Self-consistent Hybrid Functionals: What We’ve Learned So Far

Abstract

These are exciting times for computational materials science. We are witnessing a wide-spread availability of high-performance computing facilities, a huge increase in accessible computational resources, and an accompanying development of new exchange-correlation functionals within density functional theory. All this contributes to the establishment of density functional theory as an indispensable tool for materials science investigations in general. Here, we want to highlight some examples utilising a recently developed self-consistent hybrid functional, proposed by Shimazaki and Asai (J Chem Phys 130:164702, 2009 [3]) and Skone et al. (Phys Rev B 89:195112, 2014 [4]), allowing for the calculation of accurate material properties using a fully ab initio procedure. The obtained structural and electronic properties of a range of oxide semiconductors will be critically discussed with respect to experimental results, and pave the way towards open questions in the field.

1 Introduction

Density functional theory (DFT), introduced by Hohenberg and Kohn [1] and Kohn and Sham [2], provides us with an indispensable tool to perform predictive materials science investigations into the structural, electronic, and optical properties of a wide range of materials. However, every theoretical result has to be compared to experimental data, and the wisdom accumulated over the last decades show that the agreement of theory and experiment strongly depends on the underlying approximation of the unknown exchange-correlation functional utilised in the calculations. On more formal grounds we can write down the generalised Kohn-Sham (GKS) potential $v_{\text {GKS}}(\mathbf{r} ,\mathbf{r} ^{\prime })$ as

$$\begin{aligned} v_{\text {GKS}}(\mathbf{r} ,\mathbf{r} ^{\prime }) = v_{\text {H}}(\mathbf{r} ) + v_{xc}(\mathbf{r} ,\mathbf{r} ^{\prime }) + v_{\text {ext}}(\mathbf{r} ) \,. \end{aligned}$$

(5.1)

It is expressed as the sum of the Hartree $v_{\text {H}}(\mathbf{r} )$, the exchange-correlation $v_{xc}(\mathbf{r} ,\mathbf{r} ^{\prime })$, and the external $v_{\text {ext}}(\mathbf{r} )$ potential, respectively, where $v_{\text {H}}(\mathbf{r} )$ and $v_{\text {ext}}(\mathbf{r} )$ are in principle known, and only $v_{xc}(\mathbf{r} ,\mathbf{r} ^{\prime })$ requires a more or less sophisticated approximation.

Among others, the particular choice of an exchange-correlation functional may be influenced by the size of the unit cell and their respective number of atoms, the material properties we’re interested in, and the available computational resources. With the advent and general availability of local, regional, and national high-performance computing facilities, recent years witnessed a surge in the application of computationally more demanding so-called hybrid functionals, where a particular fraction $\alpha$ of the underlying semilocal exchange potential is replaced by a Hartree-Fock exact-exchange term

$$\begin{aligned} v_{xc}(\mathbf{r} ,\mathbf{r} ^{\prime }) = \alpha v_{x}^{ex}(\mathbf{r} ,\mathbf{r} ^{\prime }) + (1-\alpha ) v_{x}(\mathbf{r} ,\mathbf{r} ^{\prime }) + v_{c}(\mathbf{r} ) \,. \end{aligned}$$

(5.2)

Applying these hybrid functionals to semiconducting materials in particular, it has soon been noticed, that there exists a correlation between the amount $\alpha$ of Hartree-Fock exact-exchange and the particular band gap of a material. Based on more theoretical grounds, i.e. the screening behaviour of the Coulomb interaction, the fraction $\alpha$ of Hartree-Fock exact-exchange can be identified as approximately the inverse of the static dielectric constant $\varepsilon _{\infty }$ of a material

$$\begin{aligned} \alpha \approx \frac{1}{\varepsilon _{\infty }} \,. \end{aligned}$$

(5.3)

In particular (5.3) laid the foundation for the development and utilisation of so-called dielectric-dependent hybrid functionals. There are several possible ways to utilise (5.3). The simplest one relies on experimental input, either in the form of an experimentally determined static dielectric constant to be used with (5.3), or by adjusting $\alpha$ directly to reproduce experimentally known band gaps. Somewhat more sophisticated investigations perform a one-off calculation of the static dielectric constant based on a previously relaxed ground state geometry. The most involved way, however, is the one introduced by Shimazaki and Asai [3] and Skone et al. [4], where the amount $\alpha$ of Hartree-Fock exact-exchange is self-consistently determined via a second loop inside the structural geometry optimisation procedure. In this way, no experimental input is required, and the true ab initio character of the calculations is restored. The whole class of dielectric-dependent hybrid functionals have been applied to a large range of materials. For a broader overview we refer to [5,6,7,8,9,10,11,12].

2 Computational Methods

The calculations of the present work have been performed utilising the Vienna ab initio simulation package (VASP) [13,14,15] together with the projector-augmented wave (PAW) method [16]. We used the PAW potentials supplied with the VASP package that contributed 6, 10, 14, 12, and 14 valence electrons for the O (2$s^{2}$ 2$p^{4}$), Ti (3$p^{6}$ 4$s^{2}$ 3$d^{2}$), Fe (3$p^{6}$ 4$s^{2}$ 3$d^{6}$), Zn (3$d^{10}$ 4$s^{2}$), and Sn (5$s^{2}$ 4$d^{10}$ 5$p^{2}$) atoms, respectively.

While the focus of the present work is on the application of the self-consistent hybrid functional scPBE0, in a first step calculations based on the PBE0 functional [17] have been performed as well. Cited results from previous investigations also utilised generalised gradient approximation (GGA) functionals in the parametrisations of Perdew et al. [18] and the one revised for solids [19], as well as the hybrid functional HSE06 [20, 21].

Structural relaxations were performed for the various unit cells, with the following numerical parameters in terms of k-point grid, cut-off energy, and convergence criteria for energy and forces on the atoms: ZnO $(8 \times 8 \times 6$, 500 eV, 10$^{-5}$ eV, 10$^{-3}$ eV Å$^{-1}$), SnO$_{2}$ and TiO$_{2}$ $(6 \times 6 \times 8$, 500 eV, 10$^{-5}$ eV, 10$^{-3}$ eV Å$^{-1}$), ZnFe$_{2}$O$_{4}$ $(6 \times 6 \times 6$, 500 eV, 10$^{-5}$ eV, 10$^{-3}$ eV Å$^{-1}$), amorphous Sn-Ti oxides ($\Gamma$-only, 400 eV, 10$^{-4}$ eV, 5$\times 10 ^{-3}$ eV Å$^{-1}$), respectively. For all structural parameters this ensured well-converged results.

3 Application 1: Semiconducting Oxides

As a first application we look at the performance of the self-consistent hybrid functional for the structural and electronic properties of bulk semiconducting oxides, namely wurtzite ZnO, and rutile SnO$_{2}$ and TiO$_{2}$ [10, 32].

Table 5.1 Ground state structural parameters for wurtzite ZnO, and rutile SnO$_{2}$ and TiO$_{2}$, obtained with different approximations for the exchange-correlation potential in comparison to low-temperature experimental data

ZnO naturally crystallises in the hexagonal wurtzite structure (P6$_{3}$mc, space group 186). The calculated structural parameters at different levels of theory are summarised in the upper part of Table 5.1. For better visualisation Fig. 5.1 shows the calculated unit cell volumes at different levels of theory in comparison to the experimental unit cell volume. As expected, for plain PBE calculations the calculated unit cell volume is largely overestimated by ${\approx }4 \%$. This improves already for standard PBE0 calculations, and best agreement with respect to the experimental unit cell volume is observed for the scPBE0 calculations (Fig. 5.1).

Fig. 5.1

Left: Ground state unit cell volumes V with respect to the experimental volumes $V_{\text {exp}}$. Right: Kohn-Sham energy gaps $E_{\text {KS}}$ with respect to the experimental energy gaps $E_{\text {exp}}$. Shown are the results for different levels of theory, namely PBE (dotted), PBE0 (lined), and scPBE0 (squared). Experimental data corresponds to the dashed horizontal lines

Fig. 5.2

Electronic band structures of wurtzite ZnO (left panel), rutile SnO$_{2}$ (middle panel), and rutile TiO$_{2}$ (right panel), calculated with the scPBE0 functional. Energies are in electron volt (eV) with the valence band maximum set to zero

The direct Kohn-Sham energy band gap of wurtzite ZnO at PBE level (0.715 eV) is largely underestimated with respect to the experimental band gap of 3.4449 eV [25], thereby showing the well-known “band-gap” problem of plain GGA approaches in DFT calculations. This improves slightly when switching to the PBE0 hybrid functional approach (3.132 eV) and the best agreement with respect to experiment is observed for the scPBE0 calculations (3.425 eV).

SnO$_{2}$ and TiO$_{2}$ crystallise naturally in the tetragonal rutile structure (P4$_{2}$/mnm, space group 136). Their respective calculated structural parameters at different levels of theory are given in the middle and lower parts of Table 5.1. As can be seen from the middle parts of Fig. 5.1, SnO$_{2}$ follows the same trend as wurtzite ZnO, namely a large overestimation of the plain PBE unit cell volume with respect to the experimental one and better agreement for the plain PBE0 calculations. Again, the best agreement with experiment is observed for the scPBE0 ground state volume. However, while the plain PBE ground state volume yields worst agreement with experiment for rutile TiO$_{2}$, plain PBE0 gives a slighty smaller ground state volume compared to experiment. The scPBE0 calculations are overestimated with respect to the experimental ground state volume and are slightly worse compared to the plain PBE0 results (Fig. 5.1).

The direct Kohn-Sham energy band gaps of the plain PBE calculations are again strongly underestimated compared to the experimental results. For the hybrid functional calculations, we observe an opposite trend compared to the ground state structural properties. For rutile SnO$_{2}$ the plain PBE0 calculations yield the best agreement compared to the experimental band gap, while the scPBE0 is slightly worse. However, for rutile TiO$_{2}$, where plain PBE0 calculations yielded best agreement in terms of structural properties, the Kohn-Sham energy band gap is strongly overestimated. The slightly worse results in the scPBE0 structural properties has to be counterbalanced by the best agreement of the Kohn-Sham energy band gap compared to the experimental value.

Based on the relaxed ground state structural and electronic properties (Fig. 5.2) it is in principle possible to calculate the optical properties as well, i.e. the real and imaginary parts of the dielectric function, which subsequently would give access to experimentally measurable quantities such as the absorption coefficient or reflectivity [10, 32].

4 Application 2: Ferro(i)magnetic Oxides

As a second application we look at the performance of the self-consistent hybrid functional for bulk ferro(i)magnetic oxides in the spinel structure. In particular, spinel ferrites show a plethora of interactions due to their unique interplay of structural, electronic, and magnetic properties, e.g. CoFe$_{2}$O$_{4}$ and NiFe$_{2}$O$_{4}$ have shown huge potential as building blocks of artificial multiferroic [33, 34] or spintronics devices [35]. However, here we’re concerned with ZnFe$_{2}$O$_{4}$ [36] whose magnetic ground state is still under dispute.

Fig. 5.3

Left panel: Total and projected density of states per formula unit for ZnFe$_{2}$O$_{4}$. The octahedral $O_{h}$ (tetrahedral $T_{d}$) states of Fe (Zn) are shown as solid (dashed) lines, and the shaded grey area depicts the total density of states. Minority spin projections are shown using negative values. Right panels: Electronic band structure for ZnFe$_{2}$O$_{4}$, where the majority-spin (minority-spin) bands are shown in the left (right) panel. The zero energy is set to the valence band maximum in all panels

While a previous investigation of ZnFe$_{2}$O$_{4}$ was more concerned with the influence of the inversion degree between the two cationic sublattices on the structural, electronic, and optical properties [36], it also showed that for calculations using the hybrid PBE0 functional the ferromagnetic ground state in the normal spinel phase is favoured over the anti-ferromagnetic one, contrary to experimental findings. Applying the scPBE0 functional separately to the ferromagnetic and anti-ferromagnetic spin arrangements of normal ZnFe$_{2}$O$_{4}$ confirms this result, i.e. the ferromagnetic spin arrangement in the normal spinel phase is the ground state in both cases. The lattice constant calculated with the scPBE0 functional of $a = 8.427$ Å is slighty reduced compared to the PBE0 calculations ($a = 8.452$ Å). Figure 5.3 shows the resulting total and projected density of states (DOS) of ZnFe$_{2}$O$_{4}$ (left panel), whereas in the right panels the electronic band structure of the majority- and minority-spin states is shown. The scPBE0 DOS is nearly indistinguishable from the PBE0 calculations [36], and the electronic band gap is again between the majority-spin valence band and the minority-spin conduction band. However, due to the increased fraction $\alpha$ of Hartree-Fock exact-exchange, the band gap increases from 3.13 eV (PBE0) to 4.28 eV (scPBE0).

The ferromagnetic spin-arrangement in the normal spinel phase of ZnFe$_{2}$O$_{4}$ has recently been confirmed by Ulpe et al. [37] who applied the scPBE0 functional to a range of ternary transition metal oxides. These investigations show that the scPBE0 functional can not only be applied to investigate the structural and electronic properties of semiconductors, but is capable to go beyond and yield information on magnetic properties as well.

5 Application 3: Amorphous Oxides

As a third application we look at the performance of the self-consistent hybrid functional for the electronic and optical properties of transparent amorphous semiconducting oxides which got into focus for applications ranging from displays to solar cells [38]. Here, we’re concerned with the particular example of amorphous (SnO$_{2}$)$_{1-x}$(TiO$_{2}$)$_{x}$ (0 $\le x \le 1$) oxides (Sn-Ti oxides) [32], which have been the focus of a previous investigation. Since a full ab initio molecular dynamics generation of amorphous structure models based on the scPBE0 hybrid functional is out of scope of even nowadays computational resources we rely on the previously generated amorphous structure models for the whole range of (SnO$_{2}$)$_{1-x}$(TiO$_{2}$)$_{x}$ (0 $\le x \le 1$) based on the PBEsol functional [19] and applying a permutation operation. Reference [32] shows in detail that this combined approach yields reliable amorphous structure models which can then be utilised to analyse different structure models for a given composition x of (SnO$_{2}$)$_{1-x}$(TiO$_{2}$)$_{x}$ and moreover, the subsequent influence of different structure models on the optical properties (dielectric functions, optical gaps) can be scrutinised. While the previous investigation showed that plain PBEsol optical gaps are severely underestimated with respect to the experimental values [32], even hybrid functional calculations based on the HSE06 functional [20, 21] showed still some discrepancies.

Here, we follow a slightly different approach. In general, amorphous materials are unique in the sense that on the one hand they lack the long-range order of crystalline materials, but on the other hand a short range order is still present. This can be seen from analysing the radial distribution functions, which measure the statistical distribution of pairwise atomic distances [32]. Since the local atomic arrangement between crystalline phases and the generated amorphous structure models for SnO$_{2}$ and TiO$_{2}$ are quite similar, here we transfer the previously determined fractions $\alpha$ of Hartree-Fock exact-exchange to the respective amorphous phases, and interpolate linearly for the intermediate concentrations x of (SnO$_{2}$)$_{1-x}$(TiO$_{2}$)$_{x}$.

Figure 5.4 summarises the results. While the upper panels show the calculated real and imaginary dielectric functions utilising the scPBE0 hybrid functional (solid lines), compared to the previously reported results from PBEsol calculations (dashed lines) [32], the lower panels show the absorption coefficients suitably scaled for a subsequent Tauc-plot analysis to extract the optical band gaps. Similarly to our earlier investigation, different amorphous structure models in the intermediate range yield optical band gaps with an error margin of 0.1 eV. Moreover, the extracted optical band gaps of amorphous SnO$_{2}$ (4.2 eV) and TiO$_{2}$ (3.4 eV) are in favourable agreement with the respective experimental values of around 4.0 eV [38] and 3.2–3.5 eV [39]. Moreover, we significantly improve over the previously reported values of 2.9 eV (SnO$_{2}$) and 3.8 eV (TiO$_{2}$) determined using the hybrid HSE06 functional [32].

Fig. 5.4

Upper panels: Real (light) and imaginary (dark) parts of the dielectric functions for one particular amorphous structure model for (SnO$_{2}$)$_{1-x}$(TiO$_{2}$)$_{x}$ (0 $\le x \le 1$) calculated with the PBEsol (dashed lines) and the scPBE0 hybrid functional (solid lines), respectively. Lower panels: Tauc plots ($\sqrt{\alpha h \nu }$ vs. energy) based on the scPBE0 calculated absorption coefficients for various amorphous structure models showing the optical gaps

6 Summary and Outlook

In summary, we presented an up-to-date compilation on what we’ve learned so far about the application and performance of a new self-consistent hybrid functional, recently introduced by Shimazaki and Asai [3] and Skone et al. [4]. Applications included technologically important oxide semiconductors, and lesser known examples such as magnetic and amorphous oxides. In all cases, the knowledge gain in terms of structural, electronic, magnetic, and optical properties, outweighs the additional computational effort put into this method. Future research directions will focus on the detailed calculation of the static dielectric constant of the materials, the deep underlying connection to many-body perturbation theory, and possibly the further improvement of exchange-correlation functionals to be applied in computational materials science.