1. Introduction

Since the first demonstration of quantum-size effects in semiconductor quantum wells in 1974 [1] to the present day colloidal and epitaxial two-dimensional quantum systems have been attracting great interest of researchers, constantly opening new horizons for their use in devices of unique functionalities [2–7]. In recent years, special attention has been paid to intersubband transitions in semiconductor quantum wells, which underlie the operation of infrared detectors and quantum cascade lasers [7–10]. Quantum wells with asymmetric profiles, in particular step quantum wells, are especially appealing for such applications [11–15]. The development of quantum-well devices with superior performance heavily relies on the accurate description of their electronic structure and the ability to control their optical properties.

In this paper, we analyse the size-quantized subbands of a bilayer step quantum well with infinitely high potential barriers and obtain an analytical expression for the probability of intersubband transitions in the conduction band of the quantum well. The analysis employs the standard formalism of the solid-state physics and can be readily extended to investigate the energy structure and hole dynamics in the valence bands of bilayer quantum wells. Our results may prove useful in the development of new devices based on epitaxial quantum wells and colloidal bilayer nanoplates.

2. Theoretical framework

Consider the conduction band $\Gamma _6$ of a bilayer step quantum well in the two-band approximation (taking into account two zones degenerated by spin) [16]. We shall assume that the electron affinity of the first-layer material (A) is greater than that of the second-layer material (B), and that the work functions of both materials are the same. In the approximation of infinitely high potential barriers, the motion of the conduction electrons inside the quantum well is determined by the potential shown in Fig. 1. The growth direction of the structure coincides with the $z$ axis while the origin coincides with the outer surface of layer A.

 

Fig. 1. Confining potential for conduction electrons inside a bilayer quantum well; $E_{\Gamma _6,\mathrm {A}}^0$, $E_{\Gamma _6,\mathrm {B}}^0$, $L_{\mathrm {A}}$, and $L_{\mathrm {B}}$ are the lowest energies in the conduction bands of bulk materials and the layer thicknesses of materials A and B; $L=L_{\mathrm {A}}+L_{\mathrm {B}}$.

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According to the Bloch theorem, the electron wave function $\Psi _{\Gamma _6,s_z}(\mathbf {r})$ in the conduction band $\Gamma _6$ of the quantum well is the product of the envelope function $f_{\Gamma _6}(\mathbf {r})$ and the Bloch factor $u_{\Gamma _6,s_z}(\mathbf {r})$, where $s_z=\pm 1/2$ is the projection of the electron spin. Within the framework of the $\mathbf {k}\cdot \mathbf {p}$-perturbation theory, we characterize the motion of an electron in each zone by an effective mass calculated using the Bloch amplitudes. Since the Bloch amplitudes and the effective masses of electrons in materials A and B are different, the eigenvalue problem for calculating the subband energies $E_{\Gamma _6}$ and the envelope function $f_{\Gamma _6}(\mathbf {r})$ inside the quantum well $0<z<L$ can be written in the form

The fact that the electrons can move freely in the $xy$ plane makes possible the following factorization of the envelope function:

The general solution of the above equation inside the quantum well $(0<z<L)$ is given by

In the dipole approximation, the matrix element has the form [17]

After a similar calculation of $M_{fi}^{\mathrm {B}}$ we find the total intersubband matrix element to be given by

When the boundary condition $\mathbf {A}_\parallel ^{\mathrm {A}}=\mathbf {A}_\parallel ^{\mathrm {B}}\equiv \mathbf {A}_\parallel$ is satisfied, integration over $z$ leads to the factor $\delta _{n_f\!,n_i}$ for the term that is proportional to $\mathbf {A}_\parallel$. This term vanishes for intersubband transitions with $n_f\!\neq n_i$. Therefore, $M_{f\!i}$ takes the form

where

Finally, we calculate the rate of direct intersubband transitions per unit area of the quantum well. Using the golden rule of quantum mechanics, this rate can be represented in the form

3. Results and discussion

It is easy to show that the above expression for $M_{f\!i}$ is reduced to the well-known result for a single-layer quantum well. This result is recovered by setting $L_{\mathrm {B}}=0$,

We illustrate our results using the Al$_{0.35}$Ga$_{0.65}$As/GaAs heterojunction as an example. Figure 2(a) shows the energies of the first four subbands of a bilayer Al$_{0.35}$Ga$_{0.65}$As/GaAs quantum well plotted as functions of $q^2$. The subbands are seen to be generally nonparabolic, yet with parabolicity in certain areas — where $E_{\Gamma _6,n}(q^2)$ is approximately linear. Two dashed lines $E_{\Gamma _6}(q^2)$ on the figure correspond to $q_{\mathrm {A}}=0$ and $q_{\mathrm {B}}=0$. The slopes of these lines are determined by the effective masses $m_{\mathrm {c,A}}^\ast$ and $m_{\mathrm {c,B}}^\ast$. It is easy to see that $q_{\mathrm {A}}$ is imaginary below line $q_{\mathrm {A}}=0$ and is real above it, and similar for $q_{\mathrm {B}}$. It should be also noted that for all the energies of the subbands and any $q$

 

Fig. 2. (a) The first four subbands (colour curves labelled by $n=1,\ 2,\ 3,\ \textrm {and}\ 4$) of the conduction band of a bilayer quantum well made from GaAs (layer A) and Al$_{0.35}$Ga$_{0.65}$As (layer B); $L_{\mathrm {A}}=L_{\mathrm {B}}=10\ \textrm {nm}$; dashed are the lines where $q_{\mathrm {A}}$ and $q_{\mathrm {B}}$ vanish. (b) Effective masses of electrons in the four subbands $n=1,\ 2,\ 3,\ \textrm {and}\ 4$ as functions of wave number of electrons; dashed are the effective masses of bulk GaAs $(m_{\mathrm {c,A}}^\ast )$ and bulk Al$_{0.35}$Ga$_{0.65}$As $(m_{\mathrm {c,B}}^\ast )$. All the material parameters are taken from Ref. [20].

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We next analyse how the effective masses of electrons, defined as $m_{\mathrm {c,n}}^\ast (q)=\hbar ^2/E_{\Gamma _6}^{\prime \prime }(q)$ (here the double prime denotes the second derivative with respect to $q$), vary as functions of wave number. The effective masses for the first four subbands of the bilayer Al$_{0.35}$Ga$_{0.65}$As/GaAs quantum well are shown in Fig. 2(b). For small $q$ the following condition is satisfied for the first two subbands:

 

Fig. 3. Probability $\left |f_{\Gamma _6,n}(0;z)\right |^2$ of finding an electron with $q=0$ inside a bilayer Al$_{0.35}$Ga$_{0.65}$As/GaAs quantum well for the first four subbands $(n=1,\ 2,\ 3,\ \textrm {and}\ 4)$. All the parameters are the same as in Fig. 2.

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The opposite situation occurs at sufficiently large $q$ for which

The inequality

For all the subbands the function $m_{\mathrm {c,n}}^\ast (q^2)$ exhibits sharp peaks at points where the dispersion curves intersect with the straight lines $q_{\mathrm {A}}=0$ and $q_{\mathrm {B}}=0$. They correspond first to transitions to a state in which the electron is localized in the entire structure $(q_{\mathrm {B}}=0)$, and then to transitions to a state in which it is localized in layer B $(q_{\mathrm {A}}=0)$. For subbands whose energies at $q=0$ are greater than $\max (E_{\Gamma _6,\mathrm {A}}^0,E_{\Gamma _6,\mathrm {B}}^0)$, only the equation $q_{\mathrm {A}}=0$ can be satisfied and $q_{\mathrm {B}}$ is always real.

4. Conclusions

We have developed a theory of size quantization and intersubband optical transitions in a bilayer semiconductor quantum well with asymmetric profile. It was shown that the spatial confinement of electrons inside the quantum well leads to the appearance of size-quantized subbands, which are essentially nonparabolic due to the asymmetric profile of the well. The effective masses of electrons, defined through the second derivative of the energy with respect to the wave number, were found to depend not only on the wave number, but also on the number of the subband and be closely related to the conditions of the electron localization in the layers of the quantum well. We also derived analytical expressions for the matrix elements of optical transitions between the subbands and for the transition rates, which are needed for the design of new quantum-well-based optoelectronic devices. We believe that our results will benefit the development and implementation of such devices.