Density matrix modeling of quantum cascade lasers without an artificially localized basis: A generalized scattering approach

We derive a density matrix (DM) theory for quantum cascade lasers (QCLs) that describes the influence of scattering on coherences through a generalized scattering superoperator. The theory enables quantitative modeling of QCLs, including localization and tunneling effects, using the well-defined energy eigenstates rather than the ad hoc localized basis states required by most previous DM models. Our microscopic approach to scattering also eliminates the need for phenomenological transition or dephasing rates. We discuss the physical interpretation and numerical implementation of the theory, presenting sets of both energy-resolved and thermally averaged equations, which can be used for detailed or compact device modeling. We illustrate the theory's applications by simulating a high performance resonant-phonon terahertz (THz) QCL design, which cannot be easily or accurately modeled using conventional DM methods. We show that the theory's inclusion of coherences is crucial for describing localization and tunneling effects consistent with experiment.

Figure 1

Categories of superoperator terms (denoted by arrows) and the coupling they induce between elements of the density matrix. For clarity, the transverse momentum $\vec{k}, {\vec{k}}^{\prime }$ is suppressed in the illustration. See text for discussion.

Figure 2

Wave functions of a three-level system with (a) energy eigenstates and (b) ad hoc localized states. (c) Contributions to electron density from populations and coherence of anticrossed energy states $A$ and $B$; positive ${\rho }_{AB}$ leads to charge buildup in the “upstream” well to the left (as pictured) and vice versa. The transverse momentum $\vec{k}$ dependence is suppressed for clarity.

Figure 3

Coherence-coherence scattering pathway from initial DM element (CD) to final DM element (AB) via scattering matrix elements $V_{AC}$ and $V_{DB}$ and intermediate coherences $CB$ and $AD$. The energy-conserving delta functions are associated with the indices of the (red) intermediate coherences in the Schrodinger picture and with the indices of the (blue) scattering processes in the interaction picture.

Figure 4

(Top) GaAs/${\mathrm{Al}}_0.15{\mathrm{Ga}}_0.85\mathrm{As}$ superlattice band structure as function of barrier thickness. Thick lines denote the two wave functions comprising each module; thin lines represent the shifted wave functions associated with neighboring modules. (Bottom) Current density (left axis) and degree of charge localization (right axis) vs barrier width. An uniform doping profile $N_d={10}^{16}{\mathrm{cm}}^{-3}$ is assumed and lattice temperature is 77 K.

Figure 5

Band structure for diagonal resonant phonon THz QCL in GaAs/${\mathrm{Al}}_{0.15}{\mathrm{Ga}}_{0.85}\mathrm{As}$ for module bias of 48 mV. The device layer thicknesses are 3.7/17.2/5.1/10.3/1.7/10.7/3.7/8.8 in nanometer. Solid lines indicate the eigenstates for a given module with labels indicated at right; dotted lines indicate the eigenstates for neighboring modules. States are labeled in ascending order of energy for convenience when describing different biases.

Figure 6

Current density vs module bias for experimental resonant phonon (dotted line) compared with semiclassical, energy-resolved, and thermal averaged DM models. The lattice temperature is assumed to be 100 K in all calculations, and the electron temperature is also 100 K in the thermal averaged model.

Figure 7

Electron density at 48 mV from semiclassical, energy-resolved, and thermal averaged calculations. The position axis corresponds to that in Fig. 5; barrier regions are shaded gray.

Figure 8

Probability densities of energy eigenstates (solid lines) and density matrix eigenstates (dashed lines) at (a) 26, (b) 36.7, and (c) 48 mV module bias. Since the density matrix eigenstates do not have definite energies, they are plotted on the $y$ axis with respect to their energy expectation values.

Figure 9

(a) Energy-resolved distributions within subbands at 48 mV (subband labels correspond to wave functions in Fig. 5). (b) Small signal gain as function of frequency and bias.