(a) Assume that the electron mobility in an n-type...

(a) Assume that the electron mobility in an n-type semiconductor is given by $$ \mu_{n}=\frac{1350}{\left(1+\frac{N_{d}}{5 \times 10^{16}}\right)^{1 / 2}} \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s} $$ where $N_{d}$ is the donor concentration in $\mathrm{cm}^{-3}$. Assuming complete ionization, plot the conductivity as a function of $N_{d}$ over the range $10^{15} \leq N_{d} \leq 10^{18} \mathrm{~cm}^{-3} .$ (b) Compare the results of part ( $a$ ) to that if the mobility were assumed to be a constant equal to $1350 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s} .(c)$ If an electric field of $E=10 \mathrm{~V} / \mathrm{cm}$ is applied to the semiconductor, plot the electron drift current density of parts $(a)$ and $(b)$.

(a) Assume that the electron mobility in an n-type semiconductor is given by
$$
\mu_{n}=\frac{1350}{\left(1+\frac{N_{d}}{5 \times 10^{16}}\right)^{1 / 2}} \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}
$$
where $N_{d}$ is the donor concentration in $\mathrm{cm}^{-3}$. Assuming complete ionization, plot the conductivity as a function of $N_{d}$ over the range $10^{15} \leq N_{d} \leq 10^{18} \mathrm{~cm}^{-3} .$ (b) Compare the results of part ( $a$ ) to that if the mobility were assumed to be a constant equal to $1350 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s} .(c)$ If an electric field of $E=10 \mathrm{~V} / \mathrm{cm}$ is applied to the semiconductor, plot the electron drift current density of parts $(a)$ and $(b)$.

Key Concepts

Electron Drift Current Density

Electron drift current density refers to the current resulting from the motion of electrons under the influence of an applied electric field. It can be determined by the product of the number of free electrons, the elementary charge, the mobility, and the applied electric field. This concept is essential for understanding how electrical currents are generated and controlled in semiconductor devices.

Doping and Donor Concentration

Doping involves deliberately introducing impurities into a semiconductor to modify its electrical properties. In n-type semiconductors, donor atoms contribute extra electrons to the conduction band. The donor concentration directly influences the carrier concentration and can also affect mobility through increased scattering, thereby playing a pivotal role in the overall conductivity.

Complete Ionization

The assumption of complete ionization in a doped semiconductor means that every dopant atom donates its extra electron to the conduction band, maximizing the free carrier concentration. This simplification is particularly useful at room temperature where thermal energy is sufficient to ionize most impurity atoms, allowing for a direct relationship between doping level and carrier concentration.

Semiconductor Conductivity

Semiconductor conductivity is a measure of how easily charge carriers (electrons and holes) can move through the material when an electric field is applied. It is determined by both the carrier concentration and the mobility of those carriers. In doped semiconductors, the conductivity varies with the level of impurities introduced, which act to increase the number of free carriers available for conduction.

Electron Mobility

Electron mobility describes how quickly electrons can move through a semiconductor when subjected to an electric field. It is affected by scattering mechanisms, such as impurity scattering and phonon scattering, and can vary with doping concentration. A functional dependence of mobility on donor concentration is common and is critical in determining the performance of semiconductor devices.

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