Giant magneto-optical Raman effect in a layered transition metal compound

Significance

Raman scattering is a powerful technique to probe optical phonons in solids. It usually involves an electron-mediated three-step process, involving photon–electron, electron–phonon, and electron–photon interactions. In principle, manipulating electrons, for instance by applying a magnetic field, should affect Raman phonon intensity, yet there is no direct experimental measurement to demonstrate this. In this work we report the first realization to our knowledge of the idea in a prototype material, MoS₂. From monolayer and bilayer to bulk MoS₂ we observe a dramatic modification of Raman phonon intensity induced by magnetic field. Such a giant magneto-optical effect appearing at a monoatomic layer level and its technological implications for magnetic-optical devices should inspire a new branch of inelastic light scattering.

Abstract

We report a dramatic change in the intensity of a Raman mode with applied magnetic field, displaying a gigantic magneto-optical effect. Using the nonmagnetic layered material MoS₂ as a prototype system, we demonstrate that the application of a magnetic field perpendicular to the layers produces a dramatic change in intensity for the out-of-plane vibrations of S atoms, but no change for the in-plane breathing mode. The distinct intensity variation between these two modes results from the effect of field-induced broken symmetry on Raman scattering cross-section. A quantitative analysis on the field-dependent integrated Raman intensity provides a unique method to precisely determine optical mobility. Our analysis is symmetry-based and material-independent, and thus the observations should be general and inspire a new branch of inelastic light scattering and magneto-optical applications.

Raman scattering is an inelastic light scattering technique that has generated an enormous impact on the study of the functionality of crystalline solids for the last quarter of a century (1). In solids, Raman scattering can be used to observe the Raman active phonons as well as optical quasiparticles (plasmons and magnons). The intensity of any Raman active mode depends upon the symmetry of the crystal, the symmetry of the mode, the intermediate excited states, and the polarization of the incident and detected light (2). This has made Raman spectroscopy an extremely powerful technique for probing nascent symmetry and symmetry changes induced by internal or external perturbations. For example, Raman spectroscopy has been used to explore the effects of external pressure, temperature, field, defects, and doping (1, 3). From a phenomenological perspective, the application of a magnetic field is an effective way to manipulate electrons through Lorentz force in the intermediate state of the Raman process, thus modulating the electronic susceptibility that determines the Raman phonon intensity. In this study, we demonstrate a very large magneto-optical Raman effect in a classic layered, nonmagnetic material, MoS₂, driven by the broken symmetry induced by a magnetic field.

Layered MoS₂ is an obvious prototype material because it has demonstrated promising optoelectronic applications at nanometer scale (4–10), is a quasi-2D material, and has two well-defined and clearly distinct Raman active phonon modes (11). The layered semiconductor MoS₂, which exhibits weak van der Waals coupling between neighboring layers, is structurally similar to graphite, and atomically thin flakes can be obtained from bulk crystals through mechanical exfoliation. Monolayer MoS₂ has an direct band gap of ∼1.9 eV, whereas the thicker members of the family of transition-metal dichalcogenides are indirect-gap (∼1.2 eV) semiconductors (12, 13). This material allows us to investigate the significance on the Raman intensity due to symmetry changes going from a single layer of MoS₂ to a bulk sample.

Demonstrating a large magneto-optical Raman effect in a prototype compound is a fundamental breakthrough in the field of inelastic light scattering, allowing in principle the development of a new type of magneto-optical sensor. Magneto-optical applications are particularly important because these devices allow light intensity or polarization to be controlled using magnetic rather than electric or radiation fields. This control can greatly extend the optoelectronic application range and device-design degrees of freedom. The signal from conventional magneto-optical effects, such as the Kerr and Faraday effects, is generally weak, and an appreciable rotation of the light polarization requires sufficiently thick bulk materials. A magneto-optical coupling based on a different mechanism, such as the one demonstrated in this paper, would open new avenues for applications in atomically thin films or heterostructures.

Monolayer and bilayer MoS₂ flakes were obtained through mechanical exfoliation from a natural MoS₂ single crystal (SPI Supplies) and transferred onto silicon wafers capped with 300-nm-thick SiO₂ by a method that is analogous to that used for production of graphene (Fig. S1). The bulk samples were obtained by cleaving single crystals. Atomically thin MoS₂ flakes were first visually identified by observing their interference color through an optical microscope. The thickness was further confirmed by measuring the frequency difference between the E_2g/E′ and A_1g/A′₁ Raman modes (Fig. S1) (11).

Fig. S1.

(Upper) Optical images of MoS₂ samples in the form of (A) monolayer in the dashed region, (B) bilayer in the dashed region, and (C) bulk in the bright region. (Lower) Raman spectra from the corresponding three samples.

Confocal micro-Raman measurements were performed in a backscattering configuration using a Jobin Yvon T64000 system equipped with a back-illuminated (deep-depletion) CCD. A 532-nm diode-pumped solid-state laser (Torus 532; Laser Quantum) was used for the measurements. The laser was focused onto the samples with a spot size of 5–10 μm². The laser power was maintained at a level of 300 μW and monitored with a power meter (Coherent Inc.). Magnetic fields were generated up to 9 T using a superconducting magnet (Cryomagnetics) that has a room-temperature bore that is suitable for a microscope lens. The magnetic field direction was perpendicular to the sample surface. The 521-cm⁻¹ Raman mode of the Si substrate for supporting the mono- and bilayer of MoS₂ was also carefully monitored in the presence of magnetic fields; the intensities of the Si mode remained almost unchanged as the field was varied from 0 to 9 T in both the parallel and cross configurations (Fig. S2). Thus, the mode can be used to normalize the observed phonon intensities.

Fig. S2.

(Left) Raman spectra of monolayer and (Right) bilayer MoS₂ under magnetic fields. The 521-cm⁻¹ mode from substrate silicon is also shown here for intensity comparison.

A Stokes Raman process involving phonons is illustrated in Fig. 1A (1). Microscopically, Raman scattering is a three-step process: absorption of incoming photon for exciting the system to an intermediate excited state, phonon excitation by decaying to a lower-energy intermediate state, and emission of scattered photon by returning back to the ground state. Phonons are excited through electron–phonon interaction in the intermediate step. Naturally, the intensity of excited phonons reflected in Raman scattering spectra should be changed if the intermediate states of electrons are perturbed by magnetic field. The experimental schematic to realize such an idea is shown in Fig. 1B. A standard backscattering configuration is adopted in our experiments, that is, a linearly polarized incident light is normal to the surface of prototype material MoS₂ and the scattered light is also normal to the surface. A magnetic field is applied perpendicular to the sample surface (i.e., parallel to the incident and scattering light beam) to perturb the symmetry of electronic states.

Fig. 1.

(A) A schematic of a three-step Raman excitation process where phonons are excited through the intermediate electronic states. (B) Schematic of experimental configuration. The incident light polarization (e_i) is along the x direction whereas the scattered light polarization (e_s) can be along either the x (parallel) or y (perpendicular) direction, and magnetic fields are perpendicular to the sample plane (i.e., along the z direction). A field-driven transverse component of electron motions under radiation field is illustrated, which effectively gives rise to a rotation of the polarization plane of the scattered light. Raman spectra of monolayer MoS₂ of (C) parallel and (D) perpendicular polarization configurations with and without the magnetic field. The vibrational patterns of the corresponding E and A Raman modes are illustrated in the insets.

MoS₂ has a honeycomb-lattice structure similar to that of graphite. Mo atoms are sandwiched by neighboring S planes in a separate MoS₂ layer (Fig. 1B), with the point-group symmetry of D_6h in the bulk, which becomes D_3d in the bilayer case. In monolayer form, the symmetry is further reduced to D_3h due to the disappearance of the inversion center (Table 1). There are two distinct and well-defined Raman modes in all three forms of MoS₂. One is the stretching of S atoms along the c-axis (A-mode) and the other is the in-plane breathing motion (E-mode) as illustrated in Fig. 1C, Inset (11). The symmetries of the two modes are changed from bulk to bi- and monolayer because of the reduction of point-group symmetry mentioned above (Table 1). However, such reduction of symmetry does not induce significant changes in the two characteristic modes (11).

Table 1.

A- and E-mode symmetries with and without magnetic field in MoS₂

	Bulk/bilayer	Monolayer	Bulk/bilayer	Monolayer
Field	B = 0		B // c
Symmetry	D6h/D3d	D3h	C6h/C3i	C3h
A-mode	A1g	A′1	Ag	A′
E-mode	E2g/Eg	E′	E2g/Eg	E′

Strikingly, the application of a magnetic field results in dramatic and mode-selective change in Raman intensity, shown in Fig. 1 C and D for a monolayer case. The E-mode is almost magnetic-field-independent, but the A-mode displays a giant change in intensity with magnetic field, decreasing its intensity with field as the polarization of the measured scattered light is parallel to the incident light (${\mathit{e}}_i\parallel {\mathit{e}}_s$; Fig. 1C) but increasing its intensity as the scattered and incident light polarizations are perpendicular to each other (${\mathit{e}}_i\perp {\mathit{e}}_s$; Fig. 1D).

The dramatic modulation of the Raman intensity by magnetic field is observed in monolayer, bilayer, and bulk MoS₂. Fig. 2 displays the magnetic-field-dependent polarized Raman spectra for the three cases and two polarization configurations: parallel (${\mathit{e}}_i\parallel {\mathit{e}}_s$, left) and perpendicular (${\mathit{e}}_i\perp {\mathit{e}}_s$, right). The field dependence of the spectra for the three samples is almost identical. With increasing magnetic field strength, the frequencies of both E- and A-modes remain unchanged. The intensity of the E-mode is almost constant with increasing field for both polarization configurations, whereas for A-mode the intensity variation with the field exhibits complementary behavior between ${\mathit{e}}_i\parallel {\mathit{e}}_s$ and ${\mathit{e}}_i\perp {\mathit{e}}_s$ scattering configurations.

Fig. 2.

Raman spectra of monolayer (A and B), bilayer (C and D), and bulk (E and F) MoS₂ at room temperature and in the presence of magnetic fields in the (A, C, and E) parallel and (B, D, and F) perpendicular polarization configurations. Here e_i/e_s denote the polarization of incident and scattered light, respectively.

Fig. 3 A–C displays the evolution of A-mode intensity as a function of magnetic field for the two polarization configurations and three samples, clearly showing similar field-induced anticorrelation intensity modulation. A major difference in the three cases is that, for monolayer MoS₂, the A₁′-mode intensity in the ${\mathit{e}}_i\perp {\mathit{e}}_s$ polarization configuration reaches its maximum at ∼6 T and then decreases slightly with increasing field. The maximum shifts to higher magnetic field in the bilayer and bulk cases, which will be discussed later.

Fig. 3.

Magnetic-field dependence of the A′₁/A_1g intensity of (A) monolayer, (B) bilayer, and (C) bulk MoS₂. The colored solid curves represent the theoretically derived intensity functions. (D–F) Polarization P, which is defined as the relative intensity difference between the perpendicular and parallel polarization configurations, and its rate of change versus magnetic field.

To understand the experimental results, we first examine the prevailing explanation of magnetic-field effects on Raman spectra. Most existing studies focus on spin-dependent phonon processes in magnetic materials (14) or the phonon–plasmon coupling (15). The absence of local spin moments in MoS₂ imposes constraints on possible explanations. For example, the possibilities related to magnetic ions, such as spin-lattice interaction, magnetic polarons, or magnetic impurities, can be ruled out because there are no magnetic ions in the present material. The phonon–plasmon coupling mechanism seems unlikely because this coupling typically results in a change in phonon frequency, line width, and/or phonon line shapes due to the energy exchange between the excitations, which is not observed in our measurements. Furthermore, the fact that similar field-induced modulations are observed in both monolayer and bulk MoS₂ makes interface-related effects implausible.

Recent studies of Raman spectra under magnetic fields in graphene revealed that Landau levels are superimposed on original energy bands, which can substantially tune the inter- or intraband transitions and hence cause magneto-phonon resonances (16–18). In the present work, thermal fluctuations are sufficiently strong to melt the Landau levels formed in the presence of magnetic fields up to 9 T at room temperature (Fig. S3). This makes it unlikely that the Landau levels are responsible for the anticorrelated Raman intensity change in response to magnetic fields in the two polarization configurations.

Fig. S3.

Energy scales for (blue) Landau levels, (dashed) room temperature, and (red and green) phonon frequencies.

The above analysis suggests that a field-induced intensity effect observed in the present work requires an examination of Raman process, especially the field effect on the electrons that mediate the inelastic light scattering. Classically, Raman scattering intensity is determined by the second-order electronic susceptibility (susceptibility derivative). The electronic susceptibility α can be expressed as a function of normal coordinates and external fields. The magnetic field effect on normal coordinates is negligible because ions/anions are too heavy compared with electrons, which allows us to safely separate electronic susceptibility into two decoupled terms, that is, $\alpha =\alpha (Q,B)\approx \alpha \left(Q\right)\times \alpha \left(B\right)\approx \text{const}.\times \alpha \left(B\right)$, where B is the magnetic field and Q represents the normal coordinates.

First, let us consider α(B) in a semiclassic scenario. The Lorentz force due to the applied field in normal direction (along z direction) to the sample plane (xy plane) generates a transverse component to the in-plane electron motion (Fig. 1B), resulting in a transfer of Raman intensity between the two orthogonal polarization configurations. The magnetic field dependence α(B) can be derived by considering the Lorentz force. In a 2D case, the kinetic equations for electrons driven by a radiation field can be written classically as

$$\{\begin{array}{c}\ddot{x}+\gamma \dot{x}+{\omega }_0^{2}x+\frac{eB}{m}\dot{y}+\frac{e}{m}E{e}^{i\omega t}=0\\ \ddot{y}+\gamma \dot{y}+{\omega }_0^{2}y-\frac{eB}{m}\dot{x}=0\end{array},$$ [1]

where E is the electric field vector of incident light along the x direction (i.e., e_i || x) and Lorentz forces are taken into account. The parameters e, m, B, ω, ω₀, and γ are the charge, effective mass of electron, the magnetic field, the frequency of incident light, the restoring frequency of excited electrons, and the damping factor (scattering rate), respectively.

For electrons, a magnetic field normal to the sample plane breaks the vertical mirror symmetry and the horizontal twofold symmetry. Hence, the corresponding Raman tensors, whose elements are normally the derivatives of α, should have a lower symmetry when a magnetic field is applied. A detailed analysis of the Raman tensors can be found in Supporting Information. The Raman intensities of A- and E-modes for ${\mathit{e}}_i\parallel {\mathit{e}}_s$ (xx) and ${\mathit{e}}_i\perp {\mathit{e}}_s$ (xy) configurations can be expressed as

$$I_{xx}^{E_{2g}}=I_{xy}^{E_{2g}}\propto h^2{\left|\frac{B}{B_0^2+B^2}\right|}^2+f^2{\left|\frac{B_0}{B_0^2+B^2}\right|}^2$$ [2]

$$I_{xx}^{A_g}\mathrm{\propto }{a}^2{\left|\frac{B_0}{B_0^2+B^2}\right|}^2\mathrm{;}{\mathit{I}}_{\mathit{xy}}^{{\mathit{A}}_{\mathit{g}}}\mathrm{\propto }{\mathrm{c}}^2{\left|\frac{B_0}{B_0^2+B^2}\right|}^2,$$ [3]

where $B_0=\frac{m\gamma }{e}\left(1-i\frac{{\omega }_0^2-{\omega }^2}{\omega \gamma }\right)$, which can be viewed as a resonance-like field at which the effect of magnetic field on phonon intensity approaches a maximum. Here a, c, h, and f are the parameters of the Raman tensor elements, which are proportional to the derivatives of susceptibility with respect to lattice normal coordinates. As shown in Eq. 2, the E_2g intensities exhibit the same magnetic-field dependence in both polarization configurations. Furthermore, E-mode has twofold degeneracy (11) so that its intensity in Eq. 2 includes two terms that compensate each other with the field (see Fig. S4 for fitting results of E-mode). This explains why the observed magnetic-field dependence of the E_2g intensities is relatively weak. In sharp contrast, as shown in Eq. 3, the nondegenerate A-mode has field- and polarization-dependent intensity. For each sample, we have performed a fitting for the A-mode intensity with a single set of B₀ parameters in both ${\mathit{e}}_i\parallel {\mathit{e}}_s$ and ${\mathit{e}}_i\perp {\mathit{e}}_s$ configurations. The results are shown in Fig. 3. Excellent agreement between the theoretical curve and experimental data for all three samples is obtained (see Tables S1 and S2 for the details of fitting parameters).

Fig. S4.

Experimental (solid dots) and fitted E-mode intensity as a function of applied magnetic field for (A and B) monolayer, (C and D) bilayer, and (E and F) bulk MoS₂, and two light polarization configurations, respectively.

Table S1.

Fitting parameters for A-mode using Eq. 3

Sample	a	c
Monolayer	557	743
Bilayer	724	1,229
Bulk	1,001	1,668

Table S2.

Fitting results for the A-mode

Sample	Real part of B0 m*γ/e = 1/μ, T	Imaginary part of B0, T	Mobility μ, cm2/V·s	Effective mass, m* (me) (33)	γ (1/s)
Monolayer	5.1	3.6	1,960	0.37	2.4 × 1012
Bilayer	8.9	0	1,124
Bulk	10.5	0	950	0.47	3.9 × 1012

The real part of B₀, mγ/e, is the reciprocal of the mobility (i.e., 1/μ). Thus, optical mobility can be extracted from analyzing the Raman scattering process because the intermediate state for electron hopping in a Raman process exists in the real conduction bands (19). We have determined both the real part and imaginary part of B₀ by fitting the field dependence of Raman intensities to the model functions given in Eq. 3 (Fig. 3). The value of the real part of B₀ is 5.1, 8.9, and 10.5 T for monolayer, bilayer, and bulk MoS₂, respectively (Fig. S5 and Table S2). These results correspond to optical mobilities of 1,960, 1,124, and 950 cm²/V·s, respectively. The abrupt increase in the optical mobility in the monolayer case is attributed to the dimensionality effect, which is accompanied by a reduction in the number of effective electron scattering channels. The temperature dependence of optical mobility is consistent with the results obtained from transport measurements (20) (Fig. S6). Note that the values of the optical mobility determined in the present work are much larger than those measured using transport measurements (21–25) but similar to the values obtained in a high-k HfO₂ gated field-emission transistor (20, 26). Furthermore, our results are also greater than the calculated phonon-scattering-limited value of ∼400 cm²/V·s (27). The optical mobility data obtained with the present method should be a good measure of intrinsic scattering processes because of the effective exclusion of grain-boundary/impurity scattering (28).

Fig. S5.

Fitting the intensities of monolayer MoS₂ in parallel configuration with (purple) and without (green) the imaginary part of B₀.

Fig. S6.

Intensity evolution of the A- and E-mode in monolayer MoS₂ under magnetic fields at (A) 70 K, (B) 150 K, and (C) 300 K. (D–F) The magnetic-field dependence of A-mode integrated intensities at three temperatures is summarized, where the solid lines are the fitting curves.

Because the measured Raman intensity corresponds to the intensity of inelastically scattered light, our observations indicate that the intensity difference of the scattered light between the parallel and perpendicular polarization configurations can be precisely tuned by using magnetic field. This effect can be regarded as a “giant” magneto-optical effect. We define a polarization between the two polarization configurations, which is similar to the valley polarization (6, 7), as shown in Fig. 3 D–F, and the slope of each curve is a measure of the sensitivity of the light intensity to magnetic field. The polarization goes up to 80% at low fields and its slope reaches 25% per tesla at intermediate fields. This quantitatively demonstrates the remarkable efficiency of the polarization manipulation by magnetic fields. Because the effect described above is symmetry-based and material-independent, it should have potential technological applications in magneto-optical devices. We have proposed a prototype layout based on the effect, as illustrated in Fig. S7. In general, the weak intensity of the inelastically scattered light posts a challenge for applications. With the observed giant magneto-optical effect, this issue may be resolved by identifying high-intensity modes in a material and/or employing the surface-enhanced Raman technique and coherent Stokes/anti-Stokes Raman scattering (29).

Fig. S7.

Schematic diagram of the magnetic-field-modulated Raman effect, which shows a prototype layout for possible new magneto-optical applications in read heads, sensors, and optical switches. A band-pass filter is used to filter out non-Raman-scattered light, and a Wollaston prism is used to separate the scattered light into two polarization channels.

Finally, we would like to point out the unique behavior of monolayer MoS₂. Zeng et al. (30) showed that integrated intensity of the A-mode gradually decreases with reduced thickness from the bulk but suddenly increases in a monolayer sample. This gives an apparently lower integrated intensity of A-mode in the bilayer than in the monolayer or bulk case (Fig. 3 A–C and Fig. S1). The most likely origin of this effect is the anomalous behavior of the monolayer (30), both electronically and structurally. MoS₂ is an indirect-gap semiconductor from bilayer to bulk but exhibits a direct gap for the monolayer case. The question to be answered is why the monolayer is so different, requiring more experimental measurements and detailed density functional theory calculations.

In summary, we have measured the evolution of Raman spectra of monolayer, bilayer, and bulk MoS₂ with a magnetic field perpendicular to the layer surface. We find that the A_1g/A₁′ Raman mode exhibits a giant response to the field with anticorrelated intensity changes in two orthogonally polarized configurations of scattered light. The intensity difference between these two configurations can be controlled and fine-tuned by the magnetic field. This magneto-optical effect stems from a magnetic-field-induced symmetry breaking for the electron motion in the inelastic Raman scattering process; thus, the basic mechanism is material-independent. The present discovery lays a solid foundation for innovative magneto-optical device applications with materials from bulk crystals to single atomic form. It also provides a new approach for the precise measurements of optical mobility in atomically thin films.

Sample Characterizations

Similar to graphene, the different contrast of optical images due to optical interference in few-layer MoS₂ provides a basic identification for the layer number (Fig. S1). The frequency separation between the E- and A-mode is also a standard way to identify layer thickness in few-layer MoS₂ (31–33). The separation of 19 cm⁻¹ corresponds exactly to the single-layer case. The frequency separations are used as a secondary check on the layer thickness for our samples.

Scattered Light Intensity of MoS₂ A-Mode and Substrate Silicon

To make quantitative measurements in an accurate way, the intensities of MoS₂ A-mode relative to the silicon mode at 521 cm⁻¹ are carefully monitored. As shown in Fig. S2, the intensities of the silicon mode remain almost unchanged with increasing magnetic field, whereas the intensities of MoS₂ A-mode exhibit a systematic evolution. Thus, we use the intensity of Si mode as a reference for all of the intensity analysis of field dependence of MoS₂ modes. The present results clearly rule out the effect of extrinsic factors, such as possible laser power instability or temperature-induced fluctuations, on the magnetic-field-dependent intensities of the A-mode.

All of the reported intensities including the results presented in Fig. 3 are normalized to that of the 521-cm⁻¹ mode of the silicon substrate. Such normalization enables us to have accurate and quantitative field-dependent intensity analysis.

Comparison of Landau-Level Spacing and Phonon Energies

Generally the emergence of Landau levels (LLs) under magnetic fields will subtly modify the original band structures and perhaps heavily affect the optical transitions. It may result in a Raman resonance if the allowed optical transition matches the LLs in energy. Furthermore, the possibility of Pauli blocking becomes much higher in this case, particularly in the samples with doping.

Our measurements were performed at room temperature, at which the transitions between Landau levels are completely smeared out, as mentioned in the main text (see Fig. S3). However, the LL interaction must be taken into account at very low temperatures and the experimental observations may be significantly different from those at room temperature.

Symmetry Breaking by a Magnetic Field in a Raman Process

Based on the semiclassic scenario of a 2D case, the magnetic field dependence of electron susceptibility$\alpha \left(B\right)$ can be obtained by solving the coupled equations given in Eq. 1 in the main text as

$$\{\begin{array}{c}{\alpha }_{xx}\left(B\right)={\alpha }_{yy}\left(B\right)=\frac{-ex}{E}\propto \frac{B_0}{B_0^2+B^2}\\ {\alpha }_{xy}\left(B\right)={\alpha }_{yx}\left(B\right)=\frac{-ey}{E}\propto \frac{B}{B_0^2+B^2}\end{array},$$

where $B_0=\frac{m\gamma }{e}\left(1-i\frac{{\omega }_0^2-{\omega }^2}{\omega \gamma }\right)$. As a spinor field, a magnetic field applied perpendicular to the MoS₂ plane does not break the horizontal mirror plane symmetry, but it breaks the horizontal twofold rotation and/or vertical mirror plane symmetry for the electron motion in the MoS₂ layer. This means that the D_6h/D_3h symmetry of bulk/monolayer MoS₂ will be reduced to C_6h/C_3h when a perpendicular magnetic field is applied. The complete forms of Raman tensors under magnetic field can be derived by combining the requirements of the lattice symmetry and the magnetic field dependence of electron susceptibility (1), which yields

$$\{\begin{array}{c}{A}_g/A\prime \to {\chi }_{A_g}^{ij}:\left(\begin{array}{ccc}{\alpha }_{xx}\left(B\right)\cdot a& {\alpha }_{yx}\left(B\right)\cdot c& 0\\ -{\alpha }_{xy}\left(B\right)\cdot c& {\alpha }_{yy}\left(B\right)\cdot a& 0\\ 0& 0& b\end{array}\right)\\ E_{2g}/E\prime \to {\chi }_{E_{2g}}^{ij}\left(1\right):\left(\begin{array}{ccc}{\alpha }_{xx}\left(B\right)\cdot h& -{\alpha }_{yx}\left(B\right)\cdot f& 0\\ -{\alpha }_{xy}\left(B\right)\cdot f& -{\alpha }_{yy}\left(B\right)\cdot h& 0\\ 0& 0& 0\end{array}\right){\chi }_{E_{2g}}^{ij}\left(2\right):\left(\begin{array}{ccc}{\alpha }_{xy}\left(B\right)\cdot f& {\alpha }_{xx}\left(B\right)\cdot h& 0\\ {\alpha }_{xx}\left(B\right)\cdot h& -{\alpha }_{xy}\left(B\right)\cdot f& 0\\ 0& 0& 0\end{array}\right)\end{array}.$$

Here a, c, h, and f are the lattice symmetry restricted elements that are proportional to the susceptibility derivatives with respect to the corresponding normal coordinates [i.e., the derivatives of α (Q)]. It should be noted that the matrix elements of the twofold degenerate E_2g/E′ tensors are dependent on each other due to the symmetry requirements. The Raman tensors naturally return to the forms required by the D_6h/D_3h symmetries at zero field (1), indicating the self-consistency of the above symmetry analysis.

The Raman intensities of the E_2g/E′ modes can be derived as

$$\{\begin{array}{c}{I}_{xx}^{E_{2g}}\propto {\left|{e\rightharpoonup }_i\cdot {\chi }_{E_{2g}}^{ij}\left(1\right)\cdot {e\rightharpoonup }_s\right|}^2+{\left|{e\rightharpoonup }_i\cdot {\chi }_{E_{2g}}^{ij}\left(2\right)\cdot {e\rightharpoonup }_s\right|}^2=h^2\cdot {\left|{\alpha }_{xy}\left(B\right)\right|}^2+f^2\cdot {\left|{\alpha }_{xx}\left(B\right)\right|}^2\\ I_{xy}^{E_{2g}}\propto {\left|{e\rightharpoonup }_i\cdot {\chi }_{E_{2g}}^{ij}\left(1\right)\cdot {e\rightharpoonup }_s\right|}^2+{\left|{e\rightharpoonup }_i\cdot {\chi }_{E_{2g}}^{ij}\left(2\right)\cdot {e\rightharpoonup }_s\right|}^2=h^2\cdot {\left|{\alpha }_{xy}\left(B\right)\right|}^2+f^2\cdot {\left|{\alpha }_{xx}\left(B\right)\right|}^2\end{array}.$$

Here the E_2g intensities exhibit the same magnetic-field dependence in the (xx) and (xy) configurations. The intensity compensation offered by ${\alpha }_{xx}\left(B\right)$ and${\alpha }_{xy}\left(B\right)$ qualitatively explains why the observed magnetic-field dependence of the E_2g intensities is relatively weak.

Fitting Results

We have fitted in integrated intensities of both E- and A-mode with magnetic field to the functions of these two mode intensities as a function of field given in Eqs. 2 and 3, which allows us to obtain the parameters in the expression of B₀. The fitting results are given in Figs. S4 and S5 as well as Tables S1 and S2.

It is noted that the imaginary part of B₀ is finite in monolayer MoS₂, whereas it goes to zero in bilayer and bulk cases. This is consistent with the transition from an indirect- to a direct-gap semiconductor.

Temperature Dependence of the Intensities of A- and E-Modes

The results in Fig. S6 A–C show drastic intensity modulations by magnetic field for the A-mode at each temperature. The magnetic-field dependence of A-mode integrated intensities shown in Fig. S6 D–F indicates that the peak position of the intensity as a function of the field undergoes a monotonic decrease with lowering temperature. Fitting the results using the expression derived in the following section, we obtain the optical mobility for each temperature listed in Fig. S6 D–F. The obtained optical mobilities increase with decreasing temperature and show a saturation tendency at low temperatures. This behavior is qualitatively consistent with the results given by transport measurements (20) and can be naturally understood as a consequence of the reduction of phonon scattering, which is gradually frozen by decreasing temperature.

Possible Applications of the Observed Effect

Based on the magnetic-field-controlled effect, we propose a prototype layout as illustrated in Fig. S7, which is formally similar to the Hall effect and may serve as a basis for design of new magneto-optical applications, such as read heads, sensors, and optical logic elements. The intensity ratio of scattered light between two orthogonal polarization configurations resolved by the two photo detectors can be tuned continuously by varying the applied magnetic field.