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Two-dimensional materials arouse ever greater interest in the scientific community due to their electronic and optical properties. Among these 2D materials, the 2D family of transition metal dichalcogenides (TMDs) offers great potential for applications in optoelectronics and nanotechnology. Of these TMD nanomaterials, platinum diselenide PtSe2 has been extensively studied since the successful synthesis of a PtSe2 monolayer in 2015. In this chapter, the multilayer PtSe2 is investigated with first-principle calculations. In order to calculate the optical properties of the system, we first determine its dielectric function. From this, we can extract other optical functions, such as refractive index, extinction coefficient, absorption coefficient, and reflectivity. A good description of these properties can be enhanced by a detailed study of the material’s band structure.
Department of Physics, Tunis Faculty of Science, Tunis El Manar University, Tunisia
*Address all correspondence to: nourelhoudasafi07@gmail.com
1. Introduction
In 2004, Konstantin Novoselov and André Geim succeeded in exfoliating graphene from graphite. For this, they were awarded the Nobel Prize in Physics in 2010. Since this discovery, the scientific community has become very interested in 2D materials, due to graphene’s physical properties. Indeed, this one has shown high charge carrier mobility of 2.105cm2/Vs, excellent flexibility, transparency of 95% [1], high electrical conductivity, and thermal conductivity equal to 104Ω−1cm−1 and 3000W/mK [2], respectively. However, graphene is a zero-gap semimetal. This limits its use in optoelectronics. Other similar 2D materials, such as transition metal dichalcogenide semiconductors (TMDs), are paving the way for optoelectronic applications [3].
This present work deals with one of the TMDs PtSe2. It has a wide range of applications such as photodetectors [4, 5], gas sensing [6], and electronics. Platinum diselenide PtSe2 belongs to the transition metal dichalcogenide (TMD) family. Compared with other materials in the TMDs family, PtSe2 is characterized by high electron mobility at room temperature. The platinum diselenide material PtSe2 is centrosymmetric, stable in the 1T phase, with symmetry group D3d (space group P3¯m1), and crystallizes under the hexagonal structure [7]. The primitive cell of PtSe2 contains three atoms: one platinum (Pt) and two selenium (Se). In the same plane, the atoms are connected by covalent bonds with no charge transfer between them, whereas, out of the plane, the Se-Se atoms interact only through weak van der Waals (vdw) interaction.
Figure 1b shows a PtSe2 monolayer where the lattice parameter a is of order 3.7 Å, the Pt-Se bond length is 2.52 Å and the distance noted ’h’ separating the Sesup−Seinf atoms is equal to 2.53 Å. These parameters are measured using a scanning tunneling microscope (STM) [10].
Figure 1.
Structure of PtSe2 (a) top view where the elemental lattice is presented by a rhombus, (c) side view, atoms in blue; Se and atoms in purple; Pt, (b) monolayer PtSe2, visualized by Vesta [9].
The determination of a large band gap for multilayer PtSe2 remains a major theoretical and experimental challenge. At the experimental level, a study [11] showed by scanning tunneling spectroscopy (STS) and scanning tunneling microscopy (STM), that the value of the gap energy decreases from a value of 1.79 pm 0.04 eV for monolayer to 0.62 pm 0.02 eV for bilayer, and then, it cancels for trilayer. In contrast, another literature [12] showed that the trilayer PtSe2 is a semiconductor with a gap of 0.2 eV.
At the theoretical level, studies such as GW overestimate the gap energy, which predict that the band gap of monolayer, bilayer, and trilayer PtSe2 are equal, respectively, to 2.4, 1.5, and 0.7 eV. However, other theories based on the density functional approach give different estimates due to the accuracy of the adapted exchange-correlation energy functional. For example, the standard density functional theory (DFT) approximations (GGA-LDA) underestimate the gap energy, and they predict that the gap energy of the monolayer, bilayer and trilayer PtSe2 are equal, respectively, to: 1.2, 0.2, and 0 eV [12]. On the contrary, the hybrid functional HSE06 shows excellent accuracy in determining the prohibited bandwidths for the PtSe2 monolayer, bilayer, and trilayer: 1.7 (eV), 0.52 (eV) and 0.13 (eV) [13]. However HSE06 requires a large memory capacity and a very long computation time.
2.1 Calculation method
The electronic and optical properties are determined by the Quantum Espresso (QE) code [14]. The Perdew-Burke-Ernzerhof (PBE-GGA) [15] and hybrid functionals optB86b-vdw [16] are adopted for the exchange-correlation potential. The exchange-correlation energy (XC) between electrons is processed by the XC functional. The interaction between electrons and their nucleus is described by the projection-augmented plane wave (PAW) method, in which the valence electrons used are 10 platinum (Pt) 5d96s1 and 6 selenium (Se) 4s24p4. In order to obtain reliable and similar results to the different experimental and theoretical literatures, we proceed to choose appropriate values for specific important parameters: the cut energy Ecut and the number of k-points in the Monkhorst-Pack grid in the Brillouin zone (ZB) [2] according to a convergence criterion. Based on these tests, we applied 60 Ry for the cutoff energy and 15×15×1 for k-points for the relaxation and self-consistent calculation for multilayer PtSe2. On the contrary, the bulk PtSe2 is treated with Ecut equal to 90 Ry and k-points 15×15×15. In order to calculate the density of states, the cutoff energy is set at 90 Ry for all systems, and the k-points in the ZB chosen for the monolayer and bilayer of PtSe2 are 40×40×1, while those for the trilayer and quadrilayer are 33×33×1. For the bulk PtSe2, the k-points are kept at 33×33×33 (Figure 2).
Figure 2.
Brillouin zone, the blue triangle is the irreducible Brillouin zone of the monolayer PtSe2.
By using the «Variable Cell-relax» option of Quantum Espresso that allows relaxing both ions and the crystal lattice, the structure of the system is relaxed up to the total energy is less than 10−5Ry and the forces acting on atoms are less than 10−4Ry/bohr3. To account for van der Waals-type interactions, we used the Gimmes DT-D3(BJ) correction. We add a vacuum space on the z axis to avoid interaction with the neighboring mesh in the z direction.
2.1.1 Band structure of PtSe2 monolayer, multilayer, and bulk
The band structure calculation for multilayer PtSe2 are performed along the K−Γ−M−K path in the irreducible Brillouin zone. It is well known that PBE functionals underestimate bandgaps. We thus add the van der Waals correction, which is close to the experimental result. However, PBE-D3 predicts the semiconductor–metal transition of PtSe2 films from three layers, which disagrees with experimental data [12]. Therefore, we adopt the optB86b functional [16] for further calculations for bilayer and trilayer PtSe2.
2.1.1.1 Monolayer PtSe2
The PtSe2 monolayer is a semiconductor with an indirect gap of 1.448 eV without spin-orbit coupling (SOC) and 1.259 eV with SOC, which is in good agreement with previous theoretical studies of the same system [17, 18]. The conduction band minimum is located between the points “Γ” and “M,” and the valence band is at the point “Γ.” Taking into account the spin-orbit coupling effect, we notice that the degeneration at least of the valence band point “Γ” has disappeared, and the gap energy decreases (Figure 3).
Figure 3.
Band structure of monolayer PtSe2 without and with spin-orbit coupling (SOC).
2.1.1.2 Bilayer PtSe2
The PtSe2 bilayer is a semiconductor with an indirect gap of 0.14 eV by PBE-D3 functional and 0.30 eV by optB86b-vdw functional. OptB86b-vdw functional gives a good agreement with previous theoretical studies [17]. The conduction band minimum is between Γ and M points, analogous to the band structure of the monolayer. For comparison to monolayer, the valence band maximum is translated between the points K-Γor Γ-M (Figure 4).
Figure 4.
Band structure of bilayer PtSe2.
2.1.1.3 Trilayer PtSe2
The optB86b functional correctly predicts the semiconductor bandgap of the trilayer (0.04 eV) as observed in experiments [17]. The conduction band minimum is between the Γ point and M, and the valence band maximum is in Γ point (Figure 5).
The conduction and valence bands began to overlap (Ec−Ev=−0.56eV with Ec is the minimum of the conduction band energy and Ev is the maximum of the valence band energy) which show that the quadrilayer of PtSe2 is a semi-metal. The conduction band minimum is between the Γ point and M, and the valence band maximum is in Γ points, similar to monolayer and trilayer PtSe2 films.
2.1.1.5 Bulk PtSe2
The bulk PtSe2 is a semi-metal Dirac type II [19], represented under the hexagonal structure (Figure 7):
Figure 7.
Bulk PtSe2 structure in lateral view, visualized by VESTA [9]. Atoms in blue are: Se and atoms in purple are: Pt.
The band structure of bulk PtSe2 exhibited a semimetallic band structure along the Γ-A path. The conduction band minimum is at point K or between points H and A, and the valence band maximum is in points Γ. The conduction and valence bands began to overlap (Ec−Ev=−0.58eV with Ec is the minimum of the conduction band energy and Ev is the maximum of the valence band energy) (Figure 8).
Figure 8.
Band structure of bulk PtSe2.
2.1.2 Densities of states of PtSe2 monolayer, multilayer and bulk
Figures 9–13 represent the total and partial densities of states of multilayer PtSe2 films.
Figure 9.
Calculated partial density of states (PDOS) for monolayer PtSe2 films.
Figure 10.
Calculated partial density of states (PDOS) for bilayer PtSe2 films.
Figure 11.
Calculated partial density of states (PDOS) for trilayer PtSe2 films.
Figure 12.
Calculated partial density of states (PDOS) for quadrilayer PtSe2 films.
Figure 13.
Calculated partial density of states (PDOS) for bulk PtSe2 films.
According to the PDOS structure, the platinum (Pt) and selenium (Se) p-orbitals are strongly hybridized and are responsible for valence band and conduction band formation. The density of electronic states clearly shows Van Hove singularities (peaks). These critical points provide information on the different direct transitions between the valence and conduction bands [20]. From monolayer to bulk PtSe2, the intensity of these peaks decreases and the density structure becomes more smoother. Then, these Van Hove singularities are the result of the two-dimensional character of PtSe2.
When an electromagnetic wave propagates in a dispersive material medium at frequencies of the same order of magnitude as the electronic vibration frequencies of this material, the real dielectric constant ε becomes a function of the energy of incident photons. Thus, if this medium is absorbent, the dielectric function is written in the complex form εω [21].
3.2 Calculation method
The dielectric function was calculated by the codeturboEELS [22] that is a component of Quantum Espresso [14, 23, 24] . This code adapts the theory of perturbation of the time-dependent density functional (TDDFPT) based on the Liouville-Lanczos (LL) approach. This method has been established for electron energy loss spectroscopy (EELS) that is directly related to the dielectric response of the material. Moreover, this approach makes it possible to calculate the dielectric function for any value of the transfer moment q. For small q, long-range effects are important, while for large q short-range effects are dominant. This approach has several advantages over the conventional TDDFT approach, such as no counted empty state, which allows to extend the EELS spectra calculations from the low-loss region to the 50–100 eV region [25].
The calculation was determined by the use of Norm-conserving pseudo-potential, the GGA approximation for exchange-correlation energy (XC) and for weak photon wave vector values q. The incident field is modeled only under the direction x. We consider only interband radiative transitions.
3.2.1 Imaginary part of the dielectric function
The imaginary part of the dielectric function of multilayer PtSe2 films is shown in Figure 14.
Figure 14.
Variation of the imaginary part of the dielectric function as a function of energy for multilayer PtSe2.
The imaginary part Imε of PtSe2 is strictly positive. It is important in the incident photon energy region [1, 2, 3, 4, 5, 6]. Also, Imε rises as a function of incident photon energy toward a dominant peak, and then, it decreases to large energy values. Noting that by increasing the thickness of the material PtSe2, this peak becomes more and more intense at lower energy value (red shift). The following table shows the variation of this peak of interband transitions from the monolayer to the bulk PtSe2 (Table 1).
1T−PtSe2
Monolayer
Bilayer
Trilayer
Quadrilayer
Bulk
Photon energy (eV)
2.4
1.761
1.5
1.465
1.422
Imε (u.a)
7.9
12.286
19.189
28
48.749
Table 1.
Maximum peak variation Imε from the monolayer to the bulk PtSe2.
The imaginary part of the dielectric function Imε provides information on the absorption of the medium. According to Figure 14, the absorption tends to increase when the dimension of PtSe2 is high. As a result, the PtSe2 massif is characterized by high optical absorption compared to PtSe2 2D materials. In addition, the Imε is proportional to the sum of all transitions between the valence and conduction bands. As mentioned, the platinum orbital d (Pt) and the selenium orbital p (Se) are responsible for the formation of valence and conduction bands. Therefore, the spectrum of Imε reflects three critical peaks corresponding to the transitions between the orbitals d(Pt) and p(Se).
Variation of the real part of the dielectric function for multilayer PtSe2.
The spectrum, below, shows the decrease of the real part of the dielectric function Reε of the material PtSe2 by increasing the photon energy. From zero eV, Reε starts with positive values going toward a maximum, then it decreases toward zero for semiconductors: 1L and 2L or to a minimum of −5 (u.a) for semi-metals: 3L, 4L, and bulk. Then it is followed by oscillations near zero in the ultraviolet (UV) region. In addition, Reε tends to have a maximum at lower energy values from the monolayer to the bulk PtSe2. From these figures, we can extract the values of the static dielectric constant εω=0 which represents the dielectric response to the static electric field of PtSe2. These values are grouped in Table 2.
Variation of static ε going from the monolayer to the bulk PtSe2.
This variation of the constant ε0 can be explained by the Penn model [28]. According to Eq. (2), the dielectric constant decreases by increasing the gap energy:
ε=1+hωEgE2
In addition, according to the spectrum [15], the Reε of the trilayer, quadrilayer, and bulk PtSe2 have negative values. This means that the electromagnetic wave does not propagate in the semi-metals 3L, 4L, and the bulk PtSe2, and that the reflection and absorption processes occurs.
The shift of the maximum peaks of the Reε and Imε to very low energy values (red shift) can be explained by the quantum confinement effect that is the result of the material shrink. Moreover, under an external excitation, the photon energy emitted by nano-objects depends on their size, with increasing size, the emission tends to lower energies [29]. This is the quantum confinement effect.
3.2.3 Comparison with theoretical studies
The literatures [26, 27] calculated the linear optical response ε for the monolayer and the bulk PtSe2 using WIEN2K and VASP software, respectively. The dielectric functions determined for the monolayer and the bulk PtSe2 in the [27] study are similar to our calculations. The Imεbulk shows a very strong maximum at an energy value of 1.25 that is close to the energy value calculated by the QE code at 1.4 eV for a maximum of Imε. The Reεbulk has a very intense maximum at an energy value of 1 eV [27] that is very close to the energy value found by the QE code of 0.91 eV for maximum intensity.
3.3 Refractive index and extinction coefficient
The dielectric function is a fundamental physical parameter to study the optical properties of the material medium. But often, opticians adapt other functions are easier for a better description of properties, such as the refractive index used in the field of photovoltaics and the absorption coefficient for transparent materials.
The complex refractive index N is directly related to the dielectric function ε by the relation N2=ε, where N is defined by:
N=n−iKE3
With nω is the refractive index and Kω the extinction coefficient. So, the real part and imaginary ε are written:
Reεω=nω2−Kω2etImεω=2nKE4
We calculate the refractive index nω and the extinction coefficient Kω using the following expressions [21] :
Variation of refractive index as a function of energy for PtSe2 multilayer.
Figure 17.
Variation of extinction coefficient as a function of energy for PtSe2 multilayer.
The refractive index curve shows a light shift analogous to the dielectric function as a function of thickness. In fact, the maximum peaks of nω tend to distribute toward low energies by increasing the size of multilayer PtSe2. As previously indicated, this displacement is a consequence of quantum confinement phenomenon due to the thickness effect. From the spectra 16, the static refractive index can be extracted (Table 3).
Variation of static refractive index as a function of energy for PtSe2 multilayer.
The static refractive index measures the superiority of the speed of light c in a vacuum (or air) compared to over that in the material: n=cv. The calculated refractive index values are in good agreement with theoretical results from other literature. Moreover, they respect the relation [31] n0=ε0.
The extinction coefficient measures the fraction of light lost due to scattering and absorption at a particular wavelength: K=αλ4π.
The spectrum of Kω [17] reveals an increase in the infrared for the bilayer, trilayer, quadricayer, and bulk PtSe2 . In contrast, the extinction coefficient of the monolayer PtSe2 begins to increase in the visible region. In the ultraviolet region, the extinction coefficient of multilayer PtSe2 decreases until it stabilizes near 1.
3.4 Reflectivity and absorption coefficient
The α absorption coefficient measures the energy loss of electromagnetic radiation as it passes through the absorbing medium. This coefficient is calculated from the expression:
αω=4πKEλ=4πhcK.EE7
Reflectivity is determined by the following formula [21]:
Rω=N−1N+12=n−12+K2n+12+K2E8
According to these curves, platinum diselenide PtSe2 is a good absorber in the visible and ultraviolet (UV) region since the PtSe2 bulk shows a strong absorption with a maximum value equal to 1.8106cm−1 in the ultraviolet (UV) range.
As seen in Figure 18, multilayer PtSe2 shows transparency in the infrared range. However, the PtSe2 bulk has a very high reflectivity, covering the entire spectrum. In contrast, the reflectivity of PtSe2 in multilayers is much lower than that of the bulk, tending to zero at 10 eV. In addition, the bilayer and monolayer are less reflective than the other systems. The reflectivity of the bilayer and monolayer starts to decrease at 1.8 eV and 2.3 eV, respectively. As a result, multilayer PtSe2 is more absorbent than bulk PtSe2. This gives great potential for optoelectronic applications in the visible and ultraviolet range (Figure 19).
Figure 18.
Variation of the absorption coefficient as a function of energy for multilayer PtSe2 in the energy range [0–6] eV and [0–30] eV.
Figure 19.
Variation of reflectivity as a function of energy for multilayer PtSe2.
In this chapter, we have studied the electronic and optical properties of multilayer PtSe2. This study is based on density functional theory (DFT) and its TDDFT extension. Our results are in good agreement with both theoretical and experimental literature.
This work presents a theoretical study of the effect of thickness on the properties of PtSe2 in multilayers, such as: electronic band structure, density of states, dielectric function, extinction coefficient, refractive index, absorption coefficient, and reflectivity. Firstly, we have shown the transition of multilayer platinum diselenide PtSe2 from a semimetal (in bulk structure) to a semiconductor (in monolayer structure). We have adapted the hyride exchange and correlation functional (XC): optB86b-vdw to describe correctly the transition from the semiconducting to the semimetallic state from the fourth layer. Secondly, we studied the total and partial densities of states to identify the contributions of orbitals in the valence and conduction bands. The variation of the density of states as a function of the number of layers of PtSe2 shows Van Hove singularities due to the two-dimensional character of PtSe2. Next, we determined the influence of thickness on the optical properties of PtSe2 multilayers.
This thickness dependence of electronic and optical properties is the sign of quantum confinement in multilayer PtSe2 nanoparticles. In addition, we investigated the absorption coefficient, reflectivity, refractive index, and extinction coefficient of multilayer PtSe2. We conclude that multilayer PtSe2 is more absorbent than bulk, making it a promising candidate for optoelectronic applications.
A promising prospect of the multilayer platinum PtSe2 is the determination of its nonlinear optical properties to investigate the possibility of a THz emission.
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Written By
Nour El Houda Safi
Submitted: 14 September 2023Reviewed: 06 October 2023Published: 19 April 2024
Open access peer-reviewed chapter - ONLINE FIRST
