14.4 Quantum Dots
413
Fig. 14.22 a Theoretical transition energies of semiconducting (filled symbols) and metallic (open symbols) carbon
nanotubes as a function of tube diameter (Kataura plot). Energies are calculated from van-Hove singularities in the JDOS
within the third-order tight-binding approximation [1252]. b Experimental Kataura plot for the first two semiconducting
(S, closed symbols) and the first metallic (M, open symbols) transition. Dashed lines connect the (near-to) armchair tubes;
full lines connect tubes in a branch, ν = (n 1 − n 2 ) mod 3. Data from photoluminescence [1375] and resonant Raman
scattering [1376]. Adapted from [1377]
Fig. 14.23 Band structure and density of states (DOS) of C(3,3) and BN(3,3) nanotubes, calculated with DFT-LDA.
Adapted from [1381]
14.4 Quantum Dots
14.4.1 Quantization in Three-Dimensional Potential Wells
The solutions for the d-dimensional (d = 1, 2, or 3) harmonic oscillator, i.e. the eigenenergies for the
Hamiltonian
ˆ
H =
p
2
2m
+
d
i=1
1
2
m ω
2
0 x
2
i
(14.6)
are given by
E n =
n +
d
2
ω 0 ,
(14.7)
with n = 0, 1, 2, . . .. More detailed treatments can be found in quantum-mechanics textbooks.
Next, we discuss the problem of a particle in a centrosymmetric finite potential well with different
masses m 1 in the dot and m 2 in the barrier. The Hamiltonian and the potential are given by
413
Fig. 14.22 a Theoretical transition energies of semiconducting (filled symbols) and metallic (open symbols) carbon
nanotubes as a function of tube diameter (Kataura plot). Energies are calculated from van-Hove singularities in the JDOS
within the third-order tight-binding approximation [1252]. b Experimental Kataura plot for the first two semiconducting
(S, closed symbols) and the first metallic (M, open symbols) transition. Dashed lines connect the (near-to) armchair tubes;
full lines connect tubes in a branch, ν = (n 1 − n 2 ) mod 3. Data from photoluminescence [1375] and resonant Raman
scattering [1376]. Adapted from [1377]
Fig. 14.23 Band structure and density of states (DOS) of C(3,3) and BN(3,3) nanotubes, calculated with DFT-LDA.
Adapted from [1381]
14.4 Quantum Dots
14.4.1 Quantization in Three-Dimensional Potential Wells
The solutions for the d-dimensional (d = 1, 2, or 3) harmonic oscillator, i.e. the eigenenergies for the
Hamiltonian
ˆ
H =
p
2
2m
+
d
i=1
1
2
m ω
2
0 x
2
i
(14.6)
are given by
E n =
n +
d
2
ω 0 ,
(14.7)
with n = 0, 1, 2, . . .. More detailed treatments can be found in quantum-mechanics textbooks.
Next, we discuss the problem of a particle in a centrosymmetric finite potential well with different
masses m 1 in the dot and m 2 in the barrier. The Hamiltonian and the potential are given by