Polarization of Germanium Quantum Dots in Heterostructure Ge/Si …
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given in (31), using (26), we obtain the expression for the transition dipole moment
D
1,3
1,2 (a) of an electron between the ground (n = 1, l = 2) and (n = 1, l = 3) SIE
states
D
1,3
1,2 (a) = 2
6 3
1/2 7
−1/2 j
7/2
2 j
9/2
3 ( j 2 + j 3 )
−9
(1−K )( j 2 + j 3 )
3 + 18( j 2 + j 3 )
2 + 126( j 2 + j 3 ) + 136
(2 j
2
2 + 10 j 2 + 15)
−1/2 ( j
2
3 + 28 j 3 + 56)
−1/2 ea
(32)
5 Calculation Results and Discussion
Using formulas (28) and (12), in the interval of radii a (27) QD for QDs with an
average radius a 1 = 8.4 nm, we obtain the highest values of the transition dipole
moment D
1,1
1,0 (a 1 ) = 14.9 D 0 (where D 0 = e
◦
A in Debye units) of an electron and
oscillator strength of the transition f
1,1
1,0 (a 1 ) = 0.47 electrons between the ground (n
= 1, l = 0) and (n = 1, l = 1) by SIE states (taking into account the values j 0 = 2.04,
j 1 = 2.18, L
1,1
1,0 (a 1 ) = 0.355 taken from [5], and a
2D
ex = 2.6 nm (8)). In this case, the
polarizability (14) of QDs with a radius a 1 , caused by such a transition, takes the
value A
(a 1 ) = 2.2 . 10
−19 cm
3 .
In the interval of radii a (29) QD for QDs with an average radius of a 2 = 13.1 nm,
using (30) and (12), we find the largest values of the dipole transition moment D
1,2
1,1 (a 2 )
= 15.2 D 0 electrons and oscillator strengths of the transition f
1,2
1,1 (a 2 ) = 0.50 electrons
between the ground (n = 1, l = 1) and (n = 1, l = 2) SIE states (taking into account
the values j 1 = 2.24, j 2 = 2.36, L
1,2
1,1 (a 2 ) = 0.363 taken from [5]). Due to this
transition, the polarizability (14) of QDs with radius a 2 takes the value A
(a 2 ) = 2.3
. 10
−19 cm
3 .
From formulas (32) and (12), it follows that in the interval of radii a (31) QD
for QDs with an average radius a 3 = 18.4 nm, the maximum values of the dipole
transition moment are D
1,3
1,2 (a 3 ) = 15.8 D 0 electron and oscillator strengths of the
transition f
1,3
1,2 (a 3 ) = 0.56 electron between the ground (n = 1, l = 2) and (n = 1, l
= 3) SIE states (taking into account the values j 2 = 2.5, j 3 = 2.62,L
1,3
1,2 (a 3 ) = 0.376
taken from [5]). The polarizability (14) of the QDs with a radius a 3 , caused by such
a transition, takes the value A
(a 3 ) = 2.5 . 10−19 cm
3 .
Quantitative values of the dipole moment of transitions D
1,l+1
1,l (a) (28), (30), (32),
oscillator strengths of transitions f
1,l+1
1,l
(a) (12) and polarizabilities A
(a) (14) are
listed in the table.
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given in (31), using (26), we obtain the expression for the transition dipole moment
D
1,3
1,2 (a) of an electron between the ground (n = 1, l = 2) and (n = 1, l = 3) SIE
states
D
1,3
1,2 (a) = 2
6 3
1/2 7
−1/2 j
7/2
2 j
9/2
3 ( j 2 + j 3 )
−9
(1−K )( j 2 + j 3 )
3 + 18( j 2 + j 3 )
2 + 126( j 2 + j 3 ) + 136
(2 j
2
2 + 10 j 2 + 15)
−1/2 ( j
2
3 + 28 j 3 + 56)
−1/2 ea
(32)
5 Calculation Results and Discussion
Using formulas (28) and (12), in the interval of radii a (27) QD for QDs with an
average radius a 1 = 8.4 nm, we obtain the highest values of the transition dipole
moment D
1,1
1,0 (a 1 ) = 14.9 D 0 (where D 0 = e
◦
A in Debye units) of an electron and
oscillator strength of the transition f
1,1
1,0 (a 1 ) = 0.47 electrons between the ground (n
= 1, l = 0) and (n = 1, l = 1) by SIE states (taking into account the values j 0 = 2.04,
j 1 = 2.18, L
1,1
1,0 (a 1 ) = 0.355 taken from [5], and a
2D
ex = 2.6 nm (8)). In this case, the
polarizability (14) of QDs with a radius a 1 , caused by such a transition, takes the
value A
(a 1 ) = 2.2 . 10
−19 cm
3 .
In the interval of radii a (29) QD for QDs with an average radius of a 2 = 13.1 nm,
using (30) and (12), we find the largest values of the dipole transition moment D
1,2
1,1 (a 2 )
= 15.2 D 0 electrons and oscillator strengths of the transition f
1,2
1,1 (a 2 ) = 0.50 electrons
between the ground (n = 1, l = 1) and (n = 1, l = 2) SIE states (taking into account
the values j 1 = 2.24, j 2 = 2.36, L
1,2
1,1 (a 2 ) = 0.363 taken from [5]). Due to this
transition, the polarizability (14) of QDs with radius a 2 takes the value A
(a 2 ) = 2.3
. 10
−19 cm
3 .
From formulas (32) and (12), it follows that in the interval of radii a (31) QD
for QDs with an average radius a 3 = 18.4 nm, the maximum values of the dipole
transition moment are D
1,3
1,2 (a 3 ) = 15.8 D 0 electron and oscillator strengths of the
transition f
1,3
1,2 (a 3 ) = 0.56 electron between the ground (n = 1, l = 2) and (n = 1, l
= 3) SIE states (taking into account the values j 2 = 2.5, j 3 = 2.62,L
1,3
1,2 (a 3 ) = 0.376
taken from [5]). The polarizability (14) of the QDs with a radius a 3 , caused by such
a transition, takes the value A
(a 3 ) = 2.5 . 10−19 cm
3 .
Quantitative values of the dipole moment of transitions D
1,l+1
1,l (a) (28), (30), (32),
oscillator strengths of transitions f
1,l+1
1,l
(a) (12) and polarizabilities A
(a) (14) are
listed in the table.