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Lattice Constants

The lattice constant of a solid, $a_0$, corresponds to the size of the conventional unit cell length at the equilibrium volume, and is obtained computationally by minimising the total energy as a function of cell volume. Experimental lattice constants are usually obtained from low temperature X-ray diffraction measurements, and extrapolated to zero Kelvin.

The calculation of theoretical lattice constants is straightforward for cubic systems - single-point energy calculations are performed at several different volumes using the same kinetic energy cutoff and $k$-point sampling, and the results are fitted to the Murnaghan equation of state [110],

$\displaystyle E = \frac{B_0 V}{B_0'(B_0'-1)}
\Bigg[B_0' \Bigg(1- \frac{V_0}{V}
\Bigg) + \Bigg( \frac{V_0}{V} \Bigg)^{B_0'} - 1 \Bigg]
+ E(V_0) \, .$     (3.22)

Here $B_0$ is the equilibrium bulk modulus which is defined in Sec. 3.3.2, and $B_0'$ is the derivative of the bulk modulus. Usually $6$ to $8$ calculated energy-volume points that span the equilibrium volume by approximately $\pm5\%$ are sufficient for an accurate determination of $a_0$. An example of a Murnaghan interpolation is shown in Fig. 3.2 for Si obtained using the HCTH functional.

Figure 3.2: The open circles represent total energies calculated using the HCTH functional for different unit cell volumes for Si. The solid line shows the Murnaghan fit determined from relation (3.22), from which the equilibrium cell volume $V_0$ and bulk modulus $B_0$ are obtained.
\hspace{-0.5cm}\epsfig {file=pics/,scale=0.40,angle=-90}\end{center}\end{figure}

Table 3.1 presents the calculated lattice constants obtained with the LDA, PW91 and HCTH functionals. As expected, the LDA uniformly underestimates with a mean error of $-$0.06 Å, while PW91 uniformly overestimates them, with a mean error of $+$0.03 Å. HCTH performs worse than even the LDA with a systematic overestimation of $+$0.08 Å. For HCTH there is a clear correlation between the accuracy of the lattice constant and the number of occurrences of the constituent atoms in the G$2$ fitting data used to determine the functional. For C, Si and Ge, the lattice constant errors are $-$0.01 Å ($\sim 0.3{\%}$), 0.07 Å ($\sim 1.3{\%}$), and 0.14 Å ($\sim 2.5{\%}$) respectively - the number of systems in the fitting data containing carbon, silicon, and germanium atoms are 19, 7, and 0 respectively. The errors for the aluminium semiconductors AlN, AlP, and AlAs are $0.06$ Å ($\sim
1.4{\%}$), $0.09$ Å ($\sim 1.7{\%}$), and $0.12$ Å ($\sim 2.1{\%}$) with 8, 4, and 0 occurrences of nitrogen, phosphorus, and arsenic atoms in the fitting data. The errors for the gallium semiconductors GaN, GaP, and GaAs are $0.07$ Å ($\sim 1.6{\%}$), $0.09$ Å ($\sim 1.7{\%}$), and $0.16$ Å ($\sim 2.8{\%}$) respectively. The fitting data contains one system with aluminium but none with gallium, which is consistent with the lattice constants for GaN and GaAs being less accurate than AlN and AlAs.

Table 3.1: Optimised lattice constants (in Å) calculated using the LDA, PW91 and HCTH. Mean absolute errors (mae) from experiment are also given.
...olumn{5}{l}{^a \hbox{Reference~\cite{GaN1}}}

The significant overestimation of lattice constants made by HCTH is consistent with the findings of Kurth  et al. [108]. Their unit cell volumes for Si, Ge, and GaAs correspond to lattice constants of $5.48$, $5.80$, and $5.80$ Å respectively, which are close to the values shown in Table 3.1. It should be noted that the degradation in performance for heavier systems is also evident in molecular calculations [98]. A subset of the molecules in the HCTH fitting data have well-known experimental data. The subset includes 28 molecules containing just first-row atoms and 12 containing second-row atoms. HCTH overestimates the bond lengths of these two sets of systems by an average of 0.008 Å and 0.025 Å respectively [98].

next up previous contents
Next: Bulk Moduli Up: Properties of group IV Previous: Properties of group IV   Contents
Dr S J Clark