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

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

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

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

Light Emitting Diodes pn junction

Light Emitting Diodes pin junction

Laser Cavity: Mirrors Providing Feedback

Fabry Perot (Edge Emitting) SC Laser

Vertical Cavity SC Laser (VCSEL)

 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

 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

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

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

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

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

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

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

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

optical susceptibility Optical Response of a Dielectric

optical dielectric function Optical Response of a Dielectric

 Plane wave propagation: Optical Response of a Dielectric

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

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

 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

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

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

dipole moment Classical Oscillator

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

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

 Even at a very simple classical level: Classical Oscillator

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

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

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

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

 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

 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

k

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

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

 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

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

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

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

=+

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

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

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

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

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

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

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

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.

Absorption Spectra of GaAs Quantum Wells

Microscopic Mechanisms for Lasing in II-VI Quantum Wells

 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