Two-band finite difference method for the bandstructure calculation with nonparabolicity effects in quantum cascade lasers

Ma, Xunpeng; Li, Kangwen; Zhang, Zuyin; Hu, Hao; Wang, Qing; Wei, Xin; Song, Guofeng

doi:10.1063/1.4817795

Cited by 12 publications

(5 citation statements)

References 17 publications

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“…According to the effective two-band model in a multiple QW structure, a two-component wave function is given by [18] is the transverse wave vector, r = + ˆxx yy is the transverse position vector, and A is the area of the material, respectively. Finally, f ( )…”

Section: Effective Two-band Model For Conduction Band Npbmentioning

confidence: 99%

“…The other numerical method was to solve a two-band Schrödinger equation based on the effective two-band model, which consisted of a 2×2 Hamiltonian acting on the two-dimensional envelope function [8,9]. This two-band model of the non-parabolic Schrödinger equation in combination with the FDM was successfully applied to the calculation of electronic subband energies in QCLs, which had the advantages of an easy-understanding form, acceptable accuracy, and faster computation time compared with the other methods [18]. However, the effective two-band FDM has not yet been applied to the theoretical treatment of intersubband optical dipole moments and the polar-opticalphonon (POP) scattering times with NPB effects, which is essential in the design of QCLs.…”

Section: Introductionmentioning

confidence: 99%

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Effect of conduction band non-parabolicity on the optical gain of quantum cascade lasers based on the effective two-band finite difference method

Cho

Kim

2017

Semicond. Sci. Technol.

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We theoretically investigate the effect of conduction band non-parabolicity (NPB) on the optical gain spectrum of quantum cascade lasers (QCLs) using the effective two-band finite difference method. Based on the effective two-band model to consider the NPB effect in the multiple quantum wells (QWs), the wave functions and confined energies of electron states are calculated in two different active-region structures, which correspond to three-QW single-phonon and four-QW double-phonon resonance designs. In addition, intersubband optical dipole moments and polar-optical-phonon scattering times are calculated and compared without and with the conduction band NPB effect. Finally, the calculation results of optical gain spectra are compared in the two QCL structures having the same peak gain wavelength of 8.55 μm. The gain peaks are greatly shifted to longer wavelengths and the overall gain magnitudes are slightly reduced when the NPB effect is considered. Compared with the three-QW active-region design, the redshift of the peak gain is more prominent in the four-QW active-region design, which makes use of higher electronic states for the lasing transition.

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Section: Effective Two-band Model For Conduction Band Npbmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Effect of conduction band non-parabolicity on the optical gain of quantum cascade lasers based on the effective two-band finite difference method

Cho

Kim

2017

Semicond. Sci. Technol.

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“…More sophisticated numerical methods have been developed, which allow solutions to be located directly, for example by converting the equation into a linear (but much larger) eigenvalue problem [17] 2 or by solving for the conduction-and valence-band states simultaneously [18]. However, as with the shooting method, this is a slow process that relies on an appropriate choice of energy cell size.…”

Section: Extension To Include Band Non-parabolicitymentioning

confidence: 99%

Numerical solutions

2016

Quantum Wells, Wires and Dots

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Chapter 2 focused principally upon finding analytical solutions for the behaviour of carriers in heterostructures. In some cases, such as the infinite quantum well discussed in Section 2.1, it was possible to find closed-form solutions for the energies and wave functions. In other words, the solutions were expressed in terms of well-understood, simple mathematical functions (sin, cos, etc.), allowing the energies and wave functions to be calculated directly.In general, however, closed-form solutions cannot be found for anything beyond the simplest systems. Indeed, even the single finite well system in Section 2.7 has no closedform solution, and instead the solutions were found iteratively using numerical methods. This chapter begins by describing a few of these root-finding algorithms that allow solutions of a complicated mathematical function to be found.Modern semiconductor devices can contain exceedingly complex heterostructures. One example is the quantum cascade laser [1], which contains hundreds of layers of semiconductor materials with a range of different thicknesses, with the band structure being distorted by large internal and external electric fields. In such cases, it may be impossible to even determine an analytical expression to be solved for the states in the system. Instead, discrete mathematical techniques can be used to create a sampled approximation of the system which can be solved computationally. Although these methods do not provide the same direct insight into the behaviour of systems as the analytical solutions in Chapter 2, they are generally applicable and very flexible, and the same techniques can be used to describe a vast range of structures. This chapter focuses on a range of such methods and their application to more complex heterostructures.

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“…During past three decades, different models have been presented to describe the carrier and photon interactions in semiconductor lasers [1][2][3][4][5][6]. In different models, there is a trade off between desirable accuracy and model complexity [7,8].…”

Section: Introductionmentioning

confidence: 99%