Energy band structure (or electronic band structure) describes the law of electron movement in solid-state physics. Typically, the energy band structure is mainly divided into the conduction band, valence band, and forbidden band. Owing to the principles of lowest energy and Pauli exclusion, the band split by the inner energy level is always firstly filled by the electron, and then occupies the outer band with higher energy.

Energy band structure is a significant attribute to evaluate materials that are ubiquitous in life. Energy band structure prediction can greatly guide experiments in terms of new phenomena prediction and new materials discovery. It plays a crucial role in solid electron research. It includes band gap, Fermi energy, the width of conduction and valence bands, charge densities, effective mass of electron and hole, band structures and so on. New connections between topology and energy band structure assist in a wide variety of discoveries in condensed matter physics research. Energy band structure prediction for semiconductors provides useful information about the band gap and band-edge energies, which creates new opportunities in electronics.

Figure 1. Energy band diagram (Khan, Y.; *et al*. 2020)

### Our Methodologies

- Dynamical mean-field theory (DMFT)

The main idea of MFT is to replace all interactions to any one body with an average or effective interaction, so MFT sometimes is also called a molecular field. It can reduce any many-body problem into an effective one-body problem. The dynamical mean-field theory is a widely applicable approximation scheme for the investigation of correlated quantum many-particle systems on a lattice, such as electrons in solids and cold atoms in optical lattices, critical behavior of the mixed-spin model. At Alfa Chemistry, we apply the DMFT to calculate electronic band structures, which has led to a powerful numerical approach allowing us to explore various properties of correlated materials.

- First-principles calculation

First-principles calculation refers to an algorithm that using the principles of quantum mechanics to directly solve the Schrödinger equation after a series of approximate treatments based on the principle of the interaction between nuclei and electrons and the basic laws of motion. The first principles calculation can be divided into two methods, ab initio calculation based on Hartree-Fork self-consistent field calculation, and density functional theory (DFT) calculation.

We utilize the first principles to develop the qualitative and quantitative calculations of the energy band structure, including the width of the energy band, the analysis of impurity state and spin polarization, the diagram of energy band of substrate material. Moreover, we also study the density of states which is applied as a visualization of the energy band and can support us to predict the energy band structure more accurately.

- Free electron approximation

In the free electron approximation, the interaction between electrons can be completely ignored. This kind of approximation allows the use of Bloch's theorem, which states that electrons in periodic potentials have wave functions and energies and have periodicity in the wave vector until the phase shift between adjacent reciprocal lattice vectors is constant.

We apply the free electron approximation model in the study of metal materials to explain the valence band of metals since the distance between adjacent atoms in metals is small, and the overlap between atomic orbitals and the potential on adjacent atoms is relatively large. In that case, the wave function of the electron can be approximated by a (modified) plane wave. Therefore, we can apply the free electron model to study the motion behavior of electrons that can pass through the lattice in a near-free motion, as well as calculate the electronic energy band structure of metals.

- Generalized gradient approximation (GGA)

GGA function is developed based on the correction of Local Density Approximation (LDA). Compared to LDA, GGA has the following advantages: more accurate calculations of atomic and molecular energy, correction of excessive bonding, more accurate calculations of reaction activation energy.

Our teams use the generalized gradient approximation method in which the coulomb interaction has been taken into consideration, to calculate and predict the energy band structure of a series of alloys. In addition, the obtained information such as the density of states, dielectric function, and optical absorption coefficient is also an important supplement to the energy band structure information.

- Green's Functions methods

Green's functions can provide not only the ground state of the total energy of the system but also the observable value of its excited state. The pole of Green's function is the energy of the quasi-particle, which is also called the energy band of the solid.

We apply Green's function method to calculate the energy band which includes the many-body effect of the electron-electron interaction. Once the self-energy of the system is known, our scientists use the Green's function to perform an accurate calculation by solving the Dyson equation. Moreover, this method is more relevant in solving the calculation of the energy band diagram and spectral functions, and can support the Ab initio calculation.

- Korringa–Kohn–Rostoker (KKR) methods

Korringa-Kohn-Rostoker methods use multiple scattering theory to solve the Schrödinger equation and determine the band structure of periodic solids. This method allows the calculation of surfaces, interfaces and electronic properties of a general layered system with two-dimensional periodicity, which requires no more approximation.

At Alfa Chemistry, we combine the Green's function and KKR method to obtain the Korringa-Kohn-Rostoker coherence potential approximation (KKR-CPA) to predict the energy band structure. KKR-CPA is used to calculate the electronic states of alternative solid solution alloys. In addition, we are also capable of using Local Self-Consistent Multiple scattering (LSMS) to find a wider range of electronic states of condensed matter structures.

- Tight-binding model

The tight-binding model assumes that the electrons in the crystal behave like a collection of constituent atoms, and it also assumes that the solution of the time-independent single-electron Schrödinger equation can be well approximated by the linear combination of atomic orbitals.

We apply the tight-binding model in the structure analysis of materials where the potential overlap between atomic orbits and neighboring atoms is limited, such as the band structure of materials including Si, GaAs, SiO2, and diamond. We also use a hybrid tight-binding-near free electron approximation model to describe the wide near free electron approximate conduction band and the narrow embedded compact d-band in transition metals. In addition, we use the wave function approximation extension based on the pseudo-potential method to calculate the frequency band structure of the tight-binding model or the hybrid tight-binding-near free electron approximation model.

### Key Advantages

- A combination of topology and energy band structure is useful for deeply understanding the properties of materials.
- High-throughput energy band structure calculations facilitate your research.
- Keep pace with hot and advanced topics of materials science.
- Accurate prediction.
- Quick calculation.

### Customer Notice

### Customers provide

- Specific requirements of energy band structure prediction services
- Analysis cycle

### We deliver

- Calculation algorithms and methods
- Raw data
- Analysis results

Alfa Chemistry provides personalized and high-quality services of energy band structure prediction at competitive prices for global customers. Our energy band structure prediction services promote new breakthroughs in semiconductors, modern materials, and electron movement in crystals. Our services can satisfy fundamental and materials research. If you are interested in our services, please contact us for more details.

References

- Khan, Y.;
*et al*. Ionic moieties in organic and hybrid semiconducting devices: influence on energy band structures and functions.*J. Mater. Chem. C*2020, 8: 13953-13971. - Narang, P.;
*et al*. The topology of electronic band structures.*Nat. Mater.*2021, 20: 293-300. - Setyawan, W.; Curtarolo, S. High-throughput electronic band structure calculations: Challenges and tools.
*Comput. Mater. Sci.*2010, 49: 299-312.