Electrostatic Control of Terahertz High‐Harmonic Generation in Turbostratically Stacked Graphene

ABSTRACT

Graphene exhibits strong nonlinear optical responses across a broad spectral range, including the terahertz (THz) regime. Although THz high‐harmonic generation in graphene has been demonstrated and can be enhanced by electrostatic doping and multilayer stacking, how interlayer carrier redistribution determines the overall nonlinear response remains unclear. Here, we show that the THz nonlinearity of turbostratically stacked multilayer graphene is governed by electrostatically induced interlayer Fermi‐level redistribution. We experimentally observe gate‐dependent modulation of third‐ and fifth‐harmonic generation in top‐gated monolayer, bilayer, and trilayer graphene devices. To account for these results, we combine a thermodynamic model of graphene THz nonlinearity with a nonlinear screening model for interlayer charge redistribution under electrostatic doping. The model quantitatively reproduces the measured layer‐ and gate‐dependent harmonic responses, showing that the total THz nonlinear response of turbostratically stacked multilayer graphene is determined not only by the total carrier density but also by its distribution across the layers. These findings identify control over the layer‐resolved carrier distribution as a key design principle for optimizing nonlinear performance.

This study investigates electrostatic control of terahertz high‐harmonic generation in top‐gated turbostratic multilayer graphene. A quantitative model combining graphene's thermodynamic nonlinear terahertz response with multilayer bias screening reproduces the experimental yields. Results demonstrate that Fermi‐level redistribution across layers strongly affects the overall harmonic yield, highlighting the importance of precise layer‐resolved Fermi‐level control to optimize the nonlinear response and device performance.

1. Introduction

With recent advancements in high‐intensity terahertz (THz) sources, THz nonlinear photonics has emerged as a scientifically and technologically significant frontier with broad applications in imaging, material characterization, and high‐speed communication [1, 2, 3, 4, 5]. Among various candidate materials, graphene has emerged as a particularly compelling platform owing to its massless Dirac fermion dynamics and its atomic‐scale thickness, which enables seamless integration with silicon‐based electronics. Graphene exhibits remarkably efficient THz high‐harmonic generation, with odd‐order harmonics dominating because of its inversion symmetry [6, 7, 8, 9, 10, 11]. The intraband THz nonlinearity of graphene can be described by a thermodynamic model in which ultrafast oscillations of the carrier temperature induce a time‐dependent modulation of the transmitted field [6, 8, 12].

Within this thermodynamic picture, the incident THz field is absorbed by graphene's free carriers, which rapidly thermalize through carrier‐carrier scattering to establish a quasi‐equilibrium carrier temperature on a sub‐50‐fs timescale. The hot‐carrier distribution then undergoes an initial rapid cooling within ∼100 fs, predominantly via optical phonon emission [13, 14]. Since both processes occur within a single THz oscillation cycle, the carrier temperature can follow the sub‐cycle evolution of the absorbed THz power. Under the doped, low‐photon‐energy conditions relevant to this work, interband transitions are essentially Pauli‐blocked, and the nonlinear response is therefore dominated by intraband dynamics. With the carrier density fixed, the time‐dependent carrier temperature modulates the intraband conductivity and hence the instantaneous transmission: as the carrier temperature increases, the Drude‐like conductivity decreases and the transmission increases, and vice versa. The resulting sub‐cycle modulation of the transmitted THz field leads to high‐harmonic generation. This thermodynamic model has yielded accurate quantitative predictions of the THz nonlinear response of monolayer graphene across a broad range of Fermi levels [9].

Despite such progress, this thermodynamic framework has not yet been extended to multilayer graphene for the quantitative prediction of THz high‐harmonic generation. While a previous study experimentally demonstrated that electronically decoupled graphene stacks exhibit enhanced THz nonlinearity and preserve electrostatic tunability [15], a quantitative theoretical prediction of their gate‐dependent harmonic response was not provided. Extending the thermodynamic model to a gated multilayer graphene stack remains nontrivial even when the layers are electronically decoupled. Nonlinear electrostatic screening establishes a carrier‐density profile across the stack that strongly depends on both gate bias and layer count, and the harmonic response of each layer depends nonlinearly on its local carrier density [9]. Consequently, the overall nonlinear response cannot be predicted from the total gate‐induced charge alone.

Turbostratically stacked graphene provides a particularly suitable model system for investigating how the layer‐resolved carrier distribution governs the overall THz harmonic response of multilayer graphene. In Bernal‐stacked multilayers, coherent interlayer hopping hybridizes the electronic states of the individual layers, giving rise to collective multilayer bands. AB‐stacked bilayer graphene exhibits approximately parabolic low‐energy bands, whereas ABA‐stacked trilayer graphene contains monolayer‐like and bilayer‐like sectors that can be further hybridized by a perpendicular displacement field [16, 17, 18]. Consequently, their density of states, electronic heat capacity, and intraband conductivity differ from the corresponding monolayer quantities. Modeling their nonlinear THz response would therefore require a coupled multiband treatment, making Bernal‐stacked multilayers less suitable for isolating the role of the layer‐resolved carrier distribution in determining the overall harmonic response.

By contrast, turbostratic stacking introduces rotational misalignment between adjacent graphene layers, which strongly suppresses coherent interlayer electronic hybridization and preserves an approximately monolayer‐like electronic dispersion in each layer [19, 20, 21, 22, 23, 24]. This electronic decoupling allows each layer to be treated as a distinct THz nonlinear subsystem characterized by its own carrier temperature. At the same time, electrostatic coupling between the layers is retained, such that the gate‐induced charge is redistributed across the stack, resulting in layer‐dependent carrier densities. Accordingly, a theoretical framework that captures gate‐induced, layer‐resolved carrier redistribution is needed to quantitatively describe and optimize the nonlinear response of such multilayer systems.

Here, we investigate third‐ and fifth‐harmonic generation in monolayer graphene (1LG) and in turbostratically stacked bilayer and trilayer graphene devices (2LG and 3LG, respectively) over a wide range of electrostatic doping levels. We find that gate‐induced doping strongly modulates high‐harmonic generation, leading to either enhancement or suppression relative to the zero‐bias condition. Although turbostratic stacking enhances the overall harmonic yield, the enhancement already begins to saturate at the trilayer level, whereas the gate‐induced modulation depth decreases with increasing layer number. To account for these results quantitatively, we combine the thermodynamic model of graphene's THz nonlinearity with a nonlinear screening model that captures layer‐resolved carrier redistribution under electrostatic doping. Our model quantitatively reproduces the measured layer‐ and gate‐dependent behaviors, showing that the overall THz nonlinear response is governed by layer‐resolved carrier redistribution rather than by total doping alone.

2. Results and Discussion

2.1. Layer‐Resolved Carrier Distribution Under Asymmetric Electrostatic Doping

Figure 1 shows a schematic of the graphene device investigated in this work. Monolayer graphene grown by chemical vapor deposition (CVD) is turbostratically stacked on a polyimide (PI) substrate with a thickness of 2 µm. An ion‐gel electrolyte layer is subsequently transferred onto the top surface to serve as a gate dielectric. In addition to the dynamically tunable gate‐induced doping, the charge transfer from the substrate introduces a static Fermi‐level offset [25, 26, 27, 28]. Thus, under the single‐sided gating scheme, the multilayer system is subject to an asymmetric electrostatic doping configuration, consisting of dynamically tunable top‐gate doping and static substrate‐induced doping from the bottom interface. The layer‐resolved carrier‐density distribution is then governed by nonlinear electrostatic screening [25, 29, 30, 31, 32, 33, 34]. The equilibrium distribution results from competition between electrostatic energy, which favors charge localization near the charge‐inducing interfaces, and band‐filling energy, which favors charge spreading across the layers; because the latter depends nonlinearly on carrier density through graphene's Dirac density of states, the resulting charge profile varies nonlinearly with gate bias.

FIGURE 1.

Schematic illustration of THz high‐harmonic generation in turbostratic multilayer graphene. A narrowband, multicycle THz wave at a fundamental frequency of f = 0.4 THz is incident on the graphene. The top panel depicts the layer‐resolved doping profile: graphene layers farther from the gate electrode are screened from gate‐induced electrostatic doping by intermediate layers, while substrate‐induced doping is screened in the opposite direction along the stacking axis. In the thermodynamic picture, each electronically decoupled graphene layer generates odd‐order harmonics of the fundamental THz field. These harmonic fields add coherently, resulting in an enhanced overall THz nonlinear response of the multilayer structure.

The carrier density in each layer affects the overall harmonic response of the multilayer stack through two distinct mechanisms. First, for a given local pump field, the absorbed THz power, electronic heat capacity, and temperature dependence of the intraband conductivity all depend on the carrier density. These carrier‐density‐dependent quantities collectively govern the ultrafast sub‐cycle modulation of the carrier temperature and conductivity, and thus the harmonic field generated by each layer. Second, the carrier density of each layer influences the electromagnetic weighting of the layer‐specific harmonic contributions within the stack. In a simplified single‐pass sequential‐transmission picture, an upstream layer (closer to the incident side) attenuates the pump field reaching downstream layers, whereas a downstream layer attenuates the harmonic fields generated by upstream layers. When internal reflections are taken into account, the conductivities of all layers jointly determine the local pump field at each layer and the propagation and outcoupling of the harmonic fields generated throughout the stack. This electromagnetic weighting is incorporated into the numerical calculations using a transfer‐matrix formalism, as detailed in the Supporting Information.

2.2. Harmonic Generation Efficiency and Carrier Density Mapping

Figure 2 shows representative frequency‐domain spectra and time‐domain waveforms of the incident THz pump pulse with a central frequency of 0.4 THz, a peak electric‐field amplitude of 75 kV cm⁻¹, and a pulse duration of 14.7 ps, together with the third‐ and fifth‐harmonic signals generated by the 1LG sample. Clear spectral peaks are observed at 1.2 and 2.0 THz, corresponding to the third (3f) and fifth (5f) harmonics, respectively. To generate the narrowband fundamental excitation, a metasurface‐based band‐pass filter (BPF) with cross‐shaped apertures was placed before the graphene sample. During the harmonic measurements, an additional BPF was placed after the graphene to suppress the intense transmitted fundamental wave and isolate the desired harmonic signal. A schematic of the experimental setup is shown in Figure S1.

FIGURE 2.

(a) Frequency spectra and (b) time‐domain signals of the fundamental THz wave and the third‐ and fifth‐harmonic signals generated by the 1LG sample measured at zero gate bias and room temperature. For the third‐ and fifth‐harmonic measurements, BPFs with center frequencies of 1.2 THz and 2.0 THz, respectively, were placed after the graphene sample to suppress the intense fundamental pump at 0.4 THz. Dashed lines in (a) show control spectra measured with a bare PI substrate under otherwise identical conditions, confirming that the observed harmonic signals are not residual spectral components of the fundamental wave at the corresponding frequencies. Each harmonic spectrum was normalized by the peak transmission of the corresponding BPF. In the fifth‐harmonic measurement, residual leakage of the fundamental wave is strongly attenuated but remains visible at delays beyond 20 ps in the time‐domain trace, where it partially overlaps with the fifth‐harmonic waveform.

The thermodynamic origin of graphene's THz nonlinearity leads to an early deviation from the perturbative power‐law scaling of harmonic generation at incident THz electric field amplitudes far below the 75 kV cm⁻¹ used in our measurements (see Figure S2). Thus, to quantify the nonlinear response, we define the harmonic field conversion efficiency as:

$${\eta }_{nf}=\sqrt{\frac{I_{nf}}{I_f}}=\frac{\sqrt{n_{\mathrm{PI}}}{E}_{nf}}{\sqrt{n_{\mathrm{air}}}{E}_f}$$ (1)

where I_nf and I_f are the peak spectral intensities of the n‐th harmonic and the fundamental THz wave, respectively; n _PI and n _air are the refractive indices of PI and air, respectively; E_f is the peak spectral field amplitude of the incident fundamental wave in air; and E_nf is the peak spectral field amplitude of the n‐th harmonic inside the PI substrate. The value of E_nf is obtained by dividing the harmonic field amplitude measured in air by the coherent field‐outcoupling coefficient of the finite PI film. The refractive index of the PI substrate in the relevant THz range is taken to be 1.8 [35, 36, 37].

To compare the graphene samples on a common electrostatic scale, we convert the applied gate bias V into the total gate‐induced carrier density n _gate. This conversion accounts for the influence of sample‐to‐sample variations in the electric double‐layer capacitance between the ion‐gel gate and graphene, as well as the variations in offset voltage arising from interfacial charge transfer at the ion‐gel/graphene interface. This allows the nonlinear response of multilayer graphene samples with different layer numbers to be compared on a common carrier‐density scale. As shown in Figure 3, the conversion is obtained by fitting the measured transmission at the fundamental frequency to numerically calculated curves using the charge relation:

$$e{n}_{\mathrm{gate}}=e\sum _{i=1}^N{n}_{\mathrm{gate}}^{\left(i\right)}=C_{\mathrm{gate}}\left(V-V_0\right)$$ (2)

where e is the elementary charge, n ^{( i )} _gate is the gate‐induced carrier density in the i‐th layer, C _gate is the effective gate capacitance between the ion‐gel gate and graphene, and V ₀ is the offset voltage corresponding to zero gate‐induced charge. Note that V ₀ is the bias voltage at which the total gate‐induced carrier density vanishes and does not coincide with the charge neutrality point (CNP). Because of substrate‐induced doping, the Fermi levels of the graphene layers generally do not coincide with the Dirac point when V = V ₀. The layers are indexed from the gate side, with i = 1 denoting the gate‐adjacent layer and i = N the substrate‐adjacent layer. Using the same layer‐resolved notation, we define the substrate‐induced carrier density in the i‐th layer as n ^{( i )} _sub and the total substrate‐induced carrier density as n _sub = Σ n ^{( i )} _sub, with analogous notation for the net carrier density: n ^{( i )} _net = n ^{( i )} _sub + n ^{( i )} _gate and n _net = n _gate + n _sub. We define $n_{\mathrm{sub}}^{(i)}\equiv n_{\mathrm{net}}^{(i)}{|}_{n_{\mathrm{gate}}=0}$ as the fixed reference density at zero total gate‐induced carrier density, and n ^{( i )} _gate ≡ n ^{( i )} _net–n ^{( i )} _sub as the gate‐induced change from this reference state. This decomposition is a bookkeeping convention rather than a physical distinction, because the equilibrium carrier population cannot be distinguished according to whether the carriers were introduced by the gate or the substrate. Throughout this work, we adopt the convention that positive carrier densities denote hole doping, such that the built‐in p‐doping induced by the PI substrate corresponds to a positive carrier density [38, 39].

FIGURE 3.

THz field transmission coefficient of 1LG, 2LG, and 3LG at the fundamental frequency of 0.4 THz. Symbols represent experimentally measured transmission values normalized to that of the bare PI substrate. Solid lines show the numerically calculated transmission curves incorporating a sample‐specific baseline reduction in field transmission. This reduction was treated as an additional fitting parameter for each sample and was determined to be 1.8%, 6.3%, and 13.9% for 1LG, 2LG, and 3LG, respectively (see Supporting Information). The conversion from gate bias to ngate was performed using the following fitting parameters: C _gate = 0.65, 1.03, and 0.85 µF cm⁻² and V ₀ = 0.66, 0.61, and 0.71 V for 1LG, 2LG, and 3LG, respectively.

In our single‐sided gating geometry, the layers in multilayer graphene cannot all be brought to the CNP simultaneously, as discussed below. By contrast, 1LG can be tuned to its CNP when a negative n _gate compensates the positive n _sub, bringing the net carrier density, n _net = n _gate + n _sub, to zero, although a finite residual carrier population may remain because of thermal excitation and spatial charge inhomogeneity. The numerically calculated 1LG transmission curve shown in Figure 3 exhibits a clear maximum at n _gate ≈ −3.4 × 10¹² cm⁻² and is symmetric about this value. Over the measured n _gate range, the calculated curve agrees well with the experimental data. Because charge transfer between graphene and the PI substrate is expected to be governed primarily by work‐function mismatch [38], we adopt n _sub = 3.4 × 10¹² cm⁻² as a working value for the total substrate‐induced carrier density for all layer numbers. For 1LG, this carrier density corresponds to a Fermi‐level shift of approximately 214 meV from the CNP.

2.3. Layer‐Dependent Tuning of THz Harmonics via Electrostatic Doping

Figure 4 shows the dependence of the third‐ and fifth‐harmonic field conversion efficiencies of 1LG, 2LG, and 3LG on the total gate‐induced carrier density, n _gate. While turbostratic stacking enhances the overall harmonic response relative to 1LG, especially at large positive n _gate, this additional gain already begins to saturate at the trilayer level. The subtle enhancement from 2LG to 3LG is resolved in the third‐harmonic measurement; however, the marginal gain in fifth‐harmonic generation predicted by our numerical model is obscured by the experimental uncertainty of the inherently weaker fifth‐order signals.

FIGURE 4.

Layer‐number and electrostatic‐doping dependence of the high‐harmonic field conversion efficiencies for (a) third‐harmonic and (b) fifth‐harmonic generation in 1LG, 2LG, and 3LG. Symbols denote experimentally measured values. The measured harmonic spectra were corrected for the corresponding BPF and Fresnel transmission at the substrate‐air interface to obtain the harmonic electric field amplitudes inside the substrate. Solid lines represent the numerically calculated field conversion efficiencies.

For positive n _gate, corresponding to additional hole doping, the field conversion efficiencies increase in all samples and then gradually saturate at higher doping levels. For fifth‐harmonic generation, both the experiment and the numerical model show a decrease in efficiency at even higher n _gate for 2LG and 3LG. Within the thermodynamic model, this non‐monotonic behavior arises from the competition between two density‐dependent effects: increasing carrier density enhances THz absorption by increasing the intraband conductivity, but the absorbed energy is distributed among a larger carrier population, reducing the average energy gained per carrier and thereby suppressing the rise in carrier temperature [8, 9]. Compared to third‐harmonic generation, fifth‐harmonic generation exhibits this thermodynamic efficiency reduction at lower n _gate, bringing it within the experimentally accessible doping range.

For negative n _gate, both harmonic field conversion efficiencies initially decrease to pronounced minima before increasing again. Because the PI substrate introduces built‐in p‐type doping, negative n _gate initially reduces the net hole density. As n _gate becomes more negative, each affected graphene layer reaches the CNP and subsequently enters the n‐doped regime. The minimum conversion efficiency is lowest for 1LG, where the single graphene layer can be tuned to the CNP, while it is highest for 3LG. In multilayer graphene, the individual layers reach the CNP at different n _gate values, as discussed in Section 2.4. Therefore, the harmonic responses of the individual layers cannot be minimized simultaneously. The resulting higher, broader minimum reduces the modulation depth, defined as M_nf = (η_{nf .max}–η_{nf .min}) / η_{nf .max}, where the extrema are evaluated over the accessible gate‐density range, thereby reducing the achievable contrast between high‐ and low‐efficiency states.

The discrepancy between the measured and calculated harmonic conversion efficiencies near the predicted minima likely arises, at least in part, from the assumption that the total substrate‐induced carrier density is identical for all samples, with n _sub = 3.4 × 10¹² cm⁻². Because some of the graphene layers in 2LG and 3LG approach the CNP within a relatively narrow range of n _gate, the calculated minima are expected to be particularly sensitive to variations in n _sub. Such variations may arise from sample‐to‐sample differences in the graphene/PI interface introduced during the transfer of the first graphene monolayer onto the PI substrate.

For 1LG, the harmonic field conversion efficiency varies approximately linearly with net carrier density over the range n _gate = −3 × 10¹² to 1 × 10¹² cm⁻², corresponding to n ^{( i )} _net = 0.4 × 10¹² to 4.4 × 10¹² cm⁻². A comparable linear dependence on n _gate is observed for the multilayer samples over the interval in which the numerically calculated n ^{( i )} _net remains within this range for each layer. Moreover, near n _gate = 0, the linear trends for all samples nearly overlap, and the harmonic conversion efficiency depends only weakly on the number of graphene layers in this regime.

These observations can be understood qualitatively by treating the layer‐resolved responses as approximately additive, although the full numerical calculation accounts for more complex effects, including layer‐dependent carrier heating and the electromagnetic weighting arising from internal attenuation and reflections. To leading order, the overall harmonic field generated by a multilayer sample can be approximated as the sum of the harmonic fields from the individual layers. In the linear regime identified above, the harmonic field contribution from each layer also varies approximately linearly with its carrier density, while the layer‐dependent electromagnetic weighting varies only weakly across the stack because each layer remains relatively transmissive. Consequently, within this approximate picture, the total harmonic field depends primarily on the total carrier density n _net = Σn ^{( i )} _net, rather than on its detailed layer‐resolved distribution, providing a qualitative explanation for the weak layer‐number dependence of the harmonic field conversion efficiency in this regime.

Furthermore, the individual layers remain within the linear doping regime up to higher n _gate in thicker samples, because the total gate‐induced carrier density is distributed among a larger number of layers. Accordingly, 2LG and 3LG retain an approximately linear dependence of their harmonic field conversion efficiencies on n _gate over progressively wider ranges. The onset of saturation in the doping‐induced enhancement therefore shifts to higher n _gate, allowing multilayer stacking to increase the maximum harmonic field conversion efficiency attainable at optimal doping.

2.4. Nonlinear Screening in Turbostratic Multilayer Graphene

Figure 5 shows the calculated layer‐resolved net carrier densities, n ^{( i )} _net, in 2LG and 3LG as functions of n _gate, obtained using a nonlinear electrostatic screening model [30] in which each graphene monolayer is treated as a discrete two‐dimensional conducting layer. These layer‐resolved electrostatic trends directly explain the two main features of Figure 4: why the stacking‐induced enhancement of the harmonic response begins to saturate by 3LG and why the modulation depth decreases with increasing layer number.

FIGURE 5.

Numerically calculated layer‐resolved net carrier densities in (a) 2LG and (b) 3LG, obtained from the nonlinear screening model. Here, i = 1 denotes the gate‐adjacent layer, and increasing i corresponds to layers closer to the substrate. The vertical dashed line indicates n _gate = 0, that is, the condition of zero gate‐induced carrier density, which is distinct from zero applied gate bias. The horizontal dashed line marks the CNP. The total substrate‐induced carrier density is taken to be 3.4 × 10¹² cm⁻².

At n _gate = 0, the layer‐resolved net carrier density n ^{( i )} _net reduces to n ^{( i )} _sub, with layers closer to the substrate exhibiting higher hole densities. As n _gate varies, the carrier distribution evolves nonlinearly because of screening, with the gate‐adjacent layers accommodating a larger fraction of the total gate‐induced charge, irrespective of the sign of n _gate. For n _gate > 0, strong screening by the gate‐adjacent layer (i = 1) results in nearly identical values of n ⁽¹⁾ _net for 2LG and 3LG, particularly at high n _gate. In 2LG, the remaining gate‐induced charge is accommodated by the i = 2 layer, whereas in 3LG it is partitioned between the i = 2 and i = 3 layers. Consequently, the additional harmonic enhancement obtained by adding the third layer is limited.

For n _gate < 0, each layer crosses the CNP, where n ^{( i )} _net = 0, and enters the n‐doped regime at a different value of n _gate. In 2LG, the net carrier densities of both layers remain close to zero near n _gate ≈ −3.4 × 10¹² cm⁻². In contrast, there is no value of n _gate at which all three layers in 3LG can simultaneously approach the CNP. Near n _gate ≈ −2 × 10¹² cm⁻², the i = 1 and i = 2 layers are close to the CNP, while the i = 3 layer remains appreciably p‐doped with n ⁽³⁾ _net ≈ 2 × 10¹² cm⁻². Conversely, when the i = 3 layer approaches the CNP, the i = 1 layer has already become heavily n‐doped. This staggered approach to the CNP, caused by nonlinear screening in the presence of substrate‐induced doping, explains the reduced modulation depth observed in multilayer samples: even when some layers are tuned near charge neutrality, others remain sufficiently doped to continue contributing appreciably to the total nonlinear THz response.

These results have direct implications for the design of gate‐tunable THz devices. In a single‐sided gating geometry, nonlinear electrostatic screening concentrates the gate‐induced charge in the layers nearest the gate, causing these layers to rapidly exceed the carrier density optimal for maximizing the harmonic field conversion efficiency while leaving more distant layers only weakly tunable. This imbalance limits both the stacking‐induced enhancement and the achievable modulation depth. In addition, the heavily doped gate‐adjacent layers exhibit reduced THz transmission, resulting in stronger attenuation of the pump field reaching the other layers. A dual‐gate architecture could mitigate this limitation by controlling the carrier distribution from both sides of the stack [40, 41, 42]. More complete layer‐resolved Fermi‐level tuning could, in principle, be achieved using a multi‐gate graphene/dielectric heterostructure in which individual graphene sheets are electrically isolated, separately contacted, and individually gated through ultrathin THz‐transparent dielectric or electrolyte spacers. Such an architecture would, however, require substantially more complex fabrication and bias control while maintaining THz transparency and electronic decoupling between the graphene sheets.

3. Conclusion

In summary, we have combined experimental measurements with theoretical modeling to investigate how layer‐resolved carrier‐density distributions govern THz high‐harmonic generation in turbostratically stacked graphene under single‐sided electrostatic gating. We show that, although turbostratic stacking and gating enhance the harmonic response, the THz nonlinearity of multilayer graphene is not determined by the total carrier density alone. Instead, it is governed by the layer‐resolved Fermi‐level distribution produced by nonlinear electrostatic screening. By combining the thermodynamic model of graphene THz nonlinearity with the nonlinear screening model, we quantitatively reproduce the measured layer‐number‐ and gate‐dependent harmonic field conversion efficiencies. In particular, this framework captures both the early saturation of stacking‐based enhancement in harmonic yield and the reduced modulation depth in thicker samples.

These results reveal a limitation inherent to single‐sided gating in turbostratic multilayer graphene: nonlinear electrostatic screening concentrates gate‐induced carrier modulation primarily in the gate‐adjacent layer, while layers farther from the gate respond only weakly. Device optimization should therefore focus on controlling the layer‐resolved carrier distribution rather than simply increasing the stack thickness. Dual‐gate architectures or multi‐gate graphene/dielectric heterostructures could provide improved electrostatic control across the stack [40, 41, 42]. Such gating schemes, combined with larger numbers of graphene layers [43], substrate engineering [44], and hybridization with other two‐dimensional materials [45], may provide a pathway toward scalable enhancement and active modulation of the THz nonlinear response of multilayer graphene. More broadly, this design principle may also apply to other gate‐tunable multilayer graphene devices, including THz modulators [46, 47, 48], THz detectors [49, 50, 51], and metamaterial devices [6, 52, 53].

4. Experimental Methods and Theoretical Modeling

4.1. Experimental Setup and Sample Fabrication

Figure S1(a) shows a simplified schematic of the THz harmonic‐generation setup. To generate an intense narrowband THz pump, a broadband THz pulse was spectrally filtered using a cross‐slot metasurface BPF, producing a multicycle THz pulse centered at 0.4 THz. The narrow spectral bandwidth of the fundamental facilitates clear separation of the generated harmonics from the pump in the frequency domain. The multicycle THz pulse was focused onto the graphene sample, which generates odd‐order harmonics of the fundamental frequency. To isolate each harmonic, an additional band‐pass filter centered at the corresponding harmonic frequency was placed after the sample to suppress the much stronger fundamental wave.

The broadband THz pulse was generated in a lithium niobate crystal by optical rectification of 800‐nm, 100‐fs pulses from a regenerative amplifier. The tilted‐pulse‐front excitation technique [54, 55] using a diffraction grating was employed to improve phase matching. The transmitted THz waveform was measured in the time domain by electro‐optic (EO) sampling using a 1‐mm‐thick ZnTe crystal.

Figure S1(c) shows the schematic of the graphene device and the sample fabrication process. First, metallic electrodes consisting of 200‐nm‐thick Au on a 20‐nm‐thick Cr adhesion layer were patterned on a 2‐µm‐thick PI substrate by conventional ultraviolet lithography followed by lift‐off. Next, CVD‐grown graphene was transferred onto the PI substrate such that it directly contacted the ground electrode. Each monolayer graphene sheet was transferred using an MMA copolymer resist support layer, which was removed after transfer. This transfer process was repeated to fabricate turbostratically stacked multilayer graphene samples. Finally, an ion‐gel layer was transferred onto the graphene and brought into contact with the gate electrode, enabling large changes in graphene's carrier density at relatively low gate voltages.

4.2. Thermodynamic Model of THz Intraband Nonlinearity in Graphene

In the thermodynamic model [8, 9, 12], the free‐carrier temperature oscillates in response to the incident THz wave, and the resulting temporal modulation of graphene's THz transmission leads to nonlinear frequency conversion in the THz regime. Figure S3 shows the numerically calculated temporal evolution of the free‐carrier temperature and the transmission coefficients of the individual graphene layers.

Monolayer graphene's transmission can be calculated from the complex intraband sheet conductivity $\stackrel{\sim }{\sigma }(\omega )$ by the Tinkham equation [56]:

$${\stackrel{\sim }{E}}_t\left(\omega \right)=\frac{2n_i}{n_i+n_t+Z_0\stackrel{\sim }{\sigma }\left(\omega \right)}{\stackrel{\sim }{E}}_i\left(\omega \right)$$ (3)

where n_i and n_t are the refractive indices of the incident and transmitted media, respectively. E_i and E_t are the incident and transmitted THz electric fields, and Z ₀ is the vacuum impedance. The conductivity depends on the carrier temperature T _e and is expressed as an average sum of Drude‐like conductivities derived from the Boltzmann transport equation:

$$\stackrel{\sim }{\sigma }\left(\omega \right)=-\frac{e^2{v}_F^2}{2}\underset{0}{\stackrel{\infty }{\int }}D\left(\mathcal{E}\right)\frac{\tau \left(\mathcal{E}\right)}{1-i\omega \tau \left(\mathcal{E}\right)}\frac{\partial f_{\mathrm{FD}}\left(\mathcal{E},\mu ,T_{\mathrm{e}}\right)}{\partial \mathcal{E}}d\mathcal{E}$$ (4)

where v_F is the Fermi velocity in graphene, D($\mathcal{E}$) is the density of states of graphene, τ($\mathcal{E}$) is the energy‐dependent carrier scattering time, f _FD is the Fermi‐Dirac distribution, and µ is the chemical potential. As long‐range Coulomb scattering is the dominant electron‐scattering mechanism in CVD‐grown graphene [57], the scattering time is assumed to depend linearly on the carrier energy according to $\tau (\mathcal{E})=\gamma \mathcal{E}$, where γ is set to 47 fs/(170 meV) [8, 12]. The chemical potential µ varies with the carrier temperature T _e under a constant carrier density assumption and within a one‐band model [8].

The carrier temperature T _e is determined by the heat energy per unit area accumulated in the carrier system of each graphene layer due to the THz field:

$$\begin{array}{ccc}\hfill \mathrm{\Delta }Q\left(t\right)& =& \frac{1}{2}{\epsilon }_{0}c\left\{n_i{\left(E_i^{+}\left(t\right)\right)}^2-n_i{\left(E_i^{-}\left(t\right)\right)}^2\right.\hfill \\ & & \left.-n_t{\left(E_t^{+}\left(t\right)\right)}^2+n_t{\left(E_t^{-}\left(t\right)\right)}^2\right\}*R\left(t\right)\hfill \end{array}$$ (5)

where ϵ ₀ is the vacuum permittivity and c is the speed of light; n_i and n_t are the refractive indices of the incident and transmitted media, respectively; E_i,t ⁺ and E_i,t —denote the forward‐ and backward‐propagating electric fields, respectively; * denotes temporal convolution; and R(t) represents the carrier‐cooling response kernel associated with phonon emission. The cooling response kernel is modeled as:

$$\begin{array}{ccc}\hfill R\left(t\right)& =& 0.001\exp \left(-\frac{t}{13\mathrm{fs}}\right)+0.46\exp \left(-\frac{t}{80\mathrm{fs}}\right)\hfill \\ & & +0.539\exp \left(-\frac{t}{860\mathrm{fs}}\right)\mathrm{for}t\ge 0\hfill \end{array}$$ (6)

with R(t) = 0 for t < 0, where the 13, 80, and 860 fs components correspond to electron‐electron scattering, optical phonon emission, and acoustic phonon scattering, respectively [13]. We assume that all absorbed heat energy contributes to carrier heating:

$$\mathrm{\Delta }Q\left(t\right)=\underset{T_0}{\stackrel{T_{\mathrm{e}}}{\int }}{C}_{\mathrm{e}}\mathrm{dT}=\underset{T_0}{\stackrel{T_{\mathrm{e}}}{\int }}\frac{2\pi \left|E_F\right|k_B^2}{3{\hslash }^2{v}_F^2}\mathrm{TdT}$$ (7)

where T ₀ is the ambient temperature, C _e is the low‐temperature approximation of the electronic heat capacity per unit area of monolayer graphene [58, 59], E_F is the Fermi energy, and k_B is the Boltzmann constant. Owing to the electron–hole symmetry of graphene, n‐doped and p‐doped layers are treated equivalently.

For the multilayer calculations, each graphene sheet is modeled as a zero‐thickness conductive interface characterized by its carrier‐density‐ and temperature‐dependent complex sheet conductivity. The forward‐ and backward‐propagating fields are then solved using a transfer‐matrix method (see Supporting Information), thereby coherently accounting for multiple internal reflections. Phase accumulation across the atomic‐scale interlayer separations between adjacent graphene sheets is neglected, whereas propagation through the 2‐µm‐thick PI substrate is retained. The THz permittivity of the interlayer region between graphene layers is assumed to be 2.5 [60, 61].

4.3. Nonlinear Screening in Multilayer Graphene

In multilayer graphene, the electric fields generated by charges on the gate and substrate are screened by the individual graphene layers [30, 34]. The carrier distribution across the layers can be determined using a discrete‐layer nonlinear screening model. Carriers with densities −n_gate and −n_sub accumulate at the gate layer and the substrate, respectively, while the graphene layers collectively accumulate a total carrier density n _net = n _gate + n _sub such that the combined system consisting of the gate, multilayer graphene, and substrate remains electrically neutral. The carrier distribution can then be obtained by minimizing the total energy per unit area U of the system, which consists of the band‐filling energy U ^{( i )} _b of each graphene layer and the electrostatic energy U ^{( i )} _el stored in the interlayer regions:

$$U=\sum _i^N{U}_b^{\left(i\right)}+\sum _i^{N-1}{U}_{el}^{\left(i\right)}$$

$$\begin{array}{ccc}& =& \sum _i^N\left(\underset{-\infty }{\stackrel{\infty }{\int }}\mathcal{E}·D\left(\mathcal{E}\right)\left(f_{\mathrm{FD}}\left(\mathcal{E},{\mu }_i,T_0\right)-f_{\mathrm{FD}}\left(\mathcal{E},0,T_0\right)\right)d\mathcal{E}\right)\hfill \\ & & +\sum _i^{N-1}\frac{{\epsilon }_0{d}_0}{2}{E}_i^2\hfill \end{array}$$ (8)

where T ₀ is the equilibrium temperature at ambient conditions, d ₀ = 0.34 nm is the spacing between adjacent graphene layers, and E_i is the electric field in the interlayer region between the i‐th and (i+1)‐th layers. We take the DC permittivity of the interlayer space to be the same as the vacuum permittivity ϵ ₀. As the carrier density n ^{( i )} _net in a graphene layer is varied, both its chemical potential µ_i and the interlayer electric fields change accordingly. Note that the electrostatic energies in the region between the gate layer and the i = 1 graphene layer and in the region between the i = N graphene layer and the substrate do not depend on the carrier distribution and are therefore omitted.

Author Contributions

Seong Cheol Lee: methodology, data curation, investigation, validation, formal analysis, visualization, writing – original draft. Donghak Oh: methodology, investigation, validation, formal analysis, data curation, visualization. Soojeong Baek: methodology, investigation. Dongho Lee: methodology. Wontae Kim: methodology, investigation. Bumki Min: writing – review and editing, conceptualization, funding acquisition, supervision, resources, data curation, formal analysis, project administration, validation. Fabian Rotermund: conceptualization, writing – original draft, writing – review and editing, funding acquisition, supervision, resources, data curation, formal analysis, project administration, validation.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Supporting File: smtd71067‐sup‐0001‐SuppMat.docx.

Data Availability Statement

The data that support the findings of this study are available on request from the corresponding author.