Optical absorption of an InGaAs quantum well | 1D¶

Header¶

Files for the tutorial located in nextnano++\examples\optical_spectra

absorption_InGaAs-QW_Dumitras_PRB_2002_1D_nnp.in

Scope of the tutorial:

InGaAs quantum well
simple absorption spectrum

Main adjustable parameters in the input file:

$run_optics
$w_well
$w_barrier
$alloy_composition

Relevant output files:

bias_00000\bandedges.dat - energy profile (see Figure 2.4.328)
bias_00000\Quantum\probabilities_shift_quantum_region_kp8_00000.dat - probability distributions (see Figure 2.4.328)
bias_00000\Quantum\absorption_coeff_quantum_region_TE_y_eV.dat - absorption spectrum TE (see Figure 2.4.329)
bias_00000\Quantum\absorption_coeff_quantum_region_TM_z_eV.dat - absorption spectrum TM (see Figure 2.4.329)

Introduction¶

This tutorial presents a simple setup to calculate optical absorption coefficient as a function of photon energy for transitions in a quantum well (QW) by means of 8-band $\mathbf{k} \cdot \mathbf{p}$ method. As an example, we chose 8-nm-wide In_0.2Ga_0.8As quantum well with barriers made of GaAs, as in [DumitrasPRB2002]. The InGaAs QW is pseudomorphically strained to the GaAs (001) substrate and the temperature of the system is assumed to be 150 K.

Simulation¶

Input file¶

The input file absorption_InGaAs-QW_Dumitras_PRB_2002_1D_nnp.in is prepared to solve Schrödinger and Poisson equations without self-consistency, with included strain effects. A couple of variables defined within the input file are especially interesting to play with when trying the simulation for the first time. The first of them is $run_optics which allows turning calculation of the optical spectra on and off. When the spectra are computed, the Fermi’s Golden Rule is used. Other parameters are temperature of the system $temperature and parameters characterizing the dimensions, $w_well and $w_barrier, and content of the QW $alloy_composition. We encourage modifying other parameters as well to explore the simulation capabilities.

Note

The bandoffset bowing parameter for the In(x)Ga(1-x)As alloy has been set to 0 at the end of the input file to obtain energy profile similar with the one reported in [DumitrasPRB2002].

Eigenstates in the quantum well¶

Energy profiles together with probability densities of all states confined in the InGaAs QW (at $k_\parallel = 0$) are showed in the Figure 2.4.328. The energy profiles can be found in bias_00000\bandedges.dat while the probability densities in bias_00000\Quantum\probabilities_shift_quantum_region_kp8_00000.dat.

../../../_images/1D_InGaAs_quantum_well_band_edges.svg — Figure 2.4.328 Energy profiles and probability distributions of confined electrons and holes states at $k_\parallel = 0$. The conduction band is labeled as CB. The heavy-hole valence bands is denoted as **VB (hh)** while the light-hole valence band as **VB (lh)**. The first and the second electron states are labeled as e1 and e2, respectively. Similarly, heavy-hole states are labeled as **hh1** and **hh2**. $E_1$ is a transition energy between e1 and **hh1**. $E_2$ is a transition energy between e2 and **hh2**.¶

The prepared simulation computes 20 electron states and 40 hole states (sum of light-hole and heavy-hole states). All of these states (at each wave vector) are used for computation of the optical spectra as they contribute to the part representing continuum. However, only the bound states are crucial for the analysis of the quantum well. One can quickly compute the most relevant interband transition energies, $E_1$ and $E_2$, if omitting the exciton corrections. These transitions are the strongest ones, following the selection rule $\Delta n = 0$, between two states with the same quantum number, e.g., between e1 and h1 or between e2 and h2.

The transition energies $E_1$ and $E_2$ are defined as

\[ \begin{align}\begin{aligned}E_1 = E_{\mathrm{e1}} - E_{\mathrm{hh1}},\\E_2 = E_{\mathrm{e2}} - E_{\mathrm{hh2}},\end{aligned}\end{align} \]

where $E_{\mathrm{e1}}$, $E_{\mathrm{e2}}$, $E_{\mathrm{hh1}}$, and $E_{\mathrm{hh2}}$ are eigenenergies of the states e1, e2, hh1, and hh2, respectively. Using respective values from the output file bias_00000\Quantum\probabilities_shift_quantum_region_kp8_00000.dat one can calculate

\[ \begin{align}\begin{aligned}E_{1} = 1.028\;\mathrm{eV} - [-0.275\;\mathrm{eV}] = 1.303\;\mathrm{eV},\\E_{2} = 1.118\;\mathrm{eV} - [-0.302\;\mathrm{eV}] = 1.420\;\mathrm{eV}.\end{aligned}\end{align} \]

Note that these transition energies are calculated at $k_\parallel = 0$.

Hint

One can use Show Differences feature in nextnanomat to extract these numbers from the eigenenergies stored in bias_00000\Quantum\probabilities_shift_quantum_region_kp8_00000.dat. Also, nextnano++ can produce an output file containing all transition energies, see output_transition in optics{ quantum_spectra{} }.

Optical absorption spectrum¶

When $run_optics = 1 in the input file for this tutorial, then optical spectra are also computed. The simulation is prepared to model optical spectra for two kinds of light polarization modes.

The transverse electric (TE) mode corresponds to the optical field (could be light) polarized parallel to the plane of the QW, that is in the yz plane of the simulation. In the input file we choose the direction y. Choosing z direction for the TE mode brings the same results. The light in this mode can propagate either in the plane of the QW or perpendicular to it.

The transverse magnetic (TM) mode corresponds to the optical field polarized perpendicular to the plane of the QW, that is in the x direction of the simulation. The light in this mode can propagate only in the pane of the QW.

Figure 2.4.329 shows the optical absorption spectrum as a function of photon energy for TE and TM polarized optical field.

../../../_images/1D_InGaAs_quantum_well_optical_absorption.svg — Figure 2.4.329 Absorption spectrum for TE (turquoise) and TM (magenta) modes of optical field.¶

While optical transitions involving both heavy holes and light holes can be observed within TE mode (heavy holes are dominating), only absorption with contribution of light holes is visible in the TM mode.

Attention

The above does not hold exactly in realistic conditions because the TM modes also have a component of the electric field parallel to the plane. However, this component is small in weakly guiding structures. Therefore, typically only the transition involving the light holes is seen (e1-lh1) and the heavy hole transitions are suppressed (e1-hh1, e2-hh2) in Figure 2.4.329.

The transitions $E_1$ and $E_2$ are clearly visible in the computed TE absorption spectrum as steps at 1.303 eV and 1.420 eV, respectively. Both computed TE and TM spectra exhibit series of transitions at around 1.37 eV and 1.46 eV. These are numerical artifacts related to transitions between the states confined in the InGaAs QW and numerically limited continuum in the GaAs. To explore this aspect of the simulation one can modify the width of the barrier $w_barrier and number of computed quantum states $eigen_e and $eigen_v.

Hint

Using normalization_volume may become very helpful when comparing spectra computed for different dimensions of the structure, see optics{ quantum_spectra{} }.

Last update: 07/03/2024