Asymmetric Double AlGaAs/GaAs Quantum Well

• Asymmetric double AlGaAs/GaAs quantum wells are semiconductor structures with two coupled wells of differing potential depths, exhibiting unique tunneling dynamics and energy-level splitting.
• They employ precise engineering techniques such as layer thickness variation, alloy composition tuning, and applied electric fields to control quantum confinement and resonant conditions.
• These structures impact applications in optoelectronics and spintronics by enhancing excitonic responses, resonant tunneling efficiency, and spin lifetime through asymmetry modulation.

An asymmetric double AlGaAs/GaAs quantum well consists of two closely spaced quantum wells fabricated from alternating layers of AlGaAs and GaAs, separated by a potential barrier. In this configuration, the double well is intentionally engineered to have different potential depths or widths in each well, resulting in two potential minima with unequal energies. This asymmetry can be introduced by varying layer thicknesses, alloy composition, interface sharpness, applied electric field, or through specific growth-induced effects such as atomic segregation. The asymmetric double quantum well structure is fundamental in determining tunneling dynamics, energy level splitting, and optical and transport properties pivotal to advanced semiconductor device architectures.

1. Model Potential, Quantum Confinement, and Asymmetry Engineering

The idealized model for an asymmetric double well in AlGaAs/GaAs devices is based on a one-dimensional potential $V(x)$ with two quadratic minima at positions $x = -a$ (left well) and $x = b$ (right well). The potential depths satisfy $V(-a) = (n + \varepsilon)\hbar \omega$ and $V(b) = 0$, where $\omega$ is the characteristic frequency set by effective mass and confinement, $n$ is an integer, and $\varepsilon$ is a small asymmetry parameter (0803.3113).

Achieving quadratic minima with identical curvature and engineered energy offsets is accomplished by precise control of the growth parameters:

• GaAs well width and barrier (AlGaAs) thickness set the minima location and barrier height.
• Varying composition (Al fraction) or applying an electric field tunes the asymmetry $\varepsilon$. Compared to symmetric structures, the asymmetric configuration lifts parity protection, resulting in non-degenerate localized states with unique tunneling dynamics and energy-level patterns (Mukherjee et al., 2019, Grigoryev et al., 2016).

2. Tunneling, Energy Splitting, and Resonant Conditions

Tunneling between the wells is analyzed via the semiclassical WKB approximation. In the barrier region ($V(x) > E$), the wave function decays exponentially:

$$\psi(x) \sim \frac{1}{\sqrt{p(x)}} \exp\left( -\frac{1}{\hbar} \int p(x) dx \right),$$

where $p(x)=\sqrt{2m(V(x)-E)}$. The exact solutions in the well regions (parabolic minima) are given by harmonic oscillator eigenfunctions (parabolic cylinder functions). Matching the WKB and exact wave functions at the turning points yields the splitting formula

$$\Delta_l = \sqrt{g_l g_{l+n} \frac{\hbar \omega}{\pi}}\, \exp\left( -\frac{1}{\hbar} \int_{-a_l}^{b_l} p(x)\, dx \right).$$

Asymmetry modifies this result:

$$\Delta_l^\varepsilon = \sqrt{\Delta_l^2 + (\hbar \omega \varepsilon)^2}.$$

The tunneling amplitude, and thus the energy splitting between bonding and anti-bonding states, is exponentially sensitive to the barrier properties and asymmetry (0803.3113).

When the energy offset between the wells $(n + \varepsilon)\hbar \omega$ is close to a multiple of $\hbar \omega$, resonant tunneling occurs. This is manifest in a Lorentzian enhancement of the tunneling probability and strong hybridization of localized states (0803.3113). Device operation can exploit this resonance for high tunneling rates in diodes or polaritonic systems (Christmann et al., 2011, Sun et al., 2012).

3. Disorder Effects, Interface Broadening, and Structural Variations

Realistic quantum well interfaces are not perfectly sharp. The impact of static disorder, roughness, or atomic-scale mixing at AlGaAs/GaAs boundaries is modeled by smoothing the ideal discontinuity with convolution envelopes (Gaussian, Lorentzian, or arctan profiles) (Gavryushin, 2011, Garbaczewski et al., 2022). The smoothed interface alters the confining potential and increases the energy of localized states (blue shift), but the resonant tunneling conditions and doublet structures typically persist.

A generalized approach employs the Fokker–Planck diffusion equation to map the disorder-induced amorphization of the interfaces onto a smooth double-well confining potential used in the Schrödinger spectrum calculations. This method predicts strong influence of interface mixing on absorption edge location and tunneling rates while preserving qualitative doublet features (Garbaczewski et al., 2022).

Atomic-layer insertions (AlAs or InAs monolayers) are used to tailor local potentials and engineer interband/intersubband transition energies, with selection rules and oscillator strengths linked to the position of the insertion and symmetry reduction (Raouafi et al., 2016).

4. Excitonic Effects and Optical Properties

Exciton resonances in asymmetric double quantum wells display enhanced sensitivity to asymmetry and interface shape. Detailed reflectance measurements and theory reveal that the transition energies, radiative broadenings, and optical phase shifts depend strongly on the profile and composition of the potential well, including segregation effects during growth (e.g., indium diffusion) (Grigoryev et al., 2016). The phenomenological response incorporates:

$$r_{QW} = \sum_n \frac{i \Gamma_{0(n)} e^{i\phi_n}}{\tilde{\omega}_{0(n)} - \omega - i(\Gamma_n + \Gamma_{0(n)})},$$

with $\phi_n$ quantifying asymmetry-induced phase shifts.

The symmetry class (e.g., D$_{2d}$ vs. C$_{2v}$) defines the mixing of heavy-hole and light-hole states, leading to enhanced in-plane optical anisotropies in asymmetric structures (Ruiz-Cigarrillo et al., 2021). This enhancement, detected through reflectance anisotropy spectroscopy (RAS), directly correlates with structural asymmetry and is calculable via the overlap integrals and mixing matrix elements.

5. Quantum Transport: Tunneling, Transmission, and Four-Wave Mixing

Quantum transport properties, including electron transmission probability and nonlinear optical phenomena, are critically dependent on well asymmetry and barrier configuration. In double quantum wells with triple barrier, the transmission probability $T_{prob}$ decreases exponentially with barrier height/width and exhibits resonant peaks at specific electron energies (Magar et al., 19 Feb 2025):

• Symmetric barrier configuration: Two transmission peaks.
• Slight central barrier reduction: Single higher-value peak. This behavior reflects the sensitive dependence of quasi-bound states and resonant tunneling conditions on the potential profile.

Resonant tunneling in asymmetrical double wells also dramatically enhances four-wave mixing (FWM) efficiency through constructive interference between multiple transition pathways. Engineering the structure to allow such interference effects enables high-efficiency generation of coherent long-wavelength radiation, attractive for optoelectronic device applications (Sun et al., 2012).

6. Spin-Orbit Coupling, Spin Relaxation, and Photoluminescence

Spin relaxation and photoluminescence (PL) properties in asymmetric double quantum wells are determined by the interplay between Dresselhaus and Rashba spin-orbit coupling mechanisms, well asymmetry, and built-in electric fields. The wider wells in an asymmetric structure reduce the Dresselhaus effect, yielding longer electron spin lifetimes ($\tau_s$) (Bravo-Velazquez et al., 2022):

$$1/\tau_s = \langle \Omega\cdot\Omega \rangle \tau_p$$

where $\Omega$ is the effective Larmor frequency from spin-orbit fields.

Excitation with longer-wavelength lasers produces carriers with lower initial quasi-momentum, further decreasing $\Omega$ and increasing $\tau_s$. PL measurements show increased polarization degree and spin lifetime in asymmetric quantum wells, with Rashba contributions modulated by barrier doping. These effects can be exploited for tailored spintronic device performance.

7. Quantum Information Measures, Localization, and Device Implications

Information entropy-based analysis (Shannon entropy, Fisher information, Onicescu energy) quantifies localization-delocalization transitions in asymmetric double well systems. The asymmetry parameter governs transitions where electron probability shifts from occupying both wells to being localized in the deeper well. Diagnostic entropic measures display abrupt changes at these transition points, offering insight into tunneling, localization, and decoherence, which are essential for quantum information applications and robust electronic device operation (Mukherjee et al., 2019).

Summary Table: Key Physical Effects of Asymmetry in AlGaAs/GaAs Double Quantum Wells

Physical Effect	Role of Asymmetry	Device/Application Impact
Energy splitting $\Delta$	Modifies bonding/antibonding state gap	Tunneling rates, transport
Resonant tunneling	Enhances transmission at specific offsets	RTDs, quantum cascade lasers
Interface disorder	Blue shifts energy, preserves resonance	Optical absorption, reliability
Exciton resonance	Alters energies, broadening, phase	Optical response, filtering
Spin relaxation	Reduces Dresselhaus, tunes $\tau_s$	Spintronics, PL devices

Asymmetric double AlGaAs/GaAs quantum wells offer a platform for precisely tunable quantum transport, optical, and spintronic phenomena. Theoretical and computational techniques—ranging from analytic WKB matching and matrix diagonalization to numerical interface disorder modeling—enable predictive design and characterization. Controlled asymmetry, whether via structure, composition, or applied fields, directly manipulates tunneling, energy spectrum, excitonic, and spin properties for applications in advanced semiconductor device engineering.