Electrons, Holes, and Defect ionization The process of forming intrinsic electron-hole pairs is excitation a cross the band gap ( formation energy ). intrinsic electronic reaction : null e + h When electrons and holes are tightly bound to an ion, or otherwise localized ( trapped ) at a lattice site, the whole is considered to be one ionic defect. ex ) valence of a transition metal ion ( ( ) oxygen vacancy ) cation interstitials At a given temp. One valence state is often much more prevalent than the other. Change in valence take place via ionization reaction g 1, g 2 are effectively the ionization energy of the defect. 119
Oxidation and Reduction Reaction Equilibration of ionic solids with an ambient gas ambient gas ex 1. reduction Oxygen Halogen Metal vapor Play an important role in determining defect structure as another type of solute species. the removal of oxygen to the gas phase leaving behind oxygen vacancies 2. oxidation is a constant is the free energy of reduction oxidation and reduction are the same thermodynamic process simply reversed, the reaction are not independent. : atm or MPa [ ] : or mole fraction defect conc. 120
1 2 There are a number of ways of writing the oxidation / reduction reactions, which we may choose for convenience in order to show the formation or removal of particular defects. ex O oxidation (i) to show the formation of cation vacancies upon oxidation +) (Schottky defect) (ii) to show the formation of electrons upon oxidation +) 1 It is usually convenient ( and less confusing ) to choose just one representation that includes the prevailing defects in the system. 121
Extent of Nonstoichiometry Highly stoichiometric oxides : O, Al 2 O 3, ZrO 2. (1) have cations of a fixed valence (2) a large free energy for reduction or oxidation Nonstoichiometry oxides : TiO 2-x, BaTiO 3-x, SrTiO 3-x (1) Change in P O alone cause little change in defect conc. 2 (2) Oxides containing multivalent cations, such as the transition metals TiO 2, Ti 4+ can easily be reduced to Ti 3+, causing oxygen deficiencies of order 1 % within the limits of stability of the oxide. The transition metal monoxide series. Ni 1-x O Co 1-x O Mn 1-x O Fe 1-x O a fraction of divalent cation is easily xidized to trivalent state, resulting in a cation deficiency Ni 1-x O ( x ~ 5 10-4 ) Co 1-x O ( x ~ 0.01 ) Mn 1-x O ( x ~0.1 ) Fe 1-x O ( x~ 0.15 ) Fe1-xO can never be made stoichiometry having a minimum x 0.05 122
To illustrate the relationship between defect reaction and the extent of nonstoichiometry. ex1. O, - electrical conductivity measurement of doped O (cm -9 MPa 1 /2) P O 2 : MPa This equilibrium constant tell us that O Can be oxidized at high T, producing extrinsic and If no other defects are present, for charge neutrality e.g. T / Tm =0.8 ( T = 2478K ) The actual conc. are very small, but vary with T and P O 2 =3 10 16 cm -3 ( x in 1-x O equal to 0.6 ppm) at air = 2x10 15 cm -3 (x in 1-x O equal to 0.04 ppm ) at P O 2 =10-9 MPa In O, x is very small, highly stoichiometry ( reducing atm.) 123
Example 2 TiO 2-x electrical conductivity Ti x Ti + 2OO = Tii + 4e' + O2( g )... k R kj = i O cm 2 RT 960 4 122 15 [ Ti ] n P = 6.55 10 exp mol ( MPa. ) - If only two defects are present for electroneutrality, x = at [ Ti ] i T T [ ] n = 4 Ti i m = 0.8 = = 93ppm in air 0.27 0 0 in ( T 1690 K ) 9 PO = 10 Ma ( reducting atm) 2 Reduced TiO 2 becomes a good n-type semiconductor due to the high n concentration (highly nonstoichiometric) 124
Electronic disorder Disorder in the electronic structure of solid influence many electrical, optical and chemical properties. Electronic defects can be Thermally excited Optically (by absorbtion process) Distinction between atomic and electronic defects : 1. atomic defects are discussed with reference to physical position in a lattice and conc. Related to # of atoms. 2. electronic defect is density of electron state of given energy per volume of crystal, which is not a constant quantity but varies with T and composition. When a large number of individual atoms (each of which has its own discrete electron orbital energy level) are brought together into crystal. Interatomic bonding results when the discrete energy levels of the valence electrons are broadened into continuous energy bonds. atom atom r o crystal 125
- Order : a perfect crystal may be regarded as one in which all electron occupy their ground state or lowest energy configuration. - Disorder : any excitation of electrons to higher energy levels. Bandgaps In a metal the highest energy band is only particularly filled. There is no barrier to the excitation of electron to higher energy levels within the band with an increse in T. In intrinsic semiconductor and insulator Eg separates a completely filled valence band from a completely empty conduction band at 0 o K The distinction between an intrinsic semiconductor and insulator is based on the valence of Eg. GaAs insulator ceramic Eg = 1 ~ 1.5 ev Eg = >3 ev Eg = 2.5 ~ 3 ev 126
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Concentration of Intrinsic Electron and Holes In defect concentration. gv h v = N exp 2kt Site density The probability that the site contains a vacancy In concentration of electrons with a particular energy level E h (E) = N(E) F(E) The volume density of electron level of energy Density of state Probability that they occupied Fermi-Dirac Function N (E) as function varies with energy level 128
At the edge of conduction and valence bands, N(E) is found from band theory to vary probablically with energy away from either band edge. Integrating over energy near the conduction band edge Nc 2π m = 2 2 h Effective conduction band density of state * 3 e kt 2 19 3 10 cm ( at T = 300 o K) m e* = the effective mass of electron in conduction band h = planck s constant Similarly for valence band * 2π m kt 3 h 2 19 3 Nv = 2 10 cm ( at T = 300 2 h Effective valence band density of state o K) m h* = the effective mass of hole in conduction band m * e and m * h are generally > the mass of a free electron by factors of approximately 2 ~ 10 in oxides. on a per volume basis, the density of state is about 10 4 less than the typical atom density of solid (~10 23 cm -3 ) 129
The Fermi-Dirac Function At 0 o K, all electrons levels are occupied up to a maximum energy E f, called the Fermi energy. 0 o K, some electrons are excited to energy level above E f. Notice that at E f, the probability function remains When For an intrinsic semiconductor or insulator: conc. of conduction band electrons conc. of valence band electrons 130
To evaluate Ef, for charge neutrality in intrinsic semiconductor, n e =n h. for an intrinsic semiconductor or insulator. Writing the excitation of electrons across the bandgap as a defect chemical reaction. In contrast to the schottky constant This is because we have selected to write the electron and hole conc. In units of # per cm 3. If the semiconductor is intrinsic, n e =n h 131
In defect chemistry, ionic defect conc. are commonly given in units of mole fraction. Equilibrium constant may be written with all defect conc. in no./cm 3. The Fermi level is defined as the energy at which the probability of electron occupation is ½. In a metal, E f is the maximum occupied energy at 0 o K. In intrinsic semiconductors and insulators, F(E) = 1/2 lies very near the middle of the energy gap. The rmodynamically, the Fermi level represents the chemical potential of electrons in the system. 132
When electron donors are added to the system and increase the conduction band electron density, the chemical potential of electron is raised toward the conduction band. Accepter dopants lower the Fermi level toward the valence band edge. When dissimilar materials are joined, electrons flow from one system to the other until the Fermi levels are equilibrated. p-n junction: band bending in the vicinity of the interface. p-n junction has rectifying properties (electron conduction is energetically uphill from the n to p) Band-bending at an interface occurs when acceptor level or traps are present at the G.B. of n-type semiconductor, which result in barriers to electron conduction across the boundaries. 133
These conduction barriers, which can overcome with sufficient electric field, are the basis for nonlinear conducting devices based on semiconducting oxides Such as ZnO varistor BaTiO 3 positive temperature coefficient (PTC) thermistor Example : Intrinsic ionic and electronic defect conc. in O and NaCl. O : (intrinsic defect in Schottky pair) O, at 1400, (i) The Schottky defect conc. its value decrease with increasing temperature at a rate of about 1 mev/k. (ii) The electron and hole conc. 134
m * e ( m 0 = 0.38 m 0 = 9.11 10 a + 1673K, E n =, m g = 4.6 10 * h 31 p = (1.3 10 ~ 6.28 ev 10 = 0.77 m 20 kg) cm cm 3 3 0 )exp( 6.28 2 ev kt ) To complete the comparison, we must convert to equivalent units. 1 hs 2 12 [ V ] = [ Vo ] = K s = exp[ ] = 2.5 10 mole farction 2kT 2.5 10 [ V 12 ] = [ V ρ ONa MWO o ] = 1.4 10 g ρ O = 3.58 3 cm Na : Avogadros no WM 11 cm O 3 : molecular weight g (40.31 ) mole The schottky defect conc. is still slighly greater than that electrons and holes, due to the difference between the conduction band and valence band density of state and (N C ~N V ~10 20 cm -3 ) the density of lattice sites (5.3 10 22 cm -3 ). The conc. of both intrinsic electronic or ionic defect is so low ~ 10-12 mole fraction << impurity level nevere intrinsic 135
NaCl : h s =2.2~2.4 Ev E g =7.2 Ev at T=T m =810 [ V Na ] = [ V Cl ] 1 ppm ( due to much smaller value of the Schottky energy) intrinsic electron and hole conc. [ p] = [ n] 10 4 cm 3 (~ 10 mole fraction) [ V Na ] = [ VCl ] >> n = p ionic defect dominate model ionic solid 9 Donors and Acceptors large bandgap Ceramic with that electronic high impurity level defects mostly extrinsically determined. In elemental and compound semiconductor, impurities are (Si, Ge) (GaAs, InSb) commonly used to increase conc. of increase conc. of electrons (n-type) or holes (p-type). 136
Solutes, vacancies, and interstitials all perturb the band structure to some extent, and can introduce localized energy levels within the band gap. Dondor: a defect with energy level located near the conduction band may be able to donate an electron within the band gap. Acceptor:Defect with energy level near the valence band, which can accept an electron Ex: donor in Si are pentavalent cations (As 5+, P 5+, Sb 5+ ) Acceptor... trivalent cations (B 3+, Al 3+...) The ionization of donors and acceptors in Si As ( metal) AsSi + e B ( metal) B Si + h If the ionization energy is the same order as kt, the probability of ionization is high and the defect is known as shallow donor or acceptor. (at 25, kt=0.025 ev) Impunties with ionization energy of 0.05 ev are shallow. 137
A deep donor and acceptor refers to a solute with a high ionization energy. In an ionic solid, all ionic defects with nonzero effective charge can be viewed as either a donor or acceptor. Defects with a positive effect charge are donors (These have given up an electron in order to become ionized positive relative to the perfect lattice site. Defect with negative effective charge are acceptors (These have accepted electrons relative to the perfect lattice) O TiO 2 BaTiO 3 Al Na Nb Al Ti Al Ta Ga La Ti V O Ti V Na Ti Ti Ba, Cl Fe, V Ti V Y O Ba Ti acceptors donors donors acceptors Y is a special case, which can be either a donor or an acceptor depending on which cation it substitutes for. O O Ti Y Ti donors acceptors 138
In the band gap Scheme: For donor defect, the energy level is labeled with the charge state before ionization V V O O = = V V O O + e + e ( h ( h = 0.5 ev ) = 2 ev ) ( labeled as V ( labeled as V For aaaeptor defect, it is the charge state after ionization. ) O O ) V V + e + e = V = V ( h ( h = 0.5 ev ) = 1.5 ev ) ( labeled as V ( labeled as V ) ) 139