The electronic structure of materials is determined according to the chemical composition such as crystal structure and atomic coordinate. Graphene, a 2D material, have the Dirac point, which has a zero band gap. However, this property can affect external factors, particularly when applying an external electric field to the bilayer graphene. The band splits at the Dirac point, which allows us to adjust the band gap. ,
In this weekly tip, we will illustrate the effect of applying an electric field to an electronic structure using bilayer graphene.
1. Structure modeling and electric field setting
We used the Bernal-stacked bilayer graphene, which is one of the most common among graphene and well known as having the lowest energy. The interlayer distance is set as 3.35 Å, which is the interlayer distance of graphite.
You can adjust the electric field setting using the following keywords.
Set the electric field direction perpendicular to the basal plane and parallel to the c-axis, and the electronic structure of two layers will differ. On the other hand, the electric field direction was set along the c-axis to find the difference. The materials are located at the center of the vacuum; therefore, the position of the sawtooth potential will be added and set to 0; and the width of the sawtooth potential, to 1 Å. Furthermore, the magnitude of the electric field will be set to 0 and 0.5 V/Å.
The unit of an electric field is an atomic unit in Quantum Espresso. 1 V/Å is 0.0194469054 a.u.; hence, we need to put a value as a multiple of the value to set the magnitude of the electric field as V/Å unit. For example, 0.5 V/Å is 0.5 × 0.0194469054 = 0.0097234527.
After the modeling and electric field setting are finished, we need to determine a way to check the change of an electronic structure. Adding the Charge Density solver to see the electrostatic potential; and DOS solver, to confirms the density of states that electrons can occupy, we can visually understand from post-processing. The previous weekly tip described the method of obtaining these data. The following table will help you to get the appropriate data from Materials Square.
|Data||Solver tab||MatSQ's Weekly Tip|
|Density of states||DOS||#6 Easy to Get, but Contains a Lot of Information, Density of States(1)|
|Electrostatic potential||Charge Density||#20 Dipole Correction, an Offsetting Effect of the Artificial Electric Field|
|Band structure||Band Structure||#12 Finding the Electronic Structure of a Material Using a Band Structure (1)|
2. The electronic structure depends on the applied electric field
Add the Charge Density module to visible the electrostatic potential. The potential in a vacuum is parallel to an x-axis when an electric field is not applied. However, when an electric field is applied, the potential will have a slope because it is affected by the electric field.
The band gap of the high symmetry point K varies when an electric field is applied.
|E (V/Å)||VBM (eV)||CBM (eV)||Δ (eV)||Band gap (eV)|
If there is no electric field, the bilayer graphene has zero-gap, single-layer graphene. However, if the electric field is applied, a band gap appears.
Adding the projected density of states (PDOS) and projected band structure (Fatband) determines the state caused by what orbital of what atom. Select what you want, and add the PDOS wherein you can see PDOS’ upper and lower layers with different shapes from each other.
The electric field, which is applied perpendicularly to the basal plane of graphene, makes two layers no longer identical. Therefore, the band gap was increased because of the separated conjugated states with transformed band structure by differing the identical environment for the two layers.
In particular, the state caused by 2pz orbital around the Fermi level significantly differed. Therefore, it can be comprehended that the perpendicular electric field affects the pz orbital, which is perpendicular to the basal plane of graphene, instead of px and py orbitals, which are parallel to the basal plane.
For the electric field applied and not applied cases, draw the Charge Density Difference to see the electron moved between two graphene layers. The electron moved into the opposite direction of the electric field because it has a negative charge.
3. Precautions of the electric field applied system
Most of the DFT package, even Quantum Espresso, creates a sawtooth potential depending on the input electric field amplitude. In this example, a 0.5 V/Å of an electric field is applied to the length of the supercell that is 20 Å, creating a difference of 0.5×20 = 10 volt in the supercell. However, an applied electric field has a stronger intensity because of the screening effect produced by the material.
The actual intensity of an electric field can be determined from the slope of potential in a vacuum. In the example with a 0.5 V/Å electric field, the potential in the vacuum has a slope of 0.66, which means that the actual applied electric field has an intensity of 0.66 V/Å.
As the ratio of materials in the supercell decreases, the electric field screening effect will do as well so that the difference in the actual intensity and input intensity of the electric field will decrease. Therefore, considering this, it is better to set a sufficient vacuum size.
In this weekly tip, we determine the electronic structure that becomes different when an external electric field is applied throughout the electrostatic potential, Density of States, and band structure. The electronic structure has a huge change when the electric field is applied, and we can understand the changed electronic structure by combining the information from several data results. In particular, there are several kinds of research that focus on the properties of materials or adjust the band gap by applying an electric field. Applying the electric field DFT calculation can be useful.
Find out the electronic structure of a structure that applied an electric field in Materials Square!
 Castro, E. V., Novoselov, K. S., Morozov, S. V., Peres, N. M. R., Dos Santos, J. L., Nilsson, J., ... & Neto, A. C. (2007). Biased bilayer graphene: semiconductor with a gap tunable by the electric field effect. Physical review letters, 99(21), 216802.
 Balu, R., Zhong, X., Pandey, R., & Karna, S. P. (2012). Effect of electric field on the band structure of graphene/boron nitride and boron nitride/boron nitride bilayers. Applied Physics Letters, 100(5), 052104.
 Ramasubramaniam, A., Naveh, D., & Towe, E. (2011). Tunable band gaps in bilayer graphene− BN heterostructures. Nano letters, 11(3), 1070-1075.
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