===== Absorption Spectrum =====
With the optics features of **nextnano++**, the optical absorption spectrum can be calculated for various polarization directions.
==== Physics Model ====
The absorption rate in a semiconductor can be written as
\[
R_{ba} = \frac{2}{V} \sum_{k_a} \sum_{k_b} \frac{2 \pi}{ \hbar} |H_{ba}| ^2 \delta(E_b - E_a -\hbar \omega)(1-f_a) \cdot f_b,
\]
where the matrix element $|H_{ba}|$ depends on the polarization of light and the $\bf k$ vector.
==== Input File ====
===== Results =====
==== Transition Matrix Element ====
The transition matrix element $H_{ab}({\bf k})$ is plotted as a function of ${\bf k}=(k_x,k_y)$ for a quantum well structure (1D simulation) in figure {{ref>absk}}
;#;
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2.45359778404 3.68039679527 19.5683151298
2.86253070831 3.68039679527 20.2020374442
3.27146363258 3.68039679527 20.8508396147
3.68039679527 3.68039679527 21.5116819931
;#;
Transition matrix element in ${\bf k}$ space
==== Eigenvalues ====
The dispersion of the ground state energy is plotted with respect to $\bf k$ space in figure {{ref>edisp}} for electrons, and in figure {{ref>hdisp}} for holes, respectively.
;#;
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0 3.27146363258 -0.903598376688
0.408932954073 3.27146363258 -0.906450417517
0.817865908146 3.27146363258 -0.914984811351
1.22679889202 3.27146363258 -0.929180458738
1.63573181629 3.27146363258 -0.949012009122
2.04466485977 3.27146363258 -0.974456867107
2.45359778404 3.27146363258 -1.00550011937
2.86253070831 3.27146363258 -1.04213388878
3.27146363258 3.27146363258 -1.08435584787
3.68039679527 3.27146363258 -1.13216664005
-3.68039679527 3.68039679527 -1.17996002526
-3.27146363258 3.68039679527 -1.13216659196
-2.86253070831 3.68039679527 -1.08997156397
-2.45359778404 3.68039679527 -1.05337606782
-2.04466485977 3.68039679527 -1.02238335793
-1.63573181629 3.68039679527 -0.996999522479
-1.22679889202 3.68039679527 -0.977233995321
-0.817865908146 3.68039679527 -0.963099269308
-0.408932954073 3.68039679527 -0.954608275125
0 3.68039679527 -0.951771197426
0.408932954073 3.68039679527 -0.954608246015
0.817865908146 3.68039679527 -0.963099333448
1.22679889202 3.68039679527 -0.97723403565
1.63573181629 3.68039679527 -0.996999511551
2.04466485977 3.68039679527 -1.02238333506
2.45359778404 3.68039679527 -1.05337612503
2.86253070831 3.68039679527 -1.08997159884
3.27146363258 3.68039679527 -1.13216656366
3.68039679527 3.68039679527 -1.17995996739
;#;
Energy dispersion relation $E(k_x,k_y)$ for the lowest electron eigenvalue
;#;
-3.68039679527 -3.68039679527 5.32195115805
-3.27146363258 -3.68039679527 5.14027884292
-2.86253070831 -3.68039679527 4.97289872625
-2.45359778404 -3.68039679527 4.82150057848
-2.04466485977 -3.68039679527 4.68802902715
-1.63573181629 -3.68039679527 4.57461229112
-1.22679889202 -3.68039679527 4.48343380365
-0.817865908146 -3.68039679527 4.41654819403
-0.408932954073 -3.68039679527 4.37565907366
0 -3.68039679527 4.3618992577
0.408932954073 -3.68039679527 4.3756590303
0.817865908146 -3.68039679527 4.41654822468
1.22679889202 -3.68039679527 4.4834338311
1.63573181629 -3.68039679527 4.57461234221
2.04466485977 -3.68039679527 4.68802903982
2.45359778404 -3.68039679527 4.82150056386
2.86253070831 -3.68039679527 4.97289872152
3.27146363258 -3.68039679527 5.14027886168
3.68039679527 -3.68039679527 5.32195113308
-3.68039679527 -3.27146363258 5.1402788242
-3.27146363258 -3.27146363258 4.95002862907
-2.86253070831 -3.27146363258 4.77360208181
-2.45359778404 -3.27146363258 4.61291020168
-2.04466485977 -3.27146363258 4.47022607936
-1.63573181629 -3.27146363258 4.34811768349
-1.22679889202 -3.27146363258 4.24930149758
-0.817865908146 -3.27146363258 4.17640482466
-0.408932954073 -3.27146363258 4.13165697153
0 -3.27146363258 4.11656537092
0.408932954073 -3.27146363258 4.13165697486
0.817865908146 -3.27146363258 4.17640481493
1.22679889202 -3.27146363258 4.24930147615
1.63573181629 -3.27146363258 4.34811768765
2.04466485977 -3.27146363258 4.47022606074
2.45359778404 -3.27146363258 4.61291022306
2.86253070831 -3.27146363258 4.77360208073
3.27146363258 -3.27146363258 4.95002866071
3.68039679527 -3.27146363258 5.14027881135
-3.68039679527 -2.86253070831 4.97289875816
-3.27146363258 -2.86253070831 4.77360211484
-2.86253070831 -2.86253070831 4.58746914725
-2.45359778404 -2.86253070831 4.41662003406
-2.04466485977 -2.86253070831 4.2636719
-1.63573181629 -2.86253070831 4.13169214932
-1.22679889202 -2.86253070831 4.02404304875
-0.817865908146 -2.86253070831 3.94408567842
-0.408932954073 -2.86253070831 3.89475285163
0 -2.86253070831 3.87806893214
0.408932954073 -2.86253070831 3.89475282867
0.817865908146 -2.86253070831 3.94408567659
1.22679889202 -2.86253070831 4.02404307988
1.63573181629 -2.86253070831 4.13169211593
2.04466485977 -2.86253070831 4.26367189888
2.45359778404 -2.86253070831 4.416620012
2.86253070831 -2.86253070831 4.58746917358
3.27146363258 -2.86253070831 4.77360214124
3.68039679527 -2.86253070831 4.97289871743
-3.68039679527 -2.45359778404 4.82150059574
-3.27146363258 -2.45359778404 4.61291023153
-2.86253070831 -2.45359778404 4.41662001445
-2.45359778404 -2.45359778404 4.23492446809
-2.04466485977 -2.45359778404 4.0707758082
-1.63573181629 -2.45359778404 3.92778175488
-1.22679889202 -2.45359778404 3.81006273034
-0.817865908146 -2.45359778404 3.72190172715
-0.408932954073 -2.45359778404 3.66716493833
0 -2.45359778404 3.64858981456
0.408932954073 -2.45359778404 3.6671649589
0.817865908146 -2.45359778404 3.72190173976
1.22679889202 -2.45359778404 3.81006273843
1.63573181629 -2.45359778404 3.92778176229
2.04466485977 -2.45359778404 4.07077586784
2.45359778404 -2.45359778404 4.23492443539
2.86253070831 -2.45359778404 4.41662000735
3.27146363258 -2.45359778404 4.61291021878
3.68039679527 -2.45359778404 4.82150057518
-3.68039679527 -2.04466485977 4.68802906722
-3.27146363258 -2.04466485977 4.47022602655
-2.86253070831 -2.04466485977 4.26367184953
-2.45359778404 -2.04466485977 4.07077584514
-2.04466485977 -2.04466485977 3.89478225296
-1.63573181629 -2.04466485977 3.7398423514
-1.22679889202 -2.04466485977 3.61092352187
-0.817865908146 -2.04466485977 3.51342774863
-0.408932954073 -2.04466485977 3.45243254544
0 -2.04466485977 3.43164569442
0.408932954073 -2.04466485977 3.45243250478
0.817865908146 -2.04466485977 3.51342771103
1.22679889202 -2.04466485977 3.61092354989
1.63573181629 -2.04466485977 3.73984236044
2.04466485977 -2.04466485977 3.894782275
2.45359778404 -2.04466485977 4.07077585784
2.86253070831 -2.04466485977 4.26367189768
3.27146363258 -2.04466485977 4.47022604735
3.68039679527 -2.04466485977 4.68802907299
-3.68039679527 -1.63573181629 4.57461233686
-3.27146363258 -1.63573181629 4.3481177132
-2.86253070831 -1.63573181629 4.13169212828
-2.45359778404 -1.63573181629 3.92778173958
-2.04466485977 -1.63573181629 3.73984237907
-1.63573181629 -1.63573181629 3.57251749274
-1.22679889202 -1.63573181629 3.43165119906
-0.817865908146 -1.63573181629 3.3239276393
-0.408932954073 -1.63573181629 3.25592788016
0 -1.63573181629 3.23263615961
0.408932954073 -1.63573181629 3.25592790402
0.817865908146 -1.63573181629 3.32392762667
1.22679889202 -1.63573181629 3.43165119761
1.63573181629 -1.63573181629 3.57251750549
2.04466485977 -1.63573181629 3.7398423795
2.45359778404 -1.63573181629 3.92778175095
2.86253070831 -1.63573181629 4.13169214428
3.27146363258 -1.63573181629 4.34811773134
3.68039679527 -1.63573181629 4.57461231897
-3.68039679527 -1.22679889202 4.48343380803
-3.27146363258 -1.22679889202 4.24930149758
-2.86253070831 -1.22679889202 4.02404304149
-2.45359778404 -1.22679889202 3.81006272919
-2.04466485977 -1.22679889202 3.61092351641
-1.63573181629 -1.22679889202 3.43165120061
-1.22679889202 -1.22679889202 3.27890195817
-0.817865908146 -1.22679889202 3.16069886045
-0.408932954073 -1.22679889202 3.08534486519
0 -1.22679889202 3.05938600272
0.408932954073 -1.22679889202 3.08534484889
0.817865908146 -1.22679889202 3.16069883522
1.22679889202 -1.22679889202 3.27890196006
1.63573181629 -1.22679889202 3.43165118841
2.04466485977 -1.22679889202 3.61092356056
2.45359778404 -1.22679889202 3.81006272677
2.86253070831 -1.22679889202 4.02404306176
3.27146363258 -1.22679889202 4.24930146154
3.68039679527 -1.22679889202 4.48343381062
-3.68039679527 -0.817865908146 4.41654819403
-3.27146363258 -0.817865908146 4.17640481108
-2.86253070831 -0.817865908146 3.94408565546
-2.45359778404 -0.817865908146 3.72190171366
-2.04466485977 -0.817865908146 3.51342769674
-1.63573181629 -0.817865908146 3.32392764285
-1.22679889202 -0.817865908146 3.16069886045
-0.817865908146 -0.817865908146 3.03297203742
-0.408932954073 -0.817865908146 2.95076172742
0 -0.817865908146 2.92227889516
0.408932954073 -0.817865908146 2.95076176227
0.817865908146 -0.817865908146 3.03297207263
1.22679889202 -0.817865908146 3.16069883522
1.63573181629 -0.817865908146 3.3239275849
2.04466485977 -0.817865908146 3.51342775781
2.45359778404 -0.817865908146 3.7219016958
2.86253070831 -0.817865908146 3.94408568251
3.27146363258 -0.817865908146 4.17640480232
3.68039679527 -0.817865908146 4.41654817268
-3.68039679527 -0.408932954073 4.37565907366
-3.27146363258 -0.408932954073 4.13165696313
-2.86253070831 -0.408932954073 3.89475283183
-2.45359778404 -0.408932954073 3.66716498152
-2.04466485977 -0.408932954073 3.45243256257
-1.63573181629 -0.408932954073 3.25592789062
-1.22679889202 -0.408932954073 3.08534486519
-0.817865908146 -0.408932954073 2.95076172742
-0.408932954073 -0.408932954073 2.86350371539
0 -0.408932954073 2.8331376035
0.408932954073 -0.408932954073 2.86350373777
0.817865908146 -0.408932954073 2.95076176227
1.22679889202 -0.408932954073 3.08534484889
1.63573181629 -0.408932954073 3.25592791219
2.04466485977 -0.408932954073 3.45243256131
2.45359778404 -0.408932954073 3.66716495683
2.86253070831 -0.408932954073 3.8947528626
3.27146363258 -0.408932954073 4.13165701038
3.68039679527 -0.408932954073 4.37565902447
-3.68039679527 0 4.3618992577
-3.27146363258 0 4.1165653562
-2.86253070831 0 3.87806889003
-2.45359778404 0 3.64858982357
-2.04466485977 0 3.43164571422
-1.63573181629 0 3.23263615515
-1.22679889202 0 3.05938600272
-0.817865908146 0 2.92227889516
-0.408932954073 0 2.83313763509
0 0 2.8020631268
0.408932954073 0 2.83313763509
0.817865908146 0 2.92227889516
1.22679889202 0 3.05938600272
1.63573181629 0 3.23263615515
2.04466485977 0 3.43164571422
2.45359778404 0 3.64858982357
2.86253070831 0 3.87806889003
3.27146363258 0 4.1165653562
3.68039679527 0 4.3618992577
-3.68039679527 0.408932954073 4.37565902447
-3.27146363258 0.408932954073 4.13165701038
-2.86253070831 0.408932954073 3.8947528626
-2.45359778404 0.408932954073 3.66716495683
-2.04466485977 0.408932954073 3.45243256131
-1.63573181629 0.408932954073 3.25592791219
-1.22679889202 0.408932954073 3.08534484889
-0.817865908146 0.408932954073 2.95076176227
-0.408932954073 0.408932954073 2.86350373777
0 0.408932954073 2.8331376035
0.408932954073 0.408932954073 2.86350371539
0.817865908146 0.408932954073 2.95076172742
1.22679889202 0.408932954073 3.08534486519
1.63573181629 0.408932954073 3.25592789062
2.04466485977 0.408932954073 3.45243256257
2.45359778404 0.408932954073 3.66716498152
2.86253070831 0.408932954073 3.89475283183
3.27146363258 0.408932954073 4.13165696313
3.68039679527 0.408932954073 4.37565907366
-3.68039679527 0.817865908146 4.41654817268
-3.27146363258 0.817865908146 4.17640480232
-2.86253070831 0.817865908146 3.94408568251
-2.45359778404 0.817865908146 3.7219016958
-2.04466485977 0.817865908146 3.51342775781
-1.63573181629 0.817865908146 3.3239275849
-1.22679889202 0.817865908146 3.16069883522
-0.817865908146 0.817865908146 3.03297207263
-0.408932954073 0.817865908146 2.95076176227
0 0.817865908146 2.92227889516
0.408932954073 0.817865908146 2.95076172742
0.817865908146 0.817865908146 3.03297203742
1.22679889202 0.817865908146 3.16069886045
1.63573181629 0.817865908146 3.32392764285
2.04466485977 0.817865908146 3.51342769674
2.45359778404 0.817865908146 3.72190171366
2.86253070831 0.817865908146 3.94408565546
3.27146363258 0.817865908146 4.17640481108
3.68039679527 0.817865908146 4.41654819403
-3.68039679527 1.22679889202 4.48343381062
-3.27146363258 1.22679889202 4.24930146154
-2.86253070831 1.22679889202 4.02404306176
-2.45359778404 1.22679889202 3.81006272677
-2.04466485977 1.22679889202 3.61092356056
-1.63573181629 1.22679889202 3.43165118841
-1.22679889202 1.22679889202 3.27890196006
-0.817865908146 1.22679889202 3.16069883522
-0.408932954073 1.22679889202 3.08534484889
0 1.22679889202 3.05938600272
0.408932954073 1.22679889202 3.08534486519
0.817865908146 1.22679889202 3.16069886045
1.22679889202 1.22679889202 3.27890195817
1.63573181629 1.22679889202 3.43165120061
2.04466485977 1.22679889202 3.61092351641
2.45359778404 1.22679889202 3.81006272919
2.86253070831 1.22679889202 4.02404304149
3.27146363258 1.22679889202 4.24930149758
3.68039679527 1.22679889202 4.48343380803
-3.68039679527 1.63573181629 4.57461231897
-3.27146363258 1.63573181629 4.34811773134
-2.86253070831 1.63573181629 4.13169214428
-2.45359778404 1.63573181629 3.92778175095
-2.04466485977 1.63573181629 3.7398423795
-1.63573181629 1.63573181629 3.57251750549
-1.22679889202 1.63573181629 3.43165119761
-0.817865908146 1.63573181629 3.32392762667
-0.408932954073 1.63573181629 3.25592790402
0 1.63573181629 3.23263615961
0.408932954073 1.63573181629 3.25592788016
0.817865908146 1.63573181629 3.3239276393
1.22679889202 1.63573181629 3.43165119906
1.63573181629 1.63573181629 3.57251749274
2.04466485977 1.63573181629 3.73984237907
2.45359778404 1.63573181629 3.92778173958
2.86253070831 1.63573181629 4.13169212828
3.27146363258 1.63573181629 4.3481177132
3.68039679527 1.63573181629 4.57461233686
-3.68039679527 2.04466485977 4.68802907299
-3.27146363258 2.04466485977 4.47022604735
-2.86253070831 2.04466485977 4.26367189768
-2.45359778404 2.04466485977 4.07077585784
-2.04466485977 2.04466485977 3.894782275
-1.63573181629 2.04466485977 3.73984236044
-1.22679889202 2.04466485977 3.61092354989
-0.817865908146 2.04466485977 3.51342771103
-0.408932954073 2.04466485977 3.45243250478
0 2.04466485977 3.43164569442
0.408932954073 2.04466485977 3.45243254544
0.817865908146 2.04466485977 3.51342774863
1.22679889202 2.04466485977 3.61092352187
1.63573181629 2.04466485977 3.7398423514
2.04466485977 2.04466485977 3.89478225296
2.45359778404 2.04466485977 4.07077584514
2.86253070831 2.04466485977 4.26367184953
3.27146363258 2.04466485977 4.47022602655
3.68039679527 2.04466485977 4.68802906722
-3.68039679527 2.45359778404 4.82150057518
-3.27146363258 2.45359778404 4.61291021878
-2.86253070831 2.45359778404 4.41662000735
-2.45359778404 2.45359778404 4.23492443539
-2.04466485977 2.45359778404 4.07077586784
-1.63573181629 2.45359778404 3.92778176229
-1.22679889202 2.45359778404 3.81006273843
-0.817865908146 2.45359778404 3.72190173976
-0.408932954073 2.45359778404 3.6671649589
0 2.45359778404 3.64858981456
0.408932954073 2.45359778404 3.66716493833
0.817865908146 2.45359778404 3.72190172715
1.22679889202 2.45359778404 3.81006273034
1.63573181629 2.45359778404 3.92778175488
2.04466485977 2.45359778404 4.0707758082
2.45359778404 2.45359778404 4.23492446809
2.86253070831 2.45359778404 4.41662001445
3.27146363258 2.45359778404 4.61291023153
3.68039679527 2.45359778404 4.82150059574
-3.68039679527 2.86253070831 4.97289871743
-3.27146363258 2.86253070831 4.77360214124
-2.86253070831 2.86253070831 4.58746917358
-2.45359778404 2.86253070831 4.416620012
-2.04466485977 2.86253070831 4.26367189888
-1.63573181629 2.86253070831 4.13169211593
-1.22679889202 2.86253070831 4.02404307988
-0.817865908146 2.86253070831 3.94408567659
-0.408932954073 2.86253070831 3.89475282867
0 2.86253070831 3.87806893214
0.408932954073 2.86253070831 3.89475285163
0.817865908146 2.86253070831 3.94408567842
1.22679889202 2.86253070831 4.02404304875
1.63573181629 2.86253070831 4.13169214932
2.04466485977 2.86253070831 4.2636719
2.45359778404 2.86253070831 4.41662003406
2.86253070831 2.86253070831 4.58746914725
3.27146363258 2.86253070831 4.77360211484
3.68039679527 2.86253070831 4.97289875816
-3.68039679527 3.27146363258 5.14027881135
-3.27146363258 3.27146363258 4.95002866071
-2.86253070831 3.27146363258 4.77360208073
-2.45359778404 3.27146363258 4.61291022306
-2.04466485977 3.27146363258 4.47022606074
-1.63573181629 3.27146363258 4.34811768765
-1.22679889202 3.27146363258 4.24930147615
-0.817865908146 3.27146363258 4.17640481493
-0.408932954073 3.27146363258 4.13165697486
0 3.27146363258 4.11656537092
0.408932954073 3.27146363258 4.13165697153
0.817865908146 3.27146363258 4.17640482466
1.22679889202 3.27146363258 4.24930149758
1.63573181629 3.27146363258 4.34811768349
2.04466485977 3.27146363258 4.47022607936
2.45359778404 3.27146363258 4.61291020168
2.86253070831 3.27146363258 4.77360208181
3.27146363258 3.27146363258 4.95002862907
3.68039679527 3.27146363258 5.1402788242
-3.68039679527 3.68039679527 5.32195113308
-3.27146363258 3.68039679527 5.14027886168
-2.86253070831 3.68039679527 4.97289872152
-2.45359778404 3.68039679527 4.82150056386
-2.04466485977 3.68039679527 4.68802903982
-1.63573181629 3.68039679527 4.57461234221
-1.22679889202 3.68039679527 4.4834338311
-0.817865908146 3.68039679527 4.41654822468
-0.408932954073 3.68039679527 4.3756590303
0 3.68039679527 4.3618992577
0.408932954073 3.68039679527 4.37565907366
0.817865908146 3.68039679527 4.41654819403
1.22679889202 3.68039679527 4.48343380365
1.63573181629 3.68039679527 4.57461229112
2.04466485977 3.68039679527 4.68802902715
2.45359778404 3.68039679527 4.82150057848
2.86253070831 3.68039679527 4.97289872625
3.27146363258 3.68039679527 5.14027884292
3.68039679527 3.68039679527 5.32195115805
;#;
Energy dispersion relation $E(k_x,k_y)$ for the highest hole eigenvalue