Presentation on theme: "Defects and semiconductors"— Presentation transcript:
1Defects and semiconductors Learning Outcomes:By the end of this section you should:know about vacancies, interstitials and Frenkel defectsbe able to calculate the energy of vacancy formation from quenching databe able to describe the different types of line defect and use the Burgers’ vectorknow the difference between intrinsic and extrinsic conduction, p- and n-type silicon and donor and acceptor dopingbe able to calculate number densities of holes and electrons in doped materials
2DefectsUp 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 mmThis sounds small but is about 10 million unit cells!However, defects are very important in processing and for optical and electrical properties.
31. 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
4VacanciesVacancies 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 EV is the energy of vacancy formation and kB is Boltzmann’s constant. Applies to pairs also.
5Diffusion Similary, the diffusion coefficient, D, is given by: where ED is the energy of diffusion and DO is a diffusion constant specific to the element.Strictly this applies only to self-diffusion, that is diffusion in an elemental substance.
6QuenchingNon-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.
7Usesso:y = c mxEV can be obtained from a graph of lnR against (1/T)
92. InterstitialsPreviously 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
103. Frenkel DefectsOften 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
114. 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
12Line 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.
13Burgers VectorIf 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
14SlippingSuch 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
15Line 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.
16Burgers’ VectorIn 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
17Line 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 faultsABCABCBACBAHere C is the twin planePolymorphic compounds (i.e. ones with more than one crystal structure) are prone to twinning, e.g. YBa2Cu3Od
184. “Impurities”Preparing pure crystals is extremely difficult - often foreign atoms enter the structure and substitute for “native” atoms - often by contamination from containerThis 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.
19SiliconSilicon is a group IV element and, like carbon, bonds to four nearest neighbours:TAt elevated temperatures bonds are broken to produce a (positive) gap - known as a hole - and a conduction electron.This is known as the intrinsic effect in semiconductors
20Doped SiliconIf 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
21Doped SiliconIf we take a group III element (e.g. B) and substitute (at low levels) for Si there is a positive hole and no conduction electronBoron acts as an electron acceptor to Si.Holes can move by diffusion - “hopping” into the hole leaves behind a new hole.Again this is an extrinsic semiconductor and because the hole is positive this is a p-type semiconductor
22The band pictureRed = filled energy states, light blue = empty, white = forbidden (energy gap)Green dotted = donor states, blue dotted = acceptor statesBottom band = valence band, top band = conduction band.Points to note: Energy gap is big (>3eV) in an insulator, ~1eV in a semiconductor
23(where ni = intrinsic no. of electrons) The numbers!In the intrinsic material, electrons and holes are created as pairsIf n=no. of electrons, p=no. of holes, thenn=p=ni(where ni = intrinsic no. of electrons)We can state thatpn=ni2As we add donors, the number of holes will decrease. Thus this condition is generally true.
24Charge neutralityWe define ND and NA as the number of donor/ acceptor atoms. At 300K, these are fully ionisedAll materials, doped or intrinsic, have the condition of charge neutralityp + ND = n + NACombine the 2 equations and solve as quadratic, e.g.n = p + ND - NA = ni2/p p2 + p(ND - NA) - ni2 = 0
25Solution and similar for n. Use with care! Example Find the electron and hole densities in a semiconductor if ni = 1016 m-3, ND = 1020 m-3 and NA = 0.(a) calc n using quadratic(b) calc p using quadratic(c) calc p from pn = ni2
26Long Solution n = ½ (1020 + (1040 + 41032) ) = 1020 m-3 p = ni2/n = 1012 m-3.But if we usep = ½ ( ( 1032) )= ½ ( 1020) (!!)Could miss this
27Useful simplifications We define “effective doping” in cases where one type dominates:1) if NA > ND then we defineNA’ = NA - ND (effective acceptor doping)2) if ND > NA then we defineND’ = ND - NA (effective donor doping)
28Useful simplifications Then:1) If ND = NA dopants cancel n = p = ni2) If NA’ >> ni p = NA’ n = ni2/NA’p-type semiconductor3) If ND’ >> ni n = ND’ p = ni2/ND’n-type semiconductor
29ExampleExampleFind the electron and hole densities in a semiconductor if ni = 1016 m-3, ND = 1020 m-3 and NA = 1018 m-3.ND’ = = 9.9 1019 m-3 = np = ni2/n = 1.01 1012 m-3
30Summary Most crystals contain defects Extra vacancies can be produced by quenching; this can produce an increase in resistivity which can be calculated.Line defect formation can be described using the Burgers’ Vector, BDefects can be used to advantage, e.g. doped siliconWe can calculate numbers of holes and electrons in doped materials 4th year