Dear Muhammad,

My message consists of two parts: (1) a bad new on the use of DFT in calculating band gap and (2) overview & future directions towards solving the mentioned problem.

Part (1): http://phys.org/news/2010-11-efficiency-band-gap-solids.html

Enhanced efficiency when determining band gap in solids

November 23, 2010 By Miranda Marquit feature

(PhysOrg.com) -- "With density functional theory, we are able to put different elements in a computer simulation and do calculations based on quantum mechanics to find out about their different properties," Maria Chan tells PhysOrg.com. However, density functional theory is not entirely helpful in identifying all the properties associated with different compounds.

Chan, a post-doc at Argonne National Lab (formerly at MIT), points out that the theory is lacking when it comes to band gaps. "The band gap problem is a well known one. However, using current methods, there is quite a bit of inaccuracy when it comes to calculating band gaps." Band gaps in solids are important, especially if researchers want to identify the best materials for a variety of functions. The maximum efficiency of a solar cell, for example, is determined by the band gap of the material. "The inability to predict the band gap is holding back research in photovoltaics, as well as in semiconductors and thermoelectrics," Chan points out.

However, Chan thinks that a solution might have been found. Working with professor Gerbrand Ceder at MIT, it appears that a modified application of density functional theory – with a special generalization for solids – might hold the key to more accurate predictions of band gaps. The work is described in Physical Review Letters: "Efficient Band Gap Prediction for Solids."

“In the past, researchers have broken down individual electrons using a sort of itemized list of individual states. This has allowed for calculations revealing different properties, including band gaps. Unfortunately, the accuracy of the band gap predictions has been off,” Chan says. She points out that scientists know that silicon has a band gap of about 1.2 eV, but when current methods are employed to calculate the band gap, the answer is 0.7 eV. “You can see how that’s a problem if you are trying to gauge the suitability of a material for specific purposes,” she continues.

To get a more accurate prediction of band gaps, Chan and Ceder created a method that involves altering the use of density functional theory so that an itemized list of individualized states is not the only consideration. “We also recognize that there are a number of interactions between electrons. So we look at the total energy, which includes these interactions,” Chan explains.

Not only do Chan and Ceder make use of the total energy, but they also demonstrate that the band gap can be viewed as a property of the ground state. “This changes the way we view the band gap, seeing that it is a part of the ground state,” Chan says.

Moving forward, Chan hopes that this technique can be used to identify the band gaps of different materials with more accuracy. This could prove useful in identifying the best options when creating future technology. “This work is part of the Materials Genome project started by Professor Ceder, with a goal of predicting properties of known compounds and using the knowledge to design new ones,” Chan says. “Part of that is understanding the band gap and being able to quickly determine the band gaps of various materials.”

“Our method is relatively inexpensive, and could be useful when learning the properties of new materials,” Chan continues. “If someone came up with a new kind of material, predicting the band gap is not a question easily answered in the past. Hopefully our work will pave the way for easier answers in the future.”

Part (2): Attached is a recent review by Hasnip et al. entitled "Efficient Band Gap Prediction for Solids " published in Philos Trans A Math Phys Eng Sci. 2014 discusses the different methods used to predict band gaps and the time cost for the important ones. 

I have copied some important text for quick view:

Most extensions to the LDA include a dependence on the local density gradient as in the generalized-gradient approximations (GGAs) including the PW91 [6] and Perdew, Becke and Ernzerhof (PBE) functionals [7]. These extensions preserve the spherical average of the exchange–correlation hole, often based on the effect of weak perturbations on the homogeneous electron gas. GGAs systematically improve the atomization or cohesive energies of a wide range of molecules and solids [8] and correct the LDA's severe overbinding of hydrogen-bonded solids [9]. However, they are not a universal improvement over the LDA as in both cases their accuracy relies on some cancellation of error, which is system dependent. There are several approximations beyond the GGA; some commonly used approximations include (i) meta-GGAs [10] where the Laplacian of the density can be included (in practical terms, this is expressed in terms of the Laplacian of the wave function, i.e. the kinetic energy); (ii) hybrid functionals where an empirical fraction (often around 20–25%) of Hartree–Fock exchange is included to obviate the band-gap problem; and (iii) DFT+U where an on-site Hubbard-U potential is included to enhance localization of electrons, usually applied to the d or f shells, which improve on magnetic properties of materials. The last two of these methods will be detailed in §3.

While the key shortcomings of HF, namely the lack of screening or any correlation, can be addressed with an ad hoc mixture of HF and LDA (or GGA), there is an alternative approach. In screened exchange (SX), the screening is included by introducing an exponential damping to the Fock exchange, with a characteristic screening length. The screening length could be considered a free parameter, but as it has a physical meaning its value is usually approximated by a simple model (e.g. the Thomas–Fermi screening length). The lack of correlation in HF may be addressed by adding in the correlation contribution from an XC functional, for example LDA, leading to the SX-LDA approximation [34]. It should be noted that the computational cost of such HF-like functionals for plane-waves is significantly larger than that for semi-local functionals (see §2). While the computational time for large simulations scales approximately the same as an ordinary (semi-)local DFT calculation, the absolute time required is an order of magnitude greater.

Hoping this will be helpful,

Rafik

Thank you for your comprehensive reply.

Dear Muhammad,

Generally the term "hybrid functional" refers to functionals where a semi-local XC functional is mixed with some Fock exchange. In a plane-wave basis, applying the Fock operator is computationally demanding and will dominate the calculation time -- thus all hybrid functionals take almost the same computational time per SCF cycle.

Regarding computing band-gaps, I would advise against using most hybrid functionals. The motivation behind most hybrid functionals is the observation that Hartree-Fock overestimates band-gaps and semi-local density functionals underestimate them, so by mixing the two you can get an intermediate result -- including, of course, the "correct" one if the amount of mixing is tuned; however this completely neglects the effects on the band-structure itself. Using a functional which has been tuned to give the correct band-gap sacrifices the predictive accuracy of the simulations, and is often counter-productive.

Tuning the XC method for a single number, be it the band-gap or any other single property, runs the risk of obtaining a fortuitous cancellation of error in that one property which is not a reflection of its actual accuracy. The Fock operator is ill-behaved (in fact, pathological) at the Fermi-level, as anyone who has ever used Hartree-Fock for a metal can see, and any proportion of Fock exchange can cause problems in the band-structure. As an example, band-widths computed with hybrid functionals are often terrible, far worse than those computed with semi-local functionals.

The method used by Clark et al, which we included as figure 4 in the Phil. Trans. R. Soc. A article referred to by Rafik[1] (incidentally, the title is actually "Density functional theory in the solid state"), is screened-exchange, which uses a modified Fock operator to account for screening in the material, akin to a Yukawa modification of the Coulomb potential. This has a single parameter, the screening length, which can be determined from a physical model (e.g. Thomas-Fermi theory) to give a parameter-free functional which, as can be seen from the aforementioned paper, gives excellent agreement with experiment. Nothing in that figure has been tuned to give the correct band-gap, the band-gap emerges as an ab initio prediction just like the other properties.

Hope that helps,

Phil Hasnip

[1] http://dx.doi.org/10.1098/rsta.2013.0270