High-field terahertz response of graphene

We investigate the response of multi-layer epitaxial graphene and chemical vapor deposition (CVD)-grown single-layer graphene to strong terahertz (THz) fields. Contrary to theoretical predictions of strong nonlinear response, the transmitted fields exhibit no harmonic generation, indicating that the nonlinear response is limited by fast electron thermalization due to carrier–carrier scattering. The fast electron heating gives rise to large THz transmission enhancement (> 15%) in single-layer CVD graphene at high THz fields (ETHz > 10 kV cm−1). The nonlinear effects exhibit non-Drude behavior in the THz conductivity, where THz fields induce extreme non-equilibrium electron distributions.


Introduction
Graphene has attracted a great deal of attention from the scientific and industrial community since the initial demonstrations of isolated layers [1][2][3][4]. Among various applications, graphene is considered a promising electronic material for high speed electronic devices because of its high carrier mobility at room temperature and its high Fermi velocity (∼1/300 of the speed of light). Experimentally, high speed graphene devices such as 300 GHz transistors and photodetectors have been demonstrated [5][6][7][8]. As the operating frequency of graphene devices goes beyond 100 GHz, it becomes crucial to understand how this material behaves in the terahertz (THz) regime. In addition, due to many unique optical properties, such as strong THz absorption of one atomic layer [9] and plasmon resonances of patterned structures [10], graphene may play an important role in the so called THz gap (0.1-10 THz), where there is a need for improved sources and detectors. In a weak optical field where graphene behaves linearly, the optical properties have been studied extensively over a broad spectral range from visible to THz [11][12][13]. A universal optical conductivity of πe 2 /(2h) due to interband transitions is predicted and observed for photon energies less than ∼1 eV and greater than twice the Fermi level E F (Fermi level measured relative to the charge neutrality point) [11]. The optical conductivity of graphene demonstrates a Drude-like frequency dependence in THz frequency range [13][14][15][16].
Recent theoretical work on the high-field response of graphene indicates that the unique electronic properties of graphene lead to remarkably strong optical nonlinearities in the THz and IR regime [17][18][19][20][21][22][23][24]. The predicted strong nonlinear responses of graphene make it an attractive material for active photonic devices. Furthermore, as the internal electric field inside electronic devices could possibly reach 100 kV cm −1 , it is desirable to determine the maximum field at which graphene still behaves linearly. Transport in a strong field has also been studied theoretically, and nonlinear relationships between current and electric field are predicted [17][18][19][20]. Harmonic generation in graphene under a strong THz field has been predicted by several groups using different theoretical approaches [21][22][23][24]. For example, Mikhailov and Ziegler [21] analyzed the intraband contribution via the quasi-classical kinetic Boltzmann theory. In the limit of non-interaction quasi-particles, an electron gains a momentum p(t) = − eE 0 ω 0 sin ω 0 t in an oscillating electric field E(t) = E 0 cos ω 0 t. The electron moves in the same direction with p(t), yet its speed is constant, i.e. v(t) = −v F sgn(sin ω 0 t). Accordingly, the sheet current density is expressed as J (t) = −en 2d v(t) = en 2d v F sgn(sin ω 0 t), where n 2d is the sheet carrier density (see figure 1). This means that a sinusoidal excitation can produce a square wave-like current, and therefore odd harmonics are generated efficiently [21]. Wright et al [22]  calculated the three-photon process in intrinsic graphene from the Dirac equation, and predicted a strong frequency-tripled current. Ishikawa [23] analyzed the THz harmonic generation from the Dirac equation by using a time domain approach. The threshold electric field to observe the nonlinear harmonic generation predicted by above works ranges approximately from 1 to 10 kV cm −1 [21][22][23][24], which is easily achievable in realistic devices, and in free-space THz spectroscopy.
It is important to note that all scattering mechanisms in graphene are neglected in the theoretical works predicting harmonic generation [21][22][23][24]. In realistic graphene systems, nonlinear effects may be obscured by various scattering mechanisms. First of all, manybody Coulomb interactions in graphene are strong [25,26]. Carrier dynamics in graphene are also susceptible to defects, substrate interface interactions, and its own three-dimensional ripples [27]. Rigorous experimental studies are desirable to investigate how the intrinsic and extrinsic scatterings affect the nonlinear THz responses of graphene. Currently there are only few experiments on the high THz-field response of graphene [9,28,29]. Experimental observation of THz harmonic generation in graphene has not yet been reported. Dragoman et al [30] observed harmonic generation from graphene in the millimeter wave range, but the generation efficiency was extremely low.
In this paper, we present two experimental investigations of the high THz field response of graphene. In the first set of experiments, we searched for nonlinear harmonic generation in n-type multi-layer epitaxial graphene excited by broadband THz pulses whose peak field amplitude reaches 40 kV cm −1 ; no nonlinear response was observed, contrary to a number of theoretical predictions in the literature, indicating the role of strong electron scattering in high fields. A second series of experiments probed the response of p-type single-layer chemical vapor deposition (CVD) graphene in the presence of strong THz fields up to 70 kV cm −1 . These experiments showed that THz-induced transparency becomes observable when THz fields exceed a moderate amplitude of 10 kV cm −1 . The nonlinear absorption in doped layers can also be seen as a consequence of increased electron scattering rates at high carrier energy.

Interaction of strong terahertz (THz) pulses with multi-layer epitaxial graphene
Our measurements of multi-layer epitaxial graphene were designed to search for THz harmonic generation in graphene. Strong single-cycle THz pulses were produced by two-color ionization of air [31][32][33][34]. The resulting THz radiation was focused onto a graphene sample, and the spectrum of the transmitted THz radiation is obtained by a Michelson interferometer. The optical setup is shown in figure 2. A 500 Hz regeneratively amplified Ti-sapphire laser system (lambda cubed laser system at University of Michigan) produces 30 fs, 3 mJ pulses [35]. The beam is focused by a lens, and a BBO crystal is placed between the lens and the focal point to generate second harmonic at 400 nm wavelength. The fundamental and second-harmonic pulses add coherently to produce a symmetry-broken field at the focus [31][32][33][34], which ionizes the air at the focal point. Because of the broken symmetry, the air plasma at the focal point contains a transient directional current [33,34], which is the origin of THz radiation. THz radiation generated at the focal point is collimated and subsequently re-focused on the graphene sample by parabolic mirrors. A silicon low-pass filter blocks the unwanted optical light and a 3 THz low-pass filter (C103, IRLabs) further restricts the spectral range to below 3 THz. An epitaxial graphene sample is placed at the focal point of the second parabolic mirror. The sample has approximately ten graphene layers grown on C-face 4H-SiC substrate. The first few layers close to the SiC substrate are highly n-doped, while other layers are nearly intrinsic [36,37]. The transmitted THz radiation is analyzed by a Michelson interferometer, in which the THz energy is measured by a bolometer (HD-3, IRLabs). By scanning the delay line of the Michelson interferometer, interferograms are acquired, and their Fourier transform gives the power spectra of the THz radiation [38]. The field strength at the focal point is 40 kV cm −1 , calculated from the measured pulse energy, duration and spot size. Figure 3(a) shows the interferogram of the transmitted THz radiation, along with a reference interferogram acquired when the sample is not present. The transmission through this ten-layer epitaxial graphene sample is ∼50%, and the two interferograms are identical within the noise level when they are normalized, as shown in figure 3(b). This indicates that the graphene sample responds linearly to the incident THz field with a nearly flat spectral response. Figure 4 shows the transmission and reference spectra, calculated from the Fourier transform of the interferograms in figure 3. Most of the power of the incident THz radiation is below 3 THz  because of the presence of the 3 THz low-pass filter. However, the transmission spectrum shows no increase of power above 3 THz. In the presence of noise and nonzero transmission through the low-pass filter above 3 THz, the minimal detectable third harmonic generation efficiency is ∼2%. The fact that the transmitted power is smaller than excitation power at all frequencies indicates that no nonlinear third harmonic generation larger than ∼2% is observed. We also vary the field strength by moving the graphene sample away from the focal point. Figure 5 shows that the normalized interferograms acquired from different field strengths are nearly identical, indicating lack of nonlinear harmonic generation.
These measurements show no observable THz harmonic generation in epitaxial graphene, even though the incident field greatly exceeds the predicted threshold field strength. The most likely explanation for the lack of nonlinear harmonic generation is the existence of fast carrier-carrier scattering when graphene is excited by a strong field [28]. Scattering mechanisms are neglected in the theoretical studies that predict harmonic generation [21][22][23][24], the justification being that the low-field mobility of electrons in graphene is very high [21]. However, in a strong field, electrons gain kinetic energy from the electric field, with an average energy much higher than the lattice thermal energy. Recent experiments and calculations [28,39,40] suggest that hot electrons in graphene undergo extremely fast carrier-carrier scattering, which becomes the dominant thermalization mechanism. Furthermore, unlike materials with parabolic bands, carrier-carrier scattering in graphene does not conserve current [39,41]; in fact electron-electron scattering directly leads to damping of the current. When the electrons are energetically excited by the THz field, carrier-carrier scattering can occur in a time scale of ∼10 fs [28], much shorter than the THz pulse duration. Therefore, within the single-cycle THz pulse excitation, carrier-carrier scattering efficiently thermalizes the electrons and damps the current [28]. As a result the fast thermalization leads to a harmonic response of the entire electron gas to the applied THz field, suppressing any nonlinear harmonic generation [21]. Indeed, any theory of nonlinear response would have to include self-consistently the collective response of the electron gas to the applied field including current relaxation due to carrier scattering.

Nonlinear THz transmission spectroscopy of chemical vapor deposition-grown single-layer graphene
Unlike epitaxial multi-layer graphene, THz-induced nonlinear effects do become observable in CVD-grown single-layer graphene sheets when THz fields exceed a moderate amplitude, 10 kV cm −1 . We grew large-area, single-layer CVD graphene on Cu-foil via standard methods. Micro-Raman spectroscopy showed typical signatures of single-layer graphene [42]. The graphene layer had a p-type carrier concentration of ∼10 12 cm −2 . The graphene sheet was subsequently covered with a thin poly(methyl methacrylate) (PMMA) layer (thickness ∼100 nm) and transferred to a sample holder that allows the graphene-on-PMMA film to be suspended freely over 2 mm diameter holes. The free-standing graphene-PMMA thin film sample structure reduces parasitic substrate effects in two ways: (i) THz absorption and interference caused by the substrate are negligible and (ii) the single graphene-PMMA interface induces only small changes in the THz properties of graphene [16]. The PMMA is an uncharged polymer and contains much fewer charge traps than substrates like SiO 2 .
We carried out THz transmission spectroscopy to investigate the frequency-dependent nonlinear response in the graphene samples. Strong, broadband THz pulses (central frequency, 0.9 THz; bandwidth, 0.6 THz) were generated by optical rectification of femtosecond laser pulses (pulse energy, 1 mJ; pulse duration, 90 fs; repetition rate, 1 kHz) with tilted pulse fronts in a LiNbO 3 crystal (THz field amplitude reaches 120 kV cm −1 at an optical pulse energy of 0.6 mJ) [43][44][45]. We detected the transmitted THz pulses using a liquid-He cooled Si:Bolometer to acquire (i) spectrally integrated total THz transmitted power and (ii) transmission spectra via Michelson interferometry as shown in figure 6. Figure 7 shows power dependent measurements of spectrally integrated THz transmission through a free-standing graphene-PMMA layer. The large linear THz absorption by graphene (∼30%) at low intensities indicates that intraband transitions dominate the interactions of THz waves with graphene. From this, we obtained the sheet conductivity of the graphene sample (σ S ≈ 1.0 × 10 −3 −1 ) using the thin-film transmission coefficient based on the Drude model, where the graphene-PMMA layer was treated as an infinitely thin film [15,16]. Z 0 (376.7 ) is the vacuum impedance. This simple picture, however, does not work at high THz intensities. The relative transmission through the graphene-PMMA film gradually increases for E THz > 10 kV cm −1 , and the transmission enhancement ( T /T ) reaches ∼15% at 70 kV cm −1 . We speculate that strong THz pulses give rise to substantial changes in the electronic structure of graphene and the electron distribution via intraband transitions (inset of figure 7), and hence modulate the optical properties of the material. The frequency-dependent THz transmission measurements shown in figure 8 provide more insights into the nonlinear THz effects. Figure 8(a) shows the interferograms of transmitted THz pulses when the peak THz fields are 5, 28, 47 and 59 kV cm −1 . Figure 8(b) shows the Fourier spectra obtained from the interferogram at 59 kV cm −1 . The frequency components above 2 THz are less than 0.5% compared to the peak at 0.9 THz. The result indicates that no harmonic generation larger than 0.5% is observed. The spectrum below 2 THz, however, shows small, yet noticeable changes as shown in the normalized transmission spectrum (figure 8(c)): we observed a slight blueshift relative to the reference spectrum (δν = 0.07 THz) while there is little change in the bandwidth (0.64 THz). The THz-induced spectral modulations correspond to the frequency-dependent transmission shown in figure 8(d), which were acquired by normalizing the Fourier spectra of the interferograms with the reference spectrum of no sample. The solid lines indicate linear fit for the spectra. The transmission enhancement at high THz fields is stronger at higher frequencies, while the flat transmission spectrum at the low field amplitude of 5 kV cm −1 is as expected from a Drude model at low THz intensities over this spectral range. The frequency-dependent transmission indicates that non-Drude behavior in the THz conductivity emerges at strong THz fields.
We performed an analysis of the THz-induced transparency using a simple phenomenological model based on energy-integrating saturable absorption: where T 0 is the linear transmission, U sat is the saturation energy, I 0 (t) is the input pulse intensity at time t and U 1 (t) = t −∞ I 0 (t) dt. The transient nature of saturable absorption gives rise to pulse shortening, which leads to a slight blueshift and spectral broadening. The results are, however, not quantitatively consistent with the experimental data. A full theoretical model including microscopic processes is required to fully understand the THz-induced transparency. Here we make a rough estimation how the strong THz fields induce nonlinear carrier dynamics in graphene. Strong THz pulses drive the electrons in graphene into a non-equilibrium state (inset of figure 8(a)), which transiently modulates the THz properties. With no THz-field present, the Fermi wavenumber k F is roughly 10 6 cm −1 at a typical carrier density of the CVD graphene samples (n ≈ 10 12 cm −2 ). When the THz-driven electron momentum ( p = h k) exceeds the Fermi momentum ( p F =hk F ), the electron transport will enter an extreme non-equilibrium regime. The gain of the electron momentumh k is approximately eE THz τ , where τ is the scattering time. The far-from-equilibrium criterion, k > k F , is achieved in our experimental conditions of E THz > 10 kV cm −1 with a typical scattering time τ ≈ 10 −13 s. The carrier acceleration and subsequent carrier-carrier scattering raise the electron temperature on a subpicosecond time scale. The temperature increase shortens the scattering time, which reduces conductivity, leading to the rise in transmission.

Conclusion
Strong-field THz spectroscopy has been applied to investigate the nonlinear carrier dynamics in graphene. Our preliminary studies show no harmonic generation in multi-layer epitaxial graphene or in p-type CVD-grown single-layer graphene, implying fast electron thermalization at subpicoseond timescales due to carrier-carrier scattering. The epitaxial graphene samples exhibit only linear THz responses up to amplitudes of 40 kV cm −1 . Strong THz fields above 10 kV cm −1 , however, enhance transparency of the CVD-grown graphene by heating the electrons and reducing the THz conductivity. We anticipate that these experiments will motivate the development of full theoretical models of carrier acceleration including strong fields and self-consistently accounting for carrier scattering.