Nonlinear optical rectification in semiparabolic quantum wells with an applied electric field

doi:10.1016/j.physb.2005.06.040

Physica B: Condensed Matter

Volume 368, Issues 1–4, 1 November 2005, Pages 82-87

https://doi.org/10.1016/j.physb.2005.06.040 Get rights and content

Abstract

The optical rectification (OR) in a semiparabolic quantum well with an applied electric field has been theoretically investigated. The electronic states in a semiparabolic quantum well with an applied electric field are calculated exactly, within the envelope function and the displaced harmonic oscillator approach. Numerical results are presented for the typical Al_xGa₁₋_xAs/GaAs quantum well. These results show that the applied electric field and the confining potential frequency of the semiparabolic quantum well have a great influence on the OR coefficient. Moreover, the OR coefficient also depends sensitively on the relaxation rate of the semiparabolic quantum well system.

Introduction

In the past few years, the nonlinear optical properties of semiconductor quantum wells (QWs), superlattices (SL), and nanostructures have attracted much attention in both theoretical and the applied physics, because of their relevance in practical applications and as a probe for investigating the electronic structure of mesoscopic media [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. Among the nonlinear optical properties, more attention had been paid to the second-order nonlinear optical properties [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], such as optical rectification (OR), second-harmonic generation (SHG), electro-optic effect (EOE), because the second-order nonlinear processes are the simplest and the lowest-order nonlinear effects, and the magnitudes of second-order nonlinear coefficients are usually stronger than those of the higher-order ones, if the quantum systems demonstrate significant asymmetry.

The main aspect of these optical nonlinearities, associated with intrasubband transitions in QW structures, is the asymmetry in the potential profile of the well (QW). Such an asymmetry in the potential profile can be achieved either by applying an external electric field to the QW [6], [11], [12] or by using sophisticated material growing technology such as molecular beam epitaxy (MBE) and metal-organic chemical vapour deposition (MOCVD) [8], [9], [13]. In their pioneering work, Gurnick and DeTemple have proposed to grow Al_xGa₁₋_xAs multiple QWs with asymmetric composition gradients of Al in the growth direction, for obtaining this asymmetry [8]. In their work, these authors have considered an asymmetric Morse potential and shown that nonlinearities are 10–100 times larger than in those for bulk materials. Khurgin has considered asymmetric coupled QWs [9]. Ahn and Chuang [11] have proposed to apply a bias to a symmetric QW to obtain this asymmetry. This has been realized by Fejer et al. [12], who obtained a SHG coefficient more than 70 times higher than in those for bulk GaAs. In a more recent work, Yuh and Wang [13] have suggested the use of a step-QW structure.

In this paper, the OR coefficient in the semiparabolic QW with an applied electric field is investigated. It is obvious that the semiparabolic QW system is an asymmetrical structure, and the applied electric field can adjust the asymmetry of the potential profile. In Section 2, the electronic states in the semiparabolic QWs are described adopting the methods of envelope function and displaced harmonic oscillator. In Section 3, numerical calculations on typical Al_xGa₁₋_xAs/GaAs material are performed; the OR susceptibility as a function of the electric field, the confining potential frequency of the semiparabolic QW, the incident photon energy, and the relaxation rate of the system are plotted. Results reveal that the large OR coefficient depends explicitly on these factors.

Section snippets

Theory

Electrons in a semiparabolic QW with an applied electric field are described by the effective-mass Hamiltonian $H = - \frac{ℏ^{2}}{2 m^{*}} [\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}] + V (z) + qFz,$ where z represents the growth direction, F is the strength of the applied electric field parallel to z direction, q is the electron charge, m* is the conduction band effective mass. Here V(z) is the confining potential, $V (z) = {\begin{matrix} \frac{1}{2} m^{*} w_{0}^{2} z^{2}, & z \geq 0, \\ \infty, & z < 0 . \end{matrix}$

Under the envelope function approximation, the eigenfunctions $Ψ_{n, k} (r)$ and eigenenergies $ε_{n, k}$ are $Ψ_{n, k} (r) = Φ_{n} ($

Numerical results and discussions

Numerical calculations are carried out on Al_xGa₁₋_xAs/GaAs material. The parameters chosen in this work are: $m^{*} = 0.067 m_{0}$ , $σ_{s} = 5 \times 1 0^{24} m^{- 3}$ , and $τ = 0.14 ps$ .

In Fig. 1, the variation of OR coefficient peak value $χ_{o, \max}^{(2)}$ with the applied electric field strength is plotted. From Fig. 1, it can be seen that the OR coefficient does not vanish even in the absence of the electric field ( $F = 0 V / m$ ). This nonvanishing behaviour of the OR coefficient arises from the noncentrosymmetric charge distribution of the

Conclusion

In this study, we present an efficient way of calculating the OR susceptibility in semiparabolic QW. Numerical calculations are done on the typical Al_xGa₁₋_xAs/GaAs QW, and the calculations mainly focus on the dependence of the OR coefficient on the electric field, confining potential frequency, the incident photon energy and the relaxation rate of semiparabolic QW system. The results show that the OR coefficient almost linearly increases with the increasing of the magnitude of the electric

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Nonlinear optical rectification in semiparabolic quantum wells with an applied electric field

Abstract

Introduction

Section snippets

Theory

Numerical results and discussions

Conclusion

Int. J. High Speed Electron.

J. Non-Cryst. Solids

Phys. Rev. Lett.

Phys. Rev. B

Electron. Lett.

Appl. Phys. Lett.

SPIE Proc.

Phys. Rev. B

Phys. Rev. B

IEEE J. Quantum Electron.