414
14 Nanostructures
ˆ
H = ∇
2
2m
∇ + V (r )
(14.8)
V (r ) =
−V 0 , r ≤ R 0
0 , r > R 0
.
(14.9)
The wavefunction can be separated into radial and angular components (r) = R nlm (r ) Y lm (θ, φ),
where Y lm are the spherical harmonic functions. For the ground state (n = 1) the angular momentum
l is zero and the solution for the wavefunction (being regular at r = 0) is given by
R(r ) =
sin(k r)
k r
, r ≤ R 0
sin(k R 0 )
k R 0
exp (−κ (r − R 0 )) , r > R 0
(14.10a)
k
2
=
2 m 1 (V 0 + E)
2
(14.10b)
κ
2
= −
2 m 2 E
2 .
(14.10c)
From the boundary conditions that both R(r ) and
1
m
∂ R(r )
∂r
are continuous across the interface at
r = R 0 , the transcendental equation
k R 0 cot (k R 0 ) = 1 −
m 1
m 2
(1 + κ R 0 )
(14.11)
is obtained. From this formula the energy of the single particle ground state in a spherical quantum dot
can be determined. For a given radius, the potential needs a certain strength V 0,min to confine at least
one bound state; this condition can be written as
V 0,min =
π
2
2
8 m ∗ R
2
0
(14.12)
for m 1 = m 2 = m
∗ . For a general angular momentum l, the wavefunctions are given by spherical
Bessel functions j l in the dot and spherical Hankel functions h l in the barrier. Also, the transcendental
equation for the energy of the first excited level can be given:
k R 0 cot (k R 0 ) = 1 +
k
2 R
2
0
m 1
m 2
2+2κ R 0 +κ 2 R
2
0
1+κ R 0
− 2
.
(14.13)
In the case of infinite barriers (V 0 → ∞), the wavefunction vanishes outside the dot and is given by
(normalized)
R nml (r ) =
2
R
3
0
j l (k nl r )
j l+1 (k nl R 0 )
,
(14.14)
where k nl is the n-th zero of the Bessel function j l , e.g. k n0 = nπ . With two-digit precision the lowest
levels are determined by
k nl
l = 0 l = 1 l = 2 l = 3 l = 4 l = 5
n = 0 3.14 4.49 5.76 6.99
8.18 9.36
n = 1 6.28 7.73 9.10 10.42
n = 2 9.42
14 Nanostructures
ˆ
H = ∇
2
2m
∇ + V (r )
(14.8)
V (r ) =
−V 0 , r ≤ R 0
0 , r > R 0
.
(14.9)
The wavefunction can be separated into radial and angular components (r) = R nlm (r ) Y lm (θ, φ),
where Y lm are the spherical harmonic functions. For the ground state (n = 1) the angular momentum
l is zero and the solution for the wavefunction (being regular at r = 0) is given by
R(r ) =
sin(k r)
k r
, r ≤ R 0
sin(k R 0 )
k R 0
exp (−κ (r − R 0 )) , r > R 0
(14.10a)
k
2
=
2 m 1 (V 0 + E)
2
(14.10b)
κ
2
= −
2 m 2 E
2 .
(14.10c)
From the boundary conditions that both R(r ) and
1
m
∂ R(r )
∂r
are continuous across the interface at
r = R 0 , the transcendental equation
k R 0 cot (k R 0 ) = 1 −
m 1
m 2
(1 + κ R 0 )
(14.11)
is obtained. From this formula the energy of the single particle ground state in a spherical quantum dot
can be determined. For a given radius, the potential needs a certain strength V 0,min to confine at least
one bound state; this condition can be written as
V 0,min =
π
2
2
8 m ∗ R
2
0
(14.12)
for m 1 = m 2 = m
∗ . For a general angular momentum l, the wavefunctions are given by spherical
Bessel functions j l in the dot and spherical Hankel functions h l in the barrier. Also, the transcendental
equation for the energy of the first excited level can be given:
k R 0 cot (k R 0 ) = 1 +
k
2 R
2
0
m 1
m 2
2+2κ R 0 +κ 2 R
2
0
1+κ R 0
− 2
.
(14.13)
In the case of infinite barriers (V 0 → ∞), the wavefunction vanishes outside the dot and is given by
(normalized)
R nml (r ) =
2
R
3
0
j l (k nl r )
j l+1 (k nl R 0 )
,
(14.14)
where k nl is the n-th zero of the Bessel function j l , e.g. k n0 = nπ . With two-digit precision the lowest
levels are determined by
k nl
l = 0 l = 1 l = 2 l = 3 l = 4 l = 5
n = 0 3.14 4.49 5.76 6.99
8.18 9.36
n = 1 6.28 7.73 9.10 10.42
n = 2 9.42