Vacancies: conductivity and reactivity

Vacancies are point defects in the crystalline structure of a solid and may control many physical properties in materials such as conductivity and reactivity. In microcrystalline solids at temperatures above 0 K, vacancies invariably exist in thermal equilibrium. In the simple case of metals with one type of vacancy, the number of vacancies in a crystal consisting of N atom sites is approximated by an Arrhenius-type expression

n = N exp(- Qr/ RT)

T: absolute temperature
R: the gas constant
Qr: energy required to form one mole of vacancies

Qr = N_A qr
N_A: Avogadro number
qr: the activation energy for the formation of one vacancy

Kelsall, p 21



The Avogadro constant (symbols: L, NA), also called the Avogadro number, is the number of atoms in exactly 12 grams of 12C. A mole is defined as this number of "entities" (usually, atoms or molecules) of any material. The currently accepted value for this number is:

N_A = (6.02214179 +- 0.00000030)x 10^{23} mol^-1


Source: wikipedia


The influence of defects
The details of the arrangement of atoms in crystal lattice can be determined by analysing the diffraction effects which are produced when a beam of some appropriate radiation (X rays, electrons, or neutrons) is incident on the crystal. This is extremely unlikely that the atoms are exactly at their correct positions in the crystal, when we recall that there are about 10^22 atoms in 1 cm3 of material. There must always be some atoms which are not exactly in their right place and so the lattice will contain imperfections or DEFECTS.

There are two main types of geometrical defect in a crystal. There are those which are very localized and are of atomic dimensions – these are called point defects. The most obvious example is an impurity atom. This can be either a substitutional or an interstitial impurity depending on whether it replaces an atom on the host lattice or whether it is betweenthe host atoms on a non lattice site. In both cases there will be some distortion around it. There are also those defects which are of a more extended nature – the most important and interesting being the dislocation. In addition to these static defects, the perfection of the lattice arrangement will be continually upset by the thermal vibrations of the atoms.

The presence of all types of defect increases the disorder to the crystal, ie it increases its entropy. There is therefore a tendency for more defects to be present at higher temperatures.



Point defects: vacancies
The simplest defect to consider is the configuration which arises when an atom is missing from its site in the lattice. This vacant lattice is called a vacancy. in the region around a vacancy there will be a tendency for the atomic arrangement to readjust slightly and so the crystal lattice becomes distorted.
The presence of vacancies provides a means for atoms to diffuse fairly easily from one part of the crystal to another since an atom can move to a vacancy thereby leaving its own site vacant without producing too much disruption of the existing crystal lattice. ie. Very little energy is required. Thus diffusion can be thought of as a migration of vacancies in the opposite direction. In an ionic crystal the presence of a vacancy at a positive-ion site upsets the electrical neutrality of the region and so the vacancy has an effective negative charge associated with it. This would increase the electrostatic energy of the system: hence in order to maintain electrical neutrality there is a tendency for positive and negative ion vacancies to be produced in pairs.

Clarendon publishing


In crystallography, a vacancy is a type of point defect in a crystal. Crystals inherently possess imperfections, often referred to as 'crystalline defects'. A defect wherein an atom, such as silicon, is missing from one of the sites is known as a 'vacancy' defect.

Vacancies occur naturally in all crystalline materials. At any given temperature, up to the melting point of the material, there is an equilibrium concentration (ratio of vacant lattice sites to those containing atoms). At the melting point of some metals the ratio can be approximately 0.1%

The creation of a vacancy can be simply modeled by considering the energy required to break the bonds between an atom inside the crystal and its nearest neighbor atoms. Once that atom is removed from the lattice site, it is put back on the surface of the crystal and some energy is retrieved because new bonds are established with other atoms on the surface. However, there is a net input of energy because there are fewer bonds between surface atoms than between atoms in the interior of the crystal.

At any given temperature, the amount of energy needed to create a vacancy is diminished because creating a vacancy disorders the interior of the crystal. The measure of this disorder is called the entropy of the system. Adding vacancies to the material increases the entropy, which tends to reduce the total energy required to create the vacancy. We call this energy the free energy and this is the energy that is required to create an equilibrium concentration of vacancies at a given temperature.


Point defects are defects which are not extended in space in any dimension. There is not strict limit for how small a "point" defect should be, but typically the term is used to mean defects which involve at most a few extra or missing atoms without an ordered structure of the defective positions. Larger defects in an ordered structure are usually considered dislocation loops. For historical reasons, many point defects especially in ionic crystals are called 'centers': for example the vacancy in many ionic solids is called an F-center.

* Vacancies are sites which are usually occupied by an atom but which are unoccupied. If a neighboring atom moves to occupy the vacant site, the vacancy moves in the opposite direction to the site which used to be occupied by the moving atom. The stability of the surrounding crystal structure guarantees that the neighboring atoms will not simply collapse around the vacancy. In some materials, neighboring atoms actually move away from a vacancy, because they can better form bonds with atoms in the other directions. A vacancy (or pair of vacancies in an ionic solid) is sometimes called a Schottky defect.

* Interstitials are atoms which occupy a site in the crystal structure at which there is usually not an atom. They are generally high energy configurations. Small atoms in some crystals can occupy interstices without high energy, such as hydrogen in palladium.
* A nearby pair of a vacancy and an interstitial is often called a Frenkel defect or Frenkel pair

* Impurities occur because materials are never 100% pure. In the case of an impurity, the atom is often incorporated at a regular atomic site in the crystal structure. This is neither a vacant site nor is the atom on an interstitial site and it is called a substitutional defect. The atom is not supposed to be anywhere in the crystal, and is thus an impurity.

* Anti-site defects occur in an ordered alloy. For example, some alloys have a regular structure in which every other atom is a different species, for illustration assume that type A atoms sit on the cube corners of a cubic lattice, and type B atoms sit in center of the cubes. If one cube has an A atom at its center, the atom is on a site usually occupied by an atom, but it is not the correct type. This is neither a vacancy nor an interstitial, nor an impurity.

* Topological defects are regions in a crystal where the normal chemical bonding environment is topologically different from the surroundings. For instance, in a perfect sheet of graphite (graphene) all atoms are in rings containing six atoms. If the sheet contains regions where the number of atoms in a ring is different from six, while the total number of atoms remains the same, a topological defect has formed. An example is the Stone Wales defect in nanotubes, which consists of two adjacent 5-membered and two 7-membered atom rings.
* Also amorphous solids may contain defects. These are naturally somewhat hard to define, but sometimes their nature can be quite easily understood. For instance, in ideally bonded amorphous silica all Si atoms have 4 bonds to O atoms and all O atoms have 2 bonds to Si atom. Thus e.g. an O atom with only one Si bond can be considered a defect in silica.

* Complexes can form between different kinds of point defects. For example, if a vacancy encounters an impurity, the two may bind together if the impurity is too large for the lattice. Interstitials can form 'split interstitial' or 'dumbbell' structures where two atoms effectively share an atomic site, resulting in neither atom actually occupying the site.

source: wikipedia




Nanotopography is defined as “the deviation of a surface within a spatial wavelength of around 0.2 to 20 mm.”(2) This is a parameter that measures the front-surface, freestate topology of an area which can range in size from fractions of a millimeter to tens of millimeters. In this sense, nanotopography differs from front-referenced site flatness in that for nanotopography the wafer is measured in a free state, while for flatness it is referenced to a flat chuck. A wafer may have perfect flatness (in the classical definition of flatness), yet still have nanotopography. If a wafer has surface irregularities on the front and backside of the wafer, but front and back surfaces are parallel, the wafer has perfect flatness. However, the same wafer will exhibit nanotopography (Figure 1).




Nanotopography bridges the gap between roughness and flatness in the topology map of wafer surface irregularities in spatial frequency. Nanotopography of the silicon wafer is dictated to a large extent by the polishing process. A true planetary, freefloating, double-sided polishing process that polishes both sides of a silicon wafer simultaneously technically achieves the best nanotopography and flatness results.


http://www.memc.com/t-wafer-nanotopography.asp




Pure Cu
Defect structures in pure Cu are described here for reference:

Interstitial-type dislocation loops were formed up to 523 K, and Fig. la shows typical examples. As found previously [1], very few loops formed in the middle
temperature around 373 K, which has been referred to as 'down peak' in the temperature dependence of the loop number density. Figure 2 shows the variation of the loop number density with irradiation time at several temperatures. At room temperature, loops nucleated during the initial stage of the irradiation, but over 2/3 of them disappeared without growing larger. As the temperature was raised to 323 K, loops disappeared earlier, and the saturation number density was smaller than at
300 K. Almost no loops were observed around 373 K. Above this temperature, faulted dislocation loops again formed, and the majority of them grew larger until they
eventually unfaulted and were lost from the foil by glide. They had a larger growth rate and a smaller number density at higher temperatures, and were not formed above 523 K.

www.oxfordjournals.org



Planar defects

* Grain boundaries occur where the crystallographic direction of the lattice abruptly changes. This usually occurs when two crystals begin growing separately and then meet.

* Anti phase boundaries occur in ordered alloys: in this case, the crystallographic direction remains the same, each side of the boundary has an opposite phase: For example if the ordering is usually ABABABAB, an anti phase boundary takes the form of ABABBABA.

* Stacking faults occur in a number of crystal structures, but the common example is in close-packed structures. Face-centered cubic (fcc) structures differ from hexagonal close packed (hcp) structures only in stacking order: both structures have close packed atomic planes with sixfold symmetry -- the atoms form equilateral triangles. When stacking one of these layers on top of another, the atoms are not directly on top of one another -- the first two layers are identical for hcp and fcc, and labelled AB. If the third layer is placed so that its atoms are directly above those of the first layer, the stacking will be ABA -- this is the hcp structure, and it continues ABABABAB. However there is another location for the third layer, such that its atoms are not above the first layer. Instead, the fourth layer is placed so that its atoms are directly above the first layer. This produces the stacking ABCABCABC, and is actually a cubic arrangement of the atoms. A stacking fault is a one or two layer interruption in the stacking sequence, for example if the sequence ABCABABCAB were found in an fcc structure.

Bulk defects

* Voids are small regions where there are no atoms, and can be thought of as clusters of vacancies.

* Impurities can cluster together to form small regions of a different phase. These are often called precipitates.

wikipedia.org