Calculates complex index \(\bar{n}=\Delta n+i\kappa\), gain, and spontaneous emission for TE and TM modes in multiple quantum well structures using 4x4 k.p electronic band structure method [1-3]. The conduction band is parabolic, while heavy and light hole valence bands are mixed according to 4x4 k.p method and they are nonparabolic. \(\Delta n\) is the change in the input refractive index due to MQW gain (or absorption) and \(\kappa\) is the corresponding attenuation coefficient. This script command supports the same materials as mqwgain.
Syntax |
Description |
---|---|
result = mqwindex(stack_properties, simulation_parameters, config); |
stack_properties: struct with fields that define MQW stack geometry and material properties. simulation_parameters: struct with fields that define simulation parameters for which the output will be calculated. config: struct with fields that configure the behavior of the simulation. result: struct or a cell of structs in case of multiple partitions, where each struct contains 5 datasets: spatial band diagram, band structure in (E,k) space, spatial wave functions for each (E,k) state, emission coefficients, and complex index. |
result = mqwindex(stack_properties, simulation_parameters); |
same as above, but using all default values for the fields in the config struct. |
The input arguments to this script command are exactly the same as for mqwgain, while result is similar to mqwgain with the addition of a complex index:
Syntax |
Type |
Description |
---|---|---|
banddiagram |
dataset |
Conduction and valence band edge including strain, but not including quantum confinement effects. |
bandstructure |
dataset |
(E,kt) band diagram for conduction and valence bands. With the exciton model turned off the attributes are: conduction_band, valence_band_lo, valence_band_up, where the 4x4 k.p basis in the valence band is transformed into two 2x2 bases (lo for lower and up for upper). For more information look at references [1] and [2]. With the exciton model turned on the attributes are: conduction_band, valence_band, with the 4x4 k.p basis in the valence band (the basis is not transformed). Parameters are kt and subband. |
wavefunction |
dataset |
Spatial wavefunction for each (E,kt) point. With the exciton model turned on the attributes are: conduction_band_1, valence_band_lo_1, valence_band_lo_2, valence_band_up_1, valence_band_up_2, where the 4x4 k.p basis in the valence band is split into two 2x2 bases (lo for lower and up for upper) and the vectors in each 2x2 basis are designated with 1 and 2. For more information look at references [1] and [2]. With the exciton model turned on the attributes are: conduction_band_1, valence_band_1, valence_band_2, valence_band_3, valence_band_4, with the 4x4 k.p basis in the valence band (the basis is not transformed) and the vectors in the 4x4 basis designated with 1, 2, 3, and 4. Parameters are coordinate, kt, and subband. |
ome |
dataset |
Optical matrix element. With excitons turned off: Magnitude squared of the dipole matrix element in the units of \(nm^2\). Attributes are ome_lo_TE, ome_lo_TM, ome_up_TE, ome_up_TM, where TE and TM designate optical modes and up and lo refer to the 2x2 bases, same as for the bandstructure and wavefunction. Parameters are kt (transverse wave vector), CBsubband (conduction band subband index) and VBsubband (valence band subband index). With excitons turned on: Oscillator strength (unitless). Attributes are ome_TE and ome_TM where TE and TM designate optical modes. Parameters are the exciton orbital quantum number (orbital) and angular momentum quantum number (angularMomentum). |
emission |
dataset |
Gain and spontaneous emission coefficients in the units of [1/m]. Attributes are: spontaneous_TE, spontaneous_TM, stimulated_TE, stimulated_TM, where TE and TM stand for electromagnetic modes. Parameters are: frequency/energy/wavelength and ndensity (charge density). Emission coefficients are calculated for the total stack thickness, including barriers. If only quantum well thickness is of interest, excluding barriers, these coefficients should be scaled by multiplying with (total length)/(total qw length). It is important to ensure that emission coefficients apply only to the thickness used for the calculation of the mode overlap with the gain region. When using partitioning, there will be overlapping barriers between different partitions, e.g. simulation_parameters.stackpartitions = [1,3;3,5], where 1, 3, and 5 are barriers. In that case emission coefficients for each partition again apply to the total thickness of that partition, meaning there may be some double counting with respect to the mode overlap region thickness. To avoid this, emission coefficients can be scaled to apply to quantum wells only, or to apply to a portion of the partition that does not overlap with adjacent partitions. When the exciton model is turned on the attributes become: absorption_TE, absorption_TM. These represent the absorption coefficients (negative gain) in the units of [1/m]. The spontaneous emission is not calculated due to the assumption of the depleted carrier density in the quantum wells. |
index |
dataset |
Complex index \(\bar{n}=\Delta n+i\kappa\). Attributes are: index_TE, index_TM, where TE and TM stand for electromagnetic modes. Parameters are: frequency/energy/wavelength and ndensity (charge density). Same comments regarding index scaling for correct thickness as for emission dataset. |
ex |
dataset |
Exciton energies Ex. Exciton energies are a function of exciton orbital quantum number (orbital) and angular momentum quantum number (angularMomentum). |
phix |
dataset |
Exciton wavefunctions PhiX in the momentum (in-plane wavevector) space. The wave function coefficients are parametrized in terms of conduction band subband index (cSubband), valence band subband index (vSubband), transverse wave vector (kt), angular momentum quantum number (angularMomentum), and orbital quantum number (orbital). |
NOTE: Please note that due to a different formulation in mqwindex, where the dipole matrix element is used, compared to mqwgain, where the momentum matrix element is used, there may be a slight difference in the values of gain (or absorption) calculated from mqwindex compared to the values calculated from mqwgain. For more information please see the MQW solver reference manual. |
References
- D. Ahn et al., J. Appl. Phys. 64, 4056 (1988)
- S. L. Chuang, Physics of Optoelectronic Devices
- Chuang, Phys. Rev. B, 43, 9649 (1991)
- Vurgaftman et al., J. Appl. Phys., 89, 5815 (2001)
- C. Y.-P. Chao et al., Phys. Rev. B, 48, 8210 (1993)
Examples
See MQW product reference manual.