An algorithm is presented for the calculation of the nondegenerate two-photon absorption coefficient by using second-order perturbation theory and a Kane band-structure model, including the effects of nonparabolicity and nonzone-center wave functions. The polarization dependence is included by correctly accounting for the symmetry of the electronic wave functions. A comparison is made with degenerate two-photon absorption data in various zinc blende semiconductors, and excellent agreement is found without the use of fitting parameters. Comparisons are also made with nondegenerate two-photon absorption spectra measured in ZnSe and ZnS by using a picosecond continuum and with some polarization-dependent measurements obtained by a two-color Z-scan measurement.
Claudiu M. Cirloganu, Lazaro A. Padilha, Dmitry A. Fishman, Scott Webster, David J. Hagan, and Eric W. Van Stryland Opt. Express 19(23) 22951-22960 (2011)
W. Andreas Schroeder, D. S. McCallum, D. R. Harken, Mark D. Dvorak, David R. Andersen, Arthur L. Smirl, and Brian S. Wherrett J. Opt. Soc. Am. B 12(3) 401-415 (1995)
Sepehr Benis, Claudiu M. Cirloganu, Nicholas Cox, Trenton Ensley, Honghua Hu, Gero Nootz, Peter D. Olszak, Lazaro A. Padilha, Davorin Peceli, Matthew Reichert, Scott Webster, Milton Woodall, David J. Hagan, and Eric W. Van Stryland Optica 7(8) 888-899 (2020)
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z Components of the Scaled Momentum Matrix Element Mij = 〈i|pz|j〉ℏ/m0P as a Function of the Electronic k Vector in Polar Coordinatesa
Spin States
c α
hh α
lh α
so α
c α
2accs cos θ
0
(accl + alcc)cos θ
(accs + ascc)cos θ
c β
0
hh α
0
0
0
0
hh β
0
lh α
(accl + alcc)cos θ
0
2alcl cos θ
(alcs + ascl)cos θ
lh β
0
so β
(accs + ascc)cos θ
0
(alcs + ascl)cos θ
2ascs cos θ
so β
0
The labels α and β refer to the two (degenerate) spin states in each band [conduction (c), heavy-hole (hh), light-hole (lh), and split-off (so)]. The coefficients ai, bi, and ci are the Kane coefficients determined from Eqs. (10). Only transitions from α spin states are shown here, but to get transitions from the β spin states the relations Miβ,jβ = Miα,jα and
can be used.
Table 2
x Components of the Scaled Momentum Matrix Element Mij = 〈i|px|j〉ℏ/m0P as a Function of the Electronic k Vector in Polar Coordinatesa
Spin States
c α
hh α
lh α
so α
c α
2accc sin θ cos ϕ
0
(accl + alcc)sin θ cos ϕ
(accs + ascc)sin θ cos ϕ
c β
0
hh α
0
0
0
0
hh β
0
lh α
(accl + alcc)sin θ cos ϕ
0
2alcl sin θ cos ϕ
(alcs + ascl)sin θ cos ϕ
lh β
0
so α
(accs + ascc)sin θ cos ϕ
0
(alcs + ascl)sin θ cos ϕ
2ascs sin θ cos ϕ
so β
0
Again only transitions from α spin states are shown, and the relations Miβ,jβ = Miα,jα and
should be used for the transitions from the β spin states. Abbreviations are defined as for Table 1.
Table 3
Degenerate 2PA in a Variety of Zinc Blende Semiconductors
The theoretical values were determined by using the algorithm presented here, which includes band nonparabolicity, the contribution from the split-off band, and the use of nonzone-center wave functions.
Tables (3)
Table 1
z Components of the Scaled Momentum Matrix Element Mij = 〈i|pz|j〉ℏ/m0P as a Function of the Electronic k Vector in Polar Coordinatesa
Spin States
c α
hh α
lh α
so α
c α
2accs cos θ
0
(accl + alcc)cos θ
(accs + ascc)cos θ
c β
0
hh α
0
0
0
0
hh β
0
lh α
(accl + alcc)cos θ
0
2alcl cos θ
(alcs + ascl)cos θ
lh β
0
so β
(accs + ascc)cos θ
0
(alcs + ascl)cos θ
2ascs cos θ
so β
0
The labels α and β refer to the two (degenerate) spin states in each band [conduction (c), heavy-hole (hh), light-hole (lh), and split-off (so)]. The coefficients ai, bi, and ci are the Kane coefficients determined from Eqs. (10). Only transitions from α spin states are shown here, but to get transitions from the β spin states the relations Miβ,jβ = Miα,jα and
can be used.
Table 2
x Components of the Scaled Momentum Matrix Element Mij = 〈i|px|j〉ℏ/m0P as a Function of the Electronic k Vector in Polar Coordinatesa
Spin States
c α
hh α
lh α
so α
c α
2accc sin θ cos ϕ
0
(accl + alcc)sin θ cos ϕ
(accs + ascc)sin θ cos ϕ
c β
0
hh α
0
0
0
0
hh β
0
lh α
(accl + alcc)sin θ cos ϕ
0
2alcl sin θ cos ϕ
(alcs + ascl)sin θ cos ϕ
lh β
0
so α
(accs + ascc)sin θ cos ϕ
0
(alcs + ascl)sin θ cos ϕ
2ascs sin θ cos ϕ
so β
0
Again only transitions from α spin states are shown, and the relations Miβ,jβ = Miα,jα and
should be used for the transitions from the β spin states. Abbreviations are defined as for Table 1.
Table 3
Degenerate 2PA in a Variety of Zinc Blende Semiconductors
The theoretical values were determined by using the algorithm presented here, which includes band nonparabolicity, the contribution from the split-off band, and the use of nonzone-center wave functions.