电子工程代写|超大规模集成电路系统代写Introduction to VLSI Systems代考|The Depletion-Region Width

One important feature of the depletion region is that both sides of it have the same amount of charge but of opposite polarities. That is,
$$x_{n} N_{d}=x_{p} N_{a}$$
where $x_{n}$ and $x_{p}$ represent the widths of $n$-type and $p$-type depletion regions accounting from the junction. For simplicity, the junction area is generally omitted in both sides. Referring to Figure $2.6(\mathrm{~b})$, the depletion-region width of a $p n$ junction is equal to
$$x_{d}=x_{n}-\left(-x_{p}\right)=x_{n}+x_{p}$$ and can be represented as a function of built-in potential and impurity concentrations.
$$x_{d}=\sqrt{\frac{2 \varepsilon_{s i} \phi_{0}}{e}\left(\frac{N_{a}+N_{d}}{N_{a} N_{d}}\right)}$$
where $\phi_{0}$ is the built-in potential and the $\varepsilon_{s i}\left(=\varepsilon_{r(s i)} \varepsilon_{0}\right)$ is the permittivity of silicon. Generally, the value of $\varepsilon_{r(s i)}$ is $11.7$ and $\varepsilon_{0}$ is $8.854 \times 10^{-14} \mathrm{~F} / \mathrm{cm}$.

The depletion-region width $x_{d}$ is affected by the external voltage $V_{a}$ being applied to the $p n$ junction and can be expressed as follows.
$$x_{d}=\sqrt{\frac{2 \varepsilon_{s i}\left(\phi_{0}-V_{a}\right)}{e}\left(\frac{N_{a}+N_{d}}{N_{a} N_{d}}\right)}$$
As $V_{a}$ is greater than or equal to $0 \mathrm{~V}$, i.e., forward-biased condition, the depletionregion width $x_{d}$ decreases; as $V_{a}$ is less than $0 \mathrm{~V}$, i.e., reverse-biased condition, the depletion-region width $x_{d}$ increases.

When the acceptor concentration $N_{a}$ is much greater than donor concentration $N_{d}$, the resulting depletion region is effectively extended into the $n$-type region. This results in a junction called a $p^{+} n$ one-sided junction. Similarly, an $n^{+} p$ one-sided junction is formed when the donor concentration $N_{d}$ is much greater than acceptor concentration $N_{a}$. In this case, the resulting depletion region is effectively extended into the $p$-type region. An illustration of $p^{+} n$ one-sided junction is given in the following example.

电子工程代写|超大规模集成电路系统代写Introduction to VLSI Systems代考|Metal-Semiconductor Junctions

In the semiconductor realm, when an $n$-type material is physically joined with a $p$-type material, the resulting junction or contact is called a rectifying junction or rectifying contact. A rectifying junction is the one that provides a conduction with high resistance of current flow in one direction and with low resistance in the other direction. When the objective of a contact between a metal and a semiconductor is to connect the semiconductor to the outside world, as shown in Figure $2.5$, the contact should provide a conduction with low resistance in both directions of current flow. Such a contact is referred to as a nonrectifying or an ohmic contact.

Generally, when a metal is contacted with a semiconductor, the resulting junction or contact can be either a rectifying or an ohmic junction, depending on the work function difference between the metal and the semiconductor as well as the type of semiconductor. In theory, the resulting contact is an ohmic contact as a metal with work function $\phi_{m}$ contacts with an $n$-type semiconductor with work function $\phi_{s}$ such that $\phi_{m}<\phi_{s}$ or contacts with a $p$-type semiconductor with work function $\phi_{s}$ such that $\phi_{m}>\phi_{s}$. However, in practice, due to the imperfection of material such as surface states, the above two types of junctions are not necessary to form good ohmic contacts.
The rectifying junction formed by contacting a metal with a semiconductor is referred to as a Schottky junction. The built-in potential and current-voltage characteristic of a Schottky junction are similar to those of a pn junction. The device built with a Schottky junction is called a Schottky diode. The current-voltage characteristic of a Schottky diode is basically the same as that of a pn diode except that it has a smaller cut-in or turn-on voltage than a pn junction diode. The cut-in voltage of a typical Schottky diode is around $0.3$ to $0.5 \mathrm{~V}$ while that of a typical $p n$ diode is about $0.6$ to $0.8 \mathrm{~V}$.

Due to much more free electrons in the metal, a Schottky junction is indeed a one-sided junction in which the depletion region is extended into the semiconductor side. Remember that the depletion-region width decreases with the increasing impurity concentration. As the depletion-region width reduces to a few nanometers, the effect of barrier tunneling becomes apparent. The barrier tunneling is a physical phenomenon that a nonzero probability of finding electrons from one side of a barrier exists if there are electrons appearing on the other side of the barrier. The probability is inversely proportional to an exponential function of the barrier width. The smaller barrier width the higher tunneling probability. Hence, the tunneling probability increases profoundly with the increasing impurity concentration of the semiconductor.

Based on the aforementioned discussion, whenever an ohmic contact between a metal and a semiconductor is desired, a heavy impurity concentration, such as $n^{+}$or $p^{+}$, must be used to avoid forming an undesirable Schottky junction. For instance, both source and drain areas shown in Figure $2.5$ are heavily doped in order to avoid the formation of Schottky junctions between metals and these areas.

电子工程代写|超大规模集成电路系统代写Introduction to VLSI Systems代考|The Depletion-Region Width

$$x_{n} N_{d}=x_{p} N_{a}$$

$$x_{d}=x_{n}-\left(-x_{p}\right)=x_{n}+x_{p}$$

$$x_{d}=\sqrt{\frac{2 \varepsilon_{s i} \phi_{0}}{e}\left(\frac{N_{a}+N_{d}}{N_{a} N_{d}}\right)}$$

$$x_{d}=\sqrt{\frac{2 \varepsilon_{s i}\left(\phi_{0}-V_{a}\right)}{e}\left(\frac{N_{a}+N_{d}}{N_{a} N_{d}}\right)}$$

电子工程代写|超大规模集成电路系统代写Introduction to VLSI Systems代考|Metal-Semiconductor Junctions

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