  We present a numerically efficient density-matrix model applicable to
midinfrared quantum cascade lasers. The model is based on a Markovian master
equation for the density matrix that includes in-plane dynamics, preserves
positivity of the density matrix and does not rely on phenomenologically
introduced dephasing times. Nonparabolicity in the bandstructure is accounted
for with a three-band k.p model, which includes the conduction, light-hole, and
spin-orbit split-off bands. We compare the model to experimental results for
QCLs based on lattice-matched as well as strain-balanced InGaAs/InAlAs
heterostructures grown on InP. We find that our density-matrix model is in
quantitative agreement with experiment up to threshold and is capable of
reproducing results obtained using the more computationally expensive
nonequilibrium Green's function formalism. We compare our density-matrix model
to a semiclassical model where off-diagonal elements of the density matrix are
ignored. We find that the semiclassical model overestimates the threshold
current density by 29% for a 8.5-um QCL based on a lattice-matched
heterostructure and 40% for a 4.6-um QCL based on a strain-balanced
heterostructure, demonstrating the need to include off diagonal density matrix
elements for accurate description of midinfrared QCLs.
