Intrinsic Semiconductor

What an intrinsic semiconductor is

• An intrinsic semiconductor is a pure semiconductor with no added impurity, in which all the charge carriers — electrons and holes — are produced by thermal excitation within the material itself. Their presence is an intrinsic property of the material, not a consequence of doping.
• The standard examples are Ge and Si.

Crystal structure and covalent bonding

• Si and Ge crystallise in a diamond-like structure in which each atom is surrounded by four nearest neighbours. Lattice spacings: C $3.56$ Å, Si $5.43$ Å, Ge $5.66$ Å.
• Both have four valence electrons. Every atom shares one of its four valence electrons with each of its four nearest neighbours, and takes a share of one electron from each of them. Each shared pair forms a covalent bond (valence bond); the two shared electrons shuttle back and forth between the two atoms, binding them strongly.
• Diagram in words: a flat square grid of atom cores marked $+4$, with each core joined to each of its four neighbours by a pair of short parallel lines representing the two shared electrons of that bond. At low temperature every bond is drawn intact.

Generation of electrons and holes

• At low temperature no bond is broken, so there are no free carriers and the crystal behaves as an insulator.
• As the temperature rises, thermal energy breaks a few bonds. The electron that breaks away (charge $-q$) becomes a free electron in the conduction band and contributes to conduction.
• The neighbourhood it left behind now has a vacancy of effective charge $+q$. This vacancy is called a hole, and it behaves as an apparent free particle with an effective positive charge.
• Thermal energy ionises only a very few atoms of the lattice, so the number of carriers is small — but each event always creates one electron and one hole together, as a pair.

$$n_e = n_h = n_i$$

• $n_i$ is the intrinsic carrier concentration. This equality is the defining property of an intrinsic semiconductor. For Si at room temperature $n_i \approx 1.5\times10^{16}$ m$^{-3}$.

How a hole moves, and why holes conduct

• Suppose a hole sits at site 1. An electron in the covalent bond at a neighbouring site 2 may jump into the vacancy at site 1. After the jump, site 1 has an electron and site 2 has the vacancy — the hole has apparently moved from site 1 to site 2.
• Note carefully: the electron originally set free is not involved at all in this process. It moves independently in the conduction band as a conduction electron. Hole motion is only a convenient way of describing the motion of the bound valence electrons whenever an empty bond exists somewhere in the crystal.
• Under an applied electric field, free electrons move towards the positive end and give the electron current $I_e$; holes move towards the negative end and give the hole current $I_h$.

$$I = I_e + I_h$$

• The two currents add rather than cancel: the carriers have opposite signs of charge and move in opposite directions, so their conventional currents point the same way.

Generation and recombination

• Alongside generation of electron-hole pairs, recombination goes on continuously: a conduction electron colliding with a hole falls back into the bond and both carriers disappear.
• At equilibrium the rate of generation equals the rate of recombination, which is what fixes $n_i$ at a given temperature.

Band picture of an intrinsic semiconductor

• At $T = 0$ K: the valence band is completely filled, the conduction band is completely empty, and the semiconductor behaves as a perfect insulator. Draw two shaded blocks separated by the narrow gap $E_g$, the lower one full, the upper one empty.
• At $T > 0$ K: thermal energy excites some electrons from the valence band across $E_g$ into the conduction band. Draw a few filled dots near the bottom of the conduction band and exactly the same number of empty circles near the top of the valence band — the holes those electrons left behind. Equal numbers is the point of the diagram.

Effect of temperature — the standard assertion-reason

• Assertion: the resistance of an intrinsic semiconductor decreases as its temperature rises. Reason: the numbers of conduction electrons and of holes both increase with temperature.
• Both statements are true, and the reason is the correct explanation of the assertion: more thermal energy breaks more bonds, so more electron-hole pairs are generated, so there are more charge carriers, so the conductivity rises and the resistance falls.
• This negative temperature coefficient of resistance is opposite to a metal's behaviour, and it is the cleanest experimental way to tell a semiconductor from a conductor.

Why C is an insulator while Si and Ge are semiconductors

• C, Si and Ge have the same lattice structure and four valence electrons each, but their bonding electrons lie in the second, third and fourth orbits respectively.
• The energy needed to take an electron out of the atom (that is, the band gap $E_g$) is therefore least for Ge, more for Si and highest for C.
• Consequently the number of free electrons available for conduction is significant in Ge and Si but negligibly small in C — so carbon is an insulator and Si and Ge are intrinsic semiconductors.

Standard results

• $n_e = n_h = n_i$ in a pure semiconductor at every temperature.
• $I = I_e + I_h$.
• Every carrier is created as an electron-hole pair; the numbers of the two are always created equal in an intrinsic sample.
• At equilibrium, rate of generation = rate of recombination.
• An intrinsic semiconductor is an insulator at $0$ K.
• Resistance falls, and conductivity rises, as the temperature rises.
• Room-temperature conductivity of an intrinsic semiconductor is far too small for any practical device — which is exactly why the next section introduces doping.