energy, where

where is the permittivity of the surrounding dielectric material and

an indirect proportionality to the size of the particle. Fig. 2.1 to Fig. 2.3 compare this electrostatic energy to the other characteristic energies involved in charging an island with a small number of electrons. For the sake of simplicity, the relative permittivity, , was assumed to be 10. (For relative permittivities of different materials see the Material Properties Table .) A more accurate model for the capacitance of an island is the capacitance of two or more spheres in a line. A calculation for characteristic spatial dimensions shows an increase of about 15% of the total capacitance. See Appendix A for a detailed outline of the calculations, numerical results, and the description of an algorithm to calculate the capacitance of an arbitrary arrangement of spheres. A collection of practical analytical capacitance formulas is given in Table 2.1.

For a system of

where the

where

Systems with sufficiently small islands are not adequately described with this model. They exhibit a second electron-electron interaction energy, namely the change in Fermi energy, when charged with a single electron. One must distinguish between metals and semiconductors, because of their greatly different free carrier concentrations and the presence of a band gap in the case of semiconductors. Metals have typically a carrier concentration of several and undoped Si about at room temperature (see Material Properties Table ). The low intrinsic carrier concentration for semiconductors is in reality not achievable, because of the ubiquitous impurities which are ionized at room temperature. These impurities supply free charge carriers and may lift the concentration, in the case of Si, to around , still many orders of magnitude smaller than for metals. Clearly, doping increases the carrier concentration considerably, and can lead in the case of degenerated semiconductors to metal like behavior. We are considering only undoped semiconductors to emphasize the difference to metals.

The dependence of the Fermi energy *E*_{F} on carrier concentration *n* is
derived
in Appendix B
(see also [7]). For metals it is given by

and for semiconductors it is

where
is the net carrier concentration defined as
.
Fig. 2.1 compares the change in the Fermi energy for the addition
of one, two, and three electrons in Si to the electrostatic charging energy.

The first electron causes the biggest change in Fermi energy. The second and third electron contribute a change smaller than the thermal energy for room temperature. Hence, for single-electron devices one will try to build structures with few free carriers, to achieve a high change in energy when charging an island.

Fig. 2.2 compares Si with Al. It is clearly visible that the change
is much bigger in Si. This is caused by two reasons. First, the difference in
free carrier concentration. In metals much more carriers are available and
therefore
the addition of one electron has not such a big impact anymore. Second in
metals an additional electron finds a place slightly
above the Fermi level. In semiconductors due to the energy gap the electron
needs to be inserted considerably above the Fermi level.