Abstract
In bulk and quantumconfined semiconductors, magnetooptical studies have historically played an essential role in determining the fundamental parameters of excitons (size, binding energy, spin, dimensionality and so on). Here we report lowtemperature polarized reflection spectroscopy of atomically thin WS_{2} and MoS_{2} in high magnetic fields to 65 T. Both the A and B excitons exhibit similar Zeeman splittings of approximately −230 μeV T^{−1} (gfactor ≃−4), thereby quantifying the valley Zeeman effect in monolayer transitionmetal disulphides. Crucially, these large fields also allow observation of the small quadratic diamagnetic shifts of both A and B excitons in monolayer WS_{2}, from which radii of ∼1.53 and ∼1.16 nm are calculated. Further, when analysed within a model of nonlocal dielectric screening, these diamagnetic shifts also constrain estimates of the A and B exciton binding energies (410 and 470 meV, respectively, using a reduced A exciton mass of 0.16 times the free electron mass). These results highlight the utility of high magnetic fields for understanding new twodimensional materials.
Introduction
Atomically thin crystals of the transitionmetal disulphides (MoS_{2} and WS_{2}) and diselenides (MoSe_{2} and WSe_{2}) constitute a novel class of monolayer semiconductors that possess direct optical bandgaps located at the degenerate K and K′ valleys of their hexagonal Brillouin zones^{1,2}. The considerable recent interest in these twodimensional (2D) transitionmetal dichalcogenides (TMDs) derives from their strong spin–orbit coupling and lack of structural inversion symmetry, which, together with timereversal symmetry, couples spin and valley degrees of freedom and leads to valleyspecific optical selection rules^{3,4,5,6,7,8}: light of σ^{+} circular polarization couples to interband exciton transitions in the K valley, while the opposite (σ^{−}) circular polarization couples to transitions in the K′ valley. The ability to populate and/or probe electrons and holes in specific valleys using polarized light has renewed longstanding interests^{8,9,10,11} in understanding and exploiting such ‘valley pseudospin’ degrees of freedom for both fundamental physics and farreaching applications in, for example, quantum information processing.
The bands and optical transitions at the K and K′ valleys are nominally degenerate in energy and related by timereversal symmetry. However, in analogy with conventional spin degrees of freedom, this K/K′ valley degeneracy can be lifted by external magnetic fields B, which break timereversal symmetry. Recent photoluminescence studies of the monolayer diselenides MoSe_{2} and WSe_{2} in modest fields have indeed demonstrated this ‘valley Zeeman effect’, and revealed an energy splitting between σ^{+} and σ^{−} polarized photoluminescence from the lowestenergy ‘A’ exciton transition^{12,13,14,15,16,17}. In most cases, valley splittings in these monolayer diselenides were found to increase linearly with field at a rate of approximately −4μ_{B} (≡−232 μeV T^{−1}), where μ_{B}=57.9 μeV T^{−1} is the Bohr magneton. While this value agrees surprisingly well with simple expectations from a twoband tightbinding model (namely, that electron and hole masses are equal, and that the exciton Zeeman shifts derive solely from the hybridized atomic orbitals with magnetic moment ±2μB that comprise the K/K′ valence bands^{12,13,14,15}), it is also widely appreciated that a more complete treatment based on established k·p theory should, with proper inclusion of strong excitonic effects, also provide an accurate description. However, initial k·p models have so far had limited success accounting for the measured valley Zeeman effect in monolayer TMDs^{12,16,18,19}.
Regardless of circumstances, magnetooptical studies of these new monolayer semiconductors are still at a relatively early stage, and several outstanding questions remain. In particular, measurements of valley Zeeman effects in the monolayer disulphides WS_{2} and MoS_{2} have not been reported to date, which would provide a natural complement to the existing data on monolayer WSe_{2} and MoSe_{2}. In addition, the valley Zeeman splitting of the higherenergy ‘B’ exciton has not yet been reported in any of these 2D materials. Both of these studies would provide a more complete experimental basis against which to benchmark new theoretical approaches. And finally, the diamagnetic energy shift of these excitons, which is anticipated to increase quadratically with field and from which the spatial extent of the fundamental (1s) exciton wavefunctions can be directly inferred^{20,21,22}, has not yet been observed in any of the monolayer TMDs. Likely, this is because the diamagnetic shift, ΔE_{dia}=e^{2}〈r^{2}〉_{1s}B^{2}/8m_{r}, is expected to be very small and difficult to spectrally resolve in these materials owing to the small root mean squared (r.m.s.) radius of the 1s exciton , and large reduced mass . For example, if r_{1}≈1.5 nm and m_{e}=m_{h}≈m_{0}/2 (where m_{0} is the bare electron mass and m_{e/h} is the effective electron/hole mass), then ΔE_{dia} is only ∼20 μeV at B=10 T, clearly motivating the need for large magnetic fields. Crucially, knowledge of ΔE_{dia} can also constrain estimates of the exciton binding energy—a subject of considerable recent interest in the monolayer TMDs^{23,24,25,26,27,28,29,30,31,32,33,34,35}, wherein the effects of nonlocal dielectric screening and Berry curvature can generate a markedly nonhydrogenic Rydberg series of exciton states and associated binding energies^{36,37,38,39,40}.
In the following, we address these questions with a systematic study of circularly polarized magnetoreflection from largearea films of monolayer WS_{2} and MoS_{2} at low temperatures (4 K) and in very high pulsed magnetic fields up to 65 T. Clear valley splittings of about −230 μeV T^{−1} are observed for both the A and B excitons, providing measurements of the valley Zeeman effect and associated gfactors in monolayer transitionmetal disulphides. Moreover, due to the very large magnetic fields used in these studies, we are also able to resolve the small quadratic diamagnetic shifts of both A and B excitons in monolayer WS_{2} (0.32±0.02 and 0.11±0.02 μeV T^{−2}, respectively), permitting estimates of the r.m.s. exciton radius r_{1}. These results are compared with similar measurements of bulk WS_{2} crystals, and are quantitatively modelled within the context of the nonhydrogenic binding potential^{23,36,37} that is believed to exist in 2D semiconductors due to nonlocal dielectric screening. Within this framework, we estimate A and B exciton binding energies of ∼410 and ∼470 meV, respectively, and we show how these values scale with reduced mass m_{r}.
Results
Samples and experimental setup
Largearea samples of monolayer WS_{2} and MoS_{2} were grown by chemical vapour deposition on SiO_{2}/Si substrates^{41,42}. MoO_{3} and pure sulphur powder were used as precursor and reactant materials, respectively, and the growth was performed at a reactant temperature of 625 °C. In addition, perylene3,4,9,10tetracarboxylic acid tetrapotassium salt was loaded on the SiO_{2}/Si substrate, which acted as a seeding promoter to achieve uniform largearea monolayer crystals^{43}. The monolayer nature and high quality of these samples were confirmed by photoluminescence and Raman studies^{42} (Supplementary Figs 1–5 and Supplementary Notes 1–5). In addition, a freshly exfoliated surface of a bulk WS_{2} crystal was also prepared.
Magnetoreflectance studies were performed at cryogenic temperatures (down to 4 K) in a capacitordriven 65 T pulsed magnet at the National High Magnetic Field Laboratory in Los Alamos. Broadband white light from a xenon lamp was coupled to the samples using a 100 μm diameter multimode optical fibre. The light was focused onto the sample at nearnormal incidence using a single aspheric lens, and the reflected light was refocused by the lens into a 600 μm diameter collection fibre. A thinfilm circular polarizer mounted over the delivery or collection fibre provided σ^{+} or σ^{−} circular polarization sensitivity. The collected light was dispersed in a 300 mm spectrometer and detected with a chargecoupled device detector. Spectra were acquired continuously at a rate of 500 Hz throughout the ∼50 ms long magnet pulse.
Exciton transitions and Zeeman effects in monolayer TMDs
Figure 1a depicts a singleparticle energy diagram of the conduction and valence bands in monolayer TMDs at the K and K′ points of the hexagonal Brillouin zone, along with the A and B exciton transitions (wavy lines) and valleyspecific optical selection rules. Strong spin–orbit coupling of the valence band splits the spinup and spindown components (by Δ_{v} ∼400 and 150 meV in WS_{2} and MoS_{2}, respectively), giving rise to the wellseparated A and B exciton transitions that are observed in optical absorption or reflection spectra. As depicted, σ^{+} circularly polarized light couples to both the A and B exciton transitions in the K valley, while light of the opposite σ^{−} circular polarization couples to the exciton transitions in the K′ valley.
At zero magnetic field, timereversed pairs of states in the K and K′ valleys—for example, spinup conduction (valence) bands in K and spindown conduction (valence) bands in K′—necessarily have the same energy and have equalbutopposite total magnetic moment . Therefore, an applied magnetic field, which breaks timereversal symmetry, will lift the K/K′ valley degeneracy by shifting timereversed pairs of states in opposite directions in accord with the Zeeman energy −μ·B. This will Zeeman shift the measured exciton energy if the relevant conduction and valence band moments are unequal; viz, ΔE_{Z}=−(μ^{c}−μ^{v})·B. In the following, we consider strictly outofplane fields, .
Figure 1b depicts the fielddependent energy shifts of the conduction and valence bands in the K valley (σ^{+} polarized light), for both positive and negative fields. The various contributions to the total Zeeman shift in the monolayer TMDs have been discussed in several recent reports^{3,12,13,14,15,44}, which we summarize as follows. In general, the total magnetic moment μ of any given conduction or valence band in the K or K′ valley contains contributions from three sources: spin (μ_{s}); atomic orbital (μ_{l}); and the valley orbital magnetic moment that is associated with the Berry curvature (μ_{k}). Note that the latter two have been referred to as ‘intracellular’ and ‘intercellular’ contributions to the orbital magnetic moment, respectively^{12,13}. The spin contribution to the exciton Zeeman shift ΔE_{Z} is expected to be zero, since the optically allowed transitions couple conduction and valence bands having the same spin . In contrast, the conduction and valence bands are comprised of entirely different atomic orbitals: the orbitals of the conduction bands have azimuthal orbital angular momentum l_{z}=0 , while the hybridized orbitals that comprise the valence bands have l_{z}=±2ħ in the K and K′ valleys, respectively. This contribution is expected to generate a Zeeman shift of the K and K′ exciton of , respectively, and therefore, a total exciton splitting of −4μ_{B}B. Finally, the valley orbital (Berry curvature) contributions to the conduction and valence band moments are and in the K and K′ valleys, respectively. In a simple twoband tightbinding model where m_{e}=m_{h}, then and shifts due to the valley orbital magnetic moment do not appear in ΔE_{Z}. In more general models^{45} where m_{e}≠m_{h}, these Berry curvature contributions may play a role and cause a deviation of the exciton Zeeman splitting away from −4μ_{B}.
To selectively probe the K and K′ transitions in our magnetoreflectivity experiments, we typically fixed the handedness of the circular polarizer to σ^{+}, and pulsed the magnet in the positive (+65 T) and then the negative (−65 T) field direction. The latter case is exactly equivalent (by timereversal symmetry) to measuring the σ^{−} optical transitions in positive field (we also explicitly verified this by changing the circular polarizer). Sign conventions were confirmed via magnetoreflectance from a diluted magnetic semiconductor (Zn_{0.92}Mn_{0.08}Se)^{46}.
Valley Zeeman effect in monolayer WS_{2}
Figure 2a shows the reflection spectrum (raw data) from monolayer WS_{2} at 4 K. Both the A and B exciton transitions are clearly visible and are superimposed on a smoothly varying background. Figure 2b shows the wellresolved Zeeman splitting of the A exciton in WS_{2} at the maximum ±65 T applied magnetic field. Red, blue and (dashed) black curves show the normalized reflection spectra, 1−R/R_{0} (where R_{0} is a smooth background), at +65, −65 and 0 T respectively. A valley splitting of ∼15 meV, analysed in detail below, is observed. Moreover, because these measurements are based on magnetoreflectance spectroscopy (rather than photoluminescence), the valley splitting of the higherenergy B exciton in WS_{2} can also be observed, as shown in Fig. 2c. For both the A and B excitons, the energy of the exciton transition in positive magnetic fields (hereinafter called E^{+}) shifts to lower energy, while the exciton energy in negative fields (E^{−}) shifts to higher energy, as labelled.
The exciton resonances were fit using complex (absorptive+dispersive) Lorentzian lineshapes to extract the transition energy. The fielddependent energies of the split peaks in monolayer WS_{2}, E^{+}(B) and E^{−}(B), are shown in Fig. 2d,e for the A and B excitons, respectively. The splitting between the two valleys, E^{+}−E^{−}, is shown in Fig. 2f for both the A and B excitons. The measured valley Zeeman splitting is negative, and increases in magnitude linearly with applied field, with nearly identical rates of −228±2 μeV T^{−1} for the A exciton and −231±2 μeV T^{−1} for the B exciton. These values correspond to Landé gfactors of −3.94±0.04 and −3.99±0.04, respectively, thereby quantifying the valley Zeeman effect in the monolayer transitionmetal disulphides, and also providing a measurement of the B exciton valley splitting in monolayer TMD materials.
The A exciton valley splitting that we measure in monolayer WS_{2} is quite close to that reported recently from magnetophotoluminescence studies of its diselenide counterpart, monolayer WSe_{2} (refs 13, 16). For comparison, reported gfactors for all the monolayer TMDs are shown in Table 1. As discussed above, our measured values of g≃−4 agree surprisingly well with a simple twoband tightbinding model, wherein m_{e}=m_{h} and valley moment (Berry curvature) contributions to the exciton magnetic moment cancel out, so that the exciton Zeeman shifts derive solely from atomic orbital magnetic moments of the valence bands. However, Berry curvature contributions to the Zeeman splitting are expected in more general models^{45} where m_{e}≠m_{h}. Deviations away from g=−4, observed, for example, in refs 13, 14, have been explained along these lines (although, note that for tightly bound excitons, the total valley moment contribution can vary significantly in magnitude and sign, because this orbital moment must be averaged over a substantial portion of the Brillouin zone^{13}).
In view of the above, it is therefore particularly noteworthy that we also measure g≃−4 for the B exciton in monolayer WS_{2}, despite the fact that its reduced mass almost certainly differs from that of the A exciton, as shown below from direct measurements of the diamagnetic shift (that is, m_{h} cannot equal m_{e} for both spinup and spindown valence bands). Note that early studies of bulk MoS_{2} (refs 47, 48, 49) also indicate that the B exciton mass significantly exceeds that of the A exciton. This suggests that contributions to the orbital moment from Berry curvature effects, expected when m_{e}≠m_{h}, may not play a significant role in determining the measured exciton magnetic moment and the valley Zeeman effect.
Nonlocal dielectric screening in monolayer TMDs
In addition to the reduced mass m_{r}, the characteristic size of the A and B excitons in monolayer TMDs is an essential parameter for determining material and optical properties. This is especially relevant because of nonlocal dielectric screening in these and other 2D materials, which fundamentally modifies the functional form of the attractive potential V(r) between electrons and holes^{23,36,37}. Rather than a conventional Coulomb potential, V(r) is believed to assume the following form:
where H_{0} and Y_{0} are the Struve function and Bessel function of the second kind, respectively, and the characteristic screening length , where χ_{2D} is the 2D polarizability of the monolayer material^{23,37}. This potential follows a 1/r Coulomblike potential for large electron–hole separations , but diverges weakly as log(r) for small separations , leading to a markedly different Rydberg series of exciton states with modified wavefunctions and binding energies that cannot be described within a hydrogenlike model^{23,24,25,27}.
Diamagnetic shifts in monolayer WS_{2}
To this end, the use of very large 65 T magnetic fields allows us to measure the small diamagnetic shifts of excitons in monolayer TMDs so that the characteristic size of their wavefunctions can be directly inferred. In general^{20,21,22}, an exciton diamagnetic shift ΔE_{dia} is expressed as
Here σ is the diamagnetic shift coefficient, m_{r} is the inplane reduced mass, r is a radial coordinate in a plane perpendicular to the applied magnetic field B (here for , r is in the monolayer plane) and is the expectation value of r^{2} over the 1s exciton wavefunction . Equation (2) applies in the ‘lowfield’ limit where the characteristic cyclotron energies ħω_{c} (and also ΔE_{dia}) are less than the exciton binding energy, which is the case for excitons in TMDs even at ±65 T. Given m_{r}, σ can then be used to determine the r.m.s. radius of the 1s exciton in the monolayer plane, r_{1}:
This definition is entirely general and independent of V(r). (Note that for a standard Coulomb potential V(r)=−e^{2}/(4πɛ_{r}ɛ_{0}r) in two dimensions, , where a_{0,2D}=2πɛ_{r}ɛ_{0}ħ^{2}/m_{r}e^{2} is the classic Bohr radius for hydrogenic 2D excitons.)
Exciton diamagnetic shifts have eluded detection in recent magnetophotoluminescence studies of monolayer MoSe_{2} and WSe_{2} (refs 12, 13, 14, 15, 16), likely due to the limited field range employed (B<10 T). Here the diamagnetic shift of the A exciton in monolayer WS_{2} can be seen in 65 T fields via the slight positive curvature of both E^{+}(B) and E^{−}(B) in Fig. 2d. To directly reveal ΔE_{dia}, Fig. 2g shows the average exciton energy, (E^{+}+E^{−})/2. Overall quadratic shifts are indeed observed, indicating diamagnetic coefficients σ_{A}=0.32±0.02 μeV T^{−2} for the A exciton and a smaller value of σ_{B}=0.11±0.02 μeV T^{−2} for the B exciton. These measurements were repeated on five different regions of the monolayer WS_{2} sample, with similar results.
Exciton radii and binding energies
Importantly, knowledge of σ constrains not only the r.m.s. exciton radius r_{1} (if the mass is known) but also the exciton binding energy if the potential V(r) is known. Theoretical estimates^{3,23,31,32} for the A exciton reduced mass in monolayer WS_{2} range from 0.15 to 0.22m_{0}, from which we can then directly calculate r_{1,A}=1.48–1.79 nm via equation (3). These values are in reasonable agreement with recent ab initio calculations of the 1s exciton wavefunction in monolayer WS_{2} (ref. 25), and further support a picture of 2D Wanniertype excitons with lateral extent larger than the monolayer thickness (0.6 nm) and spanning several inplane lattice constants.
Moreover, σ, m_{r} and r_{1} can then be used to calculate the A exciton wavefunction and its binding energy, by numerically solving the 2D Schrödinger equation for describing the relative motion of electrons and holes using the potential V(r) as defined in equation (1), and taking the screening length r_{0} as an adjustable parameter to converge on a solution for that has the correct r.m.s. radius r_{1}. For example, using m_{r,A}=0.16m_{0} for the A exciton in WS_{2}, and using the measured diamagnetic shift σ_{A}, we find that r_{1,A}=1.53 nm via equation (3). A wavefunction with this r.m.s. radius, shown explicitly in Fig. 3a, is calculated if (and only if) the screening length r_{0}=5.3 nm, and the binding energy of this state is 410 meV. For comparison, this inferred screening length is somewhat larger than expected for a suspended WS_{2} monolayer (where r_{0}=2πχ_{2D}=3.8 nm; ref. 23), but is less than the value of 7.5 nm used recently by Chernikov^{24} to fit a nonhydrogenic Rydberg series of excitons in WS_{2} from reflectivity data. Similarly, the 410 meV exciton binding energy that we estimate exceeds the value inferred by Chernikov (320 meV), but is less than the 700–830 meV binding energies extracted from twophoton excitation studies^{25,26} and reflectivity/absorption studies^{28} of monolayer WS_{2}. We emphasize, however, that the exciton wavefunctions and binding energies that we calculate necessarily depend on the reduced mass m_{r} and the exact form of the potential V(r), which is sensitive to the details of the dielectric environment and choice of substrate material^{33,50}.
More generally, Fig. 3b shows a colourcoded surface plot of the exciton binding energy, calculated within the framework of the nonlocal dielectric screening potential V(r) defined in equation (1), over a range of reduced masses m_{r} and effective dielectric screening lengths r_{0}. At each point, the 1s exciton wavefunction , its binding energy, and its r.m.s. radius r_{1} were calculated, from which we computed the expected diamagnetic shift coefficient . Importantly, the solid lines on the plot indicate the contours of constant diamagnetic shift that correspond to our experimentally measured values σ_{A} and σ_{B}. At intervals along these contours, r_{1} is indicated. From this plot, it can be immediately seen that over the range of theoretically calculated masses (m_{r,A}=0.15–0.22m_{0}), excitons having the appropriate size to give the measured diamagnetic shift σ_{A} (that is, those lying along the σ_{A} contour) have binding energies in the range of 480–260 meV. Within this model, excitons with even larger binding energies (but still constrained to exhibit the correct diamagnetic shift) are anticipated if the reduced mass m_{r} is lighter and the effective screening length r_{0} is smaller.
In addition, Fig. 3b also allows us to estimate the mass, binding energy and spatial extent of the B exciton in monolayer WS_{2}, for which a smaller diamagnetic shift of σ_{B}=0.11 μeV T^{−2} was measured. Assuming that the local dielectric environment is similar for A and B excitons (that is, r_{0} is unchanged), then parameters for the B exciton lie at a point on the σ_{B} contour that is directly to the right of those on the σ_{A} contour. Thus, if m_{r,A}=0.16m_{0} and r_{1,A}=1.53 nm as discussed above, then the B exciton reduced mass is m_{r,B}=0.27m_{0}, its r.m.s. radius is r_{1,B}=1.16 nm and its binding energy is 470 meV. These values are qualitatively consistent with trends identified in early optical studies of bulk MoS_{2} crystals^{48,49}, in which B exciton masses and binding energies were found to exceed those of A excitons. These results highlight a further interesting consequence of the potential V(r), which is that exciton binding energies scale only weakly and nonlinearly with m_{r}, in contrast to the case for hydrogenic potentials.
Zeeman splitting and diamagnetic shifts in bulk WS_{2}
For direct comparison with monolayer WS_{2}, circularly polarized magnetoreflectance measurements were also performed on the exfoliated surface of a bulk WS_{2} crystal (grown by chemical vapour transport at the Tennessee Crystal Center). Figure 4a shows the wellknown^{51} A exciton resonance in bulk WS_{2}, which arises from the lowestenergy direct optical transition that is located at the K points of the Brillouin zone^{52} (this transition, with only slight modification in energy, eventually becomes the lowest overall transition when WS_{2} is thinned to a single monolayer and becomes a directgap semiconductor^{1,2}). At low temperatures and in ±60 T magnetic fields, the Zeeman splitting of the bulk A exciton is readily resolved (Fig. 4a–c) and is found to increase linearly with field at a rate of −193 μeV T^{−1} (g=−3.33). This value is in very close agreement with early magnetic circular dichroism measurements of gfactors in bulk WS_{2} (ref. 53), wherein it was suggested that deviations from g=−4 arise from the crystalfield mixing of ptype chalcogen atomic orbitals into the predominantly dtype character of the conduction and valence bands. Within this context, the value of g≃−4 that we measured in monolayer WS_{2} (Fig. 2f) may suggest that such mixing effects, if present, may be suppressed in atomically thin WS_{2}.
In addition, Fig. 4d shows that the measured diamagnetic shift of the A exciton in bulk WS_{2} is 0.64 μeV T^{−2}, which is twice as large as in monolayer WS_{2}. Assuming an inplane reduced mass of m_{r}=0.21m_{0} in bulk WS_{2} (ref. 51), we calculate via equation (3) an inplane r.m.s. radius of r_{1}=2.48 nm for the bulk A exciton, which is substantially larger than that inferred for monolayer WS_{2}. This large r.m.s. radius indicates an effective dielectric screening constant ɛ_{r}≃7.0, in agreement with early work^{51}, and from which the A exciton binding energy in bulk WS_{2} can be estimated via the standard hydrogenic formulation, × 13.6 eV=58 meV. This value agrees extremely well with early work on bulk WS_{2} (ref. 51), and is close to that found in other bulk TMDs^{54}. Therefore, we find that the binding energy of the A exciton in WS_{2} increases by approximately a factor of 7 on reducing the dimensionality of the host crystal from threedimensional to 2D. Note, however, that these estimates depend on the assumed value of the reduced mass m_{r}, which has not yet been measured independently by, for example, cyclotron resonance studies.
Valley Zeeman effect in monolayer MoS_{2}
To complete this study of the monolayer transitionmetal disulphides, we also performed highfield magnetoreflectance studies on largearea samples of monolayer MoS_{2} (Fig. 5). The A and B exciton linewidths are broader and the optical reflection contrast is lower than for monolayer WS_{2} (Fig. 5a). Nonetheless, a clear valley Zeeman splitting of both excitons is observed (Fig. 5b,c). The energies of the fieldsplit exciton peaks are shown in Fig. 5d,e for the A and B excitons, respectively. Although the reduced signals and broader features lead to increased scatter in the fitted data, Fig. 5f shows that the measured valley splitting of the A and B excitons in MoS_{2} increases approximately linearly with field at rates of −233±10 and −270±10 μeV T^{−1}, corresponding to g≃−4.0±0.2 and −4.65±0.17, respectively. For the A exciton, this value is very close to those inferred from lowfield magnetophotoluminescence studies^{12,15,16} of their diselenide counterpart, monolayer MoSe_{2} (Table 1). As discussed above for the case of monolayer WS_{2}, a gfactor of −4 for the A exciton agrees surprisingly well with expectations from a simple twoband tightbinding picture, and suggests that the valley Zeeman effect in MoS_{2}, much like MoSe_{2}, is largely uninfluenced by contributions from the valley orbital (Berry curvature) magnetic moment. We note, however, that the measured valley gfactor is somewhat larger for the B exciton in monolayer MoS_{2}. Unfortunately, the reduced signal levels from these monolayer MoS_{2} samples led to correspondingly increased scatter in the fitted exciton energies, precluding an accurate determination of exciton diamagnetic shifts in monolayer MoS_{2} (Fig. 5g).
Discussion
In summary, we have presented a comprehensive study of valley Zeeman effect and diamagnetic shifts of excitons in the archetypal monolayer transitionmetal disulphides WS_{2} and MoS_{2}. Valley gfactors of the A excitons are approximately −4, which are similar to those obtained from transitionmetal diselenides. Unexpectedly, the heavier B exciton in monolayer WS_{2} also exhibits g≃−4, suggesting that the valley Zeeman effect is largely unaffected by the exciton reduced mass. The very large magnetic fields used in these studies also allowed initial measurements of the exciton diamagnetic shifts in a monolayer TMD—specifically, WS_{2}—from which r.m.s. exciton radii were directly computed (r_{1}=1.53 and 1.16 nm for the A and B excitons, respectively). Within a picture of nonlocal dielectric screening in these 2D semiconductors, these measurements of diamagnetic shifts allowed us to constrain estimates of the exciton binding energies, which we calculate (using a reduced A exciton mass of 0.16m_{0}) to be 410 and 470 meV for the A and B excitons, respectively, in monolayer WS_{2}. These studies highlight the utility of very large magnetic fields for characterizing new 2D material systems.
Additional information
How to cite this article: Stier, A. V. et al. Exciton diamagnetic shifts and valley Zeeman effects in monolayer WS_{2} and MoS_{2} to 65 Tesla. Nat. Commun. 7:10643 doi: 10.1038/ncomms10643 (2016).
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Acknowledgements
We thank K. Velizhanin and P. Hawrylak for helpful discussions. These optical studies were performed at the National High Magnetic Field Laboratory, which is supported by NSF DMR1157490 and the State of Florida. Work at NRL was supported by core programs and the NRL Nanoscience Institute, and by AFOSR under contract number AOARD 14IOA018134141. J.K. was supported by the Air Force Office of Scientific Research under Award Number FA95501410268.
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S.A.C. and J.K. conceived and directed the experiments; K.M.M. and B.T.J. synthesized and characterized the monolayer samples; A.V.S. built and performed the highfield optical experiments; A.V.S. and S.A.C. analysed the data and wrote the paper in close consultation with all authors.
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Stier, A., McCreary, K., Jonker, B. et al. Exciton diamagnetic shifts and valley Zeeman effects in monolayer WS_{2} and MoS_{2} to 65 Tesla. Nat Commun 7, 10643 (2016). https://doi.org/10.1038/ncomms10643
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