First-principles methods and empirical models are two fundamental approaches for calculating semiconductor band structures, each with distinct advantages and limitations. The choice between them depends on the required accuracy, computational cost, and specific material properties under investigation.

First-principles methods, such as density functional theory (DFT) and the GW approximation, solve the quantum mechanical equations of a system from fundamental physical principles without empirical parameters. DFT is the most widely used first-principles method due to its balance between accuracy and computational efficiency. It approximates the many-body Schrödinger equation by mapping it onto a system of non-interacting electrons in an effective potential. DFT provides reliable predictions of ground-state properties, including lattice constants, bulk moduli, and electronic charge densities. However, standard DFT suffers from the bandgap problem—it systematically underestimates bandgaps due to the approximate treatment of exchange-correlation effects. For example, DFT often predicts the bandgap of silicon to be around 0.6 eV, significantly lower than the experimental value of 1.1 eV.

The GW approximation addresses this limitation by incorporating many-body corrections to the electron self-energy. It yields more accurate quasiparticle band structures, particularly for semiconductors and insulators. GW calculations typically reproduce experimental bandgaps within 0.1–0.3 eV for many materials, making it a gold standard for electronic structure predictions. However, GW is computationally expensive, often requiring orders of magnitude more resources than DFT. This limits its application to small or moderately sized systems.

In contrast, empirical models like k·p theory and tight-binding rely on experimentally determined parameters to simplify the band structure calculation. k·p theory is particularly useful near high-symmetry points in the Brillouin zone, where the band dispersion can be expanded using perturbation theory. It is widely employed for modeling the electronic properties of bulk semiconductors, quantum wells, and superlattices. For instance, k·p models accurately describe the valence band structure of III-V compounds like GaAs, including heavy-hole, light-hole, and split-off bands. However, k·p theory is less reliable for materials with strong intervalley scattering or when applied far from the expansion point.

Tight-binding models approximate the electronic structure by considering atomic orbitals and their overlaps. They are computationally efficient and can handle large systems, including nanostructures and disordered materials. Tight-binding parameters are typically fitted to experimental data or first-principles calculations, making the method semi-empirical. While it lacks the predictive power of first-principles methods, tight-binding is valuable for studying trends in material properties, such as strain effects or alloy disorder. For example, tight-binding has been successfully applied to model the electronic properties of carbon nanotubes and graphene nanoribbons.

The strengths and limitations of these methods can be summarized as follows:

First-principles methods (DFT, GW)
Strengths:
- Parameter-free, relying only on fundamental constants.
- Predictive for ground-state and excited-state properties (with GW).
- Applicable to new materials without prior experimental data.

Limitations:
- High computational cost, especially for GW and large systems.
- DFT underestimates bandgaps; GW is more accurate but resource-intensive.

Empirical models (k·p, tight-binding)
Strengths:
- Computationally efficient, suitable for large systems.
- Can incorporate experimental data for improved accuracy.
- Useful for device modeling and engineering applications.

Limitations:
- Require empirical parameters, limiting predictive power.
- Accuracy depends on the quality of fitted parameters.
- Less reliable for materials with complex electronic interactions.

Applications of these methods vary depending on the research goals. First-principles methods are indispensable for fundamental studies of new materials, such as high-throughput screening of novel semiconductors or investigating defect physics. GW calculations are often used to validate experimental measurements or refine theoretical models. On the other hand, empirical models are favored in device physics, where rapid calculations are needed for heterostructure design or carrier transport simulations.

In summary, first-principles methods offer high accuracy and predictive capability but at a high computational cost, while empirical models provide efficiency and practicality at the expense of some predictive power. The choice between them depends on the specific requirements of the study, balancing accuracy, computational resources, and the availability of experimental data. Combining both approaches—using first-principles methods to inform empirical models—can often yield the most comprehensive understanding of semiconductor band structures.