620
S. I. Pokutnyi
In the weak optical field, the main contribution to the polarizability A
(ω, a) (9)
of QD is made by single transitions in the discrete spectrum of SIE states. Separating
in polarizability (9), the contribution from only one resonant term corresponding to
the transition between the ground (n = 1, l) and (n = 1, l + 1) SIE states, we can
write polarizability A
(ω, a) (9) of the QD in the form [30]
A
(ω, a) = f
1,l+1
1,l
(a)
e 2
μ ex
ω 2
1,l+1 (a) − ω 2
(ω 2
1,l+1 (a) − ω 2 ) 2 +
ωΓ 1,l+1
2 + i
ωΓ 1,l+1
(ω 2
1,l+1 (a) − ω 2 ) 2 +
ωΓ 1,l+1
2
(13)
The restriction of these states does not affect the clarification of the behavior of
the polarizability A
(ω, a) (9) of the nanosystem depending on the radius a QD
when taking into account other states and for other resonant frequencies, since, as
follows from (10)–(12), the dependence of the oscillator strengths of the f
1,l+1
1,l
(a)
transitions between such states on the radius a QD is general.
Using (13), we obtain the expression for polarizability, provided that frequencies
of the external field waves ω is lower and far from the resonant exciton state (n = 1,
l + 1) frequency ω 1,l+1 (a) (i.e. ω
2 <<
ω 1,l+1 (a)
2 :
A
(a) ≈ (˜ ε
4)
E
2D
ex
E 1,l+1
2 f
1,l+1
1,l
(a)
a
2D
ex
3
(14)
For frequencies close to resonant ω ≈ ω 1,l+1 (a), the real part of polarizability is
small compared to the imaginary part and can be neglected:
A
2 (a) ≈ i(˜ ε/4)
E
2D
ex
2 /
E 1,l+1 Γ 1,l+1
f
1,l+1
1,l
(a)
a
2D
ex
3
(15)
In the case ω ≈ ω 1,l+1 , the real part of polarizability is negative and decreases
with frequency increasing more slowly than the imaginary part:
A
3 (a) ≈ −(˜ ε/4)
E
2D
ex /E ω
2 f
1,l+1
1,l
(a)
a
2D
ex
3
(16)
where E ω = ω.
The ratios of the polarizability absolute values are equal to:
K 1 = A
1 (a)
A
2 (a) = hΓ 1,l+1
E 1,l+1 1
(17)
K 2 = A
3 (a)/A
2 (a) = hΓ 1,l+1 E 1,l+1 /(E ω )
2
1
(18)
K 3 = A
3 (a)/A
1 (a) =
E 1,l+1 /E ω
2 1
(19)
The expressions (14–16) are applicable to an ensemble of noninteracting QD, i.e.,
to QD concentration N:
S. I. Pokutnyi
In the weak optical field, the main contribution to the polarizability A
(ω, a) (9)
of QD is made by single transitions in the discrete spectrum of SIE states. Separating
in polarizability (9), the contribution from only one resonant term corresponding to
the transition between the ground (n = 1, l) and (n = 1, l + 1) SIE states, we can
write polarizability A
(ω, a) (9) of the QD in the form [30]
A
(ω, a) = f
1,l+1
1,l
(a)
e 2
μ ex
ω 2
1,l+1 (a) − ω 2
(ω 2
1,l+1 (a) − ω 2 ) 2 +
ωΓ 1,l+1
2 + i
ωΓ 1,l+1
(ω 2
1,l+1 (a) − ω 2 ) 2 +
ωΓ 1,l+1
2
(13)
The restriction of these states does not affect the clarification of the behavior of
the polarizability A
(ω, a) (9) of the nanosystem depending on the radius a QD
when taking into account other states and for other resonant frequencies, since, as
follows from (10)–(12), the dependence of the oscillator strengths of the f
1,l+1
1,l
(a)
transitions between such states on the radius a QD is general.
Using (13), we obtain the expression for polarizability, provided that frequencies
of the external field waves ω is lower and far from the resonant exciton state (n = 1,
l + 1) frequency ω 1,l+1 (a) (i.e. ω
2 <<
ω 1,l+1 (a)
2 :
A
(a) ≈ (˜ ε
4)
E
2D
ex
E 1,l+1
2 f
1,l+1
1,l
(a)
a
2D
ex
3
(14)
For frequencies close to resonant ω ≈ ω 1,l+1 (a), the real part of polarizability is
small compared to the imaginary part and can be neglected:
A
2 (a) ≈ i(˜ ε/4)
E
2D
ex
2 /
E 1,l+1 Γ 1,l+1
f
1,l+1
1,l
(a)
a
2D
ex
3
(15)
In the case ω ≈ ω 1,l+1 , the real part of polarizability is negative and decreases
with frequency increasing more slowly than the imaginary part:
A
3 (a) ≈ −(˜ ε/4)
E
2D
ex /E ω
2 f
1,l+1
1,l
(a)
a
2D
ex
3
(16)
where E ω = ω.
The ratios of the polarizability absolute values are equal to:
K 1 = A
1 (a)
A
2 (a) = hΓ 1,l+1
E 1,l+1 1
(17)
K 2 = A
3 (a)/A
2 (a) = hΓ 1,l+1 E 1,l+1 /(E ω )
2
1
(18)
K 3 = A
3 (a)/A
1 (a) =
E 1,l+1 /E ω
2 1
(19)
The expressions (14–16) are applicable to an ensemble of noninteracting QD, i.e.,
to QD concentration N: