Current flow through a non degenerated semiconductor

5.4 Current flow through a non degenerated semiconductor

We investigate the charge flow in an electrical field for a non degenerated semiconductor. Combining the equations (5.18) and (5.21) we get

$\vec{j} = - \frac{q}{4 π^{3}} \int_{V_{k}} τ (\vec{k}) \vec{v} (\vec{k}) \frac{\partial f_{0}}{\partial E} ⟨ (q \vec{E} - {\vec{\nabla}}_{r} μ - (E - μ) {\vec{\nabla}}_{r} ln (T)), \vec{v} ⟩ d^{3} k .$

(5.24)

We will neglect gradients in the temperature, so $(E - μ) {\vec{\nabla}}_{r} ln (T)$ vanishes. Since the semiconductor is not degenerated we can simplify the Fermi statistics by the Boltzmann statistics, i.e.

$\frac{\partial f_{0}}{\partial E} = - \frac{f_{0}}{k T} .$

(5.25)

We take the band energies of the free electron gas

$E = E_{0} + \frac{m^{*}}{2} {| \vec{v} (\vec{k}) |}^{2}, so v_{i} = \frac{ℏ}{m^{*}} k_{i} .$

(5.26)

In addition we assume $τ$ to be independent of $\vec{k}$ . Summing up all approximations we find for the particle current

$\vec{j} = - \frac{q τ ℏ^{2}}{4 π^{3} k T {(m^{*})}^{2}} [e \vec{E} - \vec{\nabla} μ] \tilde{M},$

(5.27)

and $\tilde{M}$ is a matrix with the components

${\tilde{M}}_{i j} = \int_{V_{k}} k_{i} k_{j} f_{0} (E (k)) d^{3} k = \int_{k} d k k^{4} f_{0} (E (k)) \int_{S_{k}} d Ω \frac{k_{i} k_{j}}{k^{2}} .$

(5.28)

Using $k^{2} = k_{x}^{2} + k_{y}^{2} + k_{z}^{2}$ and integrating over the surface of a sphere we get

$\int_{S_{k}} d Ω \frac{k_{i} k_{j}}{k^{2}} = \frac{4 π}{3} δ_{i j} .$

(5.29)

Using

$d E = \frac{ℏ^{2}}{m^{*}} k d k,$

(5.30)

the current density is written as

$\vec{j} = - \frac{q τ}{3 π^{2} k T m^{*}} [q \vec{E} - \vec{\nabla} μ] (\frac{2 m^{*}}{ℏ^{2}}) \int_{E_{0}}^{\infty} d E {(E - E_{0})}^{\frac{3}{2}} exp (- \frac{E - μ}{k T}) .$

(5.31)

After partial integration we get

$\vec{j} = - \frac{q τ}{3 π^{2} k T m^{*}} [q \vec{E} - \vec{\nabla} μ] (\frac{2 m^{*}}{ℏ^{2}}) \frac{- 3 k T}{2} \int_{E_{0}}^{\infty} d E {(E - E_{0})}^{\frac{1}{2}} exp (- \frac{E - μ}{k T}) .$

(5.32)

Taking into account the density of state of free electrons

$D (E) = \frac{1}{2 π^{2}} {(\frac{2 m^{*}}{ℏ^{2}})}^{\frac{3}{2}} {(E - E_{0})}^{\frac{1}{2}},$

(5.33)

we finally get

$\vec{j} = \frac{q τ}{m^{*}} [q \vec{E} - \vec{\nabla} μ] \int_{E_{0}}^{\infty} d E D (E) f_{0} (E) = \frac{q τ n_{e}}{m^{*}} [q \vec{E} - \vec{\nabla} μ] .$

(5.34)

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