14.4 Quantum Dots
415
87.6%
81.8%
76.2%
74.6%
9.6%
13.7%
17.6%
21.0%
2.3%
3.6%
5.3%
3.0%
V0
V1
V2
V3
V0
V1
V2
V3
V0
V1
V2
V3
V0
V1
V2
V3
C0
C1
C2
C3
Electrons
Holes
(a)
(b)
(c)
(d)
(e)
Fig. 14.24 Isosurface plots (25% of maximum value) of the total probability densities a, b and valence-band projections
(c)–(e) of bound electron (a) and hole (b)–(e) states in a model pyramidal InAs/GaAs quantum dot with base length
b = 11.3 nm. The percentages are the integrals of the projections to the bulk heavy, light and split-off hole bands,
respectively, and the isosurfaces show the corresponding projection shapes. For each valence-band state the difference
from 100% is the integral
∞
−∞ |ψ s↑ | 2 +|ψ s↓ | 2 d 3 r of the s-type (conduction band) Bloch function projection (not shown).
Reprinted with permission from [1385], ©2002, Springer
The (2l+1) degenerate energy levels E nl are (V 0 = ∞, m = m 1 ):
E nl =
2
2m
k
2
nl
1
R
2
0
.
(14.15)
The 1s, 1p, and 1d states have smaller eigenenergies than the 2s state.
A particularly simple solution is given for a cubic quantum dot of side length a 0 and infinite potential
barriers. One finds the levels E n x n y n z :
E n x n y n z =
2
2m
n
2
x + n
2
y + n
2
z
π
2
a
2
0
,
(14.16)
with n x , n y , n z = 1, 2, . . . . For a sphere, the separation between the ground and first excited state is
E 1 − E 0 ≈ E 0 , for a cube and a two-dimensional harmonic oscillator it is exactly E 0 . For a threedimensional harmonic oscillator this quantity is E 1 − E 0 = 2E 0 /3.
For realistic quantum dots a full three-dimensional simulation of strain, piezoelectric fields and
the quantum-mechanical confinement must be performed [1382, 1383]. In Fig. 14.24, the lowest four
electron and hole wavefunctions in a pyramidal InAs/GaAs quantum dot (for the strain distribution see
Fig. 5.34 and for the piezoelectric fields see Fig. 16.16) are shown. The figure shows that the lowest
hole states have dominantly heavy-hole character and contain admixtures of the other hole bands. The
wavefunction in such quantum dots can be imaged using scanning tunneling microscopy [1384].
415
87.6%
81.8%
76.2%
74.6%
9.6%
13.7%
17.6%
21.0%
2.3%
3.6%
5.3%
3.0%
V0
V1
V2
V3
V0
V1
V2
V3
V0
V1
V2
V3
V0
V1
V2
V3
C0
C1
C2
C3
Electrons
Holes
(a)
(b)
(c)
(d)
(e)
Fig. 14.24 Isosurface plots (25% of maximum value) of the total probability densities a, b and valence-band projections
(c)–(e) of bound electron (a) and hole (b)–(e) states in a model pyramidal InAs/GaAs quantum dot with base length
b = 11.3 nm. The percentages are the integrals of the projections to the bulk heavy, light and split-off hole bands,
respectively, and the isosurfaces show the corresponding projection shapes. For each valence-band state the difference
from 100% is the integral
∞
−∞ |ψ s↑ | 2 +|ψ s↓ | 2 d 3 r of the s-type (conduction band) Bloch function projection (not shown).
Reprinted with permission from [1385], ©2002, Springer
The (2l+1) degenerate energy levels E nl are (V 0 = ∞, m = m 1 ):
E nl =
2
2m
k
2
nl
1
R
2
0
.
(14.15)
The 1s, 1p, and 1d states have smaller eigenenergies than the 2s state.
A particularly simple solution is given for a cubic quantum dot of side length a 0 and infinite potential
barriers. One finds the levels E n x n y n z :
E n x n y n z =
2
2m
n
2
x + n
2
y + n
2
z
π
2
a
2
0
,
(14.16)
with n x , n y , n z = 1, 2, . . . . For a sphere, the separation between the ground and first excited state is
E 1 − E 0 ≈ E 0 , for a cube and a two-dimensional harmonic oscillator it is exactly E 0 . For a threedimensional harmonic oscillator this quantity is E 1 − E 0 = 2E 0 /3.
For realistic quantum dots a full three-dimensional simulation of strain, piezoelectric fields and
the quantum-mechanical confinement must be performed [1382, 1383]. In Fig. 14.24, the lowest four
electron and hole wavefunctions in a pyramidal InAs/GaAs quantum dot (for the strain distribution see
Fig. 5.34 and for the piezoelectric fields see Fig. 16.16) are shown. The figure shows that the lowest
hole states have dominantly heavy-hole character and contain admixtures of the other hole bands. The
wavefunction in such quantum dots can be imaged using scanning tunneling microscopy [1384].