Figure 1 shows the top-view schematic (a) and atomic force microscope images (b) of H-terminated zigzag GNMs and the optical microscope image of the Hallmeasurement pattern on the fabricated GNM with an ionic-liquid gate (c), (d). The GNMs, which have honeycomb-like array of hexagonal nanopores, were fabricated on mechanically exfoliated graphenes via the nonlithographic method using a nano-porous alumina template (NPAT) as an etching mask [10,11].

The NPAT is easily fabricated through self-organization by anodic oxidation of pure (99.99%) aluminum films. The graphene was carefully etched by optimized low-power Ar gas (e.g., 200 - 600 V for 10 - 40 min) so as to avoid giving damages (Figure 1(d)) and the naomesh of the NPAT was transferred to graphene. Then, the NPAT was detached from the surface of the fabricated GNM, either mechanically or chemically.

The GNMs fabricated through these processes were annealed at a critical temperature 800˚C in high vacuum (10⁻⁶ Torr) for 0.5 - 3 days with continual pumping of gas and, then, in hydrogen gas by the field-emission-type radical CVD system under pressure >1 MPa at least for 3 h. The first annealing is for deoxidization of the pore edges with recovering all damages and defects and is the key to forming zigzag pore edges by edge atomic reconstruction [10,12-14], while the second annealing is the key for termination of the carbon atoms at the pore edges by hydrogen atoms for production of flat-band ferromagnetism.

It is to be noted that for electrical measurements, we employed ~5-layer graphenes, because monolayer GNMs exhibited poor electrical features. We have previously confirmed that even such thin-layer GNMs can exhibit ferromagnetism comparable with that of monolayer GNMs, although the amplitude reduces [1-3,22,23]. In fact, we have performed electrode fabrication and measurements after the confirmation of presence of the ferromagnetism.

The ionic-liquid gate was formed to apply V_ig following ref. [19] by dropping N, N-diethyl-N-(2-methoxyethyl)-N-methylammonium bis-(trifluoromethylsulfonyl)- imide (DEME-TFSI) (Figures 1(c) and (d)). Applied V_ig is consumed on at the interface of ionic liquid and the GNM, forming electrical-double layer within ~1 nm thickness (Figure 1(d)). Thus, it induces extremely high electric fields on the GNM surface. In the present experiment, side-gate electrode not exposed to the ionic liquid was used to apply the V_ex.

Figure 2 shows the drain current (I_ds) as a function of the back-gate voltage (V_bg) and V_ig (i.e., before and after the formation of the ionic-liquid gate). Without the ionicliquid gate, the FM-GNM exhibits only n-type semiconductive behavior; I_ds increases only with increasing V_bg (Figure 2(a)). I_ds is zero even when V_bg is as small as −20 V. In contrast, with the ionic-liquid gating, I_ds drastically increases with decreasing V_ig (Figure 2(c)). Even when V_ig is ~ −1 V, the p-type semiconductive behavior appears significantly. This can be attributed to hole doping by the partial oxygen-termination of the pore edges. Immediately before forming the ionic-liquid gate, the GNMs were exposed to an air atmosphere, because it is impossible to form the gate in high vacuum or a hydrogen atmosphere. Because the GNM is not covered by any passivation films, the pore edges are easily oxidized through this process, thereby resulting in the partial oxygen-termination and hole doping. Applying −V_ig significantly enhances this hole doping via pore edges. This occurrence is different from the case of bulk graphene without edges. Otherwise, the ionic-liquid gate itself might cause oxygen-like termination.

Figures 2(b) and (d) show the logarithmic scales of I_ds transformed from Figures 2(a) and (c). The subthreshold slope values (V_sub) are 5 V/decade for increasing I_ds in the +V_bg region (Figure 2(b)) and 270 mV/decade for increasing I_ds in the −V_ig region (Figure 2(d)). The V_sub value reflects the on-off ratio of I_ds (I_on/off) for the applied gate voltages and, hence, it is extremely important for field-effect transistor (FET) operation. The large I_on/off value directly leads to a high gain in the radio-frequency (RF) operation of FETs. The V_sub is inversely proportional to the I_on/off. Previously, we have reported on the electrical characteristics of high-quality and low-defect GNRs derived from the unzipping of carbon nanotubes (CNTs) combined with three-step annealing [8]. These GNR-FETs exhibited a large V_sub of 3 V/decade for a 20-nm width. The GNMs derived using polymer masks also exhibited a large V_sub of ~3 V/decade for a 7-nm

interpore width (i.e., GNR width) [29]. Only the GNRs obtained from the unzipping of CNTs exhibited a small V_sub of 210 mV/decade but for a 2-nm width [9]. Thus, the present V_sub of 270 mV/decade for the 20-nm interpore width is extremely small compared with those of any other GNR-FETs. This strongly suggests the effectiveness of ionic-liquid gating in producing and modulating extremely high electric fields on the GNM surface and the interpore GNR regions.

We investigate the charge density n_s achieved upon applying V_ig by observing the magnetoresistance (MR) behaviors in the FM-GNMs.

When magnetic fields (B) are applied perpendicular to the GNM plane, the electrons follow a cyclotron motion with the classical radius of the cyclotron orbit,

, where h and e denote Planck’s constant and the electron charge, respectively [28]. A variety of interesting MR phenomena has been observed depending on the correlation of 2R_c with (the pore diameter) and a (the diameter of the unit cell). For instancecommensurability MR peaks and Aharonov-Bohm (AB)- type oscillations (with an oscillation period , where S denotes the area of the unit cell) were observed around low B values at points where electrons encircle and localize around the nanopores (see Figure 3(a); 2R_c = a) for samples with large /a values and small interpore spacing (i.e., 2(a −)) that is insufficient to allow electron cyclotron motion. In contrast, at high B values, Shubnikov-de Haas (SDH) oscillations and consequent quantum Hall effects appeared for samples with small /a values and large interpore spacing that allowed for electron cyclotron motion with the small Rc values.

In a previous work on FM-GNMs with nearly identical structure parameters as those in the present one but without the ionic-liquid gate, a commensurability MR peak was observed at B ~ 1.2 T [28]. From the expression, we estimated and an elastic mean free path of (where D denotes the diffusion constant and v_F denotes the Fermi velocity) ~800 nm. Based on this result, we can estimate n_s under extremely high electric fields caused by applying V_ig to FM-GNMs with an ionic-liquid gate.

Figure 3 shows the MR (R_xx) as a function of V_ig. The commensurability MR peak can be observed at B ~ 1 T at V_ig = 0 V. As V_ig decreases, the peak shifts to larger B regions. The maximum peak position of B ~ 5 T is observed at V_ig = −12 V. This indicates an increase in n_s with decreasing V_ig, because a large n_s value requires a large B value in order to satisfy the condition for the commensurability peak:. Consequently, the maximum n_s of ~1 × 10¹³ cm⁻² can be estimated from the peak B value for V_ig = −12 V.

This value is extremely large despite the small value of the assembled 20-nm-width GNR structure of the present GNM. This large value is attributed to the large electric field caused by V_ig. Moreover, this result also implies the contribution of the induced edge-p states of the nanopores, in which high-density carriers localize at the pore edges even when B = 0 (i.e., the large peak around V_ig = −12 V in Figure 4(b), explained later). These edge-localized carriers are added to the cyclotron electrons at V_ig = −12 V, thereby resulting in extremely large values of n_s. Indeed, it is noteworthy that the peak position moves to a lower B value and n_s decreases at V_ig = −16 V. This evidences the contribution of the edge-p electrons localized around the pore at V_ig = −12 V. Furthermore, the extremely large n_s value results in a small mean free path (l_e) of ~800 nm, which is considerably smaller than the circumferential length of the pore and cell (~250 nm). Thus, the present GNM is not within a ballistic charge transport regime, and no AB-type oscillations are observed in Figure 3(b).

In addition to the extremely high value of n_s, the other strong advantage of using the ionic-liquid gate is the obtainment of a large geometrical capacitance (C_geo) [19], which allows the direct observation of the quantum capacitance (C_Q)_. The large geometry of the ionic-liquid region (Figure 1(c)) provides an extremely large C_geo. This capacitance allows the direct observation of the quantum capacitance (, where m^* denotes the effective mass of carriers and D₂ is the electronic density of states of the 2D layer), because the total capacitance is given by and the 1/C_geo value is negligible. In fact, a previous work has reported on the possible detection of D_2D at higher energy bands arising from the C_Q of individual layers in monolayer to trilayer graphenes, using this method. Here, the capacitance of 2D systems is conventionally obtained from the Hall measurements (i.e., changes in n_s as a function of). However, it is difficult to employ this method in our case because the scattering by the honeycomb-like nanopore array prevents Hall measurements.

Instead, we employ a GNR model in order to estimate C_geo, and reconfirm the possibility of the direct observation of C_Q, because the interpore regions are GNRs (Figure 1). The GNR models gives carrier mobility , where g_m denotes the mutual conductance and L denotes the GNR length8. The g_m value of ~500 mS at V_ds = 0.8 V is observed in Figure 2(d), and L corresponds to the inter-electrode distance of 2500 nm in our fabricated GNM. In a previous report, we have also estimated μ < ~100 cm²/Vs in FM-GNMs from MR measurements [28]. Consequently, C_geo for the side gate electrode is estimated to be ~12 μF/cm².

This value is in good agreement with those in previous reports on ionic-liquid (electrolytic) gates [19]. Conventional GNRs with widths of 2 ~ 20 nm placed on SiO₂ substrates have C_geo ~ 10⁻¹⁰ - 10⁻¹² F for the source electrodes. Thus, the present C_geo value is nearly 10⁵ times greater than these reported values. Therefore, we can estimate C_Q and D_2D directly from the measurement results related to C_tot also in our GNM.

Figure 4(a) shows the differential conductivity (dI/dV; R_xx⁻¹) as a function of V_ig in the low-V_ig region when B = 0 T. Even in the present ~5-layer GNM, a Dirac-pointlike dip (zero-bias anomaly) is observed in the dI/dV curves. Moreover, as the absolute value of V_ig increases, other dips and peaks appear. Such dI/dV anomalies due to V_ig variation have been reported in tri-layer graphenes with ionic-liquid gates and attributed to carrier filling in higher energy bands (i.e., interband scattering). This is a advantage unique to the electrolytic gate method for the abovementioned reasons (i.e., introducing large electric fields and the presence of a large C_geo). The curves in Figure 4(a) are qualitatively consistent with the abovementioned features. However, in Figure 4(a), the dI/dV anomalies observed at V_ig ~ ±0.3 V are considerably more defined than those observed in the tri-layer graphenes. Moreover, the plateau of the second peak in the +V_ig region is broad and consists of very small peaks. The presence of these peaks might suggest very strong interband scattering and the presence of a complicated higher-band structure in our ~5-layer FM-GNM.

In the large V_ig region (Figure 4(b)), we observe many dI/dV peaks. In particular, a significantly large and broad dI/dV peak is observable around V_ig ~ −12 V, when the side-gating V_ex is 0 V (dotted line in Figure 4(b)). Although the dI/dV properties as a whole are very complicated, the observed large dI/dV values in the -V_ig region at V_ex = 0 V are consistent with the hole-dominant property in Figures 2(c) and (d). This consistency indicates

the significant appearance of p-type semiconductive behavior.

The dI/dV values typically have a strong correlation with D_2D and n_s, conventionally. A large D_2D value around Fermi level results in a large dI/dV value [16,17]. Hence, the maximum dI/dV peak around V_ig ~ −12 V (Figure 4(b)) should reflect the maximum D_2D value of our FMGNM.

Because the edge-localized flat p band has the largest D_2D values (edge states) and strongly contributes to the appearance of the ferromagnetism also in graphites [3, 21-23], the maximum dI/dV peak should correspond to this edge-p band. Theoretical works have proposed that the edge-magnetic moment arising from the edge-p state survives even in graphites with interlayer interaction (e.g., ABA stacking with mono-hydrogenated edges) [3,21-23]. Moreover, because the present FM-GNM has only ~5layers, the influence of interlayer coupling for suppressing the emergence of ferromagnetism is very weak.

However, the localization of the edge-electrons prohibits current flow and obstructs detection of the large D_2D by dI/dV observation in the present case, even if the D_2D value is very high. Nevertheless, we have reported certain MR behaviors (e.g., periodic MR oscillations [28] and saw-tooth-like MR oscillations [10]) originating from edge-localized p-electrons in similar ~5-layer FM-GNMs operating under a constant current mode in four-probe measurements. We have argued that certain numbers of pore-edge-localized electrons can travel between electrodes because of the constant current mode (i.e., under non-thermal equilibrium). Moreover, the n_s value of ~1 × 10¹³ cm⁻² around V_ig ~ −12 V observed in our ionic-liquid-gated FM-GNMs is considerably larger than the n_s value of conventional FM-GNRs. This result supports the strong contribution of the drastically induced pore-edge p-electrons, as mentioned above. In such a case, the excess p-electrons localizing at the pore edges can be transported considerably more efficiently under a nonthermal equilibrium condition and contribute to the dI/dV. Therefore, we can assume that the maximum dI/dV value directly reflects the edge-localized p band with the highest DOS via the excess p electrons, as seen from scanning tunnel microscopy observations of the DOS.

Such maximum dI/dV peaks cannot be observed in conventional electrolytic-gated bulk graphenes without nanopores (i.e., without edge states), which showed no FM behavior, and such behavior has not thus far been observed in measurements obtained by applying V_bg. These results strongly support the correlation of the observed maximum dI/dV peak with the induced pore-edge pelectrons in our ionic-liquid gated ~5-layer FM-GNMs.

On the other hand, several other dI/dV peaks are also observed in Figure 4(b). These should originate from other D_2D states, which are sensitive to spin interference by 1) the spin alignment of both the edges of the interpore GNR regions [3], 2) large ensemble of the GNR regions, 3) the interlayer stack structures [22], and also 4) V_ig and V_ex. Thus, it is difficult to identify the origins of the individual peaks. For example, the second-largest peaks observed around V_ig ~ −5 V and −12 V at V_ex = 0 in Figure 4(b) may be associated with the σ band, because the present GNMs have ~5-layer. In the present work, we focus on the largest dI/dV peak observable around V_ig ~ −12 V at V_ex = 0 in Figure 4(b), because Figures 4(b)-(d) suggest its strong correlation with the edge-p band, as stated in next section.

As mentioned above, we interestingly find that the maximum dI/dV peak around V_ig ~ −12 V separates into two peaks upon increasing the V_ex (Figure 4(b)). The splitting voltage ΔV_ig monotonically increases with increasing V_ex and saturates at around V_ex = 8 V with ΔV_ig ~ 3 eV (Figure 4(c)). At low values of V_ex, the left peak is considerably more prominent than the right peak (Figure 4(b)), while the left peak height reduces for large V_ex values, and eventually, the two peaks exhibit similar heights. The peak ratio for the left/right peaks after peak separation becomes equal with increasing V_ex because of the decrease in the left-peak height (Figure 4(d)).

This spin splitting can be qualitatively understood by the resolving of the double spin degeneration of the edgelocalized flat p band (i.e., separating of the majorityand minority-spin bands, which correspond to the left and right peaks, respectively) by applying the side-gating V_ex, which plays role like applying in-plane V_ex. Because the contribution of the localized minority-spins to dI/dV is very small, the right-peak height is actually small throughout all the V_ex regions (Figure 4(b)). Indeed, such a spin splitting upon the application of an electric field has been reported as an anomalous spin Hall effect in FM/non-FM hybrid materials (e.g., Pt/Ni-Fe). From the theoretical viewpoint, the presence of spin bands for the majority and minority spins has also been reported in FM-GNRs [3].

One particular theory [6] has predicted the half-metallicity and spin-gap opening that can be realized upon applying the in-plane field V_ex in zigzag-GNRs with AF spin alignment at both the edges (as mentioned in the introduction). In our fabricated FM-GNMs, the spin alignment at the two edges of the interpore GNR region is FM. However, when a similar mechanism is realized for the majority and minority spins with opposite moments for FM-GNRs assembled between the electrodes, the above mentioned spin splitting can be observed (Figures 4(e)- (g)), and it can be explained by assuming that the sidegate V_ex plays the role of the in-plane field V_ex. The predicted spin-gap opening as per the theory was Δ_theory ~ 0.5 eV for V_ex = 0.1 V/Å. In contrast, the abovementioned value of ΔV_ig ~ 3 eV for V_ex = 8 V corresponds to a considerably larger ΔV_ig. As mentioned above, we have used the side-gating voltage V_ex for this observation. When V_ex = 8 V is applied from the side-gate electrode directly placed on the Si substrate, the voltage propagates via the substrate surface, thereby leading to a potential difference between both sides of the individual interpore GNR with 20-nm width. The potential difference can be estimated to be ~0.1 V/20nm, based on our previous work [8]. Thus, ΔV_ig ~ 3 eV for V_ex = 8 V corresponds to ΔV_ig ~ 600 eV for V_ex = 0.1 V/Å. This ΔV_ig value is considerably larger than the theoretical prediction Δ_theory ~ 0.5 eV.

However, because this value of ΔV_ig ~ 600 eV is not the actual spin splitting energy in the GNM but corresponds to the V_ig value applied to the side-gate electrode, the actual splitting energy ΔV_sp needs to be calculated. The ΔV_sp value corresponds to the energy in the singleparticle energy spectrum given by

Here, v_F ~ 10³ m/s denotes the Fermi velocity in our fabricated GNM, which is at least 1000 times smaller than that for bulk graphene [8] considering edge scattering. Using the previously estimated value of C_geo ~ 12 μF/cm², ΔV_sp can be estimated to be ~0.54 eV. This value is quantitatively in good agreement with the abovementioned theoretical prediction of ΔV_theory ~ 0.5 eV. This estimation also implies that the edge-localized p band (i.e., the observed dI/dV peak at V_ig ~ −12 V) exists at ~1.2 eV below the initial Fermi level position at V_ig = 0 V. Because the FM-GNM consists of the assembled FM-GNRs, this might smear the dI/dV maximum and also its splitting by averaging. However, we assume that only the dominant some GNRs (interpore regions), of which the width-direction (i.e., both edges) locates in parallel with electrodes to apply V_ex, and have perfect zigzag edges, strongly contribute to these phenomena.

Moreover, it should be noticed that the peak ratio starts to significantly increase from the critical V_ex value of 8 V due to the reduction in the height of the left peak (Figure 4(d)), at which point ΔV_ig saturates (Figure 4(c)). This result supports the conclusions of the half-metallicity model [6]. Following this model, the saturation of ΔV_ig corresponds to the gap closing for one-moment spins existing at both edges of the GNRs (Figure 4(f)). Although this gap closing facilitates the same-moment spin current flow across the GNR, the localization of the edge electrons for the majority spin is suppressed by the spin current flow. A further increase in V_ex results in a decrease in the amplitude of the edge states (i.e., resulting in a decrease in the excess n_s) for the same-moment spins and, thus, the left-peak height decreases (Figure 4(g)) despite appearance of the current flow.

Consequently, we suggest that the applied side-gate V_ex causes spin splitting of the edge-p band (for the majority and minority spins with opposite moments for FMGNRs) similar to that caused by the application of an inplane V_ex following a theory [6]. This is because extremely large values of the D_2D, which originate from the edge states of the nanopores induced by V_ig, are highly sensitive to added external fields. In fact, we could observe no such V_ex-dependent changes in the electronic states in the absence of V_ig (in this case, we applied only the back-gate voltage and, consequently, we could not detect even edge-p states, as in Figure 4). Because the present GNM is placed on a SiO₂ film, the side-gating V_ex cannot directly modulate the interpore GNR edges. However, when the ionic-liquid gate is formed, the liquid faces the pore edges (i.e., both edges of an interpore GNR) and also the surface of SiO₂ at the pore’s inner regions. We speculate that such conditions drastically change the large edge-electronic states even for a small potential difference existing between the two edges of the interpore GNR region caused by the side-gating V_ex, and the V_ex acts in a manner similar to the actual in-plane field V_ex.

Subsequent to measurements performed over nearly three days, we observed a degradation of the electrical and magnetic characteristics of the FM-GNM. In particular, the appearance of single-electron charging effect was observed in the current channel region (Figure 5).

The staircase-like feature is confirmed in the Ids vs. V_ds relationship in Figure 5. The inset shows dI_ds/dV_ds as a function of V_ds. It exhibits a periodical oscillation with a period of 180 mV. It is well known that such a phenomenon is understood as the Coulomb staircase, which originates from the charging effect of a single electron in an extremely small dot area with a charging energy of E_c = e²/2C_dot ~ 180 meV, where C_dot denotes the capacitance of the dot area [30,31]. The C_dot value corresponds to ~10⁻¹⁸ F, which is a value that is considerably smaller than C_geo and also any other capacitances of the present GNM-FET structure. This result strongly suggests the emergence of small defects in the current channel region due to degradation after the three-day measurements. Indeed, ref. [16] reported on the quick degradation of graphenes with electrolytic gates and the consequent importance of obtaining quick measurements.

The sample was placed under high-vacuum conditions in a Cryomag system (Oxford Instruments) at T = 1.5 K during the measurements. Thus, we concluded that the edge degradation is due to the chemical reaction of the (frozen) ionic liquid with the GNM during the measurement. Because the H-terminated pore edges face the ionic liquid, they are most sensitive to such degradation. In the present edge-related measurements, a more careful treatment of the GNMs will be required, because it is well known that the edge-based phenomena are highly sensitive to edge degradation.

Moreover, in our previous works, we have reported observing AB-type MR oscillations [28] and saw-tooth like MR oscillations (i.e., the spin pumping effect [10]) arising from the localized electrons at the pore edges in ~5-layer FM-GNMs without ionic-liquid gating. On the other hand, in the present structure with ionic-liquid gates, these oscillations are hardly observed. This absence of oscillations might also be due to the partial degradation caused by the chemical reaction between the pore-edge carbon atoms and the ionic liquid.