Finite Element Computation of Semiconductor Quantum States: Convergence, Sparse Eigensolvers, and Verification

Abstract This work presents and analyzes a finite element framework for the stationary Schrödinger eigenvalue problem in one to three spatial dimensions. The weak formulation yields a generalized symmetric eigenproblem Aψ = EM ψ with A = K + M V . Where the discrete form is the exact restriction of the continuous one, the conforming approximation produces eigenvalues that are upper bounds for the exact ones, and under standard regularity assumptions the eigenvalue error converges as O(h2s ); the nodal interpolation of the potential used here preserves the bound for convex V on the tensor-product meshes considered, and is verified numerically otherwise. The usual diagonal-shift treatment of Dirichlet data introduces spurious eigenvalues, whereas eliminating the constrained degrees of freedom reproduces the constrained spectrum exactly. Three eigensolution strategies are considered: dense factorization and sparse shift-and-invert Arnoldi iteration are compared on identical matrices, and the serial constraint of the in-framework solver is stated, together with the export route that lifts it. The sparse route is up to 4097× faster, replaces the O(n2f ) dense storage by sparse matrices whose memory is dominated by the factorization fill-in and the Arnoldi basis rather than by the matrices themselves, and makes a problem with 3.96 × 10 4 degrees of freedom (infeasible in dense arithmetic at 51.2 GB) solvable in 0.41 s. Position-dependent mass is supported in the conservative form − ℏ2 /2 ∇ · [(1/m ∗ )∇ψ]; on an InGaN/GaN well, where the mass contrast is 6.3%, the levels move by 3.0%. Verification is fivefold: seven analytic benchmarks, a code-to-code comparison agreeing to 4 × 10−13 , measured convergence orders against the O(h2 ) prediction, structural invariants, and the reproduction of two published quantum-confinement benchmarks.