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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|ECE1192

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

耗尽区的一个重要特征是它的两侧具有相同数量的电荷但极性相反。那是,
$$
x_{n} N_{d}=x_{p} N_{a}
$$
在哪里 $x_{n}$ 和 $x_{p}$ 表示宽度 $n$-类型和 $p$ 型耗尽区从结开始计算。为简单起见,结区一般在两侧省略。参考图 $2.6(\mathrm{~b})$, 的耗尽区宽度 $p n$ 结等于
$$
x_{d}=x_{n}-\left(-x_{p}\right)=x_{n}+x_{p}
$$
并且可以表示为内置电位和杂质浓度的函数。
$$
x_{d}=\sqrt{\frac{2 \varepsilon_{s i} \phi_{0}}{e}\left(\frac{N_{a}+N_{d}}{N_{a} N_{d}}\right)}
$$
在哪里 $\phi_{0}$ 是内在潜力和 $\varepsilon_{s i}\left(=\varepsilon_{r(s i)} \varepsilon_{0}\right)$ 是硅的介电常数。一般来说,价值 $\varepsilon_{r(s i)}$ 是 $11.7$ 和 $\varepsilon_{0}$ 是 $8.854 \times 10^{-14} \mathrm{~F} / \mathrm{cm}$
耗尽区宽度 $x_{d}$ 受外部电压影响 $V_{a}$ 被应用于 $p n$ 结,可以表示如下。
$$
x_{d}=\sqrt{\frac{2 \varepsilon_{s i}\left(\phi_{0}-V_{a}\right)}{e}\left(\frac{N_{a}+N_{d}}{N_{a} N_{d}}\right)}
$$
作为 $V_{a}$ 大于或等于 $0 \mathrm{~V}$ ,即,正向偏置条件,耗尽区宽度 $x_{d}$ 减少;作为 $V_{a}$ 小于 $0 \mathrm{~V}$ ,即反向偏置条件,耗 尽区宽度 $x_{d}$ 增加。
当受体浓度 $N_{a}$ 远大于供体浓度 $N_{d}$ ,由此产生的耗尽区有效地扩展到 $n$ 型区域。这会导致一个结点称为 $p^{+} n$ 单边结。同样,一个 $n^{+} p$ 当施主浓度形成单们结 $N_{d}$ 远大于受体浓度 $N_{a}$. 在这种情况下,得到的耗尽区有效 地扩展到 $p$ 型区域。一个揷图 $p^{+} n$ 以下示例中给出了单侧结。

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

在半导体领域,当一个n型材料物理上与p型材料,由此产生的结或接触称为整流结或整流接触。整流结是提供在一个方向上具有高电阻的电流并且在另一方向上具有低电阻的传导的结。当金属与半导体接触的目的是将半导体与外界连接时,如图所示2.5,触点应在电流流动的两个方向上提供低电阻的传导。这种接触称为非整流或欧姆接触。

通常,当金属与半导体接触时,产生的结或接触可以是整流结或欧姆结,这取决于金属和半导体之间的功函数差异以及半导体的类型。理论上,由此产生的接触是欧姆接触,作为具有功函数的金属φ米与n具有功函数的型半导体φs这样φ米<φs或与p具有功函数的型半导体φs这样φ米>φs. 但在实际应用中,由于材料表面状态等缺陷,上述两种结对于形成良好的欧姆接触并不是必需的。
通过金属与半导体接触形成的整流结称为肖特基结。肖特基结的内建电位和电流-电压特性与 pn 结相似。用肖特基结构建的器件称为肖特基二极管。肖特基二极管的电流-电压特性与 pn 二极管基本相同,只是它比 pn 结二极管具有更小的切入或导通电压。典型肖特基二极管的截止电压约为0.3至0.5 在而一个典型的pn二极管是关于0.6至0.8 在.

由于金属中有更多的自由电子,肖特基结确实是一种单侧结,其中耗尽区延伸到半导体一侧。请记住,耗尽区宽度随着杂质浓度的增加而减小。随着耗尽区宽度减小到几纳米,势垒隧穿效应变得明显。势垒隧穿是一种物理现象,如果在势垒的另一侧出现电子,则存在从势垒一侧找到电子的非零概率。该概率与屏障宽度的指数函数成反比。势垒宽度越小,隧穿概率越高。因此,隧穿概率随着半导体杂质浓度的增加而显着增加。

基于上述讨论,每当需要金属和半导体之间的欧姆接触时,重杂质浓度,例如n+或者p+,必须用于避免形成不希望的肖特基结。例如,如图所示的源极和漏极区域2.5重掺杂以避免在金属和这些区域之间形成肖特基结。

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

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