3.1 Effects of step-edge density and hydrogen intercalation on transport properties
Although SiC substrates used in this work have a nominal miscut of zero degrees from the (0001) crystal axis, the existence of a slight miscut leads to a terraced morphology as shown in Fig. 1 . During the high temperature graphitization step, significant step-bunching can occur. This step-bunching leads to a highly non-planar morphology with terraces, or plateaus, 1–10 µm wide separated by steps as high as 10 nm and as wide as 1–2 µm. Due to the unique growth kinetics present at the step-edge, graphene nucleation and subsequent growth occur rapidly, leading to multi-layer graphene over the step-edge . On the terrace plateau, EG samples exhibit primarily monolayer graphene while QFEG samples exhibit primarily bilayer graphene. The hydrogen intercalation process leads to bilayer graphene films by converting the carbon buffer layer in the EG samples into a second layer of graphene. Figure 1 shows a schematic representation of the QFEG sample post-hydrogen passivation. Raman spectroscopy of the graphitized SiC confirm the presence of bilayer graphene on the terrace and multi-layer graphene at the step-edge. Figures 1 are 2D color maps of the full width half maximum of the 2D peak of the graphene Raman signature, where bilayer graphene on the step terraces is typically ∼50 cm−1 and multi-layer graphene on the step-edges is ∼75 cm−1, identified by peak fitting [12, 13]. Across the surface of the sample, the extent of step-bunching is found to vary, with some VdP test structures showing a large change in step height (high density of step-edges) across the active region, with others showing significantly less (low density of step-edges) (Fig. 1).
Figure 1. (a) Schematic representation of SiC surface before graphitization compared to (b) after high temperature graphitization and hydrogen passivation steps showing significant step-bunching in the substrate. (c),(d) Raman maps of the 2D peak full width half maximum, showing the presence of bilayer graphene across the step terrace and multi-layer graphene at the step-edge for the hydrogen intercalated samples.
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In order to discern the effects of substrate and dielectric overlayers on the transport properties of the underlying graphene and bilayer graphene, scattering models are utilized to fit the experimental data and explain the relative temperature and carrier density dependencies of Hall mobility for samples coated with various dielectrics. Although much theoretical and experimental work has been dedicated to understanding the scattering physics in ideal monolayer and bilayer graphene samples, the terraced morphology of the EG and QFEG samples and variation in number of graphene layers across them severely complicates modeling of the scattering physics for the samples used in this work. Ideal monolayer graphene exhibits a linear dispersion relation about the Dirac point, while bilayer graphene is known to exhibit a parabolic band structure close to the Dirac point, leading to fundamentally different scattering physics. Still, the two systems have been shown experimentally to exhibit very similar carrier density-dependent conductivities, with a linear relationship between conductivity and carrier density as well as a minimum conductivity plateau about the Dirac point. In monolayer graphene, this linear relationship has been attributed singularly to charged impurity scatterers, which act as long-range unscreened Coulomb scatterers [14-16]. A sublinear response in conductivity with carrier density has also been shown in monolayer graphene [17, 18], which has been attributed to short-range scatterers and is only observable at high carrier densities, far away from the Dirac point. Importantly, Hwang et al. observed that this short-range scattering dominated regime disappears when the ratio of short-range scatterers (zero range point defects) to long-range scatterers (charged impurity defects) approaches zero (npoint/nimp ≪ 1), which is often the case for experimental graphene samples and is especially appropriate for the EG samples examined in this work . Therefore, additional scattering by short range scattering centers is not considered for samples in this work. For the case of bilayer graphene, the linear conductivity response with carrier density has been attributed to charged impurity scattering as well as the additional effect of short-range scatterers, where both contributions lead to a linear conductivity response with carrier density [19-21]. Because of the similar dependency on carrier density between these two scattering mechanisms, it remains difficult to distinguish between charged impurity and short-range scatterers in bilayer graphene. In order to investigate the bilayer system further, recent experimental works have examined the change in conductivity of bilayer graphene samples with the addition of charged impurities [22, 23]. Xiao et al. studied the effect of potassium doping on bilayer graphene and showed a decrease in the minimum conductivity that goes with which are consistent with theoretical predictions, but found that while the magnitude of charged impurity scattering in bilayer graphene was similar to that of monolayer graphene, the overall mobility in bilayer graphene was roughly an order of magnitude lower . The authors suggested that this additional reduction in mobility may be a result of short-range scatterers. Soon after, Zhang and Li demonstrated experimentally the existence of a long-range scatterer dominated and short-range scatterer dominated regime in bilayer graphene, where long-range scatterers (charged impurities) dominated at high impurity densities (low mobility samples) and short-range scatterers (point defects) dominated at low impurity densities (high mobility samples) .
In this work, the effect of short-range and long-range scatterers in bilayer graphene are grouped into a single fitting parameter that varies linearly with carrier density and follows the approach of Konar et al. in modeling the scattering time in ideal monolayer graphene due to remote charged impurity scatterers . In this way, an effective impurity density is calculated, which can be compared before and after dielectric integration on either EG or bilayer QFEG samples in order to understand the relative change in scattering due to changes in substrate or dielectric coating. It should be noted that this approach will not give absolute impurity densities for these samples unless they closely approximate ideal monolayer graphene, but are sufficient in characterizing the change in relative scattering due to charged impurities before and after dielectric integration for the case where long-range scatterers dominate (high impurity densities). For a more complete understanding of the scattering physics, the effect of short-range scatterers in bilayer graphene should be considered at low impurity densities, especially as impurity density is reduced to values below 1.5 × 1012 cm−2 .
This combined model for effective remote charged impurity scattering  is joined with a model for remote SOP  in order to explain the extrinsic scattering mechanisms introduced by substrate and dielectric coating. These extrinsic scattering mechanisms are often cited as the major contributing factors limiting charge transport in practical graphene devices . In this work, they are found sufficient to model the experimental results, although recent work by Ong et al. suggests that a full understanding of the scattering processes can only be obtained by also considering the dynamical response of graphene plasmons to the surface polar phonon modes . The consideration of coupling between graphene plasmons and surface polar phonon modes, as suggested by Ong and Fischetti, leads to the formation of interfacial plasmon–phonon modes that can lead to additional screening, or anti-screening, of the remote SOPs . In this work, the additional damping or amplification of remote SOPs is captured by a fitting parameter which scales their contribution.
Charged impurity scattering is often the result of dangling bonds or charged defects at or near the substrate–graphene and dielectric–graphene interfaces that act as long-range scatterers. As previously mentioned, remote charged impurity scattering dominated transport exhibits a constant mobility away from the Dirac point in both monolayer and bilayer graphene, which is evidenced by a conductivity that is linearly dependent on carrier concentration. Due to the unique dispersion relation of graphene, these defects lead to degradation of carrier mobility that remains constant with temperature (temperature independent). For the case of bilayer graphene, a slight temperature dependency is found at the minimum conductivity point, but little change in conductivity is found at appreciable carrier densities, away from the Dirac point. Alternatively, remote SOP scattering leads to a degradation of carrier mobility that varies with temperature (temperature dependent). This type of scattering is a result of a non-vanishing decaying electric field at the surface of the substrate or dielectric which arises due to the presence of SOP modes that propagate longitudinally along the surface of the substrate and dielectric overlayer and in close proximity to the graphene. The presence of this non-vanishing, time-varying electric field at the graphene–substrate and graphene–dielectric interfaces leads to additional scattering of the carriers, which is based on the characteristic energy of the SOP modes as well as the degree of electron–phonon coupling across the interface.
In this work, an acoustic deformation potential of 4.8 eV, optical deformation potential of 25.6 eV A−1 [26, 27], and intrinsic optical phonon energy of 160 meV  are selected based on previous results and are utilized to explain the intrinsic scattering of the samples. On the other hand, extrinsic scattering is found to depend critically on both substrate and dielectric material, where an optical phonon energy of 116 meV is used for the SiC substrate and dielectric constants of 9.7 and 1 are used for the SiC substrate and air, respectively. For the case of as-grown bilayer QFEG (no dielectric coating), room temperature Hall effect measurements show that the presence of step-edges in the SiC substrate lead to decreased mobility. Step-edges have often been cited as a source of additional scattering, with reports of degradation of bulk mobility occurring with increasing step-edge density  as well as reports of conductivity anisotropy parallel and perpendicular to step-edges [30, 31]. Recently, Ross et al. have utilized atomic scale measurements to directly measure the additional resistance introduced across the step-edge . For QFEG samples, the effect of step-edges is shown in Fig. 2, where the room temperature Hall mobility and carrier concentration are plotted as a function of absolute change in step-height [i.e., total distance traveled along the vertical axis (c-axis)]. For these samples, optical interferometry is utilized to capture the surface topology of as-grown graphene VdP structures, shown in Fig. 2. From these surface maps, the absolute change in step height across the 5 µm × 5 µm structure is calculated, shown in Fig. 2. The low and high step-density height profiles shown in Fig. 2 correspond to the low and high step-density Raman maps in Fig. 1. As can be seen in Fig. 2, there is a strong correlation between Hall mobility and step-edge density. Increased absolute change in step-height (increased step-edge density) leads to decreased Hall mobility and a slight increase in the hole mobility (∼2%), yet the expected dependency of mobility on carrier concentration at high carrier densities is found insufficient to explain the change in mobility between the low and high step-edge density samples. Figure 2 plots the measured Hall mobility versus carrier density as well as the simulated dependency of mobility on carrier concentration. These results suggest that step edges are a source of additional scattering. Although these and previous results within the literature clearly indicate step-edges as a source of additional scattering, little focus has been given as to what specific scattering mechanisms are responsible for the degradation in transport properties.
Figure 2. (a) Height map of VdP cross showing the presence of step-edges across the structure along with (b) extracted height profiles for low and high step-edge density samples. (c) Plot of room temperature Hall mobility versus extracted absolute change in step-height and (d) Hall mobility versus carrier density showing the negative impact of high step-densities on transport.
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To understand the degradation of mobility at the step-edge for as-grown bilayer QFEG samples, temperature dependent Hall measurements are used, showing an increase in remote charged impurity scattering and remote SOP scattering for high step-edge density samples. The extent of remote charged impurity scattering and remote SOP scattering is determined by applying the intrinsic and extrinsic scattering models to temperature dependent Hall effect measurements (Fig. 3). An effective remote impurity concentration, nimp, and a remote phonon scattering fitting parameter, β, as a function of step-edge density are then extracted from the simulated data. As mentioned previously, the effective impurity concentration does not take into account the effect of additional short-range scatterers in the bilayer graphene, but should be sufficient to explain the scattering behavior at high impurity densities. Furthermore, the remote phonon scattering fitting parameter allows us to account for the additional screening or anti-screening of remote phonons that occurs due the formation of interfacial plasmon–phonon modes, which are not accounted for in the models utilized in this work. Figure 3 plots the temperature dependent Hall mobility of two bilayer QFEG samples and one monolayer EG sample, while Table 1 reports the extracted values of nimp and β. The error for the fitted curve is less than 1% for the hydrogen intercalated graphene with high or low step-edge densities, but rises to ∼2% for the non-hydrogen intercalated graphene, which may be partly an effect of the overall lower mobilities for this sample. Below 200 K, the QFEG and EG samples exhibit Hall mobilities that are temperature independent and limited by remote charged impurity scattering to values ranging from 1000 to 3000 cm2 V−1 s−1 at carrier densities near 1 × 1013 cm−2. From this temperature independent region, the effective remote impurity concentration, nimp, is extracted from the scattering model. As can be seen in Fig. 3, hydrogen passivation of the SiC results in a substantial reduction of 2.4 × 1012 cm−2 in nimp, which is attributed to the effect of increased spacing between the graphene and SiC substrate as well as the passivation of defects and dangling bonds at the graphene–SiC interface after hydrogen intercalation. These results show that despite any enhanced short-range scattering as a result of moving from monolayer to bilayer graphene between the EG and QFEG samples the overall reduction in long-range scatterers leads to a substantial reduction in scattering processes and confirms that the samples studied in this work are indeed dominated by charged impurity scattering.
Figure 3. (a) Plot comparing as-grown EG to high and low step-density bilayer QFEG samples, showing the reduction in effective remote charged impurity and temperature dependent SOP scattering with hydrogen passivation and with decreasing step-edge density, indicating that step edges are a likely source of both scattering processes. (b) Temperature dependency of carrier density for the samples.
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Table 1. Fitting parameters for as grown EG and QFEG samples
|fitting parameter||EG sample 1||QFEG sample 1||QFEG sample 3|
|nsheet (cm−2)||1.2 × 1013||9.1 × 1012||9.3 × 1012|
|nimp (cm−2)||8.4 × 1012||5.8 × 1012||3 × 1012|
Importantly, for the QFEG samples it is found that there is an additional reduction in nimp of up to 1.7 × 1012 cm−2 by moving from a high step-edge density sample to one with low step-density. These results suggest that the step-edge is a primary source of remote charged impurities. Reducing the step-edge density leads directly to a reduction in charged impurities and, subsequently, remote charged impurity scattering and enhanced mobility.
Above 200 K, the effect of remote SOP scattering from the substrate (extrinsic) as well as phonon scattering within the graphene (intrinsic) leads to a decrease in Hall mobility with increasing temperature. In this temperature range, we find that there is a strong correlation between substrate morphology and the extrinsic phonon scattering fitting parameter, β, which quantifies the extent of remote SOP scattering induced by the underlying SiC substrate. As can be seen in Fig. 3, hydrogen passivation of the SiC can result in a significant reduction in remote SOP scattering, which is represented as a significant decrease in the fitting parameter (Table 1). A similar reduction in remote SOP scattering has been reported elsewhere  and is likely the combined effect of increased spacing between the graphene and SiC substrate and an alteration of the SOP dispersion after hydrogen termination. Furthermore, it is found that the reduction in remote SOP scattering with hydrogen intercalation is highly dependent on the presence of step-edges, where increasing step-edge density leads to increased contribution from phonon scattering. These results show for the first time that step-edges may be a source of additional phonon scattering or another temperature dependent scattering process. This may be a result of incomplete hydrogen passivation at the step-edge or, possibly, due to decreased phonon energy for phonons propagating in the c-axis of the SiC crystal. Alternatively, Langer et al. showed that the presence of a high step-edge density can lead to damping of plasmon modes in graphene on SiC . The damping of plasmon modes in the graphene could lead to reduced screening of remote phonons as shown by Ong and Fischetti . Further work is needed to analyze how hydrogen passivation occurs at the step-edge as well as how it effects the phonon dispersion relation at the SiC surface. Additionally it should be emphasized that the presence of step-edges leads to the growth of multi-layer graphene, which also may have an effect in enhancing remote SOP scattering for these high step-edge density samples. On the other hand, it is clear from Fig. 3 that as step-edge density is reduced for the QFEG samples, there is a dramatic reduction in β to the point that the underlying SiC substrate results in negligible additional scattering due to remote SOP scattering for the high mobility (low step-edge density) sample (QFEG sample 3, Fig. 3). For this sample, the temperature dependence of Hall mobility can be explained using only the intrinsic phonon scattering model. These results confirm that hydrogen passivation of the SiC can be extremely effective in reducing the contribution from both remote charged impurity and remote SOP scattering from the underlying substrate.
3.2 Effects of dielectric integration on transport properties
After dielectric integration the impurity density, nimp, is found to increase for both h-BN and HfO2 coated bilayer QFEG samples, indicating that both dielectrics introduce additional charged defects into the system. Yet while both dielectrics are found to increase nimp, h-BN overlayers lead to an increase in the hole concentration while HfO2 overlayers lead to a decrease in the hole concentration, indicating p-type and n-type doping of the graphene, respectively. The n-type doping and p-type doping of graphene by HfO2 and h-BN, respectively, has been reported elsewhere [7, 34, 37]. Figure 4 summarizes the results and plots the experimental data for two high step-edge density QFEG samples before and after dielectric integration as well as one low step-edge density QFEG sample after dielectric integration while Table 2 details the fitting parameters and room temperature sheet carrier densities. The percent error for the fitted curves ranges from <1 to 3% where the high mobility data for the h-BN curve exhibits the best fit. As previously mentioned, there are several possible sources of error, including the complicated morphology of the samples and the presence of multi-layer graphene at the step-edge. The lack of consideration of other short-range scattering processes, plasmon–phonon coupling, and temperature dependent traps also contribute to error in the fitting process, although some of these effects are captured by the fitting parameters.
Figure 4. (a) Plot of carrier mobility for HfO2 and h-BN coated samples on different substrates showing the benefit of h-BN for high quality (low step-density samples). (b) Temperature dependent mobility before and after dielectric integration. (c,d,e) Percent contribution of the various scattering processes contributing to total mobility as a function of carrier density, showing the dominance of remote charged impurity scattering for both dielectrics and minimal contribution from remote SOP scattering with h-BN dielectric. The plot includes acoustic phonon scattering (ADP), optical phonon scattering (OP), impurity scattering (nimp), and remote SOP scattering for the SiC substrate and dielectric overlayer.
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Table 2. Fitting parameters for dielectric coated QFEG samples and Ref. 
|fitting parameter||HfO2 on QFEG1||h-BN on QFEG2||h-BN on QFEG3||CVD graphene on SiO2 (Ref. )|
|ħωSO1 (dielec.) (meV)||75||160||160||70|
|ħωSO2 (dielec.) (meV)||–||–||–||149|
|nsheet (cm−2)||5.4 × 1012 cm−2||1.3 × 1013 cm−2||9 × 1012||1.0 × 1012|
|nimp (cm−2)||1.5 × 1013||9.2 × 1012||3.0 × 1012||3.6 × 1011|
|Δnsheet (cm−2)||3.7 × 1012||1.8 × 1012||–||–|
|Δnimp (cm−2)||6.6 × 1012||3.4 × 1012||–||–|
QFEG sample 1 is coated with O-ALD deposited HfO2 while QFEG samples 2 and 3 are coated with transferred synthetic h-BN. The results indicate that h-BN integration is most beneficial for low step-edge density samples, but shows little benefit for high step-edge density samples (Fig. 4). Integration of h-BN leads to an increase in nimp of 3.4 × 1012 cm−2 while integration of ALD deposited HfO2 leads to a ∼2× greater increase in nimp (6.6 × 1012 cm−2). These results suggest that use of h-BN can be beneficial by limiting the incorporation of additional remote charged impurities, yet Fig. 4 shows that the overall change in mobility after dielectric integration is roughly the same for HfO2 and h-BN coated QFEG samples 1 and 2 at a carrier density of ∼1 × 1013 cm−2 (20 and 21% degradation, respectively). This is explained by the additional effect of the high-k dielectric to screen the graphene from remote charged impurities, whereby the larger dielectric constant acts to offset the additional charged impurities introduced by HfO2. Alternatively, the benefit of dielectric screening will gradually lose out to increased remote SOP scattering as carrier density increases due to the lower energy surface optical modes and increased electron–phonon coupling of the HfO2 coating. Figures 4 plot the percent contribution of the various scattering processes for the different samples, indicating that remote charged impurity scattering dominates all three samples and contributes as much as 94% of the total scattering at a carrier concentration of 1 × 1013 cm−2. Additionally, it is found that remote SOP scattering constitutes a higher percentage of the total scattering processes in the HfO2 coated sample (11%) as compared to the h-BN sample (1–2%) due primarily to the fact that h-BN introduces negligible additional remote SOP scattering over that introduced by the SiC substrate. The integration of h-BN with low step-edge density QFEG (QFEG sample 3) shows minimal temperature dependency and confirms that h-BN introduces negligible additional remote SOP scattering, indicating that h-BN is excellent at preserving the mobility of high quality graphene samples. Figure 4 summarizes the results and plots the carrier mobility as a function of substrate for the two different dielectric coatings, showing the benefit of h-BN for high quality QFEG samples.
Although the integration of HfO2 and h-BN with high step-edge density QFEG shows relatively little gain in performance when utilizing h-BN, low step-edge density QFEG benefits significantly with a ∼2.6× increase in Hall mobility to values >3000 cm2 V−1 s−1, emphasizing that the overall benefit of h-BN dielectrics is highly dependent on the effective remote charged impurity density, nimp. To this end, Fig. 5 plots the simulated percent increase in mobility when using h-BN over HfO2 dielectrics on bilayer QFEG at a fixed carrier density of 1 × 1012 cm−2 along with the experimental results. The results are also compared to exfoliated graphene on SiO2, where the same fitting process was applied to the temperature dependent data collected by Chen et al. . With impurity densities above 5 × 1011 cm−2, QFEG mobility is limited to values <20 000 cm2 V−1 s−1 for sheet carrier densities of 1 × 1012 cm−2. In this regime, remote charged impurity scattering is the dominant scattering mechanism and acts to quench the benefit of h-BN dielectrics, yet high-k dielectrics are able to recover transport properties through dielectric screening and can outperform both h-BN and SiO2. However, for impurity densities below 5 × 1011 cm−2, remote SOP scattering in the HfO2 coated sample begins to contribute more significantly to the total scattering. In this regime and as impurity concentration continues to decrease, QFEG mobility for h-BN samples far surpasses that of HfO2 coated samples and is predicted to increase to values ∼70 000 cm2 V−1 s−1 for high step-edge density samples (QFEG sample 2) and >100 000 cm2 V−1 s−1 for low step-edge density samples (QFEG sample 3), similar to values reported for CVD graphene on h-BN substrates . The large difference in mobilities between the h-BN and HfO2 coated samples found at low impurity densities for this model is due to the reduced contribution of SOP scattering for h-BN relative to HfO2. The authors note that these simulations do not take into account the effect of screening and anti-screening due to graphene plasmon–surface polarized phonon coupling and that a more complete treatment may show that remote SOPs as a result of HfO2 dielectrics may be significantly damped by the dynamic response of the graphene plasmons, as shown by Ong et al. .
Figure 5. Plot of simulated Hall mobility as a function of remote charged impurity density showing the benefit of dielectric screening at high impurity densities and the benefit of low remote SOP scattering at low impurity densities at a fixed temperature of 300 K and carrier concentration of 1 × 1012 cm−2.
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