<p>We develop a finite-element computational framework for modeling fractional quantum transport in semiconductor nanostructures using the fractional effective mass Schrödinger equation. The method is applied to a singly ionized double donor (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textrm{D}_2^+$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mtext>D</mtext> <mn>2</mn> <mo>+</mo> </msubsup> </math></EquationSource> </InlineEquation>) confined in a GaAs/AlGaAs coupled quantum dot–ring structure under Aharonov–Bohm flux, Rashba spin–orbit coupling, hydrostatic pressure, and external electric fields. The fractional kinetic operator (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0 < \alpha \le 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>) is implemented via its Dirichlet integral representation within an adaptive finite-element scheme combined with shift-invert Lanczos eigensolvers. Numerical convergence is verified through mesh refinement and recovery of the classical limit as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha \rightarrow 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">→</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Simulations show that decreasing <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> enhances donor binding, increases bonding–antibonding splitting, strengthens localization, and suppresses ring-mediated tunneling. Fractional dispersion attenuates Aharonov–Bohm oscillations and modifies Rashba-induced spin splitting. Optical absorption calculated from the computed eigenstates exhibits redshifts and enhanced nonlinear response for reduced <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>. These results demonstrate the stable integration of nonlocal fractional operators in realistic nanoelectronic geometries and provide a computational tool for analyzing generalized quantum transport in confined semiconductor systems.</p>

Finite-element modeling of fractional quantum transport and optical response in coupled quantum dot–ring nanostructures with donor impurities

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发表日期：

2026年3月24日

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Lakhdar Sek,
Issam Zaiz,
Abdelmonem Miloudi

摘要

We develop a finite-element computational framework for modeling fractional quantum transport in semiconductor nanostructures using the fractional effective mass Schrödinger equation. The method is applied to a singly ionized double donor ( $\textrm{D}_2^+$ $D_{2}^{+}$ ) confined in a GaAs/AlGaAs coupled quantum dot–ring structure under Aharonov–Bohm flux, Rashba spin–orbit coupling, hydrostatic pressure, and external electric fields. The fractional kinetic operator ( $0 < \alpha \le 2$ $0 < α \leq 2$ ) is implemented via its Dirichlet integral representation within an adaptive finite-element scheme combined with shift-invert Lanczos eigensolvers. Numerical convergence is verified through mesh refinement and recovery of the classical limit as $\alpha \rightarrow 2$ $α \to 2$ . Simulations show that decreasing $\alpha $ $α$ enhances donor binding, increases bonding–antibonding splitting, strengthens localization, and suppresses ring-mediated tunneling. Fractional dispersion attenuates Aharonov–Bohm oscillations and modifies Rashba-induced spin splitting. Optical absorption calculated from the computed eigenstates exhibits redshifts and enhanced nonlinear response for reduced $\alpha $ $α$ . These results demonstrate the stable integration of nonlocal fractional operators in realistic nanoelectronic geometries and provide a computational tool for analyzing generalized quantum transport in confined semiconductor systems.