3.3 Band Structure Model

Figure 3.4: Equi-energy surfaces of the full band structure of silicon within one octant of the first Brillouin zone.

Generally, the full band structure of semiconductors incorporates an anisotropic dispersion relation $\ensuremath{\mathcal{E}}(\ensuremath{\ensuremath{\mathitbf{k}}})$ , which is only accessible by numerical methods. Fig. 3.4 illustrates iso-energy surfaces of the first conduction band of silicon within one octant of the Brillouin zone, where the lowest energy minima are located close to the $X$ -points. Around these energy minima valleys are formed, whose shape deviates strongly from the conventionally used elliptic approximation with increasing energies. Beside the valleys at the $X$ -points, further valleys are located at the $L$ -points, whose energy minima are higher than that of the $X$ -valleys. However, in order to describe the transport in the semiconductor in a closed analytical way, simplified expressions for the rather complicated full-band structure are commonly introduced. In the following derivations, isotropic bands are assumed, which imply the dispersion relation to depend only on the magnitude of the wave vector $\ensuremath{\ensuremath{\mathitbf{k}}}$ . For the sake of convenience, the dispersion relation is separated into its parabolic and non-parabolic contributions [76]

$$\ensuremath{\mathcal{E}}(k) H_\ensuremath{\mathcal{E}}(\ensuremath{\mathcal{E}}) = \frac{\hbar^2 k^2}{2 m^*}$$ (3.12)

with the non-parabolicity function $H_\ensuremath{\mathcal{E}}$ and the isotropic carrier mass $m^*$ . The simplest band structure model is the parabolic one, which is obtained for $H_\ensuremath{\mathcal{E}}=1$ as

$$\ensuremath{\mathcal{E}}(k) = \frac{\hbar^2 k^2}{2 m^*} \,.$$ (3.13)

However, the validity of the parabolic band model is restricted to low carrier energies. In order to address higher energies, the non-parabolicity of the band structure has to be taken into account by appropriate expressions for $H_\ensuremath{\mathcal{E}}$ . Kane proposed a first-order correction [77] in the form

$$\ensuremath{\mathcal{E}}(k) \left( 1 + \ensuremath{\gamma}\ensuremath{\mathcal{E}}(k) \right) = \frac{\hbar^2 k^2}{2 m^*} \,,$$ (3.14)

whose validity is limited to energies below $1\,\ensuremath{\mathrm{eV}}$ in silicon [73]. In order to accurately describe the dispersion relation for even higher energies, more sophistic models can be applied, such as tabulated data for $H_\ensuremath{\mathcal{E}}$ obtained from numerical band structure calculations. However, the carriers in thermoelectric devices as investigated in this work are not driven far from equilibrium and thus the band structure is described by the parabolic expression (3.13) in the following derivations of transport models.

M. Wagner: Simulation of Thermoelectric Devices