Tuning of nonlinear optical responses is the essence to many photonics and optoelectronics applications. Due to the low-dimensionality and dispersion of massless Dirac Fermions, the nonlinear optical susceptibilities of graphene can be readily controlled via electrical gating. Based on the quantum interference between multi-photon transition pathways, the tuning mechanism of graphene nonlinearity is intrinsically different from most other systems. The phenomenon enables investigations into some nonlinear optical processes from fundamental regards. It also exhibits appealing features contrasting conventional materials, which can be desirable for novel device applications.

Optical nonlinearity was discovered soon after the invention of the first laser1 and has enabled enormous applications in fundamental research and industrial technology, ranging from the ultrafast laser systems, nonlinear optical spectroscopies, optical switches, and communication to bio-imaging and sensing, micro-fabrication and machining, etc.2–4 In device applications, tuning of the optical nonlinear response is particularly useful and desirable, and among all tuning protocols, the electrical tuning is the most viable due to the high efficiency and easy integration of opto-electronic systems. The most widely utilized electro-optical (EO) mechanisms include the Pockels effect and the Kerr effect, through which the power, phase, and polarization of light can be controlled by applied electrical fields.5 The Pockels effect occurs in materials with broken inversion symmetry and manifests as a change in the linear refractive index n(ω). Upon the application of an external electric field Eext, there is ΔnPωrP(2)ωEext, with rP(2)ω being the third rank Pockels EO tensor.6 The Kerr effect is also called the quadratic EO effect and is not limited to non-centrosymmetric media. The induced change by the Kerr effect is proportional to χK(3)Eext2, with χK(3) being the fourth rank nonlinear susceptibility tensor.7 In practical devices, high operation voltages of a few hundred to a few thousand volts are often essential to induce a large enough change in optical constants.6,7 New concept EO devices without the high field may be achieved if susceptibilities themselves could be directly and effectively tuned by modifying material properties.

Nonlinear optical susceptibilities, in general, have two contributions: the interband and intraband (or free carrier) components. So far, most EO applications are based on semiconducting and insulating optical materials without involving the intraband component. This is largely because conventional EO devices often operate in the transmitted scheme to elongate the light–matter interaction distance, while free carriers are often not optically transparent.8 Even with semiconducting (semi-transparent) materials, upon the application of an external electrostatic field, free carriers can screen out the field and shorten the effective interaction distance.9 Alternatively, if the doping level is high enough to significantly alter the electronic band structure and chemical potential, the system nonlinearity can also be tunable via the interband contribution. Yet it is hard to dynamically dope the entire bulk phase at such a high level as the number of carriers involved would be gigantic.

The story becomes quite different in two-dimensional (2D) systems. The materials are often very thin; hence, the optical transmittance remains high even with the metallic phase. Moreover, due to the low dimensionality, a relatively small amount of additional carriers can appreciably change the chemical potential or even the electronic band structure. Hence, in 2D, it becomes much more viable to directly tune the optical nonlinearities themselves via both intraband and interband contributions. This is not only valuable for micro-optoelectronic applications but also provides a route to investigate nonlinear optical processes from a more fundamental point of view, as we shall elaborate below.

Graphene is arguably the most investigated and promising 2D materials.10,11 Through the years, many techniques have been developed for manipulating both electronic and optical properties of graphene.12 Shortly after the isolation of graphene monolayers, Yan and co-workers demonstrated that via electrical gating, which efficiently shifted the graphene chemical potential, both the strength and line-shape of graphene Raman modes could be tuned due to the Pauli blocking effect.13 Soon afterward, Wang and co-workers applied the same method to switch on and off the linear optical absorption in the mid-infrared range of both monolayer and bilayer graphene.14,15 These studies opened the venue for electrical tuning of graphene optical properties.

Most related experimental studies were carried out with a field-effect-transistor (FET) configuration to control the chemical potential (μ) in graphene [Fig. 1(a)]. Here, an external potential is applied via the gate electrode. An electric field discontinuity builds up at the interface and causes charges to accumulate (Gauss’s law). With a sheet capacitance of the gate material being Cg and a gate potential being Vg, the amount of doping in graphene is ng = Vg/Cg. In early studies, the silicon substrates covered with thin films of oxides (hundreds of nanometers-thick) were often used as the back gate, and the sheet capacitance is on the order of 10−2 µF/cm2.16 As limited by the breakdown field of silicon dioxide, the maximum doping range is often less than ∼5 × 1012 cm2.17 Other conventional oxides have also been used as gating materials such as Al2O3, Ta2O5, HfO2, and TiO2, with typical values of Cg about 10−2 µF/cm2–1 µF/cm2.18 

FIG. 1.

Schematics of gate-controlled graphene devices and the gating mechanism. (a) The device with the dielectric bottom gate (fused silica as an example). (b) The device with the ion-gel top gate. (c) The charging due to electrostatic (left) and electrochemical (right) effects at the electrolyte/material interface.

FIG. 1.

Schematics of gate-controlled graphene devices and the gating mechanism. (a) The device with the dielectric bottom gate (fused silica as an example). (b) The device with the ion-gel top gate. (c) The charging due to electrostatic (left) and electrochemical (right) effects at the electrolyte/material interface.

Close modal

Later, electrochemical (EC) gating became more often adopted, for it provides a much greater sheet capacitance compared to most oxides.19 Such EC transistors use ionic liquids or electrolytes as gate dielectrics, inside which the electric double layers (EDL) function as nanogap capacitors [Fig. 1(b)]. With huge capacitances on the order of 1 µF/cm2–10 µF/cm2, they can accumulate charges to a level as high as 8.0 × 1014 cm−2 at the electrochemical interfaces.20 Applications of the EDL gating on solid materials have attracted much interest in recent years, and the interfacial carrier accumulation could be due to either electrostatics or electrochemistry [Fig. 1(c)], enabling the tuning of various physical and chemical phenomena.21,22

Because of the small number of states in 2D materials, such a doping concentration can shift the system chemical potential quite appreciable. For graphene, due to the symmetric electron and hole bands, the direct interband optical transition would be blocked at the photon energy less than 2|μ|, thus yielding a greater linear transmittance for corresponding photons [Fig. 2(a)]. Note that off-resonant transitions also contribute to the net amplitude, but their contribution is usually diminishing and would be neglected in the following discussion. For example, using PS-PEO-PS(7 wt. %), [EMIM][TFSI](93 wt. %) ionic-liquid gating, the optical transmittance could be tuned, as shown in [Fig. 2(b)].23 Each spectrum exhibited a step function (broadened due to the finite temperature effect and sample inhomogeneity), with a higher transmittance of about 2.3%24,25 at the lower photon energy. Such measurements also allowed the extraction of the exact position of chemical potential as a function of gating [Fig. 2(c)]. Notably, the chemical potential can now be tuned into the energy range of visible photons. As we demonstrated in Ref. 26, we can effectively turn on and off the upconverted photoluminescence due to ultrafast scattering between hot carriers excited by visible photons.

FIG. 2.

Optical transitions in graphene modulated by doping. (a) The Pauli blocking of one-photon transition in graphene. (b) Spectra of transmittance change of an ion-gel gated graphene device under various gate voltages. (c) The chemical potential derived from (b) and the source-drain resistance of the device. (d) Schematics of transition diagrams with zero, one-, and two-photon resonant pathways gradually shut down at increasing chemical potential. Materials with permission from Jiang et al., Nat. Photonics 12, 430 (2018). Copyright 2018 Springer Nature.

FIG. 2.

Optical transitions in graphene modulated by doping. (a) The Pauli blocking of one-photon transition in graphene. (b) Spectra of transmittance change of an ion-gel gated graphene device under various gate voltages. (c) The chemical potential derived from (b) and the source-drain resistance of the device. (d) Schematics of transition diagrams with zero, one-, and two-photon resonant pathways gradually shut down at increasing chemical potential. Materials with permission from Jiang et al., Nat. Photonics 12, 430 (2018). Copyright 2018 Springer Nature.

Close modal

Things can get even more interesting when it comes to nonlinear. Phenomenologically, the modulation depth in linear absorption is usually small, and specific excitation and/or detection schemes such as waveguides are often required to elongate the light–material interaction distance.27,28 In contrast, nonlinear responses is often background-free and can be tuned from effectively zero to an appreciable strength, thus yielding a large modulation depth. More significantly and fundamentally, there are multiple photons and various transition pathways involved in nonlinear optical processes, so naturally one expects richer and more profound behaviors in response to the varying chemical potential. For instance, for a two-photon absorption and emission process, all transition pathways are allowed at the charge neutral point (CNP, μ = 0); when the chemical potential gradually shifts away from the CNP, the one-photon and then two-photon resonances will be sequentially shut off [Fig. 2(d)]. Since the nonlinear optical susceptibility is the superposition of all transition amplitudes, it would thus be a function of the chemical potential and tunable by the external gate voltage.

We first applied the scheme to investigate the third harmonic generation (THG) from graphene. The reason we started from the third order process, instead of the second, was because graphene is nearly perfectly centrosymmetric. Therefore, second order responses, including the second harmonic generation (SHG) and sum- and difference-frequency generations, are all forbidden under the electric-dipole approximation.29–31 The third order processes thus become the leading order responses of graphene. As pointed out by Mikhailov and co-workers,32 the non-parabolic dispersion of graphene energy bands leads to intrinsically huge odd-order nonlinear responses, which were indeed observed in THG and four-wave mixing (FWM) studies.33–35 On the other hand, it was unclear that how the reported nonlinear susceptibilities could divert over six orders of magnitudes from case to case,23 and how they would respond to the varying chemical potential in graphene.

In our experiments, the THG was excited and collected in a normal incident scheme, and an ion-gel gate was applied to the sample to control the doping level [Fig. 3(a)]. When the chemical potential is tuned away from zero, based on the above discussion, it might be expected that the THG signal would gradually diminish as more and more transition pathways get switched off. Yet counter-intuitively, the THG does not drop but rather gets stronger toward higher |μ| [Fig. 3(b)].23,36 To understand the underlying mechanism, we resorted to the full quantum mechanical calculation, through which both the amplitude and phase of each multi-photon transition pathway can be evaluated.37,38 It turned out that phases of one-, two-, and three-photon transition pathways are different from each other, being positive, negative, and positive, respectively. Consider the four cases in Fig. 3(c), with none, one-, two-, and three-photon direct transition pathways are turned off, respectively. At μ = 0, all pathways are allowed, but they beat with and largely cancel each other; when 2|μ| approaches ω, the one-photon pathway becomes switched off, all the overall magnitude increases from the initial cancellation. We called it the “Fermi-edge resonance” of one-photon transition. Similarly, this is the case when the two-photon pathway is switched off, which sees further increment in the overall THG intensity. Only when 2|μ| exceeds 3ω, with all transition pathways are off, the overall amplitude would start to drop. Overall, the THG signal is the strongest within 3ω > 2|μ| > ω.

FIG. 3.

Gate tuned THG and FWM of monolayer graphene. (a) The normal incident excitation and detection scheme of graphene THG and FWM. (b) The mapping of graphene emission spectra including the THG signal vs the chemical potential (input photon energy = 0.79 eV). (c) Schematics of transition diagrams with zero, one-, two-, and three-photon resonant pathways gradually shut down at increasing chemical potential. “+” and “−” indicate the relative sign between various transition amplitudes. (d) Calculated amplitude, real, and imaginary parts of THG χ(3) vs the chemical potential at zero temperature. (e) Transition diagrams of various FWM processes. (f) The experimentally measured and calculated SFM and DFM signals vs the chemical potential. Materials with permission from Jiang et al., Nat. Photonics 12, 430 (2018). Copyright 2018 Springer Nature.

FIG. 3.

Gate tuned THG and FWM of monolayer graphene. (a) The normal incident excitation and detection scheme of graphene THG and FWM. (b) The mapping of graphene emission spectra including the THG signal vs the chemical potential (input photon energy = 0.79 eV). (c) Schematics of transition diagrams with zero, one-, two-, and three-photon resonant pathways gradually shut down at increasing chemical potential. “+” and “−” indicate the relative sign between various transition amplitudes. (d) Calculated amplitude, real, and imaginary parts of THG χ(3) vs the chemical potential at zero temperature. (e) Transition diagrams of various FWM processes. (f) The experimentally measured and calculated SFM and DFM signals vs the chemical potential. Materials with permission from Jiang et al., Nat. Photonics 12, 430 (2018). Copyright 2018 Springer Nature.

Close modal

Besides the interference between quantum pathways, the full quantum mechanical calculation revealed more interesting features about Fermi-edge resonances. Figure 3(d) shows the calculated real and imaginary parts of χ(3)(3ω = ω + ω + ω) (at zero temperature for clarity), with the former showing sharp spikes and the latter as step functions. This is right the opposite from usual optical susceptibilities that the imaginary parts exhibit peaks upon resonances.29 The Fermi-edge resonance is truly a unique phenomenon regarding this aspect.

The Fermi resonance also has a practically useful feature. It is noticed that the strongest THG appears at 2|μ| > ω, that is, when the one-photon direct transition, or the linear absorption, is switched off. This means that the THG resonant enhancement is accompanied by the reduced linear absorption. In conventional materials, nonlinear optical susceptibility also undergoes resonant enhancement but is often accompanied by an enhanced linear absorption, which causes the materials to be easily damaged and the pump power must be lowered. However, now in graphene, the damage threshold becomes even higher upon the Fermi edge resonance, which is a rather desirable feature for photonic applications.

Upon two-color excitations, instead of one-color, we can further monitor the FWM responses from graphene. There are various types of FWM processes, which we call the sum-frequency mixing (SFM) and difference-frequency mixing (DFM) [Fig. 3(e)]. They were both observed experimentally [Fig. 3(f)].23 When the chemical potential is changed, the SFM modes behave similarly to that of THG, being the weakest at CNP and stronger toward greater |μ|. While the DFM modes behave quite differently: they are strong near CNP but decrease, instead of increase, toward greater |μ|. Again, the behavior could be reproduced by the full quantum mechanical calculations, while it can also be comprehended with a clear physical picture. Consider the transition diagrams of SFM and DFM processes [Fig. 3(e)], one of the incoming transitions became outgoing in the latter case. This causes the sign of the related transition amplitude to flip and so does the phase of the corresponding transition pathway.2 Hence, the DFM behaved distinctively from THG and SFM. The interference between optical transition pathways has been known to physicists since the establishment of nonlinear optics theory. Now with graphene, we can directly visualize and manipulate this quantum interference effect via experiments, which could help us gain deeper insight into fundamental nonlinear optical processes. Interestingly, the calculation predicted the DFM magnitude to be inversely proportional to Δω2ω = ω1 − ω2ω1, ω2) at CNP, which means that its intensity would grow steeply toward the low photon energy. Combined with the broadband nature of the graphene optical response, this property hints a highly promising route for mid-infrared and THz generation.39,40

Strong SHG responses have been observed from non-centrosymmetric 2D materials41–44 and showed gate tunability upon field induced effects.45 However, as mentioned above, second order nonlinearities of the perfectly centrosymmetric graphene are vanishing under the electric dipole approximation.46–48 Nonetheless, if we consider graphene and the excitation photon as an entity, the inversion symmetry of the combined system can be effectively broken.49 This is a simplified way for understanding the electric quadruple (EQ) type of second harmonic generation, which is through the process of P22ω=χEQ32ω=ω+ωqEωEω+, where q refers to the incident photon wave vector and χEQ(3) refers to the EQ-type of the third-order nonlinear susceptibility tensor. Here, χEQ(3) has the same construction with that of the electric dipole type χED(3) for THG and FWM, but now with one of the subscripts denoting the direction component of incident photon wavevector. For example, χEQ,xxyy3(2ω=ω+ω) is the tensor element that maps the product qx(ω)Ey2(ω) to the nonlinear polarization component Px(2ω). Since graphene belongs to the D6h point group, all its χ(3) tensor elements have the form of χiijj(3), χijij(3), or χijji(3), with i, j ∈ {x, y, z}. As the 2D lattice supports only in-plane (x–y plane) electric field components, the input photon wavevector must also contain in-plane components to contribute to the EQ response. In other words, either an evanescent or an oblique incident geometry would be essential [Fig. 4(a)]. Indeed, when we detected graphene SHG outputs under normal incidence, the signal was two orders of magnitudes weaker than that from the oblique incident case, consistent with its electric quadruple nature.

FIG. 4.

Gate tuned electric quadruple SHG of monolayer graphene. (a) The oblique incident excitation and detection scheme of graphene quadruple SHG. (b) The mapping of graphene emission spectra including both SHG and THG signals vs the chemical potential (input photon energy = 0.95 eV). The broadband signal at the left bottom of the map is the ultrafast photoluminescence. (c) The experimentally measured (left) and calculated (right) quadruple χ(3) elements vs the chemical potential. (d) Transition schematics under the operation of μ → −μ and k → −k with the electron–hole symmetry and time-reversal symmetry for q → −q, where k is the electron wave vector and q is the photon wave vector. (e) The SHG (red) intensity vs the angle between the beam incident plane and the analyzer polarization direction under 45° polarized linear excitation. The corresponding THG (gray) data are shown for comparison. Reprinted figure with permission from Zhang et al., Phys. Rev. Lett. 122, 47401 (2019). Copyright 2019 The American Physical Society.

FIG. 4.

Gate tuned electric quadruple SHG of monolayer graphene. (a) The oblique incident excitation and detection scheme of graphene quadruple SHG. (b) The mapping of graphene emission spectra including both SHG and THG signals vs the chemical potential (input photon energy = 0.95 eV). The broadband signal at the left bottom of the map is the ultrafast photoluminescence. (c) The experimentally measured (left) and calculated (right) quadruple χ(3) elements vs the chemical potential. (d) Transition schematics under the operation of μ → −μ and k → −k with the electron–hole symmetry and time-reversal symmetry for q → −q, where k is the electron wave vector and q is the photon wave vector. (e) The SHG (red) intensity vs the angle between the beam incident plane and the analyzer polarization direction under 45° polarized linear excitation. The corresponding THG (gray) data are shown for comparison. Reprinted figure with permission from Zhang et al., Phys. Rev. Lett. 122, 47401 (2019). Copyright 2019 The American Physical Society.

Close modal

Similar to THG and FWM, the quadruple SHG also sensitively depends on the chemical potential. Figure 4(b) shows the μ dependence of both graphene SHG and THG outputs. Both are the weakest at CNP and grow stronger toward increasing |μ|. The trend again could be reproduced nicely by the full-quantum mechanical calculation [Fig. 4(c)]. On the other hand, there is a marked distinction between the quadruple SHG and THG. At CNP, the THG signal is already appreciable, yet the SHG is almost completely absent [Fig. 4(c)]. Further analysis shows that it is not a coincidence but indicates rich underlying physics. While THG (as well as FWM) are even functions of μ, the quadruple SHG is an odd function of μ, meaning that it must vanish at CNP given that it is a continuous function of μ. This is a direct consequence of the time reversal symmetry and the electron–hole symmetry of graphene. A simple illustration can help us understand this physics [Fig. 4(d)]: the electric quadruple-type response involves a transition relating initial and final states separated by a photon wavevector q.49–53 If we apply the time reversal operation (k → −k) and reverse the electron and hole bands (μ → −μ), we would obtain the same diagram with q → −q, which leads to χ(μ) → −χ(−μ). Therefore, the nulling of quadruple SHG at CNP is a manifestation of the system symmetry properties. Before, SHG has long been known as a sensitive probe of structural symmetry41,54,55 and the broken of time-reversal symmetry.56,57 Now, we can further add electron–hole symmetry to this family.

In conventional materials, the quadruple SHG is usually overwhelmed by its electric dipole counterparts, either from the bulk or the surface;2,9 hence, rather little was known about it. Graphene provides a unique platform for us to study this effect. Moreover, the graphene quadruple SHG is exceptionally strong, even comparable to the electric dipole SHG of non-centrosymmetric 2D materials in the off-resonance case. Here, the large band velocity of vF plays a key role according to the calculation.51 The same physics would also apply to other graphene-like systems, and the doping dependent quadruple SHG could serve as a sensitive probe of the electron–hole band symmetry.

This tunable SHG also has potential device applications. As THG, it also has an intrinsically high damage threshold and can be pumped at high intensity. While unlike THG, the SHG response is not optically isotropic, as χEQ,xxxx3χEQ,xxyy3 [Fig. 4(c)]. Hence, the quadruple SHG is, in general, elliptically polarized upon linear polarized excitation [Fig. 4(e)]. As the ellipticity depends on both the input polarization and the chemical potential, it may serve as a miniature ellipticity modulator/analyzer that is dynamically tunable. Meanwhile, upon Fermi edge resonance, both imaginary and real parts of the susceptibility contribute, meaning the SHG phase also varies across the resonance. This implies general applications as dynamic and miniature phase modulators.58 

To briefly summarize, the electrically tunable nonlinear susceptibilities of graphene provide an unprecedent platform for us to investigate nonlinear optical processes from fundamental aspects, visualizing the switching of and interference between multi-photon transition pathways. Such phenomena are not only limited to graphene but also applicable to other low dimensional systems with similar electronic dispersions and/or small enough bandgaps that can be readily doped via electrical gating. As exhibiting many folds of unique and desirable properties, such tunable optical nonlinearities are likely to play an increasingly important role in emerging optoelectronic applications with broadband and ultrafast capabilities, as well as a compact footprint.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

This work was supported by the National Key Research and Development Program of China (Grant Nos. 2019YFA0308404 and 2016YFA0300900) and the National Natural Science Foundation of China (Grant Nos. 11991062, 11622429, and 91950201). W.-T.L. is also supported from the National Program for Support of Top-Notch Young Professionals and the Shu Guang Project (No. 18SG02). S.W. is supported by the Shanghai Municipal Science and Technology Major Project (No. 2019SHZDZX01).

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