# Extensive Benchmarking of DFT+U Calculations for Predicting Band Gaps

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Materials

#### 2.1. DFT+U

#### 2.2. Materials

## 3. Technical Details

**q**-point sampling, hybrid functional (HSE06) calculations were performed for two representative materials (TiO${}_{2}$ and NbNO) to compare the computational costs of HSE06 and U calculations. The results of this analysis are detailed in Appendix B.

## 4. Results and Discussion

#### 4.1. Determination of the States Requiring Hubbard Corrections

#### 4.2. Hubbard U Parameters from First Principles

#### 4.3. Band Gaps from DFT+U Calculations

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Convergence Tests for the Self-Consistent Calculation of Hubbard Parameters

**q**-points are equivalent to the

**k**-points for a given material and hence, the most computationally expensive option (in this case the spacing between the $\mathbf{q}$-points is also 0.04 Å${}^{-1}$). Since we enforce a consistent

**k**-point sampling density in our calculations,

**k**-points are unique for each material in this study. Thus, when we choose the number of

**q**-points for the sampling of the BZ corresponding to the primitive cell, we look at the ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$ ratio by which we divide the

**k**-points for each material and truncate down to the nearest integer value. For example, if the

**k**-point mesh is $6\times 5\times 5$ and we have a ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$ ratio of 2, then the resulting

**q**-point mesh is $3\times 2\times 2$. The ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$ ratios tested, as well as their subsequent

**q**meshes for each material are shown in Table A1. From the

**q**-point mesh testing, a ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$ ratio of 2 was selected with a convergence threshold of $\le 0.1$ eV. By selecting a higher ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$ ratio than 1, we are able to speed up Hubbard parameter calculations and reduce the computational cost.

**Figure A1.**Dependence of the Hubbard U parameter on the ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$ ratio showing the results of ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$ testing for 3 representative materials: (

**a**) TiO${}_{2}$, (

**b**) NbNO, and (

**c**) Y${}_{2}$SO${}_{2}$. For each plot, the

**q**points density increases from right to left, where at ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$=1, the

**q**-point mesh matches the

**k**-point mesh. In panel (

**c**), Y${}_{2}$SO${}_{2}$ does not have a ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$ ratio of 4 plotted because its ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$ ratio of 3 results in a

**q**mesh equivalent to that produced using a ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$ ratio of 4 (see Table A1).

**Figure A2.**Convergence of the Hubbard U parameter with respect to the number of self-consistent cycles for 3 representative materials from our dataset: (

**a**) TiO${}_{2}$, (

**b**) NbNO, and (

**c**) Y${}_{2}$SO${}_{2}$.

**Table A1.**Comparison of

**q**-point meshes for each material (TiO${}_{2}$, NbNO, Y${}_{2}$SO${}_{2}$), corresponding to ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$ ratios of 1, 2, 3, and 4. For the case where ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$=1, the

**q**-point mesh is equivalent to the

**k**-point mesh.

Formula | k-Point | ${\mathit{N}}_{\mathit{k}}/{\mathit{N}}_{\mathit{q}}$ Ratio | |||
---|---|---|---|---|---|

Mesh | 1 | 2 | 3 | 4 | |

TiO${}_{2}$ | $9\times 6\times 6$ | $9\times 6\times 6$ | $4\times 3\times 3$ | $3\times 2\times 2$ | $2\times 1\times 1$ |

NbNO | $6\times 5\times 5$ | $6\times 5\times 5$ | $3\times 2\times 2$ | $2\times 1\times 1$ | $1\times 1\times 1$ |

Y${}_{2}$SO${}_{2}$ | $8\times 8\times 4$ | $8\times 8\times 4$ | $4\times 4\times 2$ | $2\times 2\times 1$ | $2\times 2\times 1$ |

## Appendix B. Computational Cost Comparison of HSE06 and Hubbard Parameter Calculations

**q**-points relative to the

**k**-points of a given material by setting a ratio ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$ higher than 1 [139]. For the current HSE06 calculations, ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$ was set to 3 for each material [140]. The resulting $\mathbf{k}$- and $\mathbf{q}$-point meshes for the U linear-response and HSE06 calculations are listed in Table A2 (we stress that the $\mathbf{q}$-point sampling has different physical meaning in these two types of calculations). The Hubbard parameters calculations reported in Table A2 were performed using a ${N}_{\mathbf{k}}/{N}_{\mathbf{q}}$ ratio of 2, as discussed in Appendix A.

**Table A2.**Comparison of the computational costs of the one-shot U linear-response and HSE06 calculations for TiO${}_{2}$ and NbNO. The ratio between the time for HSE06 (${t}_{\mathrm{HSE}}06$) and U (${t}_{U}$) calculations is given by ${t}_{\mathrm{HSE}}06/{t}_{U}$, and the band gaps ${E}_{\mathrm{g}}$ calculated using HSE06 are also reported (in eV).

Formula | No. of Atoms | k-Points | U Calculation | HSE06 Calculation | Ratio | |||
---|---|---|---|---|---|---|---|---|

per Unit Cell | Mesh | q-Points | ${\mathit{t}}_{\mathit{U}}$ | q-Points | ${\mathit{E}}_{\mathbf{g}}$ | ${\mathit{t}}_{\mathbf{HSE}}06$ | ${\mathit{t}}_{\mathbf{HSE}}06/{\mathit{t}}_{\mathit{U}}$ | |

TiO${}_{2}$ | 6 | $9\times 6\times 6$ | $4\times 3\times 3$ | 3 h 46 m | $3\times 2\times 2$ | 3.30 | 10 h 1 m | 2.7 |

NbNO | 12 | $6\times 6\times 6$ | $3\times 2\times 2$ | 7 h 35 m | $2\times 2\times 2$ | 2.68 | 61 h 52 m | 8.2 |

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**Figure 1.**PDOS (in states/eV/cell) for all materials studied in this work obtained from standard DFT calculations. PDOS are color-coordinated such that s states are green, p states are red, and d states are blue. Note that cases where there are multiple lines of the same color indicate non-equivalent atoms. The zero of energy corresponds to the top of valence bands (except for InN which comes out to be metallic at the semilocal DFT level, and hence the zero of energy corresponds in this case to the Fermi level).

**Figure 2.**Comparison of the computationally predicted band gaps (from DFT and DFT+U) and the experimental band gaps for materials containing (

**a**) TM elements, and (

**b**) p-block (group III-IV) elements of the periodic table. The dashed black line indicates where the computational band gap equals the experimental band gap. Data was fit to first order polynomials so that data points from DFT and different DFT+U calculations could be more easily compared using the resulting solid trendlines (note that all data points were included in these fits). DFT+U calculations were performed using atomic and ortho-atomic projectors. On panel (

**a**), the legend also indicates whether U corrections are applied to only d states of the TM elements (U-TM), or to the d states of TM and p states of light elements (U-All). On panel (

**b**), the legend also indicates whether U corrections are applied to p states of light elements (U-Light), or to p states of light and p-block (group III-IV) elements (U-All). Note that in panel (

**b**), the “DFT+U Ortho-Atomic U-Light” trendline (blue) is dashed so that the “DFT+U Ortho-Atomic U-All” trendline (orange) can also be seen.

**Figure 3.**PDOS (in states/eV/cell) for all materials studied in this work obtained using DFT+U with ortho-atomic projectors. The U correction was applied both to the d states of TM elements and to the p states of the light and p-block (group III-IV) elements (i.e., U-All). PDOS are color-coordinated such that s states are green, p states are red, and d states are blue. The zero of energy corresponds to the top of valence bands.

Formula | Space Group | |
---|---|---|

1 | TiO${}_{2}$ | $P{4}_{2}/mnm$ |

2 | ZrO${}_{2}$ | $P{2}_{1}/c$ |

3 | HfO${}_{2}$ | $P{2}_{1}/c$ |

4 | V${}_{2}$O${}_{5}$ | $Pmmn$ |

5 | Ta${}_{2}$O${}_{5}$ | $C2/c$ |

6 | WO${}_{3}$ | $Pm\overline{3}m$ |

7 | SnO${}_{2}$ | $P{4}_{2}/mnm$ |

8 | PbO${}_{2}$ | $P{4}_{2}/mnm$ |

9 | InN | $P{6}_{3}mc$ |

10 | Sn${}_{3}$N${}_{4}$ | $Fd\overline{3}m$ |

11 | TiS${}_{2}$ | $P\overline{3}m1$ |

12 | ZrS${}_{2}$ | $P\overline{3}m1$ |

13 | HfS${}_{2}$ | $P\overline{3}m1$ |

14 | MoS${}_{3}$ | $P{2}_{1}/m$ |

15 | Sc${}_{2}$S${}_{3}$ | $Fddd$ |

16 | SnS${}_{2}$ | $P\overline{3}m1$ |

17 | NbNO | $P{2}_{1}/c$ |

18 | TaNO | $P{2}_{1}/c$ |

19 | Ge${}_{2}$N${}_{2}$O | $Cmc{2}_{1}$ |

20 | Y${}_{2}$SO${}_{2}$ | $P\overline{3}m1$ |

**Table 2.**Comparison of the Hubbard U parameters (in eV) as computed using DFPT with atomic and ortho-atomic Hubbard projectors. Numbered atoms (e.g., O1, O2) indicate nonequivalent atoms which require individual U values.

Formula | Hubbard U (Ortho-Atomic) [eV] | Hubbard U (Atomic) [eV] | ||||
---|---|---|---|---|---|---|

TiO${}_{2}$ | Ti($3d$): 6.10, | O($2p$): 8.23 | Ti($3d$): 3.81, | O($2p$): 15.89 | ||

ZrO${}_{2}$ | Zr($4d$): 2.72, | O1($2p$): 9.20, | O2($2p$): 8.79 | Zr($4d$): 1.07, | O1($2p$): 33.67, | O2($2p$): 27.20 |

HfO${}_{2}$ | Hf($5d$): 2.74, | O1($2p$): 9.58, | O2($2p$): 9.41 | Hf($5d$): 1.14, | O1($2p$): 47.79, | O2($2p$): 38.18 |

V${}_{2}$O${}_{5}$ | V($3d$): 5.37, | O1($2p$): 7.42, | O2($2p$): 7.65, | V($3d$): 3.70, | O1($2p$): 12.96, | O2($2p$): 10.09, |

O3($2p$): 7.06 | O3($2p$): 11.12 | |||||

Ta${}_{2}$O${}_{5}$ | Ta1($5d$): 3.06, | Ta2($5d$): 3.07, | O1($2p$): 8.99, | Ta1($5d$): 1.54, | Ta2($5d$): 1.55, | O1($2p$): 30.11, |

O2($2p$): 8.36, | O3($2p$): 8.20 | O2($2p$): 30.10, | O3($2p$): 19.03, | O4($2p$): 17.79 | ||

WO${}_{3}$ | W($5d$): 3.99, | O($2p$): 8.27 | W($5d$): 2.80, | O($2p$): 15.18 | ||

TiS${}_{2}$ | Ti($3d$): 5.61, | S($3p$): 3.85 | Ti($3d$): 4.45, | S($3p$): 6.39 | ||

ZrS${}_{2}$ | Zr($4d$): 2.61, | S($3p$): 3.96 | Zr($4d$): 1.62, | S($3p$): 8.36 | ||

HfS${}_{2}$ | Hf($5d$): 2.61, | S($3p$): 3.75 | Hf($5d$): 1.81, | S($3p$): 9.65 | ||

MoS${}_{3}$ | Mo($4d$): 3.28, | S1($3p$): 3.31, | S2($3p$): 4.61, | Mo($4d$): 3.87, | S1($3p$): 6.08, | S2($3p$): 5.62, |

S3($3p$): 4.62 | S3($3p$): 5.53 | |||||

Sc${}_{2}$S${}_{3}$ | Sc1($3d$): 3.28, | Sc2($3d$): 3.37, | S1($3p$): 3.83, | Sc1($3d$): 1.69, | Sc2($3d$): 1.77, | S1($3p$): 8.72, |

S2($3p$): 3.88 | S2($3p$): 9.15, | S3($3p$): 9.17 | ||||

NbNO | Nb($4d$): 3.48, | N($2p$): 6.47, | O($2p$): 8.51 | Nb($4d$): 2.17, | N($2p$): 9.85, | O($2p$): 19.94 |

TaNO | Ta($5d$): 3.10, | N($2p$): 6.66, | O($2p$): 8.87 | Ta($5d$): 1.88, | N($2p$): 11.51, | O($2p$): 26.04 |

Y${}_{2}$SO${}_{2}$ | Y($4d$): 1.98, | S($3p$): 4.16, | O($2p$): 9.17 | Y($4d$): 0.59, | S($3p$): 28.16, | O($2p$): 47.87 |

SnO${}_{2}$ | O($2p$): 10.19, | Sn($5p$): 1.26 | O($2p$): 95.85, | Sn($5p$): 0.67 | ||

PbO${}_{2}$ | O($2p$): 9.58, | Pb($6p$): 1.20 | O($2p$): 39.60, | Pb($6p$): 0.58 | ||

InN | N($2p$): 6.52, | In($5p$): 1.17 | N($2p$): 28.91, | In($5p$): 0.62 | ||

Sn${}_{3}$N${}_{4}$ | N($2p$): 6.56, | Sn1($5p$): 1.43, | Sn2($5p$): 1.23 | N($2p$): 24.14, | Sn1($5p$): 1.01, | Sn2($5p$): 0.81 |

SnS${}_{2}$ | S($3p$): 3.88, | Sn($5p$): 1.37 | S($3p$): 7.66, | Sn($5p$): 1.16 | ||

Ge${}_{2}$N${}_{2}$O | O($2p$): 10.35, | N($2p$): 6.62, | Ge($4p$): 1.18 | O($2p$): 103.08, | N($2p$): 31.57, | Ge($4p$): 0.72 |

**Table 3.**Comparison of the band gaps (in eV) as computed using DFT and DFT+U (using ortho-atomic and atomic projectors) and as measured in experiments for compounds containing TM elements, with the mean absolute error (MAE) given for each method. DFT+U results obtained with ortho-atomic and atomic projectors are split into two columns each: (1) band gaps predicted with U corrections applied only to the d states of TM elements (U-TM), and (2) band gaps predicted with U corrections applied to the d states of TM and p states of light elements (U-All). The computationally predicted band gaps with the closest values to the experimental band gaps are highlighted in bold.

Formula | Expt. | DFT | DFT+U (Ortho-Atomic) | DFT+U (Atomic) | ||
---|---|---|---|---|---|---|

U-TM | U-All | U-TM | U-All | |||

TiO${}_{2}$ | 3 [118] | 1.85 | 2.60 | 3.95 | 2.28 | 4.96 |

ZrO${}_{2}$ | 5.8 [119] | 3.67 | 4.16 | 5.81 | 4.04 | 11.75 |

HfO${}_{2}$ | 6 [120] | 4.11 | 4.68 | 6.55 | 4.59 | 15.44 |

V${}_{2}$O${}_{5}$ | 2.35 [121] | 2.01 | 2.69 | 3.50 | 1.69 | 2.82 |

Ta${}_{2}$O${}_{5}$ | 3.9 [122] | 3.12 | 3.41 | 4.85 | 3.22 | 8.08 |

WO${}_{3}$ | 2.6 [123] | 0.69 | 0.99 | 2.20 | 0.81 | 3.62 |

TiS${}_{2}$ | 1 [124] | 0.02 | 1.53 | 1.86 | 1.03 | 1.56 |

ZrS${}_{2}$ | 1.7 [125] | 1.13 | 1.92 | 2.27 | 1.45 | 2.81 |

HfS${}_{2}$ | 2.85 [126] | 1.30 | 2.16 | 2.51 | 1.78 | 3.73 |

MoS${}_{3}$ | 1.35 [127] | 0.62 | 0.98 | 1.52 | 0.19 | 0.58 |

Sc${}_{2}$S${}_{3}$ | 2 [128] | 1.08 | 2.04 | 2.55 | 1.61 | 3.36 |

NbNO | 2.1 [129] | 1.65 | 2.05 | 2.58 | 1.67 | 3.04 |

TaNO | 2.5 [129] | 1.95 | 2.36 | 3.03 | 2.07 | 4.05 |

Y${}_{2}$SO${}_{2}$ | 4.6 [130] | 3.11 | 3.23 | 4.38 | 3.14 | 11.88 |

MAE | 1.10 | 0.66 | 0.55 | 0.87 | 2.68 |

**Table 4.**Comparison of the band gaps (in eV) as computed using DFT and DFT+U (using ortho-atomic and atomic projectors) and as measured in experiments for compounds containing group III-IV elements (Ge, In, Sn, Pb), with the mean absolute error (MAE) given for each method. DFT+U calculations using ortho-atomic projectors are split into two columns: (1) band gaps predicted with U corrections applied only to the p states of light elements O, N, and S (U-Light), and (2) band gaps predicted with U corrections applied to the p states of group III-IV and light elements (U-All). For atomic calculations, band gaps were predicted with U corrections applied only to the p states of light elements (U-Light). The computationally predicted band gaps with the closest values to the experimental band gaps are highlighted in bold.

Formula | Expt. | DFT | DFT+U (Ortho-Atomic) | DFT+U (Atomic) | |
---|---|---|---|---|---|

U-Light | U-All | U-Light | |||

SnO${}_{2}$ | 3.4 [131] | 0.65 | 4.69 | 4.65 | 15.81 |

PbO${}_{2}$ | 1.5 [132] | 0.09 | 1.10 | 1.06 | 10.27 |

InN | 0.8 [133] | 0 | 0.85 | 0.80 | 8.87 |

Sn${}_{3}$N${}_{4}$ | 1.6 [134] | 0.21 | 2.09 | 2.08 | 9.63 |

SnS${}_{2}$ | 2.4 [135] | 1.53 | 2.00 | 1.98 | 3.00 |

Ge${}_{2}$N${}_{2}$O | 3.78 [136] | 2.61 | 5.03 | 4.99 | 15.46 |

MAE | 1.40 | 0.65 | 0.63 | 8.26 |

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Kirchner-Hall, N.E.; Zhao, W.; Xiong, Y.; Timrov, I.; Dabo, I.
Extensive Benchmarking of DFT+*U* Calculations for Predicting Band Gaps. *Appl. Sci.* **2021**, *11*, 2395.
https://doi.org/10.3390/app11052395

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Kirchner-Hall NE, Zhao W, Xiong Y, Timrov I, Dabo I.
Extensive Benchmarking of DFT+*U* Calculations for Predicting Band Gaps. *Applied Sciences*. 2021; 11(5):2395.
https://doi.org/10.3390/app11052395

**Chicago/Turabian Style**

Kirchner-Hall, Nicole E., Wayne Zhao, Yihuang Xiong, Iurii Timrov, and Ismaila Dabo.
2021. "Extensive Benchmarking of DFT+*U* Calculations for Predicting Band Gaps" *Applied Sciences* 11, no. 5: 2395.
https://doi.org/10.3390/app11052395