618
S. I. Pokutnyi
β = (ε 2 − ε 1 )/(ε 1 + ε 2 )—the nanosystem parameter, E g is the bandgap energy of
the silicon matrix.
To estimate the energy spectrum of the ground state of an exciton (n = 1, l),
a variational method was used. The average value of the Hamiltonian (1) SIE on
hydrogen-like wave functions is obtained in [5]:
E 1,l (a, j l (a)) = R 1,l (r )|H (r, a)|R 1,l (r ),
(2)
where
R 1,l (r ) = A l (r − a)
l ex p(− j l (r − a)/a),
A
2
1,l = (2 j l /a)
3+2l
(4(2l)! j
2
l + 4(2l + 1)! j l + (2l + 2)!)
−1
,
(3)
are the hydrogen-like radial wave functions, j l (a)—variational parameter. From the
results of a variational calculation of the energy E 1,l (a) (2) of the ground state of the
SIE follows, that in contrast to excitons in a bulk material when the radius of QD a
increased, then starting from the size
a
∗
c (1, l) ≤ a ≤ a c (1, l)
(4)
the band of exciton quasi-stationary states appears above the bottom of the conduction
band of the silicon matrix. With a further increase of QD radius
a > a c (1, l) > a
∗
c (1, l),
(5)
the quasi-stationary states become the stationary states in the bandgap of the silicon
matrix [5]. The critical QD radii for these states (n = 1, l ≤ 3) have corresponding
values [5]: a
∗
c (1, l) = 8.04 nm; 11.1 nm; 15.5 nm; a c (1, l) = 6.54 nm; 8.35 nm;
11.95 nm; 17.34 nm.
The results have shown that the upper boundary of the quasi-stationary
states band E max
1,l
(a) takes the maximum value for (n = 1, l = 3), and
E
max
1,l=3
a = a
∗
c (1, l = 3)
∼ = 248.4 meV. In the interval of QD radii a
a ≥ ˜
a c (1, l) = 20.8 nm
(6)
given in (6), exciton stationary states E 1,l (a) (2) approach to the states of 2D SIE
localized close to a flat interface (germanium–silicon) [5]. The energy of this exciton
[5] is equal to:
E 1,l (a) = −E
2D
ex , E
2D
ex = 2
2
/μ ex (a
2D
ex )
2
(7)
where E
2D
ex = 82 meV is the binding energy of 2D SIE. The 2D SIE Bohr radius is
defined as follows:
S. I. Pokutnyi
β = (ε 2 − ε 1 )/(ε 1 + ε 2 )—the nanosystem parameter, E g is the bandgap energy of
the silicon matrix.
To estimate the energy spectrum of the ground state of an exciton (n = 1, l),
a variational method was used. The average value of the Hamiltonian (1) SIE on
hydrogen-like wave functions is obtained in [5]:
E 1,l (a, j l (a)) = R 1,l (r )|H (r, a)|R 1,l (r ),
(2)
where
R 1,l (r ) = A l (r − a)
l ex p(− j l (r − a)/a),
A
2
1,l = (2 j l /a)
3+2l
(4(2l)! j
2
l + 4(2l + 1)! j l + (2l + 2)!)
−1
,
(3)
are the hydrogen-like radial wave functions, j l (a)—variational parameter. From the
results of a variational calculation of the energy E 1,l (a) (2) of the ground state of the
SIE follows, that in contrast to excitons in a bulk material when the radius of QD a
increased, then starting from the size
a
∗
c (1, l) ≤ a ≤ a c (1, l)
(4)
the band of exciton quasi-stationary states appears above the bottom of the conduction
band of the silicon matrix. With a further increase of QD radius
a > a c (1, l) > a
∗
c (1, l),
(5)
the quasi-stationary states become the stationary states in the bandgap of the silicon
matrix [5]. The critical QD radii for these states (n = 1, l ≤ 3) have corresponding
values [5]: a
∗
c (1, l) = 8.04 nm; 11.1 nm; 15.5 nm; a c (1, l) = 6.54 nm; 8.35 nm;
11.95 nm; 17.34 nm.
The results have shown that the upper boundary of the quasi-stationary
states band E max
1,l
(a) takes the maximum value for (n = 1, l = 3), and
E
max
1,l=3
a = a
∗
c (1, l = 3)
∼ = 248.4 meV. In the interval of QD radii a
a ≥ ˜
a c (1, l) = 20.8 nm
(6)
given in (6), exciton stationary states E 1,l (a) (2) approach to the states of 2D SIE
localized close to a flat interface (germanium–silicon) [5]. The energy of this exciton
[5] is equal to:
E 1,l (a) = −E
2D
ex , E
2D
ex = 2
2
/μ ex (a
2D
ex )
2
(7)
where E
2D
ex = 82 meV is the binding energy of 2D SIE. The 2D SIE Bohr radius is
defined as follows: