### Optical selection rules in Quantum Wells

In this example we demonstrate how the selection rules can be verified in a Quantum well structure.

We used a $4 nm$ wide $GaAs$ well, which is embedded in $Al_{0.3}Ga_{0.7}As$ barriers. The band structure is shown in Figure 1.

In order to simulate the optical properties we need to select envelope function calculation, and add the envelope function models we are interested in, and choose to calculate the spectrum. In this example we added $cG$(electron at gamma point), $HH$, $LH$ single band models.

The shifted envelope functions are plotted in Figure 2. corresponding to the band structure.

An interband transition is allowed when the envelope functions overlap $<\varphi_{electron} || \varphi_{hole}>0$. The overlap interal is calculated in the “SingleBand_Overlaps.txt” file:

Band_i	Band_j	subband_i	subband_j	Trans_energy(eV)	Energy_i	Energy_j	Overlaps
G_up	HH_up	0	0	1.60891	0.77331	-0.835601	0.996469
G_down	HH_up	0	0	1.60891	0.77331	-0.835601	0.996469
G_up	HH_down	0	0	1.60891	0.77331	-0.835601	0.996469
G_down	HH_down	0	0	1.60891	0.77331	-0.835601	0.996469
G_up	LH_down	0	0	1.66623	0.77331	-0.892918	1.00572
G_down	LH_down	0	0	1.66623	0.77331	-0.892918	1.00572
G_up	LH_up	0	0	1.66623	0.77331	-0.892918	1.00572
G_down	LH_up	0	0	1.66623	0.77331	-0.892918	1.00572
G_up	HH_up	0	1	1.73648	0.77331	-0.963171	1.38773e-008
G_down	HH_up	0	1	1.73648	0.77331	-0.963171	1.38773e-008
G_up	HH_down	0	1	1.73648	0.77331	-0.963171	1.38773e-008
G_down	HH_down	0	1	1.73648	0.77331	-0.963171	1.38773e-008
G_up	HH_up	0	2	1.94215	0.77331	-1.16884	0.135121
G_down	HH_up	0	2	1.94215	0.77331	-1.16884	0.135121
G_up	HH_down	0	2	1.94215	0.77331	-1.16884	0.135121
G_down	HH_down	0	2	1.94215	0.77331	-1.16884	0.135121

In this file the atomic orbital overlap and/or spin is not considered, just the value of the envelope function.

For an intraband transition the dipole integral of the envelope functions must be considered. $<\varphi_{electron} |x| \varphi_{hole}>$

The data values for a band are collected in the file in “/Dipoles/“bandname”.txt”. For example the values for “G_down” are collected like the following:

From	To	Trans_energy(eV)Energy_from	Energy_to	Dipole_x_r	Dipole_x_i	Dipole_y_r	Dipole_y_i	Dipole_z_r	Dipole_z_i
0	0	0	0.77331	0.77331	9.66421e-018	0	0	0	0	0
0	1	0.449528	0.77331	1.22284	1.11205e-009	0	0	0	0	0
0	2	0.79145	0.77331	1.56476	6.84954e-019	0	0	0	0	0
0	3	0.798423	0.77331	1.57173	3.33706e-011	0	0	0	0	0
1	0	-0.449528	1.22284	0.77331	1.11205e-009	0	0	0	0	0
1	1	0	1.22284	1.22284	6.02536e-019	0	0	0	0	0
1	2	0.341921	1.22284	1.56476	5.06469e-010	0	0	0	0	0
1	3	0.348894	1.22284	1.57173	5.12771e-018	0	0	0	0	0
2	0	-0.79145	1.56476	0.77331	6.84954e-019	0	0	0	0	0
2	1	-0.341921	1.56476	1.22284	5.06469e-010	0	0	0	0	0
2	2	0	1.56476	1.56476	9.39338e-017	0	0	0	0	0
2	3	0.00697297	1.56476	1.57173	5.40954e-009	0	0	0	0	0
3	0	-0.798423	1.57173	0.77331	3.33706e-011	0	0	0	0	0
3	1	-0.348894	1.57173	1.22284	5.12771e-018	0	0	0	0	0
3	2	-0.00697297	1.57173	1.56476	5.40954e-009	0	0	0	0	0
3	3	0	1.57173	1.57173	9.31352e-017	0	0	0	0	0