# Download ref.3 Optical properties in semiconductors

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Transcript

Optical properties of semiconductors Today’s topics: interaction of light with electronic states in semiconductors did this (C9S9) did this (C9S11) Fundamentals of Optical Science Spring 2008 - Class 12 slide 1 Schrödinger equation To describe interaction of light with electrons, need to describe electron states Schrödinger equation describes behavior of matter in terms of the wave function 2 2 V x x , t i x, t 2 t 2m x Hamiltonian, giving the energy density of the wave function Note that a ‘high curvature’ corresponds to a high energy (as for light waves) and that the time dependence scales with the energy of the wave function In free space and choosing V=0 we find ‘probability waves’ of the form ik r k (r ) Ae where the probability of finding matter at position x scales with ||2 or ×* Compare: probability of detecting light scales with |E(x,t)|2 or E×E* Fundamentals of Optical Science Spring 2008 - Class 12 slide 2 Momentum and energy Classical relations: Energy E = ½ mv2 Momentum p = mv = (2mE) Quantum mechanical: 2 2 V x x , t i x, t 2 t 2m x Energy 2 k e2 E V 2m Momentum p = (2mE) = or 2 ke2 E 2m ik r k (r ) Ae for V=0 ke We will see that the photon wave vector (and corresponding momentum k phot ) is often small compared to electron wave vector (e of the order of an atomic spacing) Fundamentals of Optical Science Spring 2008 - Class 12 slide 3 Valence and conduction states Inside a solid, electrons propagate in periodic potential: V(r) = V(r + a) E low bound solutions with low probability in between atoms valence electrons E high propagating solutions with significant probability in between atoms conduction electrons Fundamentals of Optical Science Spring 2008 - Class 12 slide 4 Energy vs. wave vector Shorter electron wavelength corresponds to higher energy: 2 ke2 E 2m Conduction electrons in crystal (shown for ‘very weak binding potential’) If a = 1Å E 150 eV Energy of an electron with 1 A wavelength 150 eV Energy of a photon with 1 A wavelength 12 keV Fundamentals of Optical Science Spring 2008 - Class 12 slide 5 Reduced zone scheme vs. extended zone scheme Functional form of solutions to Schrödinger equation can be written as Bloch wave: k ( r ) u k ( r )e ik r can describe high energy electrons (short waves) in terms of phase difference between adjacent atoms (unit cells) described by ‘k’, and shape of wave function within unit cell, described by function uk(r) Behavior within unit cell gives rise to energy bands We can describe electron behavior with wave vectors that lie in the first Brillouin zone 1st Brillouin zone Fundamentals of Optical Science Spring 2008 - Class 12 slide 6 Effective mass In case of larger periodic potential, ‘repulsion’ at the zone boundary: E Bands no longer periodic, but approximately parabolic near k=0 Can describe energy vs. k as before, but use effective mass m*: 2 ke2 E 2m * Note: high curvature m* must be small -2/a -/a 0 /a parabola = low effective mass’ 2/a‘sharp 3/a k Key point: even though the electron energies are large, the energy differences caused by ‘neighbor electron interactions’ are on the order of eV’s Fundamentals of Optical Science Spring 2008 - Class 12 slide 7 Important note: things normally not that simple Different direction of the wavevector: electron sees different ‘periodicity of crystal’ Example from Ashcroft and Mermin (p. 161): free electrons in FCC lattice Letters on bottom axis represent directions and magnitudes of k Fundamentals of Optical Science Spring 2008 - Class 12 slide 8 Inter-band absorption (direct gap) Finite amount of electrons results in filled and empty states In semiconductor, highest-energy filled states are in the E states are in the conduction band Lowest unoccupied valence band conduction electrons empty Egap /a -/a 0 /a valence electrons filled 2/a 3/a k Energy difference between valence and conduction band is called the bandgap of the semiconductor Material is called a semiconductor if Egap < 4eV, and insulator if Egap > 4eV Fundamentals of Optical Science Spring 2008 - Class 12 slide 9 Energy levels in real semiconductors In three dimensions, the dispersion relation depends on direction in the crystal source: Optical Properties of Semiconductor Nanocrystals, Gaponenko Fundamentals of Optical Science Spring 2008 - Class 12 slide 10 Summary of previous slides Single atoms are ‘surrounded by’ bound electrons In solids: electrons in neighboring atoms can interact: electron levels are modified, resulting in energy bands In semiconductors the highest-energy band that is ‘filled’ (occupied by electrons) is the valence band, and the lowest unoccupied states are in the conduction band Coming slides: strongest optical response if electron transitions can be induced E E conduction states E (usually empty) empty Egap k a -/a 0 /a 2/a 3/a filled Fundamentals of Optical Science Spring 2008 - Class 12 k k valence states (usually filled) slide 11 Inter-band absorption in direct gap semiconductors Direct-gap semiconductor: highest occupied and lowest unoccupied state occur at k=0 Light can induce electronic transitions if energy and momentum are conserved: E Efinal – Einitial = Ephot and k = k phot 0 (Photon: long wavelength compared to atomic spacing kphot « /a ) k Direct gap semiconductors Photons with E < Egap have insufficient energy to ‘kick a valence electron into a conduction state’ absorption starts at Ephot = Egap These band-band absorptions have the usual implications for n and (recall Kramers-Kronig relations) Fundamentals of Optical Science Spring 2008 - Class 12 slide 12 Interband absorption in indirect gap semiconductors Indirect-gap semiconductor: highest occupied and lowest unoccupied state have k≠0 Direct transitions possible for k0 strong direct interband absorption occurs at E > Egap Egap Other possibility: momentum and energy can be conserved by photon absorption and simultaneous absorption or emission of a phonon: Indirect transitions possible with ‘assistance of a phonon’ Shown here are optically induced transitions Egap - during phonon emission a phonon is generated in the process - during phonon absorption a phonon is generated in the process Fundamentals of Optical Science Spring 2008 - Class 12 slide 13 Excitons Excitons are combined electron-hole states: A free electron and a free hole (empty electronic state in the valence band) exert Coulomb force on each other: hydrogen-like bound states possible: excitonic states n=3 n=2 n=1 e Coulomb force E Eb is the exciton binding energy = h energy released upon exciton formation, or Eb k energy required for exciton breakup Wave functions of electron and hole look similar to free electron and free hole Note: exciton can move through crystal, i.e. not bound to specific atom! Fundamentals of Optical Science Spring 2008 - Class 12 slide 14 Excitonic absorption Light can excite an electron from the valence band and generate an exciton at energies slightly below the bandgap see absorption at Ephot = Egap – Eb (absorption slightly below Egap) n=3 n=2 n=1 e Coulomb force E h Eb k Exciton binding energy on the order of a few meV Thermal energy at room temperature: kT ~ 25 meV exciton rapidly dissociates at room temperature absorption lines broaden / disappear for higher temperatures Fundamentals of Optical Science Spring 2008 - Class 12 slide 15 Optical transitions related to dopant atoms Ga: 3 valence electrons Si: 4 valence electrons As: 5 valence electrons Fundamentals of Optical Science Spring 2008 - Class 12 slide 16 Donor levels Substitute Si atom with As atom (impurity atom in the Si lattice): weakly bound extra valence electron Low T Low T: donors neutral, electron weakly bound lew energy light can excite donor electron in to conduciton band Binding energy Ed similar to kT at room temperature (‘RT’): At room temperature the bound electron is quickly released impurity mostly ionized at RT : Arsenic is a donor in Si RT At RT such transitions are typically too broad to observe Fundamentals of Optical Science Spring 2008 - Class 12 slide 17 Acceptor levels Substitute Si atom with Ga atom : empty electronic state just above the Si valence band: at finite temperature, Si valence electron may fill acceptor level location of unoccupied valence state (hole) can orbit the charged Ga dopant ‘hole’ = available electron state Binding energy Ea similar to kT at room temperature (‘RT’): At room temperature the hole can leave the dopant, producing a ‘free charge’ Fundamentals of Optical Science Spring 2008 - Class 12 slide 18 Infrared absorption due to dopants Dopant binding energies low: donor level related absorptions invisible at RT, but observable at low temperatures Example: direct valence band → acceptor level absorption in boron doped Si Transition at ~40 meV absorption at 30 m : infrared Fundamentals of Optical Science Spring 2008 - Class 12 slide 19 Dopant related transitions Possible dopant related transitions: Typically visible at low T, but not clearly observable at RT Fundamentals of Optical Science Spring 2008 - Class 12 slide 20 Free carrier absorption (1/2) At RT, predominant dopant related absorption is free carrier absorption in which a photon excites an electron into a higher lying state Example: p-type semiconductors: filled states in the conduction band: optical transitions possible at Ephot < Egap ! Free electrons: absorption typically indirect phonon-assisted transition Free holes can make direct transitions form the heavy-hole band to the light-hole band holes cause stronger free carrier absorption than electrons Fundamentals of Optical Science Spring 2008 - Class 12 slide 21 Free carrier absorption (2/2) Free electron absorption can be described by the Drude model Dopant levels in semiconductors range from ~1014 - 1018 /cm3 which is ~108 – 106 lower than free electron densities in metals Plasma frequency of doped semiconductors 104 - 103 lower than of metals: IR At frequencies above plasma frequency,εr and described by p2 r ' ( ) 1 2 , p2 p2 r " ( ) 3 3 p2 2 ( ) "( ) 2 c c c 2p Electron FCA up for lower energies Free hole absorption less well defined Fundamentals of Optical Science Spring 2008 - Class 12 slide 22