In this example, we show how to use the stand-alone MQW GUI. The results of the Lumerical MQW solver are benchmarked to published results.
Overview
Understand the simulation workflow and key results
The Multiple Quantum Well (MQW) solver can be used to obtain several important properties of a 1-dimensional Quantum Well structure, such as the Conduction Band (CB) and Valence Band (VB) wave functions, CB and VB band diagram, optical matrix elements, refractive index perturbation and spontaneous/stimulated emission.
The materials supported are Zinc Blende and Wurtzite type crystals that can be solved with a 4×4, 6×6 and 8×8 K dot P (k.p) method. The input parameters to the MQW solver are the temperature, carrier density and the electric field inside the structure.
In this example, we show how to use the stand-alone MQW GUI (version 2022 and later) to compute the described results above for quantum wells. The results of the Lumerical MQW solver are benchmarked to published results.
Step 0: Setting up the MQW simulation (Please refer to the Appendix section of this page)
In the appendix section of this manual (at the bottom of the page), the explanation of how to set up correctly the MQW simulation using the GUI interface and how to visualize the results from the MQW is shown. Users that are already familiar with the MQW solver, can skip this step and focus on the benchmark results.
Step 1: Single quantum well of zincblende materials using the 4×4 k.p method
In this step we benchmark Lumerical MQW’s 4×4 k.p result to the published paper of Ahn et. al. [1]. The script will adjust some options and plot the results showing excellent accuracy.
Step 2: Multiple quantum wells of nitride materials using 6×6 k.p, method
In this step, we benchmark Lumerical MQW’s 6×6 k.p result to the published paper of Chuang et. al. [2]. We show excellent agreement with the paper and the script is used to automatically plot and label the results.
Step 3: Multiple quantum wells of zincblende materials using 8×8 k.p method
In this step, we benchmark Lumerical MQW’s 8×8 k.p result to the published paper of Franceschi et. al. [3]. We show excellent match with the reference paper.
Run and Results
Instructions for running the model and discussion of key results
Step 1: Single quantum well of zincblende materials using 4×4 k.p method
- Open and run [[mqw_4x4.ldev]] where the external electric field is set to zero across the heterostructure.
- Open and run [[mqw_4x4.lsf]] where the external electric field is set to 50 kV/cm.
The layout of materials is demonstrated in the following figure
The structure is composed of one single GaAs quantum well (10 nm thick) sandwiched between Al _{ 0.25 } Ga _{ 0.75 } As barriers (10 nm thick on each side). The temperature is set at 300 K and the carrier concentration is set at 1×10 ^{ 24 } 1/m ^{ 3 } .
The valence bands are plotted below and the MQW result is compared to that of Ahn et. al. [1]. With external electrical field being turned on (at magnitude F = 50 kV/cm), the two-fold valence-band degeneracy is removed due to lack of spatial inversion symmetry, as can be seen in the following plot.
The conduction band that results from MQW are plotted below
The probability distributions of eigen-states at k=0 together with the underlying band diagrams are shown below for the valence and conduction band respectively. Each curve is shifted vertically according to the eigen-energy of the underlying wavefunction. With the external electrical field being turned on, the band diagram becomes tilted.
The gain profiles of both TE and TM modes are shown below, with comparison to the reference results from Ahn et. al. [1]. With the external electric field being turned on, we observe a red shift in terms of the peak energy in the gain spectra and a reduction in the gain values. This is commonly known as the Stark effect.
The relative change in the refractive index (the real part) can be extracted from the index data structure of the MQW result. To validate this result, we use the general Kramers-Kronig relation, which relates the real part of a linear response spectrum to its imaginary part, so that we can compute the refractive index from the MQW emission result. In the following plot, the solid curves represent the data directly taken from the MQW index result, and the dashed curves represent results computed with the Kramers-Kronig relation.
Step 2: Multiple quantum wells of nitride materials using 6×6 k.p method
- Open [[mqw_6x6.ldev]] and [[mqw_6x6.lsf]]
- Run the script [[mqw_6x6.lsf]]
The layout of materials is demonstrated in the following figure.
The structure is composed of three GaN quantum wells (6 nm thick) separated by Al _{ 0.2 } Ga _{ 0.8 } N barriers (10 nm thick). The kdotp order is changed to 6×6 in the Configuration Tab of the MQW GUI.
The in-plane band dispersion of electronic eigen-states residing in the quantum wells can be read and visualized from the bandstructure result data structure of MQW solver. The script produces the following band dispersion figures of the valence and conduction band respectively. The dashed lines in the valence band plot represent the result reported in Chuang et. al. [2], whereas the solid lines are from the MQW result.
The following figures show the probability distributions (module square of wavefunction normalized to fit the plot) of the eigen-states at k=0 in valance and conduction bands respectively. Each curve is shifted vertically according to the eigen-energy of the underlying wavefunction. The probability distributions are plotted over the band diagram. As can be seen, the probability distributions are mainly confined within the individual quantum wells, and the wavefunctions contain more nodes as their eigen-energies increase.
The optical spectra of the multiple quantum wells at 300 K and 350 K with a fixed carrier concentration of 4×10 ^{ 24 } m ^{ -3 } are shown in the following figure. TE and TM denote different polarization modes of the incident electromagnetic field. As can be seen from the plot, the spectra peak is around 357 nm (842 THz), which corresponds to the energy gap between the conduction and the valence bands. Spontaneous emission rates are positive by definition, whereas stimulated emission rates can change signs. A positive emission coefficient due to optical stimulation is also known as the gain coefficient, and a negative value indicates optical absorption. This transition takes place when the optical energy exceeds the difference between the electron and hole quasi-Fermi levels. For the higher temperature of 350 K, due to the narrowing of the band gap with the increase in temperature, a red shift in terms of the peaks of the spectra is observed.
On the other hand, if we fix the temperature at 300K and increase the carrier concentration from 4×10 ^{ 24 } m ^{ -3 } to 1×10 ^{ 25 } m ^{ -3 } , a blue shift in the spectra is observed due to band filling, namely the upshift (downshift) of electron (hole) quasi-Fermi level. This effect is demonstrated in the following figure.
The momentum matrix elements that enter the calculations for emission coefficients are recorded in the result data structure named ome. The following figure plots, as a function of in-plane wavenumber, the momentum matrix element between the lowest conduction band and the highest valence band. Note that all degenerate band indices are summed over to get the momentum matrix element.
The change in complex index of the material due to light absorption or emission by quantum wells is plotted below:
Step 3: Multiple quantum wells of zincblende materials using 8x8 k.p method
- Open [[mqw_8x8.ldev]] and [[mqw_8x8.lsf]]
- Run the script [[mqw_8x8.lsf]]
The layout of materials is demonstrated in the following figure.
The structure is composed of three GaAs quantum wells (4.8 nm thick) separated by AlAs barriers (12 nm thick). The kdotp order is changed to 8×8 in the Configuration Tab of the MQW GUI.
The script produces the following band dispersion figures of the valence and conduction band respectively. The dashed lines in the valence band plot represent the result reported in Franceschi et. al. [3] whereas the solid lines the MQW result from Lumerical. Here only a small segment of the reference result (dashed lines) is shown, because at large wavenumbers the reference result is anisotropic in the quantum well plane, while MQW leverages the axial approximation such that the band dispersion is averaged over all directions in the quantum well plane.
The probability distributions of eigen-states are shown below for the valence and conduction band respectively at k=0. Each curve is shifted vertically according to the eigen-energy of the underlying wavefunction.
The spectra of emission coefficients, the momentum matrix element for TE and TM optical modes as function of wavenumber, and the change in complex index of the material due to light absorption or emission by quantum wells are plotted below:
Important Model Settings
- Material parameters: to match specific references we customize some k.p and band gap parameters in our database to enter the values used in those references.
- Band offsets: it is a good practice to check the band diagram for band offsets and override them in the MQW layer table if default values are incorrect compared to the expected band offset. These are often reported in literature as band offset ratios (e.g. ∆Ec/∆Ev).
- Strain: strain should be included either directly in the MQW layer table or the user should check the option “calculate strain” in the Layers tab, which calculates strain as the relative error between the layer lattice constant and the reference material lattice constant.
- Configuration options: frequency range for emission spectrum, Brillouin zone ratio for band structure calculation, number of transverse wave vector points, mesh spacing (1 angstrom increment), and k.p order should be appropriately set in the Configuration tab.
- Simulation parameters: Temperature, carrier density (averaged over the total layer table thickness), and electric field should be appropriately set in the Parameters tab.
- Boundary conditions: Hard wall or PML (where supported) boundary conditions and the corresponding cut-off values to filter out unbound states should be set.
- MQW partitioning: For multiple quantum wells where barriers are thick enough (decoupled wells) the layer table can be partitioned to decouple quantum wells and speed up simulation. Alternatively, for periodic layers, it is possible to simulate a single quantum well and scale the results afterwards to the full number of wells. Check our KB docs for MQW and laser examples on the next slide for more info.
Updating the Model With Your Parameters
For the case where the Quantum Wells are de-coupled, simulating a single Quantum Well is enough as it was shown in this example. However, for other cases, where the Quantum Wells are coupled, it is required to simulate all of them. Please notice that this will take more time than simulating a single Quantum Well.
Taking the Model Further
The gain curves obtained for different carrier densities can be exported to a text file. Then, in Lumerical FDTD, those gain curves can be fitted such that they provide the coefficients for the rate equations such that VCSEL simulations are possible.
Additional Resources
For examples how to use MQW solver in laser and microLED applications check:
For examples how to use MQW solver in electro-absorption applications check:
Related Publications
- Ahn, D., Chuang, S. L., and Chang, Y., Valence band mixing effects on the gain and the refractive index change of quantum well lasers. J. Appl. Phys., 4056, 1988.
- Chuang, S. L., and Chang, C. S. A band-structure model of strained quantum-well wurtzite semiconductors. Semiconductor Science and Technology, 12(3), 252, 1997.
- Franceschi, S. D., Jancu, J.-M., and Beltram, F. Boundary conditions in multiband kp models: A tight-binding test. PHYSICAL REVIEW B, 59, 9691, 1999.
See Also
- Grating coupler
- SOI taper design
- Spot size converter
- Convergence testing process for EME simulations
- Python API
- S-parameter simulator (SPS)
- S-parameter matrix sweep feature in KB
- S-parameter file formats
Related Ansys Innovation Courses
Appendix
Additional background information and theory
Step 1: Adding the MQW material
Setting up the correct parameters in the k.p material model is crucial, and this can be accomplished by using the internal parameter calculator. The composition of the ternary or quaternary alloy needs to be defined, as well as the base materials and the calculator will return the required values for the k.p model. The structure to simulate is shown below and consists of a single quantum well (GaAs) and two quantum barriers (AlGaAs; x =0.25).
Therefore, the first step is to create the material’s k.p model to be used for the gain calculations. This can be done in Lumerical Multiphysics solutions by
- Opening the material Database
- Selecting the alloy to be used (AlGaAs)
- Inputting the molecular concentration in x.
After clicking create, the new material is added to the “materials” folder, and it is a good practice to double check that the k.p model tab has been correctly populated with the values calculated internally. The procedure must be repeated one more time for the Well compositions.
Step 2: MQW setup
In this step, the optical frequency range of interest is determined as well as the carrier density inside the structure and the electric field distribution. The new GUI is useful to keep track of the parameters as well as the rest of the MQW settings. Since the 2022 R1 release, a graphical MQW interface has been added to the Lumerical Multiphysics product. This wizard can be added by clicking the MQW solver in the Design Tab. The main information to input regarding the materials is the thickness of each layer, the base refractive index of the material. The explanation of the other options can be found in the MQW reference manual.
- Open the MQW wizard and in the Layers tab, input the required thickness / Material layers. For our case, 2 Barriers of 0.01 microns, and 1 Well with the same thickness are required. The materials were created in Step 1 of this article.
- Add the neff of the material, which is 3.6, and if overriding the Valence Band offset or Strain to match the values published in reference papers is required, it can be done in this window.
The required input parameters for the simulation are the total carrier density and the electric field inside the geometry that was defined above.
- In the Parameters tab, set the carrier density (cden) to uniform with a density of 1e24 (1/m3), and an efield of 0 V/m. This means that there is no surface charge generation due to strain. Optionally, these values could be calculated with CHARGE in another step.
- In the Configuration tab, input the frequency range of interest (380 nm to 1000 nm).
Step 3: MQW results
The results that are returned are the spectral gain profile, band diagram, band structure and wavefunctions. Furthermore, temperature analysis, carrier density dependence analysis can be performed using the sweep functionalities in the GUI. Before running the simulation, it is important to verify that the band diagram is correct by clicking on “banddiagram” in the geometry tab. (It is necessary to click OK and re-open the MQW editor window to be able to visualize the results).
If the band diagram is correct, we can run the MQW simulation by clicking the run button. When the simulation completes, the results are organized in the Result View window.
We can plot the band structure with the transverse wave vector in the horizontal axis for the valence band showing non-parabolic curves for different sub bands. In the same dataset, we can plot the results for the conduction band, showing two set of similar curves, each set for each Well. Futhermore,the wavefunction results can be plotted to verify the probability amplitude. Notice that the sub band to be plotted is at the bottom of the “parameters” section in the visualizer. We can also plot the emission coefficients and can see the gain curves for Spontaneous and Stimulated emissions, for each TE and TM polarization (a total of 4 curves). Finally, the refractive index perturbation can also be plotted for the TE and TM polarization.