Characterization and physical modeling of MOS capacitors in epitaxial graphene monolayers and bilayers on 6H-SiC

The electron transport properties of graphene monolayers and bilayers have generated significant amount of interest and competitive high speed field effect devices have been demonstrated in both materials.^1,2 Intercalated monolayers and bilayers grown by epitaxy on SiC are particularly promising as they routinely demonstrate the excellent transport properties and material uniformity required for the fabrication of microwave integrated circuits.^3,4 However, field effect devices in graphene often demonstrate poor current modulation which significantly compromises high frequency performance.^5–7 In metal-oxide-semiconductor (MOS) systems, current modulation is strongly affected by dielectric quality and charge trapping effects. As graphene is a gapless semiconductor, devices in graphene are expected to demonstrate subdued current modulation relative traditional semiconductor devices. For this reason, graphene devices are particularly sensitive to dielectric charging and interface trapping effects as they can easily screen current modulation. This trade-off between exceptional material properties and non-ideal dielectrics warrant an investigation of charge control in metal-oxide-graphene systems.

Capacitance-voltage (CV) and conductance-voltage (GV) measurements are commonly used to investigate interface states and trapping effects in MOS systems such as silicon, SiC,⁸ and III/V heterostructures.⁹ The CV/GV technique indirectly probes the interaction of charge carriers with other aspects of the MOS system such as interface states (D_it), surface potential fluctuations (δϵ_f), material non-uniformity, and substrate polarization (ΔP).

Charge control in graphene MOS systems has been investigated previously, and recent studies have sought to quantify the quantum capacitance (C_q) of monolayers and bilayers in top gated field effect devices.¹⁰ Surface potential fluctuations (δϵ_f) were later addressed in the context of graphene monolayers and bilayers, and results were treated phenomenologically as a broadening of the density of states in graphene.¹¹ In exfoliated monolayers on SiO₂, Dröscher et al. attribute poor current modulation in top gated structures to surface potential fluctuations of order 100meV.¹² In Ref. 13, charge control is investigated in monolayers transferred onto SiO₂ with an Al₂O₃ gate dielectric grown by atomic layer deposition (ALD). Results demonstrate dispersion in the CV curves associated with interface states (D_it), and temperature dependence is attributed to thermally activated charge trapping in the dielectric.

It is also necessary to consider substrate induced effects in graphene MOS systems. In epitaxial graphene on 4H(6H)-SiC in particular, the spontaneous polarization of the substrate (ΔP) is responsible for the hole conductivity observed in intercalated monolayers and bilayers.¹⁴ Additionally, ΔP is known to open a narrow energy gap (ϵ_g) in epitaxial bilayers, which has important consequences for interpreting CV data.^15,16

In this work, a quantitative physical model of charge control in graphene monolayers and bilayers is presented in conjunction with temperature dependent CV/GV measurements performed on planar MOS capacitors.The devices are fabricated in hydrogen intercalated epitaxial monolayers and bilayers grown on 6H-SiC with a thin Al₂O₃ gate dielectric. The dielectric is prepared by repeated deposition and subsequent thermal oxidation of thin layers of Al metal, a technique which frequently appears in the literature as an alternative to atomic layer deposition (ALD).^17–19 

With accurate modeling, a number of relevant device parameters including the density of interface states, the magnitude of surface potential fluctuations, and the presence of a narrow energy gap in bilayers induced by the spontaneous polarization of the substrate are assessed in a single experiment. Supporting measurements addressing surface potential fluctuations, hysteresis, and charge injection are also discussed in order to facilitate a deeper understanding of the CV/GV data. While developed in the context of epitaxial graphene on SiC, a straightforward application of the modeling methods outlined in this work should be sufficient to describe CV/GV data in a wide variety of graphene MOS systems.

An analysis of charge control in a MOS capacitor begins by considering the modulation of the Fermi energy ϵ_f by an applied voltage v. The total capacitive response observed in a CV measurement may be expressed as

In Eq. (1), C_ox represents the oxide capacitance, C_q = e²ρ(ϵ) the quantum capacitance in graphene, and C_it = e²D_it(ϵ) the capacitance due to interface states.^20,21 Eq. (1) implies the following relation between ϵ_f and v²² 

Integrating equation Eq. (2) over ϵ ∈ [0, ϵ_f] and v ∈ [v_D, v] yields the following expression.

In Eq. (3), Δv = v − v_D where v_D is the Dirac voltage (ϵ_f = 0). Eq. (3) is the equivalent of the Berglund integral in graphene MOS.²³ The electron density may be calculated via the Fermi-Dirac integral.

In Eq. (4), k_b is Boltzmann’s constant and T is the absolute temperature. The occupation statistics are given by the Fermi-Dirac distribution f(ϵ : ϵ_f, k_bT) = [1 + e^{(ϵ−ϵ_f)/k_bT}]⁻¹. The hole density n_h(ϵ_f) may be obtained by transforming ϵ_f → −ϵ_f and integrating over ϵ ∈ [0, − ∞). The total carrier density is then given by n = n_e + n_h. When ϵ_f ≫ 0 electron density dominates and when ϵ_f ≪ 0 hole density dominates. An ambipolar condition occurs near ϵ_f ≈ 0, as both electron and hole density contribute to the total carrier density. The monolayer and bilayer density of states relations are

where, g_s(g_v) are the twofold spin(valley) degeneracies respectively, v_f ≈ 1 ⋅ 10⁸cm/s is the Fermi velocity in graphene, and ħ is the reduced Plank constant. In the case of bilayer graphene, the density of states in Eq. (5) is approximated as the sum of the density of states at low and high energy. The quantity γ_⊥ ≈ 0.4 eV represents the interlayer coupling constant in Bernal stacked bilayers.²⁴ 

In order to accurately model CV curves, it is necessary to account for interface states (D_it). Generally, the effect of a large D_it is to compromise charge control in the channel by screening C_q. A common approach to estimate D_it is to compare the capacitive response of the MOS structure at low and high frequency.⁹ 

When a MOS capacitor is biased at low frequency, the total capacitance $C tot 0$ will contain contributions from C_ox, C_q, and C_it. As the frequency of the test signal is increased, interface states will contribute less to the total capacitance observed. In the case of very high frequencies only C_q and C_ox will contribute to the observed capacitance $C tot ∞$⁠. This dispersive effect in the C_tot is due to the finite capture and emission lifetimes (τ_c,e) of trap states. In the majority of dielectric/semiconductor systems, τ_e ≫ τ_c such that the dominant contribution to frequency dependence in C_tot is τ_e.

Eq. (6) tends to underestimate D_it especially when C_q ≫ C_it. In order to account for this, the effective D_it may be estimated by multiplying Eq. (6) by a scaling factor D₀. The dispersive effect due to the finite lifetimes of trap states is well described by a simple exponential where ω = 2πf is the angular frequency.

The movement of charge in and out of interface states gives rise to a small signal conductance G_it such that D_it can be estimated by examining the frequency dependence of G_it.

Eq. (8) exhibits a maximum in conductance when interface states are in resonance with the test signal.

When analyzing CV data, it is also necessary to account for surface potential fluctuations (δϵ_f). Surface potential fluctuations describe a spatial variation in ϵ_f due to charge inhomogeneities at the graphene/substrate and graphene/oxide interfaces. In graphene, surface potential fluctuations are especially relevant near ϵ_f = 0 as they generate localized islands of electron and hole conduction.^25–27 In order to model surface potential fluctuations, it is useful to introduce a random variable to describe the Fermi energy $ϵ ̃ f$⁠.

The distribution $N$ represents the statistics which describe the spatial variations of ϵ_f. Typically, $N$ may be assumed to be normally distributed.

In Eq. (10) the terms ϵ_f and δϵ represent the mean standard deviation of the Fermi energy statistics $ϵ ̃ f$⁠.

δϵ_f represents the root mean square (RMS) value of surface potential fluctuations near the Dirac point. Eqs. (10) and (11) describe a case where the magnitude of the surface potential fluctuations decays with standard deviation δσ_f as one moves further from ϵ_f = 0. Generally the term δσ_f is found to be of order 100meV such that δϵ ≈ δϵ_f near the Dirac point. When |ϵ_f| ≫ 0, surface potential fluctuations have little effect on the behavior of the CV characteristic.

In this work, we propose the following method to model CV-curves in graphene. First, $D i t ∗$ may be estimated via Eq. (6). As this is known to be an underestimation, the scale parameter D₀ is then introduced and the corresponding D_it may be included. If a negative D₀ is required to obtain accurate high frequency capacitance curves, then the measurement data must be corrected for inductance. Typically, an inductance correction is only needed for measurement frequencies exceeding 1MHz. Using D_it(v, ω), one may obtain ϵ_f(v, ω) via Eq. (3) via nonlinear optimization methods. In order to obtain proper capacitance curves, it is necessary to account for surface potential fluctuations. This is accomplished via a kind of Monte Carlo simulation in which noisy ϵ_f(v, ω) curves are generated via Eqs. (10) and (11). These are then used to calculate noisy capacitance curves via Eq. (1). Results are then averaged in order to obtain a final model.

CV(GV) measurements are performed as a function of temperature on 10000μm² planar MOS capacitors using an Agilent E4980A LCR meter. The geometry of the MOS capacitors is shown in Fig. 1. In the CV measurements, the applied bias is swept quasistatically from -2 to 2V, and the capacitive(conductive) responses of the device to a 10mV test signal are measured at several frequencies f ∈ [1, 10, 100, 200, 500, 1000kHz]. All measurements consist of a forward and reverse sweep in order to track hysteretic effects in the devices.

The monolayer and bilayer samples were grown on semi-insulating (SI) 6H-SiC by chemical vapour deposition (CVD) and in-situ intercalated with hydrogen.^28,29 Upon intercalation, both monolayers and bilayers exhibit hole conduction (ϵ_f < 0) as a consequence of the spontaneous polarization of the substrate.¹⁴ Prior to device fabrication, the samples were characterized via microwave reflectivity measurements and scanning electron microscopy (SEM) in order to assess material quality and the number of layers. The microwave reflectivity measurements yielded mobilities of 4500(3000)cm²/V ⋅ s and carrier densities of 0.95(0.87) ⋅ 10¹³ cm⁻² for the monolayer(bilayer) samples.

Dielectric deposition on graphene is challenging owing to the fact that low temperature processes are required. For this reason, high-κ dielectrics such as Al₂O₃ are often deposited on graphene via atomic layer deposition (ALD). However, several studies document the difficulty of achieving uniform layers with ALD as the Al₂ (CH₃)₆ precursor does not effectively wet pristine graphene.³⁰ To circumvent this, a nucleation layer is often used (2-3nm) in order to facilitate the growth of the subsequent ALD layer.^31,32 The nucleation layer is usually thermally oxidized aluminium (as is shown in our study), though polymer functionalization has also been shown to be effective. When thin dielectric layers are needed, thermally oxidized aluminium is usually sufficient with regard to leakage thus precluding the need for the subsequent ALD step.

The Al₂O₃ dielectric was deposited by repeated evaporation and subsequent hotplate oxidation at 200°C of 1nm aluminium metal films. In both samples, a target oxide thickness (t_ox) of 15nm was chosen in order to ensure adequate coverage of the terraced morphology of the SiC substrate. The thermal oxidation method was chosen in part because the resulting oxide demonstrated excellent leakage characteristics on the large area MOS devices (<1nS at 1kHz). Similar tests on nucleated 15nm ALD layers demonstrated nonuniform coverage on terrace edges along with considerable leakage (>1μ S at 1kHz) such that a reliable extraction of D_it via the CV method was not feasible. However, as the interface is identical in both systems, there should little difference between the two methods with regard to (D_it) provided a high quality ALD layer with uniform coverage is achieved.

In addition to the planar MOS devices, ancillary van der Pauw (vdP) structures and Transfer Length Method (TLM) structures are included to assess the low field transport properties and contact resistance after processing. From these structures, mean mobilities of 1601(2028) cm²/V ⋅ s and carrier densities of 1.05(0.79)⋅10¹³ cm⁻² are obtained for the monolayer(bilayer) samples. Measurements on the TLM structures indicated a contact resistance of 300(200)Ω ⋅ μm for the monolayer(bilayer) samples.

The temperature sweep is carried out in a liquid N₂ cryostat, and the temperature is swept linearly from 77K to 280K. Additional measurements are performed at room temperature in order to investigate charge injection effects in connection with the hysteresis observed in the devices. In all CV curves presented in this work, a low frequency conductance of <1nS is observed.

In order to assess material uniformity, work function (ϕ_g), and surface potential fluctuations (δϵ_f) in epitaxial graphene, frequency modulated Kelvin Probe Force Microscopy (KPFM) is performed on small 25μ m² van der Pauw (vdP) structures.³³ As KPFM is only sensitive to the surface of a material, it was necessary to fabricate samples without the gate oxide present. Prior to performing KPFM, AFM cleaning was performed in contact mode to remove contaminants and residues from the surface. As graphene is sensitive to atmospheric and polymer contaminants, it is necessary to perform the KPFM measurements in a controlled atmosphere.^34–36 Prior to scanning, the chamber was evacuated and then subsequently filled with N₂ at room temperature. Finally, the atmosphere was saturated to a relative humidity of 30% to approximate ambient conditions. The work function calibration was done using ϕ_g = ϕ_probe − eV_cpd where V_cpd is the measured surface potential. The probe work function (ϕ_probe) was calibrated against the an Au contact electrode within the scan area.

Modeling the CV curves is computationally difficult as a combination of Monte Carlo methods, nonlinear methods, and parameter optimization is required. For this reason, an efficient CV simulation kernel was implemented on a graphics processor (GPU). GPU processing offers the flexibility of a massively parallel computation scheme in a highly threaded environment allowing for efficient Monte Carlo simulations [supplementary material].

The measured and modeled low frequency CV characteristics at 77K are shown in Fig. 2 for both monolayer and bilayer material. ϵ_f(v_g) as calculated from Eq. (3) is also shown. Both materials exhibit a minimum in capacitance which corresponds to ϵ_f = 0. As both materials are intercalated, v_D > 0 indicating hole density at zero bias. Moving away from v_D in either direction, the capacitance increases and then saturates indicating accumulation of carriers at the graphene/oxide interface. In the saturation regions, C_q ≫ C_ox such that the oxide capacitance and dielectric constant may be estimated κ = C_oxt_ox/ε₀. Additionally, the CV curves are approximately symmetric around v_D, which reflects the symmetric behavior of ρ_m,b(ϵ_f) around ϵ_f = 0 (see Eq. (5)).

All parameters for the modeled monolayer and bilayer capacitance curves of Fig. 2 are shown in Table I. In the following sections, details are presented with respect to the implementation and interpretation of modeling results. First, a commentary on D_it is provided. Next, surface potential fluctuations and material non-uniformity are addressed in the context of SEM and KPFM imaging. Energy gap modeling in bilayers then described, and a band diagram for the graphene MOS systems is proposed. Finally, charge injection and hysteresis are discussed alongside temperature dependent measurements.

Both monolayer and bilayer material exhibit significant dispersion when measuring CV(GV) curves as a function of frequency. By fitting the CV(GV) measurement data to Eqs. (7) and (8), independent estimates of D_it and τ_e may be made. The estimation of τ_e and D_it via the 77K CV data of Fig. 2 is shown in Fig. 3.

A D_it of 3.75(1.51)⋅10¹² eV⁻¹ cm⁻² is extracted from the monolayer(bilayer) material from the CV curves via Eq. (7). Similar values of 1.51(1.50)⋅10¹² eV⁻¹ cm⁻² are obtained from modeling GV curves via Eq. (8). This should be compared with a D_it of 10¹²-10¹³ eV⁻¹ cm⁻² reported for 30nm ALD layers prepared on CVD graphene transferred to SiO₂/p-Si substrates.³⁷ Similar values of 10¹² eV⁻¹ cm⁻² have been reported in AlGaN/GaN heterostructures with low temperature ALD Al₂O₃ gate dielectrics.³⁸ For comparison, values as low as 5 × 10¹⁰ eV⁻¹ cm⁻² and 10¹¹ eV⁻¹ cm⁻² have been achieved in silicon and SiC MOS devices respectively with high temperature SiO₂ dielectrics.^39,40

From Fig. 2, the monolayer material exhibits the larger swing in Fermi energy with ϵ_f  ∈  [ − 0.32, 0.28eV] over the applied bias range. Thus, the measurement only probes an energy interval at the dielectric/graphene interface over 0.60eV near the middle of the dielectric band gap. The D_it for such a limited energy range are in most cases rather flat (e.g for SiO₂ and Al₂O₃ on SiC and Si) such that the peak in D_it near v_D is an artifact of the extraction. When v is far from v_D, C_q is large such that a estimation of D_it by Eq. (7) is difficult. For this reason, the maximum density of interface states $D i t m a x$ occurring at ϵ_f = 0 is taken to estimate the true D_it.

The effect of surface potential fluctuations is to generate a distributed capacitance minimum in the CV characteristics. In the case of a monolayer, C_q(ϵ_f) → 0 when ϵ_f = 0 such that $C tot ∞$ should sharply approach zero near v_D. The fact that such a minimum is not seen in measurement data demonstrates the effect of surface potential fluctuations (δϵ_f). Modeling the CV characteristics in monolayers and bilayers yields values of 92(78)meV for δϵ_f. This should be compared with values of 100meV, 25-40meV and 30-100meV in graphene, Si, and SiC MOS devices respectively.^12,41,42

The results of KPFM imaging are shown in Fig. 4. The magnitude of the surface potential fluctuations in pristine graphene may be compared with those extracted from CV measurements. The work function data is normally distributed for the monolayer(bilayer) regions with mean of ϕ_g ≈ 4.82(4.73)eV. Equating the surface potential fluctuations as the work function RMS for the entire active area, one has δϵ_f ≈ δϕ_g ≈ 80 meV in relative agreement with what is obtained from CV modeling. The KPFM data indicates that monolayer(bilayer) inclusions contribute significantly to magnitude of surface potential fluctuations.

The SEM and KPFM images of Figs. 1 and 4 show that the large area monolayer MOS capacitors have bilayer inclusions which have an effect on the CV characteristics. These inclusions are a consequence of the growth mechanism. During epitaxy, graphene growth nucleates at step edges and propagates over the terrace. On monolayer(bilayer) samples, bilayer(multilayer) graphene is common on terrace edges respectively.⁴³ Additionally, inclusions of monolayer(bilayer) material in bilayer(monolayer) samples may also appear on terraces.⁴⁴ 

It is straightforward to account for inclusions in CV modeling by considering the density of states as linear combination of the monolayer and bilayer relations (Eq. (5)).

In Eq. (12), the quantity p represents the mixing ratio of monolayer area to the total area of the device. In order to estimate p, SEM imaging was performed on monolayer and bilayer material. Terraces and terrace edges are clearly visible on the surface of the substrate. In Fig. 1, low contrast regions correspond to monolayer material while high contrast regions correspond to bilayer material. Values of 0.8(0.1) were obtained from imaging monolayer(bilayer) material respectively.

The energy band diagram for the graphene MOS system as shown in Fig. 5 provides a useful context to understand CV measurements. The mean work function of ϕ_g = 4.8 eV for graphene estimated from the KPFM measurements is in relative agreement with literature values.^36,45,46 The estimation of the ϕ_g from KPFM is calibrated relative to the work function of the Au contact metalization.

As the amount of mobile charge in the semi-insulating SiC is negligible, there should be minimal band bending in the SiC bulk. Thus, ϵ_f passes through the midgap such that the band offset between the conduction band in the SiC and the ϵ_f in the graphene is $ϵ g S i C /2$⁠. The band gap in 6H-SiC is $ϵ g S i C ≈3.0$ eV resulting in a band offset of 1.5 eV.^14,47

The band gap in Al₂O₃ oxide $ϵ g o x$ has been shown vary with the phase of the material and its quality. Values for high quality crystaline films range from 8.8 eV in α-Al₂O₃ to 7.1-8.0 eV in γ-Al₂O₃. For lower quality amorphous films, values of 5.1-7.1 eV are reported.^48,49 Measurements for the conduction band offset between in the amorphous Al₂O₃/SiC system yield values of 2.06 eV such that the charge neutrality point in the graphene lies near the midgap in the Al₂O₃.^50–52 For this reason, Al₂O₃ is an ideal dielectric for graphene MOS on SiC.

Although the CV characteristic observed in monolayers and bilayers is qualitatively similar, the physical origin of the capacitance minimum is different. This may be seen by returning to the expression for the total capacitance (Eq. (1)). At high frequency, C_it ≈ 0 such that the total capacitance is simply $C tot = [ C q − 1 + C ox − 1 ] − 1$⁠. In both monolayer and bilayer material, a minimum in C_tot is expected at ϵ_f = 0. However, in a bilayer C_q ≠ 0 when ϵ_f = 0. Evaluating the theoretical quantum capacitance in a bilayer, a value of 4.170 μF/cm² is obtained at ϵ_f = 0. For the A = 10000μm² bilayer capacitors C_qA = 471 pF. As the observed oxide capacitance is C_oxA ≈ 33 pF, the minimum expected capacitance at high frequency in the bilayer MOS pads is $C tot m i n A≈30.8$ pF.

The minimum high frequency capacitance observed in the bilayer data (23.2pF) is significantly lower than the expected 30.8pF (see Fig. 2). In order to account for the anomalous behavior, it is necessary to introduce an energy gap ϵ_g into the density of states relation near ϵ_f = 0.

The notion of an energy gap in graphene bilayers is well understood, and results in a symmetry breaking of the bilayer Hamiltonian which occurs when the individual layers are at different potential energies.^15,16 In bilayer MOS, there are two sources of such potential which function to open a gap: the high density of interface states at the graphene/oxide interface D_it, and the spontaneous polarization of the 6H-SiC substrate ΔP. In both cases the sheet charge density involved is of order 10¹²cm⁻² at minimum, such that a symmetry breaking of the bilayer Hamiltonian is realistic. The notion of an energy gap is additionally supported by the fact that the 1 kHz CV curve in bilayer material exhibits a significantly deeper capacitance minimum than the monolayer case despite comparable D_it and δϵ_f.

The presence of surface potential fluctuations (80-90meV) reflects that the charge densities involved are not uniform. In this case, the magnitude of the energy gap will vary locally from point to point within the bilayer MOS structure such that an empirical model is needed for ρ_g(ϵ : ϵ_g, σ_g).

The effect of Eq. (14) is to cut a smoothed notch out of the bilayer density of states relation in (Eq. (5)). Here ϵ_g represents the mean value of the energy gap, while σ_g characterizes its dispersion. Results from CV modeling suggest an energy gap of 260meV in the case of the bilayer sample, and a value of 274meV for the bilayer component of the monolayer sample. These values are in qualitative agreement with experiments in dual gated field effect transistors, in which a narrow energy gap of ϵ_g = 250 meV has been observed.^53,54 Polarization induced gaps of order ϵ_g = 150 meV have also been observed epitaxial bilayers on SiC.⁵⁵ 

The quantitative nature of the CV model becomes evident when considering sensitivity with regard to the parameters of Table I. A particular sensitivity is observed with respect to δϵ_f and ϵ_g as summarized in Fig. 6. Further, all parameters introduced into the model are physical with the possible exception of δσ_f. In the Si and SiC cases, the magnitude of the surface potential fluctuations is typically independent of bias such that δσ_f → ∞. In the context of the CV model, δσ_f effectively corrects for the artificial profile of D_it obtained from Eq. (6). The strong low frequency dispersion near the Dirac point in the monolayer and bilayer samples presented in this work suggests that the electron traps are physically located at the graphene/oxide interface or within the graphene sheet itself. Physically, vacancies in native Al₂O₃ dielectric located at the graphene/oxide interface and inclusions in the graphene likely account for the observed dispersion.

The dispersion observed in this work is in contrast to what is observed in transferred layers on SiO₂ in which the low frequency dispersion is observed under accumulation.¹³ In the devices investigated by Lin et al., dispersion is observed under accumulation suggesting that trapping occurs above the graphene/oxide interface (i.e. in the dielectric itself). To account for this behavior, the authors suggest two trap bands for their ALD oxides which are spaced somewhat symmetrically around ϵ_f = 0. This is different than our case, in which the dispersion suggests one continuous trap band of approximately constant D_it. Thus, the interpretation of dispersive characteristics highlights important differences between in interface trapping mechanisms between native Al₂O₃ and ALD Al₂O₃ dielectrics on graphene.

When measuring the CV characteristic as a function of temperature, several additional effects are observed which are not considered in the CV model (Fig. 7). First, a hysteresis of anti-clockwise orientation opens in both samples for temperatures(thermal energies) greater than 160K(13.7meV). Hysteresis is a common problem in the context of graphene field effect transistors and has been attributed to a plurality of mechanisms.^56–58 The orientation of the hysteresis is significant and suggests a charge injection effect.^59,60

By comparing the extracted forward (⁠$v D f$⁠) and reverse (⁠$v D r$⁠) sweep Dirac points a similar trend appears in both materials. Generally, $v D f$ is constant, while $v D r$ increases suggesting that charge injection occurs only when ϵ_f > 0. Electron conduction in amorphous Al₂O₃ layers is supported by current-voltage measurements in p-Si MOS structures with an ALD Al₂O₃ dielectric.⁶¹ In Ref. 61, Novikov et al. show a strong temperature dependence in the leakage current through their ALD Al₂O₃ layers above 77K. In order to account for electron conduction a multiphonon ionization mechanism for deep levels is described.⁶² A bulk trap density(activation energy) of 2⋅10²⁰ cm⁻³ (1.5 eV) is estimated from the multiphonon ionization model. The high density of deep levels is relevant as it implies temperature dependent transport through the amorphous Al₂O₃ layer via the ionization of deep levels. Transport is determined to be monopolar(electron), and the temperature dependence described is qualitatively consistent with that of the hysteresis observed in the CV measurements presented in this work. For this reason, it is reasonable to suggest that deep levels in the oxide are responsible for hysteresis shown in Fig. 7.

In monolayer(bilayer) a $v D r − v D f$ of 0.43(0.52V) is observed at 280K indicating similar levels of charge injection in both materials. Repeated CV sweeps of increasing amplitude reveal a drift of the capacitance minima towards positive bias indicating permanent injection of negative charge into the oxide layer. In addition to hysteresis, a monotonic increase in the zero bias capacitance is observed with increasing temperature. In Ref. 13, similar trends are also attributed to a thermally activated trap mechanism. The effect of oxide charging is deleterious with regard to effective charge control as the lagging of ϵ_f behind v generates a more shallow slope and additional broadening of the CV characteristic on either side of v_D. The charge injection hysteresis is observed at all frequencies owing to the fact that it is a DC effect. The frequency relevant to charge injection is the sweep rate of the applied bias rather that of the test signal.

The effectiveness of charge control can be estimated by considering the ratio of the carrier density in accumulation n^acc to the intrinsic carrier density. From modeling CV data, the maximal ϵ_f is approximately -320(-220)meV for monolayer(bilayer) material. This corresponds to a carrier density of approximately 0.78(1.03)⋅10¹³ cm⁻². The intrinsic (i.e. ϵ_f = 0) electron densities are given by the following relations.⁶³ 

Counting both electrons and holes, the above relations evaluate to 0.16(2.79)⋅10¹² cm⁻² at 77K. This suggests a ratio n^acc/n^tot(0) of 469(36) in monolayer(bilayer) material suggesting that charge control should be much more effective in the monolayers. However, in the presence of surface potential fluctuations, the RMS carrier densities become 0.51(2.86)⋅10¹² cm⁻² such that a reduced ratio of 15(3.6) is expected at 77K. Modulation of the carrier density in both cases is further limited by the presence of D_it ≈ 2 ⋅ 10¹² eV⁻¹cm⁻². In the case of monolayer material, D_it destroys the remaining modulation of carrier density at low frequency. In bilayer material, some charge control is preserved due to ϵ_g ≈ 260meV.

A method to model measured CV data in graphene MOS structures has been described. With accurate models, it is possible to estimate the density of interface states D_it, the magnitude of surface potential fluctuations δϵ_f, the effect of material anisotropy, and the presence of a narrow energy gap ϵ_g in bilayer material. The density of interface states is significant in both materials, and values of order 2 ⋅ 10¹² eV⁻¹cm⁻² are extracted from measurement data. An analysis of the D_it results yields an emission lifetime τ_e of several hundred nanoseconds for the trap states. In both materials, surface potential fluctuations of order 80-90meV are found to generate a distributed capacitance minimum. Similar values are obtained from KPFM measurements, and surface potential fluctuations are found to be correlated with inclusions of monolayer(bilayer) material. An narrow energy gap of order 260meV is obtained for the bilayer constituents of both materials consequent to the spontaneous polarization of the substrate. An anti-clockwise hysteresis effect is observed due to a thermally activated trap in the dielectric. The hysteresis is found to be temperature dependent, and a thermal barrier of about 160K(13.7meV) is deduced from temperature dependent CV data. The hysteresis has a deleterious effect on charge control, and generates considerable broadening in the CV characteristics of both MOS systems.

These results are of interest from a physical and technological perspective as they suggest a need to improve dielectric quality and material uniformity in graphene MOS devices. The effect of D_it and δϵ_f substantially compromise charge control in graphene MOS systems. Monolayer material exhibits poor charge control characteristics as a direct consequence of these effects. In bilayers, some degree of charge control is maintained due to the opening of a narrow energy gap indicating that bilayers may be more applicable in a transistor context.

See supplementary material for a GPU implementation of the GMOS CV model (2016).

This work was supported by the European Science Foundation (ESF) under the EUROCORES Program EuroGRAPHENE, and by the EU Graphene Flagship (No. 604391). We also acknowledge support from the Swedish Foundation for Strategic Research (SSF), and the Knut and Alice Wallenberg Foundation (KAW).