3.225 9 Diffraction Picture of the Origin of Band Gaps Probability Density=probability/volume of finding electron=|ψ| 2 x a x a s a π ψ π ψ 2 2 2 2 cos4 sin4 = = a a • Only two solutions for a diffracted wave • Electron density on atoms • Electron density off atoms • No other solutions possible at this wavelength: no free traveling wave © E. Fitzgerald-1999 3.225 10 • Assume electrons with wave vectors (k’s) far from diffraction condition are still free and look like traveling waves and see ion potential, U, as a weak background potential • Electrons near diffraction condition have only two possible solutions – electron densities between ions, E=E free -U – electron densities on ions, E= E free +U • Exact solution using HΨ=EΨ shows that E near diffraction conditions is also parabolic in k, E~k 2 Nearly-Free Electron Model © E. Fitzgerald-1999 5 6 3.225 11 Nearly-Free Electron Model (still 1-D crystal) m p m k E 22 222 == h E k ∆ k=2 π /L Quasi-continuous k m k E m k dk dE ∆=∆ = 2 2 h h states π/a-π/a 0 E g =2U Diffraction, k=n π /a Away from k=n π /a, free electron curve ∆ k=2 π /a=G=reciprocal lattice vector Near k=nπ/a, band gaps form, strong interaction of e- with U on ions © E. Fitzgerald-1999 3.225 12 Electron Wave Functions in Periodic Lattice • Often called ‘Bloch Electrons’ or ‘Bloch Wavefunctions’ E k π/a 0 Away from Bragg condition, ~free electron m k Ee m U m H ikx o 2 ; ; 22 22 2 2 2 2 hhh =≈∇ − ≈+∇ − = ψ Near Bragg condition, ~standing wave electron () () () xUExuGxGxxUU m H ooo ==≈≈+∇ − = ;sinor cos ; 2 2 2 ψ h Since both are solutions to the S.E., general wave is () xue ikx latticefree ==Ψ ψψ termed Bloch functions © E. Fitzgerald-1999 7 3.225 13 Block Theorem • If the potential on the lattice is U(r) (and therefore U(r+R)=U(r)), then the wave solutions to the S.E. are a plane wave with a periodic part u(r) that has the periodicity of the lattice () () () ( ) Rruru ruer rik += =Ψ ⋅ Note the probability density spatial info is in u(r): () )( * 2 * ruru o ⋅Ψ=ΨΨ An equivalent way of writing the Bloch theorem in terms of Ψ: ( ( ( ( () ( () reRr e r eRrueRr Rik rik RrikRrik Ψ=+Ψ Ψ =+=+Ψ ⋅ ⋅ ++ © E. Fitzgerald-1999 3.225 14 Reduced-Zone Scheme • Only show k=+-π/a since all solutions represented there π/a −π/a © E. Fitzgerald-1999 ) ) ) ) ) 8 3.225 15 Real Band Structures • GaAs: Very close to what we have derived in the nearly free electron model • Conduction band minimum at k=0: Direct Band Gap © E. Fitzgerald-1999 3.225 16 Review of H atom () () () ψψ φθψ EH rR = ΦΘ= Do separation of variables; each variable gives a separation constant φ separation yields m l θ gives r gives n l After solving, the energy E is a function of n ( 2 22 2 42 6.13 24 n eV n eZ E o − = − = hπε µ m l and Φ and Θ give Ψ the shape (i.e. orbital shape) l The relationship between the separation constants (and therefore the quantum numbers are:) n=1,2,3,… =0,1,2,…,n-1 m l =- , - +1,…,0,…, , (m s =+ or - 1/2) l l l l 0 -13.6eV U(r) © E. Fitzgerald-1999 ) in -1 9 3.225 17 Relationship between Quantum Numbers s p d Origin of the periodic table s s p © E. Fitzgerald-1999 3.225 18 Bonding and Hybridization • Energy level spacing decreases as atoms are added • Energy is lowered as bonding distance decreases • All levels have E vs. R curves: as bonding distance decreases, ion core repulsion eventually increases E E R s p Debye-Huckel hybridization NFE picture, semiconductors © E. Fitzgerald-1999 1 3.225 1 Properties of non-free electrons • Electrons near the diffraction condition are not approximated as free • Their properties can still be viewed as free e- if an ‘effective mass’ m* is used π/a −π/a 2 2 2 * * 22 2 k E m m k E ec ec ∂ ∂ = = h h 2 2 2 * * 22 2 k E m m k E ev ev ∂ ∂ = = h h Note: These electrons have negative mass! m k E 2 22 h = © E. Fitzgerald-1999 3.225 2 Band Gap Energy Trends © H.L. Tuller-2001 Note Trends: 1. As descend column, MP decreases as does Eg while a o increases. 2. As move from IV to III-V to II-VI compounds become more ionic, MP and Eg increase while a o tends to decrease II B III IV V VI B NO Al Si P S Zn Ga Ge As Se Cd In Sn Sb Te MP ( ° K) Eg (eV) a o A 6 / 10 3.56 / 3.16 1685 / 1770 1.1 / 3 5.42 / 5.46 1231 / 1510 / ? 0.72 / 1.35/ ? 5.66 / 5.65 / ? 508 / 798 / ? 0.08 /0.18 / 1.45 6.45 / 6.09 / ? IV / III-V / II-VI* ∗ Fill in as many of the question marks as you can. C 3.225 3 Trends in III-V and II-VI Compounds Band Band Gap Gap ( ( eV eV ) ) Lattice Constant (A) Lattice Constant (A) SiGe SiGe Alloys Alloys Larger atoms, weaker bonds, smaller U, smaller E g , higher µ, more costly! © E. Fitzgerald-1999 3.225 4 Energy Gap and Mobility Trends Material GaN AlAs GaP GaAs InP InAs InSb Eg(eV)°K 3.39 2.3 2.4 1.53 1.41 0.43 0.23 µn(cm 2 /V·s) 150 180 2,100 16,000 44,000 120,000 1,000,000 Remember that: * m e τ µ = and 2 2 2* 11 k E hm ∂ ∂ = © H.L. Tuller-2001 2 3.225 5 Metals and Insulators •E F in mid-band area: free e-, metallic •E F near band edge –E F in or near kT of band edge: semimetal –E F in gap: semiconductor •E F in very large gap, insulator © E. Fitzgerald-1999 3.225 6 Semiconductors • Intermediate magnitude band gap enables free carrier generation by three mechanisms – photon absorption –thermal – impurity (i.e. doping) • Carriers that make it to the next band are free carrier- like with mass, m* © E. Fitzgerald-1999 3 4 3.225 7 Semiconductors: Photon Absorption • When E light =hν>E g , an electron can be promoted from the valence band to the conduction band E c near band gap E v near band gap E k E=hν Creates a ‘hole’ in the valence band © E. Fitzgerald-1999 3.225 8 Holes and Electrons • Instead of tracking electrons in valence band, more convenient to track missing electrons, or ‘holes’ • Also removes problem with negative electron mass: since hole energy increases as holes ‘sink’, the mass of the hole is positive as long as it has a positive charge Decreasing electron energy Decreasing electron energy Decreasing hole energy © E. Fitzgerald-1999 5 3.225 9 Conductivity of Semiconductors • Need to include both electrons and holes in the conductivity expression * 2 * 2 h h e e he m pe m ne pene ττ µµσ=+= p is analogous to n for holes, and so are τ and m* Note that in both photon stimulated promotion as well as thermal promotion, an equal number of holes and electrons are produced, i.e. n=p © E. Fitzgerald-1999 3.225 10 Thermal Promotion of Carriers • We have already developed how electrons are promoted in energy with T: Fermi-Dirac distribution • Just need to fold this into picture with a band-gap E F f(E) 1 E g(E) g c (E)~E 1/2 in 3-D g v (E) Despite gap, at non-zero temperatures, there is some possibility of carriers getting into the conduction band (and creating holes in the valence band) E g © E. Fitzgerald-1999 + . 798 / ? 0.08 /0.18 / 1 .45 6 .45 / 6.09 / ? IV / III-V / II-VI* ∗ Fill in as many of the question marks as you can. C 3.225 3 Trends in III-V and II-VI Compounds Band Band Gap Gap ( ( eV eV ) ). = and 2 2 2* 11 k E hm ∂ ∂ = © H.L. Tuller-2001 2 3.225 5 Metals and Insulators •E F in mid-band area: free e-, metallic •E F near band edge –E F in or near kT of band. Fitzgerald-1999 3.225 4 Energy Gap and Mobility Trends Material GaN AlAs GaP GaAs InP InAs InSb Eg(eV)°K 3.39 2.3 2 .4 1.53 1 .41 0 .43 0.23 µn(cm 2 /V·s) 150 180 2,100 16,000 44 ,000