Light–matter interaction in transition metal dichalcogenides and their heterostructures

TMDs are layered materials with a triple X-M-X layer as the smallest unit, where the transition metal M is surrounded by two chalcogen atoms X with strong covalent in-plane bonds, whereas the individual layers hold together by weak van der Waals interaction. The metal atoms are condensed either in octahedral or trigonal prismatic coordination, resulting in semiconducting or metallic behavior [20]. An individual layer can be seen as hexagonally arranged MX₂ molecules [83]. Different stacking sequences in multilayer structures result in a variety of polytypes with diverse properties. Here we focus on the semiconducting polytype of MoS₂, WS₂, MoS₂ and WSe₂ that condenses in a 2H-configuration in an ABAB stacking sequence with the second layer superimposed on the first one but rotated by 180° around the c-axis as depicted in figure 1(a) [20]. The unit cell of a single layer consists of one MX₂ molecule and hence three atoms. The lattice symmetry of a single layer is D_3h [84] without an inversion center as visualized in figure 1(a). The unit cell of bulk materials consists of two MX₂ molecules and hence six atoms (see figure 1(a)) and exhibits D_6h symmetry with an inversion center. Generally, crystals with an odd number of layers have D_3h symmetry and those with an even number of layers have D_3d symmetry, but are often treated as a bulk material that has D_6h symmetry [84].

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In 1923 Dickinson and Pauling studied the crystal structure of MoS₂ by x-ray diffraction and determined the lattice constants of bulk material to be a  =  3.15 Å and c  =  12.30 Å [85] in the 2D plane and perpendicular to it, respectively. The distance between Mo and S atoms constitutes d_Mo-S  =  2.42 Å and between two sulfur atoms d_S–S  =  3.17 Å [86]. The interlayer distance is given by d  =  6.5 Å and is commonly regarded as the thickness of a single layer [20].

The electronic band structures of the four SC-TMDs in the focus of this review exhibit very similar signatures with a remarkable dependence on the number of layers due to quantum confinement [15]. The Bloch states defining the single-particle band structure are defined by the electron configuration of the atoms together with the bonding structure. In figure 1(c) the band structure determined from density functional theory (DFT) calculations is plotted for bulk, bilayer and monolayer MoS₂ [16]. Bulk and bilayer MoS₂ exhibit an indirect band gap close to the Γ-point that is increased by decreasing the number of layers. Nearby is a direct transition located both at the K- and the K'-points that remain almost unaffected by changing the number of layers. The transitions at the K- and K'-points are the lowest-energy transitions for a monolayer MoS₂, because the valence band maximum at the indirect transition is further lowered and simultaneously the conduction band minimum increased so that the indirect transition holds a larger gap compared to the direct transition in the monolayer limit. The transition from an indirect to a direct semiconductor by reducing the number of layers from a bi-layer to a monolayer manifests in a significant increase in the photoluminescence quantum efficiency for monolayers compared to bi- and multilayers, ranging from just one order of magnitude up to several orders [15, 16] depending on sample and experimental parameters such as temperature, doping density, defects, strain and environment.

The origin of this transition is explained by changes in the hybridization between chalcogen p_z-orbitals and transition metal d-orbitals together with quantum confinement phenomena. The valence band and conduction band states near the Γ-point are predominantly formed by a linear combination of transition metal d-orbitals and antibonding p_z-orbitals from the chalcogen atoms. As a consequence, these states are significantly affected by interlayer interaction as well as quantum confinement [87]. In turn, the energy of the conduction band increases near the Γ-point, whereas the energy of the valence band decreases with lowering the number of layers [16]. In contrast, both the conduction and valence band at the K- and K'-points are dominated by localized transition metal d-orbitals that lie in the middle of each X-M-X triple unit. Therefore, the energies of the electronic bands at the K- and K'-points are almost unaffected by interlayer coupling effects, and therefore mostly independent of the number of layers. The excitonic transitions at the K- and K'- points are much brighter in optical emission experiments compared to the indirect transitions. This qualitative description holds for all SC-TMDs [20, 87]. The bandgaps for monolayers MoS₂, WS₂, MoSe₂ and WSe₂ constitute 1.8–1.9 eV, 1.8–2.1 eV, 1.5–1.6 eV and 1.6–1.7 eV, respectively, determined from DFT band structure calculations [13]. Whereas DFT calculations might underestimate the single-particle band gap from TMDs, the numbers agree reasonably well with the lowest energy transitions in optical absorption or photoluminescence measurements with the transition energy significantly lowered by excitonic effects compared to the single-particle band gap. The single-particle electronic bands, as well as the excitonic effects, are highly sensitive to environmental and doping-induced screening and renormalization effects [24, 42, 73, 76, 88–90] and they can be also altered by strain [70, 71, 91, 92] which enables us to engineer the optoelectronic properties.

Generally, the d-orbitals of the transition metal cause strong spin–orbit coupling (SOC). As a direct consequence, the valence band is spin-split due to SOC by ~150 meV for the MoX₂ and by ~450 meV for the WX₂ compounds [13, 47, 93–96]. The two spin-split optical transitions are called A- and B-excitonic transitions in optical emission and absorption spectra. The band structure for the monolayer in the close vicinity of the K- and K'- valley can be modelled by an effective 2D massive Dirac Hamiltonian consisting of two additional terms taking into account the finite band gap due to broken symmetry and SOC due to the transition metal d-orbitals [94]. The two degenerate high-symmetry points in the Brillouin zone at the K- and K'- valley and the lack of an inversion center for monolayers locks the spin and valley pseudospin degree of freedom [30, 32, 33, 93, 94]. The valley pseudospin can be controlled by excitation with circularly polarized light, resulting in interesting valley and spintronic studies [33, 93] such as valley polarization [30, 31] and the observation of the valley Hall effect [32]. Furthermore, the valley pseudospin can be controlled by applied magnetic fields [97, 98] and the degeneracy can be broken either by the magnetic field [99, 100] or the optical Stark effect [101, 102].

For fundamental studies, mono- and few-layer TMD flakes are mostly prepared by micromechanical cleavage from bulk crystals using an adhesive tape [3, 4]. The flakes are then transferred to an appropriate flat substrate material, e.g. Si/SiO₂, sapphire or glass. The thin crystals can then be identified by optical microscopy, utilizing light interference contrast, as done for graphene [103–107]. An example of an optical micrograph of a MoS₂ flake deposited on a Si/SiO₂ substrate with 300 nm SiO₂ is shown in figure 1(d). The contrast is highlighted by a false-color representation demonstrating that the mono-, bi-, tri- and few-layer regions are clearly distinguishable. This exfoliation method is applicable for almost all 2D materials existing as bulk crystals and results in high-quality and ultra-pure samples perfectly suited for fundamental studies. Exfoliated or grown flakes can be precisely positioned on a substrate using several transfer methods for van der Waals assembly [12, 14] e.g. by colamination [19, 108, 109], PDMS stamping [110] or by a pick-and-place method [111, 112]. The various stacking methods also allow the preparation of vertically stacked van der Waals heterostructures with precise control over the individual layers as well as the rotational alignment [113]. However, the micromechanical cleavage method is not scalable; the yield on large single-layer flakes is small and lacks control over size and thickness of the flakes.

Regarding scalable fabrication methods, there is great development in both bottom-up and top-down approaches to produce large amounts of high-quality 2D materials for different applications [20, 114]. Two typical top-down approaches use bulk materials or powder for liquid exfoliation in solvents [115] or for chemical delamination by chemical intercalation [116] or galvanostatic discharging [117] to produce solutions with a large quantity of TMD nanosheets. Thin films can be prepared from such solutions, e.g. by inkjet printing, spray or dip-coating and filtration [114]. These approaches are scalable and the achieved nano-meshes are suitable for some applications. However, these methods do not produce large ultra-pure single crystalline materials.

The synthesis of large-area wafer scale and uniform layers of single-layer TMDs is approached by bottom-up methods such as chemical vapor deposition (CVD) [118–121], metal organic chemical vapor deposition (MOCVD) [122], physical vapor phase growth (PVG) [123, 124] or dip-coating of a precursor and thermolysis crystallization by annealing in sulphur gas [125]. Great progress has been made, and it is possible to grow single-layer, high-quality flakes up to 300 µm in size covering a cm-sized substrate surface [126]. It is still challenging to avoid large gradients across the substrate and to grow homogenously connected single-layer films with high crystal quality [20]. Nevertheless, lateral and vertical van der Waals heterostructures with SC-TMDs that are optically and electrically active have been realized by means of CVD [127–130], indicating the realistic potential to overcome the scalability issue by the fabrication of single-layer SC-TMDs and their heterostructures for future applications.

In a recent study, we investigated the complex dielectric function of mechanically exfoliated mono-, tri- and few-layer MoS₂ on a transparent sapphire substrate by imaging ellipsometry in the spectral range from 1.7 eV–3.1 eV. The used angle of incidence of 50° allowed us to access the optical response of MoS₂ parallel (x, y) and perpendicular (z) to the plane of the flake. A lateral resolution of better than 2 µm enabled us to probe ultra-high-quality crystalline regions. The complex dielectric functions for different numbers of layers were extracted from the measured ellipsometric angles Δ and ψ as input to a multilayer model using Lorentz as well as Tauc–Lorentz fit approaches [48]. In the model, the thickness of the MoS₂ monolayer was kept at the theoretical value of d  =  6.15 Å [78, 85]. The resulting anisotropic complex dielectric functions ε_x  =  ε_y  ≠  ε_z with real part ε₁(E) and imaginary part ε₂(E) are plotted for mono-, tri- and few-layer MoS₂ in figure 2 [48]. The in-plane component of the dielectric function ε_x(E)  =  ε_y(E) is best described by five Lorentz profiles and the out-of-plane component ε_z(E) by one Tauc-Lorentz profile. As visualized in figures 2(a) and (b) the in-plane component exhibits three well pronounced critical points and two rather sharp peaks centered at 1.9 eV and 2.05 eV assigned to the A and B spin-split excitonic transition at the K- and K'-points in the Brillouin zone. The prominent but rather broad resonance at around 2.9 eV is called the C excitonic transition and indicates a high joint density of states [23, 78]. This energy range comprises a combination of several individual bright and dark excitonic transitions between 2.2 eV and 3 eV that are located near the Γ-point [23] in addition to a band nesting region—a region with nearly parallel conduction and valence bands between the M- and Γ- points with a high joint density of states. The individual transitions are significantly broadened by the interaction of electrons with optical phonons [23, 136].

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The out-of-plane component plotted in figures 2(c) and (d) exhibits one critical point that is located for the monolayer MoS₂ at 1.9 eV and hence slightly above the A excitonic transition at the direct band gap. The energy of the critical point is reduced in energy for increasing number of layers to 1.7 eV for a trilayer and to 1.67 eV for few-layer flake. Consequently, this optical transition is below the direct optical transition at the K- and K'- points. Therefore, it is most likely related to the states in the Brillouin zone that form the indirect band gap in agreement with the trend that the band gap is reduced for increasing number of layers. The magnitude of the out-of-plane contribution of the dielectric functions is significantly reduced in amplitude compared to the in-plane component and seems to be inversely proportional to the number of layers. This behavior can be intuitively understood such that the interaction of a light field perpendicular to the 2D film is much stronger for a thicker film compared to an atomically thin monolayer. Independent of the number of layers, the dielectric functions of the out-of-plane component approach constant values of ε₁(z)  ≈  1 and ε₂(z)  ≈  0 above approximately 2 eV [48]. An alternative explanation for the weak out-of plane component of the dielectric function compared to the in-plane component might be also found in the fact that in monolayer TMDs the conduction and valence bands are symmetric with respect to the horizontal mirror plane of the crystal lattice [93, 137]. As a direct consequence dipolar transitions due to out-of plane electric (light) fields are only allowed by breaking the symmetry, e.g. by placing a monolayer TMD on a substrate or by covering the layer with a medium with a different dielectric function. Similar symmetry arguments might be developed for multilayers.

The absorbance α of the isolated MoS₂ mono-, tri-, and few-layer films can be directly calculated from the dielectric function, taking the relation between the extinction coefficient κ  =  2ε₁ε₂ and the absorbance α  =  4πκ/λ at a light wavelength λ. The in-plane and out-of-plane absorbance for mono- and trilayer MoS₂ is plotted in figure 3. A monolayer MoS₂ with a thickness of less than 1 nm absorbs about 4% of the parallel incoming light field at the energy of the A- and B-exciton and almost 15% at the C-exciton [15, 46–48]. In the whole visible range, the absorbance is reduced by increasing the number of layers. In particular, the excitonic resonance at the A- and B-transitions are broadened and reduced in intensity, indicating an increased dielectric screening [138]. While the energy positions of the A- and B-excitons are less affected by the number of layers, the energy position of the C-resonance shifts to lower energies by increasing the number of layers, demonstrating that the electronic band structure and many-body interactions, such as the Coulomb interaction responsible for the exciton formation, are located at different k-values in the Brillouin zone. These effects are unequally affected by changing the number of layers depending on the atomic orbitals dominating the Bloch states.

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The absorbance of a light field perpendicular to the 2D film is 0.2% at the excitonic resonance at 1.9 eV and zero elsewhere. Only 1% of an out-of-plane light field is absorbed by the excitonic peak for trilayer and about 4% for few-layer MoS₂.

The described characteristics of the in-plane dielectric functions ε_x(E)  =  ε_y(E) are in good agreement with the absorbance spectra for exfoliated and CVD-grown ultrathin MoS₂ layers determined either by spectroscopic ellipsometry with isotropic or anisotropic fit approaches or extracted from reflectance measurements [47, 48, 131, 132, 139, 140]. The universality of the main characteristics of the dielectric functions underlines the robustness of the strong light–matter interaction in atomically thin MoS₂ layers. However, we would like to emphasize that the strength as well as the specific spectral position of the A-, B- and C-excitonic transitions are unique for each sample and highly affected by the individual situation such as environment, substrate, doping or strain just to name some major sources modifying the light–matter interaction. As described above, the different critical points in the electronic band structure with a high joint density of states are formed by different atomic orbitals of the transition metal and sulfur atoms, respectively. The A- and B-excitons can consequently be affected differently by perturbations and changes compared to the higher-energy transitions at the C-excitonic resonance or to the spectral region between the B- and C-exciton transitions.

All four SC-TMDs are characterized by strong light–matter interaction, resulting in similar values and overall characteristics in absorbance spectra as well as dielectric functions in the visible range as demonstrated in figure 4 for MoSe₂, WSe₂, MoS₂ and WS₂ monolayers [47]. The 2D crystals were prepared from bulk crystals by micromechanical exfoliation and placed on fused silica substrates. The optical parameters were extracted from reflection measurements with the incident focused light beam perpendicular to the 2D plane. The light field in such a configuration is parallel to the sample plane and therefore only the in-plane light–matter interaction can be probed. The in-plane components of the dielectric functions ε_x(E)  =  ε_y(E) are derived from the reflection spectra by a Kramers–Kroning constrained analysis [47].

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As already introduced above, the absorbance can be directly extracted from the dielectric functions. The absorbance spectra for quasi free-standing and supported monolayers are displayed in figure 4(b), respectively. The authors of [47] have calculated the absorbance spectra for the supported monolayers starting from the identical dielectric function (shown in figure 4(a)), but taking into account a local field correction factor for the light intensity above the substrate that depends on the refractive index of the substrate material [47]. The absorbance of the free-standing material constitutes in the visible range up to 15% and is reduced by about 1/3 for the monolayers on fused silica [47]. The high absorbance values underline the strong light–matter interaction of SC-TMDs even in the monolayer limit with a thickness of less than 1 nm, as has been reported by several theoretical and experimental studies [15, 23, 27, 28, 46–49, 131, 133, 139, 141]. The sizeable reduction of the absorbance for monolayers on a substrate points to the fact that the optical response can be significantly altered just by modification of the substrate or environment by dielectric engineering and screening effects [39, 63, 64, 66, 74, 90, 138, 142–144]. The strong influence of the environment on the light–matter interaction of atomically thin semiconducting membranes motivates the great potential not only for sensing applications, but also for novel device architectures with precisely tailorable optical properties.

The critical points in the optical response found in the complex dielectric functions and absorbance spectra are similar for MoSe₂, WSe₂, MoS₂ and WS₂ monolayers. The lowest-energy transitions, labeled as A- and B-excitons, belong to the direct optical transitions at the K- and K'- points of the band structure. The energy of the A-exciton is the single-particle band gap minus the exciton binding energy which is of the order of several hundred meV [23, 26]. Since both the single-particle band structure and the Coulomb interaction determining the exciton binding energy are reduced by screening effects such as doping [24, 73, 75, 76], the A-excitonic transition energy is only minimally affected by screening. However, the oscillator strength in both absorption and photoluminescence is significantly altered by screening and doping [64, 73, 88, 90, 142]. The B-exciton belongs to the K-, K'- points in momentum space and is the transition from the lower-energy spin-split valence band state to the conduction band [93]. Since the Coulomb interaction is comparable for A-and B-excitons, the energy splitting between A- and B-excitons in the absorbance is a direct measure for the spin-splitting in the valence band due to spin–orbit coupling. The spin-splitting of the valence bands at the K- and K'-points constitutes between 150 meV and 200 meV and between 400 and 450 meV for the MoX₂ and WX₂ (X  =  S, Se) monolayers, respectively. The increase in the spin-splitting of the tungsten-based compounds is due to the larger atomic number of tungsten and, hence, an increase in the spin–orbit coupling strength of the tungsten d-orbitals. Similar to the discussion of the broad C-resonances for MoS₂ monolayers above, the broad C-excitonic signatures for the other compounds belong to higher-lying interband transitions with a high joint density of states at different areas in momentum space. The same holds for the so-called D-resonance observable for WSe₂ in the plotted spectral range [47].

MoSe₂, WSe₂, MoS₂ and WS₂ monolayers exhibit superior light–matter interaction properties and exciton-dominated optical response in the visible range with maxima in the peak absorbance for freestanding materials in the order of 15% for less than 1 nm thick crystals. As expected for 2D materials, the optical responses described by the complex dielectric functions are highly anisotropic for the interaction of the light field parallel or orthogonal to the plane of the 2D materials. The light–matter interaction is reduced for increasing number of layers and the excitonic signatures are broadened presumably due to a decreased Coulomb interaction and hence decreased excitonic phenomena. However, further decay and relaxation channels of the carriers by interlayer scattering exist. In particular, the C-excitonic transition is redshifted with an increasing number of layers. The same observation holds for the only critical point in the out-of-plane component of the dielectric function ε_z(E). It is found that this resonance is redshifted with increasing number of layers. The amplitude, however, is enlarged with the number of layers, as intuitively expected since the thickness in the z direction increases.

As introduced in 2010 by Lee et al [82] for MoS₂, the frequencies of the first-order phonon modes with $E_{\text{2g}}^{1}$ and A_1g symmetry change monotonously with the number of layers with opposite trends. Raman spectra taken on exfoliated MoS₂ on Si/SiO₂ substrate are plotted in figure 5(b). The in-plane mode $E_{\text{2g}}^{1}$ softens and the Raman signal is therefore redshifted with increasing number of layers, whereas the out-of-plane mode A_1g is significantly stiffened and the Raman frequency is blueshifted. Therefore, the energy difference between the two modes ΔE  =  |ω(A_1g)  −  ω($E_{\text{2g}}^{1}$)| allows us to determine the number of layers with very high precision, as demonstrated in figure 5(c). In particular, mono-, bi- and trilayers can be unambiguously determined.

The stiffening of the out-of-plane A_1g mode with increasing number of layers is well understood considering the classical model for coupled harmonic oscillators [82]. The unusual behavior of the in-plane $E_{\text{2g}}^{1}$ mode in softening with increasing number of layers is controversially discussed in the literature and was first explained by a modification of long-range Coulomb interlayer interactions induced by the additional layer, causing an enhanced dielectric screening between the effective charges in MoS₂ [82, 153]. The unusual layer dependence of the $E_{\text{2g}}^{1}$ mode can be more likely explained by surface effects with a larger Mo-S force constant at the surface of thin film compared to the inner layers [170, 171].

These two interlayer phonon modes are useful to determine the number of layers for other SC-TMDs, too. In table 1, the relevant mode frequencies are summarized for monolayers and bulk materials of MoS₂, WS₂, MoSe₂ and WSe₂. The same trend as for MoS₂ holds for the two interlayer phonon modes of WSe₂, MoSe₂ and WS₂ [172]. Only for monolayers of WSe₂, the E' and $A_{1}^{\prime}$ modes are degenerate with a frequency of ω  ≈  250 cm⁻¹ [152] and cannot be separated in experiments. However, they are split for WSe₂ bulk and the (degenerate) mode frequencies change with the number of layers so that an estimation of the number of layers is also possible for WSe₂ [152]. For completeness we would like to mention that the low-frequency interlayer shear modes and layer breathing modes are very sensitive to the number of layers in TMD materials [149, 173–175].

Table 1. Energies for the most prominent first-order Raman modes E'/$E_{\text{2g}}^{1}$ and $A_{1}^{'}$/A_1g belonging to the Raman active optical phonon branches of the phonon dispersion at the Γ-point for MoS₂, WS₂, MoSe₂ and WSe₂ monolayers and bulk materials, respectively. Notations are given with respect to the D_3h/D_6h point groups for monolayer/bulk materials condensed in 2H symmetry. All values are extracted from measurements on mechanically exfoliated flakes excited at room temperature with a laser wavelength of λ  =  514.5 nm.

	MoS2 [82]	WS2 [150]	MoSe2 [152]	WSe2 [152]
E' (1L)	384.2 cm−1	355.2 cm−1	287.2 cm−1	≈250 cm−1
$E_{\text{2g}}^{1}$ (bulk)	382.0 cm−1	355.3 cm−1	285.9 cm−1	248.0 cm−1
$A_{1}^{'}$ (1L)	403.0 cm−1	417.2 cm−1	240.5 cm−1	≈250 cm−1
A1g (bulk)	407.1 cm−1	420.1 cm−1	242.5 cm−1	250.8 cm−1

Figures 6(a) and (b) show the E' ($E_{\text{2g}}^{1}$) and $A_{1}^{\prime}$ (A_1g) phonon mode frequencies for monolayer and bilayer MoS₂ as a function of the top gate voltage using a solid electrolyte gate electrode [74]. The flakes have been placed on a highly p-doped Si back electrode covered with 300 nm SiO₂ and contacted by two Ti/Au contacts. A solid electrolyte gate using polyethylene oxide (PEO) and cesium-perchlorate is utilized because its high capacity allows us to tune the charge carrier density over a wide range by applying a relatively small voltage [159]. With the top gate, the 2D electron density in the MoS₂ flakes can be tuned by two orders of magnitude from approximately 10¹¹ to 10¹³ cm⁻² by applying a top gate voltage of V_TG  =  ±1 V [74]. Raman measurements are carried out at room temperature with an excitation wavelength of λ  =  488 nm. For both, mono- and bilayer MoS₂ the phonon frequencies of the in-plane modes are almost unaffected by changing the gate voltage and hence the carrier concentration, whereas the out-of-plane modes are redshifted by Δω(A')  ≈  3 cm⁻¹ for the monolayer and Δω(A')  ≈  1.5 cm⁻¹ for the bilayer by changing the charge carrier density by approximately two orders of magnitude. In both cases, the change in the mode frequencies seems to be directly proportional to the applied gate voltage and therefore to the 2D charge carrier density. A line-shape analysis using two Lorentz profiles to describe one Raman spectrum reveals that simultaneously the FWHM of the out-of-plane modes increases with increasing charge carrier densities. By increasing the charge carrier density from ~10¹¹ to ~10¹³ cm⁻², the FWHM increases from approximately 6.5 cm⁻¹ to 10 cm⁻¹ for the monolayer and from 8.8 cm⁻¹ to 11.6 cm⁻¹ for the bilayers (figures 6(c) and (d)). The in-plane modes are again unaffected by changing the charge carrier density and the FWHM for the mono- and bilayers is approximately 5.5 cm⁻¹.

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Very similar qualitative and quantitative findings in Raman investigations on gated MoS₂ monolayers have been reported in a comparable range of charge carrier densities by Chakraborty et al [159]. The authors calculated the charge carrier dependence of the electron–phonon coupling from first-principles DFT. They found an excellent agreement between theory and experiments for doped MoS₂ monolayers: theoretically the strengthened electron–phonon coupling with increasing charge carrier density results in a significant softening and broadening of the out-of-plane phonon. The in-plane mode is hardly affected by doping [159]. The different reactions of the two optical phonon modes on doping result from their different symmetry. The A' (A_1g) mode shares a symmetry with the lattice and consequently structural distortion in this mode does not break the symmetry of MoS₂, resulting in a large electron–phonon coupling strength since all electronic states in the electron–phonon coupling matrix elements can have non-zero expectation value [159]. The conduction band states at the K- and K'- points in momentum space have the character of the Mo d_z2 state. By doping with electrons, the antibonding states with the Mo d_z2 orbital character are filled, which reduces the bond strength. As a result the out-of-plane phonon mode is softened. The occupation of the conduction band minimum states at the K- and K'-points induces an increase in the electron–phonon coupling [159]. Such a doping-induced phonon renormalization is absent for the in-plane mode with E' ($E_{\text{2g}}^{1}$) symmetry due to its orthogonality to the A' (A_1g) symmetry. Therefore, the matrix element that couples electrons and phonons vanishes for the in-plane mode because of this orthogonality [159].

It is interesting to see that the general behavior of the two optical phonon modes is also valid for bilayers (compare figures 6(b) and (d)) and seemingly also for tri- and four-layer crystals [74] indicating that the conduction band minimum at the K- and K'-points matters for thin 2D crystals with an indirect bandgap. The symmetry argument that the phonon renormalization is only effective for the A_1g modes is valid also for bi-, tri- and four-layer MoS₂.

Overall, the sensitivity of the out-of-plane phonon mode with A_1g symmetry empowers Raman experiments to identify the charge carrier density in atomically thin SC-TMDs without the need to fabricate Ohmic contacts and to apply magnetic fields to conduct Hall measurements—the standard probe to determine the charge carrier density in doped semiconductors. Since the explanation of the doping-induced phonon renormalization holds also for other SC-TMDs with identical lattice symmetry and direct band gaps at the K- and K'-points, the determination of the charge carrier density from Raman measurements is expected to be also suitable for all four TMDs in the focus of this review.

An example of this convenient method to determine the change in the charge carrier density of monolayer MoS₂ is provided by the results shown in figure 7(a). Non-resonant Raman spectra (λ_laser  =  488 nm at room temperature) taken on a micro-mechanically exfoliated MoS₂ monolayer on a Si/SiO₂ substrate are contrasted for measurements in vacuum (p  ⩽  5  ×  10⁻⁵ mbar), ambient conditions and with the flake immersed in DI-water (water measurements are performed using a water-dipping objective as described elsewhere [54]). The frequency and linewidth of the in-plane E'-mode are relatively constant for all environments. The frequency of the out-of-plane $A_{1}^{\prime}$-mode is highest for measurements in water, but reduced by almost 3 cm⁻¹ and broadened for spectra taken in vacuum. As shown by us in an earlier work [74], the $A_{1}^{\prime}$-mode frequency and its broadening in ambient conditions strongly depend on the laser power and vary between the two limits for measurements in water and vacuum. The mode frequencies for lowest laser power approach the values measured in water, whereas those observed with higher laser power are close to the mode frequencies observed in vacuum. The experimental observations are consistent with a change of the charge carrier density. The free electron density for measurements in vacuum is up to two orders of magnitude higher compared to those with the MoS₂ flakes completely immersed in DI water. We attribute this change to the effect of molecular doping. Due to their dipolar moments, some molecules, such as O₂ and H₂O, can act as molecular gates causing the transfer of a fraction of the charge from the 2D material to the molecules [64, 88], effectively depleting the 2D electron system in MoS₂. By illuminating the flakes in vacuum, the surface of the 2D material is cleaned and physisorbed molecules are removed. In vacuum, we assume that the intrinsic charge carrier density is measured. Intrinsically, MoS₂ is unintentionally n-type doped with electron densities exceeding 10¹³ cm⁻² most likely because of a high density of sulfur vacancies [17, 25]. Immersed in water, the surface of a MoS2 flake is completely covered with H₂O and O₂ molecules, as sketched in figure 7(b), resulting in the most efficient molecular gating effect and a depletion of the MoS₂ flake by up to two orders of magnitude. The laser power dependence is explained by the gradual removal of physisorbed molecules with a steady state between adsorption and removal depending on the laser power. Since this process is highly reversible, the charge carrier density of MoS₂ in ambient conditions can be tuned just by the illumination intensity [74].

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In contrast to the doping that affects only the homopolar out-of-plane phonon with $A_{1}^{\prime}$ (A_1g) symmetry, uniaxial strain predominantly affects the in-plane phonon with E' ($E_{\text{2g}}^{1}$) symmetry [70, 156]. In particular, uniaxial strain induces a reduction of the mode frequency per % tensile strain of Δω/ε  =  −2.1 cm⁻¹ for the E' phonon mode and only Δω/ε  =  −0.4 cm⁻¹ for the $A_{1}^{\prime}$ phonon mode [156], and another group reports no measurable shift of the $A_{1}^{\prime}$ phonon mode due to uniaxial tensile strain [70]. Bilayer MoS₂ is affected in a similar way [70]. For the application of biaxial tensile strain to monolayer MoS₂, only minor softening of the $A_{1}^{\prime}$ mode is reported; however the E'-mode is substantially softened and a mode splitting appears for tensile strain larger than 0.8% [157]. The splitting of the doubly degenerate E' mode into the LO and TO branch is assigned to symmetry breaking of the crystal caused by biaxial strain [157]. We note that the application of strain not only affects the zone center phonon modes but also the electronic band structure. The bandgap is lowered with increasing tensile strain and the strained material can even undergo the transition from a direct to an indirect semiconducting material [70, 157].

The effect of disorder on the Raman spectrum of monolayer MoS₂ is reproduced in figure 8 [158], where the defects are induced by a bombardment with Mn⁺ ions with varying dose, enabling a systematic study of the Raman response with respect to the interdefect distance L_D. The observed changes in the Raman spectra contrasts the modifications in the spectra induced by strain or doping. With increasing defect density, the E'-phonon frequency is lowered, and the frequency of the $A_{1}^{\prime}$ mode is simultaneously increased [158]. Both modes are broadened with increasing defect density and the LA(M) mode, a longitudinal acoustic mode at the M point at the Brillouin zone boundary, is activated. A combination of doping caused by ion bombardment together with strain cannot explain the experimental signatures since the $A_{1}^{\prime}$ mode frequency not only increases with increasing ion bombardment but also significantly broadens, excluding a reduction of the electron density to be the main source for the modification of the out-of-plane phonon mode. Mignuzzi et al [158] adopted an already established phonon confinement model [176, 177] to explain the defect-induced changes of the first-order phonon modes. The authors find excellent agreement between experiment and numerical calculations of the phonon lineshape utilizing the phonon confinement model together with the phonon dispersion from DFT calculations [158]. Furthermore, the intensity ratios between the defect-activated phonon modes such as the LA(M) and the first-order E' and $A_{1}^{\prime}$ phonons increases monotonically with decreasing interdefect density L_D [158] serving as a very suitable estimate for the defect density from Raman measurements in addition to the opposite frequency shifts of the E' and $A_{1}^{\prime}$ phonons.

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Temperature-induced changes in the lattice constant of a crystal have immediate consequences on the phonon mode frequencies. Typically, the lattice constant increases with increasing temperature. As a direct consequence the phonon modes are softened. Therefore, Raman measurements provide a fast, non-contact method to monitor local changes in the temperature with the spatial resolution of the laser spot that can be better than 1 µm. A rise in temperature can be directly induced by light irradiation, by electric currents in electronic devices and other heat sources. Raman measurements with a high spatial resolution can not only be utilized to measure the local temperature but also to investigate the thermal conductivity of a material. It has been shown that both optical phonon modes with E'/$E_{\text{2g}}^{1}$ and $A_{1}^{\prime}$/A_1g symmetry of SC-TMDs are lowered with increasing temperature [160–163]. The first-order temperature coefficients χ_T for MoS₂ are reported to be linear for both phonon modes, and to be very similar for supported and suspended monolayers. The temperature coefficients for suspended MoS₂ mono-layers are χ_T (E')  =  −0.011 cm⁻¹ K⁻¹ and χ_T ($A_{1}^{\prime}$)  =  −0.012 cm⁻¹ K⁻¹ for the E' and $A_{1}^{\prime}$ modes, respectively and, for sapphire-supported MoS₂ monolayers, χ_T (E')  =  −0.017 cm⁻¹ K⁻¹ and χ_T ($A_{1}^{\prime}$)  =  −0.012 cm⁻¹ K⁻¹ [160]. With the help of Raman measurements, the thermal conductivity κ of a monolayer MoS₂ at room temperature was determined to be κ  =  34.5 W m⁻¹ K⁻¹ [160], which is significantly smaller than the thermal conductivity of graphene with κ  =  2000 W m⁻¹ K⁻¹ [178].

Overall, Raman investigations provide unique access to both number of layers and doping, as well as strain, defect densities and temperatures. Particularly, for monolayers and ultrathin SC-TMDs, the changes in the high-frequency, first-order Raman modes with $A_{1}^{\prime}$(A_1g) (out-of-plane) and E'($E_{\text{2g}}^{1}$) (in-plane) symmetries which are easily experimentally accessible, provide unique and unequivocal access to the above-mentioned parameters. A temperature increase shifts both modes to lower frequencies, whereas increasing defect densities shifts the in-plane mode to lower frequencies and the out-of-plane mode to higher frequencies, with both modes becoming broadened. A decrease in the charge carrier density also shifts the out-of-plane mode to higher frequencies; however the mode is then getting narrower and the in-plane mode is unaffected. This character of the two higher-frequency phonon modes enables us to clearly separate between temperature-, defect- and doping-related effects on the Raman response. Moreover, the effect of uni- and biaxial strain on the Raman modes can be clearly distinguished since strain mainly affects the in-plane mode, while the out-of-plane mode is mostly unchanged. With increasing tensile strain, the in-plane mode frequency is reduced, broadened, and eventually the degeneracy of the TO and LO phonon branches lifts, resulting in a splitting of the in-plane mode.