Numerically stable and flexible method for solutions of the Schrodinger equation with self-interaction of carriers in quantum wells
Tóm tắt
A numerically stable method to calculate the quantum states of carriers based on the variational principle is proposed. It is especially effective for the carriers confined in the quantum wells under the influence of self-interaction of the carriers. In this treatment, a wave function is defined as a set of scalar numbers based on the finite-difference approach. An action defined as the expectation value of a Hamiltonian becomes a multivariate function of the wave function. Application of numerical multidimensional minimization procedures to the action can achieve stable convergence even under the conditions where the conventional self-consistent approach to Schrodinger and Poisson equations fails to give solutions. Application to the calculations of ground states in modulation-doped single quantum wells is demonstrated, and quantitative comparison to the conventional method is also presented. This method has implications not only for numerical procedures, but also for the numerical realization of the variational principle, a fundamental concept in physics.
Từ khóa
#Schrodinger equation #Wave functions #Carrier confinement #Potential well #Finite difference methods #Multidimensional systems #Convergence of numerical methods #Poisson equations #Stationary state #Epitaxial layersTài liệu tham khảo
jackson, 1998, Classical Electrodynamics
10.1063/1.337788
10.1103/PhysRevB.53.9993
10.1103/PhysRevB.38.9945
10.1103/PhysRev.152.683
10.1063/1.357906
10.1063/1.98097
10.1063/1.362895
10.1103/PhysRevB.5.4891
10.1103/PhysRevB.28.3241
harrison, 2000, Quantum Wells Wires and Dots
10.1143/JPSJ.47.1518
10.1103/PhysRevLett.54.1279
10.1063/1.335563
10.1103/RevModPhys.64.1045
mailhiot, 1986, <formula><tex> ${\bf{k}} \cdot {\bf{p}}$</tex></formula> theory of semiconductor superlattice electronic structure. ii. application to ga<formula><tex>$_{1-x}$</tex></formula>in<formula> <tex>$_x$</tex></formula>as–al<formula><tex>$_{1-y}$</tex></formula>in<formula> <tex>$_{y}$</tex></formula>as[100] superlattices, Phys Rev B, 33, 8360, 10.1103/PhysRevB.33.8360
10.1063/1.346357
conte, 1981, Elementary Numerical Analysis An Algorithmic Approach
10.1103/PhysRevB.61.4461
10.1063/1.365396
10.1088/0268-1242/2/9/007
10.1063/1.354921
10.1063/1.97428
trellakis, 1997, comparison of iteration schemes for the solution of the multidimensional schrödinger–poisson equations, Proc 5th Int Conf Numerical Analysis of Semiconductor Devices and Integration Circuits, 270
10.1103/PhysRevB.56.15752
10.1063/1.349389
10.1103/PhysRevB.42.5166
landau, 1958, Quantum Mechanics Non-Relativistic Theory
negele, 1987, Quantum Many-Particle Systems
sakurai, 1985, Modern Quantum Mechanics
10.1103/PhysRevB.49.10533
10.1103/PhysRevB.38.10057
10.1088/0953-8984/4/32/003
10.1103/PhysRevB.24.5693