Quantum well infrared photodetector (QWIP) structures have been developed since 1990s [1]. There are many different types of QWIP structures. QWIPs can be categorized by their electrical properties: photovoltaic or photoconductive, or by their layer thicknesses: multi-quantum wells (MQW) or superlattice structures. They can also be categorized by having optical responsivity at a single or multiple wavelengths. Multi-color QWIPs can be composed of double barriers [2], stepped quantum wells [3], and stepped barriers. The structures with stepped barriers are also called as staircase-like QWIPs in the literature [4].

In this work, photomodulated reflectance (PR) and photoluminescence (PL) experiments were carried out on two different staircase-like QWIP structures at room temperature. PR is a powerful characterization method to determine optical transitions in both bulk and low-dimensional multilayer semiconductor structures. Its absorption-like character and high sensitivity makes it possible to observe optical transitions between ground and excited states, even at room temperature. PR spectroscopy utilizes the modulation of the built-in electric field at the semiconductor surface or at the interfaces through photo-injection of electron–hole pairs generated by a chopped incident laser beam. This technique produces sharp spectral features related to the critical points of the band structure. This provides a more explicit comparison of experimental results with theoretical models. However, PL only gives information about ground state transitions in QWs at room temperature. PR spectra were analyzed using the third derivative functional form (TDFF) in order to fit the optical transition energies, and the results were compared to the theoretical values calculated using transfer matrix method.

### Theory

Transfer matrix technique is a common method for solving Schrödinger equation for MQW structures which consist of layers having different band gaps and effective masses. By virtue of this technique, energy levels, wave functions under zero or constant electric field can be calculated in complex structures [5–7]. In this work, we had employed this technique to calculate the energy levels in each QW at 300 K.

In order to determine the band gap of GaAs at room temperature, Varshni equation [

8] was used:

${E}_{g}\left(T\right)={E}_{g}\left(0\right)-\frac{\alpha {T}^{2}}{T+\beta}\text{,}$

(1)

where

*E*_{g}(0) is the band gap of GaAs at

*T* =

*0* K;

*α* = 5.405 × 10

^{−4} eV/K and

*β* = 204 K are Varshni parameters at the

*Г* point. For Al

_{
x
}Ga

_{1−x}As ternary alloys. Temperature dependence of the band gap for

*x* <

*0*.

*4* can be estimated by:

${E}_{g}\left(x,T\right)=1.519+1.155x+0.37{x}^{2}-\frac{\alpha {T}^{2}}{T+\beta}\text{,}$

(2)

where

*α* and

*β* are Varshni parameters of Al

_{
x
}Ga

_{1−x}As. Adachi showed that compositional dependence of Varshni parameters becomes significant in Al

_{
x
}Ga

_{1−x}As ternary alloys for

*x* > 0.4 [

9]. However, since

*x* < 0.4 for Al

_{
x
}Ga

_{1−x}As in our structures, we used the same values as GaAs [

9,

10]. The conduction and the valance band offsets were chosen as 60% and 40%, respectively. In the calculations of energy levels, the effective mass for each layer was considered separately. The effective masses of electrons in AlAs and GaAs were taken as 0.15 and 0.067, respectively. Using these values, the effective mass of electrons in Al

_{
x
}Ga

_{1−x}As layers was calculated by applying Vegard’s law:

${m}_{{\text{Al}}_{x}\text{Ga}{1-x}_{}\text{As}}=\frac{{m}_{\text{AlAs}}\phantom{\rule{0.12em}{0ex}}{m}_{\text{GaAs}}}{x\phantom{\rule{0.12em}{0ex}}{m}_{\text{GaAs}}+\left(1-x\right){m}_{\text{AlAs}}}\text{.}$

(3)

Similarly, the effective masses of holes in the Al_{
x
}Ga_{1−x}As layers were also calculated using Equation 3, taking the density of states heavy hole effective masses as 0.81 and 0.55, and the averaged light hole effective masses were taken as 0.16 and 0.083 in AlAs and GaAs, respectively [9].

PR spectra were fitted using the linear combination of several Aspnes’ TDFFs [

11], expressed as:

$\frac{\Delta R}{R}=\mathrm{Re}{\displaystyle \sum _{j=1}^{n}\left[{A}_{j}{e}^{-i{\phi}_{j}}{\left(E-{E}_{\mathit{gj}}+i{\Gamma}_{j}\right)}^{-{m}_{j}}\right]+{f}_{j}\left(E\right)}\text{,}$

(4)

where *n* is the number of spectral features to be fitted; *E* is the photon energy; *A*_{
j
}, *φ*_{
j
}, *E*_{
gj
}, and *Г*_{
j
} are the amplitude, phase, band gap energy, and line broadening of the *j*_{
th
} feature, respectively. *m*_{
j
} represents the type of critical point depending on the dimensionality of the structure, and its value is 2.5 or 3 for 3-D (bulk) or 2-D cases, respectively. The background signal in the measurements was simulated and suppressed from Equation 4 by a linear *f*(*E*) function.