Exciton absorption in infinite quantum well

Input files:

1D_InterbandExcitonAbsorption_InfiniteWell_GaAs_8kp_nnp.in 1D_InterbandExcitonAbsorption_InfiniteWell_GaAs_effective_mass_nnp.in

Scope of the tutorial:

In this tutorial, we show how excitonic correction affects the absorption in infinite quantum well.

The most relevant keywords:

• optics{ quantum_spectra{} }

• quantum{ region{ excitons{} } }

Relevant output files:

bias_xxxxx\bandedges.dat bias_xxxxx\Optics\absorption_quantum_region_TE_eV.dat bias_xxxxx\Quantum\probabilities_shift_quantum_region_kp8_00000.dat

This tutorial presents calculation of interband absorption spectrum in a quantum well including excitonic effects. The tutorial aims to provide a comprehensive explanation of how excitonic correction significantly influences the optical absorption characteristics in a quantum well.

In this tutorial we calculate the absorption spectrum of a 10 nm GaAs quantum well. The purpose is to calculate the absorption spectrum for a simple model and model that includes excitonic effects on the absorption spectrum.

The tutorial is structured into two parts. The first part involves the computation of valence and conduction states using simple parabolic dispersion models, also known as the “single-band” model. In the second part, the states will be computed using an 8-band kp Hamiltonian.

Theory of optical excitonic correction

An exciton is a bound state of an electron and a hole in a solid material, resulting from the Coulomb attraction between them. The exciton eigenvalue is computed using variational approach with the wave function

\[ \begin{align}\begin{aligned}F (r, x_h, x_e) = f(x_e) g(x_h) \phi(r)\\\phi(r) = \frac{2}{\pi} \frac{1}{\lambda} \exp (-r/\lambda)\end{aligned}\end{align} \]

where \(f(x_e), g(x_h)\) – electron and hole wave functions, \(r\) – radial variable in plane orthogonal to growth direction, \(\lambda\) – variational parameter.

The exciton correction to absorption consists of 2 terms: exciton peak and Sommerfeld enhancement factor (also known as Coulomb enhancement). The exciton peak is located few \(meV\) below the absorption edge of corresponding electron-hole pair (i.e. transition energy is reduced by binding energy of exciton) The intensity of the peak is dependent on the parameter \(\lambda\).

\[\alpha_{ex} \propto \frac{2}{\pi \lambda ^2}V(E_{ij}-E_b, \hbar \omega)\]

where \(V\) is Voigt profile, \(E_{ij}\) is the transition energy between electron i and hole j, \(E_{b}\) is binding energy of exciton.

The second contribution is enhancemnt of the absorption above transition energy by the Sommerfeld factor

\[S_{2D} = \frac{ \exp(\pi/\sqrt{\Delta})} {\cosh(\pi/\sqrt{\Delta})}\]

where \(\Delta\) is the total excess energy of the electron-hole pair normalized to \(E_{b}/4\)

Input File

In order to include excitonic correction to absorption, excitons section should be present both in quantum{region{}} and optics{quantum_spectra{}}.

In quantum, methods to compute excitons from conduction and valence band eigenstates are defined (see details in keywords documentation “quantum {region {excitons} }”)

quantum{
    region{
        ...
        excitons{
            density_averaged_masses = yes
            energy_cutoff = 2.5
            accuracy = 1e-5
        }
    }
}

In optics, the corrections to optical absorptions is defined. Setting coulomb_enhancement = no and num_exciton_levels = 0 will output absorption without exciton correction (so called single-particel model).

optics{
    quantum_spectra{
        ...
        excitons{
            coulomb_enhancement = yes
            num_exciton_levels = 1
        }
    }
}

The input files provided for this simulation have three modes, depending on the value of the variable $calculation, defined at the top of the input file.

– $calculation=1 – computes single-particle absorption (no exciton correction)

– $calculation=2 – the computed absorption includes Coulomb enhancement

– $calculation=3 – the computed absorption includes both Coulomb enhancement and exctiton peaks

Simulation 1: single-band model

For this simulation, 1D_InterbandExcitonAbsorption_InfiniteWell_GaAs_effective_mass_nnp.in input file is used.

The parameters used in the calculation are the following

Property	Symbol	unit	Value
quantum well width	\(L\)	nm	10.0
barrier height	\(E_b\)	eV	1000
Electron effective mass	\(m_e\)	\(m_0\)	0.065
Heavy hole effective mass	\(m_{hh}\)	\(m_0\)	0.51
refractive index	\(n_r\)		3.3
linewidth (FWHM) Lorentzian	\(\Gamma_{\rm L}\)	meV	3
linewidth (FWHM) Gaussian	\(\Gamma_{\rm G}\)	meV	5
temperature	\(T\)	K	300

To simplify the calculation, only heavy hole states are computed in the valence band. To include light hole and split off, set $compute_LH_and_SO variable to 1 in the input file.

The eigenstates from the calculation are shown in the Figure 2.5.12.93

Figure 2.5.12.93 Computed eigenstates in the GaAs infinite quantum well with effective mass Hamiltonian in conduction and valence bands. The colored dashed line are band edges, the solid lines are eigenstates.

In the figure below, the computed absorption in the quantum well is shown (Figure 2.5.12.94). The figure shows the absorption without exciton correction, absorption including Sommerfeld enhancement factor and total excitonic absorption (i.e. both exciton peak and Coulomb enhancement).

Figure 2.5.12.94 Absorption in infinite quantum well computed with effective mass Hamiltonians. The figure shows absorption with and without exciton correction.

Simulation 2: 8-band kp model

For this simulation, 1D_InterbandExcitonAbsorption_InfiniteWell_GaAs_8kp_nnp.in input file is used.

The parameters used in the calculation are the following

Property	Symbol	unit	Value
quantum well width	\(L\)	nm	10.0
barrier height	\(E_b\)	eV	1000
8-band kp parameters for GaAs	\(Eg, Ep, L, M, N\)	n/a	from default database
refractive index	\(n_r\)		3.3
linewidth (FWHM) Lorentzian	\(\Gamma_{\rm L}\)	meV	3
linewidth (FWHM) Gaussian	\(\Gamma_{\rm G}\)	meV	5
temperature	\(T\)	K	300

The eigenstates from the calculation are shown in the Figure 2.5.12.95

Figure 2.5.12.95 Eigenstates in the GaAs infinite quantum well computed with 8-band kp Hamiltonian. THe colored dashed line are band edges, the solid lines are eigenstates.

In the figure below, the computed absorption in the quantum well is shown (Figure 2.5.12.96). Similarly to the Simulation 1, the figure shows the absorption with and without exciton correction.

Figure 2.5.12.96 Absorption in infinite quantum well computed with 8-band kp Hamiltonian. The figure shows absorption with and without exciton correction.

In both simulations, exciton correction increase the absorption significantly above the absorption edge and also gives rise to a sharp peak at energy few meV below absorption edge.

Acknowledgment

This tutorial is based on the nextnano GmbH collaboration in the scope of the SiPho-G Project aiming at development of ultrahigh-speed optical components for next-generation photonic integrated circuits, and it is funded by the European Union’s Horizon 2020 research and innovation program under grant agreement No 101017194.

Last update: nn/nn/nnnn