An intrinsic Si sample is doped with donors from one side...

An intrinsic Si sample is doped with donors from one side such that $N_{d}=N_{0} \exp (-a x) .$ (a) Find an expression for the built-in electric field at equilibrium over the range for which $N_{d} \gg n_{i}$. (b) Evaluate the field when $a=1(\mu \mathrm{m})^{-1}$. (c) Sketch a band diagram such as in Fig. 4-15 and indicate the direction of the field.

Key Concepts

Equilibrium in Doped Semiconductors

Thermal equilibrium in a semiconductor requires that all net currents vanish, which implies that the diffusion of carriers due to concentration gradients is exactly balanced by drift currents induced by the electric field. This balance ensures a static distribution of carriers and establishes the built-in electric field. Understanding equilibrium conditions is critical for analyzing device behavior and for estimating electrical properties in devices with non-uniform doping.

Built-in Electric Field

A built-in electric field refers to an internal electric field established within a semiconductor material under thermal equilibrium due to spatial variations in doping or heterojunctions. This internal field arises to balance the diffusion of charge carriers, ensuring that there is no net current flow in the absence of an external voltage. The built-in field is a central concept in devices like p–n junctions and heterostructures, influencing charge carrier distribution and device behavior.

Doping Gradient

A doping gradient describes a spatially varying concentration of dopants within a semiconductor. Such gradients create non-uniform charge carrier densities that lead to localized regions of net charge. This non-uniformity is fundamental to establishing a built-in electric field because the difference in dopant concentration drives carrier diffusion, ultimately leading to equilibrium characterized by an internal electric potential variation.

Poisson’s Equation in Semiconductors

Poisson’s equation is a fundamental relation that links the spatial distribution of electric potential within a semiconductor to the local charge density. In the context of doped semiconductors, it is used to relate the built-in electric field to the distribution of ionized impurities and free carriers. By solving Poisson’s equation with an appropriate charge density (which, in many cases, is dominated by the ionized dopants), one can obtain expressions for the electric field and potential across the semiconductor.

Energy Band Diagrams

Energy band diagrams graphically represent the spatial variation of the semiconductor’s conduction and valence band edges. These diagrams are crucial for visualizing how built-in electric fields influence the energy levels, leading to band bending. This band bending is a direct consequence of the internal electric field and the doping profile, and it plays a central role in determining carrier dynamics, junction behaviors, and overall device performance.

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