Towards improved exact exchange functionals relying on $GW$ quasiparticle methods for parametrization

We use fully self-consistent GW calculations on diamond and silicon carbide to reparametrize the Heyd-Scuseria-Ernzerhof (HSE) exact exchange density functional for use in band structure calculations of semiconductors and insulators. We show that the thus modified functional is able to calculate the band structure of bulk Si, Ge, GaAs, and CdTe with good quantitative accuracy at a significantly reduced computational cost as compared to $GW$ methods, and also gives significantly improved band gap predictions in wide-gap ionic crystals as compared to the HSE06 parametrization. We discuss the limitations of this functional in low dimensions by calculating the band structures of single-layer hexagonal BN and ${\mathrm{MoS}}_2$, and by demonstrating that the diameter scaling of curvature induced band gaps in single-walled carbon nanotubes is still physically incorrect using our functional; we consider possible remedies to this problem.

Figure 1

The electronic density of states for bulk diamond and $\beta$ silicon carbide calculated with HSE06, the modified HSE functional, and the $GW$ approximation. HSE06 considerably underestimates the gap, while our modified functional is in close agreement with the $GW$ calculations.

Figure 2

The root mean square of the least square fit between the band structure obtained from the modified HSE functional and $GW$ calculations using data points spanning the entire Brillouin zone in bulk diamond and $\beta$ silicon carbide. The minimum is found at $AEXX=0.46$ and $\mu =0.24$ Å${}^{-1}$ where the RMS is 0.10 eV; this is a substantial improvement over HSE06 and HSE12. The standard deviation of data points in the full range of parameters considered is 0.71 eV.

Figure 3

The electronic band structure for bulk diamond (top left), bulk silicon (middle left), bulk germanium (bottom left), $\beta$ silicon carbide (top right), gallium arsenide (middle right), and cadmium telluride (bottom right) calculated with the modified HSE functional and compared to results obtained with the HSE12 functional.

Figure 4

The electronic band structure for single-layer hexagonal boron nitride (top) and molybdenum disulphide (bottom) calculated with the modified HSE functional and compared to results obtained with the HSE12 functional.

Figure 5

The primary (top) and secondary (bottom) band gaps of SWCNTs according to our modified HSE functional and the HSE12 functional. The primary gaps correctly scale with the inverse diameter ($E_{\mathrm{gap}}=a/d$ where $a_{HSE12}=9.44\mathrm{eV}$ Å and $a_{HSE\mathrm{mod}ZK}=9.76\mathrm{eV}$ Å; compare with the LDA result of $a_{LDA}=7.1\mathrm{eV}$ Å) and the previously reported buckling caused by the trigonal warping effect is also observable. The curvature induced secondary gaps follow a $1/d^3$ scaling ($E_{\mathrm{gap}}=b/d^3$ where $b_{HSE12}=43.8$ eVÅ${}^3$ and $b{}_{HSE\mathrm{mod}ZK}=46.6$ eVÅ${}^3$; compare with the LDA result of $b{}_{LDA}=34.1$ eVÅ${}^3$), which is unphysical considering that curvature effects should scale with an even power of $1/d$.