Optical rectification and second harmonic generation in CdSe/MgSe asymmetric double quantum wells

Abstract

In this study, we report theoretically the effect of well, barrier widths and polarization on optical properties of intersubband transitions (ISBT) in CdSe/MgSe asymmetric quantum wells (ADQWs). Eigenenergies and their corresponding wave functions have been calculated by solving numerically the Schrödinger equation. The second harmonic generation and the optical rectification including intersubband transition energies have been discussed. Obtained results revealed that intersubband transition depends strongly on the quantum wells and the barrier widths as well as the stark effect. With appropriate intensity, optical rectification can reach great magnitude. We hope that the numerical results of our research are valuable theoretically and experimentally to our scientific community in nonlinear optics.

1. Introduction

In recent years, considerable attention has been paid to the nonlinear optical properties of low-dimensional semiconductor structures from theoretical or applied point of view [[1], [2], [3], [4]]. In fact, a stronger nonlinear optical effect can be obtained from such structures namely quantum wires [[5], [6], [7], [8]], quantum wells QWs [[9], [10], [11]] and quantum dots (QDs) [12,13]. The progress in nanoscience permits to investigate QWs and double quantum wells (DQWs) which are very practical in electronic devices such as light-emitting diodes, electro-optical modulators [14,15], far-infrared photo-detectors [16,17] and semiconductor optical amplifiers [[18], [19], [20], [21], [22], [23], [24]]. Up to now, a great focus has been devoted to second-harmonic generation (SHG) [24,25], third-harmonic generation (THG) [26,27] and optical rectification (OR) [14,15,17,28]. For example, Zhang et al. [24] have reported the effect of electric field (EF) and the geometry factors on the nonlinear optical rectification (NOR) for asymmetrical and symmetrical Gaussian potential QWs. Karimi et al. [29] studied the SHG in GaAs/Ga_1-xAl_xAs ADQW and they obtained resonant peaks which are blue-shifted when increasing barrier height. The authors in Ref. [30] have investigated the OR with applied polarization in Al_xGa_1-xN/GaN and their numerical results show that the magnitude of the OR coefficient is stable and the peak position is red-shifted with the increasing of the intensity. Moreover, the NOR was not only investigated in asymmetric double quantum well but also in asymmetric double quantum dots. Indeed, the authors in Refs. [31,32] have investigated theoretically the effect of QDs size, temperature and pressure in the NOR, they found that the eigenvalues, eigenfunctions and NOR coefficient are tuned by the change of these parameters.

Recently, the group II–VI materials QW structures have been investigated for several devices [[33], [34], [35], [36], [37], [38]] such as quantum cascade structures [33,34], short wavelength QW infrared photodetector [35], yellow laser [36], and yellow emitters [37]. Among the others, MgSe/CdSe/ZnCdMgSe QW structures have been recommended as an alternative to existing systems for many devices functioning in the optical communication wavelength of 1.55 μm [38]. Compared to the GaAs/AlGaAs structure, the QW formed by MgSe and CdSe materials have a direct band gap with strong confinement (as much as 1500 meV) and direct band gap which can be used for possible SHG devices based on the ISBTs in the conduction band mainly in the mid-infrared spectral region. The small lattice mismatch between CdSe/MgSe semiconductors ($\left|\left(a_{CdSe}-a_{MgSe}\right)/a_{MgSe}\right|\approx 2\%$) allows these structures to be grown properly epitaxially. CdSe/MgSe QW are very suitable for short wavelength optoelectronic device designs due to their low defect densities, high quantum efficiencies and wide band gaps ($E_g\left(CdSe\right)=1.74eV$ and $E_g\left(MgSe\right)=3.6eV$) [39].

In this paper, the OR and SHG coefficients in the CdSe/MgSe ADQW is investigated under the framework of compact density matrix (CDM) formalism. The eigenenrgies and their corresponding wave functions have been determined by solving numerically the Schrödinger equation. We have examined the variation of electronic energies and OR coefficient versus the width of both right and left wells (W₁ and W₂ respectively). The influence of polarization on the OR susceptibility has been studied for different wells’ widths. After the introduction, the theoretical framework is given in section 2. In section 3, the numerical results and discussions are performed. Finally section 4 presents the main conclusions of the investigation.

2. Theory and calculation

In the ADQW, the electronic state is described by the solution of the Schrödinger equation under the effective mass approximation:

$$\left[\frac{-{\hslash }^2}{2}\frac{d}{dz}\frac{1}{m^{*}\left(z\right)}\frac{d}{dz}+V_B\left(z\right)-E_v+qFz\right]{\phi }_v\left(z\right)=0$$ (1)

Where z represents the growth direction of this QW, $v$ is the subband index, m*(z) is the layer-dependent electron effective mass, ℏ is Planck's constant, V_B(z) is a step-wise function representing the conduction band offset at the interface, $E_v$ and φ_υ are respectively the eigenvalues and its wave electronic function situated in the conduction band. The wave function of the electron confined in asymmetric double quantum wells may be given as:

$${\phi }_{\mathrm{\upsilon }}\left(z\right)=\sum _{n=1}^{\infty }{c}_{n}f\left(z\right)$$ (2)

Where

$$f\left(z\right)=\sqrt{\frac{2}{L}}\sin \left(\frac{n\pi z}{L}\right)$$ (3)

With $L$ corresponding to the effective length of the QW (the well lies between $z=0$ and $z=L$).

Now, we will present an approach for SGH and OR coefficient in ADQWs. We consider an incident electromagnetic field $E\left(t\right)=\tilde{E}\mathit{exp}\left(-i\omega t\right)+\tilde{E}\mathit{exp}\left(i\omega t\right)$ is applied to the system with a polarization vector normal to the QWs. Let's give ρ as the one-electron DM for this regime. Then, the evolution of the one-electron DM obeys the following [40]:

$$\frac{\partial {\rho }_{ij}^{\left(n+1\right)}}{\partial t}=\frac{1}{i\hslash }{\left[H_0-qzE\left(t\right),\rho \left(t\right)\right]}_{ij}-{\mathrm{\Gamma }}_{ij}{\left(\rho -{\rho }^{\left(0\right)}\right)}_{ij}$$ (4)

Where $H_0$ is the Hamiltonian for this system without the incident field $E\left(t\right)$, $q$ is the electronic charge, ${\rho }^{\left(0\right)}$ is the unperturbed DM. Eq. (4) is solved by using the usual iterative method:

$$\rho \left(t\right)=\sum _n{\rho }^{\left(n\right)}\left(t\right)$$ (5)

With:

$$\frac{\partial {\rho }_{ij}^{\left(n+1\right)}}{\partial t}=\frac{1}{i\hslash }\left\{{\left[H_0,{\rho }^{\left(n+1\right)}\left(t\right)\right]}_{ij}-i\hslash {\mathrm{\Gamma }}_{ij}\right\}-\frac{1}{i\hslash }{\left[qz,{\rho }^{\left(n\right)}\left(t\right)\right]}_{ij}E\left(t\right)$$ (6)

The QW electronic polarization can be extended as Eq. (5). We will restrict ourselves to considering the first two orders, i.e:

$$P\left(t\right)={\epsilon }_0\left({\chi }_{\omega }^{\left(1\right)}\tilde{E}\mathit{exp}\left(i\omega t\right)+{\chi }_{2\omega }^{\left(2\right)}{\tilde{E}}^2\mathit{exp}\left(i\omega t\right)\right)+c.c+{{\epsilon }_0\chi }_0^{\left(2\right)}{\tilde{E}}^2\mathit{exp}\left(i\omega t\right)$$ (7)

Where ${\chi }_{\omega }^{\left(1\right)}$ , ${\chi }_{2\omega }^{\left(2\right)}$ and ${\chi }_0^{\left(2\right)}$ denote the linear, SHG, and OR susceptibility, respectively. Based on the density-matrix method, the analytical expression of the optical rectification coefficient [28,41] in two energy states is given as follows for this model:

$${\chi }_0^{\left(2\right)}=\frac{8q^3{\sigma }_v}{{\epsilon }_0{\hslash }^2}{M}_{21}^2{\delta }_{21}\frac{{\omega }_{21}^2}{\left[{\left({\omega }_{21}-\omega \right)}^2+{\Gamma }_0^2\right]\left[{\left({\omega }_{21}+\omega \right)}^2+{\Gamma }_0^2\right]}$$ (8)

Where $M_{12}=⟨{\Psi }_1\left|z\right|{\Psi }_2⟩$ is the dipole matrix element per electronic charge, ${\epsilon }_0$ is the vacuum permittivity, ${\delta }_{21}=\left|M_{22}-M_{11}\right|$, $E_{12}=E_2-E_1$ represents the energy difference between the ith and $j^{th}$ subbands, ω is the frequency of the incident radiation, ${\sigma }_v$ is the electron density in the system called carrier density and Γ₀ (=1/τ) denotes the phenomenological relaxation rate from state 1 to 2.

The ${\chi }_0^{\left(2\right)}$ takes a largest value at exact resonance where $\hslash ⍵=E_2-E_1=ћ{\omega }_{12}$ then it amounts to:

$${\chi }_{0,\mathit{max}}^{\left(2\right)}=\frac{q^3{\sigma }_v}{{\epsilon }_0{\hslash }^2{\Gamma }_0^2}{M}_{21}^2{\delta }_{21}$$ (9)

From Eqs. (8), (9) we can remark that.

• (i)The OR coefficient has a single resonant peak which is near the E₂₁ energy, where ${\omega }_{12}=⍵\text{.}$.
• (ii)The resonant peak value of the OR, ${\chi }_{0,\mathit{max}}^{\left(2\right)}$, is proportional to the geometrical factor $M_{21}^2{\delta }_{21}$. It's well known that the electron quantum confinement is strongly affect the geometric factor $M_{21}^2{\delta }_{21}$.

The analytical expression of the SHG for the ADQW is taken from Ref. [42]:

$${\chi }_{2\omega }^{\left(2\right)}=\frac{q^3{\sigma }_v{M}_{12}{M}_{23}{M}_{31}}{{\epsilon }_0}\frac{1}{\left[E_{31}-2\hslash \omega +i\hslash {\Gamma }_0\right]\left[E_{21}-\hslash \omega +i\hslash {\Gamma }_0\right]}$$ (10)

The SHG at the resonant peak is taken from Refs. [42,48]:

$${\chi }_{2\omega ,\mathit{Max}}^{\left(2\right)}=\frac{q^3{\sigma }_v{M}_{12}{M}_{23}{M}_{31}}{{\epsilon }_0{\left(\hslash {\Gamma }_0\right)}^2}$$ (11)

3. Results and discussion

Our modeled structure consists of a left CdSe QW (3.2 nm) (W₂) and a right one (3.0 nm) (W₁), those wells are coupled by a very thin MgSe barrier with W_B = 1.0 nm width, and the whole is surrounded by two MgSe barriers (Fig. 1).

Fig. 1.

mSchematic diagram of the confined potential profile, the three lowest eigenenergies and their related wavefunctions in an asymmetric coupled MgSe/CdSe QWs with W₁ = 2.0 nm, W_B = 1.2 nm and W₂ = 2.7 nm respectively.

The different parameters used in our simulation are m*_CdSe = 0.13m₀ [43] and m*_MgSe = 0.23m₀ [44]. The relative dielectric constant ε_r is 6.2 [45] for CdSe and 3.8 for MgSe [46]. The conduction band offset is ${\Delta E}_c=1.5eV$ [46] at CdSe/MgSe interface. The electron density ${\sigma }_v$ in the QW is 5 × 10²⁴cm⁻³ [47] and τ = 0.14 ps [47].

The confined potential profile and the electronic states E₁, E₂ and E₃ with their related wave functions distribution are shown in Fig. 1 for the modeled structure with W₁ = 3.0 nm, W₂ = 3.2 nm and W_B = 2.7 nm.

The evolution of the electronic states and the transition energy E₂₁ are shown in Fig. 2(a), with W_B for different values of W₁ with W₂ = 3.2 nm. It is straight forward to observe that for a fixed value of W₁, E₁ increases with increasing W_B. For W_B ≥ 0.1 nm, E₁ and E₂ converge to the same limit corresponding to the known case of two single QWs without any interaction. In Fig. 2(b), the intersubband transition energy E₂₁ is depicted with W_B for different values of W₁. One can notice that E₂₁ decreases monotonically as W_B augments. For larger values of W_B, E₂₁ remains constant and tends to zero corresponding to the case of two separated single QWs. For the fixed value of W_B, E₂₁ diminishes with increasing W₁ due to the decrease of both E₁ and E₂. Our findings are similar to those reported in Refs. [49,50].

Fig. 2.

The electronic states E₁ and E₂ with the barrier width W_B for several values of the right well width W₁ with a fixed left well width W2 = 3.2 nm (a), and the transition energy E₂₁(b).

In Fig. 3 (a), the OR susceptibility ${\chi }_0^{\left(2\right)}$ is plotted with the incident photon energy ℏ⍵ for several values of W₁ while W₂ = 3.2 nm and V = 0 V. It is clear that the value and the position of ${\chi }_0^{\left(2\right)}$ peak depends strongly on W₁. The peak of ${\chi }_0^{\left(2\right)}$ rises to a maximum value and then decreases with a redshift as W₁ augments. Note that the peak of ${\chi }_0^{\left(2\right)}$ in CdSe/MgSe ADQW reaches 10⁻⁶ m/V, this important OR feature is principally related to the strong coupling between the double QWs. Fig. 3 (b) illustrates ${\chi }_0^{\left(2\right)}$ with the pump photon energy ℏ⍵ for several values of W_B with W₁ = 2.8 nm, W₂ = 3.2 nm and V = 0 V ${\chi }_0^{\left(2\right)}$ shows the same behavior as before. According to Fig. 3 (a) and (b), we can conclude that the OR susceptibility attains a maximum value of 6.45 × 10⁻⁶ m.V⁻¹ when W_B = 0.5 nm and W₁ = 3.0 nm when W₂ = 3.2 nm and V = 0 V. From the figure we also obviously find that the NOR does not monotonously change with W₁ and W_R. This figure shows that if W₂ and W_B are fixed, W₁ plays a significant role in getting a large amplitude of ${\chi }_0^{\left(2\right)}$. The same conclusion has been established by the authors in refers [10,51].

Fig. 3.

The OR susceptibility ${\chi }_0^{\left(2\right)}$ with the pump photon energy $\hslash \omega$ for five several W₁ = 2.8, 2.9, 3.0, 3.1, 3.2 nm under the condition W₂ = 3.2 nm and V = 0 V with W_B = 0.4 nm (a). The OR susceptibility ${\chi }_0^{\left(2\right)}$ as a function of barrier width W_B with W₁ = 2.8, W₂ = 3.2 nm and with V = 0 V (b).

In Fig. 4, ${\chi }_{0,\mathit{max}}^{\left(2\right)}$ is presented with W_B under the condition W₁ = 2.8 nm and V = 0 V. With the absence of an applied polarization, increasing W_B leads to an increase of ${\chi }_{0,\mathit{max}}^{\left(2\right)}$ magnitude for W_B = 0.5 Å. This behavior is directly attributed to the coupling between the two quantum wells which becomes weaker as W_B gets larger.

Fig. 4.

OR susceptibility ${\chi }_{0\mathit{max}}^{\left(2\right)}$ with the barrier width for W₁ = 2.8 nm, W₂ = 3.2 nm and under condition V = 0 V.

In Fig. 5, the OR susceptibility ${\chi }_0^{\left(2\right)}$ is depicted versus incident photon energy ℏω for several polarization values V with W₁ = 2.8 nm and W₂ = 3.2 nm. Increasing V leads to the diminishing of the intensity of peaks of ${\chi }_0^{\left(2\right)}$ with a blueshift. Physically, when F increases, the energy intervals between energy states become bigger and this gives a blue shift of peak. Also, raising V can reduce the interaction between the two coupled wells making the geometric factor decrease which has a first consequence of reducing the $M_{21}^2{\delta }_{21}$ and then reducing the peak values of the OR coefficient.

Fig. 5.

NOR susceptibility ${\chi }_0^{\left(2\right)}$ versus the pump photon energy $\hslash \omega$ for four several values of polarization V = 0, 0.2, 0.5, 0.75 V with W₁ = 2.8 nm and W₂ = 3.2 nm and W_B = 0.5 nm.

Fig. 6 reflects the geometric factor $M_{21}^2{\delta }_{21}$ with the applied polarization with W₁ = 2.8 nm, W₂ = 3.2 nm and W_B = 0.5 nm. From the figure, the $M_{21}^2{\delta }_{21}$ is not a monotonic function of F. With the increase of V, the $M_{21}^2{\delta }_{21}$ increases, reaches rapidly a maximum value and then diminishes. This means that as a result of applied F in ADQW, OR susceptibility is principally affected through the geometric factor. For example, if V = 0, the geometric factor $M_{21}^2{\delta }_{21}$ is equal to 7.5 × 10⁻²⁶C.m.

Fig. 6.

Geometric factor $M_{21}^2{\delta }_{21}$ with the polarization for W₁ = 2.8 nm, W₂ = 3.2 nm and W_B = 0.5 nm.

The OR coefficient ${\chi }_0^{\left(2\right)}$ with the pump photon energy ℏω for several values of V with W₁ = 3.12 nm and W₂ = 3.2 nm is presented in Fig. 7 (a). For these selected values, our structure is considered a quasi-symmetric DQW. It seems that applying V affects strongly ${\chi }_0^{\left(2\right)}$. In fact, when V augments, the peak value of ${\chi }_0^{\left(2\right)}$ diminishes considerably and a blue shift is observed. ${\chi }_0^{\left(2\right)}$ ≈5.5 × 10⁻⁶ m.V⁻¹ is obtained for V = 0.25. All these features agree with the results presented in Fig. 5 where the OR susceptibility's maximums are blue-shifted when V increases. In Fig. 7(b) the OR susceptibility ${\chi }_0^{\left(2\right)}$ versus the pump photon energy ℏω for several values of V with W₁ = 3.2 nm and W₁ = 3.2 nm is displayed. In this case, our heterostructure is actually a symmetric DQW. We notice that a weak V can induce a larger ${\chi }_0^{\left(2\right)}$ magnitude and by increasing V, peaks experience a blue shift. In Fig. 7(c), the OR coefficient ${\chi }_0^{\left(2\right)}$ with the incident photon energy ℏω for various strengths of V with W₁ = 3.0 nm and W₁ = 3.2 nm is reported. From the plot, we can conclude that a maximum value of the resonant peak values of ${\chi }_0^{\left(2\right)}$ can be found by an appropriate choice of polarization. Comparing these Figures, we also obviously find that the optimal OR coefficient can obtained with quasi-symmetric and symmetric QW with weak polarization is larger than that obtained with asymmetric QW without polarization. For V = 0 V, ${\chi }_{0,opt}^{\left(2\right)}$ = 5.4 × 10⁻⁶m/V⁻¹. Our study is similar to those reported in Ref. [10], when the authors studied the effect of EF intensity on the NOR coefficient in double triangular quantum wells (DTQWs), they proved that with the rise of F the ${\chi }_0^{\left(2\right)}$ peak value quickly rises to a maximum and then diminishes gradually. Moreover, with an appropriate EF, a larger magnitude can be obtained with quasi-symmetric and symmetric shapes as compared with that obtained in the same quantum systems without EF (${\chi }_0^{\left(2\right)}$ = 0 in symmetric DTQW with F = 0 kV/cm). This f can be explained as follows: the applied EF breaks down the symmetry of the QW and locates the lowest energy levels in different QWs, which leads to a rise in the mean electron displacement.

Fig. 7.

Evolution of OR coefficient with the pump $\hslash \omega$ for several polarization values V = 0, 0.2, 0.5, 0.75 V under the condition W₂ = 3.2 nm, W₁ = 3.12 nm and W_B = 0.5 nm (a), for W₂ = 3.2 nm, W₁ = 3. 2 nm and W_B = 0.5 nm (b), with W₂ = 3.2 nm, W₁ = 3. 0 nm and W_B = 0.5 nm (c).

Fig. 8 demonstrates the variation of the SHG coefficients ${\chi }_{2\omega }^{\left(2\right)}$ with the incident photon energy ℏω for several strengths of polarization V. For each selected value of V, the curve presents two peaks situated exactly at λ and 2λ and this can be explained by a transition from a two-photon to two single-photon resonances will appear with appropriate. The increase of V can enlarge the magnitude of ${\chi }_{2\omega }^{\left(2\right)}$. With V = 0.25 V, ${\chi }_{2\omega }^{\left(2\right)}$ = 2.67 × 10⁻⁶m/V⁻¹ while with V = 0.75 V, ${\chi }_{2\omega }^{\left(2\right)}$ = 1.7 × 10⁻⁶m/V⁻¹. The peak of ${\chi }_{2\omega }^{\left(2\right)}$ decreases and their peak intensity move towards to higher energies is induced as the strength of polarization increases. This result has been reported in Refs. [41,47] when the author investigated the nonlinear optical properties in semi-parabolic with the applied EF, they proved that the confinement frequency ω₀ has great influences on SHG with the effect of applied EF intensity.

Fig. 8.

The Second harmonic generation $\left|{\chi }_{2⍵}^{\left(2\right)}\right|$ with incident photon energies $\hslash \omega$ for four different values of applied polarization V = 0, 0.2, 0.5, 0.75 V for W₁ = 2.8 nm and W₂ = 3.2 nm and W_B = 0.5 nm.

4. Conclusions

This paper investigated the NOR and SHG for a typical asymmetric CdSe/MgSe double quantum well using the CDM formalism and wave function approximation. Numerical results on CdSe/MgSe materials show that NOR and SHG in double quantum wells depend on the confined potential and the external EF. According to our results, the NOR does not monotonously change with the right-well width. We have been able to conclude that only when an appropriate W₁ is chosen, the largest ${\chi }_0^{\left(2\right)}$ can appear. In addition, increasing the polarization V can induce a blue shift of the ${\chi }_0^{\left(2\right)}$ peak. We think that our theoretical results may make a contribution to experimental studies, and may open new opportunities for optical exploitation of the quantum-size effect in optoelectronic devices such as ultrafast optical switches, and infrared waveguide systems.

CRediT authorship contribution statement

Nouf Ahmed Althumairi: Writing – review & editing, Writing – original draft.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.