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Objectives By the end of this section you should: know how the Lennard-Jones [12,6] potential describes the interaction between atoms be able to calculate the van der Waals radius, the distance between the atoms that minimises their energy know about vacancies, interstitials and Frenkel defects be able to calculate the energy of vacancy formation from quenching data know the difference between intrinsic and extrinsic conduction, p- and n-type silicon and donor and acceptor doping be able to describe the different types of line defect and use the Burgers vector

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Each atom exludes other from the space it occupies Attraction? Electrons are moving so that, at some instant, distribution is uneven Positively (electron-deficient) and negatively (electron- rich) charged regions electrical dipole Dipole induces an opposing dipole in neighbouring atom attraction Close Packing

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Attractive force is known as: van der Waals interaction London interaction induced dipole-induced dipole interaction

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The total potential energy for two atoms a distance, r, apart can be written as: This is called the Lennard-Jones (12,6) potential function First term is repulsive, second term is attractive. We want to find a minimum - so differentiate w.r.t. r

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This is the van der Waals radius, the distance between the atoms that minimises their energy

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Substituting back in to the (12-6) potential gives the minimum energy:

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Energy has a minimum value of - at the van der Waals radius

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Defects Up to now we have considered perfect crystals, i.e. crystals with perfect periodic arrangements. Most good crystals show very little departure from this idea, e.g. silicon single crystals can be grown without defects over a range of several mm This sounds small but is about 10 million unit cells! However, defects are very important in processing and for optical and electrical properties.

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1. Vacancies A vacancy is the absence of an atom in the lattice. In ionic crystals (e.g NaCl) vacancies occur in pairs (Na + Cl) so that charge balance is maintained. Also called a Schottky Defect. Vacancies allow diffusion through the crystal: Vacancy : point defect - associated with a point in the crystal

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Vacancies Vacancies are not energetically favourable - the number of vacancies increases with temperature (i.e. putting energy into the system) Mathematically, for a crystal containing N atoms, there is an equilibrium number of vacancies, n, at temperature T (in K) given by: where E V is the energy of vacancy formation and k B is Boltzmanns constant. Applies to pairs also.

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Diffusion Similary, the diffusion coefficient, D, is given by: where E D is the energy of diffusion and D O is a diffusion constant specific to the element. Strictly this applies only to self-diffusion, that is diffusion in an elemental substance.

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Quenching Non-equilibrium concentrations of vacancies may be obtained by rapidly cooling (quenching) metals from high temperatures. These defects can cause additional resistivity proportional to the number of defects: where C is a proportionality constant. R is the relative increase in resistance at low temperature after quenching from the temperature T.

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Uses so: y = c + mx E V can be obtained from a graph of ln R against (1/T)

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Example - Gold

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2. Interstitials Previously we discussed small tetrahedral and octahedral interstitial atoms within the close packed structure. If the interstitial atom is the same size as the close packed atoms, then considerable disruption to the structure occurs. Again, this is a point defect and requires much energy

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3. Frenkel Defects Often a vacancy and interstitial occur together - an ion is displaces from its site into an interstitial position. This is a Frenkel Defect (common in e.g. AgCl) and charge balance is maintained. Frenkel defects can be induced by irradiation of a sample

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4. Impurities Preparing pure crystals is extremely difficult - often foreign atoms enter the structure and substitute for native atoms - often by contamination from container This can have a large effect (either detrimental or beneficial) on the properties of the crystal. We can also add impurities (or dopants) deliberately. An important example is that of silicon.

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Silicon Silicon is a group IV element and, like carbon, bonds to four nearest neighbours: At elevated temperatures bonds are broken to produce a (positive) gap - known as a hole - and a conduction electron. T This is known as the intrinsic effect in semiconductors

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Doped Silicon If we take a group V element (e.g. As) and substitute (at low levels) for Si there is a spare electron for conduction and no positive hole: This process is known as doping. Arsenic acts as an electron donor to Si, making it easier to conduct electricity. Si doped with As is an extrinsic semiconductor and because the electron is negative this is an n-type semiconductor

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Doped Silicon If we take a group III element (e.g. B) and substitute (at low levels) for Si there is a positive hole and no conduction electron Boron acts as an electron acceptor to Si. Electrons can move by diffusion - hopping into the hole leaves behind a new hole. Again this is an extrinsic semiconductor and because the hole is negative this is a p-type semiconductor

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Line Defects - 1. Stacking Faults We discussed h.c.p which has sequence ABABABA and c.c.p. which has sequence ABCABCA. A stacking fault occurs when the sequence goes wrong, e.g. ABCBCABCABC (A missing) or ABCABACABC (extra A) Often these do not extend right across the plane, e.g. This is also known as a partial dislocation

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Line Defects - 2. Edge dislocations Originally proposed to account for mechanical strength in crystals. Consists of an extra plane of atoms which terminates within the crystal. This distorts the local environment.

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Burgers Vector If the dislocation was not present, then atom at A would be at A We define a vector B which shows the displacement of A due to the dislocation. B is known as the Burgers Vector. For an edge dislocation, the Burgers vector is perpendicular to the dislocation

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Slipping Such defects are produced by part of the crystal slipping with respect to the rest. Consider a close packed structure: For the top layer to slip to the right, to another close packed position, it must pass through a non-equilibrium position

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Line Defects - 3. Screw dislocations Here there are no extra planes - the defect appears as though part of the crystal has been cut in two, then shifted down on one side of the cut.

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Burgers Vector In this case, A would have been at A had the dislocation not occurred. The Burgers Vector B is hence parallel to the direction of the screw dislocation. Screw dislocations are important in the growth of single crystals since they provide nucleation sites for the growth of a new layer

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Line Defects - 4. Twinning Crystals are often grown with a fault in which one region of the crystal is a mirror image of the other: In c.p. structures, twins are produced by stacking faults ABCABCBACBA Here C is the twin plane Polymorphic compounds (i.e. ones with more than one crystal structure) are prone to twinning, e.g. YBa 2 Cu 3 O d

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Summary the attraction/repulsion between two atoms of size, r, can be adequately described by the Lennard-Jones [12,6] potential the point of minimum energy in the LJ potential is the van der Waals radius Most crystals contain defects Extra vacancies can be produced by quenching; this can produce an increase in resistivity which can be calculated. Defects can be used to advantage, e.g. doped silicon Line defect formation can be described using the Burgers Vector, B

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