1 Lecture contents Burstein shift Excitons Interband transitions in quantum wells Quantum confined Stark effect
Absorption edges in semiconductors Offset corresponds to bandgap Abs. coefficient is orders of magnitude higher for direct transitions Abs. coefficient roughly follows density of states
Burstein-Moss shift Shift of absorption edge in degenerate semiconductors 3 Usually in direct n-type semiconductors with low effective mass k Due to occupation of band energy states up to:, the edge shifts: n 4kBTe m * e k 1 1 E g * * me mh Burstein edge in degenerate n-type semiconductor Absorption edge shift in doped n-insb From Seeger, 1973
Wannier Excitons 4 Similar to hydrogen-like impurity: electron and hole bound by screened coulomb interaction V i e ( r) r Solution for discrete energy levels: E ex 4 e 1 1 1 Ry n m n With reduced effective mass (electron and hole orbiting around their center of mass): 1 1 m 1 * * e m h Exciton Ry* Envelope function of the ground state (hydrogen-like): F( r) 1 exp 3 1 a B B r a Bohr radius: a B m 0 e m0 Free exciton can move in the crystal as a quasiparticle with a mass * * M m e m h For excitons in GaAs (m e *=0.07m: and = 1.6 ): Ry* = 6 mev, a B = 100 A
Exciton absorption Exciton absorption red-shifts the absorption edge by the exciton binding energy 5 Exciton edge absorption is higher than for band absorption (Sommerfeld Enhancement) Exciton peaks at room temperature are difficult to resolve in most materials (notable exception - quantum wells) Excitons in bound states are fragile e.g., broken by colliding with phonons (e.g., in a few hundred femtoseconds). By the uncertainty principle, they must then have broad linewidth Exciton absorption edge in GaAs From Harris, 004
Sommerfeld Enhancement Even without excitonic peaks, bandedge absorption is enhanced due to Coulomb interaction between electrons and holes The reason is an increased density of states of excitons over the band edge DOS ex dn de n Ry 3 ex Absorption edge in direct band semiconductors 6 This results in increasing of the absorption coefficient at the band edge: ex E g free Ry 1 ex E 1 Above the bandedge exciton contribution is due to mobile excitons with nonzero wavevector k: ex E g free xe g x sinhx Lots of bound states near the onset of continuum sum together to give Sommerfeld enhancement. with Ry x 1 ex E 1 g
Exciton absorption in forbidden direct band edges 7 Exciton absorption in Cu O at 4 K Forbidden direct band-to-band transition Cu O due to even parity of electron and hole wavefunctions (momentum operator has odd parity) Higher order transition (quadruple instead of dipole) and dipole transition for non-zero k in confined states are allowed From Seeger, 1973
Absorption in Si 8 Low absorption (indirect) High absorption (direct)
Interband transitions in quantum wells 9 Calculated absorption spectrum of 100A GaAs/Al 0.3 Ga 0.7 As without exciton effects Strong exciton effects are present Absorption spectra of GaAs/Al 0.3 Ga 0.7 As and In 0.53 Ga 0.47 As/n 0.5 Ga 0.48 As QWs Heavy-hole exciton binding energy as a function of well size Alloy broadening From Singh, 003
Modulation of interband transitions in bulk semiconductors: Franz-Keldysh effect 10 Concept of Franz-Keldysh effect: solution for electron and hole envelope wavefunctions with constant field are Airy functions. Wavefunctions now "tunnel" into the bandgap region allowing overlap of electron and hole wavefunctions even for photon energies less than the bandgap energy, hence allowing optical absorption below the bandgap energy. Franz-Keldysh effect Franz-Keldysh effect in GaAs From Seeger, 1973
11 Franz-Keldysh effect Franz -Keldysh effect is a central-force problem with perturbation: Absorption spectrum due to Franz-Keldysh effect Absorption spectrum reduce to the familiar square root energy when field 0 Airy function Ai(Z) Z>0: electron-hole energy < electric field potential Z<0: electron-hole energy > electric field potential, i.e. above bandgap oscillation wavefunction
Modulation of interband transitions in quantum wells 1 With applied field, electron and hole wavefunctions are distorted (second order perturbation) The intersubband separation decreases with electric field (dominant term) Binding energy of excitons decreases with field; carriers are separated by the field (few mev effect) QW no electric field QW in electric field Calculated variation of ground state intersubband transition in W= 100A GaAs/Al 0.3 Ga 0.7 As QW E () 1 1 4 15 m* e 1 W 4
Modulation of intersubband transitions in quantum wells 13 Absorption edge red-shifts with electric field Exciton absorption strength reduces with field because the electron and hole wavefunctions are separated by electric field Electric field modulation of transmission spectra of 100 A GaAs/AlGaAs QW at two polarizations Polarization rules apply due to symmetry of electron-radiation matrix elements i H er f e mc i A p f From Miller, 1986