Crystal parameters extraction from mix structure of cubic and hexagonal MnAs/InAs/GaAs(111)B heterostructure system.

 X-ray diffraction (XRD)

    Crystal structures are very important for device technology. Specifically, single-crystal or epitaxial grown material systems play a key role to develop device technology on the right track. So, studying crystal structures of novel materials systems is quite important. It is well known that the crystal structure of solid is widely examined by x-ray diffraction (XRD) techniques. The wavelength of the x-ray is around 0.1 to 10 Å. The material structure study is carried out based on the interference principle of x-ray with the materials system. Through the interaction, we can extract crystal parameters. The interference obeying a famous law defined by Bragg is known as Bragg law. We consider a simple crystal structure where crystal parameters (a=b=c) are connected with miller indices (hkl) and interplanar spacing d. It is seen that a monochromatic x-ray beam has a coherent radiation incident on a crystal as shown in the schematic figure 1. 

Figure 1: Diffraction relation examines.

The atomic lattice which constitutes diffraction is from parallel planes in symmetric measurement. For the simple cubic system, the d and a, and hkl are related by the following relations,

Similarly for a hexagonal unit cell d is connected as follows. 

Bragg's law derivation

The diffraction peaks are visible from constructive interference of diffracted beams from parallel planes. It is necessary to have an in-phase of diffracted beams after leaving the crystal planes. We labeled beams 1 and 2 in figure 1. Path difference between the two diffracted beams (EF+FG) will be equal to an integral number (n) of the wavelength of the incident x-ray.

EF+FG= nλ (3)

EF=FG then, Sinɵ=EF/d

Therefore, EF=dsinɵ, then equation (3) becomes

nλ=2dsinɵ (4)

It is known as Bragg’s law. This explains the angular relation of diffracted x-ray with their wavelength. For first-order diffraction, (n=1) Bragg’s law becomes,

λ=2dsinɵ (5)

At the diffraction conditions incident angle ω is equal to the diffraction angle ɵ. Through this measurement, we only can have information about interplanar distance and hence lattice constants. The basic XRD measurement setup schematic is shown in figure 2.

Figure 2: Schematic of symmetric XRD measurement 

Example of lattice parameter extraction

Now, we consider MnAs/InAs/GaAs heterostructure where MnAs is a hexagonal ferromagnetic metal, InAs and GaAs are cubic compound semiconductors. The structure is very important for spintronic device applications. Anyway, we have carried out XRD symmetric diffraction, where incident angle ω and diffraction angle ɵ maintain the same angle during the 2ɵ/ω measurement. Figure 3 shows the measured XRD result. From figure 3, we see diffraction peaks from different planes from respective materials. Hence with angle position information and corresponding Miller indices (hkl), we can extract the lattice parameters using equations (1), (2), and (5). The corresponding lattice parameters of cubic GaAs and InAs are 5.65 and 6.05 Å respectively. The lattice constant of hexagonal MnAs is around 5.71 Å. 

Figure 3: Symmetric ω/2ɵ XRD results of MnAs/InAs/GaAs(111)B

However, for the early rising researcher, this kind of basic study is quite important to develop themselves in science and technology. The above discussion is somehow different from a usual discussion about crystal structure study.  I believe that my little effort will be helpful to understand the mix-crystal structure study. Therefore, I would like to invite people who are really interested in studying crystallography. You may also have an interest in TLM devices, ferromagnetic phase transition-related problem solving, or Schottky diode evaluation for power device applications. These are available in my blog also please check here.