## 17Calculus Electronics - Doping

Doping is a process where, essentially, we add a small percentage of foreign atoms to the silicon lattice, which replaces some of the silicon atoms. The type of foreign atoms we add determine whether we are n-doping or p-doping.

Before we discuss the details of the types of doping, let's watch a quick video. In this video they use the terms trivalent atoms and pentavalent atoms. Here is an outline of the terms.

trivalent atoms

3 valence electrons

boron, aluminium, gallium

pentavalent atoms

5 valence electrons

antimony, arsenic, phosphorus

### All About Circuits - Doping [video - 2min 24secs]

Now that you have an overview of the atoms and their valence electrons, let's discuss the types of doping, n-doping and p-doping.

n-Doping

Figure 1 - Antimony Atom

Figure 2 - n-Doping

When we add a small quantity of atoms that each have 5 valence electrons (called pentavalent atoms), we perform what is called n-doping. Some possible pentavalent atoms we can use are phosphorous, arsenic or antimony. Figure 1 shows the simplified structure of an antimony atom. Notice that it has five valence electrons.

When we modify a semiconductor by adding a small number of pentavalent atoms we get an n-doped material, whose structure is shown in Figure 2. The impurity atom, antimony in this case, introduces an extra electron called a donor. This donor electron is freer to move about and is therefore available to produce current requiring only small amounts of energy to move.

The resulting material that we get by n-doping is called an n-type semiconductor. It is still considered a semiconductor since most of it's atoms are silicon. But it's conductivity is changed, moving it closer to the conductivity of the donor atom.

### All About Circuits - n-Doping [video - 1min 31secs]

p-Doping

Figure 3 - Boron Atom

Figure 4 - p-Doping

Similarly, when we introduce atoms that have 3 valence electrons (one less than silicon), the resulting material is called a p-type semiconductor. Boron (see Figure 3) is one of the elements used (aluminum or gallium may be used instead of boron) and the resulting structure is shown in Figure 4. When an atom with a missing electron is added, we get a hole, i.e. an open slot for an electron, and the impurity is called an acceptor impurity.

### All About Circuits - p-Doping [video - 1min 49secs]

Learn how electronics really work.

 basic chemistry semiconductors

### 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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