Michigan Tech PhD. Defense
by Kevin Waters
Follow along @ kwaters4.github.io/Presentation/Defense/
Materials with at least one dimension in the sub-micron range.
Unpublished work from Bhandari et. al.
Unpublished work from Bhandari et. al.
2D | 3D |
---|---|
Coverage (%) | Band Gap (eV) | Binding Energy (eV) |
---|---|---|
0 | 4.25 | |
16 | 3.34 | -0.74 |
25 | 3.11 | -0.70 |
50 | 2.25 | 0.72 |
N/m | Graphene^{1} | This Work | BN^{2} | This Work | |
---|---|---|---|---|---|
C_{11} | 358.1 | 353.7 | 293.2 | 290.5 | |
C_{12} | 60.4 | 61.7 | 66.1 | 64.4 | |
C_{22} | |||||
C_{66} | 148.9 | 144.9 | 113.5 | 113.1 |
N/m | Graphene^{1} | BN^{2} | BN_{2} |
---|---|---|---|
C_{11} | 358.1 | 293.2 | 368.8 |
C_{12} | 60.4 | 66.1 | 47.2 |
C_{22} | 153.3 | ||
C_{66} | 148.9 | 113.5 | 58.7 |
Exact-Exchange in Real Space $$ \frac{1}{2} \sum_{\uparrow,\downarrow} \sum^{N^{\sigma}_{occ}}_{n=1} \sum^{N^{\sigma}_{occ}}_{m=1} \int\int_V \frac{\rho^{\sigma}_{nm}(\mathbf{r}) \rho^{\sigma}_{nm}(\mathbf{r})}{|\mathbf{r}-\mathbf{r'}|} d\mathbf{r} d\mathbf{r'} $$
Exact-Exchange in Reciprocal Space $$ \frac{1}{2\Omega} \sum_{\uparrow,\downarrow}\frac{1}{V_{BZ}^2}\int_{V_{BZ}} d\mathbf{k} \int_{V_{BZ}} d\mathbf{l} \sum^{N^{\sigma}_{occ}}_{n=1} \sum^{N^{\sigma}_{occ}}_{m=1} \sum_{\mathbf{G}} \frac{4\pi}{|\mathbf{G}-\mathbf{k}+\mathbf{l}|^2} \rho^{\sigma}_{m\mathbf{l};n\mathbf{k}}(-\mathbf{G})\rho^{\sigma}_{m\mathbf{k};n\mathbf{l}}(\mathbf{G}) $$
Filtered Potential $$ V_f{(\mathbf{G})} = \frac{1}{{V_{BZ}}^2} \int\int_{V} \frac{4\pi}{|\mathbf{G}-\mathbf{k}+\mathbf{l}|^2}d\mathbf{k}d\mathbf{l} $$
$$ V_f{(\mathbf{G=0})} = \frac{1}{{V_{BZ}}^2} \int\int_{V_{BZ}} \frac{4\pi}{|\mathbf{l}-\mathbf{k}|^2}d\mathbf{k}d\mathbf{l} $$
$$ = \frac{4\pi}{{V_{BZ}}^2} \int\int_{V_{BZ}} \frac{1 - e^{\alpha|\mathbf{l} - \mathbf{k}|^2}}{|\mathbf{l}-\mathbf{k}|^2}d\mathbf{k}d\mathbf{l} + \int\int_{V_{BZ}} \frac{e^{\alpha|\mathbf{l} - \mathbf{k}|^2}}{|\mathbf{l}-\mathbf{k}|^2}d\mathbf{k}d\mathbf{l} $$
$$ \approx \frac{4\pi}{{V_{BZ}}^2} \int\int_{V_{BZ}} \frac{1 - e^{\alpha|\mathbf{l} - \mathbf{k}|^2}}{|\mathbf{l}-\mathbf{k}|^2}d\mathbf{k}d\mathbf{l} + \frac{8\pi^2}{V_{BZ}}\sqrt{\frac{\pi}{\alpha}} $$