the
heliums quite accurately. The possibility therefore exists that theory
OCR for page 137
DEFECTS AND DIFFUSION 137
may take the leading role in the exploration of these point-defect
properties.
Quantum crystals may also have unusual extended defects. For
example, the liquid-solid interface shows exaggerated lateral mobility
in certain circumstances. Also, melting waves are observed at per-
turbed liquid-solid interfaces, owing to the high heat conduction
permitted by the superfluid. Finally there is an expectation that
dislocations may become delocalized. Measurements nevertheless
show that the string model of dislocations, pinned at vacancies,
describes the mechanical behavior down to temperatures of 1.5 K for
solid 4He and to about 0.2 K for solid 3He. For both point and extended
defects, therefore, the promise of remarkable defect behavior remains
mostly to be verified in future work.
COMMENTS ON ACTIVE AREAS
What follows are brief descriptions of additional areas that seem
particularly noteworthy in the light of past developments or potential
future interest. More material concerning surfaces and interface prop-
erties will be found in Chapter 7.
Point Defects in Simple Solids
Steady progress has been made over the past decade in collecting
and interpreting values of the formation and migration properties of
simple defects in prototypical solids. In some cases the known activa-
tion energies, typically ~1 eV, have changed by less than 10 percent
from values quoted two decades ago. The intervening years have
nevertheless been filled with careful effort. In many cases, activation
energies for self-diffusion and for vacancy formation have been mea-
sured over large temperature ranges using new techniques. Curvatures
of Arrhenius plots are now commonplace; the interpretation in terms of
the temperature dependence of mechanisms, hopping parameters, or
defect clustering nevertheless remains a difficult and unsettled area.
The assistance of definitive theories will probably be required before
these subtleties are resolved. To convey the difficulty, one may note
that for many bee metals the Arrhenius plot for diffusion has been
known for many years to exhibit strong curvature, yet the cause
remains imperfectly resolved. One reasonable possibility is the anom-
alous behavior of phonons in these metals.
Persistent efforts are necessary in this area. The correct character-
ization of defect processes in simple metals, salts, and valence solids is
OCR for page 138
138 A DECADE OF CONDENSED-MATTER PHYSICS
essential if progress toward a predictive framework for complex
materials of interest for engineering applications is eventually to be
forthcoming. In this connection the realm of definitive information is
now advancing from noble metals and alkali halides to refractory and
other metals, covalent solids, and refractory oxides. This must be
recognized as a major achievement of the field.
An indicator of future progress is the healthy arsenal of techniques
now applied to quantitative characterization of defect structure in
metals. In the early 1970s, for example, the existence of the self-
interstitial in Cu as a dumbbell configuration was established using
ultrasonic attenuation, neutron scattering, and diffuse x-ray scattering.
To these techniques have been added positron annihilation, muon spin
rotation, perturbed angular correlation spectroscopy, various special
NMR techniques, ion channeling, the Mossbauer effect, and other
specialized probes. These advance far beyond the resistivity measure-
ments and occasional specific heat and Bragg x-ray scattering experi-
ments available before 1970, and the prospects for steady future
progress are improved accordingly.
Surface Diffusion
Only at temperatures typically below half the melting temperature
does the diffusion of atoms on clean surfaces usually resemble the
site-to-site hopping of atoms in the bulk crystal. Most available
information for lower temperatures pertains to refractory metals that
can be cleaned in ultrahigh vacuum to secure reproducible results.
Field ion microscopy provides a microscopic probe of adatom mobility
and clustering in the low-mobility regime, and fluctuation spectroscopy
of adatom Auger signals is a second, more macroscopic, probe.
Reasonable Arrhenius behavior has been observed in a number of
experiments covering limited temperature ranges; evidence for quan-
tum behavior at low temperatures has recently been reported for H and
D adsorbates. Phase transitions of the bulk and surface reconstructions
are both expected to change the diffusion characteristics, but as yet
these have not been investigated. Anisotropic diffusion is seen to
occur, and interesting mechanisms have been deduced, for anisotropic
surfaces containing ridges, for example. These have been simulated in
computer dynamical calculations.
At higher temperatures, simulations indicate fast surface diffusion.
Moreover, so many mobile point defects and surface ledges are
activated that the surface becomes quite rough and is subject to rapid
fluctuations. Interestingly enough, the surface layer itself is observed
OCR for page 139
DEFECTS AND DIFFUSION 139
in simulations to exhibit liquidlike behavior below the melting point of
the solid. These novel dynamics are, of course, confined to a skin on
the crystal surface since the bulk cannot melt. None of these high-
temperature processes has yet been observed for real crystals.
Photochemical Processes
It has been known for several decades that optical interband
transitions in salts can create point defects. In alkali halides the
products are H centers (a negative halogen molecule at an anion site)
and F centers (an electron trapped at an anion vacancy). The excitor
created in the optical event self-traps, and a nonradiative decay
channel leads to the defect production. A number of similar processes
warrant mention here. For example' the Jahn-Teller instability of the
vacancy in Si transforms electronic recombination energy into the
motion required to surmount the barrier to atomic migration; thus,
excitation promotes fast migration. Another example is dislocation
glide, which occurs at high excitation levels in solid-state quantum-well
lasers made, for example, from GaAs and Ga'_~.Al`As. Yet a further
example is photodesorption, in which the surface of a crystal absorbs
a photon and an ion is subsequently ejected from the surface. It is
believed that Auger processes convert what was formerly a negative
ion into a positive ion, which then desorbs under the repulsive field of
its neighbors. Angle-resolved effects from molecules oriented on the
surface may be expected; site symmetries have been elucidated from
the paths of desorbing ions.
Photochemical processes of this type offer new opportunities for
investigation. Interesting progress has been made in recent years using
picosecond pulse-probe laser techniques to monitor the decay of
excitors into F and H centers. It appears that a V`. center forms after
an unresolvable short time and that it evolves into the H center.
Nowadays, laser pulses can be created in the 10-fs (1o-~4 S) time
domain, which is faster than most lattice vibrations. A special oppor-
tunity therefore exists in the future to examine photochemical pro-
cesses, including photon-induced point-defect migration, on the time
scale of the atomic jump process itself.
Molecular Dynamics
Computer simulation of dynamical processes in solids has led to
vivid insight into complex mechanisms, to the discovery of qualita-
tively new processes, and to quantitative mimicking of processes that
OCR for page 140
140 A DECADE OF CONDENSED-MATTER PHYSICS
occur in real solids. As mentioned above, molecular dynamics has
served as the only source of numerous insights into the dynamics and
stability of defects, for example in surface structure and diffusion.
More recent successes that warrant mention here include the accu-
rate treatment of ionic motion in fast ion conductors such as AgI, in
which the Ag sublattice disorders, and in CaF', where the F sublattice
undergoes large fluctuations involving defects and mobility. Model
interatomic forces have reproduced observed diffusion rates quantita-
tively. Also of major interest are dynamical studies of the early stages
of precipitation. In these investigations the embryo is found to nucleate
from the disordered state, possessing from the earliest stages the
appropriate symmetry as a precursor of the eventual lattice structure.
It is not clear what other approaches could possibly provide such direct
access to these important phenomena.
The future of computer simulation and dynamics contains research
problems to match whatever complexity new generations of computers
can handle. Direct modeling of the mechanical properties introduced
by dislocations may require consideration of ~106 particles in place of
today's 103. Bulk mechanical behavior, including grain-boundary
structure, may require still more. Future dynamics programs may be
coupled to fast quantum chemistry routines to replace pair forces by
more realistic solid-state modeling of the crystalline potential energy.
Surface treatments, including particle-beam mixing, the resulting
nonequilibrium structure in alloys, and the influence of directed heat
input by laser processing, all appear readily susceptible to investigation
by simulation. It seems highly probable that applications of this type
will ensure a significant future role for computer studies of defect
properties.
Dislocation Motion in Glasses
The glide of dislocations in crystals has long been recognized as a
determining factor in plastic behavior. It has recently been recognized
that many of the same effects occur in glasses also, despite the fact that
the geometrical characteristics are somewhat less clear owing to the
amorphous structure of the solid. The experimental fact is that slip
bands are observed after plastic How in certain metallic glasses. They
occur as expected on planes defined by the maximum shear stress and,
in general, resemble similar processes in crystals. Computer modeling
of dislocations introduced into a Lennard-Jones glass shows that the
core structure and the long-range elastic field remain stable after the
atoms are allowed to relax. By way of comparison, a vacancy in the
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DEFEC TS A ND DI FF USI ON 1 4 1
Lennard-Jones glass disappears into the structure when relaxation is
permitted; however, bond models with forces that vary with bond
angle can lead to stable vacancies.
These initial discoveries establish that glassy materials can, in some
cases, support defect structures resembling those in crystals. The area
warrants further effort in the future to determine the range of phenom-
ena that occur and the way in which such defects move.
Defect Imaging at Atomic Resolution
A notable development over the last decade has been the refinement
of experimental methods that can image crystals and defect structures
with spatial resolution at the atomic level. These are not universally
applicable probes but instead generally require particular sample
characteristics for successful detection of atomic locations and defect
geometry. They are nevertheless extremely powerful techniques when
handled well. Two such probes that are surface sensitive are the
scanning vacuum tunneling microscope and the field ion microscope.
These are described in Chapter 7. A third, the high-resolution electron
microscope, can image defects on the surface or within the bulk of the
crystal.
Over the period from 1970 to 1982 the resolution of commercial
transmission electron microscopes improved from 4 to about 1.5 A.
Since the beam passes through the entire sample, which must therefore
be quite thin (~1000 A thick), these methods are naturally adapted to
linear and planar defects aligned with the beam. Vivid patterns of
atomic distributions in the perfect crystal can be obtained for appro-
priate thin films. Dislocation and defect structures at properly aligned
grain boundaries can also be imaged. Early high-resolution successes
concerned the planar defect structures of stacking faults and polytypes
and the defects accommodating nonstoichiometry in oxides. More
recent applications involve grain-boundary structures and the geome-
try of heterojunctions, for example of semiconducting materials in
device configurations (Figure 6.1~. In all cases the apparently clear
imaging of atomic positions is at least partly illusory; careful theoretical
modeling is required to obtain precise interpretation of the relevant
diffraction processes. Most of the necessary theoretical machinery is
now widely available.
The past decade has also seen the parallel development of scanning
transmission electron microscopes, which form an electron probe
o
focused to about 3 A. Images with atomic resolution are then formed
by monitoring such properties as scattered intensity as the beam is
OCR for page 142
142 A DECADE OF CONDENSED-MATTER PHYSICS
FIGURE 6.1 Atomic resolution image of the interface (lines) between Si (left) in the
(110) projection and epitaxial NiSi2 (right), which has the fluorite structure. Each dark
blob is the image of two projected rows of atoms in a sample about 100 A thick.
(Courtesy of J. C. H. Spence, University of Arizona.)
rastered in a suitable pattern. Heavy atoms located on carbon films
have been imaged individually by these means; other direct uses at
atomic resolution include the analysis of small precipitates; chemical
analysis at high resolution can be completed by energy-loss methods.
SOME DIRECTIONS FOR FUTURE RESEARCH
A qualitative understanding of phase microstructure and phase
generation in radiation fields is developing, but detailed model descrip-
tions and even the basic theoretical framework remain largely to be
developed. One phenomenon, radiation-induced homogeneous precip-
itation in undersaturated solid solutions, has been described using a
simplified quasi-thermodynamic theory. This is a research area in its
infancy. It adds a new dimension to the currently active field of phase
relations, phase transformations, and the stability of phases.
At present there exist no precise methods by which the steady-state
nonuniform substrate concentration profiles that develop while atoms
are being sputtered away from the solid surfaces can be predicted
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DEFECTS AND DIFFUSION 1 43
quantitatively, so accurate analysis awaits improved understanding of
the damage, displacement, and diffusion processes in the subsurface
region.
The structures and energies of point defects are apparently coming
within the grasp of ab initio theoretical calculations. Indeed, electronic
aspects of calculations for point defects can already be treated compre-
hensively, but the rather complicated problem of lattice relaxation and
its effect on the electronic system is not generally tractable at present.
As with density functional methods, the new capabilities provided by
the use of the unrestricted Hartree-Fock approximation, corrected by
the use of many-body perturbation theory, open a wide range of
problems to future exploration, including chemical pathways, equilib-
rium solid-state lattice geometries, and defect properties, although
most investigations to date involve surface and adsorbate problems. It
is probable that, over the next few years, accurate research on spectro-
scopic applications in both pure and defective solids, but particularly
when some degree of electronic localization occurs, will make use of this
approach. For example, ab initio calculations on the F and FA center
excitations of salts are apparently yielding excellent predictions. Recent
results of cluster calculation on metals indicate that local electronic
excitations of defects may be predictable to within an uncertainty of 0.1
eV. This accuracy is adequate for most practical purposes, so the oppor-
tunities for new applications in the future appear particularly inviting.
It is fairly clear that the discrepancies between the results of
statistical theories and those of molecular-dynamics calculations for
atomic jump rates in systems in which diffusion is occurring will be
resolved over the next 5 years for model crystals with reasonably
simple potential functions. Detailed properties of atomic jumps in
model crystals, including isotope effects and thermodynamic deriva-
tives of jump rates, will thus become accessible for the first time. Atten-
tion will then focus on the ability of the assumed potential functions to
simulate that of the real crystal, much as for defect formation.
For the future it seems assured that atomic resolution investigations
of defect structure in solids will continue to build momentum. Applica-
tions of these methods to surface studies in ultrahigh vacuum currently
remain in their infancy but hold extraordinary promise. The real-time
recording of solid-state reactions at atomic resolution under controlled
conditions can be expected to reveal a wealth of detailed mechanisms
over the next decade. Therefore seminal contributions may be ex-
pected in areas such as precipitation and phase transitions, where the
atomic mechanisms and motions at the reaction front are of central
interest.
Representative terms from entire chapter:
defect structures
