Our daily life takes place in a three-dimensional space. We are so familiar with the three-dimensional space that we call the two-dimensional sub-dimension one and zero dimensions low. In a way, it seems like an unfair designation. For those who are accustomed to two-dimensional planes, one-dimensional lines, and zero-dimensional points, what images come to mind when they first hear the word low-dimensional matter?
For example, a possible misunderstanding when you first hear the word two-dimensional material is to imagine a two-dimensional material as a plane with zero thickness. But we know that all matter is made up of atoms. So even extremely thin matter is thicker than an atom. The same is true for one-dimensional matter and zero-dimensional matter. A line without thickness is not a point without width. No matter how low-dimensional matter is, it has “volume.”
What is low-dimensional matter?
In solid physics, the criterion for the dimension of a matter having a crystalline structure is periodicity. Stacking the same Lego blocks over and over will only increase the height. Continuing to build blocks that are larger than the size of the Lego blocks themselves creates a one-dimensional cycle that repeats in only one direction.
Figure. 1 (Left) A single Lego block, which is a repeating unit. (Right) By repeatedly stacking the same Lego blocks in height, a one-dimensional cycle occurs. The red arrow indicates the grid vector.
(Figure 1. To facilitate mathematical modeling, periodic boundary conditions are introduced, but every solid crystal has a surface. However, a piece of metal with the width, length, and height of 1 cm each and a volume of 1 cm3 is a three-dimensional metal that may look small in a macroscopic dimension.) Such matters are called one-dimensional materials and are zero-dimensional materials if they do not repeat in either direction.
You can ask these questions here. If 100 identical Lego blocks are stacked high, is this a zero-dimensional material or a one-dimensional material? If the properties of the material inside do not change as the thickness of the block is further increased in the repeating direction, it can be said that it is a one-dimensional crystal structure. On the other hand, a two-dimensional material has two-dimensional periodicity because one unit repeatedly appears in two directions. (Figure 2)
Figure. 2 Two-dimensional grid. The unit cell repeatedly appears in two directions.
Carbon, a familiar element, has allotropes of various dimensions. Low-dimensional materials include soccer-ball-shaped fullerenes (zero-dimensional materials, also called buckyballs) composed of 60 carbon atoms, carbon nanotubes with one-dimensional repeating periods, graphene with two-dimensional repeating periods, etc. and three-dimensional materials, including diamond and graphite.
Various two-dimensional materials
The era of two-dimensional material research has begun, starting with graphene which is one layer removed from graphite with a stacked structure. There are materials, such as hexagonal boron nitride (h-BN), stanene, germanene, and silicene, that have a monoatomic layer structure like graphene and transition metal chalcogen compounds (TMDs) consisting of three atom layers and slightly thicker Mxene (a material in which carbon or nitrogen is bonded to a transition metal) are being actively studied. The first principle calculation method plays a major role in research to find new two-dimensional materials. [1–3]
First principle calculation of two-dimensional matter
The first step in starting the first principle calculation of the crystal structure is to establish the unit cell, which is a repeating unit. In the case of three-dimensional material, since it has three lattice vectors that are linearly independent of each other—whether it is a principal lattice vector or not—there is nothing to worry about.
However, since three-dimensional unit cells are still used when calculating two-dimensional materials, there is a degree of freedom in determining the size of the lattice vector in the third direction (direction perpendicular to the two-dimensional plane) in which the crystal structure does not have periodicity, and how to determine it is ambiguous. Naturally, if the length is too short, there may be interactions between neighboring images that should not have existed.
However, if the length is increased, the calculation cost increases because the number of FFT (Fast Fourier Transform) grid points increases. Since there are two-dimensional materials with several layers of atoms, such as TMDs and Mxene, with a certain thickness, set the lattice vector so that the gap between nearby planes, commonly called vacuum gap size, is 12-20 Å, not the lattice vector itself. (Figure 3)
Figure. 3 A side view of a unit cell of two-dimensional material MoS2. There is a degree of freedom in determining the vacuum gap size.
Meanwhile, the vacuum gap size does not only affect the calculation time but also changes the total energy of the calculated material. Therefore, it is important to always consistently calculate when comparing the relative energies between different materials or calculating the energy of formation after adsorbing atoms or molecules onto two-dimensional material.
Also, remember that even though the absolute value of the total energy varies greatly depending on the size of the unit cell, the value obtained from the “difference” of energy, such as relative energy or formation energy, converges relatively quickly. Note that not only the vacuum gap size but also the fact that the energy difference value converges relatively quickly when running energy cutoff or k-point convergence tests.
Finally, I think doping calculations should be particularly careful when calculating two-dimensional materials. Doping electrons or holes into a system in an electronic structure calculation program is functionally very easy. All you have to do is to write in the input file how many electrons to add or subtract per unit cell.
However, it is also important to keep in mind that depending on the amount of charge on the electrons we add or subtract, a compensating background charge is automatically created, making the entire unit cell electrically neutral. This is because the total energy of the lattice structure having periodicity diverges when the unit cell is not electrically neutral. There would be no big problem if a physically reasonable amount of electrons were doped, depending on the target system.
However, if too many electrons are doped, the additional electrons may prefer to settle in a vacuum gap where a uniform positive background charge spreads rather than entering a two-dimensional material that is already filled with other electrons. This can be noticed quickly by drawing the energy band after calculating the electronic structure calculation.
Comparing the electronic structure of undoped and doped materials, s-like bands (a band in which the energy level increases in a parabolic form as it goes from the zone center toward the zone boundary) appear near gamma points, and it should not be prematurely judged that a new state has emerged after doping. It is likely to be embedded between comfortable vacuum gaps.
An easy way to check this is to draw a picture of the charge density where the state is spatially located. This is what I experienced while doing research and as a reviewer of a journal. It was pointed out in the process of reviewing a paper.
For undergraduate students who are new to low-dimensional matter research, here are some general points to note in the low-dimensional matter and first-principle calculations. It will be a good study to compare the energy band obtained from the tight-binding model in a simple system, such as a one-dimensional grid or a two-dimensional square grid, with the first principle calculation results.
1) A. K. Geim and I. V Grigorieva, Van Der Waals Heterostructures, Nature 499, 419 (2013).
2) B. Anasori, M. R. Lukatskaya, and Y. Gogotsi, 2D Metal Carbides and Nitrides (MXenes) for Energy Storage, Nat. Rev. Mater. 2, 16098 (2017).
3) N. Mounet, M. Gibertini, P. Schwaller, D. Campi, A. Merkys, A. Marrazzo, T. Sohier, I. E. Castelli, A. Cepellotti, G. Pizzi, and N. Marzari, Two-Dimensional Materials from High-Throughput Computational Exfoliation of Experimentally Known Compounds, Nat. Nanotechnol. 13, 246 (2018).
Prof. Jeongwoon Hwang | Chonnam National University