17Calculus Electronics - Semiconductors

A semiconductor is an element that falls between a conductor and an insulator with respect to how easily it allows electricity to flow through it. For example, metals like silver and copper conduct electricity easily, i.e. they have low resistance and are considered conductors. Materials like glass and rubber do not allow electricity to flow easily and, therefore, are considered good insulators.

The property of these materials that determines into which category they fall is how closely bound the valence electrons are held. As you know, electricity 'flows' when electrons are allowed to move. Insulators hold onto their valence electrons tightly, so these electrons are difficult to move. Whereas conductors hold onto their valence electrons loosely, so the electrons are freer to move. But what does it mean to 'hold' onto valence electrons?

To release or free an electron from an atom, some amount of energy is required. Insulators require a lot of energy to release a valence electron while a conductor does not require much energy. This energy comes from a potential or a voltage placed across the material. As you would expect, in a semiconductor a moderate amount of energy is required to release a valence electron when compared to a conductor or insulator.

Figure 1 - Copper Atom

src - livescience

Take copper for example. You know that copper is a good conductor. Copper has 29 electrons (and protons). Figure 1 shows the distribution of electrons (grey dots) in a copper atom. The closer to the nucleus an electron is, the tighter the electron is held and consequently greater amounts of energy is required to remove an electron. Conversely, the electrons furthest away from the nucleus are not held as tightly and consequently require less energy to free from the atom. Notice that in the valence band (the ring furthest from the nucleus) only one electron exists. This electron is easily freed from the atom making it available to move and contribute as moving current.

Table 1

resistivity ($$\Omega$$-m) at 20oC

Conductors

Silver

$$1.59\times10^{-8}$$

Copper

$$1.68\times10^{-8}$$

Semiconductors

Germanium

$$4.6\times10^{-1}$$

Silicon

$$6.4\times10^2$$

Insulators

Glass, Rubber

$$1\times10^{13}$$

Air

$$2.3\times10^{16}$$

src: wikipedia

Let's take a look at Table 1, which gives some approximate resistivity values for various materials. Resistivity is a measure of how tightly an atom holds onto its valence electrons. Small numbers indicate that it does not take much energy to pull away a valence electron. Notice that conductors have low (small value) resistivity, i.e. they do not require much force to remove a valence electron (they hold onto them loosely). However, with glass and air (insulators), a lot of force is required to remove a valence electron since their resistivity values are very high (they hold onto their valence electron very tightly).

In the middle of the table we show two of the most common semiconductors used to build transistors, silicon (not silicone, a quite different material) and germanium. Silicon is, by far, used more than germanium. One reason is that silicon can withstand higher temperatures than germanium.

Silicon

Figure 2 - Simplified Structure of the Silicon Atom

Figure 3 - Simplified Silicon Lattice

Now let's focus on Figure 2, which shows a simplified structure of the silicon atom. Silicon has four valence electrons in its outer shell.
When we set up a silicon lattice, as shown in Figure 3, the silicon atoms arrange themselves to form covalent bonds between shared electrons. The result is actually a crystal structure called an intrinsic semiconductor, which can conduct a small amount of current.

Learn how electronics really work.

 basic chemistry

For more in-depth discussion, we highly recommend HyperPhysics at Georgia State University. They have more discussion including band theory which will help you understand semiconductors more completely.

wikipedia: semiconductor

HyperPhysics at Georgia State University

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

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