A recently discovered semiconductor has Nc=10^19 cm^-3, Nv=5 ×10^18 cm^-3, and Eg=2 eV. If it

step 1 In this question we have to find the electron diffusion current density, the whole drift current

A recently discovered semiconductor has Nc=10^19 cm^-3,...

A recently discovered semiconductor has $N_{c}=10^{19} \mathrm{~cm}^{-3}, N_{v}=5 \times 10^{18} \mathrm{~cm}^{-3}$, and $E_{g}=2 \mathrm{eV}$. If it is doped with $10^{17}$ donors (fully ionized), calculate the electron, hole, and intrinsic carrier concentrations at $627^{\circ} \mathrm{C}$. Sketch the simplified band diagram, and specify the value of $E_{F}$ and $E_{i}$ with respect to the band edges. If we apply $5 \mathrm{~V}$ across a piece of this semiconductor $8 \mu \mathrm{m}$ long, what is the current? The piece is $2 \mu \mathrm{m}$ wide and $1.5 \mu \mathrm{m}$ thick. The diffusion coefficient of holes and electrons is $25 \mathrm{~cm}^{2} / \mathrm{s}$ and $75 \mathrm{~cm}^{2} / \mathrm{s}$, respectively.

Key Concepts

Einstein Relation

This relation connects the diffusion coefficient of a carrier (electron or hole) to its mobility via the thermal voltage (kT/q). It is essential in linking the random thermal motion (diffusion) of carriers with their response to an electric field (drift), providing a unified picture of carrier transport in semiconductor materials.

Drift Current

Drift current in a semiconductor occurs due to the motion of charged carriers (electrons and holes) under an applied electric field. It is determined by the carrier concentration, mobility, and the magnitude of the field, and is critical for analyzing the conduction properties of semiconductor devices when a voltage is applied.

Semiconductor Band Diagram

The band diagram is a graphical representation of the energy bands (conduction and valence bands) and the positions of crucial energy levels such as the Fermi level and intrinsic level. It visually summarizes how doping and temperature affect the energy landscape of the semiconductor, making it an important tool for understanding carrier dynamics and device behavior.

Fermi Level and Intrinsic Level

The Fermi level is the chemical potential for electrons and indicates the probability of occupancy of energy states at thermal equilibrium. In intrinsic semiconductors the intrinsic level (E_i) is a midpoint between the conduction and valence bands, but doping shifts the Fermi level towards the conduction or valence band. The relative positions of E_F and E_i provide insight into the type and degree of doping.

Effective Density of States

The effective density of states (N_c for the conduction band and N_v for the valence band) quantifies the number of available states for electrons and holes respectively at the band edges. These values are essential for calculating carrier concentrations, as they take into account the effect of the semiconductor’s band structure and the temperature, linking to the distribution of carriers via the Boltzmann statistics under non-degenerate conditions.

Doping and Ionization

Doping introduces impurities (donors or acceptors) into the semiconductor, deliberately altering carrier concentrations. For example, adding a donor atom provides extra electrons to the conduction band upon full ionization. Doping shifts the Fermi level closer to the conduction band in n-type materials and affects the balance of electron and hole concentrations in accordance with overall charge neutrality.

Mass Action Law

In semiconductors, the mass action law states that under thermal equilibrium, the product of the electron and hole concentrations remains constant and equals the square of the intrinsic carrier concentration (n_i^2). This law is used to link the majority and minority carrier concentrations, especially when the semiconductor is doped.

Intrinsic Carrier Concentration

This concept refers to the number of electron–hole pairs generated in a pure (undoped) semiconductor solely by thermal energy. It is strongly temperature dependent and can be calculated using effective density of states (for both conduction and valence bands) and the bandgap energy via an exponential relationship. The product of electron and hole concentrations in intrinsic material is constant at a given temperature, following the mass?action law.

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