Absorption of quantum wells¶
For modelling the optical properties of QWs we use the method described by S. Chuang ([1]). The absorption coefficient at thermal equilibrium in a QW is given by:
where is the overlap integral between the holes
in level
and the electrons in level
;
is a
step function,
= 1 for
, 0 and 0 for
,
is the 2D joint density of states,
a proportionality constant dependent on the energy, and
the excitonic contribution, which will be discussed later.
Here, is the refractive index of the material,
the reduced,
in-plane, effective mass and
an effective period of the
quantum wells. The in-plane effective mass of each type of carriers is
calculated for each level, accounting for the spread of the wavefunction
into the barriers as ([2]):
This in-plane effective mass is also used to calculate the local density
of states shown in Figure [fig:qw]b. In Eq. [eq:QW_abs2],
is the momentum matrix element,
which depends on the polarization of the light and on the Kane’s energy
, specific to each material and determined experimentally.
For band edge absorption, where
= 0, the matrix elements for
the absorption of TE and TM polarized light for the transitions
involving the conduction band and the heavy and light holes bands are
given in Table [tab:matrix_elements]. As can be deduced from this
table, transitions involving heavy holes cannot absorb TM polarised
light.
TE | TM | |
---|---|---|
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0 |
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Table: Momentum matrix elements for transitions in QWs.
is the bulk matrix element.
In addition to the band-to-band transitions, QWs usually have strong
excitonic absorption, included in Eq. [eq:qw_abs] in the term
. This term is a Lorenzian (or Gaussian) defined by an
energy
and oscillator strength
. It
is zero except for
where it is given by Klipstein
et al. ([3]):
Here, is a constant with a value between 0 and 0.5 and
is the width of the Lorentzian, both often adjusted to
fit some experimental data. In Solcore, they have default values of
= 0.15 and
= 6 meV.
is the exciton
Rydberg energy ([1]).
Fig. [fig:QW_absorption] shows the absorption coefficient of a range of InGaAs/GaAsP QWs with a GaAs interlayer and different In content. Higher indium content increases the depth of the well, allowing the absorption of less energetic light and more transitions.

-
solcore.absorption_calculator.absorption_QW.
exciton_rydberg_energy_2d
(me, mh, eps_r)[source]¶ Parameters: - me – electron effective mass (units: kg)
- mh – hole effective mass (units: kg)
- eps_r – dielectic constant (units: SI)
Returns: The exciton Rydberg energy
-
solcore.absorption_calculator.absorption_QW.
exciton_bohr_radius
(me, mh, eps)[source]¶ Parameters: - me – electron effective mass (units: kg)
- mh – hole effective mass (units: kg)
- eps – dielectic constant (units: SI)
Returns: Exciton Borh radius
-
solcore.absorption_calculator.absorption_QW.
alpha_c_hh_TE
(E, z, E_e, E_hh, psi_e, psi_hh, well_width, me, mhh, Ep, nr)[source]¶ Absortion coefficient for incident light forming a transision between hh and c band of a quantum well
NB. Assumes that valence band is a zero energy, might need to manually apply an offset.
Parameters: - E – photon energy (units: J)
- z – mesh points along growth direction (units: m)
- E_e – Electron state energy (units: J)
- E_hh – Heavy hole state energy (units: J)
- psi_e – Electron envelope function (psi_e^2 must be normalised)
- psi_hh – Heavy hole envelope function (psi_hh^2 must be normalised)
- well_width – (units: m)
- me – electron effective mass (units: kg)
- mhh – heavy hole effective mass (units: kg)
- Ep – the Kane parameter “Optical dipole matrix elemet”, sometimes “Momentum matrix element”, e.g. Ep for GaAs ~ 28eV
- nr – refractive index
Returns:
-
solcore.absorption_calculator.absorption_QW.
alpha_c_lh_TE
(E, z, E_e, E_lh, psi_e, psi_lh, well_width, me, mlh, Ep, nr)[source]¶ Absortion coefficient for incident light forming a transision between hh and c band of a quantum well
NB. Assumes that valence band is a zero energy, might need to manually apply an offset.
Parameters: - E – photon energy (units: J)
- z – mesh points along growth direction (units: m)
- E_e – Electron state energy (units: J)
- E_hh – Heavy hole state energy (units: J)
- psi_e – Electron envelope function (psi_e^2 must be normalised)
- psi_hh – Heavy hole envelope function (psi_hh^2 must be normalised)
- well_width – (units: m)
- me – electron effective mass (units: kg)
- mhh – heavy hole effective mass (units: kg)
- Ep – the Kane parameter “Optical dipole matrix elemet”, sometimes “Momentum matrix element”, e.g. Ep for GaAs ~ 28eV
- nr – refractive index
Returns:
-
solcore.absorption_calculator.absorption_QW.
alpha_exciton_ehh_TE
(exciton_index, E, z, E_e, E_hh, psi_e, psi_hh, well_width, me, mhh, Ep, nr, eps, hwhm=9.6e-22, dimensionality=0.15, line_shape='Lorenzian')[source]¶ Parameters: - exciton_index –
- E –
- z –
- E_e –
- E_hh –
- psi_e –
- psi_hh –
- well_width –
- me –
- mhh –
- Ep –
- nr –
- eps –
- hwhm –
- dimensionality –
- line_shape –
Returns:
-
solcore.absorption_calculator.absorption_QW.
alpha_exciton_elh_TE
(exciton_index, E, z, E_e, E_lh, psi_e, psi_lh, well_width, me, mlh, Ep, nr, eps, hwhm=9.6e-22, dimensionality=0.15, line_shape='Lorenzian')[source]¶ Parameters: - exciton_index –
- E –
- z –
- E_e –
- E_lh –
- psi_e –
- psi_lh –
- well_width –
- me –
- mlh –
- Ep –
- nr –
- eps –
- hwhm –
- dimensionality –
- line_shape –
Returns:
-
solcore.absorption_calculator.absorption_QW.
sum_alpha_c_hh_TE
(E, z, E_e, E_hh, psi_e, psi_hh, well_width, me, mh, Ep, nr)[source]¶ Parameters: - E –
- z –
- E_e –
- E_hh –
- psi_e –
- psi_hh –
- well_width –
- me –
- mh –
- Ep –
- nr –
Returns:
-
solcore.absorption_calculator.absorption_QW.
sum_alpha_c_lh_TE
(E, z, E_e, E_lh, psi_e, psi_lh, well_width, me, mh, Ep, nr)[source]¶ Parameters: - E –
- z –
- E_e –
- E_lh –
- psi_e –
- psi_lh –
- well_width –
- me –
- mh –
- Ep –
- nr –
Returns:
-
solcore.absorption_calculator.absorption_QW.
sum_alpha_exciton_c_hh_TE
(E, z, E_e, E_hh, psi_e, psi_hh, well_width, me, mh, Ep, nr, eps, hwhm=9.6e-22, dimensionality=0.5, line_shape='Lorenzian')[source]¶ Parameters: - E –
- z –
- E_e –
- E_hh –
- psi_e –
- psi_hh –
- well_width –
- me –
- mh –
- Ep –
- nr –
- eps –
- hwhm –
- dimensionality –
- line_shape –
Returns:
-
solcore.absorption_calculator.absorption_QW.
sum_alpha_exciton_c_lh_TE
(E, z, E_e, E_lh, psi_e, psi_lh, well_width, me, mlh, Ep, nr, eps, hwhm=9.6e-22, dimensionality=0.5, line_shape='Lorenzian')[source]¶ Parameters: - E –
- z –
- E_e –
- E_lh –
- psi_e –
- psi_lh –
- well_width –
- me –
- mlh –
- Ep –
- nr –
- eps –
- hwhm –
- dimensionality –
- line_shape –
Returns:
-
solcore.absorption_calculator.absorption_QW.
calc_alpha
(QM_result, well_width, kane_parameter=4.48e-18, refractive_index=3.5, hwhm=9.6e-22, dimensionality=0.5, theta=0, eps=1.1421902283930001e-10, espace=None, line_shape='Lorenzian')[source]¶ Calculates the absorption coeficient of a quantum well structure assuming the parabolic approximation for the effective masses.
Parameters: - QM_result – The output of the Schrodinger solver, incldued in the ‘quantum_mechanics’ package
- well_width – The well width
- kane_parameter – The Kane parameter
- refractive_index – Refractive (effective) index of the QW
- hwhm – Full width at half maximum of the excitonic lineshape
- dimensionality –
- theta –
- eps –
- espace –
- line_shape –
Returns:
-
solcore.absorption_calculator.absorption_QW.
NonBlackBodyEmission
(E, voltage=0, nr=3.5, T=300)[source]¶ Parameters: - E –
- voltage –
- nr –
- T –
Returns:
-
solcore.absorption_calculator.absorption_QW.
calc_emission
(QM_result, well_width, voltage=0, theta=0)[source]¶ Parameters: - QM_result –
- well_width –
- voltage –
- theta –
Returns:
References¶
[1] | (1, 2) Chuang, S.L.: Physics of Optoelectronic Devices. Wiley- Interscience, New York (1995) |
[2] | Barnham, K., Vvedensky, D. (eds.): Low-Dimensional Semi- conductor Structures: Fundamentals and Device Applications. Cambridge University Press, Cambridge (2001) |
[3] | Klipstein, P.C., Apsley, N.: A theory for the electroreflectance spec- tra of quantum well structures. J. Phys. C Solid State Phys. 19(32), 6461–6478 (2000) |