Chapter 2 Fundamental Theory
2.1 Theory for Exciton
An exciton is formed from an electron in conduction band and an empty hole in valence band. The electron and hole wave functions are written as
, ( ) ( ) ( )
denote Bloch functions for describing the microscopic observation at Γ-point while , ( )
present envelope functions that satisfy the Schrödinger equation.
The electronic structure in effective mass approximation is carried out by
denotes the electron confinement potential. m*represents electron effective mass. Eieerefers to the eigenenergy of iethe state. εij is the strain tensor, and a refers to a specific strain constant for conduction deformation c potential.
The electronic structure of the valence hole is computed by four-band k p⋅ model. [4]
6 strain constants for conduction deformation potential.
Appendix II lists the Hamiltonian elements.
Exchange interaction for multi-bands theory
The whole single exciton Hamiltonian in a quantum dot is
ˆ ˆ
( , ) ( , ) ( , )
X e e e h h h eh e h
H =H p rv + H p rv −V r rv v
(2.1.10)
7
ˆ ( , )
e e e
H p rv
(
H p rh(ˆh, )vh)
denotes electron(hole) Hamiltonian, and
2 energy spectrum can be obtained by solving the Schrödinger equation
X X X
X n n n
H ψ =E ψ (2.1.11)
An electron ie and a hole state ih form an exciton ie ih , in which the exciton state is expressed as
† † orbitals. The Hamiltonian can be further written as
' ' ' '
s j denotes the electron (hole) orbitals. †,
e z
ci s and ci se,z (h†j jh, z and hj jh,z ) are the electron (hole) creation and annihilation operators.
, , ,
e h h e
eh i j k l
V and Vi j k leehxc,h,h,e are electron-hole Coulomb interaction and electron hole exchange interaction where
2
8
Model analysis
Eigenstates of 4-band k ⋅pHamiltonian without HH-LH coupling are taken as the basis for further expanding the envelope function , ( )
. Moreover, to rearrange the order of the basis of Hamiltonian in Eq. (2.1.5). The reduced basis of single exciton ie ih
becomes
{
1/2 3/2 1/2 3/2 1/2 1/2 1/2 1/2}
{sz jz }= f ue e− fHHu+h , f ue e+ fHHu−h , f ue +e f uLH +h , f ue −e f uLH −h
(2.1.17) and the exciton Hamiltonian[13-15] can be expressed as
,(0)
,(0)
∆HL denotes the energy separation of heavy hole and light hole
(
Fig. 2.1.1 A diagram illustrates the hole potential and the relative position of HH- and LH- states and the
VQD represents arbitrary confining potential, while long-range part of e-h exchange interaction.
1 1 and light-hole coupleing.
)
the energy separation of heavy hole and light hole
) ( )
2.1.1 A diagram illustrates the hole potential and the relative position of states and the HH-LH- splitting.
trary confining potential, while ∆1 and h exchange interaction.
1 1 2 the energy separation of heavy hole and light hole
(2.1.22)
2.1.1 A diagram illustrates the hole potential and the relative position of
and ∆0 denote the
0 0 BD
δ + ∆ ≡ ∆
∆BD refers to the bright exciton and dark exciton levels (BX which is induced mainly
δ0 denotes the short-range
Riand Rjrepresent positions of the WS cells.
The eigenenergies are obtained of the two lowest eigenvalues
Eπx or Eπy, as introduce
positioned as the dark exciton state
Fig. 2.1.2 The zero energy level locates at dark exciton state. The bright exciton was
separated due to the exchange interaction.
Asymmetric 3-D parabolic model The confining potential of
10
the bright exciton and dark exciton levels (BX mainly by long-range interaction.
range portion of the e-h exchange interaction.
0 1 2
positions of the WS cells.
are obtained by diagonalizing the matrix,
two lowest eigenvalues is defined as EnX and EnX′, which correspond to introduced later. The zero energy level of the Hamiltonian is positioned as the dark exciton state, as shown below.
Fig. 2.1.2 The zero energy level locates at dark exciton state. The was originally a two-fold degenerate level, but separated due to the exchange interaction.
D parabolic model
The confining potential of a quantum dot is described using asymmetric 3 (2.1.25) the bright exciton and dark exciton levels (BX-DX) splitting,
h exchange interaction.
2 1/2 2 2 3/2 2 1 3/2 1 1 1/2 1
and the separation which correspond to . The zero energy level of the Hamiltonian is
Fig. 2.1.2 The zero energy level locates at dark exciton state. The fold degenerate level, but is
is described using asymmetric 3-D
11
parabolic model, and is written into three parts as
2 2 2 2 2 2
Where the effective masses are
0 0 hole orbitals are thus
2 2 2
are the characteristic length as well as the wave function extended.
12
QDs with electron-hole exchange interaction
With the wave function of 3D parabolic potential as the basis, the elements of Hamiltonian in Eqs. (2.1.14) and (2.1.15) thus become
2 Notable, according to the thesis of Chang[18], the quantity of HH- LH- splitting in a four-band model is over estimated, while the strain effect is involved.
Therefore, the HH- LH- splitting that effected by strain of six-band model is employed in our numerical calculation. The geometry dependence part(k part) remains using the 4-band model.
For the six-band model, ∆kHL remains Eq. (2.1.40) but
13
where
ε
d, Ep, EgQDdenote static dielectric constant, optical matrix parameter and energy gap of QD. The energy gap of QD is determined byQD exchange energy. Moreover, the energy is obtained by Hamiltonian matrix diagonalization.
Simplified exciton two-level system
A simpler formula is derived to analyze and observe the basic physical picture by reducing the 4 4× to 2 2× one by using the Löwden perturbation method.
The basis now are the two bright states The eigen energies are
,( 0)
; ( HH ( ))
X X BD eff
E ± = E + ∆ − ± ∆
Subsequently leading to the splitting between two bright states which is known as fine structure splitting
1
while corresponds to the polarization along
Fig. 2.1.3 (a) Optical polarization polarization chart corresponding energy
to the splitting between two bright states which is known fine structure splitting (Sπ πx, y).
Optical polarization polarization chart, and corresponding energy spectrum. to the splitting between two bright states which is known
(2.1. 49)
(2.1. 50) (2.1. 51) (2.1. 52)
, and (b) the
15
For the initial shear strain
ε
xy =0, this thesis considers only the −∆ + ∆1 ′ portion, which illustrates the splitting of the competition among the two components. The greater equality the two components implies a smaller splitting.Once the size is determined, the long-range term, ∆1 is also derived. However, the ∆′ term can be further tuned via strain effect to achieve small or even zero splitting.
Optical polarization of single exciton in QDs
An electron is excited to conduction band, creating a hole in valence band;
the electron then recombines with a hole and emits light to release energy. This process is called spontaneous emission. For the PL polarization emission spectrum, the intensity in each angle is manipulated using Fermi’s Golden state; and Pev− is refer to the polarization operator which can be performed as
, annihilation operators.
16 Based on (2.1.16) to (2.1.18), the emission intensity is proportional to the dipole matrix element, the dipole matrix element
( )
Epdenotes the optical matrix element.
Further use the Lödwen perturbation theory, the two lowest HH-LH-mixed wavefunctions are written as
1 1/2 3/2 1/2 1/2
17
By returning the functions back to Eq. (2.1.50) and using Table I, the polarization is obtained as
1( 0) 2( 0)
For another definition
and also sketched as
Fig. 2.1.4 A Gaussian function along
Assume that the wave function boundary L/2. The height at
18 is always positive.
wave function described by the Gaussian function in x
A Gaussian function along x-direction.
the wave function is reduced to A percent of the peak ( The height at L/2 can be expressed as
(2.1. 65)
19
from Eq. (2.3.2.4), we obtain the relation between the geometric length and its characteristic length can be obtained via the ratio A
2
as in the case for other directions.
Consider a case of InAs/GaAs cubic shaped quantum dot.
A Lx =20(nm), Ly =18(nm) and Lz =5(nm) InAs/GaAs cubic quantum dot wave functions are obtained by one-band (six-band) for electron (hole) finite difference numerical method which was programmed in the thesis of Ku[12].
The initial strains are considered. The strain distributions in and out of dots are computed by using finite element software package Comsol multiphysics® .
(a)
Fig.2.1.5 Electron wave function along (a) 20 nm, A = 0.09
x x
L =
According to Eq. (2.1.68), Or Lx =4.39lxe,Ly =4.39lx ×
20
(b)
Fig.2.1.5 Electron wave function along (a) 20 nm, A = 0.09, (b) z-direction, Lz =5 nm, A = 0.47z .
Eq. (2.1.68), lxe =4.56 (nm) and ley = ×ξ 4.56 (nm) 4.39 e
y x
L = l ×ξ ,Lz =2.5lz
x-direction, 5 nm, A = 0.47.
4.56 (nm) lze =2.03 (nm).
21