Physics in a nutshell

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Diamond Structure

In this article we will have a look at the crystal structure which is formed by many elements of the 4th main group of the periodic table.[1] [2] Besides carbon these are germanium and silicon which are both very important for semiconductor physics. Remember that the common feature of these elements is the electron configuration of the outer shell:

One could in principle expect that these atoms have a filled s orbital and two half-filled p orbitals. However, it is possible that these orbitals merge and form four new equivalent so-called sp3 hybrid orbitals all being only half-filled. This rearrangement entails initially some energy expense but afterwards the atoms are able to form four very strong covalent bond which compensate this expense by far.

Therefore it is evident that such atoms try to form a three-dimensional structure in which every atom has four uniformly distributed nearest neighbours as binding partners.[3]

The tetrahedrical structure of diamond
Tetrahedrical structure of diamond: Each atom forms bonds with four nearest neighbours (enclosed angles are 109.47°). There are no spare bonds.

Conventional Unit Cell

How can this structure be classified in our previous classification (14 Bravais lattices)? The structure is not a Bravais lattice by itself because there are two types of lattice points with different environments. But when we choose a proper perspective, we can see that the underlying structure is actually a fcc structure with a two-atomic basis. Thus there are two atoms attached to each fcc lattice point: One located just at the position of the lattice point and one being shifted by the vector ${\left( \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \right) }$. Thereby the number of atoms per conventional unit cell is doubled from 4 to 8.[4] [5] [6]

Conventional unit cell of the diamond structure: There are four additional atoms, shifted by 1/4 of the cube diagonal with respect to the positions of the four bottom left atoms.
Conventional unit cell of the diamond structure: The underlying structure is fcc with a two-atomic basis. One of the two atoms is sitting on the lattice point and the other one is shifted by $\frac{1}{4}$ along each axes. This forms a tetrahedrical structure where each atom is surrounded by four equal-distanced neighbours.

Packing Density

To calculate the packing density of a crystal structure one thinks of the atoms as inflated spheres (of volume $V_\text{sph}$) which just touch each other, i.e. they cannot be increased any further without overlapping. The packing density $\varrho$ is then defined as the ratio of the volume filled by the spheres to the total volume.

The easiest way to calculate $\varrho$ is to consider the conventional unit cell: There are $n=4$ lattice points per unit cell with $N=2$ atoms sitting on each such lattice point. Neighboured atoms are shifted by a vector of length $d = \sqrt{3} \cdot \frac{a}{4}$. Thus the atoms are assigned a radius of $r = \frac{d}{2}$. Then the packing density reads \begin{align} \varrho &= \frac{n \cdot N \cdot V_\text{sph}}{V_\text{uc}} \nonumber \\ &= \frac{ 2 \cdot 4 \cdot \frac{4}{3} \pi \left( \frac{\sqrt{3}}{8} a \right)^3 }{ a^3} = \frac{\sqrt{3}}{16}\pi \nonumber \\ &\approx 34\% \end{align} with $V_\text{uc} = a^3$ being the volume of the unit cell.

This value is really small compared to the close-packed structures (74%). But even though there are not many neighbours to form bonds with, the diamond structure is very resistant because the few existing bonds are extremely tight.[7] [8]

Coordination number

The atoms in the diamond structure have $c_1 = 4$ nearest neighbours (coordination number) at a distance of $d_{c_1} = 2r = \frac{\sqrt{3}}{4}a$ as discussed above and $c_2 = 12$ next-nearest neighbours at the neighboured faces of the cube with a distance of $d_{c_2} = \frac{1}{\sqrt{2}}$.[9] [10]

References

[1] N. W. Ashcroft, N. D. Mermin Festkörperphysik Oldenbourg 2001 (pp. 96-97)
[2] Ch. Kittel Einführung in die Festkörperphysik Oldenbourg 2006 (p. 20)
[3] R. Gross, A. Marx Festkörperphysik De Gruyter 2014 (ch. 1.2.8)
[4] R. Gross, A. Marx Festkörperphysik De Gruyter 2014 (ch. 1.2.8)
[5] N. W. Ashcroft, N. D. Mermin Festkörperphysik Oldenbourg 2001 (pp. 96-97)
[6] Ch. Kittel Einführung in die Festkörperphysik Oldenbourg 2006 (p. 20)
[7] R. Gross, A. Marx Festkörperphysik De Gruyter 2014 (ch. 1.2.8.)
[8] Ch. Kittel Einführung in die Festkörperphysik Oldenbourg 2006 (p. 20)
[9] N. W. Ashcroft, N. D. Mermin Festkörperphysik Oldenbourg 2001 (pp. 96-97)
[10] Ch. Kittel Einführung in die Festkörperphysik Oldenbourg 2006 (p. 20)

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