Band structures always play a
very important role in materials application or devices design. It reveals the
basic electronic properties like bandgaps, effective masses, etc. For decades,
as a computer method of predicting the materials properties, the first
principle calculation can give us very satisfied results such as total energy, Fermi
levels, showing good agreements with the experiments. However, the theoretical
description of band structures by the first principle calculation remains a
very challenging problem, mainly due to the inaccuracy of density function exchange
correlations.
Generalized Gradient Approximations
(GGA) and Local Density Approximations (LDA) are the most commonly used
exchange correlations. For most of materials, they can present reasonable
bandgaps giving a good agreement with the experiments. But for some materials,
like III-Nitride semiconductors, their bandgaps are always underestimated by GGA
or LDA. Under the Kohn-Sham DFT theory, multiple electrons interactions will be
simplified as a non-interaction system with the same electron density, as a
result, those exchange correlations will give the exact total energy and
properties related to the electron density. However, the wave functions and
orbital energies are given as byproducts. Those byproducts can lead to the
similar band structures under the average field. Although the byproducts
present accurate qualitative diagrams of the energy bands, they can’t predict
bandgaps quantitatively because of the ignorance of multiple particles
interactions, especially for semiconductor systems.
Several solutions have been
introduced to solve this problem. One is the so-called GW approximation, which
is different from the DFT at the exchange correlation part. For DFT, we
introduce the exchange correlation operator to simplify the multiple particles
interactions; for GW approximation, we derive the Hedin equations to get the
operator through the Green’s function multiplying the shielding effect of
coulomb’s interaction. Mathematically, GW method will require several orders of
magnitude higher computing resources than the normal DFT calculation. That’s
the reason it’s only applied to the small system. Another solution is the
hybridization of Hatree-Fork theory and DFT, we will not discuss the HF theory
any further, but applying the method will overestimate the bandgaps. So if we
can combine HF and DFT together somehow, we can make a better description of
bandgaps without costing so much resources like GW approximation.
In 2016, Jason. M et al. made
a comparison with different calculation methods describing the bandgaps, from
the figure below, we can know that the hybrid functions are more efficient than
others, although all of them had some disadvantages for certain materials.
1.
Crowley J
M, Tahir-Kheli J, Goddard III W A. Resolution of the Band Gap Prediction
Problem for Materials Design[J]. The journal of physical chemistry letters,
2016, 7(7): 1198-1203.