ELEMENTARY BAND THEORY PHYSICIST Solid state band Valence band, VB Conduction band, CB Fermi energy, EF Bloch orbital, delocalized n-doping p-doping Band gap, Eg Direct band gap Indirect band gap Phonon or lattice vibration Peierls distotion CHEMIST Molecular orbital HOMO LUMO Chemical potential Molecular orbital, localized Reduction, pH scale base Oxidation, pH scale acid HOMO-LUMO gap Dipole allowed Dipole forbidden Vibrational mode Jahn-Teller effect l Electronic Structure of Solids Valence electrons from the atoms spread throughout the entire structure Molecular orbitals are extended over all the constituent atoms A large number of overlapping atomic orbitals lead to molecular orbitals with very similar energies = continuous band. The bands are separated by band gaps (energy values where there are no available levels) 2 Band Theory 1 atom N a atoms 4s Energy levels allowed forbidde allowed Erergy Enefgy forbidde bands one two atom atoms three atoms many atoms allowed Band Band Gap ] Band Band Gap Band Molecular Orbitals as Bands Energies of electrons are quantized = can possess only allowed energies, can occupy only allowed levels, cannot enter forbidden band gaps. 3 Molecular orbital ^^^^^^^^^ Electronic band N atomic orbitals combine to form bonding and antibonding molecular orbitals, N energy levels. Large rings - cyclic boundary condition - v n o O O A rough rule of thumb: the separation of the energy levels in the dimer corresponds to about half width of the energy band. 4 Electronic Structure of Solids Bands arise from many MO's of slightly different energies -different degree of bonding The bottom of the band - the lowest energy MO, all bonding character The top - the highest energy MO with all anti-bonding character The rest of the band is formed from all the MO's with intermediate bonding character between the two extremes 5 n= 0 . m 1 2 Band Theory Bloch functions, crystal orbitals simple example: infinite one-dimensional array % « ? of s-orbitals Xg X-| %2 Antibonding = x0-x1+x2-x3 + k = wavevector, a = lattice constant, n = orbital counter t e Large number of discreet levels = band k = 0 i=Ie°xn = 2xn u n n = Xo + X, + X2 +■ X3 + Bonding Band Theory Antibonding orbitals = x0-x1+x2-x3 + k = 0 4/= 2e° xn = 2 xn O IT "IT = Xo+ X,+ X2+ X3 + - ■• Bonding orbitals TT/C Crystal Orbitals in ID f\j rt/ nt ai f\f AO AI Jtj JL3 JU dntibonding 0-Q 0 Q~Q 4 Zi X: Nonbonding Q—•—Q—•—Q 2 CK) GO+OO i Bonding CK>0-0-0 ° # of Nodes 8 Crystal Orbitals in ID N atoms in the chain = N energy levels and N ^0 Xi Xa electronic states (MO) 00~0~0~0 k=Tr/G The wavefunction for each electronic state: -oo Where: OO • # # a is the lattice constant (spacing between atoms) n identifies the individual atoms within the chain O^O^C^C^O k=0 Xn represents the atomic orbitals k is a quantum number that identifies the wavefunction and the phase of the orbitals -A - h 9 Crystal Orbitals in ID Filling Bands by Electrons N atoms, 1 electron on each N levels in a band Occupied by pairs of electrons N/2 levels filled n N/2 levels empty Bands in Metals Bandwidth Bandwidth or band dispersion: energy difference between the highest and lowest level Bandwidth increases with better orbital overlap •shorter interatomic distance •closer energy match •topology •density, oxides more diffuse than halides, wider bands •localization of electrons - narrow bands Bandwidth arising from sigma > pi > delta overlap Core orbitals - narrow bands (0.1 eV), 4f in lanthanides Valence orbitals, s, p - wide bands (10 eV) 13 Bandwidth degree of orbital overlap between building units Wide bands = Large intermolecular overlap = delocalized Narrow bands = Weak intermolecular overlap = localized Bandwidth or band dispersion -o- o-o-o-o-o- energy difference between the highest and lowest level. k = 0 k = w/a Different types of orbitals (symmetry) form separate bands s, p, d bands distinct bands with a band gap overlaping bands depends on the separation of the orbitals and strength of the interaction between the atoms Strong interaction = wide bands and a greater overlap. xy xz zz 15 k — ir/a Two dimensional lattice Two dimensional lattice Density of states Density of states (DOS, N(E) dE) Number of levels available for electrons at different energies per unit volume of the solid. DOS is zero in the band gap Rutile Ti02 Band Structure r x m r z dos - (b) 20 band structure (a) and DOS (b) 1 -2 -5 Contributions to the total DOS of rutile -20- (a) Ti and O _23. -26--29- (b) Ti d-orbitals, t2g and eg -32 -35 (a) 1 -2 -8 t -11 f -14 [eV] 14 -17 -20--23--26--29--32 - -35 0» Classification of solids Molecular solids - N2, C6H6,I2,... Van der Waals forces, little change from the gas phase, electronic bands correspond to empty and filled MOs of the individual molecules. Ionic solids - NaCl, CaF2, NiO,... Charge transfer from cations to anions, energy bands made up from the atomic orbitals of cations and anions. NaCl: 3p of CI is the top filled band, 3s of Na is the lowest empty band. Covalent solids - diamond, Si,..... Overlap of orbitals and electron sharing between adjacent atoms. Filled bands are made up from bonding MOs, empty bands are made up from antibonding MOs. Metallic solids - Cu, Mg, W, TiO,.... Simple metals - Na Very strong overlap of atomic orbitals on adjacent atoms, arising bands are very broad, 3s, 3p, and 3d merge into a single wide band, electrons move freely, little attraction to the atomic cores. Ionic solids Example NaCl, E = 9 eV i = ions in the gas phase ii = ions in the lattice, Madelung potential, filled levels stabilized by positive potential of cations, empty levels destabilized iii = polarization energy ID OH o 00 O QC (eV) -5 C.B Na+3s ,f < t I CL"3p 5 Na+3s A i 10 Cl'3p 15 t Band gap V.B (i) (ii) (iii) (iv) Metallic and Non-metallic Solids The distinction between metallic and non-metallic solids the orbitals filling Metallic behavior - a partially filled band, no gap between the top filled level (Fermi level) and the lowest empty one Non-metallic behavior - a completely filled level (the valence band) and an empty one (the conduction band) separated by a band gap 24 Fermi level EF = the thermodynamic chemical potential for electrons in the solid Metals - boundary between filled and unfilled levels The Fermi-Dirac distribution function: f(E) = 1/[1 + exp{(E - EF)/kT}] The Fermi level cuts a band in a metal Fermi Level Ef occupation probability lA Levels E < Ef occupied E > Ef empty p = l t ■■a> ■c ■ LU (E-Ef)/kT+ 1 o 7=0 1 0 Population Fermi level Metallic and Non-metallic Solids In the filled band every electron is matched by another - no overall net motion of electric charge For conduction to occur electrons have to be excited up to the conduction band by overcoming an activation energy and hence, the conduction of these compounds increases with temperature 27 Metallic and Non-metallic Solids 28 The band gap size determines a semiconductor or an insulator Band gap = the minimum photon energy required to excite an electron up to the conduction band from the valence band Insulators - a completely filled valence band separated from the next energy band, which is empty, by a large, forbidden gap Diamond = insulator, a very large band gap of 6 eV very few electrons have sufficient energy to be promoted and the conductivity is negligibly small When conductivity of insulators is measured it is found to increase with temperature 29 Electrical Conductivity 1 _ Metal Superconductor fused, silica 10 -20 Electrical Conductivity insulators semiconductors diamond germanium glass silicon. 1 0 -16 -12 -8 10 10 u 10 Conductivity 1 o -cm 1) metals copper iron. 1 0 1 0 10- Se mi conductor 10 100 1000 77K 30 M.O.'s M.O.'s Bands in Diamond 32 Carbon Covalent atoms bonds -'ii'jhi 1999 John Wiley end Bob, Ire. All li^na rsaemed Semiconductors Semiconductors - a similar band structure to insulators but the band gap is not very large and some electrons have sufficient thermal energy to be promoted up to the empty conduction band. Two types of conduction mechanism in semiconductors: - Electrons promoted into the conduction band = negative charge carriers, move towards a positive electrode under an applied potential. - The holes these electrons leave behind = positive holes. Holes move when an electron enters them - new positive hole. The positive holes move in an opposite direction to the electrons. 33 Semiconductors E Fermi level a gap between the filled and empty states in a semiconductor/insulator Semiconductors INTRINSIC Intrinsic semiconductors are pure materials with the bandstructure. The number of electrons in the conduction band is determined only by the size of the band gap and the temperature (more electrons with small band gap and high temperature). EXTRINSIC Extrinsic semiconductors are materials where the conductivity is controlled by adding dopants with different numbers of valenece electrons to that of the original material. 35 Two fundamental differences between extrinsic and intrinsic semiconductors: 1) At standard temperatures extrinsic semiconductors tend to have significantly greater conductivities than comparable intrinsic ones. 2) The conductivity of an extrinsic semiconductor can easily and accurately be controlled by controlling the amount of dopant. Materials can be manufactured to exact specifications of conductivity. 36 Metals Insulators 38 Intrinsic Semiconductors T>0 39 Extrinsic Semiconductors Semiconductors: n- and p-Type Electrons in conduction band Donor levels " Fermi Level" o o 0 □ □ □ □ □ □ o o o Fermi Level Acceptor levels Electron holes in valence band n-type doping p-type doping 40 Extrinsic Semiconductors Doping of semiconductors - introducing atoms with more or less electrons than the parent element. Doping is substitutional, the dopant atoms directly replace the original atoms. Very low levels of dopant are required, only 1 atom in 109 of the parent atoms. 41 Extrinsic Semiconductors n-type 42 Extrinsic Semiconductors n-type Silicon - phosphorous atoms introduce extra electrons (one extra valence electron for each dopant atom introduced as P) The dopant atoms form a set of energy levels that lie in the band gap between the valence and conduction bands, but close to the conduction band. The electrons in the dopant levels cannot move directly - there is not enough of them to form a continuous band. The levels act as donor levels because the electrons have enough thermal energy to get up into the conduction band where they can move freely. n-type semiconductors, the negative charge carriers or electrons. Extrinsic Semiconductors p-type ■v p-type Si 44 Extrinsic Semiconductors p-type Doping with an element with one less valence electron such as Ga For every dopant atom - an electron missing, form a narrow, empty band consisting of acceptor levels which lie just above the valence band, discrete levels if the concentration of gallium atoms is small. Electrons from the valence band have enough thermal energy to be promoted into the acceptor levels, electrons in the acceptor levels cannot contribute to the conductivity of the material. The positive holes in the valence band left behind by the promoted electrons are able to move - p-type semiconductors, the positive holes. 45 Controlled Valency Semiconductors Some transition metal compounds can be conductors due to the presence of an element in more than one oxidation state. NiO On oxidation - turns black and becomes a relatively good conductor Some of the Ni2+ ions oxidized to Ni3+ and some Ni2+ ions diffuse out to maintain charge balance leaving cation holes. The reason for the conduction is the ability of electrons to transfer from Ni2+ to Ni3+ ions. This basically allows the Ni3+ ions to move and black NiO is therefore a p-type semiconductor. 46 Hopping Semiconductor the transfer process is thermally controlled and therefore highly dependent on temperature, makes controlling the conductivity difficult. Controlled valency semiconductors rely on control of the concentration of Ni3+ ions by controlled addition of a dopant (such as lithium). NiO Li+xNi^.2xNi3^0 the concentration of Li+ ions controls the conductivity 47 Delocalized Bands - Localized Bonds Molecules: Mulliken overlap population cl5 c2 same sign = bonding cl5 c2 opposite sign = antibonding S12 overlap integral H 2 Cj c2 Sj2 Solids: Overlap population-weighted density of states = crystal orbital overlap population (COOP) - for a specific bond ■H—H — H—H — H—H—H- -®-o-@-o@- V anti- ••-bonding bonding—* t E DOS— COOP curves Sign - positive = bonding, negative = antibonding Amplitude - depends on DOS, orbital overlap, MO coefficients DOS and COOP for the Ti-O bonds in rutile Peierls distortion 51 Peierls distortion - maximizing bonding, lowering the DOS at the Fermi level, bonding states down in energy, antibonding states up, band gap opens at the Fermi level u 1 2 3 4 5 6 Destabilization -@#tO- Stabilization