2. Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials and Engineering Research Institute Sheffield Hallam University S1 1WB Sheffield, United Kingdom M.Pereira@shu.ac.uk

3. Outline  Introduction to Semiconductor Lasers and Interband Optics  Interband vs Intersubband Optics  Fundamentals and Applications  Intersubband Antipolariton - A New Quasiparticle

4. Introduction to Semiconductor Lasers  From classical oscillators to Keldysh nonequilibrium many body Green’s functions.  Fundamental concepts:  Lasing = gain > losses + feedback  Wavefunction overlap  transition dipole moments  Population inversion and gain/absorption calculations  Many body effects  Further applications: pump and probe spectroscopy – nonlinear optics

5. Laser = Light Amplification by Stimulated Emission of Radiation Stimulated emission in a two-level atomic system.

6. Light Emitting Diodes pn junction

7. Light Emitting Diodes pin junction

8. Laser Cavity: Mirrors Providing Feedback

9. Fabry Perot (Edge Emitting) SC Laser

10. Vertical Cavity SC Laser (VCSEL)

11.  In multi-section Distributed Bragg Reflector (DBR) lasers, the absorption in the unpumped passive sections may prevent lasing.  Simple theories predict that forward biasing leading to carrier injection in the passive sections can reduce the absorption. Many-Body Effects on DBR Lasers: the feedback is distributed over several layers

12.  Forward biasing is not a solution! A. Klehr, G. Erbert, J. Sebastian, H. Wenzel, G. Traenkle, and M.F. Pereira Jr., Appl. Phys. Lett.,76, 2653 (2000).  On the contrary, the absorption increases over a certain range due to Many Particle Effects!! Many-Body Effects on DBR Lasers

15.  A classical transverse optical field propagating in dielectric satisfies the wave equation: Semiclassical Optical Response

16.  A classical transverse optical field propagating in dielectric satisfies the wave equation: Fourier Transform Semiclassical Optical Response

17.  A classical transverse optical field propagating in dielectric satisfies the wave equation: Fourier Transform Semiclassical Optical Response

18.  A classical transverse optical field propagating in dielectric satisfies the wave equation: Fourier Transform Optical Response of a Dielectric

19.  A classical transverse optical field propagating in dielectric satisfies the wave equation: Fourier Transform Displacement field Optical Response of a Dielectric

20.  A classical transverse optical field propagating in dielectric satisfies the wave equation: Fourier Transform Electric field Optical Response of a Dielectric

21.  A classical transverse optical field propagating in dielectric satisfies the wave equation: Fourier Transform Polarisation Optical Response of a Dielectric

23. optical susceptibility Optical Response of a Dielectric

24. optical dielectric function Optical Response of a Dielectric

25.  Plane wave propagation: Optical Response of a Dielectric

26.  Plane wave propagation: wavenumber refractive index Optical Response of a Dielectric

27.  Plane wave propagation: extinction coefficient absorption coefficient Optical Response of a Dielectric

28.  Usually, in semiconductors, the imaginary part of the dielectric function is much smaller then the real part and we can write: Optical Response of a Dielectric

29.  Microscopic models for the material medium usually yield Kramers-Kronig relations (causality) Optical Response of a Dielectric

30. - + d ……. A linearly polarized electric field induces a macroscopic polarization in the dielectric Classical Oscillator

32. dipole moment Classical Oscillator

33.  Electron in an oscillating electric field: Newton’s equation: damped oscillator. Classical Oscillator

34.  Electron in an oscillating electric field: Newton’s equation: damped oscillator. Retarded Green function Classical Oscillator

36.  Even at a very simple classical level: Classical Oscillator

37.  Even at a very simple classical level: optical susceptibilityGreens functions Classical Oscillator

38.  Even at a very simple classical level: optical susceptibilityGreens functions Classical Oscillator

39.  Even at a very simple classical level: optical susceptibilityGreens functions renormalized energydephasing Classical Oscillator

40.  Even at a very simple classical level: optical suscpetibilityGreens functions renormalized energydephasing  Current research: Nonequilibrium Keldysh Greens Functions  Selfenergies: energy renormalization & dephasing Classical Oscillator

41.  The electrons are not in pure states, but in mixed states, described, e.g. by a density matrix  The pure states of electrons in a crystal are eigenstates of Free Carrier Optical Response in Semiconductors

42.  The electrons are not in pure states, but in mixed states, described, e.g. by a density matrix  The pure states of electrons in a crystal are eigenstates of n band label k crystal momentum Free Carrier Optical Response in Semiconductors

43. k

44.  The optical polarization is given by k Free Carrier Optical Response in Semiconductors

45.  The optical susceptibility in the Rotating Wave Approximation (RWA) is Free Carrier Optical Response in Semiconductors

46.  sum of oscillator transitions, one for each k-value.  Weighted by the dipole moment (wavefunction overlap) and by the population inversion: k Each k-value yields a two-level atom type of transition Free Carrier Optical Response in Semiconductors

47.  The Keldysh Greens functions are Greens functions for the Dyson equations: Keldysh Greens Functions

48.  The Keldysh Greens functions are Greens functions for the Dyson equations: =+ Keldysh Greens Functions

49.  Semiconductor Bloch Equations can be derived from projections of the GF’s =+ Keldysh Greens Functions

50. =+

51.  Start from the equation for the polarization at steady-state Semiconductor Bloch Equations: Projected Greens Functions Equations

52.  Start from the equation for the polarization at steady-state renormalized energies from Semiconductor Bloch Equations: Projected Greens Functions Equations

53.  Start from the equation for the polarization at steady-state dephasing from Semiconductor Bloch Equations: Projected Greens Functions Equations

54.  Start from the equation for the polarization at steady-state Screened potential Semiconductor Bloch Equations: Projected Greens Functions Equations

55.  Introduce a susceptibility Semiconductor Bloch Equations: Projected Greens Functions Equations

56. quasi-free carrier term with bandgap renormalization and dephasing due to scattering mechanims Semiconductor Bloch Equations: Projected Greens Functions Equations

57. Coulomb enhancement and nondiagonal dephasing Sum of oscillator-type responses weighted by dipole moments, population differences and many body effects! Semiconductor Bloch Equations: Projected Greens Functions Equations

58. Pump-Probe Absorption Spectra Semiconductor Slab Strong pump laser field generating carriers Weak probe beam. Susceptibility can be calculated in linear response in the field and arbitrarily nonlinear in the resulting populations due to the pump.

59. Absorption Spectra of GaAs Quantum Wells

60. Microscopic Mechanisms for Lasing in II-VI Quantum Wells

61.  Coulomb and nonequilibrium effects are important in semiconductors and can be calculated from first principles with Keldysh Greens functions.  It is possible to understand the resulting optical response as a sum of elementary oscillators weighted by dipole moments, population differences and Coulomb effects.  The resulting macroscopic quantities can be used as starting point for realistic device simulations. Summary