Numerical Methods

Numerical Modeling for Semiconductor
Quantum Dot Molecule Based on the Current
Spin Density Functional Theory
Jinn-Liang Liu
Jen-Hao Chen
O. Voskoboynikov
Department of Applied Mathematics, NUK
Department of Applied Mathematics, NCTU
Department of Eletronic Engineering, NCTU
Outline
1. Introduction
2. The Current Spin DFT
3. Numerical Methods and Algorithms
4. Numerical Results
5. Conclusion
Introduction : motivation
Quantum Computer
 Fermionic Qubits
 Electronic Excitations of Coupled QDs
 Artificial Molecule (QDM)
 Forster-Dexter Energy Transfer

Introduction : model
6 electrons
 Hard-wall confinement potential
 An external magnetic field
 Effective-mass approximation with band
nonparabolicity
 Exchange-correlation energy ( by
Saarikoski et al. )

Introduction : model

Three vertically aligned InAs/GaAs QDs

A cubic eigenvalue problem
Self-consistent algorithm
Schrodinger-Poisson system
Jacobi-Davidson method and GMRES



The CSDFT : ground state energy

Electron number : N
 Total spin : S
 Spin-up and spin-down :
 Total density :
 Constraint :
The CSDFT : noninteracting kinetic energy



Kohn-Sham (KS) orbitals and eigenvalues :
The CSDFT : effective mass




Energy-band gap :
Spin-orbit splitting in the valence band :
Momentum matrix element :
The CSDFT : Hartree potential

Permittivity of vacuum :
 Dielectric constant :
The CSDFT : energy of magnetic field

Lande factor :
 Bohr magneton :

Paramagnetic current density :
The CSDFT : xc energy

xc energy per particle depends on the
magnetic field

Vorticity :
The CSDFT : KS Hamiltonian

To minimize the total energy under the
constraint of the orbitals being normalized
The CSDFT : KS Hamiltonian
where
The CSDFT : xc energy functional

Spin polarization :
 Wigner-Seitz radius :
 Saarikoski et al. :
The CSDFT : xc energy functional
where
 Levesque, Weis, and MacDonald :

Perdew and Wang :
Numerical Methods : 2D problem


Principal quantum number :
Quantum number of the projection of angular
momentum :
Numerical Methods : 2D problem

KS equations are then reduced to a 2D problem :
where
Numerical Methods : 2D problem

Interface conditions :

Boundary conditions :
Numerical Methods : Hartree potential


(3D) is solved by Poisson equation
By cylindrical symmetry :
where
Numerical Methods : Hartree potential

Separating variables :
 Substituting it into (3.11) :
Numerical Methods : Hartree potential

By setting

is a particular sol of (3.14)
satisfying

The corresponding homogeneous
general solution is
satisfying
Numerical Methods : Hartree potential

The general solution of the nonhomogeneous
equation (3.14) is therefore of the form
Numerical Methods : Hartree potential

Interface conditions :

Boundary conditions :
Numerical Methods : Hartree potential

By imposing these boundary conditions to the
general solution (3.17),
is in fact a general solution of
(3.14) and thus of (3.11), i.e.,
Numerical Methods : cubic EVP

Since the mass and the Lande factor are
energy dependent :

Poisson equation :
Numerical Algorithm : self-consistent
(1)
Set k = 0.
At B=0, first three lowest energies :
we therefore must solve (3.20) six times.
At B=15, first three lowest energies :
we thus solve (3.20) two times.
Numerical Algorithm : self-consistent
(2) Evaluate
If converges then stop.
Otherwise set
(3) Solve (3.21) for the Hartree potential
by using GMRES.
(4)
Numerical Algorithm : JD method

Eigenvalues are embedded in the interior of
the spectrum.
 Nonsymmetric system
 Degenerate eigenstates
 In stead of using deflation scheme in JD
solver, we compute several eigenpairs
simultaneously and several corrections are
incorporated in search subspace at every
iteration.
Numerical Algorithm : JD method
Numerical Algorithm : JD method
Numerical Algorithm : JD method
Numerical Results

Energy differences between the parabolic and
nonparabolic dispersion relations :
Numerical Results

All energy components at B=0 :

Accuracy of the exchange energies
Numerical Results

All energy components at B=15 :

Accuracy of the exchange energies
Numerical Results
Conclusion

A new mathematical model :
nonparabolicity + magnetic field + CSDFT
+ advanced xc energy + QDMs

A new Jacobi-Davidson method in cubic EVP

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