Skip to main content

Main menu

  • Home
  • Articles
    • Current
    • Special Feature Articles - Most Recent
    • Special Features
    • Colloquia
    • Collected Articles
    • PNAS Classics
    • List of Issues
  • Front Matter
    • Front Matter Portal
    • Journal Club
  • News
    • For the Press
    • This Week In PNAS
    • PNAS in the News
  • Podcasts
  • Authors
    • Information for Authors
    • Editorial and Journal Policies
    • Submission Procedures
    • Fees and Licenses
  • Submit
  • Submit
  • About
    • Editorial Board
    • PNAS Staff
    • FAQ
    • Accessibility Statement
    • Rights and Permissions
    • Site Map
  • Contact
  • Journal Club
  • Subscribe
    • Subscription Rates
    • Subscriptions FAQ
    • Open Access
    • Recommend PNAS to Your Librarian

User menu

  • Log in
  • My Cart

Search

  • Advanced search
Home
Home
  • Log in
  • My Cart

Advanced Search

  • Home
  • Articles
    • Current
    • Special Feature Articles - Most Recent
    • Special Features
    • Colloquia
    • Collected Articles
    • PNAS Classics
    • List of Issues
  • Front Matter
    • Front Matter Portal
    • Journal Club
  • News
    • For the Press
    • This Week In PNAS
    • PNAS in the News
  • Podcasts
  • Authors
    • Information for Authors
    • Editorial and Journal Policies
    • Submission Procedures
    • Fees and Licenses
  • Submit
Research Article

Emergence of electron coherence and two-color all-optical switching in MoS2 based on spatial self-phase modulation

Yanling Wu, Qiong Wu, Fei Sun, Cai Cheng, Sheng Meng, and Jimin Zhao
  1. Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

See allHide authors and affiliations

PNAS September 22, 2015 112 (38) 11800-11805; first published September 8, 2015; https://doi.org/10.1073/pnas.1504920112
Yanling Wu
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Qiong Wu
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Fei Sun
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Cai Cheng
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Sheng Meng
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
  • For correspondence: jmzhao@iphy.ac.cn smeng@iphy.ac.cn
Jimin Zhao
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
  • For correspondence: jmzhao@iphy.ac.cn smeng@iphy.ac.cn
  1. Edited by Roberto Merlin, University of Michigan, Ann Arbor, Michigan, and accepted by the Editorial Board July 25, 2015 (received for review March 11, 2015)

  • Article
  • Figures & SI
  • Info & Metrics
  • PDF
Loading

Significance

Generating electron coherence is nontrivial in that most sophisticated electronic experimental methods are noncoherent or cannot be used to induce and detect collective states. By using coherent spatial self-phase modulation (SSPM) (a nonlinear optical property) we observed the emergence of electron coherence in a gapped quantum material, MoS2. By observing gap-dependent SSPM we discovered that it is a ubiquitous property of two-dimensional layered quantum materials. Furthermore, we demonstrate that this ac electron coherence can be harnessed to realize two-color all-optical switching with superb performance.

Abstract

Generating electron coherence in quantum materials is essential in optimal control of many-body interactions and correlations. In a multidomain system this signifies nonlocal coherence and emergence of collective phenomena, particularly in layered 2D quantum materials possessing novel electronic structures and high carrier mobilities. Here we report nonlocal ac electron coherence induced in dispersed MoS2 flake domains, using coherent spatial self-phase modulation (SSPM). The gap-dependent nonlinear dielectric susceptibility χ(3) measured is surprisingly large, where direct interband transition and two-photon SSPM are responsible for excitations above and below the bandgap, respectively. A wind-chime model is proposed to account for the emergence of the ac electron coherence. Furthermore, all-optical switching is achieved based on SSPM, especially with two-color intraband coherence, demonstrating that electron coherence generation is a ubiquitous property of layered quantum materials.

  • electron coherence
  • transition metal dichalcogenide
  • self-phase modulation
  • optical switching
  • emergent phenomena

Recently 2D layered quantum materials have attracted tremendous interest since the discovery of graphene a decade ago (1). Various layered materials, ranging from boron nitride sheets to transition metal dichalcogenides and from topological insulators to high-temperature superconductors, have been intensively investigated (2⇓⇓⇓⇓⇓⇓⇓⇓–11). Strict 2D atomic crystals can now be produced at a macroscopic scale, using a variety of methods (11, 12). Among them molybdenum disulfide (MoS2) and related layered quantum materials are particularly interesting due to their novel optical properties and potential valleytronics applications (4, 5, 8, 9) at a thickness of monolayer and few layers (2, 10, 13). Layered materials share common physical properties rooted in their ubiquitous 2D quantum nature, for which achieving pure coherence among electrons (lattices) is of particular interest (14⇓⇓⇓⇓–19). The presence of multiple domains is ubiquitous in many known 2D quantum materials, ranging from stripe-order cuprate superconductors to polycrystalline strongly correlated systems. For example, phase locking between different layers of stripe orders is crucial for enhancing the superconducting phase in layered high-temperature superconductors (20, 21).

In this work we demonstrate unambiguously that nonlocal and intraband ac electron coherence, of which the electronic wave function oscillates at an optical frequency of 1014 Hz, can be generated in separate MoS2 flakes, using spatial self-phase modulation (SSPM). The SSPM is a coherent third-order nonlinear optical process systematically investigated decades ago (22), where the nonlinear optical susceptibility χ(3) is uniquely determined by the laser-intensity-dependent refractive index n = n0 + n2I, where n0 and n2 are linear and nonlinear refractive indexes, respectively. If this effect is strong enough in a material, the phenomenon of self-focusing can be directly observed. The SSPM is also frequently referred to as the optical Kerr effect or the ac Kerr effect (note the difference from the regular Kerr effect). It is parallel to the other third-order nonlinear optical processes, such as third harmonic generation (THG) and four-wave mixing (FWM).

Our investigations show that gap-dependent SSPM is a general method for inducing electron coherence in 2D layered materials. Monolayer and few-layer MoS2 (and other similar layered materials) have finite bandgaps, which lift the obstacle to versatile electronic and optical applications (2, 23, 24). Because SSPM using below-gap photons has not been reported, MoS2 provides an ideal test platform for observing SSPM in finite-bandgap materials. A model has been developed to account for the coherence emergence process and theoretical calculations have been carried out to unravel the underlying mechanism. Moreover, we demonstrate all-optical switching based on SSPM, which employs intraband electron coherence. This optical switch has multiple advantages, including weak-control-strong performance, cascade-possible, and high-contrast two-color switching.

Results and Discussion

Fig. 1 displays the experimental setup and typical SSPM patterns obtained from the flake domains of MoS2 in a suspension (Methods). It can be seen from Fig. 1B that the sample is mainly composed of MoS2 flakes with a thickness of 30 ∼ 55 layers. In Fig. 1 D–F we show the SSPM patterns obtained using 400-nm ultrafast laser pulses, 532-nm continuous wave (cw) laser beams, and 800-nm ultrafast laser pulses, respectively. The SSPM is a nonlinear optical process in that the phase (rather than the intensity) of the output beam depends on the input intensity (Methods).

Fig. 1.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. 1.

(A) Schematic lattice structure of MoS2. (B) Scanning electron microscopy image of the MoS2 flakes. (C) Schematic experimental setup. (D–F) The SSPM patterns generated using 400-nm, 532-nm, and 800-nm laser beams, respectively.

To unambiguously verify that we have observed SSPM, we performed intensity-dependence measurements. Fig. 2 shows that the number of rings N increases nearly linearly with the laser intensity, until the saturation is reached. By this linear dependence the optical nonlinear coefficient n2, hence χ(3), can be obtained directly (Fig. 2 and Methods). Because nearly all of the light is diffracted off the incidence direction and the Rayleigh scattering is low, ideally it is a purely coherent third-order nonlinear optical process. This pure coherence must originate from the coherence within the sample—the electron coherence, where the electronic wave function preserves a definite phase (see discussion below). In the case that photoexcited carriers become out of phase due to collisions, impurity scatterings, and boundary reflections, they reassume their coherence transiently (at a timescale of femtoseconds) under the external light pulse.

Fig. 2.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. 2.

Intensity dependence of ring numbers and bandgap dependence of χ(3). The slopes directly illustrate the nonlinear coefficients χ(3). The solid lines are guides to the eye. (Inset) The thresholds to observe the SSPM patterns at different photon energies.

Next we consider nonlocal electron coherence. In our sample, each flake is a domain. Charge carriers in any form (including photocarriers, excitons, free electron–hole pairs, and intrinsic electrons and holes) in different and the same domains are initially completely out of phase. Besides, each domain has an arbitrary orientation. Upon light interaction, photoinduced quasiparticles interact with the electric field (Fig. 3A) and assume its local phases.

Fig. 3.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. 3.

Wind-chime model and the emergence of electron coherence. (A, Left) Collective response of the MoS2 flakes to the laser beam electric field, being aligned to form a “wind chime.” (A, Right) SSPM pattern formed. (B and C) Snapshots of the pattern formation at 473-nm and 532-nm laser beam excitations, respectively. The whole process takes ∼0.20 s, which is consistent with the wind-chime model prediction (main text).

We propose a wind-chime model to describe the emergence of electron coherence from the nonlocal domains driven by SSPM (Fig. 3A). Initially there exists an arbitrary orientation angle θ between a MoS2 flake (thus the electrical polarization it contains) and the electric field. Due to energy relaxation, the electric field reorients the flakes so that each domain contains an axis parallel to the polarization of the external field, as if each domain is “hung” by a vertical “thread.” This image mimics that of a wind chime (Fig. 3A). Once the wind chime is formed, the electron coherence is completely set up within and among each of the different domains. This scenario depicts the emergence of nonlocal electron coherence as a complex collective behavior.

We first show that the model is reasonable. Assuming a circular disk shape for the MoS2 flake domains, we derived (SI Text, section S1) that the rotation torque generated by the laser pulse M=∫|P×E|dV has a magnitude ofM=14sin 2θ(εr−1)εrε0E02πR2he−(2(ct−z)2/c2τ2),[1]where R is the disk radius, h is the disk thickness, and τ is the pulse width. Taking the Newton fluid with constant viscosity coefficient η and linear velocity v at the interface, the rotation torque V due to the viscous force isV=η∫0πdvdr(R⁡sin⁡φ)(ξ⋅2πR⁡sin⁡φ)(Rdφ)=πηΩξR3,[2]where Ω is the rotation velocity and ξ is the portion of the fluid globe that is rotating together with the disk. Hence the rotation angle accumulated due to each single light pulse is (SI Text, section S1)δθ=∫0TΩdt≈∫0∞Ω0e−(πηξR3/(JMoS2+JSolution))tdt=(εr−1)εr2TIh4ηξRcsin 2θ,[3]where T is the repetition period. The time needed for the pattern formation is (SI Text, section S1)T=εrπηξRc1.72(εr−1)Ih=0.3 s,[4]where we have used the values εr = 3.33 for MoS2, η = 3.2 × 10−4 Pa∙s for acetone, R ∼ 1 μm, h ∼ 10 nm, I = 250 W/cm2, and ξ ∼ 0.03 in our experiment. As a result, T does not depend on T. Thus, for both pulsed and cw excitations the results are identical, which has been verified in our experiment. To compare with the experiment, we recorded the formation processes of the ring patterns (see snapshots in Fig. 3 B and C). The ring number and diameter both increase monotonously with time until the maximum is reached after 0.20 s. This compares well with the predicted value by the wind-chime model. Thus, the observed time for the ring formation is a strong experimental support to the wind-chime model.

We then discuss the emergence of the ac electron coherence. Writing the electronic wave function at rA as ψA(rA)=ρA(rA)eiϕ(rA), with ψA∗(rA)⋅ψA(rA)=ρA(rA) being the local electron density, the phase ϕ(rA)=k⋅rA−ω t+ϕ0(rA) of the electronic wave function is completely determined by the external light field (Fig. 3A). The wave function oscillates at a frequency of ∼1014 Hz (i.e., optical frequency). Classically, this is a forced oscillation, where electrons have to assume the enforced local phase by the external field, even surviving scatterings and collisions. Here ϕ​0(rA) represents the phase lag in response to the external field. It is not the initial random phase, which is wiped off by dissipation. The same applies to the electronic wave functions at rB in a different (or the same) flake. Because the two wave functions are both coherent with the light wave, they are coherent to each other too. This is a dynamic or ac electron coherence, which is different from the commonly known steady-state or dc electron coherence in transport measurements. The electrons do not move far away from their equilibrium positions. After the wind-chime formation the electron coherence becomes stronger because more electrons get correlated and fewer collisions get involved, owing to the aligned geometry. At the far field, the intensity is I(rC)=E∗(rC)⋅E(rC), where E(rC) is the electric field of the light wave at rC. It becomes E(rC)=ς(χ(3))Ψ(rC), with Ψ(rC)=ei(k⋅rC−ωt+ϕ 0)[ρA(rA)e−i k⊥⋅rA+ρB(rB)e−i k⊥⋅rB] being the optical phase except for a normalization factor. Here k⊥=dΔϕ/dr is the wavevector generated due to the nonlinear response term of n2 (Methods). The exact coefficient ς(χ(3)) is determined by the optical nonlinearity χ(3), which is comprehensively determined by the carrier properties in the material.

We found that SSPM can be used to measure easily the χonelayer(3) value for many of the layered quantum materials (we have also observed SSPM in dispersion of MoSe2 flakes and obtained the χ(3) value). In other coherent methods (e.g., THG or FWM), the signal is much weaker than the input beam (25). Instead, in SSPM the diffracted beam is strong, owing to the intrinsic electron coherence. Significantly, SSPM can be used to measure χ(3) from near IR to UV, compared with THG working only for the region where the 3 ω signal can be easily detected. This is essential for materials with a relatively large gap. In this sense we have found a powerful and ubiquitous (in some cases unique) way to measure χ(3) for many of the 2D gapped quantum materials.

To quantitatively investigate the gap-dependent property, we obtained χ(3) values for excitations at multiple wavelengths (Fig. 4A). For example χonelayer(3) at 532 nm is 1.6 × 10−9 electrostatic units (e.s.u.) (i.e., 2.23 × 10−17 m2/V2; Methods), which is orders of magnitude larger than that of conventional semiconductors, owing to the high carrier density and mobility in 2D layered materials. In parallel we calculated the absorption as a function of wavelength for few-layer MoS2, where the dipole transition is proportional to the occupation probabilities of valence and conduction bands. As shown in Fig. 4A, the experimental results and theoretical calculations compare well. A sharp change in χ(3) and threshold values can be seen at 1.72 eV (722 nm), which is exactly the gap value of MoS2. This gap-dependent property is a manifestation of the electronic band structure of MoS2. Thus, SSPM can be used to investigate band structures of layered quantum materials and can be conveniently compared with other experimental methods such as absorption and photoluminescence. The SSPM method applies especially to those materials where large single crystals are not available and absorption is hard to measure.

Fig. 4.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. 4.

Bandgap-dependent χ(3) and Ith and the SSPM mechanisms. (A) Bandgap dependence of χ(3) (solid spheres) and Ith (open circles) for ultrafast (blue) and cw (red) excitations, respectively. The calculated photoabsorption spectrum of MoS2 is also shown (black and red curves) for comparison. The below-gap (purple) and above-gap (orange) regions are marked. (B) Scenarios of the SSPM formation mechanisms: Ⓐ Single-photon absorption-induced SSPM; Ⓑ two-photon absorption-induced SSPM; Ⓒ indirect-bandgap phonon-assisted SSPM. The bandgap structure is adopted from ref. 2. (C) Two-photon absorption leads to SSPM of the fundamental beam.

In Fig. 4A the experimental results compare well with the calculated absorption along x and y, but not that along the z direction (SI Text, section S2). This is strong evidence that, in our experiment, light propagation is perpendicular to the x or y but not the z axis, which confirms the validity of the wind-chime model. Note that this is an even stronger restriction than the wind-chime model. Considering the plasmonic interaction due to free carriers (26), it is indeed possible that the collective behavior leads to a “frozen” wind chime, where all domains become parallel to each other and perpendicular to the direction of laser propagation. The gap-dependent nature displayed in Fig. 4 shows again that SSPM is mainly due to the various charge carriers.

The below-gap SSPM has never been reported before. Three mechanisms might be responsible for it (Fig. 4 B and C): Ⓐ single-photon SSPM with Fermi distribution smearing, Ⓑ two-photon SSPM, and Ⓒ phonon-assisted indirect-gap SSPM. For Ⓐ, it can be seen that Fermi smearing relevant for significant photoabsorption occurs only at zones near the H and K points in momentum space. For Ⓑ, the carriers generated through two-photon absorption build up an electrical polarization, which diffracts not only those photons above the gap, but also those below the gap (Fig. 4B). As such, even though it is a two-photon nonlinear process, one expects a linear power dependence relation because the change in refractive index is proportional to the light intensity I. Heuristically, the excited carriers diffract both excitation photons (i.e., two output photons). In comparison there is only one output photon in second harmonic generation and two-photon photoluminescence. Two-photon SSPM is the most likely mechanism responsible for below-gap SSPM. For Ⓒ, the indirect gap photon excitation requires assistant phonon injection (Fig. 4C), which usually has a very small probability. In-gap defect states are relatively sparse (resonant in-gap states are more sparse) and localized (whereby charge carriers cannot move freely and scattering occurs frequently, which ruins the acquired uniform phase), so they are unlikely a major reason for SSPM.

We further demonstrate that, in forming SSPM, free carriers generated by photons at one wavelength can diffract photons at another wavelength, as if they were generated by the latter photons. The experimental setup is illustrated in Fig. 5C, which demonstrates a two-color (473 nm and 532 nm) SSPM. When we fix the power of one beam (even below threshold) and increase that of the other, the ring number and diameter for each beam both increase simultaneously. The brightness decreases, with the total intensity remaining unchanged, and energy conservation is preserved (Fig. 5D). In Fig. 5B the ring number of the 473-nm pattern exhibits linear dependence on the sum of the two beam powers, although the intensity of the 473-nm beam is fixed. The same is true for the 532-nm ring (SI Text, section S3). Once the coherence is correlated, each beam experiences the SSPM—as if the free carriers are created by the beam itself. Fig. 5A is an overall illustration of this property. This indistinguishability helps explain why in two-photon SSPM one can still have linear dependence and still obtain a relatively large χ(3).

Fig. 5.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. 5.

All-optical switching based on SSPM. (A and B) Dependence of the 473-nm beam ring number on the sum intensity. (C) Schematic of the switch. (D) Generating the 473-nm pattern by increasing the 532-nm beam intensity. Both ring numbers (and diameters) increase simultaneously, with fixed 473-nm intensity below threshold.

Significantly, our experiment is to our knowledge the first demonstration of all-optical switching based on SSPM. Because the switching beam is controlling the phase of the other one, a small change in intensity will result in the shift of the strong signal pattern in real space, where a detector is placed. The detector can be made to detect the whole rings. Thus, a weaker beam can switch on and off a stronger beam (27, 28). Quantitatively taking the intensity of the signal beam (strong beam) at 473 nm to be Istrong = 120 W/cm2, the intensity of the control beam (weak beam) at 532 nm needs only to be Iweak = 2.0 W/cm2 to change the phase of the strong beam by π (i.e., ΔN = 0.5, Fig. 2). Hence to make an all-optical switch with full contrast ratio, Iweak:Istrong = 1:60. This is characteristic of weak-control–strong all-optical switching, which is apparently cascade possible (27, 28).

It is interesting that in the experiment intraband electron coherence has been correlated between photocarriers of 473-nm and 532-nm excitations. We note that similar coherence is also induced among the holes, the excitons, and the free electron–hole pairs. The two-color switching can further enable high-contrast-ratio operations, because in two-color operation optical filters can be used to block the signal beam. Two-color switching also makes it possible for multichannel mixed switching and thus enables very broad-band optoelectronic devices. We leave additional details of the SSPM-based all-optical switching to future investigations. In summary, our experiment demonstrates an all-optical switching based on SSPM, with the advantages including a weak-control–strong, cascade-possible, high-contrast-ratio, room-temperature device; broadband functioning; condensed state; and particularly broad-band-accessible quantum materials.

SI Text

S1) Derivation of the Wind-Chime Model

Each pulse generates an angular momentum on each MoS2 flake that makes them rotate unless they develop an axis parallel to the laser beam polarization. This is also associated with rotation of a small portion of the liquid nearby the flake surface. Then, the viscous force at the flake edge reduces the angular velocity exponentially until it becomes static. When the next pulse arrives, the same process repeats, where the integrated result is the macroscopic angular motion of the flake (i.e., the formation of the wind chime).

Taking E(z,t)=E0e−((ct−z)2/c2τ2)sin(kz−ω0t), where τ is pulse width, we have I=εE2¯c=(c/2T)ε0E02τπ/2, where T is the period of pulse repetition and we have assumed 2π/ω0 << τ. The electrical polarization in the flake domain is P = (εr − 1)ε0E cos θ. In molybdenum disulfide the electrical field is E/εr. Thus, assuming a circular disk shape for the flake domains, the rotation torque generated by the laser pulse M=∫|P×E| dV has a magnitude ofM=14sin 2θ(εr−1)εrε0E02πR2he−(2(ct−z)2/c2τ2),[S1]where R is the disk radius and h is the disk thickness. Thus, when a pulse traverses the disk, the initial angular velocity acquired isΩ0=∫MdtJMoS2+JSolution=14sin 2θ(εr−1)εrε0E02πR2h(π/2)τJMoS2+JSolution.[S2]Considering Newton’s fluid with constant viscosity coefficient η and linear velocity v at the interface between the rotating and nonrotating parts, the rotation torque V due to the viscous force isV=η∫0πdvdr(R⁡sin⁡φ)(ξ⋅2πR⁡sin⁡φ)(Rdφ)=πηΩξR3,[S3]where Ω is the rotation velocity and ξ is the portion of the globe that is rotating together with the disk. Here ξ assumes a value between 0 and 1 and we estimate it to be closer to 0, because for each pulse the rotation has to be slow and not much fluid will sense the rotation before it reaches the static state. Hence in the relaxation process induced by the viscous force we obtain Ω=Ω0e−(πηξR3/(JMoS2+JSolution))t. With Eq. S2, the rotation angle accumulated due to each single light pulse is thusδθ=∫0TΩdt≈∫0∞Ω0e−(πηξR3/(JMoS2+JSolution))tdt=(εr−1)εr2TIh4ηξRcsin 2θ.[S4]The time for the pattern formation is∑i=1Nδθi=(εr−1)εr2TIh4ηξRc⋅N⋅sin2θi¯,[S5]where N is the number of laser pulses. Assume that the pattern formation time is approximately that needed for a flake at 3π/8 to rotate to π/8. Thus, we have sin 2θi¯ ∼ 0.86 and ∑i=1NΔθi=π/4. Therefore, the time needed for the pattern formation isT=N⋅T=εrπηξRc1.72(εr−1)Ih=0.3 s,[S6]where we have used the values εr = 3.33 for MoS2, η = 3.2 × 10−4 Pa⋅s for acetone, R ∼ 1 μm, h ∼ 10 nm, I = 250 W/cm2, and ξ ∼ 0.03 in our experiment. As a result, T does not depend on the laser repetition rate. In the extreme condition when the rate is approaching infinity, the above derivation approximately accounts for the case of a continuous wave laser beam excitation. So we have actually considered situations of both pulsed and cw laser beam excitations and the results are identical. This has been verified in our experiment. Note that apparently T is inversely proportional to the laser intensity. However, the value of ξ is somehow positively correlated to the intensity at a certain range of I. So the actual time T is not tightly dependent on the laser intensity either, which has also been confirmed in our experiment.

S2) Comparison of χ(3) with Calculated Absorption Along z

From Fig. S1 it can be seen that our experimental results do not compare well with the theoretical absorption along z. At below 2 eV, the measured values are a few times higher than that of the theoretical calculation.

Fig. S1.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. S1.

Comparison of the gap-dependent χ(3) with the calculated absorption. The blue and red spheres represent the pulsed and cw laser excitation results.

S3) Ring Number of the 532-nm Pattern as a Function of the Total Laser Intensity

In the main text we discussed the ring number of the 473-nm pattern. Here we show the other aspect of the same data (Fig. 5A) by illustrating the ring number of the 532-nm pattern (Fig. S2).

Fig. S2.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. S2.

Two-color all-optical switching based on SSPM. (A and B) Ring number of the 532-nm pattern, which is proportional to the sum intensity of the 532-nm and 473-nm laser beams. (A) Three-dimensional plotting of the results in B. (B) Data of multiple 473-nm laser fluences. The wide diagonal red line is a guide to the eye.

Conclusions

In conclusion, nonlocal ac electron coherence among different flake domains of MoS2 has been optically correlated based on SSPM. A wind-chime model is proposed to account for this emergent collective behavior, which is verified by experimental observation of the pattern formation time. Significantly, gap-dependent SSPM has been observed, where the χ(3) value and the threshold Ith both show abrupt changes across the MoS2 bandgap. Mechanisms including two-photon SSPM have been discussed. Furthermore, we have experimentally demonstrated all-optical switching based on SSPM, which employs the intraband electron coherence. Our investigation signifies a ubiquitous property of 2D layered quantum materials and extends their potentials in optoelectronic applications by realizing all-optical switching.

Methods

Ultrafast and cw Experiment of SSPM in MoS2.

Multiple ultrafast and cw laser beams were used to investigate the SSPM from suspended MoS2 flakes. In the ultrafast experiment, 800-nm light pulses of 200-fs pulse width and 80-MHz repetition rate were focused onto the sample. The solvent is acetone and the container is a 1-cm-thick cuvette. The distance between the focusing lens (f = 200 mm) and the center of the cuvette is kept fixed at 150 mm. The beam spot size after focusing is 0.2 mm for the 1/e2 intensity radius. In the cw experiment, single transverse-mode 532-nm and 473-nm laser beams were used instead, having a similar beam spot size on the sample. The relation between the fringe thickness and the flake concentration is investigated and discussed in SI Text, section S4. Micrometer-sized powder of MoS2 crystals (LSKYD, www.kydmaterials.com/en/index.php) was added into the solvent and then ultrasonicated for 30 min before the measurement. The optimum concentration is 0.14 g/L (i.e., 8.5 × 10−4 mol/L). From our Raman and absorption characterization there are very few monolayer MoS2 flakes in our sample (SI Text, section S5). From Fig. 1B, it can be seen that our sample is mainly composed of MoS2 flakes with a thickness of 30 ∼ 55 layers. Being different from the linear optical phenomena of Newton’s rings, the SSPM pattern has a finite number of fringes and the thickest fringe has the largest diameter. The critical proof of observing a SSPM pattern is that the number of rings, N, increases linearly with the input laser intensity.

Fundamentals of SSPM and the Calculation of the χonelayer(3) in MoS2.

The nonlinear optical response SSPM is characterized by n = n0 + n2I, where n0 and n2 are the linear and nonlinear refractive indexes, respectively. Assuming a Gaussian beam, the optical phase accumulated after propagation length l is Δψ(r)=(2πn0/λ)∫−l/2l/2n2I(r,z)dz, where z is the propagation direction of the incident laser beam. Thus, we have Δψ(0) − Δψ(∞) = (2πn0/λ)n2·2I·l, with I(0, z) = 2I. The number of rings N is proportional to the phase change in the output beam (Δψ(0) − Δψ(∞) = 2πN), leading to n2 = (λ/2n0l)(N/I), where n0 = 1.36 and N is proportional to I for a nearly constant n2. By the slope in Fig. 2 we can straightforwardly obtain n2 from the experimental data. With n2=(12π2/(n02c))103χ(3) (in the SI unit) we estimate that χ(3) = 1.44 × 10−4 e.s.u. for the 532-nm cw laser beam excitation. The number of effective single layers, M, that the laser beam passes through is estimated to be about 300, using the concentration of the sample (SI Text, section S6). As Itotal ∼ M2Ionelayer, we estimate that χ(3) = M2χonelayer(3). The estimated χonelayer(3) for 532 nm is 1.6 × 10−9 e.s.u. (i.e., 2.23 × 10−17 m2/V2).

Conveyance of the ac Electron Coherence to the Optical Coherence.

The diffraction of the beam off the z axis is due to a finite perpendicular wavevector, which is, for a Gaussian beam Δψ(r)=Δψ(0)exp(−2r2/a2), explicitly written as k⊥=dΔψ(r)/dr=(−16πn0n2Il/a2λ)r⋅exp(−2r2/a2). After the wind chime is formed, the optical coherence forming the fringes is conveyed from the electron coherence:Ψ(rC)=[ρA(rA)eiϕ(rA)ei[kz⋅(rC−rA)]ei [∇r⊥(Δϕ)|rA]r⊥^⋅(rC−rA)+ρB(rB)eiϕ(rB)ei[kz⋅(rC−rB)]ei [∇r⊥(Δϕ)|rB]r⊥^⋅(rC−rB)+...]=ei(kz⋅rC−ωt)[ρA(rA)eiϕ0(rA)ei [∇r⊥(Δϕ)|rA]r⊥^⋅(rC−rA)+ρB(rB)eiϕ0(rB)ei [∇r⊥(Δϕ)|rB]r⊥^⋅(rC−rB)+...]=ei(k⋅rC−ωt+ϕ0)[ρA(rA)e−ik⊥⋅rA+ρB(rB)e−i k⊥⋅rB+...].

Here we have taken that ϕ0 is identical for rA and rB, and only identical k┴ can produce interference rings.

S4) Relation Between the Fringe Thickness and the Flake Concentration

As shown in Fig. S3A (Upper panel is adopted from ref. 22 and Lower panel is adopted from ref. 14), the fringes are interfering results of the output beams having an identical wavevector. The width of the ring peak is determined by, for a given angle change in the wavevector, how much phase change is occurring (k⊥(r)=dΔψ(r)/dr). The larger χ(3) is (or equivalently n2), the more phase change, and hence the narrower the fringe. It is worth noting that both the bright and dark fringes change simultaneously. If more MoS2 flakes are under illumination, the width becomes narrower due to enhanced χ(3)=χtotal(3).

Fig. S3.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. S3.

Sample density dependence of the fringe thickness. (A) Illustration of the parallel wavevector and fringe formation. Note the difference in the fringe thickness. (B) The fringe thickness, depending on the solution density. B, Right corresponds to a sample with higher MoS2 density. The white lines are used to mark the absolute fringe thickness.

In Fig. S3B we show the results from two different samples. Fig. S3B, Left corresponds to a sample that has 85% MoS2 concentration of that of Fig. S3B, Right; thus there are 85% MoS2 flakes illuminated. All other factors of the experiment, including the screen and camera positions, are identical. Thus, χleft(3)=(Mleft/Mright)2χright(3), number of rings Nleft=(Mleft/Mright)2Nright, and ring diameter Dleft=(Mleft/Mright)2Dright, where M is the effective number of MoS2 layers [note that in Methods we have χ(3)=M2χonelayer(3)]. In Fig. S3B, the ring diameter is enlarged by (Mright/Mleft)2=(1/0.85)2=1.4× and the number of rings is also enlarged by 1.4×. Therefore, the absolute ring thickness is unchanged (parallel white straight lines in Fig. S3B). However, the relative fringe thickness is narrowed: Because the number of rings is enlarged by 1.4×, the relative fringe thickness (absolute thickness divided by the ring diameter D) is narrowed by 1.4×, too. Conversely, if one takes the jth ring counted from the ring center, both its absolute and relative thicknesses are narrowed.

S5) Absorption and Raman Spectra of the MoS2 Flakes

The absorbance of our MoS2 suspension is shown in Fig. S4A. It is different from that in ref. 2. Rather, it can be understood based on the results in ref. 29. In ref. 29 red shift has been observed for the multilayer samples. The red shift in the absorption peak of our result demonstrates that most of the flakes in our sample are not monolayer MoS2 flakes. Here the suspension modifies the background tendency of the absorption. We also measured the Raman scattering of the flakes from the suspension (Fig. S4B). Comparison with that in ref. 30 shows that our sample is composed mainly of few-layer MoS2 flakes.

Fig. S4.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. S4.

Sample characterization. (A) Absorption spectra of the MoS2 flake suspension. (B) Raman shift of the MoS2 flakes after the evaporation of the solution.

S6) Estimation of the Sample Concentration

The concentration of our MoS2 suspension solution is ρ = 0.00085 mol/L. The volume is V = 4 × 10−3 L. Thus, the total number of molecules in our solution is M = ρ × V × NA = 2 × 1018. The space group of bulk MoS2 is P63/mmc hexagonal and the lattice constants are a = 3.161 Å and c = 12.295 Å, respectively. Thus, a single effective layer contains m = S/unit cell area in ab section = 1 × 4 cm2/(sin 60°) × (3.161)2 Å2 = 4.6 × 1015 molecules. Hence the layer number is n = M/m = 444. Due to the instability of the suspension, we estimate the effective layer number to be 300.

Acknowledgments

We acknowledge useful discussion with C.-K. Shih, James Hinton, and Joseph Orenstein. This work was supported by the National Basic Research Program of China (2012CB821402 and 2012CB921403), the National Natural Science Foundation of China (11274372 and 11222431), and the External Cooperation Program of the Chinese Academy of Sciences (GJHZ1403).

Footnotes

  • ↵1To whom correspondence may be addressed. Email: jmzhao{at}iphy.ac.cn or smeng{at}iphy.ac.cn.
  • Author contributions: J.Z. designed research; Y.W., Q.W., F.S., and J.Z. performed research; C.C. and S.M. obtained theoretical absorption; Q.W. and J.Z. derived wind chime model; Y.W., S.M., and J.Z. analyzed data; and S.M. and J.Z. wrote the paper.

  • The authors declare no conflict of interest.

  • This article is a PNAS Direct Submission. R.M. is a guest editor invited by the Editorial Board.

  • This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1504920112/-/DCSupplemental.

Freely available online through the PNAS open access option.

References

  1. ↵
    1. Novoselov KS, et al.
    (2004) Electric field effect in atomically thin carbon films. Science 306(5696):666–669
    .
    OpenUrlAbstract/FREE Full Text
  2. ↵
    1. Mak KF,
    2. Lee C,
    3. Hone J,
    4. Shan J,
    5. Heinz TF
    (2010) Atomically thin MoS2: A new direct-gap semiconductor. Phys Rev Lett 105(13):136805
    .
    OpenUrlCrossRefPubMed
  3. ↵
    1. Xu XD,
    2. Yao W,
    3. Xiao D,
    4. Heinz TF
    (2014) Spin and pseudospins in layered transition metal dichalcogenides. Nat Phys 10(5):343–350
    .
    OpenUrlCrossRef
  4. ↵
    1. Kim JH, et al.
    (2014) Ultrafast generation of pseudo-magnetic field for valley excitons in WSe2 monolayers. Science 346(6214):1205–1208
    .
    OpenUrlAbstract/FREE Full Text
  5. ↵
    1. Xiao D,
    2. Liu GB,
    3. Feng W,
    4. Xu X,
    5. Yao W
    (2012) Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides. Phys Rev Lett 108(19):196802
    .
    OpenUrlCrossRefPubMed
  6. ↵
    1. Mak KF, et al.
    (2013) Tightly bound trions in monolayer MoS2. Nat Mater 12(3):207–211
    .
    OpenUrlCrossRefPubMed
  7. ↵
    1. Jones AM, et al.
    (2014) Spin–layer locking effects in optical orientation of exciton spin in bilayer WSe2. Nat Phys 10(2):130–134
    .
    OpenUrlCrossRef
  8. ↵
    1. Zeng H,
    2. Dai J,
    3. Yao W,
    4. Xiao D,
    5. Cui X
    (2012) Valley polarization in MoS2 monolayers by optical pumping. Nat Nanotechnol 7(8):490–493
    .
    OpenUrlCrossRefPubMed
  9. ↵
    1. Xiao D,
    2. Yao W,
    3. Niu Q
    (2007) Valley-contrasting physics in graphene: Magnetic moment and topological transport. Phys Rev Lett 99(23):236809
    .
    OpenUrlCrossRefPubMed
  10. ↵
    1. Basov DN,
    2. Fogler MM,
    3. Lanzara A,
    4. Wang F,
    5. Zhang YB
    (2014) Colloquium: Graphene spectroscopy. Rev Mod Phys 86(3):959–994
    .
    OpenUrlCrossRef
  11. ↵
    1. Novoselov KS, et al.
    (2005) Two-dimensional atomic crystals. Proc Natl Acad Sci USA 102(30):10451–10453
    .
    OpenUrlAbstract/FREE Full Text
  12. ↵
    1. Coleman JN, et al.
    (2011) Two-dimensional nanosheets produced by liquid exfoliation of layered materials. Science 331(6017):568–571
    .
    OpenUrlAbstract/FREE Full Text
  13. ↵
    1. Fei Z, et al.
    (2012) Gate-tuning of graphene plasmons revealed by infrared nano-imaging. Nature 487(7405):82–85
    .
    OpenUrlPubMed
  14. ↵
    1. Wu R, et al.
    (2011) Purely coherent nonlinear optical response in solution dispersions of graphene sheets. Nano Lett 11(12):5159–5164
    .
    OpenUrlCrossRefPubMed
  15. ↵
    1. Hendry E,
    2. Hale PJ,
    3. Moger J,
    4. Savchenko AK,
    5. Mikhailov SA
    (2010) Coherent nonlinear optical response of graphene. Phys Rev Lett 105(9):097401
    .
    OpenUrlCrossRefPubMed
  16. ↵
    1. Singh A, et al.
    (2014) Coherent electronic coupling in atomically thin MoSe2. Phys Rev Lett 112(21):216804
    .
    OpenUrlCrossRef
  17. ↵
    1. Kumar N, et al.
    (2013) Third harmonic generation in graphene and few-layer graphite films. Phys Rev B 87(12):121406
    .
  18. ↵
    1. Ge S, et al.
    (2014) Coherent longitudinal acoustic phonon approaching THz frequency in multilayer molybdenum disulphide. Sci Rep 4:5722
    .
    OpenUrlPubMed
  19. ↵
    1. Cheng JL,
    2. Vermeulen N,
    3. Sipe JE
    (2014) Third order optical nonlinearity of graphene. New J Phys 16:053014
    .
    OpenUrlCrossRef
  20. ↵
    1. Mankowsky R, et al.
    (2014) Nonlinear lattice dynamics as a basis for enhanced superconductivity in YBa2Cu3O6.5. Nature 516(7529):71–73
    .
    OpenUrlCrossRefPubMed
  21. ↵
    1. Berg E, et al.
    (2007) Dynamical layer decoupling in a stripe-ordered high-T(c) superconductor. Phys Rev Lett 99(12):127003
    .
    OpenUrlCrossRefPubMed
  22. ↵
    1. Durbin SD,
    2. Arakelian SM,
    3. Shen YR
    (1981) Laser-induced diffraction rings from a nematic-liquid-crystal film. Opt Lett 6(9):411–413
    .
    OpenUrlPubMed
  23. ↵
    1. Kappera R, et al.
    (2014) Phase-engineered low-resistance contacts for ultrathin MoS2 transistors. Nat Mater 13(12):1128–1134
    .
    OpenUrlCrossRefPubMed
  24. ↵
    1. Sie EJ, et al.
    (2014) Valley-selective optical Stark effect in monolayer WS2. Nat Mater 14(3):290–294
    .
  25. ↵
    1. Wang R, et al.
    (2014) Third-harmonic generation in ultrathin films of MoS2. ACS Appl Mater Interfaces 6(1):314–318
    .
    OpenUrlCrossRefPubMed
  26. ↵
    1. Bian F, et al.
    (2010) Laser-driven silver nanowire formation: Effect of femtosecond laser pulse polarization. Chin Phys Lett 27(8):088101
    .
    OpenUrlCrossRef
  27. ↵
    1. Han XF, et al.
    (2008) Single-photon level ultrafast all-optical switching. Appl Phys Lett 92(15):151109
    .
    OpenUrlCrossRef
  28. ↵
    1. Dawes AMC,
    2. Illing L,
    3. Clark SM,
    4. Gauthier DJ
    (2005) All-optical switching in rubidium vapor. Science 308(5722):672–674
    .
    OpenUrlAbstract/FREE Full Text
  29. ↵
    1. Shi H, et al.
    (2013) Exciton dynamics in suspended monolayer and few-layer MoS2 2D crystals. ACS Nano 7(2):1072–1080
    .
    OpenUrlCrossRefPubMed
  30. ↵
    1. Lee C, et al.
    (2010) Anomalous lattice vibrations of single- and few-layer MoS2. ACS Nano 4(5):2695–2700
    .
    OpenUrlCrossRefPubMed
PreviousNext
Back to top
Article Alerts
Email Article

Thank you for your interest in spreading the word on PNAS.

NOTE: We only request your email address so that the person you are recommending the page to knows that you wanted them to see it, and that it is not junk mail. We do not capture any email address.

Enter multiple addresses on separate lines or separate them with commas.
Emergence of electron coherence and two-color all-optical switching in MoS2 based on spatial self-phase modulation
(Your Name) has sent you a message from PNAS
(Your Name) thought you would like to see the PNAS web site.
CAPTCHA
This question is for testing whether or not you are a human visitor and to prevent automated spam submissions.
Citation Tools
Electron coherence in MoS2 induced by SSPM
Yanling Wu, Qiong Wu, Fei Sun, Cai Cheng, Sheng Meng, Jimin Zhao
Proceedings of the National Academy of Sciences Sep 2015, 112 (38) 11800-11805; DOI: 10.1073/pnas.1504920112

Citation Manager Formats

  • BibTeX
  • Bookends
  • EasyBib
  • EndNote (tagged)
  • EndNote 8 (xml)
  • Medlars
  • Mendeley
  • Papers
  • RefWorks Tagged
  • Ref Manager
  • RIS
  • Zotero
Request Permissions
Share
Electron coherence in MoS2 induced by SSPM
Yanling Wu, Qiong Wu, Fei Sun, Cai Cheng, Sheng Meng, Jimin Zhao
Proceedings of the National Academy of Sciences Sep 2015, 112 (38) 11800-11805; DOI: 10.1073/pnas.1504920112
del.icio.us logo Digg logo Reddit logo Twitter logo CiteULike logo Facebook logo Google logo Mendeley logo
  • Tweet Widget
  • Facebook Like
  • Mendeley logo Mendeley

Article Classifications

  • Physical Sciences
  • Physics
Proceedings of the National Academy of Sciences: 112 (38)
Table of Contents

Submit

Sign up for Article Alerts

Jump to section

  • Article
    • Abstract
    • Results and Discussion
    • SI Text
    • S1) Derivation of the Wind-Chime Model
    • S2) Comparison of χ(3) with Calculated Absorption Along z
    • S3) Ring Number of the 532-nm Pattern as a Function of the Total Laser Intensity
    • Conclusions
    • Methods
    • S4) Relation Between the Fringe Thickness and the Flake Concentration
    • S5) Absorption and Raman Spectra of the MoS2 Flakes
    • S6) Estimation of the Sample Concentration
    • Acknowledgments
    • Footnotes
    • References
  • Figures & SI
  • Info & Metrics
  • PDF

You May Also be Interested in

Reflection of clouds in the still waters of Mono Lake in California.
Inner Workings: Making headway with the mysteries of life’s origins
Recent experiments and simulations are starting to answer some fundamental questions about how life came to be.
Image credit: Shutterstock/Radoslaw Lecyk.
Depiction of the sun's heliosphere with Voyager spacecraft at its edge.
News Feature: Voyager still breaking barriers decades after launch
Launched in 1977, Voyagers 1 and 2 are still helping to resolve past controversies even as they help spark a new one: the true shape of the heliosphere.
Image credit: NASA/JPL-Caltech.
Drop of water creates splash in a puddle.
Journal Club: Heavy water tastes sweeter
Heavy hydrogen makes heavy water more dense and raises its boiling point. It also appears to affect another characteristic long rumored: taste.
Image credit: Shutterstock/sl_photo.
Mouse fibroblast cells. Electron bifurcation reactions keep mammalian cells alive.
Exploring electron bifurcation
Jonathon Yuly, David Beratan, and Peng Zhang investigate how electron bifurcation reactions work.
Listen
Past PodcastsSubscribe
Panda bear hanging in a tree
How horse manure helps giant pandas tolerate cold
A study finds that giant pandas roll in horse manure to increase their cold tolerance.
Image credit: Fuwen Wei.

Similar Articles

Site Logo
Powered by HighWire
  • Submit Manuscript
  • Twitter
  • Facebook
  • RSS Feeds
  • Email Alerts

Articles

  • Current Issue
  • Special Feature Articles – Most Recent
  • List of Issues

PNAS Portals

  • Anthropology
  • Chemistry
  • Classics
  • Front Matter
  • Physics
  • Sustainability Science
  • Teaching Resources

Information

  • Authors
  • Editorial Board
  • Reviewers
  • Subscribers
  • Librarians
  • Press
  • Cozzarelli Prize
  • Site Map
  • PNAS Updates
  • FAQs
  • Accessibility Statement
  • Rights & Permissions
  • About
  • Contact

Feedback    Privacy/Legal

Copyright © 2021 National Academy of Sciences. Online ISSN 1091-6490