Electronic structure of Calcium hexaborides Page: 4 of 14
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6 x 6 x 6 k-point grids. The kinetic energy cutoff is 60 Ry and we used Troullier-Martins norm-
conserving pseudopotentials. The variational solution of Eq.(1) is solved self-consistently.
We first study the band structure of this system. In order to compare the results of
different methods, we choose a set of crystal parameters used in previous studies, a = 4.15
A and x = 0.207, which is inside the range of experimental measurements. Our resulting
LDA and sX-LDA band structures are shown in Fig. 1. The band gap appears at the X
point. While there is a small LDA band gap of about 0.1 eV, the sX-LDA band gap is
about 0.9 eV. Note the sX-LDA result is very close to the pseudopotential GW calculations
using similar crystal structure parameters . Besides the band gaps, our valence band
width and the second and third subvalence band overlaps are also extremely close to the
GW results as reported in Table I. This closeness of our sX-LDA results with the GW
results indicates that the dynamic screening, which is missing in the sX-LDA formalism,
might not be important here. We would also like to point out that a recent weighted density
approximation (WDA) calculation by Wu et al. for the same system also produces a similar
band gap . The opened band gap is attributed to the corrected self-interaction in the
WDA treatment. In our sX-LDA formalism, the erroneous self-interaction has also been
corrected to some extent. This explains the increased band gap in our sX-LDA result and
also the increased bond charge as will be discussed below.
The equilibrium crystal structure is found by minimizing the total energy in the two
dimensional parameter space of a and x. The sX-LDA relaxed minimum is at a = 4.15
A and x = 0.212, while the LDA results are a = 4.11 A and x = 0.202 as shown in
Table I. The sX-LDA lattice constant is closer to the experimental results, while LDA
underestimates it by ~ 1% in accordance with previous calculations . A more pronounced
difference between LDA and sX-LDA predictions arises from the internal position of the B
atoms. While LDA predicts a shorter inter-octahedral distance [A-C in Fig. 2 (b)] than the
intra-octahedral bond length b [A-B in Fig. 2 (b)], the sX-LDA predicts the opposite. The
total charge density difference between the sX-LDA and LDA results is shown in Fig. 2(a).
For comparison purpose, we used the same lattice constant, 4.15 A, and internal parameter,
0.207, for both sX-LDA and LDA. Interestingly, the sX-LDA has more B-B bond charge both
within the octahedra and between the octahedra. This is consistent with the results found
in other conventional semiconductors, such as Si , where the enhancement of bonding
charge is found in sX-LDA calculations and is consistent with experimental results .
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Lee, Byounghak & Wang, Lin-Wang. Electronic structure of Calcium hexaborides, article, June 15, 2005; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc891739/m1/4/: accessed January 17, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.