**2. Theoretical methods**

We have mainly used *ab initio* density functional theory-based methods to calculate various properties of defected 2D materials such as graphene and its derivatives in general. In this section, we will provide a brief introduction to the theoretical methods used.

#### **2.1. Density functional theory**

Various different properties of these many-body systems are described by the wave func‐ tions associated with it. These wave functions are governed by the time-dependent Schrö‐ dinger equation

$$
\hat{H}\Psi = \mathbb{E}\Psi\tag{1}
$$

where *H* ^ is the Hamiltonian of the many-body systems and represents the energy operator, and *E* is the total energy of the system. However, one needs various approximations to solve the Schrödinger equation for all kinds of systems.

To exploit these various properties, the samples have to be made in a scalable way. Chemi‐ cal vapor deposition (CVD) has become a very common method for large-scale fabrication. Nonetheless, the CVD samples inevitably contain defects, for example, edges, hetero struc‐ tures, grain boundaries, vacancies, and interstitial impurities [13–15]. These defects can be seen very easily in transmission electron microscopy (TEM) experiments [16] or scanning tunnel‐ ling microscopy (STM) experiments [17]. **Figure 1a**, **b** shows experimental STM and TEM images of an isolated single vacancy in graphene. In the STM image, the single vacancy can be seen as a blob because of increased local density of states. These states appear due to the

**Figure 1.** (a) Experimental STM image of single isolated vacancy in graphene. Reprinted with permission from Ugeda et al. [17], copyright (2010) by the American Physical Society. (b) Experimental TEM image of reconstructed single vacancy with atomic configurations. Reprinted (adapted) with permission from Meyer et al. [16], copyright (2008) by

In general, these defects manipulate the properties of the materials and hence their avoid‐ ance or deliberate engineering requires a thorough understanding. In one hand, defects can be detrimental to device properties [13], but on the other hand, especially at the nanoscale,

In this book chapter, we address a few cases of defects in 2D materials such as graphene and its derivatives. We show how one can tune the various properties of the pristine materials with

We have mainly used *ab initio* density functional theory-based methods to calculate various properties of defected 2D materials such as graphene and its derivatives in general. In this

Various different properties of these many-body systems are described by the wave func‐ tions associated with it. These wave functions are governed by the time-dependent Schrö‐

defects can bring new functionalities which could be utilized for applications [18, 19].

the control insertion of defects in these systems and use them in various applications.

section, we will provide a brief introduction to the theoretical methods used.

presence of dangling bonds around the single vacancy.

the American Chemical Society.

216 Recent Advances in Graphene Research

**2. Theoretical methods**

**2.1. Density functional theory**

dinger equation

In density functional theory (DFT), the electron density *n*(*r* ⇀ ) is used to obtain the solution of the Schrödinger equation. The core concept of the DFT is given by two theorems of Hohen‐ berg and Kohn [20], where they showed that the properties of interacting systems can be obtained exactly by the ground state electron density, *n*0(*r* ⇀ ). Following the two theorems, the total energy of the system can be written as follows:

$$E\left[n\left(\bar{r}\right)\right] = F\left[n\left(\bar{r}\right)\right] + \int V\_{ext}\left(\bar{r}\right)n\left(\bar{r}\right)\,d\bar{r} \tag{2}$$

Where functional *F* represents kinetic energy and all electron-electron interactions. Function‐ al *F* does not depend on the external potential, and hence, it is same for all the systems. However, Hohenberg-Kohn theorem does not provide any solution toward the exact form of the functional *F*.

Kohn and Sham [21] gave a way around to obtain the functional *F* by replacing the interact‐ ing many-body system with a non-interacting system consisting of a set of one electron functions (orbitals) while keeping the same ground state. According to the Kohn-Sham formalism, the total energy functional can be written as follows:

$$E\left[n(\bar{r})\right] = T\_s\left[n(\bar{r})\right] + \left[V\_{ext}(\bar{r})n(\bar{r})\right]d\bar{r} + \frac{1}{2}\left[\int \frac{n(\bar{r}\_1)n(\bar{r}\_2)}{|\bar{r}\_1 - \bar{r}\_2|}d\bar{r}\_1d\bar{r}\_2 + E\_{\text{xc}}\left[n(\bar{r})\right] \right] \tag{3}$$

Where *TS* is the kinetic energy term of the non-interacting electrons, and *Vext* is the external potential. The third term in the above equation is the Hartree term representing the classical Coulomb interactions between electrons, and the last term is known as exchange-correlation energy (*EXC*), which contains all the many-body effects. The formalism of Kohn-Sham is an exact theory. If the form of the *EXC* is exactly known, then using this formalism, one can calculate the exact ground state of the interacting many-body system.

In reality, the exact form of the exchange-correlation is not trivial, and hence, it is necessary to model the form of the exchange-correlation. Different forms of exchange-correlation can be constructed depending upon various level of approximation, for example, local density approximation (LDA) [20, 22, 23], generalized gradient approximation (GGA) [24–26], hybrid functionals (a mixture of Hartree-Fock and DFT functionals) etc. It is also important to remember that the implementation of single-particle Kohn-Sham equation is not trivial due to the complex behavior of wave functions in different spatial region, for example, in the core and in the valence region. To describe this complex wave function, a complete basis function is needed which can be of different form, for example, plane waves, localized atomic-like orbitals, Gaussian functions etc.
