Graphene is conceivably the most nonlinear optoelectronic material we know. Its nonlinear optical coefficients in the terahertz frequency range surpass those of other materials by many orders of magnitude. Here, we show that the terahertz nonlinearity of graphene, both for ultrashort single-cycle and quasi-monochromatic multicycle input terahertz signals, can be efficiently controlled using electrical gating, with gating voltages as low as a few volts. For example, optimal electrical gating enhances the power conversion efficiency in terahertz third-harmonic generation in graphene by about two orders of magnitude. Our experimental results are in quantitative agreement with a physical model of the graphene nonlinearity, describing the time-dependent thermodynamic balance maintained within the electronic population of graphene during interaction with ultrafast electric fields. Our results can serve as a basis for straightforward and accurate design of devices and applications for efficient electronic signal processing in graphene at ultrahigh frequencies.
The development of efficient broadband electronic frequency multipliers and modulators operating at the technologically important terahertz frequency range and under normal ambient conditions is of great relevance, but also challenging (1–5). These technologies require the integration of a highly nonlinear material in an electronic device with the possibility to control its nonlinear behavior. In addition, compatibility between the device constituents is crucial. These requirements are not easy to meet, especially in simple electronic circuits (4–6). Here, we demonstrate the possibility of accomplishing these requirements in a graphene-based device. A straightforward and efficient wide-range control of the terahertz nonlinearity of graphene is enabled by electrical gating of only a few volts.
One of the essential technological consequences of the Dirac-type electronic bandstructure of graphene (7, 8) is its extremely nonlinear response to electric fields in the terahertz frequency range, which persists at room temperature and under normal ambient conditions. Graphene is a highly efficient terahertz nonlinear absorber, capable of displaying a nonlinear power transmission modulation of about 50% per single monolayer (9–12). It is also an extremely efficient terahertz frequency multiplier, allowing for the straightforward generation of multiple terahertz harmonics (12–15). Such a nonlinear response is directly attainable with quite moderate input electric fields, oscillating at frequencies not exceeding a few hundred gigahertz and with a field strength in the range of 10 to 100 kV/cm, which is even lower than typical channel fields in the current generation of high-speed transistors (4, 16. Conveniently, graphene is also compatible with complementary metal-oxide semiconductor (CMOS) technology (6, 17). This suggests the possibility to develop hybrid graphene-CMOS technology, with the subterahertz CMOS device acting as a reliable and cost-effective pump source, providing the input signal for the efficient nonlinear processing in graphene at terahertz frequencies. One of the straightforward applications of such a technology is all-electronic terahertz frequency mixing or upconversion of the subterahertz CMOS–generated signals into the terahertz range, with the potential to outcompete the current generation of diode-based ultrahigh frequency mixers (1–3, 18, 19) in terms of conversion efficiency, operation bandwidth, compactness, and cost.
Recent experiments on terahertz high-harmonic generation (HHG) in graphene directly provided the measurement of its nonlinear coefficients up to the seventh order (14). These nonlinear coefficients were found to exceed the respective coefficients of all other known electronic materials by many orders of magnitude (12, 14). In a nonoptimized terahertz HHG experiment in (14), the field conversion efficiency from the driving subterahertz electric signal to terahertz higher harmonics of η ~ 1% was demonstrated in just a single–atomic layer graphene sample. Such a strong terahertz nonlinearity of graphene is attributed to the collective thermodynamic response of its background free-carrier population (electrons or holes) to the driving terahertz field (11, 12, 14). Increasing the density of these free carriers in graphene (or equivalently, its Fermi energy EF) enhances the power absorption of the terahertz driving field and consequently provides a control knob that allows for tuning the nonlinear response. On the other hand, too high carrier density and hence the strongly metallic phase of graphene with increased electronic heat capacity will diminish the material’s thermodynamic nonlinearity. This suggests the existence of optimal doping in graphene, favoring its terahertz nonlinearity, which can be controlled externally. Here, we demonstrate such an efficient control of graphene nonlinearity by varying its free carrier density by electrical gating.
Being only one atomic layer thick and having a gapless bandstructure, graphene is a material for which wide tuning of the Fermi energy is easily achievable using a small gating voltage of only a few volts, not only in one band but also between the valence and conduction bands across the Dirac point. This offers a simple route to control the terahertz nonlinearity of graphene. Here, we demonstrate the effects of the carrier density (or the material Fermi energy, EF) on terahertz saturable absorption (SA) and HHG, as well as the associated nonlinear carrier dynamics in gated graphene. We achieve this using nonlinear terahertz spectroscopy using single- and multicycle terahertz driving fields with a peak electric field strength reaching 80 kV/cm. Within the experimentally tested range of EF, increasing from ~50 to 200 meV relative to the Dirac point, transient changes in the terahertz field transmission of graphene in response to a terahertz driving field of 80 kV/cm are enhanced by a factor of ~70. For the same range of increasing EF, an enhancement in the power conversion efficiency in third terahertz harmonic generation by almost two orders of magnitude has been achieved.
Physical picture of gate-tunable terahertz nonlinearity
The nonlinear interaction between free carriers in graphene and the field of a driving terahertz signal resembles a thermodynamic process exhibiting an interplay of heating-cooling dynamics (11, 12, 14), as illustrated in Fig. 1 for two different terahertz pulse shapes: single-cycle (Fig. 1A) and a multicycle (Fig. 1B) pulses. The terahertz energy absorbed in graphene by the free carriers via the intraband optical conductivity mechanism is quasi-instantaneously (on a sub–50-fs time scale) converted into electronic heat, corresponding to a rise in the carrier temperature and consequently to a suppression of both the THz conductivity and power absorbance (i.e., SA) of graphene (see Fig. 1, C to F). This heating process is then followed by a slower, picosecond-timescale carrier cooling process, occurring via emission of optical and acoustic phonons, which then restores the initial high conductivity of graphene. The interplay of the carrier heating and cooling processes during the interaction of graphene with the driving terahertz signal results in a temporal modulation of both the terahertz conductivity and the instantaneous terahertz absorbance (see Fig. 1, C to F). This leads to a nonlinear temporal deformation of the driving terahertz field passing through graphene, which eventually results in a spectral broadening for the single-cycle pulse, as shown in Fig. 1G, and a highly efficient terahertz frequency multiplication, i.e., terahertz higher odd-order harmonic generation for the multicycle pulse, as shown in Fig. 1H (see also the Supplementary Materials for further information).
(A and B) Driving single-cycle and multicycle terahertz fields, respectively. (C and D) and (E and F) The associated terahertz field–induced heat and SA, respectively, for two-doping levels corresponding to Fermi energies of 80 and 180 meV. The terahertz-induced heat (or equivalently the rise in the electron temperature) leads to a reduction and a temporal modulation of both the intraband conductivity, σ(t) and the power absorption coefficient α(t), as shown in (C) and (D). This results in a nonlinear terahertz field–induced current J(t) = σ(t, ETHz)ETHz(t) in the graphene layer, yielding terahertz field–induced transparency (SA) and electromagnetic reemission at higher odd-order harmonics. Both the terahertz-induced heat and SA become more pronounced when the doping concentration (the Fermi energy, EF) increases, scaling nearly with
. (G and H) The spectral amplitudes of the terahertz fields transmitted through graphene relative to the pump field (gray background), for the tested Fermi energies of 80 and 180 meV. a.u., arbitrary units.
Because the electronic terahertz nonlinearity of graphene is facilitated by the absorption of the driving terahertz field by the free carriers and the subsequent thermodynamic process involving the electronic and phononic subsystems, it is the density of these free carriers that provides the key to controlling the terahertz nonlinearity in this material. That is, the terahertz nonlinearity relies on the presence of free carriers, and increasing the free carrier density increases the power absorption and leads to a stronger nonlinear response (scaling approximately with the square of the Fermi energy, EF), as shown in Fig. 1 (C to H). However, in the regime where the density becomes extremely high (likely above EF = 200 meV), the rise in the carrier temperature becomes modest, as the absorbed terahertz energy is to be shared among a very large number of carriers, leading to a weaker nonlinear response. Hence, there is an optimal carrier density that provides the largest nonlinearity (see the Supplementary Materials for further information).
To examine the degree of control over the nonlinearity that we can expect, we first consider two levels of carrier densities corresponding to EF of 80 and 180 meV and simulate two nonlinear experiments. We use two different terahertz waveforms, single-cycle broadband and multicycle quasi-monochromatic terahertz pulses, as shown in Fig. 1 (A and B), respectively, similar to the terahertz signals used in our experiments, with identical peak field strengths of 80 kV/cm. We see from Fig. 1 (C to H) that for both cases of terahertz driving signals, the nonlinear terahertz field–induced effects in graphene are more pronounced for the larger EF.
Another important feature shown in Fig. 1 is that for the case of multicycle quasi-monochromatic driving terahertz signal, the modulation effect in graphene is generally stronger both in amplitude and in temporal dynamics, as compared to the excitation with the single-cycle pulse, at the same value of EF and peak electric field strength in the signal. This is clearly visualized in Fig. 1 (G and H) through the spectra of the terahertz signals transmitted through the graphene layer. The reason for the stronger nonlinear modulation of terahertz properties of graphene in case of multicycle driving field is that (i) the power carried by the multicycle pulse is larger, assuming the same peak field as for the single-cycle and (ii) the accumulated electronic heat persists more than a few hundred femtoseconds, which, in turn, allows for enhancing the electronic interaction with the longer-lasting multicycle driving signal. This presents novel opportunities for the efficient nonlinear signal processing of quasi-monochromatic terahertz electronic signals.
Our calculations based on the thermodynamic model presented here use a single-band picture, in which only intraband free carrier dynamics, subjected to the conservation of carrier density within the terahertz-excited band, is considered (i.e., either electrons in the conduction band or holes in the valence band). Whereas this approach is strictly valid for graphene with a large EF, it is expected to be less accurate for the situation with EF < kBTe, where a two-band thermalization picture should instead be considered. Here kB is the Boltzmann constant and Te is the carrier temperature. However, we use the single-band approach universally for the whole range of tested Fermi energies, because it very well reproduces the entirety of our experimental observations using a minimum of adjustable parameters (namely, only the proportionality constant between the electron energy and electron scattering time). At the same time, the two-band model would require the introduction of additional adjustable parameters (see the Supplementary Materials).
Sample and nonlinear terahertz spectroscopy
On the basis of our simulations that predict a large gate tunability of the nonlinear response in both single-cycle and multicycle cases, we have conducted two types of nonlinear terahertz spectroscopy experiments on our electrically gated graphene sample: (i) table top laser–based intense terahertz pump–terahertz probe (TPTP) spectroscopy using single-cycle broadband terahertz pulses, as shown in Fig. 2A, and (ii) accelerator-based nonlinear terahertz time-domain spectroscopy (nonlinear terahertz-TDS) with multicycle quasi-monochromatic terahertz fields, as shown in Fig. 2B. The former provided the measurements of the terahertz field–induced SA and the associated carrier (temperature) dynamics, while the latter enabled the observations of terahertz HHG. The graphene sample is composed of a chemical vapor deposition (CVD)–grown single-layer graphene on a 1-mm-thick infrasil quartz substrate and is connected at two opposite edges to two electrodes acting as source and drain. The gating is enabled by another pair of electrodes and a polymer electrolyte [LiClO4:polyethylene oxide (PEO)] deposited on the graphene layer and the gate electrodes, as shown in Fig. 2C. The Fermi energy of graphene is then varied by a small gate voltage in the range between −2 to +2 V (see Materials and Methods for further information about the experiments and the sample preparation).
(A) TPTP experiment with gated graphene. The electric field of the terahertz pump is vertically polarized and has a peak field strength up to 80 kV/cm, while the electric field of the terahertz probe is horizontally polarized with a peak field strength of less than 1 kV/cm. (B) Nonlinear terahertz-TDS, using multicycle quasi-monochromatic terahertz pulses oscillating at a fundamental central frequency of 0.3 THz and having a peak electric field up to 80 kV/cm. The transmitted field through the graphene sample consists of higher odd-order harmonics in addition to the fundamental frequency. Note that EF refers to the Fermi energy of the graphene sample, while Ef refers to the peak electric field of the driving THz signal oscillating at the fundamental frequency f. (C) The gated graphene sample device in which the graphene film acts as a channel between source and drain electrodes subjected to a constant potential difference of 0.2 mV. The graphene film is covered on top by an electrolyte subjected to a varying gating voltage to tune the Fermi level of the graphene layer. (D) Experimentally determined gating response of the graphene sample given by the sheet resistance of the graphene film as a function of the gating voltage relative to the voltage V0 corresponding to the minimum Fermi energy in the vicinity of the Dirac point, exhibiting a maximum resistance. Positive V − V0 leaves electrons in the graphene layer with the Fermi level elevated in the conduction band, while hole doping with the Fermi level pinning the valence band is induced by negative V − V0.
Figure 2D shows the gating response of the graphene sample, indicated by the graphene sheet electrical resistance, retrieved by measuring the source-drain current through the graphene layer at a drain voltage of 10 mV, as a function of the gating voltage. The initial doping of the graphene layer (doping at zero gate voltage) was p-type, as usually provided unintentionally by chemical residues during the preparation process and due to interaction with the substrate (12, 20, 21), and the deposited polymer electrolyte. Here, a nonzero gate voltage V0 has compensated for this hole doping and shifted the Fermi energy to a minimum near the Dirac point, which corresponds to the peak resistance of the sample in Fig. 2D. We note here that EF is considered relative to the Dirac point and is never uniformly zero at ∣V − V0∣ = 0 due to inhomogeneity in the doping and gating field across the graphene layer, as well as a finite thermal broadening of the electron distribution, given kBT = 26 meV at room temperature T = 300 K, resulting in the finite peak resistance at ∣V − V0∣ = 0. Increasing the absolute gating voltage away from V0 leads to an increase in the absolute value of EF, whether for electrons or holes, as shown in Fig. 2D. Quantitatively, EF is related to the free carrier density, Nc, through
, where ℏ is the reduced Planck’s constant and vF = 1 × 106 m/s is the Fermi velocity in graphene.
RESULTS AND DISCUSSION
Figure 3 shows both the as-measured and processed results of the single-cycle TPTP experiment, demonstrating the efficient control of the nonlinear terahertz transmission of graphene using a small gating voltage of 2 V at maximum. Here, we refer to ∣V − V0∣ simply as the gating voltage. The graphene sample was excited by terahertz pump pulses with peak electric field strength ranging from 1 to 80 kV/cm. The effects induced by the terahertz pump were probed by a weak terahertz probe pulse, at various values of EF controlled by the gating voltage. Figure 3 (A and B) shows the terahertz probe pulses transmitted through the graphene sample with and without the intense terahertz pump at 80 kV/cm for two different gating voltages. After the pump, the probe transmission increases, indicating terahertz absorption bleaching (or equivalently SA) in graphene, which corresponds to a reduction of the graphene terahertz conductivity. This effect becomes more pronounced with increasing the gating voltage (the increase in transmission of the “red” pulse relative to the “black” pulse is more pronounced in Fig. 3A compared to the situation in Fig. 3B). This is further clarified by the difference pulses shown in blue (∆E = Ered − Eblack, multiplied by four for clarity) that show a larger pump-induced change in transmission for the case of the larger gating voltage.
(A and B) The terahertz probe fields before (black) and after (red) intense terahertz pump excitation, normalized to the peak field after excitation with pump peak electric field (80 kV/cm), and their difference (blue) multiplied by four for clarity, for gating voltages of 1 and 0.3 V, respectively. (C) The as-measured peak field transmission of the probe pulses at various gating voltages when the graphene is pumped at 80 kV/cm. T0 is the background probe transmission of the graphene sample before the pump, which increases by decreasing the doping (gating) level. (D) The terahertz probe differential transmission ΔT/T0 = (T − T0)/T0, where T is the probe transmission after the pump, as a function of the pump-probe delay time, obtained by analysis of the experimental data of (C). The dots represent the experimental results, while the solid lines are guides for the eye. (E) The peak of the probe differential signal ΔT/T0 as a function of the terahertz pump field at various doping levels and (F) the peak of ΔT/T0, extracted from (E), as a function of the gating voltage at some selected pump peak electric fields. The pink arrows indicate that the data in (D) and (F) are obtained by analysis of data from (C) and (E), respectively, while the violet arrows in (E) and (F) indicate increase in ΔT/T0 with the gate voltage and the driving field, respectively.
Figure 3C shows the as-measured transmission of the probe peak field as a function of the pump-probe delay time for various gating voltages. We note that the background probe transmission T0 before the pump decreases with the gating voltage due to increasing absorption when the carrier density increases by gating. With the intense terahertz pump, a transient increase in the transmission occurs. This nonlinearity becomes much more pronounced at a larger gating voltage. The secondary smaller features observed at ~16.5 ps in Fig. 3C are due to internal round-trip etalon reflection within the quartz substrate. Figure 3D shows the analyzed terahertz differential transmission ΔT/T0 of the terahertz probe obtained from the data of Fig. 3C, revealing that the nonlinear terahertz field transmission of graphene is modulated from a minimum of less than 0.1% to ~7%, i.e., by 70 times, when a gating voltage of only 2 V is applied. This demonstrates an efficient, straightforward control of the graphene nonlinearity with a very modest gating voltage.
We further observe from Fig. 3D that the transient increase in transmission reaches a peak after ~2 ps and is then followed by an exponential decay over a few picoseconds. The dots represent the experimental data from Fig. 3C, while the solid lines guide the eye (see the Supplementary Materials for further information). This temporal evolution of ΔT/T0 in Fig. 3D is attributed to the nonlinear thermodynamic modulation of the graphene properties illustrated in Fig. 1 and its related discussion above. It is also consistent with observations reported in numerous studies where the electron temperature dynamics were ascribed to heating-cooling effects using various techniques, including time-resolved angle-resolved photoemission spectroscopy (22), TPTP (9), optical pump–terahertz probe spectroscopy (23–25), and time-resolved photocurrent microscopy (26, 27). We note here that the time resolution of our measurement, which is of the order of 2 ps, is limited by the noncollinear pump-probe geometry used in the experiment, as shown in Fig. 2A. The dependence of the peak values of ΔT/T0 on both the terahertz pump electric field and the gating voltage is shown in Fig. 3 (E and F). The as-measured data of Fig. 3E were obtained, for each gating voltage, via systematic variation of the terahertz peak field of the pump pulses using a pair of wire-grid polarizers. We see that ΔT/T0, i.e., the nonlinear response of graphene, increases with both the terahertz pump field and the gating voltage. We further see the saturation of ΔT/T0 setting in for gating voltages exceeding 0.7 V, which mimics the behavior of the graphene sheet resistance shown in Fig. 2D. We also note here that the data shown in Fig. 3 were collected for the negative branch of the gating voltage. On the positive branch of the gating voltage, the sample maintains a very similar (but not perfectly the same) behavior.
Terahertz high harmonics
In Fig. 4, we show the results of the nonlinear multicycle terahertz-TDS experiment using the same gated graphene sample. In this experiment, the sample was excited by a multicycle terahertz field with a central frequency of 0.3 THz, as shown in the inset of Fig. 4A. The peak electric field of the driving pulse was as high as 80 kV/cm, i.e., the same as the pump peak field in the single-cycle TPTP experiment. The response of the graphene sample was revealed through transmission measurements using calibrated free-space electro-optic sampling detection of the transmitted terahertz field. In Fig. 4A, we show the frequency-domain spectra of the terahertz pulses transmitted through the graphene sample at various gating voltages. The spectrum includes odd-order higher harmonics of the fundamental frequency f1 = 0.3 THz up to the seventh order, namely centered at the higher-harmonic frequencies f3 = 0.9 THz, f5 = 1.5 THz, and f7 = 2.1 THz. We note here that a conservative estimate of the corresponding nonlinear electronic susceptibilities for HHG from graphene in the perturbative regime (namely at lower driving terahertz fields below 30 kV/cm) are as high as χ(3)~10−9 m2/V2, χ(5)~10−22 m4/V4, and χ(7)~10−38 m6/V6, for the third-, fifth-, and seventh-order harmonics, respectively (14). These values are several orders of magnitude larger than the analogous coefficients of other known materials investigated in many different frequency ranges (12, 14). As we see from Fig. 4A, the amplitudes of the generated harmonics, normalized to the peak at the fundamental frequency, are strongly enhanced by increasing the gating voltage from 0.14 to 0.56 V.
(A) The terahertz amplitude spectra of the terahertz fields of the incident driving field and the transmitted fields through the graphene sample exhibiting generation of higher odd-order harmonics up to the seventh order for two doping levels, with the driving terahertz signal shown in the inset, and (B) the peak electric field of the generated harmonics as a function of the gating voltage.
Figure 4B shows the dependence of the actual peak electric fields of the generated harmonics on the gating voltage, for a terahertz driving field of 80 kV/cm at the peak. Here, increasing the gating voltage from 0.14 to 0.56 V leads to a magnification of the generated harmonic fields by factors of 9 for the third harmonic, 4 for the fifth harmonic, and 3 for the seventh harmonic, which correspond to enhancement factors of 81, 16, and 9 for the power conversion efficiencies of these harmonics, respectively. These results display a wide range of tunability of the graphene nonlinearity and can pave ways for practical graphene-based ultrahigh-frequency electronics.
Modeling the experimental results
We now reproduce the experimental results through simulations based on the thermodynamic model discussed above [see the Supplementary Materials and (12, 14) for further information]. In our calculations, we simulate the nonlinear propagation of the driving terahertz signals, in the time domain, through the graphene layer at various Fermi energies corresponding to the experimentally tested gate voltage. In Fig. 5 (A and B), we show the model calculations (solid lines) along with the experimental results (symbols) of Figs. 3F and 4B, respectively, after replacing the gating voltage ∣V − V0∣ by the corresponding EF shown in Fig. 5C. We see a good agreement between the experiment and the model calculations for the assumed range of EF up to 200 meV, in which the graphene nonlinearity (both differential transmission and HHG) is enhanced substantially with increasing EF. Above this doping level and up to EF~260 meV, we see from Fig. 5A that the experimental data show saturation in the nonlinearity, which is also captured by the model calculations. We attribute this saturation behavior to the fact that the electronic heat capacity of graphene increases with EF at high doping levels (28–30), restricting the increase in the carrier temperature with further increase in terahertz field excitation, which leads to a smaller increase in the graphene nonlinearity at these high Fermi energies (see the Supplementary Materials for further information). These effects are automatically included in the thermodynamic model by respecting the conservation of energy and total carrier density (11, 12, 30).
(A and B) The thermodynamic model calculations (solid lines) and the experimental results (solid circles) of the peak of ΔT/T0 and the electric field of the generated harmonics up to the seventh order, respectively, as functions of the EF and various terahertz pump fields, (C) EF as a function of the gating voltage, and (D) the carrier momentum scattering time as a function of EF.
In our model calculations, we considered a range of EF from 50 to 260 meV (as a variable parameter) to be equivalent to the values corresponding to the experimentally applied gating voltage ∣V − V0∣ in the range from 0 to 2 V, as shown in Fig. 5C. As previously noted, EF is never uniformly zero at ∣V − V0∣ = 0. Therefore, we assigned a value of EF= 50 meV to the gating voltage ∣V − V0∣ = 0, which is consistent with the puddle energy estimates reported in previous works on similar graphene samples (25, 31, 32). The initial linear increase in EF with the gating voltage is consistent with Hall measurements (see the Supplementary Materials). Another free parameter used in our calculations is the momentum scattering time τ0 associated with the linear response of graphene to low terahertz fields and is considered to be a function of EF (24, 25, 33), as shown in Fig. 5D. Here, we considered the linear terahertz peak transmission T0 indicated in Fig. 3C, which is related to the “frequency-integrated conductivity” of the graphene layer, to estimate τ0 and EF as carefully chosen pairs of values that reproduce the nonlinear response to the intense driving signals as well. We note that choosing other pairs of τ0 and EF in our calculations resulted in a significant deviation between the model and the experimental data. With the obtained range of EF and τ0, we extract a carrier mobility of ~3000 cm2/Vs, which is consistent with values reported for typical CVD graphene (34, 35). Moreover, the increase in τ0 with EF indicates that long-range Coulomb scattering dominates the carrier scattering process (24, 25, 33).