The test materials with the structural methods you want are all here

In solid-state physics, the band structure of a solid (also known as the electronic band structure) describes the energy levels that are either allowed or forbidden for electrons. This is caused by the diffraction of quantum mechanical electron waves in a periodic lattice. The band structure of a material determines various properties, particularly its electronic and optical characteristics. Today, we will walk you through the methods for testing band structures. The experimental group plays to their strengths by directly using experimental instruments for measurement, while the theoretical group prefers to use first-principles calculations.

Figure 1:Band Structure of Graphene

Method 1: Angle-Resolved Photoemission Spectroscopy (ARPES)

When photons are incident on a sample, electrons within the sample absorb the photons and undergo transitions. If the energy exceeds the surface barrier (Φ, the work function of the material, which is the surface barrier that prevents valence electrons from escaping the sample, typically around 4-5 eV for metals), there is a certain probability that the electrons will escape from the surface of the sample. The maximum escape energy is ( hν - Φ ) (where ( hν ) is the energy of the incident photon).

Photons, typically from a gas discharge lamp, synchrotron radiation, or a laser, are incident on the sample, exciting electrons. The escaped electrons are then collected by an energy analyzer with a finite acceptance angle. During this process, the kinetic energy of the photoelectrons, the work function of the material, and the binding energy of the electrons together equal the energy of the incident photons.

By measuring the kinetic energy of photoelectrons at different emission angles, ARPES can obtain the momentum components of electrons parallel to the sample surface. By correlating the obtained energy and momentum, the electron dispersion relationship in the material can be determined. Additionally, ARPES can also provide the density of states and momentum density curves and directly reveal the Fermi surface of the solid. This is an experimental method.

Figure 2:Schematic of ARPES Experimental Geometry

Method 2: Cyclotron Resonance

Cyclotron resonance refers to the phenomenon of diamagnetic resonance that occurs when charge carriers in a semiconductor are subjected to both a constant magnetic field and a high-frequency electric field. When a semiconductor is placed in a uniform, constant magnetic field with an induction strength of ( B ), and if an electron in the semiconductor has an initial velocity ( v ) at an angle ( \Theta ) to ( B ), the magnetic force ( f ) acting on the electron is given by: ( f = -e (v \times B) ).

In a cyclotron resonance experiment, in addition to applying a constant magnetic field to the semiconductor sample, an alternating electromagnetic field is also applied, with the electric field component ( E ) perpendicular to the magnetic field. Under these conditions, the electron undergoes spiral motion around the magnetic field while also being affected by the alternating electromagnetic field. When the frequency of the electromagnetic field matches the electron’s cyclotron frequency, the electron is continuously accelerated by the alternating electric field, gaining energy and causing resonance absorption.

By measuring the frequency of the electromagnetic field and the magnetic induction strength ( B ) at resonance absorption, the effective mass of the charge carriers can be determined. Additionally, by changing the direction of the magnetic field and measuring the number and position changes of the resonance absorption peaks, the distribution of band extrema in the Brillouin zone and the shape of the energy band surface can also be determined. This is an experimental method.

Figure 3:Electron Motion in a Constant Magnetic Field

Method 3: Orthogonalized Plane Wave Method

The Orthogonalized Plane Wave (OPW) method is a simple approach that uses plane waves to expand the electron states in the valence and conduction bands. The basis for the expanded wave function consists of a set of plane waves orthogonal to the eigenenergy wave functions. Therefore, this method is called the Orthogonalized Plane Wave method (OPW). This method overcomes the challenge of describing wave functions that change rapidly near the atomic nuclei. Using a similar approach, normalized OPW functions can be combined to describe the plane waves at the symmetry points of the Brillouin zone, forming basis functions for the irreducible representations of the crystal space group. This is a computational model.

Figure 4:Schematic of the Muffin-Tin Potential

Method 4: Augmented Plane Wave (APW) Method

The Augmented Plane Wave (APW) method uses a so-called Muffin-Tin potential, which consists of a spherically symmetric potential centered at the ion positions, combined with a constant potential in the interstitial regions. In the spherical symmetric potential, the Schrödinger equation for an electron's motion can be solved in spherical coordinates. The augmented plane wave is equivalent to a plane wave outside the region of the spherical potential and is a linear combination of spherical harmonic solutions and the radial function in the spherical symmetric potential.

To satisfy the requirements of an acceptable wave function, appropriate parameters are chosen. At the boundary of the spherically symmetric potential, the wave functions in the two regions must match in both value and logarithmic derivative. This method was initially proposed by Slater in 1937 and, with the development of computer technology, has become more conveniently applicable in calculations. The number of APW functions depends on the crystal structure and the type of bands involved. Typically, the s- and p-bands converge faster, while the d-band converges more slowly. This is a computational model.

Figure 5: Plane Wave

Method 5: Green's Function Method

The Green's Function Method, also known as the KKR method (Korringa, Kohn, and Rostoker), assumes a Muffin-Tin potential with a constant potential outside the region of spherical symmetry. The wave function is considered to be scattered by the potential itself. Korringa (1947) divided the wave function into incoming and outgoing components. Kohn and Rostoker (1954) introduced an integral equation approach, where the integral is taken over all Muffin-Tin spheres. This method is very similar to the APW method.

In the KKR method, a summation over all reciprocal lattice vectors is required, leading to the derivation of the secular equation, which captures contributions from different spherical harmonic functions. This is a computational model.

Figure 6: Band Structure of a Topological Insulator

Method 6: Pseudopotential Method

A pseudopotential replaces the true potential within the core of an ion with a fictitious potential, but without altering the eigenvalues or the wave function in the region between the ions when solving the wave equation. The wave function derived from the pseudopotential is called the pseudo-wave function. In the region between ions, the true potential and the pseudopotential yield the same wave function.

The pseudopotential method substitutes the real potential with an effective potential. For certain crystals, calculating the Muffin-Tin potential distribution can be particularly complex. A simplified approach is to replace the atomic potential with a weaker potential, which for conduction band electrons yields the same discrete amplitude as in the KKR method. This is a computational model.

Method 7: k·p Method

For narrow-bandgap semiconductors, the k·p perturbation method is highly effective. This method assumes that all states ( U_n(0, r) ) and energies ( E_n(0) ) at ( k = 0 ) are known. Then, using crystal symmetry, it applies perturbation theory to determine the expression for ( E_n(k) ) and the wave functions near ( k = 0 ). The band structure at special points in k-space can be determined by experimentally obtained band parameters, such as the bandgap width, electron, and hole effective masses. From the k·p perturbation method, the band structure at other points in k-space can then be determined.

Since semiconductors with narrow bandgaps have strong conduction and valence band interactions and strong electron spin-orbit interactions, perturbation theory is well-suited for handling these effects. Therefore, the k·p perturbation method is especially important for narrow-bandgap semiconductors.