1 Design, Modeling and Simulation of Optoelectronic Devices A course on: Device Physics Processes Governing Equations Solution Techniques Result Interpretations

2 Course Outline Introduction Optical equations Material model I – single electron band structure Material model II – optical gain and refractive index Carrier transport and thermal diffusion equations Solution techniques Design, modeling and simulation examples –Semiconductor lasers –Electro-absorption modulators –Semiconductor optical amplifiers –Super-luminescent light emitting diodes –Integrated optoelectronic devices

3 Introduction - Motivation Increased complexity in component design to meet the enhanced performance on demand Monolithic integration for cost effectiveness - similarity to the development of electronic integrated circuits Maturity of fabrication technologies Better understanding on device physics Maturity of numerical techniques –which leads to the recent rapid progress on the computer-aided design, modeling and simulation of optoelectronic devices

4 Introduction - Motivation New idea “Back of envelop” design Experiment (costly) Works? End Yes No (very likely) New idea Computer-aided design and modeling Simulation Works? Yes Experiment (costly) Works? End Yes No (less likely) No (very likely) Conventional Approach Effective Approach

5 Introduction - Physics Processes in Optoelectronic Device Bias I(t) or V(t) Ambient temperature T(t) Potential and carrier distribution (The Poisson and continuity equations) Temperature distribution (The thermal diffusion equation) Material gain and refractive index change (The Heisenberg equation) Band structure (The Schr Ö dinger equation) Optical field distribution (The Maxwell equations) Output Recombinations Saturation and detuning

6 Introduction - Description of Physics Processes From external bias to electron and hole generation – carrier transport model (Maxwell’s equations in its quasi- static electric field form for bulk, or Heisenberg equation for low-dimension materials such as QW and QD) From electron and hole recombination to optical gain generation – semiconductor material model (Schr Ö dinger equation) From optical gain to photon generation and propagation – Maxwell’s equations (in its full dynamic form) Unwanted accompanying thermal process – thermal diffusion model

7 Introduction - Course Organization Lectures: –Optical equations: 8 hours –Material model I: 8 hours –Material model II: 4 hours –Carrier transport and thermal diffusion: 2 hours –Solution techniques: 6 hours –Design, modeling and simulation examples: 4 hours –Total: 32 hours Textbook: –Optoelectronic Devices: Design, Modeling and Simulation, X. Li, Cambridge University Press Assessment: –Working groups of 3-5 people –Minor project on modeling (governing equation extraction for given components) 40% –Major project on simulation (problem solving on the extracted governing equation) 60% –Open-book, take-home (open-discussion) for minor and major projects

8 Optical Equations 1 Maxwell’s equations – A historical review for a better understanding

9 Electrostatic Field Coulomb’s law Feature 1: why inversely proportional to the distance square? Implication – flux conservation in a 3D space Hence we have the Gauss’ law (electric)

10 Electrostatic Field Feature 2: centered force Implication – zero-curled field (swirl free) Hence we can introduce the scalar potential to obtain the Poisson’s equation Advantages of using scalar potential instead of vectorial field: 1. only single variable is involved; 2. with both features in the electrostatic field captured. Disadvantage: PDE order is raised

11 Electrostatic Field Summary: the electrostatic field is divergence driven, curl free, and fully described by the Poisson’s equation.

12 What happens for the dielectric media in the electrostatic field? Methodology to treat the dielectric media: Dipole forms inside the dielectric media if we move this media into an electrostatic field generated by a single charge; Dipole generates a new (extra) electrostatic field which can be calculated by the Poisson’s equation; Equivalent this extra electrostatic field to a field generated by another (equivalent) single charge; Sum up the electrostatic fields generated by the two charges; Using the linear superposition theory, we can treat the dielectric media (with many dipoles) in any electrostatic field (formed by an arbitrary charge distribution).

13 What happens for the conductive media in the electrostatic field? Inside the conductive media, the electrostatic field is zero (due to the motion of free electrons, which must distribute in such a way that makes the field generated by the redistribution cancelled out with the original field applied to this conductive media). Consequently, inside the conductive media, the scalar potential is identical everywhere.

14 What happens if the charge moves? Current forms as The total number of charges must be conserved, hence the current flow through the surface of a closed region equals to the reduction of the charge density rate inside the closed region, i.e., (carrier continuity equation) The current is driven by the Coulomb force, hence the Ohm’s law holds

15 Static Motion (DC Current) Inside the current flow region, the carrier continuity leads to a continuous current flow Outside the current flow region, the field is still electrostatic as it is formed by the constant charge distribution inside the DC current flow region. Therefore, the electric field originated from a DC current is still swirl free, i.e., It can be mapped to the electrostatic field inside a dielectric media if we view J as D, σ as ε 0 ε r, and E the same. (Known as the electrostatic imitation, this mapping is widely used for field measurement, we can always take the field measuring of a DC current source rather than a charge distribution, or vice versa, once a proper mapping between the DC current source and the charge distribution is established.) What is the new effect of the charge motion then?

16 Static Motion (DC Current) Outside of the current flow region, if there is another charge stream in a static motion (i.e., a DC current), this charge stream “feels” a new force that is described by the Biot- Savert’s law in a form analogous to the Coulomb’s law, except for the two involved scalar charges must be replaced by the two involved vectorial DC currents: A swirl is generated, we name it the magnetic flux. Since this swirl acts on the moving charge only, without any effect to the stay still charge, it differs from the electric field.

17 Static Motion (DC Current) Therefore, the magnetic interaction between two moving charges is a reflection of a purely “derivative” effect; Whereas the electric interaction between two charges, regardless of their status, in motion or at rest, is a reflection of a “static” effect.

18 Magneto-static Field Feature 1: the flux is a closed vectorial flow Implication – the flux is continuous in a 3D space, known as zero-diverged (source/drain free) Hence we can introduce the vector potential and the Gauss’ law (magneto) holds

19 Magneto-static Field Feature 2: non-centered force, otherwise, the magneto- static flux has neither divergence nor curl, which makes it zero everywhere according to the Helmholtz’s theorem; the flux is, again, inversely proportional to the distance square Implication – flux conservation in a 3D space Hence we have the (derivative form of) Ampere’s law

20 Magneto-static Field Summary: the magneto-static flux is divergence free, curl driven, and fully described by the vectorial Poisson’s equation However, unlike the scalar potential introduced in the electrostatic field, the vectorial potential introduced in the magneto-static field is not so popular as it doesn’t have much pronounced advantages compared to the flux description. One advantage it bears, however, is that, unlike in the flux description where a flux component is usually a mix up of different source component, the vectorial potential component and the source component has one-to-one correspondence.

21 Summary on the Electro- and Magneto- Static Fields

22 Summary on the Electro- and Magneto- Static Fields