Empirical pseudo-potentials

As we have seen beofre, we can write the pseudopotential as a Fourier series:

$$V({\bf r}) = \sum_{\bf K}V({\bf K}) e^{i{\bf K}\cdot {\bf r}... ...\sqrt{\Omega}} \int d^3r V({\bf r}) e^{-i{\bf K}\cdot {\bf r}}$$ (226)

In crystal structures that consist of more than one atom per unit cell, we need to introduce a structure factor $S_{\bf K}$, defined as

$$S_{\bf K} = \frac{1}{N}\sum_{i=1}^N e^{i{\bf K}\cdot {\bf r_i}},$$ (227)

where the sum runs over all teh $N$ atoms in the unit cell, at positions ${\bf r}_i$. The pseudopotential is teh expressed as a Fourier series, with the coefficients corrected by the structure factor as:

$$V_{\bf K} \rightarrow V_{\bf K}S_{\bf K}.$$ (228)

In crystals with a diamond structure there are two atoms at the positions ${{\bf {r}}}_1$ and ${{\bf {r}}}_2$ in the primitive unit cell. By taking the midpoint between the two atoms in the unit cell as origin, the positions of the atoms are given by ${{\bf {r}}}_1 = \frac{a_0}{8} (1,1,1) = \tau$ and ${{\bf {r}}}_2 = -\frac{a_0}{8} (1,1,1) = -\tau$. Thus, the structure factor is given by

$$S_{\bf K}=\frac{1}{2} \left( \exp(-i{\bf K}\cdot {\bf\tau})+... ...{\bf K}\cdot {\bf\tau})\right) = \cos({\bf K} \cdot {\bf\tau})$$ (229)

In unstrained diamond structures the reciprocal lattice vectors in order of increasing magnitude are (in units of $\frac{2\pi}{a_0}$):

$${\bf K }_0 = (0,0,0)$$ (230)

$${\bf K }_3 = (1,1,1), (\hphantom{-}1,-1,1) ,\dots ,(-1,-1,-1)$$ (231)

$${\bf K }_4 = (2,0,0), ( -2,\hphantom{-}0,0), \dots ,(\hphantom{-}0,\hphantom{-}0,-2)$$ (232)

$${\bf K }_8 = (2,2,0), (\hphantom{-}2,-2,0) ,\dots ,(\hphantom{-}0,-2,-2)$$ (233)

$${\bf K }_{11} = (3,1,1), ( -3,\hphantom{-}1,1) ,\dots ,(-3,-1,-1)$$ (234)

Form factors with reciprocal lattice vectors larger than $K^2 > 11(\frac{2\pi}{a_0})^2$ are neglected, since typically $V_{{{\bf {K}}}}$ decreases as $K^{-2}$ for large ${{\bf {K}}}$. Assuming that the atomic pseudopotentials are spherically symmetric $V({{\bf {r}}}) = V({{\bf {\vert r\vert}}})$, the form factors only depend on the absolute value of the reciprocal lattice vector. The form factor belonging to ${{\bf {K}}}_0$ shifts the entire energy scale by a constant value, and can therefore be set to zero. The form factors belonging to the reciprocal lattice vectors ${{\bf {K}}}_3$ have an absolute value of $\sqrt 3 \cdot \frac{2\pi}{a_0}$ and are conventionally labeled $V_3$. Since the structure factor of the reciprocal lattice vectors ${{\bf {K}}}_4$ with magnitude $2 \cdot \frac{2\pi}{a_0}$ vanishes,

$$\cos\left ( \frac{2\pi}{a_0} {{\bf\tau }} \cdot (\pm 2,0,0)\right ) = \cos\left ( \pm \frac{\pi}{2} \right ) = 0 ,$$ (235)

the respective form factor $V_4$ does not enter the pseudopotential. Thus, only three pseudopotentials form factors $V_3$, $V_8$ and $V_{11}$ are required to calculate the band structure.

Form Factor (Ry)	Si	Ge
$V_3$	-0.2241	-0.2768
$V_8$	0.0551	0.0582
$V_{11}$	0.0724	0.0152

In Table 5.4.2 the parameters employed in the empirical pseudopotential calculations are listed. They consist of three local form factors $V_3,V_8,V_{11}$.