3.5.1 Microscopic and Macroscopic Quantities

While microscopic quantities represent a certain state in $(\ensuremath{\ensuremath{\mathitbf{r}}},\ensuremath{\ensuremath{\mathitbf{k}}})$ -space, their macroscopic counterparts are averages over $\ensuremath{\ensuremath{\mathitbf{k}}}$ -space. As a consequence, their dependency restricts to $\ensuremath {\ensuremath {\mathitbf {r}}}$ -space. Macroscopic quantities are obtained by the integration of the according microscopic quantity multiplied by the distribution function $\ensuremath{f}$ . The spin degeneracy is implied by a factor of two, a further factor of $1/(2\pi)$ per degree of freedom results from the transition from discrete states to a continuum distribution function. Thus, a general macroscopic density $\ensuremath{x}$ reads from its microscopic, scalar-valued counterpart $\ensuremath{X}$ and the distribution function $\ensuremath{f}$

$$\ensuremath{x}(\ensuremath{\ensuremath{\mathitbf{r}}}) = \frac{2}... ...\mathrm{d}}k_x \, \ensuremath{\,\mathrm{d}}k_y \, \ensuremath{\,\mathrm{d}}k_z}$$ (3.19)

$$= \frac{2}{\left( 2 \pi \right)^3} \ensuremath{\iiint\limits_{-\i... ...\mathrm{d}}k_x \, \ensuremath{\,\mathrm{d}}k_y \, \ensuremath{\,\mathrm{d}}k_z}$$

$$= \ensuremath{\nu}(\ensuremath{\ensuremath{\mathitbf{r}}}) \ensur... ...ath{\ensuremath{\mathitbf{r}}},\ensuremath{\ensuremath{\mathitbf{k}}}) \rangle}$$

$$= \ensuremath{\langle \! \langle \ensuremath{X}(\ensuremath{\ensuremath{\mathitbf{r}}},\ensuremath{\ensuremath{\mathitbf{k}}}) \rangle \! \rangle}$$

with $\ensuremath{F}$ as the normalized distribution function and $\ensuremath{\nu}$ the carrier density. The short forms $\ensuremath{\langle \cdot \rangle}$ and $\ensuremath{\langle \! \langle \cdot \rangle \! \rangle}$ denote the normalized statistic average and the statistic average, respectively. Analogously, macroscopic current densities are defined from vector-valued microscopic quantities $\ensuremath{\ensuremath{\mathitbf{X}}}$ and the distribution function $\ensuremath{f}$

$$\ensuremath{\ensuremath{\mathitbf{j}}_\ensuremath{x}}(\ensuremath... ...\mathrm{d}}k_x \, \ensuremath{\,\mathrm{d}}k_y \, \ensuremath{\,\mathrm{d}}k_z}$$ (3.20)

$$= \ensuremath{\nu}(\ensuremath{\ensuremath{\mathitbf{r}}}) \ensur... ...ath{\ensuremath{\mathitbf{r}}},\ensuremath{\ensuremath{\mathitbf{k}}}) \rangle}$$

$$= \ensuremath{\langle \! \langle \ensuremath{\ensuremath{\mathitb... ...{\mathitbf{r}}},\ensuremath{\ensuremath{\mathitbf{k}}}) \rangle \! \rangle} \,.$$

Macroscopic densities occurring in the following derivations are the carrier density $\ensuremath{\nu}$ and the energy density $\ensuremath{w}$ . The corresponding fluxes are a particle flux $\ensuremath{\ensuremath{\mathitbf{j}}_\nu}$ and the energy flux $\ensuremath{\ensuremath{\mathitbf{j}}_\nu^\mathrm{u}}$ , respectively. The formulation used within this work consequently implies the particle flux, which differs from the electric current by the elementary charge. Important microscopic quantities and their macroscopic counterparts are outlined in Table 3.1.

Table 3.1: Some important macroscopic quantities for transport models with their definition from the microscopic counterparts.

Macroscopic quantity	Symbol	Definition
general macroscopic density	$\ensuremath{x}$	$\ensuremath{\langle \! \langle \ensuremath{X} \rangle \! \rangle}$
general macroscopic flux	$\ensuremath{\ensuremath{\mathitbf{j}}_\ensuremath{x}}$	$\ensuremath{\langle \! \langle \ensuremath{\ensuremath{\mathitbf{X}}} \rangle \! \rangle}$
carrier density	$\ensuremath{\nu}$	$\ensuremath{\langle \! \langle 1 \rangle \! \rangle}$
carrier flux density	$\ensuremath{\ensuremath{\mathitbf{j}}_\nu}$	$\ensuremath{\nu}\ensuremath{\langle \ensuremath{\ensuremath{\mathitbf{v}}} \rangle}$
average energy density	$\ensuremath{w}$	$\ensuremath{\langle \ensuremath{\mathcal{E}} \rangle}$
energy flux density	$\ensuremath{\ensuremath{\mathitbf{j}}_\nu^\mathrm{u}}$	$\ensuremath{\langle \ensuremath{\ensuremath{\mathitbf{v}}}\ensuremath{\mathcal{E}} \rangle}$

In the following, important averages are given for a heated, displaced Maxwellian (3.17) and parabolic bands (3.13). The carrier concentration evaluates as

$$\ensuremath{\nu}= \ensuremath{\langle \! \langle 1 \rangle \! \ra... ...u}}}} \,\, \ensuremath{\,\mathrm{d}}^3 \ensuremath{\ensuremath{\mathitbf{k}}}'}$$ (3.21)

$$= \frac{2A}{(2\pi)^3} 4\pi \ensuremath{\int\limits_{0}^{\infty} {... ...math{\mathrm{B}}\ensuremath{T_\ensuremath{\nu}}}{\hbar^2} \right)^{\frac{3}{2}}$$

and is used to normalize further averages in the sequel. In order to derive the average energy $\ensuremath{w}$ , the average of the carrier energy $\ensuremath{\langle \! \langle \ensuremath{\mathcal{E}} \rangle \! \rangle}$ has to be evaluated

$$\ensuremath{w}= \frac{1}{\ensuremath{\nu}} \ensuremath{\langle \!... ...{\nu}}}} \frac{\hbar^2 k^2}{2 m^*} \,\, \mathrm{d}^3 \ensuremath{\mathitbf{k}}}$$ (3.22)

$$= \frac{1}{\ensuremath{\nu}} \frac{2A}{(2\pi)^3} \ensuremath{\int... ... m^*} \,\, \ensuremath{\,\mathrm{d}}^3 \ensuremath{\ensuremath{\mathitbf{k}}}'}$$

$$= \frac{1}{\ensuremath{\nu}} \frac{2A}{(2\pi)^3} \ensuremath{\int... ...) \,\, \ensuremath{\,\mathrm{d}}^3 \ensuremath{\ensuremath{\mathitbf{k}}}'} \,.$$

The second term within the parenthesis vanishes due to the product of an odd and an even term in the integrand. Furthermore, the transformation to polar coordinates yields

$$\ensuremath{w}= \frac{1}{\ensuremath{\nu}} \frac{2A}{(2\pi)^3} 4\... ...{T_\ensuremath{\nu}}}} \frac{\hbar^2 k_0^2}{2 m^*} \ensuremath{\,\mathrm{d}}k'}$$ (3.23)

$$= \frac{1}{\ensuremath{\nu}} \left( \frac{3A}{(2 \pi)^3} k_\ensur... ...thrm{B}}\ensuremath{T_\ensuremath{\nu}}}{\hbar^2} \right)^{\frac{3}{2}} \right)$$

$$= \frac{3}{2} k_\ensuremath{\mathrm{B}}\ensuremath{T_\ensuremath{\nu}}+ \frac{\hbar^2 k_0^2}{2 m^*} \,.$$

The resulting average energy consists of two parts comprising the thermal energy by random movement and the drift energy corresponding to the average carrier movement. For comparison, the average energy is evaluated within the diffusion approximation, whereby the heated Maxwellian is expressed by its first-order Taylor approximation (3.18)

$$\ensuremath{w}= \frac{1}{\ensuremath{\nu}} \ensuremath{\langle \!... ...{\nu}}}} \frac{\hbar^2 k^2}{2 m^*} \,\, \mathrm{d}^3 \ensuremath{\mathitbf{k}}}$$ (3.24)

$$= \frac{1}{\ensuremath{\nu}} \frac{2A}{(2\pi)^3} 4\pi \ensuremath... ...math{\mathrm{B}}\ensuremath{T_\ensuremath{\nu}}}{\hbar^2} \right)^{\frac{3}{2}}$$

$$= \frac{3}{2} k_\ensuremath{\mathrm{B}}\ensuremath{T_\ensuremath{\nu}} \,.$$

In contrast to the full displaced Maxwellian, the first-order approximation leads to a neglection of the drift component in the average energy expression. This result underlines the range of validity of the diffusion approximation. For a slowly drifting, hot carrier gas, the drift term in (3.23) is negligibly small compared to the thermal energy. The formulation of the distribution function approximation (3.18) has been motivated exactly by this assumption.

M. Wagner: Simulation of Thermoelectric Devices