Intrinsic and extrinsic semiconductors

One foreign atom in a million

Take a crystal of silicon and replace one atom in every million by an atom of arsenic. The conductivity grows by a factor of tens of thousands. A few foreign atoms change the crystal more than heating it by a hundred degrees. This is doping, and every transistor depends on it.

The silicon crystal

Silicon is in group 14. Each atom has $4$ valence electrons, and it shares one with each of $4$ neighbours. These shared pairs are covalent bonds. At $0$ K every electron sits in a bond. There are no free carriers, so pure silicon is an insulator. Germanium has the same structure.

n-type material

Replace a silicon atom by an atom of group 15, such as phosphorus, arsenic or antimony. It has $5$ valence electrons. Four make bonds with the four neighbours. The fifth has no bond to join. It is held only weakly, and a small energy of about $0.05$ eV or less sets it free.

• The impurity atom is a donor, because it gives an electron.
• On the energy diagram, the donor level lies just below the conduction band. At room temperature nearly every donor has given its electron to the conduction band.
• The crystal stays neutral. The donor atom becomes a fixed positive ion, which cannot move.
• The material is n-type. Electrons are the majority carriers and holes are the minority carriers.

p-type material

Replace a silicon atom by an atom of group 13, such as boron, aluminium, gallium or indium. It has only $3$ valence electrons, so one bond lacks an electron. This is a hole. An electron from a neighbouring bond hops in easily, and the hole moves.

• The impurity atom is an acceptor, because it takes an electron.
• On the energy diagram, the acceptor level lies just above the valence band.
• The atom becomes a fixed negative ion. The crystal stays neutral.
• The material is p-type. Holes are the majority carriers and electrons are the minority carriers.

How many carriers?

Doping adds carriers of one kind. It does not stop heat from making pairs, so recombination now has more of the majority carriers to meet. The result is the law of mass action:

The product stays at its intrinsic value. For an n-type crystal with donor number $N_D \gg n_i$, nearly every donor gives an electron, so $n_e \approx N_D$. The hole number is

The full condition of neutrality is $n_h + N_D = n_e + N_A$, where $N_A$ is the number of acceptors. For silicon at $300$ K, we use $n_i = 1.0 \times 10^{16}\ \text{m}^{-3}$. NCERT prints $1.5 \times 10^{16}\ \text{m}^{-3}$, and older books use $1.45 \times 10^{16}\ \text{m}^{-3}$. The newer measurements are close to $10^{16}\ \text{m}^{-3}$.

A worked example. Silicon has $5.0 \times 10^{28}$ atoms in each cubic metre. One phosphorus atom is added for every $5.0 \times 10^7$ silicon atoms. Find the electron and hole numbers.

1. The donor number. $N_D = \dfrac{5.0 \times 10^{28}\ \text{m}^{-3}}{5.0 \times 10^7} = 1.0 \times 10^{21}\ \text{m}^{-3}$.
2. The electrons. $n_e \approx N_D = 1.0 \times 10^{21}\ \text{m}^{-3}$.
3. The holes. $n_h = \dfrac{n_i^2}{n_e} = \dfrac{(1.0 \times 10^{16}\ \text{m}^{-3})^2}{1.0 \times 10^{21}\ \text{m}^{-3}} = 1.0 \times 10^{11}\ \text{m}^{-3}$.

The electrons outnumber the holes by $10^{10}$ to $1$. Doping has made the crystal almost a one-carrier conductor. The crystal as a whole is still neutral.

Conductivity

Each kind of carrier adds to the conductivity. If the electron mobility is $\mu_e$ and the hole mobility is $\mu_h$ (drift speed per unit field), then

For silicon, take $\mu_e = 0.135\ \text{m}^2/\text{V s}$ and $\mu_h = 0.048\ \text{m}^2/\text{V s}$. Pure silicon has $\sigma = 1.6 \times 10^{-19}\ \text{C} \times 10^{16}\ \text{m}^{-3} \times 0.183\ \text{m}^2/\text{V s} = 2.9 \times 10^{-4}\ \text{S/m}$.

Slide the donor power from $14$ to $25$. At which power does the electron curve start to rise away from the hole curve? Does the product ever change?

What temperature does

Heating a doped crystal adds more electron-hole pairs, but they matter little while $n_i \ll N_D$. At a high enough temperature $n_i$ grows to the size of the doping, and the crystal turns intrinsic again. A device must work well below that temperature. For silicon this is a few hundred degrees above room temperature, and for germanium it is lower. This is one reason silicon replaced germanium.

Practise with new numbers

Try it in Python

The program makes a table. Each row has a dopant (none, a donor or an acceptor) at some ratio to the silicon atoms. It finds the electron and hole numbers and the conductivity.

Common mistakes

• Saying an n-type crystal is negatively charged. It is neutral. The extra electrons are balanced by the positive donor ions.
• Saying the majority carriers are fixed ions. Ions are fixed. Carriers are electrons and holes.
• Forgetting that minority carriers exist. They are few, but they give the reverse current of a diode and they grow fast with temperature.
• Using $n_e = n_i$ in a doped crystal. Use $n_e \approx N_D$, and then $n_h = n_i^2/n_e$.
• Mixing up donor and acceptor. A donor is from group 15 and gives n-type. An acceptor is from group 13 and gives p-type.

Remember these

• n-type: donors (P, As, Sb), $n_e \approx N_D$, electrons are the majority carriers. p-type: acceptors (B, Al, Ga, In), $n_h \approx N_A$, holes are the majority carriers.
• The law of mass action: $n_e\, n_h = n_i^2$.
• Conductivity: $\sigma = e\,(n_e \mu_e + n_h \mu_h)$.
• The crystal is neutral. Donor ions are positive and acceptor ions are negative, and none of them move.

References

• NCERT Physics Class 12, Semiconductor Electronics, the sections on extrinsic semiconductors.
• OpenStax University Physics Volume 3, the chapter Condensed Matter Physics, on semiconductors and doping.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Phosphorus is added to silicon. What kind of material results, and what are the majority carriers?

2. A p-type crystal has $N_A = 2.0 \times 10^{21}\ \text{m}^{-3}$ and $n_i = 1.0 \times 10^{16}\ \text{m}^{-3}$. How many electrons are there in each cubic metre?

3. Is an n-type crystal charged?

4. Why is a doped crystal a better conductor than a pure one even though doping adds very few atoms?