NUSOD 5 Tutorial MA Introduction to Optoelectronic Device Simulation by Joachim Piprek Outline:. Introduction: VCSEL Example. Electron Energy Bands 3. Drift-Diffusion Model 4. Thermal Model 5. Gain/Absorption Model 6. Optical Model Main source: Semiconductor Optoelectonic Devices: Introduction to Physics and Simulation by J. Piprek, Academic Press, San Diego, 3 Example: 5µm Vertical-Cavity Surface-Emitting Laser (VCSEL) GaAs/AlGaAs mirrors InGaAsP MQW oxide p-inp GaAs/InP fused interfaces Optoelectronic Device Modeling Electronic Model n-inp Gain/Absorption Model Thermal Model GaAs substrate Optical Model 3 Academic Press, used by permission 3 4
Electron Energy Bands based on Schrödinger Equation HΨ=EΨ Ψ - electron wave function H Hamiltonian E electron energy. Energy Bands Fermi Energy Position material parameters: lattice constants electron effective mass Luttinger parameters energy gaps, offsets elastic stiffness constants deformation potentials Energy [ev]..5. band edge E c band edge E v Position Fermi energy E F band gap E g Density of States T=4K T=3K..5. Fermi function Energy (ev).6.4.. E c E F for n-doping E F for p-doping E c -E D =. ev,. ev E A -E v =. ev,. ev output: E(x,y,z, k x,k y,k z ) x,y,z = position in space k x,k y,k z momentum in space. E v -. E4 E5 E6 E7 E8 E9 Doping Density (cm -3 ) 5 3 Academic Press, used by permission 6 Conduction and Valence Bands: E(k) Electron Distribution n(e) 35. Conduction Band Energy Density ( 8 cm -3 ev - ) E F E c 3.8 5 f(e) D c (E).6 5.4 n(e). 5...5..5. Electron Energy (ev) Fermi Function Electron Concentration n ( 8 cm -3 ) 5 4 3 Fermi vs. Boltzmann Boltzmann Approximation Fermi Statistics - -5 5 E F -E c (mev) Heavy-Hole Valence Band Light-Hole Valence Band E g (L) E g (Γ) E g (X) Wave vector Split-Off Valence Band 3 Academic Press, Reprinted by permission 7 3 Academic Press, Reprinted by permission 8
Influence of Strain (GaAs) Drift-Diffusion Model % compression no strain % tension based on Semiconductor Transport Equations... input: geometry, materials, doping. C HH. C HH. C LH material parameters: carrier mobility carrier generation & recombination dielectric constant energy band offsets -.5 LH -.5 LH -.5 HH output: density of electrons (n) and holes (p) current density distribution (j) current vs. voltage (IV) -. -.... -. -.... -. -.... k x [π/a ] k z k x [π/a ] k z k x [π/a ] k z 3 Academic Press, Reprinted by permission 9 Drift-Diffusion Equations p-n-junction Doping ( 8 cm -3 ) Carriers ( 8 cm -3 ) p N A w p Charges ( 8 cm -3 ) NA-p N D w n N D -n n Field (kv/cm) Potential (V) Bands (ev) E v E F F max = -qn A w p / εε φ bi =F max (w p +w n )/ E -q φ c bi - -4 - -.4.5.6 Vertical Distance (µm) 3 Academic Press, Reprinted by permission
Mobility of Electrons and Holes Recombination Mechanisms Carrier Mobility (cm /Vs) electrons holes T = 3 K T = 4 K InP a spontaneous stimulated recombination recombination E c E v photon photon photons SRH recombination R n R p phonon(s) defect level E t phonon(s) Auger recombination C n C p.. Doping Density ( 8 cm -3 ) radiative recombination nonradiative recombination 3 3 Academic Press, Reprinted by permission Recombination Parameters Multi-Quantum Well Active Region Radiative Coefficient B (cm 3 /s) x -9 x - spontaneous emission Auger recombination R spon =B (np-n p ) R Aug =(nc n +pc p )(np-n p ) InAsSb InSb InAs InGaSb GaSb InGaAs InGaAsP InP GaAs x -...4.6.8...4 Energy Band Gap (ev) Auger Coefficient C (cm 6 /s) x -3 x -4 x -5 x -6 x -7 x -8 x -9 x -3 x -3 InAsSb InSb InAs InGaSb InGaAs InGaAsP InP GaAs...4.6.8...4 Energy Band Gap (ev) Electron Energy [ev]...5. -.5 n-inp SCL MQW SCL p-inp electron injection non-radiative recombination R SRH +R Aug leakage leakage radiative recombination R sp +R st quantum well bandgap hole injection -. 4.4 4.5 4.6 4.7 4.8 4.9 Vertical Position [µm] 3 Academic Press, Reprinted by permission 5 3 Academic Press, Reprinted by permission 6
Semiconductor Heterojunctions Energy Band Gap vs. Composition for III-V Compounds.4 AlP x Sb -x-y Ga -x E E A c E A v E A v,av Semiconductor A Semiconductor B χ B affinity χ A E c band gap E g A E g B split-off A / 3 E v B / 3 E c B E v B E v,av B Direct Energy Band Gap (ev).. As -y Ga -x.8.6.4 Ga x. Ga y -y As. Ga x As -y...4.6.8. Composition x lattice matched to InP 3 Academic Press, Reprinted by permission 3 Academic Press, Reprinted by permission 8 Example: VCSEL Defects at fused InP/GaAs interface GaAs/AlGaAs mirrors InGaAsP MQW oxide p-inp n-inp GaAs substrate GaAs/InP fused interfaces Energy Band Diagram InP/GaAs band offset E c =.ev E v =.5eV donor-type interface traps density 3 cm - energy.7ev + E v cross section -7 cm Electron Energy [ev].5.. V= no interface charges.5 E donor-type interface traps c. Fermi level -.5 -. E v MQW InP GaAs AlGaAs -....3.4.5.6.7 Vertical Position [µm] npnp Shockley diode switch (thyristor) 3 Academic Press, Reprinted by permission 9
Electron Energy [ev] 3..5...5. V th =V active +V fusion +V rest V active =.8V VCSEL Threshold Voltage V fusion = V V=V th =V active +V fusion +V rest InP GaAs quasi Fermi levels -.5....3.4.5.6.7 Vertical Position [µm] Bias [V] 4. 3.5 3..5...5 threshold voltage turn-on voltage. 3 4 5 Current [ma] donor-type defects create high interface resistance.4...8.6.4.. Light Power [mw] VCSEL axis InP/GaAs Trap Effect on D Current Distribution 3 cm - interface traps no interface traps p-algaas/gaas DBR oxide confinement layer fused interface InP radius Thermal Model Heat Flux Equation (r,z) based on Heat Flux Equation input: geometry, materials heat source distribution P(r,z) Joule heat nonradiative recombination absorption etc. material parameters (steady state) thermal conductivity k(t,r,z) output: temperature distribution T(r,z) thermal resistance 3 4
InP Alloy Thermal Conductivity Thin Layer Thermal Conductivity Thermal Conductivity κ L (W/cmK).8.7.6.5.4.3 Ga -x As -y Ga x As -y AlP x Sb -x-y Ga x Ga y -y As κ InP =.68 W/Kcm Ga -x....4.6.8. Alloy Composition x AlAs GaAs AlAs bulk mean free path DBR interfaces restrict phonon mean free path 3 Academic Press, Reprinted by permission 5 6 VCSEL Self-Heating VCSEL Heat Flux two problems: heat flux heat power Geometry Temperature Heat Flux 5 T(r,z) contours thermal resistance [ K/W] temperature rise [K] 5 threshold current (dashed: constant) 4 6 8 4 6 8 4 Oxide Aperture Diameter [µm] 3 Academic Press, Reprinted by permission 7 8
Gain/Absorption Model Quantum Well Energy Bands based on (quantum well) energy band structure input: geometry, materials material parameters: band structure parameters (incl. strain) transition matrix element dielectric constant E c,barrier E 3 E Φ 3 Φ Energy E t E 3 t well C C "allowed" transitions Energy "forbidden" transitions output: band-to-band gain/absorption function g(t, λ, N) E E c,well Φ E t bulk HH LH HH LH d z z k x or k y Density of States In-plane wave vector k t 9 3 Academic Press, Reprinted by permission 3 basic gain calculation g = C g M D r (f c -f v ) Spontaneous Emission and Gain C g M D r f c -f v pre-factor transition matrix element reduced density of states inversion factor < absorption > gain bulk semiconductor f c (-f v ) f c -f v D r f c -f v quantum well gain D r spontaneous emission r sp = C sp M D r D opt f c (-f v ) E g spontaneous emission C sp D opt pre-factor photon density of states gain longer wavelength λ=c /ν E F higher energy hν longer wavelength λ=c/ν gain higher energy hν 3 3 Academic Press, Reprinted by permission 3
Quantum Well Gain Spectrum with Energy Broadening Gain / Absorption vs. Carrier Density [ 8 cm -3 ] Quantum Well Gain [/cm].5...5. no broadening Asada model for τ s (N,T) E F =hν C-LH C-HH τ s =.5 ps Γ s =.6 mev τ s =. ps Γ s =3. mev.78.8.8.84.86.88 Photon Wavelength [µm] Lineshape (/ev) single-particle gain with Lorentz broadening. Gaussian Lorentzian (πγ s ) -/ exp[- E /Γ s ] (Γ s /π)(γ s + E ) -. -.3 -. -.....3 Energy Difference E (ev) Quantum Well Gain [/cm] 6 4 - -4-6 -8 - - 8 6 4 C-HH C-LH C-HH.76.78.8.8.84.86.88 Photon Wavelength [µm] 3 Academic Press, Reprinted by permission 33 34 Gain [cm - ] 8 6 4 8 6 4 Temperature Effects on VCSEL Gain emission wavelength, N=3x 8 cm -3 o C o C 4 o C 6 o C 8 o C o C o C.46.48 4 6 8.6.6.64 Wavelength [µm] Gain [cm - ] 4 3 T[ o C]=..().. λ e =54nm 3 4 5 6 7 8 Carrier Density [cm -3 ] Optical Model based on Maxwell Equations input: geometry, materials material parameters: refractive index n(t,z) absorption coefficient α(t,z) output: standing wave (mode) emission wavelength threshold gain optical quantum efficiency => gain reduction triggers carrier density increase at higher temperatures 35 36
Maxwell Equations Dielectric Constant ε = ε + i ε ε = n k ε = n k k = λα / 4π n refractive index α - absorption const. λ - wavelength 37 3 Academic Press, Reprinted by permission 38 Helmholtz Equations VCSEL: Effective Index Method within uniform, isotropic, and passive layers reduced scalar waveguide equations for harmonic wave in z-direction k wave vector, β propagation constant 39 4
VCSEL: Transversal Optical Intensity VCSEL: Vertical Optical Intensity Bessel Functions Near Field Measurement. bottom DBR MQW top DBR. J (x).8.6 J (x) J (x).4.. -. -.4 3 4 5 x Normalized Optical Intensity..8.6.4.. - -8-6 -4-4 6 8 Lateral Position r (µm) Normalized Optical Intensity.8.6.4.. 4 5 6 7 8 9 Vertical Position (µm) 3 Academic Press, Reprinted by permission 4 3 Academic Press, Reprinted by permission 4 VCSEL: D Mode Matching 98nm Vertical-Cavity Laser top: GaAs/ O(.8µm) bottom: GaAs/AlGaAs Conclusion. self-consistent combination of relevant models. careful adjustment of material parameters Electronic Model Gain/Absorption Model Thermal Model Optical Model questions: piprek@ieee.org [ Noble et al., IEEE J. Quantum Electronics 34, 89 (998)] 43 44