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Theoretical investigation of electronic and optical properties of 2D transition metal dichalcogenides MoX2 (X = S, Se, Te) from first-principles
Physica E 126 (2021) 114416
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Physica E: Low-dimensional Systems and Nanostructures journal homepage: http://www.elsevier.com/locate/physe
Theoretical investigation of electronic and optical properties of 2D transition metal dichalcogenides MoX2 (X = S, Se, Te) from first-principles Razieh Beiranvand Physics Group, Department of Basic Science, Ayatollah Boroujerdi University, Boroujerd, Iran
A B S T R A C T
To use the Full-Potential Linearized Augmented Plane Wave (FP-LAPW) method in density functional theory (DFT), we have calculated the electronic structure and linear optical properties of 2D Transition metal dichalcogenides (TMDs), MoX2 (X = S, Se, Te). The predicted optical gap of MoX2 monolayers are in good agreement with the available experimental and theoretical data. The real and imaginary parts of dielectric function, optical absorption coefficient, reflectivity and energy loss function are also discussed in details for two directions of applied external electric field (E‖x and E‖z). It is found that the dielectric functions are highly anisotropic in low energy range (below 9 eV) and becomes isotropic in high energy range (above 9 eV). The most promising feature of MoX2 monolayers is high value absorption coefficient with a wide absorption spectrum making them very suitable material for solar cells and optoelectronic applications. Also, the wide band gap of TMDs which is thickness dependent makes them promising candidates for optoelectronic devices. The results of our systematic study are expected to guide the experi mentalists to achieve suitable properties related to band gap modifications via optical absorption measurements at the nanoscale.
1. Introduction The separation of graphene from graphite in 2004 demonstrated that two-dimensional crystals can be found in the isolating form [1–3]. This single layer of carbon atoms has since become one of the most promising materials, attracting the interest of physicists [4–7]. As early as 2005, it was also clear that other layered materials could be laminated down to a monolayer. However, graphene remained the only two-dimensional crystal to receive widespread attention as yet. The electronic [4,8–10], optical [11,12], mechanical [13–16], magnetic [5,17–21] and thermal [22,23] properties of graphene-based systems is still an interesting field especially for condensed matter physicists. In the past few years, graphene-like inorganic monolayer materials such as SiC [24,25], ZnO [26,27], BN [28,29], AlN [30–32], GaN [29] and transition-metal dichalcogenides (TMDs) [33,34] have been synthesized and reported in many aspects [35,36]. Comparing to graphene, these two-dimensional materials differ in the linear energy spectrum and zero-bandgap of graphene. Among these materials, the two-dimensional TMDs, MoX2 (X = S, Se, Te), show a wide range of electronic, optical, mechanical, chemical and thermal properties [37,38]. In contrast to the zero-gap graphene, these 2D materials possess sizable band gap which is very applicable in field-emission transistors (FETs) and other optical devices [39–41]. They could also complement graphene in applications that require thin transparent semiconductors, such as opto-electronics and energy harvesting. The TMDs are indirect-gap semiconductors in their bulk form [42].
Upon thinning down to a monolayer, they sustain a transition from indirect-to-direct gap [37]. Due to the interesting band gap modulation, thermal stability, high degree of immunity to short channel effects, high current on/off ratio and reliable dielectric properties, TMDs are pro spective 2D materials for photovoltaic applications, solar cells, sensing, biomedicine and catalysis [43–46]. Bulk unit cell structure of MoS2 belongs to space group P63/mmc. Its hexagonal-layered structure is similar to graphene, in which Mo and S are bonded. In each layer, covalent force has attached Mo atom to six S atoms, and every S atom is attached to three Mo atoms. There are some experimental and theoretical studies which demonstrated that decreasing in thickness of MoS2, causing a transition from an indirect band gap of about 1.29 eV for bulky MoS2 to direct band gap of about 1.84 eV and strong photoluminescence for monolayer MoS2 [47,48]. Because of similarity, we expect that other TMDs, such as MoSe2 and MoTe2 have similar properties like MoS2. All of the above actually motivate us to investigate the electronic and optical properties of MoX2 monolayers. There are a lot of reported on physical properties of three dimen sional structure of MoX2 in both theoretical and experimental view, but there is a lack of systematic calculation on the optical properties of TMD monolayers although it had been fabricated and studied experimentally in the past several years. Complex dielectric function, the most basic description of light-matter interactions, provides a meeting point be tween experiment and theories of excited-state properties of TMD monolayers. Knowledge of the dielectric function is also crucial for the
E-mail address: [email protected]. https://doi.org/10.1016/j.physe.2020.114416 Received 5 April 2020; Received in revised form 31 May 2020; Accepted 14 August 2020 Available online 30 September 2020 1386-9477/© 2020 Elsevier B.V. All rights reserved.
R. Beiranvand
Physica E: Low-dimensional Systems and Nanostructures 126 (2021) 114416
Fig. 2. Band Structure (right panel) and total density of state (left panel) of MoS2 monolayer calculating by DFT method.
Fig. 1. Unit cell (side view (a) and top view (b)) and super cell (side view (c) and top view (d)) of single layer hexagonal MoS2. A vacuum space of about 15 Å was used along c axis to avoid interactions between layers.
inside and outside the muffin-tin sphere, respectively. Inside the atomic spheres, the charge density and the potential are expanded in spherical harmonics up to lmax = 10. The density plane wave cut-off is RKmax = 9, while the potential cut-off extends up to 500 eV, so no shape approxi mation to the potential is necessary. The core electrons are treated fully relativistically, while the valence electrons are treated scalar relativis tically. The structures are fully relaxed until the forces on the atoms reach values less than 2 mRy/au. Considering the complete charge of the system, the convergence of the whole energy in the self-consistent cal culations occurs with a tolerance of 0.001 electron charges.
Table 1 The calculated band gap and static dielectric constant of monolayer MoX2 along with the available experimental and theoretical data. Material MoS2
MoSe2
MoTe2
Band gap (eV)
ε (0)
Our results (GGA)
Experiment
Others
E ll x
E ll z
1.83
1.88 1.85 1.90
1.679, 1.866, 2.89, 1.60 (PBE), 2.05 (HSE), 2.82 (GW) 1.444, 1.613, 1.35 (PBE), 1.75 (HSE), 2.41 (GW) 0.95 (PBE), 1.30 (HSE), 1.75 (GW)
5.74
2.70
1.52
1.11
1.57
–
3. Results and discussions 8.01
8.00
4.51
3.1. Electronic properties The physics of TMD monolayers are essentially the same. So we can use MoS2 as an example. Structurally, MoS2 can be regarded as strongly bonded 2D S–Mo–S layers that are coupled to each other by Van der Waals forces. The Mo and S atoms in each layer, form a two-dimensional hexagonal lattice in a trigonal prismatic geometry. For the calculations of monolayer systems, we have used a periodic supercell, leaving enough distance between adjacent sheets which is sufficient to ensure minimal interlayer coupling (approximately 15 Å). The lengths of the Mo–S bonds are uniformly 2.41 Å, Mo–S–Mo (or S–Mo–S) bond angles are 82.31 and the calculated binding energy is 5.20 eV per atom. In Fig. 1, the unit cell and super cell of the hexagonal monolayer MoS2 are shown (Other MoX2 structures are similar to MoS2). Utilization of MoX2 monolayers in optoelectronic devices relies upon their electronic features and thus the computed band structure and corresponding total density of states of these materials have been re ported in this part. The monolayers of TMDs have a hexagonal BZ with two inequivalent K and K′ points (valleys). Because of the lack of inversion symmetry and the strong spin-orbit coupling, the conduction and valence bands are spin split in these valleys. However, in this work we do not focus on band-splitting. This interesting issue addressed in many theoretical studies before that is beyond the scope of this work. One of the most accurate approaches to calculate the electronic structure is the density functional theory of Kohn-Sham. This theory is extensively employed in the solid state community. Despite being exact for ground-state properties, DFT results are highly dependent on the
4.33
characterization of these advanced materials in opto-electronics. The purpose of this paper is to present a detailed characterization of the electronic band structure and linear optical properties of MoX2 mono layers via first-principles calculations. Experimental techniques, namely photoemission, photoabsorption, and photoluminescence spectroscopy, have been utilized to predict quasi-particle band structures and optical spectra precisely. A wide range of optical properties such as dielectric tensor, reflectivity, loss function, optical conductivity and reflection and extinction coefficients of monolayer MoX2 are studied by first principles and compared with available experimental results. 2. Calculation method Standard Kohn-Sham DFT calculations [49] using Wien2k package [50] were first performed for structural relaxation of the TMD mon layers. The exchange-correlation potentials approximated by semi-local density approximation (GGA) with Perdew et al. functional [51]. In the self-consistent field potential and total energy calculations a set of (15 × 15 × 1) k-point sampling is used for Brillouin Zone (BZ) integration in k-space. Spherical harmonic expansion and plane wave basis set are used 2
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Physica E: Low-dimensional Systems and Nanostructures 126 (2021) 114416
Fig. 3. The calculated dielectric function spectrum of MoX2 monolayer for perpendicular (Blue lines) and parallel (Red lines) directions of polarization (E‖x and E‖z). Inside graph from Ref. [55]. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
approximation of the exchange correlation (xc) potential. These results explain the many-body interaction between electrons. The approxima tion that is used in the calculation, is the most popular one for the xc potential, generalized gradient approximation (GGA). GGA has been successful in predicting structural, mechanical and optical properties of different materials, however, DFT in GGA method has a great deal of limitations on predicting the electronic features of strongly correlated systems such as 2D semiconductors, but nonetheless it should be noticed that the recent GW quasi-particle calculations explain the band structure in 2D semiconductors better. We note, however, that recent GW quasiparticle calculations, give a
better description of the band structure in 2D semiconductors [52,53]. An important character in low-dimensional systems is their strong exciton binding due to the weak screening compared to bulk cases. Theoretical studies employed DFT predicted monolayer MoS2 to have a direct gap of 1.78 eV. Because of the systematic errors of DFT method, the good band gap agreement between theoretical and experimental results for monolayer MoS2 may be a mere coincidence. The calculated band gap of MoS2 monolayer varies from 1.6 to 1.9 eV in the literature due to different approximations for the XC functionals. The single-layered MoS2 is a semiconductor with a direct band gap of 1.69 eV at the Γ point on the basis of the DFT-PW91 level. In Table 1, the 3
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Physica E: Low-dimensional Systems and Nanostructures 126 (2021) 114416
same methodology (Fig. 2). 3.2. Optical properties When optical absorption spectra are considered, Electron-hole in teractions should be accounted. Technically, the simulation of optical features needs the integration over the irreducible BZ using sufficiently dense k-point mesh. So, the convergence of k-point sampling is impor tant in the case of optical properties. We used a set of (25 × 25 × 1) kpoint sampling for calculating the linear optical properties. We have also calculated ϵ2(ω) up to 50 eV to produce convergence in the Kra mers− Kronig transformation [54]. By taking the appropriate transition matrix elements into account, the imaginary part of the dielectric function ϵ2(ω) is given by random phase approximation (RPA) as follow: ∫ ∑ Ve2 2 ′ ϵ2 (ω) = d3 k | < kn|p|kn > | f (kn) 2πℏm2 ω2 ′ (1) nn ′
′
×(1 − f (kn ))δ(Ekn − E(kn ) − ℏω) in which ℏω present the energy of the incident photon, p is the mo mentum operator applied on the eigenfunction |kn > with eigenvalue Ekn and f(kn) is the Fermi distribution function. Dielectric properties of TMD monolayers can be calculated with electric field vector (E) perpendicular or parallel to c axis. The complex dielectric function describes the interactions between light and matter in these materials as best as possible. Regarding the characterization of TMD monolayers in the opto-electronic applications, this function is essential. The complex dielectric functions ϵ⊥(ω) and ϵ‖(ω) are given by [54] ϵ⊥ (ω) =
ϵxx (ω) + ϵyy (ω) 2
(2)
and (3)
ϵ‖ (ω) = ϵzz (ω) xx
yy
zz
where ϵ (ω), ϵ (ω) and ϵ (ω) are diagonal elements of the dielec tric matrix ϵij(ω). The imaginary part of dielectric function has been calculated from the band structure and real part from imaginary part using Kramerse− Kronig relation. The calculated real and imaginary parts of dielectric functions are shown in Fig. 3. It is found that the dielectric functions are highly anisotropic in low energy range (below 9 eV) and becomes isotropic in high energy range (above 9 eV). Comparing with available experimental spectra for MoS2 show similar peaks in the ϵ2(ω) with energy up to 10 eV. The real component of the dielectric function at zero frequency (ϵ⊥,‖(0)) is called the static dielectric constant that is an important quantity in optical measurements. For E‖z and E‖z polarizations, the calculated ϵ(0) are mentioned in Table 1. To enable more direct comparison with experiments, we consider next the absorption spectrum of monolayer MoX2. In Fig. 4, the calcu lated absorption spectra of the three structures of MoX2 are compared with the available experimental results. The spectra are calculated with a Lorentzian broadening of 0.025 eV. The in-plane graphs show the result of experiments that commonly have 0.1 eV broadening value. The broad peaks above 9 eV are entirely due to the continuum of inter-band transitions between the π and π* bands. Comparing to bandgap energy, photons with smaller energy or longer wavelength cannot be absorbed, because the wavelength is inversely proportional to bandgap energy. If the photon energy of an incident light is smaller than the band gap energy of a material, the material is transparent for that light. Before the light is entirely absorbed by the material, the penetration depth of light of a certain wavelength to a material is defined by the absorption co efficient. Fundamental absorption has caused the fast increase of the absorption coefficient for the light that its wavelength is shorter than the
Fig. 4. Absorption coefficients of MoX2 for perpendicular (Blue lines) and parallel (Red lines) directions of polarization (E‖x and E‖z). The z-axis is considered as the optical axis. In the inset of figures, the absorption coefficients in the energy range of 0–5 eV are also shown. (For interpretation of the ref erences to colour in this figure legend, the reader is referred to the Web version of this article.)
calculated and experimental band gap of bulk and monolayer structure of MoX2 is reported. Our calculated results show that the monolayer MoS2 has semiconductor behavior with a direct band gap of about 1.8 eV from Γ to K which in good agreement with previous reports with the 4
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Fig. 5. Calculated electron energy LOSS function of MoX2 along with the available experimental results (Ref. [56]) for perpendicular (Blue lines) and parallel (Red lines) directions of polarization (E‖x and E‖z). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
bandgap. Unlike the bulk material, the MoX2 monolayers emit light strongly. The excitation energies of them are predicted to lie in the range of 1–2 eV. The freestanding monolayer of MoS2 shows an increase in lumi nescence quantum efficiency by more than a factor of 104 compared with the bulk material. From the dielectric functions, we obtain the absolute absorbance of the TMD monolayers. It was shown
experimentally that there are two adsorption peaks for MoS2 layers, i.e., peaks at 1.85 eV (670 nm) and 1.98 eV (627 nm), corresponding to the direct excitonic transitions at the Brillouin zone K point. These absorp tion peaks can be changed by small fraction with sample thickness. In the monolayer MoS2, a strong photoluminescence (PL) peak at about 1.90 eV indicated that MoS2 undergoes an indirect to direct band gap transition when its bulk or multilayer form replaced by a monolayer. 5
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Physica E: Low-dimensional Systems and Nanostructures 126 (2021) 114416
monolayers. EELS is an important quantity that can be calculated from dielectric functions: Im(
1 ϵ2 (ω) )= 2 ϵ(ω) ϵ1 (ω) + ϵ22 (ω)
(4)
As it can be seen from the Fig. 5, there is a close match between our calculated results and the experimental data in the low energy region of MoS2 EELS. For other structures (monolayer MoSe2 and MoTe2), there are not suitable experimental results in the literature, but from the MoS2 EELS, we can conclude that the calculated spectrum of monolayer MoSe2 and MoTe2 will be very closed to the experimental values. The most prominent peak(s) in the EELS spectrum is identified as the plasmon peak. The plasmon peaks show the energy of excitations of the electronic charge density in the material. On analyzing the data presented in Fig. 5, we can see that all spectra consist of two prominent resonance features. The first dominant peaks (a,b for E ‖x and a′ ,b’ for E ‖z) lie below 9 eV that called the π plasmon peaks. Two maximums (c,d for E ‖x and c′ ,d’ for E ‖z) above 9 eV are also exist that called the π + σ plasmon peaks. The π plasmon feature arises due to collective π − π* transition, while π + σ plasmon results from the π − σ * and σ − σ * excitations. These findings are well supported by experiments for MoS2 monolayer. We can also express the material response in terms of the real con ductivity σ(E) = σ1(E) + iσ 2(E). When a beam of light can pass through the material with no significant propagation, real conductivity provides a full description of the optical response. In Fig. 6 we plot the frequency dependence real conductivity of TMD monolayers. Similar to the dielectric function spectra, the optical conductivities are highly aniso tropic in low energy region (below 9 eV) and becomes isotropic in high energy range (above 9 eV). The data reported in Table 1 are also proved by the optical conductivity spectra. Finally, we calculate the refractive index (n) and extinction coeffi cient (k) of TMD monlayers that can be considered as the fingerprint of the material. There are a lot of experimentally reported on the optical gap and the extinction coefficient of TMDs in the literature. The nonlinear refractive index of the MoX2 nanosheets was measured to be ≈ 10− 7cm2W− 1. As it can be seen in Fig. 7, the refraction and extinction coefficients of MoX2 monolayers are highly anisotropic in the energy range below 10 eV. Increasing energy cause full isotropic pattern in the range over 20 eV. 4. Conclusion Briefly, first-principles theoretical investigations on the electronic and optical features of MoX2 monolayers are presented as sample of TMD monolayers in a broad energy range. We, also, have studied the real and imaginary parts of the dielectric function, optical absorption coefficient, reflectivity and energy loss function in detail in two di rections of applied external electric field (E‖x and E‖z). The calculated optical gap of MoX2 monolayers are in good agreement with the avail able experimental and theoretical data. It is found that the dielectric functions are highly anisotropic in low energy range (below 9 eV) and becomes isotropic in high energy range (above 9 eV). The most prom ising feature of MoX2 monolayers is high value absorption coefficient with wide absorption spectrum making them very suitable material for solar cells and opto-electronic applications. We hope our results will motivate future experiments in constructing layered structures and studying their unique opto-electronic properties.
Fig. 6. Calculated real conductivity of MoX2 monolayers for perpendicular (Blue lines) and parallel (Red lines) directions of polarization (E‖x and E‖z). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
These peak positions for MoSe2 and MoTe2 is about 1.3 eV and 1.15 eV, respectively. In solar cell applications, the energy region for solar spectrum is mainly considered within 0–5 eV. To check the applicability of MoX2 thin films in solar cells, we have also presented the total absorption (area under the absorption coefficients curve) from 0 to 5 eV. It is clear that in the energy range 1–5 eV, all the intense peaks show a similar trend, except their energy values. Next we calculated electron energy loss spectra (EELS) for TMD
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Physica E: Low-dimensional Systems and Nanostructures 126 (2021) 114416
Fig. 7. Calculated refraction and extinction coefficients of MoX2 monolayer for perpendicular (Blue lines) and parallel (Red lines) directions of polarization (E‖x and E‖z). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
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Appendix A. Supplementary data
[27] [28] [29] [30]
Supplementary data to this article can be found online at https://doi. org/10.1016/j.physe.2020.114416.
[31] [32] [33]
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Physica E: Low-dimensional Systems and Nanostructures journal homepage: http://www.elsevier.com/locate/physe
Theoretical investigation of electronic and optical properties of 2D transition metal dichalcogenides MoX2 (X = S, Se, Te) from first-principles Razieh Beiranvand Physics Group, Department of Basic Science, Ayatollah Boroujerdi University, Boroujerd, Iran
A B S T R A C T
To use the Full-Potential Linearized Augmented Plane Wave (FP-LAPW) method in density functional theory (DFT), we have calculated the electronic structure and linear optical properties of 2D Transition metal dichalcogenides (TMDs), MoX2 (X = S, Se, Te). The predicted optical gap of MoX2 monolayers are in good agreement with the available experimental and theoretical data. The real and imaginary parts of dielectric function, optical absorption coefficient, reflectivity and energy loss function are also discussed in details for two directions of applied external electric field (E‖x and E‖z). It is found that the dielectric functions are highly anisotropic in low energy range (below 9 eV) and becomes isotropic in high energy range (above 9 eV). The most promising feature of MoX2 monolayers is high value absorption coefficient with a wide absorption spectrum making them very suitable material for solar cells and optoelectronic applications. Also, the wide band gap of TMDs which is thickness dependent makes them promising candidates for optoelectronic devices. The results of our systematic study are expected to guide the experi mentalists to achieve suitable properties related to band gap modifications via optical absorption measurements at the nanoscale.
1. Introduction The separation of graphene from graphite in 2004 demonstrated that two-dimensional crystals can be found in the isolating form [1–3]. This single layer of carbon atoms has since become one of the most promising materials, attracting the interest of physicists [4–7]. As early as 2005, it was also clear that other layered materials could be laminated down to a monolayer. However, graphene remained the only two-dimensional crystal to receive widespread attention as yet. The electronic [4,8–10], optical [11,12], mechanical [13–16], magnetic [5,17–21] and thermal [22,23] properties of graphene-based systems is still an interesting field especially for condensed matter physicists. In the past few years, graphene-like inorganic monolayer materials such as SiC [24,25], ZnO [26,27], BN [28,29], AlN [30–32], GaN [29] and transition-metal dichalcogenides (TMDs) [33,34] have been synthesized and reported in many aspects [35,36]. Comparing to graphene, these two-dimensional materials differ in the linear energy spectrum and zero-bandgap of graphene. Among these materials, the two-dimensional TMDs, MoX2 (X = S, Se, Te), show a wide range of electronic, optical, mechanical, chemical and thermal properties [37,38]. In contrast to the zero-gap graphene, these 2D materials possess sizable band gap which is very applicable in field-emission transistors (FETs) and other optical devices [39–41]. They could also complement graphene in applications that require thin transparent semiconductors, such as opto-electronics and energy harvesting. The TMDs are indirect-gap semiconductors in their bulk form [42].
Upon thinning down to a monolayer, they sustain a transition from indirect-to-direct gap [37]. Due to the interesting band gap modulation, thermal stability, high degree of immunity to short channel effects, high current on/off ratio and reliable dielectric properties, TMDs are pro spective 2D materials for photovoltaic applications, solar cells, sensing, biomedicine and catalysis [43–46]. Bulk unit cell structure of MoS2 belongs to space group P63/mmc. Its hexagonal-layered structure is similar to graphene, in which Mo and S are bonded. In each layer, covalent force has attached Mo atom to six S atoms, and every S atom is attached to three Mo atoms. There are some experimental and theoretical studies which demonstrated that decreasing in thickness of MoS2, causing a transition from an indirect band gap of about 1.29 eV for bulky MoS2 to direct band gap of about 1.84 eV and strong photoluminescence for monolayer MoS2 [47,48]. Because of similarity, we expect that other TMDs, such as MoSe2 and MoTe2 have similar properties like MoS2. All of the above actually motivate us to investigate the electronic and optical properties of MoX2 monolayers. There are a lot of reported on physical properties of three dimen sional structure of MoX2 in both theoretical and experimental view, but there is a lack of systematic calculation on the optical properties of TMD monolayers although it had been fabricated and studied experimentally in the past several years. Complex dielectric function, the most basic description of light-matter interactions, provides a meeting point be tween experiment and theories of excited-state properties of TMD monolayers. Knowledge of the dielectric function is also crucial for the
E-mail address: [email protected]. https://doi.org/10.1016/j.physe.2020.114416 Received 5 April 2020; Received in revised form 31 May 2020; Accepted 14 August 2020 Available online 30 September 2020 1386-9477/© 2020 Elsevier B.V. All rights reserved.
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Physica E: Low-dimensional Systems and Nanostructures 126 (2021) 114416
Fig. 2. Band Structure (right panel) and total density of state (left panel) of MoS2 monolayer calculating by DFT method.
Fig. 1. Unit cell (side view (a) and top view (b)) and super cell (side view (c) and top view (d)) of single layer hexagonal MoS2. A vacuum space of about 15 Å was used along c axis to avoid interactions between layers.
inside and outside the muffin-tin sphere, respectively. Inside the atomic spheres, the charge density and the potential are expanded in spherical harmonics up to lmax = 10. The density plane wave cut-off is RKmax = 9, while the potential cut-off extends up to 500 eV, so no shape approxi mation to the potential is necessary. The core electrons are treated fully relativistically, while the valence electrons are treated scalar relativis tically. The structures are fully relaxed until the forces on the atoms reach values less than 2 mRy/au. Considering the complete charge of the system, the convergence of the whole energy in the self-consistent cal culations occurs with a tolerance of 0.001 electron charges.
Table 1 The calculated band gap and static dielectric constant of monolayer MoX2 along with the available experimental and theoretical data. Material MoS2
MoSe2
MoTe2
Band gap (eV)
ε (0)
Our results (GGA)
Experiment
Others
E ll x
E ll z
1.83
1.88 1.85 1.90
1.679, 1.866, 2.89, 1.60 (PBE), 2.05 (HSE), 2.82 (GW) 1.444, 1.613, 1.35 (PBE), 1.75 (HSE), 2.41 (GW) 0.95 (PBE), 1.30 (HSE), 1.75 (GW)
5.74
2.70
1.52
1.11
1.57
–
3. Results and discussions 8.01
8.00
4.51
3.1. Electronic properties The physics of TMD monolayers are essentially the same. So we can use MoS2 as an example. Structurally, MoS2 can be regarded as strongly bonded 2D S–Mo–S layers that are coupled to each other by Van der Waals forces. The Mo and S atoms in each layer, form a two-dimensional hexagonal lattice in a trigonal prismatic geometry. For the calculations of monolayer systems, we have used a periodic supercell, leaving enough distance between adjacent sheets which is sufficient to ensure minimal interlayer coupling (approximately 15 Å). The lengths of the Mo–S bonds are uniformly 2.41 Å, Mo–S–Mo (or S–Mo–S) bond angles are 82.31 and the calculated binding energy is 5.20 eV per atom. In Fig. 1, the unit cell and super cell of the hexagonal monolayer MoS2 are shown (Other MoX2 structures are similar to MoS2). Utilization of MoX2 monolayers in optoelectronic devices relies upon their electronic features and thus the computed band structure and corresponding total density of states of these materials have been re ported in this part. The monolayers of TMDs have a hexagonal BZ with two inequivalent K and K′ points (valleys). Because of the lack of inversion symmetry and the strong spin-orbit coupling, the conduction and valence bands are spin split in these valleys. However, in this work we do not focus on band-splitting. This interesting issue addressed in many theoretical studies before that is beyond the scope of this work. One of the most accurate approaches to calculate the electronic structure is the density functional theory of Kohn-Sham. This theory is extensively employed in the solid state community. Despite being exact for ground-state properties, DFT results are highly dependent on the
4.33
characterization of these advanced materials in opto-electronics. The purpose of this paper is to present a detailed characterization of the electronic band structure and linear optical properties of MoX2 mono layers via first-principles calculations. Experimental techniques, namely photoemission, photoabsorption, and photoluminescence spectroscopy, have been utilized to predict quasi-particle band structures and optical spectra precisely. A wide range of optical properties such as dielectric tensor, reflectivity, loss function, optical conductivity and reflection and extinction coefficients of monolayer MoX2 are studied by first principles and compared with available experimental results. 2. Calculation method Standard Kohn-Sham DFT calculations [49] using Wien2k package [50] were first performed for structural relaxation of the TMD mon layers. The exchange-correlation potentials approximated by semi-local density approximation (GGA) with Perdew et al. functional [51]. In the self-consistent field potential and total energy calculations a set of (15 × 15 × 1) k-point sampling is used for Brillouin Zone (BZ) integration in k-space. Spherical harmonic expansion and plane wave basis set are used 2
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Fig. 3. The calculated dielectric function spectrum of MoX2 monolayer for perpendicular (Blue lines) and parallel (Red lines) directions of polarization (E‖x and E‖z). Inside graph from Ref. [55]. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
approximation of the exchange correlation (xc) potential. These results explain the many-body interaction between electrons. The approxima tion that is used in the calculation, is the most popular one for the xc potential, generalized gradient approximation (GGA). GGA has been successful in predicting structural, mechanical and optical properties of different materials, however, DFT in GGA method has a great deal of limitations on predicting the electronic features of strongly correlated systems such as 2D semiconductors, but nonetheless it should be noticed that the recent GW quasi-particle calculations explain the band structure in 2D semiconductors better. We note, however, that recent GW quasiparticle calculations, give a
better description of the band structure in 2D semiconductors [52,53]. An important character in low-dimensional systems is their strong exciton binding due to the weak screening compared to bulk cases. Theoretical studies employed DFT predicted monolayer MoS2 to have a direct gap of 1.78 eV. Because of the systematic errors of DFT method, the good band gap agreement between theoretical and experimental results for monolayer MoS2 may be a mere coincidence. The calculated band gap of MoS2 monolayer varies from 1.6 to 1.9 eV in the literature due to different approximations for the XC functionals. The single-layered MoS2 is a semiconductor with a direct band gap of 1.69 eV at the Γ point on the basis of the DFT-PW91 level. In Table 1, the 3
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same methodology (Fig. 2). 3.2. Optical properties When optical absorption spectra are considered, Electron-hole in teractions should be accounted. Technically, the simulation of optical features needs the integration over the irreducible BZ using sufficiently dense k-point mesh. So, the convergence of k-point sampling is impor tant in the case of optical properties. We used a set of (25 × 25 × 1) kpoint sampling for calculating the linear optical properties. We have also calculated ϵ2(ω) up to 50 eV to produce convergence in the Kra mers− Kronig transformation [54]. By taking the appropriate transition matrix elements into account, the imaginary part of the dielectric function ϵ2(ω) is given by random phase approximation (RPA) as follow: ∫ ∑ Ve2 2 ′ ϵ2 (ω) = d3 k | < kn|p|kn > | f (kn) 2πℏm2 ω2 ′ (1) nn ′
′
×(1 − f (kn ))δ(Ekn − E(kn ) − ℏω) in which ℏω present the energy of the incident photon, p is the mo mentum operator applied on the eigenfunction |kn > with eigenvalue Ekn and f(kn) is the Fermi distribution function. Dielectric properties of TMD monolayers can be calculated with electric field vector (E) perpendicular or parallel to c axis. The complex dielectric function describes the interactions between light and matter in these materials as best as possible. Regarding the characterization of TMD monolayers in the opto-electronic applications, this function is essential. The complex dielectric functions ϵ⊥(ω) and ϵ‖(ω) are given by [54] ϵ⊥ (ω) =
ϵxx (ω) + ϵyy (ω) 2
(2)
and (3)
ϵ‖ (ω) = ϵzz (ω) xx
yy
zz
where ϵ (ω), ϵ (ω) and ϵ (ω) are diagonal elements of the dielec tric matrix ϵij(ω). The imaginary part of dielectric function has been calculated from the band structure and real part from imaginary part using Kramerse− Kronig relation. The calculated real and imaginary parts of dielectric functions are shown in Fig. 3. It is found that the dielectric functions are highly anisotropic in low energy range (below 9 eV) and becomes isotropic in high energy range (above 9 eV). Comparing with available experimental spectra for MoS2 show similar peaks in the ϵ2(ω) with energy up to 10 eV. The real component of the dielectric function at zero frequency (ϵ⊥,‖(0)) is called the static dielectric constant that is an important quantity in optical measurements. For E‖z and E‖z polarizations, the calculated ϵ(0) are mentioned in Table 1. To enable more direct comparison with experiments, we consider next the absorption spectrum of monolayer MoX2. In Fig. 4, the calcu lated absorption spectra of the three structures of MoX2 are compared with the available experimental results. The spectra are calculated with a Lorentzian broadening of 0.025 eV. The in-plane graphs show the result of experiments that commonly have 0.1 eV broadening value. The broad peaks above 9 eV are entirely due to the continuum of inter-band transitions between the π and π* bands. Comparing to bandgap energy, photons with smaller energy or longer wavelength cannot be absorbed, because the wavelength is inversely proportional to bandgap energy. If the photon energy of an incident light is smaller than the band gap energy of a material, the material is transparent for that light. Before the light is entirely absorbed by the material, the penetration depth of light of a certain wavelength to a material is defined by the absorption co efficient. Fundamental absorption has caused the fast increase of the absorption coefficient for the light that its wavelength is shorter than the
Fig. 4. Absorption coefficients of MoX2 for perpendicular (Blue lines) and parallel (Red lines) directions of polarization (E‖x and E‖z). The z-axis is considered as the optical axis. In the inset of figures, the absorption coefficients in the energy range of 0–5 eV are also shown. (For interpretation of the ref erences to colour in this figure legend, the reader is referred to the Web version of this article.)
calculated and experimental band gap of bulk and monolayer structure of MoX2 is reported. Our calculated results show that the monolayer MoS2 has semiconductor behavior with a direct band gap of about 1.8 eV from Γ to K which in good agreement with previous reports with the 4
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Fig. 5. Calculated electron energy LOSS function of MoX2 along with the available experimental results (Ref. [56]) for perpendicular (Blue lines) and parallel (Red lines) directions of polarization (E‖x and E‖z). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
bandgap. Unlike the bulk material, the MoX2 monolayers emit light strongly. The excitation energies of them are predicted to lie in the range of 1–2 eV. The freestanding monolayer of MoS2 shows an increase in lumi nescence quantum efficiency by more than a factor of 104 compared with the bulk material. From the dielectric functions, we obtain the absolute absorbance of the TMD monolayers. It was shown
experimentally that there are two adsorption peaks for MoS2 layers, i.e., peaks at 1.85 eV (670 nm) and 1.98 eV (627 nm), corresponding to the direct excitonic transitions at the Brillouin zone K point. These absorp tion peaks can be changed by small fraction with sample thickness. In the monolayer MoS2, a strong photoluminescence (PL) peak at about 1.90 eV indicated that MoS2 undergoes an indirect to direct band gap transition when its bulk or multilayer form replaced by a monolayer. 5
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monolayers. EELS is an important quantity that can be calculated from dielectric functions: Im(
1 ϵ2 (ω) )= 2 ϵ(ω) ϵ1 (ω) + ϵ22 (ω)
(4)
As it can be seen from the Fig. 5, there is a close match between our calculated results and the experimental data in the low energy region of MoS2 EELS. For other structures (monolayer MoSe2 and MoTe2), there are not suitable experimental results in the literature, but from the MoS2 EELS, we can conclude that the calculated spectrum of monolayer MoSe2 and MoTe2 will be very closed to the experimental values. The most prominent peak(s) in the EELS spectrum is identified as the plasmon peak. The plasmon peaks show the energy of excitations of the electronic charge density in the material. On analyzing the data presented in Fig. 5, we can see that all spectra consist of two prominent resonance features. The first dominant peaks (a,b for E ‖x and a′ ,b’ for E ‖z) lie below 9 eV that called the π plasmon peaks. Two maximums (c,d for E ‖x and c′ ,d’ for E ‖z) above 9 eV are also exist that called the π + σ plasmon peaks. The π plasmon feature arises due to collective π − π* transition, while π + σ plasmon results from the π − σ * and σ − σ * excitations. These findings are well supported by experiments for MoS2 monolayer. We can also express the material response in terms of the real con ductivity σ(E) = σ1(E) + iσ 2(E). When a beam of light can pass through the material with no significant propagation, real conductivity provides a full description of the optical response. In Fig. 6 we plot the frequency dependence real conductivity of TMD monolayers. Similar to the dielectric function spectra, the optical conductivities are highly aniso tropic in low energy region (below 9 eV) and becomes isotropic in high energy range (above 9 eV). The data reported in Table 1 are also proved by the optical conductivity spectra. Finally, we calculate the refractive index (n) and extinction coeffi cient (k) of TMD monlayers that can be considered as the fingerprint of the material. There are a lot of experimentally reported on the optical gap and the extinction coefficient of TMDs in the literature. The nonlinear refractive index of the MoX2 nanosheets was measured to be ≈ 10− 7cm2W− 1. As it can be seen in Fig. 7, the refraction and extinction coefficients of MoX2 monolayers are highly anisotropic in the energy range below 10 eV. Increasing energy cause full isotropic pattern in the range over 20 eV. 4. Conclusion Briefly, first-principles theoretical investigations on the electronic and optical features of MoX2 monolayers are presented as sample of TMD monolayers in a broad energy range. We, also, have studied the real and imaginary parts of the dielectric function, optical absorption coefficient, reflectivity and energy loss function in detail in two di rections of applied external electric field (E‖x and E‖z). The calculated optical gap of MoX2 monolayers are in good agreement with the avail able experimental and theoretical data. It is found that the dielectric functions are highly anisotropic in low energy range (below 9 eV) and becomes isotropic in high energy range (above 9 eV). The most prom ising feature of MoX2 monolayers is high value absorption coefficient with wide absorption spectrum making them very suitable material for solar cells and opto-electronic applications. We hope our results will motivate future experiments in constructing layered structures and studying their unique opto-electronic properties.
Fig. 6. Calculated real conductivity of MoX2 monolayers for perpendicular (Blue lines) and parallel (Red lines) directions of polarization (E‖x and E‖z). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
These peak positions for MoSe2 and MoTe2 is about 1.3 eV and 1.15 eV, respectively. In solar cell applications, the energy region for solar spectrum is mainly considered within 0–5 eV. To check the applicability of MoX2 thin films in solar cells, we have also presented the total absorption (area under the absorption coefficients curve) from 0 to 5 eV. It is clear that in the energy range 1–5 eV, all the intense peaks show a similar trend, except their energy values. Next we calculated electron energy loss spectra (EELS) for TMD
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Fig. 7. Calculated refraction and extinction coefficients of MoX2 monolayer for perpendicular (Blue lines) and parallel (Red lines) directions of polarization (E‖x and E‖z). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
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Appendix A. Supplementary data
[27] [28] [29] [30]
Supplementary data to this article can be found online at https://doi. org/10.1016/j.physe.2020.114416.
[31] [32] [33]
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