There are various kinds of approximation methods in band theory. In my opinion, Bloch theorem implies the existence of energy band. From nearly-free electron approximation or tight binding method, we can calculate the energy band. They can tell us the information of band gap and band width. However, I am quite confused with k.p method and envelop function method. Starting from some particular point in the band, we can calculate the effective mass from k.p method. But why do we need to do so? Also, it seems hard to calculate energy band from k.p method. Lost in all these approximation methods, i want to ask the following questions: What is the major task in energy band theory? To calculate band structure and band gap? What is the main differences in application (not in theory, I know how to derive these methods. ) between these methods: neatly-free, OPW, PP, TB, k.p, envelop function...?
Typically, in semiconductors, only electron or hole levels near the band minimum or maximum, respectively, are occupied. Hence in these situations it is sufficient to calculate the effective mass from kp perturbation theory. This yields good qualitative insights into the properties of electrons in semiconductors. Hence it is also often introduced in pedagogical texts. This may not be sufficient in other compounds and e.g. for spectroscopy where also transitions far from the band gap are considered. Some of the other methods you are mentioning are simply choices of the basis functions for expansion of the Bloch waves in numerical calculations. Which one to choose depends largely on the problem at hand. E.g. OPW will be a good starting point for nearly free electrons, like metals, while tight binding is good for systems with rather strongly localized electrons, like molecular cystals, organic semiconductors and the like.
It is very helpful. Still, I have a question. In numerical methods such as OPW and TB, how do we determine the potential V, which is generally unknown. I know there are some empirical methods to determine V, but do we have other choice? Also, it seems that Hartree approximation and Hartree-Fock approximation are only valid for ground state. Why only ground state is of interest?
I am not sure about the potential. I think there exist ab initio methods for both bases which don't require to guess a potential. The Hartree Fock method also yields excited orbitals which can be used to create excited states. I think one of the most accurate method to go beyond Hartree Fock in solids i the so called GW approximation. This can also be used to improve on excited states.
As you have pointed out, there exists a wide variety of approximations for the calculation of electronic band structure. The choice of method is typically a matter of scale, available computational resources and desired level of accuracy. In an ideal world, everything could be done using ab initio methods, but that is not practical at present. Hartree Fock (typically more common in the computational chemistry community) and DFT (typically favoured by physicists and also common in chemistry) can deal with systems with perhaps something on the order of 100s of atoms (some groups are pushing the boundaries and working with molecules of 1000s of atoms, but with minimal basis sets) of perfect bulk crystals, where symmetry can be utilised. These are ground state theories and therefore cannot on their own accurately predict band gaps, but that is a matter for another thread perhaps. At the same time, although the conduction and/or valence band edges are predicted to be at the wrong energy, the band structure topolgy is predicted quite well and can be calculated throughout the entire Brillouin zone. There are methods for improving on the limitations, often using some form of many-body perturbation theory, including post-Hartree Fock methods or GW, or using so-called hybrid functionals (controversial in the solid state physics community but effective in certain circumstances). However, semiconductor device modelling, for example, is out of the question at present using ab initio techniques. That is where empirical or semi-empirical models such and tight binding and k.p perturbation theory come in. In the case of the k.p model, the Hamiltonian is constructed using parameters from experiment and (sometimes entirely from) ab initio calculations. This is often referred to as multiscale modelling, which the group I work with uses in the modelling of semiconducting quantum dots. As I alluded to before however, there is a trade-off in terms of accuracy. The most commonly used 8 band model and even the higher 14 and 16 band models that I use cannot reproduce the band structure through the whole Brillouin zone. In fact, the region accurately calculated is often only around 0.1 reciprocal Angstroms either side of the critical point calculated at, typically Gamma. However, this is often enough for device engineering purposes. Higher band models show some promise in improving on this. I have seen one example for a 30 band model from the band structure of GaAs is reasonably predicted throughout the entire Brillouin zone.
Thanks a lot! Why semiconductor device modelling cannot be solved with ab initio method? What is the difference? You mentioned that tight binding and k.p are empirical or semi-empirical. Is it just because we do not know the exact form of potential? Also, I thought methods like Hartree-Fock, which solving equations iteratively, get better and better wavefunctions and energy. Is that we insert the wavefunction back into the Schrodinger equation to get the form of potential?
I think tight binding is semi empirical because it requires prior knowledge of the crystal lattice. Ab initio modelling of semiconductors (i.e. calculating the band gap) is one of the big open questions in condensed matter physics. Solve it, and you'll become famous. (Hehehehe, that should keep the young whipper snapper busy for a bit).
Tight binding means only that you use some set of atomic orbitals as a basis. Further steps can be done both ab-initio and semi-empirically. Quite accurate values for the band gap can be obtained using e.g. the GW method mentioned below. I think the ab initio description of semiconductors is not very much of a problem today.