1. Numerical Modeling for Semiconductor Quantum Dot Molecule Based on the Current Spin Density Functional Theory
Jinn-Liang Liu Department of Applied Mathematics, NUK
Jen-Hao Chen Department of Applied Mathematics, NCTU
O. Voskoboynikov Department of Eletronic Engineering, NCTU

2. Outline 1. Introduction 2. The Current Spin DFT
3. Numerical Methods and Algorithms
4. Numerical Results
5. Conclusion

3. Introduction : motivation
Quantum Computer
Fermionic Qubits
Electronic Excitations of Coupled QDs
Artificial Molecule (QDM)
Forster-Dexter Energy Transfer

4. 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. )

5. Introduction : model Three vertically aligned InAs/GaAs QDs
A cubic eigenvalue problem
Self-consistent algorithm
Schrodinger-Poisson system
Jacobi-Davidson method and GMRES

6. The CSDFT : ground state energy
Electron number : N
Total spin : S
Spin-up and spin-down :
Total density :
Constraint :

7. The CSDFT : noninteracting kinetic energy
Kohn-Sham (KS) orbitals and eigenvalues :

8. The CSDFT : effective mass
Energy-band gap :
Spin-orbit splitting in the valence band :
Momentum matrix element :

9. The CSDFT : Hartree potential
Permittivity of vacuum :
Dielectric constant :

10. The CSDFT : energy of magnetic field
Lande factor :
Bohr magneton :
Paramagnetic current density :

11. The CSDFT : xc energy
xc energy per particle depends on the magnetic field
Vorticity :

12. The CSDFT : KS Hamiltonian
To minimize the total energy under the constraint of the orbitals being normalized

13. The CSDFT : KS Hamiltonian
where

14. The CSDFT : xc energy functional
Spin polarization :
Wigner-Seitz radius :
Saarikoski et al. :

15. The CSDFT : xc energy functional
where
Levesque, Weis, and MacDonald :
Perdew and Wang :

16. Numerical Methods : 2D problem
Principal quantum number :
Quantum number of the projection of angular momentum :

17. Numerical Methods : 2D problem
KS equations are then reduced to a 2D problem :
where

18. Numerical Methods : 2D problem
Interface conditions :
Boundary conditions :

19. Numerical Methods : Hartree potential
(3D) is solved by Poisson equation
By cylindrical symmetry :
where

20. Numerical Methods : Hartree potential
Separating variables :
Substituting it into (3.11) :

21. Numerical Methods : Hartree potential
By setting
is a particular sol of (3.14)
satisfying
The corresponding homogeneous
general solution is satisfying

22. Numerical Methods : Hartree potential
The general solution of the nonhomogeneous equation (3.14) is therefore of the form

23. Numerical Methods : Hartree potential
Interface conditions :
Boundary conditions :

24. 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.,

25. Numerical Methods : cubic EVP
Since the mass and the Lande factor are energy dependent :
Poisson equation :

26. 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.

27. Numerical Algorithm : self-consistent
(2) Evaluate
If converges then stop.
Otherwise set
(3) Solve (3.21) for the Hartree potential
by using GMRES.
(4)

28. 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.

29. Numerical Algorithm : JD method

30. Numerical Algorithm : JD method

31. Numerical Algorithm : JD method

32. Numerical Results
Energy differences between the parabolic and nonparabolic dispersion relations :

33. Numerical Results All energy components at B=0 :
Accuracy of the exchange energies

34. Numerical Results All energy components at B=15 :
Accuracy of the exchange energies

35. Numerical Results

36. Conclusion A new mathematical model :
nonparabolicity + magnetic field + CSDFT
+ advanced xc energy + QDMs
A new Jacobi-Davidson method in cubic EVP