**2. Density functional theory study of layered PtSe2**

We employed density functional theory (DFT) to calculate the electronic band structure and density of states of mono layer, bi-layer, tri-layer, and bulk 1T -phase PtSe2. This section deals with the theoretical calculations. DFT is a quantum modeling method used to investigate the properties of chemical systems, including atoms and molecules (i.e., many body systems), particularly the electronic structure properties. It is an ab inito method for solving the Schrodinger equations for many-electron systems which are defined by the electron density. The approach taken is, instead of using a many-body wave function, one-body density is used as the fundamental variable. Since the electron density *n*(*r*) is a function of only three spatial coordinates, rather than 3N coordinates of the wave function, DFT is computationally feasible for small to large systems. The root of DFT comes from two theorems given by Hohenberg and Kohn who considered interacting electrons in an external field [16, 17]. The theorems state that the ground state energy is a unique function of electron density, allowing us to work in three dimensions than in 3N dimensions. Only the electron density that minimizes the energy of functional is taken, with the assumption that the function is known. HK theorems sound simple but can be applied only under certain circumstances but they are not considered in practice as densities of atoms do not obey these constraints. Taking into consideration some of the fundamental issues, Kohn and Sham reduce the problem to noninteracting electrons moving in an effective potential, leading to a set of self-consistent, single particle equations known as Kohn-Sham equations that contains exchange correlation potential. The result of DFT calculations depends on the choice of exchange-correlation functional. In terms of increasing accuracy, we have LDA, GGA, meta-GGA, and hybrid functionals. Some functionals give good results for one system and some for another. More accurate functionals consume more computational resources with a trade-off between accuracy and speed of calculation.

#### **2.1 Computational methods**

All the computational calculations were performed using the DFT with the projector augmented wave (PAW) pseudopotentials available with quantumespresso [18]. Which is an integrated suite of open-source codes for the electronic *Thickness Dependent Spectroscopic Studies in 2D PtSe2 DOI: http://dx.doi.org/10.5772/intechopen.103101*

structure calculation and materials modeling at the nanoscale. We used the generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof (PBE) functional for the calculation of exchange and correlation potential. The van der Waals correction for the layered structures was taken into account using the DFT-D2 method as proposed by Grimme [19]. The arrangement of layers was taken such that it has the lowest ground state energy value and the top-to-top (AA) stacking order is the most favorable one with the interlayer distance calculated to be 2.44 Å. The kinetic energy cut-off for a plane-wave basis set was taken to be 800 eV.

The convergence criteria for self-consistent calculations for electronic structures were set to 10−6 eV. For the optimized geometrical configurations, the energy convergence criterion was set to 10−5 eV, structure relaxations were conducted until the residual force acting on each atom is less than 0.01 eV/Å and pressure values less than 1 kbar. The sampling of the first Brillouin zone was done using Г-centered *k*-point mesh of 15 × 15 × 15 and 15 × 15 × 1 for bulk and thin film structures, respectively. A region of at least 14 Å vacuum space was added in the *z*-direction to minimize the interaction between the neighboring atomic layers.

#### **2.2 Structure and electronic properties**

The bulk phase of PtSe2 has 1T phase with tetragonal symmetry having space group P-3 m1 and the lattice constant, after optimization, found to be *a* = 3.77 Å, *c* = 5.52 Å, a significant close to the experimental value and similar other reported values as discussed earlier. The monolayer structure is composed of three atomic sublayers with a Pt layer sandwiched between two Se layers. The lattice constant is calculated to be 3.70 Å which is in agreement with previously reported data, and the vertical distance between the upper and lowermost layer of Se is about 2.65 Å close to the reported value of 2.53 Å [12]. While moving from bilayer to bulk structure, the interlayer distance also decreases, along with the trend of increasing layer-layer interaction which enlarges the covalent Pt-Se bond. The lattice constant of monolayer, bilayer and trilayer is found to be 3.70 Å, 3.73 Å and 3.74 Å respectively and for bulk it is 3.78 Å.

**Figure 4** shows the electronic band structure of 1L,2L, 3L and bulk PtSe2.The electronic structure calculations using the PBE functional show the transition from semiconductor to (semi)metal behavior of the material. The calculated bandgap of monolayer is around 1.39 eV, close to the experimental value [20]. While moving from monolayer to bilayer, the electronic band gap rapidly decreases to 0.38 eV. It is also found that PtSe2 crystals having a thickness larger than two layers exhibit metallic behavior. Looking at the band structures, one can see that 1T-PtSe2 monolayer has its valence band maximum (VBM) at Г point and conduction band minimum (CBM) within Г-M point. While going from monolayer to bilayer and higher layers, we observe that position of CBM at Г-M point is fixed and VBM shifted from Г point to within K-Г high symmetry point. The reason behind this shift might be due to the non-periodicity of layered structure along the growth direction which is different from the band structure of bulk 3D structure of same material, as the energy band structure is strongly dependent on the crystal periodicity. The decrease in band energy of CB states and increase in VB states leads to metallization starting from trilayer [21, 22].

The DOS variation with different layers is shown in **Figure 5a–d**. In the DOS plots, for monolayer (**Figure 5a**), peak appears for the VBM, depicting a greater number of bands near the VBM, and flat near 0 eV. For the bilayer (**Figure 5b**), the

**Figure 4.** *Calculated electronic band structure of (a) 1L, (b) 2L, (c) 3L and (d) bulk PtSe2.*

**Figure 5.** *Calculated density of state of (a) 1L, (b) 2L, (c) 3L and (d) bulk PtSe2.*

*Thickness Dependent Spectroscopic Studies in 2D PtSe2 DOI: http://dx.doi.org/10.5772/intechopen.103101*

peak disappears near the VBM, where the band involves two peaks near Г point. This remains until peak appears around fermi energy when it comes to more than three layers. It is also noticed that DOS under VBM is small for multilayers due to the large splitting between the first valence band with the second valence band [23, 24]. In this study, we were able to observe an increase in band gap with the decrease in layer numbers from bulk down to monolayer structures. Unlike other TMDs like MX2 (M = Mo and W; X = S and Se) which are direct bandgap semiconductors at monolayer, there is no shift from indirect-to-direct band gap with decrease in number layers from bulk to monolayer limit. This may be due to the difference in crystal structure i.e. – MX2 has 2H structure and PtSe2 has 1T structure. We were able to observe the inverse relationship between the band gap and number of layers, which is governed by factors such as quantum confinement effect and interlayer interaction.
