Effect of weak positional disorder on the miniband structure of spherical quantum dot chains

Technical Summary: Effect of Weak Positional Disorder on the Miniband Structure of Spherical Quantum Dot Chains

Problem Statement
While ordered arrays of quantum dots (QDs) offer significant advantages for optoelectronic devices—such as predictable optical properties and engineered miniband structures—perfect periodicity is technologically unattainable. Real-world fabrication processes (e.g., Stranski-Krastanov growth or colloidal synthesis) inevitably introduce inhomogeneities in QD size and spatial distribution. Although the miniband structure of perfectly ordered QD superlattices has been extensively studied using methods like the plane wave and tight-binding approximations, the specific impact of weak positional disorder on the electron miniband structure in one-dimensional (1D) chains of spherical QDs requires further elucidation. This study addresses the gap in understanding how small, random deviations in inter-dot spacing modify the electronic band structure.

Methodology
The authors develop a theoretical framework combining the tight-binding approximation with an effective-medium approach to model an infinite 1D chain of spherical QDs.

• System Model: The system consists of spherical GaAs QDs embedded in an AlAs matrix, treated as a spherically symmetric rectangular potential well. Under a single-level approximation, each QD is constrained to possess only one electronic bound state.
• Disorder Modeling: Weak positional disorder is introduced as small, random deviations ($\delta_n$) from ideal lattice positions ($R_n = an$). These deviations are modeled as independent random variables following a Gaussian distribution with zero mean and standard deviation $\sigma$, where $\sigma \ll a$ (the mean inter-dot distance).
• Analytical Derivation: The inter-dot distance $d_n$ is expressed as $a + \delta$, where $\delta$ follows a Gaussian distribution with variance $2\sigma^2$. The key tight-binding parameters—hopping integral ($B$), overlap integral ($Q$), and on-site energy shift ($M$)—are expanded in a Taylor series with respect to these small deviations.
• Ensemble Averaging: By performing an ensemble average over the Gaussian distribution, the authors derive analytical expressions for the disorder-renormalized parameters ($\langle B \rangle, \langle Q \rangle, \langle M \rangle$). This process retains terms up to order $\sigma^2$, effectively mapping stochastic fluctuations onto the renormalization of Hamiltonian parameters.
• Numerical Implementation: Numerical calculations are performed for GaAs/AlAs systems at $T \to 0$ K to isolate disorder effects from thermal broadening. The QD radius is fixed at 20 Å, and the inter-dot distance $a$ is varied between 40 and 60 Å, with disorder strength set to $\sigma = 0.1a$.

Key Results

1. Miniband Broadening: The primary finding is that weak positional disorder leads to a broadening of the electron minibands. The generalized dispersion relation, derived by substituting the renormalized parameters into the tight-binding energy equation, shows that the miniband width increases compared to the ideal periodic case.
2. Quantitative Impact: For typical fabrication fluctuations where $\sigma = 0.1a$, the miniband width increases by approximately 8–12%, depending on the specific mean inter-dot distance.
3. Distance Dependence: The sensitivity of the miniband width to disorder decreases rapidly as the lattice period ($a$) increases. This is attributed to the exponential decay of electron wave functions outside the QDs, which causes the overlap integrals ($B, Q, M$) and their derivatives to diminish with larger separation. Consequently, the discrepancy between the disordered and ideal miniband widths narrows as $a$ grows.
4. Localization Regime: In the considered weak-disorder regime, the Anderson localization length ($\xi_{loc}$) significantly exceeds the lattice constant ($a$). Therefore, the miniband states remain delocalized. The study notes that true Anderson localization would require much stronger disorder ($\sigma/a \gtrsim 0.3$), which lies outside the scope of this work.
5. Physical Mechanism: The broadening is physically interpreted as a result of two competing effects: the suppression of coherent tunneling due to phase mismatches in wave functions caused by random spacing (affecting the hopping integral $B$), and the modification of the effective onsite energy and overlap due to changes in the confinement potential profile (affecting $M$ and $Q$).

Significance and Claims
The paper claims to provide a theoretical tool for assessing the impact of manufacturing tolerances on the electronic properties of QD-based devices. By deriving explicit formulas for disorder-renormalized parameters, the work offers a method to predict how weak positional fluctuations alter the miniband structure without requiring complex numerical simulations for every specific disorder realization.

The authors state that these findings serve as practical recommendations for enhancing the efficiency of quantum dot-based optoelectronic components, specifically citing cascade lasers and infrared detectors. The study emphasizes that while disorder broadens the bands, the fundamental band structure remains intact under weak perturbations, suggesting that devices can be engineered to function effectively even with realistic fabrication imperfections. The work does not propose new experimental setups or claim to solve the problem of strong localization, but rather quantifies the limits of weak disorder in maintaining delocalized transport in QD chains.