Modeling of optoelectronic devices with one-band effective mass equation: nonequilibrium Green’s function approach

Abstract
One-band one-dimensional effective mass model useful in the simulations of layered n-type devices is proposed. The model preserves nonparabolicity both in transport and in-plane directions and enables calculations of intersubband absorption. It is integrated into nonequilibrium Green’s function method which is used to simulate quantum cascade laser.


Introduction
When modeling unipolar devices, a one-band effective mass equation (EME) is the first choice. The main benefit of this approach is the saving of computer resources which can be then used for more complex description that takes into account quantum coherence and scattering. For optoelectronic devices utilizing intersubband optical transition in lm range, the energy levels are raised well above the band bottom, where they are strongly influenced by the remote bands. One-band description can account for this effect, e.g., by the use of energy-dependent effective mass. For layered n-type devices, an efficient approximation is one dimensional (1D) energy-dependent EME 2 þ 2 h 2 k 2 m Ã E g s 8 < : 9 = ; ; for a subband originated at energy level E z ¼ E i can be obtained inserting mðE; z aw Þ ¼ m Ã ½1 þ ðE À E c Þ=E g into Eq. (3b). The plots E(k) predicted by Eq. (4) for an exemplary quantum well with two confined states at E 1 ffi 0:15 eV and E 2 ffi 0:413 eV are shown in Fig. 1. The deviation from the dispersion relation predicted by the 8-band kp model for the lower state is even smaller than that for Eq. (1). Moreover, there is no evidence which approximation is better because the 8-band kp model is also an approximation which introduces inaccuracy at high in-plane momentum. This is evident when compared to more exact results predicted by a 14-band kp model (Ekenberg 1989) (see Fig. 1). The subscript 'aw' in Eq.
(3) abbreviates 'active wells', so z aw points at the wells where a major optical transition takes place. Fixing z at z aw in Eq. (3b) makes longitudinal energy loose its spatial dependence. Then E z becomes a real number and Eq. (3a) can be rewritten in the form which has the energy-independent TB counterpart Equation (6) provides the orthonormal solutions (f, g) which can be used in the calculations of the momentum matrix elements (Leavitt 1991;Sirtori et al. 1994).

Nonequilibrium Green's function (NEGF) implementation
Equation (3) defines a k-dependent Hamiltonian which can be used in Nonequilibrium Green's Function (NEGF) method. In the real space implementation, the conduction band Green's functions (GFs) G R , G \ are the four-parameter functions of positions z, z 0 , energy E, and in-plane momentum modulus k. For discretized Hamiltonians, useful in numerical simulations and offering a compact form of integral equations, the retarded GF, G R ðE; kÞ, is the solution of the Dyson equation (Kubis et al. 2009) where the self-energy term R R comprises scatterings and couplings to the leads. The valence component of the retarded GF is given by Kolek (2015) where H ct is the discretized differential operator Lesser Green's functions can be obtained from the Keldysh equation where R \ is the conduction band lesser self-energy. Equations (7-9) together with the Poisson equation should be iterated. In this iteration, the GFs valence components G R t , G \ t should be included as they contribute to the density of states (DOS) and the density of electrons (DOE) and thus influence the GFs through the Poisson equation. Once the self-consistency is achieved, the optical absorption can be evaluated making use of the approach developed by Wacker (2002) applied to the TB Hamiltonian (Kolek 2015).

Quantum cascade laser
The model described in Sect. 2 is especially useful for a quantum cascade laser (QCL)-a device consisting of tens of layers with carefully tuned width what calls for a very dense discretization mesh in the simulations. This generates huge matrices and makes multiband modeling hardly possible. On the contrary, the one-band approach keeps the matrices in reasonable size and enables the use of the NEGF method, which treats quantum tunneling and scattering on equal footing (Kubis et al. 2009) what is necessary for this type of devices. The model of Sect. 2 and its implementation of Sect. 3 were used to calculate electronic transport and optical gain in QCL emitting at ffi 5 lm (hm ffi 0:25 eV), designed by Evans et al. (2007). The equations of the NEGF formalism were solved for one QCL module with the contact self-energies which mimic periodic boundary conditions (Haldas et al. 2011;Kolek et al. 2012). Other self-energies included into formalism represent electron-phonon (LO, LA), interface roughness, alloy disorder and ionized impurity scatterings. Carrier-carrier interaction was included as the mean-field through the Poisson equation. The discretization mesh spacing was 0.6 nm. In Fig. 2, which shows exemplary results, the focus is paid on the k-resolved quantities which demonstrate in-plane nonparabolicity inherently contained in the model. Preserving this feature is crucial for realistic modeling of intersubband gain. As discussed by Faist et al. (1996), in QCLs the optical gain can emerge due to the local (in k-space) population inversion which is not destroyed by the absorption at higher k-values due to the in-plane nonparabolicity. Indeed, our simulations show that substantial gain emerges (see Fig. 2 inset) (due to 4 ! 3 transitions) at the energy hm ffi 0:26 eV that agrees very well with the experimental lasing wavelength. This is in sharp contrast with purely one-band model which predicts both wrong lasing energy and almost no gain: for this case, the gain peak of merely 0:8 cm À1 occurs at hm ¼ 0:32 eV! Results of one-band simulations are also shown in Fig. 2 for the comparison.
When gain emerges due to local population inversion (which is our case) the most important is an accurate modeling of the in-plane dispersion at low and medium k-values, where intersubband transitions occur at nearly the same values of the photon energy and thus contribute to the gain spectrum. The photons emitted at remote k's have the energies well apart from the energy emitted in the transitions at k ffi 0 and so hardly influence the value of the gain peak. Then, going beyond the accuracy of 8-band description, which one may concern looking at the deviations in Fig. 1, seems to be both hard and impractical, at least in the case of QCLs. The model defined by Eq. (3) preserves the accuracy of 8-band model at low k-values which, as shown, is sufficient enough to provide reliable results.
The gain in Fig. 2 inset was calculated as the negative absorption coefficient averaged over one laser period. Both the conduction and the valence components of GFs were involved in these calculations. The amount of the valence component can be estimated, e.g., from the contribution to the total electron density. The plots in Fig. 3  In summary, a one-band effective mass model was proposed which allows for k-resolved calculations which preserve correct nonparabolicity both in transport and in-plane directions and allows for rigorous calculations of intersubband absorption which takes these (nonparabolicities) into-account.