Carrier Transport Modeling in Quantum Cascade Lasers

Transcription

1 Carrier Transport Modeling in Quantum Cascade Lasers C Jirauschek TU München, TU München, Germany

2 Overview Methods overview Open two-level model Monte Carlo simulation Quantum transport 2

3 Typical QCL Structure V GaAs/Al 05 Ga 085 As z 3

4 Carrier Transport Simulation Methods Rate equations Monte Carlo method (semiclassical) Quantum transport (eg, density matrix, NEGF) E k in E k in Complexity, accuracy 4

5 Overview Methods overview Open two-level model Monte Carlo simulation Quantum transport 5

6 Quantum Cascade Ring Laser Op ptical Pow wer Propagation coordinate 6

7 Extended Maxwell-Bloch Equations Gain recovery Gain Los ss Propagation coordinate Propagation coordinate 7

8 Rate Equations Δ p /T T T 00 8

9 Coherent Effects T 2 E 9

10 Coherent Effects Standard amplification: Rate equations Coherent effects (Maxwell-Bloch equations) Pulse on In nversion Rabi frequency Rabi oscillations: i Quantum beat between two states Oscillatory driving field Time 0

11 Coherent instability = Rabi splitting Gain Frequency -Ω Rabi +Ω Rabi Slow gain Fast gain Mode locking Q switching Coherent instability [Spatial hole burning] Opt tical Powe er Optical Frequency (THz) Frequ uency (TH Hz) xRabi frequency, theory Spectrum splitting, experiment Wavenumber (cm - ) Power /2 (mw /2 )

12 Overview Methods overview Open two-level model Monte Carlo simulation Quantum transport 2

13 Simulation V Schrödinger-Poisson solver Monte Carlo simulation z 3

14 Monte Carlo Solver k, E kin k, E kin Carrier distribution function f(k) 4

15 Boltzmann equation carrier distribution function f(r,k,t) q f = v f c ( E+ v B) f + f t r k t scatt Acoustic phonons E Optical phonon absorption/emission (polar LO / non-polar TO phonons) E Optical phonon E Carrier-carrier scattering E k Optical phonon k k k 5

16 Evaluation of Scattering Rates n E m 5 2ΔE 3ΔE 5 ΔE 2ΔE ΔE 5 5 2ΔE 0 ΔE 4 ΔE 5 3 2ΔE 2 3ΔE 2ΔE ΔE k, E kin e ates Tabulate sca attering r on stic phono attering Acous sca LO phonon n scattering 5 ΔE 0 5 2ΔE 3ΔE 5 2ΔE ΔE ΔE Rando om selec ction

17 Ensemble Monte Carlo Method Particle N Optical phonon scattering Acoustic phonon scattering Carrier-carrier scattering 3 2 Subinterval Δt 7 Time t

18 Ensemble Monte Carlo Method State m Carrier N Carrier Carrier 2 Carrier Subinterval Δt 8 Time t

19 Ensemble Monte Carlo Method State m Carrier N Carrier Carrier 2 Carrier Subinterval Δt 9 Time t

20 Monte Carlo Solver Start Import parameters and wavefunctions Calculate scattering rate Wavefunctions, eigenenergies Schrödinger-Poisson solver (effective mass) Initialize carrier distribution Next subinterval Next particle No New free flight Scattering process Stationary state? Yes Output t of scattering rates, carrier distribution, etc No Update state of next particle End of subinterval? Yes No Last particle? Yes Evaluation of subinterval, update of carrier distribution End 20

21 Monte Carlo Simulation - Example THz QCLs Ene ergy (ev) ic carrier dis stribution fu ucntion Kineti ' 5 ' Position (nm) Upper laser level T L =00 K T L =200 K 3 T L =300 K 6 x 0-3 Lower laser level Kinetic energy (ev) Kinetic energy (ev) In nversion A/cm 2 ) Current dens (k T L =00 K T=200 K L T=300 K L Applied bias (kv/cm) 2

22 Overview Methods overview Open two-level model Monte Carlo simulation Quantum transport 22

23 Carrier Transport Simulation Methods Monte Carlo method Non-equilibrium Green's function method (NEGF) E kin E kin semiclassical; no quantum correlations (eg, dephasing) most general scheme for incoherent quantum transport modest computational effort 23 huge computational effort neglect e-e scattering,

24 Current Density for 34 THz Structure (00 K) ensity (ka A/cm 2 ) (ev) Current C d Eigene energies NEGF Monte Carlo Applied field (kv/cm) 24

25 Electron Resolved Density of States Energy resolved electron density [0 8 cm -3 ev - ] T Kubis et al, phys stat sol (c) 5, 232 (2008) 25

26 Optical Gain at Current Peak Absorption coefficient α [/cm]

27 Results (Summary) Open two-level model Includes rate equation elements and quantum effects Description of optical instabilities/mode-locking in QCLs C Y Wang et al, Phys Rev A 75, 03802(R) (2007) Monte Carlo simulation of QCLs Takes into account kinetic electron distribution Analysis of experimental results C Jirauschek et al, J Appl Phys 0, (2007); C Jirauschek and P Lugli, phys stat sol (c) 5, 22 (2008); C Jirauschek and P Lugli, J Comput Electron 7, 436 (2008) Quantum transport Includes quantum correlations/dephasing Allows for simulation of spectral gain 27

28 Acknowledgment Collaborations Monte Carlo simulations: Prof P Lugli, TUM Scamarcio group, Bari (experimental) Maxwell-Bloch model: Kärtner group, MIT Capasso group, Harvard Further collaborations: Vogl group/tillmann Kubis, TUM Financial support