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2.1.2 Quantum Confinement Energies
With decreasing island size the energy level spacing of electron states
increases indirectly proportional to the square of the dot size. Taking an
infinite
potential well as a simple model for a quantum
dot, one calculates by solving Schrödinger's equation
(see Appendix C.1) the
energy levels to
Because of the lower effective mass in Si compared to
Al, the quantum
confinement energy is bigger in Si (see Fig. 2.3).
Figure 2.3:
Comparison of the quantum confinement energy in Si and Al.

For very small particle diameters (around 1 nm) the formulas
(2.6), (2.7), and
(2.8) cease to be valid. The concept
of the effective mass is based on a periodic lattice and starts to break down
for small islands. One would have to apply cluster theory to calculate
the energy levels more precisely. In reality slightly smaller confinement
energies than predicted in (2.8) are observed, which may
partly be attributed to the model of an infinite well. The energy levels
slightly decrease in the case of a finite well
(see Appendix C.2). Furthermore, the lattice
constant for metals is about 0.4 nm whereas for semiconductors it is almost
0.6 nm. Thus the concept of carrier densities is not valid anymore.
Room temperature operation is achievable with structures smaller than
10 nm. Cooling singleelectron devices with liquid nitrogen reduces thermal
fluctuations to some extent, but relaxes the size constrains only by a factor
of two.
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Christoph Wasshuber