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Harold

A hetero-structure laser diode model

Harold EAM

Electro-Absorption Modulator with Quantum-Confined Stark Effect (QCSE) model

The Harold EAM module will allow you to model Electro-Absorption Modulators (EAM) and Electro-Refractive Modulators, calculating the absorption spectrum and refractive index spectrum of your device for any range of reverse bias values. Plotting the evolution of absorption and refractive index spectra as a function of bias will allow you to fully characterise how light gets modulated when it travels through the electro-absorption modulator.

The model includes a full physical description of the Quantum-Confined Stark Effect (QCSE). You can find a validation with experimental results here for a AlGaAs electro-absorption modulator and a SiGe electro-absorption modulator.

In terms of interface Harold EAM is fully integrated with Harold and the epitaxial structure and QWs are defined using Harold’s layers editor.

The EAM model

Unlike other electro-absorption modulator models, Harold EAM is a rigorous first-principle physical model which relies on a rigorous modelling of the physics at a fundamental level.

The EAM model consists of four solvers that are run in sequence:

  • Poisson-Drift-Diffusion Solver

  • Schrödinger Solver

  • Exciton Electrical

  • Permittivity Solver  

Harold EAM allows you to follow and control each step of the complex solution. In a multi-stage model like this, it is sometimes difficult to track the origin of problems. In Harold EAM, the user can inspect the results of the intermediate solvers, obtain physical insight into specific trends of the device’s behaviour and spot problems at early stages.

Poisson-Drift-Diffusion 
  • Calculates the band edges of the
    biased epitaxial layer structure 
Poisson Solver
Schrödinger 
  • Calculates electron, light-hole and heavy-hole eigenvalues and eigenfunctions in the
    quantum-wells
  • Can handle leaky modes in biased structures  
Poisson Solver
Exciton 
  • Calculates the energy levels of the exciton transition for each electron-hole pair (which
    make a very strong contribution to the absorption)
Poisson Solver
Electrical Permittivity 
  • Calculates the real and imaginary part of the permittivity as a function of wavelength
  • Generates the refractive index and absorption spectra  
Poisson Solver
Simulations results

Poisson-Drift-Diffusion Solver

For a defined range of reverse biases, Harold EAM can produce the following physical values as functions of vertical position  

  • conduction and valence band edges of the structure

  • quasi-Fermi levels

  • carrier and charge densities  

Poisson Solver results

Poisson Solver results

Conduction and valence band edges for a bias of 0V   Conduction and valence band edges for a bias of 5V

Conduction and valence band edges for a bias of 0V (left) and 5V (right).

Schrödinger Solver

The Schrödinger Solver of Harold EAM can produce  

  • electron and hole wavefunctions and their energy levels in the QW structure

  • number of electron-hole pairs with an overlap integral above a defined cutoff value  

This tool allows the user to observe the Quantum-Confined Stark Effect (QCSE) in a specific QW structure as a function of reverse bias.  

Electron and hole wavefunctions and their energy levels for a bias of 0V Electron and hole wavefunctions and their energy levels for a bias of 5V Electron and hole wavefunctions and their energy levels for a bias of 10V

Electron and hole wavefunctions and their energy levels for a bias of 0V (top), 5V (middle) and 10V (bottom).

 

Absorption and refractive index spectra 

The final results, the absorption and index spectra, are produced for TE and TM polarizations and can be plotted in different units:  

  • wavelength or photon energy in the x-axis

  • absorption, real and imaginary parts of refractive index or electrical permittivity  

Schroedinger Solver results

Results panel for absorption and permittivity spectra

TM absorption spectra for various values of bias TE absorption spectra for various values of bias

Absorption spectra for various values of bias for the TM (left) and TE (right) polarisations.

The user can chose the temperature, the bias and spectrum range. 

Validation: comparison with experimental data for AlGaAs and SiGe modulators

We used Harold EAM to model a AlGaAs electro-absorption modulator and a SiGe electro-absorption modulator. The results are shown below alongside published experimental data; as you can see the simulations are in good agreement with the experiments, with all main features reproduced. We expect that the differences in absolute absorption coefficients are due to scaling ambiguities in the experimental papers.

Comparison with experiment for AlGaAs modulator
AlGaAs modulator: absorption spectra for various values of the bias voltage
(left) Experimental data from [1] and (right) Harold EAM results for (top) TE polarisation and (bottom) TM polarisation

Comparison with experiment for SiGe modulator

SiGe modulator: absorption spectra for various values of the bias voltage
(left) Experimental data from [2] and (right) Harold EAM results for TE polarisation

References

[1] S.-L. Chuang, S. Schmitt-Rink, D. A. B. Miller and D. S. Chemla, “Exciton Green’s function approach to optical absorption in a quantum well with an applied electric field”, Phys. Rev. B, 43, 2, pp. 1500-1509 (1991)

[2] Y.-H. Kuo, Y. K. Lee, Y. Ge, S. Ren, J. E. Roth, T. I. Kamins, D. A. B. Miller, and J. S. Harris, “Quantum-confined stark effect in Ge–SiGe quantum wells on Si for optical modulators”, IEEE J. Sel. Topics Quantum Electron., 12, pp. 1503–1513 (2006)