Polarization of Germanium Quantum Dots in Heterostructure Ge/Si …
621
a
2D
ex N
1/3
1
( 2 0 )
4 Dipole Moments of Transitions Between SIE States
in a Quantum Dot
Because the electron is out of QD, the transition dipole moment D
1,l+1
1,l (a) consists
of dipole moments (i) of polarization the QD by electron, (ii) of the electron relative
to the hole localized in QD. Within the dipole approximation framework, when the
electromagnetic field on the scale of a QD does not vary and can be considered as
uniform, the operator of transition dipole moment to the SIE state takes the following
form [31]:
D(r ) =
1−K (a/r )
3
er,
(21)
where radius vector defines the distance between an electron and a hole localized in
QD center, the coefficient K = ((ε 2 – ε 1 )/(ε 2 + 2ε 1 ).
For estimating the D
1,l+1
1,l (a) value, we consider the electron transition between
(n = 1, l) and (n = 1, l + 1) ground SIE states in nanosystem containing the QD
with average radii a satisfying the condition (4) A transition between such states is
allowed by the selection rules (in this case, the principal quantum number n changes
arbitrarily, whereas the orbital quantum number l changes by unity). If the direction
of the external radiation electric field vector ε(ω,t) is along the Z-axis, then:
(a) = Ψ 1,l+1 (r, θ)|D
1,l+1
1,l 1−K (a / r )
3
)er|Ψ 1,l (r, θ)
(22)
The exciton wave functions Ψ 1,l+1 (r, θ) and Ψ 1,l (r, θ) will be chosen as follows:
Ψ 1,l (r, θ) = R 1,l (r )Y l,0 (θ ), Ψ 1,l+1 (r, θ) = R 1,l+1 (r )Y l+1,0 (θ )
(23)
where Y l,0 (θ ) and Y l+1,0 (θ ) are the exciton spherical wave functions, is the azimuthal
angle defining the position of electron radius vector, R 1,l (r ) and R 1,l+1 (r ) are the
hydrogen-like radial wave functions defined by (3). The wave function Y l,0 (θ ) can
be expressed by the Legendre polynomial:
Y l,0 (θ ) = ((2l + 1)/4π )
1/2 P l (cos θ)
(24)
It should be noted that only projection of transition dipole moment on the vector
ε(ω,t) direction is not equal to zero because of the QD spherical shape.
Using (22), (23), (24) expressions we can to write:
D
1,l+1
1,l (a) = (2l + 1)
1/2
(2l + 3)
1/2 2
−1 e
621
a
2D
ex N
1/3
1
( 2 0 )
4 Dipole Moments of Transitions Between SIE States
in a Quantum Dot
Because the electron is out of QD, the transition dipole moment D
1,l+1
1,l (a) consists
of dipole moments (i) of polarization the QD by electron, (ii) of the electron relative
to the hole localized in QD. Within the dipole approximation framework, when the
electromagnetic field on the scale of a QD does not vary and can be considered as
uniform, the operator of transition dipole moment to the SIE state takes the following
form [31]:
D(r ) =
1−K (a/r )
3
er,
(21)
where radius vector defines the distance between an electron and a hole localized in
QD center, the coefficient K = ((ε 2 – ε 1 )/(ε 2 + 2ε 1 ).
For estimating the D
1,l+1
1,l (a) value, we consider the electron transition between
(n = 1, l) and (n = 1, l + 1) ground SIE states in nanosystem containing the QD
with average radii a satisfying the condition (4) A transition between such states is
allowed by the selection rules (in this case, the principal quantum number n changes
arbitrarily, whereas the orbital quantum number l changes by unity). If the direction
of the external radiation electric field vector ε(ω,t) is along the Z-axis, then:
(a) = Ψ 1,l+1 (r, θ)|D
1,l+1
1,l 1−K (a / r )
3
)er|Ψ 1,l (r, θ)
(22)
The exciton wave functions Ψ 1,l+1 (r, θ) and Ψ 1,l (r, θ) will be chosen as follows:
Ψ 1,l (r, θ) = R 1,l (r )Y l,0 (θ ), Ψ 1,l+1 (r, θ) = R 1,l+1 (r )Y l+1,0 (θ )
(23)
where Y l,0 (θ ) and Y l+1,0 (θ ) are the exciton spherical wave functions, is the azimuthal
angle defining the position of electron radius vector, R 1,l (r ) and R 1,l+1 (r ) are the
hydrogen-like radial wave functions defined by (3). The wave function Y l,0 (θ ) can
be expressed by the Legendre polynomial:
Y l,0 (θ ) = ((2l + 1)/4π )
1/2 P l (cos θ)
(24)
It should be noted that only projection of transition dipole moment on the vector
ε(ω,t) direction is not equal to zero because of the QD spherical shape.
Using (22), (23), (24) expressions we can to write:
D
1,l+1
1,l (a) = (2l + 1)
1/2
(2l + 3)
1/2 2
−1 e