**About the Editors**

#### **Rositsa Yakimova**

Rositsa Yakimova is a professor emerita in material science at the Department of Physics, Chemistry, and Biology, Linkoping University. She is an internationally recognized expert in the field of ¨ semiconductor crystals and nanostructure growth of wide bandgap materials, e.g., SiC, AlN, ZnO, and gapless graphene. She has made substantial contributions to the development of the sublimation growth process. Her major efforts since the demonstration of graphene have been in the field of epitaxial graphene on SiC. Yakimova has pioneered a novel method for the fabrication of large areas of uniform graphene, which is patent-protected. Since 2008, she has led the research of graphene on SiC at Linkoping University, and for a long time, she was a member of Graphene Flagship. In ¨ total, she has authored over 500 publications and has received 9000 citations. Prof. R. Yakimova is a cofounder and a member of the Board of Graphensic AB, a spin-off company from Linkoping ¨ University.

#### **Ivan Shtepliuk**

Ivan Shtepliuk is a researcher at the Department of Physics, Chemistry, and Biology, Linkoping ¨ University. He received his Ph.D. in Solid State Physics (2013) at the Institute for Problems in Materials Science (Kyiv, Ukraine). During the years 2013 to 2015, he worked as a senior researcher at the Institute for Problems in Materials Science, National Academy of Sciences of Ukraine (Kyiv, Ukraine), where he was engaged in the development of ZnO-based light-emitting diodes. He has since moved to Linkoping University in 2014. His research focus has shifted to graphene-based ¨ materials, such as graphene quantum dots, epitaxial graphene on SiC, and metal-decorated graphene, which are applicable in various technological devices, including heavy metal sensors, gas sensors, and Schottky diodes. He is also an expert in theoretical modelling using density functional theory methods. He has authored more than 60 papers published in reputable peer-reviewed journals.

## *Editorial* **Special Issue "Fundamentals and Recent Advances in Epitaxial Graphene on SiC"**

**Ivan Shtepliuk \* and Rositsa Yakimova**

Semiconductor Materials, Department of Physics, Chemistry and Biology-IFM, Linköping University, SE-58183 Linköping, Sweden; rositsa.yakimova@liu.se

**\*** Correspondence: ivan.shtepliuk@liu.se

**Abstract:** The aim of this Special Issue is to provide a scientific platform for recognized experts in the field of epitaxial graphene on SiC to present their recent studies towards a deeper comprehension of growth mechanisms, property engineering and device processing. This Special Issue gives readers the possibility to gain new insights into the nature of buffer layer formation, control of electronic properties of graphene and usage of epitaxial graphene as a substrate for deposition of different substances, including metals and insulators. We believe that the papers published within the current Special Issue develop cumulative knowledge on matters related to device-quality epaxial graphene on SiC, bringing this material closer to realistic practical applications.

**Keywords:** epitaxial graphene; sublimation; SiC; buffer layer; electronic properties; material engineering; deposition

#### -- **-**

**Citation:** Shtepliuk, I.; Yakimova, R. Special Issue "Fundamentals and Recent Advances in Epitaxial Graphene on SiC". *Appl. Sci.* **2021**, *11*, 3381. https://doi.org/10.3390/ app11083381

Received: 12 March 2021 Accepted: 30 March 2021 Published: 9 April 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**1. Introduction**

For more than a decade, investigations of epitaxial graphene on SiC have gained special urgency in view of its possible applications in many fields, including metrology, electronics and sensorics. Further progress in the development of related technologies requires both rethinking of already existing knowledge and discovery of innovative solutions. This was the primary motivation for opening the call for papers within the Special Issue "Fundamentals and Recent Advances in Epitaxial Graphene on SiC".

In total, the Special Issue encompasses four research papers and three review papers. Two research works touch on crucial aspects of early stage of graphene growth, namely buffer layer formation [1], and graphene quality estimation [2]. Kaushik et al. [3] reported on a principal possibility to tune structural and electronic properties of epitaxial graphene through nitrogen ion implantation. Fundamental knowledge on both copper electrodeposition and atomic layer deposition of high-*k* insulators on epitaxial graphene/SiC is provided by the authors of [4,5]. These results suggest that epitaxial graphene is a stable support for metal and metal oxides, which is important in the context of metal contacts, gating, etc. Concomitantly, the interaction between epitaxial graphene and its environment, including metal contacts may limit, to some extent carrier transport in epitaxial graphene and therefore needs to be considered in detail. The role of such interaction has been a research subject of the review paper by Pradeepkumar et al. [6]. Finally, Wu et al. [7] critically reviewed recent advances in graphene twistronics and identified epitaxial graphene on SiC as the most promising platform for twistronics.

#### **2. Critical Aspects of Epitaxial Graphene Growth: Recipes, Properties, and Quality**

The quality of the buffer layer (also known as C-rich surface reconstruction of SiC and zero graphene layer) is identified as one of the most important factors determining the quality of the epitaxial graphene monolayer on SiC grown via Si sublimation approach. In other words, the fabrication of a large-area epitaxial graphene layer with high thickness uniformity requires pre-formation of high-quality continuous buffer layer on large areas. Thus, an optimization of the growth regime with respect to the buffer layer formation during early stage graphenization process as well as a complete understanding of the growth mechanism are key ingredients to fabricate device-quality graphene. In fact, a successful graphenization process may occur only in a very narrow operational temperature/pressure window which imposes significant restrictions on the growth regime tunability. This makes the optimization of growth conditions quite a challenging task. Despite the large research efforts to tackle this task, it still requires more systematic consideration. In this regard, the critical study by the authors of [1] on optimizing the formation conditions of buffer layers through control of the graphite crucible temperature and varying the Ar gas pressure is a recent contribution to the process. It was revealed that the buffer layer coverage is strongly dependent on the temperature at which Ar gas is introduced, demonstrating a gradual decrease as the temperature increases. The mechanism behind this behavior has been discussed. In the same paper, the relationship between the growth temperature and electronic properties (carrier mobility, carrier density) of quasi-freestanding graphene monolayer and epitaxial graphene monolayer, respectively, were discussed. It was also illustrated that the conductivity type and free carrier density for graphene are extremely sensitive to ambient conditions which was observed by many researchers earlier. In line with this, the review paper by Pradeepkumar et al. [6] provides a more general picture of the effect of epitaxial graphene-ambient interaction on the carrier transport in SiC-supported epitaxial graphene. The authors highlight the adsorption of different molecules (O2, H2O, NO2, H2O2, CO2, NH3, CO, NO, N2O4) as a main reason that underlie conductivity type flipping, transport properties fluctuations and carrier density saturation.

Apart from the unintentional doping of epitaxial graphene by environmental gases and other molecules, the electronic properties of epitaxial graphene on SiC can be modulated by the intentional incorporation of external dopants, as was demonstrated in another study [3]. Nitrogen ion implantation was proposed as an instrumental approach to stabilize the *n*-type conductivity in epitaxial graphene without serious structural damage. However, a balance between graphene quality and implantation dose must be reached. In that light, the mentioned paper dealt with finding the correlation between the fluence value and epitaxial graphene properties such as fragmentation degree, and defect density.

It is instructive that all mentioned works exploit Raman Spectroscopy to estimate the quality of epitaxial graphene. More specifically, the relationship between the intensities of 2*D* and *G* characteristic peaks is used to determine the number of graphene layers, while *D*/*G* amplitude ratio is employed to calculate the defect density. Although the graphene Raman spectroscopy is a mature field, it continues to evolve especially in the direction of signal processing (for example, peak fitting quality). In this regard, the work by Kunc and Rejhon [2] originally offers a Voigt line shape fitting approach for analysis of 2*D* peak line shape for epitaxial graphene, which includes both the inhomogeneous and homogeneous broadening. They also interpreted the physical nature of each term by ascribing the homogeneous broadening to intrinsic lifetime and inhomogeneous broadening to strain fluctuations, respectively.

#### **3. Epitaxial Graphene as a Host for Material Deposition**

Epitaxial graphene on SiC is of great interest because it not only has extraordinary intrinsic properties but also can be used as an atomically flat robust support for nonhybridized growth of different materials, especially metals and metal oxides. Such an integration may expand the functionality of epitaxial graphene and boost the development of innovative technologies in conceptually new fields, like catalysis, plasmonics, and spintronics. Thus, research efforts to contribute to this field and to enrich the existing knowledge capital are in high demand. In response to this demand, the group at Linköping University [4] launched a systematic study of metal electrodeposition on epitaxial graphene on SiC, choosing copper as a model metal at the first stage. This work sheds light on fundamental aspects of copper electrochemistry on epitaxial graphene and shows that copper electroreduction occurs via two subsequent single-electron transfer steps. The

instantaneous nucleation mechanism was identified as a dominating mechanism during copper electrodeposition. The present results provide a deep understanding of the nature of copper–epitaxial graphene interaction, thereby facilitating the design of novel copper– graphene nanohybrid materials.

At the same time, Giannazzo et al. [5] in their work gave an overview of the recent results on the growth of high-k insulators on epitaxial graphene on SiC, focusing on atomic layer deposition of Al2O3 thin layers, which are important for fabrication of epitaxial graphene-based devices. It was argued that the monolayer epitaxial graphene uniformity is a key factor to achieve a homogeneous Al2O3 coverage via direct deposition, the latter has not been successful before in other studies. The role of different seeding layers and surface pre-functionalization in atomic layer deposition processes on epitaxial graphene is critically discussed. Finally, the authors explained the effect of pre-treatment and grown layers on the quality and electronic properties of epitaxial graphene. As was discussed in [6], this issue requires careful consideration, since the interaction between graphene and deposited layers may significantly affect the electron transport in graphene.

#### **4. A New Look at Possible Applications of Epitaxial Graphene on SiC**

Although the epitaxial graphene on SiC is nowadays reasoned to be utilized in electronics, quantum metrology, and gas/liquid sensing, the unique properties of this material make it promising for use in other non-conventional fields. Wu et al. [7] claim that epitaxial graphene on SiC could be regarded as an excellent platform for formation of twisted few-layer graphene with a magic twist angle that might be useful to control spin orders, ferromagnetism, and superconductivity. The authors have substantiated this claim by the fact that owing to its natural compatibility with the semiconductor technologies, epitaxial graphene-based device processing requires no intermediate graphene transfer steps and thus is more attractive from a technological point of view in comparison to transferred graphene. In this regard, there is a plenty of room for manipulation of the twist angle and for formation of twisted graphene layers on SiC with the desired angle through adjusting the sublimation growth conditions.

#### **5. Concluding Remarks**

The Guest Editors consider the current collection of papers as an important piece of the puzzle needed to boost both the more rational implementation of epitaxial graphene into traditional devices and the development of non-conventional innovative technologies. Furthermore, the new results reported in the frame of the Special Issue complement the existing knowledge on buffer layer formation, material preparation–property relationships, and growth mechanisms of different materials on epitaxial graphene. We believe that this information input will provide the driving force behind future experimental efforts to improve the epitaxial graphene quality and to design sophisticated devices exploiting epitaxial graphene as active and passive components.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** Guest editors are grateful to all authors, reviewers, and Applied Sciences editors for their important contributions to prepare the current Special Issue.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Critical View on Buffer Layer Formation and Monolayer Graphene Properties in High-Temperature Sublimation**

**Vallery Stanishev 1, Nerijus Armakavicius 1,2, Chamseddine Bouhafs 1, Camilla Coletti 3, Philipp Kühne 1,2, Ivan G. Ivanov 4, Alexei A. Zakharov 5, Rositsa Yakimova <sup>4</sup> and Vanya Darakchieva 1,2,\***


**Citation:** Stanishev, V.; Armakavicius, N.; Bouhafs, C.; Coletti, C.; Kühne, P.; Ivanov, I.G.; Zakharov, A.A.; Yakimova, R.; Darakchieva, V. Critical View on Buffer Layer Formation and Monolayer Graphene Properties in High-Temperature Sublimation. *Appl. Sci.* **2021**, *11*, 1891. https://doi.org/ 10.3390/app11041891

Academic Editor: Vasili Perebeinos

Received: 31 December 2020 Accepted: 13 February 2021 Published: 21 February 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**Abstract:** In this work we have critically reviewed the processes in high-temperature sublimation growth of graphene in Ar atmosphere using closed graphite crucible. Special focus is put on buffer layer formation and free charge carrier properties of monolayer graphene and quasi-freestanding monolayer graphene on 4H–SiC. We show that by introducing Ar at higher temperatures, *T*Ar, one can shift the formation of the buffer layer to higher temperatures for both *n*-type and semi-insulating substrates. A scenario explaining the observed suppressed formation of buffer layer at higher *T*Ar is proposed and discussed. Increased *T*Ar is also shown to reduce the *sp*<sup>3</sup> hybridization content and defect densities in the buffer layer on *n*-type conductive substrates. Growth on semi-insulating substrates results in ordered buffer layer with significantly improved structural properties, for which *T*Ar plays only a minor role. The free charge density and mobility parameters of monolayer graphene and quasi-freestanding monolayer graphene with different *T*Ar and different environmental treatment conditions are determined by contactless terahertz optical Hall effect. An efficient annealing of donors on and near the SiC surface is suggested to take place for intrinsic monolayer graphene grown at 2000 ◦C, and which is found to be independent of *T*Ar. Higher *T*Ar leads to higher free charge carrier mobility parameters in both intrinsically *n*-type and ambient *p*-type doped monolayer graphene. *T*Ar is also found to have a profound effect on the free hole parameters of quasi-freestanding monolayer graphene. These findings are discussed in view of interface and buffer layer properties in order to construct a comprehensive picture of high-temperature sublimation growth and provide guidance for growth parameters optimization depending on the targeted graphene application.

**Keywords:** epitaxial graphene on SiC; buffer layer; quasi-free-standing graphene; monolayer graphene; high-temperature sublimation; terahertz optical Hall effect; free charge carrier properties

#### **1. Introduction**

Epitaxial graphene on SiC substrates [1–4] holds promise for myriad of future electronic and sensing applications [5–9]. In particular, on the Si-face of SiC, the number of graphene layers can be well controlled and uniform monolayer graphene (MLG) can be obtained. Epitaxial graphene grown in ultra-high vacuum (UHV) on Si-face SiC consists of small domains with a typical size of 200–500 nm [10–15]. In such instances the surface roughens during the graphitization even when growth starts from an atomically-flat surface. If the graphitization is performed in argon (Ar) atmosphere, smoother surface and

large-size MLG domains can be obtained [1,2,15]. However, small inclusions of bi-layer graphene (BLG) are typically present, most often formed on the step edges (due to the small miscut of nominally on-axis wafers) or in association with surface defects [15,16]. Hydrogen pre-treatment has widely been used to provide step-like surface morphology with atomically flat terraces and typical step height of 0.75 nm. Consequently, BLG always forms on the step edges of hydrogen etched SiC and giant step bunching is observed in the graphitization process [2,17]. The two layers in the BLG are AB-stacked, hence possessing a parabolic band structure in contrast to the linearly dispersing bands (Dirac cones) at the K points of the first Brillouin zone of MLG. As a result, BLG inclusions may degrade significantly the transport properties of graphene on the Si-face of SiC and limit its applications [18,19].

Several approaches dispensing with H etching have been explored to eliminate giant step bunching. For example, we have shown that high-temperature sublimation (*T* > 1800 ◦C) in Ar atmosphere in closed graphite crucible delivers wafer-scale MLG with negligible BLG inclusions and without hydrogen pre-treatment [1,15,20–24]. Other openreactor strategies involve pre-conditioning of the SiC wafer by annealing in Ar and/or use of polymer layer, which enables smooth and uniform BLG-free MLG [4,17,25].

Formation of MLG on the Si-face SiC is preceded by consecutive surface reconstructions as the wafer is heated up [26]. The surface undergoes reconstruction from the Si-enriched (3 <sup>×</sup> 3) phase to the C-enriched (6√<sup>3</sup> <sup>×</sup> <sup>6</sup> <sup>√</sup>3)-R30◦ phase. The latter phase is often called "buffer layer" or "zero-layer" graphene because it has the same honeycomb lattice structure as graphene. About 1/3 of the C atoms in this initial layer are covalently bound to the SiC surface and thus the buffer layer is devoid of the electronic properties of graphene [27]. Hydrogen intercalation may be employed to decouple the buffer layer from the substrate turning it into quasi-free-standing (QFS) MLG as the former covalent bonds are broken and the Si dangling bonds at a SiC surface are saturated with hydrogen [27,28].

In UHV conditions the surface reconstructions up to the (6√<sup>3</sup> <sup>×</sup> <sup>6</sup> <sup>√</sup>3)-R30◦ phase occur in the temperature range of 800–1200 ◦C [29]. Upon heating to a higher temperature, the buffer layer decouples from the SiC to form a graphene sheet and another buffer layer forms underneath. Tropm and Hannon [26] have shown that the temperature range within which the surface reconstructions occurs can be shifted up by as much as 200 ◦C in comparison to the case of an ultrahigh vacuum by increasing the Si background pressure to ∼<sup>8</sup> × <sup>10</sup>−<sup>7</sup> Torr using disilane. Ar atmosphere efficiently enhances the Si pressure at the substrate surface since Ar atoms act as a diffusion barrier that limits the Si desorption from the surface. As a result, in Ar atmosphere graphene starts to form at higher temperatures as compared to growth in UHV. It has been shown that in an open Ar atmosphere with a pressure of ∼900 mbar graphene starts to form at temperatures above 1550 ◦C and the buffer layer forms between 1400 ◦C and 1550 ◦C [2,4].

Forming the buffer layer at higher temperature has been theoretically suggested to be the key to grow high-quality graphene [30]. Experimentally it has also been shown that forming a smooth buffer layer at a temperature of *T* 1400 ◦C prevents giant step bunching and consequently it is possible to obtain a smooth surface covered with uniform MLG [17] even on wafers with a large miscut angle of 0.37◦[4]. Introducing Ar at different temperatures during the graphitization process may provide an alternative pathway to influence the phase transition temperature between different surface reconstructions, and hence enable the growth of smooth MLG without the need of special pre-treatment. However, this approach has not been explored despite the intense investigation of buffer layer properties and optimization [4,31–34].

In this work, we report a comprehensive study of the effect of introducing Ar at different temperatures on the buffer layer formation and its properties in high-temperature sublimation for both *n*-type doped and high-purity semi-insulating (SI) 4H–SiC. The free charge carrier density and mobility parameters of the corresponding MLG and QFS-MLG are determined for different environmental conditions and discussed. A combined analysis of free charge carrier and structural properties provides insights into the graphitization processes in an enclosed environment and basis to design growth strategies depending on graphene targeted application.

#### **2. Experimental Details**

Buffer and MLG samples were prepared on the Si-face (0001) of on-axis SI and *n*-type doped 4H–SiC substrates (Cree, Inc., Durham, NC, USA) by high-temperature sublimation in Ar atmosphere [35] using the sublimation growth facilities at Linköping University. The thickness and miscut angle of the SI and *n*-type doped wafers were 360 μm and 0.09◦, and 340 μm and 0.05◦, respectively. The substrates were chemical–mechanical polished (CMP) on the Si-face and optically polished on the C-face. Samples with different sizes of 10 mm × 7 mm, 10 mm × 10 mm or 15 mm × 10 mm were fabricated. The substrates were first cleaned with acetone and ethanol, followed by the standard RCA1 and RCA2 cleaning procedures. Prior to transfer into the growth chamber, the substrates were treated with a hydrofluoric acid solution to remove the native oxide on the surface.

A graphite crucible with a closed inner cavity has been designed with the Virtual Reactor software (http://www.str-soft.com/products/Virtual\_Reactor/ (accessed on 1 February 2021)) to provide uniform (within ∼0.5 ◦C) temperature distribution over 2-inch diameter wafer. The inner cavity design was optimized to minimize the lateral temperature variation resulting in a relatively complex shape. A sketch of the crucible is shown in Figure 1. A special graphite holder is used to position the SiC substrate in the crucible cavity. The crucible was placed into thermally-isolating porous graphite insulation and loaded into the growth chamber. The chamber is pumped down to vacuum level of ∼10−<sup>6</sup> mbar and the crucible was inductively heated. Initially, the temperature is ramped up in vacuum at a rate of ∼16 ◦C per min until the crucible temperature, measured with pyrometer on its surface, has reached 1300 ◦C. During this initial temperature ramp-up, Ar gas with pressure *P*Ar = 850 mbar was introduced into the chamber when the crucible temperature, *T*Ar, was between 640 ◦C and 1300 ◦C. At the moment Ar was introduced the typical vacuum level was ∼<sup>5</sup> × <sup>10</sup>−<sup>5</sup> mbar and it took about 5 min for the Ar pressure to reach *P*Ar = 850 mbar. During this time the temperature typically increased by about 100◦. Above 1300 ◦C, the temperature ramp-up continues at an increased rate of ∼70 ◦C per min until the targeted growth temperature, *T*gr, is reached. The temperature is then kept constant for 0 min or 5 min, which we refer to as growth time, *t*gr. During this final temperature ramp-up, *P*Ar slightly increased to *P*Ar = 880 mbar. Once the growth is finished, the inductive heating is switched off and the sample cools down passively at a rate of ∼65 ◦C per min. The MLG and buffer layer samples were grown at *T*gr = 2000 ◦C and *T*gr = 1600 ◦C, respectively. The growth conditions for all samples are listed in Table 1.

**Figure 1.** A schematic of the crucible with the distribution of the temperature overplotted. Note that the SiC substrate is placed within a tightly closed inner cavity and it is completely surrounded by graphite.


**Table 1.** Growth conditions of the samples studied in this work: *T*Ar, *T*gr and *t*gr of buffer layer (BL), monolayer graphene (MLG) and quasi-free-standing (QFS)-MLG grown on *n*-type and semiinsulating (SI) substrates.

Micro-reflectance and micro-Raman scattering spectroscopy (μ-RS) maps were measured using the set-up described in Ref. [36]. A diode-pumped semiconductor laser with a wavelength of 532 nm (photon energy *E*<sup>L</sup> = 2.33 eV) was used for the excitation. The full-width at half-maximum (FWHM) of the focused laser spot is ∼0.4 μm using a <sup>100</sup>× objective. Typically, 30 × <sup>30</sup> <sup>μ</sup>m<sup>2</sup> reflectance maps with step sizes of 0.3 <sup>μ</sup>m were measured at different locations of the sample. The typical size of the Raman maps was <sup>10</sup> × <sup>10</sup> <sup>μ</sup>m2. For each Raman spectrum, the micro-reflectance was also simultaneously measured. To obtain clean Raman spectra of MLG and buffer layers, a Raman spectrum of a bare 4H–SiC substrate was subtracted. Furthermore, all Raman spectra are normalized to the 4H–SiC substrate.

The surface morphology of the MLG and buffer layers was characterized by tapping mode atomic force microscopy (AFM) (Veeco Dimension 3100). Microprobe low-energy electron diffraction (μ-LEED), low energy electron microscopy (LEEM), X-ray photoelectron emission microscopy (XPEEM) and micro-focused X-ray photoelectron spectroscopy (micro-XPS) were used to investigate the structural properties and chemical composition of the buffer layer samples. The experiments were performed using the ELMITEC-LEEM III instrument at the I311 beamline of the MAX-Lab synchrotron radiation facility in Lund, Sweden.

Contactless terahertz (THz) cavity-enhanced (CE) optical Hall effect (OHE) measurements were performed for the determination of graphene-free charge carrier properties using the custom-built ellipsometry instrumentation at the THz Materials Analysis Center [37]. The OHE describes the magnetic field induced optical birefringence generated by free charge carriers under the influence of the Lorentz force, and can be measured by Mueller matrix ellipsometry [38]. The CE-OHE measurements were performed at room temperature by placing the sample on either of the two sides of a permanent neodymium magnet with a field strength of *B* = 0.548 T and an external cavity of ∼100 μm [39]. Insitu environmental control gas cell was employed to measure the samples in different gases and relative humidity (RH) [37,40]. Mueller Matrix data collected at magnetic fields *B* = +0.548 T and *B* = −0.548 T and their differences were simultaneously analyzed using a stratified optical model with parameterized model dielectric functions (MDFs) assigned to each layer, following the methodology described in Ref. [38]. The model consists of a perfect mirror (magnet), air gap, 4H–SiC substrate and an MLG or a QFS-MLG layer. The dielectric function of 4H–SiC was first determined from measurements of a bare substrate. The substrate MDF parameters were then kept fixed during the analysis of the graphene samples. The MDF of graphene was described by Drude contribution in the presence of magnetic field [37,38]. The free charge carrier mobility μ and sheet density *N*<sup>s</sup> of graphene were determined by non-linear least-squares fit of the calculated Mueller matrix data to the experimental data. The effective mass *m*∗ was parametrized as *m*∗ = - (*h*2*N*s)/(4*πv*<sup>2</sup> F) following Ref. [41], where *<sup>v</sup>*<sup>F</sup> = 1.02 × 106 m s−<sup>1</sup> is the Fermi velocity and *<sup>N</sup>*<sup>s</sup> is the carrier sheet density.

#### **3. Results and Discussion**

#### *3.1. Buffer Layer Formation*

Figure 2 shows μ-Raman spectra of buffer layers on *n*-type 4H–SiC, for which the Ar gas was introduced at *T*Ar = 800 ◦C, 900 ◦C, 1150 ◦C and 1300 ◦C, respectively (BL1-BL4, Table 1). The Raman spectra reveal features in the range of 1200–1700 cm−1, typical for the buffer layer [31,33,42]. The band around 1330 cm−<sup>1</sup> appears to be on par in terms of intensity with the band around 1580 cm−<sup>1</sup> for all samples. It has been argued that the buffer layer Raman spectrum is not composed of discrete peaks but rather reflects the vibrational density of states [42]. The integrated intensity ratio of the D-band around 1330 cm−<sup>1</sup> (DBL) and the G-band 1580 cm−<sup>1</sup> (GBL) can be used to evaluate the content of *sp*<sup>3</sup> hybridization [31] or discuss correlations associated with buffer structure in general [33]. We will come back to this question when comparing buffer layers grown on *n*-type and SI 4H–SiC. However, what is important to the present discussion is the observation that the intensities of the two bands scale down with increasing *T*Ar (see Figure 2). The analysis of the Raman scattering maps shows that the areas with lower reflectivity are associated with lower intensity of the DBL and GBL bands, which we attribute to lower buffer layer coverage. Furthermore, we estimate that the difference of the reflectance between regions that are barely covered with buffer and those with full coverage is ∼1%. Hence, reflectance mapping can also be employed to obtain information on the buffer layer uniformity on a large-scale.

**Figure 2.** Normalized average μ-Raman scattering spectra obtained over 3 μm × 3 μm maps for the buffer layer samples with different *T*Ar, indicated in the inset.

The μ-LEED patterns and the respective 30 μm × 30 μm reflectance maps of the buffer layer samples from Figure 2 are shown in Figure 3. The μ-LEED pattern of the sample with *<sup>T</sup>*Ar = 800 ◦C (Figure 3a) displays well resolved (6√<sup>3</sup> <sup>×</sup> <sup>6</sup> <sup>√</sup>3)-R30◦ surface reconstruction [11]. The uniform buffer layer coverage, for this sample, is corroborated by LEEM I(V) (not shown) and the reflectance map (Figure 3e), which reveals uniform intensity distribution. A clear buffer layer can also be inferred from the μ-LEED pattern of the buffer layer with *T*Ar = 900 ◦C (Figure 3b), however, some charging on the surface is observed. The latter could be associated with oxidized SiC areas not covered by the buffer

layer. For *T*Ar = 1150 ◦C even stronger charging is observed in μ-LEED and patches of oxidized Si are identified by XPEEM (Figure 4). A mixture of the buffer layer and oxidized Si is inferred for this sample. Further confirmation of the suppressed buffer layer formation in the case of *T*Ar = 900 ◦C and *T*Ar = 1150 ◦C comes from the respective reflectance maps (Figure 3f,d), which show nonuniform intensity distribution with dark and bright areas. The size of the dark areas with suppressed buffer layer formation increases with increasing *T*Ar up to 1150 ◦C. This sample also shows the highest RMS of 0.7 nm as compared to 0.35 nm and 0.5 nm for the buffer layers with *T*Ar = 800 ◦C and *T*Ar = 900 ◦C, respectively. Note the resemblance between the XPEEM image (Figure 4) and the reflectance map (Figure 3g). We have previously reported a decrease in the relative reflectance of MLG with respect to the SiC substrate due to the presence of the oxide layer at the interface [43]. Finally, the sample with *T*Ar = 1300 ◦C is severely charging and consists mostly of SiC substrate with the buffer layer just beginning to form, as revealed by μ-LEED (Figure 3d). In this case, the reflectance map (Figure 3h) appears quasi-uniform as the buffer layer nuclei are significantly smaller in comparison with the laser spot size.

**Figure 3.** (**a**–**d**) Microprobe low-energy electron diffraction (μ-LEED) patterns taken at electron energy of 50 eV (**a**,**b**) and 40 eV (**c**,**d**) and (**e**–**h**) 30 μm × 30 μm normalized reflectance maps of buffer layer samples with *T*Ar = 800 ◦C (**a**,**e**),*T*Ar = 900 ◦C (**b**,**f**), *T*Ar = 1150 ◦C (**c**,**g**) and *T*Ar = 1300 ◦C (**d**,**h**). *T*Ar are indicated in the up right corner of the respective images. The difference in reflectance between bare substrate and fully covered with buffer layer is ∼0.01.

Based on the Raman scattering spectroscopy, reflectance mapping as well as μ-LEED results, we can conclude that with the increasing temperature at which Ar is introduced, the formation of the buffer layer is suppressed and shifted to a higher temperature. The same trend is also consistently observed when the buffer layers are formed on SI 4H–SiC substrates. Our investigations further indicate that the SiC substrate areas not covered by the buffer layer are oxidized. There are three possible scenarios: (i) oxidation occurs after the buffer layer formation due to ambient exposure when the samples are removed from the reactor; (ii) oxidation occurs after the buffer layer formation during cooling down and (iii) oxidation occurs during the annealing process. Scenario (ii) and (iii) necessitate residual oxygen in the growth system. Oxidation of buffer and MLG samples as a result of residual oxygen has been previously observed for both conventional and high-temperature sublimation growth [44,45]. It has been suggested that since the graphitization process does not take place in ultra-high vacuum (oxygen-free) conditions, oxygen may be present as a result of oxygen-containing adsorbates on graphite parts and/or inner walls of the reactor. Different growth strategies to obtain high-quality MLG and/or buffer layer (e.g., for QFS-MLG applications) should be employed depending on whether scenario (i), (ii) or (iii) transpires.

**Figure 4.** Si 2p oxide X-ray photoelectron emission microscopy (XPEEM) image taken at photon energy of 133 eV and electron energy of 26 eV with 40-μm field-of-view for the buffer layer sample with *T*Ar = 1150 ◦C. The bright areas correspond to higher content of SiOx but even the dark areas of the image have some oxide component.

In order to elucidate which of the above scenarios takes place, we will discuss in the following the structural evolution of SiC during the sublimation process in Ar atmosphere. Both SiC restructuring and surface reconstruction are expected to be affected by the presence of Ar, which influences the gas pressure at the crystal-vapor interface and the mean free path length. Ar atmosphere effectively enhances the Si pressure since it leads to a reduced Si evaporation rate. The stability of steps on the SiC surface at a given temperature is also affected by Si pressure since the surface Si is in equilibrium with the gas phase Si as well as the bulk SiC. At higher Si pressures higher temperatures are needed to initiate Si decomposition from the terrace [30] and decomposition proceeds rather from the step resulting in smoother surface morphology as compared to ultrahigh vacuum [1,2]. Ar atmosphere also influences the mass transport of various species. Another consequence of the enhanced Si pressure in Ar is that Si depletion close to the SiC is slowed down and a higher temperature is needed to trigger and complete the buffer layer formation (consequently graphene formation). Indeed, it has been demonstrated that the phase transformation temperatures associated with different surface reconstructions on the Si-face SiC can be shifted by several hundred degrees Celsius by balancing the rate of Si evaporation with an external flux of Si [26]. In our experiments, when Ar is introduced at 800 ◦C the entire surface reconstruction process up to 1600 ◦C proceeds under enhanced Si pressure, which should shift the formation of the buffer layer to higher temperatures. In contrast, for *T*Ar = 1300 ◦C the reconstruction occurs in vacuum up to this temperature and the formation of buffer layer should already take place [29]. We have previously shown that no etching by Ar occurs in the sublimation process in closed crucible [45] as confirmed here by step height distribution (See supplementary information Figure S1). Therefore, one would expect a better developed buffer layer for *T*Ar = 1300 ◦C compared to *T*Ar = 800 ◦C. Surprisingly, we find the opposite trend from the Raman scattering spectroscopy, reflectance mapping and μ-LEED results. These findings are not compatible with scenarios (i) and (ii) in which oxidation of uncovered areas occurs after buffer layer formation. A potential explanation for the observed suppression of buffer layer formation at higher *T*Ar is provided by scenario (iii) in which the observed oxidation occurs during the annealing process.

It has been shown that intermediate SiOx on the Si-face of SiC is stable up to a temperature of 1200 ◦C and it is difficult to be fully eliminated even at 1400 ◦C [46]. Thus, if oxidation occurs during annealing and Ar is introduced at temperatures higher than 1200 ◦C the oxide layer will prevent the buffer layer formation. As the oxide layer starts to gradually be removed above 1200–1400 ◦C Ar effectively enhances the Si gas pressure and suppresses the phase transformation to (6√<sup>3</sup> <sup>×</sup> <sup>6</sup> <sup>√</sup>3)-R30◦ surface reconstruction. As a results after heating up to 1600 ◦C, the sample with *T*Ar = 1300 ◦C (BL4) shows only the initial stage of the buffer layer and is mostly uncovered SiC (Figure 3d). At *T*Ar lower than 1200 ◦C (BL1, BL2, BL3), Ar reduces the mean free path of oxygen suppressing oxide

formation and allowing complete (partial) buffer layer formation for *T*Ar = 800 ◦C (900 ◦C– 1150 ◦C.) We note that no charging or any indication of oxidation is observed in the buffer layer sample with *T*Ar = 800 ◦C, which may be understood in view of the reduced mean free path of oxygen at lower temperatures.

Scenario (iii) has several important implications for the growth strategies to obtain high-quality graphene by high-temperature sublimation. As the buffer layer becomes the first graphene layer upon annealing, forming the buffer layer and, consequently, graphene at higher temperatures should be favorable in terms of surface roughness and uniform restructuring as they affect positively free charge carrier mobility. At the same time, one can argue that if the buffer layer forms at lower temperatures it can be conditioned during the annealing process until the temperature of graphene formation is reached, reducing the density of defects such as vacancies or/and *sp*3-defects. Another interesting question is to compare the properties of QFS-MLG obtained from buffer layers grown using different *T*Ar and understand which mechanism has a decisive role. To address these questions we have investigated the free charge carrier properties of MLG and QFS-MLG samples for which the Ar was introduced at different *T*Ar (Table 1). The MLG and QFS-MLG were grown on SI substrates in order to reliably measure the free charge carrier properties. Interestingly, a difference between the Raman scattering spectra grown at the same conditions on *n*-type and SI 4H–SiC is observed.

#### *3.2. Comparison between Buffer Layers Grown on n-Type and SI 4H–SiC*

A comparison of the Raman spectra of buffer layers on *n*-type and SI 4H–SiC obtained at *T*Ar = 800 ◦C is presented in Figure 5a. The Raman spectrum of the buffer layer grown on n-type substrate displays DBL (around 1330 cm<sup>−</sup>1) and GBL (around 1580 cm<sup>−</sup>1) bands with similar intensities. The latter is slightly asymmetric due to a band at around 1530 cm−<sup>1</sup> (see also Figure 2). Such Raman spectrum is typical for carbon-rich graphitic clusters bonded to SiC [27] and can be associated with a large degree of disorder [47]. On the other hand, the buffer layer grown at the same conditions but on SI substrates exhibits blue shift of the DBL and the GBL bands, and the band at around 1530 cm−<sup>1</sup> becomes more pronounced. These are typical vibrational characteristics of a well-connected buffer layer domains [4]. Further information about disorder and the content of *sp*<sup>3</sup> hybridization can be obtained from the histograms of the GBL band position (Figure 6a,c) and the ratios of the DBL and GBL bands areas, *A*DBL /*A*GBL , (Figure 6b,d). The GBL band energy changes from 1583 cm−<sup>1</sup> to 1606 cm−<sup>1</sup> and the *<sup>A</sup>*DBL /*A*GBL changes from 2.0 to 1.3 comparing the buffer layers grown on *n*-type and SI substrates, respectively. A similar trend is also found for the case of *T*Ar = 1300 ◦C (Figure 6e,g). According to the amorphization trajectory presented for nano-crystalline graphite in Ref. [48], these changes can be associated with a significant reduction of the *sp*<sup>3</sup> hybridization content for the case of the SI 4H–SiC. The *A*DBL /*A*GBL is further related to the degree of disorder introduced by the presence of *sp*<sup>3</sup> defects, which is proportional to the average distance between the defects [47]. Accordingly, the density of defects in the buffer layer grown on the SI substrate is 46% lower and the crystallite size is 35% larger. Again, very similar trend is found for the buffer layer with *T*Ar = 1300 ◦C (Figure 6f,h). The observed differences between the two types of substrates could be understood considering the fact that electron concentration generally enhances thermal conductivity. Hence, temperature variations should occur slower for the SI substrates during the heating up, bringing the graphitization process closer to thermodynamic equilibrium and allowing the formation of a well-connected buffer layer with a lower density of defects. It is interesting to note that the vibrational features of the buffer layer formed underneath MLG, grown at *T*gr = 2000 ◦C for 0 s, (Figure 5b) become even finer and bear closer resemblance with the buffer vibrational density of states [42]. Note that the spectral features are identical for the buffer layers on conductive and SI substrates. This further highlights the important roles of the carbon-rich environment and the high temperature for the formation of high-quality buffer layer.

**Figure 5.** A comparison between the average μ-Raman scattering spectra for buffer layer samples with *T*Ar = 800 ◦C: (**a**) on n-type and SI 4H–SiC, and (**b**) the buffer layer features in fully-formed MLG at *T*Gr = 2000 ◦C for 0 s on *n*-type and SI 4H–SiC.

**Figure 6.** Histograms of the GBL band position and the ratio of the DBL and GBL band areas, *A*DBL /*A*GBL , for the buffer layers grown with *T*Ar = 800 ◦C (**a**–**d**) on *n*-type (**a**,**b**) and SI (c,d) 4H– SiC; and for the buffer layers grown with *T*Ar = 1300 ◦C (**e**–**h**) on *n*-type (**e**,**f**) and SI (**g**,**h**) 4H–SiC. The histograms are obtained over Raman maps of 3 μm × 3 μm. Three Lorentzian lineshapes centered around GBL of 1585–1600 cm<sup>−</sup>1, DBL of 1330–1530 cm−<sup>1</sup> and a band centered at 1340 cm−<sup>1</sup> were used for the fitting.

Comparing the buffer layers grown on *n*-type substrates and different *T*Ar, a moderate blue-shift of the G-like band position for *T*Ar = 1300 ◦C to 1593 cm−<sup>1</sup> with respect to the sample with *T*Ar = 800 ◦C (1583 cm−1) can be seen (Figure 6a,e). This can be explained by a reduced *sp*<sup>3</sup> hybridization content as expected due to the higher temperature at which the reconstruction occurs. At the same time, the *A*DBL /*A*GBL increases from 2.0 to 2.7 (Figure 6b,f), which could be related to a reduced crystallite size with 30%. This finding is in accordance with our μ-LEED results showing that the buffer layer with *T*Ar = 1300 ◦C has just begun to form. We now turn our attention to the buffer layers grown with different *T*Ar on SI 4H–SiC substrates. The same trend of suppressed reconstruction with increasing *T*Ar is found. In fact, for the case of *T*Ar = 1300 ◦C heating up to 1600 ◦C did not result into a buffer layer formation and heating up to 1800 ◦C was needed for a clear buffer layer Raman spectrum to be obtained. Interestingly, the buffer layers grown with *T*Ar = 800 ◦C and *T*Ar = 1300 ◦C exhibit very similar GBL positions (Figure 6c,g) and *A*DBL /*A*GBL ratios (Figure 6d,h), indicating similar *sp*<sup>3</sup> hybridization contents and densities of defects. A slightly broader distribution is observed for the case of *T*Ar = 1300 ◦C for both n-type and SI 4H–SiC substrates, reflecting a slightly larger variation of the crystallite size.

Based on these results we can conclude that the temperature at which Ar is introduced has a determining role in the formation of the buffer layer in high-temperature sublimation in closed crucible independently of the 4H–SiC substrate conductivity. As a result of an interplay between oxidation and restructuring in Ar atmosphere, the formation of the buffer layer is shifted to higher temperatures for increased *T*Ar of 1300 ◦C. Increasing *T*Ar also leads to reduction of *sp*<sup>3</sup> hybridization contents and densities of defects on *n*-type 4H–SiC. However, *T*Ar has a less pronounced effect for SI substrates, where ordered buffer layers form with similar structural properties.

#### *3.3. Free Charge Carrier Properties of MLG and QFS-MLG*

It is well-known that MLG on SiC is intrinsically *n*-type doped [49–51]. However, exposure to ambient can cause environmental doping of graphene via an acceptor redox reaction at the surface of the graphene involving various environmental gases (O2, H2O, and CO2), which results in electron withdrawal [52]. Consequently, MLG can exhibit *p*-type conductivity depending on sample history [24,53]. We have previously shown that the THz OHE is an excellent tool to precisely determine free charge carrier density and mobility parameters of graphene and monitor their in-situ variation under the influence of different gases [24,37,40,54]. In order to determine the intrinsic properties of MLG and QFS-MLG, prior to the measurements they were annealed in vacuum (10−<sup>6</sup> mbar) at 1000 ◦C and 500 ◦C, respectively. The annealing temperature was confirmed to not cause deintercalation or any changes in the QFS-MLG structural properties by LEEM, AFM, and μ-LEED. The samples were kept in dry N2 during the measurements and storage. In addition, we have performed measurements after purging with dry N2 for several days and air with RH of 45% for several hours. Both transient and static measurements were carried out. Finally, the samples were measured after being stored in ambient conditions for several months. We have selected for these investigations samples with the following *T*Ar: (i) *T*Ar = 800 ◦C, for which the surface reconstruction happens entirely in Ar atmosphere and that shows completed buffer layer after heating to *T*gr = 1600 ◦C (0 s); (ii) *T*Ar = 1300 ◦C, for which the surface reconstruction happens entirely in vacuum, and which needed heating to *T*gr = 1800 ◦C for the buffer layer to form. Although no indications of surface oxidation were observed for the buffer layer sample with *T*Ar = 800 ◦C, a nanoscale oxidation cannot be excluded. Furthermore, the graphitization process is shifted to higher temperatures in comparison to *n*-type substrate as pointed out above. We, therefore, included in our investigation MLG and QFS-MLG samples, for which the Ar was introduced at (iii) *T*Ar = 640 ◦C. Growth temperature *T*gr = 1600 ◦C was employed to produce the buffer layer sample in this case. The QFS-MLG samples were obtained by hydrogen intercalation of the respective buffer layers as described in Ref. [28]. The MLG samples were fabricated using our optimized conditions of *T*gr = 2000 ◦C for 0 s growth time, which results in less than 1% BLG inclusions. The sample with *T*Ar = 1300 ◦C required a longer growth time of 5 min for a homogeneous MLG to form leading to increased BLG inclusions of 8%.

Figure 7 shows the free charge carrier density (left panel) and mobility (right panel) of MLG (filled symbols) and QFS-MLG (open symbols) with different *T*Ar for different environmental conditions. The mobility parameters were found to be slightly anisotropic in accordance with our recent study [24]. The anisotropy, which is caused by the substrate step edges, does not have any bearing on the results discussed in the current work. Consequently, for brevity we present here the averaged mobility between the parameters determined along and perpendicular to the step edge. The freshly annealed MLG samples show *<sup>n</sup>*-type conductivity, as expected, with values in the range of 3.9 × <sup>10</sup> <sup>12</sup> cm−<sup>2</sup> to 6.6 × 1012 cm<sup>−</sup>2. Due to the semi-insulating nature of the substrates, the MLG doping should be entirely governed by charge transfer due to surface donor states [55]. All three free electron density values are below the saturation density of n-type doping of MLG of 10<sup>13</sup> cm−<sup>2</sup> [55], indicating successful efficient annealing of donors on and near the SiC surface. The observed differences with *T*Ar, albeit small, are significantly below the error bar of 0.3 ×

1012 cm<sup>−</sup>2. Since the MLG with *T*Ar = 1300 ◦C was obtained for a considerably longer time (5 min as compared to 0 s) it is tempting to speculate that the longer annealing may have a positive effect on reducing the interface dangling bonds effectively reducing the density of the surface state and leading to a lower free electron density. We have previously shown that purging with N2 (or inert gases) effectively removes the ambient acceptor dopant, which may require up to several days of purging [37,40]. The free electron densities in the MLG samples with *T*Ar = 640 ◦C and *T*Ar = 800 ◦C after purging in dry N2 for 9–10 days increased slightly to 5.1 × <sup>10</sup><sup>12</sup> cm−<sup>2</sup> and 7.0 × <sup>10</sup><sup>12</sup> cm−2, respectively, remaining below the the saturation density of n-type doping. The electron mobility parameters in these two cases slightly decreased in comparison to the freshly annealed samples, most likely as a result of the slightly increased charge density. The MLG with *T*Ar = 1300 ◦C shows the opposite behavior with slightly decreased charge density and slightly increased mobility parameter. Overall the purging with dry N2 led to very small changes in the MLG electron density and mobility, which can be considered as the intrinsic free-electron parameters of MLG.

**Figure 7.** Free charge carrier density (left panel) and mobility (right panel) of MLG (filled symbols) and QFS-MLG (open symbols) with *T*Ar = 640 ◦C (black circles), *T*Ar = 800 ◦C (red squares) and *T*Ar = 1300 ◦C (blue triangles) for different environmental conditions: after annealing in vacuum (Annealed), after being purged with dry N2 for several days (N2 RH 0%), after being purged with moist air (RH 45%) for several hours (Air RH 45%), and after being exposed to the ambient for several months (Ambient).

As expected after purging with moist air (RH of 45%) the electron density in the MLG samples decreased due to the acceptor redox reaction at the graphene surface. The samples with different *<sup>T</sup>*Ar show very similar electron density of ∼<sup>2</sup> × 1012 cm−<sup>2</sup> after ∼20 h of purging. We have measured the in-situ variations of free charge carrier properties and found that approximately 45 h purging in moist air are needed to flip the conductivity of MLG from *<sup>n</sup>*-type to *<sup>p</sup>*-type with free hole density of 1.4 × 1012 cm<sup>−</sup>2. Long-term exposure in ambient conditions (several months) leads to only a very small increase of free hole density to ∼<sup>2</sup> × <sup>10</sup><sup>12</sup> cm−<sup>2</sup> indicating saturation of p-type ambient doping in MLG. Again, very similar free hole densities are found for the samples with different *T*Ar = 1300◦. On the other hand, the free charge carrier mobility of the ambient doped MLG with *T*Ar = 1300 ◦C is more than 50% larger than the respective values of MLG with *T*Ar = 800 ◦C and *T*Ar = 640 ◦C. This is true for both the cases of free electrons and free holes (see Figure 7 right panel results for Air RH 45% and Ambient). This finding is very interesting considering that the samples with *T*Ar = 640 ◦C and *T*Ar = 800 ◦C have better MLG coverage of 99% and lower RMS 0.4 nm, compared with the *T*Ar = 1300 ◦C sample, which has 92% MLG coverage and RMS 0.75 nm. It was previously suggested that dominant scattering mechanisms at room temperature in graphene on SiC are the remote interface phonon scattering, as a result of coupling to the polar modes in the substrate, and scattering by impurities [56–58]. Since the MLG samples are grown at the same *T*gr and have a similar history we do not

anticipate a difference in impurity levels. It is thus plausible to suggest that in the MLG with *T*Ar = 1300 ◦C the interface phonon scattering is reduced as a result of different interface properties. We recall that the buffer layers grown at different *T*Ar on SI substrates exhibit very similar *sp*<sup>3</sup> contents and defect densities (Figure 6). Furthermore, the Raman scattering spectral features associated with the buffer layer in the respective MLG samples with different *T*Ar are practically identical. Hence, the reduced interface phonon scattering is likely a result of a different interface between MLG and the buffer layer rather than between buffer layer and SiC substrate. This suggestion is further supported by the similar free charge carrier density in the ambient doped MLG with different *T*Ar indicating similar surface state densities. To gain further insight into the origin of the different interface properties between MLG and the buffer layers we turn now our attention to the free charge carrier properties of the QFS-MLG samples.

In QFS-MLG the intercalated hydrogen saturates the Si dangling bonds passivating the interface donor states. Consequently, QFS-MLG exhibits *p*-type doping induced by the spontaneous polarization of the SiC substrate [28,59,60]. The resulting free hole density in QFS-MLG on SI 4H–SiC was reported to be 8.6 × 1012 cm−<sup>2</sup> as determined by angular resolved photo-electron spectroscopy (ARPES) [60]. As expected our freshly annealed QFS-MLG samples show *p*-type conductivity (see Figure 7 left panel). We find very similar free hole densities in the QFS-MLG with *<sup>T</sup>*Ar = 640 ◦C and *<sup>T</sup>*Ar = 800 ◦C of 1.2 × <sup>10</sup><sup>13</sup> cm−<sup>2</sup> and 1.5 × <sup>10</sup><sup>13</sup> cm−2, respectively. These values are slightly higher than the free hole density expected from pure polarization doping [60]. It is possible that some residual ambient doping is present as the annealing temperature for the QFS-MLG samples was relatively low in order to prevent deintercalation. Purging in dry N2 for several days lead to a small reduction of the free hole density in these two samples to ∼1.0 × <sup>10</sup><sup>13</sup> cm−2, which is suggested to be the intrinsic value for our QFS-MLG resulting from polarization doping. We consider this to be a good agreement with the previously reported value of 8.6 × <sup>10</sup><sup>12</sup> cm−<sup>2</sup> [60] given the different experimental techniques used in the two works and the various fitting parameters employed to deduce the free hole concentration from ARPES. Both the freshly annealed and the dry N2 purged QFS-MLG with *T*Ar = 1300 ◦C show significantly lower free hole density of 4.4 × <sup>10</sup><sup>12</sup> cm−<sup>2</sup> and 2.1 × 1012 cm−2, respectively. According to the polarization doping model, the negative pseudo-polarization charge, which is a constant parameter for the 4H–SiC, is balanced by the free holes in the QFS-MLG and the positive space charge in the substrate depletion layer [60]. Since the bulk doping in the SI substrate is the same for all three samples leading to a similar positive space charge in the substrate depletion layer, the observed lower free hole density in QFS-MLG with *T*Ar = 1300 ◦C indicates the presence of donor surface states. As noted earlier, the buffer layers grown at different *T*Ar on SI substrates exhibit very similar *sp*<sup>3</sup> contents and defect densities (Figure 6). We also confirmed by μ-Raman scattering spectroscopy mapping that no structural changes occur as a result of the intercalation process. Recall that in comparison to lower *T*Ar the buffer layer with *T*Ar = 1300 ◦C is incomplete. We speculate that this incomplete buffer layer formation may be the cause of the surface donor states, likely dangling bonds. Interestingly, purging with moist air (RH 45%) for ∼18 h leads to small increase of free hole density in QFS-MLG with *T*Ar = 800 ◦C and *T*Ar = 640 ◦C while for *T*Ar = 1300 ◦C the hole density remains unchanged. This can be potentially explained by the above-mentioned scenario since the purge with moist air has different effects: for the polarization doped QFS-MLG it leads to chemical acceptor doping of graphene while for the sample with *T*Ar = 1300 ◦C it leads to passivation of surface donor states. The two processes will naturally have different dynamics. This proposal is also consistent with the results for prolonged exposure to ambient. The free hole density in QFS-MLG with *<sup>T</sup>*Ar = 1300 ◦<sup>C</sup> increases to 9.2 × <sup>10</sup><sup>12</sup> cm−<sup>2</sup> nearing the intrinsic polarization doping since most (all) surface donor states have been passivated. For *T*Ar = 640 ◦C and *T*Ar = 800 ◦C the free hole densities increase to 2.3 × <sup>10</sup><sup>13</sup> cm−<sup>2</sup> and 1.9 × <sup>10</sup><sup>13</sup> cm−2, respectively, as a result of chemical acceptor doping. In all cases, except for the freshly annealed samples, the largest hole mobility parameters are found for the QFS-MLG with *T*Ar = 1300 ◦C . This

is most likely related to the generally lower free hole density parameters. Note that the free charge mobility (and density) parameters represent average parameters obtained over the entire sample area of 10 mm × 10 mm.

#### **4. Conclusions**

We have critically reviewed the processes in high-temperature sublimation growth of graphene in Ar atmosphere using closed graphite crucible with emphasis on buffer layer formation and free charge carrier properties of MLG and QFS-MLG on 4H–SiC. We have explored the effect of introducing Ar at different temperatures, *T*Ar. We have found that the buffer layer coverage decreases with increasing *T*Ar with well-developed buffer layer for *T*Ar = 800 ◦C, while for *T*Ar = 1300 ◦C the buffer layer is just beginning to form. The observed suppression of buffer layer formation at higher *T*Ar is accompanied by surface oxidation of the uncovered regions of the SiC substrates. A scenario in which oxidation occurs during the annealing process is proposed to explain the peculiar shift of the buffer layer formation to higher temperatures. The latter leads to reduced *sp*<sup>3</sup> hybridization content and defect densities in the buffer layer when grown on *n*-type conductive substrates. Growth on SI substrates results in significantly improved structural properties of the buffer layers, which is attributed to a slower graphitization process closer to equilibrium due to the reduced thermal conductivity of the substrate. For SI substrate *T*Ar plays a minor role for the *sp*<sup>3</sup> hybridization content and defect densities in the buffer layer. A comprehensive study of the free charge density and mobility parameters of MLG and QFS-MLG with *T*Ar = 640 ◦C, *T*Ar = 800 ◦C and *T*Ar = 1300 ◦C and four different environmental conditions: freshly annealed in vacuum, after purging with dry N2 (RH 0%) for ∼20 h, after purging with moist air (RH 45%) for ∼18 h and after ambient exposure for several months, allows us to draw the following conclusions:

(i) successful efficient annealing of donors on and near the SiC surface can be inferred for MLG grown at 2000 ◦C independent of *T*Ar;

(ii) approximately 45 h purging with moist air (RH 45%) is needed to flip the conductivity of MLG from *n*-type to *p*-type and long term exposure to ambient leads to a saturation of the free hole density at ∼<sup>2</sup> × <sup>10</sup><sup>12</sup> cm<sup>−</sup>2;

(iii) the highest mobility of MLG is determined for *T*Ar = 1300 ◦C in both intrinsically *n*-type and ambient *p*-type doped situations. It is suggested that this is a result of reduced interface phonon scattering due to improved interface between MLG and the buffer layer rather than between the buffer layer and the SiC substrate;

(iv) a free hole density of ∼1.0 × <sup>10</sup><sup>13</sup> cm−<sup>2</sup> is suggested to be the intrinsic value for our QFS-MLG resulting from polarization doping in good agreement with the previously reported value of 8.6 × <sup>10</sup><sup>12</sup> cm−<sup>2</sup> [60];

(v) *T*Ar is found to have a profound effect on the free hole parameters of QFS-MLG. A significantly lower free hole density of ∼<sup>2</sup> × 1012 cm−<sup>2</sup> is found in intrinsic QFS-MLG with *T*Ar = 1300 ◦C, which is attributed to additional surface donor states associated with incomplete buffer formation.

Our findings contribute to establishing a comprehensive picture of high-temperature sublimation growth and provide guidance for growth parameters optimization depending on the targeted application of QFS-MLG and MLG on SiC.

**Supplementary Materials:** The following are available online at https://www.mdpi.com/2076-3 417/11/4/1891/s1, Figure S1: Representative AFM images and step height distributions of buffer layers grown on SI 4H–SiC with *T*Ar = 800 ◦C and *T*Ar = 1300 ◦C.

**Author Contributions:** individual contributions of the authors are as follows: conceptualization, V.S. and V.D.; methodology, N.A. and P.K.; software, V.S.; validation, V.S.; formal analysis, V.S. and V.D.; investigation, V.S., N.A., A.A.Z., C.C., I.G.I. and C.B.; resources, R.Y. and V.D.; writing—original draft preparation, V.S. and V.D.; writing—review and editing, V.S., N.A., R.Y., C.C. and V.D.; visualization, V.S. and V.D.; supervision, V.D.; project administration, V.D.; funding acquisition, R.Y. and V.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors would like to acknowledge financial support from the Swedish Research Council (VR Contract 2016-00889), the Swedish foundation for strategic research (SSF) under Grants No. FFL12-0181 and No. RIF14-055, the Swedish Government Strategic Research Area in Materials Science on Functional Materials at Linköping University (Faculty Grant SFO Mat LiU No 2009 00971). RY is grateful for financial support by SSF via grant RMA 15-0024.

**Acknowledgments:** We thank Valdas Jokubavicius for his help with annealing the MLG and QFS-MLG samples in vacuum.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


### *Article* **Raman 2D Peak Line Shape in Epigraphene on SiC**

#### **Jan Kunc \* and Martin Rejhon**

Institute of Physics, Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, CZ-121 16 Prague 2, Czech Republic

**\*** Correspondence: kunc@karlov.mff.cuni.cz

Received: 29 February 2020; Accepted: 23 March 2020; Published: 30 March 2020

**Abstract:** We measured a 2D peak line shape of epitaxial graphene grown on SiC in high vacuum, argon and graphene prepared by hydrogen intercalation from the so called buffer layer on a silicon face of SiC. We fitted the 2D peaks by Lorentzian and Voigt line shapes. The detailed analysis revealed that the Voigt line shape describes the 2D peak line shape better. We have determined the contribution of the homogeneous and inhomogeneous broadening. The homogeneous broadening is attributed to the intrinsic lifetime. Although the inhomogeneous broadening can be attributed to the spatial variations of the charge density, strain and overgrown graphene ribbons on the sub-micrometer length scales, we found dominant contribution of the strain fluctuations. The quasi free-standing graphene grown by hydrogen intercalation is shown to have the narrowest linewidth due to both homogeneous and inhomogeneous broadening.

**Keywords:** epitaxial graphene; silicon carbide; Raman spectroscopy; 2D peak line shape; G peak; charge density; strain

#### **1. Introduction**

The Raman spectroscopy of graphene is a well-established technique [1–3] to determine number of graphene layers [4], strain [5,6], charge density [6–9], grain size [10–14], graphene functionalization [15], misorientation of graphene layers [16] or degree of hydrogen intercalation of epitaxial graphene on SiC [17,18]. The graphene's most prominent Raman spectral features are the D peak, G peak and 2D peak. The D peak reflects the amount of defects, or the graphene grain size. The G peak is related to the in-plane bond-stretching optical vibrations of sp<sup>2</sup> hybridized carbon atoms in the graphene lattice [19,20]. The 2D peak is recognized as a combination mode of lattice and electronic excitations. The predicted unique property of the 2D peak is its line shape. The 2D peak is predicted to have a Lorentzian line shape in the case of the single layer graphene (SLG) [21]. The 2D peak is predicted to have four components in the case of bilayer graphene. The single and four-component nature of the 2D peak was proved experimentally [22,23]. It is also known that the spectral position of the 2D peak is determined by the uniaxial [24–27] and biaxial [5,6,26] mechanical strain and charge density [7,8] of the graphene layer. The scaling of the 2D peak position with charge and strain was studied extensively. The 2D peak line shape is influenced by the strain uniformity [28] and strain fluctuations on the nanometer scale [29]. Also, different line shape was identified for graphene on SiC terraces and step edges [30].

However, beside the known spectral line shape and parameters determining the position of the 2D peak, there is little known about the combined effect of the homogeneous and inhomogeneous broadening. The knowledge of the mutual effects of the homogeneous and inhomogeneous broadening can provide deeper insight into the formation of the graphene layers and it can be used to further optimize graphene growth. Hence, we propose here to analyze in detail the spectral line shape of the 2D peak. Instead of describing the 2D peak line shape only by the Lorentzian broadening, we assume also the contribution by the inhomogeneous broadening. The homogeneous and inhomogeneous

broadening have to be taken into account simultaneously. As the inhomogeneous broadening describes the random nature of the parameters determining the position of 2D peak, it is described by the normal (Gaussian) distribution. The mutual effect of the homogeneous and inhomogeneous broadening thus leads to the convolution of the Lorentzian and Gaussian broadening, also called the Voigt broadening. We show here that the Voigt profile describes the 2D peak line shape better than the Lorentzian profile. We show that the Voigt broadening, though it has one more fitting parameter, it does not show any signatures of over-parametrized model. We test the better fit quality by the F-test and the results are compared for three different samples and four different locations on each sample.

#### **2. Materials and Methods**

The epitaxial graphene was grown on a Si-face of 6H-SiC by thermal decomposition [31–34]. We grew three samples, each under different growth conditions. The sample grown in vacuum [31,32] at 10−<sup>5</sup> mbar was heated to 1600 ◦C for 5 min. The sample grown in argon [33] at 1050 mbar was heated at 1650 ◦C for 5 min. The quasi free-standing monolayer graphene (QFMLG) [35,36] was grown in two steps. First, the graphene buffer layer was grown in 1050 mbar of argon at 1550 ◦C for 5 min. The second growth step was the hydrogen intercalation at 1120 ◦C for 5 min followed by a 2 hours long cooling to 600 ◦C. The growth temperature was adjusted with respect to the sublimation rate of silicon from the heated SiC wafer. The growth in high vacuum allows high silicon sublimation rates, hence, the temperature is reduced to 1600 ◦C in contrast to growth in 1050 mbar of argon, where the growth temperature has to be increased by 50 ◦C to 1650 ◦C to promote silicon sublimation comparable to the sublimation rate in high vacuum. The growth of QFMLG requires to grow so called buffer layer first. The buffer layer grows at even lower temperature than the single layer graphene. For this reason, the first growth step is performed at 1050 mbar of argon at 1550 ◦C for 5 min. The buffer layer is the graphene lattice, where about 30% carbons are sp3 bonded to the underlying SiC substrate [37]. This sp<sup>3</sup> bonding was switched into sp<sup>2</sup> bonding by hydrogen intercalation [17,38]. More details on graphene growth and hydrogen intercalation can be found in our previous works [17,18,39].

Raman spectra were measured by WITec alpha300 (WITec, Ulm, Germany) micro-Raman confocal microscope. The Raman spectra were excited by 532 nm laser light. We used 25 mW laser power and the laser spot diameter was 1 μm in the focal plane. The spectra were acquired in two 30 s accumulations to achieve low level of noise in the tails of the 2D peak. The spatial Raman maps were measured on the area 4 × <sup>4</sup> <sup>μ</sup>m<sup>2</sup> and we accumulated Raman spectra twice 5 s to optimize lateral resolution, noise level and the total measurement time. The Atomic Force Microscopy (AFM) and Lateral Force Microscopy (LFM) were measured by WITec alpha300 (WITec, Ulm, Germany) AFM. We measured AFM and LFM in the contact mode, and, the scanned area was 20 × <sup>20</sup> <sup>μ</sup>m2.

The 2D peak was fitted by Lorentzian and Voigt line shapes. The Lorentzian line shape was taken in the form of Equation (1)

$$G\_L = \frac{\gamma}{\pi[(\mathbf{x} - \mathbf{x}\_0)^2 + \gamma^2]},\tag{1}$$

where *x*<sup>0</sup> is the spectral position of the peak, *γ* determines the width of the 2D peak. The Equation (1) is scaled by a factor *I*<sup>0</sup> describing the intensity of the Lorentzian peak. Parameter *γ* is related to the more experimentally accessible Full Width at Half Maximum (FWHM) of the Lorentzian peak *fL* by Equation (2).

$$f\_{\mathcal{L}} = 2\gamma \tag{2}$$

The Voigt line shape is given by the convolution of the Lorentzian Equation (1) and Gaussian Equation (3) line shape

$$G\_G = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-x\_0)^2}{2\sigma^2}}.\tag{3}$$

The Gaussian broadening *σ* relates to the FWHM *fG* by Equation (4)

$$f\_{\mathbb{G}} = 2\sigma \sqrt{2\ln(2)}.\tag{4}$$

As the convolution of the Gaussian and Lorentzian function is numerically demanding, we used a common approximation of the Voigt line shape [40,41] given by Equation (5)

$$G\_V = l\_0 \left[ \eta G\_L + (1 - \eta) G\_G \right],\tag{5}$$

where *η* is a function of the total FWHM *f* , *fG* and *fL*

$$\eta = 1.36603 \frac{f\_L}{f} - 0.47719 \left(\frac{f\_L}{f}\right)^2 + 0.11116 \left(\frac{f\_L}{f}\right)^3. \tag{6}$$

The total Voigt broadening *f* is given by Equation (7)

$$f = (f\_{\rm G}^5 + 2.69269 f\_{\rm G}^4 f\_{\rm L} + 2.42843 f\_{\rm G}^3 f\_{\rm L}^2 + 4.47163 f\_{\rm G}^2 f\_{\rm L}^3 + 0.07842 f\_{\rm G} f\_{\rm L}^4 + f\_{\rm L}^5)^{1/5}. \tag{7}$$

#### **3. Results**

We show in Figure 1 the typical Raman spectra of graphene grown in high vacuum, argon and the intercalated buffer, so called QFMLG. All spectra show the typical graphene characteristics as the D peak, G peak and 2D peak. The Raman spectra also show typical characteristics of the single-layer graphene. These are the ratio of the integrated 2D peak to G peak intensity larger than 2, and, the single-component line shape of the 2D peak. The characteristic distinction between the hydrogen intercalated and non-intercalated samples can be also found by the absence/presence of the peaks labeled (1), (2) and (3) in inset of Figure 1. These peaks were attributed to the buffer layer [18,42]. The hydrogenated (QFMLG) and non-hydrogenated (SLG-vac, SLG-Ar) samples can be also distinguished by a low intensity background in the spectral range of the D peak. This background was assigned to the buffer layer, too [42]. We observe rather similar Raman spectra of graphene grown in argon and high vacuum. Their differences are discussed in the following detailed analysis.

**Figure 1.** Raman spectra of quasi free-standing monolayer graphene (QFMLG), single layer graphene grown in vacuum (SLG-vac) and single layer graphene grown in argon (SLG-Ar). Graphene related D, G and 2D peaks are labeled by arrows. Inset shows details of Raman spectra including three buffer related Raman modes, labeled by arrows (1), (2) and (3).

The 2D peaks of the three samples are shown in Figure 2 by black circles. We obtained the spectra by two 30 s integration periods. The 30 s integration reduced the noise level and the two accumulations allowed us to remove spikes in recorded spectra. The low level of the noise is essential to fit the 2D peak line shape at low-energy and high-energy tails from the central peak position. Before we analyzed the 2D peak line shape, we subtracted a linear background. The background was determined using five experimental points at 2530 and at 2870 cm−1. The 2D peaks are fitted by the Lorentzian (green curves) and Voigt (red curves) line shapes in Figure 2. The residuals are displayed in the top insets and the corresponding histograms of the residuals are shown in the bottom insets of Figure 2.

**Figure 2.** Line shape of the 2D peak in (**a**) QFMLG, (**b**) SLG-Ar and (**c**) SLG-vac. The black dots are the experimental data, green curve is a Lorentzian fit and red curve is the Voigt fit to the experimental data. The upper insets show the residuals for the (green point) Lorentzian and (red points) Voigt line shape. The bottom insets show the histograms of thee residuals.

The fit improvement by the Voigt line shape is tested by the F-test. The Degrees of Freedom (DOF) and Residual Sum of Squares of the Lorentzian (model 1) and Voigt (model 2) line shape are summarized in Table 1. The null hypothesis for the F-test is: the Lorentzian and Voigt line shapes are the same. The F-number in Table 1 results from the comparison of the two models. The p-number determines the cumulative probability that the expected F number is lower than the experimentally determined F-number.

**Table 1.** Parameters used to perform the F-test. Degrees of freedom (DOF), Residual Sum of Squares (RSS1) of the Lorentzian model and RSS2 of the Voigt model. F is the value of the F-statistics. Number *p* is the p-number to reject the null hypothesis.


The F-test shows that the Voigt line shape describes the 2D peak line shape better. As the Voigt line shape has four fitting parameters (one more with respect to the Lorentzian broadening), we need to verify that the Voigt line shape model is not overparametrized. We verify the overparametrization by determining the *χ*<sup>2</sup> statistics of the fitted models. The *χ*<sup>2</sup> statistics requires the experimental error *σexp*. We determine the experimental error from the high-energy tails of the 2D peak, as depicted in Figure 3. The high-energy tail is fitted by the second order polynomial to describe the trend of the experimental data. The residuals are considered as a random experimental error. The normality of these residuals is tested by the Kolmogorov-Smirnov test, as depicted by the cumulative distribution function in the inset of Figure 3. The normality is verified at the 95% confidence level. The experimental error is estimated to be *<sup>σ</sup>exp* ≈ 10 cm−1. The centered and normalized *<sup>χ</sup>*<sup>2</sup> for the Lorentzian and Voigt model is 1000 and 260, respectively.

**Figure 3.** The high-energy tail of the 2D peak (black points) was fitted to the polynomial of the 2nd order (red curve) to determine the experimental error. The normality of the residuals is tested by the Kolmogorov-Smirnov test (inset).

We fitted the 2D peak spectra at four different positions on each sample. The statistics for the three samples is shown in Figure 4 and summarized in Table 2. The FWHM was obtained directly from the experimental data without using any fitting procedure. The parameters *γ* and *σ* are the fitting parameters of the Voigt line shape. We observe that the QFMLG shows the narrowest broadening. The QFMLG also shows the smallest contribution of the homogeneous and inhomogeneous broadening. It can be seen in Figure 4 that the homogeneous and inhomogeneous broadening are similar in QFMLG. However, the inhomogeneous broadening is larger by 2–3 cm−<sup>1</sup> with respect to the homogeneous broadening in SLG-Ar and SLG-vac. The role of inhomogeneous broadening is thus larger in SLG-Ar and SLG-vac than in the QFMLG.

**Figure 4.** The (**a**) FWHM and (**b**,**c**) Voigt fitting parameters of (**b**) homogeneous (*γ*) and (**c**) inhomogeneous (*σ*) broadening are compared for the three samples (QFMLG, SLG-Ar, SLG-vac).


**Table 2.** Fitted parameters *γ* and *σ* of Voigt line shape and FWHMs for three samples (QFMLG, SLG-Ar, SLG-vac). The line shape was determined at four different positions on each sample.

We study sources of inhomogeneous broadening by measuring spatial Raman maps. We show the Raman maps of the 2D peak position and FWHM of 2D peak in Figure 5. We select three representative points (marked by black, red and blue circles in Figure 5a–f) and Raman spectra taken at these points are plotted in Figure 5g–i. The SLG-vac sample is the least homogeneous. The broad and blue-shifted 2D peak (red spectrum in Figure 5g) is a fingerprint of bilayer graphene. We observe similar small areas of broad and blue-shifted 2D peak in QFMLG, too (red spectrum in Figure 5i). Though SLG-Ar sample appears to be more homogeneous than SLG-vac and QFMLG, this could be due to the larger steps of SiC, or, due to the specific area chosen for the Raman map. To verify further the origin of these inhomogeneities, we measured also AFM and LFM, see Figure 6. The topography of SLG-vac shows circular-like SiC terraces of 1–4 μm in diameter. The SLG-Ar and QFMLG show regular SiC step bunching. The regular terraces are 1–4 μm broad in SLG-Ar, and, they are 6–7 μm broad in QFMLG.

**Figure 5.** Maps of Raman scattering. The map of (**a**–**c**) 2D peak position and (**d**–**f**) 2D peak FWHM measured in (**a**,**d**,**g**) SLG-vac, (**b**,**e**,**h**) SLG-Ar and (**c**,**f**,**i**) QFMLG. We also show 2D peak line shape at three locations marked by black, red and blue circles. The spectra are plotted using corresponding black, red and blue curves in (**g**–**i**).

To identify these different areas in topography, we measured also LFM. The LFM can distinguish different materials if their friction with an AFM tip is different. The LFM images are shown in Figure 6d–f. We also plot friction force profiles in Figure 6g–i. We observe reduction of the friction force at the edges of homogeneous SiC areas in SLG-vac. Some edges show similar reduction of the friction force in SLG-Ar, too. In both cases, the friction force is reduced ≈ 2×. Such reduction of the friction force was shown between single and bilayer graphene [43]. The friction force is increased in the step edge areas in QFMLG. The increase of friction force was related to the buffer layer [43], however; we observe increase by only ≈ 2.5×. The expected increase is ≈ 10× for buffer layer. We note, that the step edge, or, sidewall area does not have the same structure as buffer, as shown in literature [44,45]. We interpret the graphene at the SiC step edges as the sidewall graphene ribbons [46,47]. We also observe in LFM that the friction displays large homogeneous areas, and, it also shows considerable amount of sub-micrometer sized inhomogeneities. These patches of different friction/material can contribute to the overall line shape of the 2D peak, too.

**Figure 6.** The topographies (20 <sup>×</sup> <sup>20</sup> <sup>μ</sup>m2) of investigated samples are depicted in graphs (**a**–**c**) and corresponding friction force maps are shown in graphs (**d**–**f**). The profiles (**g**–**i**) of friction force maps (green lines) demonstrate the SLG, BLG, QFMLG and sidewall area.

Another contribution to the 2D peak broadening is the charge density and strain variation on the sub-micrometer length scale. We analyzed the positions of the G and 2D peaks measured by Raman mapping. The correlation of the G and 2D peak positions are plotted in the inset of Figure 7. We describe the relation between the G and 2D peak position *ωG*, *ω*2*<sup>D</sup>* and mechanical strain and charge density by the following set of two equations.

$$
\omega\_{\rm G} = \omega\_{\rm G0} + \mathfrak{a}\_{\rm G}|n\_{2D}| - 2\gamma\_{\rm G}\omega\_{\rm G0}\varepsilon \tag{8}
$$

$$
\omega\_{2D} = \omega\_{2D0} - \mathfrak{a}\_{2D}\mathfrak{n}\_{2D} - 2\gamma\_{2D}\omega\_{2D0}\mathfrak{e}\_{\prime} \tag{9}
$$

where proportionality constants are *α<sup>G</sup>* = 6.8 cm−1/1013 cm−<sup>2</sup> and *α*2*<sup>D</sup>* = 2.7 cm−1/1013 cm−2. The effective Grüneisen parameters are *γ<sup>G</sup>* = 1.8 and *γ*2*<sup>D</sup>* = 3.5. The G and 2D peak positions in the charge neutral unstrained graphene are *ωG*<sup>0</sup> = 1582 cm<sup>−</sup>1, and *ω*2*D*<sup>0</sup> = 2680 cm<sup>−</sup>1. We chose the linear dependence of G and 2D peak position on strain in agreement with previous experimental works studying the G peak [7,26] and 2D peak [24–27]. We chose the functional dependence of the G peak position on the charge density as an approximation to the theoretically predicted dependence [7–9,48]. The expected charge density dependence of G peak position is smoothed at room temperature [48] in comparison to *T* = 0 K. The measured dependence can be clearly approximated by absolute-value function, when compared to theory [48] and experiment [5,7–9]. Our fitting parameters lead to 2.5× stronger charge density sensitivity of G peak than the sensitivity of 2D peak. The absolute values and relative strength of G to 2D peak charge sensitivity is in agreement with previous works [7,9]. Other fitting parameters lead to the sensitivity of G peak position to strain <sup>Δ</sup>*ω<sup>G</sup>* = −<sup>57</sup> cm<sup>−</sup>1/%,

and, to the sensitivity of 2D peak position to strain <sup>Δ</sup>*ω*2*<sup>D</sup>* = −<sup>188</sup> cm<sup>−</sup>1/%. These G and 2D peak sensitivities are in very good agreement with work of Mohiuddin [26], where authors found the G peak sensitivity to the biaxial strain <sup>Δ</sup>*ω<sup>G</sup>* = −<sup>63</sup> cm<sup>−</sup>1/% (page 4 in Ref. [26], 2nd column, 1st paragraph). The 2D peak sensitivity to the biaxial strain was found <sup>Δ</sup>*ω<sup>G</sup>* = −<sup>191</sup> cm<sup>−</sup>1/%, see Ref. [26], page 5, 1st column, 4th paragraph. These parameters are also in good agreement with an experimental results of Schmidt [5] (Δ*ω<sup>G</sup>* = −<sup>63</sup> cm<sup>−</sup>1/%, <sup>Δ</sup>*ω*2*<sup>D</sup>* = −<sup>149</sup> cm<sup>−</sup>1/%), and, Density Functional Calculations (DFT) <sup>Δ</sup>*ω<sup>G</sup>* = −58 cm<sup>−</sup>1/% and <sup>Δ</sup>*ω*2*<sup>D</sup>* = −144 cm<sup>−</sup>1/%, see Ref. [26], page 5, 2nd column.

**Figure 7.** The variation of the charge density and strain as determined from the spectral position of the G and 2D peaks in the three samples (QFMLG, SLG-Ar, SLG-vac). The inset shows the correlation between the G and 2D peak positions. The left and bottom histograms depict the distribution of the charge density and strain, respectively.

We solve the set of Equations (8) and (9) numerically for each point of the Raman map and we obtain the correlation between the charge density and strain, as depicted in Figure 7. We observe that QFMLG shows the largest variation of the charge density. At the same time, the QFMLG sample shows the smallest inhomogeneous broadening, as we showed in spectral analysis of the 2D peak line shape fitted by the Voigt function. The anticorrelation between the observed charge density and inhomogeneous broadening in QFMLG led us to conclude that the main contribution to the sub-micrometer inhomogeneous broadening is the strain variation.

#### **4. Discussion**

The detailed analysis of 2D peak line shape shows that the Voigt line shape describes the 2D peak better than the Lorentzian line shape. Although the improvement is significant, the *χ*<sup>2</sup> statistics suggests that further corrections are needed. These deviations from the Voigt model are probably due to the larger areas of non-normally distributed strain, or, due to the patches of side wall graphene at the SiC step edges. Such areas cause side bands or local extrema in the normal distribution of the strain. We propose these deviations from the normal strain distribution can be reduced by defocusing the laser spot. However, defocusing will also reduce the Raman signal, and, it will also cause broader 2D peak. We assumed only biaxial strain contributing the total broadening of 2D peak. Though the uniaxial strain was neglected, it could contribute to the 2D peak broadening, too. The uniaxial strain splits the 2D peak [24–27]. If the 2D peak splitting is smaller or comparable to the spectral resolution (2 cm−1),

the 2D peak could be misinterpreted as a single-component peak. We assume that the uniaxial strain would effectively contribute to the 2D peak broadening only. Also, since the 2D peak splitting also leads to unequal intensities of the two split components, we expect contribution to the asymmetry of the 2D peak.

Another contribution to the inhomogeneity of epitaxial graphene on SiC is the stacking alignment between graphene and underlying SiC substrate [49]. These stacking domains can lead to strain inhomogeneity. However, since the size of these domains is well below the laser spot size (1 μm) we assume these stacking domains will mainly contribute to the Gaussian broadening, and, they could be studied by the Voigt line shape fitting.

We note it is necessary to fit the data of the 2D peak in the large enough spectral range. We used a 300 cm−<sup>1</sup> broad spectral window. The spectral window broader than 300 cm−<sup>1</sup> can not be used due to the presence of the combination Raman mode at ≈2450 cm<sup>−</sup>1. The narrower spectral range can lead to better fit [17,50], however, the fitting parameters might not be as reliable. As the Raman signal is low in the low and high energy tails, the signal might become weaker than the noise level. For the reason of high signal-to-noise ratio the Raman spectra have to be collected with a high numerical aperture objective and the data have to be integrated for at least 60 s.

Our analysis also shows a promise for growth improvements of the epitaxial graphene. The correlation between the width of the 2D peak and carrier mobility has been demonstrated [22]. Hence, we assume, if the strain inhomogeneity is reduced, the graphene carrier mobility could be improved, too.

Contrary to improving graphene, the effect of inhomogeneous broadening can also result in large deviations from the here presented line shapes. We have studied samples where the inhomogeneous broadening is comparable to the homogeneous broadening. However, if the inhomogeneous broadening dominates, the Gaussian component is expected to be prevalent in the 2D peak line shape.

#### **5. Conclusions**

We show the 2D peak line shape is given by a convolution of the inhomogeneous and homogeneous broadening, so called Voigt broadening, rather than just by a single Lorentzian line shape. We interpreted the inhomogeneous broadening to be mostly given by a sub-micrometer length scale strain variations. The hydrogen intercalated buffer layer is shown to have the smallest homogeneous and inhomogeneous 2D peak broadening.

**Author Contributions:** Individual contributions of the two authors are following: conceptualization, J.K.; methodology, J.K.; software, J.K. and M.R.; validation, J.K. and M.R.; sample preparation, J.K. and M.R.; formal analysis, J.K. and M.R.; investigation, J.K. and M.R.; resources, J.K.; data curation, J.K. and M.R.; writing—original draft preparation, J.K. and M.R.; visualization, J.K. and M.R.; supervision, J.K.; project administration, J.K.; funding acquisition, J.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Czech Science Foundation grant number 19-12052S. The MEYS project VaVpI CZ.1.05/4.1.00/16.0340 is also gratefully acknowledged.

**Acknowledgments:** We acknowledge helpful comments of J. Bok. These comments led us to the data analysis presented here.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **Abbreviations**

The following abbreviations are used in this manuscript:



#### **References**


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