Lecture Electron states and optical properties of semiconductor nanostructures
Bulk semiconductors Band gap E g Band-gap slavery: only light with photon energy equal to band gap can be generated. Very few semiconductors are suitable Oscillator strengths, selection rules cannot be changed Low density of states, low dg/dn Near-infrared, red, blue Just recently green Mid-infrared: low-t operation, bad quality
Quantum-confined electron gas E g E g E g z
Envelope function approximation (a) Add quantum-well potential U(r) to the bulk Hamiltonian H 0 (b) Seek the solution as ikxxik y y ( r ) fn( z) e un0( ) A r n f n (z) slowly varying envelope functions (c) Replace k z with i z and solve the resulting differential matrix equation for the vector f(z) For a single band we may obtain effective mass approximation: m eff ( z, E) d f ( z) k dz m eff U ( z) y x f ( z) z Ef ( z) Continuity of f and its flux Particle-in-a-box intuition
Quantum wells y x z f (z) Bulk: f e ikr u k (r) Quantum well: ik xik y u k x y ( z) e ( r) E E z k z k n ; f (z) n,,... E n E n k 0 ; n,,... m 50 5 50 00 40 75 30 50 0 5 0 E k m e 0.0 0.0 0.0 0.0 k, k A -, A - 0.0 0.0 0.0 0.0 n
Envelope functions f(z) and optical transitions.54.539 E, ev E (ev).078 0.67 AlInAs GaInAs 80 A AlInAs 0.6 0.56-0.3-0.305 0 40 80 0 60 00 z (Å) 00.6..595.9 0 00 z, A Subband dispersion E(k ) Interband transitions: similar to bulk materials, but better performance E (ev) E, ev 0.785 0.4 0.38-0.05 4 6 8 0 K k, (0 cm 6 - ) 0 7
Optical transitions in quantum wells Intersubband transitions: sharp atomic-like lines No cross-absorption.6.595..9 E, ev E (ev) 0.785 0.4 0.38-0.05-0.05 4 6 8 0 0 K, k cm (0 6 cm - - ) 0 7 Line broadening ~ 0 mev due to interface roughness and non-parabolicity (in narrow-gap semiconductors)
Intersubband transitions: dipole moment.65.5.37 E (ev).3.09 Dipole matrix element: 0.95 0 40 80 0 60 00 z (Å) * zmn fm ( z) fn( z) dz z f ~ ~ cos kzz, f ~ sin kz z; z Lz Lz Typical values ~ 0-00 A Compare with atomic transitions ~ 0.-0.5 A L z
Intersubband transitions: selection rules.65.5.37 E (ev).3.09 0.95 0 40 80 0 60 00 z (Å) - Only TM-polarization (E QW plane) - Dipole matrix element: * zmn fm ( z) fn( z) dz z f and f 3 are even -> z 3 = 0 E, z
Build your own nanostructure: E-field V ( z) V ( z) eez Sharp resonances Tunable frequencies and oscillator strengths High-quality materials Indirect-gap semiconductors Coupling to other excitations: phonons, plasmons
Superlattices Periodic super potential superimposed on periodic lattice potential Keldysh 964; Esaki and Tsu 970
600 500 Energy (mev) Energy (mev) Energy (mev) Wavefunctions From discrete to quasicontinuous spectrum E(q) 400 300 00 minigap 00 miniband 0 0 00 00 300 400 500 600 Z-axis (Angstrom)
Molecular Beam Epitaxy Growth rate m/hr or atomic layer in sec A. Cho, Bell Labs.
III-V semiconductor grown on Ge
Only materials with closely matching lattice periods and thermal expansion coefficients can be grown on top of each other without defects.5.0 GaP AlAs AlAsSb 55 nm Energy Gap (ev).5.0 0.5 GaAs InP GaInAs AlInAs AlInAs AlSb GaSb InAs InSb 0.0 5.4 5.6 5.8 6.0 6. 6.4 Lattice Constant (A) Ga 0.47 In 0.53 As/Al 0.48 In 0.5 As/InP GaAs/Al x Ga -x As/GaAs
GaAs/Al x Ga -x As; Ga x In -x As y P -y /Al x In -x As on InP; InAs -x Sb/AlGa -x Sb on GaSb
Quantum wires and dots http://www.nanonet.go.jp/english/mailmag/ 003/00a.html
http://www.mpi-halle.mpg.de/
Magnetic quantum wires and dots Landau levels: E n n B k m z n ; n 0,,,... B z 0 00 80 y x z E n+> n> c 60 40 0 0.0 0.0 0.0 0.0 k z, A - DOS (E)
Electron states in semiconductor nanostructures Blackboard derivation
Density of states
Quantum-confined electron gas has sharp, tunable resonances in optics (from terahertz to visible light) How can we use it? Determine material parameters: effective masses, band offsets, g-factors, scattering rates Study new phenomena: Bloch oscillations, huge optical nonlinearities, BEC of excitons, entangled states, Make new devices: lasers, detectors, transistors, memory, computers, etc.
How to get lasing between intersubband transitions? Problem: ultrafast relaxation due to phonon emission
Superlattice laser: Kazarinov and Suris 97 Electric field F = 40 kv/cm Energy (mev) F = 00 kv/cm Energy (mev) Does not work due to domain formation and insufficient population inversion
J. Faist, F. Capasso, et al. Science 64, 553 (994) Quantum cascade lasers e injector (n-doped) active region 3 injector (n-doped) active region From sawtooth to staircase potential 50 mev 60 nm Control of lifetimes: phonons, tunneling; need t3 > t E = E phonon k w, jlw, j kb, jlb, Cascading: high power when t_stim approaches T j
55 nm Vertically stack 0-30 stages; sandwich them into the waveguide supporting a low-loss transverse EM mode
Mid-infrared ( ~ 4-0 m) Quantum cascade lasers are Extremely powerful (P max > 0 W) Operate at high temperatures T max ~ 00 C Reliable, stable, etc.
Problems with lasers: Lasers are not widely tunable, do not cover all wavelengths of interest, can operate CW at room-t only in the narrow spectral range, cryogenic at very short and very long wavelengths Nonlinear optical sources (OPO etc.) flexible and tunable, but they are bulky and expensive Is it possible to combine the advantages of both types of sources??
Nonlinear Optics Nonlinear crystal () () P NL E EE Pump Nonlinear signal Need high-power external laser pump Nonlinearity is small in the transparency region Bulky and costly lab equipment
Resonant nonlinear optics with nanostructures Energy, mev Coupled quantum well structures can be designed to have huge resonant optical nonlinearity (known for 30 years) 3 () 3 3 N e d d3d )( ~ 3 ( 3 3 d ij ij ij detunings linewidths dipole moments () ~0 4 0 6 pm/v ) z, A Compare with -0 pm/v for bulk crystals
However, these advantages are usually inaccessible () () P NL E EE 3 3 Double resonance: () N e d d3d )( ~ 3 ( 3 3 ) E SH E p Resonance in absorption for both pump and the nonlinear signal: () N Im pump ~ e d ( )
4 3 Conventional nonlinear optics 3 Pump All detunings are large: ~ >> ; I SH I p All frequencies are in the transparency region of the NLO crystals Absorption and nonlinearity are small; Need high power pump
A way to get around resonant absorpti Resonant optical nonlinearity is accompanied by resonant absorption Energy, mev 3 z, A () N e d d3d )( ~ 3 ( 3 3 Solution: create the nonlinear medium with gain This leads to nonlinear quantum cascade lasers )
Integration of injection lasers with resonant electronic nonlinearities We deal with semiconductors 5 4 3 I. II. Let s try to inject electrons, create population inversion and generate the optical pump right inside the nonlinear structure Active nonlinear medium: Laser field serves as a coherent optical pump for the nonlinear process One can approach resonance since resonant absorption is compensated by laser gain The tightest possible confinement and mode purity No problem with external pump; an injection-pumped device
Monolithic integration of quantum-cascade lasers with resonant optical nonlinearities 5 4 I 5 4 energy z g active region 3 3 I. II. Second harmonic generation Maximizing the product of dipoles d 3 d 34 d 4 Quantum interference between cascades I and II Milliwatt power in SHG: O. Malis et al. 004 () ~ 0 5 pm/v in the mid-ir () ~ 0 6 pm/v in the THz This is NOT sequential photon absorption/reemission!
Single-mode and tunable SH emission Fabry-Perot Laser Single-mode Laser 3.5 Intensity (a.u.) 0 7. 7. 7.3 7.4 7. 7.3 Intensity (a.u.) 0 0 6.7 6.8 6.9 7.0 7. 6.8 7.0 3.50 3.49 3.48 Wavelength (m) Intensity (a.u.) 0 0 3.55 3.60 3.65 3.70 3.6 3.65 Wavelength (m) Intensity (a.u.) 0 3.35 3.40 3.45 3.50 3.55 3.4 3.5 Wavelength (m) 3.47 0. nm/k 3.46 0 00 00 300 Temperature (K) APL 84, 75 (004)
Third Harmonic Generation 000 Power (arb. units) 00 0 0. 0.0 50 SHG THG 300 350 400 450 500 Energy (mev) 5 4 3 Triple resonance 3 P( 3 ). m 5.5 m 3.7 m P( ) ~ 0 I. II. (3) ~ 0-7 esu T. Mosely, A. Belyanin, C. Gmachl, Optics Express, 97 (004)
Difference frequency generation in two-wavelength QCLs (a) Electron population (b) 3 THz THz Make a powerful mid-ir QCL emitting at two modes Provide strong nonlinearity for frequency mixing process Design a low-loss, phase-matched waveguide for all three modes THz, k k k THz
Raman lasing and other coherent nonlinear phenomena 3 Triply resonant Raman lasing E s Lasing without inversion E p Slow light, intersubband polaritons, mixing with phonons, plasmons, t Beyond semiclassical picture: squeezing, entanglement Beyond rate approximation: instabilities, superfluorescence Density matrix: 3 3 3 3 33 E s E p E p Quantum coherence E s 0 t
In most Raman amplifiers and lasers, both pump and Raman fields are very far from one-photon resonance p p, s Very large detuning to avoid absorption No real transitions to upper state 3 Raman shift is fixed to be the phonon frequency Signal gain 3-00 - 80-60 - 40-0 0 40-0 s - 3, mev - 0 E p E s - 30-40 - 50 Gain at two-photon resonance: p - s =
Stimulated Raman scattering Raman inversion Raman gain I p ( N ) N ~ s Raman decoherence rate 3 Raman coherence << (Except experiments by Sokolov, Harris et al.) I p I s Propagation of coupled Raman and pump fields Manley-Rowe relation: I p I s p s const
Approaching resonance Both good and bad effects get enhanced Real one-photon processes become important Raman coherence also increases Raman gain increases strongly Absorption is increased 3 ~ E p E s t Es E p E p E s
E p 3 E s Stokes gain at arbitrary detuning Gain, cm - 5 0 5-30 - 0-0 0 0 30-5 ~ s - 3, mev 3 3 3 3 i( i( i( 3 3 ) s ) p ( )) Rabi frequency Raman gain is enhanced, but resonant absorption of the pump limits the interaction length p One-photon absorption d ( n n ) 3 3 g Re s p ( n ) n * 3 * * / 3 p 3 s Two-photon gain ij p ~ 5 s 0 mev d 3 E p d 3 E s
Mid-IR Raman injection laser Raman shift is determined by intersubband transition and is tunable Energy 5 4 Miniband 6 6 3 3 3 0 mev Resonant -scheme Miniband Optical power (arb. units).5.0 0.5 0 Power (mw) 6.6 6.7 6.8 Wavelength (µm) 7.5 8.0 8.5 9.0 9.5 0.0 0.5 Wavelength (µm) 00 0 Laser Stokes 0. 0 3 4 5 Current (A) Pump Stokes Very large Raman gain at resonance: ~ 0-4 cm/w Position M. Troccoli, A. Belyanin, F. Capasso, Nature 433, 845 (005) 40 mw Raman threshold 6 mw Stokes power