KAHIBARO
Table of Contents
Pure semiconductor behavior
An intrinsic semiconductor is a semiconductor in its pure form, without intentional addition of impurity atoms. The most common examples are silicon and germanium. In an intrinsic semiconductor, electrical behavior comes only from the atoms of the material itself.
This makes intrinsic semiconductors different from conductors and insulators. A conductor has many charges that can move easily. An insulator has very few mobile charges. An intrinsic semiconductor lies between these two cases. At low temperature it behaves almost like an insulator, but as temperature rises it can conduct electricity better because some electrons gain enough energy to move.
Energy bands in an intrinsic semiconductor
In a solid, electrons do not usually occupy isolated atomic energy levels. Instead, they occupy energy bands. For intrinsic semiconductors, the two most important bands are the valence band and the conduction band.
The valence band is the highest band that is mostly filled with electrons at low temperature. The conduction band is the next higher band, where electrons can move through the crystal and carry current. Between these two bands there is a forbidden energy region called the band gap.
If the band gap is small enough, thermal energy can excite some electrons from the valence band into the conduction band. When this happens, the material begins to conduct.
In an intrinsic semiconductor, electrons must gain at least the band-gap energy $E_g$ to move from the valence band to the conduction band.
Electron-hole pair creation
When an electron jumps into the conduction band, it leaves behind an empty state in the valence band. This empty state is called a hole. A hole behaves like a positive charge carrier.
So, in an intrinsic semiconductor, charge transport involves two kinds of carriers. There are electrons in the conduction band and holes in the valence band. They are created together, so every excited electron is associated with one hole.
This is called an electron-hole pair.
Equal carrier concentrations
Because intrinsic semiconductors are pure, electrons and holes appear in equal numbers. If $n$ is the electron concentration and $p$ is the hole concentration, then for an intrinsic semiconductor
$$
n = p = n_i
$$
where $n_i$ is called the intrinsic carrier concentration.
This is one of the defining features of an intrinsic semiconductor.
For an intrinsic semiconductor,
$$
n = p = n_i
$$
Electrons and holes are present in equal concentrations.
Conduction mechanism
Current in an intrinsic semiconductor can be carried by both electrons and holes. Electrons move through the conduction band. Holes move through the valence band as neighboring electrons fill empty states, making the hole appear to move in the opposite direction.
Although a hole is not a physical particle like an electron, it is very useful to describe semiconductor behavior as if holes were positive charge carriers.
The total conductivity depends on both kinds of carriers. In a simple form,
$$
\sigma = q \left( n \mu_n + p \mu_p \right)
$$
where $q$ is the magnitude of the electron charge, $\mu_n$ is electron mobility, and $\mu_p$ is hole mobility.
For an intrinsic semiconductor, since $n = p = n_i$,
$$
\sigma = q n_i \left( \mu_n + \mu_p \right)
$$
This shows that conductivity increases when the intrinsic carrier concentration increases.
The conductivity of an intrinsic semiconductor is
$$
\sigma = q \left( n \mu_n + p \mu_p \right)
$$
and for a pure intrinsic material,
$$
\sigma = q n_i (\mu_n + \mu_p)
$$
Temperature dependence
One of the most important properties of intrinsic semiconductors is that their conductivity increases strongly with temperature. As temperature rises, more electrons gain enough energy to cross the band gap. That creates more electron-hole pairs.
This behavior is opposite to that of ordinary metals, whose resistance usually increases with temperature.
The intrinsic carrier concentration depends strongly on temperature and band gap. A commonly used form is
$$
n_i \propto e^{-E_g/(2kT)}
$$
where $E_g$ is the band-gap energy, $k$ is Boltzmann's constant, and $T$ is absolute temperature.
This exponential dependence means that even a moderate temperature increase can produce a large increase in carrier concentration.
For intrinsic semiconductors, increasing temperature usually increases conductivity because more electron-hole pairs are thermally generated.
Silicon and germanium as intrinsic semiconductors
Silicon and germanium are classic intrinsic semiconductors. Both have four valence electrons per atom, which allows them to form covalent bonds with neighboring atoms in the crystal.
At very low temperature, nearly all electrons are bound in these covalent bonds, so there are almost no free carriers. At higher temperature, some bonds break effectively through thermal excitation, producing conduction electrons and holes.
Silicon has a larger band gap than germanium, so at the same temperature germanium generally has a larger intrinsic carrier concentration and therefore higher intrinsic conductivity.
| Material | Typical band gap at room temperature | Intrinsic behavior |
|---|---|---|
| Silicon | about $1.1 \, \text{eV}$ | moderate intrinsic conductivity |
| Germanium | about $0.66 \, \text{eV}$ | higher intrinsic conductivity |
| Insulator, for comparison | much larger than a few eV | very low thermal carrier generation |
Generation and recombination
Electron-hole pairs are not only created, they can also disappear. When a conduction electron loses energy and falls back into a hole in the valence band, the process is called recombination.
In a pure intrinsic semiconductor at thermal equilibrium, generation and recombination balance each other. That means the average number of electrons and holes stays constant over time, even though microscopic events continue.
This balance is essential for understanding why intrinsic carrier concentration has a stable value at a given temperature.
Why intrinsic semiconductors matter
Intrinsic semiconductors are the starting point for semiconductor physics. They show how a crystal can conduct through thermally generated electrons and holes, even without impurities. This basic behavior explains why semiconductors are neither true conductors nor true insulators.
They are also the reference case for understanding more advanced semiconductor materials. Once the pure material is understood, it becomes easier to see how adding impurities changes its electrical properties.
Key ideas to remember
An intrinsic semiconductor is a pure semiconductor material. Its conduction comes from electron-hole pairs created by thermal excitation across the band gap. Electrons and holes exist in equal numbers, so $n = p = n_i$. Both types of carriers contribute to current. As temperature increases, the intrinsic carrier concentration increases strongly, so conductivity also increases.
Essential facts about intrinsic semiconductors:
$$
n = p = n_i
$$
$$
\sigma = q \left( n \mu_n + p \mu_p \right)
$$
Higher temperature, more thermally generated carriers, higher conductivity.