We investigate the charge flow in an electrical field for a non degenerated semiconductor. Combining the equations (5.18) and (5.21) we get
We will neglect gradients in the temperature, so \((E - \mu) \vec{\nabla}_r\ln(T)\) vanishes. Since the semiconductor is not degenerated we can simplify the Fermi statistics by the Boltzmann statistics, i.e.
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| \begin{equation*} \frac{\partial f_0}{\partial E}= - \frac{f_0}{k T} \qquad . \end{equation*} | (5.25) |
We take the band energies of the free electron gas
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| \begin{equation*} E = E_0 + \frac{m^*}{2}\left|\vec{v}(\vec{k}) \right|^2 \; \mbox{, so} \quad v_i = \frac{\hbar}{m^*}k_i \quad . \end{equation*} | (5.26) |
In addition we assume \(\tau\) to be independent of \(\vec{k}\). Summing up all approximations we find for the particle current
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| \begin{equation*} \vec{j}= - \frac{q \tau \hbar^2}{4 \pi^3 k T(m^*)^2} \left[e\vec{E}-\vec{\nabla}\mu \right] \tilde{M} \qquad , \end{equation*} | (5.27) |
and \(\tilde{M}\) is a matrix with the components
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| \begin{equation*} \tilde{M}_{ij}=\int_{V_k} k_i k_j f_0(E(k))d^3k = \int_k dk k^4 f_0(E(k)) \int_{S_k}d\Omega\frac{k_i k_j}{k^2} \qquad . \end{equation*} | (5.28) |
Using \(k^2 = k_x^2 + k_y^2 + k_z^2\) and integrating over the surface of a sphere we get
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| \begin{equation*} \int_{S_k}d\Omega\frac{k_i k_j}{k^2} = \frac{4 \pi}{3} \delta_{ij} \qquad . \end{equation*} | (5.29) |
Using
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| \begin{equation*} dE = \frac{\hbar^2}{m^*} k dk \qquad , \end{equation*} | (5.30) |
the current density is written as
After partial integration we get
Taking into account the density of state of free electrons
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| \begin{equation*} D(E) = \frac{1}{2\pi^2} \left(\frac{2m^*}{\hbar^2} \right)^{\frac{3}{2}} (E-E_0)^{\frac{1}{2}} \qquad , \end{equation*} | (5.33) |
we finally get
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© J. Carstensen (Stat. Meth.)