17.2. Semiconductors in Equilibrium#
17.2.1. Structure of a semi-semiconductor#
Symbol |
Concentration of |
|---|---|
\(p\) |
Holes in the valence bands of \(Si\)-atom lattice |
\(n\) |
Free electrons in the conduction band |
\(N_A\) |
Acceptor atoms, usually class \(\text{III}\) |
\(N_D\) |
Donor atoms, usually class \(\text{V}\) |
\(N_A^-\) |
Ionized acceptor atoms, having collected \(e^-\) (usually from neighboring \(Si\) atoms) |
\(N_D^+\) |
Ionized donor atoms, having released \(e^-\) |
Usually, \(N_A\), \(N_D\), \(N_A^-\), \(N_D^+\) only depends on the space coordinate, as a consequencen of doping. The concentration of free electrons and holes may be time dependent instead (…).
17.2.1.1. Intrinsic semiconductor#
Lattice of \(\text{Si}\) atoms. Some free electrons have left the valence bands of the atoms and entered the conduction band. For a new free electron, a new hole is left behind in the lattice. If the medium is electrically neutral the numbers of free electrons and holes are locally equal, and thus their concentrations
17.2.1.2. Doped semiconductors#
17.2.1.2.1. \(p\)-type#
The semiconductor is doped with atoms of class \(\text{III}\), replacing some of the \(\text{Si}\) atoms in the lattice. These atoms introduce a hole in the lattice and are prone to collect \(e^-\) from neighboring \(\text{Si}\) atoms.
Under full-ionization condition, all the doping atoms are ionized, and thus \(N_A^- = N_A\). In a electrically netural region with no donor atoms, the concentration of the holes is equal to the sum of the concentration of the ionized acceptor atoms and the free electrons,
In a standard \(p\)-type semiconductor, \(N_A >> n_i\), and thus \(p \sim N_A^-\).
17.2.1.2.2. \(n\)-type#
The semiconductor is doped with atoms of class \(\text{V}\), replacing some of the \(\text{Si}\) atoms in the lattice. These atoms introduce a “extra” loosely bound electron in the lattice and are prone to release it in the conduction band of the semiconductor.
Under full-ionization condition, all the doping atoms are ionized, and thus \(N_D^+ = N_D\). In a electrically netural region with no acceptor atoms, the concentration of the free electrons is equal to the sum of the concentration of the ionized donor atoms and holes,
In a standard \(n\)-type semiconductor, \(N_D >> p_i\), and thus \(n \sim N_D^+\).
17.2.1.2.3. Full ionization#
todo
17.2.2. Electric charge density#
being
\(q\) the elementary charge, the charge of the electron (here the opposite to get a positive numerical value of \(q\) in Coulomb)
\(n(\mathbf{r},t)\) the number volume density of the mobile electrons
\(p(\mathbf{r},t)\) the number volume density of the holes in the valence bands of the lattice
\(N_D^+(\mathbf{r},t)\) the number volume density of the positive donor ions
\(N_A^-(\mathbf{r},t)\) the number volume density of the negative acceptor ions
Usually, \(N_D^+(\mathbf{r})\), \(N_A^-(\mathbf{r})\), corresponding to the density of the donor and acceptor atoms in the lattice, that have fixed positions.
Remark. todo Discuss the values of \(N_D^+\) and \(N_D\) in terms of energy levels at different temperatures. Add this discussion to full-ionization section or in another section and then point to it?