We propose a model of transport in one period based on a rate equation approach. We assume that electrons are in the upper detector state |up

through absorption of a photon. Current as a function of lifetimes involved in this structure can be written:

The parameters α and

*N*_{down }are, respectively, the absorption factor and sheet density of |down

and are constant. The subscribe

*c *stands for the whole cascade. The quantum efficiency QE is the ratio of the lifetime τ

_{up-down }divided by τ

_{up-down }+ τ

_{up-c }and corresponds to the fraction of electrons on the level |up

that contributes to the photocurrent. In our model we suppose that any incident photon generates an absorption between the levels |down

and |up

.

We present in Table the calculated scattering rates of the different processes at

*B *= 0 T. For interface roughness, we used a Gaussian autocorrelation of the roughness, with an average height of Δ= 2.8 Å and a correlation length of Δ = 60 Å. LO phonon emission scattering rate has been calculated as in ref. [

10]. In our structure impurities scattering is the most efficient process [

11]. Usually in GaAs quantum cascade structures this mechanism is neglected because the doped layers are not in the active region. In order to take into account the main scattering process we calculate ionized impurities scattering as a function of magnetic field. The details of the calculation are presented elsewhere [

12].

| **Table 1**Calculated scattering rates in s^{-1}. |

Figure represents a comparison between experimental data and electron-ionized impurities scattering time as a function of magnetic field. Figure shows the two lifetimes involved in Equation 1 as a function of *B *calculated with electron-ionized impurities scattering. Figure shows the calculation of the related quantum efficiency.

The oscillating behavior at high magnetic field (

*B *> 9T) is a result of the electronic transfer from |up

to |down

. This transfer leads to minima in the current which fit well with

and QE. The long period oscillating behavior of

as a function of

*B *enhances the peak at

*B *= 14 T in QE in agreement with experimental data. QE, which describes the performance of the detector, is oscillating between 74 and 85% under

*B*. By extrapolating, at

*B *= 0T, QE is equal to 75%, a value that should be increased to improve the detector performance. An optimized structure should take these results into account by shifting the ionized impurities from the active region, where they are enhancing

, to a position where they would only enhance

. The series of peak at

*B *< 9T corresponds to a characteristic energy of 37 meV. This energy is attributed to transitions in the cascade involving an elastic scattering mechanism.