Question

The compound gallium arsenide is a commonly used semiconductor, having an energy gap $E_g$ of $1.43eV$. Its crystal structure is like that of silicon, except that half the silicon atoms are replaced
by gallium atoms and half by arsenic atoms. Draw a flattened-out sketch of the gallium arsenide lattice, following the pattern of Fig.What is the net charge of the $\left(a\right)$ gallium and $\left(b\right)$ arsenic ion
core? $\left(c\right)$ How many electrons per bond are there? (Hint: Consult the periodic table in Appendix $G$.)

Answer 1

Each arsenic atom is connected (by covalent bonding) to four gallium atoms, and
each gallium atom is similarly connected to four arsenic atoms. The “depth” of their very
nontrivial lattice structure is, of course, not evident in a flattened-out representation such
as shown for silicon in Fig.
Still we try to convey some sense of this (in the $[1,0,0]$ view shown — for those who
might be familiar with Miller indices) by using letters to indicate the depth: A for the
closest atoms (to the observer), $b$ for the next layer deep, $C$ for further into the page, d for
the last layer seen, and $E$ (not shown) for the atoms that are at the deepest layer (and are
behind the A’s) needed for our description of the structure. The capital letters are used for
the gallium atoms, and the small letters for the arsenic.
Consider the arsenic atom (with the letter b) near the upper left; it has covalent bonds
with the two A’s and the two C’s near it. Now consider the arsenic atom (with the letter d)
near the upper right; it has covalent bonds with the two C’s, which are near it, and with
the two E’s (which are behind the A’s which are near :+).
$\left(a\right)$ The $3p,3d$, and $4s$subshells of both arsenic and gallium are filled. They both have
partially filled $4p$ subshells. An isolated, neutral arsenic atom has three electrons in the $4p$ subshell, and an isolated, neutral gallium atom has one electron in the $4p$ subshell. To
supply the total of eight shared electrons (for the four bonds connected to each ion in the
lattice), not only the electrons from $4p$ must be shared but also the electrons from $4s$. The
core of the gallium ion has charge $q=+3e$ (due to the “loss” of its single 4p and two 4s
electrons).
$\left(b\right)$ The core of the arsenic ion has charge $q=+5e$ (due to the “loss” of the three $4p$ and
two $4s$ electrons).
$\left(c\right)$ As remarked in part $\left(a\right)$, there are two electrons shared in each of the covalent bonds.
This is the same situation that one finds for silicon see $Fig.$