An Exact Analysis for Freeze-out and Exhaustion in Single Impurity Semiconductors

摘要

In this paper, a complete analytical description for an exact expression for temperature dependence of the majority carrier in a single-impurity, equilibrium semiconductor is proposed. Analysis establishes that the problem is solvable exactly by identifying the only physically possible root to a cubic equation. This model provides an attractive alternative to approximate standard classroom approaches for this topic covered in senior and first year graduate level solid state courses in physics and electrical engineering. Integrated circuits (ICs) are specified to operate between designated temperature limits. The circuit designer selects the doping level or levels and typically assumes that the dopants are approximately 100% ionized, i.e., exhaustion of dopant and the temperature is not too high. There can be a significant impact on the values for a plethora of device parameters, such as depletion width or a field effect transistor (FET) threshold voltage, if the assumption is violated. If the temperature is too low, the percentage ionization of dopant or dopants will be significantly less than 100%. This reversal of the high percentage of ionization of the dopant is commonly referred to as freeze-out. Most semiconductor devices and ICs are designed to be operated in exhaustion regime also known as "extrinsic regime" see Figure 1, borrowed from B. Streetman’s classic undergraduate text book, for which the majority carrier concentration is approximately equal to the dopant concentration e.g. N{sub}D=10{sup}15/cm{sup}3. Again, in reference to Fig. 1, if the temperature is too high, the thermal generation effect causes the majority carrier concentration to become excessively higher than the dopant in what is called the intrinsic temperature regime. In the intrinsic regime, the majority carrier concentration is approximately the intrinsic concentration, n{sub}i. The exhaustion regime lies between theses two extremes, intrinsic and freeze-out. The semiconductor designer will be interested in the temperature dependence of the majority carrier. As on Fig. 1, this dependence is often represented as a log-plot of the majority carrier concentration versus reciprocal of the temperature, T.